Week 2: Over(leaf) the moon for Theory

Kira A -

Greetings from the world of LaTeX! This week, along with participating in my last Arizona Music Educators Association West Region Honor Band, I shifted my focus for my project from instrumentation and design of my spacecraft toward theory and writing for my final paper.

This past summer, while participating in the Astrophysics Summer Science Program (SSP) at CU Boulder I learned to use Overleaf typesetting with LaTeX. LaTeX is a document preparation system where one can code the content of their documentation producing a final product that is far more conducive to professional and scientific papers. This is why I have opted to use Overleaf to write my paper. Below is an example from my SSP Orbit Determination Paper for asteroid 66269 (1999-JN3) of how graphs and equations can seamlessly and cleanly integrate into the text.

Page 3 Of SSP Orbit Determination Final Paper – Includes graphs, equations, and flow charts.

Additionally, here is an example of the Overleaf typesetting interface for the title page of my paper as well as a sneak peek of the page itself. 🙂

Title page LaTeX typesetting window

 

Title page – Moon Bound: Trajectory Analysis & Mission Design in the Cislunar Region

 

To make my paper digestible and accessible to all readers, I am dedicating the theory section of my paper to explaining the most commonly used concepts, or fundamental equations, and their derivations in my project. This includes everything from the definitions of concepts such as the orbital elements which will help in defining the orbit and trajectory of my spacecraft, to the final orbit-raising maneuver which is the culmination of everything in my project. This theory section will also reinforce my knowledge of the concepts within my project and ensure I have a solid understanding of them to build upon for more advanced concepts later down the line as I transition into my college classes this fall.

Beyond this, to integrate the theory needed for the final orbit-raising maneuver into my paper, I have been hand-deriving the equations to grasp the process of obtaining them so as to not merely pull equations from textbooks and implement them into my final code without a clear understanding of what they mean and how they fit into the bigger picture. One such equation is the modified rocket equation. This equation plays a major role in my calculations as it will allow me to calculate the mass expelled from the rocket as exhaust during each orbit-raising maneuver the rocket completes. This mass can be equated to the fuel used during the maneuvers which I aim to reduce through this project and the orbit-raising maneuver. Below is a picture of the derivation for the rocket equation I completed on a whiteboard.

Rocket equation derivation on whiteboard

Looking towards this upcoming week, I am excited to be able to participate in the senior trip as well as a snowboarding vacation with my family; however, I will not have access to my laptop at all times to work on LaTeX, so I will use this coming week to focus on spacecraft instrumentation as well as mission objectives.

As, always, I am forever grateful to Mr. Joseph and Dr. Farooq who are willing to offer their time to guide me throughout this project, and to all of you who take the time to read and comment on my posts. I hope you will all tune in next time.

 

Ad Lunam!



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Comments:

All viewpoints are welcome but profane, threatening, disrespectful, or harassing comments will not be tolerated and are subject to moderation up to, and including, full deletion.

    Akash Joseph
    Hi Kira, INCREDIBLE WORK with the derivations! From the picture you uploaded, it looks like you have attained a ratio of masses equivalent to the exp(-delta v/(Isp*g0)) - 1 rather than the expected 1- exp(-delta v/(Isp*g0)). Can you provide us with more insight as to why this is? Super excited to see your progress in the coming weeks! Ad Victoriam, Ad Astra!!
    kira_a
    Welcome back, Mr. Joseph. Thank you for the support and for allowing me to use your whiteboard for the derivation shown in the picture :). While deriving the modified rocket equation, the intended solution is indeed (delta m)/m0 = 1 - exp (-(delta v)/(Isp*g0)), however, I attained a similar, yet incorrect result. I later discovered that my notation for delta m was incorrect, as the delta m in the final answer represents the mass of the fuel expended by the rocket as exhaust, not the final mass of the rocket after expending the fuel, which is what I had it notated as. After realizing this, I corrected my mistake and obtained the correct equation! Thank you for your curiosity.
    Sid Vasireddy
    Hi! This is exciting work!! Are you going to explicitly derive all equations to be used or would you be using a reference of some sort to justify all of your calculations? Look forward to following your progress!!
    Jeremiah Larson
    Great walkthrough of your thought processes during the week, it was easily comprehensible! The success of this project is highly dependent upon your organization, and it seems you have nailed it. The organization of your various derivations throughout this research project must be difficult. What system(s) of organization have you used to ensure you don’t lose any information in the process? Do you believe it is the most efficient method or do you plan to change it in the future?
    Vignesh Nagarajan
    Love the scope of your project! Will you be using sample values from a textbook (or elsewhere), or will you be using actual data collected from previous mission launches to the moon such as ISRO's Chandrayaan-3? Looking forward to the next blog post!
    Sara Simon
    Hi Kira, your project is insane, especially for a high schooler - very inspiring! Quick question, how is your project different from previous moon missions? Specifically, is your mission’s trajectory based on any previous moon missions? Can’t wait to see you present in May!
    Ishaan Suthar
    Dear Kira, I hope this message finds you well. I noticed the equation: 𝐹𝑡=𝐼 on the right whiteboard, located directly below the drawing of the two rockets. I wanted to inquire whether this equation is intended to represent the formula for impulse, where force is multiplied by the time interval, typically denoted as 𝐽. As you may know, 𝐼 is often used to represent conventional electric current, rotational inertia, luminous intensity, or isospin in other contexts, and I was curious to understand the conventions being followed in your project, particularly in relation to the symbols and equations used. I just wanted to clarify whether these align with standard engineering notations. Thank you very much for your time and consideration. I look forward to hearing from you.
    Inesh Singh
    Hi Kira, great work!! I see you chose Overleaf with LaTex for your final paper due to its suitability for professional and scientific documentation. What are the specific advantages of LaTex over traditional word processors like Microsoft Word when it comes to integrating complex mathematical equations, graphs, and flowcharts into a document? How does LaTex enhance the readability and reproducibility of scientific work? As a junior, your work is very inspiring and I hope to pursue something as ambitious next year!
    Jason Hsu
    Hello Kira! It sounds like the orbit raising maneuver could reduce the amount of fuel needed to send your rocket to the moon. I'm curious on how this could reduce the barrier of entry for companies and for the average consumer. Also, would reducing the amount of fuel needed help with potential environmental impact in the future? Looking forward to seeing more of your project!!!
    Bhavin Dang
    Hello Kira and Ishaan, I would like to direct your attention to the mathematical statement below the one referred by Ishaan. It states: F = dp/dt/t . This would imply that force equals ratio with the derivative of momentum with respect to time as the numerator and time as the denominator. However, this is an invalid assertion as the net force on a body rather equals the derivative of momentum of that body with respect to time, according to the second law formulated by Sir Isaac Newton.
    kira_a
    Hi Sid, I will be using the Howard D. Curtis Orbital Mechanics for Engineers textbook as a guide for my derivations. However, while I plan on using it as a reference, I will be deriving each equation fully myself to understand the concepts and application of the equations more deeply. Furthermore, I will be deriving the orbit-raising maneuver equation with no reference as it is novel in the field of orbital mechanics and is not explained in any texts that I am currently able to comprehend.
    kira_a
    Thank you, Jeremiah! I am definitely trying to stay as organized as possible in order to focus as much of my time on the contents of the project as opposed to trying to organize them. As far as how I am keeping organized, I have a notebook that I have been using to keep track of every derivation as well as a Google Drive folder with each reference I have used in my paper and a spreadsheet to keep track of my hours.
    kira_a
    Thank you for the comment Vignesh! I will mainly be using values from the Howard D. Curtis textbook I mentioned in response to Sid's comment to cross-check my code. The Chandrayaan-3 lunar mission is a big inspiration for my project, so I would like to compare the values for that mission as well once I have calculated my orbit-raising maneuver. This may be difficult because ISRO has not released much information on how they carried out the orbit-raising maneuver, but I will keep digging to see what I find.
    kira_a
    Hi Sara, thanks for the inquiry! My mission strays from the typical lunar mission because of its unique approach to establishing a trajectory to the Moon - the orbit-raising maneuver. I talk more about the orbit-raising maneuver and its benefits in my week 5 post if you would like to learn more about it!
    kira_a
    Hi Ishaan, good catch! Yes, the equation you cited above is indeed meant to represent the formula for impulse. As for my notation for derivations, orbital mechanics is a bit different than traditional physics, math, and engineering conventions, so some of the equations may look slightly different. I do believe, however, that J is supposed to represent impulse in this case, so that was just a slip-up on my part while deriving.
    kira_a
    Hey Inesh! Overleaf allows for more seamless integration of equations and diagrams as you said because of its unique code window that more traditional word editors like Google Docs and Microsoft Word don't feature. This window allows for the messy editing and formatting to be done behind the scenes. For example, when I want to include an equation in my text, I can simply type "\begin{equation}" and then the next few lines use a different syntax that spits out a properly formatted equation. When I am done with the equation I simply type "\end{equation}" and recompile the document, then Overleaf automatically numbers the equation in the text and displays the final product. This automatic equation numbering system is just one of the features that makes Overleaf more scientifically oriented as for scientific texts, equations are typically numbered for ease of reference. I hope this helped. If you have any more questions, please let me know.
    kira_a
    Hi Jason, very insightful question! Yes, the goal of the orbit-raising maneuver is to reduce the fuel cost of a cislunar mission. This would lower total mission costs to get beyond geosynchronous orbit and ideally allow more space agencies to get involved in the cislunar region and establish a permanent human presence on and around the Moon. As far as environmental impact goes, this reduced fuel use could possibly lower the resource extraction needed to obtain these fuels, because less is needed, but I do not see much of an environmental impact beyond that.
    kira_a
    Great comment, Bhavin. You are correct in your assertion that F =dp/dt, not dp/dt/t.

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