- #1
FrankPlanck
- 31
- 0
Hi all, this is the problem:
A galaxy shows a rotation curve with a given velocity [tex] v(r) [/tex].
[tex] r [/tex] is the distance from the center, [tex] c [/tex] is the speed of light and [tex] r_{c} = [/tex] 1 kpc is constant.
I have to find:
1) the mass density profile of the galaxy [tex] \rho(r) [/tex]
2) the total mass [tex] M [/tex]
3) Since mass/luminosity is constant [tex] M/L = 2 [/tex] what is the radius of the sphere that contains half total luminosity of the galaxy
[tex] v(r)=c \sqrt{r/(r^2+r_{c}^2)} [/tex]
I try...
1)
from Newton
[tex] \frac{v(r)^2}{r} = G \frac{M(r)}{r^2} [/tex]
hence
[tex] \frac{G M(r)}{r} = \frac{r c^2}{r^2+r_{c}^2} [/tex]
hence
[tex] M(r) = \frac{c^2}{G}\ \frac{r^2}{r^2+r_{c}^2} [/tex]
and so
[tex] \rho (r) = \frac{M(r)}{V(r)} = \frac{c^2}{G}\ \frac{1}{(r^2+r_{c}^2)(4/3 \pi r)} [/tex]
I could rewrite it, but it doesn't really matter.
2)
From point 1)
[tex] M(r) = \frac{c^2}{G}\ \frac{r^2}{r^2+r_{c}^2} [/tex]
hence
[tex] M_{tot} = \frac{c^2}{G}\ \int^R_0 \frac{dr}{1+ \frac{r_{c}^2}{r^2}} [/tex]
I don't know... damn integral...
3)
I need point 2)I'm not a native speaker, so I apologize for any possible misunderstanding.
Thank you!
Homework Statement
A galaxy shows a rotation curve with a given velocity [tex] v(r) [/tex].
[tex] r [/tex] is the distance from the center, [tex] c [/tex] is the speed of light and [tex] r_{c} = [/tex] 1 kpc is constant.
I have to find:
1) the mass density profile of the galaxy [tex] \rho(r) [/tex]
2) the total mass [tex] M [/tex]
3) Since mass/luminosity is constant [tex] M/L = 2 [/tex] what is the radius of the sphere that contains half total luminosity of the galaxy
Homework Equations
[tex] v(r)=c \sqrt{r/(r^2+r_{c}^2)} [/tex]
The Attempt at a Solution
I try...
1)
from Newton
[tex] \frac{v(r)^2}{r} = G \frac{M(r)}{r^2} [/tex]
hence
[tex] \frac{G M(r)}{r} = \frac{r c^2}{r^2+r_{c}^2} [/tex]
hence
[tex] M(r) = \frac{c^2}{G}\ \frac{r^2}{r^2+r_{c}^2} [/tex]
and so
[tex] \rho (r) = \frac{M(r)}{V(r)} = \frac{c^2}{G}\ \frac{1}{(r^2+r_{c}^2)(4/3 \pi r)} [/tex]
I could rewrite it, but it doesn't really matter.
2)
From point 1)
[tex] M(r) = \frac{c^2}{G}\ \frac{r^2}{r^2+r_{c}^2} [/tex]
hence
[tex] M_{tot} = \frac{c^2}{G}\ \int^R_0 \frac{dr}{1+ \frac{r_{c}^2}{r^2}} [/tex]
I don't know... damn integral...
3)
I need point 2)I'm not a native speaker, so I apologize for any possible misunderstanding.
Thank you!
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