Final answer:
The orbital period Tplanet of the planet can be calculated using Kepler's Third Law, which relates the period of orbit to the orbital radius and the mass of the star. Converting the radius to meters and plugging in the known values into the formula, then taking the square root and converting the result to Earth days will yield the planet's orbital period.
Step-by-step explanation:
To calculate the orbital period Tplanet of a planet, we use Kepler's Third Law of Planetary Motion, which relates the period of orbit (T) to the semimajor axis of the orbit (r) and the mass of the star (M) the planet is orbiting around.
Kepler's law in terms of period and radius is shown by the equation:
T² = (4π²/GM)r³
Where:
- T is the orbital period,
- r is the orbital radius,
- M is the mass of the star,
- G is the gravitational constant (6.674×10⁻¹± N·m²/kg²).
However, in this case, we need to convert the radius from kilometers to meters (1 km = 1,000 m). The radius in meters (r) is 8.85 x 10⁷ km × 1,000 = 8.85 x 10¹± m. Plugging in the values:
T² = (4π²/6.674×10⁻¹± N·m²/kg²)(6.75 x 10¹¹ kg)r³
T² = (4π²/6.674×10⁻¹±)(6.75 x 10¹¹ kg)(8.85 x 10¹± m)³
This will give us T² in seconds squared. Taking the square root will give us T in seconds, which we then convert to Earth days by dividing by the number of seconds in one day (86,400 seconds).
Calculating T and doing the conversions will provide us with the orbital period of the planet in Earth days.