The KSP Math and Calculation Thread!

[Olsson]

*This thread is for asking about all the relevant math that concerns KSP. Wheter this might be Delta-V or something else feel free to ask a question and feel free to respond to it!*

With the the new version of KSP coming out (.17) that will contain many new planets we'll need some more accuracy with our landings, orbit transfers and rocket designing. The only way to do this is by using math or the kerbal way which we all know is trial and error. Though as the planets will be far away, hard to get into orbit around, hard to land on and probably even harder to get off from I think that trial and error will not be the best approach to this anymore. I've seen alot of math questions scattered around the forums lately and most of them have been without response, probably because they were abit misplaced and most wouldn't guess they were there. That's why I created this thread which is a collected thread of all math, calculations and theorems that concern space and KSP. Anyone can share their calculations, anyone can ask, anyone can help, anyone can add to the post to make it bigger. There's another similiar thread which is sort of a 101 guide to learning the most important math aspects to KSP. If any mod finds that this thread isn't needed and that the previous mentioned thread is enough then feel free to lock this.

Relevant Threads

A Guide to Basic Kerbal Rocket Design Through Science

by VincentMcConell

Contains Delta-V, Hohmann Transfer, Fuel Flow in Mass, Delta-V Map, Body Orbital Velocity Calculation, Weight to Thrust-ratio.

(I might add previous content into the OP or other posts if I'm given permission by the person who posted the content.)

[Anomalous_Matter]

Could somebody please walk me through this like I'm five:

How would I know when to burn to get from a circular orbit at Kerbin's height to, say, a body with a radius of 500KM orbiting at 20,000,000,000 meters up?

I realize this probably involves a good deal of maths and equations, so walk me through them like I'm a little kid trying to learn addition or whatever.

[Olsson]

Could somebody please walk me through this like I'm five:

How would I know when to burn to get from a circular orbit at Kerbin's height to, say, a body with a radius of 500KM orbiting at 20,000,000,000 meters up?

I realize this probably involves a good deal of maths and equations, so walk me through them like I'm a little kid trying to learn addition or whatever.

Sorry to dissapoint you. I want to learn this too. Basically it has to do with Kepler's Third Law but I still don't understand it all. Though I could post what maltesh sent to me yesterday even though I still have a few questions.

Crap, I did the calculation wrong in the post. (gotta halve the final result, we're only travelling a half orbit during the transfer) Calculation should be:

t_f = (a_t/a_f)^1.5 * 0.5

Will go fix that now. Anyway...

When you're doing a Hohmann transfer, you're going from one orbit around a body, to another orbit around that same body. If you're doing Hohmann transfers between two objects in orbit around Kerbin (say, the Mun, and Minmus,) the body everything is in orbit around is Kerbin.

Let me work an example going the opposite direction from the one in the thread: Going from Minmus to the Mun. One nice thing about the equations is as long as we use the same units for everything, it still works out.

Minmus Orbital Altitude: 46400km.

Mun's Orbital Altitude: 11400km.

Diameter of Kerbin: 1200km.

Semimajor Axis of transfer orbit: A_t = (46400+11400+1200)/2 = 29500

Semimajor axis of the destination body's orbit (the Mun) A_f = 12000 km (since the Mun is in a circular orbit, its semimajor axis is its orbital altitude + kerbin's radius (600km)).

t_f = (29500/12000)^(1.5) * 0.5= 1.92722

This means that, in the time it takes you to transfer from Minmus's orbit to the Mun's orbit, the Mun will go around 1 full times, and another 0.9722 of a full orbit. Translating that to degrees (multiply by 360), means that you'll want to leave Minmus when the Mun is about 349° away from reaching the opposite side in its orbit (or you can alternatively think of it as leaving when the Mun is about 11° ahead of the opposite side in its orbit).

Correcting for orbital inclination on the way is left as an exercise for the pilot.

The calculation will get you close when flying interplanetary distances. Navigating /into/ a planetary SOI is going to be tricky, simply because of how small targets they are on that scale. And targets in non-circular mean that your target orbit won't be moving at the same rate throughout its orbit, which makes the calculation rather more difficult, should that turn out to be the case..

And this was the starting post which made me send him a question and the reply is above:

If you're going for the Hohmann Transfer, figuring the angle is fairly easy for objects in circular orbits. You've just got to remember Kepler's Third Law.

Define a_t as the semimajor axis of your transfer orbit, equal to (starting orbital altitude + final orbital altitude + diameter of body being orbited by everything)/2.

Define a_f as the semimajor axis of the orbit of your target object.

t_f =0.5 * (a_t/a_f)^1.5 (Edited to fix mistake in calculation)

t_f is the fraction of your target's orbital period that will pass during your Hohmann Transfer travel time.

Your transfer window is thus when your target object is t_f of its orbital period away from the point directly opposite your starting point.

For instance, Hohmann transferring from the Mun to Minmus, t_f = 0.25, and you should launch directly into a Hohmann transfer when Minmus will have to travel 90 degrees along its orbit to get to a point directly opposite Kerbin from where the Mun is at your time of launch.

I guess what my questions are: If you want to go from Minmus-Mun what are you supposed to burn towards, your apoapsis and try to push it towards the mun? Also if it orbits 1.9722 this means that it's one orbit and almost another full orbit (about 97% of the orbit). Though am I supposed to do the burn when it has completed 97% of its orbit or when it just has started it and is at 3% of its orbit? In other words, do I do the burn when it has orbited 0.9722 or when it has orbited 1-0.9722=0.0278?

Also if you make it into degrees (multiply by 360 which maltesh also taught me) it says that 0.9722 is about 350 degrees and 0.0278 is about 10 degrees. Can I use these degrees for determining more specifically when to do the burn or am I just supposed to use the map and then kind of guess when to burn when the planet is at the mentioned degree?

Edited August 5, 2012 by Olsson

[Dimitri_Kerman]

I wish I was math smart. Only history smart. I need a calculator for simple math. Good luck you math geniuses!