# What is the spectral power output of a typical galaxy?

I'm looking for a function which gives the typical isotropic spectral power of light emitted by an entire galaxy on the whole electromagnetic spectrum (in watts per frequency unit), as a function of frequency: $$\tag{1} \mathrm{d}\mathcal{P} = \mathcal{Q}(\omega) \, \mathrm{d}\omega = \; ?$$ The total bolometric power emitted (in watts) is then $$\tag{2} \mathcal{P}_{\text{tot}} = \int_0^{\infty} \mathcal{Q}(\omega) \, \mathrm{d}\omega.$$ What would be the function $\mathcal{Q}(\omega)$, for an "ideal" (theoretical) spherical galaxy?

As a candidate, how can we justify a function like the following one, where $\alpha$ and $\omega_0$ are two adjustable positive parameters, and $\mathcal{P}$ is a constant (the total bolometric power)? $$\tag{3} \mathcal{Q}(\omega) = \frac{\mathcal{P}}{\Gamma(\alpha + 1)} \; \Big( \, \frac{\alpha \, \omega}{\omega_0} \, \Big)^{\alpha} \; e^{-\, \alpha \, \omega / \omega_0} \; \frac{\alpha}{\omega_0}.$$

• I found an old paper which gives something very similar to equation (3), with $\alpha = 2$. See page 687 : thesis.library.caltech.edu/3219/1/Shectman_sa_1974.pdf
– Cham
Nov 2 '16 at 0:52
• I can't say I have a definitive answer for you, but why does Planck's law not work for you? That is literally exactly the isotropic spectral power of light of a black body. You can modify the function to apply to an entire galaxy with some simple changes. I think its also worth pointing out that you equation 3 is basically just a scaled version of Wien's approximation to Planck's law, when $\alpha = 3$. Nov 2 '16 at 13:16
• @zephyr, yes, I knew that the function (3) is a generalisation of Wien's law (when $\alpha = 3$). But I don't think that a galaxy could be modelized as a blackbody. There are lots of absorption and diffusion caused by dust and gaz. And you have a mix of many kind of stars inside (i.e. a mix of small blackbodies, each with different temperatures). Actually, this is the sense of my question : what should be the best representation of the spectral power output of a galaxy ?
– Cham
Nov 2 '16 at 13:47
• I found a detailled paper about this subject. The spectral luminosity of galaxies is extremely complex, and is so far away from the Planck's distribution, it's not even funny anymore ! Equation (3) above is not even close ! :-( arxiv.org/pdf/1008.0395v1.pdf
– Cham
Nov 3 '16 at 13:13
• Great! Feel free to post an answer to your own question. Nov 3 '16 at 13:38