# Apparent Flux and number of stars

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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You are wrong in final equation.
\begin{equation}
N=n.A.l
\end{equation}
You know
\begin{equation}
n = \int_{L_{min}}^{L_{max}} n(L)dL.
\end{equation}
If we put this to first equation we get
\begin{equation}
N = \int_{L_{min}}^{L_{max}} n(L)A.LdL.
\end{equation}

Other equations are true, i think

You are wrong in final equation.
\begin{equation}
N=n.A.l
\end{equation}
You know
\begin{equation}
n = \int_{L_{min}}^{L_{max}} n(L)dL.
\end{equation}
If we put this to first equation we get
\begin{equation}
N = \int_{L_{min}}^{L_{max}} n(L)A.LdL.
\end{equation}

Other equations are true, i think
Where did you determine that first equation from?

Also, do we therefore conclude that $$A = \frac{N}{\int_{L_{min}}^{L_{max}} L \cdot n(L) dL}?$$

In my equations A is area. In my equation
\begin{equation}
A=4.\pi.r^2=S
\end{equation}
I should have S for this for doesn't mixing the question provided and asked constant.
And also i had mistake in the above post
\begin{equation}
N=n.V
\end{equation}
where is V volume, n tota number of star per unit volume.
And you have to express n(L) depends on variables we know. But i didn't found these method i think in this problem there aren't enough knowledge to get this. This question from a textbook? If yes you may look up the issues maybe n(L) defined by in the textbook.

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