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Equate the free-fall time of a uniform sphere $\sqrt{3\pi /(32G\rho )}$ to a radius-based sound-crossing time $R/c_s$, then substitute $R$ in $M=4\pi \rho R^3/3$. This gives the displayed mass. It has the scaling of a Jeans mass, but its coefficient differs from the convention using a sphere of diameter equal to the Jeans length. Equality of these two times is an approximate support comparison, not the exact marginal mode of an infinite-medium Jeans instability.

The physical Jeans length is ${\ell }_J=c_s\sqrt{\pi /(G{\rho }_0)}$. A pressureless uniform sphere of initial density ${\rho }_0$ has free-fall time of a uniform sphere $t_{\mathrm{ff}}=\sqrt{3\pi /(32G{\rho }_0)}$, giving the title relation. Thus a full-wavelength sound-crossing time and this exact free-fall convention agree in scaling, not numerical value. Using the scale $k_J^{-1}$ instead gives $1/(c_s{k}_J)=1/\sqrt{4\pi G{\rho }_0}$, the inverse long-wave static Jeans growth-rate spectrum limit. At the threshold itself pressure balances gravity and the net growth rate vanishes; the pressureless free-fall time must not be confused with that marginal mode's infinite growth time.

Take positive $\dot{Q}$ to mean net cooling per unit mass. At fixed pressure for an ideal gas of unchanged composition, $\rho \propto T^{-1}$, so $\dot{Q}=C{T}^{\alpha -1}-H$ with $C>0$. At thermal equilibrium, $C{T}_0^{\alpha -1}=H$. Because expansion work is included at fixed pressure, the perturbed temperature obeysThis isobaric stability criterion for density-dependent power-law cooling gives stability for $\alpha >1$ and instability for $\alpha <1$. For optically thin bremsstrahlung, $\alpha =1/2$, soA slightly cooler parcel becomes denser and cools faster; a slightly hotter parcel has reduced cooling and heats further. The conclusion assumes the sound-crossing time is short enough to maintain nearly common pressure, with conduction and other stabilizing processes omitted.

Using a center-to-edge sound-crossing time $t_s=R/c_s$ and the pressureless uniform-sphere free-fall time of a uniform sphere $t_{\mathrm{ff}}=\sqrt{3\pi /(32G\rho )}$, equality givesOn scales larger than this estimate, pressure information cannot cross the region before its gravitational dynamical evolution. It therefore identifies the order of the Jeans length and separates gravity-dominated from pressure-supported behavior in the simple comparison. It is not an exact normal-mode stability calculation: with the standard Jeans wavelength ${\lambda }_J=c_s\sqrt{\pi /(G\rho )}$, this radius is $\sqrt{3/32}{\lambda }_J$. Using the diameter rather than the radius as the crossing distance changes the numerical coefficient. Boundary conditions, density structure, turbulence and magnetic support also matter for an actual cloud.

A local thermal imbalance can amplify a gas perturbation even without an initial displacement of an interface. Let $\mathcal{L}(\rho ,T)$ denote net cooling per unit mass, including heating with a negative sign, and suppose $\mathcal{L}({\rho }_0,T_0)=0$. When the sound-crossing time is short compared with the thermal evolution time, pressure equilibrates and an ideal gas perturbation is approximately isobaric. The linearized thermal equation isThus a negative constant-pressure derivative produces isobaric thermal instability: a cooler parcel experiences a net loss that cools it further, while a warmer parcel heats further. For example, isobaric thermal stability of a heated ideal gas with $\mathcal{L}=A\rho T^{\alpha }-H$ gives $\sigma =(1-\alpha )H/(c_p{T}_0)$ and instability for $\alpha <1$. Different heating laws and fixed-density perturbations give different criteria; thermal conduction damps sufficiently short-scale temperature contrasts. Such feedback helps a gas separate into denser cool and rarer warm regions when stable thermal branches exist.

The Rayleigh-Taylor instability instead releases gravitational potential energy when a heavier fluid lies above a lighter fluid in a downward gravitational field. For two inviscid incompressible half-spaces with no surface tension or magnetic field, a horizontal perturbation of wavenumber $k$ hasThe Atwood number measures the density contrast. A descending heavy-fluid finger and an ascending light-fluid bubble both lower the total gravitational potential energy, increasing the original displacement. Interchanging the layers gives an oscillatory stable interface. An accelerating or decelerating astrophysical shell has an analogous effective acceleration, allowing mixing at suitable ejecta interfaces in a supernova remnant. Finite interface width, viscosity, surface tension and magnetic tension modify the short-wavelength behavior; the ideal sharp-interface formula does not select a finite fastest-growing wavelength.

For small isentropic perturbations of a fixed-composition gas, linearizing its equation of state gives $p_1=(\partial p/\partial \rho {)}_s{\rho }_1=c_s^2{\rho }_1$. The adiabatic sound speed is evaluated at the uniform background, so it is spatially constant at this order. If fast heat exchange instead enforces constant temperature, use the isothermal sound speed consistently. A completely arbitrary entropy perturbation would require an additional thermodynamic variable.

The uniform static infinite medium requires the Jeans swindle: its background gravitational field is subtracted or externally balanced, and only the perturbation sources the Poisson equation for Newtonian gravity. Linearizing the continuity equation, Euler equation and gravitational equation then givesTaking one more time derivative of the first equation and substituting the other two yieldsA Fourier mode proportional to $e^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}$ therefore obeys the Jeans instability relationFor $k>k_J$, the mode oscillates as a sound wave. For $0<k<k_J$, $\omega =\pm i\sqrt{4\pi G{\rho }_0-c_s^2{k}^2}$ gives a growing and a decaying mode; the static Jeans growth-rate spectrum has no preferred finite fastest-growing wavelength in the infinite-medium approximation. The threshold $k=k_J$ is marginal at linear order. The Jeans length is ${\lambda }_J=2\pi /k_J=c_s\sqrt{\pi /(G{\rho }_0)}$. Defining the Jeans mass as a sphere whose diameter is this wavelength givesOther radius conventions alter its numerical coefficient; the mass scale compares self-gravity with pressure communication.

For the pressureless finite cloud, the shell theorem gives an edge acceleration $\ddot{r}=-GM/r^2$, with fixed enclosed $M=4\pi r_0^3{\rho }_0/3$. Multiplication by $\dot{r}$ and integration from rest givesThe negative root describes collapse. Substituting $r=r_0{\sin }^2\theta$ gives the free-fall time of a uniform sphereThe surface and inner shells reach the origin together in pressureless homologous spherical collapse; pressure, rotation or shell crossing would change this ideal solution.

Equating this time to the specified radius-based sound-crossing time $r_0/c_s$ gives $r_0=c_s\sqrt{3\pi /(32G{\rho }_0)}$. The resulting crossing-time cloud support mass isThis is the same dimensional mass scale as $M_J$, with a different coefficient because the adopted length and time criteria differ. In particular, $t_{\mathrm{ff}}=r_0/c_s$ is not the exact marginal dispersion-relation condition.

With the requested isotropic support approximation, replace $c_s$ by the Alfvén speed $v_A=B/\sqrt{{\mu }_0{\rho }_0}$. In the crossing-time convention the Magnetic Jeans mass isUsing the wavelength-based Jeans mass convention instead replaces $C_{\mathrm{cross}}$ by ${\pi }^{5/2}/6$. The scaling, rather than a convention-independent coefficient, is the shared prediction.

For a fixed cloud mass undergoing isothermal homologous contraction, $\rho /{\rho }_0=(r_0/r{)}^3$ and $c_s$ is constant. Its number of thermal Jeans masses therefore grows asA cloud initially close to the support threshold has $N_{J,0}$ of order one, so smaller pieces can subsequently become independently unstable: this is isothermal fragmentation.

For approximately isotropic contraction with magnetic flux freezing, $B{r}^2$ stays constant, so $B\propto {\rho }^{2/3}$. Hence $M_B\propto B^3/{\rho }^2$ stays constant andThis ideal magnetic support therefore does not create progressively more unstable masses in the way thermal isothermal support does. The same conclusion follows from the conserved mass-to-flux ratio. A magnetically subcritical cloud cannot initiate unrestricted cross-field collapse under ideal flux freezing alone; flux loss or initially supercritical conditions are needed. A real magnetic field also gives anisotropic support: motion and fragmentation along its field lines are not excluded by this scalar-speed estimate.

The vertical acceleration in the thin disk is of order ${\Omega }^{2}H$. Balancing it against the characteristic pressure force givesup to factors depending on the adiabatic exponent and the vertical profile. A displacement of size $\delta z$ has a restoring acceleration of order ${\Omega }^2\delta z$, so $\delta \ddot{z}\sim -{\Omega }^2\delta z$. Equivalently the sound-crossing time is $H/c_s$. Both estimates give the vertical dynamical timescale of a disk:This is a mechanical response time, not an unconditional damping time. An inviscid stable layer can exhibit a vertical breathing mode of an astrophysical disk without settling; its adiabatic frequency satisfies ${\omega }_H^2=(\gamma +1){\Omega }^2$. Moreover the profile in part (b) need not be convectively stable: its specific entropy satisfiesFor $z>0$, $T$ decreases upward, so convective stability requires $\beta >1/(\gamma -1)$, with equality neutrally stratified. Thus interpreting the timescale as re-establishment of a stable equilibrium presupposes suitable stability and damping; the stated equations alone do not guarantee either.

The time for a vertical pressure or gravity response. A sound-crossing time and a small vertical oscillation period have this scale in a thin disk. It need not equal a damping time.