Is the usual Escape Velocity eqn an approximation?

[David Baker]

Text books ordinarily give the escape velocity of a mass-M body (in the center of mass frame of the system of the body and the escaping projectile, whose mass I'll label m) as

(*) v² = 2GM/r

where r is the distance between the body and the escaping projectile.

it doesn’t seem to me that (*) can be right except as an approximation when M>>m. For let w be the escape velocity of the mass-M body in the center-of-mass coordinate system. w is
-(m/M)v,
so this would give
w² = 2Gm²/Mr, which isn’t of the same form as (*). Am I making a mistake here?

When I try to derive the escape velocity equation, what I get is put most naturally just using the variable r. Letting z be the rate of change of r, I get

(a) z² = 2G(M+m)/r.

Put in terms of the mixed variables, that’s

(b) v² = (2GM²)/(M+m)r.

It is (b) (and hence (a)) rather than (*) that the energy argument seems to yield. The kinetic energy at escape velocity is half of mv² + Mw², i.e. half of mv² + M(mv/M)², i.e. half of v² times (M+m)m/M. This must be the negative of the potential energy, i.e. GMm/r. This yields (b), hence its equivalent (a).

Another reason for suspecting that it’s (a)/(b) rather than (*) that’s correct is that in the r formulation M and m should appear only in the combination M+m. For the 2-body problem is equivalent to a 1-body problem with reduced mass Mm/(M+m):

[Mm/(M+m)]dz/dt = -GMm/r²,

i.e. dz/dt = -G(M+m)/r².

So in the variable r, there’s no separate role of M and m besides in M+m.

Is this right? I've tried checking some orbital mechanics books, and none of them mention that the escape velocity equation is an approximation. But I really don't think I'm making a mistake here.

 

[PeroK]

(b) v² = (2GM²)/(M+m)r.

That looks right for the velocity of one mass in the original rest frame of the two bodies.

The usual escape velocity equation is a very minor approximation based on the assumption that the larger body does not move. How much does the Earth move if a space probe is launched?

 

[mfig]

The usual escape velocity equation is a very minor approximation based on the assumption that the larger body does not move.

Minor indeed! The ratio of the two velocities differs from 1 by only 1 part in 10¹⁶ when considering the escape velocity of a rocket 1000 times more massive than the Saturn V.