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Escape velocity and orbital velocity
Posted: Tue Jul 01, 2014 3:46 pm
by Stargate38
I'm trying to find a formula for escape velocity, as well as a formula for orbital velocity. Could someone please help? So far, I've only found one for escape velocity at r=1: The integral from r to ∞ of f(r) dr*sqrt(m/100). In TEX: [tex]$\int_{r}^{\infty} f(r) dx*\sqrt \frac{m}{100}]$[/tex]. The orbital velocity formulas that I showed you earlier are only for forces of the form r^n or r^-n.
Grr... WHY don't you support TEX? It should look like the attached image:
Re: Escape velocity and orbital velocity
Posted: Sun Jul 06, 2014 11:25 am
by A Random Player
Orbital velocity at radius r and force law G*M*m*f(r) is a straightforward application of a = v^2/r:
Code: Select all
a = G * M * f(r)
v = sqrt(a * r)
v = sqrt(G * M * f(r) * r)
For f(r) = r^a this simplifies to
Code: Select all
v = sqrt(G * M * r ^ a * r)
v = sqrt(G * M * r ^ (a + 1))
In the case of r^-2 this simplifies to
Code: Select all
v = sqrt(GM * r ^ -1) or
v = sqrt(GM/r)
There's some weird stuff with the scale though, so you'd have to correct for that. (eg. f(r) = r - 1 balances out at 100 instead of 1)
Re: Escape velocity and orbital velocity
Posted: Mon Jul 07, 2014 12:03 am
by testtubegames
A few things:
a) Great explanation, random!
b) Scale-wise, there is indeed some weird stuff afoot. Namely, the 'r' that is used in the equations is actually a dimensionless value... the Distance (in units) divided by 100 units. That is just a helpful way to do it -- allowing you to write statements like sin(r), which wouldn't make sense if r had a dimension. The constant 100 is nice, since that means all the basic, to-a-power force laws (r^10, r^-2, r^-20) will be equal to one another at 100 units, which is a couple inches on your screen. A nice scale.
One extra thing I'll note is that in order to make this scaling work, I needed to tweak G, too. So G actually gets divided by 100*100. (since G*M/r^2 = G*100*100*M/Distance^2) If that makes sense.
c) If you really want latex in the forums, I could try to add it in. Could be useful. Though when I tried to add it earlier to day, I broke the forums temporarily... so I'll tread a bit more cautiously this next time.