new planet is discovered orbiting a distant star. Observations have confirmed that the planet has a circular orbit with a radius of 21 AU and takes 118 days to orbit the star. Determine the mass of the star. State your answer with appropriate mks units. [NOTE: AU stands for "astronomical unit". It is the average distance between Earth & the Sun. 1 AU ≈ 1.496 × 1011 m.]
I would use Keplers Law here.
To determine the mass of the star, we can use Kepler's Third Law, which relates the orbital period and radius of a planet to the mass of the star it orbits. Kepler's Third Law is given by the equation:
T^2 = (4π^2/G) * (r^3/m)
Where:
T is the orbital period of the planet (in seconds),
π is a mathematical constant approximately equal to 3.14159,
G is the gravitational constant (approximately 6.67430 × 10-11 m^3 kg^(-1) s^(-2)),
r is the radius of the circular orbit (in meters), and
m is the mass of the star (in kilograms).
First, let's convert the given values to the appropriate units:
Orbital period (T) = 118 days = 118 * 24 * 60 * 60 seconds
Radius (r) = 21 AU = 21 * (1.496 × 10^11) meters
Plugging in these values into the equation, we can solve for the mass (m):
(118 * 24 * 60 * 60)^2 = (4π^2/G) * (21 * (1.496 × 10^11))^3 / m
After rearranging the equation and plugging in the known values for π and G, we get:
m = (4π^2/G) * (21 * (1.496 × 10^11))^3 / (118 * 24 * 60 * 60)^2
Now, let's calculate the mass using this formula.
m = (4 * (3.14159^2) / (6.67430 × 10^-11)) * (21 * (1.496 × 10^11))^3 / (118 * 24 * 60 * 60)^2
After performing the calculation, the mass of the star is approximately 1.905 x 10^30 kilograms.