Past exam of the exoplanets and Planetary Systems course of the University of Cambridge / 2015 / Paper 1 / 3

A sufficiently self-gravitating disk can fragment directly, bypassing the slow assembly of a massive solid core. In the simplest thin gas-disk model the competition is between self-gravity, pressure and rotational shear. Axisymmetric perturbations havewhere $\kappa$ is the radial epicyclic frequency, not the opacity. In a nearly Keplerian disk, $\kappa \simeq \Omega$. This is Toomre's stability criterion; finite thickness and global/nonaxisymmetric modes modify the precise boundary. The fastest local wavelength is $\lambda =2c_s^2/(G\Sigma )$, of order the disk scale height times $2\pi$ when $Q_T\simeq 1$.

Instability alone does not ensure bound planets. Compression and shocks heat the gas, raising $Q_T$; a slowly cooling disk can maintain self-regulated spiral turbulence. Cooling-limited disk fragmentation requires energy loss on a sufficiently short dynamical timescale, conventionally expressed as $\beta =\Omega t_{\mathrm{cool}}$ below a model-dependent threshold. The often quoted $\beta \sim 3$ belongs to a particular local razor-thin calculation, not every real disk.^[ref] Cold massive outer disks are promising sites, but fragment contraction, tidal survival, further accretion and migration determine whether the result remains a planetary-mass body or grows into a brown dwarf.

For gravitational collapse of a settled particle layer, if “core collapse” instead denotes a solid-particle layer, its random-velocity dispersion replaces gas sound speed in an appropriate particle-disk stability condition. Settling and concentration can enable collapse into planetesimals; gas turbulence and particle stirring oppose it. This route makes solid seeds for later core accretion and is physically distinct from direct gas fragmentation. In both cases one must establish self-gravity against velocity support and tides, rather than infer collapse from high surface density alone.

The Hill radius compares planetary gravity with the differential stellar tide. Expanding the stellar tide near a circular orbit gives $G{M}_p/r_H^2\simeq 3{\Omega }^2{r}_H$, soIt sets the size of the planet's tidal neighbourhood, controls encounter and feeding zone scales, and limits the region within which material can remain bound against stellar shear. It is not itself a solid surface or a guarantee of capture.

The Bondi radius compares planetary potential energy per unit mass with the gas thermal scale,This convention uses the ambient sound speed; definitions based on escape speed can differ by a factor of two. Inside $r_B$, gravity can substantially perturb thermal gas. Gas still needs a way to dissipate energy and angular momentum to accrete; a geometric boundary alone does not supply a gas-capture rate.

For an embedded hydrostatic envelope a useful outer scale is $r_{\mathrm{out}}\sim \min (r_B,r_H)$, subject to disk thickness and gas recycling. Let $h=H/a=c_s/(a\Omega )$ and $q=M_p/M_{\star }$. The Bondi-Hill crossover in a protoplanetary disk obeysSmall cores are usually thermally limited, whereas larger ones become tidally limited. These radii also organize pebble accretion: gas drag can turn an encounter into capture, with drift-dominated and shear-dominated regimes depending on particle stopping and encounter times. There is no universal accretion efficiency fixed by $r_H$ or $r_B$ alone.^[ref]

Type I migration is orbital angular-momentum exchange between an embedded low-mass planet and its disk without a deep gap. The planet excites spiral density waves near Lindblad resonances; the disk's reaction torques act back on the planet. The outer wake usually supplies a negative torque and the inner wake a positive one, with their small imbalance often driving inward migration. Gas on horseshoe orbits in a planetary disk near corotation adds a corotation torque that depends on entropy and vortensity gradients.

For a thin disk, define $q=M_p/M_{\star }$, $h=H/a$ andThe coefficient $C$ is signed and depends on disk gradients, thermodynamics and diffusion; it is not a universal constant. For a circular orbit with fixed planet mass, $L_p=M_p{a}^2\Omega \propto a^{1/2}$. The Type I migration timescale from disk torque follows from ${\dot{L}}_p=\Gamma$:Thus a more massive embedded core generally migrates faster within this regime, unlike a picture in which migration starts only after giant-planet formation.

In a simple locally isothermal smooth disk the net torque is commonly negative, potentially removing cores before they acquire gas. In a non-isothermal disk an unsaturated entropy-related corotation torque can balance or exceed the negative wave torque, giving a planet trap or outward migration. Viscosity and thermal diffusion help maintain the background gradients across the horseshoe region; without adequate replenishment corotation-torque saturation can suppress that contribution. The linear low-mass scaling requires a small disturbance, roughly $q\ll h^3$; larger planets, gap formation and strong disk feedback require a different treatment. Type I migration is therefore neither universally inward nor literally viscous advection of the planet.^[ref]

In transit spectroscopy, starlight passes tangentially through the planet's terminator atmosphere during primary transit. Wavelength-dependent opacity changes the effective occulting radius. For a thin atmosphere with an absorption feature spanning $N_H$ atmospheric scale heights,Here $\mu$ is the mean particle mass in units of $m_H$, not a chemical potential. Large planets with warm low-gravity, low-molecular-weight atmospheres against small stars have larger signals. A spherical isothermal atmosphere has tangential column enhancement of order $\sqrt{2\pi R_p/H}$ over a vertical column, explaining sensitivity to tenuous high-altitude gas. Molecular bands, atomic lines, scattering slopes and clouds constrain the terminator, but cloud opacity, reference pressure and abundance-temperature degeneracies limit unique retrievals.^[ref]

Occultation spectroscopy instead subtracts the star-only spectrum during secondary eclipse from the star-plus-planet spectrum immediately outside eclipse. It measures the dayside planet/star flux ratio. In the blackbody approximation the thermal eclipse depth isLonger infrared wavelengths generally improve contrast for a cooler planet; molecular absorption or emission depends on the vertical temperature gradient. At optical wavelengths reflected light can dominate, with full-phase contrast $A_g(\lambda )(R_p/a{)}^2$, where $A_g$ is geometric albedo. Eclipse spectra therefore probe dayside emission or reflection, rather than the transit's slant absorption. Clouds and temperature inversions affect interpretation.^[ref]

For either measurement, the photon-limited signal-to-noise ratio scales as the spectral feature amplitude times $\sqrt{N_{\gamma }}$, with a penalty for subtracting finite in- and out-of-event samples. Finer spectral bins reduce photons per bin; bright hosts, repeated events and stable instruments improve sensitivity. Detector systematics, telluric absorption for ground observations, and variable starspots can limit the practical gain. Space observations remove telluric contamination, not stellar variability or instrumental drift. Neither technique directly measures a unique bulk composition or samples all longitudes of the planet.