Why are these equations valid despite seemingly inconsistent units? I am having quite a difficult time in trying to understand what units are used in this paper and how to convert things to SI. For example, look at equation (1):
$$T_M \approx 1500 \rho^{1/3}\ \mathrm{K}\tag{1}$$
It seems to be showing that temperature is measured in units of $\mathrm{g\,cm^{-3}\,K}$. Then look at equation (2),
$$T_M \approx 2800 \rho^2\ \mathrm{K}\tag{2}$$
which seems to be showing that temperature is measured in $\mathrm{g^2\,cm^{-6}\,K}$. Equation (10) doesn't make sense with these either:
$$\sigma \approx \frac{5\times 10^{20}\rho^{4/3}}{T(1 + 3x)}\mathrm{esu}\tag{10}$$
How are these consistent?
 A: Each equation contain a different arbitrary constant:  1500, 2800, and 5 E20. 
It can be assumed that each arbitrary constant has exactly the right units to make everything come out right...
It is sloppy to not specify the units of these constants...
Edited for example:
I could conduct experiments on the dynamics of falling objects, and publish that the distance of fall from rest, in metres (D), and the time of fall, in seconds, (t) seem to be related, and the best fit gives:$$D \approx4.9 \times t^2$$without implying that the units of distance are time squared...
A: The units are not consistent. Or in less precise terms, wrong.
Here's the only way I can think of for this to make some sense: just after equation (1), the paper says

...where $\rho$ is the density in $\mathrm{g\,cm^{-3}}$.

My guess is that they intend you to take $\rho$ as a pure number. For example, if the density is $0.1\ \mathrm{g\,cm^{-3}}$, then you should take $\rho = 0.1$. But that's inconsistent where the part just above equation (2) where it says

For $\rho \gtrsim 0.4\ \mathrm{g\,cm^{-3}}$...

which requires that $\rho$ actually have units in it.
I suppose the intent is that you always consider $\rho$ to be either $\text{density}$ or $\frac{\text{density}}{\mathrm{g\,cm^{-3}}}$, as needed. One would hope that, especially in modern times, this sort of sloppiness with units becomes increasingly rare, because it is confusing.
