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EART162: PLANETARY INTERIORS

EART162: PLANETARY INTERIORS. Last Week. Gravity and the potential Bulk density inferred from gravity. This Week – Shapes and Moments of Inertia. Gravity gives us the mass/density of a planet. How? Why is this useful? Density provides constraints on interior structure

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EART162: PLANETARY INTERIORS

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  1. EART162: PLANETARY INTERIORS

  2. Last Week • Gravity and the potential • Bulk density inferred from gravity

  3. This Week – Shapes and Moments of Inertia • Gravity gives us the mass/density of a planet. How? • Why is this useful? Density provides constraints on interior structure • What is the gross structure of a spinning planet? • We can obtain further constraints on the interior structure from the moment of inertia • How do we obtain it? • What does it tell us? • We can also use gravity to investigate lateral variations in the subsurface density • See Turcotte and Schubert chapter 5

  4. What is the shape of a spinning planet? f = (a-c)/a • Why do we care? • f(internal structure, strength) • Spinning produces large topography relative to Earth’s geologic topography. • As the largest deformation, it’s an important reference point for all gravity and topography measurements. • Example: Satellite motions. • So what is the shape in equilibrium with gravity? Steel vs. water planet? Newton/ Late 1600s.

  5. a Newton’s Method for Shape f = (a-c)/a • The pressure under each water well (or column of mass) at the pole and equator must be equal, if the sphere is in equilibrium. • g varies with distance differently in each well. c a

  6. Spinning Water Balloon Demonstration • Problems with comparing this to planets?

  7. Reference Ellipsoid • A longitude-independent ellipsoid that serves as a reference for the shape of a planet (e.g. WGS 84 for the Earth). • Defined by a = 6,378.137 km at the equator and f = 1/298.257223563.

  8. Back to potentials • Recall: • Gravity potential extends out to infinity. • Obeys Laplace’s equation. • Its negative gradient is gravity. • Equipotential surfaces help define undulations in gravity (drawing of spherical potential with mountain perturbation) • Oceans form an equipotential surface.

  9. The geoid and reference ellipsoid • Geoid: the observed equipotential that coincides with sea level (inferred under continental mass, but equal to the water level occupied by a thin canal connecting the oceans) • Geoid “height”: height of geoid above /below reference ellipsoid. • Gravity is everywhere perpendicular on the geoid.

  10. Disturbances to ocean surfaces/geoid • Hydrostatic figure: the hypothetical equipotential figure that would be assumed if a planet was fluid (assuming you know its density structure and spin rate). E.g. ocean surface E.g. seamount From MIT OCW

  11. Moments of Inertia • Gravity tells you about the density distribution, non-uniquely • Moments of inertia tell you about the radial distribution of density (also non-uniquely).

  12. Moment of Inertia (1) • The moment of inertia (MoI) is a measure of an object’s resistance to being “spun up” or “spun down” • In many ways analogous to mass, but for rotation • MoI must always be measured about a particular axis (the axis of rotation) • The MoI is governed by the distribution of mass about this axis (mass further away = larger MoI) • Often abbreviated as I; also A,B,C for planets • In the absence of external forces (torques), angular momentum (Iw) is conserved (ice-skater example) R w Linear acceleration: F Rotational acceleration: F (T is torque (=2 F R))

  13. Moment of Inertia (2) • MoI is useful because we can measure it remotely, and it tells us about distribution of mass (around an axis) • This gives us more information than density alone Same density Different MoI • Calculating MoI is straightforward (in theory): r dm

  14. R a a Calculating MoI • Some simple examples (before we get to planets) Uniform hoop – by inspection I=MR2 R Uniform disk – requires integration I=0.5 MR2 Uniform sphere – this is one to remember because it is a useful comparison to real planets I=0.4 MR2

  15. Moments of Inertia y • Arbitrary 3D object • Can calculate the moment of inertia about any axis. x z I r dm

  16. Define the inertia tensor y x z

  17. Principal moments of inertia y I2 • There are three axes, that diagonalize this tensor and distribute mass evenly along the three axes: • Eigenvectors provide the axis directions. • The eigenvalues are the principal moments of inertia, I1, I2, I3 x I1 z I3

  18. In planetary science: C • Planets are mostly spherical, but have slight deviations. • Principal moments: A < B < C A B

  19. Aside: Relation to gravity spherical harmonics • Principal moments of inertia are related to the 2-degree gravity spherical harmonics coefficients (r radius, M mass):

  20. Tri-axial ellipsoid • An ellipsoid that represents the short, long, and intermediate axes of a body. • Recall WGS 84 represents the present standard ellipsoid c C b B a A Length axes. Which are the moment of inertia axes?

  21. True Polar Wander • Changes in a spinning planet’s moments of inertia produce an orientation change known as “true polar wander.” Spin axis does NOT change

  22. First, add spin y C • Regular motion results when angular momentum vector L and spin vector w align with a principal moment. L ω x A z B

  23. Torque free precession y C • If the angular momentum is not along one of the principal moments, the object precesses • Complex motion • w and C precess around L. • Lstays fixed in place L x ω A z L w C B Thrown plate example

  24. Example 1: Toutatis • Rotation is not along a principal axis, free precession. • Damping to principal axis rotation? Chang’e 2 lunar mission (Dec. 2012):

  25. Precession damping: Reorientation y C • If the body is not rigid, precession will damp • Orientation will change. L x ω A z B

  26. Precession damping: Reorientation y • If the body is not rigid, precession will damp • Orientation will change. • Angular momentum vector stays fixed! C L x ω A z B

  27. Aside: Damping timescale • Time to damp out precession in a non-rigid body • Depends on Q (quality factor) of the whole-body oscillations. • m is bulk rigidity (similar to Young’s modulus) • K3 is ratio of average to maximum strain (~0.01-0.1) See Burns and Safronov 1973

  28. True polar wander (1) New C! C Initially L, w, C aligned. ω L My house (small) Density anomaly forms. A B A = B = C

  29. True polar wander (2) L ω Initially L, w, C aligned. C Density anomaly forms. A w, C precess around L (w, Lbarely change) Eventually, precession damps, the anomaly moves to the equator: True polar wander B A ≠ B ≠ C

  30. True polar wander (3) C Initially L, w, C aligned. ω L Density anomaly forms. w, C precess around L (w, Lbarely change) A Eventually, precession damps, the anomaly moves to the equator: True polar wander B A < B = C Final, equilibrium configuration

  31. Example: Mars Tharsis • Why is this terrain centered at the equator?

  32. Example: Mars Tharsis • Why is this terrain centered at the equator? • Probably formed at a different latitude, driven to the equator by true polar wander.

  33. Other examples • Pluto and the Moon Center of farside Center of long axis of shape ellipsoid See Garrick-Bethell et al. 2014

  34. Other examples • Pluto and the Moon Nimmo et al. 2016

  35. Torque-free precession (summary) L w n • Precession: rotation of the rotation axis, about some other axis. • Rotation about a principal axis of inertia is stable. • Angular velocity vector (w) and angular momentum (L = Iw) vectors will be aligned. • If misaligned: • Torque-free precession. Precession of w (and n in the same plane) about L. Example: thrown plate/spinning disk. • If the body is at least at least semi-fluid, it will slowly rearrange to align the two (principal axis rotation), possibly over millions of years for planetary bodies. • Example: Lunar polar wander. Asteroid Toutatis is in a state of free precession

  36. True polar wander (summary) • Moment of inertia changes happen (geology)  misaligns C and L. • C prefers energetically to align with L, and does so over long times if the body is semi-fluid. • Hence, the planet’s orientation changes, while L stays fixed in space. Red blob represents a positive density anomaly that just appeared. It shifts the C moment axis away from the spin axis. Eventually Earth shifts the mass to the equator, realigning C and L.

  37. Moments of inertia + gravity • Planets are flattened (because of rotation - centripetal) • This means that their moments of inertia (A,B,C) are different. By convention C>B>A • C is usually the moment about the axis of rotation A C • Differences in moments of inertia are indications of how much excess mass is concentrated along which axes, example: (C-A) represents how much mass is in the equator. Related to the flattening f = (a-c)/a.

  38. Mass deficit at poles Mass excess at equator Moment of Inertia Difference • Because a moment of inertia difference indicates an excess in mass at the equator, there will also be a corresponding effect on the gravity field • So we can use observations of the gravity field to infer the moment of inertia difference • The effect on the gravity field will be a function of position (+ at equator, - at poles) How do we use the gravity to infer the moment of inertia difference?

  39. P b r a f R Point source: Extra term from bulge: Corresponding increase in C : mR2 increase in A: 0 So now we have a description of the gravity field of a flattened body, and its MoI difference (C-A = mR2) Relating C-A to gravity (1) • Here is a simple example which gives a result comparable to the full solution • See T&S Section 5.2 for the full solution (tedious) We represent the equatorial bulge as two extra blobs of material, each of mass m/2, added to a body of mass M. We can calculate the resulting MoI difference and effect on the gravitational acceleration as a function of latitude f. M m/2

  40. Gravity field of a flattened planet • The full solution is called MacCullagh’s formula: MoI difference Contribution from bulge Point source • Note the similarities to the simplified form derived on the previous page • So we can use a satellite to measure the gravity field as a function of distance r and latitude f, and obtain C-A • We’ll discuss how to get C from C-A in a while • The MoI difference is often described by J2, where J2 is dimensionless, a is the equatorial radius. This is a the second degree spherical harmonic coefficient, with l = 2, m = 0, can be written C2,0

  41. Examples • MESSENGER mission to Mercury: Early flyby produced C-A estimates. • MESSENGER mission goal was to improve these estimates via the second degree spherical harmonics (J2) • C-A estimates can tell us about the strength and internal structure of a body: is the flattening equal to the hydrostatic value (more later)?

  42. Radial component Radial component of acceleration: r cosf Outwards acceleration where w is the angular velocity So the complete formula for acceleration gon a planet surface is: Effect of rotation (on surface) • Final complication – a body on the surface of the planet experiences rotation and thus a centrifugal acceleration • Effect is pretty straightforward: w r f

  43. Recall: Gravitational Potential • Gravitational potential is the work done to bring a unit mass from infinity to the point in question: • For a spherically symmetric body, U=-GM/r • For a rotationally flattened planet, we end up with: • This is useful because a fluid will have the same potential everywhere on its surface – so we can predict the shape of a rotating fluid body

  44. a c a Rotating Fluid Body Shape • For a fluid, the grav. potential is the same everywhere on the surface • Let’s equate the polar and equatorial potentials for our rotating shape, and let us also define the ellipticity (or flattening): • After a bit of algebra, we end up with: Note approximate! Remember that this only works for a fluid body! • Does this make sense? • Why is this expression useful? • Is it reasonable to assume a fluid body?

  45. Pause & Summary • Moment of inertia depends on distribution of mass • For planets, C>A because mass is concentrated at the equator as a result of the rotational bulge • The gravity field is affected by the rotational bulge, and thus depends on C-A (or, equivalently, J2) • So we can measure C-A remotely (e.g. by observing a satellite’s orbit) • If the body has no elastic strength, we can also predict the shape of the body given C-A (or we can infer C-A by measuring the shape)

  46. How do we get C from C-A? • Recall that we can use observations of the gravity field to obtain a body’s MoI difference C-A • But what we would really like to know is the actual moment of inertia, C(why?) • Two possible approaches: • Observations of precession of the body’s axis of rotation • Assume the body is fluid (hydrostatic) and use theory

  47. Torque-induced Precession (1) • Application of a torque (T) to a rotating object causes the rotation axis to move in a circle - precession wp T=mgrsin() wp=mgrsin()/Iw r  • The circular motion occurs because the instantaneous torque is perpendicular to the rotation axis • The rate of precession increases with the torque T, and decreases with increasing moment of inertia (I) • An identical situation exists for rotating planets . . .

  48. w Precession (2) North Star planet • So the Earth’s axis of rotation also precesses • The rate of precession depends on torque and MoI (C) • The torque depends on C-A (why?) • So the rate of precession gives us (C-A)/C Sun summer winter

  49. Putting it together • If we can measure the rate of precession of the rotation axis, we get (C-A)/C • For which bodies do we know the precession rate? • Given the planet’s gravitational field, or its flattening, we can deduce J2 (or equivalently C-A) • Given (C-A)/C and (C-A), we can deduce C • Why is this useful? • What do we do if we can’t measure the precession rate?

  50. Cassini (1693) Ecliptic normal 1.5º Spin pole M The lunar spin axis processes around the ecliptic pole with a period of 18.6 years.

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