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.