--- title: "Magnetic support depends on direction" description: "Derive the linear gravity–MHD modes and compare compression along, across and oblique to a magnetic field. Direction changes the Jeans response." canonical_url: "https://sandboxphysics.com/stories/molecular-clouds/magnetic-support-has-a-direction/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/molecular-clouds/magnetic-support-has-a-direction/" last_modified: "2026-11-04" --- [Molecular cloud collapse and magnetic support](https://sandboxphysics.com/stories/molecular-clouds/) Molecular clouds · L2 # Magnetic support depends on direction Sandbox Physics · 2026-11-04 Gas gathering along a magnetic field can avoid squeezing the field strongly. Compression across it crowds magnetic flux and brings in a magnetic restoring force. A field strength alone cannot specify the support; the direction of compression matters too. ## Before you begin Read the L1 Jeans derivation. You need vector components and sinusoidal waves. Complex amplitudes are introduced as a compact way to keep sine and cosine phases together. Perturbations remain small on a uniform background. ## Compress along the field or crowd it sideways Imagine a bundle of lines marking the direction of a uniform magnetic field. They show a vector field; they are not separate physical ropes around the gas. In ideal magnetohydrodynamics, magnetic flux moves with the fluid. Cross-field compression changes field strength, and bending introduces another restoring effect. A difference in magnetic pressure pushes from stronger-field regions toward weaker ones. Magnetic tension acts along curved fields and tends to straighten them. Together these effects give the magnetic force. A compression parallel to the field couples to that force differently from a perpendicular compression. Choose one density-wave direction, then rotate the background field relative to it. The angle in this comparison refers to that specified wave. A cloud can permit additional motions along other directions. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field · Molecular-cloud Jeans standalone scientific reproduction ## Put six disturbances in one coordinate system Let the wavevector be $\mathbf k=k\hat{\mathbf x}$. The background field lies in the $x,y$ plane at angle $\theta$ to $x$. Divide the SI magnetic field by $\sqrt{\mu_0\rho_0}$, where $\mu_0$ is vacuum permeability and $\rho_0$ is background volume density. The normalized field has velocity units, and its magnitude is the Alfvén speed $v_A$. The background components are $B_x=v_A\cos\theta$ and $B_y=v_A\sin\theta$. Here $B_x,B_y$ already use velocity units; the disturbances $b_y,b_z$ use the same normalization. Add relative density $\delta$ and velocities $v_x,v_y,v_z$ to obtain six time-dependent variables. The perturbed field obeys $\nabla\cdot\delta\mathbf B=0$. A nonzero Fourier wave varying only along $x$ therefore has $b_x=0$. A spatially constant background $B_x$ can still be nonzero. Write the spatial variation as $e^{ikx}$ and take its real part to recover the physical field. The number $i$ obeys $i^2=-1$. Multiplication by $i$ shifts sine and cosine phases. Complex amplitudes keep those phases together while all plotted fields come from the same state. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field · Molecular-cloud Jeans standalone scientific reproduction ## Induction and magnetic force act together Define $\omega_J^2=4\pi G\rho_0$. Continuity still relates density to longitudinal flow. Longitudinal momentum now contains a magnetic term as well as pressure and perturbed gravity. The background field along the wave connects transverse magnetic disturbances to transverse velocities. The field disturbance evolves too. Fluid motion changes it through induction, and the changed field accelerates the fluid. Integrating both sets of equations is what lets a density map, velocity arrows and magnetic directions describe one physical process. The longitudinal magnetic sign has a direct interpretation. After normalization, the first-order magnetic-pressure change divided by background density is $B_yb_y$. Its negative $x$ gradient is $-ikB_yb_y$. A more crowded transverse field pushes outward. The longitudinal background field separately connects transverse tension, retained in the other component equations. Prepare a small longitudinal displacement $\xi_x$. Continuity gives $\delta=-\partial_x\xi_x$, and crowding transverse magnetic flux gives $b_y=-B_y\partial_x\xi_x$. Thus $b_y=B_y\delta$ initially. Oblique evolution can subsequently excite transverse velocity, so this initial relation cannot replace the full induction equation at all times. $$ \dot\delta=-ikv_x,\qquad \dot v_x=-ik\left(c_s^2-\frac{\omega_J^2}{k^2}\right)\delta-ikB_yb_y $$ Density couples to longitudinal velocity. Poisson’s equation has eliminated the gravitational potential. $$ \dot v_y=ikB_xb_y,\qquad \dot v_z=ikB_xb_z $$ The transverse magnetic forces depend on the background field along the wave. $$ \dot b_y=ik(B_xv_y-B_yv_x),\qquad \dot b_z=ikB_xv_z $$ Induction evolves the magnetic disturbances, completing the six-variable system. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field ## Keep the compressive and Alfvén branches distinct Seek time dependence $e^{-i\omega t}$ and eliminate the component amplitudes. Disturbances involving density compression satisfy the quartic below. Treating it as a quadratic in $\omega^2$ gives two branches. A negative frequency squared supplies a growing and a decaying time solution. A separate pair of Alfvén modes exchanges transverse kinetic and magnetic disturbance energy. Their frequency depends on the field component along the wave. Omitting this pair would leave two eigenvalues missing from the six-variable operator. For $\theta=0$, the compressive branch recovers $\omega^2=c_s^2k^2-\omega_J^2$ from L1. Increasing the field leaves that parallel Jeans threshold unchanged. At exactly $90^\circ$, the nonzero compressive frequency contains $c_s^2+v_A^2$ because magnetic pressure participates in the restoring force. Take field strength to zero as another check. The quartic separates a $\omega^2=0$ branch from the ordinary Jeans branch, and the Alfvén frequency vanishes. Transverse velocity may stay constant; density compression returns to L1. This limit can reveal an extra field-strength factor in an implementation. $$ \omega^4-\left[(c_s^2+v_A^2)k^2-\omega_J^2\right]\omega^2+v_A^2k^2\cos^2\theta(c_s^2k^2-\omega_J^2)=0 $$ The compressive dispersion relation explicitly depends on angle. A universal scalar replacement for sound speed would remove that information. $$ \omega_A^2=k^2v_A^2\cos^2\theta $$ The separate Alfvén factor and the compressive quartic together supply all six time eigenvalues. Reading: Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field · Molecular-cloud Jeans standalone scientific reproduction ## Calculate a strong-field example Use $J=1.5$, a box-spanning fundamental wave, and $v_A/c_s=4$. Solve the quadratic in frequency squared. The parallel growth rate is $0.745356/t_0$. At $45^\circ$ it falls to $0.537226/t_0$, and at $89^\circ$ it is $0.0135481/t_0$. These values are recomputed from the quartic in the worked-example program. The last rate is small and positive. A finite-duration record near the perpendicular direction can therefore show very little slow growth. The exact $90^\circ$ endpoint includes neutral and oscillatory branches. A tiny positive real part left by floating-point arithmetic needs a scale-aware neutral classification. The saved field example starts from a longitudinal displacement with $b_y=B_y\delta$, zero velocity and density amplitude $0.02$. This generally excites a combination of modes. Each listed rate belongs to a mode; the integrated six-component state determines how their contributions combine in the density history. ![Growing-mode rates vary with field direction; along-field growth remains at all three field strengths.](https://sandboxphysics.com/_astro/magnetic-direction-en.mMODLPZG.png) Compressive growing-mode rates from the chapter’s dispersion relation at Jeans ratio 1.5. A strong transverse field can stop this compression, while along-field growth remains. These are eigenmode rates, separate from the mixed initial conditions used in the Lab. Reading: Molecular-cloud Jeans standalone scientific reproduction ## Check every disturbance before extending a linear calculation A small density contrast alone does not guarantee that the entire disturbance remains small. Density may briefly pass through zero contrast even though velocity or magnetic disturbance has grown. The examples track density, relative field disturbance and normalized velocity, stopping before a specified small-amplitude cap is crossed. Velocity is normalized by $\sqrt{c_s^2+v_A^2}$, and the relative cap is $0.15$. This speed defines a stopping scale. Propagation and growth still come from the six equations with their directional magnetic force. The nonmagnetic gas patch evolves finite-amplitude mass and momentum. This calculation evolves linear magnetic disturbances on a uniform background. Their distinct assumptions let you choose the calculation that answers the particular question you are asking. Reading: Molecular-cloud Jeans standalone scientific reproduction ## A directional wave test differs from a cloud’s magnetic criticality A question about supporting an entire cloud also needs a shape, a surface through which to measure magnetic flux, and a specification of the external field. Published critical mass-to-flux ratios carry those geometric assumptions. One periodic wave does not define a universal critical cloud mass. In this uniform model, allowing a wavevector along the field leaves the original Jeans-unstable compressive direction available. The paper guide later examines three-dimensional turbulent simulations, where flows in many directions accompany a separate question about characteristic masses and resolution. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 · Federrath & Klessen 2012, arXiv:1209.2856v1 ## Exercises and explanations 1. Does a field with Alfvén speed four times the sound speed guarantee stability against compression in every direction? Read the explanation The parallel branch still obeys $\omega^2=c_s^2k^2-\omega_J^2$. This compression does not crowd a transverse field, so it retains the original Jeans threshold. Specify the wave direction; a strong field can substantially change perpendicular and oblique responses. 2. Why can a nearly flat finite record at $89^\circ$ not simply be assigned to the neutral perpendicular endpoint? Read the explanation The slow branch still has growth rate about $0.0135481/t_0$. Growth can be slow over the record, and the initial state excites multiple modes. Calculate the eigenvalues and state a numerical neutral tolerance. At the exact endpoint set the background longitudinal component to zero instead of assigning physics to a floating-point cosine residual. 3. Does $b_x=0$ for an $x$-varying wave prohibit a background field along $x$? Read the explanation No. The divergence-free condition restricts this nonzero-wavenumber perturbation. A uniform background $B_x$ remains possible and connects transverse velocity to magnetic disturbance through tension and induction. ## Try the prediction ### [Watch pressure and gravity change the same gas](https://sandboxphysics.com/labs/molecular-cloud-jeans/) 1. In the linear magnetic branch, compare parallel, oblique and perpendicular responses at the same Jeans ratio and field-to-sound ratio. Predict which compression has the strongest restoration. 2. Drag the field-direction handle and compare density, velocity and magnetic arrows. The changed direction prepares a new initial state that evolves through the same six-variable equations. 3. Save a record and compare its density history with the mode growth rates. Use the initial-condition discussion to explain how a mixed state combines mode contributions. [Previous: How the Jeans scale separates oscillation and growth](https://sandboxphysics.com/stories/molecular-clouds/jeans-length-and-a-clouds-mass/) [Next: When a grid can no longer resolve contraction](https://sandboxphysics.com/stories/molecular-clouds/when-a-cloud-grid-stops-being-evidence/) ## 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. [Federrath & Klessen 2012, arXiv:1209.2856v1](https://arxiv.org/abs/1209.2856v1) Pinned v1, sections 2.2/2.4, 3.1–3.3 and 7: isothermal driven three-dimensional MHD, sink criteria, the statistical magnetic-pressure approximation and its limitations. Their driving, sinks and numerical dynamo requirement are not part of the gas-patch calculation. 3. [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. 4. [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.