Suppose that a spiral galaxy is located at the center of a spherically symmetric dark matter halo

(a) if the orbital speed of a star in this galaxy, v, is constant at any radius, how does the mass density rho(r) within the halo vary with radius r?

(b) suppose the rotation curve is fitted by a single relation v x r, answer the same question as in part (a).

(c) if the total mass interior to any radius r is a constant, m(r) m, as if it is dominated by a central point mass, find out how the orbital speed v of a stair varies with raditus r.

To solve the problem it is necessary to use the concepts of Orbital Speed considering its density, and its angular displacement.

In general terms the Orbital speed is described as,

PART A) If the orbital speed of a star in this galaxy is constant at any radius, then,

PART B) This time we have, where is the angular velocity (constant at this case)

PART C) If the total mass interior to any radius r is a constant,

increase masses, decrease separation

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min