Astronomy 6570 Physics of the Planets Planetary Rotation, Figures, and Gravity Fields Topics to be covered: 1. Rotational distortion & oblateness 2. Gravity field of an oblate planet 3. Free & forced planetary precession
Rotational distortion Consider a spherical, rigid planet rotating at an angular rate. Point P on the surface experiences a centrifugal acceleration: a c = 2 r sin ˆx = 2 x ˆx By symmetry, a similar point located on the surface in the yz plane feels an acceleration a c = 2 y ŷ, so that an arbitrary point on the surface at position (x, y, z) feels an acceleration ( ) a c = 2 x ˆx + y ŷ It is convenient to think of this centrifugal acceleration in terms of a centrifugal potential, V c, such that a c = V c. Clearly we must have ( ) V c = 1 2 2 x 2 + y 2 = 1 2 2 r 2 sin 2
Now consider an ocean on the surface of the rotating planet. the fluid experiences a total potential V T ( r,)= GM + V r c ( r,). In equilibrium, the surface of a fluid must lie on an equi-potential surface (i.e., the surface is locally perpendicular to the net gravitational and centrifugal acceleration). Assuming the distortion of the ocean surface from a sphere to be small, we write r ocean = a + r ( ), a = r equator so V T ( surface) GM a + GM a 2 ( ) r 1 2 2 a 2 sin 2 2 a sin 2 r = constant the last term is << the second (see below), so r const. + 2 a 4 2GM sin2 As expected, the ocean surface is flattened at the poles. The oblateness of the ocean surface is defined as = r eq r pole r eq i.e., 2 a 3 2GM 1 2 q Note that the dimensionless ratio q is just the ratio of centrifugal acceleration at the equator to the gravitational acceleration. Our analysis will only be reasonally correct so long as q << 1.
Some observed* values of q and for planets: We see that q << 1 as we assumed above, and that and q 2 are comparable in size, but that in general is 1.5-2 times greater than predicted. Why? The answer is that we have neglected the feedback effect that the rotational distortion has on the planet's gravity field, which is no longer that of a sphere. [Aside: Rotational disruption. Clearly a planet becomes severly distorted if q approaches unity. This sets an upper limit on the rotational velocity of a planet. Roughly, max GM 2 2 G a 3 1 ( ) 1 2 where is the planet's average density. For the Earth, = 5.52 g cm 3, so max 1.2 10 3 rad sec 1 or P min 1.4 hrs. For Jupiter, P min 2.9 hrs.] *values in ( ) are uncertainties in the last figure.
Planetary Gravity Fields Because of rotational flattening, most planets (but not most satellites) may be treated to a good approximation as oblate spheroids; i.e., ellipsoids with 2 equal long axes and 1 short axis. A basic result of potential theory is that the external potential of any body with an axis of symmetry can be written in the form V G ( r,)= GM 1 J R r n n=2 r where M = total mass ( ) n P n μ ( ) R = equatorial radius J n = dimensionless constants P n (μ) = Legendre polynomial of degree n μ = cos (Note that there is no n = 1 term, as long as the origin of co-ordinates is chosen as the body's center of mass.) The first few P n 's are: P 0 ( μ)= 1 P 1 ( μ)= μ P 2 P 3 P 4 ( ) ( μ)= 1 2 3μ 2 1 ( ) ( μ)= 1 2 5μ3 3μ ( μ)= 1 ( 8 35μ 4 30μ 2 + 3).
The J n 's reflect the distribution of mass within the body, and must be determined empirically for a planet. J 2 is by far the most important, and has a simple physical interpretation in terms of the polar (C) and equatorial (A) moments of inertia: J 2 = C A MR 2 [In general, J n is given by the integral R J n = 1 1 r n P MR n n ( μ) ( r,μ)2 r 2 dμ dr 1 0 where ( r,μ) is the internal density distribution. Since P n μ whose northern and southern hemispheres are symmetric. In fact, only the Earth has a measured non-zero value of J 3.] Relation between rotation, J 2, and oblateness ( ) is an odd function for odd n, J 3 = J 5 = J 7 = = 0 for a planet Let us now return to the question of the oblateness of a rotating planet, including the effect of the non-spherical gravity field. Recall that the centrifugal potential is given by V c = 1 2 2 r 2 sin 2 = 1 3 2 r ( 2 P 2 ( μ) 1) We can ignore the term which is independent of μ, so V c has the same angular dependence as the J 2 term in the planetary gravity field. The total potential at the surface of the planet is V T ( r,)= V G ( r,)+ V c ( r,) = GM + GMR2 J r r 3 2 + 1 3 2 r 2 P 2 μ * where we neglect J 4, J 6 etc. As before, we require the surface to be an equipotential (this is true even for solid planets) and write r = R + r ( ): r = const. J 2 + 1 3 q RP μ 2 ( )
Relation of J n to rotation rate The connection between J 2, J 4, J 6, and the rotation rate is not straightforward to derive, involving a selfconsistent solution along the lines: So long as the rotation parameter, q, is small, it can be shown that J n q n 2 Where the proportionality constants are usually of order unity. Since q << 1 generally, the higher-order J n s rapidly become small. For J 2, the above relation is usually written J 2 = 1 3 k 2 q (*Note that V J 2 Vc = 3J 2 q = k 2, which is the real definition of k 2.) and k 2 is known as a 2 nd order Love number. With k 2 defined this way, we have q = 3J 2, i.e., V J2 = V c, when k 2 = 1 For a uniform density planet it can be shown that k 2 = 32 so J 2 = q 2. Examples of Planetary J n s and k 2 s: J 2 J 3 J 4 k 2 = 3J 2 /q J 4 /q 2 Note that the expression for J n on the previous slide implies that J n becomes smaller as the mass of the planet is more concentrated toward the center. Based on the values for k 2 in the Table, we conclude that Jupiter is the most centrally-concentrated, and Mars the least. Even Mars, however, has a k 2 appreciably less than that of a uniform-density planet, i.e., 1.50.
Weighting functions for J n : r n+2 (r) (Uranus)
Let s return to the question of the planet s equilibrium shape: Since P 2 ( μ) varies from 1.0 at the poles ( μ =±1) to 0.5 at the equator ( μ = 0) the planet's oblateness is = r eq r pole r eq i.e., = 3 2 J 2 + 1 2 q (This replaces our earlier result, = 1 2 q.) For our hypothetical uniform-density planet, J 2 = 1 q, so 2 = 5 4 q.) In general, in terms of the Love number k 2 = 3 J 2 q, we have = ( 1+ k 2 ) q 2 q h 2 2, k 2 is the "response coefficient" of the planet to V c where we introduce another Love number, h 2. As we have shown, for a planet in hydrostatic equilibrium, h 2 = 1+ k 2 Let's compare the observed oblateness with this result (note that if both J 2 and q are known for a planet, no further knowledge of the planet's density distribution is needed to predict ):
Voila! almost perfect agreement with our predicted values, confirming our assumption of hydrostatic equilibrium. C In the last line we give the polar moment of inertia factor, MR, derived from these observational values. This is an 2 important constraint on planetary interior models, and is obtained from the approximate formula, derived by Clairaut 5 h 2 1+ 5 2 15 4 i C MR 2 ( ). (Note that this yields the correct value of h 2 = 52 for a uniform density planet with c = 25MR 2 ; but gives h 2 = 20 / 29 for C = 0, rather than the correct value of 1.)
Review of Love numbers: centrifugal potential, V c = 1 2 2 r 2 sin 2 = 1 3 2 r 2 P 2 ( cos) 1 gravitational potential (quadrupole term), V 2 = GMR2 J r 3 2 P 2 ( cos) ( ) potential Love number, k 2 V 2 r = R V 3 = 3 GM 2 R 3 J 2 i.e., J 2 = 1 3 k 2 q hyrdrostatic figure: = 1 2 q 1+ V 2 V c = 1 2 q ( 1+ k 2 ) 1 2 qh 2 where the shape Love number h 2 = 1+ k 2 = 1 2 q + 1 2 qkq = 1 2 2 q + 3 2 J 2 In general, k 2 and h 2 depend on the internal density profile ( r). For a reasonably uniform planet, Clairaut &Radau showed that: 5 h 2 = 1+ k 2 1+ 5 2 15 2 C ( 4 MR 2 ) C if = constant, MR = 2 and we have: 2 5 h 2 = 5 2, k 2 = 3 2
R2
J 2 1 3 k q 2 f 1 2 h q 2 J 2 f 2 3 k 2 = 2 k 2 = 2 h 2 3 1+ k 2 5 for uniform planet Figure after Dermott (1984) D-R relation: J 2 f = 8 15 5 6 1 3 2 C MR 2 2
Experimental Determination of, J 2, and Rotation Rate, Interior rotation rate: Magnetic field rotation via radio emission Surface/atmospheric rotation rates: Feature tracking (surfaces & clouds) Doppler Shifts (radar) Gravity Field: GM, J 2, J 4, etc. Natural satellites: Periods & semi-major axes GM Rings & vs. a J 2, J 4 & vs. a J 2, J 4, J 6 resonance locations GM, J 2 Spacecraft: Doppler tracking GM, J 2, J 4 Oblateness, : Solid surface/cloud tops: Imaging (Earth-based or s/c) Surface pressure variations Radio occultations Upper atmosphere Radio occultations (p ~ 100 mb) Stellar occulations (1 μb < p < 1 mb)
(1) (2) Mercury s spin is in a 3:2 resonance with its orbit. Venus spin (solar day) is near a 5:1 resonance with Earth conjunction, though this may be coincidence. *Cassini results suggest a longer, and variable, period
-Voyager measurements (1979 1981) Note that 0 wind speed is measured wrt the radio (System III) rotation period, which is assumed to match the magnetic field and thus the interior rotation period. Lewis 1995
Sanchez-Lavega (2005) Science 307, 1223.
Hammel et al., (2001) Icarus 153, 229
Hammel et al., (2000) Icarus 175, 534
F. Stacey: Physics of the Earth
to 0( J 4 ) = GM R 3 1 2 R a 1 2 R 7 2 3 2 J 15 2 4 J R 4 a 7 2 3 = GM R 3 a 2 J 9 2 4 J 2 + 15 2 4 J R 4 a where a = geometric mean radius R = planet's equatorial radius M = planet's mass 2 2
Lindal et al. (1985), A.J. 90, 1136.