Derivation of gravity assist equations

[ChickenPress]

I've been wanting to tryout a aerogravity assist for a while now. So I've been scouring the internet looking for some concise explanation on gravity assist equations and their derivation. I've found this paper by Bob Johnson, and Scott Manley's video How Gravity Assist Work is somewhat useful, but I'm looking for the nitty-gritty. The 20 pages of pure uncut derivation. Can anyone direct me to some articles or recommend a good textbook? I'd much appreciate it.

[Bill Phil]

From my understanding you should be able to treat it like a collision between two particles where the relative velocity between the two is the same after collision, but the velocities relative to their center of mass or another body will be different.

Momentum is a conserved quantity so it shouldn’t be too hard to derive a reasonable approximation.

That said it can get complex and eventually approximations break down.

[K^2]

To Bill's point, the collision will not be perfectly elastic. You should treat it as a collision with an energy loss. Also, there are going to be severe limits on the angle of incidence if you want to skip out of the atmo.

[wumpus]

From your linked article

Quote

A detailed calculation appears in the book by Barger and Olsson.  This deals with a gravity-assist at Jupiter en route to Uranus.  However the formidable length of the 13-page manipulation only reinforces perception of the slingshot's subtlety.

This sounds like what you are asking for.  Looking up the footnote we get:

https://www.amazon.com/Classical-Mechanics-Modern-Perspective-2nd/dp/0070037345

And checking this link:

https://www.worldcat.org/title/classical-mechanics-a-modern-perspective/oclc/357081

I found it was supposed to be in most nearby colleges with pretensions of teaching physics.  Unfortunately, only one of them had the book on the shelves, but the nearest one (admittedly in a different city) had two copies.  YMMV.

[ChickenPress]

@wumpus Thanks for the heads up. I was lucky enough to get a copy of the textbook from my library today.

 

 @Bill Phil & @K^2Thanks for feedback. Drag losses I can figure out  with a path integral.

I'm a bit confused on how to add lift into the equation though, since that will affect my asymptotic angle. Any ideas?

 

[K^2]

7 hours ago, ChickenPress said:

I'm a bit confused on how to add lift into the equation though, since that will affect my asymptotic angle. Any ideas?

You don't. Your first order estimate is going to be that the ship just collided with hard surface and bounced off. You don't care about the details of the interaction. You do care about energy lost, which is a function of angle of incidence and impact velocity. To get that, you might want to run a bunch of simulations for with different starting conditions and fit an approximation formula to the results. This is sufficient to start solving for fly-by trajectories and come up with candidates for near-optimal solutions. Once you have these, you start actually numerically solving for a flight through atmosphere and adjusting parameters - basically, it's just a multi-variate optimization problem at this point, so you'd use a standard method.

You will need to factor in lift in your simulation for energy lost estimate function and in final optimization. Aerodynamic forces are conventionally split into drag and lift. Drag is component of aerodynamic forces matching direction of relative wind and lift is component perpendicular to relative wind. The equations for both are very similar Lift = (1/2)C_Lρv², Drag = (1/2)C_Dρv², where C_L and C_D are lift and drag coefficients respectively. v is velocity of relative wind. ρ is density of air. Both C_L and C_D are functions of angle of attack and of mach number (more generally, velocity, density, and temperature of air stream, but modeling it as just mach number is surprisingly accurate). Generally, very non-trivial functions, so you'll have to have some fit formula. Typically, these would be obtained from full aerodynamic simulations, but there are a lot of ways to get a back of the napkin estimates that aren't bad. That's a whole topic in itself, though.

So for the energy-lost simulation, I would probably select alpha for given mach number that gives you best glide ratio = C_L/C_D for given mach number and integrate flight through atmosphere with that for each choice of incident velocity and angle. Get your data, fit a formula, and use that to estimate energy-loss in your fly-by optimizer, treating each encounter with atmosphere as a hard, elastic impact. Once you have an approximate plane that works, you start simulating in earnest and optimizing all of your parameters, as well as making sure you don't exceed structural limits.

Edit: If you're doing this in KSP, your best bet for getting accurate simulation is just spawning the ship under conditions of interest and going for it. If you get enough data points, you should have a pretty good estimate.

Edited July 5, 2020 by K^2

[sevenperforce]

16 hours ago, K^2 said:

You will need to factor in lift in your simulation for energy lost estimate function and in final optimization. Aerodynamic forces are conventionally split into drag and lift. Drag is component of aerodynamic forces matching direction of relative wind and lift is component perpendicular to relative wind. The equations for both are very similar Lift = (1/2)C_Lρv², Drag = (1/2)C_Dρv², where C_L and C_D are lift and drag coefficients respectively. v is velocity of relative wind. ρ is density of air. Both C_L and C_D are functions of angle of attack and of mach number (more generally, velocity, density, and temperature of air stream, but modeling it as just mach number is surprisingly accurate). Generally, very non-trivial functions, so you'll have to have some fit formula. Typically, these would be obtained from full aerodynamic simulations, but there are a lot of ways to get a back of the napkin estimates that aren't bad. That's a whole topic in itself, though.

Yep, this is the crux of it. Assume things like AoA are going to be optimized mathematically for a given application, then model lift as a function of drag with the velocity envelope you're using.

Because the drag-force equation can be derived from Newton's penetration approximation, you can also use the cross-sectional path through the atmosphere to approximate the problem.