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## Planetary Interiors

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**Planetary Interiors**• We can see the surface • We cannot see the interior • The interior is MOST of the planet! • The deepest borehole is 12 km on Earth (0.2% of the radius) • How can we tell anything? • What effects do the interior’s physical properties and processes have on the surface?**Light cannot penetrate solid rock**What can? Sound – Seismology Potential Fields Gravity Magnetism RADAR How do we see inside?**Seismic waves bounce off interfaces in the interior**Detected on the Surface Tell us the structure of the interior Only have this for Earth and Moon) Shear waves cannot travel through liquid Seismology**Magnetism**Dipole Earth on Field Indicates a Convecting Liquid Layer Outer core on Earth Ocean on Europa Magnetic Stripes on Mars Indicates Magnetized Crust But no deeper signal**Higher Gravity over regions of high density**All Potential Field Methods yield non-unique solutions Depth hard to constrain But can be used from Orbit! Gravity GRACE map of the Earth APOD 23 July 2003**RADAR**Phillips et al. LPSC (2007) SHARAD Radargram of Mars RADAR can penetrate the surface, give us the crustal structure. Cannot go deep Tradeoff between depth and wavelength**P**a r Mass • How do we know? • Observe: • Period, P and distance, a of orbiting object (e.g. satellite) • Planetary radius, r • Relationship between these and mass? • Kepler’s Third Law! • Satellite mass unimportant! Orbital Frequency, w = 2p/P**Bulk Densities**• So for bodies with orbiting satellites (Sun, Mars, Earth, Jupiter etc.) M and r are trivial to obtain • For bodies without orbiting satellites, things are more difficult – we must look for subtle perturbations to other bodies’ orbits (e.g. the effect of a large asteroid on Mars’ orbit, or the effect on a nearby spacecraft’s orbit) • Bulk densities are an important observational constraint on the structure of a planet. A selection is given below: Data from Lodders and Fegley, 1998**What do the densities tell us?**• Densities tell us about the different proportions of gas/ice/rock/metal in each planet • But we have to take into account the fact that bodies with low pressures may have high porosity, and that most materials get denser under increasing pressure • A big planet with the same bulk composition as a little planet will have a higher density because of this self-compression (e.g. Earth vs. Mars) • In order to take self-compression into account, we need to know the behavior of material under pressure. • On their own, densities are of limited use. We have to use the information in conjunction with other data, like our expectations of bulk composition**Bulk composition**• Four most common refractory elements: Mg, Si, Fe, S, present in (number) ratios 1:1:0.9:0.45 • Inner solar system bodies will consist of silicates (Mg,Fe,SiO3) plus iron cores • These cores may be sulfur-rich (Mars?) • Outer solar system bodies (beyond the snow line) will be the same but with solid H2O mantles on top**Example: Venus**• Bulk density of Venus is 5.24 g/cc • Surface composition of Venus is basaltic, suggesting peridotite mantle, with a density ~3 g/cc • Peridotite mantles have an Mg:Fe ratio of 9:1 • Primitive nebula has an Mg:Fe ratio of roughly 1:1 • What do we conclude? • Venus has an iron core (explains the high bulk density and iron depletion in the mantle) • What other techniques could we use to confirm this hypothesis?**Distribution**• Bulk density only gets us so far • Is the planet homogeneous or differentiated? • How to tell the difference? • Moment of Inertia**Moment of Inertia**L = IωL: Angular momentum, ω: angular frequency, I moment of inertia for rotation around an axis, r is distance from that axis I is a symmetric tensor. It has 3 principal axes and 3 principal components (maximum, intermediate, minimum moment of inertia: C ≥ B ≥ A.) For a spherically symmetric body rotating around polar axis**The maximum moment of a nearly radially symmetric body is**expressed as C/(Ma2), a dimensionless number. C provides information on how strongly the mass is concentrated towards the center. C/(Ma2)=0.4 2/3 →0 0.347 for c=a/2, ρc=2ρm 0.241 for c=a/2, ρc=10ρm Homogeneous sphere Hollow shell Small dense core thin envelope Core and mantle, each with constant density Symbols: L – angular momentum, I moment of inertia (C,B,A – principal components), ω rotation frequency, s – distance from rotation axis, dV – volume element, M – total mass, a – planetary radius (reference value), c – core radius, ρm – mantle density, ρc –core density • Rotating planet flattens at the poles. • At the same spin rate, a body will flatten less when its mass is concentrated towards the center. • From the flattening, and spin rate, we can determine the MOI, IF the planet is in Hydrostatic Equilibrium**Gravity field of a planet is not simply a point source.**That is an approximation that yields Keplerian motion. • Gravity field can be completely described by spherical harmonic expansion. Each higher order term has an associated coefficient (Jn). • J2 is the second zonal harmonic due to the equatorial bulge a spinning planet will develop and the resulting extra tug of gravity that it causes. • J2 is determined from the moments of inertia of a planet, specifically the difference between the moment of inertia at the pole and at the equator. • J2 and moment of inertia are measured by observing the precession of orbits.**Determining planetary moments of inertia**McCullagh‘s formula for ellipsoid (B=A): In order to obtain C/(Ma2), the dynamical ellipticity is needed: H = (C-A)/C. It can be uniquely determined from observation of the precession of the planetary rotation axis due to the solar torque (plus lunar torque in case of Earth) on the equatorial bulge. For solar torque alone, the precession frequency relates to H by: When the body is in a locked rotational state (Moon), H can be deduced from nutation. For the Earth TP = 2π/ωP = 25,800 yr (but here also the lunar torque must be accounted) H = 1/306 and J2=1.08×10-3 C/(Ma2) = J2/H = 0.3308. This value is used, together with free oscillation data, to constrain the radial density distribution. Symbols: J2 – gravity moment, ωP precession frequency, ωorbit – orbital frequency (motion around sun), ωspin – spin frequency, ε - obliquity**Centrifugal force**Extra gravity from mass in bulge Determining planetary moments of inertia II For many bodies no precession data are available. If the body rotates sufficiently rapidly and if its shape can assumed to be in hydrostatic equilibrium [i.e. equipotential surfaces are also surfaces of constant density], it is possible to derive C/(Ma2) from the degree of ellipsoidal flattening or the effect of this flattening on the gravity field (its J2-term). At the same spin rate, a body will flatten less when its mass is concentrated towards the centre. Darwin-Radau theory for an slightly flattened ellipsoid in hydrostatic equilibrium measures rotational effects (ratio of centrifugal to gravity force at equator). Flattening is f = (a-c)/a. The following relations hold approximately: Symbols: a –equator radius, c- polar radius, f – flattening, m – centrifugal factor (non-dimensional number)**MOI of Planets**• All planets have C/Ma2 < 0.4 • Mercury: 0.33 • Venus: 0.33 • Earth: 0.3308 • Moon: 0.394 • Mars: 0.366 • What can we say about core sizes?