To find the orbital period of the faster planet, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit.
The formula for Kepler's Third Law is::
T^2 = (4 * π^2 * a^3) / (G * M)
Where:
T = Orbital period of the planet (in years)
a = Semi-major axis of the planet's orbit (in astronomical units, AU)
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = Mass of the central star (in kilograms)
Since the two planets follow circular orbits around the same star, we can assume that the mass of the central star (M) is the same for both planets.
Now, Let's calculate the semi-major axis (a) for the slower planet first:
Given data for the slower planet:
Orbital speed of the slower planet (v1) = 43.3 km/s
To calculate the semi-major axis (a) in AU, we need to convert the orbital speed (v) to AU/year:
1 AU/year ≈ 29.78 km/s
a1 = (v1 / 29.78)
Now, let's find the semi-major axis (a2) for the faster planet using its orbital speed:
Given data for the faster planet:
Orbital speed of the faster planet (v2) = 58.6 km/s
a2 = (v2 / 29.78)
Now, we can use the information about the slower planet's orbital period (T1 = 7.60 years) to find its semi-major axis (a1) in AU:
T1^2 = (4 * π^2 * a1^3) / G
a1^3 = (T1^2 * G) / (4 * π^2)
a1 = ((T1^2 * G) / (4 * π^2))^(1/3)
Similarly, we can find the semi-major axis (a2) for the faster planet using the equation:
a2 = ((T2^2 * G) / (4 * π^2))^(1/3)
Since both planets are orbiting the same star, we can assume the mass of the central star (M) is the same for both equations.d
Now, we have the values of a1 and a2 in AU, and we can use them to find the orbital period (T2) of the faster planet using Kepler's Third Law:
T2^2 = (4 * π^2 * a2^3) / G
T2 = √((4 * π^2 * a2^3) / G)