Journal of Research in Progress Vol. 3

At Howard Community College

JRIP Editorial Team

Howard Community College

Columbia, MD

Journal of Research in Progress Vol. 3 by Howard Community College is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Contents

- Front Cover

- Inside Front Cover

- A Note from the Editorial Team...

- Editorial Team

- Volume 3

- Salivary Alpha-Amylase and the Effects of Legume-based Inhibitors

- Formal Logic and the Structure of Music

- Artwork - “Mercury’s Passage”

- Looking Closely: The Transit of Mercury

- Artwork - “Van Allen Belts in Impressionism”

- Variations of Particle Motion in the Van Allen Belts

- Artwork - "Worms"

- Feeding Behaviors of Manduca sexta

- Artwork - “Behind the Stars”

- Exoplanet Photometry and False Positives

- Equivalence Classes of Complex Numbers Modulo a Natural Number

- So, You Want to Win Plinko?

- Artwork - “The Cycle”

- The Relationship Between Soil Texture, Land Use, and Passive Versus Out-competing Species in Urban Forest Patches

- Up and Away! Rebound Height and Energy Changes in a System of Collisions

- Artwork - “Entrapped”

- Ghosts of Parking Lots Past: The Effects of Legacy Pollution on Stream Health

- Artwork - “Urban Forest”

- Previous Volumes

- Inside Back Cover

- Back Cover

1

Front Cover

2

Inside Front Cover

On the Cover: In spring 2020, Digital Methods: Illustration & Compositing students had the exciting and unique opportunity to work in partnership with Howard Community College’s Journal of Research in Progress. Exploring the development of their own unique visual voices, in conjunction with STEM research projects, sparked innovation in artistic, illustrative, and scientific investigations showcased in this year’s Volume 3 of the Journal of Research in Progress. Each piece of student artwork featured on the cover can be seen in its entirety within the pages of this journal.

3

A Note from the Editorial Team…

A Note from the Editorial Team…

The editorial team for the Journal of Research in Progress at Howard Community College (JRIP) hopes you enjoy the third volume of our publication. The team began as a STEM Faculty Professional Learning Community (FPLC) in the Fall of 2017 and has expanded to include many academic areas within HCC. This third edition of the journal includes original artwork inspired by the articles inside. The team wishes to thank the ARTS 250 (Digital Methods: Illustration and Compositing) students who created the artwork as well as their instructor Professor Jeremy Bomberg.

Our editorial team knew that there were students who had produced great research and artwork while here at HCC, and our goal is to celebrate that work in print. Students began the journey to publication back in November when they submitted abstracts to be reviewed by the editorial team. Students whose abstracts were approved based on the rigor, originality, and timeline to completion of their research were invited to submit complete articles for peer review. Students greeted this challenge with excitement and worked with their mentors to craft their articles. In a similar way, the students in ARTS 250, worked enthusiastically with Professor Bomberg to create original works of art that took their inspiration from the content in the research articles. Each student research article underwent peer review by at least one member of the editorial team, another student and a volunteer faculty and/or staff member at HCC. We want to extend a special thank you to all of the faculty mentors who advised and encouraged their students, and to the peer review volunteers for donating their time to the project.

In addition to serving as mentors for some of the student researchers, the editorial team was responsible for reviewing and editing articles as well and making design and layout decisions for the journal as a whole. The work began in earnest during the fall 2019 semester when faculty and students were on campus. Despite having to shift to collaborating remotely in response to the COVID-19 pandemic, the team was able to stay on schedule. As has been the case on past volumes, this has been an all-out team effort of which we are very proud.

Our team would like to send a special thanks to the HCC Foundation for their support and particularly our donor, The Kalhert Foundation, who contributed funds to allow this publication to happen. The editorial team and the students are grateful for your vision in recognizing the importance of undergraduate research at HCC and that public presentation of research work is a critical component of the research process.

Best wishes,

The JRIP Editorial Team

Students interested in submitting articles for next year’s volume of JRIP
should contact the editorial team at JRIP@howardcc.edu.
An electronic copy of this journal is available at
https://pressbooks.howardcc.edu/jrip3

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Editorial Team

Alex M. Barr, Ph.D.
Associate Professor of Physics

Andrew L. Brown
Assistant Professor of Mathematics

Cheryl J. Campo, Ph.D., J.D. Adjunct Professor;
Science, Engineering & Technology

Floyd J. Csir
OER/Instructional Design Librarian

Bridget Holmes, RDH, MS, MPA
Professor of Dental Hygiene

Stephen Horvath
Professor of English

Carol Howald, Ph.D.
Associate Professor of Mathematics

Kathryn S. Jones, Ph.D.
Adjunct Professor of Biology

James Kaiser, M.D.
Tutor, Learning Assistance Center

Adrian Kuhlman
Assistant Professor of Mathematics

Kathy Lilly, Ph.D.
Professor of Chemistry

Mike Long, Ed.D.
Dean of Mathematics

Michael O’Donnell
Adjunct Professor of Mathematics

Loretta FitzGerald Tokoly, Ph.D
Professor of Mathematics
Director of STEM Scholars Program

Patti J. Turner
Dean of Science, Engineering, & Technology

Jane H. Wenstrom, Ph.D
Associate Professor of Mathematics

Laura Yoo
Professor of English

Download a pdf of the full journal

1. Salivary Alpha-Amylase and the Effects of Legume-Based Inhibitors
Abdelrahman Abdelaziz, Ayushi Dave, and Roshae Roberts

2. Formal Logic and the Structure of Music
Divinevictory Amayo

Artwork – “Mercury’s Passage”

Shree Krishnan

3. Looking Closely: The Transit of Mercury
Grant Bunyard

Artwork – “Van Allen Belts in Impressionism”

Dahwula Suah

4. Variation of Particle Motion in the Van Allen Belts
Ariana Bussio

Artwork – “Worms”

Rebekah Jones

5. Feeding Behaviors of Manduca sexta
Hartley Carlson, Abigail Livingston, Rohaan Mahmud, Mingqian Huang, Madhav Fadadu, and Kimia Samieniejad

Artwork – “Behind the Stars”

Janae Bucksell

6. Exoplanet Photometry and False Positives
Shila Deljookorani, Veronica Kunzle, and Dominic Leger

7. Equivalence Classes of Complex Numbers Modula a Natural Number
Jonathan Kraus and David Ryan Johnson

8. So, You Want to Win Plinko?
Benjamin Marasco and Naod Assefa

Artwork – “The Cycle”

Charise Pelan-Maclean

9. The Relationship Between Soil Texture, Land Use, and Passive Versus Out-competing Species in Urban Forest Patches
Charis Pelan-Maclean

10. Up and Away! Rebound Height and Energy Changes in a System of Collisions
Zainah Wadi, Morin French, Nian Liu

Artwork – “Entrapped”

Heather J. Sun

11. Ghosts of Parking Lots Past: The Effect of Legacy Pollution on Stream Health
Laura Wortman and Sky Erickson

Artwork – “Urban Forest”

Shuyi Cen

Salivary Alpha-Amylase and the Effects of Legume-based Inhibitors

Download a pdf of this paper

Abdelrahman Abdelaziz , Howard Community College

Ayushi Dave, Howard Community College

Roshae Roberts, Howard Community College

Mentored by Hannah Pie, Ph.D.

Abstract

Alpha-amylase is an essential digestive enzyme found in humans. In our bodies, it is responsible for starch hydrolysis into dextrins and eventually maltose within saliva and the small intestines. Maltase then hydrolyzes maltose into glucose that can be utilized by the cells. The functionality and reaction rate of alpha-amylase like all other enzymes is related to pH, temperature and initial concentration of substrate. In the body, an additional factor to consider is the presence of inhibitors that prevent enzymes from binding with their substrates. Various legumes have been shown to include compounds that inhibit alpha-amylase activity by preventing the hydrolysis of starch and limiting the release of glucose. The goal of this study was to identify the inhibiting activity of various legumes on human salivary alpha-amylase utilizing a starch-iodine detection test. The inhibitory compounds of the following legumes: black beans, kidney beans, black eyed beans and white beans, were extracted using ethanol fractionation and tested to determine which legume has the most inhibitory effect. Our results showed that kidney beans had the highest imbibition activity on salivary alpha-amylase. This was followed by black beans, white beans and black-eyed peas, respectively. The fact that kidney beans display strong inhibition on salivary alpha-amylase could make them a strong candidate for possibly replacing diet pills as a healthier alternative with reduced side effects.

Introduction

Carbohydrates, lipids, proteins and nucleic acids are the four basic macromolecules found in the foods we ingest. These molecules need to be broken down further into their monomers in order to be absorbed into the cells of gut lumen. This is done primarily through the chemical action of water and hydrolytic digestive enzymes. Each macromolecule has different hydrolytic enzymes that cleave the bonds between its monomers.  Amylase, produced within saliva and the pancreas, is the hydrolytic enzyme responsible for the hydrolysis of starch. Starch is a polysaccharide comprised of thousands of glucose monomers. Digestion of starch begins with the actions of salivary alpha-amylase to break down starch into dextrins (shorter polysaccharides) and then to maltose (a disaccharide). To further break it down, the enzyme maltase hydrolyzes it to glucose.

The amount of alpha-amylase produced and the number of gene copies that express alpha-amylase (AMY1) varies among individuals [2] The ability of some people to have multiple copies of AMY1 is thought to have evolved as populations increased starch intake within their diet [2]. Populations with diets rich in starch such as rice and potatoes correlate to more copies of AMY1 and higher levels of alpha amylase activity [3]. Inversely, populations with diets low in starch have less copies of AMY1 and lower levels of alpha-amylase activity [3].

Various legumes from the Fabaceae family have been shown to include compounds that inhibit alpha-amylase activity by preventing the hydrolysis of starch and limiting the release of glucose [4]. Phaseolus vulgaris subspecies beans, such as kidney beans, black beans, great northern white beans, have shown to exhibit moderate to high levels of pancreatic alpha-amylase inhibitory activity [4]. In contrast, the black-eyed peas, a subspecies of the cowpea Vigna unguiculata, displayed lower inhibition activity for pancreatic alpha-amylase [1]. While pancreatic alpha-amylase inhibition by these beans has been examined before, salivary alpha-amylase has not, so by conducting this study we are contributing to current literature. Current research is focusing on the use of beans and their inhibitory compounds for anti-obesity and diabetic treatments.

This study sought to examine our own salivary alpha-amylase activity to affirm the correlation between diet and concentration. All students expected to exhibit a moderate to high amount of alpha-amylase as all had diets rich in starch. Additionally, this study sought to examine the inhibiting activity of various legumes on human salivary alpha-amylase utilizing a starch-iodine detection test. The inhibitory compounds of the Phaseolus vulgaris beans (black beans, kidney beans, great northern white beans) and the Vigna unguiculata subsp.bean (black-eyed peas) were extracted using ethanol fractionation and tested to determine which has the most inhibitory effect. Based on previous research in human pancreatic alpha-amylase, we expect kidney beans to have the highest inhibitory effect on human salivary alpha-amylase.

Methods and Materials:

Procedure for Individual Salivary Alpha-Amylase Starch-Iodine Assay [4]:

In order to test how fast one’s alpha amylase breaks down carbohydrates like starch, alone and in the presence of inhibitory compounds from the beans, we used the starch iodine test. During these tests we maintained a constant temperature and pH to maintain a controlled environment to test for the functionality of the enzyme. In order to quantitatively analyze the absorbance of the starch hydrolysis, we used a spectrophotometer. This allowed us to quantify the presence of starch numerically, which  indicates how much enzyme was present.

We started by preparing 96-well plates containing 50 μL Lugol’s iodine and 10 μL 0.1 M HCl in each well to stop the reaction and for visible detection of starch. Each student collected 2 ml of saliva for testing and then prepared a 10% dilution of the saliva in NaCl (pH 7.0). Testing solutions of 5.0 mL of 1% starch and 4.0 mL of NaCl (pH 7.0) were prepared, then 1 mL of the 10% saliva was added to start the reaction. Every 30 seconds, 50 ul of testing solution was added to a well to stop the reaction until a visible achromatic point (the point at which all the starch had been completely hydrolyzed to maltose) was reached (Fig 1.). A positive control was used to show what the color looks like in the presence of starch with no enzyme, while a negative control was used to see the color of the iodine test with the presence of no starch or completely hydrolyzed starch. Spectrophotometric analysis at 595 nm was conducted to determine the achromatic point at which all the starch has been hydrolyzed using the negative control as the blank.

Figure 1: Reaction results of alpha-amylase assay with a positive control in D11 and a negative control in D12.

Procedure for Salivary Alpha-Amylase Standard Curve:

The starch-iodine assay was used for determining the achromatic points of multiple concentrations of a standard human salivary alpha-amylase (10, 25, 50, 75, 100, 125 U/mg alpha amylase) to produce a standard curve (Fig 3.). The human salivary alpha-amylase used for the standard was purchased from Sigma-Aldrich (Sigma-A0521).

Procedure for Preparation of Legume Inhibitors [1]:

Dried legumes (kidney beans, black beans, great northern white beans, and black-eyed peas) were purchased from a local market. 4.0 g of each legume was ground using a mortar and pestle to a fine meal. 25 mL of 10 mM PB buffer (7 pH) was added and solutions stirred for 24 hours at 4°C. Solutions were centrifuged for 30 minutes at 8,500 rpm and then 1 hour at 5,000 rpm. The inhibitory compounds were extracted with 48% v/v ethanol (Fig 2.) and centrifugate for 1 hour at 5000 rpm. The supernatant containing the inhibiting compounds was removed and used for the inhibition assay.

Procedure for Testing Inhibition of Alpha-Amylase by Legumes:

The starch-iodine assay was used for determining the achromatic points of the four extracted inhibitors in the presence of 40 U/mg human salivary alpha-amylase.

Figure 2: Legume Extractions with 48% v/v ethanol before removal of supernatant from pelleted material.

Results

Figure 3: This graph was constructed by gathering the achromatic points of six reactions of varying alpha-amylase concentrations (125, 100,75,50,25, and 10 U/mg). The linear regression line produced was used to calculate enzyme concentrations given time to achromatic points (y = -0.0874x+2.1517,  R^2 = 0.9789)

The standard curve (Fig 3.) shows that there is an inverse relationship between enzyme concentration and time it takes to reach the achromatic point. The quicker the reaction takes to reach its achromatic point, the higher the concentration of enzyme.

Figure 4: The achromatic points of a 40 U/mg alpha-amylase reaction only and in the presence of various beans to reveal the inhibitory effects of each bean.

The legume reactions reveal that some of the beans have more inhibitory effects than others (Fig 4.). The higher the inhibitory effect, the lower the enzymatic activity. Kidney beans took the longest time to reach achromatic point thus displaying the highest inhibition (lowest enzymatic activity) on alpha-amylase relative to other beans tested. Black-eyed peas have the most similar achromatic time to the enzyme only negative control indicating a low inhibitory effect (high enzymatic activity).  Black beans show the second most inhibitory effect (second lowest enzymatic activity) with great northern white beans following.

Figure 5: Each participant’s personal salivary alpha-amylase concentration was calculated using the reaction time to achromatic point.

Both participant 1 and participant 4 had an alpha-amylase concentration of 42.4 and 85.7 U/mg/mL saliva, respectively. However, participants 1 and 2 had an alpha-amylase concentration of 11.5 and 28.3 U/mg/mL saliva, respectively. While all the participants had alpha amylase concentrations much lower than expected, there are other factors besides diet which contribute to the number of alpha-amylase copies and overall amount of the enzyme produced in the saliva.

Discussion and Conclusions

Our results indicate that kidney beans had the highest inhibition followed by black beans, white beans and finally black-eyed peas. This means that at the same concentration of alpha-amylase, kidney beans will inhibit the activity of the enzyme the most; as evidenced by the fact that they took the longest to reach the achromatic point. Out of all the beans tested from the Fabaceae family, black eyed peas were the only one from the genus Vigna, while the other three were from the genus Phaseolus. This possibly explains why black-eyed peas had the least inhibition activity. Our results also indicate that the best candidates for bean-based diet pills are those that are subspecies of Phaseolus vulgaris  as beans from this species were the most efficient and blocking the activity of alpha-amylase. These results were previously supported by a study [5] which showed that an alternative to low glycemic index diet could be products which slow down the rate at which carbohydrates are absorbed. This is done by inhibiting alpha-amylase and glucosidase activity. In beans that are subspecies of Phaseolus vulgaris, specifically great white kidney beans, a compound known as Phase 2 Carb Controller has shown the ability to cause weight loss with doses ranging from 500-3000 mg per day [5]. Besides the potential of Phase 2 as a weight loss product, it also showed the ability to reduce post-prandial spike in blood glucose levels [5]. All these benefits are due to the ability of Phase 2 to block the activity of alpha-amylase which brings about the possibility of identifying more efficient alpha-amylase inhibiting compounds in beans of the same or other families.

For the assay testing the concentration of alpha-amylase in our four participants, the results for both participant 2 and participant 3 were unexpected. All four of the participants come from areas that consume high amounts of starch and legumes. So, the expected result was that they would all have multiple copies of the alpha-amylase gene, which would positively correlate with the amount of alpha-amylase in their saliva. The effect of consuming large amounts of legumes on the alpha-amylase concentration in individuals is still unclear, and there might be many other contributing factors effecting the number of alpha-amylase gene copies or expression of those genes within individuals.  The known range of salivary alpha-amylase activity is 1-371 U/mL saliva with an average of 93 U/ml (±62), so all participants displayed below average salivary alpha-amylase concentrations [1].

Our next steps will be to examine factors that affect the activity of alpha-amylase in our saliva to have a better understanding of the effect inhibitors have on that activity . Furthermore, we plan to test how well diet pills inhibit the activity of salivary alpha-amylase to know whether they are effective or not. By testing different kinds of beans and comparing their inhibition activity to those of diet pills, we would possibly be able to provide better alternatives.

Acknowledgements

Thank you for the STEM Scholars Program for providing us with the opportunity to do this project, Dr. Hannah Pie for her wonderful guidance, patience and time commitment, and to Professor Darling for providing the supplies needed for the experiments.

Contacts: abdelrahman.abdelaziz@howardcc.edu , adave6283@howardcc.edu, roshae.roberts@howardcc.edu, hpie@howardcc.edu

References

[1] Ghorbani, F., Sadeghi, M., Aghaei, A., Ghahghaei, A., Jalili, C., & Khodarahmi, R. (2018). Study of α-Amylase Inhibitors among Different Bean Cultivars and Evaluation of their Effectiveness Compared with a Commercial Product using In Vitro/In Vivo Experimental Systems. Journal of Research in Medical and Dental Science,6(1), 381-391. doi:10.5455/jrmds.20186162

[2] Mandel, A. L., des Gachons, C. P., Plank, K. L., Alarcon, S., & Breslin, P. A. (2010). Individual differences in AMY1 gene copy number, salivary α-amylase levels, and the perception of oral starch. PloS one, 5(10).

[3] Perry, G. H., Dominy, N. J., Claw, K. G., Lee, A. S., Fiegler, H., Redon, R., … Stone, A. C. (2007). Diet and the evolution of human amylase gene copy number variation. Nature Genetics, 39(10), 1256–1260. http://doi.org/10.1038/ng2123

[4] Xiao, Z., Storms, R., & Tsang, A. (2007). Corrigendum to “A quantitative starch–iodine method for measuring alpha-amylase and glucoamylase activities” [Anal. Biochem. 351 (2006) 146–148]. Analytical Biochemistry,362(1), 154. doi:10.1016/j.ab.2006.12.021

[5] Barrett, Marilyn L, and Jay K Udani. “A proprietary alpha-amylase inhibitor from white bean (Phaseolus vulgaris): a review of clinical studies on weight loss and glycemic control.” Nutrition journal vol. 10 24. 17 Mar. 2011, doi:10.1186/1475-2891-10-24

Formal Logic and the Structure of Music

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Divinevictory Amayo, Howard Community College, 2019,UMBC

Mentored by Dr. Mike Long, Howard Community College

Abstract

Formal logic provides a framework for formal mathematical proof in higher-level mathematics.  On the other side is music, one of the fine arts, where vocal and instrumental sounds are combined in ways to entertain audiences while challenging those performing.  For some time, music and mathematics have been connected, particularly through analysis of sound waves, which requires trigonometry.  It turns out that music and formal logic may be related as well.  This led to the question: When are musical passages equivalent in the same way that mathematical statements are logically equivalent?  Through musical analysis and use of the basic formal logic concepts of inverse, converse, and contrapositive, we explore when the equivalences exist.  It turns out that in a few cases the equivalences do exist and so passages of music are equivalent in the way that logical statements are equivalent.  But there are many instances where musical passages that might seem to be logically equivalent are not and we explore and mathematically explain why they are not.

Introduction

“Mathematics” is a word that strikes fear in the hearts of many students.  The guitar is an instrument with a rich history and an even richer sound that is loved by millions of people around the world. What do these two things have in common? In the spring of 2018, I didn’t know and I didn’t care. I was a student trying to get all of the dreadful mathematics required for his major out of the way, and a newly initiated guitarist. So, when I opened my mouth to tell my discrete mathematics professor (who is my mentor on this project), about my dislike of his field, I did not foresee that he would rope me into a challenge that would try to change my mind about mathematics. The challenge took the form of a research project that would delve into formal logic topics that were touched upon in my discrete mathematics class and combine those topics with my growing knowledge and appreciation for the music world through guitar.

Background

There are some common structures in formal logic that must be introduced to begin a discussion of this research.  The first is the conditional statement “if / then” or “if p then q” where each of p and q represent statements.  Statements in formal logic are identified as sentences that are either true or false.  Symbolically, “if p then q” is represented as .  The second structure to consider simply reverses the order of these statements, which we would write as .  This is known as the “converse.”  The third structure to consider involves adding negations to both statements, which we would write as .  This is known as the “inverse,” where the squiggly line in front of the p and the q represents the negation of the statement or, more informally, “not.”  The fourth structure reverses the order of the statements and also adds negations, which we would write as .  This is known as the “contrapositive.”  The original conditional and the contrapositive have an important characteristic in formal logic and that is that they are logically equivalent.  Similarly, the inverse and the converse are logically equivalent.  What exactly does it mean to be logically equivalent?

- The two conditional (if / then) statements have the same truth tables, or tables used in mathematical logic to determine the validity of statements

- The two conditional statements (if / then) can be proved from each other using techniques of formal mathematical proof

Table 1 compares the conditional statements.

Table 1:  Comparison of conditional statements

Conditional Statement Name:

Symbols:

Example:

Logically Equivalent To:

Original Statement

If the light is red, then I will stop.

Contrapositive

Converse

If I stop, then the light is red.

Inverse

Inverse

If the light is not red, then I will not stop.

Converse

Contrapositive

If I do not stop, then the light is not red.

Original Statement

Something similar exists in the structure of music.  These structures were seen in music composed since the 1500s in Europe, but came to prominence with serial composition in the late 1800s and early 1900s, again in Europe [1].

We first consider a passage or line of music as our baseline structure, which we might view as similar to the original conditional statement in formal logic (Figure 1 and Table 2).  The second musical structure to consider is the “retrograde.”  In music, the retrograde of the original line or passage is simply the original line or passage played in reverse starting with the last note or rest of the original passage and ending with the first note or rest of the original passage. This would be similar to the converse in formal logic where the order is reversed. The third musical structure to consider is the “inversion.”  In music, the inversion of the original line or passage is created by literally flipping the notes upside down on the musical staff, changing the contour of the music.  In this structure, the first note is the same as that of the original line or passage of music (some composers have modified this definition a bit and used a different starting note, but in this research the starting note will not be changed).  This structure would be similar to the inverse in formal logic, which is the negation or opposite of the original statements.  The final structure to consider is the “retrograde inversion.” In the retrograde inversion, the original line or passage is flipped on the musical staff, like the inversion, and then the order reversed, like the retrograde (the retrograde inversion is most often the retrograde of the inversion, but some composers convert in the opposite order, which has different results).  In this structure, the first note is the same as the last note of the inversion while the last note is the same as the first note of the original passage.  This structure would be similar to the contrapositive in formal logic, which is the negation or opposite of the original statements with the order changed.

Hypothesis

We hypothesized that the original musical statement and the retrograde inversion would be the same in rhythm and in tonality, an identical match in the same way that the original statement and the contrapositive in formal logic are logically equivalent.  This idea emerged because the original line or passage of music and the retrograde inversion were seemingly equivalent in a similar way that the original statement and contrapositive in formal logic were equivalent.

Initial Results

We have already shown the parallels between the original statement and the original line or passage of music and the retrograde inversion and the contrapositive.  However, when examining our first pieces of music, we quickly realized that our hypothesis was not correct.  The rhythm and tonality of the original line or passage of music and the retrograde inversion were different.  As a result, we changed the idea of this research work to come up with a working definition of “musical equivalence” to parallel “logical equivalence.”

Identifying Musical Equivalence

A simple passage of music with three quarter notes and a quarter rest was created (Figure 1) to better describe the musical structures:  the original musical line or passage, followed by the retrograde, the inversion, and finally the retrograde inversion.  The circles with vertical lines attached are the notes that tell the musician which pitch to sound and for how long.  The French bracket type symbols are the rests that tell the musician not to play for a period of time.  This simple passage contains four measures.  The vertical lines indicate where each new measure begins.

Figure 1: Musical passage consisting of four measures

Table 2 breaks the simple passage of music up into four parts, by measure.  The table also adds comparisons to formal logic.  This table also makes it easier to understand why the first hypothesis was made.

Table 2:  Comparison of conditional statements and music structures. Note that symbols are added in the inversion and retrograde inversions.  This is to keep the steps between the notes the same. The concept of “step” is discussed in the text.

Music

Name

Original Line or Musical Statement

Retrograde

Inversion

Retrograde Inversion

Parallel in formal logic

Conditional Statement

Converse

Inverse

Contrapositive

Symbols

Equivalence

Contrapositive

Inverse

Converse

Conditional Statement

To understand the discussion of “musical equivalence, it is useful to understand how “steps” are defined in music. The easiest way to understand steps in music is to look at the piano keyboard.  Two white keys that have no black keys in between are separated by a half step.  Two white keys that have a black key in between are separated by a whole step.  To break this down more, movement from a white key to an adjacent black key is a half step and then from the black key to the next white key is also be a half step.  Similarly, two black keys that have a white key in between are separated by a whole step.  Breaking this down, movement from a black key to the next white key is a half step and then from the white key to the next black key is also a half step.

In the original line or passage of music, notice how the notes go up a step each time on the staff and the notes are then followed by the quarter rest.  In the retrograde, where the order is changed, the quarter rest comes first and then is followed by the three notes in the opposite order.  Also, since each of the notes in the original statement goes up a step each time, in the retrograde, each of the notes goes down a step each time.  In the inversion, the quarter rest follows the three quarter notes, which is identical to the original statement.  However, each of the notes goes down a step each time, which is the opposite of the original line or passage of music where the three notes each go up by a step.  In the retrograde inversion, the order of the inversion is changed and so the quarter rest is followed by the three quarter notes.  However, the three notes each go up by one step as in the original statement, but the pitches are different.

As mentioned, the original idea for this research was that since the original statement in formal logic and the contrapositive of the original statement were logically equivalent, the original line or passage of music and the retrograde inversion would be the same passage of music in some way.  However, this simple passage shows this is not the case.  It was quite obvious that a truth table could not be used to show that the original line or passage of music and the retrograde inversion were equivalent in the same way, similar to the relationship shared by the original statement and the contrapositive in formal logic.  Also, a truth table could not be used to show that the retrograde and inversion were equivalent in the same way, similar to the converse and inverse.  Therefore, a new direction for the research was taken to identify another equivalence, a “musical equivalence,” that can be explored and defined, which does not require a truth table or formal mathematical proof to show the equivalence.

Identifying “Musical Equivalence”

There are some interesting characteristics that show up in the simple piece of music above that perhaps lead to a definition of “musical equivalence.”  In this simple example, the original musical line or passage and the retrograde inversion are not exactly the same.  Other than the first note of the original passage and the last note of the retrograde inversion, the notes are not the same.  The rhythmic structure of the original line or passage of music is not the same as the retrograde inversion since in the original musical passage the three quarter notes were followed by the quarter rest, while the retrograde inversion has a quarter rest followed by three quarter notes.  The same can be said for the retrograde and the inversion and their “musical equivalence.” They are not the same.  Other than the first note of the inversion and the last note of the retrograde, the notes are not the same.  Also, the rhythmic structure of the inversion is not the same as the retrograde since in the inversion, the three quarter notes were followed by the quarter rest, while the retrograde has a quarter rest followed by three quarter notes.  From this, it appears there is only one characteristic that might contribute to the notion of “musical equivalence,” and that is a guarantee they have one note in common, which may not be in the same position.  In fact the same notes will be the first note in one structure and the last in the second structure when the two structures are “musically equivalent,” and that only works in one direction.  For example, the first note of the original line or passage of music and the last note of the retrograde inversion are the same, but vice versa cannot be added here since this idea only works in one direction.

A deeper dive is needed to develop or dispel the idea of  “musical equivalence,” and that dive is in the actual musical structure or patterns in the pitches of the notes.  Let’s consider the following scenarios from a more complex original line of music.

- The original musical line or passage jumps up by a whole step, and then up another whole step, and then down a half step, while the retrograde will go up a half step, and then down by a whole step, and then down another whole step:

Original Musical Line                     Retrograde

- The original musical line or passage jumps up by a whole step, and then up another whole step, and then down a half step, while the inversion will drop down by a whole step, and then down another whole step, and then back up another half step:

Original Musical Line                         Inversion

- The original musical line or passage jumps up by a whole step, and then up another whole step, and then down a half step, while the retrograde inversion will go down a half step, then up a whole step, and then up another whole step:

Original Musical Line                     Retrograde Inversion

The original music line here is a little more complex than the original simple passage that was created, as the movement goes up and down and there are whole and half steps.  With the original passage, it was thought that patterns in the pitches of the notes would be critical in defining the notion of “musical equivalence.” However, that was not the case.  We started by comparing the original musical line or passage and the retrograde inversion since they are similar to the original statement and the contrapositive in formal logic, which are logically equivalent.  In the original simple musical passage, the original musical line or statement and the retrograde inversion both had movement up by two steps, but the notes or pitches were not the same.  Here, the idea of the movement between the notes was still promising.  However, in the later more complex original music line, the original musical line or passage and the retrograde inversion had very different movement.  In the original musical line or passage, the movement was two whole steps up and a half step down, while in the retrograde inversion, the movement of the notes was a half step down followed by two whole steps up.  This example alone dispels the idea that the patterns in the pitches of the notes would be critical in defining “musical equivalence.”

Summary and Conclusions

While there were ideas about a way to define “musical equivalence,” each time counterexamples proved that the ideas were not true.  The only result was that the first note of one of the structures is the same as the last note of the other structure and that this only happens in one direction.  Further research should be done toward defining characteristics for “musical equivalence.”  Another opportunity for research would be to develop a structure for determining “musical equivalence,” similar to the truth table in formal logic, which cannot be used here.  Additionally, research might examine when the notes or pitches of the “musically equivalent” structures might be the same or when the rhythmic structures are the same.

For me personally, I have realized that mathematics is not a word to be feared, but is a discipline that should be appreciated more.  I have learned that mathematics is all around in places I never thought.  I conclude also that math is ……

Contacts: divinevictoryamaya@howardcc.edu, mlong@howardcc.edu

References

[1] Smith, R. B.  (1966).  Serial Composition.  Oxford University Press.  1966.

Artwork - “Mercury’s Passage”

“Mercury’s Passage” by Shree Krishnan

Looking Closely: The Transit of Mercury

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Grant Bunyard, Howard Community College
Mentored by: Brendan Diamond, Ph.D.

Abstract

On November 11, 2019, the transit of Mercury across the sun was observed through a Celestron C11 SCT and captured with a Nikon D300. Approximately 80% of the transit and the egress was captured. The research goal was to investigate what we could determine about the solar system from the data gathered during this transit. The diameter of Mercury was calculated within 1% using the data and known distances to Mercury and the Sun. Other applications included calculating the angular separation of Mercury and the Sun, calculating the angle of Mercury’s orbit, and analyzing a historical calculation of the distance from Earth to the Sun using Kepler’s Laws, first done in 1769. Other optical phenomena including the light curve, the black drop effect, and limb darkening were observed.

Introduction

Mercury is the closest planet to our sun, and the smallest planet in our solar system; it is only slightly larger than Earth’s Moon. About 13 or 14 times every century, Mercury’s orbit takes it in between the Earth and the Sun [2]. This transit can be observed from Earth. On November 11, 2019, Mercury transited across the sun for the first time in over 3 years, and for the last time in over 13 years [3]. Many conclusions about our solar system can be drawn from the transit. Throughout the day of November 11, 2019, a small team on the observatory of Howard Community College captured over 70 gigabytes of data in the form of pictures and videos of the transit. The data was analyzed mathematically and analyzed using multiple software programs. During this analysis of the images taken of Mercury, the diameter of Mercury was calculated, the angular separation of Mercury and the Sun was calculated, and the inclination of Mercury’s orbit was calculated. A historical calculation of the distance from Earth to the Sun that was first theorized by Johannes Kepler and Edmond Halley was repeated. The brightness of the Sun as a function of time during the transit, referred to as a light curve, was also analyzed. In addition, the optical phenomena of limb darkening, and the black drop effect were observed.

Diameter of Mercury

All measurements taken for calculations used in finding the diameter of Mercury were taken from image DSC_3320. This image was captured with a Nikon D300 camera through a Celestron C11 SCT telescope using a Helios Solar Glass filter. The filter used blocks over 99.999% of all incoming light and causes the Sun to appear orange rather than white [4].

Figure 1: Image DSC_3320. 10:22 am EST Nov. 11, 2019. The faint outline of Mercury can be seen close to the middle of the Sun as it transits.

The relative size of Mercury compared to the Sun’s image was calculated from this photograph using the software program AstroImageJ. AstroImageJ is a software program that is often used for analyzing data in the field of astrophotography [1]. After the relative size of Mercury compared to the Sun’s image is calculated, the ratio between the diameter of the Sun and the diameter of Mercury can be calculated. This would then lead to obtaining a value for the diameter of Mercury.  The following diagram is a two-dimensional representation of the layout.

Figure 2: Two-dimensional representation of the layout of the transit of Mercury.

The figure shows the observer, the Sun, Mercury, and the Sun’s apparent image at the same distance as Mercury. The diagram has the following variables:

DS1 = Diameter of Sun

DS2 = Diameter of Sun’s Image

DM = Diameter of Mercury

L_1 = Distance from Earth to Sun

L_2 = Distance from Earth to Mercury

L_3 = Distance from Mercury to Sun

Because both Mercury and the Sun are spheres, the following diagram is geometrically equivalent.

Figure 3: Mathematically equivalent diagram of two-dimensional layout.

This diagram is necessary because it enables the law of similar triangles to be performed in order to establish relationships between various distances and diameters. The values of L1 and L2 for November 11, 2019 were obtained through a solar system simulator [10], and are as follows:

L_1 = 0.990 AU

L_2 = 0.675 AU

L_3 can be defined as the difference between L_1 and L_2 and was determined to be as follows:

L_3 = L_1 – L_2

L_3 = (0.990 AU) – (0.675 AU)

L_3 = 0.315 AU

The photosphere is the outermost layer of the Sun that is visible to the naked eye. According to the Stanford Solar Center, it has an approximate diameter of 1.391×10^9 m [13]. This is the value for DS1.

DS1 = 1.391×10^9 m

Now that L_1, L_2, and DS1 are known values, the law of similar triangles can be applied to Figure 3 in order to solve for DS2.

(DS1 / L_1) = (DS2 / L_2)

DS2 = (DS1 × L_2) / L_1

DS2 = [(1.391×10^9 m) × (0.675 AU)] / (0.990 AU)

DS2 = 9.4841×10^8 m

The ratio of DS2 to DM was then analyzed using AstroImageJ. The diameter of the Sun was measured to be 2908 pixels as seen in Figures 4 and 5. The diameter of Mercury was measured to be 15 pixels as seen in Figures 6 and 7.

Figure 4: The aperture photometry settings used to measure the diameter of the Sun. The radius of the object aperture is set to 1454 pixels. This would create a ring with a diameter of 2908 pixels. This blue ring is superimposed onto the image as seen in Figure 5.

Figure 5: An analysis of image DSC_3320 using AstroImageJ. The blue ring around the sun has a diameter of 2908 pixels.

Figure 6: The aperture photometry settings used to measure the diameter of Mercury. The radius of the object aperture is set to 7.5 pixels. This would create a ring with a diameter of 15 pixels. This blue ring is superimposed onto the image as seen in Figure 7.

Figure 7: An analysis of image DSC_3320 using AstroImageJ. The blue ring around Mercury has a diameter of 15 pixels.

The ratio of DS2 to DM was determined as the following:

(DS2) / (DM) = (2908 pixels) / (15 pixels)

(DS2) / (DM) = 193.87

Subsequently, the value of DS2 can be substituted in, and DM can be solved for as following:

(DS2) / (DM) = 193.87

DM = (DS2) / (193.87)

DM = (9.4841×10^8 m) / (193.87)

DM = 4.8920×10^6 m

The diameter of Mercury was calculated to be 4892 km. NASA’s measurement describes Mercury with a diameter of 4879 km [7]. This is a percent error of 0.27% for this calculation.

Angular Separation of Mercury and the Sun

The angular separation of Mercury and the Sun was calculated at the point when Mercury was at the midpoint of its transit. NASA’s Catalog of Transits of Mercury lists the start of the transit as 12:35 UTC and puts the end of the transit at 18:04 UTC [2]. This puts the midpoint of the transit at 10:20 EST. Image DSC_3314 was taken at 10:21 EST and was used for the calculation of angular separation.

Figure 8: Image DSC_3314. 10:21 am EST Nov. 11, 2019. The midpoint of the transit.

AstroImageJ was used to superimpose two concentric rings onto this image using aperture controls. The first ring was the exact size of the Sun’s disk and outlined the Sun’s border. The second ring, having the same center point as the first ring, had a radius such that the ring intersected the midpoint of Mercury.

Figure 9: An analysis of image DSC_3314 with two concentric blue rings. The outer ring outlines the border of the Sun and the inner ring has a radius that is the distance between the center of the Sun and the center of Mercury.

Figure 10: The aperture photometry settings used in Figure 9. The radius of the outer ring had a diameter of 1454 pixels, the same as the ring in Figure 5. The inner ring had a diameter of 111 pixels. This is the distance from the midpoints of the Sun and Mercury.

Because the rings are concentric, both rings were centered at the Sun’s midpoint. This means that the radius of the inner ring is the distance between the midpoint of the Sun and the midpoint of Mercury. If the diameter of the Sun’s disk was measured to be twice the radius, or 2908 pixels, and the separation of the midpoints of the two bodies was measured to be 111 pixels, the ratio of these distances can be represented as the following:

Angular Separation Ratio = (111) / (2908)

Angular Separation Ratio = 0.03817

This means that the angular separation of Mercury and the Sun is 3.817% of the angular diameter of the Sun. The angular diameter of the Sun can be found using the following equation:

[(2) × (π) × (L_1)] / DS1 = (360 degrees) / DA

DA = Angular Diameter of Sun

Substituting the previous values for DS1 and L_1, the following value is obtained for DA.

DA = 0.538 degrees

Multiplying this value by the Angular Separation Ratio obtained earlier, the following value is obtained for AM:

AM = (DA) × (0.03817)

AM = (0.538) × (0.03817)

AM = 0.0205 degrees

AM = Angular Separation of Mercury and Sun

The angular separation of Mercury and the Sun was calculated to be 0.0205 degrees, or 1.23 arcminutes at the midpoint of Mercury’s transit.

Inclination of Mercury’s Orbit

The angle of Mercury’s orbital location with respect to Earth’s orbital plane was calculated. The following diagram is a two-dimensional representation.

Figure 11: Two-dimensional representation of Mercury’s orbital angle during the transit.

The following are new variables used:

θ_2 = Mercury’s angular separation from Earth’s orbital plane

DO = Distance from Mercury to Earth’s orbital plane

Because L_2, L_3, and AM are known, DO, and subsequently, θ_2 can be calculated. The calculation for DO is as follows:

tan(AM) = (DO) / (L_2)

(L_2) × tan(AM) = DO

(0.675 AU) × tan(0.0205 degrees) = DO

DO = 5.3535 × 10^7 m

Now that DO is known, θ_2 can be solved for as follows:

tan(θ_2) = (DO) / (L_3)

θ_2 = tan-1[(DO) / (L_3)]

θ_2 = tan-1[(5.3535 × 10^7 m) / (0.315 AU)]

θ_2 = 0.06508 degrees

The angle of Mercury’s orbital location with respect to Earth’s orbital plane was calculated to be 0.06508 degrees. This value is different than the inclination of Mercury’s orbit. It was determined that not enough information was gathered during the transit to determine the inclination of Mercury’s orbit. In order to calculate the inclination of Mercury’s orbit, the value used for DO must be the maximum distance between Mercury and Earth’s orbital plane. Because Mercury was not at this point during the transit, the maximum distance cannot be determined from the data gathered during the transit. NASA’s calculations put the inclination of Mercury’s orbit at 7.01 degrees, significantly more than the 0.065 degrees observed [8]. This large difference is one of the reasons that transits of Mercury are rare. A significant amount of the times that the orbits of Earth and Mercury pass each other, the angle of Mercury’s orbital location with respect to Earth’s orbital plane is greater than the small angle necessary for a transit to be observable.

Historical Calculation Using Kepler’s Laws

In the 17th century, Johannes Kepler created a series of astronomical laws that govern orbiting bodies. He used these laws to establish relationships between the orbital distances of the planets in our solar system. Because of these laws, astronomers knew that the distance between Mercury and the Sun was about 35% to 40% of the distance between Earth and the Sun, but nobody knew how far either of these distances were [15]. It wasn’t until 1716 that astronomer Edmond Halley theorized a way to use the transit of Venus to calculate the exact distance from Earth to the Sun, also called an Astronomical Unit. He did this using the orbital laws that Kepler had created over 100 years beforehand. Unfortunately, by the next time Venus transited in 1769, both Kepler and Halley were dead. Fortunately, Halley’s theories were known well enough that many astronomers used the 1769 transit of Venus to calculate the exact value of an AU. Using these calculations, the distance from Earth to the Sun was calculated to be around 95 million miles, about 2% different from the 92.96 million miles that defines an AU today. Halley’s idea to calculate the distance from Earth to the Sun was based on the principle of parallax, the shift in apparent position that results from viewing an object from a different location. The following diagram shows the effect parallax has on planetary transits.

Figure 12: The effect of parallax on planetary transits.

As exaggerated by the diagram, someone viewing the transit from viewing point 1 would observe the planet transiting across the lower portion of the Sun, while someone viewing the transit from viewing point 2 would observe the opposite. The transit, as observed from these 2 points, would also begin and end at different times, and the length of the transit would be observed to be shorter from viewing point 2. It is the difference in these times that allowed astronomers to calculate the distance from the Earth to the Sun. This same principle can be applied to the transit of Mercury and the value of an AU could be calculated. This was not done with this data because it requires the transit to be observed from two different viewing points a significant distance apart from one another.

Light Curve

Analyzing the light curve produced by planets transiting stars that are outside of our solar systems is a common method for determining the characteristics and existence of these exoplanets. This type of analysis was attempted with the data obtained during this solar transit. Using AstroImageJ, the average brightness for all pixels included in the disk of the sun was analyzed and plotted for a series of 200 images taken during the egress of Mercury.  In other words, the amount of light reaching the Earth from the Sun was measured to see if there was a difference between when Mercury was in between Earth and the Sun, and when it was out of the way. In theory, the light curve should be consistent as Mercury transits, but rise slightly and steadily as Mercury leaves the disk of the Sun. In the data analyzed, this begins around the 120th image analyzed, or slice, and finishes around the 185th image analyzed. During this increment, the light curve should be increasing slowly and consistently. Before and after this increment, the light curve should remain constant.

Figure 13: The light curve of the egress of Mercury. Created by analyzing images DSC_3597 to DSC_3797 (1:00 pm to 1:04 pm EST Nov. 11, 2019).

As seen in Figure 13, the results that were obtained are not consistent with the results that were expected.  The variations and unpredictability of the curve are most likely due to changing atmospheric conditions and inconsistencies in cloud coverage, not because of the change in the amount of light obstructed by Mercury. These results dictate that this method is ineffective at determining characteristics of Mercury using the software programs and imaging hardware that were used for this observation.

Black Drop Effect

The Black Drop Effect is an optical phenomenon that occurs when Mercury or Venus transit across the Sun. It is characterized by the teardrop shape that the planet’s disk makes as it makes contact with the edge of the Sun. The Black Drop Effect has historically caused confusion and frustration among astronomers seeking to make precise measurements about the solar system [14]. The exact cause is unknown, but it is probably a result of multiple factors including distortion of the light from Earth’s atmosphere, and a blurred effect from the point spread function of telescopes. Images DSC_3679, DSC_3718, DSC_3663, DSC_3709, DSC_3720, and DSC_3804 show a progression of Mercury’s egress that partially displays the black drop effect.

Figure 14: Enhanced parts of images DSC_3679 and DSC_3718 taken from 1:02 pm to 1:03 pm on November 11, 2019 showing the Black Drop Effect as Mercury egresses. A slight deformation of the edge of the Sun’s disk can be seen in image DSC_3679 (left), and a clear teardrop shape can be seen in Mercury’s disk in image DSC_3718 (right).

Figure 15: Parts of Images DSC_3663, DSC_3709, DSC_3720, and DSC_3804 taken from 1:01 pm to 1:04 pm on November 11, 2019 showing the Black Drop Effect as Mercury egresses. The progression of images shows the formation of a deformation on the edge of the Sun’s disk, and then Mercury’s disk becoming a teardrop shape.

Limb Darkening

Limb Darkening is an optical effect where the brightness of the disk of the Sun gradually decreases from the center to the edges. Limb darkening occurs because the solar atmosphere increases in temperature with depth [15]. In the center of the solar ring, the observer sees the thickest and deepest layer of atmosphere that produces the most light. Towards the Sun’s limbs, only the upper layers of the solar atmosphere can be seen, and less light reaches the observer.

Figure 16: Image DSC_3232. 9:20 am. The decrease of luminosity towards the edges of the Sun is apparent as the color shifts from yellow in the center, to dark orange near the edges.

There are multiple mathematical models used today to describe the intensity of light as a function of the angle of incidence of the line of sight. These mathematical models help astrophysicists predict information about the Sun’s atmosphere and composition.

Conclusion

In conclusion, the purpose of this project was to investigate how much knowledge could be gained about the solar system using the imaging hardware and software available. The diameter of Mercury was calculated, the angular separation of Mercury and the Sun was calculated, Mercury’s orbital angular inclination was calculated, a historical calculation was repeated, the light curve of the transit was obtained and analyzed, the black drop effect was observed, and the limb darkening effects were analyzed. This was all done using equipment and software available to Howard Community College. The undergraduate research program at HCC has enabled students to complete projects like this, and many others. In the future, similar and more advanced projects for students can be envisioned, giving students the opportunity to learn how to conduct research, and further progress scientific knowledge.

Figure 17: Image DCS_3015. 7:29 am Nov. 11, 2019. The Sun as clouds pass between it and the telescope.

Acknowledgements

I would like to thank Kenny Diaz, whose personal observing equipment was used during the transit operation.

Contacts: gbunyard7@gmail.com, bdiamond@howardcc.edu

References

[1] “AstroImageJ,” AstroImageJ (AIJ) – ImageJ for Astronomy. [Online]. Available: https://www.astro.louisville.edu/software/astroimagej/. [Accessed: 17-Mar-2020].

[2] “Catalog of Transits of Mercury,” NASA. [Online]. Available: https://eclipse.gsfc.nasa.gov/transit/catalog/MercuryCatalog.html. [Accessed: 17-Mar-2020].

[3] F. Espenak, “2019 Transit of Mercury,” EclipseWise. [Online]. Available: http://www.eclipsewise.com/oh/tm2019.html. [Accessed: 17-Mar-2020].

[4] “Helios Solar Glass® Filters,” Seymour Solar LLC. [Online]. Available: https://seymoursolar.com/product/helios-solar-glass-filters/. [Accessed: 17-Mar-2020].

[5] H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099 nm),” Solar Physics, vol. 153, no. 1-2, pp. 91–114, 1994.

[6] “November 11–12, 2019 Mercury Transit,” timeanddate.com. [Online]. Available: https://www.timeanddate.com/eclipse/transit/2019-november-11. [Accessed: 17-Mar-2020].

[7] “Overview,” NASA, 28-Jan-2020. [Online]. Available: https://solarsystem.nasa.gov/planets/mercury/overview/. [Accessed: 17-Mar-2020].

[8] “Planetary Fact Sheet,” NASA. [Online]. Available: https://nssdc.gsfc.nasa.gov/planetary/factsheet/. [Accessed: 17-Mar-2020].

[9] “Size of the sun (article),” Khan Academy. [Online]. Available: https://www.khanacademy.org/partner-content/nasa/measuringuniverse/measure-the-solarsystem/a/size-of-the-sun. [Accessed: 17-Mar-2020].

[8] “Solar Model,” Solar System Live. [Online]. Available: https://www.fourmilab.ch/cgi-bin/Solar. [Accessed: 17-Mar-2020].

[9] G. StefanssonI and Pennsylvania State University, “AstroImageJ: A Simple and Powerful Tool for Astronomical Image Analysis and Precise Photometry,” astrobites, 15-Apr-2016. [Online]. Available: https://astrobites.org/2016/04/15/astroimagej-a-simple-and-powerful-tool-for-astronomical-image-analysis-and-precise-photometry/. [Accessed: 17-Mar-2020].

[10] The Editors of Encyclopaedia Britannica, “Limb darkening,” Encyclopædia Britannica, 20-Apr-2018. [Online]. Available: https://www.britannica.com/science/limb-darkening. [Accessed: 17-Mar-2020].

[11] “The Sun’s Vital Statistics,” Stanford Solar Center. [Online]. Available: http://solar-center.stanford.edu/vitalstats.html. [Accessed: 17-Mar-2020].

[12] “What is the Black-Drop Effect?,” Sky & Telescope, 26-Feb-2020. [Online]. Available: https://www.skyandtelescope.com/astronomy-news/what-is-the-black-dropeffect/. [Accessed: 17-Mar-2020].

[13] “Why Is It Important?,” Exploratorium. [Online]. Available: https://www.exploratorium.edu/venus/question4.html. [Accessed: 17-Mar-2020].

Artwork - “Van Allen Belts in Impressionism”

“Van Allen Belts in Impressionism” by Dahwula Suah

The expressive line element of Van Gogh’s “Starry Night” was one of the main inspirations for how to approach depicting the Van Allen belts and I wanted to highlight the movement of kinetic energy in the radiation zones.

Variations of Particle Motion in the Van Allen Belts

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Ariana Bussio, University of California Los Angeles 2013, Howard Community College

Mentored by: Alex M. Barr, Ph.D

Abstract

Earth’s magnetic field traps charged particles from the solar wind in the Van Allen belts. These trapped ions execute three types of motion: helical motion around magnetic field lines, longitudinal motion between North and South mirror points, and latitudinal drift motion around Earth. Each is associated with an adiabatic invariant, which expresses restrictions on each type of motion. This study examines the effect on proton motion and the three adiabatic invariants when the strength of Earth’s magnetic moment varies with time. Using a Python 3 code developed in Google Colaboratory, each invariant was calculated and graphed for a static magnetic field, for magnetic fields that varied with periods shorter than the invariant period, and for magnetic fields that varied with periods longer than the invariant period. The results demonstrate that if Earth’s magnetic moment changes quickly enough with time, the adiabatic invariant is no longer constant, allowing a wider variety of ion movement. Further study of these invariants and how they change in time-varying magnetic fields could help predict ion motion and changes to the magnetosphere during events that affect space weather, like solar storms or pole reversals.

Introduction

Earth’s magnetic field plays an important role in deflecting radiation from solar wind and protecting Earth’s atmosphere. The interaction of Earth’s magnetic field and the Sun’s solar wind causes “space weather” like geomagnetic storms and aurora phenomena. Fluctuations and extreme interactions caused by events like solar flares or Coronal Mass Ejections cause often unpredicted changes to Earth’s protective magnetic field and damage to equipment both in orbit and on the ground. These events can also pose threats to space travel, exposing astronauts to harmful radiation [1].

Earth’s Van Allen radiation belts are zones of the magnetosphere which capture and retain charged particles. Physical evidence for the Van Allen belts began interesting researchers in the late 1950s with data from Sputnik II and Explorer I, and it became clear in the 50s and 60s that ions are trapped in belts by Earth’s magnetic field [2]. Two Van Allen belts are always present: an inner belt consisting mostly of protons with energies from 10 MeV to 50 MeV [3] and an outer belt consisting mostly of electrons with energies from .04 MeV to around 7 MeV [4]. The regions are separated by a “slot.” A third temporary belt produced by major solar events and their subsequent geomagnetic storms has recently been observed in this slot between the two belts [2].

The motion of ions in the Van Allen belts is restricted by three adiabatic invariants, each of which restricts a type of motion: “helical motion,” the spiral motion an ion engages in around a magnetic field line; “bounce motion,” where an ion travels along a field line between North and South magnetic poles; and “drift motion,” where the ion travels equatorially (perpendicular to bounce motion) around Earth in a shell [5]. However, these invariants are defined for a static magnetic field. Earth’s magnetic field varies during geomagnetic storms and during pole reversals, diminishing the constraints associated with these invariants and allowing a wider range of ion motion. This paper examines each adiabatic invariant in a static magnetic field and in magnetic fields that vary periodically with time in order to help identify the role of adiabatic invariants in constraining ion motion during periods of calm and during space weather events.

Particle Motion

A Python 3 code was developed in Google Colaboratory in order to graph and examine the motion of ions and the adiabatic invariants that restrict motion. Through the fourth order Runge-Kutta method of numerical approximation, the force, acceleration, velocity, and position of each particle was calculated. Earth’s magnetic field was modeled as a magnetic dipole:

(1)

where μ_0 is the permeability of free space, M is the vector of Earth’s magnetic moment and r is the ion’s position vector from Earth’s center [6]. The acceleration of the protons was calculated using proton mass and the Lorentz equation for force, which states that the force acting on a particle is equal to the cross product of the product of its charge and velocity (qv) and the local magnetic field (B) [7]. The effects of the electric field in Earth’s magnetosphere are small enough that the electric field can be modeled as zero in an examination of adiabatic invariants and their restrictions on particle motion [5].

(2)

The kinetic energy of the ions is expressed by the relativistic equation:

(3)

This can also be expressed in terms of the relativistic factor 𝛾:

(4)

Adiabatic Invariants

Using the Python 3 code, the three adiabatic invariants which restrict particle motion were calculated and graphed with respect to time. The program was first run with the following initial conditions:

Ion 1

Ion 2

Mass (m)

1.672621777 kg

1.672621777 kg

charge (q)

+1.602176565 C

+1.602176565 C

initial kinetic energy (KE_0)

q×10^7 J (10 MeV)

q×10^7 J (10 MeV)

initial velocity (v)

m/s

m/s

initial position (r)

[2Re,0,0] m

[4Re,0,0] m

angle between B and v (θ)

π/6 radians

π/6 radians

Earth Radius (Re)

6378137 m

Equatorial strength of Earth’s magnetic dipole field (B_0) [8]

3.07×10-5 T

Strength of Earth’s magnetic moment (pz)

4πB_0Re^3/μ_0 = ~ 8×10^21 J/T

Earth’s magnetic moment (M)

pz[0,0,-1]

Table 1: Constants and initial values.

Each invariant can be calculated by integrating

(5)

over one period of the relevant motion, where A is the vector potential of the magnetic field [7]. For the first adiabatic invariant, this is one gyro-orbit, estimated at 0.1 seconds [5]. For the second adiabatic invariant, the relevant motion is one half period of bounce motion. These half-periods of bounce motion were individually calculated in the Google Colaboratory program and ranged from 0.5 to 2 seconds. For the third adiabatic invariant, the period of relevant motion is one rotation around Earth [5], each of which were also calculated in the Google Colaboratory program at between 155 and 160 seconds for Ion 1 and between 70 and 77 seconds for Ion 2. Öztürk provides a useful simplification of the integrals used to calculate each adiabatic invariant [9].

The first invariant (μ_1) expresses a conservation of the magnetic moment of a charged particle moving in a circular path. This also implies a conservation of magnetic flux through the region traced by the gyro-orbit while μ1 is constant [7].The first invariant can be calculated at each moment in time using 𝛾, the value of momentum perpendicular to B at that time, and the magnitude of B at that time [9].

(6)

If the first invariant is constant and an ion enters regions of stronger magnetic field, the perpendicular velocity must increase. To conserve energy, the parallel component of velocity must decrease, resulting in a mirror point. The second adiabatic invariant (I) is related to longitudinal motion and the mirror points at each end of the “bounce” arc. The second invariant is calculated by integrating the component of particle momentum parallel to the B field over a half period of “bounce” along a field line: for example, from a position directly above the equator (z=0) to the North mirror point and back to the position above the equator (z=0) [9].

(7)

The third adiabatic invariant (Φ) is related to a particle’s drift motion, or its latitudinal movement around Earth, and expresses that the net magnetic flux enclosed by a particle’s drift orbit is constant [5]. Therefore, particles are expected to remain in their drift shells and not increase or decrease in average altitude. The net flux within the drift orbit is equal to the net flux on the exterior of the orbit, so flux (Φ) can be calculated as an integral from the particle’s equatorial distance (R_0) to a point infinitely far away [7].

(8)

In a static B field, or in a B field that changes slowly with respect to the relevant period of motion, all three quantities (μ_1, I and Φ) will remain approximately constant [7]. When these invariants are constant, the motion associated with each of them is predictable. For example, the radius of an ion’s gyro-orbit remains constant and the altitude of the particle’s drift shell will remain constant.

This study compares two protons of equal initial kinetic energies, though electrons are more commonly found in the outer Van Allen belt and their energies typically differ from protons in the inner belt, in order to more clearly see differences in particle motion which could not be caused by differences in mass, initial kinetic energy, or charge. Earth’s magnetic moment (M) was modeled to point directly in the -z direction, toward Earth’s magnetic North Pole (geographic South Pole). Öztürk’s calculations were used as a guide for initial kinetic energy, θ, and velocity values, though it is important to note that Öztürk’s initial velocity calculations contain an error. Values for the invariants will thus vary slightly compared to those in their study [5].

After an initial data set was established with the conditions listed in Table 1, pz was altered to vary sinusoidally with time, making the strength of Earth’s magnetic moment (M) and Earth’s dipole field (B) no longer constant, but time dependent as they might be during a solar storm or magnetic pole reversal. While the variation during such an event is likely to be much more complex, this simple model nevertheless serves to illustrate the impact of a magnetic field that varies with time. The time-varying magnetic moment is given by

(9)

where A was set to 3×10^22 J/T which was chosen after some trial and error to show effects on particle motion. This resulted in a B field which varied by up to 3000 nT. This maximum variation is about 1 to 2 orders of magnitude larger than recorded changes in Earth’s magnetic field during previous powerful geomagnetic storms [10]. Similar but less drastic results can be seen with a smaller A value and therefore smaller changes in Earth’s magnetic field, however a larger magnitude of variation exaggerates the resulting changes in motion, making them easier to examine. The period of this variation was then adjusted by changing b to examine the effects of relatively “long” or “short” periods of variation on each of the adiabatic invariants and on particle motion.

Each adiabatic invariant was calculated and graphed under several conditions:

a. B does not vary with time

b. B varies with a period greater than the period of the third adiabatic invariant for Ion 2 (T_1=105 s).

c. B varies with a period between the periods of the second and third adiabatic invariant for both ions (T_2=20 s)

d. B varies with a period between the periods of the first and second adiabatic invariant for both ions (T_3=0.3 s)

e. B varies with a period shorter than the period of the first adiabatic invariant for both ions (T_4=0.05 s)

Results

Figure 1a: Motion of Ion 1 and Ion 2 over 85 seconds. Three types of motion can be observed: gyromotion around a field line, bounce motion in z and -z directions along field lines, and drift motion around Earth (light grey). Axes range from -4Re to 4Re in each dimension.

Figure 1b: Motion of Ion 1 and Ion 2 in the x-y plane over 85 seconds, which shows that the shell traced by the proton further from Earth is wider than the shell traced by the proton closer to Earth. Axes range from -4Re to 4Re in each dimension.

Figure 1a shows the motion of two protons over 85 seconds. Ion 1 (inner particle) began at an initial position of (2Re,0,0) and Ion 2 (outer particle) began at an initial position of (4Re,0,0). Earth is centered at (0,0,0) with a radius of 6378137 m (1 Re). These positions were chosen to show two distinct particles at two different altitudes (in two different “shells”) which are qualitatively similar to the inner and outer Van Allen belts. They are not separated by the physics governing the separation of the two Van Allen belts which are not included in the current model, but are separated simply by their initial assigned position.

First Invariant (μ_1)

Figure 2a: Constant magnetic moment (M).

Figure 2b: Period of magnetic moment variation: T_1=105 s.

Figure 2c: Period of magnetic moment variation: T_2=20 s.

Figure 2d: Period of magnetic moment variation: T_3=0.3 s.

Figure 2e: Period of magnetic moment variation: T_4=0.05 s.

Figure 2: First adiabatic invariant of Ion 1 (bottom line) and Ion 2 (top line) over 10 seconds. The period of relevant motion (one gyro-orbit) is about 0.1 seconds. Figures are listed from top left to bottom right.

The first invariant was evaluated for each data point. Figure 2 compares first adiabatic invariant values over 10 seconds. Figure 2a shows that μ1 for both ions is constant on average when M and B do not vary with time. Figures 2b and 2c show that with a period as low as 20 seconds, μ1 is still relatively constant, but does change minutely over time. Figures 2d and 2e show that when the magnetic field varies with shorter periods over time (0.3 s and 0.05 s respectively), μ1 is no longer constant.

Second Invariant (I)

Figure 3a: Constant magnetic moment (M).

Figure 3b: Period of magnetic moment variation: T_1=105 s.

Figure 3c: Period of magnetic moment variation: T_2=20 s.

Figure 3d: Period of magnetic moment variation: T_3=0.3 s.

Figure 3e: Period of magnetic moment variation: T_4=0.05 s.

Figure 3: Second adiabatic invariant for constant and varying magnetic moments. The period of relevant motion (one half period of bounce) is about 0.5 – 2 seconds. Figures are listed from top left to bottom right.

The second invariant was evaluated over each half period of bounce motion. Figure 3 compares the second adiabatic invariant values over 30 seconds. Figures 3b and 3c show that with a period of variation 20 s and above, I is constant. Figure 3d shows that with a period of 0.3 s, the second adiabatic invariant of Ion 1 is constant, while the second adiabatic invariant of Ion 2 shows variations. Figure 3e shows that with a period of 0.05 s, slight variations appear in Ion 1, and larger variations appear in the second adiabatic invariant of Ion 2. One value was calculated for each interval of integration, so there are fewer data points per second for the second adiabatic invariant compared to the first invariant.

Third Invariant (Φ)

Figure 4a: Constant magnetic moment (M).

Figure 4b: Period of magnetic moment variation: T_1=105 s.

Figure 4c: Period of magnetic moment variation: T_2=20 s.

Figure 4d: Period of magnetic moment variation: T_3=0.3 s.

Figure 4e: Period of magnetic moment variation: T_4=0.05 s.

Figure 4: Third adiabatic invariant for constant and varying magnetic moments. The period of relevant motion (one orbit around Earth) is between 70 and 77 seconds for the outer particle and between 155 and 160 seconds for the inner particle. Figures are listed from top left to bottom right.

The third adiabatic invariant was calculated using Eq. (8) using the equatorial altitude of each bounce as R_0, so the intervals used were also the half periods of bounce motion used in the second invariant. These intervals were used in order to obtain several data points for this shorter simulation where the total run time was less than two orbits around Earth. For a longer simulation, the period of orbit around Earth should be used as the interval and the bounds of integration. Figure 4 compares the third adiabatic invariant values over 125 seconds. There is an increase in invariant fluctuation from 4a (no time variation) to 4b (period of variation of 105 seconds) and these fluctuations in third invariant values continue to increase as period decreases (graphs 4c-4e).

Conclusions

The period of variation of the strength of Earth’s magnetic moment affects each adiabatic invariant differently. However, each invariant is similar in that as the period continues to decrease, the values vary more and the invariant is less constant. This suggests that particle motion gradually becomes less constrained as the rate at which the strength of the magnetic moment changes, rather than suddenly becoming less regular and predictable at a certain period length.

In general, when the period of variation of Earth’s B field is greater than the period of motion, the adiabatic invariant remains constant. When the period is below the period of variation, the adiabatic invariants are no longer constant and their values fluctuate. If the period is even lower and the frequency even higher, then the adiabatic invariants vary even more.

The invariant with the longest period of motion is the third, therefore it is the most easily breached. The third invariant limits ions to their drift shells, so it is most likely that ions will drift radially from their shell. The second invariant and the bounce motion it limits is the next most easily breached, while the first invariant requires the highest frequency of magnetic field variation to breach. Measuring the period of magnetic field fluctuation could help predict particle motion by predicting what type of motion will or will not be constrained by the three adiabatic invariants, and at which altitudes the effect will be greatest as the adiabatic invariants of ions at different altitudes varied in different ways.

The strength of Earth’s dipole moment and the frequency at which it varies affects the threshold at which the three adiabatic invariants are no longer constant for ions in the Van Allen belts. When B is modeled to vary with time, particle motion becomes less regular, especially in the outer Van Allen belt. This suggests that if the frequency of variation is high enough, particle motion could become unpredictable and particles could possibly escape their magnetic field lines and the magnetosphere entirely.

Further Explanations and Questions

While most particles in the outer Van Allen Belt are electrons, it was useful to model both ions we examined as protons. Comparing identical particles in two different radiation belts allows conclusions to be drawn about their motion without the uncertainty of added discrepancies between the ions like charge, mass, and kinetic energy. For further study, it would be interesting to compare the movements of a proton in the inner belt and an electron in the outer belt by changing initial mass, charge, and energy values in the program.

Experimenting with the periods of magnetic moment variation could also provide more information on particle motion in changing magnetic fields. For example, setting the period of variation to approach different periods of motion could result in resonance and possibly identify a specific “breaking point” where adiabatic invariants would no longer be useful in calculations or predictions of particle motion. Adjusting particle energy and position could also affect where adiabatic invariants no longer restrict motion. Finding these “breaking point” values may allow us to answer questions like: “How strong would a solar storm have to be to allow 10% of electrons in the outer Van Allen belt to escape?” or “How will Earth’s magnetosphere change during a magnetic pole reversal?”

Other important elements to consider for future study are the bowing effect and asymmetry of the magnetosphere created by solar wind and the accelerating effects of waves in the magnetosphere on ions along field lines. Ideally these would be included in the study of particle motion to create more accurate predictive models.

Contacts: arianabussio@gmail.com, abarr@howardcc.edu

References

[1] A. Fraknoi, D. Morrison, and S. C. Wolff, Astronomy. Hong Kong: Samurai Media Limited, 2017.

[2] D. N. Baker and M. I. Panasyuk, “Discovering Earth’s radiation belts,” Physics Today, vol. 70, no. 12, pp. 46–51, 2017.

[3] V. Markevich, R. P. Anthony, and A. N. Larsen, “Radiation Performance of Ge Technologies,” Germanium-Based Technologies, pp. 211–232, 2007.

[4] J. I. Vette, The AE-8 trapped electron model environment. Greenbelt, MD: National Space Science Data Center (NSSDC), World Data Center A for Rockets and Satellites (WDC-A-R & S), National Aeronautics and Space Administration, Goddard Space Flight Center, 1991.

[5] S. R. Elkington, Simulating electron radiation belt dynamics. Hanover, NH: Dartmouth College, 2000.

[6] D. J. Griffiths, Introduction to electrodynamics. Cambridge, United Kingdom: Cambridge University Press, 2019.

[7] M. Walt, Introduction to geomagnetically trapped radiation. Cambridge: Cambridge University Press, 2005.

[8] K. R. Lang, The Cambridge guide to the solar system. Cambridge, U.K.: Cambridge University Press, 2003.

[9] M. K. Öztürk, “Trajectories of charged particles trapped in Earth’s magnetic field,” American Journal of Physics, vol. 80, no. 5, pp. 420–428, 2012.

[10] J. J. Love, H. Hayakawa, and E. W. Cliver, “On the Intensity of the Magnetic Superstorm of September 1909,” Space Weather, vol. 17, no. 1, pp. 37–45, 2019.

[11] S. J. Ling, J. Sanny, and W. Moebs, University physics. Houston, TX: OpenStax, Rice University, 2016.

Artwork - "Worms"

“Worms” by Rebekah Jones

Growing up gardening, I never was a fun of the worms that would find their way into every plant. Horn worms were no exception. But, creating a miniature forest in my backyard was still rewarding. I wanted to create the feeling of a lush forest, focusing on native flora. It’s about the worms and their environment- inside and out.

Feeding Behaviors of Manduca sexta

Download a pdf of this paper

Hartley Carlson, Howard Community College

Abigail Livingston, Howard Community College

Rohaan Mahmud, Howard Community College

Mingqian Huang, Howard Community College

Madhav Fadadu, Howard Community College

Kimia Samieniejad, Howard Community College

Mentored by: William Gretes, Ph.D.

Abstract

Larval Manduca sexta, commonly known as the tobacco hornworm, has a natural affinity for plants in the family Solanaceae. It has been previously demonstrated that this preference for specific host plants must be induced by prior feeding experience on those plants. However, it is not yet clear whether effects of these behavioral modifications are isolated to the rearing plant or extend to other closely related plants. Cohorts of larvae were reared on a single solanaceous plant (tobacco) or one of several non-solanaceous plants to the 5th instar, and later tested in an isolated, no-choice experiment. Survivability of the larvae (from 1st to 5th instar) in each cohort was recorded. To determine the palatability of a specific plant, data was gathered on how much leaf area the caterpillars consumed at hourly intervals. This information was then graphed as a percentage of the leaf remaining over time which was used to determine the rate of feeding.

Experimental results showed that there was an overall higher larval acceptance for tobacco regardless of their rearing history. The results obtained for larvae feeding on non-solanaceous plants were considerably more varied. Cabbage, for example, appeared surprisingly palatable across all rearing backgrounds (despite extremely low survivability on cabbage). Consumption of rapeseed and turnip varied more depending on diet history. Survivability was highest on tobacco, moderate on rapeseed and turnip and low in cabbage and soybean, where only a single larva for each survived to the fifth instar.

The survivorship data demonstrates that the nutritional value and general suitability of non-solanaceous plants varied greatly. The consumption data suggests that while the acceptance of tobacco was universally high and inflexible, the acceptance of non-solanaceous plants (even within the same family) were much more heterogeneous, with varying degrees of flexibility.  Increases in the palatability of tobacco were therefore not observed. In contrast, some evidence of both inductions for the rearing plant and for other related plants were observed within the non-solanaceous plants tested. The latter would be the first such observation in the plant family Brassicae.

Introduction

Manduca sexta, more commonly known as the Tobacco Hornworm, is an insect that in nature primarily feeds on plants in nature within the one family of plants called Solanaceae. Insects that feed on a limited variety of food like Manduca sexta are refered to as oligophagous.

The family of plants preferred by the Tobacco Hornworm includes well-known plants like tomato, potato, tobacco, and other nightshades. The tobacco hornworm is known to eat certain plants outside of the solanaceous family. These include cowpea, soybean, and corkscrew vine from the family Fabaceae and rapeseed, turnip, and cabbage from the family Brassicae.

Some previous studies have shown that tobacco hornworms are willing to feed on, and can even be raised on, non-solanaceous plants [1, 2]. This raises the question as to whether these larvae are truly oligophagous. Nearly every behavioral study shows that the oligophagous nature of the tobacco hornworms can be influenced by the plant they were reared on [1,2,4,5]. When reared on a specific plant, larvae induce a preference towards that plant (induction of preference) or induce a preference towards other plants that are closely related to that reared plant (induction of oligophagy).

Yamamoto wanted to determine if Tobacco Hornworms were inherently oligophagous insects, as seen in nature, or if they were induced to oligophagy based on the plants they were reared on. To test this, he conducted an experiment under no-choice conditions: the insects were feed only one type of plant until they were first instar larvae. This included an array of solanaceous and non-solanaceous plants. The test concluded that first instar larvae ate a variety of non-solanaceous plants within 24 hours showing polyphagous characteristics. This supported the idea that Tobacco Hornworms were not born oligophagous but instead are polyphagous (able to feed on various plant species and families) at birth and become oligophagous (induction of oligophagy) when reared on solanaceous plants [1, 2].

Thus, behavioral patterns, even when increasing the variety of plants, is still consistent with Yamamoto’s results in which acceptable non-solanaceous reared larvae exhibit a loss of oligophagy that solanaceous reared larvae have and even induce a preference towards certain acceptable non-solanaceous plants. Although rearing might influence the strength of oligophagy and induce a preference, the reason for which Manduca sexta larvae strongly prefer solanaceous to begin with is still unknown.

del Campo and Renwick suggested that the host (solanaceous) preferability could be from the larvae becoming dependent on the chemical constituents of the solanaceous plants. Therefore, to induce a host specificity, the Tobacco Hornworms must become dependent on solanaceous chemical extracts [3]. To directly investigate the role of chemical constituents of the solanaceous plants on the induction of influence preferences, larvae were given a choice of a non-solanaceous disk (cowpea) treated with extract from a solanaceous plant (potato) or cowpea disk alone. The no-choice test showed that larvae reared on a solanaceous diet accepted any solanaceous foliage offered but rejected non-solanaceous foliage (cowpea). In contrast, larvae reared on a non-solanaceous diet equally accepted solanaceous and non-solanaceous foliage [3]. This further reinforced the idea that rearing influenced the strength of preferability because rearing on non-solanaceous foliage decreased the preferability for solanaceous foliage. The choice test showed that a much higher percentage of larvae reared on the solanaceous diet took their first bite on the cowpea disc treated with potato extracts than the control (cowpea). However, larvae reared on the non-solanaceous diet did not discriminate between the control disk and the cowpea treated with potato extract disk. Also, the percentage of larvae taking their first bite of a potato disk with cowpea extract was relatively the same regardless of their reared diet being solanaceous or non-solanaceous [3]. Therefore, the researchers concluded Manduca sexta larvae tend to be attracted to chemical constituents found in solanaceous foliage if reared on a solanaceous diet due to becoming dependent [3].

In this experiment, we aim to better understand feeding behavior (induction of preference and induction of oligophagy) and whether or not induction can cause increased feeding on the same plant species or related species. We also investigate hierarchy in food acceptability from consumption trials for various non-host (Brassicae and Fabaceae) and host (Solanaceae) plants.

Methods

The greenhouse used to cultivate both plants and larvae was an environmentally controlled setting that maintained light, temperature, air, and water. The greenhouse had windows that permit UV light to reach the plants during the day and during the night provided multi-spectrum LED lights that emit a purple wavelength which enables plants to absorb the most amount of light. The greenhouse was also a temperature-controlled environment that consistently remained at 80℉, humidity was also maintained, and water was provided to plants regularly. Larval eggs were imported from Carolina Biological Supply ® by road. A group of five first instar larvae were isolated into clear plastic containers with a piece of foliage from a single plant species like Tobacco or one from either the Brassicaceae or Fabaceae family. Foliage was obtained by removing a healthy piece of leaf from a plant grown in the greenhouse. Then a piece of wet paper towel was wrapped around the stem to prevent the leaf from desiccation. The foliage was replaced every day and thoroughly cleaned. Enough leaves were provided to prevent competition among larvae for food. The survivorship was recorded daily noting the number of larvae alive, the number of larvae dead (removing the dead larvae in the process), the number of larvae reaching premolt, and the total number of larvae up until the fifth instar before testing.

Before reaching the fifth instar, larval premolts were isolated and placed into individual containers. Freshly molted fifth instars were later placed into an arena which was grounded by a plastic grid. The grid was painted and necessary for the larvae to have a matrix to grip to. A fresh leaf from either a member of the Brassicaceae or Fabaceae family, or a tobacco leaf was placed into the arena. The leaf was attached using a pin to prevent movement of the leaf during its consumption by larvae. The arena was covered by a petri dish. A calibration square was placed near each arena for analyzing data. Each arena was labelled including their test plant, the plant the larvae were fed on and the date when the experiment was conducted. A GoPro camera was placed on top of every arena (a total of 11 cameras) and took a picture at an interval of every 30 seconds over a 24-hour period (raw images). Adequate light was provided to avoid reflection.

The raw images from each experiment were collected and categorized into hourly images by selecting images at every hour. Hourly images were then uploaded to Adobe Photoshop where they were cropped with lasso and highlighted with the paint bucket feature. The portion of leaf remaining in every image was highlighted manually in a neon green shade (Analyzed images). The analyzed images were transferred to ELA (Easy Leaf Analysis application) to calculate the area of the leaf by each pixel. The software calculated the number of pixels in the leaf using variation in the ratio of red and blue light present. The pixels were then converted into cm2 to measure the area of the leaf consumed. After calculating the area, the data was used to create graphs which indicated the percentage of leaf consumed over a period of time (Results).

Image 1: Manduca sexta life cycle including each instar. © The Worm Lady [7].

Results

Survivorship-

Figure 1: Mortality of different larval cohorts over 21 days. This data only displays the timing of those larvae that died before reaching the 5th instar. Larvae that survived to reach the 5th instar premolt are omitted from this figure and are instead in Figure 2. N is equal to the number of larvae that died.

Survivorship results showed a noticeable difference in the rearing capabilities of the tested plants. Figure 1 shows the percentage of larval survivability over a given day combined of each replica of larvae that were tested. Tobacco-reared larvae had the lowest mortality rate with only 4 out of 35 (about 11%) larvae mortality rate. In contrast, cabbage-reared larvae and soybean-reared larvae had the highest mortality rate as 44 cabbage-reared larvae out of 45 died (99.98%) and 14 soybean-larvae out of 15 died (99.94%). Of the other two non-solanaceous plants we tested (rapeseed and turnip) showed high survival rates of 64% and 69% respectively (Figure 1).

Figure 2: Number of days it took larvae to reach the premolt stage. N is equal to the number of larvae that reached the premolt stage.

Figure 2 shows how long it took (in days) to reach the 5th instar for those larvae that survived.  The single cabbage-reared larvae that survived took the fewest number of days to reach premolt (11 days). Cowpea-reared larvae took the largest number of days (26 days) for the 7 surviving larvae to reach premolt. The 31 suriving tobacco-reared larvae reached premolt on average in 14 days, which was the second fastest followed by rapeseed (18 days, n=35), soybean (19 days, n=1), and turnip (20 days, n=38). While it may appear that larvae reared on cabbage develop fastest, it should be noted that this data is based on the one surviving cabbage-reared lava, and is clearly an outlier that should not be used to make any conclusions.

Plant consumption-

Figure 3a: Consumption of rapeseed over time (17 hours) by larvae of various rearings.

Larvae of different rearings were tested on one of the four plants and the resulting data were used to create figures 3a-d. Hourly increments of data were mapped for each cohort of larvae and separated by consumption on the plants (cabbage, rapeseed, tobacco, and turnip). A high consumption rate can be seen as a steep sloped line and low consumption rates were seen as flatter slopes. Data was collected for up to 24 hours. In some instances, the larvae stopped feeding prior to the 24-hour data collection period and did not fully consume the plant. For those data, the graph only shows the data collected during the time (15-18 hours) the larvae were still feeding. It was expected that larvae, no matter the rearing, would consume tobacco the fastest since it is a solanaceous plant and was seen to be highly palatable in previous research [2,4,5]. Consumption of tobacco was slightly faster on average than consumption of the other plants.  However, there is not enough data to strongly draw conclusions due to the lack of larval rearings that were tested (Figure 3c). Since rapeseed is known as a very palatable non-solanaceous plant [5], we tested larvae raised on different plants including cowpea for their consumption of rapeseed. Surprisingly, rapeseed-reared larvae did not consume rapeseed the fastest: on average, both turnip-reared and tobacco-reared larvae consumed it faster (Figure 3a). A similar observation is shown up in Figure 3b, where the cabbage-reared larvae did not consume cabbage as fast as the larvae reared on other plants. There were two strong outliers in this graph: one turnip-reared larvae that stopped feeding at 93% and one cowpea-reared larvae that stopped feeding at 75% of the total leaf area (Figure 3a).

Figure 3b: Consumption of cabbage over time (17 hours) by larvae of various rearings.

Figure 3c: Consumption of tobacco over time (15 hours) by larvae of various rearings

In Figure 3b, a trend similar to that shown in Figure 3a was observed.  Cabbage-reared larvae did not consume cabbage the fastest: some rapeseed-reared, turnip-reared, and tobacco-reared larvae finished faster. Two cabbage-reared larvae were included in this Figure as the data from one cabbage-reared larvae was pulled from data obtained during similar experiments performed in our laboratory in 2018 and analyzed using our updated procedures. Figure 3c shows that tobacco was consumed at a faster both rapeseed-reared and turnip-reared larvae, which is consistent with previous observations. One rapeseed larva did not finish within 15 hours during these trials which was the only outlier in this Figure. Consumption of turnip vs. time (Figure 3d) showed the only evidence of induction of preference, where a turnip-reared larva finished a wide margin faster than other caterpillars on rapeseed. Overall tobacco was appeared to be the fastest-consumed plant (Figure 3c), even the larvae reared on turnip or rapeseed.

Figure 3d: Consumption of rapeseed over time (18 hours) by larvae of various rearings.

Discussion

Increasing Resolution

Here, we describe our efforts to better understand feeding behavior of Manduca sexta by making significant improvements in the methods of data collection and data mapping. Major problems that were previously faced included the data load and difficulties in the software. ELA requires specific ranges of color ratios, something that needs to be precisely customized for each leaf so that the software can differentiate between the leaf and the background, and between the leaf and the caterpillar. Due to lighting, glare, and the fact that the caterpillars and the leaves are very similar in color,, finding the right setting was very difficult and often resulted in a compromise which resulted in regions of the leaf being left out, or the caterpillar being partially covered— incorrectly registering pixels.

To improve the fidelity of our experiments and further prevent misrepresentation of the data, we developed a new method of data collection. Our current method involves using the XX in Adobe Photoshop to manually selected hourly images of the leaf. This approach provided a precise mapping of the leaf area with a single bright green color that ELA can easily differentiate. This color overpowered all the glare and worked with the auto pixel tool, so we did not have to adjust the sliders for each batch of images. Another benefit of this approach was that it allowed us to correctly map data in circumstances where the caterpillar covered parts of the leaf that was already consumed. Using the Adobe Photoshop, we can ould interpret the part of the leaf the larvae consumed, and then add color over the caterpillar— something that previously was just left misrepresented. This technique did add additional work to our data analysis period, but additional improvements were made to the previous procedure, which relived the overall workload. To select data points, the previous group first cropped the thousand (or more) pictures collected from the individual trial, and then imported those images into ELA while adjusting sliders and making sure the images were mostly accurate. This process took hours for the system to process alone, taking up many gigabites of space o the computer, and was so much time-consuming that they weren’t able to thoroughly analyze the data using ELA. By only analyzing 24 hourly images, we were able to analyze all the data collected since this approach saved hours of work, hard drive space, and had fewer images that needed to be analyzed.

Challenges with our current experiments primarily resulted from the survival of our larvae from certain rearings (for example soybean) and limitations when trying to grow particular plants species (for example corkscrew vine). Through the duration of these experiments, we had only 1 out of the 45 larave raised on cabbage larvae and only 1 out of the 15 larvae raised on soybean survive long enough to be tested for feeding preferences. Rearing more larvae to the 5th instar on these two plants would have allowed us to obtain more data from which we could then make conclusions.

Another group of larvae that we would have liked to see in our results more was tobacco-reared larvae. Although we raised many tobacco-reared larvae to the 5th instar and tested them on every plant, we were not able to analyze these data in enough time for it to be included in this paper. Corkscrew vine was a plant that we would have liked to test, but shipments of seeds did not make it to our laboratory in time for them to be tested.

Future plans include continuing to collect larval consumption data, especially from larvae reared on the less suitable plants (i.e., soybean and cabbage). We would also like to test a wide variety of solanaceous and non-solanaceous plants that have been examined in previous studies in order to determinin preference hierarchies and strengthening induction claims. Improvements to the data collection pipeline can now allow future groups to spend more time collecting larval data, leading to a bigger study.

Host plant suitability

The survivorship data (Figure 1) shows that certain plants are more suitable for larval development than others. Solanaceous tobacco yielded the highest survivability at 89%, which was expected due to the hornworms’ natural inclination to feed on tobacco plants. There was a great variation in the levels of host plant survivability of larvae on non-solanaceous plants. Specifically, larvae raised on rapeseed and turnip showed increased survivability compared to cabbage, soybean, and cowpea-reared larvae, which had fewer than 20% of the larvae reach the 5th instar. These results are somewhat unexpected because cabbage is in the same family (Brassicaceae) as turnip and rapeseed.  However, only one larva reared on cabbage made it to the 5th instar (out of 44) and therefore would be tested, so it is not yet clear whether or not this is a real difference. If true, this result could suggest that there are subtle differences between the plants in either composition or nutrition that causes change in survivability.

Test plant acceptability

In our consumption data, determining if there is any induction of preference or induction of oligophagy we were able to observe several trends. The data obtained from larvae tested on rapeseed (Figure 3a) had two outliers (turnip-reared and cowpea-reared), but the rest of the larvae all finished rapeseed plants around the same time. These data conflict with our original hypothesis that rapeseed-reared larvae would finish rapeseed faster than larvae raised on other plants (induction of preference). Figure 3b shows similar trends where cabbage-reared larvae did not finish significantly faster than other rearings of larvae and were surprisingly slower than tobacco, rapeseed, and even some turnip-reared larvae. This data also rejects our hypothesis that the cabbage-reared larvae would consume the fastest. In addition, it is not clear whether any induction of oligophagy occurred since larvae raised on a solanaceous plant consumed cabbage faster than larvae reared on the same family of plant (turnip and rapeseed). Tobacco was seen to be a very palatable plant for larvae despite their rearing (Figure 3c) and no consumption difference was seen between turnip or rapeseed-reared larvae when tested on tobacco.

The only evidence of induction of preference, shown in Figure 3d, was when turnip-reared larvae consumed turnip faster than rapeseed-reared larvae. This was seen in only one larva and when the data is averaged, the difference between the two rearings of larval consumption is insignificant and almost identical. The previous study [4], conducted a similar test using turnip-reared larvae tested on turnip and discovered that turnip-reared larvae consumed turnip slower than they do on other plants like tomato (solanaceous) and rapeseed (non-solanaceous). Thus, no induction of feeding preference was observed in this dataset. Additional experiments will be required to determine whether feeding preferences are induced or not.

Conclusions

Overall, the survivorship data reported in this study provides strong evidence that certain plants are much more suitable for rearing Manduca Sexta larvae than others. In addition, the consumption rate data presented here indicates that certain plants are much more palatable for tobacco hornworms than other plants.

The consumption rates for non-solanaceous plants (turnip and rapeseed) were higher than expected. Interestingly, these highly palatable non-solanaceous plants all belong to the same family of plants: Brassicae. More data are required to examine the palatability of other non-solanaceous families like Fabaceae (cowpea and soybean). Previous studies, along with our research, suggests that there are underlying mechanisms affecting feeding preferences in hornworms. Better characterization of these mechanisms requires many additional studies and combinations of controlled research. Finally, this study provides major improvements in the protocol for measuring leaf consumption involving improvement of the fidelity of measurement when using ELA— something that can greatly improve research on this topic in the future.

Acknowledgements

This material is based upon work partially supported by the National Science Foundation under Grant No. 1458149.

Contacts: hcarlson@howardcc.edu, Abigail.livingston@howardcc.edu, Mingquian.huang@howardcc.edu, Rohaan.mahmud@howardcc.edu, Mahdav.fadadu@howardcc.edu Kimia.samieniejad@howardcc.edu, wgretes@howardcc.edu

References

[1] R. T. Yamamoto, “Induction of Hostplant Specificity in the Tobacco Hornworm, Manduca Sexta”, F. Insect Physiology, vol. 20, pp. 641-650, 1974.

[2] G. de Boer and F. E. Hanson, “Foodplant selection and induction of feeding preference among host and non-host plants in larvae of the tobacco hornworm Manduca sexta,” Entomologia Experimentalis et Applicata, pp. 177-193, 1984.

[3] M. L. del Campo and A. A. Renwick, “Dependence on host constituents controlling food acceptance by Manduca sexta larvae,” Entomologica Experimentalis et Applicata, pp. 209-215, 1999.

[4] C. Frimpong, R. C. Roberts, T. R. Gongs, A. Faeque and W. C. Gretes, “Feeding Inductions on Non-solanaceous Plants in Larval Manduca sexta,” Journal of Research in Progress, pp. 74-83, 2018.

[5] W. C. Gretes, E. A. Stanwyck and F. E. Hanson, “Innate and acquired components of oligophagy in the herbivorous lepidopteran, Manduca sexta,” Entomological Experimentalis et Applicata, pp. 259-271, 2016.

[6] C. Berkley, “Teach Life Cycles with the Tobacco Hornworm”.Carolina Biological Supply Company.

[7] The Worm Lady. 2013. http://thewormlady.ca/hornworm-life-cycle.php.

Artwork - “Behind the Stars”

“Behind the Stars” by Janae Bucksell

I took inspiration from the subject matter of what the students were studying and chose to highlight this subject matter in my artwork. I knew nothing about binary stars but learned a lot from them.

Exoplanet Photometry and False Positives

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Shila Deljookorani, Howard Community College

Veronica Kunzle, Howard Community College

Dominic Leger, Howard Community College

Mentored by: Brendan Diamond, Ph.D.

Abstract

The goal of our research is to assist in the confirmation of exoplanets, which are planets outside our solar system. This is accomplished through participation in the Transiting Exoplanet Survey Satellite (TESS) Follow-up Observation Program. The satellite detects a temporary drop in brightness in a star which then requires further observations with ground-based telescopes. One of our primary objectives with the TESS collaboration is to identify false positives, primarily due to eclipsing binary stars where two stars are orbiting one another in close proximity. Over a single night, a few hundred images are taken using a 0.36m telescope at Howard Community College. The time series of images can corroborate any dip in the brightness of the target star or seek alternative sources for the initial signal. We will present examples of false positives detected with our telescope, and how to differentiate between a transiting exoplanet and an eclipsing binary star system.

Introduction

Exoplanet transit photometry, also known as the transit method, is the measurement of a drop-in brightness of a star over a period of time, due to an orbiting planet around the star. Any planet outside of our solar system is called an exoplanet. Each time an exoplanet passes in front of its host star (transits), the planet blocks a small fraction of the star’s light. The graph of brightness as a function of time is created from a series of images. This graph is known as a light curve. A false positive is any event that can mimic an exoplanet transit. Our research group at Howard Community College is part of an international collaboration of scientists working to identify false positives from new exoplanet discoveries. We joined the TESS Follow-Up Observation Program (TFOP) in July of 2019. As a member of TFOP, we assist with observational evidence to indicate if the Transiting Exoplanet Survey Satellite (TESS) has detected an exoplanet candidate or a false positive. By using smaller research telescopes such as at HCC with high magnification, the TFOP is able to obtain higher resolution images then the wide-angle all-sky survey camera on board TESS. This also prevents waste of valuable large telescope time on follow-up observations by large telescopes such as the James Webb Space Telescope (JWST) if false positives are removed from planet candidate databases [1].

False Positives

A drop in a star’s brightness is not primarily due to a transiting exoplanet; the drop could be an indication of another object passing in front of the star also known as a false positive. Identification of false positives are made primarily through the shape of a light curve, alternating shape, percent depth, or using different colored filters.

One primary indicator that distinguishes an observation as an exoplanet or a false positive is the shape of the light curve. The different shapes are produced due to the varying sizes of the objects passing in front of the star. Figure 1 below compares the differences between a v-shape light curve and exoplanet light curve. The v-shape light curve (left) would have a larger drop in brightness; and is commonly due to an eclipsing binary star system (EB). An eclipsing binary system is a two-star system that orbits each other; as they orbit when the smaller star passes in front of the bigger star, it produces a more significant drop in brightness. Although both stars are emitting light, the transiting star, relative to the observer, is still blocking out a portion of the light from the other. The exoplanet light curve (right), has a much smaller drop in brightness, this is due to the transiting object being much smaller than its host star, thus blocking out a much smaller fraction of light. The exoplanet graph will also have a faster decline in light, with an essentially constant base, then finally increase once again, thus creating a bucket-shaped curve.

Figure 1: Representative diagrams of two different light curves. To the left is a v-shaped false positive, and to the right is a bucket shaped graph which could be from a planet much smaller than the host star.

The alternating v-shaped false positives are due to multiple transits occurring on the observed star. Figure 2 below is a representation of how a v-shape alternating light curve would appear. The primary causation of these dips are also caused by eclipsing binaries (EB); these star systems are orbiting around a center of mass. When both a dimmer star and a brighter star are in the field of view, their emitting light combines creating a constant light graph. However, due to the position of the observer, we see the star system orbiting each other. As these stars orbit around each other, they produce a variation in a drop brightness. As the dimmer star passes in front of the brighter star (secondary eclipse), we have a smaller drop in brightness. As the dimmer star occults (passes behind), the larger star (primary eclipse), it will result in a more significant drop in the brightness. Looking at Figure 3 below, the first decline in brightness is representing the secondary eclipse, and the second drop is representing the primary eclipse.

Figure 2: Representative diagram of a light curve with alternating V-shaped decreases in brightness. These may be caused by eclipsing binary stars as they alternate which star is in back.

Another factor we look for when differentiating between an exoplanet transit and a false positive, is the percent drop in brightness. As mentioned previously, an exoplanet light curve would have a much smaller dip in brightness compared to a false positive. As we observe and produce a light curve graph, our data will tell us the percent drop during the transit. Planets are smaller than stars, and there is a maximum dip in the brightness that would indicate that we are looking at a false positive. When we have a transit that has over a 2% drop in brightness, we can predict that the observation is a false positive, rather than an exoplanet transit. The 2% dip is far too large for the causation to be an exoplanet. In Figure 3 below is a representation of the dip in brightness we could observe.

Figure 3: 2% depth graph.

Obtaining images with different filters help us to distinguish an exoplanet transit from an eclipsing binary system; since different stars give off distinct colors due to their temperature and sizes their depths will be different once data is processed to create a light curve. Depth variations in when using various filters gives an indication of a possible EB. This is another reliable indicator for another type of false positive. In Figure 4 below, there is a light curve represented in two different filters.  The red filter has a much shorter depth than the blue filter, indicating that there are two different stars being observed. If the event had been caused by the silhouette of an exoplanet, the drop-in brightness would be a consistent depth across all photometric filters.

Figure 4: Example of how curves can differ based on different filter usage. This variable brightness drop is otherwise known as chromaticity.

The TFOP program concerns follow-up observations with the data given to us by TESS. TESS uses larger pixels since it has such a wide field imaging range; thus, it is unable to differentiate between different light coming from different stars. Figure 5 shows how one TESS pixel (21 arc-seconds) is significantly larger compared to our pixels (0.72 arc-seconds). Ground-based telescopes have a much higher resolution; we can distinguish between the light coming from different stars. Without the blending of light and the higher resolution, we also assist in producing evidence of whether a transit is an exoplanet or false positive.

Figure 5: TESS pixel size with (inner box) compared to our images.

Materials and Methods

Our primary observation setup consists of a Planewave 0.36-meter telescope, SBIG 6303 CCD (camera), and a computer in a control room that runs the observation software. We use a confidential database of potential planet candidates from the TFOP to select our targets for our observations based on position, brightness, depth of transit and other factors. The main observation software consists of the programs MaximDL and Maxpoint, used for telescope controls and image tracking. In order to process data, we use AstroImageJ, an image processing program designed explicitly for exoplanet photometry [2]. The hardware setup is the same as the previous exoplanet research team featured in JRIP 2019 [3]. Previous observations following a similar set of methods can be seen in the Astronomical Journal article below [4].

MaximDL software allows us to control the telescope and move it into the target position to start capturing images. One of the main functions of this program is to resolve the position of the telescope, to allow precision tracking using a secondary guiding camera. Maxpoint is software that further improves tracking as well as the pointing of the telescope. Maxpoint models the telescope’s gears in the mount based on 50 to 100 stars evenly distributed across the full sky from a calibration night [5]. To limit systematic errors in the response of different parts of the CCD, tracking is maintained on a target star within a couple arcseconds of angle, corresponding to only a couple of pixels position.

In order to collect accurate data, there are parameters of the chosen target start given to us from TFOP that we need to take into consideration. These parameters involve the duration of the transit, the magnitude of the star, and the projected depth of the light curve. These variables control how we set up our telescope, as well as how long we would be taking images. MaximDL needs to be configured one more time to automatically take images and save them to a specific file location. We use a feature from MaximDL termed autosave, which allows the automatic taking of images to be saved to a specific file. Once autosaving is set up, the next thing to do is make sure tracking is maintained, and image saving is working throughout the observation period. The image taking process must begin prior and end later than the actual transit time in order to ensure there is a baseline of data.

A useful component of our imaging system is the tip/tilt autoguiding, also known as adaptive optics. This is a hardware upgrade to the SBIG CCD that is controlled through MaximDL in order to compensate for vibrations in the telescope and atmospheric turbulence. Autoguiding is done by rapidly moving a thick window in front of the CCD to offset the image, allowing for precise images to be taken even when the target star is moving [6]. Figure 6 shows the tracking menu of the SBIG AO controls, which allows for several options to be modified in order to accommodate varying observation conditions. Tracking and adaptive optics increase the precision in our observations and allow us to take better images and look at more difficult targets with greater confidence.

Figure 6: The SBIG AO Control menu [6].

In order to process our data, we use AstroImageJ, which is a derivative of the program ImageJ that is explicitly designed to be used with exoplanet photometry. Prior to processing our images, we first sort them and remove images with poor quality. This low quality can be caused by a variety of factors, such as a temporary loss of tracking, a power loss to one or more components, or an object getting in the way. Some images may also be taken without specific FITS header (image metadata) components, which requires them to be processed again through AstroImageJ to add necessary FITS header information. Once all the poor-quality images are removed from the rest of the image stack, the images can be appropriately calibrated as described by the exoplanet research team featured in JRIP 2019 [3].

While the private TFOP database gives us coordinates for a predicted transit event occurred according to TESS’s data, it does not guarantee there is a transit event on the listed star. In order to locate the event, we need to select several target stars and comparison stars. These target and comparison stars should all be of similar brightness and size, but they can vary slightly between each star if needed, as seen in Figure 7. Once our target and comparison stars are selected, we use AstroImageJ to automatically repeat the star selection for all the other images within the calibrated image stack, and we can then resolve a graph from it. AstroImageJ compares the comparison and target stars using an aperture and annulus to measure the brightness, as shown in Figure 8. Additional information regarding this process can be found in an article published by Cornell University [2]. The graphs developed from this process still need to be adjusted in order to match TFOP standards. These standards ensure a high degree of clarity in each graph, and all submissions to the TFOP database must be raised to them prior to submission. Once those options are modified, and the graphs are raised to TFOP standards, the image processing is completed, and a report can be sent to the collaboration. For a more detailed analysis on the methods to find false positives, see the article by Dennis Conti below [7].

Figure 7: The target and comparison star selection view in AstroImageJ. Target stars are labeled with T prefix while comparison stars are labeled with C prefix.

Figure 8: Up-close aperture (inside circle) and annulus (outside ring) of the circle. The annulus measures the background light while the aperture measures the inside light. The ring between the two is not measured.

Results

Throughout our time participating in the TFOP program, we have contributed ten successful observations to the TFOP database of exoplanets maintained by MIT [8]. Our observations are limited to a minimum depth of 4 parts per thousand (ppt), meaning that objects that block very little light of their host star cannot be observed effectively using our equipment.

Of the ten observations listed below in Table 1, seven were labeled as false positives. These results are categorized as NEB, PC, BEB, and EB. Ingress refers to the spot at which the whole planet enters the star’s disc (start of the transit), and Egress refers to the point at which the planet reaches the opposite side of the star’s disc and exits from it (end of the transit) [9].

Table 1: Table of submitted observations to TFOP database.

Target

Obs. Date

Depth (ppt)

Result

Observation Summary

395921960

Jul. 27, 2019

35

EB

Known EB used to verify our data reduction procedures. Full transit.

176675077

Jul. 30, 2019

300

NEB

Known EB used to verify our data reduction procedures. Full transit.

434105231

Aug. 11, 2019

20

BEB

Full transit. Combined observations showed chromaticity.

379931600

Sept. 26, 2019

80

NEB

Ingress +70%.

232967440

Oct. 23, 2019

8

PC

Full transit. Possible large planet (9 REarth)

199688472

Oct. 24, 2019

10

EB

Ingress +20%. Later observations showed V-shape curve.

468845640

Nov. 02, 2019

80

NEB

Full Transit.

219852584

Nov. 07, 2019

8

PC

Egress +40%. Possible large planet (16 REarth)

17005768

Dec. 22, 2019

10

BEB

Full transit. Confirmed BEB with TFOP spectroscopic observation.

375506058

Dec. 24, 2019

3-6

PC

Full transit in g’ and r’ filters. Depth is near our equipment uncertainty so requires high precision follow up observations.

Figure 9: The flux or brightness of the target and comparison stars obtained on October 23th, 2019 (universal time). The graphed lines represent the target star and comparison starts. Each set of vertical points are from a 45 second exposure. The graphs were also shifted down on the Y axis for easy viewing.

Figure 9 is an additional graph for our observation on October 23rd, 2019. After analysis in AstroImageJ, we found that the graph showed signs of being an exoplanet candidate. The drop-in brightness of the target star was calculated at 0.08%. In order to ensure that this was not caused by an eclipsing binary and was a possible exoplanet transit, follow up observations were conducted by other groups using different light filters to look for chromaticity such as in Figure 4. The follow up observations showed no change in the brightness drop, leading to the target star being labeled as a planet candidate in Table 1.

Figure 10: The flux or brightness of the target, nearby eclipsing binary and comparison stars (both shifted vertically for clarity) obtained on November 11th, 2019. Red points represent the target star brightness in each image, blue triangles represent the nearby eclipsing binary and comparison stars are shown with gray crosses. In order to make seeing the graph easier, it was shifted down by .3.

As shown above in Figure 10, the target star (in red) did not have a brightness drop, but the nearby star, TIC468845640-0 (in blue), unexpectedly did. The star dimmed as expected of an eclipsing binary throughout observation, with a deep V curve and rounding towards the bottom. As the target star and nearby eclipsing binary were orbiting each other, a portion of nearby eclipsing binary’s brightness was blocked due to their transit, mimicking a light curve of an exoplanet. This drop was measured using a Sloan r’ light filter and AstroImageJ calculated the peak depth of the drop at 7.92%. Once this high drop was found, the target was ruled as a false positive caused by an NEB.

Challenges

Our time spent working with the TFOP collaboration has yielded a lot of useful observations, however these only came after overcoming many challenging and unsuccessful experiences. We began to use remote observation and tracking this year, which were big steps to take for our group. While advancing into new territory between this year and last year has brought new progress, it also brought new issues unique to our project that we have learned from. For every observation we submitted to the exoplanet database, we had to terminate at least one other observation due to either bad weather, hardware issues, or software issues.

Poor weather conditions were a major limiting factor in our observations. There were many instances where the observation had been canceled before we had begun the observation process, due to a sudden overcast of clouds. Another weather factor that hindered our research was wind. Winds that we would experience every so often, would overwhelm the adaptive optics, ruining the images by causing the telescope to shake during image capture.

Hardware issues were also a limiting factor in our observations. While our availability was greatly increased by using remote observation, so was the potential for problems we could not solve without being on site. One of these issues was internet disconnection, causing the telescope to sit idle after finishing a meridian flip. Meridian flips are required in order to continue tracking on a target that is moving around our telescope, since the mount does not rotate completely. While this reorientation does normally cause us to lose one image, we couldn’t remotely restart tracking on the telescope, resulting in most of the observation being lost. Another issue that caused premature termination of observations was intermittent battery failures on the camera. While it was an effective method to protect the camera from a power surge, the battery failing ended up causing multiple observations to fail due to the inability to track or image with the CCD.

Software issues, while the rarest, were the most severe. In one of the two cases, the telescope was calibrated to the wrong orientation, which caused a failure in tracking. This ended up causing the telescope to rotate into the base of the pier. If it were not for the telescope automatically stopping, this would have caused serious damage to the telescope’s motors. Another problem was not noticed until after data processing. During the calibration phase, the camera’s shutter was open while taking some images where it needed to be closed. This created improper calibration images, which limited the quality of the final light curve.

Conclusion

Throughout the duration of our research project, our group has learned how to distinguish between a false positive and exoplanet candidate using exoplanet photometry. We used the telescope, CCD, and other equipment to observe target stars gathered from the TFOP database, and currently we have contributed 10 successful observations to the program. Through these, our group has been able to observe the major aspects of eclipsing binary false positive stars, including the large brightness drop, v-shaped graph, and chromaticity of the target star. Our November 2nd, 2019 observation in Figure 10 above is an example of all three characteristics, with the characteristic v-shape being the most prevalent of the three. Alongside that, the measured brightness drop of 7.92% added more evidence of this target being a false positive. Further follow-up observation showed chromaticity as well. The results of analyzing these false positive observations demonstrate that there is a difference between them and a normal exoplanet. Exoplanet brightness drops lack any chromaticity and do not have the characteristic v-shaped brightness drop seen in all false-positive binary stars.

Acknowledgements

This material is based upon work partially supported by the National Science Foundation under Grant No. 1458149.

Contacts: shila.deljoo@yahoo.com, verkun6430@gmail.com, dominicleger98@gmail.com, bdiamond@howardcc.edu

References

[1] G. R. Ricker and et al., “Transiting Exoplanet Survey Satellite (TESS),” Journal of Astronomical Telescopes, Instrument, and Systems, vol. 1, no. 2015, 2015.

[2] K. A. Collins, J. F. Kielkopf, K. G. Stassun and F. V. Hessman, “AstroImageJ: Image Processing and Photometric Extraction for Ultra-Precise Astronomical Light Curves (Expanded Edition),” Cornell University, Louisville, Kentucky, 2017.

[3] J. Hinchey, W. Jeffreys and P. Samadani, “Fantastic Exoplanets and Where to Find Them,” Journal of Research in Progress, vol. 2, pp. 2-11, 2019.

[4] K. A. Collins and et al, “The KELT Follow-up Network and Transit False-positive Catalog: Pre-vetted False Positives for TESS,” The Astronomical Journal, vol. 156, no. 5, p. 234, 2018.

[5] Diffraction Limited, “Maxpoint,” Diffraction Limited, [Online]. Available: https://diffractionlimited.com/product/maxpoint/. [Accessed 2020].

[6] Diffraction Limited, “SBIG AO Control,” Diffraction Limited, [Online]. Available: https://diffractionlimited.com/help/maximdl/MaxIm-DL.htm#SBIG_AO_7.htm. [Accessed 2020].

[7] D. M. Conti, “Exoplanet False Positive Detection with Sub-meter Telescopes,” Annapolis, MD, 2018.

[8] S. Seager, N. Guerrero, D. Latham and C. Burke, “TESS Observations,” Massachusetts Institute of Technology, 2018. [Online]. Available: https://tess.mit.edu/observations/. [Accessed 2019].

[9] D. M. Conti, “A Practical Guide to Exoplanet Observing,” 2018. [Online]. Available: https://astrodennis.com/Guide.pdf. [Accessed 2019].

Equivalence Classes of Complex Numbers Modulo a Natural Number

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Jonathan Kraus, Howard Community College

David Ryan Johnson, Howard Community College, 2019,Towson University

Mentored by Dr. Mike Long, Howard Community College

Abstract

Equivalence classes of natural numbers are sets of numbers that have a common characteristic.  One well-known equivalence class involves the modulus operator, or the value of the remainder when a natural number is divided by “n,” which is also a natural number.  For example, an equivalence class modulo 4, would be a set of numbers that have the same remainder when divided by 4.  Since there are four possible remainders when dividing by 4, there would be four equivalence classes in this case.  The question that emerges is what are the characteristics of equivalence classes of complex numbers modulo a natural number, “n”?  In this project, we explore those mathematical characteristics that unfold in the equivalence classes and also the geometry of the classes in the complex plane.   Further exploration of the geometry of the classes yielded intriguing, yet well-known, mathematical patterns.  This leads to another question, what patterns exist in the geometric structures of the equivalence classes with complex numbers modulo n?  We will describe those patterns and offer proof or rationale for the existence of the patterns.

Introduction

The complex numbers in mathematics are numbers that have the form a + bi with the “a” being the real part of the number and the “b” being the coefficient of the imaginary part “i” [1].  The “i” is the mathematical representation of the square root of negative one.  When studying the algebra of the real numbers, the square root of negative one is undefined, but the complex numbers allow for the square root of negative one to be defined and, with algebra, the square root of any negative number can be defined.  By defining the square root of negative one, equations that could not be solved over the real numbers can be solved.

The second part of this research involves equivalence classes.  In elementary abstract algebra or discrete mathematics, one learns about the idea of mathematical equivalence classes.  To begin defining an equivalence class (and equivalence relations), consider a set of elements and subsets of those elements that have a common characteristic or relationship [2, 3].  The equivalence relation mod n where “n” is a positive integer or natural number, is the important part of this paper.  To better understand this notion, let’s consider the set of positive integers {1, 2, 3, 4, 5, 6, 7, 8, 9} and the equivalence relation mod 3.  Another way of thinking about this is that the difference of any two elements in the equivalence classes is a multiple of three.  More formally, x – y is equal to a multiple of three where x and y are any random elements of the subset.  In order to be an equivalence relation on a set, three properties must exist: reflexive, or in this case x – x has to be divisible by three; symmetric, or in this case if x – y is divisible by three then y – x must be divisible by three; and transitive, or in this case if x – y and y – z are divisible by three, then x – z must be divisible by three.  (Here z is also an element of the original set.)  It can be shown with a simple proof that mod 3 is indeed an equivalence relation.

Proof that mod 3 is an equivalence relation

First, it must be shown that the reflexive property holds.  So every element has to be related to itself or x – x has to be divisible by three, where x  is a random element of the set being considered.  Since x – x = 0, which is also equal to three times 0,  x – x is divisible by three.  This shows the reflexive property holds.  Next, it must be shown that the symmetric property holds.  This means that if x is related to y, where x and y are random elements of the set being considered, then y is related to x or, equivalently,  if x – y is divisible by three then y – x must be as well.  We start by saying that x – y  is divisible by three and so there exists an integer, call it k, such that  x – y = 3k.  If both sides are multiplied by negative one then the equation becomes -x + y = -3k which can be rewritten as y – x = 3(-k).   The (-k) will still be an integer and so x is divisible by three.  This shows the symmetric property holds.  Finally, it must be shown that the transitive property holds.  This means that if x  is related to y and y is related to z then x is related to z, where x,  y, and z are random elements of the set being considered or if x – y and y – z are divisible by three then x – z is divisible by three.  We start by saying that x – y and y – z are divisible by three and so there exists integers  j and k  such that x – y = 3k  and y – z = 3j.  Then if x – y and y – z are added together (x – y) + (y – z) = 3k +3j and then x – z = 3(k + j), where k + j is an integer.  Therefore x – z is a multiple of three.

Another way to think about this same relation is which elements in the original set {1, 2, 3, 4, 5, 6, 7, 8, 9} have the same remainder when divided by three.  So one equivalence class would be the elements of the original set that have a remainder of zero when divided by three {3, 6, 9}.  Another class would be the elements of the original set that have a remainder of one when divided by three {1, 4, 7}.  And the final class would be the elements of the original set that have a remainder of two when divided by three {2, 5, 8}.  It should also be noted that this equivalence relation forms the partition of the original set.  In other words, the three subsets have no common elements and when the three subsets are put together, they form the entire original set.

These two ideas, complex numbers and equivalence classes, lead to the research here.

First Research Question

What do the equivalence classes of the complex numbers modulo n or mod n  (where n is a natural number) with integer values for the real and imaginary part, look like algebraically, but more important, what do they look like geometrically?  Two immediate considerations emerge:

- First, a definition must be established for complex numbers modulo n and in this case the conventional definition that both the real and imaginary parts will be independently evaluated modulo n or mod n, will be used.

- Second, the geometry will require two dimensions in order to account for the real and imaginary parts when graphing the complex numbers. This second consideration leads to a major part of the research.

In order to understand the idea of the equivalence classes, a table of complex numbers with integral values for the real and imaginary parts will be examined mod 3.  This means looking at the remainders when the real and imaginary parts of the complex number are independently divided by three.  Negative numbers need not be addressed here as we are simply looking at the remainders when dividing by “n.”  Table 1 shows this.

Table 1: Table of complex numbers with integer values for the real and imaginary parts mod 3.

Complex Number

Complex Number
mod 3

Complex Number

Complex Number
mod 3

0 + 0i

0 + 0i

3 + 0i

0 + 0i

0 + 1i

0 + 1i

3 + 1i

0 + 1i

0 + 2i

0 + 2i

3 + 2i

0 + 2i

0 + 3i

0 + 0i

3 + 3i

0 + 0i

0 + 4i

0 + 1i

3 + 4i

0 + 1i

0 + 5i

0 + 2i

3 + 5i

0 + 2i

0 + 6i

0 + 0i

3 + 6i

0 + 0i

0 + 7i

0 + 1i

3 + 7i

0 + 1i

0 + 8i

0 + 2i

3 + 8i

0 + 2i

1 + 0i

1 + 0i

4 + 0i

1 + 0i

1 + 1i

1 + 1i

4 + 1i

1 + 1i

1 + 2i

1 + 2i

4 + 2i

1 + 2i

1 + 3i

1 + 0i

4 + 3i

1 + 0i

1 + 4i

1 + 1i

4 + 4i

1 + 1i

1 + 5i

1 + 2i

4 + 5i

1 + 2i

1 + 6i

1 + 0i

4 + 6i

1 + 0i

1 + 7i

1 + 1i

4 + 7i

1 + 1i

1 + 8i

1 + 2i

4 + 8i

1 + 2i

2 + 0i

2 + 0i

5 + 0i

2 + 0i

2 + 1i

2 + 1i

5 + 1i

2 + 1i

2 + 2i

2 + 2i

5 + 2i

2 + 2i

2 + 3i

2 + 0i

5 + 3i

2 + 0i

2 + 4i

2 + 1i

5 + 4i

2 + 1i

2 + 5i

2 + 2i

5 + 5i

2 + 2i

2 + 6i

2 + 0i

5 + 6i

2 + 0i

2 + 7i

2 + 1i

5 + 7i

2 + 1i

2 + 8i

2 + 2i

5 + 8i

2 + 2i

In this example of the equivalence relation mod 3, there are nine different equivalence classes that emerge and they are represented by the complex numbers in the “Complex Number mod 3” column.  Let’s consider the equivalence class 1 + 2i to solidify this idea.  The numbers highlighted all belong to the same equivalence class, 1 + 2i, since they all have the same remainders when the real and imaginary parts of the complex number are independently divided by three.  The complex numbers within this equivalence class have the three properties already mentioned:  reflexive, symmetric, and transitive and that is proved here for a generic complex number of the form a + bi.

Proof that mod 3  is an equivalence relation

First, it must be shown that the reflexive property holds.  This means that a + bi needs to be related to itself or (a + bi) – (a + bi) has to be a multiple of three.  (a + bi) – (a + bi) = 0 = 3 x 0.  Zero is a multiple of three, and therefore a + bi is related to itself.  This shows the reflexive property exists.  Next, it must be shown that the symmetric property holds.  This means that a + bi and c + di need to be symmetric or if (a + bi) – (c + di) is a multiple of three, then (c + di) –  (a + bi) is also a multiple of three.  Since (a + bi) – (c + di) is a multiple of three there exists an integer k such that (a + bi) – (c + di) = 3k.  If both sides are multiplied by negative one then (-1)((a + bi) – (c + di)) = (-1)3k = 3(-k) = (c + di) –  (a + bi).  This shows the symmetric property exists.  Finally, it must be shown that the  transitive property holds.  This means that a + bi, c + di,  and f + gi need to be transitive.  In other words, if (a + bi) – (c + di) is a multiple of three, and (c + di) – (f + gi) is a multiple of three, then (a + bi) – (f + gi) is a multiple of three. Since (a + bi) – (c + di) is a multiple of three and (c + di) – (f + gi)  is a multiple of three, there exists integers j and k such that (a + bi) – (c + di) = 3k and  (c + di) – (f + gi) = 3j.  Now ((a + bi) – (c + di)) + ((c + di) – (f + gi)) = 3k + 3j = 3(k + j) = (a + bi) – (f + gi).  This shows that (a + bi) – (f + gi) is a multiple of three.  This shows that the transitive property exists and completes the proof.

What’s interesting to note is that there are nine equivalence classes, which happens to be a perfect square number, when considering the complex numbers mod 3.  This makes sense because when dividing by three there are three possible remainders:  0, 1, and 2.  Since we have to consider both real and complex (imaginary) parts, it is like considering the cross product of A and B where A is the possible remainders for the real part:  0, 1, and 2, and B is the possible remainders for the imaginary part:  0, 1, and 2.  So there would be nine possibilities where the first coordinate of the ordered pair is the remainder for real part and the second is the remainder for the imaginary part mod 3: . This could be easily extended to the complex numbers, with integral values, , where n is a positive integer.  And so in all cases, there would be , or a square number of equivalence classes.  Algebraically, we can say that the number of equivalence classes of the complex numbers with integral coefficients mod n, where n is a natural number, is a perfect square.

Now let’s move forward and investigate the geometry of these equivalence classes.  Remember that the complex numbers require two dimensions to be represented graphically.  Think of it as the x-axis and y-axis.  The nine equivalence classes of the complex numbers mod 3 are graphically represented in Figure 1 below.  The arrangement of the equivalence classes forms a square lattice, which seems fitting given that nine is a perfect square number.  Again, this can be easily extended to mod n where the equivalence classes would form an square and with ordered pairs (real and imaginary parts) with values from zero to n – 1, where the zero to n – 1 show all of the possible remainders when the real and imaginary parts are divided by n.  So geometrically, equivalence classes of the complex numbers with integral coefficients mod n, where n is a natural number, form square lattices.

Figure 1:  Equivalence classes of complex numbers, with integral real and imaginary parts, mod 3.

Second Research Question

A second question involving the complex numbers and their equivalence classes emerged after exploring the basics of the geometry. Suppose that you consider a complex number involving integral real and imaginary parts and all possible positive and negative values for those real and imaginary parts. Is there anything unique that happens in the geometry?  In order to begin answering this question we need to identify the real and imaginary parts of the complex number.  For simplicity, “a” will be the real part and “b” will be the imaginary part.  So the four complex numbers to be considered will be:  a + bi, a – bi ,-a + bi, and  -a – bi. Again, we want to consider each of these complex numbers mod n where n is a natural number or positive integer.  And both the real and imaginary parts will be independently evaluated mod n as before.  It turns out an interesting pattern emerges.

We will use the example along with the positive and negative combinations of the number as suggested above to begin to form a pattern:

We will also take all of the values to be positive in the equivalence classes.  Again, negative numbers need not be addressed here as we are simply looking at the remainders when dividing by “n.”  So in this example, a = 4 and b = 5 and n = 3.  Figure 2 organizes this example for simplicity.

Figure 2: This shows the geometry that results from graphing the four complex numbers above mod 3.

A rather simple geometric figure emerges, a rectangle, and in this particular case it is a square.   Figure 3 shows the graph of the results in Figure 2.

Figure 3: Graph of (4 + 5i), (4 – 5i), (-4 + 5i), and (-4 – 5i) mod 3.

This notion can be generalized as seen below and in this generalization a rectangle again forms in all cases given the coordinates of the equivalence classes of the complex numbers, with integral real and imaginary parts, mod n.  Figure 4 organizes this generalization for simplicity.

Figure 4: This shows the geometry that results from graphing the four generalized complex numbers above mod n.

Figure 5 shows the graph of the generalization in Figure 4.

Figure 5:  Geometry of (a + bi) mod n = s + ti, (a – bi) mod n = s + (n – t)i, (-a + bi) mod n = (n – s) + ti, and (-a – bi) mod n = (n – s ) + (n – t)i.

Third Research Question

When looking at the geometry of these four complex numbers, (a + bi) mod n =  s + ti, (a – bi) mod n = s + (n – t)i, (-a + bi) mod n = (n – s) + ti, and (-a – bi) mod n = (n – s) + (n – t)i, a question that emerges is whether or not the angles between the complex numbers in these rectangles that were formed when the equivalence classes were graphed, have any patterns?  One has to consider these complex numbers as vectors to do these explorations.  It was thought some patterns might emerge here, but exhaustive research using spreadsheets did not yield anything specific.

Fourth Research Question

While exploring the rectangles that are formed when a + bi, a – bi, -a + bi, and –a – bi (with integer values for a and b) are considered mod n, another question emerged.  What are the sizes and corresponding areas of the rectangles that are formed when the modulus is changed and what is the area of those rectangles?  There are some limitations on the size of the rectangles, depending on the modulus, but some patterns do emerge.  The following table, Table 2, shows the different rectangles that can be formed and the areas of these rectangles for an odd modulus.

Modulus

Sizes of Possible Rectangles

Area of Rectangles

3

1X1

1

5

1X1

1

1X3

3

3X3

9=10-1

7

1X1

1

1X3

3

1X5

5=6-1

3X3

9=10-1

3X5

15

5X5

25=28-3

9

1X1

1

1X3

3

1X5

5= 6-1

1X7

7=10-3

3X3

9=10-1

3X5

15

3X7

21

5X5

25=28-3

5X7

35=36-1

7X7

49=55-6

11

1X1

1

1X3

3

1X5

5=6-1

1X7

7=10-3

1X9

9=10-1

3X3

9=10-1

3X5

15

3X7

21

3X9

27=28-1

5X5

25=28-3

5X7

35=36-1

5X9

45

7X7

49=55-6

7X9

63=66-3

9X9

81=91-10

Table 2:  Rectangles that can be formed for an odd modulus.

If we consider the modulus in the first column to be generated by the closed formula 2k + 1 with k as a positive integer greater than or equal to one, then the number of rectangles that can be generated turns out to be the kth triangular number, i.e., the sum of the first “k” positive integers.  In other words, in mod 3, where  3 = 2(1) + 1, there is one rectangle that can be formed; in mod 5, where 5 = 2(2) + 1,  there are three rectangles that can be formed;  in mod 7, where 7 = 2(3) + 1, there are six rectangles that can be formed.  So the triangular numbers:  1, 3, 6, 10, etc. , with closed formula, , where n is a natural number, emerge to represent the number of rectangles that can be formed for larger and larger odd moduli.  It should be noted that the indices of the triangular numbers are the numbers being substituted into the closed formula mentioned above.  The structure of the table above allows for the triangular numbers to be seen.  For example, in mod 9, where the number substituted into the closed formula is four, there are ten different rectangles and ten is the fourth triangular number (index of four).  There are 4 rectangles with the first dimension listed as 1, 3 rectangles with the first dimension listed is as 3, 2 rectangles with the first dimension listed as 5, and 1 rectangle with the first dimension listed as 7 and 4 + 3 + 2 + 1 = 10.

There are some other patterns that appear to emerge, but identifying conjectures for the areas has been more challenging.  In the table above, one can see a conjecture that is left for further research: it appears that the areas of the rectangles are either triangular numbers or are differences of non-adjacent triangular numbers.

Before moving on, consideration must be given to the different rectangles that can be formed and the areas of these rectangles for an even modulus.  Table 3 below shows this.

Modulus

Sizes of Possible Rectangles

Area of Rectangles

4

2X2

4

6

2X2

4

2X4

8

4X4

16

8

2X2

4

2X4

8

2X6

12

4X4

16

4X6

24

6X6

36

10

2X2

4

2X4

8

2X6

12

2X8

16

4X4

16

4X6

24

4X8

32

6X6

36

6X8

48

8X8

64

Table 3:  Rectangles that can be formed for an even modulus.

Again the triangular numbers appear in the number of different sized rectangles that exist for larger and larger even moduli.  However, there is no conjecture that can be made about the areas and that can be left for further research.

Conclusion

This research set out to explore the algebra and geometry of equivalence classes of the complex numbers with integral coefficients mod n, with n as a natural number or positive integer.  Algebraically, it was determined that the number of equivalence classes is a perfect square and geometrically the equivalence classes form an square lattice when graphed on the complex plane.  This research also set out to identify patterns that exist in the geometric structures of the equivalence classes with complex numbers mod n.  The equivalence classes of all possible positive and negative combinations of a + bi, namely:  (a + bi) mod n = s + ti, (a – bi) mod n = s + (n – t)i, (-a + bi) mod n=  (n – s) + ti, and (-a – bi) mod n = (n – s) + (n – t)i were explored and it was determined that geometrically a rectangle would be formed in all cases. When the angles between the complex numbers of the equivalence classes above (when the complex numbers were considered as vectors) were explored, nothing was found.  Finally, the number of rectangles that can be formed by a different modulus, both even and odd, as the size of the modulus increases, turns out to form a pattern, the triangular numbers.

Contacts: djohns77@students.towson.edu, jonathan.kraus@howardcc.edu, mlong@howardcc.edu

References

[1] Mardsen, J. E. and Hoffman, M.  (1987).  Basic Complex Analysis.  W.H. Freeman.

[2] Ensley, D. E. and Crawley, J. W. (2006).  Discrete Mathematics.  John Wiley and Sons, Inc.

[3] Burton, D. M. (1991).  Elementary Number Theory.  McGaw Hill.

So, You Want to Win Plinko?

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Benjamin Marasco, Howard Community College

Naod Assefa, Howard Community College

Mentored by: Dr. Mike Long

Abstract

On January 3rd, 1983 the world was introduced to the Price is Right’s newest game: Plinko (CBS Interactive, n.d.) A contestant stood on the stage at the Plinko board, ready to drop chips in the pachinko-like board, for a chance to win different cash prizes depending on which chamber the chip fell into. Each time the contestant went to drop a chip into the board the roaring voices of the audience would attempt to guide the contestant to where to drop the chip to win the most money. Where should the contestant drop the chip to win the most money? What does the voice of mathematical reason tell the contestant? Most importantly, will the contestant listen to the quiet voice of mathematical reason, or the overpowering voices of the audience? We set out to determine what it was that the quiet voice of mathematical reason was telling the contestant all along. We created computer models to simulate the game and determine where a chip is likely to land depending on where it was dropped. From that, we started asking questions about how much the show’s producers realistically thought they might have to pay the contestant. While building the computer model, we found ourselves asking more questions about what might happen if the board was modified. As it turns out, Plinko type minigames have been popping up in numerous places, with dimensions different from the original game. The computer model was modified to determine how the different dimensions of the board would impact where the chip would land depending on where it was dropped. We revealed what it was that the voice of mathematics should be saying to each contestant to help win the best prize. Now come on down, and take your chances at Plinko, and let your mathematical voice be your guide.

Introduction

Over sixty years ago, the world was introduced to the long-running game show, the Price is Right [1]. In 1972, the game show was revamped into its current form and has been running ever since [1]. On January 3rd of 1983, the contestant who had just bid the closest price on an item won the opportunity to play a new pricing game and that game was Plinko [2]. The game offered the opportunity for a contestant to win up to $25,000 [3]. Although, it was and still is possible that the contestant may not win anything at all. The contestants win monetary values by dropping chips from the top of a large pachinko-like board that has nine possible slots to drop into, thirteen rows of pegs, with every other row offset, and nine chambers at the bottom where the chips can land. The chips fall down the board into the nine chambers across the bottom of the board that are marked with dollar values, including $0. Since these zero value chambers exist, there is a possibility that the contestant does not win anything. The contestants are given one chip and can win up to four other chips by guessing the price of four products.

Some fun facts about Plinko on the Price is Right:

- No contestant has ever landed all five chips in the center chamber, which has the highest dollar value [3]

- To date, the largest amount of money ever won at Plinko is $39,200 [4]

- To maintain consistency, there are only ten Plinko chips, and they are locked up after each use [2]

- The chips were designed to make the “plink” sound on the board [3]

Background Mathematics

The mathematics of the basic Plinko game is pretty straightforward and has been researched. The board itself looks like Figure 1 below.

Figure 1: A standard Plinko board.

As the chips fall through the pachinko-like board, there are two possibilities each time a chip hits a peg. That leads to some interesting mathematics in the number of paths to reach the chambers across the bottom. It might be easier to examine the number of paths on a smaller and simpler Plinko board such as the one in Figure 2 below.

Figure 2: A smaller look at the Plinko board.

The tree diagram that follows, Figure 3 below, shows the probabilities of landing in each of the chambers across the bottom based on the different possible paths through the board. The assumption is also made that the chip is dropped from slot B.

Figure 3: The paths, and probabilities, of chips landing in a certain output slot.

These probabilities make it easier to see the number of paths to the bottom of the actual Plinko board. And it turns out the number of paths involves a simple mathematical structure, Pascal’s Triangle, scene in Figure 4.

Figure 4: Pascal’s triangle.

The important row here is the line that starts: 1, 12, 66, etc., since that is the 13th row of the triangle. That row would define the number of paths from the middle slot at the top of the board to the bottom of the board except for the fact that the board has only nine chambers across the bottom or is nine columns wide. It is easy to correct for this when it is considered that the chips can bounce off the sides. Table 1 shows the corrections that need to be made given that the board is taller, 13 rows, than it is wide, 9 columns. The 12 that is added to the 66 and the 1 that is added to the 220 account for the chips bouncing off the sides, and we show this mathematically by creating a fold on Pascal’s Triangle after the 66’s in row 13. Think of this as physically folding the left side of the triangle on the rest of the triangle with a fold line on the vertical line between the 12 and the 66. Then note that the 12 lines up with the 66 and the 1 lines up with the 220. The same process can be done on the right side of the triangle where physically the right side of the triangle is folded on rest of the triangle,

1

12

66

220

495

792

924

792

495

220

66

12

1

+12

+1

+1

+12

78

221

495

792

924

792

495

221

78

A’

B’

C’

D’

E’

F’

G’

H’

I’

Table 1: Corrections for the possible paths on the Plinko board

So there are 78 ways to get into the $100 chambers, (A’ and I’), 221 ways to get into the $500 chambers, (B’ and J’), 495 ways to get into the $1,000 chambers, (C’ and G’), 792 ways to get into the $10 chambers, (D’ and E’), 924 ways to get into the $5,000 or $10,000 chamber, (E’). The difference in the dollar values depends on the year of production. Again, it should be repeated that these are the probabilities when the chips are dropped from the middle slot. Accounting for dropping the chips in different slots across the top simply means moving the fold lines.

Research Question 1

The first research question involves a very interesting concept of mathematics, expected value, and that question is: what do the producers expect to give away on the Price is Right each time that Plinko is played? Even though there are offers of $25,000 or $50,000, depending on whether there is a $5,000 or a $10,000 chamber in the middle of the board respectively, how much do the producers expect to give away? The following table, Table 2, describes the probabilities necessary for the calculation. This table shows the probabilities of falling into each of the nine chambers across the bottom when the chip is dropped from the middle slot at the top.

P(A’)

P(B’)

P(C’)

P(D’)

P(E’)

P(F’)

P(G’)

P(H’)

P(I’)

0.019

0.054

0.121

0.193

0.226

0.193

0.121

0.054

0.019

Table 2: Plinko probabilities for landing in each chamber

A simple expected value calculation, seen below, can be used to determine the expected value that the producers expect to give away when a chip is dropped from the middle chamber.

This means the expected value is $2,555.32 per chip for a total of $ 12,766.60  if the player had all five chips. This is the expected value when the middle chamber has a $ 10,000  value and the chips are dropped from the middle slot. This is also when the announcer says there is a chance to win up to $ 50,000. In the real game the contestants are given one chip and have a probability of 0.5 of winning the remaining four chips. That means the actual expected value is $2555.32 +(0.5)($2555.32)+(0.5)($2555.32)+(0.5)($2555.32)+(0.5)($2555.32) = $ 7665.97.   Again, all of this only accounts for dropping the chip from the middle chamber. If the contestant moves to one side or the other, the expected value drops and the simulations that are part of this research clearly showed this. This means that the expected value that the producers expect to give away is roughly 15% of what the announcer says the contestants might win.

Research Question 2

The second research question involves the Plinko or pachinko-type boards that have been showing up in various other settings. For example, at some amusement parks, individuals pay for the opportunity to drop a chip down a Plinko or pachinko-type board for the chance to win prizes such as candy, stuffed animals, or passes to the front of the line for a popular ride. The various prizes are listed in the chambers at the bottom, similar to the dollar values on the Price is Right board, with the best prize usually in the middle chamber. The major difference with these boards is the size. These differences in size lead to the research question: what happens to the probabilities or distributions for where the chips land in the chambers across the bottom as the dimensions of the board are changed?

In order to study this question, a Plinko simulator was created. This virtual gameboard was then used to run multiple iterations of the game based on various test parameters. In particular, the virtual gameboard allowed for the height and width of the board to be changed and also allowed for the chip to be dropped from different slots across the top. After each simulation was complete, a graph was displayed that included the distribution of the chips across the bottom of the board, along with the exact numeric values.

The underlying algorithm for the Plinko simulator is based on a two-dimensional array with the width and height of the board being simulated. The values in the array correspond with the slots between a row of pegs, an empty position is represented by a zero, and the current position of the chip is represented by a one. Figures 5 shows this.

Figure 5: A 3×3 Plinko board represented by an array of zeros.

When the chip falls through the board, the probabilities of it moving left or right are determined by a collision system. A collision occurs when the chip falls down and hits one of the pegs on the board. At each peg, the chip has a probability of 0.5 of moving either left or right. This is determined is by a Random Number Generator (RNG) generating a number between 0 and 99, if the number falls between 0 through 49 then the chip moves left, if the number falls between 50 and 99 then the chip falls to the right.

Plinko boards do not typically have a square shape, which is all that can be represented in an array. The rows are staggered and alternate in size to reflect the side walls of the board squeezing in and out (Figure 1). So, the algorithm for the simulation had to use logic to account for the staggered rows. The simulator observes the width of the row that the chip is currently in to determine how the chip can fall to the next row. If the chip’s horizontal position, represented by “n,” was in a row with an odd length then multiple conditions had to be checked:

1. If the chip was in the rightmost position, then the chip could only move to the nth column in the next row down

2. If the chip was in the leftmost position, then the chip could only move to the n – 1st column in the next row down

3. If the chip was in neither of the prior positions, then it could move to either the n – 1st  or nth position. It would move to the n – 1st if the output of the RNG was between 50 and 99, or the nth column if the RNG was between 0 and 49.

The rule set for odd length rows allows for the last column to be ignored in rows of even length and thus creating the desired staggered rows as seen in Figure 6 below. The black number one represents the current position of the chip, and, using the odd row-length rules that were defined above, the bold number one represents all possible positions that the chip could move to. The grayed-out position below is the positions that can never be used in order to create the staggered effect of the board.

Figure 6: (Leftmost) An odd row-length simulation, from a chip dropped on the rightmost edge. (Leftmost) An odd row-length simulation dropped from center.

Simulating the even length rows had a similar set of rules, however, the only thing determining which direction the chip moved was the random number generator, RNG. Letting the current position of the chip be represented by n, the chip could move according to following rules:

1. If the RNG output was between 0 and 49, the chip would move left to the nth position in the next row;

2. If the RNG output was between 50 and 99, the chip would move right to the n + 1st position in the next row.

After the simulator completed one full iteration of the game, it recorded the final position of the chip into the result set and began running the next iteration of the game in the simulation.

Data Collection

For each different size board that was simulated, 1,000 chip drops were run to obtain the distribution data. This simulation was repeated four more times before the size of the board was changed. The first simulation was for a board the size of the Plinko board on the Price is Right: 13 rows by 9 columns. The output from the simulation is in Figure 7 while the distribution can be seen in Table 3 below.

Table 3: Drop distribution for a standard sized Plinko board.

Figure 7: The output of the Plinko simulator, for a 9 column by 13 row standard sized Plinko board.

As the height of the board changes the number of chips falling into each chamber at the bottom, when dropped from the enter chamber, changes. Table 4 shows the results of dropping the chips from the center when the board has different heights.

Table 4: The number of chips falling into each camber at the bottom of the Plinko board, when dropped from the center, as the board heights are modified.

As the height of the board is changed, an interesting pattern emerges. The seven inner chambers start to have roughly the same number of chips falling into them. The two outer chambers have roughly the same as well, but they have far fewer chips than the inner chambers. This change in distribution is mostly due to the chips bouncing off the side of the board and appears to occur when the height of the board is three times the width. This was determined through exhaustive simulations. Figures 8 shows the distribution of a 9 column-width Plinko board, with a height of 27, three times the width.

Figure 8: This shows the distribution of a 9 column-width Plinko board, with a height of 27 units.

Conclusion

In this research, there were two major questions that were addressed. After introducing the basic mathematics of Plinko, particularly the number of paths that a chip can take to get from the middle chamber at the top to each of the nine chambers across the bottom, the probabilities for landing in each chamber were determined. The probabilities were then used to compute the expected value or the value the producers of the Price is Right might expect to give away, which was the first research question. This was then compared to what the announcer says a contestant has the possibility of winning. The second research question was to determine how the chips would fall into the chambers across the bottom when the height was modified. An interesting pattern emerged in that a roughly uniform distribution emerges for the chambers that are not on the outside while the outer chambers have roughly the same number of chips, but far fewer than the inner chambers.

Contacts: benjamin.marasco@howardcc.edu, nassefa@howardcc.edu, mlong@howardcc.edu

References

[1] Plinko. (n.d.). Retrieved from The Price Is Right Wiki: https://priceisright.fandom.com/wiki/Plinko

[2] Lewis, D. (2019, December 2). The One Thing You Can’t Take Home from The Price is Right. Retrieved from Now I Know: https://nowiknow.com/the-one-thing-you-cant-take-home-from-the-price-is-right/

[3] CBS Interactive. (n.d.). The Price is Right (Official Site). Retrieved from CBS TV Network: https://www.cbs.com/shows/the_price_is_right/photos/1004546/9-secrets-of-the-price-is-right-s-plinko/3/

Artwork - “The Cycle”

“The Cycle” by Charise Pelan-Maclean

Seeing a visual representation of how much bacteria was alive (green) compared to dead (red) in the pond sample gave a good baseline of our overall sample. This helped us determine what type of bacteria was common in the pond (rods), as well as how the bacteria community was changing.

The Relationship Between Soil Texture, Land Use, and Passive Versus Out-competing Species in Urban Forest Patches

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Charise Pelan-Maclean, Howard Community College

Mentored by: Cheryl J. Campo, PhD.

Abstract

Urban forest patches are in every large scale city, yet little is known about them. Being in such close contact with human interactions could have huge effects on species growth type and soil that is found in the forest patch. These small interactions built over a long time could change the landscape of the entire forest patch. Focusing on urban forest patches in Baltimore will help establish a relationship between land use history and the type of plant species that have grown in response to human interaction stressors. In addition, to how soil texture has changed throughout the forest patches. Twelve urban forest patches in Baltimore, Maryland were investigated as part of this study. Each forest patch was overlaid with a grid and at 40-meter intervals a point location was determined. At each point location, soil texture, ground cover species, number of trees, and tree species was recorded. Focusing on Herring run 2 and Southwest Area Park which were identified as opposing urban forest patches. Herring Run 2 had the least amount of land use history and Southwest Area Park had the most. This study focused on the presence of passive versus out-competing ground cover and tree species in relation to soil texture and land use history.

It was hypothesized that both 1) soil texture, specifically higher clay content, and 2) increased human use would increase the presence of out-competing species. The data only half supported this. There was no apparent correlation between soil textures and the relative abundance of passive or out-competing ground cover or tree species. The most predominant soil texture type throughout the two parks was sandy loam and loamy sand. In the instance of Herring Run 2, the site with a less disturbed land use history, both ground cover and tree species were more of the passive variety than out-competing whereas the site with a more disturbed land use history exhibited the opposite, both observations of which are in agreement with the above hypothesis. Also, the data suggests that land use may have a lower impact on ground cover than tree species with respect to relative abundances of passive versus out-competing species.

Introduction

Twelve site locations throughout Baltimore were selected as urban forest patches. These urban forests had a thick tree canopy that encompassed an area of at least 200 square meters. The site locations of the urban forest patches are as follows: Fendall Road, Shortline Park, Carroll Park, Morrell Park, Southwest Area Park, Harper Street, Patterson Park, Holabird Park, Herring Run 3, Clifton Park, Herring 2, and Barbara Parkwood Park as shown in Figure 1. This study will focus on Herring Run 2 and Southwest Area Park.

Figure 1: Map of Baltimore, Maryland with forest patch sites that were sampled.

According to a study done for the city of Baltimore, in 1973 the two parks being focused on—Herring Run 2 and Southwest Area Park—were considered forest and barren land, respectfully [1]. The city of Baltimore purchased portions of the land for Herring Run 2 in 1826, making it one of Baltimore’s oldest parks. In the 1980s, the city of Baltimore bought the land for Southwest Area Park, developing it into a park. Before Baltimore had expanded to its current size, the land went through many transformations. Before pre-European settlers arrived (5000 B.C.-1600 A.D.), the land around Baltimore was predominantly oak-hickory, closed canopy forests. After European settlers arrived (1600-1800), 20-40% of the land was cleared for small farms, tobacco, iron furnaces, and construction. From 1800-1900, large plots of land were being cleared for agriculture to support a growing population surrounding the city as well as metropolitan expansion [2].

When restoring previously human-impacted areas that were used for agriculture or industry, it is important to note that these processes degrade the soil. If urban forests are planned for areas with degraded soil the forests will not be able to sustain themselves without human intervention leading to mass species death [3]. This may lead to park developers using certain plant species to restore soil. These species can have a detrimental impact on the area, due to having an open space in which they can freely grow with little competition from other species [4].

This study will use the terms out-competing and passive to differentiate between two different plant species categories. Out-competing plants are comparable to invasive species and will include plants that can outperform in resource gain and growth in an area. Generally, once present, out-competing species will become over abundant and suppress other plant growth in the same area [5]. Passive species are those that grow well in ideal conditions; however, their growth can be hindered when conditions are no longer ideal. Many native species will be considered passive however some do out-compete their native members given the chance [6].

Soil texture has an influence on plant growth by determining how much a plant can grow. Soil texture can influence plant growth via water-holding capacity, soil workability, and permeability. Soil texture is separated into three different particle sizes: sand, silt, and clay. Sand is the largest particle while clay is the smallest [7].

Materials and Methods

The twelve locations were first surveyed by aerial imaging that ascertained how thick the canopy was. The process of measuring the canopy was termed as a blob which was measured by lasers on a helicopter. Once the parks with blobs were selected, using Google Earth, an overlaying grid was placed on top of them and at each cross section of the grid a point location was placed, thereby creating approximate 40m x 40m squares. The coordinates were recorded for each point location.

Figure 2: Point locations for the urban forest patches (a- Herring Run 2, b- Southwest Area Park) in a grid layout.

Upon reaching the coordinates of a given point and placing a flag down to mark it, one person stood at the point with a 1m x 1m PVC pipe square and gently tossed it behind them randomly. The ground cover species that were inside of the pipe square was recorded as a representative area of the 40m x 40m square. An example of what a 1m x 1m area might look like is shown in Figure 3a. In addition, above the flag one person would hold a rectangular wedge prism approximately 4.5 meters above the ground to determine what trees would be recorded. If a tree trunk appearing in a rectangular wedge prism overlapped with the trunk outside of the prism, the tree was considered “in”. If the trunks in and out of the prism did not overlap at all then the tree was considered out. See Figure 3b.

In addition to plant species being identified at each point, a soil texture was also collected. Soil texture (e.g., sand, loamy sand, sandy loam, sandy clay, clay loam, sandy clay loam, silty clay loam, silty clay, silt loam, loam and clay) was found by feel at each point. The soil was manipulated with a small amount of water to determine the approximate amount of sand, silt and clay that the soil contained. The soil was tested for how long of a ribbon it could make as well as how gritty or smooth it was before breaking.

Figure 3a: A representation of a 1m x 1m point sampling with 2 different ground cover species appearing inside.

Figure 3b: A rectangular wide prism view though  of tree trunks that are considered in or out. Species that are in were recorded in the study.

Results and Discussion

Figure 4: Soil texture dominance in each site determined by the topsoil at Herring Fun 2 and Southwest Area Park as illustrated by the number of points characterized to have soil of that type.

Figure 5: Number of trees found at each point location in Herring Run 2.

Figure 6: Number of trees found at each point location in Southwest Area Park.

Figure 7: Percentage of passive vs. out-competing tree species in relation to the dominant soil texture at each sample point (L-Herring Run 2 and R- Southwest Area Park).

Figure 8: Percentage of passive vs. out-competing ground cover in relation to the dominant soil texture at each sample point (L-Herring Run 2 and R- Southwest Area Park).

Herring Run 2

In 2010, Herring Run 2 was still defined as a forest [1], signifying that the city had done little to change the landscape of the area since 1973. The soil in Herring Run 2 is diverse with many different soil textures (see figure 4). Ten out of the twelve soil texture were identified in the park. No dominate soil texture in relation to out-competitive species was recognized in either tree or ground cover species. However, sandy loam and loamy sand was the dominating soil texture found throughout the park at 10 point locations each.

For tree species, 21 out of 54 point locations were 100% passive species, while only 2 out of the 54 point locations were 100% out-competing species (see Figure 7). In addition, 12 of the point locations had above 50% passive species and 14 points had above 50% out- competing species. The number of trees species found at each point location can show how abundant the passive or out-competing species are in relation to their presence in the overall park (see Figure 5). The average tree count for the whole park is 7.13. The average tree count for 100% passive tree species is 7.81 and 7.25 for points above 50% whereas the average tree count for the 100% out-competing tree species is 3 and 7.67 for points above 50%. This suggests that when the species are mixed together, out-competing tree species are more abundant than by themselves. For ground cover, out of 57 point locations, only three point locations had 100% passive species while nine point locations had 100% out-competing species (see Figure 8). Ten point locations had an equal ratio of passive to out-competing species. Eight points had above 50% passive species and 25 had above 50% out-competing species.

Overall, tree species were 76% passive and 24% out-competing whereas ground cover species were 53% passive and 47% out-competing.

Southwest Area Park

In 2010, Southwest area park was defined as a wet land [1], signifying that the city had spent effort to change the landscape from its barren appearance designation in 1973. The soil in Southwest Area Park is less diverse and only features seven of the twelve soil textures (see figure 4). No dominate soil texture in relation to out-competitive species was recognized in either tree or ground cover species. However, sandy loam was the dominant soil texture found throughout the park at twelve point locations.

For tree species, 2 out of 33 point locations were 100% passive species, while 20 out of the 33 points had 100% out-competing species (see Figure 7). In addition, 5 point locations had an equal ratio of passive to out-competing species. Zero of the points had above 50% passive species and 6 points had above 50% out-competing species. As with the Herring Run 2 analysis, the numbers of tree species found at each point location can show how abundant the passive or out-competing species are in relation to the overall park (see Figure 6). The average tree count for the whole park is 5.00. The average tree count for 100% passive tree species is 4.5 whereas the average tree count for the 100% out-competing tree species is 5.16. The average tree count for the above 50% out-competing species is 5.17; therefore, sites that have a higher percentage of out-competing species also had a higher tree count throughout the park suggesting that predominantly out-competing tree species are more likely to be found in the park. For ground cover, 6 out of 39 point locations were 100% passive species, while 18 out of the 39 points had 100% out-competing ground cover species (see Figure 8). In addition, 4 point locations had an equal ratio of passive to out-competing species. Four of the points had above 50% passive species and 7 points had above 50% out-competing species.

Overall tree species were 15% passive and 85% out-competing whereas ground cover species were 23% passive and 77% out-competing.

Conclusions and Future Directions

Herring Run 2 and Southwest Park are two urban forest patches that have a different land history use. Soil texture seemed to have no correlation with passive or out-competing species as the soil at locations was highly variable. For tree species it was observed that higher disturbed land use will lead to a larger percentage of the population that is out-competing. For ground cover species the same is also true, however, the data showed that land use may have less of an impact on the percentage of the population that is out-competing as evidenced by the smaller differential between passive and out-competing species.

In the future, determining if there is a relationship between certain tree species and ground cover species could lead to further insights in the population trends of certain forest patches. In addition, how the surrounding fauna population interacts with the flora population could give insights as to why certain forest patches have an over or under abundance of certain flora species.

Notes

Only the topsoil was considered when calculating soil texture. Minor data input errors could have occurred leading to some species being counted twice or not at all.

Six student researches were present throughout the study that provided subjective data (soil texture). This could have resulted in different textures to be recorded throughout the study. Analyzing soil samples is one of the next steps to be taken in this study.

Acknowledgements

Project was in conjunction with UMBC and HCC undergraduate environmental science research partnership.

Contacts: charise.pelanmaclean@howardcc.edu, ccampo@howardcc.edu

References

[1] Marland GIS, “MDP Land Use / Land Cover Interactive Map,” [Online]. Available: http://mdpgis.mdp.state.md.us/landuse/imap/index.html?dynTheme=dynExample2&amp;webmap=92f9de5c81354ec48164352923527d76&amp;extentBBox=-8867121.453183487,4570166.319597734,-8389848.648571078,4834638.437464317&amp;extentSR=102100. [Accessed November 2019].

[2] T. Foresman, “The Baltimore-Washington Regional Collaboratory Land-Use History Research Program,” December 2018. [Online]. Available: https://archive.usgs.gov/archive/sites/landcover.usgs.gov/luhna/chap5.html. [Accessed November 2019].

[3] B. Simmons, H. Richard, N. S. Nancy and A. D. S. N., “Long-term outcomes of forest restoration in an urban park,” Restoration Ecology, vol. 24, no. 1, pp. 109-118, 2015.

[4] D. Lamb, P. Erskine and J. Parrotta, “Restoration of Degraded Tropical Forest Landscapes,” American Association for the Advancement of Science, vol. 310, pp. 1628-1632, 2005.

[5] National Ocean service , “What is an invasive species?,” National Ocean service , 7 July 2019. [Online]. Available: https://oceanservice.noaa.gov/facts/invasive.html. [Accessed November 2019].

[6] R. Colautti and M. Hugh, “A neutral terminology to define ‘invasive’,” Diversity and Distributions, vol. 10, pp. 135-141, 2004.

[7] I. Krupenikov, B. Boincean and D. Dent, The black earth: ecological principles for sustainable agriculture on chernozem soils, Springer, 2011.

[8] D. Tilman, C. L. Lehman and K. T. Thomson, “Plant diversity and ecosystem productivity: Theoretical considerations,” PNAS, vol. 94, no. 5, pp. 1857-1861, 1997.

Up and Away! Rebound Height and Energy Changes in a System of Collisions

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Zainah Wadi, Howard Community College
Morin French, Howard Community College
Nian Liu, Howard Community College
Mentored by: Alex M. Barr, Ph.D.

Abstract

We investigate a vertical collision of two stacked balls experimentally, algebraically, and numerically to determine how various factors influence the rebound height. We calculated the predicted rebound height for both an elastic collision as well as an inelastic collision where the percent of kinetic energy each ball loses was determined experimentally using Tracker video analysis to analyze the stacked ball drop. We also modeled the experiment numerically in GlowScript where the upper ball is modeled as a system of two masses connected by a spring. The oscillations in the two-mass system act as a limited representation of the mechanical energy of the tennis ball converting to internal energy during each collision.

Introduction

Oftentimes simple experiments can be conducted to reveal explanations to seemingly complex phenomena. Supernovas and gravitational assist orbits can be better understood by investigating conservation of energy and momentum in a stacked ball drop. A stacked ball drop is when two or more balls are stacked vertically and dropped, and the top ball (ball 1) has a rebound height greater than the initial drop height. In a scenario with two balls being dropped, the bottom ball’s (ball 2) collision with the floor changes its velocity from the downwards direction to upwards. Ball 1 is traveling downwards when it collides with ball 2 which is traveling upwards. In this collision, ball 2 transfers energy to ball 1, changing the direction and magnitude of the velocity of ball 1. Using equations of conservation of energy and momentum, we can calculate the rebound height.

This phenomenon relates to a supernova because the star has a dense core that transfers a shock wave of energy outward. The transfer of energy from the dense core outward to the less dense layers causes the less dense layers to accelerate, resulting in a large velocity [1]. Collisions are typically thought of as two or more objects making physical contact; however, the same principle can be applied to a spacecraft utilizing a gravity assist maneuver. When a spacecraft enters a planet’s gravitational field some of the planet’s orbital energy can be transferred to the spacecraft, increasing the velocity of said spacecraft [2]. The sum of kinetic energy of the planet and spacecraft is preserved, however, so the interaction can be considered an elastic collision.

Mellen explored the behavior of a stacked collision that uses 7 different balls and compared the experimental data to his projected theoretical outcomes [2] . To perform the experiment with such a high number of balls he built a custom ball aligner, which he describes in detail in his paper. To determine the theoretical rebound height, Mellen used conservation of momentum with the coefficient of restitution. The coefficient of restitution is the ratio of relative velocity after the collision to relative velocity before the collision. The kinetic energy lost from each object is not distinguished, rather, the coefficient of restitution is accounting for the kinetic energy lost in the system as a whole.

Alternatively, we examined the kinetic energy lost from each ball as a separate entity. We use this along with the equations of conservation of momentum and energy to calculate theoretical rebound heights. We also modeled the collision in Glowscript to show how the kinetic energy is transformed into other forms of energy, a process we will discuss later in the paper.

Algebraic Solutions

To understand how a larger rebound height occurs, we begin by examining the scenario as an elastic collision.

Figure 1: The image to the left is ball 1 and ball 2 at the initial height (h). The center image depicts the direction of velocity for each object at moment a (before the collision) and b (after the collision). The image to the right depicts the rebound height (H) of ball 1.

Figure 1 depicts the stacked ball drop, collision, and rebound of ball 1. Balls 1 and 2 both fall a distance of h. Ball 2 collides with the floor, changing direction before the collision and ball 1 rebounds to a height H measured from the point of collision. The equations for conservation of kinetic energy and momentum can be manipulated to find the rebound velocity of ball 1.

(1)

(2)

We can simplify the equations by canceling out the ½’s in equation (1) and introducing the mass ratio

(3)

This results in the equation:

(4)

To determine the velocity of ball 1 and 2, we know that the gravitational potential energy at the starting position is equal to the kinetic energy the instant right before the ball collides with the ground. This relationship can be rewritten to obtain velocity.

(5)

The sign of velocity is determined by the direction before the collision, down is negative and up is positive. This results in and . With the velocities before the collisions defined, there are now two unknowns and two equations. Equations (4) and (5) can be combined to have the single unknown .

(6)

To determine the ratio of the rebound height with respect to the original height,  is written

(7)

Using kinetic energy and gravitational potential energy, H can be solved for as

(8)

In equation (8),x^2 is the ratio of the rebound height to the initial height. Equation (6), however, is only true in an elastic collision. In the real-world there is a percentage of kinetic energy lost during the collisions of ball 2 with the ground and ball 1 with ball 2. The energy ball 1 loses can be accounted for by multiplying the pre-collision kinetic energy by a factor of .

(9)

When ball 2 collides with the ground, the energy lost can be accounted for in the value of .

(10)

signifies the percentage of kinetic energy remaining after the collision. Equations (9) and (10) can now be used to solve for the rebound velocity of ball 1 in an elastic collision () or in a collision where each ball loses a specified percentage of kinetic energy.

Experimental Results

To determine the kinetic energy lost from the collision between ball 1 and 2, Tracker [4] was used to analyze a video of the collision between a tennis ball (ball 1) and basketball (ball 2) frame by frame to measure the velocity before and after the collision.

Figure 2: (Left) Tennis ball’s position for a sequence of video frames. (Right) Vertical velocity vs. time graph for the tennis ball bouncing off a stationary basketball. The vertical velocity of the tennis ball before the collision is -3.229 m/s and the vertical velocity after the collision is 2.116 m/s.

The percent kinetic energy remaining can be found by using the tennis ball velocity before and after it collides with the basketball.

(11)

This value is used as the value in equation (9). This process is repeated for ball 2 bouncing off the floor and that value is recorded as . These values were used to create three curves displaying the rebound ratio (H/h) with respect to the mass ratio (r); an elastic collision, a collision where only ball 1 experiences energy loss, and a collision where both ball 1 and ball 2 experience energy loss. The graph shows that as the r value approaches zero, the energy lost from the ball 2 has a greater impact on the rebound height than the energy loss of ball 1 alone. As r approaches one, the impact of the energy lost from the ball 2 decreases. This is plausible because momentum and energy are quantities calculated using mass and velocity. In this scenario, ball 1 and 2 have the same magnitude of velocity but different masses, therefore, the object with the greater mass is contributing more energy and momentum to the system. When r approaches zero, the mass of ball 1 is negligible compared to the mass of ball 2 resulting in a greater decrease in rebound height when accounting for the energy lost from ball 2. As r approaches 1, the difference in mass of ball 1 and ball 2 is decreasing until they become the same mass at r = 1 causing the energy lost from ball 1 and 2 to have equal impacts on the rebound height.

Figure 3: Rebound height ratio (H/h) vs. the mass ratio of ball 1 to ball 2 (r) for the three scenarios.

When comparing the algebraic solution and the experimental results, we begin by examining the mass ratio of the tennis ball to the basketball, which is approximately 0.1. Figure 3 illustrates that in a collision where r = 0.1, and the final height of the tennis ball when the system is dropped from 1 meter should be approximately 5 meters. Our experimental data does not support this claim. Figure 4 shows that the tennis ball only reaches 3 meters.

Figure 4: Experimental results for the stacked ball drop. (Left) The system at rest at the initial drop height. (Right) The tennis ball at its maximum rebound height.

Numerical Model

In the experiment, the mechanical energy of the tennis ball–basketball system decreases during the collision. Our algebraic solutions account for a percentage energy reduction but are unable to model the mechanism or possible forms to which the mechanical energy may be converted. To explore these questions, we modeled the collision in Glowscript, an adaptation of VPython, where we explicitly calculate the forces exerted on each ball at each moment. The change in forms of energy of the tennis ball was our primary focus; assuming that a significant amount of the mechanical energy was converted to internal energy, we modeled the tennis ball as two masses connected by a spring. The compression of the spring represents the deformation of the tennis ball during the collision. To investigate how the stiffness of that spring impacts the amount of energy transformed from mechanical to internal, we chose various spring constants and ran separate iterations of the program for each spring constant.

The lower ball was a necessary component of the simulation, but we were less interested in its behavior. Therefore, it was modeled as a single mass with an associated spring constant, whose primary purpose was to emulate the impact of the basketball colliding with the floor.

Figure 5: The tennis ball modeled as 2 spheres connected by a spring.

The greater the spring constant k, the greater the “stiffness” of the spring. A greater k constant should yield a more elastic collision, because stiffer springs do not easily transfer energy. A lack of energy transfer or transformation leaves no opportunity for energy loss, so the collision would conserve mechanical energy; ergo it would be an elastic collision. Taking the average forward deformation of a tennis ball (the amount it “squishes” upon impact), we calculated a minimum possible k constant for an elastic collision using conservation of energy [5]. By relating the gravitational potential energy before the drop to the elastic potential energy in the instant the tennis ball stops during the collision, we find our minimum k:

(12)

When our tennis ball and basketball are dropped from 1 meter and k = 27,370.4142 N/m we ought to see a significant rebound height. Instead we see a rebound of less than 1.5 times the initial drop height, despite what the algebraic results would suggest.

Figure 6: Graph showing the y-position of center of mass of the tennis ball. After the bounce the rebound height should be considerably greater than the initial drop height, but that is clearly not the case.

If we substitute lesser and lesser k constants into the Glowscript model the collision should become more inelastic. Just as a greater k constant meant a stiffer spring, a lesser k constant means a less stiff spring. Decreasing the stiffness of the spring allows more energy to be transferred to elastic potential as the spring compresses, which in turn means we cannot achieve an elastic collision. In our simulation, we struggled to work with such reduced k constants. We reduced k from ~27,000N/m to 270N/m to 2.7N/m to model increasing amounts of mechanical energy being converted to elastic potential energy. However, in a low k simulation with just the tennis ball we see the two mass halves exchange position, which is physically impossible. Returning to equation (13) for conservation of energy we see that if GPE = EPE at low k values we, in turn, get a large :

The average diameter of a tennis ball at rest is approximately 0.067m [5]. During the course of a collision, it is not possible for the tennis ball to stretch or compress beyond its initial length. Unfortunately, that is the behavior exhibited by the simulation. The smaller k constants were needed to produce a model that showed percent energy loss consistent with experimental data, but the behavior of the tennis ball at low k constants means that the model cannot be accurate. Although the intent of the numerical model was to create a simplified version of the vertical collision, the position and energy graphs from our simulations indicate that the model was too simplistic.

A fundamental problem underlying all other quirks of our numerical model is that it was built with the assumption that mass is distributed evenly across the tennis ball, and that the k remains constant across the ball and throughout an event such as a collision. This oversimplification fails to capture how the tennis ball would behave before, during, and after a collision. If one regards the tennis ball as a series of cross-sections, akin to Rod Cross’ analysis of the dynamics of a sphere, it becomes apparent that not all cross-sections have the same mass and that changes the stiffness of each section [6].

Figure 7: Illustration of cross sections of a tennis ball. Uneven mass distribution means m3 will have the greatest mass.

Using this more detailed model of a ball’s mass distribution, we can incorporate Young’s Modulus to predict the different k values for each cross section within the sphere:

(13)

where A = area of the cross-section, w = thickness of the cross-section, and E = Young’s Modulus, i.e. the force per unit surface along the bounce axis divided by the strain (proportional deformation). According to Cross, the end sections along the bounce axis will be considerably less stiff (smaller k values) because their cross-sectional area goes to zero at the edges.

With this representation of a spring constant, we find that k would stiffen as the sphere compresses on impact. The non-uniform distribution of mass also means that our system of only two masses and a spring will not be enough to accurately model the behavior of a ball during collision. Cross found some success modeling an elastic collision with a system of five masses and five springs, but even this would be insufficient to model an inelastic collision [6].

Conclusions

We investigated a vertical collision of two stacked balls algebraically to determine the rebound height of the top ball in both an elastic collision and where there is a percentage of energy loss in each ball. We gathered experimental data using Tracker and also modeled the experiment in Glowscript.

The algebraic model shows the significance the mass ratio holds for the rebound height. In order to have a greater transfer of energy to ball 1, it is imperative to have as small a mass ratio as possible. The algebraic model also demonstrates how energy loss from the more massive ball contributes greater to the energy loss of the whole system, decreasing the rebound height significantly. While conducting the experiment, it was quite difficult to get ball 1 and 2 to collide at a 90o angle. The subtle inconsistency in drop angle could have an impact on the results for kinetic energy loss calculations from ball 1 and 2 as well as the rebound height of ball 1 during the experiment. The introduction of a ball aligner could decrease the effects of horizontal velocity.

To expand upon this project, the effects of drag can be incorporated into the calculation of the theoretical rebound height to determine if it is the cause of inconsistency between the experimental and theoretical rebound height.

Our numerical model proved too limited to accurately portray the stacked collision of a tennis ball and basketball. The tennis ball model was built utilizing the perspective of point particle physics employed in early physics classes; this led to such assumptions as that mass and spring constants would be uniform throughout each sphere. A more realistic approach could incorporate ideas more aligned with mechanics of materials, such as the application of Young’s Modulus as previously discussed.

Although our numerical model failed to meet our stated objective, we have stumbled across a potential exercise to help students make the leap from point particle physics to more advanced physics concepts. When tasked to create a simulation of a stacked ball drop, many early physics students would likely make the same erroneous assumptions we have made. Building (and subsequently troubleshooting) a model such as this, prompts students to identify for themselves the discrepancies and shortcomings of early physics lessons when discussing more complex concepts. In turn, this exercise creates an avenue through which students can begin to explore the shift in thinking required to move to higher-level physics and engineering courses.

Contacts: zainahwadi@gmail.com , morin.french@gmail.com, nian.jasmine@gmail.com, abarr@howardcc.edu

References

[1]Physics Girl. Stacked Ball Drop, (2015). Retrieved from

https://www.youtube.com/watch?v=2UHS883_P60 .

[2] Huebner, J. S., & Smith, T. L. Multi‐ball collisions. The Physics Teacher ,

30 (1), 46–47 (1992). doi: 10.1119/1.2343467

[3] Mellen, W. R., Aligner for Elastic Collisions of Dropped Balls. The Physics

Teacher , 33 (1995) .

[4] Tracker Video Analysis https://physlets.org/tracker/ (2019).

[5] 2018 ITF Ball Approval Procedures, (2019). Retrieved from

https://www.itftennis.com/media/2236/2020-itf-ball-approval-procedures.pdf.

[6] Cross, R., Differences between bouncing balls, springs, and rods . American Journal of Physics,
76, 908 (2008). https://aapt.scitation.org/doi/10.1119/1.2948778

Artwork - “Entrapped”

“Entrapped” by Heather J. Sun

Ghosts of Parking Lots Past: The Effects of Legacy Pollution on Stream Health

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Laura Wortman, Howard Community College

Sky Erickson, Howard Community College 2019, University of Maryland College Park

Mentored by: William Straube, Ph.D.

Abstract

High conductivity is often reported in urban streams with the ion source commonly attributed to the application of road salt in the winter. Salinization of freshwater streams poses a threat to the overall health of the stream system. After reviewing a recent two-year study monitoring the health of a Howard Community College campus stream, new research was conducted with a focus on identifying the possible ion sources responsible for elevated conductivity. Stream samples were tested for conductivity and chloride concentration, with results revealing a positive correlation between the two. The stream’s baseline remains relatively constant, suggesting that newly applied road salt is not the primary source of the stream’s high conductivity. The distribution of ions within the soil at each stream site suggest the presence of existing salt in addition to the newly applied road salt. Past site surveys indicate the presence of several gravel parking lots impacted by antiquated salting methods within the stream’s watershed. This legacy pollution is most likely the primary contributor of Scholarship Drive stream’s high conductivity.

Introduction

Samples collected from the Scholarship Drive stream display significantly high conductivity readings compared to other campus water bodies as well as Howard County and Maryland streams [1], [2], [3]. Both conductivity and chloride readings of soil and streams are increasing due to road salt, an increase in impervious land cover, and other anthropogenic influence within watersheds. Salinization of urban streams due to road salts and other contributing factors has been documented extensively [4], [5]. Saline water can damage infrastructure, increase erosion, and reduce water quality [6]. When salt infiltrates the soil, it can enter groundwater and make its way into rivers and streams [6]. As freshwater streams increase in conductivity and salt content, the aquatic life is compromised. Salt in streams will result in increased conductivity as ions from the salt become suspended in the water. The ideal conductivity for a freshwater stream ranges from 150 – 500 μS/cm [7]; this stream ranges from 545 – 7654 μS/cm.

Variation in stream conductivity is expected as each stream differs in geological makeup, stream flow, impervious ground cover, and exposure to pollutants. When compared to data collected from other Maryland streams, the conductivity and chloride readings from this stream not only show elevated readings, but surpass the ranges detected in others [3]. A closer examination into this stream’s conductivity and chloride content was collected and contributing factors were analyzed to identify possible sources of pollution. This analysis includes a close comparison of Scholarship Drive’s stream and Hickory Ridge’s stream revealing impacts of similar factors with historical evidence suggesting the source of the elevated levels may be due to legacy pollution from gravel parking lots. Legacy pollution is defined as the effect of contaminants introduced in the past that are no longer actively being deposited but continue to impact the natural environment.

The Scholarship Drive Stream is a tributary of the Little Patuxent River, which in turn flows into the Chesapeake Bay. Pollutants dumped in its watershed eventually make their way into the Bay. While the health of this tiny stream is important, it is critical to remember the impact that it, and streams like it, will have on larger ecosystems.

Figure 1: HCC campus map depicting the watersheds of each stream and soil site.

Methods

The two sites along the Scholarship Drive stream from the previous study [8] in which the elevated levels of conductivity were discovered were selected for continued monitoring [Fig. 1]. Another site near Hickory Ridge Road, Stream Site 3, was selected because of the stream’s similarity to Scholarship Drive’s. Both streams are located on HCC’s campus, are made up of the same soil composition, have forest buffers, and share proximity to impervious land cover, a road and sidewalk [9].

Stream water samples were collected approximately twice a week and following winter storm events in which road salt was applied. The Scholarship Drive sites were monitored for this study beginning October 18, 2019 and are referred to as Stream Site 1 and Stream Site 2. Stream Site 3 was added on December 20, 2020 to conduct a stream and soil conductivity comparison. Stream water was collected in 60 mL plastic containers with a plastic screw top. The samples were brought into the lab and allowed to reach room temperature to remove ambient and water temperature as variables in conductivity readings [10]. The room temperature samples were then tested for conductivity utilizing the LabQuest 2 with Vernier conductivity probe at the 20,000 μS/cm range. A random assortment of 22 samples were tested with the Vernier chloride probe, following the manufacturer’s protocol, to determine whether chloride and conductivity levels were correlated.

Soil sites A, C, and HR were chosen in each stream site’s watershed [Fig. 1]. The bounds of each watershed were determined using the USGS StreamStats tool [11] and HCC site plans [12]. Topsoil was scraped from the top 3 cm with a trowel and placed in a glass beaker. Approximately 100 g of soil from each site was added to 400 mL of DI water and mixed thoroughly in a glass beaker. The soil solution was left to sit for at least two days in order to allow soil particles to settle. The water was then tested with the Vernier conductivity probe.

In order to investigate the possibility that salt concentrates in certain subsurface soil strata, core samples were collected with a steel core probe between parking lot A and C and Stream Sites 1 and 2. Parking lots A and C receive road salt applications prior to winter storm events and are within Scholarship Drive stream’s watershed. The soil layers of each core were identified and separated into tins to be measured. Approximately 30 g of each layer were added to 120 mL of DI water (a ratio proportional to the topsoil test) in glass beakers. Each solution was then tested utilizing the method used for the topsoil sites.

Throughout the study, HCC faculty and staff were consulted or interviewed to provide relevant information. Personal accounts along with both aerial [13] and ground photographs [14] provided evidence of historical gravel lots which were either paved over or tilled into the earth to allow for new construction. These gravel lots received excessive amounts of road salt during winter storm events.

Results

The conductivity of Stream Sites 1 and 2 remain high throughout the testing period [Fig. 2], with medians of 1943 μS/cm and 1733.5 μS/cm respectively, which are much higher than other area streams. For example, Stream Site 3 is significantly lower, 944 μS/cm, though still high compared to healthy streams. Dips in conductivity correspond with rainy days, which dilute the concentration.

Figure 2: Stream conductivity obtained from water samples at three sites from October 2019 to January 2020.

Three sample dates are noted as having had snow in the previous 48 hours: 11/14/2019, 12/12/2019, and 1/8/2020. The Hickory Ridge site was added after the first two snow dates, and so only includes the last. The first two snow dates did not appear to have a great impact on the conductivity of sites 1 and 2. It is possible that less salt was put down in anticipation of the first two snow dates than the last. According to local stations of the National Operational Hydrologic Remote Sensing Center [15], [16] the first two snow dates produced less than a millimeter of snow each. There was about 7.5 mm of precipitation 1/8/2020, the third snow date. Almost half of it was snow. A spike on January 8th corresponds with a rainy event following a winter road salt treatment [Fig. 3].  Preparation for this winter weather event is the most likely cause for the spike in conductivity. While all three sites showed a substantial increase in conductivity on January 8th, Stream Site 3 jumped significantly higher than Stream Sites 1 and 2. This suggests that roads in the Hickory Ridge site watershed are more heavily salted than the watersheds for Stream Sites 1 and 2.

Figure 3: Stream conductivity obtained from water samples at each site after the winter weather event on January 8th, 2020.

Several ion sources were discussed as possible causes of high conductivity. A reaction between calcium carbonate and acid rain would result in the presence of calcium and sulfate ions. Rock salt used for de-icing can also result in the presence of sodium and chloride ions, and is often found to be a major contributor to high conductivity in streams. Presence of chloride was investigated first due to availability of the proper equipment. Conductivity and chloride concentration of 22 samples were compared [Fig. 4], showing a strong correlation between the two, indicating that the cause of the high conductivity is NaCl from road salt. The mean chloride concentration threshold for chronic toxicity to freshwater life in 250 mg/L [4]; this stream has a mean of 506 mg

Figure 4: A comparison of chloride concentration and conductivity level in stream water samples.

A comparison of the topsoil from pertinent locations in each watershed showed that a site at Lot C in Site 1 and 2’s watershed was much more conductive than others [Fig. 5]. If urban soil becomes salinized because of road salt [4], why does the stream that peaks higher during salting events have less conductive soil? While road salt may be a contributor, another source might explain the extreme levels.

Figure 5: A comparison of soil conductivity at soil sites C, A, and HR.

Additionally, a Lot C core sample taken from the same day showed that conductivity remained high in lower layers of the soil. The surface layer of the soil core collected near Lot A showed much lower conductivity, and a deeper layer showed even lower conductivity [Fig. 6].

Figure 6: Soil core samples from the watersheds of Site 1 and 2 at Lots A and C. Figure 7, Right: A comparison of conductivity by soil layer in the watersheds of Sites 1 and 2.

Figure 7: A comparison of conductivity by soil layer in the watersheds of Sites 1 and 2.

There are currently no obvious ion sources on the HCC campus that would not also be present in other local watersheds. Two campus streams were compared. Both streams were composed of similar geological makeup, but were at different slopes [9]. Both contained large areas of impervious ground cover in their watersheds including an adjacent road and sidewalk [9]. Forest buffers of similar size are located along each stream [9]. Although we are not able to qualify the stream flow due to lack of tools, they appear to be similar. When asked, several long-time HCC employees recalled changes in the parking situation. Lot C was once a gravel lot; another gravel lot was replaced by the campus quad [Fig. 8a, 8b, 9]. This anecdotal evidence was confirmed in aerial photographs retrieved from the Howard County Interactive Map [13]. Bob Marietta, the college’s Environmental Health and Safety Supervisor, described a de-icing method in which gravel lots were oversalted because gravel cannot be plowed. Leftover salt from this method would explain the abnormally high conductivity level of the soil in Lot C, even in deeper soil layers. The watershed of Stream Site 3 is not impacted by former gravel sites in the same manner Stream Sites 1 and 2 are.

Figure 8a: HCC Campus Maps depicting historic and present day parking lots and their relation to the stream site watersheds. Sites 1 & 2.

Figure 8b: Bottom: Site 3 (HR)

Consideration of Possible Sources of Error

Some instrument error may exist due to calibration issues with the LabQuest.

The sample size of snow dates is limited due to a particularly warm winter. More data from the Hickory Ridge site during the warmer months would give a more accurate picture.

Some samples were lost due to spillage.

Suggestions for Continued Monitoring and Research

More extensive soil testing throughout each watershed would give a more comprehensive story. Samples from the quad may prove valuable because it served as the historical parking lot mentioned in the study. Additionally, a measurement of the stream’s flow would indicate whether the data is proportional to a stream of its size. Continued monitoring could lead to further understanding of HCC’s environmental effect on the stream’s health and determine whether its protective measures are adequate or improving the natural environment. Though no historical soil composition data exists for much of the campus, Howard EcoWorks has records of the soil compositions used in the rain gardens on campus prior to application. This historic record could be used to observe the effect of regular salting on campus rain gardens and how the soil conductivity changes over time.

Conclusion

Conductivity is abnormally high in one particular campus stream year-round. The source of the high conductivity was linked to NaCl, salt. High conductivity in urban streams has been commonly reported; however, conductivity levels on salt days compared to our control stream suggested that our watershed is subjected to less salting while year-round levels remain consistently higher. Anecdotal and archival evidence suggests that legacy pollution is the source of the majority of the salt. Soil data supports this, though more thorough sampling is needed.

When preparing the campus for human use, it is important to remember the impact human activities will have on the environment. HCC is currently utilizing traditional rock salt in conjunction with a brine treatment to minimize its adverse environmental effect. While current campus salting protocols take precautions to prevent polluting the environment, the careless methods used over a decade ago continue to affect nearby streams. This is the crux of the problem of legacy pollution; actions taken now can continue to impact the environment for years to come.

Contacts: wortman.laura.j@gmail.com, sky.r.erickson@outlook.com, wstraube@howardcc.edu

Figure 9: An historic photograph of a HCC parking lot, which is now the quad. Provided by Dave Hinton [14]

References

[1] Howard County Stream Waders, “HoCo Stream Wader Info ‘19,” [Online]. Available: https://www.howardwsa.org/hoco-stream-wader-info/

[2] Stream Health, Maryland Department of Natural Resources, n.d. [Online]. Available: https://geodata.md.gov/streamhealth/

[3] Results from high-frequency chloride and specific conductance analysis, Towson University, R. M. Fanelli, A. J. Sekellick, and J. Moore, Nov. 4, 2019. [Online]. Available: https://tigerweb.towson.edu/moore/chloride_map_12062019.html

[4] Kaushal, G. Likens, M. Pace, R. Utz, S. Haq, J. Gorman, and M. Grese, “Freshwater salinization syndrome on a continental scale,” Proceedings of the National Academy of Sciences of the United States of America, 23 January 2018. [Online]. Available: https://www.pnas.org/content/115/4/E574.full?. [Accessed February 2, 2020].

[5] Stranko, R. Bourquin, J. Zimmerman, M. Kashiwagi, M. McGinty, and R. Klauda, “Do Road Salts Cause Environmental Impacts?” Maryland Department of Natural Resources, April 2013 [Online]. https://dnr.maryland.gov/streams/Publications/RoadSalt2013.pdf. [Accessed February 2, 2020].

[6] Podmore, “Urban salinity – causes and impacts,” Primefacts 938, October 2009. [Online]. Available: https://www.dpi.nsw.gov.au/__data/assets/pdf_file/0004/309316/Urban-salinity-causes-and-impacts.pdf. [Accessed February 2, 2020].

[7] Friends of Sligo Creek, “Definition of Water Quality Parameters,” [Online]. Available: http://fosc.org/WQData/WQParameters.htm.

[8] Beckjord, and W. Straube, “Long Term Monitoring of a Howard Community College Stream,” Journal of Research in Progress, vol. 2, pp. 28-38. 2019.

[9] Natural Resources Conservation Service, “Soil Survey of Howard County, Maryland,” [Online]. Available: https://www.nrcs.usda.gov/Internet/FSE_MANUSCRIPTS/maryland/MD027/0/MDHoward5_08.pdf

[10] “Volunteer Stream Monitoring: A Methods Manual,” United States Environmental Protection Agency Office of Water, EPA 841-B-97-003, November 1997. [Online]. Available: https://www.epa.gov/sites/production/files/2015-06/documents/stream.pdf

[11] USGS, “Stream Stats”, v4.3.11. [Online]. Available: https://streamstats.usgs.gov/ss/

[12] Patton Harris Rust & Associates, “Overall SWM Drainage Area Map,” June, 2005, unpublished.

[13] Howard County, Maryland, “Howard County Interactive Map,” Howard County, MD, 2015. [Online]. Available: https://data.howardcountymd.gov/InteractiveMap.html

[14] Hinton, collection of photographs, “HCC Past,” unpublished.

[15] “Interactive Snow Information, Station MD-HW-11” National Operational Hydrologic Remote Sensing Center, https://www.nohrsc.noaa.gov/interactive/html/graph.html?station-MD-HW-11. [Accessed February 2, 2020].

[16] “Interactive Snow Information, Station MD-HW-31” National Operational Hydrologic Remote Sensing Center, https://www.nohrsc.noaa.gov/interactive/html/graph.html?station-MD-HW-31. [Accessed February 2, 2020].

Artwork - “Urban Forest”

“Urban Forest” by Shuyi Cen

Volume 2: 2019

Volume 1: 2018

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Thank You

We would especially like to thank The Kalhert Foundation for their support of our students and specifically this journal. The Kalhert Foundation is interested in Education in general and specifically Education in the STEM fields.

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