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The electron-acoustic recoil length compares electronic recoil with the acoustic quantum energy associated with the same momentum. For transferred wave number $q$,The dispersions cross at $q=1/\lambda$; there the reduced electron de Broglie wavelength is $\lambda$ and its free-particle group velocity is $2c$. Thus $\lambda$ is the length scale at which an electron's localization/recoil energy and a corresponding acoustic excitation energy become comparable. At $q\lambda \gg 1$ the electron recoil energy is large compared with the phonon energy; this explains the separation of electronic and acoustic energy scales when lattice wave numbers exceed $1/\lambda$.

Formally it is half the Reduced Compton wavelength with the speed of light replaced by the speed of sound. That is an analogy of scales, not a relativistic electron mass or a pair-production threshold: sound does not replace the limiting speed in electron dynamics. Also it is a reduced-wavelength convention; an ordinary de Broglie wavelength $2\pi /q$ at the crossing is $2\pi \lambda$. No scattering lifetime appears, so $\lambda$ is not an electron mean free path.

An intrinsic semiconductor has its carrier populations set primarily by thermal excitation across the band gap; in the ideal pure, neutral case its conduction-electron and hole densities are equal, $n=p=n_i$. An extrinsic semiconductor has carrier populations significantly controlled by donor or acceptor impurities, producing predominantly n-type semiconductor or p-type semiconductor behavior. Chemical purity alone does not fix conductivity independently of temperature and gap size.

The positive charge carrier mobility is the magnitude of mean drift speed per unit weak electric field. For electrons ${\mathbf{v}}_e=-{\mu }_e\mathbf{E}$, for holes ${\mathbf{v}}_h={\mu }_h\mathbf{E}$, and its SI unit is ${\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. In an isotropic relaxation-time model, $\mu =|q|\tau /m^{\ast }$.

A band effective mass characterizes how an electron wave packet accelerates in an energy band, rather than replacing its vacuum rest mass. In one dimension and its tensor generalization,These follow by differentiating $v_i={\hslash }^{-1}\partial E/\partial k_i$ and using $\hslash {\dot{k}}_j=F_j$. Effective masses can be anisotropic and electron band curvature can be negative.

An electron hole is a missing electron in an otherwise almost filled valence band. A completely filled band has no net transport current: its velocities cancel over the Brillouin zone. Describing a few missing states as mobile excitations is therefore simpler than following almost every occupied electron state.

Near an isotropic valence maximum, write the electron dispersion as $E_v(\mathbf{k})=E_v(0)-{\hslash }^2{k}^2/(2m_h)$ with $m_h>0$. The electron curvature mass is negative there. Removing that electron creates an excitation with momentum $\hslash {\mathbf{k}}_h=-\hslash \mathbf{k}$ and energy measured upward from the filled reference,ThusThe sign refers to the positive excitation energy of a missing electron near a maximum, not to negative vacuum mass. A general maximum gives the corresponding positive hole-curvature tensor.^[ref]

In the simple transport model, ${\mu }_e=e{\tau }_e/m_e^{\ast }$ and ${\mu }_h=e{\tau }_h/m_h^{\ast }$. Conduction bands in many common semiconductors have lighter transport masses than the dominant valence hole bands. For comparable momentum-relaxation times, this gives larger electron mobility. It is a common trend, not a theorem: band anisotropy, light and heavy hole populations and different scattering times can alter the comparison. The hole's mass is not the mass of an atom, and crystal translational symmetry does not make its mobility infinite.

For the intrinsic germanium sample, electrons and holes both contribute positive conventional conductivity:ThereforeThis is the density of each species; the combined carrier number density is $2n_i$. Assigning separate densities from the two mobilities would violate intrinsic charge neutrality.

Take one conduction electron per atom and a three-dimensional free electron gas, with spin degeneracy two and electron mass $m_e$. State counting inside the occupied Fermi sphere givesThe highest occupied energy at zero temperature is $E_F={\hslash }^2{k}_F^2/(2m_e)$, soThis relates the atomic number density to the Fermi temperature only because monovalency and the free-electron dispersion have been assumed. A real energy band can require a different density of states and band effective mass.

For the isotropic Debye model, count three acoustic polarizations up to one spherical wavevector cutoff $q_D$. The number of vibrational modes per volume isWith $\omega =cq$ and $\hslash {\omega }_D=k_B{\theta }_D$,The effective $c$ is the Debye sound speed: matching the low-frequency mode density of one longitudinal and two transverse branches requires $3/c^3=1/c_L^3+2/c_T^3$. It is not generally the arithmetic average of their speeds. The cutoff preserves the mode count; it need not coincide with the actual nonspherical Brillouin zone boundary.

Assume nondegenerate three-dimensional parabolic band edges, a fixed gap and approximately temperature-independent density-of-states effective masses. Let $\mu$ be the chemical potential, $E_c$ and $E_v$ the band edges, and $N_c,N_v$ their effective density of states. Boltzmann occupation givesFor one spin-degenerate spherical conduction extremum, direct integration of $g_c(E)\propto \sqrt{E-E_c}$ givesand similarly for the valence band with its appropriate density-of-states mass. Valley and additional band degeneracies can be included in these prefactors. The $T^{3/2}$ factor follows from the integral ${\int }_0^{\infty }\sqrt{ϵ}{e}^{-ϵ/(k_{B}T)}dϵ\propto T^{3/2}$.

Multiplication removes the chemical potential:The semiconductor mass-action law combined with intrinsic charge neutrality, $n=p$, therefore givesThe intrinsic chemical potential is not necessarily exactly midgap: ${\mu }_i=(E_c+E_v)/2+(k_{B}T/2)\ln (N_v/N_c)$. Equal effective masses are unnecessary for the displayed carrier-density law. A changing gap, effective mass or degeneracy would make the prefactor or activation law temperature dependent.

The sign must become positive, so use p-type acceptor impurities, such as boron substituting for tetravalent silicon. Assume uncompensated, fully ionized acceptors of density $N_A$, nondegenerate carriers and equilibrium throughout the interval. Charge neutrality and mass action giveFor a nearly constant hole-dominated Hall coefficient, the dopant plateau must persist up to the highest temperature. Its necessary concentration scale is thereforeThe order-of-magnitude onset is $N_A\sim 4\times 10^{20}{\mathrm{m}}^{-3}$, but equality is not a flat plateau: at $N_A=n_i(500)$, $p/N_A=(1+\sqrt{5})/2=1.618$, changing the hole-only Hall coefficient by about 38 percent. The word approximately does not define a unique sharp minimum without a tolerance.

For $N_A\gg n_i$, $p\simeq N_A[1+(n_i/N_A{)}^2]$, so the hole-only $R_H\simeq 1/(e{N}_A)$ varies by a fraction of order $(n_i/N_A{)}^2$. An illustrative one-percent tolerance in that approximation is obtained with $N_A\simeq 10n_i(500)=4.0\times 10^{21}{\mathrm{m}}^{-3}$. This gives $R_H\simeq +1.56\times 10^{-3}{\mathrm{m}}^3{\mathrm{C}}^{-1}$ and less than one percent hole-density-induced variation. It is an explicit plateau criterion, not an exact minimum deduced from the unspecified word approximately.

At 300 K the minority electron concentration isThus the onset-scale estimate gives about $2.5\times 10^{11}{\mathrm{m}}^{-3}$, whereas the explicit ten-times-larger plateau choice gives $2.5\times 10^{10}{\mathrm{m}}^{-3}$.

The acceptor density for a stable Hall sign can be more restrictive if both carriers matter. With mobility ratio $r_{\mu }={\mu }_e/{\mu }_h>1$, positive Hall sign requires $p>r_{\mu }{n}_i$, or $N_A>n_i(r_{\mu }-r_{\mu }^{-1})$. In the strongly extrinsic limit the full two-carrier Hall correction scales as $(1+r_{\mu }{)}^2(n_i/N_A{)}^2$, not just $(n_i/N_A{)}^2$. Hence a true measured-Hall tolerance can require a larger dopant density than the hole-only illustration. Mobilities and their temperature dependence are not supplied, so they cannot be used to manufacture a unique sharper minimum.

The mass per lead atom isLet $u_n$ be its longitudinal displacement. Newton's second law gives the harmonic chain equationInsert the normal mode $u_n=u{e}^{i(kna-\omega t)}$ to obtainAt long wavelength, $\omega \simeq a\sqrt{\kappa /m}|k|$. The sound speed is the corresponding low-frequency group velocity:The relevant inertia in this chain equation is one atomic mass. The reduced mass $m/2$ belongs to the relative vibration of an isolated pair and should not be substituted for $m$ in the chain's dispersion relation.

Quantum statistics matter when thermal energy is comparable with a characteristic vibrational level spacing. The chain's natural angular frequency is ${\omega }_0=\sqrt{\kappa /m}=5.7471\times 10^{12}{\mathrm{s}}^{-1}$, givingFactors of order unity depend on which vibrational scale is used. The largest chain phonon angular frequency is $2{\omega }_0$, corresponding to $87.8\mathrm{K}$. An isolated two-atom bond has reduced mass $m/2$ and angular frequency $\sqrt{2}{\omega }_0$, corresponding to $62.1\mathrm{K}$. Thus quantum corrections become appreciable on the scale of several tens of kelvin, and all modes are classical only when $T$ is comfortably above the largest relevant vibrational scale. Below it, high-frequency phonons are progressively frozen out and equipartition fails.

For the isolated bond, the zero-temperature root-mean-square relative displacement isIt is small enough for a local bond expansion. The classical weakly anharmonic estimate in part d therefore has a useful parametric windowwhere the lower condition means comfortably above the vibrational scale, rather than a sharp threshold. The model's upper bound concerns local anharmonicity and binding, not a prediction of the melting temperature.

In the diamond cubic structure of silicon, the longitudinal acoustic phonon branch and longitudinal optical phonon branch meet at $X=(2\pi /a,0,0)$. The two atomic basis sites are identical, and the associated crystal symmetry protects their boundary degeneracy of normal modes. In aluminium phosphide, the two sites contain different chemical atoms. The corresponding zinc blende structure lacks the site-exchange symmetry, and these two longitudinal branches generally split at $X$.

A simple way to see the splitting is a diatomic-chain dispersion relation with equal springs, representing successive $[100]$ atomic planes by alternating masses $M_1,M_2$ at spacing $d=a/4$. With effective spring constant $K$, its angular frequencies obeyAt $X$, $kd=\pi /2$, so the inter-sublattice coupling cancels and the two squared angular frequencies are $2K/M_1$ and $2K/M_2$. Equal masses give a contact; unequal masses giveIn aluminium phosphide, the heavier phosphorus plane moves in the lower-frequency edge mode, and the lighter aluminium plane in the higher-frequency mode in this model. Their similar atomic masses explain why the splitting is relatively small. Different electronic environments can also change the force constant matrix. The chain calculation illustrates the mechanism; a quantitative three-dimensional phonon spectrum requires its full dynamical matrix. It does not assert that all transverse acoustic and optical branches meet in silicon.

For the degenerate electron gas, let $D_g=Am/(\pi {\hslash }^2)$ include both spin states. The Sommerfeld expansion givesFor the gapless symmetric intrinsic semiconductor, part (d) gave the same coefficient with $D_s=A{m}^{\ast }/(\pi {\hslash }^2)$. Thus for equal sample areas and equal spin and valley degeneracies,The equality often intended in this comparison requires the same density of states. The source sets $m_e^{\ast }=m_h^{\ast }$ but does not explicitly equate them to the electron gas mass $m$. Without that additional assumption the ratio is $m^{\ast }/m$, or more generally $D_s/D_g$ if sample areas or degeneracy factors differ. For example, $m^{\ast }=2m$ with equal areas satisfies the stated equal-effective mass condition in the semiconductor and gives ratio two. The filled Fermi sea does not add a factor $N_e/N_c$: only the thermal shell near its Fermi energy contributes to its heat capacity.

Integrate the conservation law over $[-R,R]$: $d{\int }_{-R}^R\rho dx/dt=-t[{\rho }^2{]}_{-R}^R$. The Gaussian data and their finite-time characteristic/entropy evolution have vanishing far-field flux. Taking $R\to \infty$ yields $\boxed{M^{\prime }(t)=0}$ and hence $\boxed{M(t)=\sqrt{2\pi }}$. After a shock forms, its Rankine-Hugoniot condition preserves the same integral mass; the formal multivalued characteristic surface itself must be replaced by the conservative entropy solution.

Before intersection, the characteristic map is $x=\xi +{\rho }_0(\xi {)}^{3}t$, with derivative $1+3{\rho }_0^2{\rho }_0^{\prime }t$. Because the initial profile decreases, higher-density characteristics behind move faster than lower-density characteristics ahead. Compression therefore forms shocks, which absorb more characteristics. The far states remain one and zero, so a shock joining their limiting traces has speed tending to $[q]/[\rho ]=1/4$.

The precise large-scale interpretation uses $\zeta =x/t$. The rescaled initial transition tends to a step, and entropy comparison for the convex flux givesOne way to see the comparison is to choose distant initial points beyond which the profile is within $\delta$ of each far state. Step profiles with those nearby states bracket it; their shock speeds tend to $1/4$ as $\delta \downarrow 0$, while their fixed initial locations disappear after division by $t$.

A fixed translation of the shock can persist. If the deviation from a reference step is integrable, its signed mass fixes a phase shift. Without such a tail assumption one should assert convergence on the $x/t$ scale and $S(t)/t\to 1/4$, rather than claim that the exact line $x=t/4$ is the unshifted asymptote for every smooth initial transition.

Mass preservation of a nonnegative finite-mass profile requires $\alpha +\beta =0$. Balancing ${\rho }_t$ and ${\partial }_x({\rho }^4/4)$ requires $\alpha -1=4\alpha -\beta$. ThereforeThe similarity ODE isFor a source-type profile adjoining vacuum, the integration constant is zero. The positive branch is $f={\eta }^{1/3}$ for positive $\eta$, terminated by a shock into zero. Normalize the mass:Thus a unit-mass similarity solution isAt the moving right edge, $S^{\prime }={\eta }_F{t}^{-3/4}/4$ equals ${\rho }_L^3/4$, so the Rankine-Hugoniot condition is satisfied. The characteristic speeds satisfy $0<S^{\prime }<{\rho }_L^3$, making it an entropy shock. The left edge meets vacuum continuously. As $t\downarrow 0$ the mass concentrates at the origin; this is a source solution, not a claim that every finite-mass initial profile equals it at finite time.

Test a single discontinuity connecting the two constant initial states. With state 1 below the discontinuity and state 2 above it, their jumps areThe first Rankine-Hugoniot condition gives $F^{\prime }=\sqrt{3}$, and the second independently gives $(-\sqrt{3})F^{\prime }=-3$, the same value. Since the initial jump is at the origin,Values on the shock itself do not affect the weak solution. No intermediate state or rarefaction is needed: the prescribed states already lie on the same compressive shock curve.

For admissibility, the upper-family characteristic speed is $\sqrt{3}/2+\sqrt{2}$ behind the shock and $1$ ahead of it. They satisfyBoth lower-family speeds, $-1$ and $\sqrt{3}/2-\sqrt{2}$, are less than the shock speed. These are the Lax shock admissibility conditions for a family-plus shock. This is the entropy-admissible solution for the mass/momentum conservative extension specified by the preceding jump law; imposing an additional velocity conservation law across the same jump would describe a different problem.

For a smooth solution, the first equation is $u_x+(uv{)}_y=0$. Multiplying it by $v$ and adding $u$ times the second equation givesThus the appropriate shallow water equations conservation variables and flux areUse this conservative extension to define a weak solution across $y=F(x)$. A jump contributes $-F^{\prime }[W]\delta (y-F(x))$ to $W_x$ and $[H]\delta (y-F(x))$ to $H_y$. Cancellation of the singular parts gives the Rankine-Hugoniot conditionsEquivalently this follows by integrating each conservation law over a thin space-time control region crossing the discontinuity.

The conservative extension is essential here. Piecewise smoothness by itself does not imply any jump law. Moreover the smooth second equation can also be written $v_x+(u+v^2/2{)}_y=0$; imposing that different conservation law distributionally would give a different shock condition. Weak conservation laws depend on conserved variables. The requested jump law specifically selects mass $u$ and momentum $uv$, and this is the extension used below.

In the current configuration, balance of linear momentum is the Cauchy momentum equationHere $\tau$ is the Cauchy stress tensor, $\rho$ is current mass density, $b$ is body force per unit mass, and $v(x,t)$ is the velocity of the material particle currently at $x$. The expression in parentheses is the material acceleration. The divergence is taken in current spatial coordinates, with $(\mathrm{div}\tau {)}_i={\partial }_j{\tau }_{ij}$.

Equivalently the right side is $\rho {\partial }_t^{2}x(X,t)$ evaluated at the particle, because this is the same material acceleration in Lagrangian coordinates. For the subsequent static inflation with negligible body force, the equation reduces to $\mathrm{div}\tau =0$.

A stable sinusoidal radial-velocity semi-amplitude can be stellar reflex motion caused by a companion on a nearly circular orbit. The amplitude is compatible with a giant planet; a low-mass star or brown dwarf viewed nearly face-on can mimic the same projected motion, but requires an increasingly unlikely orientation. Starspots and magnetically suppressed convection can produce a rotational stellar activity signal on a ten-day timescale; contemporaneous photometry, activity indicators and stellar spectral-line bisector and other profile changes distinguish these from a clean common Doppler shift. Solar-like stellar oscillations ordinarily have much shorter periods, so a coherent ten-day oscillation of an otherwise ordinary solar-type dwarf is less plausible. Instrumental drift, sampling aliases and blended spectra also need checking. A sinusoid alone does not establish a planetary origin, and the data do not assign numerical probabilities to these alternatives.

For the minimum companion mass from radial velocity, take a circular two-body orbit and initially $M_{\star }=M_{\odot }$. The exact relation and its low-mass limit areWith $\sin i=1$,Here $a=[G{M}_{\star }{P}^2/(4{\pi }^2){]}^{1/3}$ is the relative orbital semi-major axis. Nonzero orbital eccentricity multiplies the inferred $M_p\sin i$ by $\sqrt{1-e^2}$ at fixed measured $K,P$, but the sinusoidal model motivates $e=0$ here.

For the radial-velocity sensitivity limit at fixed planet mass, use $K=M_p\sin i\sqrt{G/(M_{\star }a)}$ and interpret the stated detection scale as a semi-amplitude threshold $K_{\min }$. The largest distance occurs for $\sin i=1$:For an inclination $i$, this boundary is smaller by ${\sin }^{2}i$. The result assumes adequate time coverage and neglects the stipulated stellar noise. If the detection scale instead denotes a full peak-to-peak excursion, set $K_{\min }$ to half that scale; $a_{\max }$ then increases fourfold.

The stellar-mass scaling of radial-velocity planet estimates changes these inferences because Kepler's third law and the stellar reflex response depend on the stellar mass. At fixed $K,P$, $M_p\sin i\propto M_{\star }^{2/3}$ and $a\propto M_{\star }^{1/3}$, whereas the fixed-mass detectability boundary scales as $M_{\star }^{-1}$. ThereforeThe observed period and amplitude are unchanged. The stellar radius does not enter these leading dynamical estimates, but is crucial for the exoplanet transit inference.

For the transiting companion, assume an opaque spherical planet, negligible transit dilution and negligible limb darkening. Its radius ratio is $k=R_p/R_{\star }=\sqrt{\delta }=0.03873$, henceUsing the revised stellar mass and a circular orbit gives $a=0.01850\mathrm{au}$ and $A=a/R_{\star }=7.957$. Interpret the duration as the total first-to-fourth-contact duration $T_{14}$. The impact parameter from total transit duration follows from the total duration of a circular planetary transit:This is noncentral but not grazing: $b<1-k$. The small-planet straight-chord approximation gives $b\simeq 0.845$; retaining finite $R_p$ and orbital curvature gives the stated value. A duration measured between other contacts needs the corresponding contact formula. Unknown orbital eccentricity and argument of periapsis change the transverse speed and prevent a unique unconstrained $b$; the numerical estimate is conditional on the stated circular model.^[ref]

Because the transit constrains $\sin i\simeq 0.994$, the RV nondetection now bounds the actual planetary mass, rather than only its projected mass. The bulk-density upper limit from a transit and an RV nondetection isThese bounds assume the instrumental threshold can be applied to a fitted signal at the transit period; no statistical confidence level is specified. Such a large radius at this low mass is inconsistent with an ordinary compact rock-and-iron body. A low-density volatile-rich planet, possibly with an extended H/He envelope, is indicated under the planetary interpretation. The observations do not uniquely determine interior composition, atmospheric abundance or formation history; blended eclipses and activity still require validation. There is no luminosity or effective temperature supplied for the host, so a numerical planetary equilibrium temperature cannot be inferred from these data alone.

The numerical conversions above use externally documented $(GM{)}_{\odot }=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, $R_{\odot }=6.957\times 10^8\mathrm{m}$, $M_{\oplus }=5.9722\times 10^{24}\mathrm{kg}$, $R_{\oplus }=6.371\times 10^6\mathrm{m}$, $M_J=1.89813\times 10^{27}\mathrm{kg}$, $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $1\mathrm{au}=149597870700\mathrm{m}$. They are conversion constants, not additional measured system parameters.^[ref][ref]

Core accretion builds a solid core and then a massive gaseous envelope before protoplanetary disk dispersal. The solid and gas stages overlap; this is not direct collapse of a whole disk patch.

In a protoplanetary disk, refractory grains and, beyond a snow line, condensed ices provide solids. Settling and protoplanetary dust coagulation build aggregates, but fragmentation, bouncing and radial drift of solids in a gas disk can interrupt simple sticking. Concentration processes such as the streaming instability can produce dense particle structures that collapse into planetesimals. Subsequent collisions build planetary embryos; alternatively, embryos can efficiently capture drifting particles through gas-assisted pebble accretion. Neither the initial planetesimal population nor unrestricted growth through every size should be assumed without a mechanism.

For gravitationally focused growth of a planetary core, the two-body capture cross-section isAn illustrative thin-disk estimate is ${\dot{M}}_c\sim {\Sigma }_s\Omega {\sigma }_{\mathrm{coll}}$, up to geometry and velocity-distribution factors. At fixed solid density and random speed, strong gravitational focusing gives ${\dot{M}}_c\propto M_c^{4/3}$: runaway planetesimal growth. As large embryos stir the smaller bodies, encounter speeds increase and focusing weakens, giving oligarchic growth. Consuming the local feeding zone leads toward an isolation mass; radial supply, migration and mergers can extend growth. Snow lines can increase the solid reservoir, but a distant core also grows more slowly because orbital times are longer.^[ref]

A growing core binds gas over a scale limited by the smaller of its Bondi radius and Hill radius. The envelope approximately satisfiesAn equation of state, disk boundary conditions, and radiative or convective energy transport close the structure. In an optically thick radiative region,Here $\kappa$ is opacity; an unstable radiative gradient instead drives convection. Solid accretion releases roughly $L_s\sim G{M}_c{\dot{M}}_c/R_c$, heating and supporting the envelope. Gas capture consequently depends on cooling, not just whether the potential can bind gas instantaneously.

The envelope loses energy on a Kelvin-Helmholtz cooling time, $t_{\mathrm{KH}}\sim |E_{\mathrm{env}}|/L$. Lower grain opacity, higher core mass, or reduced solid-accretion heating can accelerate contraction. The critical core mass for gas accretion is therefore model dependent; “of order ten Earth masses” is a common scale, not a universal switch. As envelope self-gravity becomes important and the gas and solid masses become comparable, contraction permits still more gas capture, producing runaway gas accretion. The classical sequence has a rapid solid-growth phase, a slower envelope-growth phase, and accelerated gas accumulation.^[ref]

The decisive race is $t_{\mathrm{core}}+t_{\mathrm{envelope}}\lesssim t_{\mathrm{disk}}$. Failure to cool or obtain a sufficiently large core before gas removal leaves a smaller solid or ice-rich planet. Successful growth is ultimately limited by disk supply, gap opening by a planet, and dispersal; a gap does not completely seal off accretion. Angular-momentum exchange drives planetary migration, so the observed orbit need not be the formation radius. Accretion-shock radiation and subsequent contraction set the young giant's entropy and luminosity; final core mass, envelope enrichment and orbital location depend on the whole growth history, not on one fixed threshold. The necessary cooling, supply and orbital constraints make the model testable rather than an assertion that every embryo must become a gas giant.

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.