A star of mass 2.0 1031 kg that is 4.8 1020 m from the center of a galaxy revolves around that center once every 4.2 108 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.)

stars

Use ^ in front of exponents please.

Express the galactic mass in terms of the number of equal-size stars, N.

Set the centripetal force equal to the gravitational frce on the edge star, and sziolve for N.

To estimate the number of stars in the galaxy, we can use the concept of gravitational attraction and the given information about the mass of the star and its distance from the center of the galaxy.

1. Start by calculating the gravitational force between the star and the galactic center using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects (in this case, the star and the galactic center), and r is the distance between them.

2. Since the star is essentially at the edge of the galaxy, we can assume that the force of gravity is provided only by the stars within its distance from the center. Therefore, we can equate the gravitational force due to the star itself to the cumulative gravitational force due to all the stars within that distance. This assumption allows us to estimate the number of stars in the galaxy.

3. Setting up the equation, we have:
F = G * [(m1 * m2) / r^2] = G * (n * m1^2) / r^2,
where n is the estimated number of stars in the galaxy.

4. Rearranging the equation to solve for n:
n = (F * r^2) / (G * m1^2).

5. Substituting the given values into the equation:
m1 = 2.0 * 10^31 kg,
r = 4.8 * 10^20 m,
F = gravitational force between the star and the galactic center,
G = gravitational constant (6.67430 × 10^-11 N m^2 / kg^2).

6. Calculate the gravitational force:
F = G * (m1 * m2) / r^2,
where m2 is the mass of the star at the center of the galaxy.

7. Since the stars in the galaxy have the same mass as the star at the edge, we have:
m2 = m1 = 2.0 * 10^31 kg.

8. Plugging in the values, we have:
F = G * (m1 * m2) / r^2,
F = 6.67430 × 10^-11 * (2.0 * 10^31)^2 / (4.8 * 10^20)^2.

9. Calculate the estimated number of stars:
n = (F * r^2) / (G * m1^2).

By following these steps and substituting the given values into the equations, you can calculate an estimated number of stars in the galaxy.

To estimate the number of stars in the galaxy, we can use the concepts of gravitational force and centripetal force.

First, we need to find the gravitational force acting on the star due to the galaxy's mass. The gravitational force can be calculated using Newton's law of gravitation:

F_grav = G * (m1 * m2)/r^2

where F_grav is the gravitational force, G is the gravitational constant, 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, the mass of the star (m1) is given as 2.0 * 10^31 kg, the mass of each star in the galaxy (m2) is also given as the same value (since each star in the galaxy has the same mass), and the distance from the star to the center of the galaxy (r) is given as 4.8 * 10^20 m.

Next, we need to calculate the centripetal force acting on the star, which is required for circular motion. The centripetal force can be calculated using the following equation:

F_centripetal = (m * v^2)/r

where F_centripetal is the centripetal force, m is the mass of the star, v is the velocity of the star, and r is the distance from the star to the center of the galaxy.

In this case, the velocity of the star can be found from the time period for one revolution around the galactic center, which is given as 4.2 * 10^8 years. The velocity can be calculated as the circumference of the orbit divided by the time period:

v = (2 * π * r)/T

where v is the velocity, π is a mathematical constant (approximately 3.14), r is the distance from the star to the center of the galaxy, and T is the time period.

Now, equating the gravitational force and the centripetal force, we get:

G * (m1 * m2)/r^2 = (m1 * v^2)/r

We can rearrange this equation to solve for the mass of each star in the galaxy (m2):

m2 = (v^2 * r^3) / (G * T^2)

Using the given values, plug them into the equation to calculate the mass of each star in the galaxy.

After finding the mass of each star, we can then estimate the number of stars in the galaxy by dividing the total mass of the galaxy by the mass of each star. The total mass of the galaxy can be found by multiplying the mass of one star by the number of stars in the galaxy.

Finally, substitute the values into the equation to calculate the number of stars in the galaxy.

So using Keplers third law,

T^2=(4pi^2/GM)r^3
T=4.2 x 10^8
r=4.8 x 10^20
G=6.67 x10^-11
solve for M, then divide by 2.0x10^31?
which would give 1.855 x 10^25 stars