How would you estimate the number of stars in the Galaxy given the following information?

List of Assumptions:
[1] All the mass inside milkyway are in the form of stars,
[2] All stars are of the same mass as out sun,
[3] Sun goes around the galactic centre in a circular orbit,
[4] Mass is distributed uniformly in all direction (spherically symmetric distribution),

Symbols:

#M# - mass of the galaxy in kilograms,
#M_{sun} = 1.99\times10^{30} kg# - Mass of the Sun,
#r = 2.2\times10^{20} m# - orbital radius of Sun,
#T = 2.5\times10^8 Yrs = 7.884\times10^{15} s# - Orbital period of Sun,
#G=6.67\times10^{-11} (Nm^2)/(kg^2)# - Gravitational constant.
#N_s# - Number of stars in the galaxy,

Step 1 : Write the expression for the centripetal force (#F_c#) required to keep the Sun in a circular orbit of radius #r# and time period #T#.

Centripetal Force: #F_c=4\pi^2\frac{M_{sun}r}{T^2}#,

Step 2: Calculate the gravitational force exerted on the Sun by the galaxy, assuming that all of the galactic mass #M# are concentrated at the galactic centre.

Gravitational Force: #F_g=\frac{GMM_{sun}}{r^2}#

Is the assumption made in Step 2 right? : Yes! Since the mass is distributed in a spherically symmetric way, Gauss's theorem ensures that the gravitational force on the Sun by these masses is the same as the force if all of the mass were located at the galactic centre.

Step 3: Recognise that the centripetal force required to keep Sun in a circular orbit is the gravitational force exerted on it by all the masses inside its orbit.

Equating #F_c# to #F_g# we find #M#,

#F_g = F_c \qquad \rightarrow \frac{GMM_{sun}}{r^2} = 4\pi^2\frac{M_{sun}r}{T^2} #

#M=\frac{4\pi^2}{G}\frac{r^3}{T^2}=1.01\times10^{41} kg#

Step 4: If all the stars are of the same mass as the Sun then the total mass of the galaxy is related to the number of stars and the mass of Sun as : #M=N_s\timesM_{sun}#.

Now solve for #N_s#

#N_s = M/M_{sun}=(1.01\times10^{41}kg)/(1.99\times10^{30}kg)=51\times10^9 = 51 Billions#