Gravity-Assists in Reverse

Gravity assist maneuvers are incredibly useful for sending spacecraft far into space. The maneuver is like a skiing parent using a pole to give their kid a boost. The parent loses a bit of momentum, but leverage their larger size to quickly speed their kid up. The same is true for a gravity assist. There, instead of firing boosters to go directly to some far away place, the spacecraft flies towards a closer-by planet. Then, the spacecraft orbits around the planet, leveraging its gravity to launch it further into deep space. New Horizons was able to gain an additional 14,000 km/hr in velocity with a gravity assist around Jupiter.

But the laws of physics dictate that energy and momentum must be conserved. So, Jupiter slows down very slightly as the spacecraft speeds up. Effectively, the spacecraft is stealing some of Jupiter’s momentum and its energy. But this raises an interesting question: what would happen if you tried to do this in reverse? Gravity assists typically have the spacecraft and planet travel in the same direction (counter-clockwise). But if the spacecraft is orbiting in the opposite direction as the planet, can you send the spacecraft faster in one direction and the planet faster in the other? That would preserves momentum. But, if both increase their velocities, both would seem to have more energy, which would violate the laws of physics.

Unfortunately, there aren’t many good public-source orbital simulators that involve inter-planetary transfers. So I had to use the next best thing: Kerbal Space Program. KSP is a video game that uses a very basic model to simulate orbits. It assumes that every object is orbiting only one other object, and ignores the effect of other nearby planets. After all, in real life, the Moon’s distance from the Sun is very similar to the Earth’s distance from the Sun. So, you can get a good estimate of the Moon’s location by looking at its position relative to Earth as a function only of the Earth’s gravity, and then tracking the Sun’s effect on Earth separately.

So, using KSP, I did my best to plan a maneuver to do a reverse-gravity assist. Here are a series of pictures showing the gravity-assist I attempted:

This was the beginning orbit, on course to be intercepted by the Moon (or, in KSP, the “Mun”). Important to note are its velocity and altitude: 2799.2m/s at an altitude of 243,190m. Source: KSP
This is the spacecraft in its orbit, about to enter the Moon’s sphere of influence. Source: KSP
This was the orbital trajectory of the spacecraft as its was captured by the Moon. Source: KSP
This is the final orbit. Notably, the velocity increased (∆v = 6.9m/s) even at the higher altitude (∆d = 158m). Source: KSP

The maneuver did, technically, work. The spacecraft went faster despite its higher altitude. But it wasn’t close to the magnitude of effect from a traditional gravity-assist. And with a change in velocity of only 6.9m/s, KSP’s physics engine may have just shown something that a more accurate engine would show couldn’t happen. But it is worth noting that the spacecraft is further from the planet than the Moon when it enters the Moon’s sphere of influence. This yields a possibility: that the spacecraft accelerated towards the planet and the Moon accelerated away from the planet. This would decrease the Moon’s orbital velocity, conserving energy and momentum.

Fundamentally, though, I wanted to answer a simple question: could you do the sort of gravitational assist that’s typically possible if the spacecraft and planet orbit in opposite directions? And I got an answer: basically, no.

3 thoughts on “Gravity-Assists in Reverse

  1. Awesome post, Andrew! Upvoted for KSP. Viewing your test results made me realize that the physics behind gravity-assists is probably a bit more complicated than one or two formulas. Yet, we expect the spacecraft’s orbit to obey conservation laws. I wonder what calculations KSP made for the velocity-increasing maneuver?

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  2. I love KSP! I’m especially interested in KSP’s treatment of multi-body gravity simulation. While it’s fairly easy to treat a system of two gravitationally-interacting bodies, it’s notoriously difficult to do with even one additional body (see the “three-body problem”). The numerical methods that are required for 3+ body problems are much more computationally intensive than the closed-form solution of the 2-body problem, which makes me wonder about which kinds of shortcuts KSP takes to do its simulations.

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  3. From a TA: I LOVE KSP! So stoked to see you use it. Gravity assists are awesome and efficient uses of gaining momentum with minimal fuel usage. There are different types of these kind of orbital transfers like leaving the sphere of influence (SOI) of the body you are gravitationally bound to or entering an SOI of a body you want to be bound to. You can read about these kind of transfers here: https://en.wikipedia.org/wiki/Hohmann_transfer_orbit

    The Apollo missions also used these kind of transfers to fly by, orbit and eventually land on the Moon. https://en.wikipedia.org/wiki/Trans-lunar_injection

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