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This post comes from physics exchange forum, I have transfered it here, maybe I will be luckier since I have not received answers on the other forum.

Here is the issue :

I am asked to give the formal expression of the total number density of galaxies and explain why is this expression problematic in practice.

From what I saw from my research and into my lectures, I have found the follwing relation which gives the number of galaxies $N$ with magnitude $(m < M)$ (number counts) :

$\text{log}\,N(m<M)\,\propto\,0.6 M + \text{constant}\quad(1)$

$N(m<M)\,\propto\,\text{exp}(0.6 M)\,\text{exp(constant)}\quad(2)$

I don't know what the teacher wants to highlight, i.e by saying that the formal expression of the total number density of galaxies is problematic in practice, given that I even haven't the kind of expression or equation which gives the estimation of galaxy density.

1) Concerning this distribution to use, do you advise me to use a distribution as a function of magnitude, mass or a distribution as function of luminosity (like Schechter) ?

2) I have seen on web the HMF (Halo Mass function) but it seems to be about the dark matter halos : can we count one dark matter halo per galaxy ?

3) But the problem is here is that a dark matter halo is more massive than the galaxy hosted in this halo : how can we deal with this ?

If someone could help me, this would be kind.

PS: I have also to explain why the estimation of galaxies density is difficult and problematic.

4) What are the issues when we want to estimate density of galaxies ?

Concerning the distribution as a function of luminosity, I have found the Schechter luminosity function :

$$N(L)\ \mathrm {d} L=\phi^{*}\left({\frac{L}{L^{*}}}\right)^{\alpha}\mathrm {e}^{-L/L^{*}}{\frac{\mathrm {d} L}{L^{*}}}$$

UPDATE 1: my apologies, $m$ and $M$ represent apparent magnitude and not masses.

But the questions remain about the counting of galaxies and the computation of total density at large scale.

In practise, what are the major problems when we want to estimate this density ?

UPDATE 2: people here not really motivated by this bounty ... feel free to ask for precisions about this post .

For the moment, I am faced to know at which scale I can compute a mean density : have I got to take into account this scale since it seems to have a dependance between density and volume considered, doesn't it ?

Is there a general formula to express this dependance ?

Regards

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    $\begingroup$ The problem with the form "N(m<M)" is that you have to define when a clump is so small that you no longer consider it a galaxy. Usually one considers N(m>M), since the number density decreases rapidly with halo mass / galaxy mass. I don't know where you got Eqs. 1 & 2 from, but usually one is interested in a average number density, not an absolute number, unless you're talking specifically about some particular cluster or whatever. I think you'll find the answers in the links I give below to other astro.SE posts. $\endgroup$ – pela Dec 12 '18 at 18:16
  • $\begingroup$ -@pela . Thanks for your quick answer. For the moment, I have found the Schechter Luminosity function. I still try to find out what are the main problems where one wants to estimate the density of galaxies : I have proposed different hypothesis into my UPDATE 1 : feel free to tell me if you think these issues prevent a good estimation for this density of galaxies in universe, or if you know relevant criteria which make things harder to calculate. regards $\endgroup$ – youpilat13 Dec 12 '18 at 23:04

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