Week 5: Raising The Project To Greater Heights
Kira A -
Welcome back everyone to your favorite orbital mechanics blog! Thanks again to all of you who have been tuning in since the beginning of my journey and those who jumped in on the way. This week, as I briefly mentioned in my last blog, I have been looking into the orbit-raising maneuver and its use in cislunar navigation, specifically for cargo transport. However, my offsite mentor, Dr. Farooq, offered me some great insight into my process for analysing the different orbital trajectories to the moon. Instead of limiting myself to the orbit-raising maneuver (ORM), I should test variations that include plane changes, multiple impulses, and/or different initial conditions, and verify and validate that the ORM is, in fact, the most efficient.
I will be continuing to work on the ORM and the accompanying maneuvers throughout the rest of this week and next week, however, I would like to take some time to briefly explain the concept to the ORM and how it might help to reduce the fuel needed to send a spacecraft beyond Earth’s orbit, and therefore reduce mission costs. To begin, we must define our initial conditions. We have a spacecraft orbiting Earth at about 250 kilometers from the surface in a circular orbit (or orbital eccentricity of zero). By the end of our maneuver we want our spacecraft to have enough energy to escape the orbit it has around Earth right now and be on its merry way to the destination, which is, in our case, the Moon. Traditionally, this is done with a large singular burn, however, this requires a lot of fuel to be expended in establishing that trajectory, and therefore, can be very expensive. With the orbit-raising maneuver, the goal is to use a series of small burns to gradually increase the spacecraft’s energy so it can eventually reach its targeted escape trajectory toward the Moon. Below is a picture of the traditional method using one large burn.
The reason that the ORM should theoretically allow my mission to conserve costs on fuel is due to the Oberth effect. Let’s go back to the system we set up. Recall our spacecraft orbiting the Earth at 250 km. At a user-inputted and desired point (let’s call this the point of action) in the orbit, we will perform a thrust maneuver where we expel a targeted amount of fuel (say 2%) over a small period of time (Δt). The impulse will cause the point of action to now act as a new perigee point (closest point to the Earth) of our new orbit, which is now elliptical and has an apogee point (furthest point from the Earth) that is much farther out (orbital eccentricity is now greater than 0) – ergo, we have now transformed our original circular orbit into an elliptical one! From the apogee point, the spacecraft accelerates back toward the perigee point due to the gravitational field of the Earth – a simple consequence of our friend Newton and his Law of Universal Gravitation (NLUG), which we are all familiar with at BASIS Phoenix from all the way back in Physics 7!
Fg =GMm/R^2
Following the above equation, we can see that the gravitational force acting on an object increases as the distance from the source decreases due to the inverse-square relationship between force and distance.
This acceleration, combined with a second impulse maneuver of the same magnitude (another 2%) will increase our orbital eccentricity even further than the first go around, effectively moving the apogee point further out than before. The extra energy that we gain in terms of an increased apogee point, is where the Oberth effect comes into play. In the context of our system, the Oberth effect states that with a greater velocity, the same impulse (or 2% fuel burn) will result in a greater change in kinetic energy than that same impulse at a lesser velocity. We can equate this to a more grounded example. Similar to the Oberth effect, if your car broke down and you had to push it out of the road, you can imagine that it would be far easier to push the car once it was already in motion than when it is still at rest. Now, this isn’t a perfect analogy, but it works for the sake of understanding the concept.
In theory, this should allow the spacecraft to obtain the required energy to achieve a Trans Lunar Injection Orbit (TLIO) but with less total fuel spent as the sum of the small burns will be less than one large burn. However, this requires more than just conservation of energy to solve and that is why it is the main focus of my project. This is why I will also spend most of next week working on it.
I am excited to keep you all updated next week on the progress I will make in solving the orbit-raising maneuver as well as analyzing the other methods of reaching the moon. I know that next week is spring break for many of you so I do not expect you all to be able to tune back in right away but I do hope you will stop by when you get the chance!
Thank you again to Mr. Joseph for providing me with a workspace to dive into this project and offering me guidance along the way. I would also like to thank Dr. Farooq for taking time out of his busy schedule to help oversee this project and offer his insights. Thank you to Dr. Goodwin for always being willing to troubleshoot any issues I might face when it comes to my paper or method. Lastly, a big thank you to everyone who is reading this blog and to those who interact with my blogs as well. I hope you all have a great spring break!
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