Need to derive a formula for the number of red giants

[Rageuke]

Homework Statement

Give an expression for the number of these red giants that can be observed with a given apparent
brightness 𝑚 can be observed within a solid angle Ω.
Express this as ΔN/Δ𝑚 as a function of 𝑚. Assume that this type of red giant represents a certain fraction 𝑓 of the total star population. Further assume also that for this calculation there is no
extinction of the interstellar medium. (Hint: use the given Taylor series that is valid when 𝑥 ≪ 1).

Relevant Equations

Taylorseries: log(1 + 𝑥) = 1/ln(10)*( 𝑥 − 𝑥^2/2 + 𝑥^3/3 − 𝑥^4/4 + ⋯ )
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Assume that the density distribution of stars in the Galaxy can be represented by the following exponential function:
---
𝑛(𝑟) = 𝑛0*𝑒^−((𝑟−𝑅0)-(𝑅𝑑))
---
Herein:
𝑟 represents the distance from the center of the Galaxy,
𝑅0 the distance from the Sun to the center of the Galaxy,
𝑅𝑑 the radius of the disk of the Galaxy,
𝑛0 is the density of stars in the vicinity of the Sun.
All distances in the above model are given in 𝑘𝑝𝑐.
An astronomer observes a certain type of red giants in the center of our Galaxy that can be used as standard candles with a constant absolute brightness of 𝑀 = -0.2.

My hypothesis:
The number of red giants is equal to the number of stars times the given fraction f.
The number of stars in a solid angle omega, is given by the density distribution of stars in the Galaxy times the volume of the observed solid angle:
#RG = f*n(r)*V
where V = (d^3*omega)/3.
I assume the maxium distance at which a telescope can see red giants with a given apparent magnitude m is equal to distance d mentionned in the equation above, which can also be calculated with the following formula (relating the apparent magnitude to the absolute magnitude):
M = m + 5 - 5*log(d)
where the base of log is 10
which gives
d = 10^((M-m-5)/(-5))
If you substitute everything, you become an expression with the only variable being the apparent magnitude...
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My questions are:
The expression I computed is not expressed as ΔN/Δ𝑚, what could I change to my formul to achieve this?
Shouldn't I use an integral to integrate n(r) to the volume V, to determine the number of stars?
What does the expression ΔN/Δ𝑚 stand for? The number of red giants with a given apparent magnitude m?
It is said I must use the given Taylorseries, I don't know where or when...
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Thx in advance for your help! :D

 

[haruspex]

Assume that the density distribution of stars in the Galaxy can be represented by the following exponential function:
---
𝑛(𝑟) = 𝑛0*𝑒^−((𝑟−𝑅0)-(𝑅𝑑))
---
Herein:
𝑟 represents the distance from the center of the Galaxy,
𝑅0 the distance from the Sun to the center of the Galaxy,
𝑅𝑑 the radius of the disk of the Galaxy,
𝑛0 is the density of stars in the vicinity of the Sun.

I do not understand that. What has the density of stars at distance r from galactic centre to do with the Sun's distance from it?

Also, to be asking about some arbitrary solid angle, I assume d is somewhat less than the thickness of the galactic disc.

 

[Rageuke]

Well, they use the density of the stars near the Sun and its distance from the centre of the galaxy as an estimate, you could say a standard, for the density in the rest of the Milkyway. As you go further and further from the centre, the density decreases.

 

[haruspex]

Well, they use the density of the stars near the Sun and its distance from the centre of the galaxy as an estimate, you could say a standard, for the density in the rest of the Milkyway. As you go further and further from the centre, the density decreases.

Doh! Yes, I see how that works now.

You didn't answer my second question, so I'll stick with my assumption .

What does the expression ΔN/Δ𝑚 stand for?

If m is a continuous variable, there are, in principle, no stars with apparent magnitude exactly m. The number $\Delta N$ with apparent magnitude between $m$ and $m+\Delta m$ is a function of m.

To put it another way, if $N(m)$ is the number with apparent magnitude less than or equal to m then you are to find $\frac{dN}{dm}$.

Please post the expression you got.

 

[Rageuke]

Hey, thanks a lot!! I'll post the equation I found once I get home!

 

[Rageuke]

Hey, thanks a lot!! I'll post the equation I found once I get home!

The equation I found: dN/dm = f*n(r)*omega*r^2*ln(10)*10^((-M+m+5)/5)
It's a simple and elegant equation, and everything seems to make sense, except the part where I didn't use the given Taylor series, but it isn't explicitly said that I must use it, it's just a hint... (idk why I'm promoting my equation so much btw, it's like I'm making an ad of it)

 

[haruspex]

Well, they use the density of the stars near the Sun and its distance from the centre of the galaxy as an estimate, you could say a standard, for the density in the rest of the Milkyway. As you go further and further from the centre, the density decreases.

Wait, there's still a problem with that equation. First, it makes no sense dimensionally: the exponent should be dimensionless. Secondly, substituting $r=R_0$ should give $n(R_0)=n_0$. So the subtraction of "Rd" should be a division? I'll assume so.

The difficulty I have with this problem is that no direction is specified for the viewed angle. That means we should take the density as constant, no? Otherwise we would need to use a higher density when looking towards galactic centre than when looking away.
So we can take the density as just $n_0$, and the number of red giants within the view is $Vfn_0$.
The remaining work is to figure out the distribution according to apparent magnitude.
In your equation in post #6, I think there is some confusion over the meaning of r.

 

[Rageuke]

Wait, there's still a problem with that equation. First, it makes no sense dimensionally: the exponent should be dimensionless. Secondly, substituting $r=R_0$ should give $n(R_0)=n_0$. So the subtraction of "Rd" should be a division? I'll assume so.

The difficulty I have with this problem is that no direction is specified for the viewed angle. That means we should take the density as constant, no? Otherwise we would need to use a higher density when looking towards galactic centre than when looking away.
So we can take the density as just $n_0$, and the number of red giants within the view is $Vfn_0$.
The remaining work is to figure out the distribution according to apparent magnitude.
In your equation in post #6, I think there is some confusion over the meaning of r.

It is not specified wether we are looking from our solar system or any random point in our Milky Way, I don't think you can just substitute r with r0...

 

[haruspex]

It is not specified wether we are looking from our solar system or any random point in our Milky Way,

First, I'm pretty certain it means an observer on Earth, or it would have said. Secondly, that does not fix the problem with your equation. All it means is that the r in $n(r)$ in your equation stands for $r_{observer}$, but what is the other r?

I stand by what I wrote in post #7.

Fwiw, I get $dN/dm=n_0f\Omega \frac 15\ln(10)10^{\frac 35(m+5-M)}$.