Measuring Gravity

[mrx89_07]

I'm interested in measuring the gravity on the various celestial bodies on the game. Does anybody have any sort of idea about how one might go about doing that? One thing I thought about was determining the force exerted by a Kerbal Astronaut when he jumps, and then I could land on the various celestial bodies and time the amount of time it takes a karbal to reach apoapsis and retern to the body. I'm not sure that would work or even be accurate enough.

[PakledHostage]

I'd do it by calculating the mass of the celestial body and then calculating the surface gravity. This will be trivial if you know the body's radius, but it will only be a little bit more complicated if you don't.

If you don't know the body's radius, you'll need to solve for two unknowns (radius and mass) simultaneously. The easiest way to do that within reasonable experimental error would be to enter a circular orbit around the body and record the orbital period (from the map view) and orbital speed (from the navball), then enter the values in the equations for orbital speed and orbital period and solve the system of equations for M and r. Once you know those two values, you can easily calculate the gravity at a given distance from the centre of mass.

You can find the equations for orbital speed and orbital period of circular orbits online. You can also readily find the equation for gravitational force at a given distance from the centre of mass. (I’d post the equations myself, but imgur isn’t working for me this evening.)

You could achieve higher accuracy by using the equations for elliptical orbits together with data acquired by flying elliptical orbits in the game (because it is very difficult to achieve a nearly circular orbit and a circular orbit may not be a good approximation), but the algebra is more complicated in that case.

[EndlessWaves]

Are you after measuring the gravity as a project or do you just want the final values for spaceship design?

[LewisTherin]

If you don't know the body's radius, you'll need to solve for two unknowns (radius and mass) simultaneously. The easiest way to do that within reasonable experimental error would be to enter a circular orbit around the body and record the orbital period (from the map view) and orbital speed (from the navball), then enter the values in the equations for orbital speed and orbital period and solve the system of equations for M and r. Once you know those two values, you can easily calculate the gravity at a given distance from the centre of mass.

You can find the equations for orbital speed and orbital period of circular orbits online. You can also readily find the equation for gravitational force at a given distance from the centre of mass. (I’d post the equations myself, but imgur isn’t working for me this evening.)

Basically this, for a circular orbit solve T^2 = 3*pi*V/(GM) and v = sqrt(GM/r) simeltaneously for GM and r (T the period, V = volume of the sphere enclosed by the circular orbit, v the velocity of the thing orbiting, r the radius from the centre of the body at which you're orbiting). Then just use F = GM/(r^2) to find gravity at some r. If you want to know the surface gravity you can again use the data from your orbiting object, just subtract the value of r printed on the screen from the value of r from the centre of the body that you calculated, this should give you the radius of the body, which you then chuck into F = GM/(r^2) to get surface gravity. (I hope that makes sense xD)

The game will print out a value of velocity for your orbit, I assume to find the period, you could land one object on the planet to use as a stationary reference point, then time how long it takes for your orbiting craft to make one circuit back over the object (but the error would be on the order of minutes most likely). I'm not sure how badly the errors would compound through the calculation, really depends on how accurately you can get your orbital period timing, hopefully the overall result wont be more than 10-20km out for the body radius of something the size of kerbin. If you want to cheat for a nice circular orbit just use mechjeb, it does an amazing job.

[edit] Actually ignore that stationary reference point bit, given the planet is also rotating the relative velocity would throw off the result, probably significantly.

[edit 2] I suppose if you have a stationary reference point on the surface, and 2 orbiters, one going in a prograde orbit, the other a retrograde orbit, and you measured the period of both (both must be at the same altitude / velocity for this to work), then you would measure two periods T1 (prograde) and T2 (retrograde), which would correspond to the 'true' period of rotation T by T1 = T + dt, T2 = T -dt, and get a somewhat accurate result for the rotational period for the rest of the analysis.

Edited October 11, 2012 by LewisTherin

[melaus]

measuring mass and the planet's radius with a larger orbital radius may reduce the magnitude of some of the sources of error due to slightly non-eliptical orbit and your measurement of time. Although I dont imagine the error would be very significant even with a smaller orbit as long as you make the measurements accurately.

[PakledHostage]

I assume to find the period, you could land one object on the planet to use as a stationary reference point, then time how long it takes for your orbiting craft to make one circuit back over the object (but the error would be on the order of minutes most likely). I'm not sure how badly the errors would compound through the calculation, really depends on how accurately you can get your orbital period timing, hopefully the overall result wont be more than 10-20km out for the body radius of something the size of kerbin. If you want to cheat for a nice circular orbit just use mechjeb, it does an amazing job.

[edit] Actually ignore that stationary reference point bit, given the planet is also rotating the relative velocity would throw off the result, probably significantly.

[edit 2] I suppose if you have a stationary reference point on the surface, and 2 orbiters, one going in a prograde orbit, the other a retrograde orbit, and you measured the period of both (both must be at the same altitude / velocity for this to work), then you would measure two periods T1 (prograde) and T2 (retrograde), which would correspond to the 'true' period of rotation T by T1 = T + dt, T2 = T -dt, and get a somewhat accurate result for the rotational period for the rest of the analysis.

Or you could look at the value given in the map view. Hover your mouse over the Pe and Ap icons to see how many seconds you are away from each. The difference is half the orbital period.