Past exam of the exoplanets and Planetary Systems course of the University of Cambridge / 2016 / Paper 1 / 3

A protoplanetary disk is a flattened, rotating reservoir of gas and solids around a young star. Its radial gravity is usually dominated by the host, making its rotation close to Keplerian rotation, ${\Omega }_K=(G{M}_{\star }/r^3{)}^{1/2}$. Gas pressure provides vertical support and a small radial correction to the orbital speed; it does not make the disk a nonrotating spherical atmosphere.

At height $|z|\ll r$, vertical hydrostatic equilibrium in the stellar gravitational field givesFor a locally vertically isothermal ideal gas, $P_g={\rho }_g{c}_s^2$ with $c_s^2=k_{B}T/(\mu m_H)$, where $\mu$ is mean molecular weight. Integration givesHere ${\Sigma }_g$ is the vertically integrated gas surface density of a disk, and $H$ is the isothermal disk scale height. The disk aspect ratio $H/r=c_s/v_K$ is small in a geometrically thin disk. For $T\propto r^{-q}$ it scales as $r^{(1-q)/2}$, so $q<1$ produces disk flaring. This illustrative local vertical model does not determine a unique radial surface-density law or temperature normalization from the information given.

The gas is predominantly molecular hydrogen and helium. A much smaller component consists of heavier elements, partitioned between vapor, solid refractory grains and condensed volatile ices. A gas-to-solid mass ratio of order $100$ is a common initial fiducial model, not a universal measurement applying at every disk radius or evolutionary stage. Solids can grow and settle while gas accretes or disperses, changing the local ratio substantially.^[ref]

Thermal structure organizes composition. In the hottest inner region, grains can sublimate and much of their material remains in the gas. Refractory silicate minerals and metallic grains survive at lower temperatures; farther out, water and then other sufficiently volatile species can condense when their local condensation conditions are met. The water snow line can raise the solid inventory available to planet formation. Its location depends on stellar irradiation, accretion heating, opacity and gas pressure, and evolves with time; it is not fixed merely by naming a star or adopting a solar-system distance.

Stellar radiation heats the exposed surface of a passive disk. Beneath it, shielding can leave a cooler, molecule-rich interior with a higher mass density. In the inner actively accreting disk, dissipation can instead make the optically thick midplane the hottest region. These are distinct heating regimes; a vertically isothermal profile is an analytic approximation rather than a claim that every real layer has the same temperature. Dust opacity controls how radiation is absorbed and re-emitted. Infrared and millimetre emission probe solid material, while spectral lines probe gas; neither observable alone supplies a model-free inventory of the whole disk.

Dust settling in an astrophysical disk makes larger, less tightly coupled particles more concentrated toward the midplane, while turbulence and mixing oppose that settling. For an aerodynamic stopping time $t_s$ short compared with ${\Omega }_K^{-1}$, terminal vertical balance gives $v_z\simeq -{\Omega }_K^2{t}_{s}z$. Small grains remain better coupled to the gas. Radial drift also changes the distribution of solids and their access to snow lines and embryos.

The disk evolves through accretion and angular momentum transport, potentially involving turbulent stresses, magnetorotational instability or magnetically driven winds where conditions permit. Protoplanetary disk dispersal eventually removes gas through accretion and mass loss, including photoevaporation. In standard evolutionary pictures this occurs on a scale of a few million years, with substantial system-to-system variation. That finite reservoir lifetime is central to whether an embryo acquires a massive gaseous envelope.^[ref]

The essential structure is a nearly Keplerian gas disk with vertical pressure support, radial and vertical thermal gradients, and an evolving solid population concentrated by settling, growth and drift. Its composition is set jointly by the original elemental inventory, local condensation conditions and transport.

Sub-Keplerian rotation means that the gas rotates more slowly than a pressure-free particle on a circular orbit at the same cylindrical radius. Let $v_K=(G{M}_{\star }/r{)}^{1/2}$. Neglecting radial acceleration, radial force balance in the gas isAn outwardly decreasing pressure, $d{P}_g/dr<0$, supplies part of the inward-gravity balance through an outward pressure force. Less centrifugal support is needed, so $v_{g\varphi }<v_K$. Using $c_s^2=P_g/{\rho }_g$, define the pressure-gradient parameter of a protoplanetary disk byThenFor a thin disk and a smooth radial pressure profile, $\eta$ is small, of order $(H/r{)}^2$. The factor of two is part of this explicit convention; changing the convention for $\eta$ changes the drift formula correspondingly.

A solid particle feels essentially no continuum gas-pressure support. Left alone, it would orbit at $v_K$, so in the gas it feels a headwind. Drag removes angular momentum, causing radial drift of solids in a gas disk. Define Stokes number $\mathrm{St}={\Omega }_K{t}_s$ using the aerodynamic stopping time $t_s$. In a dilute test-particle limit with linear drag, negligible radial gas flow and small departures from a circular Keplerian orbit, let $u_{\varphi }=v_{p\varphi }-v_K$. The steady local radial and azimuthal balances areSolving givesThe sign is inward for the usual negative pressure gradient. Very small particles, $\mathrm{St}\ll 1$, closely follow the gas and have a small relative drift $v_{pr}\simeq -2\eta v_K\mathrm{St}$. Very weakly coupled large bodies, $\mathrm{St}\gg 1$, also drift slowly, with $v_{pr}\simeq -2\eta v_K/\mathrm{St}$. The fastest pressure-driven inward drift occurs at $\mathrm{St}=1$, where $|v_{pr}|=\eta v_K$. The corresponding physical grain or aggregate size depends on gas density, orbital radius and the drag regime; “metre-size barrier” is a conventional description of some disk models, not a universal size.

The drift timescale in this model isIt can be much shorter than the disk gas lifetime, threatening loss of the material needed to make planetesimals. Growth through the vulnerable coupling range must compete with this loss, while differential drift generates collisions that may stick, bounce or fragment. Radial redistribution also changes solid-to-gas ratios and moves material across condensation fronts. These are consequences for planet formation, rather than merely a small correction to an orbital frequency.

The same mechanism can help concentrate solids. At a local radial pressure maximum, $d{P}_g/dr=0$, so this component of drift vanishes. On the inside, a positive pressure gradient gives outward drift; on the outside, a negative gradient gives inward drift. Particles therefore converge toward a dust trap. Turbulent diffusion and inward gas advection can limit retention. With a nonzero radial gas speed $v_{gr}$, the test-particle result generalizes to $v_{pr}=[v_{gr}-2\eta v_K\mathrm{St}]/(1+{\mathrm{St}}^2)$, so gas advection need not vanish even at a pressure maximum.

Finally, if local solid loading becomes large, backreaction on the gas can no longer be neglected. Coupled gas-particle drift can support streaming instability and strong solid concentrations, potentially enabling gravitational collapse into planetesimals. The simple test-particle drift formula is a useful starting point, but not a description of that nonlinear high-loading regime. Pressure support creates both a drift obstacle to growth and opportunities for concentration where the pressure structure changes.

Oligarchic growth is the stage in which a relatively small number of planetary embryos dominate the dynamics of many smaller planetesimals. It follows runaway planetesimal growth. The embryos remain separated, grow to broadly comparable masses within similar local disk conditions, and regulate their accretion by stirring the population that supplies their material. The primary numerical study describes this embryo–planetesimal structure and typical separations of about ten individual Hill radii.^[ref]

The key feedback is gravitational focusing. In a dispersion-dominated encounter with approach speed $v$ and a small impactor, conservation of energy and conservation of angular momentum give the collision cross-sectionAt fixed internal mass density, $R\propto M^{1/3}$ and $v_{\mathrm{esc}}^2\propto M^{2/3}$. If the background random speed initially is controlled by small planetesimals and varies little with embryo mass, strong focusing gives ${\sigma }_{\mathrm{coll}}\propto M^{4/3}$. For a disk with planetesimal thickness of order $v_z/{\Omega }_K$ and a fixed ratio $v/v_z$, the associated estimate isup to geometric factors. With nearly fixed $v$, this gives $\dot{M}\propto M^{4/3}$ and $\dot{M}/M\propto M^{1/3}$: the largest bodies have the shortest fractional growth times, the essence of runaway growth.

As embryos grow, their gravitational encounters increasingly excite the random eccentricities and inclinations of nearby planetesimals. This is viscous stirring of planetesimals, not molecular fluid viscosity. Higher encounter speeds lower the focusing factor. The largest local embryo heats its feeding population most strongly and thereby suppresses its own relative growth advantage. Gas drag and dissipative collisions oppose stirring; dynamical friction tends to keep the more massive embryos' random motions relatively small while transferring random energy to lighter bodies. The balance between these processes controls the accretion rate.

For an illustrative self-regulated scaling $v\propto M^{1/3}$, the ratio $v_{\mathrm{esc}}/v$ is approximately fixed. The previous growth estimate becomesLarger embryos now grow more slowly in fractional terms, allowing neighboring embryos in comparable environments to approach similar masses. This scaling illustrates the feedback; it is not a universal exact exponent for every planetesimal size distribution, damping law or disk surface-density profile. In particular, a two-body focusing cross-section must be modified in the shear-dominated regime, where encounters are governed by the stellar tidal field.

The relevant gravitational interaction scale is the Hill radius,Close embryo encounters and orbital repulsion keep adjacent embryos separated by several Hill-scale lengths. The mutual Hill radius uses their combined mass; for equal neighboring masses it is $2^{1/3}{R}_H$. Stating which definition is used matters when quoting a spacing coefficient. The spacing is a dynamical scale, not an exact universal integer.

Each feeding zone of a planetary embryo contains a finite solid reservoir. To estimate its isolation mass, let the full annular width be $\Delta a=B{R}_H$, where the dimensionless coefficient $B$ includes the stated choice of width and Hill-radius convention. For local solid surface density ${\Sigma }_s$,Solving givesThus denser solid disks and wider feeding annuli allow larger embryos. If ${\Sigma }_s\propto a^{-p}$, the estimate scales as $M_{\mathrm{iso}}\propto a^{3-3p/2}$ at fixed host mass and $B$. This is a local exhaustion estimate, not an absolute maximum planet mass: radial delivery of solids, pebble accretion, migration and embryo mergers can replenish or enlarge a feeding zone. No solid surface-density normalization is given, so a numerical isolation mass would require an invented disk model.

As the local supply is depleted, growth slows further. After gas damping weakens, embryo orbits may cross and giant collisions can assemble larger terrestrial planets. Farther out, sufficiently massive cores may instead acquire gas before protoplanetary disk dispersal. Oligarchic growth explains why a few comparable embryos share a disk while many smaller planetesimals remain: gravitational stirring weakens focusing and makes accretion self-regulating.