1. Understand the Relationship
The relationship between the orbital period of a planet (Earth in this case), its distance from the star (Sun), and the star's mass is governed by Kepler's Third Law of Planetary Motion and Newton's Law of Universal Gravitation.
2. Kepler's Third Law
Kepler's Third Law states:
* *T² ∝ a³*
Where:
* T = orbital period (in seconds)
* a = average orbital radius (in meters)
* ∝ means "proportional to"
3. Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation states:
* F = G * (m1 * m2) / r²
Where:
* F = force of gravity
* G = gravitational constant (6.674 x 10⁻¹¹ N m²/kg²)
* m1 = mass of the Sun (what we want to find)
* m2 = mass of the Earth
* r = distance between the Sun and Earth (average orbital radius)
4. Combining the Laws
We can combine these laws to solve for the mass of the Sun:
* Step 1: The gravitational force between the Sun and Earth is the centripetal force that keeps Earth in orbit. So, we can equate the two:
* F = (m2 * v²) / r (centripetal force)
* F = G * (m1 * m2) / r² (gravitational force)
* Step 2: Equate the two forces and simplify:
* (m2 * v²) / r = G * (m1 * m2) / r²
* v² = G * m1 / r
* Step 3: Substitute the orbital velocity (v) with the relationship v = 2πa/T:
* (2πa/T)² = G * m1 / r
* (4π²a²) / T² = G * m1 / r
* Step 4: Solve for the mass of the Sun (m1):
* m1 = (4π²a³) / (GT²)
5. Calculate the Mass of the Sun
* Earth's Orbital Period (T): 365.25 days = 31,557,600 seconds
* Earth's Average Distance from the Sun (a): 149.6 million kilometers = 1.496 x 10¹¹ meters
* Gravitational Constant (G): 6.674 x 10⁻¹¹ N m²/kg²
Substitute these values into the equation:
* m1 = (4π² * (1.496 x 10¹¹ m)³) / (6.674 x 10⁻¹¹ N m²/kg² * (31,557,600 s)²)
* m1 ≈ 1.989 x 10³⁰ kg
Therefore, the mass of the Sun is approximately 1.989 x 10³⁰ kilograms.
