The astronomical unit (AU, equal to the mean radius of the Earth’s orbit) is 1.4960E11 m, and a year is 3.1557E7 s. Newton’s gravitational constant is G=6.6743E-11 m^3kg^-1s^-2. Calculate the mass of the Sun in kilograms. (Recalling or looking up the mass of the Sun does not constitute a solution of this problem.)
To calculate the mass of the Sun, we can use Newton's law of gravitation. The formula is:
F = (G * M * m) / r^2
where:
- F is the gravitational force between two objects
- G is the gravitational constant
- M is the mass of the Sun
- m is the mass of the Earth
- r is the distance between the centers of the Sun and the Earth (equal to 1 AU, which is 1.4960E11 m)
We can rearrange the formula to solve for the mass of the Sun (M):
M = (F * r^2) / (G * m)
The gravitational force (F) between the Sun and the Earth can be calculated using the centripetal force formula:
F = (m * v^2) / r
where:
- m is the mass of the Earth
- v is the velocity of the Earth around the Sun
The velocity (v) can be calculated using the distance traveled in one year (1 AU) and the duration of a year (3.1557E7 s):
v = (2 * π * r) / T
where:
- π is a mathematical constant (approximately 3.14159)
- r is the distance between the Sun and the Earth (1 AU)
- T is the duration of a year (3.1557E7 s)
Substituting the values back into the equations and solving step by step will give us the mass of the Sun in kilograms.