Planetary Interiors. Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology

Transcription

1 Planetary Interiors Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology EAS 4803/ CP 22:1

2 Planetary Interiors In order to study the interiors of planets, we first assume that the planet is large enough to be in hydrostatic equilibrium. We then require several pieces of information: Gravitational Field Equation of State -- Relates temperature, density and pressure of the planet s interior - Composition - Thermal Profile: Sources, Losses & Transport A template and a few simplifying assumptions don t hurt either! EAS 4803/ CP 22:2

3 Hydrostatic Equilibrium First order for a spherical body: Internal structure is determined by the balance between gravity and pressure: ( ) = g P r " P r R $ ( )#( r ") d r " r 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 C = 3GM 2 8" R 4 * Which is really just a lower limit since we know generally density decrease with radius. EAS 4803/ CP 22:3

4 Hydrostatic Equilibrium Alternatively you can approximate the planet as a single slab with a pressure peak in the middle, 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 they do not have uniform densities. 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. EAS 4803/ CP 22:4

5 Planetary Interiors Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology EAS 4803/ CP 22:5

6 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 EAS 4803/ CP 22:6

7 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 hydrogren behaves like a metal in terms of conductivity, and it s believed that this region is where the convective motion drives the magnetic dynamo Jupiter Interior Phase Diagram for Hydrogen: Wikipedia EAS 4803/ CP 22:7

8 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 fluid or a gas) with higher conductive properties Supoercritical Fluid Uranus/Neptune Phase Diagram for Water, Modified from UCL EAS 4803/ CP 22:8

9 Equation of State The Equation of State relates density, temperature and pressure of a material. 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. EAS 4803/ CP 22:9

10 Planetary Interiors Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology EAS 4803/ CP 22:10

11 Gravity Fields Gravitational Potential: ( ) = %) GM ( r " g r,#,$ ' ( ) + &" g r,#,$ Where the first term represents a non-rotating fluid body in hydrostatic equilibrium, and the second term deviations from that idealized scenario. *, + "# g ( r,$,% ) = GM r & ( ' - n,, n =1 m =0 R r ) + * n ( C nm cos m$ + S nm sinm$ )P nm ( cos% ) Where C nm and S nm are Stokes coefficients determined by internal mass distribution and P nm are the Legendre polynomials EAS 4803/ CP 22:11

12 Gravity Fields "# g ( r,$,% ) = GM r & ( ' - n,, n =1 m =0 R r ) + * n ( C nm cos m$ + S nm sinm$ )P nm ( cos% ) This can be greatly simplified for giant planets with a few assumptions: Axisymmetric about the rotation axis -- (reasonable since most departures from a sphere are due to the centrifugally driven equatorial bulge) S nm = 0, also C nm = 0 when n is odd or when m is not = 0. Read up more on approximations for Terrestrial Planets in Section EAS 4803/ CP 22:12

13 Planetary Interiors Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology EAS 4803/ CP 22:13

14 Isostacy Courtesy of U of Leeds Now apply this idea to topography and the crust EAS 4803/ CP 22:14

15 Airy Model Courtesy of U of Leeds EAS 4803/ CP 22:15

16 Credit: Wikipedia The Earth s Crust 1. Airy Scheme: Accommodate topography with crustal root (assumes same ρ for all of the crust) 2. Pratt Scheme: Lateral density variation causes topography 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. EAS 4803/ CP 22:16

17 Planetary Interiors Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology EAS 4803/ CP 22:17

18 Planetary Thermal Profiles Necessary to have a thermal profile, which requires knowing sources of heat: Gravitational accretion Radioactive Decay Tidal and Ohmic Heating As well as needing to understand properties of heat transport: Conduction/Radiation Convection EAS 4803/ CP 22:18

19 Planetary Interiors Hydrostatic Equilibrium Constituent Relations Gravitational Fields Isostatic Equilibrium Heating Seismology EAS 4803/ CP 22:19

20 Earth s s Interior - Our Terrestrial Template Determined through a combination of observation and modeling and validated/refined via seismology. EAS 4803/ CP 22:20

21 Earth s s Interior - Seismology Liquid Outer Core Solid Inner Core Shadow Zone Shadow Zone 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. P - waves S - waves EAS 4803/ CP 22:21

22 Earth s s Interior - Seismology Change in shape & in volume v P = Change in shape only v S = K m µ rg " µ rg " The velocities of these body waves are described above, where K m is the bulk modulus, or the messure of stress needed to compress a material. µ rg is the shear modulus, or the messure of stress needed to change the shape of a material without changing the volume. EAS 4803/ CP 22:22

23 Earth s s Interior - Seismology The ratio of K m /ρ can be obtained from combining the velocity equations (and is hence an observable quantity): K m " = v 2 P # v S And by its definition, K m of a homogeneous material governed by the pressure of overlying layers: K m " # dp d# Hence a density profile can be obtained: d" dr = # GM"2 r K m r 2 EAS 4803/ CP 22:23 ( )

24 Earth s s Interior - Seismology You can use this and integrate to model the mass of the Earth over prescribed shells: M r R ( ) E = M E " 4# $ r % r ( )r 2 dr 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). EAS 4803/ CP 22:24