EART160 Planetary Sciences

Francis Nimmo

Last Week• Elliptical orbits (Kepler’s laws) are explained by

Newton’s inverse square law for gravity• In the absence of external torques, orbital angular

momentum is conserved (e.g. Earth-Moon system)• Orbital energy depends on distance from primary• Tides arise because gravitational attraction varies

from one side of a body to the other• Tides can rip a body apart if it gets too close to the

primary (Roche limit)• Tidal torques result in synchronous satellite orbits• Diurnal tides (for eccentric orbit) can lead to heating

and volcanism (Io, Enceladus)

Giant Planets• What determines their internal structure?• How did they form and evolve?• What controls their atmospheric dynamics? (Week 5)• What about extra-solar planets?• Rings not discussed

Image not to scale!

Giant Planets

Basic Parameters

Data from Lodders and Fegley 1998. Surface temperature Ts and radius R are measured at 1 bar level.

a (AU)

Porb

(yrs)

Prot

(hrs)

R

(km)

M

(1026 kg)

Obli-quity

Density g/cc

Ts

K

Jupiter 5.2 11.8 9.9 71492 19.0 3.1o 1.33 165

Saturn 9.6 29.4 10.6 60268 5.7 26.7o 0.69 134

Uranus 19.2 84.1 17.2R 24973 0.86 97.9o 1.32 76

Neptune 30.1 165 16.1 24764 1.02 29.6o 1.64 72

Compositions (1)• We’ll discuss in more detail later, but briefly:

– (Surface) compositions based mainly on spectroscopy

– Interior composition relies on a combination of models and inferences of density structure from observations

– We expect the basic starting materials to be similar to the composition of the original solar nebula

• Surface atmospheres dominated by H2 or He:

(Lodders and Fegley 1998)

Solar Jupiter Saturn Uranus Neptune

H283.3% 86.2% 96.3% 82.5% 80%

He 16.7% 13.6% 3.3% 15.2%

(2.3% CH4)

19%

(1% CH4)

Pressure• Hydrostatic approximation• Mass-density relation• These two can be combined (how?) to get the

pressure at the centre of a uniform body Pc:

)()( rgrdrdP

2)( )(4 rrdrrdM

4

2

8

3

R

GMPc

• Jupiter Pc=7 Mbar, Saturn Pc=1.3 Mbar, U/N Pc=0.9 Mbar

• This expression is only approximate (why?) (estimated true central pressures are 70 Mbar, 42 Mbar, 7 Mbar)

• But it gives us a good idea of the orders of magnitude involved

Temperature• If parcel of gas moves up/down fast enough that it doesn’t

exchange energy with surroundings, it is adiabatic

• In this case, the energy required to cause expansion comes from cooling (and possible release of latent heat); and vice versa

• For an ideal, adiabatic gas we have two key relationships:

RT

P cP Always true Adiabatic only

Here P is pressure, is density, R is gas constant (8.3 J mol-1 K-1), T is temperature, is the mass of one mole of the gas, is a constant (ratio of specific heats, ~ 3/2)

• We can also define the specific heat capacity of the gas at constant pressure Cp:

• Combining this equation with the hydrostatic assumption, we get:

dPdTC p

pdzdT

C

g

•At 1 bar level on Jupiter, T=165 K, g=23 ms-2, Cp~3R, m=0.002kg (H2), so dT/dz = 1.4 K/km (adiabatic)

Hydrogen phase diagram

• Jupiter – interior mostly metallic hydrogen

Hydrogen undergoes a phase change at ~100 GPa to metallic hydrogen (conductive)

It is also theorized that He may be insoluble in metallic H. This has implications for Saturn.

Interior temperatures are adiabats

• Saturn – some metallic hydrogen

• Uranus/Neptune – molecular hydrogen only

Compressibility & Density• As mass increases, radius

also increases

• But beyond a certain mass, radius decreases as mass increases.

• This is because the increasing pressure compresses the deeper material enough that the overall density increases faster than the mass

• The observed masses and radii are consistent with a mixture of mainly H+He (J,S) or H/He+ice (U,N)

mass

radius

Con

stan

t den

sity

From Guillot, 2004

Magnetic fields

How are they generated?• Dynamos require convection in a conductive medium

• Jupiter/Saturn – metallic hydrogen (deep)

• Uranus/Neptune - near-surface convecting ices (?)• The near-surface convection explains why higher-order terms

are more obvious – how? (see Stanley and Bloxham, Nature 2004)

Summary• Jupiter - mainly metallic hydrogen. Rock-ice core

~10 ME.• Saturn - mix of metallic and molecular hydrogen;

helium may have migrated to centre due to insolubility. Similar rock-ice core to Jupiter. Mean density lower than Jupiter because of smaller self-compression effect (pressures lower).

• Uranus/Neptune – thin envelope of hydrogen gas. Pressures too low to generate metallic hydrogen. Densities (and moment of inertia data) require large rock-ice cores in the interior.

• All four planets have large magnetic fields, presumably generated by convection in either metallic hydrogen (J,S) or conductive ices (U,N)

Giant Planet Formation (see Week 1)• Initially solid bodies (rock + ice; beyond snow line)• When solid mass exceeded ~10 Me, gravitational

acceleration sufficient to trap an envelope of H and He• Process accelerated until nebular gas was lost• So initial accretion was rapid (few Myr)• Uranus and Neptune didn’t acquire so much gas

because they were further out and accreted more slowly• Planets will have initially been hot (gravitational

energy) and subsequently cooled and contracted • We can investigate how rapidly they are cooling at the

present day . . .

Energy budget observations• Incident solar radiation much less than that at Earth• So surface temperatures are lower• We can compare the amount of solar energy absorbed

with that emitted. It turns out that there is usually an excess. Why?

5.4

8.1

13.5

48

2.0

2.6

4.6

14

0.6

0.6

3.5

0.3

0.3

0.6

1.4

incidentreflected

After Hubbard, in New Solar System (1999)All units in W/m2

Jupiter Saturn Uranus Neptune

Sources of Energy • One major one is contraction – gravitational energy

converts to thermal energy. Helium sinking is another.• Gravitational energy of a uniform sphere is

• So the rate of energy release during contraction is

RGMEg /6.0 2 Where does this come from?

dt

dR

R

GM

dt

dEg

2

2

6.0

e.g.Jupiter is radiating 3.5x1017 W in excess of incident solar radiation.This implies it is contracting at a rate of 0.4 km / million years

• Another possibility is tidal dissipation in the interior. This turns out to be small.

• Radioactive decay is a minor contributor.

Puzzles• Why is Uranus’ heat budget so different?

– Perhaps due to compositional density differences inhibiting convection at levels deeper than ~0.6Rp .May explain different abundances in HCN,CO between Uranus and Neptune atmospheres.

– This story is also consistent with generation of magnetic fields in the near-surface region (see earlier slide)

• Why is Uranus tilted on its side?– Nobody really knows, but a possible explanation is an

oblique impact with a large planetesimal (c.f. Earth-Moon)

– This impact might even help to explain the compositional gradients which (possibly) explain Uranus’ heat budget

Atmospheric Structure (1)• Atmosphere is hydrostatic:• Gas law gives us:

• Combining these two (and neglecting latent heat):

)()( zgzdzdP

RT

gP

dz

dP

Here R is the gas constant, is the mass of one mole, and RT/gis the scale height of the (isothermal) atmosphere (~10 km) which tells you how rapidly pressure increases with depth

RT

P

Atmospheric Structure (2)• Lower atmosphere (opaque) is dominantly heated from below

and will be conductive or convective (adiabatic)

• Upper atmosphere intercepts solar radiation and re-radiates it

• There will be a temperature minimum where radiative cooling is most efficient; in giant planets, it occurs at ~0.1 bar

• Condensation of species will occur mainly in lower atmosphereTemperature (schematic)

tropopause

troposphere

stratosphere

mesosphere

~0.1 bar

radiation

adiabat

CH4 (U,N only)

NH3

NH3+H2S

H2O

80 K

140 K

230 K

270 K

Theoretical cloud distribution

clouds

Giant planet atmospheric structure

• Note position and order of cloud decks

Extra-Solar Planets• A very fast-moving topic• How do we detect them?• What are they like? • Are they what we would have expected? (No!)

How do we detect them?• The key to most methods is that the star will move

(around the system’s centre of mass) in a detectable fashion if the planet is big and close enough

• 1) Pulsar Timing

• 2) Radial Velocity

planet

pulsarA pulsar is a very accurate clock; but there will be a variable time-delay introduced by the motion of the pulsar, which will be detected as a variation in the pulse rate at Earth Earth

planet

star

Earth

Spectral lines in star will be Doppler-shifted by component of velocity of star which is in Earth’s line-of-sight. This is easily the most common way of detecting ESP’s.

How do we detect them? (2)• 2) Radial Velocity (cont’d)

The radial velocity amplitude is given by Kepler’s laws and is

23/2

3/1

1

1

)(

sin2

eMM

iM

P

Gv

ps

p

orb

Earthi

Ms

Mp

Does this make sense?

From Lissauer and Depater, Planetary Sciences, 2001

Note that the planet’s mass is uncertain by a factor of sin i. The Ms+Mp term arises because the star is orbiting the centre of mass of the system. Present-day instrumental sensitivity is about 3 m/s; Jupiter’s effect on the Sun is to perturb it by about 12 m/s.

How do we detect them? (3)• 3) Occultation

Planet passes directly in front of star. Very rare, but very useful because we can:

1) Obtain M (not M sin i)2) Obtain the planetary radius3) Obtain the planet’s spectrum (!)Only one example known to date.

Light curve during occultation of HD209458.From Lissauer and Depater, Planetary Sciences, 2001

• 4) Astrometry Not yet demonstrated.

• 5) Microlensing Ditto.

• 6) Direct Imaging Brown dwarfs detected.

What are they like?• Big, close, and often highly eccentric – “hot Jupiters”• What are the observational biases?

From Guillot, Physics Today, 2004

Jupiter Saturn

HD209458b is at 0.045 AU from its star and seems to have a radius which is too large for its mass (0.7 Mj). Why?

Note the absence of high eccentricities at close distances – what is causing this effect?

What are they like (2)?• Several pairs of planets have been observed, often in

2:1 resonances• (Detectable) planets seem to be more common in

stars which have higher proportions of “metals” (i.e. everything except H and He)

Sun

Mean local value ofmetallicity

From Lissauer and Depater, Planetary Sciences, 2001

There are also claims that HD179949 has a planet with a magnetic field which is dragging a sunspot around the surface of the star . . .

Puzzles• 1) Why so close?

– Most likely explanation seems to be inwards migration due to presence of nebular gas disk (which then dissipated)

– The reason they didn’t just fall into the star is because the disk is absent very close in, probably because it gets cleared away by the star’s magnetic field. An alternative is that tidal torques from the star (just like the Earth-Moon system) counteract the inwards motion

• 2) Why the high eccentricities?– No-one seems to know. Maybe a consequence of scattering

off other planets during inwards migration?

• 3) How typical is our own solar system?– Not very, on current evidence

Consequences• What are the consequences of a Jupiter-size planet

migrating inwards? (c.f. Triton)• Systems with hot Jupiters are likely to be lacking any

other large bodies• So the timing of gas dissipation is crucial to the

eventual appearance of the planetary system (and the possibility of habitable planets . . .)

• What controls the timing?• Gas dissipation is caused when the star enters the

energetic T-Tauri phase – not well understood (?)• So the evolution (and habitability) of planetary

systems is controlled by stellar evolution timescales – hooray for astrobiology!

Summary• Giant planets primarily composed of H,He with a ~10

Me rock-ice core which accreted first

• They radiate more energy than they receive due to gravitational contraction (except Uranus!)

• Clouds occur in the troposphere and are layered according to condensation temperature

• Many (~200) extra-solar giant planets known• Many are close to the star or have high eccentricities

– very unlike our own solar system• Nebular gas probably produced inwards migration

Key concepts• Adiabat, lapse rate, scale height• Hydrogen phase diagram• Hydrostatic assumption• Compressibility, mass-radius relationship• Gravitational contraction• Radial velocity• Inwards migration• Orbital circularization

End of lecture

Temperature (2)• At 1 bar level on Jupiter, T=165 K, g=23 ms-2, Cp~3R,

=0.002kg (H2), so dT/dz = 1.4 K/km (adiabatic)

• We can also use the expressions on the previous page to derive how the adiabatic temperature varies with pressure

11 10

1

1

/1

0 )1(

PP

C

cTT

p

This is an example of adiabatic temperature and density profiles for the upper portion of Jupiter, using the same values as above, keeping g constant and assuming =1.5

Note that density increases more rapidly than temperature – why?

Slope determined by

(Here T0,P0 are reference temp. and pressure, and c is constant defined on previous slide)