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An isopycnal is a surface of constant mass density. In a Boussinesq approximation, it is equivalently a surface of constant buoyancy.

The regular spherical isothermal inflow from mass density ${\rho }_{\infty }$ at rest at infinity has sonic mass density $e^{3/2}{\rho }_{\infty }$. Combining it with the Bondi sonic point and mass conservation gives $\dot{M}=\pi e^{3/2}{G}^2{M}^2{\rho }_{\infty }/c_s^3$.

For $\beta =16c_s^2/c^2$, the exterior sonic point of the barotropic spherical accretion equation lies outside $r_G=2GM/c^2$. Its mass density is ${\rho }_c={\rho }_{\infty }\exp [4/(1+\sqrt{1+\beta })-1/2]$. The selected smooth transonic accretion rate is $4\pi r_c^2{\rho }_c{c}_s$. Subsonic formal solutions are not assigned this rate without transonic selection.

Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.

Dividing the mass flux, momentum flux, and density-weighted buoyancy flux by ambient mass density gives the kinematic fluxes $q=\pi b^{2}V$, $m=\pi b^2{V}^2$, and $f=\pi b^{2}Vg({\rho }_0-\rho )/{\rho }_0$. This convention makes dimensional similarity solutions independent of the arbitrary ambient density unit.

For constant mass density $\rho$, kinematic viscosity $\nu$ and magnetic permeability ${\mu }_0$, the incompressible momentum equation isIt is the Navier-Stokes equation with Lorentz force density. The magnetic force decomposes into magnetic tension and a magnetic pressure gradient. A uniform streamwise pressure gradient need not make ordinary pressure independent of transverse coordinates.

For spherical planetesimals with mass density $\rho$ and total-number distribution $n(D)=K{D}^{-\alpha }$ between $D_{\min }$ and $D_{\max }$,Thus $K=6M_{\mathrm{tot}}(4-\alpha )/[\pi \rho (D_{\max }^{4-\alpha }-D_{\min }^{4-\alpha })]$ for $\alpha \ne 4$. At $\alpha =4$, use $K=6M_{\mathrm{tot}}/[\pi \rho \log (D_{\max }/D_{\min })]$. Divide $n(D)$ by belt volume to obtain the number-density distribution used in a planetesimal collision rate.

Continuity of temperature, pressure and mass density matches the ideal gas envelope pressure $\mathcal{R}{\rho }_b{T}_c$ to the nonrelativistic electron degeneracy pressure $K_{\mathrm{NR}}{\rho }_b^{5/3}$. Substituting this base mass density into the radiative-zero white-dwarf envelope law gives the luminosity-temperature scaling behind Mestel's cooling law. The matched point is a transition approximation; the cold degeneracy law does not represent finite-temperature corrections there exactly.

A material curve consists of the same fluid elements at all times. Differentiating their flow map gives $D\mathbf{\ell }/Dt=(\mathbf{\ell }\cdot \nabla )\mathbf{u}$ for its tangent. In ideal magnetohydrodynamics a magnetic-field tangent obeys this same transport law after division by mass density.

The mean molecular weight per electron is the mass carried per free electron in units of atomic mass $m_u$. Fully ionized hydrogen has ${\mu }_e=1$, while helium, carbon, and oxygen have approximately ${\mu }_e=2$. It converts electron number density to bulk mass density in the equation of state of a cold electron gas.

The reflection-symmetric Newtonian gravitational potential $\Phi =v_0^2\log [(\sqrt{R^2+z^2}+|z|)/R_{\ast }]$ is harmonic off the plane. Its normal-derivative jump is $2v_0^2/R$, so the Poisson equation for Newtonian gravity gives surface density of a disk $S=v_0^2/(2\pi GR)$ and volume mass density $\rho =S\delta (z)$. The midplane circular speed is constant. The scale-free model has infinite total mass and does not use a finite-mass boundary condition at infinity.

A uniform solid sphere of mass $M$ and radius $a$ has the indicated moment of inertia about any diameter. Integrating the squared perpendicular distance against uniform mass density gives $I=8\pi \rho a^5/15=2M{a}^2/5$. It is different from the value for a thin spherical shell and determines the rotational contribution to kinetic energy during rolling without slipping.

A finite-amplitude ideal-MHD Alfvén wave can rotate the transverse magnetic field at constant magnitude, with constant mass density, pressure and longitudinal components. Its transverse velocity obeys ${\mathbf{u}}_{\perp }=\mathbf{C}-s{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0\rho }$ and its speed is constant along the wave. The constant magnetic pressure prevents compressive acceleration.

For the logarithmic-potential two-integral distribution, $q^2>1$ makes $A<0$. Accessible phase space satisfies $L_z^2{e}^{-2E/V^2}\le V^2/e$, attained by equatorial circular motion. Hence the galactic distribution function is nonnegative precisely when $B+A{V}^2/e\ge 0$, giving the displayed upper bound. A larger prolate flattening parameter produces negative phase-space density even though the spatial mass density remains positive.

The ocean mixed layer is a near-surface layer with relatively weak vertical variations of temperature and mass density, maintained by mixing. Beneath it, stable density stratification can isolate colder water from recent surface heating. The mixed-layer depth is not necessarily the full local water depth.

Multiply encounter mass flux $\Sigma (x)Sxdx$ by specific angular-momentum gain $r_0\Delta v_y$. The net disk torque replaces $\Sigma (x)$ by $\Sigma (x)-\Sigma (-x)$, so a symmetric mass density gives zero net torque. For constant surface density, Keplerian shear and the full-flyby value $\chi =2$, the one-sided coefficient is $8/27$ in $q^2\Sigma r_0^4{\Omega }^2(r_0/H{)}^3$, matching the primary coplanar impulse normalization.

If a spherical parent and its fragments have the same mass density, substitute $m=\pi \rho D_c^3/6$ into the endpoint-dominated cross-section of a power-law fragment population. The diameter obeysThis inversion requires specified fragment cutoffs. If $D_{\max }=\eta D_c$ instead, solving that dependence changes the power of $D_c$ to $\alpha -1$. It is useful for estimating the parent comet needed to supply an observed debris disk, subject to the assumed fragment model.