Dark matter density in fucntion of radius

[Fabio010]

Homework Statement

Evidence for dark matter comes from “flat” rotation curves of galaxies. Assume
that the observed matter moves in circular orbits about the center of the galaxy
and that the velocity of the matter as a function of the radius v(r) is a constant.


Also assume (mass of luminous matter is negligible) and the dark matter is
distributed with spherical symmetry about the center of the galaxy.




What is the density ρ(r) of the dark matter as a function of radius?



2. Attempt

Critic density = ρ(r) = 3Ho^2 / 8piG Ho= Hubble constante

because Ho = v/d then

ρ(r) = 3v^2 / 8piGr^2 r= distance from the center of galaxy


So we can conclude that the dark matter density in a galaxy is proportional to 1/r^2.


But i found this wrong. Can somebody the me what is wrong?

 

[mfb]

There is no expansion of the universe involved in the problem, why do you use the Hubble constant and what is d?

Just use v as a constant and Newtonian gravity.
You could calculate M(r) (the total mass up to radius r) as intermediate step.

 

[Fabio010]

There is no expansion of the universe involved in the problem, why do you use the Hubble constant and what is d?

Just use v as a constant and Newtonian gravity.
You could calculate M(r) (the total mass up to radius r) as intermediate step.

like this:

[itex]\vec{}[/itex]

[itex]F=\frac{GMm}{R^{2}}[/itex]

[itex]a=\frac{v^{2}}{R}[/itex]

The Attempt at a Solution

[itex]\frac{F}{m}=\frac{GM}{R^{2}}[/itex]

[itex]4\pi \frac{F}{m}=\frac{4\pi GM}{R^{2}}[/itex]

[itex]\frac{v^{2}}{R}=\frac{GM}{R^{2}}[/itex]

[itex]dM=4\pi\rho (R)R^{2}dR[/itex]

[itex]dM=\frac{v^{2}}{G}dR[/itex]

[itex]4\pi\rho (R)R^{2}dR=\frac{v^{2}}{G}dR[/itex]

[itex]\rho (R)=\frac{v^{2}}{4\pi GR^{2}}[/itex]


so [itex]\rho (R)[/itex] is propitiational to [itex]\frac{1}{r^{2}}[/itex]

 

[Fabio010]

anyone?

 

[mfb]

Sorry, missed that post somehow.
I don't see an error.

 

[Fabio010]

Sorry, missed that post somehow.
I don't see an error.

So with my result, i can conclude that the dark matter density decreases with the distance from the center of the galaxy.

But is not the dark matter suppose to increase with the distance? To maintain the galaxy rotation velocity constant?

 

[mfb]

The total amount of dark matter increases (M~R), the density decreases. If density would be constant everywhere, our galaxy would not have any border in terms of its gravitational attraction.

 

[Fabio010]

The total amount of dark matter increases (M~R), the density decreases. If density would be constant everywhere, our galaxy would not have any border in terms of its gravitational attraction.

Thanks for the help mfb!