A star of mass 2.0 1028 kg that is 1.6 1020 m from the center of a galaxy revolves around that center once every 3.4 108 years. Assuming that this star is essentially at the edge of the galaxy, each of the stars in the galaxy has a mass equal to that of this star, and the stars are distributed uniformly in a sphere about the galactic center, estimate the number of stars in the galaxy. (Do not round your answer to an order of magnitude.)
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To estimate the number of stars in the galaxy, we can use the concept of gravitational force between two objects.
The gravitational force between two objects is given by the equation:
F = G * ((m1 * m2) / r^2)
Where:
F is the gravitational force,
G is the gravitational constant (6.67 × 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, we can consider a star at the edge of the galaxy (of mass 2.0 × 10^28 kg) and the center of the galaxy. The gravitational force acting on the star can be equated to the centripetal force keeping it in orbit around the galactic center:
F_gravity = F_centrifugal
So, we have:
G * ((m_star * m_galaxy) / r^2) = (m_star * v^2) / r
Where:
m_star is the mass of the star,
m_galaxy is the mass of the galaxy (assumed to be the total mass of all stars in the galaxy),
r is the distance between the star and the galaxy's center, and
v is the velocity of the star.
Given that the star revolves around the galaxy's center once every 3.4 × 10^8 years, we can calculate its orbital velocity using the formula:
v = (2 * π * r) / T
Where:
T is the period of revolution in seconds.
Substituting this value for v in the earlier equation and rearranging, we get:
m_galaxy = (G * r) / (v^2 / m_star)
Now, if we assume each star in the galaxy has the same mass as the star at the edge, we can estimate the number of stars in the galaxy by dividing the total mass of the galaxy by the mass of each star:
number_of_stars = m_galaxy / m_star
Let's plug in the given values and calculate the number of stars in the galaxy: