TITLE: Proportionality and units
QUESTION [1 upvotes]: This might be very easy, but I'm not 100% sure how it's done.
Lets say I have this equation:
$$R = R_{0} \cdot \left[1 - \frac{P_{0}R_{0}}{GM_{0}\rho_{0}}\right]^{-1},$$
where I know $P_{0}$, $M_{0}$ and $\rho_{0}$. I'm then told to find $R$, and that $M_{0} \propto R_{0}^{-3}$.
Now, how do I do this ? From the proportionality I can also say $R_{0} \propto 1/\sqrt[3]{M_{0}}$, and then just insert it into the above equation, right ? But let's say that $M_{0} = 10^{28} \text{kg}$, how does that translate into $R_{0}$, with units and all ? I don't end up with $R$ being a length, if $R_{0} = 1/\sqrt[3]{10^{28}\text{kg}}$.
Yeah, I'm just a little bit confused. But I can't seem to get my head around this right now.

REPLY [2 votes]: It seems that that you're converting from a volume ($R_0^3$) (is the minus sign a typo?) to a mass($M_0$). 
The constant of proportionality is a density, presumably the $\rho_0$ that you're given.
In more general terms, the constants that are hidden by the $\propto$ symbol can have units too.
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