Solar System Formation & Dynamics

The gas and dust in the protoplanetary disk can be heated by two distinct sources of energy,

• Stellar irradiation: some fraction of the starlight will hit the disk (at the top and bottom), be absorbed by the dust, and heat it. A completely flat disk of negligible thickness absorbs 1/4 of the total emitted stellar radiation, and a thicker or “flared” disk (one with a bowl-like shape) absorbs even more.
• Accretion energy: if gas is flowing through the disk toward the star, a fraction (half, by the virial theorem) of the potential energy goes into heating the gas. This is sometimes described as “viscous” heating, since the transport of angular momentum within the disk that leads to accretion is somewhat akin to the viscosity in a fluid.

If the disk extends inward to radius , the accretion energy if the accretion rate is (units grams per second) is just,

,

where the factor of two comes from the fact that only half of the liberated potential energy goes into heat – the other half goes into kinetic energy which is higher for the faster orbital speeds close to the star. Plugging in typical numbers (e.g. an accretion rate of , which is a fairly typical number for a young Solar mass star with a disk) one can derive the accretion luminosity. The result is that, for most disks, stellar irradiation is the dominant energy source. The temperature scales with orbital radius as something like,

.

Knowing the temperature within the disk (and also the pressure, which we can obtain following the arguments given in the previous lecture), we can try and work out what types of solid material ought to be present at different locations within the disk. This is the concept known as the condensation sequence. First, we measure the abundances of the different elements within the Sun, using spectroscopic observations of the Solar surface (these measurements may be supplemented by lab measurements of the abundances within primitive meteorites). If we assume that these abundances reflect what the initial composition of the protoplanetary disk was, they tell us (e.g.) how much carbon was present relative to the amount of oxygen, hydrogen, magnesium etc. They do not, however, specify the form that carbon was in. Was it elemental carbon (e.g. graphite), a molecule like carbon dioxide or methane, or some mineral such as calcium carbonate? To determine this, we calculate the thermodynamically preferred mix of chemical compounds that would be present at given temperature and pressure for the known set of elemental abundances (for the physicists, we minimize the Gibbs free energy of the system). This mix defines the condensation sequence. To give some examples, methane is predicted to be present for temperature T < 40 K, important minerals such as perovskite below about 1400 K, and the hardiest materials such as aluminum oxide below about 1700 K. At higher temperatures, all elements are predicted to be in the gas phase.

Water deserves special attention, both because of its critical role in planetary habitability and because ice is a major component of the disk by mass in regions where it’s cold enough for ice to be present (recall that both oxygen and hydrogen are abundant elements in the Sun). The phase diagram for water is described in the Wikipedia article “Properties of Water”. It looks like,

At atmospheric pressure on Earth (1 bar, 100,000 Pa) water can exist in any of its three phases: vapor, liquid and solid. The protoplanetary disk, however, has much lower pressures that are far below the triple point (“TP” in the above diagram). At low P, the only stable phases are water ice and water vapor. Under disk conditions, we thus expect water to be in the form of ice for temperatures below about 150-170 K, and to be in the form of vapor for higher temperatures. The radial location where the transition from vapor to ice occurs is known as the snow line. In the Solar System, meteorites that originated from asteroids interior to about 2.7 AU are found to be water-poor, while those that came from further out are quite water-rich. We thus estimate that the Solar system snow line was at about 2.7 AU (this is a potentially misleading statement, as presumably the location of the snow line moved around as the Solar Nebula evolved and dissipated, but it’s reasonable as an empirical estimate).

The most important thing to note here is the distinction between the location of the snow line (in the disk), and the “habitable zone” defined as the range of distances from the star where a planet could sustain liquid water on its surface. A glance at the water phase diagram suffices to convince one that the snow line is invariably further from the star than the outer edge of the habitable zone, because at the low pressures in the disk water remains a vapor down to much lower temperatures than on the surface of a planet. This reasoning leads one to think that, when the Earth formed, the bulk of material at its location would have been “dry” minerals with very little water content. If so, then the water that is so critical to life on Earth must have been delivered later, from bodies that were initially further out… such as asteroids or potentially comets.