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Put $\Psi =-\Phi =GM(r^p+a^p{)}^{-1/p}$. The spherical Poisson equation gives the hypervirial model density

For the proposed hypervirial distribution function, write $\mathcal{E}=-E=\Psi -v^2/2$ and $n=(3p+1)/2$. Direct velocity integration givesMatching this expression to the density fixesThis coefficient is positive for $0<p\le 2$, so the distribution function self-consistently generates the stated density and potential.

For $P=K{\rho }^{1+1/n}$,The pseudo-enthalpy definition therefore integrates towhere the constant makes $w=0$ at the surface. Thus

Hydrostatic balance is $c_0^{2}dw/dz=-{\Omega }^{2}z-d{\Phi }_d/dz$. Differentiate it, use the Poisson equation, and set $\zeta =\Omega z/c_0$:Since the definitions in the question give $Q_0={\Omega }^2/(\pi G{\rho }_0)$,

For $n=1$, put $\lambda =2/\sqrt{Q_0}$. The solution isThe surface $z=H$ is the first zero, so, using $c_0^2=2K{\rho }_0$,

The first equation is vertical hydrostatic equilibrium: the pressure gradient balances the vertical gravity of the central mass, ${\Omega }^{2}z$, and the disk's own potential ${\Phi }_d$. The second is the plane-parallel Poisson equation for disk self-gravity. The third balances the vertical increase of radiative flux against local viscous heating in a Keplerian alpha disk, and the fourth is the optically thick radiative diffusion law.

With $P/\rho \sim c_s^2$, $\Sigma \sim \rho H$, and all terms in hydrostatic balance comparable,The first comparison gives $H\sim c_s/\Omega$; inserting it into the second gives $\Omega c_s\sim G\Sigma$. Henceup to the order-one constants deliberately omitted by the scaling argument.

Take the divergence of the linear Euler equation, use $\delta P=c_s^{2}m\bar{n}\delta$, and differentiate the continuity equation. Eliminating $\nabla \cdot \delta \mathbf{v}$ givesThe perturbed Poisson equation isThusHere $k_J$ is the comoving Jeans wavenumber.

The magnetic field is $B=\nabla \times A$. The magnetostatic Ampère-Maxwell equation is $\nabla \times B={\mu }_{0}J$, so the curl of the curl identity givesA gauge transformation $A\mapsto A+\nabla \chi$ does not change $B$. We may therefore choose the Coulomb gauge $\nabla \cdot A=0$ by solving the Poisson equation ${\nabla }^2\chi =-\nabla \cdot A$. It follows component by component in a Cartesian coordinate system thatThe free-space Green function of the Laplacian then givesFor a localized steady current density, conservation of electric charge gives ${\nabla }^{\prime }\cdot J=0$; integration by parts confirms directly that this integral has zero divergence and hence obeys the Coulomb gauge.

Stationarity changes $\square$ to the spatial Laplace operator. The Green function of the Poisson equation therefore givesHere the bar denotes the trace-reversed metric perturbation; this is the variable denoted by $h$ in the displayed field equation of the question.

Let $k_{\mathrm{eq}}$ be the horizon scale at matter-radiation equality. At a fixed time soon after recombination, the cold-dark-matter transfer function defined as $T_c={\Delta }_c/\mathcal{R}$ has the schematic behaviorwith a smooth turnover near $k_{\mathrm{eq}}$. Large-scale modes enter the horizon during matter domination and the Poisson equation converts an almost scale-independent primordial potential into a density contrast proportional to $k^2$. Small-scale modes enter during radiation domination; radiation controls the potential and pressure prevents it from clustering, so cold-dark-matter growth is only logarithmic until equality. Subsequent matter-era growth multiplies all these modes by the same linear growth factor.

Equivalently, if the conventional matter transfer function is normalized to one as $k\to 0$, it is constant for $k\ll k_{\mathrm{eq}}$ and falls approximately as $\log (k/k_{\mathrm{eq}})/k^2$ for $k\gg k_{\mathrm{eq}}$. Multiplication by the Poisson factor $k^2$ gives the behavior of the definition used here.

The collapse is perpendicular to the initial field, so magnetic flux freezing preserves the mass-to-flux ratio of each material flux tube:With no toroidal field, radial magnetostatic equilibrium isDefine the effective polytropic constantThen gas and magnetic pressure combine as $P+B_z^2/(8\pi )=K_{\mathrm{eff}}{\rho }^2$. Dividing equilibrium by $\rho$, differentiating, and using the cylindrical Poisson equationgivesPut $a^2=K_{\mathrm{eff}}/(2\pi G)$. This is the order-zero Bessel differential equation, and regularity on the axis together with $\rho (0)={\rho }_1$ yields

SetThe spherical Poisson equation givesThe numerator identitysplits this into two scale-free density-potential components:Thus

At small radius, $D\sim 2b$ and the first term dominates:The cusp has finite enclosed mass because $r^2\rho \sim r^{1/2}$. Moreover,so the circular speed tends to zero as $r^{1/4}$. At large radius, $D\sim r$ andSince $r^2\rho \sim r^{-3/2}$ is integrable at infinity and $\Phi \sim -GM/r$, the total mass is finite and equals $M$.

In the razor-thin disk approximation, the three-dimensional Poisson equation isFor a horizontal Fourier mode with wavevector $\mathbf{k}$ and $k=|\mathbf{k}|>0$, the Fourier transform of this equation isThe solution that decays away from the disk isConsequently the required razor-thin disk Poisson kernel in the midplane isThe spatially uniform $k=0$ background is excluded from this local perturbation formula.

During matter domination, pressureless matter has ${\bar{P}}_m=\delta P_m=0$. On subhorizon scales the time derivative of the Newtonian gauge in cosmology potential is negligible, so the continuity and Euler equations reduce toTaking the divergence of the second equation, differentiating the first, and using the Poisson equation givesSince $d/d\tau =a,d/dt$, one hasDivision by $a^2$ yields the standard linear cosmological density perturbation equation

The axisymmetric razor-thin astrophysical disk equations areFor the stationary background, mass and azimuthal momentum conservation are automatic, while radial force balance and the Poisson equation giveHere $d{p}_0/dR=0$ under the stated constant-pressure assumption.

Retaining first-order perturbations and using $c_s^2=dp/d\Sigma$ giveswhereis the squared radial epicyclic frequency. In the local $|k|R\gg 1$ limit, a Fourier mode $e^{i(kR-\omega t)}$ obeysEliminating ${\Sigma }^{\prime }$, $u_{\varphi }^{\prime }$, and ${\Phi }^{\prime }$ yields the local dispersion relation

For a Keplerian orbit, $\Omega \propto R^{-3/2}$, soInstability requires ${\omega }^2<0$, equivalently a mode with positive imaginary frequency. Treating the right-hand side as a quadratic in $x=|k|$, its two roots areReal distinct roots, and hence an unstable interval $x_{-}<|k|<x_{+}$, exist exactly when the Toomre stability criterion has $Q<1$.

With $h=c_s/\Omega$,The constant positive term is epicyclic restoration by rotation and stabilizes long wavelengths. The negative term is the razor-thin disk's self-gravity and drives collapse. The positive $k^2$ term is gas-pressure restoration and stabilizes short wavelengths. Gravitational instability can therefore survive only on an intermediate band of scales.

For $P=K{\rho }^2$, the specific enthalpy isHydrostatic equilibrium says $\nabla (H+\Phi )=0$. Taking a Laplacian and using the gravitational Poisson equation gives the Helmholtz equation

For a spherical star, regularity at the centre selects the $n=1$ stellar polytropeIts first zero is $kR=\pi$, henceDirect integration gives $M=4{\rho }_c{R}^3/\pi$, and therefore

On the cube, the separated positive solutionvanishes on all six faces. It solves the same Helmholtz equation whenThe mean of each sine over $[0,L]$ is $2/\pi$, so

The interior fields formally solve the local structure equations, but an isolated fluid surface must be an equipotential and its interior gravitational field must match a decaying exterior solution with continuous normal derivative. A cube does not satisfy the global free-boundary conditions for a nonrotating self-gravitating barotrope. Such sharp planar faces and edges are also not observed in stars; ordinary pressure and gravity smooth the body toward a sphere.

For a slowly moving particle in a static weak field, the spatial part of the geodesic equation reduces toTo first order in the perturbation,and henceNewtonian acceleration is $-{\delta }^{ij}{\partial }_j\Phi$, so $h_{00}=-2\Phi$. Substitution into part (a), followed by the Poisson equation ${\nabla }^2\Phi =4\pi G\rho$, givesThus the Newtonian limit of general relativity fixes

Multiply the Poisson equation by $v\in V$ and apply Green's first identity. The Dirichlet boundary condition makes the trace of $v$ vanish on ${\Gamma }_1$, while the Neumann boundary condition makes the boundary flux vanish on ${\Gamma }_2$. Thus the weak formulation is: find $u\in V$ such thatwhere

Differentiate the linearized cosmological continuity equation, $\dot{\delta }=-\theta /a$, and use $\dot{a}/a=H$:Substitution of part (a), followed by the Poisson equation ${\nabla }^2\Phi =4\pi G{a}^2\bar{\rho }\delta$, yields

The second estimate in part (a) bounds $({\omega }_m)$ in $H_{\mathrm{per}}^2$. The periodic Poisson equation $-\Delta {\Psi }_m={\omega }_m$ and the supplied curl identity then bound $({\Psi }_m)$ in $H_{\mathrm{per}}^4$ and $(u_m)$ in $H_{\mathrm{per}}^3$. After passing to a subsequence,The Rellich-Kondrachov compactness theorem also gives strong convergence in the corresponding spaces with one fewer derivative. In particular, $u_m\to u$ in $L^{\infty }$ and $\nabla {\omega }_m\to \nabla \omega$ in $L^2$, soPassing to the limit in the Galerkin equations givesThese identities have the claimed Sobolev regularity, and the first holds in $L_{\mathrm{per}}^2$. Finally, the weak lower semicontinuity of the Hilbert norm preserves the estimates

A continuous function $v$ is a Perron subfunction for $\Delta u=f$ when it lies below every classical solution of that Poisson equation on a relatively compact open ball whenever it lies below that solution on the ball's boundary. For $C^2$ functions, this is equivalent to $\Delta v\ge f$. A superfunction reverses the comparison and has $\Delta v\le f$ when $C^2$. Adding a boundary upper bound gives the family whose supremum defines the Perron method for the Dirichlet problem.

A mean-field coupling of the Poisson equation to an ideal Boltzmann distribution of mobile ions. In Gaussian units with uniform dielectric constant $ϵ$, ${\nabla }^2\varphi =-(4\pi /ϵ)[{\rho }_f+{\sum }_i{q}_i{n}_{i,0}{e}^{-\beta q_i\varphi }]$. The potential reference and chemical potentials determine the normalization factors. It neglects discrete-ion correlations and assumes local thermal equilibrium. A counterion-only model is not the same equation as a symmetric salt reservoir.

For mass density $\rho$, the Newtonian gravitational potential satisfies the Poisson equation ${\nabla }^2\Phi =4\pi G\rho$. The gravitational acceleration is $-\nabla \Phi$. An isolated system also requires the exterior vacuum solution and decay at infinity; solving the interior equation alone does not fix a globally physical field.