In the lecture it was shown that the speed of a body on a circular orbit decreased as a function of distance from the object it orbits (e.g. the Sun) as v(r)∝1/r√. This followed from Newton's gravitational law
F=GmMr2
and that for circular motion ma=mv2/r. It was discussed how it was discovered that galactic rotation curves are flat, i.e. v(r)∝constant instead of v(r)∝1/r√, and this is evidence for dark matter. What is the distribution of dark matter that would lead to such an effect? Let the density of a dark matter halo decrease with r as a power law:
ρ(r)=arn,a is a constant
What is the power n that leads to a flat rotation curve? To do this, you can replace the mass that the body orbits M with the total mass within the orbit M(r):
F=GmM(r)r2
To find the total mass M(r) within a given radius r, you simply add up all the mass within the volume of a sphere of radius r. This "adding up'' is done by taking an integral of the density within the volume M(r)=∫ρ(r)dV.
Answer for Question 9
To find the power n that leads to a flat rotation curve, we need to evaluate the integral of the density function ρ(r) over the volume of a sphere of radius r.
The density function is given as ρ(r) = ar^n, where a is a constant and n is the power we are trying to determine.
The volume of a sphere is given by V = (4/3)πr^3.
To find the total mass M(r) within the radius r, we integrate the density function ρ(r) over the volume V:
M(r) = ∫ρ(r)dV
Substituting the expression for ρ(r) and V, we have:
M(r) = ∫(ar^n)(4/3)πr^3 dr
Simplifying the expression, we have:
M(r) = (4/3)πa ∫r^(n+3) dr
Integrating this expression, we add 1 to the power of r and divide by the new power:
M(r) = (4/3)πa * [r^(n+4)/(n+4)] + C
Here, C is the constant of integration.
For a flat rotation curve, the mass M(r) should be proportional to r, i.e., M(r) ∝ r.
Therefore, we can equate the expression for M(r) to kr, where k is a constant:
kr = (4/3)πa * [r^(n+4)/(n+4)] + C
To find the power n that leads to a flat rotation curve, we need to solve this equation for n.
This equation can be used to derive the power n that leads to a flat rotation curve in the distribution of dark matter. However, the exact value of n will depend on the specific data and observations of galactic rotation curves.