Solar System Formation & Dynamics

The goal of this class is to provide an introduction to the formation and dynamical evolution of planetary systems (“dynamical evolution” here basically means “evolution due to the action of gravity”). For the most part, we don’t directly observe planets forming, and the dynamics we’re most interested in plays out over time scales far longer than any human observer (on observable time scales, we generally see a very good approximation to simple Keplerian motion). Our indirect knowledge derives from three main sources:

• Observations of protoplanetary disks around young stars – the initial conditions or raw material for planet formation.
• Observations of the Solar System.
• Observations of extrasolar planetary systems.

We will be discussing each of these during the course of the semester. Together with theory (and, to a limited extent, lab experiments) we will try to outline what we know about planet formation and what are the open questions that remain to be answered. We will start close to home.

A great deal is known about the Solar System. Spacecraft have visited all of the planets, and we have samples that originate from the Moon, perhaps 10-100 different asteroids, and Mars (the last two from meteorites). I hardly ever attend a research seminar on planetary science without learning new facts about the Solar System. For example, surprisingly recently (about a decade ago) it was discovered that the distribution of rayed craters on the surface of the Moon is notably asymmetric, with more craters on the “leading” side of the Moon along its orbit than on the “trailing” side. The reasons are not fully understood. For planet formation, the key is to identify what are the key facts that should inform our theories, and which are merely incidental.

An incomplete list of “interesting” properties of the Solar System might include:

• The planets orbit in approximately the same plane. Historically, this observation motivated the nebular hypothesis that the planets formed from a flattened disk – the Solar Nebula. That the planets formed from a disk is no longer in doubt, as we will see such disks are observed around the majority of sufficiently young stars. A curiosity, perhaps an important one, is that the orbital plane of the planets is not exactly the same as the equatorial plane of the Sun as defined by its rotation. The misalignment between these planes is about 7 degrees.
• There are two broad classes of planets, the giants and the terrestrial planets. The giant category includes the true gas giants (Jupiter and Saturn) which are primarily composed of light elements (hydrogen and helium), and the ice giants (Uranus and Saturn) which have cores made up of a mixture of water, ammonia, methane and rocks, atop which sit substantial envelopes of H and He. The terrestrial planets likewise split into two large terrestrial planets (the Earth and Venus) and two much smaller bodies in Mercury and Mars.
• None of the planets have the same composition as the Sun. This is obviously true of the terrestrial planets, but even Jupiter is enormously enriched in heavy elements (i.e. not H and He) as compared to the Sun.
• The terrestrial planets all lie interior to the giant planets.
• The “major” planets (let’s exclude Mercury and Mars for the time being) all have very nearly circular orbits.
• There are two main reservoirs of smaller bodies, the main asteroid belt, between Mars and Jupiter, and the Kuiper belt, beyond Neptune. The Kuiper belt includes many objects with interesting orbital properties, including Pluto (and many other objects) that occupy a 3:2 resonance with Neptune (i.e. Neptune orbits the Sun three times while Pluto orbits twice). In the case of Pluto, the orbit in fact crosses that of Neptune.
• The long period comets arrive in the inner Solar System on trajectories that suggest they originated from a reservoir at very large distances, known as the Oort cloud.
• Most of the planets have satellites, and some of the satellite systems are very extensive. Around the giant planets there are both regular satellites, which orbit in their planets’ equatorial plane (e.g. the Galilean satellites of Jupiter: Io, Europa, Ganymede and Callisto), and irregular satellites whose orbital planes are randomly distributed. There is also the Earth’s Moon, which is only a few times less massive than Mercury.
• The Sun is not part of a binary system.

A couple of other properties of the Solar System, which are less obviously relevant, are also worth mentioning:

• None of the planets are in mean-motion resonances with any of the others. A mean-motion resonance occurs when the orbital periods of two planets are close to an integer ratio, i.e. for two planets with orbital periods and , , with and integers. There are, on the other hand, numerous examples of such resonances among satellite orbits.
• The Solar System is “packed”, a loose term that in this context means (a) that almost all locations where bodies could in principle orbit stably for billions of years are, in fact, occupied, and (b) that we could not, in most cases, add another planet without destroying the long term stability of the system. One exception: we probably could add another terrestrial planet as long as its orbit was well inside that of Mercury.

With the benefit of the hindsight afforded us by the discovery of extrasolar planets, some of these properties now seem more surprising and interesting than they once did. The formerly unremarkable circularity of the gas giant planets’ orbits, for example, is an uncommon feature of known extrasolar planetary systems. Likewise, I don’t recall anyone ever finding it noteworthy that there are no Solar System planets with orbital periods measured in mere days, but we now know (after NASA’s Kepler mission) that many stars that seem similar to the Sun have such short-period systems of super-Earths or mini-Neptunes.

The mass budget of the Solar System

The mass of the Sun is . The mass of Jupiter, much the most massive planet, is . Obviously, most of the mass is in the Sun. A little less trivially, most of the mass of heavy elements (i.e. all those that are not H or He) is also to be found in the Sun. One interpretation of this fact is that, if we assume that most of the current mass of the Sun was once in a disk around a smaller protostar, the planet formation process need not be terribly efficient at converting the heavy elements present within the disk into planets.

The angular momentum budget of the Solar System

Although most of the Solar System’s mass is in the Sun, most of the angular momentum is in the orbital motion of the planets. Recall that the angular momentum of a particle with mass , moving with tangential velocity at distance , is,

.

The velocity of a circular Keplerian orbit around a star of mass is,

,

where is Newton’s gravitational constant. Combining these formulae, we can calculate the angular momentum associated with Jupiter’s orbital motion about the Sun,

.

This is a large but meaningless number, until we put it into context by comparing it to the angular momentum associated with the rotation of the Sun. That’s about , i.e. a hundred times smaller. The large “lever arm” of the planets’ orbits, together with the rather slow rotation rate of the Sun, mean that despite their low masses the planets have the lion’s share of the Solar System’s angular momentum.

When we discuss protoplanetary disks, we’ll discuss how it can be that the process of star and planet formation results in most of the mass going to the Sun, while the angular momentum ends up in the planetary orbits.

The Minimum Mass Solar Nebula (MMSN)

Knowing the masses, orbital radii and compositions of the planets (which we do, at least roughly), it’s possible to take a stab at estimating the surface density distribution of the gas in the Solar Nebula that would have just sufficed to form the planets in their current orbital configuration. This is called the Minimum Mass Solar Nebula. The basic procedure is:

• For each planet, we estimate the mass of some heavy element (e.g. iron) within that body. We then multiply that mass by the ratio of the mass of light elements to iron in the Sun. This gives us, for each planet, the mass the planet would have if it had its current mass of iron but the Solar composition.
• We then imagine spreading this augmented mass across an annulus that extends inward halfway to the orbit of the next planet in, and outward halfway to the next planet out. We divide the mass by the area of this annulus to get a surface density (units g per square cm) at the location of each planet.
• Finally we plot as a function of orbital radius.

Following this recipe, we get a version of the famous plot from Weidenschilling’s paper The distribution of mass in the planetary system and solar nebula:

This may not look like much of a power-law to you, but if we ignore Mars and the asteroid belt the result is that between Venus and Neptune,

.

This radial distribution of gas (with a specified normalization, often taken to be at 1 AU) is called the Minimum Mass Solar Nebula. If we integrate to find out how much mass is enclosed within radius we have,

which means that most of the mass in the MMSN is in the outer part of the disk.

As we will discuss in class, what the MMSN really means is a bit unclear. It’s probably not the real mass distribution of gas in the Solar Nebula. It’s important, however, both for historical reasons and as a benchmark or fiducial mass profile that’s often used in discussion of planet formation. Even if it’s not the “right” profile, it sometimes helps for different people thinking about planet formation to compare their models using the same assumptions for how the gas is distributed with distance from the star.