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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 following relation which gives the number of galaxies $N$ with magnitude $(m < M)$ (number counts) :

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

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

UPDATE 1: both equations $(1)$ and $(2)$ may be valid, with a constant appropriate, from a dimensional point of view.

I still 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 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 here is that a dark matter halo is more massive than the galaxy hosted in this halo : how can we deal with this ?

UPADTE 2: I have also to explain why the estimation of galaxies density is difficult and problematic**. 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 3: My apologizes, I realized that $m$ and $M$ don't represent the mass but the apparent magnitude in equations $(1)$ and $(2)$ ?

But the different issues that I talked about remains, i.e knowing the limits (lower or upper) that I have to take into account or the backgrounded/fronted others galaxies which changes the counting of objects.

Any help is welcome.

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  • $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$ – Ben Crowell Dec 12 '18 at 14:47
  • $\begingroup$ The dimensionality issue doesn't matter because the relationship is proportional to, and not equal to, so a proportionality constant with the necessary unit adjustment can fix it. I suspect that the issue your professor is getting at is that the definition of what constitutes a galaxy, especially in dense galactic clusters and with respect to smallish collections of stars is ill defined, so number density is tricky to define. Especially at the small end tail, a definition that you need 1000 v. 100000 stars to count as a galaxy could make a huge difference. It could also be about Z v. volume. $\endgroup$ – ohwilleke Dec 12 '18 at 23:01
  • $\begingroup$ -@ohwilleke. thanks for your quick answer. Have you ever seen this kind of proportionality relation like $(1)$ and $(2)$ relations ? Unfortunately, I don't know the sources of my teacher. The problem is that it defines galaxy of mass $m<M_{max}$, so one could take into account of a small object ($m$ small) which is not a galaxy, so there would be an over-estimation in counting. Hence we could take the opposite, i.e galaxy counting for $m>M_{min}$ but I can't integrate up to $+\infty$, right ? You talk about a second problem, the Z v. colume) : can you explain more precisely this issue ? $\endgroup$ – youpilat13 Dec 12 '18 at 23:15

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