- Practical Exercise 1 RADIANS OR DEGREES?
- Practical Exercise 2 MARS IN OPPOSITION AND QUADRATURE
- Practical Exercise 3 MEASURING MERCURY AND VENUS
- Practical Exercise 4 “MERCUAN”
- Practical Exercise 5 THE EARTH FROM MARS
- Practical Exercise 6 HOW BIG IS THE MOON?
- Practical Exercise 7 THE MOON AGAIN
- Practical Exercise 8 PARAMETERS OF PLANET TRAJECTORIES
- Practical Exercise 9 LIKE FROM ANOTHER PLANET...
- Practical Exercise 10 FEET FIRMLY ON THE EARTH...
- Practical Exercise 11 GREEK, HOW BIG IS THE EARTH?
- Practical Exercise 12 THE MOON IN ACTION FOR THE THIRD TIME

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

Planet | Synodic period [day] |
Synodic period [year] | Sidereal period [year] | Main half-axis [au] |

Mercury | 116 | |||

Venus | 584 | |||

Mars | 780 | |||

Jupiter | 399 | |||

Saturn | 378 | |||

Uranus | 370 | |||

Neptune | 367 |

**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

Planet | Synodic period [day] | Synodic period [year] | Sidereal pe-riod [year] | Main half-axis [au] |

Mercury | 116 | 88,0 | 0,241 | 0,387 |

Venus | 584 | 225 | 0,615 | 0,723 |

Mars | 780 | 678 | 1,88 | 1,52 |

Jupiter | 399 | 4320 | 11,8 | 5,19 |

Saturn | 378 | 10800 | 29,6 | 9,56 |

Uranus | 370 | 28500 | 78,0 | 18,3 |

Neptune | 367 | 76600 | 210 | 35,3 |

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*= 369,65 days and of Neptune

_{♅synod}*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*= 369,65 days and of Neptune

_{♅synod}*T*= 367,49 days. dwe would get the following results

_{♆synod}*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*= 369,65 days and of Neptune

_{♅synod}*T*= 367,49 days , we would get the following results

_{♆synod}*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*= 369,65 days and of Neptune

_{♅synod}*T*= 367,49 days, we would get the following results

_{♆synod}*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*= 369,65 days and of Neptune

_{♅synod}*T♆synod*= 367,49 days, we would get the following results

*T*= 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*= 369,65days and of Neptune

_{♅synod}*T*= 367,49 days, we would get the following results

_{♆synod}*T*= 84,1 years,

_{♅}*a*= 19,2 au;

_{♅}*T*= 164 years,

_{♆}*a*= 30,0 au.

_{♆}