In recent years, a number of nearby stars have been found to possess planets. Suppose, the orbital radius of such a planet is found to be 3.3 1011 m, with a period of 1080 days. Find the mass of the star
To find the mass of the star, we can use Kepler's Third Law, which relates the orbital period of a planet around a star to the mass of the star. The formula for Kepler's Third Law is:
T^2 = (4π^2 / GM) r^3
Where:
T = Orbital period of the planet (in seconds)
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = Mass of the star (in kilograms)
r = Orbital radius of the planet (in meters)
First, let's convert the given values into SI units:
Orbital radius of the planet (r) = 3.3 × 10^11 m
Orbital period of the planet (T) = 1080 days = 1080 × 24 × 3600 seconds
Now we can plug in these values and solve for the mass of the star (M):
(T^2) × (GM) = (4π^2) × (r^3)
M = [(4π^2) × (r^3)] / [(T^2) × G]
Let's calculate the mass of the star using this formula:
M = [(4π^2) × (3.3 × 10^11)^3] / [(1080 × 24 × 3600)^2 × (6.67430 × 10^-11)]
After evaluating this expression, we get the mass of the star in kilograms.