Calculate the length R of the semimajor axis of a planet whose period is 129 years.
To calculate the length of the semimajor axis (R) of a planet with a given period (T) in years, you can use the third law of Kepler:
T^2 = (4π^2 / G * M) * R^3
Where:
T = period in years
G = gravitational constant (6.67430 x 10^-11 m^3/kg/s^2)
M = mass of the central body (in this case, we assume M is the mass of the Sun which is 1.989 x 10^30 kg)
R = semimajor axis in meters
Given that the period T is 129 years, we can calculate the length of the semimajor axis R:
129^2 = (4π^2 / (6.67430 x 10^-11 * 1.989 x 10^30)) * R^3
16641 = (39.47842 / 1.33 x 10^19) * R^3
16641 = 2.967 x 10^-18 * R^3
R^3 = 5.609 x 10^21
R ≈ 1799020000 meters
Therefore, the length of the semimajor axis of the planet is approximately 1,799,020,000 meters or approximately 1.8 billion meters.