Two newly discovered planets follow circular orbits around a star in a distant part of the galaxy. The orbital speeds of the planets are determined to be 43.6 km/s and 63.5 km/s. The slower planet's orbital period is 8.36 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?

a. for the slower planet, start with speed and period to get radius.

speed=2PI r/period solve for r

then centripet force = gravitational force mv^2/r=GMm/r^2 solve for M

Can there be a one for OCPS? If not then should i make one?

To solve this problem, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet is proportional to the cube of its average distance from the star. We can also use the fact that the orbital speed is related to the mass of the star.

(a) To find the mass of the star, we need to use the orbital speed of the slower planet. The orbital speed of a planet can be calculated using the formula:

v = √(G * M / r)

Where v is the orbital speed, G is the gravitational constant, M is the mass of the star, and r is the distance between the star and the planet.

We know the orbital speed (43.6 km/s) and the orbital period (8.36 years) of the slower planet. The orbital period is related to the distance using the formula:

T = 2Ï€ * r / v

Rearranging this formula, we can solve for r:

r = (T * v) / (2Ï€)

Substituting the given values:

r = (8.36 years * 43.6 km/s) / (2Ï€)

Now that we have the distance, we can find the mass of the star using the orbital speed formula:

M = (v^2 * r) / G

Substituting the given values:

M = (43.6 km/s)^2 * r / G

Where G is the gravitational constant, which is approximately 6.674 * 10^-11 m^3/kg/s^2.

Converting the distance to meters and solving for the mass:

r = [(8.36 * 365.25 days * 24 hours * 3600 seconds) * (43.6 * 1000 meters/second)] / (2Ï€)

M = (43.6 km/s)^2 * r / (6.674 * 10^-11 m^3/kg/s^2)

After calculating, the mass of the star should be obtained in kilograms.

(b) To find the orbital period of the faster planet, we can use the same approach. We know the faster planet's orbital speed (63.5 km/s) and the mass of the star obtained in part (a). Using the orbital speed formula, we can determine the distance between the star and the faster planet. Then we can use Kepler's Third Law to find the orbital period:

v = √(G * M / r)

r = (v^2 * r) / G

T = [ (4 * π^2) / (G * M) ] * r^3

Substituting the values, we can solve for T, the orbital period of the faster planet.