The stars in our Galaxy have luminosities ranging from $L_{\text{min}}$ to $L_{\text{max}}$. Suppose that the number of stars per unit volume with luminosities in the range of $L$, $L+dL$ is $n(L)dL$. The total number of stars per unit volume if clearly $$n = \int_{L_{min}}^{L_{max}} n(L)dL.$$ Show that the total number of stars with apparent flux $f \geq f_0$ is $$N(f \geq f_0) = \frac{A}{f_0^{3/2}}$$ and find $A$ in terms of $n(L)$.

We have that the flux $f$, is given by $$f = \frac{L}{4 \pi r^2}.$$ Therefore, take $L_{min} = 4\pi r^2 f_0$ and $L_{max} = 4\pi r^2 f$. We thus have that $$N = \int_{4\pi r^2 f_0}^{4 \pi r^2 f} n(L) dL.$$ Is this on the right track?


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