Question

A star, which is 2.0 x 10²⁰ m from the center of a galaxy, revolves around that center once every 2.4 x 10⁸ years. Assuming each star in the galaxy has a mass equal to the Sun's mass of 2.0 x 10³⁰ kg, the stars are distributed uniformly in a sphere about the galactic center, and the star of interest is at the edge of that sphere, estimate the number of stars in the galaxy.

Answer 1

Answer:

Let us go to the basics first.

If the mass of the star is m, then the total mass of the stars in the galaxy is Nm, where N is the number of stars in the galaxy.

Assume that the star is moving in a circle and we use the formula for centripetal acceleration.

Then, the centripetal force of the galaxy on the star = mv²/r……………Eqn.1

[where v is the star's velocity around the center of the galaxy and r is the galactic radius.]

From here, we use Newton's law of gravity:

F = Gm₁m₂/r² …………………Eqn.2

(where the m₁ = mass of star=m; m₂=mass of the galaxy =Nm. and r will be the galactic radius).

Set these two forces equal to each other.

Thus, Gm(Nm)/r² = mv²/r     

=> Gm(N)/r = v²

=> N = v²r / (Gm)

The period of the orbit P = 2πr/v, or, v = 2πr/P. Substituting this in to the previous expression,

N = 4π²r³/(GmP²)      

[where, r=2.0 x 10²⁰ m; G=6.67*10^-11; m=2.0 x 10³⁰ kg;

P=2.4 x 10⁸ years = 2.4 x 10⁸ *365*24*3600 seconds]

=> N = 4(3.14)²(2.0 x 10²⁰)³/[6.67*10^-11*2.0 x 10³⁰ (2.4 x 10⁸ *365*24*3600)²]

=> N = 4.13x10¹⁰ stars = (41.3 billion stars)           (Answer)