A Visualization of the Efffects of Elliptical and Inclined Orbits on your Transfer

[Free Market]

I’ve been somewhat obsessed with the science behind Kerbal Space Program for some time. Disclaimer: I’m not a rocket scientist. But I enjoy browsing various webpages and wikis to learn cool things. Lately, my kick has been trying to write an interplanetary calculator in Python that accounts for incline and eccentricity in orbits. I’d like to release that someday, but as of right now it has no front-end GUI and isn’t ‘feature complete’. But it does make pretty graphs!! So I decided to stick a few pretty graphs together that show a few of the ways ‘reality’ might differ from predictions made on a circular model.

Inclination

Inclination is probably the easiest to understand. If your intercept point with the target planet is highly inclined relative to your departure planet’s orbital plane, you have to make an adjustment to raise or lower the latitude of the intercept point. Here’s a graph of latitude relative to Kerbin’s orbital plane for Moho (the most inclined orbit to date):

Latitude relative to Kerbin varies over the entire orbital period of the target body and is at a minimum at the ascending node and descending nodes. Coincidentally, many planets have ascending nodes at periapsis and descending nodes at apoapsis, something you will probably be targeting for an intercept point anyway.

Inclination adjustments are very expensive. Even considering Eve, a planet with an inclination of merely 2.1 degrees, an inclination change can increase your Delta V costs by almost 50%:

(note that in a cruel twist, the first launch window has the highest incline costs)

The cost to make an inclination change is independent of whether you are raising or lower latitude, and merely depends on magnitude, repeating with half the orbital period of the departure body. Remember that your intercept point in the target's orbit is decided by your departure planet's position.

Elliptical Orbits

One feature of elliptical orbits is that the orbital radius of the body changes over time. This will increase or decrease your travel time, as your Hohmann transfer will have a larger or smaller semimajor axis. Here’s a graph of travel times Kerbin-to-Jool with the launch dates highlighted:

Note how it varies with your departure planet's orbital period, not the target planet's period. This is because your intercept point along the target's orbit is set by your departure planet's position.

A larger semimajor axis on the Hohmann transfer also means more Delta V visiting outer planets, but less Delta V visiting inner planets (because the transfer orbit is actually a change from your current orbit). Thus, try to visit inner planets near apoapsis and outer planets near periapsis. Here’s a graph of a Kerbin-to-Duna Mission injection Delta V costs (ignoring the need to reach escape velocity) showing the fluctuations as the projected intercept point moves between periapsis and apoapsis:

Now, if your travel time changes based on the orbit of the target planet, won’t that also change your desired phase angle? Yes! A longer journey gives the planet more time to move on its orbit and could mean you over- or undershoot the target. Here’s a look at the desired phase angle for a Kerbin-to-Jool mission:

Now, here’s where things are probably getting a little too nitpicky to actually matter in game. But what if we are leaving a planet with a significantly elliptical orbit? Well, our Hohmann transfer calculations generally assume we are at apo/periapsis when we conclude the burn. This would imply all our velocity vector is directed perpendicular to Kerbol. This isn’t really an issue with Kerbin’s circular orbit. But for a planet with an elliptical orbit, there are points where the planet has a velocity component collinear with Kerbol. Take a look at the two components of Moho’s orbital velocity, broken down into the component collinear with Kerbol and perpendicular to Kerbol in the orbital plane:

(an alternate version for Duna that I accidentally uploaded)

If you desire to be at peri/apoapsis in your Hohmann transfer orbit at the point you leave Moho’s sphere of influence, you would need to eliminate the collinear velocity component with part of your hyperbolic excess velocity. This would affect your ejection burn velocity and angle.

Ellipitical orbits also affect the direction of travel of an orbiting body. Prograde on the departure body may not actually orient toward prograde on your transfer orbit. Here’s a graph of flight path angle, basically the deviation from perpendicular to Kerbol, for Duna:

This means that plotting your ejection angles based on prograde/retrograde may cause as much as a 3 degree error leaving Duna. It might be useful to convert a prograde relative ejection angle to a Kerbol relative ejection angle.

Conclusion

This guide is not meant to persuade you to throw out your circular predictions. If you’ve been getting to your target planets without worrying about this stuff, you probably still will! Nor is it meant to intimidate you with dozens of variables to consider. I merely wrote it as a nifty visualization for all those forum-goers asking, “But how does eccentricity and inclination affect my numbers?â€Â

Also, I want to warn you not to lift any numbers directly from these graphs. I haven’t alpha tested the code generating them. I remember a quite hilarious bug involving arcos and cos that literally zeroed a key variable on the interval pi-2pi. Anything little thing like that could make these numbers inaccurate.

Edited October 9, 2012 by Free Market