Unit Error in Oppenheimer-Snyder 1939 Paper?

[RogueBanana]

Is there an error, perhaps typographical, in the famous 1939 Oppenheimer-Snyder paper "On Continued Gravitational Contraction" (Phys Rev v56 Sept 1, 1956 pp 455-459)?

Do the units balance in equation 37? The last term inside the last parantheses

$\frac{3r_0^{1/2} \tau}{2R_b^2}$

where $r_0$, $\tau$ and $R_b$ all have units of length (or time), isn't unitless, as it should be to be subtracted from 1.

 

[fzero]

First of all, the curly brackets means that the 2nd line of (37) is still under the logarithm. The factor

$$\frac{3r_0^{1/2} \tau}{2R_b^2}$$

should be

$$\frac{3r_0^{1/2} \tau}{2R_b^{3/2}}.$$

This term comes from the $R_b r/(r_0 R)$ term in the expression for $M(y)$ in (32). We are to set $R=R_b$ and use (27) for $r$, which gives

$$ \frac{ R_b r}{r_0 R_b} = \frac{ 1}{r_0 } \left( R_b^{3/2} - \frac{3}{2} r_0^{1/2}\tau \right)^{2/3} = \frac{ R_b}{r_0 } \left( 1- \frac{3r_0^{1/2}\tau}{2 R_b^{3/2}} \right)^{2/3}.$$