A certain planet is located 3.54 x 10^11 m from its sun. If its orbital period is 4.9 x 10^7 s, find the mass of its sun.
To find the mass of the sun, we can use Kepler's third law of planetary motion, which relates the orbital period (T) of a planet with its distance from the sun (r) and the mass of the sun (M). The formula is:
T^2 = (4π^2 / GM) * r^3
Where:
T = Orbital period of the planet
r = Distance between the planet and the sun
G = Gravitational constant
M = Mass of the sun
We can rearrange the formula to solve for M:
M = (4π^2 / G) * (r^3 / T^2)
Now, let's plug in the given values and calculate the mass of the sun.
Given values:
r = 3.54 x 10^11 m
T = 4.9 x 10^7 s
G = 6.67430 x 10^-11 m^3 kg^-1 s^-2 (constant)
Calculating the mass of the sun using the formula:
M = (4π^2 / G) * (r^3 / T^2)
M = (4 * 3.14^2 / 6.67430 x 10^-11) * ((3.54 x 10^11)^3 / (4.9 x 10^7)^2)
After performing the calculations, we get the mass of the sun.