Orbital Velocity due to a Dark Matter Halo

In summary, the conversation discusses calculating the gravitational force from a dark matter halo and equating it to the centripetal force to find the velocity. The density of the halo is given by ρ(r) = ρ0r02/r2, and the equations for mass and gravitational force are also provided. The attempt at a solution involves integrating the density, but this leads to an infinite mass due to a missing r2 term in the integration measure. Ultimately, the correct answer is found to be v2 = 4∏Gρ0r02.
  • #1
Dovahkiin
7
0

Homework Statement



I am struggling to calculate the gravitational force from a dark matter halo given that its density is given by ρ(r) = ρ0r02/r2. Once i have found the force i plan to equate it to the centripetal force, F = mv2/r, to find the velocity, v.

Homework Equations



Density of DM halo: ρ(r) = ρ0r02/r2

Centripetal Force: F = mv2/r

Gravitational force: F = GMm/r2

Mass of DM: M = ∫ρ dV = 4∏∫(from 0 to R) ρ(r) dr

The Attempt at a Solution



I (think) i can see what i need to do but using a density ρ(r) = ρ0r02/r2 and trying to integrate from 0 (centre of halo) to R (radial position of the orbiting object) obviously gives and infinite mass?

However the answer is given to be:

v2 = 4∏Gρ0r02

I can't see how you could get to this answer without modifying the density, or am I incorrectly assuming that the force of a mass distribution is the same as the total mass centred at the "origin"?

Any help will be much appreciated... this question seems like it should be pretty trivial but I'm stumped!
 
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  • #2
Dovahkiin said:
ρ(r) = ρ0r02/r2

M = ∫ρ dV = 4∏∫(from 0 to R) ρ(r) dr

... using a density ρ(r) = ρ0r02/r2 and trying to integrate from 0 (centre of halo) to R (radial position of the orbiting object) obviously gives and infinite mass?
Shouldn't your integration measure include a r2 (thus cancelling that pesky badly behaved explosion at r=0)?
(doh!)
 
  • #3
turin said:
Shouldn't your integration measure include a r2 (thus cancelling that pesky badly behaved explosion at r=0)?
(doh!)

I knew I would have missed something simple ;) Thanks!
 

Related to Orbital Velocity due to a Dark Matter Halo

What is orbital velocity due to a dark matter halo?

Orbital velocity due to a dark matter halo refers to the speed at which an object must travel in order to maintain a stable orbit around a galaxy, taking into account the gravitational pull of the galaxy's dark matter halo.

How is orbital velocity due to a dark matter halo calculated?

The orbital velocity can be calculated using the equation V = √(GM/r), where G is the gravitational constant, M is the mass of the galaxy's dark matter halo, and r is the distance between the object and the center of the galaxy.

What factors influence the orbital velocity due to a dark matter halo?

The main factor that influences orbital velocity due to a dark matter halo is the mass of the dark matter halo. A larger mass will result in a higher orbital velocity. Other factors such as the distance from the center of the galaxy and the distribution of dark matter within the halo can also affect the orbital velocity.

Why is the concept of orbital velocity due to a dark matter halo important?

Understanding orbital velocity due to a dark matter halo is important because it helps us to better understand the dynamics of galaxies and the role that dark matter plays in their formation and evolution. It also has implications for our understanding of gravity and the structure of the universe.

Can orbital velocity due to a dark matter halo be measured?

Yes, orbital velocity due to a dark matter halo can be measured using various methods such as observing the rotation curves of galaxies or studying the gravitational lensing effects of dark matter. However, accurately measuring the orbital velocity of dark matter is still a challenging task and continues to be an active area of research.

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