I was deriving the Lane-Emden equation from the hydrostatic equation and the polytrope. I was following the procedure presented by Carroll & Ostlie's book. I was stuck on this part, it said that the collective constant

$$\left[(n+1)\left(\dfrac{K \rho_c^{(1-n)/n}}{4\pi G}\right)\right]$$

has the unit of distance squared. I can't understand this because $n$ is the polytropic index that changes with respect to the cases.

Could someone explain this why this term has a unit of distance squared?


The constant $K$ is defined by the polytropic equation of state


so $K$ has strange dimensions that cancel out the strange dimensions of $\rho_c^{(1-n)/n}$.





one has


Then, since,


one has



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