Geodesy of Terrestrial Planets

• 8 Planets and their satellites

• asteroids, comets, meteoroids

• sun

A planet is a celestial body that

• orbits around the sun

• has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equillibrium shape -> meaning its mass is large enough for it to become (nearly) spherical (if the mass is small, internal strength keeps it irregular) & the material behaves like a fluid over large timescales, allowing it to reshape

• has cleared the neighborhood around its orbit -> meaning there’s nothing of comparable size in its orbital path

  • if that’s not fulfilled but the other criteria are -> Dwarf Planet (like Pluto, which is part of the Kuiper Belt)

All planets, almost all asteroids & many comets orbit the sun in the same rotational sense (prograde; in the same direction as the sun rotates! This is counterclockwise viewed from the sun’s north pole). Most planets also rotate in the same rotational sense as they move around the sun (prograde) - exceptions are Venus (“upside down”, obliquity of 177°), Uranus (obliquity of 98°) & Pluto (obliquity of 120°) as well as some asteroids and comets.

The orbits are near-circular ellipses (low eccentricity) with an inclination of the ecliptic of almost zero (as if they were on a plate). Long-period comets like Hale-Bopp are an exception of this.

In general, all planetary motion follows the Kepler laws, which are:

• The orbit of a planet is an ellipse with the Sun at one of the two foci

• A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time (the closer the planet is to the sun, the faster it will travel)

• The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit (T² ~ a³) -> the bigger the distance is to the sun, the longer it takes the planet to travel around the sun

The Earth’s mean solar distance is 1 AU (around 149 million km).

Mercurys distance to the sun is 0.39 AU, while Pluto orbits at around 39.5 AU from the sun (Neptun: 30).

When a planet is between the Earth and the sun on the imaginary line that connects sun with earth, it’s in conjunction (inferior). When the planet is on the other side (sun is between Earth and planet) it’s also in conjunction (superior).

Opposition is referred to then the planet is on the opposite side of the Earth from the sun (Earth is in between planet and sun). That’s when the planet appears the brightest and clostest to earth.

Obliquity, also called axial tilt, is the angle between the planet’s rotational axis and the normal vector to the ecliptic.

For the Earth, this is 23°.

Rotation is the movement of the planet around the rotational axis.

Precession is when the rotational axis “wobbles” (change of orientation of the rot. axis, axis traces out a cone, Earth: ~25800 years) and nutation is when the rotational axis “nods” on top of the precession:

The rotation of planets is caused by a leftover “spin” from the early solar system (when planets form by gathering material around them, they started to spin similar to an ice skater pulling in their arms).

Both precession and nutation on Earth are caused by the gravitational forces of the sun and moon.

Precession is caused by the earth being a little flattened on the poles and a little bulged at the equator (due to its rotation). Gravity acts stronger on closer parts of the object. Equatorial bulge is closer, but due to Earth’s tilt, the bulge is not aligned with the like between Earth and the Sun/Moon. This misalignment causes twisting forces - the sun & moon pull more on one side than on the other. This causes the change of the direction of the rotational axis, similar to if you’d push a spinning top slightly. So, if a planet spins fast -> large bulge & has close moons, they will experience more precession.

Nutation is caused by the moon’s orbit being tilted slightly relative to the ecliptic (of the Earth). On top of that, the moons tilt is not fixed and the tilt direction changes due to precession. This causes the torque on the equatorial bulge to change slightly, causing nutation.

By the way: Other planets also have an effect on Earth’s precession! But minor.

• 3:2 Orbital Resonance between Neptune and Pluto: Pluto orbits the sun twice, whil Neptun orbits the sun three times in the same time

• 3:2 Spin/Orbit Resonance of mercury: Mercury orbits sun twice, and rotates three times

Mercury, Venus, Earth and Mars are considered Terrestrial Planets. They are composed mainly of iron and rock and are differentiated in a crtust, mantle & core.

The Asteroid Belt is the region between Mars and Jupiter,a belt of millions of objects, including asteroids 200 > 100km, which is between the orbits of Mars and Jupiter. There are more asteroid belts like the Kuiper Belt, but this one is just called The Asteroid Belt.

The largest of the asteroids is called Ceres with a diameter of 950 km, now considered a dwarf planet.

A Theory is that the Asteroid Belt is the remains of a planet never formed due to the gravitational influence of Jupiter.

The Asteroid belt doesn’t include Jupiter’s Trojan clouds, which sit in his Lagrangian points L4 & L5 and are called “Greeks” and “Trojans”.

The Solar System formed 4.568 billion years ago when a large molecular cloud experienced gravitational collapse.

This contracting nebular began to spin -> formed protoplanetary disk. The center became hot due to spin -> protostar.

By accretion of material, the protoplanets were formed.

Inside the frost line, temperatures were to high to form ices, therefore only rock & metal could condense, forming terrestrial planets. Beyond the frost line, ices could form, leading to gas planets.

Beyond the soot line, there’s not enough heat to vaporize carbon -> stays solid, forms planets that are more carbon-rich -> my have to do with habitability

The protoplanets that didn’t manage to become planets congregated int the Asteroid belt, Kuiper belt & Oort cloud

• Sidereal period, the time required for a planet or a satellite, to return to the same state (rotation or position) relative to the stellar reference frame. For Earth: siderial day (stellar day): 23h 45m

• Synodic period, the time required for a planet or a satellite, to return to the same state (rotation or position) relative to some observer

  • synodic rotational period: time between successive recurrences of the same phase with respect to the Sun; e.g., for the Moon: the time between one and the next full moon. Earth synodic day: 24h (solar day)

  • synodic orbital period: time for the two planets to return to the same geometric configuration, as both go around the Sun, e.g. opposition or conjugation. Earth/Mars synodic period: 2.2 Earth years (for one opposition to the next)

For the synodic rotation period, some extra rotation is needed to complete the rotation, therefore synodic rotation periods are longer (for prograde orbits) but shorter (for retrograde orbits).