Tides

• Tidal deformation of a planet (Earth)• Tidal distribution• Tidal friction• Satellite orbit evolution• Roche limit• Lagrange Points• Hill sphere• Other tidal effects

Reference:

Physics of the Earth, F. D. Stacey & P. M. Davis, Cambridge University Press, 2008

What are tides?

• Tides are the rise and fall of sea (and land) levels due to the gravitational effects of the Moon and the Sun and the rotation of the Earth

Images: Samuel Wantman © GFDL, Bay of Fundy, New Brunswick

History

• Pytheas inspired by the high tides at the British islands at 325 BC related the tides to the position of the Moon and its phases.

• Seleucus in Babylon used tides to support his heliocentric model in 2nd century BC.

• First tide table in 1065 in China (for tourism!)• Laplace formulated tidal equations as sheet flow model in 1776.• Harmonic analysis by William Thompson aka Lord Kelvin in 1867.

Tidal potential

• Rotation in the gravity field of another object (Satellite, Sun) causes a deformation of the gravitational potential (Geoid).

• The solid, liquid and gaseous material follows the change of the potential.

• For large bodies of water this is commonly called “Tide”.

• The potential W at point P can be written as:

– M is the mass of the planet– m is the mass of the satellite

22

2

1

'r

R

GmW

L

After substituting some trigonometric relations and

RmM

mb

from Kepler’s third law we get

The rotation of earth ωL is for the moment assumed to be fixed to the lunar orbital period!

mMGRL

32

• The potential W at point P due to the Moon gravitational pull is then

• The first term is the gravitational potential at the centre of earth due to the Moon. It is constant and independent of the position of P.

• The third term is the rotational potential at P due to the rotation of the Earth about its own centre.

• The second term is the tidal potential (W2). It represents the deformation of the equipotential surface to an ellipsoidal aligned with the Earth-Moon axis.

• At the pole (Ψ=90) a fixed value W2=Gma2/2R3 exists

• In the orbital plane W2 oscillates between two maxima

• The difference between polar and orbital value contributes to the ellipticity of the earths figure.

2222

3

2

sin2

1

2

1cos2

3

2

11

2

aR

Gma

mM

m

R

GmW L

W

2

1

2

33

2

2 R

GmaW

Tidal force

• The tidal force is the gradient of the potential W2

• Tidal acceleration– Moon aM=1.1 x 10-7g– Sun aS=0.52 x 10-7g– Venus aV~ 0.000113 as

• Tidal forces of separate sources overlap each other

• Tidal lift of water can be up to several metres

• Tidal lift of continents ~cm range– Corrections for GPS, CERN etc.

necessary• Atmospheric tidal forcing is low and

dynamics dominated by thermal effects

Periodicity

• The main periodicity of the tides is:– ~12h 25”Lunar semidiurnal– 12h Solar semidiurnal– + harmonic & long periodic

constituents (yrs)• Maximum: Spring tide

– Bay of Fundy Δ17m– Mont St. Michel Δ 15m

• Minimum: Neap tide

• Maxima and minima not always at the same amplitude

• Wind and atmospheric pressure effects can enforce tidal effects leading to “Storm surges”

Tidal distribution

• The shape of the continents and local shorelines impede the free flow of the water

• The characteristics of the sea floor (Bathymetry) is also shaping the local water distribution

• The Coriolis effect and interferences create a system of wave patterns (Amphidromic System)

• Places with equal tidal phases are called “cotidal”• For large enough basins (Oceans) the cotidal locations point radially

to a central location with zero tidal motion (amphidromic point)• The amphidromic system rotates counterclockwise in the northern

hemisphere and clockwise in the southern hemisphere

Tidal distribution

Image: R. Ray, NASA TOPEX/Poseidon

Tidal “bore”

• The incoming tide can sometimes be high and fast enough to travel upstream in river mouths or a bay that is narrow enough to locally constrain the water flow (Quiantang River 9m)

• Occurrence as single breaking wavefront or a smooth front and a train of secondary waves

Image: NOAA Turnagain-bore Alaska, 2m bore

Tidal prediction

• Since sea tides are a very localised phenomenon there was no easy way to predict tides without extensive computing power

• For ports long time tables were established

• After harmonic analysis mechanical tide prediction machines were built on the basis of:

William Thompson, Proceedings of the Institution of Civil Engineers, vol.65, 1881

3

222

111

3cos3

cos

cos

tA

tA

tAh

Tidal friction: Earth

• The lunar tidal potential W2 causes a deformation with the added potential k2W2.

• Turbulent drag in the sea and anelastic response of the solid part cause a delay of the tidal bulge by δ=2.9° or 12”.

• Thus the actual tidal potential at the surface of a rotating earth has to be corrected for this angle.

• There is an extra torque LT exerted on the mass m* in the bulge.

2

1cos2

3 23

22

, R

GmakW aE

sincos3

33

5*2

,*

rR

aGmmk

WmL rE

T

sincos3

33

5*2

,*

rR

aGmmk

WmL rE

T

Tidal friction: Moon

• There is also a torque exerted on the moon (m*=m, r=R and Ψ=0)

• This torque tries to realign the tidal bulge and the forcing body (Moon)

• Due to the frictional loss of the tides a exchange of angular momentum between Earth and the Moon occurs.

• The effect is slowing the rotation of Earth ω and accelerates the Moon along its orbit which increases the orbital radius R over time

• The net energy dissipation of the lunar tide is 3.06 x 1012 W

6

522

522

,

3sincos

36 R

aGmk

R

aGmkL MoonT

The change of orbital period for the Moon:

yearcmmsdt

dR

centuryarcsraddt

d L

/7.3107.1

sec25102.1

19

2123

The change of the rotation period for Earth:

centurymsradsdt

d

radsdt

d

tideTotal

tideLunar

/4.2105.6

104.5

222

_

222

_

Tidal locking

• The tidal torque between two objects can decelerate the rotation of a planet and a moon or a star and a planet

• At some point the rotation period is equilibrated with the orbital revolution and the same side is facing the other object permanently

• This is called “tidal locking” and is usually mutual• Examples:

– Moon– Phobos & Deimos– Pluto – Charon– Astroids (Contact Binaries)– Moons of giant planets– Binary stars and exoplanets

• Mercury has an excentric orbit and is locked to the sun in a 3:2 resonance

Satellite orbit evolution

• Tidal interaction causes rotation locking and exchange of angular momentum.

• The orbital motion can be accelerated or also decelerated when the orbit is inside the synchronous orbit i.e. their orbital period is shorter than their rotation.

• In the case of deceleration the orbit is decaying and the satellite is spiralling inward and will impact the planet.Examples:

• Mars: Phobos• Jupiter: Metis and Andrasteas• Saturn: ice particles in the rings• …

• Tidal deceleration and the tidal influence of the sun could be the reason for the absence of moons on the inner planets Mercury and Venus

Roche limit

• When a satellite will orbit inside a certain limiting distance around a larger object, at some point the gravitational force will exceed the internal gravitational and tensile coherence and the satellite will disintegrate

Example:– Comet D/1993 F2 Shoemaker-Levi 9 disintegrated during a

close approach to Jupiter in July 1992 and impacted on Jupiter in July 1994

– Saturn rings

Lagrange Points

• At some points the gravitational potential and the centripetal force of two objects are in equilibrium

• These points are called the “Lagrange Points”

• A smaller object can maintain a stable and stationary position relative to the two other objects if unperturbed by other forces

• The L4 and L5 points are stable insofar as a disturbed object will be pulled back into the point e.g. Trojan Asteroids

• L1, L2, L3 are only metastable i.e. only in the orbital plain. However, it is possible to maintain “halo” or Lissajous orbits which are used for space missions e.g. Herschel, Plank or SOHO

Hill sphere

• The region in which an astronomical object is gravitationally dominating it’s environment despite the presence of a larger body is called the Hill sphere (sometimes Roche sphere)

• Any smaller object inside the Hill sphere will orbit this object (e.g. artificial satellites)

• A satellite partially outside the limit of the hill sphere of the smaller object will be perturbed away by the gravitational pull of the larger object and will start orbiting this objectExample:

• Mars: Deimos will become an earth crossing asteroid

Other tidal effects

• The tidal energy dissipated inside a satellite can be in the order of 10x12 W

• Because of this, enough heat can be generated to partially melt material inside a satellite

Example:

– Volcanism on Io is triggered by tidal forces due to the elliptic orbit (observed heat flow 0.6 to 1.6×1014 W).

– Cryo-volcanismus on Enceladus due to tidal heating of Saturn Images: NASA New Horizon Probe 2007