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

Why are planets spherical

• A large 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)

Asteroid 433 Eros Asteroid 4 VestaComet 67P/CG Phobos

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

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:

02

2

2

2

2

22

zV

yV

xVV

2

2

22222

sin1sin

sin11

V

rV

rrVr

rrV

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 smallerDescribing a geoid that way is analogous to a Fourier synthesis only for

spherical coordinates

coscos 2

2

21100 PraJP

raJPJ

rGMV

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

Images: ESA GOCE

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:

• This is the potential at a stationary point without rotation• The geopotential U on a rotating planet including centrifugal forces

is:

• 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)

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 = 1.082626x10-3

• The surface of the rotating potential– (r = aequator, Φ = 0) and (r = cpole, Φ = 0)

22

23

/21cos

23

MaACJ

ACrG

rGMV

ACcG

cGMU

aACaG

aGMU

30

2230 2

12

Flattening of the ellipsoid

• The flattening f (ratio of equatorial radius to polar radius)

• For earth f = 3.3528 ˟ 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 = 6378.1 km– Pole: c = 6356.8 km

m is the ratio of the centrifugal component to total gravity at equator

GMca

ac

ca

MaAC

acaf

22

2

2

2 21

2

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

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).

222

22

2

2sin0000059.0sin0053024.01780327.9

2sin85

8sin

1417

251

msg

mffmffmgg

gg

ggequator

Determining heights above sea level

• Local deviations from the equipotential surface (geoid)1. Ocean level – follows

equipotential line2. Reference ellipsoid3. Local plump line - follows

local gravity normal4. Continent – causes local

deviations of normal5. 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

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

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

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