Something better than a patched conic approximation?

[Alexrey]

Hi guys,

I've made a Mathematica n-body simulation of the first few planets in our Solar System and thought it would be a good idea to try and simulate a spacecraft transfer from Earth to Mars. I've thought about using a patched conic approximation, but I was wondering if there is anything more along the lines of an "n-body approximation" where, say for example, the gravitational acceleration from the Earth, Mars and the Sun influence the transferring spacecraft all at the same time (i.e. no disjoint 2-body approximations and spheres of influence). Has anyone developed such a method?

By the way, how accurate is the patched conic approximation? I know it was used during the Moon missions, but what about getting to planets such as Uranus or Neptune? Does its accuracy drop off with distance, and is this drop linear?

 

[mfb]

By the way, how accurate is the patched conic approximation?

That depends on the velocity and the trajectory. It is more accurate for faster velocities (significantly above the escape velocity of one of the objects in a transition).
For a transfer from Earth to mars, I would expect better results if you simulate this numerically (taking both sun and planet into account) if the spacecraft is close to one of the planets. Or just take all significant objects (Sun, Earth, Mars, Moon, Jupiter, ...) into account. Computing power for such a simple task should not be an issue today.

 

[Alexrey]

Currently I've added the gravitational acceleration of the Sun, Earth and Mars into the simulation, which have been calculated numerically using Mathematica's NDSolve. What I really wanted to know was if there are a set of equations, similar to those for calculating planetary transfers using the patched conic approximation, that will allow me to calculate the delta-v, and phase angle etc. required, in order to hit or arrive in an orbit around the target planet.

 

[mfb]

If that would be possible in general, there would be no need for the patched conic approximation.
You will have to try some different trajectories. If they are close to what you want, check parameter values nearby, and optimize your trajectory in that way. This is just a numerical optimization problem.

 

[D H]

What I really wanted to know was if there are a set of equations, similar to those for calculating planetary transfers using the patched conic approximation, that will allow me to calculate the delta-v, and phase angle etc. required, in order to hit or arrive in an orbit around the target planet.

Of course not. Most real world problems don't have analytic solutions. This is yet another of those.

The patched conic approximation comes from a day where the most powerful supercomputer in the world had about 1/30th of the processing power of the typical laptop of today. There were better uses for the very small number (< 100) of late 1960s era supercomputers other than solving a boundary value optimal control problem for which a very good approximate solution (the patched conic approximation) exists. Nowadays you can apply numeric techniques to solve and optimize this problem on your laptop. You don't need that patched conic approximation.

On the other hand, do you need that perfect answer? An answer whose error is less than the errors that result just from firing the thrusters is good enough. Even with your ideal solution at hand, you're still going to need midcourse corrections because a real vehicle will never execute your ideal solution perfectly.

 

[Alexrey]

Thanks for the answers guys. So patched conic it is :)

I've been trying to figure out how to get the departure $\triangle v$ equation into its x- and y-components (since I'm using a Heliocentric x-y coordinate system), but have had no luck so far. This is what I did:

I took the original vector equation:
$\triangle v=\sqrt{\frac{\mu_{sun}}{r_{1}}}\left(\sqrt{\frac{2r_{2}}{r_{1}+r_{2}}}-1\right)$

And then attempted to get it into its x and y scalar components by doing the following:
$\triangle v_{x}=\sqrt{\frac{\mu_{sun}}{r_{1x}}}\left(\sqrt{\frac{2r_{2x}}{r_{1x}+r_{2x}}}-1\right)$, $\triangle v_{y}=\sqrt{\frac{\mu_{sun}}{r_{1y}}}\left(\sqrt{\frac{2r_{2y}}{r_{1y}+r_{2y}}}-1\right)$

Where $r_{1x}$, $r_{1y}$ and $r_{2x}$, $r_{2y}$ are the x- and y-components of the radial position of the Earth and Mars from the Sun respectively, and have been obtained from Mathematica's AstronomicalData function (which returns a list of 3 numbers of position, one for each coordinate). But when using these two equations to give the orbiting spacecraft the required instantaneous $\triangle v$ kick to get to Mars, I find that it has been given far too much speed and flies off into deep space instead of making it just to Mars' orbital radius. Clearly there is something drastically wrong with how I broke the original equation into its components, but cannot seem to figure out where I went wrong.

If anyone knows how I can fix this, I would greatly appreciate their help. Thanks!