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I tried to sum up two weight ranges of the IMF which wouldn't not work so my question is, if I'm doing something wrong.

Let's say my weight ranges are $\left[X M_{\mbox{sun}}, Y M_{\mbox{sun}}\right)$ and $\left[Y M_{\mbox{sun}}, Z M_{\mbox{sun}}\right)$.

For both ranges the number of stars in a particular space are

$\xi_{[X,Y)} = \xi_0 X^{-\alpha}\left(Y-X\right)$ and

$\xi_{[Y,Z)} = \xi_0 Y^{-\alpha}\left(Z-Y\right)$

which should sum up to

$\xi_{[X,Z)} = \xi_0 X^{-\alpha}\left(Z-X\right)$

but obviously

$\xi_0 X^{-\alpha}\left(Y-X\right) + \xi_0 Y^{-\alpha}\left(Z-Y\right) \neq \xi_0 X^{-\alpha}\left(Z-X\right)$ as

$X^{-\alpha}\left(Y-X\right) + Y^{-\alpha}\left(Z-Y\right) \neq X^{-\alpha}\left(Z-X\right)$

$X^{-\alpha} Y + Y^{-\alpha}\left(Z-Y\right) \neq X^{-\alpha}Z$

$Y^{-\alpha}\left(Z-Y\right) \neq X^{-\alpha}\cdot \left(Z-Y\right)$

$Y^{-\alpha} \neq X^{-\alpha}$ (if $Y \neq X$)

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1 Answer 1

This is an integral, the number of stars of mass between $XM_{SUN}$ and $YM_{SUN}$ is :

$N(XM_{SUN} \to YM_{SUN}) = \int_{XM_{SUN}}^{YM_{SUN}} \xi_0 (\frac{m}{M_{SUN}})^{-\alpha} \frac{dm}{M_{SUN}} = \int_X^Y \xi_0 (m')^{-\alpha} dm'$

So, it is clear that, with the additive properties of the integral, we have :

$N(XM_{SUN} \to ZM_{SUN}) = N(XM_{SUN} \to YM_{SUN}) + N(YM_{SUN} \to ZM_{SUN})$

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