Planetary Interiors - Georgia Institute of Technologyshadow.eas.gatech.edu/~cpaty/courses/PhysicsPlanets2013/...Hydrostatic Equilibrium First order for a spherical body: Internal structure

Planetary Interiors

Read chapter 6!!

Planetary Interiors

We’d like to know: •  Composition (bulk, and

how it varies w/ depth)

•  State of matter (function of temperature, pressure)

•  Sources of internal energy

What we can measure: •  Surface/atmospheric

composition •  Mass, radius (density) •  Gravity field •  Rotation and oblateness •  Magnetic field •  Temperature heat flux •  Seismic wave

propagation •  Topography, surface

morphology

Credit: Phil Nicholson

Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism

Earth’s Interior Structure

Earth’s Interior - Our Terrestrial Template

Determined through a combination of observation and modeling, validated/refined via seismology.

Earth’s Interior - Our Terrestrial Template

Earth’s Interior - Seismology

Solid Inner Core

Liquid Outer Core

Shadow Zone

Shadow Zone

P - waves S - waves

P - waves can travel through the liquid outer core, while S - waves cannot. Waves are also deflected away from areas of higher density, creating shadow zones where no S - or P - waves are observed.

Wave Phenomena: Snell’s law

n = c/v (by definition)

therefore

sinθ1/sinθ2 = v1/v2

Propagation through smoothly varying densities smoothly curved trajectory


vP =Km + 4

3 µrg

ρ

vS =µrg

ρ

The velocities of these body waves are described above, where Km is the bulk modulus, or the messure of stress needed to compress a material. µrg is the shear modulus, or the messure of stress needed to change the shape of a material without changing the volume.

Change in shape & in volume

Change in shape only


de Pater & Lissauer (2010)


de Pater & Lissauer (2010)

Many seismic stations allow us to watch these propagate…


€

Km

ρ= vP

2 − 43 vS

2

dρdr

= −GMρ2 r( )Kmr

2

The ratio of Km/ρ can be obtained from combining the velocity equations (and is hence an observable quantity):

And by its definition, Km of a homogeneous material governed by the pressure of overlying layers:

€

Km ≈ ρdPdρ

Hence a density profile can be obtained:


€

M r( ) = ME − 4π ρ r( )r2drr

RE∫

You can use this and integrate to model the mass of the Earth over prescribed shells:

And combined with observed seismic waves speeds from earthquakes (or synthetic EQs) to obtain a mass (or density) profile.

The assumption of a homogeneous material breaks down where discontinuities arise in the model based on

wave arrival time observations, hence more detailed structure is obtained (such as the liquid outer core which

results in a lack of S - wave arrivals at reproducible distances from the epicenter).

Bulk density continued: KBOs

Pluto

Triton

Eris Haumea

Quaoar

http://www.mikebrownsplanets.com

Bulk density continued: exoplanets

Szabó & Kiss (2011)


Hydrostatic Equilibrium

Hydrostatic Equilibrium First order for a spherical body: Internal structure is determined by the balance between gravity and pressure:

Which is solvable if the density profile is known. If we assume that the density is constant throughout the planet, we obtain a simplified relationship for the central pressure:

€

P r( ) = gP " r ( )ρ " r ( )r

R

∫ d " r

€

PC =3GM 2

8πR4

* Which is really just a lower limit since we know generally density is higher at smaller radii.

Hydrostatic Equilibrium Alternatively you can approximate the planet as a single slab with constant density and gravity, which gives a central pressure 2x the the previous approach. Generally speaking, the first approximation works within reason for smaller objects (like the Moon) even though their densities are not entirely uniform. The second approximation overestimates the gravity, which somewhat compensates for the constant density approximation and is close to the solution for the Earth. It is not enough to compensate for the extreme density profile of planets like Jupiter. ~Mbar pressures for terrestrial planets!

Moment of Inertia I = 0.4MR2 for a uniform density sphere, but most planets are centrally concentrated, so I < 0.4MR2 For planets at hydrostatic equilbrium, can infer I from mass, radius, rotation rate and oblateness


Heating

Planetary Thermal Profiles A planet’s thermal profile is driven by the sources of heat:

Accretion Differentiation Radioactive decay Tidal dissipation as well as the modes and rates of heat transport: Conduction Radiation Convection/advection


Constituent Relations

Material Properties Knowing the phases and physical/chemical behavior of materials that make up planetary interiors is also extremely important for interior models:

However, these properties are not well known at the extreme temperatures and pressures found at depth, so we rely on experimental data that can empirically extend our understanding of material properties into these regimes:

Courtesy: Synchrotron Light Laboratory

Equation of State To predict the density of a pressurized material, need Equation of State relating density, temperature and pressure of a material. At low pressure, non-interacting particles “ideal gas” For the high temperature/pressure interiors of planets, it’s often written as a power law: P = Kρ(1+1/n)

Where n is the polytropic index, and at high pressures ~ 3/2. These power laws are obtained empirically from measurements in shock waves, diamond anvil cells.

Material Properties For Gas Giants like Jupiter and Saturn understanding how hydrogen and helium behave and interact is critically important for modeling their interiors.

At > 1.4 Mbar, molecular fluid hydrogen behaves like a metal in terms of conductivity, and it’s believed that this region is where the convective motion drives the magnetic dynamo

Phase Diagram for Hydrogen: Wikipedia

Jupiter Interior

Material Properties For Ice Giants like Uranus and Neptune, understanding how water and ammonia behave and interact is critically important for modeling their interiors.

At high pressures, the liquid state of water becomes a supercritical fluid (i.e. not a true gas nor liquid) with higher conductivity Phase Diagram for Water,

Modified from UCL

Uranus/Neptune

Supoercritical Fluid


Gravitational Fields

How do we know about the Earth's mantle/crust dynamics?

Seismology - gives density, temperature, viscosity (assuming an equation of state)

Gravity - large scale gravity field reveals underlying mantle flows

Gravity Fields Gravitational Potential: Where the first term represents a non-rotating fluid body in hydrostatic equilibrium, and the second term represents deviations from that idealized scenario. €

Φg r,φ,θ( ) = −GMr

+ ΔΦg r,φ,θ( )'

( )

*

+ ,

€

ΔΦg r,φ,θ( ) =GMr

Rr

&

' (

)

* + n

Cnm cos mφ + Snm sinmφ( )Pnm cosθ( )m=0

n

∑n=1

∞

∑

Where Cnm and Snm are determined by internal mass distribution, and Pnm are the Legendre polynomials

In the most general case:

Gravity Fields

€

ΔΦg r,φ,θ( ) =GMr

Rr

&

' (

)

* + n

Cnm cos mφ + Snm sinmφ( )Pnm cosθ( )m=0

n

∑n=1

∞

∑

This can be greatly simplified for giant planets with a few assumptions: Axisymmetric about the rotation axis Snm, Cnm= 0 for m > 0 (reasonable since most departures from a sphere are due to the centrifugally driven equatorial bulge)

For N-S symmetry (not Mars!), also Jn=0 for odd n

Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism Isostasy

Isostasy

Courtesy of U of Leeds

Now apply this idea to topography and the crust…

Airy Model

Courtesy of U of Leeds

The Earth’s Crust

The Earth’s granitic (lower density) continental crust varies from < 20 km under active margins to ~80 km thick under the Himalayas.

The basaltic (higher density) oceanic crust has an average thickness of 6 km with less near the spreading ridges.

1.  Airy Scheme: Accommodate topography with crustal ‘root’ (assumes same ρ

for all of the crust)

2.  Pratt Scheme: Lateral density variation causes topography

Credit: Wikipedia

Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism Magnetism

Magnetosphere (Chapter 7) Earth’s Magnetic Field

Similar to a “bar magnet”, the Earth’s intrinsic field is roughly dipolar.

+ =

The solar wind deforms the magnetic field, and creates both a magnetopause and bow shock.

Solar Wind

The Earth’s Magnetic Field Changes in the Earth’s magnetic field

Drift of the magnetic pole Reversal of the field direction recorded in the sea floor

The Earth’s Magnetic Field Changes observed in the paleomagnetic record of

the Earth’s magnetic field indicate it can not be a ‘permanent magnet’

The highly conductive liquid outer core has the

capacity to carry the electric currents needed to support a geodynamo

Earth’s Magnetic Field

The Geodynamo and Magnetic Reversals

Freezing of liquid iron onto the solid inner core combined with buoyancy of lighter alloys provides free energy to set up convection. The Coriolis force causes helical fluid flows. This prevents fields from canceling each other out. Non-uniform heat transfer through Earth’s mantle results in possibility of field reversals. The reversal rate seems to be controlled by the solid inner core.

From Glatzmeier and Roberts

Other Planetary Magnetic Fields Mercury, Earth, Jupiter, Saturn, Uranus and Neptune all have confirmed global magnetic fields sourced internally. To date, Ganymede is the only moon with a dynamo driven magnetic field.

Nature

Nature

Neptune

Uranus

Mars currently has no global magnetic field, but clues to a past dynamo are locked in the remnant magnetization of the crust. •  Is this lack of global field responsible for the atmospheric loss? •  Can we date the dynamo ‘turn-off’ point?

D. Brain

Mars Remnant Magnetism

Next: Solar System/Planet Formation

Read chapter 13!!

Planetary Interiors - Georgia Institute of Technologyshadow.eas.gatech.edu/~cpaty/courses/PhysicsPlanets2013/...Hydrostatic Equilibrium First order for a spherical body: Internal structure

Documents