8.3 Formation and evolution of protoplanetary disks

Formation and Structure of Protoplanetary Disks

Protoplanetary disks are the birthplaces of planets. They form around young stars from the remnants of collapsing molecular clouds, and their internal structure dictates what kinds of planets can form and where. Temperature gradients, density profiles, and dust dynamics all play roles in shaping the planetary systems that eventually emerge.

This section covers how these disks form and evolve, how dust grows into planetesimals, and what observational techniques reveal about disk structure.

Structure of protoplanetary disks

A protoplanetary disk forms as a natural consequence of angular momentum conservation during the collapse of a molecular cloud core. Here's the sequence:

1. A region of a molecular cloud becomes gravitationally unstable and begins to collapse.
2. As the cloud contracts, it spins faster (conservation of angular momentum, just like a figure skater pulling in their arms).
3. Material along the rotation axis falls freely onto the protostar, but material in the equatorial plane is supported by centrifugal force. The result is a flattened, rotating disk.

This process is fast. Disk formation takes roughly $10^4$ to $10^5$ years, but the disk itself persists for $1$ to $10$ million years before dissipating through accretion, photoevaporation, and planet formation.

Radial structure. The disk has a strong radial temperature gradient, hottest near the star and cooling outward. The inner disk (within a few AU) contains hot gas and refractory dust, the region where rocky planets like Mercury and Venus form in our solar system. The outer disk (beyond the snow line, roughly $\sim 3$ AU for a solar-type star) is cold enough for volatile ices like $H_2O$ and $CO$ to condense, providing the extra solid mass needed to build giant planet cores like Jupiter and Saturn.

Vertical structure. The disk isn't a flat sheet. It has a dense, thin midplane where most of the dust settles, surrounded by a more extended gaseous atmosphere. Density drops off steeply with height above the midplane.

Density profile. Surface density generally follows a power-law distribution, $\Sigma(r) \propto r^{-p}$, with $p$ typically between 0.5 and 1.5. The inner regions are significantly denser than the outer regions.

Instabilities in disk evolution

Disks aren't static. Several instabilities drive their evolution by transporting angular momentum, redistributing mass, and sometimes even forming planets directly.

Gravitational instability is quantified by the Toomre parameter:

$Q = \frac{c_s \, \Omega}{\pi G \Sigma}$

where $c_s$ is the sound speed, $\Omega$ is the orbital frequency, $G$ is the gravitational constant, and $\Sigma$ is the surface density. When $Q < 1$, the disk is unstable to its own self-gravity. This can produce spiral arms and density fluctuations. In extreme cases (very massive, rapidly cooling disks), the disk can fragment directly into bound clumps, potentially forming gas giant planets on wide orbits. Note that $Q \approx 1$ marks marginal stability; most disks hover near this value in their outer regions.

Magnetorotational instability (MRI) is the dominant source of turbulence in regions where the gas is sufficiently ionized (typically the inner disk and disk surface layers). MRI generates turbulence that acts as an effective viscosity, enabling angular momentum to be transported outward. This allows mass to accrete inward onto the star. In poorly ionized regions near the midplane (the "dead zone"), MRI is suppressed, and other mechanisms take over.

Other turbulence sources include:

• Vertical shear instability (VSI), which arises from the radial temperature gradient in the disk
• Convective overstability, driven by buoyancy oscillations in regions with radial entropy gradients

All of these instabilities contribute to angular momentum transport, which is the fundamental process driving disk evolution. As angular momentum moves outward, mass moves inward, feeding the star and gradually depleting the disk. The turbulence also generates heating, which affects dust sublimation, ice line locations, and disk chemistry.

Dust Dynamics and Observational Signatures

Dust growth and planetesimal formation

Planets start as microscopic dust grains (sub-micron sized) embedded in the gas disk. Getting from dust to planets requires growth across roughly 13 orders of magnitude in size, and several barriers make this surprisingly difficult.

Early growth: coagulation. Dust grains collide and stick together through van der Waals forces and surface charges. Whether grains stick depends on their composition (icy grains are stickier than silicates) and their relative collision velocity (low-speed collisions favor sticking).

The bouncing barrier. Once particles reach roughly millimeter sizes, collisions tend to result in bouncing rather than sticking. Growth stalls at this stage for compact grains.

The fragmentation barrier. At higher collision velocities (typically above $\sim 1 \, \text{m/s}$ for silicates, $\sim 10 \, \text{m/s}$ for ices), particles shatter instead of growing. This limits how large grains can get through direct collisional growth alone.

The radial drift problem. This is one of the most serious challenges. Gas in the disk orbits slightly slower than the Keplerian speed because it's partially pressure-supported. Solid particles, which don't feel pressure support, experience a headwind from the gas. This drag causes them to lose angular momentum and spiral inward. The effect is strongest for meter-sized bodies, which can drift into the star in as little as $\sim 100$ years at 1 AU. This is sometimes called the "meter-size barrier."

Dust settling. Gravity pulls dust grains toward the midplane, while turbulence lofts them back up. The balance between these determines the dust scale height. A thinner dust layer means higher midplane dust densities, which increases collision rates and sets the stage for collective gravitational effects.

Planetesimal formation via streaming instability. The leading model for bypassing the growth barriers is the streaming instability. When the local dust-to-gas ratio approaches or exceeds unity (which can happen through settling and radial drift pile-ups), the dust and gas interact in a way that spontaneously concentrates particles into dense filaments. These filaments can become gravitationally bound and collapse directly into planetesimals of $\sim 10$ to $100 \, \text{km}$ in size, skipping the problematic intermediate sizes entirely.

Observational signatures of disks

We can't watch planet formation in real time, but several observational techniques reveal the structure and evolution of protoplanetary disks.

Spectral energy distributions (SEDs). Dust in the disk absorbs starlight and re-emits it thermally at infrared wavelengths. The shape of the SED tells you about the dust temperature distribution and, by extension, the disk geometry. Excess infrared emission above the stellar photosphere is the classic signature of a circumstellar disk.

Molecular line emission. Gas in the disk emits at specific frequencies corresponding to molecular transitions:

• CO rotational lines are the most commonly used tracers of gas temperature, density, and kinematics (rotation curves)
• Water vapor lines help locate snow lines and probe disk chemistry in the planet-forming region

High-resolution imaging. Two techniques have transformed the field:

• Submillimeter interferometry (ALMA) resolves thermal dust emission at angular resolutions below 0.1 arcseconds, revealing detailed disk structure
• Scattered light imaging with adaptive optics traces small dust grains in the disk surface layers at near-infrared wavelengths

Disk substructures. ALMA observations have revealed that substructure is ubiquitous in protoplanetary disks, even around very young stars. The main features include:

• Gaps and rings, which may be carved by forming planets, or produced by snow lines and dust evolution processes
• Spiral arms, which can indicate gravitational instability or gravitational perturbation by a massive companion
• Azimuthal asymmetries, which may trace dust-trapping vortices or eccentric disk features

Variability. Young stars with disks often show brightness variations on timescales of days to years. These can result from:

• Unsteady accretion onto the star (changes in the inner disk)
• Shadowing of the outer disk by warps or misalignments in the inner disk structure