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
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 - 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
Earth’s Interior - Seismology
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
Earth’s Interior - Seismology
de Pater & Lissauer (2010)
Many seismic stations allow us to watch these propagate…
Earth’s Interior - Seismology
€
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:
Earth’s Interior - Seismology
€
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).
Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism
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
Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism
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
Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism
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
Planetary Interiors Earth’s Interior Structure Hydrostatic Equilibrium Heating Constituent Relations Gravitational Fields Isostasy Magnetism
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
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
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