<p>I think there's a minus sign missing in the exponent of the expression for <span class="math-container">$\Phi(L)$</span> (see <a href="https://en.wikipedia.org/w/index.php?title=Luminosity_function_(astronomy)&amp;oldid=1067072852#Schechter_luminosity_function" rel="nofollow noreferrer">Schechter luminosity function</a>).  I will be including it here:
<span class="math-container">$$\Phi(L) = \frac{\Phi^*}{L^*}\left(\frac{L}{L^*}\right)^\alpha \exp\left( -\frac{L}{L^*} \right)$$</span></p>
<p>The integration in question is:
<span class="math-container">$$L_{tot} = \int_{0}^{\infty} L\Phi(L) \,dL$$</span></p>
<p><span class="math-container">$$L_{tot} = \int_{0}^{\infty} L\frac{\Phi^*}{L^*}\left(\frac{L}{L^*}\right)^\alpha \exp\left( -\frac{L}{L^*} \right) \,dL$$</span></p>
<p><span class="math-container">$$L_{tot} = \frac{\Phi^*}{(L^*)^{\alpha+1}} \int_{0}^{\infty} L^{(\alpha+1)} \exp\left( -\frac{L}{L^*} \right) \,dL$$</span></p>
<p><span class="math-container">$$L_{tot} = \frac{\Phi^*}{(L^*)^{\alpha+1}} \left[ (L^*)^{\alpha+2} \Gamma(\alpha+2) \right]$$</span></p>
<p><span class="math-container">$$L_{tot} = \Phi^* L^* \Gamma(\alpha+2)$$</span></p>
<p>This seems to be what you have, assuming <span class="math-container">$\Phi_0=\Phi^*$</span>.</p>

CC BY-SA 4.0
2022-01-21T22:04:36.700Z
2022-01-24T01:03:32.243Z
2022-01-24T01:03:32.243Z
GrapefruitIsAwesome
<p>I'm trying to get the total luminosity of a total population of galaxies.
I have the <a href="https://en.wikipedia.org/wiki/Luminosity_function_(astronomy)#Schechter_luminosity_function" rel="nofollow noreferrer">Schechter Function</a> like </p>

<p><span class="math-container">$$\Phi(L) = \frac{\Phi^{*}}{L^{*}} \left(\frac{L}{L^{*}}\right)^{\alpha} \exp \left(\frac{L}{L^{*}}\right)$$</span> </p>

<p>and I need to the total luminosity:</p>

<p><span class="math-container">$$L_{tot} = \int_0^\infty L\Phi(L)dL = \Phi_{0}L^{*}\Gamma(2+\alpha)$$</span></p>

<p>but when I substitute the Schecter functions I don't get the <span class="math-container">$\Gamma$</span> function form. Could you have any tips for that?</p>

<p>And I need to show that the galaxies that more contributed to the luminosity are those have <span class="math-container">$L^{*}$</span></p>

<p>Thank you in advance.</p>

<galaxy><luminosity><mathematics>
Total luminosity of total population of galaxies!
Emir Hernandez
# Total luminosity of total population of galaxies!

(Asked by: username_0 on 2020-05-20)

I'm trying to get the total luminosity of a total population of galaxies.
I have the [Schechter Function](https://en.wikipedia.org/wiki/Luminosity_function_(astronomy)#Schechter_luminosity_function) like

$$\Phi(L) = \frac{\Phi^{\*}}{L^{\*}} \left(\frac{L}{L^{\*}}\right)^{\alpha} \exp \left(\frac{L}{L^{\*}}\right)$$

and I need to the total luminosity:

$$L\_{tot} = \int\_0^\infty L\Phi(L)dL = \Phi\_{0}L^{\*}\Gamma(2+\alpha)$$

but when I substitute the Schecter functions I don't get the $\Gamma$ function form. Could you have any tips for that?

And I need to show that the galaxies that more contributed to the luminosity are those have $L^{\*}$

Thank you in advance.

**Tags:** galaxy, luminosity, mathematics


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## 1 answers

### Answer by username_1 on 2022-01-21. Score: 2

I think there's a minus sign missing in the exponent of the expression for $\Phi(L)$ (see [Schechter luminosity function](https://en.wikipedia.org/w/index.php?title=Luminosity_function_(astronomy)&oldid=1067072852#Schechter_luminosity_function)). I will be including it here:
$$\Phi(L) = \frac{\Phi^\*}{L^\*}\left(\frac{L}{L^\*}\right)^\alpha \exp\left( -\frac{L}{L^\*} \right)$$

The integration in question is:
$$L\_{tot} = \int\_{0}^{\infty} L\Phi(L) \,dL$$

$$L\_{tot} = \int\_{0}^{\infty} L\frac{\Phi^\*}{L^\*}\left(\frac{L}{L^\*}\right)^\alpha \exp\left( -\frac{L}{L^\*} \right) \,dL$$

$$L\_{tot} = \frac{\Phi^\*}{(L^\*)^{\alpha+1}} \int\_{0}^{\infty} L^{(\alpha+1)} \exp\left( -\frac{L}{L^\*} \right) \,dL$$

$$L\_{tot} = \frac{\Phi^\*}{(L^\*)^{\alpha+1}} \left[ (L^\*)^{\alpha+2} \Gamma(\alpha+2) \right]$$

$$L\_{tot} = \Phi^\* L^\* \Gamma(\alpha+2)$$

This seems to be what you have, assuming $\Phi\_0=\Phi^\*$.


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