Solar System Formation & Dynamics

Protoplanetary disks – like those seen above in HST images of the Orion Nebula – are a natural consequence of the broader star formation process. Stars within the Galaxy form within molecular cloud cores, relatively dense regions of gas that are embedded within much larger structures known as giant molecular clouds. The Orion Nebula, which can readily be seen through a small telescope, is part of one of the nearest GMCs.

Observationally, molecular cloud cores have scales of around 0.1 pc, and masses of perhaps 1 to a few Solar masses (these are obviously very rough numbers). It is thus plausible that one such core forms one star, or perhaps a binary or small multiple system. We can estimate how long collapse of a core to much higher densities ought to take, under the assumption that there is no persistent source of support against gravity (e.g. by magnetic fields). By equating kinetic energy to gravitational potential energy,

,

we obtain the escape velocity,

.

The collapse time scale can then be estimated as,

.

By noting that the mass of the cloud, , with the mean density, this can be rewritten in the more transparent form,

,

i.e. the time to collapse under gravity depends solely on the density of the collapsing cloud. Numerically, for a Solar mass cloud that is 0.1 pc in radius, we estimate that the collapse time scale is about 300,000 yr. This is the characteristic time scale on which the formation of low mass stars like the Sun proceeds.

One tenth of a parsec is about 20,000 AU, or 4 million times the current radius of the Sun. Since the specific angular momentum,

,

the very large disparity between molecular cloud core scales and those of stars means that even vanishingly small amounts of core rotation translate into too much angular momentum to form a star directly. Instead, we expect a disk to form. To be a bit more quantitative, we can equate the specific angular momentum of a core, with scale and rotation speed ,

,

to the specific angular momentum at distance from a protostar of mass ,

.

Doing so allows us to estimate a characteristic disk size, which works out to be,

.

If we take for a core of radius 0.1 pc and mass one Solar mass, we get,

.

In other words, even a very small rotational component on cloud core scales implies that infalling gas will form a disk much larger than stellar scales. There are a couple of caveats to this conclusion. If a binary forms, then much of the angular momentum of the initial core could end up in the orbital angular momentum of the binary, with correspondingly less ending up in disks. If angular momentum is lost on the collapse time scale, for example because magnetic fields act to brake the rotation as the cloud collapses, then again disks may be smaller or in principle even non-existent.

Once a star and a disk form, the specific angular momentum of gas orbiting within the disk,

,

is an increasing function of orbital radius. This has important consequences. It implies that absent either,

• Angular momentum transport, i.e. redistribution of angular momentum within the disk, or
• Angular momentum loss from the disk as a whole

the disk ought to be stable. Since both angular momentum transport and loss processes are thought to be rather slow, protoplanetary disks are able to persist for a very large number of orbital times, and in particular to outlast the dynamical collapse phase of the cores that form them.

These theoretical considerations are closely related to the scheme used to observationally classify Young Stellar Objects. Four basic stages are envisaged:

• Class 0 – this is the dynamical collapse phase, characterized by a deeply embedded protostar that is so shrouded by surrounding dust that it is only visible in the far-IR and sub-mm wavebands.
• Class I – at this stage a star and disk (perhaps accompanied by an outflow or jet) have formed, but the remnants of the cloud core are still being accreted. The system becomes visible in the near-IR.
• Class II – a star and disk only… by now the envelope has all been accreted. The stellar photosphere is now visible even in the optical.
• Class III – the disk is dispersed, leaving a pre-main-sequence star only

A cartoon of these stages looks like this,

How long the disk-bearing phases of YSO evolution last is an observational question that is not easy to answer. Young stars do not display clocks, and estimating their ages is notoriously tricky. The standard approach is to measure the disk fraction in clusters, by looking for the excess near-IR emission indicating warm dust in a surrounding disk, and then estimate the age of the cluster by fitting the position of the stars in a Hertzprung-Russell diagram to evolutionary tracks for young stars. Following this procedure, one obtains a plot of disk fraction vs age,

…from which one can read off the median disk lifetime. The standard estimate is about 3 Myr, though very recent work suggests that this may be a modest underestimate… 5 Myr may be a better number. In any event, the estimated lifetimes of protoplanetary gas disks have two important implications for planet formation,

• We have less than 10 Myr to form gas giants, before the gas is gone.
• If the Moon formation is correctly dated at 50-100 Myr after the formation of the earliest solids in the Solar Nebula, the final assembly of the terrestrial planets in the Solar System must have occurred in a gas-free environment.

The transit method is conceptually the simplest way to detect extrasolar planets. The “night side” of a planet emits essentially no optical radiation, so if the planet passes directly in front of the disk of the star as seen from Earth it acts as a black disk that temporarily blocks some fraction of the star light. Transits can be seen in the Solar System. In June 2012 the transit of Venus was visible from Earth:

For other stars, of course, we are generally unable to resolve the stellar disk. The signature of an extrasolar planetary transit is thus not an image of a small black circle crossing the face of the star, but rather a temporary drop in the brightness of the star as some of its light is blocked by the planet. (A more subtle difference is that the Earth is quite close to Venus, so the disk of Venus transiting the Sun is larger than it would appear for a distant observer seeing the same event from interstellar distances.)

The fractional drop in stellar brightness during a planetary transit can be calculated from simple geometry. The stellar disk has an area , while the planetary disk has area . The fraction of star light that is blocked is thus,

.

Jupiter has a radius that is about 10% that of the Sun (i.e. ). The same is true, roughly speaking, for all gas giant planets with masses between about a Saturn mass and up to 10 Jupiter masses. For Jupiter and other gas giant, then, we expect,

.

For the Earth (radius 6600km), the depth of a transit as it crosses the Sun (radius 696,000km) is,

.

The cartoon below shows what we expect the light curve to look like in the specific case of a gas giant that produces a roughly 1% dip in the stellar flux:

The most basic observables of a transit survey are:

• The depth of the transit – this is a direct measure of the relative size of the planet as compared to its host star.
• The period of the orbit , which can be measured directly if a series of successive transits are observed. For a planet orbiting a star of mass at semi-major axis , .

Inspection of the above formulae shows that measuring the period and transit depth are not sufficient to tell us all that much about the planet and its orbit. If, however, we have a good estimate of the stellar properties – its mass and radius – then the transit measurements give us the physical size of the planet and the semi-major axis of its orbit. The importance of good knowledge of the host star is relevant for the Kepler mission, whose ultra-precise photometry means that uncertainty in stellar parameters is a significant source of overall uncertainty in measured planetary properties.

For ground-based observatories, the primary limitation on the planets that can be detected via the transit method comes from atmospheric fluctuations, which limit the precision to which the stellar flux can be measured. The attainable precision is not a strong function of the size of the telescope, and the most successful surveys to date have used dedicated arrays of small telescopes to scan wide areas of the sky in search of planets orbiting relatively bright stars. (These offer the best targets for follow-up once planets have been found.) The SuperWASP search, for example, uses an array of 8 telephoto camera lenses, which have a field of view of 8 degrees on a side:

The atmospheric limitations mean that ground-based transit surveys are able to detect the 1% dips caused by gas giants, but are unable to detect the two orders of magnitude smaller signals produced by transiting Earth-like planets. (At least in the visible for Solar-type stars – the MEarth project targets smaller stars, around which even terrestrial planets can produce transit depths accessible from the ground.)

From space there are no atmospheric limitations, and the main barrier to detecting transits of small planets is instead the intrinsic variability of the stars themselves. Even stars that we don’t think of as being “variable” – like the Sun – vary in brightness by small amounts as star spots and other surface features form and rotate through the visible disk. NASA’s Kepler mission was designed to be able to detect transits of Earth-like planets at around 1 AU (i.e. in the habitable zone) around stars whose intrinsic noise properties were similar to those of the Sun. In practice, the realized transit detection efficiency for such planets around the actual sample of Kepler stars was not quite as good as predicted, and with the failure of the Kepler spacecraft’s pointing it is not clear whether any true extrasolar Earth analogs will be recovered from the data. Nonetheless, Kepler discovered a large number of small – probably terrestrial – planets, including Kepler-37b that has a size comparable to the Moon.

The radial velocity (or Doppler) method relies on the fact the orbital motion of a planet induces a counter-balancing (or “reflex”) motion in the star. In general, for a binary system the two orbiting bodies describe elliptical orbits about the center of mass of the binary. If the binary is made up of two equal mass stars, the center of mass will lie midway between them. In the case of a star-planet system, the center of mass will typically be inside the physical radius of the star, but not at the center. As the planet orbits, the star will then execute a much smaller orbit around the center of mass,

The idea of the radial velocity method is to indirectly detect the planet by observing the reflex motion of the star. This can be done by measuring the line of sight (“radial”) component of the stellar motion by observing the Doppler shift of spectral lines produced in the atmosphere of the star. We can estimate the size of the effect by calculating the orbital velocity of the star as it executes its small orbit about the center of mass. We first note that the orbital velocity of a planet on a circular orbit at distance from a star of mass is,

.

Conservation of linear momentum in the binary implies that when the planet is moving toward us with this velocity, the star must be moving away at a velocity such that the star and the planet have equal and opposite momenta, i.e.

.

Substituting, we get the stellar velocity as,

.

This would be the maximum radial velocity we would see if the orbit was exactly edge-on to us, at the moment when the star was coming directly toward us. If instead, the orbit is inclined at an angle (as shown in the figure above) we see only the component of the stellar orbital velocity that is projected along our line of sight. The semi-amplitude of the radial velocity curve is then,

.

Substituting numbers, we find that the radial velocity signature that the planets in the Solar System induce on the Sun is,

(Jupiter)

(Earth).

Currently, the most precise measurements of stellar radial velocities attain roughly 1 m/s precision. A wide range of planets can be detected via this technique, though it is not yet possible to find Earths within the habitable zone of Solar-type stars.

The basic observables of a radial velocity survey are,

• The period of the orbit. As with a transit survey, this gives us the semi-major axis of the planetary orbit provided we have good knowledge of the stellar mass.
• The semi-amplitude of the radial velocity oscillation . If we know the orbital inclination , then together with the stellar mass this gives us the planetary mass. In the more normal case where we don’t know the inclination, we can only deduce a minimum mass for the planet.
• The time-dependence of the radial velocity oscillation. If the orbit of the planet is circular this will be a simple sinusoid. For larger eccentricities, the curve will look more like a sawtooth, reflecting the fact that both the planet and the star are moving fastest during the brief period of closest approach. Some examples of curves for eccentric orbits are shown below,

  

  By fitting the shape of such curves, the eccentricity can be measured.

Transits and radial velocity measurements furnish independent information about extrasolar planetary systems. The transit method gives the most direct information on planetary sizes, while the radial velocity method gives information about planetary masses. If, fortuitously, we can observe transits and measure the radial velocity signal for the same planet, then combining the information gives the mass, the radius, and hence the density.