1. Figure and Gravity Figure of a planet (Earth) Geoid Gravity field
Gravity anomalies
Isostatic principle
Reference: Physics of the Earth, F. D. Stacey & P. M. Davis, Cambridge University Press, 2008

2. Why are planets spherical
A celestial body will assume the lowest state of potential energy
Given enough mass any rocky body will form into a nearly spherical object. More obviously this is valid for gaseous Objects too.
Larger objects will start to differentiate internally over time.
Deviations from an ideal sphere can be caused by
Rotation
Tidal forces
Internal movements of large masses
Plate tectonics
Mantle convection
Glaciation, Polar caps
Oceanic flows
Large impactors (Lunar Mare)

3. Comet 103P/Hartley Asteroid 433 Eros Asteroid 4 Vesta Phobos
Mercury km radius,
Venus 6,051.8 km radius ,
Earth 6,371 km radius,
Mars 3,396.2 km radius,
Vesta 525km,
Phobos 26.8 × 22.4 × 18.4 km,
Eros 34.4×11.2×11.2 km
Comet 103P/Hartley
Asteroid 433 Eros
Asteroid 4 Vesta
Phobos

4. Figure of the earth
For large distances spherical approximation sufficient
Interaction with other astronomical bodies
Sphere not accurate enough for local applications
Can not satisfy e.g. precession of the rotation axis, tides
Satellite orbits
Navigation
Geoid as standard figure description
First order approximation as oblate ellipsoid
Available for most of planets and larger objects
For earth
Solutions available from satellite geodesy
CHAMP, GRACE, GOCE Missions
Reference stations on ground

5. Gravitational potential
The shape of a planet is determined by the gravity
Gravity is a potential field V
Potential fields can be described by Laplaces’s equation
In spherical polar coordinates it is:

6. Solutions to the potential equation
Solutions to Laplace’s equation are the spherical harmonic functions
Gravitational potential V(r,Θ) as a function of
radius r and co-latitude Θ =90° - Φ (Latitude)
G is the gravitational constant
M mass of the object (Earth)
a is the equatorial radius
P0, P1, P2 are the Legendre polynomials
J0, J1, J2 represent the distribution of mass
J0 = 1 because P0 = 1 and dominates at large distances
J1 = 0 because P1 = cosΘ and that would be an off centre potential
J2 describes the oblate ellipsoid, all higher terms are much smaller
Describing a geoid that way is analogous to a Fourier synthesis only for spherical coordinates

7. Deriving the true figure of a planet
The true figure is approximated by:
precisely measuring the gravitation gradient in low orbit
orbit perturbations of satellites
For other planets only J2 or J4 can be determined with an orbiter
Images: ESA GOCE

8. Gravitational potential of a planet
Neglecting higher orders of Jn for the moment we can describe the gravity field of a planet by a standard ellipsoid:
𝑉=− 𝐺𝑀 𝑟 + 𝐺 𝑀 2 𝑎 𝐽 2 𝑟 𝑐𝑜𝑠 2 − 1 2
This is the potential at a stationary point without rotation
The geopotential U on a rotating planet including centrifugal forces is:
𝑈=𝑉− 1 2 𝜔 2 𝑟 2 𝑠𝑖𝑛 2 Θ
The surface of the earth (Geoid) is defined as the surface of constant potential U0 at the equator (r = a, Φ = 0) and A and C are the moments of inertia at the equator (x,y axis) and the pole (z)
𝑈 0 =− 𝐺𝑀 𝑟 + 𝐺 𝑟 3 𝐶−𝐴 − 1 2 𝜔 2 𝑎 2 𝑐𝑜𝑠 2 Φ
J2 = (C-A)/Ma

9. Surface of constant potential
Calculating the non rotating gravitational potential of a flattened planet
J2 describes the principal form of the geoid
for Earth J2 = x10-3
The surface of the rotating potential
(r = aequator, Φ = 0) and (r = cpole, Φ = 0)

10. Flattening of the ellipsoid
The flattening f (ratio of equatorial radius to polar radius)
For earth f = ˟ 10-3
If f and the rotation ω of a planet can be determined, we can estimate the moments of inertia, which can give information on core mantle ratios and other internal mass distributions
Earth radius
Equator: a = km
Pole: c = km
𝐽 2 = 2 3 𝑓 1− 𝑓 2 − 𝑚 3 1− 3 2 𝑚− 2 7 𝑓
m is the ratio of the centrifugal component to total gravity at equator

11. The geoids of Earth, Mars and Venus
For earth exists different geoid standards:
ED50, ETRS89, NAD83, NAVD88, SAD69, SRID, UTM, WGS84,…
Earth geoid derived from GRACE data.
Mars topographic map from the MOLA experiment of MGS
Venus geoid model from Magellan data

12. Gravity
The gravity on the geoid (or any equipotential surface U) can be derived from:
g = - grad U
The international gravity formula is
Departures from the reference value are regarded as gravity anomalies
On earth gravity anomalies are very small
The standard unit for gravity measurements is 1mGal ≡ 10-5 ms-2
In practice topographic features have also an influence on the local gravity (Mountains, sea level, ore deposits, large artificial lakes) and have to be corrected (e.g. Bouguer correction for altitude above topographic features).

13. Determining heights above sea level
Local deviations from the equipotential surface (geoid)
Ocean level – follows equipotential line
Reference ellipsoid
Local plump line - follows local gravity normal
Continent – causes local deviations of normal
Geoid
Satellites (GPS) give heights relative to the geocentric reference ellipsoid
Local height measurements are always based on the geoid
Connection between both are via reference stations and a correction grid for the satellites
Image: NOAA
Image: Wikipedia based on WGS84

14. Isostatic principle
Gravity measurements over mountains reveal that they are mostly in static equilibrium
This is called the isostatic principle
The crustal material floats on the denser mantle material
A mountain has to have a root that is approximately as deep as it protrudes above the surface (iceberg principle) to be stable
Ocean floors are thinner
Airy-Heiskanen or Pratt-Hayford models

15. Isostatic adjustments
The (fast) removal of large masses or the accumulation of them will cause a non-equilibrium state (glaciers)
This can be seen in gravity anomalies
The local crust will react by either an up- or down—movement
Image: NASA GRACE

16. Platte tectonics and isostasy
The creation of mountains on continental plates upon collision is related to isostatic adjustments
The rim of the colliding plates is thickening and thus following the isostatic principle this thicker part of one plate is lifted up whereas the other plate is subducted
Image: USGS