Encyclopedia of the Solar System 2nd ed

Interiors of the Giant Planets 405

TABLE 1 Observed Properties of Jovian Planets

Quantity Jupiter Saturn Uranus Neptune

M(kg) 1.8986× 1027 5.6846× 1026 8.683× 1025 1.024× 1026

a(km) 71,492± 4 60,268± 4 25,559± 4 24,766± 15

Ps(hours) 9.92492 10.78 17.24 16.11

J 2 × 106 14,697± 1 16,332± 10 3,516± 3 3,539± 10

J 4 × 106 − 584 ± 5 − 919 ± 40 − 35 ± 4 − 28 ± 22

J 6 × 106 31 ± 20 104 ± 50 — —

q 0.0892 0.151 0.0295 0.026

 2 0.1647 0.108 0.1191 0.136

ρ(g cm−^3 ) 1.328 0.688 1.27 1.64

Y 0.238±0.007 0.18−0.25 0.26±0.05 0.26±0.05

T 1 (K) 165 135 76 74

2. Constraints on Planetary Interiors

2.1 Gravitational Field

A variety of observations yield information about the
makeup and interior structure of the jovian planets. The
mass of each of the four jovian planets (Table 1) has been
known with some precision since the discovery of their nat-
ural satellites. The masses range from 318 times the Earth’s
mass (M⊕) for Jupiter to 14.5M⊕for Uranus. A second fun-
damental observable property is the radius of each planet
measured at a specified pressure, typically the 1-bar pres-
sure level. Radii are most accurately measured by the oc-
cultation technique, in which the attenuation of the radio
signal from a spacecraft is measured as the spacecraft passes
behind the planet. Jovian planet radii range from 11 times
the Earth’s radius (R⊕) for Jupiter to 3.9R⊕for Neptune.
The combination of mass and radius allows calculation of
mean planetary density,ρ. Although a surprising amount
can be learned about the bulk composition of a planet from
justρ(as we will later see for Extra-solar Giant Planets),
more subtle observations are required to probe the detailed
variation of composition and density with radius.
If the jovian planets did not rotate, they would assume a
spherical shape, and their external gravitational field would
be the same as that of a point of the same mass. No in-
formation about the variation in density with radius could
be extracted. Fortunately, the planets do rotate, and their
response to their own rotation provides a great deal more in-
formation. This response is observed in their external grav-
itational field.
For a uniformly rotating body in hydrostatic equilibrium,
the external gravitational potential,,is

=−

GM

r

(

1 −

∑∞

n= 1

(a

r

) 2 n

J 2 nP 2 n(cosθ)

)

where G is the gravitational constant,Mis the planetary

mass,ais the equatorial radius,θis the colatitude (the angle

between the rotation axis and the radial vectorr),P 2 nare the

Legendre polynomials, and the dimensionless numbersJ 2 n

are known as the gravitational moments. The assumption of

hydrostatic equilibrium means that the planet is in a fluid

state, responding only to its rotation, and there are no per-

manent, nonaxisymmetric lumps in the interior. This as-

sumption is believed to be quite good for the jovian planets.

The gravitational harmonics are found from observations

of the orbits of natural satellites, precession rates of ellipti-

cal rings, and perturbations to the trajectories of spacecraft.

As a spacecraft flies by a planet, it samples the gravitational

field at a variety of radii. Careful tracking of the spacecraft’s

radio signal reveals the Doppler shift due to its acceleration

in the gravitational field of the planet. Inversion of these

data yields an accurate determination of the planet’s mass

and gravitational harmonics (see Table 1). In practice, it

is difficult to measure terms of order higher thanJ 4 , and

the value ofJ 6 is generally quite uncertain. Progressively

higher order gravitational harmonics reflect the distribution

of mass in layers progressively closer to the surface of the

planet. Thus, even if they could be measured accurately,

terms such asJ 8 would not contribute greatly to an under-

standing of the deep interior.

A planet’s response to its own rotation is characterized

by how much a surface of constant total potential (includ-

ing the effects of both gravity and rotation) is distorted. The

amount of distortion on such a surface of constant potential,

known as a level surface, depends on the distribution of mass

inside the planet, the mean radius of the level surface, and

the rotation rate. The distortion, or oblateness, of the out-

ermost level surface is measured from direct observations

of the planet and is given byε=(a−b)/a, wherebis the

polar radius. The equatorial and polar radii can be found

from direct telescopic measurement or, more accurately,