How to calculate time to reach a target orbit for a transfer maneuver with constant acceleration?

[Virkon]

People in reddit AskScience were not able to help me with this, but maybe a fellow Kerbonauts more well versed in orbital dynamics can.

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Lets say there is a spacecraft with nigh-unlimited fuel, but its acceleration is limited to 9.81 m/s² for passenger comfort. The craft is constantly accelerating from the start to the point when it rendezvous and matches speeds with the target.

In a scenario where both objects start at rest relative to each other, the vessel would accelerate towards the target for half the trip, turn, and accelerate in the opposite direction until it has reached the target. In this case where the trajectory is just a straight line, I know the travel time can be calculated by:

T = 2 * sqrt[ D/A ]

Where:

T = travel time

D = total distance to target

A = acceleration

However, I have no idea how to calculate this in the context of an orbital transfer. According to what I've read, I think it would be similar to a one-tangent burn, but I'm actually not sure about it, and I don't think I can figure out the math myself. I'm trying to run this in a computer simulation (not Kerbal; but similar), so the equation above is not good enough. I would like to be able to compute not only the transit time but also the elements of the transfer orbit. Assume all orbits are on the same plane. You can also assume the initial and target orbits are both circular if it makes things a lot less complex, but I would like to get a solution that works with eccentric orbits.

TL;DR If a Hohmann transfer is thrifty this is "making it rain" with delta-v.

[K^2]

In general, orbital intercepts are absurdly complex class of problems. They are inherently non-linear, unstable, and even your strategy changes entirely depending on your initial conditions. There isn't a formula. There isn't even a simple algorithm. Off the top of my head, I couldn't even tell you for sure that engines-always-on will guarantee shortest time. You have to bite the bullet and solve the numerical optimization problem for any given starting condition.

[mrfox]

You are looking for a brachistochrone trajectory using a torch ship.

[Virkon]

You are looking for a brachistochrone trajectory using a torch ship.

Yeah, that's where I got that formula from the original post. But that only works for a straight line with two objects that start at rest relative to each other. I need to apply it to orbits.

[mrfox]

https://youtu.be/gMfuLtjgzA8?t=13m29s

Interesting relevant bit from a talk by Andy Weir, author of "The Martian", at 13:30.

He did it via numeric iteration.

[Virkon]

https://youtu.be/gMfuLtjgzA8?t=13m29s

Interesting relevant bit from a talk by Andy Weir, author of "The Martian", at 13:30.

He did it via numeric iteration.

I guess there's no "easy" way to solve my problem :/ ... though thanks for that video! It was pretty interesting and still a bit helpful.

[Bill Phil]

What's the problem? You want orbital data?

[Virkon]

What's the problem? You want orbital data?

I want the time to intercept the target (and maybe the eccentricity of the transfer orbit). I can calculate time quite easily for 2 objects that start at rest with T = 2 * sqrt[ D/A ]. However, when it is two objects orbiting the same body I don't know how to account for their orbital velocities.

[K^2]

You are looking for a brachistochrone trajectory using a torch ship.

Gravity isn't constant. You have a central potential of the planet/star and the thrust of the ship, which may change direction as necessary during trajectory. Solution will not be anything like a brachistochrone.