Physics 320
Astrophysics I:  Lecture #17
Prof. Dale E. Gary

Giant Planets (Jupiter, Saturn, Uranus, Neptune)

Physical Characteristics

Jupiter's orbit (a = 5.2 AU) is beyond the "gap" of the asteroids.  It takes amost 12 Earth years to circle the Sun once.  Its eccentricity is a typical value for planets of the solar system, at e = 0.0484.  Jupiter's axis is tilted only 1.3o out of the ecliptic, and it has the fastest rotation rate of all of the planets, at 9h 50m.  Jupiter has four very large moons, all greater than 1500 km in diameter.  These are called the Galilean moons because they were discovered by Galileo as soon as he completed his telescope and turned it on the planet in 1610.  Jupiter also has a large number (at least 20) additional moons none of which are greater than 100 km in diameter.  These are likely captured asteroids.
Jupiter's radius is 71,398 km, or 11.2 times Earth's radius.  That makes its volume (11.2)3 = 1400 times that of Earth.  Its bulk density, however is only about 1330 kg/m3.  This is considerably less dense than Earth (5500 kg/m3) and only slightly denser than water (1000 kg/m3).  Jupiter is almost entirely made of gas, about 75% by mass of hydrogen and 24% helium, with only 1% of heavier elements.  This is almost exactly the composition of the Sun, giving an important clue to how Jupiter and the solar system formed.  Jupiter probably has a rocky core amounting to several Earth masses.

The next layer out from the core is liquid metallic hydrogen. This exotic form of hydrogen can only exist at pressures exceeding 4 million bars.  Let's use our method of estimating pressure from Lecture 15 to get the core pressure of Jupiter:

Pc = (2/3) pG <r>2 R2 = (1.4 x 10-10) <r>2 R2 = (1.4 x 10-10) (1330 kg/m3)2 (7.14x107 m)2
    = 1.2 x 1012 Pa,  or about 12 million atmospheres.
Liquid metallic hydrogen consists of individual protons and electrons, where the electrons are able to flow freely like those in a metal.  At the temperature and pressure of Jupiter's interior hydrogen is a liquid, not a gas.  The temperature in this layer is relatively high, between 11,000 and 40,000 K depending on depth.  This metallic layer is the source of Jupiter's strong magnetic field -- 4x10-5 T at Jupiter's surface -- about the same as that at Earth's surface, but remember that Jupiter's surface is much farther out from the planet's center.  Earth's magnetic field at the distance of Jupiter's surface, RJ = 71,400 km, is only
B = B0(r/RE)-3 = (4 x 10-5 T)(71400/6387)-3 = 2.86 x 10-8 T.
So Jupiter's magnetic field is some 1400 times larger than Earth's.  This mammoth magnetic field traps huge numbers of energetic particles, so the particle radiation environment at Jupiter is extreme.  Spacecraft have to be carefully hardened to withstand the radiation environment.

Above this metallic layer is a layer of "ordinary" liquid hydrogen and helium, where the electrons are bound to the protons.  This ordinary liquid extends all the way to near the surface, with a relatively thin layer of gas forming clouds in Jupiter's atmosphere.

The albedo of Jupiter is quite high, at 0.52, giving an equilibrium temperature of

Tss = 279 (1-A)1/4 (rp)-1/2 = 279 (0.84) (5.2)-1/2 = 103 K.
The observed temperature of the cloud tops, however, is 130 K up to as much as 150 K.  This leads to the conclusion that Jupiter has an internal heat source, probably left over from its gravitational contraction.  The "self-gravitational" potential energy of a body is
U = -GM2/R ,
which means that as a body contracts (R decreases), the potential energy goes down (gets more negative).  According to the virial theorem, which we have mentioned briefly before, 1/2 of this energy must be radiated away, and the rest goes into heating the body.  During Jupiter's formation, it must have become extremely hot, and it is now slowly cooling down, but it appears from what we know that Jupiter may still be generating heat, perhaps continuing to contract slightly.  How rapidly must it contract to produce the excess heat that we see?  Way back in lecture 8 we discussed the flux of a black body, F = sT4 W m-2.  This flux of energy, when considered over the entire surface area of a planet (4pR2), gives a quantity called the luminosity,
L = 4pR2sT4 (watts).
We will be using this quantity a lot next semester.  It describes the total power (energy per unit time) emitted by a spherical body.  We expect Jupiter to have a surface temperature of about T0 = 103 K, but it really has a temperature of T = 130 K, so the excess energy per second is
L = 4pR2s(T4 -T04) = 4p(7.14 x 107 m)2(5.67 x 10-8)[(130)4 - (103)4] = 6 x 1017 W.
We speculate that excess power comes from a release of potential energy due to a slow contraction.  From the expression for potential energy above, the power released is
L = dU/dt = GM2/R2 dR/dt,
dR/dt = L R2 / GM2, = (6 x 1017 W)(7.14x107 m)2 / (6.67x10-11)(1.9x1027 kg)2
         = 0.4 mm/year
Thus, it does not require much contraction at all to release the observed amount of energy.  Still, over a billion years Jupiter would have to contract by 400 km, which seems like a lot.  It may be that the truth is some combination of contraction and left-over heat.
More on Jupiter's magnetic field
On the side of the planet towards the Sun (the daylight side), the solar wind compresses the magnetosphere of the planet and limits its size.  The size is determined by a balance in pressure in the solar wind and in the magnetosphere--the magnetosphere is compressed until an equilibrium is reached.  The solar wind is mostly a gas pressure, P = nkT, while inside the magnetosphere it is mostly a magnetic pressure, PB = B2/2m0. where m0 = 4p x 10-7 H/m is the magnetic permeability of free space.  In the solar wind, we find that the gas pressure falls off as one over distance from the Sun squared:  P ~ 1/r2. From this, we can estimate how much larger Jupiter's magnetosphere is than Earth's, by considering the ratio of gas pressures at Earth's and Jupiter's distance, and the ratio of magnetic pressures of Earth's and Jupiter's magnetic fields.  This is one of the homework problems.  Using a rough estimate from Fig. 4-20 of the text, the Earth's magnetic field extends to about 10 RE.  We find that Jupiter's magnetic field is about 17 RJ in size.
Missions: Auroras -- This image shows auroras at the north and south poles of Jupiter.  If you look closely, you will see a small dot just to the left of the auroral ovals, in the left-hand image.  This small dot is due to Io, the innermost moon of Jupiter.  High-energy particles accelerated at Io rain down on these spots.
Comet Collision with Jupiter

Shoemaker-Levy 9

Physical Characteristics

Moving now out to the far reaches of the solar system, the separations of the planets get much larger.  Jupiter is at 5.2 AU, but Saturn is almost twice as far, at 9.54 AU.  This means it takes almost 30 years for Saturn to circle the Sun once.  Uranus is again twice as far from the Sun as Saturn, at 19.2 AU, and takes 84 years to circle the Sun.  It is so far from the Sun that it cannot be seen easily with the naked eye.  Therefore, it was not discovered until 1781, by William (Wilhelm) Herschel.  He originally named the planet George! (Georgium Sidus, for King George III of England). Likewise, Neptune was discovered only in 1846, at a distance of 1.5 times the distance to Uranus, 30.1 AU.  It takes 165 years to circle the Sun once.

The discovery of Uranus was accidental, but the discovery of Neptune was a triumph of celestial mechanics.  By studying the orbit of Uranus, an extremely tiny perturbation was discovered independently by J.C. Adams (1843) and U.J. Leverrier (1846).  By using Newtonian celestial mechanics, they were able to predict the mass and orbit of the perturbing body.  In 1846, Johann G. Galle found Neptune within 1o of its predicted location.  However, Neptune's orbit diverged rapidly from the orbits predicted by the mathematicians, so if it had been searched for some years earlier or later, it would have been far from its predicted location.  The text has a very interesting story about Galileo in this context.  In 1613, some 234 years before Neptune's discovery, Galileo's drawings of Jupiter showed an object near Neptune's predicted position, and he even detected a small motion of the object with respect to a nearby star.  However, he failed to follow up on the discovery.

All three of these gas giants have orbits of low eccentricity (Saturn and Uranus are nearly identical to Jupiter, with e = 0.056 and 0.048, respectively, while Neptune is even lower at 0.009), and low orbital inclination (2.49o, 0.77o and 1.77o, respectively).  The inclination of their spin axes, however, are large: 26.7o, 98o, and 29o.  The 98o for Uranus, in fact, means that it is nearly lying flat in the ecliptic plane, as shown in the image below.  This means that the Sun shines on one pole for 1/2 of its year (that's 42 Earth years), and then on the other pole for 42 Earth years.

This is an infrared image of Uranus, its rings and moons,
taken by the Hubble Space Telescope.

All of these gas giant planets, including Jupiter, have ring systems, although only Saturn's rings are substantial enough to be seen from Earth.  Jupiter's rings come directly from its moons, and are outside its Roche Limit.  For Uranus and Saturn the rings are within the Roche Limit, and may be tidally disrupted bodies, although surprisingly small.  The situation for Neptune is not so clear.  We will discuss planetary ring systems in a later lecture.

Comparison of Gas Giant Sizes and Interiors
As the figure below shows, Jupiter and Saturn are far larger than Uranus and Neptune.  The interiors of Jupiter and Saturn are similar, with a thick layer of molecular hydrogen, then a layer of metallic hydrogen at the depth where the pressure is sufficient (> 4 million bars).  The solid inner core is rock in the case of Jupiter, but is a mixture of ice and rock in the case of Saturn.  The overall composition of Saturn is nearly the same as Jupiter, and hence the same as the Sun, but the lower pressure of Saturn means that the bulk density is actually less than that of water, at only 680 kg/m3.  Uranus and Nepture have bulk densities of 1240 kg/m3 and 1600 kg/m3, respectively.  It is interesting that the density falls at Saturn and then rises again as we go out from the Sun.  This probably reflects a lower amount of hydrogen in the outermost planets, which instead are made up more of heavier ices.  What scenario might cause this?


Each of these outer gas giant planets has a magnetosphere.  Saturn's is quite strong, again likely due to the metallic liquid hydrogen layer, as we saw with Jupiter.  The surface magnetic field is 2.1 x 10-5 T, which is considerably weaker than Jupiter's, but certainly stronger than Earth's.   At the distance of Saturn, the solar wind pressure can be quite variable, which greatly changes the size of the magnetosphere in a range from about 20 to 30 RS.  Let's derive a general expression for the size of a magnetosphere, given the surface magnetic field of the planet and assuming that the solar wind pressure drops as 1/r2, where r is the distance from the Sun.  Remember that a dipole field drops with distance from the center of a planet as
B = BS(r/Rp)-3            (1)
where BS is the surface magnetic field strength, and Rp is the radius of the planet.  We know that the magnetosphere will extend out to a distance where the gas and magnetic pressures (proportional to B2) are equal, and that Earth's magnetosphere extends out to a distance of about 10 REarth.  The magnetic field at this distance is B = BS(10Rp/Rp)-3 = 10-3BEarth.  Since the gas pressure falls as 1/r2, the gas pressure at a distance rp (expressed in AU) from the Sun will be reduced by rp-2, so the magnetic pressure needed to match this is also lower by the same amount.  The magnetic field strength needed is then reduced by the square root of this, or B = 10-3BEarthrp-1.  This is the field strength needed at a distance rp (expressed in AU) in terms of the Earth's surface magnetic field.  We need it in terms of the planet's surface magnetic field, which we get by equating this expression with equation (1), above, which expresses how the field falls off with distance from a planet.  So we have
BS(r/Rp)-3 = 10-3BEarthrp-1
and solving for r, we get the final expression:
r = 10 (rp BS/BEarth)1/3Rp.
This remarkably simple expression gives the size of the magnetosphere for a planet of size Rp, a distance rp from the Sun, in terms of the planet's surface magnetic field strength, BS, and the Earth's surface magnetic field strength BEarth= 4 x 10-5 T.  For Earth, of course, BS= BEarth, rp= 1, and Rp= REarth so we have r= 10REarth as required.  At Jupiter, the surface field happens to be about the same as Earth's so we have BS= BEarth, rp= 5.2, and Rp= RJ so we have r= 17 RJ as was mentioned in the previous lecture.  For Saturn, we have BS~ BEarth/2, rp= 9.4, and Rp= RS so we have r= 16.7 RS similar to the minimum distance mentioned above.

Uranus' magnetic field is odd in that it is not centered on the center of the planet and is tilted almost 60 degrees with respect to the axis of rotation. It is probably generated by motion at relatively shallow depths within Uranus.  It's surface field strength is about the same as Saturn's, which predicts a magnetosphere of size, r= 21 RU.

Likewise, Neptune's magnetic axis is tilted about 46.8 degrees from the axis of rotation.

Planetary Probe Missions: