# How to best patch my conics?

I thought I'd try to use patched conics to see what it's like. I'll go from LEO to LXO (low Planet-X orbit), co-planar all the way.

If I understand correctly, I'll have (at least) five conics. The initial and final circles around the two planets, the big ellipse in the middle, and two hyperbolae within each planets sphere of influence.

This question is about the second one — the hyperbolic escape from Earth. When I patch it to the interplanetary ellipse, I must give the endpoint of the hyperbola the Earth's velocity first. I can think of several velocities to use, the velocity of the Earth at that moment in its conic orbit around the Sun, or the velocity of a satellite in its own conic orbit around the Sun at the new distance, but there could be other options.

Wikipedia puts Earth's sphere of influence distance at 145 Earth radii, or about 0.01 AU, so the choice would not have a huge effect but it would not be so small either.

Is there a generally accepted best way to add a velocity to the endpoint of the first hyperbola to most accurately patch to the following conic?

Assuming:

• non-spinning SOIs - they only undergo translational motion along the planet's orbit, but direction of their axes relative to each other is always constant. That's the trivial approach; sun-synchronous orbits are impossible and tidally locked bodies will need spin equal to orbital period, instead of a flat 0, but calculations are made easier.
• The "root" frame of reference is the Sun's SOI (Sun is assumed to be immobile).
• We're not concerned about the motion of the craft, just motion of the conics.

The hyperbola exists in Earth's SOI and is stationary relative to it (speed relative to Earth SOI = 0); its endpoint is its part and so, the endpoint's velocity in the "root" frame of reference will be equal to speed of Earth's SOI at that point - that is, Earth velocity.

This would become more complex in case of spinning SOIs, requiring to trace the velocity of the SOI edge relative to Sun, but as I understand this is not our case.

• There's nothing in my question related to spinning. Are you choosing one of the two velocity options named in the question or suggesting an alternate? In any event can you add a link to the source if at all possible? Thanks! – uhoh Jun 8 '17 at 9:27
• @uhoh: Maybe I don't understand what you're asking for. Speed of satellite in the (any) conic going through Earth's SOI is the speed of satellite relative to Earth. As tautological as it sounds, that's all there is to that. Speed of satellite in solar SOI is its speed relative to the Sun. Translating speed at escape/capture points is just adding or substracting relative speeds of SOIs, as trivial as that. The only way to make it any more complex is to make the SOIs spin ('cause, say, you want to make all the orbits around Earth sun-synchronous in your model.) – SF. Jun 8 '17 at 9:42
• ..and if you're asking about the speed of the endpoint of the hyperbola, (as opposed to speed of the craft at the endpoint of hyperbola), since the hyperbola exists within Earth's SOI, and is static relative to it (speed 0) the endpoint is a part of it and so moves at the same speed as SOI - or Earth. – SF. Jun 8 '17 at 9:48
• Thanks for the update - looks like you've gone with my first option. Let's see if we can find a source to see if it's the "generally accepted best way" or at least "one of the..." – uhoh Jun 8 '17 at 12:13
• @uhoh: wish I could find one, but I guess other than source code of KSP it's going to be hard - because this approach and result is so trivial. Sure if you were defining SOIs in some more fancy way... – SF. Jun 8 '17 at 12:24

The widely used rule is to use the Laplace sphere rather than the Hill sphere, which is what you are using.

I sicced an intern on a rather related problem a decade ago, which was to find the best place to switch the frame of reference in which gravitation is computed in a simulation that integrates the equations of motion.

I had him use an arbitrary precision arithmetic package to develop reference trajectories for a vehicle on an outbound trans-lunar trajectory and an inbound trans-earth trajectory, using a variety of integration techniques. I had him calculate two reference trajectories for each leg, one Earth-centered inertial and the other Moon-centered inertia. The goal here was to get consensus reference trajectories that agreed to over 20 decimal places of accuracy.

Then I had him use standard IEEE double precision numbers for the integration, switching frames of reference when the vehicle became closer than (trans-lunar trajectory) / further than (trans-earth trajectory) some prescribed distance of the center of the Moon. Rinse and repeat with different distances, rinse and repeat again with different integration techniques.

The goal: For a given integration technique, find the best place to switch integration frames when using standard IEEE double precision numbers for numerical integration. Not surprisingly, switching to MCI immediately after leaving low Earth orbit yielded lousy results. Waiting to switch to MCI until the vehicle is extremely close to the Moon also yielded lousy results. Also not surprising, switching somewhere near the Hill sphere or Laplace sphere generally yielded the best results.

Somewhat surprisingly, it didn't seem to matter all that much which of the two (Hill sphere vs Laplace sphere) was used as a switching point (or some other similar value). Switching near the Laplace sphere gave marginally better results than switching near the Hill sphere for most techniques, but it was the other way around for a few techniques. The distinction is small, not quite how many angels can dance on the point of a pin small, but close to it. Just pick one. I generally pick the Laplace sphere, but that's a bit arbitrary. The space is rather flat in the vicinity of the Hill sphere or Laplace sphere.

You are using patched conics as opposed to numerically integrated trajectories, but the same concept applies. Just pick one, and then be consistent.

• I find your "teaching style" of answers really helpful, thanks! You can probably guess that I'll do the numerical integration after I do the patched conic search to see qualtitatively how they differ, then probably try a search using numerical solutions, and see how much benefit the patched conic technique really has for contemporary laptops, so I'll also hold out for a specific answer to this question. Thank you for the Laplace Sphere, I'll look it up! – uhoh Jun 8 '17 at 5:16
• By "Laplace sphere" do you mean the 2/5 power SOI described in en.wikipedia.org/wiki/Sphere_of_influence_(astrodynamics) ? – Russell Borogove Jun 8 '17 at 16:42