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 41.3 km/s and 52.7 km/s. The slower planet's orbital period is 7.74 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?
i used v=2pi(radius)/T to find the radius, and converted the km to m and the years to seconds, but how do i get the mass from this?
To find the mass of the star, you can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit.
Let's first find the semi-major axis, or the average distance between the star and each planet's orbit.
To find the semi-major axis, we can use the formula: radius = (velocity * period) / (2π)
For the slower planet:
radius = (41.3 km/s * 7.74 years) / (2π)
Convert both the velocity and period to SI units:
radius = (41.3 * 10^3 m/s * 7.74 * 365.25 * 24 * 3600 s) / (2π)
Simplify the expression to find the radius in meters.
Repeat the same process for the faster planet, using its velocity and the mass you just found.
Once you have the semi-major axis for both planets, you can apply Kepler's Third Law to find the mass of the star.
Kepler's Third Law states that the square of the orbital period (T) is equal to the cube of the semi-major axis (a), scaled by a constant (K), which is the same for all planets orbiting the same star:
T^2 = K * a^3
You can rewrite this equation as:
M = (4π^2 * a^3) / G
Where M is the mass of the star, G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), and a is the semi-major axis of the orbit.
You can now substitute the value of the semi-major axis that you found for the slower planet into this equation to calculate the mass of the star.
To find the orbital period of the faster planet, use the same equation with the newly found mass of the star and the semi-major axis of the faster planet's orbit. Rearrange the equation to solve for the period (T).
After finding the semi-major axes for both planets, substitute the values into the equation to find the mass of the star, and use the mass to calculate the orbital period of the faster planet.