A) Meteors and Asteroids
 

1. History 1.1 In 1766 Johann Titius noticed an interesting arithmetic regularity in the distances of the planets.  This was presented in an astronomy book published in 1772 by Johann Bode, and has been called Bode's law ever since (although more careful citations now call it the Titius-Bode law).  The law is presented in the table below: 1.2 In fact the Titius-Bode law is really an exponential relation, and can be written as  0.3 x 2n + 0.4                   n = -∞ 0, 1, 2, ... The question is whether this near exponential relation is real or coincidental.  There are many cases of physical processes that give exponentials as a result, and there have been attempts to see this relation as the result of an actual physical process associated with the formation of the solar system.  One example is the vortex theory of von Weizacker, another is the ring theory of Prentice.  Similar relationships can be found for the orbits of the moons of the giant planets. 1.3 In any case, this relation led to the discovery of Ceres (radius = 512 km), which was later called a minor planet or asteroid.  Its distance from the Sun is 2.8 AU; just what would be expected for the missing planet.  Other asteroids were soon found and now we know of several classes of asteroids.

2. The asteroid belt 2.1 There is a band at around 2.8 AU between Mars and Jupiter where asteroids are found.  2.2 They range in size from Ceres at 512 km to bodies much smaller.  There are about 20 bodies larger than 125 km, some 200 larger than 50 km, and around 4000 larger than 5 km that have been discovered and named.  The total mass of these asteroids is less than the mass of the Moon. 2.3 Measuring the size of an asteroid is tricky.  They are too small for us to resolve a disk, so we must estimate their albedo.  This is done by measuring the temperature in the infrared and assuming that the albedo is independent of wavelength.  In this case (and assuming an efficiency of 1) the temperature is related to the albedo via A measurement of the temperature will give the albedo, and this can then be used to relate the observed brightness to the radius of the object. 2.4 In addition to their albedo, we can measure their brightness in different wavelength bands.  This is a way of defining the asteroid color, and is related to their composition. 2.5 There are several composition classes of asteroids distinguished by their albedo and color.  More recently radar reflection studies provide information at much longer wavelengths, and even give some information regarding the structure of the regolith.  2.6 C-type asteroids are the most common, representing about 75% of the asteroids seen from Earth.  They are believed to be made of carbonaceous material with many volatiles (water of hydration is observed), and are expected to be among the least evolved objects in the solar system. 2.7 S-type asteroids make up about 15% of the population, and appear to be related to stony-iron meteorites. 2.8 M-type asteroids may be related to nickel-iron meteorites, and may represent the core of a differentiated body. 2.9 Vesta (radius 277 km) seems to show an igneous surface, and may have undergone melting in the past.  For such a small body to melt there must have been a rather intense heat source available (26Al?). 2.10 Asteroids are no longer regarded as remains of a planet that was destroyed, but rather as a planet that never got a chance to form.  They collide with each other and most meteorites that hit the Earth are believed to originate in these collisions.

3. Other asteroid populations 3.1 Trojan asteroids orbit the Sun in Jupiter's orbit, 60° in front and 60° behind.  These are the L4 and L5 Lagrange points.  We can understand the nature of these points from the following considerations.  Think of a small body under the influence of two much larger bodies in orbit around each other (i.e. asteroid, Jupiter, and Sun).  The energy of the small body is simply the sum of its potential energy in the field of each larger body, plus its own kinetic energy.  It turns out to be more convenient to do this computation in a system where the origin is fixed on the center of mass, and the x-axis connects the two massive bodies (i.e. the axes are rotating).  The potential energy of the small body is then given by 3.2 The kinetic energy is mv2/2, but this is not all.  We have to add the energy of the rotating system as well.  If the frame is rotating with some mean angular velocity n, then the kinetic energy of an object at a distance R from the center will be mR2n2/2.  Thus if a body is at rest in the inertial frame, it will appear to have this energy in the rotating frame.  We must therefore subtract it from the total.  The total energy of the small body is then given by This total energy has to be negative for a bound orbit.  Define a positive constant C by C = -2E/m.  The velocity is then given by v2 must be positive, but this will be true only if C is small enough.  For each value of C there are limits on r1, r2, and R where the small body is allowed to be.  At the edges of these limited areas, the velocity is zero.  The five points of equilibrium are the five Lagrangian points. The other three are between the Sun and the planet (L1), outside the planet on the planet-Sun line (L2), and outside the Sun on the same line (L3).  Only L4 and L5 are stable and this is where the Trojans are found.  3.3 Amor asteroids have orbits intersecting the orbit of Mars.  There are about 100 known. 3.4 Apollo asteroids have orbits that cross the orbit of the Earth.  These are the dangerous ones!  They may be extinct comets. 3.5 Chiron orbits between Saturn and Uranus.  It may be connected with the recently discovered Kuiper belt objects.

4. Meteorites 4.1 Most meteorites are believed to come from asteroids. 4.2 The most primitive meteorties are chondrites.  These are stony meteorites containing millimeter-sized spheres called chondrules.  These have a composition similar to that of the surrounding matrix, but have undergone a brief intense heating event, followed by a rapid cooling.  There is no good mechanism for producing the heating in a short time, of the right intensity, and affecting so much mass.  Many not-so-good mechanisms have been suggested.  4.3 Chondrites, because they contain many volatiles, are believed to represent the most primitive solar system material.  The abundances found in these meteorites are used to help determine solar abundances. 4.4 There are several classes of chondrites (carbonaceous, ordinary, and enstatite).  Carbonaceous chondrites are further divided into three classes (CI, CM, and CV).  The CI are the most primitive.  Ordinary chondrites are classified by iron content (H, L, and LL).  Enstatite chondrites are made almost entirely of enstatite. 4.5 Irons and stony-irons make up the remaining types.  There are interesting variations in isotope ratios and minerology that contain clues to the conditions during their formation. 4.6 There are SNC meteorites that are believed to come from Mars.  Their composition tells us something about conditions on the Martian surface.  There are also meteorites that are believed to come from the moon.

B) Jupiter
 

1. Data: Radius - equatorial 71,492 km, polar 66,854 km. Mass - 1.899 x 1030 g. Density - 1.33 g/cm3. Albedo - 0.34 T1 bar - 170K Distance from Sun - 5.2 AU. Rotation Period - 9 h 55 min

2. How are these measured?    a. Radius via parallax.    b. Mass from moons.    c. Rotation period from: Cloud features, but these are different at different latitudes.  In addition, cloud features are affected by local winds, and do not necessarily describe the rotation of the body of the planet. Magnetic field, but this is generally only possible from spacecraft.  For the case of Jupiter, there was synchrotron radiation measurable which gave the rotation period of the body of the planet. Spectroscopy can also be used to measure the rotation rate.  You put the slit of the spectroscope along the equator of the planet.  The spectral line will be composed of light coming towards the telescope at some points and going away from it at others.  This will give the line some tilt.  The angle of the tilt will depend on the speed of rotation, which, in turn, gives the rotation rate of the planet.

3. What does the low density tell us about the composition? 3.1 The central pressure is greater than 10 Mbar (much greater) so that the effect of pressure will raise the density of any material considerably.  You can show that there must be considerable amounts of hydrogen inside the planet. 3.2 Spectroscopy does indeed show that there is a great deal of hydrogen in the atmosphere. 3.3 We also see some NH3 and some CH4 as well as small amounts of many other compounds. 3.4 The beautiful colors we see on Jupiter are due to small amounts of possibly organic material probably in aerosols.

4. How do we get hydrogen into a planet? 4.1 One way is to gather up solids and have the hydrogen be in them like in H2O ice or NH3 or CH4 ice. The trouble is that bringing in so much heavy stuff with the hydrogen will raise the density above the observed value. 4.2 Another way is to attract gas gravitationally.  This means a fairly large gravitational field and it means that helium gas will be attracted as well.  This has two implications for Jupiter's structure.  First there must be a core of heavy material.  Second there should be a solar ratio of helium to hydrogen. 4.3 Helium is hard to observe because it doesn't have any rotation - vibration bands, so its spectrum is mostly in the UV.

5. Computing the internal structure: How can we tell if there is a heavy element core? We need to make theoretical models of the planet.  This means solving the equations of planetary structure. 5.1 We have equations for the pressure and the mass, and we can get an equation of state if we make an assumption about the composition.  We still need to know something about the temperature distribution. 5.2 In principle, this can also be computed if we know the method of heat transport in the interior. If the interior is conducting, and we can compute the conductivity as a function of pressure, temperature and composition, then we can integrate the equation for conductive heat transport to find the temperature distribution. If the interior is radiating, and we can compute the opacity as a function of pressure, temperature, and composition, then we can integrate the equation for radiative heat transport to find the temperature distribution. In the late 1960's infrared measurements of radiation from Jupiter (in the region where the black body curve has its maximum) showed that Jupiter's effective temperature is 125 K, whereas in thermal balance with the sun, and an albedo of 0.34, the temperature would be 110 K.  The explanation is that Jupiter has an internal heat source. The heat is from the original gravitational contraction of the planet. Such a high flux of heat can only be transported by a gradient larger than the adiabatic gradient.  This means that the interior is convecting through most of the volume. Convection is so efficient that it keeps the gradient very close to adiabatic.  This allows us to find the temperature distribution without needing to calculate the conductivity or opacity. 5.3 Computation:  From basic thermodynamics dE = TdS - PdV But for an adiabatic process, dS = 0, so dE = -PdV.  If the heat capacity of the material is C, then the internal energy can be related to the temperature by  dE = CdT so CdT = -PdV The density is related to the volume and the mean molecular weight by V = μ/ρ For an ideal gas Substituting gives Integrating gives We can define a quantity which is related to the number of ways energy can be deposited in the molecule.  Without taking quantum mechanical details into consideration, C = N0k/2 for each degree of freedom.  For pure translational motion, there are 3 degrees of freedom, and a = 1.5, for rotation and vibration there are additional degrees of freedom and a can be larger.  Complicated molecules can have a = 3.5.  What we have done here is computed an adiabat through an ideal gas.  In Jupiter the material is not ideal, so the computation is more complicated, but the idea is the same.  Knowledge of the equation of state allows us to compute an adiabat for the material, and then a variation of temperature with density.  We thus have four equations for the pressure, mass, density, and temperature as a function of radius.  With the appropriate boundary conditions we can integrate these and get the internal structure.

6. Computing the composition:  The other problem is to decide on a composition.  There are many combinations of materials that will give the right total mass.  How can we decide which composition is correct when our only measurements are from outside?  6.1 We can only measure integral properties of the density distribution.  One such integral is the mass. 6.2   Another possibility is the moment of inertia The only problem with this is that there is no good way to measure the moment of inertia.  So it is not terribly useful.  It turns out that there is a set of quantities that are related to the moment of inertia that can give more information: 6.3 When a body is rotating it becomes oblate because in addition to the force of gravity, there is also a centrifugal force.  This is not trivial to compute, since as the centrifugal force works, it changes the mass distribution that, in turn, changes the gravitational field, which affects the centrifugal force, etc.  The problem is nonlinear.  There are methods to compute the shape of a rotating body, and its resultant gravitational field by successive approximations.  In such a case, instead of the usual inverse distance relation, we get that the potential can be written as where Re is the equatorial radius of the planet, Pn is the n'th Legendre polynomial, q is the colatitude and Jn is the n'th gravitational moment.  These moments are integrals of the density distribution and can be determined from measurements of the external gravitation field of the planet (using the orbits of its satellites, for example).  In this way some additional limits can be put on the density distribution. In practice, with the Voyager flybys, we have good measurements of J2 and J4, and some information on J6.  A mass sitting on the surface of the planet will not only feel this gravitational potential, but will also feel a centrifugal force.  This can be written in terms of a potential as well and combined with the gravitational term to give Now the second Legendre polynomial is given by P2(cos θ) = (3 cos2 θ - 1)/2, so Furthermore, since the Jn's get smaller as n increases, let us limit ourselves to the first approximation where n = 1.  In this case the potential becomes Now the planet will assume a shape such that the surface is an equipotential.  In such a case, the shape of the planet can be approximated by where f is the flattening or oblateness, given by f = (Re - Rp)/Re.  If we rewrite this in terms of Legendre polynomials we get The potential on this surface is then given by substituting Rs for r in the expression for the total potential.  This gives a very messy expression, but if we assume that any terms of the order of J2 or f are small compared to 1, and terms of the order of J22 or f2 or J2f or higher are negligible, then we can get a much simpler (though approximate) expression which looks like But since Rs is an equipotential, it cannot depend on θ so the second bracket must be exactly zero.  This gives a relation between J2, f, and ω that must always be satisfied: Note that the second term is basically the ratio of the centrifugal to gravitational forces.  So long as this term is small our approximation works.  In any case, if we can measure any two of these three quantities, we can compute the third. 6.4 Results: What we learn is that the density distribution corresponds to a core of around 5-10 Earth masses, surrounded by an envelope of hydrogen and helium in the solar ratio with an admixture of about 30 Earth masses of heavier material.

7. How do we determine the compositions of the envelope and core? 7.1 We have already seen that for high pressures, where the material behaves like a dense fluid, the density of a mixture is given by  Now, suppose I assume that the envelope is composed of a mixture of hydrogen, helium, and water, and that the ratio of hydrogen to helium is solar.  The solar ratio of hydrogen to helium is 2.7 by mass, so XH = 2.7 XHe.  If there is an additional mass fraction XH2O of water, then 3.7XHe + XH2O = 1, and we get XH = 2.7 XHe = (1 - .73 XH2O).  The density of the mixture then becomes Now I can compute models with different values of XH2O in order to find the best fit to observations.  This will give the water mass in Jupiter's envelope. 7.2 The difficulty is that the hydrogen term contributes the most to the sum.  It has the smallest denominator and the largest numerator.  As a result a small uncertainty in the value of the hydrogen density will result in a large uncertainty in the water abundance.  For example, if, for a given region I need a density of 0.650 at a pressure of 0.5 Mbar, then the densities of H2, He, and H2O respectively are 0.533, 1.5, and 2.666 g cm-3.  This will give the correct density for the mixture with XH2O = 0.01.  But if I made a 5% error in the density of H2, and the value is really 0.506 then the value of XH2O goes up to 0.042.  So a 5% error in the H2 density translates into a factor of 4 in the H2O abundance!  So you need to be really careful. 7.3 There are several pitfalls.  The first is that we assumed that the H2/He ratio is solar.  This needs to be checked. The best check to date is that of the Galileo mission's probe into Jupiter's atmosphere. The results are not completely understood yet, but it seems that the H2/He ratio is very close to solar and may even be a bit higher.  Possibly some of the helium in the outer atmosphere has sunk to lower levels.  The second is that we assumed the third substance is water.  This is not such a serious limitation, since the mass of this component will not be substantially affected.  Finally, we have assumed that there are no surprises in the equations of state for these components at high pressure.  In fact hydrogen does transform to an atomic phase at a pressure of around 2 Mbar (depending on temperature), with an associated jump in density.  This must be treated correctly in order to assess the abundance of heavier materials in the envelope. 7.4 The core is harder to deal with.  It does not contribute significantly to the moment of inertia (i.e. J2), but it does contribute to the mass.  As a result, we can make a good estimate of the mass of the core, but not of its radius (i.e. density and hence composition).  The best models for Jupiter indicate a core of about 5 to 10 Earth masses.  An additional 20 to 40 Earth masses of heavy material is mixed into the hydrogen-helium envelope.

8. Jupiter's magnetic field. 8.1 The transformation to atomic hydrogen occurs for the following reason.  The H2 molecule is held together by the electrons that the two atoms share.  When the density is high enough, an electron can no longer tell which pair of atoms it is associated with, and moves between adjacent pairs.  As a result the pairs are no longer bound, and the molecules break up.  In addition, the electrons move more freely and the material behaves like a metal. 8.2 This metallic hydrogen has important consequences for the planet because it conducts electricity.  This is one of the components of the dynamo needed to generate a magnetic field.  Another component is rotation, and Jupiter certainly rotates ... about once in 10 hours.  The third component is convection in the conducting region.  We have already seen that Jupiter convects, so all the conditions are there for producing a magnetic field.  Such a field is, indeed, observed.

C) Saturn
 

1. Data: Radius equatorial = 60,268 km, polar = 54,364 km Mass = 5.69 x 1029 g. Density = 0.69 g cm-3 Albedo = 0.34 Year = 29.46 yrs. Day = 10 h 39 m Distance from the Sun = 9.53 AU Internal structure

2. Internal structure: 2.1 In general terms the internal structure of Saturn is very similar to that of Jupiter.  There is a dense central core of some 10 Earth masses, surrounded by an envelope of hydrogen and helium with an additional 20 Earth masses or so of heavy material mixed in. 2.2 There are some differences as well.  The H2/He ratio measured for Saturn is much higher than the solar value.  He is present as only 6% of the atmosphere, compared to 24% in Jupiter.  This has been explained by the fact that at high pressures hydrogen and helium are immiscible.  It is expected that the heavier helium sank out, leaving an excess of hydrogen in the envelope.  This argument is strengthened by calculations of the cooling rate for the planet.  It is observed to be emitting more heat than the models predict.  The additional heat source provided by helium rainout would explain the difference.  There is some evidence that the H2/He ratio on Jupiter is somewhat higher than solar.  Possibly rain-out has begun there as well, but because of Saturn's lower temperatures it has progressed further there. 2.3 Saturn shows fewer features on its disk.  This too is because of the lower temperatures.  The temperature in the interior is high, and as you go up towards the surface, you hit regions where the temperature is low enough so that clouds can form.  Since the temperatures are lower in Saturn, these points occur deeper in the planet, so the clouds are harder to see. 2.4 In other respects the interior structure of Saturn is similar to that of Jupiter.  The core is estimated to be about 20 Earth masses, and there are about 20 to 40 Earth masses of heavy material mixed into the envelope.  Since Saturn's envelope is only 1/3 the mass of Jupiter's, the percent of additional material is higher.

3. The system of rings - the most impressive feature of Saturn.  3.1 Galileo discovered the rings in 1610, but it was only in 1656 that Christian Huygens realized that they were actually rings.  In 1857, J. C. Maxwell showed that they could not be solid, since tidal effects would cause them to break up.  If they were stationary, they would be unstable since a small perturbation would bring one side closer to the planet and the additional pull would cause it to continue in that direction until it eventually crashed into the planet.  Giving it enough spin to stay in orbit doesn't help, since different parts of the ring have to move at different speeds to stay in orbit.  This is not possible for a solid ring.  In fact they must be made up of many smaller bodies. 3.2 As a body approaches a planet, the part nearer to the planet feels a slightly stronger gravitational pull than the further side.  If the body is small, the difference is not large, and the material strength of the body is enough to keep it together.  For larger bodies, the difference in force is larger, but the body's own gravity helps to hold it together.  When a moon-sized body gets too close to the parent, the tidal force can overcome the moon's gravity, and rip the moon apart.  This critical distance is called the Roche limit, and depends on the sizes of the interacting bodies, but a good rough guess is about 2.5 times the radius of the planet.  3.3 The rings are very thin:  Only about 2 km thick with a width of some 20,000 km.  They are optically thin, and stars can be seen through them.  Spectra show that they consist mostly of water ice particles with sizes ranging from centimeters to meters. 3.4 Originally the rings were labeled A, B, C, and D, with the A ring being furthest out.  Voyager discovered an additional E ring, beyond the A ring, and then an F and G rings between the A and the E.  The order is thus (going outward from the planet) D, C, B, A, F, G, E. 3.5 It was originally thought that mutual collisions between rings’ particles would keep them spread out, but in 1675 Giovanni Cassini discovered an empty region between the B and the A rings, now called the Cassini division.  This empty region is at a distance of about 120,000 km (about 2 Saturn radii) from the center of the planet.  From Kepler's laws, the period of an orbit varies like the distance to the 3/2 power, so the ratio of the periods of two bodies at distances R1 and R2, should vary like (R1/R2)3/2.  Saturn's moon Mimas sits at a distance of 186,000 km from the planet, so it circles Saturn in almost exactly twice the time a Cassini division particle would complete an orbit.  Since such a particle would see Mimas in the same place every second orbit, it would eventually get pulled out of the region by this 2:1 resonance. 3.6 A similar empty region, the Encke gap, was discovered in the A ring itself by Johann Encke in the 19th century.  This would put it into nearly a 5:3 resonance with Mimas.  Until the Voyager flyby of Saturn, this was thought to be the basic structure of the rings: a thin disk with two gaps caused by resonances with Mimas.  Voyager showed that the rings had much more structure, however, consisting of a very large number of fine ringlets with gaps between them.  It was soon realized that one needed a much more sophisticated approach to understand their structure.

4. Some ring dynamics: 4.1 Ring particles collide with each other, not just in actual physical collisions, but also through longer range gravitational interactions. Such collisions redistribute energy and momentum between the particles, and tend to bring all the particles in any particular region closer to some average value of energy.  This tends to circularize the orbits. 4.2 If an orbit is inclined to the equator, this averaging tends to reduce the inclination to zero.  Even those particles that are in orbits that avoid collisions don't do so for long.  The oblateness of Saturn (i.e. J2) causes the orbits to precess, so that they eventually encounter other particles and lose their momentum perpendicular to the equatorial plane.  That is why the rings are flat. 4.3 The presence of small moonlets (called shepherd satellites) inside the rings also affects the motion of the ring particles, and causes them to remain in tightly restricted areas.  This is a part of the explanation of the many ringlets that are seen.  The rest of the explanation lies in the behavior of groups of particles whose mutual interactions influence the motion of the group.  This spiral wave theory, originally developed to explain the structure of spiral galaxies, has been successfully applied to explain much of the structure of Saturn's ring system. 4.4 There are still some things not completely understood.  The braided structure of the F-ring is one example.  Electric charging effects on the dust, due to UV radiation or interactions with the magnetosphere is another example.

D) Uranus
 

1. Statistics Radius: equatorial = 25,559 km, polar = 24,973 km Mass = 8.66 x 1028 g Density = 1.27 g/cm3 Albedo = 0.30 Year = 84.01 yr Day = 17h 14m Distance from Sun = 19.2 AU

2. Atmosphere 2.1 Uranus atmosphere displays very few features, and it has an essentially featureless disk.  As a result it is very difficult to measure a rotation rate.  We have already seen three methods for measuring the rotation rate.  Timing features as they cross the disk is not practical for Uranus, since we cannot resolve any features from the ground.  Instead one can use photometry to measure the overall brightness of the disk.  If there are any features, even though we cannot resolve them, they will affect the total brightness, and by monitoring this over time we can hope to measure a periodic change from which the rotation period can be derived.  Early attempts at such measurements around 60 years ago yielded a period of about 11h.  Spectroscopic observations by Moore and Menzel in 1929 also gave a period of 10.8h, and this was the number quoted in the literature for many years. 2.2 Above, we see still another method for measuring the rotation period based on measuring the flattening and J2.  The difficulty is that the flattening is also difficult to measure, since it involves the difference between two numbers that are quite close to each other.  Observations of the oblateness by stratoscope balloon-bearing telescopes indicated a period of about 18h.  A number of attempts were made to measure the period after 1974, but the results varied widely (12h to 24h).  The current value comes from the magnetospheric measurements of Voyager. 2.3 Spectroscopy has revealed that the atmosphere consists of hydrogen and helium in solar proportions, with a rich admixture of methane and ammonia.

3. Interior 3.1 There are no completely satisfactory models of Uranus' interior, but most models agree that there is a deep atmosphere overlying a planet made up mostly of a mixture of compounds of C, N, O, Si, Fe, and other elements common in the solar system.  These may be roughly classified as ice and rock. 3.2 Like Jupiter and Saturn, we would expect Uranus and Neptune to have internal heat sources.  Neptune does, but Uranus doesn't.  The reason is not completely clear, but it may be due to the fact that Uranus has cooled enough so that it is now in equilibrium with the Sun. 3.3 Uranus has a magnetic field that is tilted from the rotation axis by 60°.  The reason for this is not clear either, but may be connected with the fact that the field-forming region is not at the center of the planet, but in a conducting shell.  The conducting material might be something like H3O+, NH4+, and similar ions. 3.4 Uranus has its own system of rings (as does Jupiter).  They too are narrow and mediated by shepherd satellites.  They are also of extremely low reflectivity; some tens of centimeters in radius and on average a few meters apart from each other.