[Tip] A List of Useful Equations

[Nemoricus]

I've found myself referring to a number of equations while playing KSP, and I thought I'd provide them in case others would find them useful as well.

KSP's mass units are assumed to be kg, for clarity.

a = semi-major axis, m

ApR = Orbital radius at apoapsis, m

delta-V = m/s, the total amount you can change your velocity by

G = 6.674e-11, the gravitational constant

g = 9.8 m/(s^2), the acceleration at Kerbin's surface

Isp = specific impulse, the number of seconds the engine can hold itself up against Kerbin's surface gravity per kg of fuel.

M = mass of planet/moon/sun

m0 = Initial mass of a vessel, kg

m1 = Mass of a vessel at stage burnout, kg

PeR = Orbital radius at periapsis, m

R = Orbital radius, altitude + planetary radius

r = planetary radius

TWR = Thrust to weight ratio. If greater than or equal to one, the vessel can support itself against the body's gravity, which is necessary for launches and landers.

v = velocity, m/s

delta-V = 9.8 * Isp * ln(m0/m1)

For multiple stages, calculate the delta-V for each stage separately, where Isp is the specific impulse of that stage's engines, m0 is the mass at the start of the stage, and m1 is the mass at the end. Keep in mind that m0 is after all mass of the previous stage is ejected, and m1 is prior to the mass of the current stage being ejected.

For stages with mixed Isp, the delta-V depends on the details of your setup.

acceleration = total thrust/total vessel mass

TWR = acceleration* R^2 / (G*M)

Fuel consumption rate (kg/s) = Thrust / (Isp * g)

a = (ApR + PeR) / 2, for closed orbits only

Circular orbital velocity = sqrt(G*M/R)

Escape velocity at R = sqrt(2*G*M/R)

Velocity in an eccentric orbit = sqrt(G*M*((2/R) - (1/a))

Period = 2*pi*sqrt(a³/(G*M))

Target Angle = 180 degrees / (semimajor axis of target orbit/semimajor axis of transfer orbit)^(3/2)

For circular target orbits, this is the angle that the target body will travel through in the time it takes the vessel to go from apoapsis to periapsis or vice versa. So, this is the angle that the target body needs to be from the transfer orbit's apoapsis or periapsis for the two to meet.

For sufficiently low targets, this can go above 360 degrees, in which case the target will lap the vessel at least once before meeting. Subtract 360 as many times as necessary to get a number between 0 and 360, which gives you the angle the target body needs to away from the periapsis for your vessel to meet it.

delta-V for inclination change = sqrt(2*(v^2)*(1-cos(itarget inclination - initial inclination))

Use this equation when you want to figure out how much delta-v you need to go from an equatorial orbit to a polar orbit, for example

Thoughts, suggestions, and comments are welcome. If I've missed anything, let me know and I'll put it in here.

Edited October 17, 2012 by Nemoricus
Equation for period.

[power5000]

Really nice info here I'd add a few like

orbital velocity= sqrt(G*M/R) and a few others

also here is a really nice online calculator I use http://web2.0calc.com/

and a website the starts of orbital mechanics http://www.braeunig.us/space/orbmech.htm

[AmpsterMan]

I would think the Rocket Equation would be fitting as well. It is the one I use the most.

[Nemoricus]

Both are already in there, and were in fact the first two I put in my list.

Any others that might be useful?

[AmpsterMan]

Both are already in there, and were in fact the first two I put in my list.

Any others that might be useful?

Oh snap sir you are correct! I am sorry that I missed that part.

To be honest I think you got the ones that are most useful down.

Maybe fuel consumption rate?

[Nemoricus]

Okay, I've added that one.

[martscht]

I'd guess you could add the orbital period for a vessel (in seconds): 2*pi*sqrt(a³/(G*M)) because this allows you to compute the time required to wait for transfers as (phi[final] - phi[initial]) / ( sqrt(G*M/a[target]³) - sqrt(G*M/a[origin]³) ) and the time required to wait for the correct ejection angle as ejection angle / 360 * period

Edited October 17, 2012 by martscht

[Nemoricus]

Okay, added.

[Bounce Ginalf]

is there an equation to figure out when to start a burn to facilitate hitting the correct ejection angle?

[EndlessWaves]

KSP seems to use 9.81m/s for gravity rather than 9.80 (at least for the fuel consumption rates).

[Nemoricus]

is there an equation to figure out when to start a burn to facilitate hitting the correct ejection angle?

No. However, I've seen a recommendation to time your burn so that you hit the angle when you're 2/3rds of the way down with the burn. Find the acceleration of your vessel and the delta-V required for the maneuver, and use that to find the time to burn correctly.

EndlessWaves: Do you have confirmation for that?

[maltesh]

KSP seems to use 9.81m/s for gravity rather than 9.80 (at least for the fuel consumption rates).

That's because Specific Impulse has nothing to do with Kerbin's gravity. When measured in seconds, it's the amount of time that(for example), the amount of fuel that weighs 1 N measured under a standard gravity will produce 1 N of thrust with the specified engine.

[AmpsterMan]

That's because Specific Impulse has nothing to do with Kerbin's gravity. When measured in seconds, it's the amount of time that(for example), the amount of fuel that weighs 1 N measured under a standard gravity will produce 1 N of thrust with the specified engine.

Huh. I always wondered why that specific factor was in the rocket equation. Thanks for the explanation

[Bounce Ginalf]

No. However, I've seen a recommendation to time your burn so that you hit the angle when you're 2/3rds of the way down with the burn. Find the acceleration of your vessel and the delta-V required for the maneuver, and use that to find the time to burn correctly.

EndlessWaves: Do you have confirmation for that?

Thanks for the info.. It is very much appreciated.

[Nemoricus]

That's because Specific Impulse has nothing to do with Kerbin's gravity. When measured in seconds, it's the amount of time that(for example), the amount of fuel that weighs 1 N measured under a standard gravity will produce 1 N of thrust with the specified engine.

Kerbin's gravity is approximately 1 g, likely to make things a little more intuitive and familiar for players.

Bounce Ginalf: You're quite welcome.

[CommanderLOL]

What's stinks is that my iPod doesn't allow me to open the tabs to see the equations.