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For laboratory velocity $v\hat{r}+w\hat{\theta }$ and field $B\hat{z}$, $F_{B,r}=qBw$ and $F_{B,\theta }=-qBv$. Their powers are opposite. The field-dependent guide-reaction power is $-qBvw$; the total ideal radial reaction for bead mass $m_b$ and constant expansion speed additionally contributes the ordinary centripetal term $-m_{b}v{w}^2/R$.

Magnetic rigidity measures the resistance of a charged particle's momentum to bending by a magnetic field. The relativistic cyclotron orbit gives $p_{\perp }=|q|Br$, where $r$ is the transverse gyroradius. For motion perpendicular to a uniform magnetic field, $p_{\perp }=p$ is the whole momentum. A measured magnetic rigidity fixes momentum per magnitude of electric charge; it does not by itself fix mass, kinetic energy, or the sign of the electric charge. This distinction makes the Anderson positron track experiment an especially revealing application of the Lorentz force.

For main-sequence stars, luminosity increases steeply with mass. A local power-law approximation uses an exponent that varies with the stellar mass range and composition; $\alpha \simeq 4$ is a useful rough near-solar estimate. Evolved stars need not follow that same calibration.

The main-sequence turnoff is the location in a coeval population's Hertzsprung-Russell diagram where stars are leaving the main sequence after exhausting their core hydrogen. More massive stars generally have larger luminosity per unit mass and shorter nuclear lifetimes, so the most massive stars still on the main sequence constrain the population's age. In a simple model, equating the stellar nuclear timescale $t\sim qM/L(M)$ to the common age determines a turnoff mass; detailed calibration requires composition and stellar models.

A regional planetary mass concentration producing a gravity anomaly; lunar examples are associated with large impact basins.

Mass measures inertia and acts as the source of Newtonian gravity.

Under mass action, random pairwise encounters between populations of sizes $X$ and $Y$ occur at a rate proportional to $XY$.

If birth counts obey $dN/dm\propto m^{-s}$ and individual luminosity scales as $m^q$, multiply by mass or luminosity and by $dm=md\ln m$ to obtain the displayed logarithmic weights. With $s=2.35$ and $q=4$, birth mass has weight $m^{-0.35}$ while light has weight $m^{2.65}$. Finite mass cutoffs are essential; stellar lifetimes also alter the luminosity weighting of an evolved population.

For an imposed equilibrium cutoff $m{c}^2=P{k}_B{T}_f$ and density $n_f=6[m{c}^2{k}_B{T}_f/(\hslash c{)}^2{]}^{3/2}{e}^{-P}$, dilution by $(T_0/T_f{)}^3$ cancels the explicit mass in the final number density. The rest-energy density still scales with particle mass. This result assumes constant entropy degrees of freedom and no post-cutoff annihilation; entropy conservation adds a factor $g_{\ast s,0}/g_{\ast s,f}$ when these assumptions are relaxed.

For fixed spring constant $k$ and linear damping coefficient $b$, replacing mass $m$ by $\lambda m$ changes the exponential envelope rate from $\gamma =b/(2m)$ to $\gamma /\lambda$. The damped angular frequency becomes ${\omega }_{d,\lambda }^2=({\omega }_d^2+{\gamma }^2)/\lambda -{\gamma }^2/{\lambda }^2$. Thus envelope decay times scale exactly as mass, while the familiar inverse-square-root frequency scaling is a weak-damping approximation. A constant gravitational force shifts equilibrium without changing either rate.

The mass of a specified constituent divided by the total volume of a sample.

Mass conservation states that matter is neither created nor destroyed. For mass density $\rho$ and velocity $\mathbf{v}$, its local form is ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{v})=0$.

A global warming potential compares equal-mass pulses. If ${ϵ}_i$ is the greenhouse gas radiative efficiency per ppb, an emitted mass produces a mixing-ratio increment inversely proportional to molar mass $M_i$. For two gases with exponential pulse removal, their GWP ratio at horizon $H$ is $({ϵ}_i/{ϵ}_j)(M_j/M_i){\tau }_i(1-e^{-H/{\tau }_i})/[{\tau }_j(1-e^{-H/{\tau }_j})]$. The common carbon-dioxide reference integral cancels. Comparing only the per-ppb coefficients would instead compare unequal emitted masses.

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.

At fixed incident energy and fixed potential-energy profile, the forbidden-region WKB action is $\int \sqrt{2m(V-E)}dx$. Its leading transmission exponent grows as the square root of mass. Equal electrostatic voltages need not define equal potential-energy barriers for particles with different charges. The leading exponential comparison assumes a sufficiently opaque barrier; prefactors and the WKB turning-point conditions remain relevant.

A description of how mass is assigned to positions, expressed for example by point masses, a mass density, or a measure. Writing $dm$ allows centre of mass and moment of inertia formulas to cover both discrete and continuous cases.

A mass eigenstate diagonalizes the Hermitian mass operator or mass matrix and has a definite mass. In a flavor basis with off-diagonal mass terms, it is a linear combination of flavor states. For unstable particles, decay propagation instead involves an effective complex matrix; its eigenvectors need not inherit the orthogonality of a Hermitian mass problem.

Institutional educational reference, last updated 13 May 2021 in the inspected page. Tom Benson is named as editor and NASA official; the institutional issuer is retained as author. Equations 1–10 express isentropic mass flow, and equation 19 identifies its sonic maximum.

Mass flux is the mass crossing a unit area per unit time. In a one-dimensional flow it is $\rho u$. Integrating its normal component over a surface gives the mass flow rate.

A compact astronomical object proposed as a constituent of a galactic dark matter halo. It can be sought through gravitational microlensing even if its emitted light is undetectable. Stellar objects and remnants consume the cosmological baryon budget; primordial black holes do not obey that particular formation-budget argument. Exclusion bounds depend on the object's mass, the halo model and survey selection.