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The fraction $f$ of stars with velocities $v$ between $v$ and $v + \Delta v$ is given by \begin{equation} f(v) \propto 4\pi v^2 e^{-\frac{m \phi(\vec{x}) + \frac{mv^2}{2}}{k_\mathrm{B}T}} \end{equation}

What does it mean when we integrate this function, please? I integrated from 0 to 1000 and obtained about 60 000. Is it the total number of stars?

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You need to divide 60,000 by the integral of this function from zero to infinity. That will then give you the fraction of stars with a velocity (I think it is probably speed, not velocity?) between 0 and 1000.

The reason you need to do this is that the distribution function as you have given is given as a proportionality and does not have the correct normalisation constant at the front.

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    $\begingroup$ Thank you very much. $\endgroup$
    – Elena Greg
    Commented Aug 3, 2022 at 9:13

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