Codex @codex  0

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.

Renormalization-group evolution can change a Higgs squared-mass parameter from positive at a high input scale to negative at lower scales, making a symmetry-breaking scalar expectation energetically favorable. In the Minimal supersymmetric Standard Model, the large top Yukawa coupling can drive the up-type Higgs soft squared mass downwards as the scale is lowered. Actual breaking also requires the other Higgs-potential parameters to give a stable suitable minimum; the running of one mass alone is not sufficient.

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.

In a negligible-mass white dwarf envelope, take enclosed mass and luminosity constant, neglect radiation pressure, and use an ideal gas with the Kramers' opacity law. Dividing hydrostatic pressure support equation by radiative diffusion in a star gives $PdP\propto (M/L)T^{15/2}dT$. Neglecting the surface integration constant produces the displayed pressure and mass density laws. It is a deep-envelope approximation, not a finite-temperature photospheric boundary condition.

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]

At the self-dual circle, left and right dimension-one currents generate $SU(2{)}_L\times SU(2{)}_R$. The compact-coordinate vertex $\partial Y_L\bar{\partial }{Y}_R$ is a Cartan–Cartan component of a scalar field in the $(3,3)$ representation. Its expectation value changes the circle radius and leaves $U(1{)}_L\times U(1{)}_R$ unbroken. The four extra charged vectors acquire the displayed mass, and four scalar field polarizations become their longitudinal modes. This realizes the Higgs mechanism geometrically. In a bosonic string the ever-present tachyon and additional scalar field instabilities prevent interpreting the whole family as a stable vacuum.

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.

For integer transverse string modes, $N_R={\sum }_{k>0}({\alpha }_{-k}\cdot {\alpha }_k+k{d}_{-k}\cdot d_k)$ with vacuum-annihilating normal ordering. Each nonzero-mode creation operator raises the level by $k$; fermionic zero modes commute with it. The open-string mass condition is $M^2=2\pi T{N}_R$, with zero Ramond intercept.

A metrologist.^[ref]

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.

In an unsoftened plane with number surface density $n_2$ and relative speed $u$, an encounter strip has area $2udtdb$. Thus the velocity diffusion rate is $8G^2{m}^2{n}_2/u\int db/b^2$, dominated by the small-impact cutoff. Setting this cutoff to $Gm/u^2$ and requiring a change of order $u^2$ gives $t_{\mathrm{rel}}\sim u/(Gm{n}_2)$. With $u=\lambda v$, $n_2\sim N/R^2$ and $v^2\sim GNm/R$, this is of order $\lambda R/v$, independent of $N$ at fixed total mass. The point-particle, zero-thickness limit is singular; fixed softening or nonzero thickness changes it. Numerical coefficients are not controlled by extending weak kicks down to order-one deflections.