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An oscillatory feature with finite width in a source's variability power spectrum, rather than a perfectly periodic clock. Orbital, precession and disc-mode interpretations connect different frequencies to the same compact object. Identifying an observed peak with an innermost stable circular orbit orbital frequency is therefore a model assumption, not a direct black-hole mass or spin measurement.

The Q value of a reaction or particle decay is the initial total rest energy minus the final total rest energy. Positive $Q$ releases kinetic energy; a negative $Q$ requires energy supplied by incident motion or another source. Atomic and bare nuclear mass conventions must be distinguished when counting electrons in beta decay.

For $V=m^2{\varphi }^2/2+\lambda ({\varphi }^2{)}^2/4$, $m^2<0$, the vacuum manifold is a sphere of radius $v=\sqrt{-m^2/\lambda }$. At a chosen vacuum the Hessian matrix is $2\lambda v^{2}n{n}^T$, where $n$ is its unit direction. The radial scalar has mass squared $2\lambda v^2$; the $n-1$ angular modes are physical Goldstone bosons when the symmetry is global.

For the two-component case, write $\mathbf{\varphi }=(v+\rho )(\cos \theta ,\sin \theta )$. The kinetic term becomes $\frac{1}{2}({\partial }_{\mu }\rho {)}^2+\frac{1}{2}(v+\rho {)}^2({\partial }_{\mu }\theta {)}^2$. At quadratic order the canonically normalized angular field is $\pi =v\theta$, with no mass term. Its Euler-Lagrange equation is the wave equation, so ${\omega }^2=|\mathbf{k}{|}^2$ in units where the propagation speed is one. The radial field instead obeys ${\omega }^2=|\mathbf{k}{|}^2+2\lambda v^2$. Thus a common rotation costs no energy, whereas a slowly varying rotation propagates as a collective wave. With the selecting source of continuous vacuum selection by an external source, use $\pi =w\theta$ and obtain ${\omega }^2=|\mathbf{k}{|}^2+h/w$. This elementary dispersion relation separates a Goldstone boson, a pseudo-Goldstone boson and a massive radial excitation within one example.

Infer a spherically averaged density profile using seismic travel times, normal-mode frequencies, mass, moment of inertia, hydrostatic structure and material constraints. Seismic speeds alone measure elastic-modulus-to-density ratios, so they do not uniquely supply density.

A particle constrained to a stationary smooth surface of revolution $z=f(r)$ under uniform gravity conserves $h=r^2\dot{\theta }$ and mechanical energy per unit mass. Substitute $\dot{z}=f^{\prime }(r)\dot{r}$ and $\dot{\theta }=h/r^2$ into that energy to obtainThis proves the radial reduction. Simple roots of $E-U_h$ are turning points; a stationary point $U_h^{\prime }(r_0)=0$ supports an exact circular orbit. At a local maximum of $U_h$, another trajectory at the same energy can approach the orbit asymptotically as a separatrix. Existence of roots elsewhere is insufficient for confinement: their dynamically accessible ordering relative to the launch radius matters.

Let a test particle start at rest at radius $r_0$ outside a fixed spherically symmetric source. The spherical shell theorem gives the exterior Newtonian gravitational potential $\Phi (r)=-\mu /r$, where $\mu =GM$ is the source's standard gravitational parameter. Zero initial transverse velocity makes the motion radial. Conservation of mechanical energy per unit mass givesThe minus sign selects inward motion; the positive square root is its speed. For a fall from $r_0=2R$ to the surface $r=R$, that speed is $\sqrt{\mu /R}$. In the limit of release from rest infinitely far away, the familiar free-fall speed is $\sqrt{2\mu /r}$, equal in magnitude to the local escape velocity. A finite starting radius gives a smaller speed.

The arrival time follows from the same first integral. Set $r=r_0{\cos }^2\eta$. ThenFor arrival at $R>0$ this is finite, and the field formula is used only outside the source. For $r_0=2R$, $\eta =\pi /4$ givesKeeping the smaller initial gravitational magnitude $g(r_0)=\mu /(4R^2)$ constant would predict $\sqrt{8}\sqrt{R^3/\mu }$, about $10.0\%$ longer. The actual acceleration increases during the fall.

The curves compare a shallow fall, $h/R=0.02$, with a fall from $r_0=2R$, so $h/R=1$. Each comparison holds the release height and initial gravitational magnitude fixed. The constant-gravity model is close for the shallow fall and increasingly inaccurate for the larger one. This is also a practical use of nondimensionalization: the normalized trajectories depend on $h/R$, while $\sqrt{R^3/\mu }$ supplies the dimensional time scale.

In static coordinates on anti-de Sitter spacetime with curvature radius $L$ and $c=1$, put $f=1+r^2/L^2$. The conserved energy per unit rest mass is $E=fdt/d\tau$. Timelike normalization for a radial geodesic gives $(dr/d\tau {)}^2=E^2-1-r^2/L^2$, hence $d^{2}r/d{\tau }^2=-r/L^2$. An outward launch from the origin at proper time ${\tau }_0$ gives the displayed harmonic solution until its first return, after proper time $\pi L$, independently of $E>1$. Beyond that return, use a signed diameter coordinate or change the radial direction to keep the areal radius nonnegative. This is a coordinate realization of Anti-de Sitter timelike geodesic refocusing.

For a circular low-mass companion and a semi-amplitude threshold $K_{\min }$, $a_{\max }=G{M}_p^2{\sin }^{2}i/(M_{\star }{K}_{\min }^2)$. The edge-on case is the largest geometrically accessible distance. Adequate baseline and cadence are separate requirements, and a full peak-to-peak detection threshold is twice the corresponding semi-amplitude.

For a hydrostatic monatomic ideal gas with negligible external pressure, $2U+{\Omega }_f=0$, where gravitational potential energy is negative. The final total energy is therefore ${\Omega }_f/2$, giving the displayed radiated energy when no mass or other energy carrier escapes. The common $-{\Omega }_f/2$ result additionally takes the initial total energy to be negligible. Initial rest does not remove initial self-gravitational binding energy.

A 2013 experiment by Paul A. Williams and collaborators mounted a 25-mm dielectric mirror on a commercial force-restoration balance. Removing its pan-weight compensation spring allowed horizontal force sensing with a horizontal laser path. They measured mirror optical reflectance and varied incidence from about $7^{\circ }$ to $45^{\circ }$. A 1071-nm laser, capable of 530 W, was switched at 0.1 Hz. Digitized mass-equivalent readings were converted to force using $g$; quadrature demodulation isolated the switched signal from drift. ^[ref]

Measurement calibration used a 20-g mass, with checks down to 1 mg. The optical forces lay below that checked range, limiting the accuracy claim. An angle scan tested the predicted cosine dependence at fixed intercepted power. Replacing the mirror with a carbon nanotube coated absorber increased absorbed power while reducing the predicted optical force; no modulation peak was detected. This bounded gas-heating artifacts under those conditions, rather than proving their universal absence. The fitted slope was about 7% below $2/c$. Inferring laser power requires calibrated force, optical reflectance, radiometric absorptance and angle of incidence, with measurement uncertainty for each. ^[ref]

The radiative efficiency is the escaping radiative power divided by an adopted accreted rest-mass energy rate. Its meaning depends on where the mass rate is measured and which emission is counted. For efficient Newtonian accretion onto a stellar surface it scales as $GM/(R_{\ast }{c}^2)$, whereas advection or outflows can reduce luminosity for the same supplied mass.

For gas-pressure-supported stellar homology with opacity $\kappa ={\kappa }_{0}Z\rho T^{-3}$ and stellar energy-generation rate $ϵ={ϵ}_0{X}^2\rho T^5$, radiative diffusion in a star gives $L\propto R{T}_c^7/(Z{\rho }_c^2)\propto {\mu }^7{M}^5/Z$. Integrating the heating rate instead gives $L\propto X^2{\mu }^5{M}^7/R^8$. Equality in stellar thermal equilibrium yields the displayed radius law and $T_c\propto {\mu }^{5/4}{M}^{3/4}{X}^{-1/4}{Z}^{-1/8}$. This fixes the dimensionless profiles and coefficients, and neglects radiation pressure; it is a model comparison rather than a universal law for all main sequence stars. At fixed mass and metal mass fraction, homogeneous hydrogen burning gives the homogeneous fuel-depletion luminosity feedback through $L\propto {\mu }^7$.

Section 62.1 distinguishes inner bremsstrahlung from structure-dependent radiative pion decay and their interference. Equations (62.3)–(62.4) give the double differential decay width in terms of the scaled photon and charged-lepton energies and the squared lepton-to-meson mass ratio.

For unattenuated isotropic radiation from a point source, the radiation pressure force on each absorbing particle is $\sigma L/(4\pi c{r}^2)$. Dividing by its mass $\bar{m}$ changes the effective inverse-square inward acceleration to $[GM-\sigma L/(4\pi \bar{m}c)]/r^2$. The wind equation therefore places the maximum of a subsonic isothermal breeze at the displayed radius, provided the bracket is positive. If it is nonpositive, the positive-temperature subsonic outflow decreases in speed throughout its domain and has no interior speed maximum. A cross section per particle requires that particle's mass; an opacity per unit mass instead absorbs this conversion.

A shock wave with significant radiation losses in its downstream evolution. Often a narrow dissipative jump initially heats the gas according to adiabatic Rankine-Hugoniot conditions, followed by a much longer radiative cooling layer. The gas's specific entropy increases at the jump and may then fall as radiation carries entropy away. If both distant states have equal temperature, the combined mass and momentum balances give an isothermal shock; the gas's energy flux is not conserved across the whole radiating region.

The nickel-56 to cobalt-56 to iron-56 decay chain supplies gamma rays and charged particles that heat supernova ejecta. Early expansion and photon diffusion delay the escaping optical light; after a peak on a scale of weeks, the luminosity falls as decay power declines and gamma rays escape more readily. With comparable diffusion times and deposition efficiencies, peak luminosity scales approximately with the nickel-56 mass. Using one fixed absolute luminosity without accounting for composition-dependent yields can therefore bias standard-candle distances.

Place two equal point masses of $1\mathrm{kg}$ each on opposite sides of a fixed origin, at equal radius $r$, and let them share a signed angular velocity $\omega =\dot{\theta }$ in a plane. Their total mass is $M=2\mathrm{kg}$ and their moment of inertia about the perpendicular axis is $I=M{r}^2$. Radial pulling forces have zero torque about that origin, soTake an initial radius $r_i=1\mathrm{m}$ and angular velocity ${\omega }_i=1rad/s$, and a final radius $r_f=0.5\mathrm{m}$. ThenandThese are endpoint states with ${\dot{r}}_i={\dot{r}}_f=0$. During the pull, the full kinetic energy also contains radial motion: $K=M{\dot{r}}^2/2+K_{\mathrm{rot}}$. Assume the pulling apparatus supplies all net work to the two point masses; the work-energy theorem then fixes its net work at $+3\mathrm{J}$, including any radial acceleration and braking. Each mass spirals inward during the pull, allowing its inward force to do positive work. Conservation of angular momentum therefore predicts the final angular velocity without requiring kinetic energy conservation. For an apparatus included in a larger system, its stored energy supplies the increase consistently with conservation of energy. OpenStax uses the same mechanism to explain a skater pulling her arms inward. ^[ref]

A logarithmic comparison of specific power against specific energy for storage or conversion devices. With consistent mass and energy units, constant-duration lines satisfy $p_s=e_s/t$. The plotted capability depends on operating conditions and mass convention.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

Rate-zonal centrifugation resolves an initially narrow sample zone by differences in sedimentation coefficient during a finite interval. Its bands remain transient: extending the interval can carry material into a centrifugation pellet. Thus a density gradient need not imply an equilibrium buoyant density measurement.^[ref]

A typical preparation layers a narrow sample above a preformed density gradient, often containing sucrose. The initial zone must be narrow enough that differences in starting position do not overwhelm the later separation. The run ends while useful bands remain inside the liquid. In the simple case where all particles are denser than every part of the column, none has an interior isopycnic centrifugation position.^[ref][ref]

The sedimentation coefficient $s=(m-{\rho }_{f}V)/f$ explains why separation is not an ordering by mass alone. At fixed local centrifugal acceleration, increasing displaced volume can increase buoyancy, while a less compact shape can increase the linear friction coefficient and reduce velocity. Even for a sphere, Stokes's law gives $s=2b^2({\rho }_p-{\rho }_f)/(9\eta )$, so both the density difference and the dynamic viscosity matter. The resulting sedimentation coefficient can change along a density gradient.