Understanding Kepler's Third Law
Kepler's Third Law states that the square of the orbital period of a planet (or comet) is proportional to the cube of the semi-major axis of its elliptical orbit.
Formula:
T² = (4π²/GM) * a³
Where:
* T is the orbital period (in years)
* G is the gravitational constant (6.674 x 10⁻¹¹ m³/kg s²)
* M is the mass of the Sun (1.989 x 10³⁰ kg)
* a is the semi-major axis of the elliptical orbit (in meters)
Steps:
1. Find the semi-major axis (a):
* The semi-major axis is the average of the comet's closest and farthest distances from the Sun.
* a = (1 AU + 7 AU) / 2 = 4 AU
* Convert AU to meters: 1 AU ≈ 1.496 x 10¹¹ meters
* a ≈ 4 * 1.496 x 10¹¹ meters ≈ 5.984 x 10¹¹ meters
2. Plug the values into Kepler's Third Law:
* T² = (4π² / (6.674 x 10⁻¹¹ m³/kg s² * 1.989 x 10³⁰ kg)) * (5.984 x 10¹¹ meters)³
* T² ≈ 1.137 x 10¹⁷ s²
* T ≈ 3.37 x 10⁸ seconds
3. Convert seconds to years:
* T ≈ 3.37 x 10⁸ seconds * (1 year / 3.154 x 10⁷ seconds) ≈ 10.7 years
Therefore, the orbital period of the comet is approximately 10.7 years.
