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The sine-product cubic polytropic interior has zero mass density gradient at every corner. Its hydrostatic potential $C-2K\rho$ therefore has zero gradient there. In contrast, at a corner of a cube containing a positive mass density, every interior mass element contributes a gravitational-potential gradient pointing into the same coordinate octant. Each component of the gradient is strictly nonzero. This directly proves failure of global isolated gravitational matching, even though the interior hydrostatic and Poisson equations hold.

With vanishing surface pressure/mass density and regular admissible displacements, integration by parts makes the axisymmetric adiabatic displacement operator symmetric in the mass density-weighted inner product. Completing the pressure square gives the displayed energy form. Nonnegative form for every admissible displacement excludes exponentially growing axisymmetric adiabatic modes in the Cowling approximation. A negative trial Rayleigh quotient proves negative spectrum by the Rayleigh-Ritz variational principle. Nonnegative stratification and epicyclic coefficients give a simple sufficient stability condition.

For spherical fragments of common mass density $\rho$, with $n(D)=K{D}^{-\alpha }$ between specified diameter cutoffs, mass normalization of a power-law size distribution and integration of the geometric cross-section giveIndeed $m=(\pi \rho K/6)\int D^{3-\alpha }dD$ and ${\sigma }_{\mathrm{tot}}=(\pi K/4)\int D^{2-\alpha }dD$; eliminating $K$ proves the formula. If both cutoff ratios raised to $\alpha -3$ and $4-\alpha$ are small, respectively, this reduces to $3m(4-\alpha )D_{\min }^{3-\alpha }{D}_{\max }^{\alpha -4}/[2\rho (\alpha -3)]$. Approaching either endpoint requires the corresponding logarithmic integral.

Curvature depresses the equilibrium melting temperature of solid convex into liquid. Here $\mathcal{K}$ is the sum of principal curvatures, equal to twice a common convention for mean curvature, and $T_m$ is absolute temperature. The coefficient uses solid-liquid surface energy, mass density and latent heat. This stabilizes fine corrugations of a solidification front.

Fresh water reaches maximum mass density near $4{}^{\circ }\mathrm{C}$ at atmospheric pressure. A useful local equation of state isso water becomes less dense when its temperature moves either above or below $T_m$.

Applying the Poisson equation for Newtonian gravity to a flattened power-law gravitational potential yieldsFor $0<\beta <1$, nonnegative mass density everywhere outside the origin requires and is ensured by $\beta \le {\lambda }^2\le 2/(\beta +1)$. The central density singularity is locally integrable and carries no point mass, although the total mass diverges at large radius in this scale-free model.

The Poisson equation for Newtonian gravity givesFor $V>0$, nonnegative mass density away from the origin is equivalent to $q^2\ge 1/2$. Strict positivity requires $q^2>1/2$. This condition does not by itself guarantee a nonnegative galactic distribution function.

A local odd mass density slope $\Sigma (x)-\Sigma (-x)=2{\Sigma }_0\beta x/r_0$ changes the cancelling one-sided torques into this net torque. A local outer cutoff $L$ adds the factor $1-(H/L{)}^2$. Positive slope transfers net angular momentum from satellite to disk. An infinite linear mass density law is only a formal local extension, since it cannot remain nonnegative on both sides.

Density stratification is variation of a fluid's mass density with height. Its stable or unstable character depends on whether the buoyancy force restores or amplifies a displaced parcel.

At a material velocity jump with continuous background mass density, vertical displacement $\eta$ and fluid pressure are continuous, while $w=ikU\eta$ need not be. For a stationary component, $p=U^2{\eta }_z$ and the signed upward-wave impedance is $Z=U^{2}m$. Reflection and vertical-displacement transmission are $R=(Z_1-Z_2)/(Z_1+Z_2)$ and $T=2Z_1/(Z_1+Z_2)$.

Replacing domain-scale viscous stresses by a local linear drag density $-\bar{\eta }\mathbf{v}$ gives the scales $L_1=(\sigma \rho /{\bar{\eta }}^2{)}^{1/3}$ and $t_1=\rho /\bar{\eta }$. The single-scale force model isFor a regular power-law asymptotic $g\sim C{u}^y$, $g^{\prime }\sim Cy{u}^{y-1}$, $g^{\prime \prime }=Cy(y-1)u^{y-2}+o(u^{y-2})$, $y>0$, the two inertial-to-drag ratios scale as $(y-1)/u$ and $y/u$. Thus drag dominates inertia at late time, and drag-capillary balance integrates to $g^3\sim 3(c_s/c_d)u$. This proves the displayed $1/3$ exponent and its independence of mass density. The scaling assumes a homogenized stationary network and dominant advective transport. Diffusive Ostwald ripening can have the same exponent, so its omission must be justified separately; order-parameter mobility can otherwise enter the prefactor.

An effective Proton–proton chain network can track protons, deuterium, helium-3, helium-4 and lithium-7 after eliminating a rapidly adjusting beryllium intermediate. Identical-reactant event rates include a factor one half. Stoichiometric changes must count both helium-4 products of lithium-7 proton capture. The weighted baryon mass density $n_1+2n_2+3n_3+4n_4+7n_7$ is conserved by the nuclear reactions at fixed volume.

Small displacements in an isotropic elastic solid satisfy a vector wave equation governed by density and the two Lamé moduli.

A nucleus can capture an electron when energetically allowed, converting a proton into a neutron and emitting an electron neutrino. Beryllium-7 electron capture participates in one of the proton-proton chain branches. Stellar capture rates depend on ionization, mass density and electron phase space.

Electron number density is the number of free electrons per unit volume. In fully ionized stellar matter, $n_e=\rho /({\mu }_e{m}_u)$, with mass density $\rho$ and mean molecular weight per electron ${\mu }_e$.

For smooth ideal magnetohydrodynamics of constant mass density, define $I_{\pm }={\int }_V|{\mathbf{z}}^{\pm }{|}^{2}dV$. Taking the dot product of the Elsässer variable equation with $2{\mathbf{z}}^{\pm }$ givesTherefore each integral is constant when its outward boundary flux vanishes, for example when both velocity and magnetic field are tangent to the fixed boundary, or for periodic boundary conditions. The kinetic energy plus magnetic energy is $\rho (I_{+}+I_{-})/4$, while the cross-helicity is $\sqrt{{\mu }_0\rho }(I_{+}-I_{-})/4$.

For an incompressible flow with constant mass density, the Elsässer variables are ${\mathbf{z}}^{\pm }=\mathbf{u}\pm {\mathbf{v}}_a$, where ${\mathbf{v}}_a$ is the Alfvén velocity. Adding and subtracting the Euler momentum equation and ideal magnetohydrodynamic induction equation yields ${\partial }_t{\mathbf{z}}^{\pm }+({\mathbf{z}}^{\mp }\cdot \nabla ){\mathbf{z}}^{\pm }=-\nabla \psi$, with $\psi =p/\rho +\Phi +|{\mathbf{v}}_a{|}^2/2$ and $\nabla \cdot {\mathbf{z}}^{\pm }=0$. Thus each field is transported by the other, a useful way to expose interactions of oppositely directed Alfvén waves.

For constant positive order-parameter mobility $M$, dynamic viscosity $\eta$ and mass density $\rho$, deterministic Model H dynamics dissipates the sum of composition free energy and kinetic energy:Assume smooth fields and periodic or closed, no-work boundaries. Using the chemical potential of a composition field, integration by parts and incompressible flow gives $\dot{F}=\int \varphi \mathbf{v}\cdot \nabla \mu -M\int |\nabla \mu {|}^2$. Dotting the momentum equation with velocity gives ${\dot{E}}_{\mathrm{kin}}=-\int \varphi \mathbf{v}\cdot \nabla \mu -\eta \int |\nabla \mathbf{v}{|}^2$. Pressure and convective work vanish under the boundary assumptions; the two capillary transfer terms cancel. This proves both the sign of the Korteweg force and its reversible exchange of energy with the composition field.