Density

Mass density is mass per unit volume.

The dimensionless ratio of a material's density to a specified reference density, with reference conditions stated. Gases are commonly compared with dry air and liquids with water.

A nonnegative mass density proportional to $r^p$ has finite local mass near a straight axis if ${\int }_0^{\epsilon }{r}^{p+1}dr$ converges, namely $p>-2$. Near a three-dimensional center the corresponding condition is convergence of ${\int }_0^{\epsilon }{r}^{p+2}dr$, namely $p>-3$. These follow from the cylindrical volume element and spherical volume element. At equality the divergence is logarithmic. An unbounded density does not by itself imply infinite mass.

The spatial dependence of density, such as $\rho (r)$ for spherical matter. Hydrostatic equilibrium relates a gas profile and temperature to its confining gravitational mass.

The flux of mass carried by a continuum moving with velocity $\mathbf{v}$. In the nonrelativistic limit, ${\partial }_t\rho +\nabla \cdot {\mathbf{j}}_m=0$ is local mass conservation. A stationary mass current can produce a stationary gravitomagnetic field without emitting gravitational radiation.

The mean mass density of a body is its total mass divided by its volume. A nonuniform density distribution can have the same mean as a uniform one; orbital information that determines only $M/V$ does not determine the internal density profile.

For a spherical surface of radius $R$ with exterior mass density $\rho (r)={\rho }_0{e}^{-(r-R)/h_0}$, integrate the spherical volume element to obtain $M=4\pi {\rho }_0{\int }_0^{\infty }(R+h{)}^2{e}^{-h/h_0}dh$. The moments ${\int }_0^{\infty }{h}^n{e}^{-h/h_0}dh=n!h_0^{n+1}$ give the displayed expression. Its leading thin-atmosphere approximation is surface area times column mass, $4\pi R^2{\rho }_0{h}_0$; the two positive correction terms account for the growth of shell area with altitude.

Gas density is the mass density of a gas. For an ideal gas sample of amount of substance $n$, mean molar mass $\overline{M}$ and volume $V$, combine $m=n\overline{M}$ with $pV=nRT$ to obtainHere $R$ is the molar gas constant. For a homogeneous mixture of gases, $\overline{M}={\sum }_i{x}_i{M}_i$, where $x_i$ is the mole fraction of component $i$. Equal pressure and temperature make a gas density ratio equal to a mean-molar mass ratio within this model. Argon detection in a binary gas mixture gives a worked example and a temperature confounder.

In an idealized rigid-globe weighing, subtract the evacuated assembly's mass from the filled assembly's mass, then divide by the internal volume. Establish the filled sample's pressure and temperature before comparing gases. The subtraction removes the container's large contribution, making a small gas contribution accessible. A real measurement also needs corrections or bounds for residual gas, adsorption, leaks, changing vessel volume and external buoyancy. The Rayleigh-Ramsay argon density experiment shows why preparation-route comparisons add information beyond repeated weighings of one sample.

Linear mass density is mass per unit length, measured in $\mathrm{kg}{\mathrm{m}}^{-1}$. A uniform string of mass $m$ and length $L$ has $\mu =m/L$; with constant tension $T$, its transverse wave speed is $\sqrt{T/\mu }$. Mass density instead measures mass per unit volume.

The gas clumping factor measures unresolved inhomogeneity in electron number density within a chosen averaging region. Nonnegative variance gives $C_e\ge 1$. At a uniform temperature, thermal bremsstrahlung scales with $⟨n_e^2⟩$, whereas the thermal Sunyaev-Zeldovich effect scales with $⟨n_e⟩$. A homogeneous cluster distance from Sunyaev-Zeldovich and X-ray signals inference then gives $d_{A,\mathrm{inferred}}=d_{A,\mathrm{true}}/C_e$, for the same geometry and temperature. Correlated temperature variations require additional weighted averages.

Projected surface mass density is mass per unit projected area, obtained by integrating mass density along the line of sight. It determines the enclosed cylindrical mass used by a thin gravitational lens equation. It need not equal a physical disk's density, and its aperture mass differs from a spherical three-dimensional enclosed mass.