A galaxy can be described as a sphere with homogeneous distributed density. The mass at any point of the space is $M(r)$, that is described as:

$M(r)$= $kr^3$ when $r\le R$

$M(r)$= $k'r$ when $r > R$

a) Find the velocity of a body in a circular orbit around a more massive one.

b) Find the circular velocity of a particle around the galaxy at a distance $r$ from the center at any point.

c) Relate $k$ and $k'$

d) Draw the fuction of the circular velocity in function of the distance to the center for all $r$.

These are my solutions.

**a)** The centripetal force on a circular orbital path equals to the gravitational force. Therefore:

$$ \frac{mv^2}{r} = \frac{GMm}{r^2}$$

$$ v=\sqrt{ \frac{GM}{r}} $$

Which is the circular velocity of the smaller body around the more massive one.

**b)** As the galaxy has a spherical distribution with homogeneous density, the sphere (galaxy) can be regarded as a point mass (point particle) placed at the origin of coordinates (galaxy's center).

We have two different cases:

Case 1: when $r\le R$, we have $M(r)$= $kr^3$. Then:

$$ \frac{mv^2}{r} = \frac{Gkr^3m}{r^2}$$

$$ v=\sqrt{Gkr^2} $$

Case 2: when $r > R$, we have $M(r)$= $k'r$. Then:

$$ v=\sqrt{Gk'} $$

**c)** We can relate both $k$ and $k'$ at the limit case $r=R$. Therefore:

$$k'=R^2k$$

$$or$$

$$k'=r^2k$$

**d)** Plugging $k'=R^2k$ into $ v=\sqrt{Gkr^2} $ and $ v=\sqrt{Gk'} $,

we obtain $ v=r\sqrt{Gk} $ and $ v=R\sqrt{Gk} $ respectively.

I represented it graphically as follows:

At **c)** I had doubts if both $$k'=R^2k$$ $$and$$ $$k'=r^2k$$ were right. I gave a thought to the matter, and I concluded $$k'=R^2k$$ is more suitable as R is constant and r is not.

At **d)**, I did the representation in function of $\sqrt{Gk}$, instead of $\sqrt{Gk'}$. I guess it does not matter.

What do you think about these results?

Thanks

edited May 1, 2018 by sammy gerbil