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In a chemically homogeneous adiabatic spherical region, $d\rho /dr=-\rho g/\Phi$, coupled to $dm/dr=4\pi r^2\rho$ and $g=Gm/r^2$. Seismic velocities supply $\Phi$; density normalization and jumps require further data and assumptions.

The pressure response to compression at fixed entropy, $K_S=\rho (\partial P/\partial \rho {)}_S$. Seismic compressions are commonly approximately adiabatic; a background density gradient follows the corresponding derivative only when composition and entropy assumptions also apply.

Agarose gel electrophoresis uses a porous agarose polymer network. For linear double-stranded DNA within a suitable separation range, shorter fragments generally move farther in a fixed run. Raising the agarose concentration changes the mesh and the range of sizes that can be resolved.^[ref] A compact circular DNA molecule need not migrate like linear DNA of the same length: geometry affects passage through the mesh.^[ref]

A concrete laboratory example is Addgene's documented DNA separation protocol. Its standard 1% gel contains 1 g of agarose in 100 mL of TAE buffer. Dissolved agarose sets in a casting tray around a comb; removing the comb leaves sample wells. The gel is covered with the same buffer solution in a box connected to a voltage source. Loading buffer contains glycerol to increase sample density and a tracking dye to show run progress. A DNA ladder occupies a separate lane beside the samples. With wells near the negative cathode, DNA migrates toward the positive anode. After separation, bound stain makes the DNA visible by fluorescence under a UV transilluminator. The DNA ladder supplies the size reference; the tracking dye monitors migration but does not identify sample DNA.^[ref]

In the manufacturer's controlled comparison, the same DNA ladder runs in 1%, 2%, and 3% agarose gels with the same run time and other stated conditions. The changed gel band spacing shows why a calibration belongs to its particular gel conditions, rather than establishing a universal distance law.^[ref]

In interpreting an unknown sample, a blank lane tests for detectable background or carryover in that blank, while the DNA ladder checks migration and detection of its known fragments. An empty sample lane cannot by itself distinguish absent DNA from unsuccessful loading or insufficient detection. These controls answer different questions; neither makes matching migration a proof of matching nucleotide sequence.

The flat barotropic matter-vacuum solution has $H_{\Lambda }/H_0=\sqrt{{\Omega }_{\Lambda 0}}$ and therefore the displayed age, measured from its Big Bang. At fixed $H_0$ and density fractions it is $1/(1+w)$ times the dust-plus-vacuum age. Its $w=0$ case is the age of a flat matter-Lambda universe.

By Hagai Netzer. Section2.2 explains collisional excitation, critical density and high-density thermalized populations.

A local compensation model with approximately fixed crustal density and variable crustal thickness. A height $h$ is supported by a crustal root $b={\rho }_{c}h/({\rho }_m-{\rho }_c)$ in a simple constant-density, air-loaded column model.

The bulk photon mode density per volume per angular frequency is ${\omega }^2/({\pi }^2{c}^3)$, including two transverse polarizations. Multiply it by the thermal oscillator energy $\hslash \omega /(e^{\hslash \omega /(k_{B}T)}-1)$ to obtain the displayed spectrum. Its integral is $u={\pi }^2{k}_B^4{T}^4/(15{\hslash }^3{c}^3)$. This is volume energy density rather than radiance or surface exitance; isotropic equilibrium radiation has outward black-surface flux $cu/4$. zero-point energy is excluded from this thermal spectrum.

The fraction $f_V$ of a geometric volume occupied by a specified emitting gas phase is its volume filling factor. For approximately uniform density within that phase, a density-squared luminosity scales as $n_e^2{f}_{V}V$. Without knowledge of $f_V$, emission gives an rms-density combination rather than a unique mean gas density. This filling factor is distinct from the fraction of a telescope beam occupied on the sky.

Official 2009 material table: pdg.lbl.gov/2009/AtomicNuclearProperties/HTML_PAGES/074.html . It gives the reference density $19.3\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. It is an external reference, not the historical NST examination Data Book and not a measured hot-filament density.

For point ions of charge $\pm e$ in potential $\psi$, equilibrium electrochemical potentials give $n_{\pm }=n_0\exp [\mp e\psi /(k_{B}T)]$, with $n_0$ the reservoir density of each species. The charge density is $-2e{n}_0\sinh [e\psi /(k_{B}T)]$. A negative wall enriches positive counterions and depletes negative co-ions.

An infinite uniform line with proper charge per unit length ${\lambda }_0$ moving axially at speed $u$ has Lorentz-contracted density $\lambda =\gamma {\lambda }_0$ and current $I=\lambda u$. Transverse distance $r$ is unchanged by the boost. The rest electric field ${\lambda }_0\hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ transforms into $\mathbf{E}=\lambda \hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ and $\mathbf{B}=\mathbf{u}\times \mathbf{E}/c^2$. This equals the magnetic field ${\mu }_{0}I\hat{\mathbf{\phi }}/(2\pi r)$ from Ampère's law. The formulas concern a uniform stationary current pattern, not an accelerating finite filament's radiation.

Compare uniform three-dimensional ideal gases at the same number density n, with fermion mass $m_F$, spin degeneracy $g_F$, boson mass $m_B$ and a single boson spin component. The Fermi temperature has $k_B{T}_F={\hslash }^2(6{\pi }^{2}n/g_F{)}^{2/3}/(2m_F)$, while Bose saturation gives $k_B{T}_B=2\pi {\hslash }^2[n/\zeta (3/2){]}^{2/3}/m_B$. Eliminating density yieldsThe degeneracy is essential; using a spinless Fermi sea changes the numerical estimate. This is an ideal-gas comparison and does not remove interaction effects in a dense liquid.

A transit radius $R_p$ and an inclination-constrained mass upper limit $M_{\max }$ imply ${\bar{\rho }}_p<M_{\max }/(4\pi R_p^3/3)$. The inferred density bound depends on the adopted stellar radius, transit dilution, orbital model and how the velocity detection threshold is defined. A nondetection threshold without a likelihood model does not by itself specify a confidence level.

Buoyant density is the surrounding liquid density at which a species has no net centrifugal drift under the stated conditions. It depends on interactions with the medium, including associated solvent, rather than necessarily equaling the dry substance's density. Equal buoyant densities can give overlapping bands even for differently sized proteins or DNA fragments.^[ref]

A circular loop of radius $R$ has constant curvature $1/R$ and contour length $2\pi R$. Integrating the quadratic bending energy density $K_b/(2R^2)$ gives the displayed energy. With $K_b=k_{B}TA$ this is $\pi k_{B}TA/R$. Applied-force work, axial strain, twist and contact energies are separate terms; the loop formula alone does not determine an actual loaded loop's radius.

Chapter 2 of Statistical Physics by David Tong. Sections 2.1–2.2 give ideal classical partition functions; section 2.5 gives the density virial expansion and Mayer f-function. No separately contributed chapter author is established.

The cold predominantly atomic neutral phase of interstellar gas, with characteristic temperatures of order $10^2\mathrm{K}$. It is denser than the warm neutral phase at comparable thermal pressure. Its existence depends on radiation, grain heating and atomic-line cooling; one universal density or temperature does not define every environment.

Relaxation of a density or composition fluctuation involving many constituents. Its coefficient can depend on osmotic compressibility and collective mobility, and need not equal the tagged-particle self-diffusion coefficient at nonzero concentration.

The positive mean loss of projectile kinetic energy per path length through electronic excitation and ionization, conventionally $S_{\mathrm{col}}=-dT/dx$. Dividing by absorber density gives mass stopping power.

A common reference depth at which lateral pressure differences are assumed relaxed in an isostatic model. It must lie below relevant lateral density anomalies; it need not be a sharp material interface.