Payton M -

Hello, welcome back to my blog!

We are now at the first stop on our journey to Eris– Jupiter, the final destination of the Hohmann Transfer (HT). The third stage of this first section is called the Jovian Rendezvous with stage one being the Orbit Raising Maneuvers (ORMs) and stage two being the HT. The rendezvous can be thought of as the departure from the last orbit in the ORM but in reverse. The first step is to change the focus of the spacecraft’s orbit from the Sun (the focus of the interplanetary HT trajectory) to Jupiter (the arrival planet and focus of the capture orbit). The last leg of the heliocentric HT is the beginning of the Jupiter-focused rendezvous, and the speed of the spacecraft at this point is known as the approach velocity VA. The speed of Jupiter needs to be subtracted to find the speed of the spacecraft with respect to Jupiter to find the hyperbolic approach velocity (V), since with respect to Jupiter, it is now in a hyperbolic trajectory. This value is used in multiple equations to solve for the ideal capture orbit perijove radius at point P (rp). 

Planetary Rendezvous. Source: Orbital Mechanics for Engineering Students 5th Edition by Howard Curtis


In MATLAB, I created a for loop to determine the ideal capture orbit eccentricity and rp value, essential for finding the total change in velocity and fuel needed to attain the capture orbit. Using calculus, an equation for the optimal rp value can be derived, and is dependent on a capture orbit eccentricity value. However, we don’t know this eccentricity value, or which value would be best, so the for loop runs different eccentricity values from 0.1 -> 1 in 0.1 increments through each equation to ultimately output the delta-v needed to transfer from the hyperbolic path to the capture orbit for each eccentricity option. Then, it takes the minimum value and calculates the amount of fuel needed. 

This week I finished the next section– the plane change– and am working on finding the correct orientation needed to depart for Eris! 

Thanks for reading and following along on this journey to Eris!

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    Hey Payton! What methods or considerations did you employ to ensure the efficiency and accuracy of your calculations in MATLAB for determining the ideal capture orbit eccentricity and any other values you calculate?
    Amazing work, Payton! I'd love to hear more about how your work with your mentor is supporting your understanding of these concepts. Your knowledge seems quite in-depth!
    Payton Miller
    Thank you, Ms. Bennet! Both Joskua and Akash have been amazing mentors who have guided and supported me. I began working with them over the summer right after Junior year, and it has been a challenging yet rewarding experience to get a deep understanding of astrodynamics. Both Akash and Joskua provided me with reading materials like the “Orbital Mechanics for Engineering Students” textbook that is used at the undergraduate level. Additionally, I learned the fundamentals of orbital mechanics in the capstone course: Aerospace Engineering Sciences, and the Multivariable Calculus capstone has helped me with understanding some of these advanced concepts. Akash also taught me the fundamentals of MATLAB early on in the school year and showed me how I can create my own algorithms for the trajectory analysis. Learning MATLAB took up a significant amount of time in the early stages of my project, but now I’m able to code and debug/troubleshoot on my own.
    Payton Miller
    Hi Tejasvi, thank you for your question! Our algorithm in the code follows the derivations and analysis done in the textbook, but modified and optimized for our application. Once we implement our code, we compare our results and our numbers to similar missions done in the real world. For instance, the Orbit Raising Maneuvers (ORMs) have been done multiple times by the Indian Space Research Organization (ISRO) in their missions to the Moon and to Mars. We compare our results to these missions to see the effectiveness of our code and our results. We also double check the physics to make sure our numbers make sense. At the moment, we are only able to numerically (MATLAB) and analytically (Physics) validate our results. Due to the time constraints on this project, we have made some assumptions that simplify our model, but the general idea is still valid!

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