Mismatching gravitational values on the wiki

[Yoha]

WARNING: I had forgotten to take into account the radius of Kerbin as Claw noticed. I leave the original post for consistency. The correct way of computing the period can be found my latter post

I want to use the predict the orbital period of my future satellites. For instance, I currently have a satellite around Kerbin (triple checked) with apoapsis 758,815m, periapsis 740,092m and orbital period 1:27:21.2. According to [1], the formula for the orbital period is:

T = 2À √(a³/µ)

with

• T being the orbital period in seconds → (1×60 + 27)×60 + 21.2 = 5,241.2s
  
• À the usual mathematical constant ~3.14159265358979323842…
  
• a being the semi-major axis, namely the mean between apo- and periapsis → a = (758815+740092) ∕ 2 = 749,453m
  
• µ being the standard gravitational parameter, namely G×M with G the universe's gravitational constant and M the mass of the planet
  
  • G = 6.67×10^{−11} N·(m/kg)² IRL but it may be different in the Kerbal universe
    
  • M = 5.2915793×10²² kg according to [2]
    
  • µ = 3.5316000×10¹² m³/s² again according to [2] (implying G is actually the same as IRL)
    

Keeping in mind that 1N = 1kg⋅m/s² it is not hard to check the homogeneity of the formula and to have the right orders for our variables.

Now, fire up Python and compute T according to the formula:

>>> from math import *
>>> pi
3.141592653589793
>>> T=(1*60+27)*60+21.2
>>> T
5241.2
>>> a=(758815+740092)/2.
>>> mu=3.5316e12
>>> 2*pi*sqrt(a**3/mu)
2169.2555530227874

The only tabulated value is µ, so let us assume that it is not right and instead compute it from what is measured:

µ = a³ / ( T/(2À) )²

or, for readability:

µ = 4 À² a³ / T²

>>> mu2=4 * pi**2 * a**3 / T**2
>>> mu2
604966712992.4885

So µ = 6.0497×10¹¹ m³/s². Then, again, at either:

• M = µ/G = 9.06×10²¹ and the wiki is wrong here too
  
• G = µ/M = 1.14×10^-11 which is roughly six times less than IRL
  
• both the wiki is wrong and G is not the same as IRL
  

Well, to check this, we need more Science! So, next experiment: another satellite probe around the Mün.

>>> T=(11*60+19)*60+37.8
>>> T
40777.8
>>> a=(1215e3+1185e3)/2
>>> mu=6.5138398e10 #according to [3]
>>> 2*pi*sqrt(a**3/mu)
32361.839579790507

Again, different result; again, let us compute µ according to the experimental measures:

>>> mu2=4 * pi**2 * a**3 / T**2
>>> mu2
41025689794.07226

Which gives us M = µ/G = 6.15×10²⁰ (not so much off) or G = µ/M = 4.20×10^-11 which is very different than found earlier.

From this, I conclude either that:

• the tabulated values for µ on the wiki are wrong and, most likely, so are the masses of the celestial bodies (at least for Kerbin and the Mün)
  
• the game does not use the right values/formulas (we can see it either way even if the game fixes the rules after all)
  
• I missed something trivial and am utterly stupid (orbital physics are not funny if you do not forget a zero sometimes)
  

Again, I did proof read this myself, but I may have forgot something so if you see something strange, please make me now!

[1] https://en.wikipedia.org/wiki/Orbital_period#Small_body_orbiting_a_central_body

[2] http://wiki.kerbalspaceprogram.com/wiki/Kerbin

[3] http://wiki.kerbalspaceprogram.com/wiki/Mun

Edited April 12, 2014 by Yoha
Adding warning for miscalculation

[Claw]

Ahh, okay.

So I thought it was a units problem at first, but then I spotted the issue.

You're using the wrong value for your axis. You're using your AP and PE that's read out in the game. However, those values are not from the center of the ellipse. They are from the surface of Kerbin. You have to add 600,000m (the radius of Kerbin) to your 749,453m. So if I use 1,349,453m in the equation, I calculate an orbital period of 5241.26s.

Edited April 12, 2014 by Claw
My Own Stupidity

[Claw]

Updated, in case you read it already.

[Yoha]

Oh right! I did that *AGAIN* (did it too for satellite placement taking communication range into account). Thanks for taking the time of noticing this!

For the record:

>>> from math import *
>>> a=(758815+740092)/2. + 600e3
>>> mu=3.5316e12
>>> 2*pi*sqrt(a**3/mu)
5241.203234619301
>>> 2*pi*sqrt(a**3/mu) / T # relative error (closer to one = better; between 0.95 and 1.05 is pretty good)
1.0000006171524272

I'll put a warning in my original post and fix the tag.

Edit: I guess it's up to moderators to update the tag

Edited April 12, 2014 by Yoha

[Kasuha]

Edit: I guess it's up to moderators to update the tag

Maybe also mark the thread as "no bug". You need to edit the first post in advanced mode to be able to do that.

[Yoha]

Maybe also mark the thread as "no bug". You need to edit the first post in advanced mode to be able to do that.

Oh, I had not seen there was an advanced mode for editing post as well. Thanks for the tip.

[Vanamonde]

By the way, there is a sub-forum for Wiki issues. And this thread has been moved there.