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 the formula for the gravitational force between two objects:
F = (G * M1 * M2) / r^2
Where:
F is the gravitational force between the objects,
G is the gravitational constant (6.67430 x 10^-11 N m^2/kg^2),
M1 and M2 are the masses of the two objects, and
r is the distance between the two objects.
In this case, we want to find the mass of the sun (M2), so we rearrange the formula:
M2 = (F * r^2) / (G * M1)
We are given the distance between the planet and its sun (r = 3.54 x 10^11 m), but we need the gravitational force (F) to solve for the mass of the sun.
The gravitational force can be calculated using the equation:
F = (M * a)
Where:
F is the gravitational force,
M is the mass of the planet, and
a is the acceleration due to gravity on the planet.
We can find the acceleration due to gravity using the formula:
a = (V^2) / r
Where:
V is the orbital velocity of the planet.
To find the orbital velocity (V), we can use the formula:
V = (2 * π * r) / T
Where:
T is the orbital period of the planet.
Now we have all the necessary formulas to calculate the mass of the sun:
1. Calculate V (orbital velocity) using the formula V = (2 * π * r) / T.
2. Calculate a (acceleration due to gravity) using the formula a = (V^2) / r.
3. Calculate F (gravitational force) using the formula F = (M * a).
4. Finally, calculate M2 (mass of the sun) using the formula M2 = (F * r^2) / (G * M1), where M1 is the mass of the planet.