Geodesy of the Terrestrial Planets

Earth Rotation

• Earth rotates from West to East, known as "Prograde" rotation.

• The Sun, Moon, and stars rise in the East and set in the West.

• Earth rotation: 360° / 24 hours = 15° / hour.

• Satellites in Low-Earth Orbit (LEO) have an orbital period of approximately 90 minutes.

• In 90 minutes, Earth rotates 22.5°, causing the satellite track to shift westward by 22.5° with each orbit.

• Sidereal day (stellar day): 23h 56m 4.1s

• Synodic day (solar day): 24h 00m

Sidereal and Synodic Periods

• Sidereal period: Time for a planet or satellite to return to the same state (rotation or position) relative to the stellar reference frame.

• Synodic period: Time for a planet or satellite to return to the same state relative to an observer.

• Synodic rotational period: Time between successive recurrences of the same phase with respect to the Sun (e.g., full moon to full moon).

• Synodic orbital period: Time for two planets to return to the same geometric configuration as they orbit the Sun.

Synodic Rotation Period Computation

• For prograde orbits, synodic periods are longer than sidereal periods; the opposite is true for retrograde orbits.

• Prograde Rotation Case: \frac{T{syn}}{T{sid}} = \frac{T{syn}}{T{orb}} + 1 => \frac{1}{T{orb}} - \frac{1}{T{sid}} = \frac{1}{T_{syn}}

• Retrograde Rotation Case: \frac{T{syn}}{T{sid}} = \frac{T{syn}}{T{orb}} - 1 => \frac{1}{T{orb}} + \frac{1}{T{sid}} = \frac{1}{T_{syn}}

Application Examples

• Synodic rotational period of the Earth:

  • Sidereal day: 23h 56m 4.1s

  • Synodic day: 24h 00m

  • Earth orbital period: T_{orb} = 365.2563604167 days

• Synodic period of the Moon:

  • Sidereal period: T_{sid} \approx 27.3 days

  • Earth orbital period: T_{orb} \approx 365.25 days

  • Synodic period: T_{syn} \approx 29.5 days

• Time between two Earth/Mars oppositions:

  • Earth orbital period: ~1 year

  • Mars orbital period: ~1.84 years ≈ 670 days

  • Earth/Mars synodic period: ~2.2 Earth years ≈ 800 days

Earth Seasons

• Seasons are caused by the tilt of Earth’s rotational axis (23.4°).

• Northern summer coincides with Southern winter, and vice versa.

• Equinoxes (Mar 20/21, Sep 22/23): 12-hour daylight everywhere; sub-solar point moves along the Equator.

• Northern Solstice (~June 21/22): Long daylight hours in the north, short in the south; Arctic circle in permanent daylight, Antarctic circle in permanent darkness; sub-solar point moves along the Northern Tropic.

• Southern Solstice (~Dec 21/22): Long daylight hours in the south, short in the north; Arctic circle in permanent darkness, Antarctic circle in permanent daylight; sub-solar point moves along the Southern Tropic.

• Equatorial areas always have 12-hour daylight.

• Seasons are not equally long due to Earth’s orbit eccentricity and varying orbital speed.

• Northern Hemisphere Season Lengths:

  • Spring: 92.75 days

  • Summer: 93.65 days

  • Fall: 89.85 days

  • Winter: 88.99 days