Total number density of galaxies and problematic expression

[fab13]

Hello,

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 mass ##(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)##

1) But If want to express the density, I need the volume for this count of galaxies above, i.e the volume in which the number of galaxies ##N(m<M)## is located.

2) Moreover, have I got to set a maximum value ##M_{max}## for ##m## in the expression ##(1)## ? It should be necessary since otherwise, the number of galaxies diverges, should'nt it ?

3) Maybe for the volume, I have to take into account just the size of observable universe since beyond this limit, we can't detect galaxies, do you agree ?

If someone could give me indications or tracks to anwser to these 2 questions ,Regards.

 

[haruspex]

give the formal expression of the total number density of galaxies

This sounds to me as though it is asking how the concept is defined, not a formula for estimating a value.

 

[fab13]

@haruspex

Do you know or see the kind of formula expressing this total number density of galaxies ?

Is the relation that I gave in my first post correct ? I mean this one :

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

PS: I have also to explain why this kind of definition for galaxies density is problematic. The track to follow is that after, the teacher told us to use an alternative way by using a luminosity distribution instead of a mass distribution.

So for this question here, this would mean that the relation searched is a function of mass (like with the equation ##(1)## above).

Maybe this density could be calculated by 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^{*}}}##

If this is the case, what the equivalent function to describe density of galaxies as a function of the mass (which is the original topic of this post) ?

 

[haruspex]

Do you know or see the kind of formula expressing this total number density of galaxies ?

You don't seem to have understood my reply.
It sounds to me as though the question is how might you define the number density. Suppose you were to define it as the total number of galaxies in the universe divided by the volume of the universe. Do you see a practical problem with estimating that?
If we define it in terms of mass thresholds (your N(m<M)) does it help? Are there still issues?

an alternative way by using a luminosity distribution instead of a mass distribution.

Ok, so does defining it in terms of luminosity thresholds make it more practical?

You eqn (1) is not a definition. It appears to be a formula that provides an estimation, but I do not understand it. It doesn't seem to make sense dimensionally. M is a mass, right?, so 0.6M is also a mass. But your eqns 1 and 2 imply it is dimensionless.
Can you provide a link for these?

 

[fab13]

@haruspex , thanks for your quick answer.

equation ##(1)## comes from the lectures of my teacher. As you have noticed, this doesn't make sense from dimensional point of view, he must have made a mistake.

Same thing for ##(2)##, we don't have the dimension of Volume taken for the count of galaxies.

So, I don't know what he 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 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 ?

 

[haruspex]

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.

Your "given that" does not make logical sense. It doesn't matter what estimation formula has been proposed, such as eqns 1 and 2. Forget those. The question is:
1. In what ways might you define number density, e.g. number of galaxies per unit volume of space? Number below a given mass threshold per unit volume? Total number below a given mass threshold? Etc.
2. For each such definition, how would you go about making observations to discover a formula? What practical difficulties arise?

But as I wrote, I am basing all this on the way the problem is worded in post #1. I cannot be sure this is what the teacher intends.

 

[fab13]

@haruspex , thanks for your quick answer.

equation ##(1)## comes from the lectures of my teacher. As you have noticed, this doesn't make sense from dimensional point of view, he must have made a mistake.

Same thing for ##(2)##, we don't have the dimension of Volume taken for the count of galaxies.

I come back on this affirmation : ##(1)## and ##(2)## are proportionality expressions, so I can introduce a constant that makes things dimensionally correct.

I think my teacher wants to higlight the fact that I have to fix a lower limit for the relation $N(m<M)$, since otherwise I would over-estimate the number of galaxies by counting small objects which are not galaxies. If this is the issue, a simple cutoff would prevent this overvalue, wouldn't it ?

In practice, what is the counting method the most used in general ? , i.e :

1) using a luminosity galaxy distrubution ?

2) using a mass galaxy distribution ?

Thanks for your help

 

[haruspex]

this affirmation : (1)(1)(1) and (2)(2)(2) are proportionality expressions, so I can introduce a constant that makes things dimensionally correct.

True, but why the 0.6?

I think my teacher wants to higlight the fact that I have to fix a lower limit for the relation $N(m<M)$,

Perhaps, but as I keep posting it does not seem to me that the question is connected with those equations. Maybe you have some other reason to believe that it is.

In practice, what is the counting method the most used in general ?

I have no idea, but I can think of difficulties with different schemes.

 

[fab13]

UPDATE: My apologizes, in my lectures, 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.

 

[haruspex]

UPDATE: My apologizes, in my lectures, 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.

What about observation bias?

 

[fab13]

What about observation bias?

Please, could you give me documentation if you have it, links or insights, about all the problems which appears in the estimation of density of galaxies into universe (over/sub estimation by background/foreground, bias with observations etc ... ?

I would really appreciate

 

[haruspex]

Please, could you give me documentation if you have it, links or insights, about all the problems which appears in the estimation of density of galaxies into universe (over/sub estimation by background/foreground, bias with observations etc ... ?

I would really appreciate

Having discovered that m and M refer to magnitudes, not masses, the issue I saw goes away in respect of the statistic N(m<M). There would be no observation bias. Magnitude is a matter of observation.

But the question as posed does not refer to number density by magnitude, just "number density". If that is intentional, then the problem remains. How do you allow for the galaxies you cannot see?

As for a formal expression, it depends what is meant by density here. Is it per unit volume of space or per unit solid angle patch of sky? If per volume then it would be "the number of galaxies per unit volume of space". You should be able to see how such a definition would be problematic.

Whether relevant or not, your eqns 1 and 2 are not consistent. It makes no sense to have the proportionality sign in eqn 1 as well as the constant factor .6. I would think it is supposed to be an equals sign.
In eqn 2 you can keep the proportionality sign but drop the *exp(constant).

 

[fab13]

the question as posed refers to number density per unit volume of space.

Have you got an example of distribution expression for number density per unit volume space ?

 

[haruspex]

the question as posed refers to number density per unit volume of space.

Have you got an example of distribution expression for number density per unit volume space ?

No.

 

[TEFLing]

Schechter luminosity function :

##N(L)\ \mathrm {d} L=\phi^{*}\left({\frac{L}{L^{*}}}\right)^{\alpha}\mathrm {e}^{-L/L^{*}}{\frac{\mathrm {d} L}{L^{*}}}##
The leading factor carries all of the units you're desiring

 

[jim mcnamara]

There has been a lot of confusion in this thread. We seem to have a definite answer. I'm closing the thread. Thanks for you help everyone.