--- title: "How the Jeans scale separates oscillation and growth" description: "Derive the Jeans dispersion relation from mass, pressure and gravity, then calculate growth, initial velocities and the separately defined spherical Jeans mass." canonical_url: "https://sandboxphysics.com/stories/molecular-clouds/jeans-length-and-a-clouds-mass/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/molecular-clouds/jeans-length-and-a-clouds-mass/" last_modified: "2026-11-04" --- [Molecular cloud collapse and magnetic support](https://sandboxphysics.com/stories/molecular-clouds/) Molecular clouds · L1 # How the Jeans scale separates oscillation and growth Sandbox Physics · 2026-11-04 Sound speed, mean density and wavelength determine whether a small density wave oscillates or grows. Pressure restores short waves more strongly, and self-gravity amplifies density differences. Their competition gives the Jeans threshold when both forces enter the same motion equation. ## Before you begin Read Why cold gas can gather into dense structures. You need derivatives and sine/cosine waves. We explain the second derivatives used here and hold the gas temperature fixed. ## Define the background and the disturbance Start with uniform density $\rho_0$ and a wave along $x$. Positive velocity $u$ points toward increasing $x$. The relative density disturbance is $\delta=(\rho-\rho_0)/\rho_0$; it is dimensionless and much smaller than one. The sound speed $c_s$ has velocity units. For isothermal gas, pressure obeys $p=c_s^2\rho$. Holding temperature fixed supplies the pressure law; we are not solving another equation for heating and cooling. The gravitational potential $\phi$ has units of velocity squared. Its negative spatial derivative gives gravitational acceleration. Sound speed squared here is the pressure derivative with respect to density at fixed temperature. The official-input chapter also encounters an adiabatic derivative. Specify the temperature response during compression before assigning a numerical sound speed. A uniform, static, infinite self-gravitating background needs a convention for its mean gravity. Here a periodic box responds only to density minus the mean, and its uniform component produces no force. This defines the perturbation problem. An isolated sphere requires its own finite-boundary gravity problem. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field · Molecular-cloud Jeans standalone scientific reproduction ## Join mass conservation to the forces Mass conservation says that density decreases where more gas leaves than enters. Keep only first-order disturbances: products of a density disturbance and a velocity are second order and are omitted for now. The momentum equation adds the pressure and gravitational accelerations. A derivative measures change over a small interval. When $\partial_xu>0$, gas on the right moves faster than gas on the left, stretching a small region. The minus sign in continuity makes its density contrast fall. Converging gas has a negative velocity gradient and an increasing density. Differentiate continuity once more in time. Substitute the momentum equation for the time derivative of velocity, then substitute Poisson’s equation for the second derivative of $\phi$. The result describes the acceleration of the density contrast. Pressure acts through spatial curvature; the gravity term has the same sign as the density contrast. Track the signs explicitly: $\partial_t^2\delta=-\partial_x(\partial_tu)$. Substitution produces two pairs of minus signs, yielding $c_s^2\partial_x^2\delta+\partial_x^2\phi$. Poisson’s equation then replaces the potential curvature with density contrast. Its positive contribution is the mathematical expression of gravity strengthening an overdensity. The density amplitude is not a particle moving along a trajectory, but its evolution can still obey an oscillator equation. A second time derivative describes how the rate of change itself accelerates. $$ \partial_t\delta=-\partial_xu,\qquad \partial_tu=-c_s^2\partial_x\delta-\partial_x\phi $$ Linear continuity and momentum with fixed background density and sound speed. $$ \partial_x^2\phi=4\pi G\rho_0\delta,\qquad \partial_t^2\delta=c_s^2\partial_x^2\delta+4\pi G\rho_0\delta $$ $G$ is the gravitational constant. The quantity $4\pi G\rho_0$ has inverse-time-squared units, matching the other term after multiplication by the dimensionless disturbance. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field · Molecular-cloud Jeans standalone scientific reproduction ## A cosine wave gives the critical wavelength Separate position and time by writing $\delta(x,t)=D(t)\cos(kx)$. The dimensionless amplitude is $D$, the angular wavenumber is $k=2\pi/\lambda$, and $\lambda$ is wavelength. Two spatial derivatives of a cosine supply $-k^2$. Short wavelengths therefore have a stronger pressure-restoring term. When $\omega^2$ is positive, the amplitude oscillates with angular frequency $\omega$. When it is negative, define $\gamma=\sqrt{-\omega^2}$; the solution contains a growing exponential and a decaying exponential. The threshold occurs at $\omega^2=0$. Both oscillation and growth become slow near the threshold. A nearly unchanged finite-time image can mean that the rate is small. Calculate the condition before interpreting an apparently quiet patch as stable. $$ \ddot D+\omega^2D=0,\qquad \omega^2=c_s^2k^2-4\pi G\rho_0 $$ The dispersion relation associates each wavelength with an oscillation frequency or growth rate. $$ \lambda_J=c_s\sqrt{\frac{\pi}{G\rho_0}} $$ Small disturbances with $\lambda>\lambda_J$ can grow; those with $\lambda<\lambda_J$ oscillate. Doubling sound speed doubles this length. Quadrupling density halves it. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field ## Starting at rest selects a different curve Use box length $L$ as the length unit and $t_0=(4\pi G\rho_0)^{-1/2}$ as the time unit. Define the Jeans ratio $J=L/\lambda_J$. For a fundamental wave spanning the box, $\omega^2t_0^2=J^{-2}-1$. With $J=1.5$, the growth rate is $\gamma t_0=0.745356$. Choose initial amplitude $D_0=10^{-4}$. If the initial velocity has exactly the inward growing-mode pattern, the amplitude follows $D_0e^{\gamma t}$. At $t=2t_0$ it reaches about $4.4403\times10^{-4}$. Initial velocity decides how much of the growing and decaying solutions is present. For gas initially at rest, $\dot D(0)=0$. Equal growing and decaying contributions cancel the initial slope. Their average is a hyperbolic cosine, giving amplitude $2.3327\times10^{-4}$ at the same time. The growing contribution dominates only later. The small-amplitude numerical growth test deliberately prepares the growing velocity. The finite-amplitude gas patch starts at rest. Comparing their early slopes directly would confuse different initial conditions with numerical error. $$ D_{\rm rest}(t)=D_0\cosh(\gamma t)=\frac{D_0}{2}(e^{\gamma t}+e^{-\gamma t}) $$ The initially resting linear solution. A pure growing mode requires an initial amplitude derivative equal to growth rate times initial amplitude. $$ u(x,0)=-\frac{\gamma D_0}{k}\sin(kx) $$ Continuity fixes the inward velocity of a pure growing wave. Setting this velocity to zero changes the solution. Reading: Molecular-cloud Jeans standalone scientific reproduction ## State a geometry before converting length to mass A common three-dimensional Jeans mass encloses the background density in a sphere of radius $\lambda_J/2$. Stating the radius matters: a different convention changes the geometric coefficient. A critical wavelength does not assign a unique mass to every shape. For an illustrative physical scaling, specify a total thermal-particle density of one thousand per cubic centimeter, mean particle mass $2.33$ times an adopted hydrogen-atom mass, box length $1\,\mathrm{pc}$, and $J=1.5$. These choices give sound speed about $187\,\mathrm{m\,s^{-1}}$, temperature $9.90\,\mathrm K$, $t_0=0.554\,\mathrm{Myr}$, and $\lambda_J=0.667\,\mathrm{pc}$. The particle density here is not specifically a count of hydrogen molecules. The stated spherical definition gives about $8.94M_\odot$. The two-dimensional density picture is invariant in the third direction and integrates mass per unit depth. Assigning the spherical number to its bright central region would combine two different geometries. An isolated homogeneous sphere’s free-fall time is another separately defined reference. Try a scaling example with no new constants. Double sound speed at fixed density: the critical length doubles, its reference sphere has eight times the volume, and the Jeans mass rises eightfold. Instead quadruple background density at fixed sound speed: the length halves, the sphere volume becomes one eighth, and four times the mass per volume makes the reference mass one half. The comparison separates a critical length from a critical mass. A smaller reference mass means that a smaller amount of gas can become unstable under the stated assumptions. The total mass of a particular evolving region still follows its own geometry and boundary conditions. $$ M_J=\frac{4\pi}{3}\rho_0\left(\frac{\lambda_J}{2}\right)^3=\frac{\pi^{5/2}c_s^3}{6G^{3/2}\rho_0^{1/2}} $$ Background-density mass in a three-dimensional sphere whose diameter equals the Jeans length. $$ \frac{M_{\rm patch}}{\ell_z}=\int\rho\,dx\,dy $$ The invariant-direction calculation supplies mass per unit $z$ depth. A finite depth $\ell_z$ must be specified separately. Reading: Molecular-cloud Jeans standalone scientific reproduction · Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 ## Follow the approximation as the wave grows The derivation discarded second-order terms. Once the density contrast becomes large, density and velocity affect each other, and a cosine can develop additional spatial structure. The linear dispersion relation still tests early evolution; later motion requires the actual mass and momentum equations. A numerical grid introduces another length. Isothermal compression shortens the local Jeans length as density rises, leaving fewer cells across it. The L3 chapter recalculates the same case with a finer grid and an independent method so that a change in the gas can be distinguished from a change caused by discretization. Reading: Molecular-cloud Jeans standalone scientific reproduction ## Exercises and explanations 1. For the fundamental wave with $J=0.75$, decide whether the mode grows or oscillates and calculate $\omega t_0$. Read the explanation Substitution gives $\omega^2t_0^2=0.75^{-2}-1=0.777778$, which is positive. The wave oscillates with $\omega t_0=0.881917$. Angular frequency differs from cycles per unit time by $2\pi$; the period is $2\pi/\omega$. 2. Can the slope of $\ln D$ immediately measure $\gamma$ for an unstable wave that starts at rest? Read the explanation No. The resting solution is $D_0\cosh(\gamma t)$, with logarithmic slope $\gamma\tanh(\gamma t)$. It starts at zero. Prepare a pure growing velocity, or wait until the growing term dominates while the perturbation is still linear. 3. Can the bright concentration in the invariant-direction patch be assigned the spherical Jeans mass calculated above? Read the explanation Its mass requires a specified integration region and depth. The spherical reference integrates a defined three-dimensional volume; the patch saves mass per unit invariant depth. Specify those geometric choices before comparing the masses. ## Try the prediction ### [Watch pressure and gravity change the same gas](https://sandboxphysics.com/labs/molecular-cloud-jeans/) 1. Choose a small initial contrast and compare fundamental waves below and above a Jeans ratio of one. Predict their motion from the dispersion relation before inspecting the history. 2. In the standalone package, compare the pure growing small-wave test with the initially resting gas patch. Explain their different early slopes using the worked example. 3. Compare the same initial condition at 64 and 128 cells. State whether your comparison concerns amplitude, the full density field or a separately defined spherical mass. [Previous: Why cold gas can gather into dense structures](https://sandboxphysics.com/stories/molecular-clouds/why-cold-clouds-gather/) [Next: Magnetic support depends on direction](https://sandboxphysics.com/stories/molecular-clouds/magnetic-support-has-a-direction/) ## Sources and calculation scope 1. [Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field](https://adsabs.harvard.edu/pdf/1953ApJ...118..116C) Section V, pages 138–141, equations 158–173. The original uses an adiabatic sound derivative and Gaussian magnetic units; the chapters state their isothermal derivative and SI normalization. Original paper linked, not redistributed. 2. [Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1](https://arxiv.org/abs/2002.01421v1) Pinned v1, sections 2–4, Figures 8–10 and equations 8–11/17/22–23. Read the mass-weighted percentiles and the number distribution separately. The Figure 8 model-name pairing and approximate dimensional example are preserved in context; no author correction is assumed. 3. [Molecular-cloud Jeans standalone scientific reproduction](https://sandboxphysics.com/stories/molecular-clouds/materials/molecular-cloud-jeans-science.zip) Independent finite-volume and oversampled pseudospectral producers; analytic Jeans and full linear MHD references; all ten support cases, a fixed longer case, original official inputs and retained failures. Default replay actually recomputes its scientific steps offline after ordinary Python dependencies are installed.