Practical Exercise 8: PARAMETERS OF PLANET TRAJECTORIES

In aable 1, for each planet there is given its synodic orbital period as measured by a terrestrial astronomer. Complete the table with the calculated value of the sidereal orbital periods of the planets and derive the main half-axes of the planets.

Table 1 Synodic orbital periods of the planets and other parameters

Answer:

ahe planets move around the Sun in the same direction, the mutual angular velocity of the Earth and the planets is calculated as the difference of their angular velocities: , resp. v, resp. . From the definition of angular velocity we get:

We calculate the main half-axis from Kepler’s third law:

, where , where , , where , .

Table 1 – answer

ahe calculated values in the table are almost identical to the actual values, with the exception of Uranus and Neptune. ahe actual values of the main half-axes and periods for Uranus are: T_♅ = 84,1 years T_♅ = 84,1 years, a_♅ = 19,2 au; for Neptune they are: T_♅ = 84,1 years, a_♅ = 19,2 au; for Neptune they are: T_♆ = 165 years, T_♅ = 84,1 years, a_♅ = 19,2 au; for Neptune they are: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅ = 84,1 years, a_♅ = 19,2 au; for Neptune: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic orbital period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅synod = 369,65 days and of Neptune T_♅ = 84,1 years, a_♅  =  19,2 au; for Neptune: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic orbital period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅synod = 369,65 days and of Neptune T_♆synod = 367,49 days. dwe would get the following results T_♅ = 84,1 years, a_♅ = 19,2 au; for Neptune: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic orbital period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅synod = 369,65 days and of Neptune T_♆synod = 367,49 days , we would get the following results T_♅ = 84,1 years, T_♅ = 84,1 years,  a_♅ = 19,2 au; for Neptune: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic orbital period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅synod = 369,65 days and of Neptune T_♆synod = 367,49 days, we would get the following results T_♅ = 84,1 years, a_♅ = 19,2 au; T_♅ = 84,1 years, a_♅ = 19,2 au; for Neptune: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic orbital period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅synod = 369,65 days and of Neptune T♆synod = 367,49 days, we would get the following resultsT_♅ = 84,1 days, a_♅ = 19,2 au; T_♆ = 164 days, T_♅ = 84,1 days, a_♅ = 19,2 au; for Neptune: T_♆ = 165 years, a_♆ = 30,1 au. ahe more significant difference between the calculated and actual values lies in the similarity between the synodic orbital period and the period of the Earth’s orbit around the Sun. If we wanted to achieve a more accurate result, we would have to indicate the synodic orbital period for a larger number of valid places. If we substituted for the synodic period of Uranus T_♅synod = 369,65days and of Neptune T_♆synod = 367,49 days, we would get the following resultsT_♅ = 84,1 years, a_♅ = 19,2 au; T_♆ = 164 years, a_♆ = 30,0 au.