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For an Itô process $d{X}_t=b_{t}dt+{\sigma }_{t}d{B}_t$ and a twice differentiable function $f$,The second-order term reflects the nonzero quadratic variation of Brownian motion.

A Prony brake measures shaft power using a resisting band: its force difference gives torque $\tau =r(F_1-F_2)$, and rotation gives $P=\tau \omega$, where $\omega$ is angular velocity. The Stirling engine experiment above uses this independent load measurement to distinguish shaft output from gas-cycle work.

For a spatially homogeneous minimally coupled scalar field with zero scalar potential, the stress-energy tensor has $\rho =P={\dot{\varphi }}^2/2$. Its field equation is $\ddot{\varphi }+3H\dot{\varphi }=0$, so $a^3\dot{\varphi }$ is constant and its energy density decreases as $a^{-6}$. Thus its homogeneous stress is precisely that of a stiff fluid. Spatial gradients would introduce directional stresses, so this equivalence requires homogeneity.

Thermosteric sea level change is the thermal expansion part of steric sea level change. For a uniform layer of depth $H$, conservation of mass gives $\rho H=(\rho +\Delta \rho )(H+\Delta \eta )$, hence $\Delta \eta \simeq -H\Delta \rho /\rho =H\alpha \Delta T$.

The optical fiber acceptance angle is the largest launch half-angle admitted by the specified guide model. For a flat entrance face perpendicular to the optical axis, external refractive index $n_0$, and a straight step-index optical fiber, Snell's law at the entrance givesAt the core–cladding interface, the critical angle for total internal reflection $i_c$ satisfiesFor a meridional ray, strict total internal reflection requires $i>i_c$. At the limiting geometry $i=i_c$, the refracted light ray is tangent to the interface, soThis identifies the numerical aperture with an entrance cone under these assumptions. With $n_0=1$ for air, the conventional inclusive boundary condition isfor a nonnegative launch angle $\theta$; a practical launch stays inside the boundary. If $\mathrm{NA}>n_0$, this ideal model imposes no further restriction within the forward hemisphere, and $\arcsin (\mathrm{NA}/n_0)$ must not be used. The light ray must also intersect the optical fiber core: a suitable angle alone does not correct a misplaced beam.

For an original example, take $n_1=1.50$, $n_2=1.48$ and air at the entrance. ThenThe internal limiting angle is only ${\varphi }_{\max }\simeq 9.37^{\circ }$, while the critical angle for total internal reflection is $i_c\simeq 80.63^{\circ }$. A $10^{\circ }$ launch has $\varphi \simeq 6.65^{\circ }$ and $i\simeq 83.35^{\circ }$, so it satisfies total internal reflection. A $20^{\circ }$ launch has $\varphi \simeq 13.18^{\circ }$ and $i\simeq 76.82^{\circ }$, so it fails this test.

This cone is a geometrical optics result for meridional rays. A single-mode optical fiber requires the spatial overlap of the incident electromagnetic wave with its guided normal mode; its numerical aperture does not impose an exact angular cutoff on arbitrary beams. Diffraction, finite cladding, and bends require a physical optics description.^[ref]

Using an axisymmetric hydrodynamic mass flux function, set $s=s(\psi )$, $\ell =R{u}_{\varphi }=\ell (\psi )$ and $\epsilon =\epsilon (\psi )$. The azimuthal vorticity is ${\omega }_{\varphi }=-R\nabla \cdot (\nabla \psi /(\rho R^2))$. The meridional projection of the Crocco form of the unsteady Euler equation gives the displayed equation. The mass density also satisfies the Bernoulli relation $\epsilon =|\nabla \psi {|}^2/(2{\rho }^2{R}^2)+{\ell }^2/(2R^2)+w+\Phi$, so an equation of state is part of closing the system.

A magnetized stellar wind extracts angular momentum through matter transport and Maxwell stress tensor. The Alfvén radius supplies an effective lever arm larger than the stellar radius, with geometry factors absorbed into an averaged lever arm. Dynamo action couples the torque to rotation by supplying the magnetic field. For a simplified radial wind of fixed speed, $R_A^2$ scales as $B_{\ast }^2{R}_{\ast }^4/({\mu }_0{\dot{M}}_w{v}_w)$, so unsaturated $B_{\ast }\propto \Omega$ gives a torque proportional to $-{\Omega }^3$. Magnetic topology and wind thermodynamics can alter this scaling.

An episode of particularly strong mass loss near the end of Asymptotic giant branch evolution, capable of removing the remaining envelope rapidly compared with its prior evolution. Radiation acting on dust can transfer momentum to the gas; pulsations help lift material into cooler dust-forming layers. Envelope removal terminates sustained hydrogen-burning shell and helium-burning shell activity.

In spherical stellar structure, a shell of radius $r$, thickness $dr$ and mass density $\rho$ has mass $dm=4\pi r^2\rho dr$. Integrating outward gives the enclosed mass, $m(r)=4\pi {\int }_0^r\rho (s)s^{2}ds$.

The regular $n=5$ Lane-Emden equation solution has no finite first zero. Its Lane-Emden mass formula gives $m(\xi )=4\pi {\rho }_c{\alpha }^3{\xi }^3/[3(1+{\xi }^2/3{)}^{3/2}]$, so its infinite-radius total mass is $4\pi \sqrt{3}{\rho }_c{\alpha }^3$. Its enclosed mean mass density is ${\rho }_c(1+{\xi }^2/3{)}^{-3/2}$ and tends to zero. A finite externally confined core instead requires a specified truncation pressure.

In stellar homology, a fixed dimensionless density profile gives ${\rho }_c=C_{\rho }M/R^3$ through the mass integral. Integrating the stellar hydrostatic equation through the same profile gives $P_c=C_{P}G{M}^2/R^4$ when surface pressure is negligible. The constants depend on the shape, rather than the mass and radius. For an ideal gas, $T_c=(C_P/C_{\rho })\mu GM/(\mathcal{R}R)$.

For radiative, gas-pressure-supported homologous stars with constant opacity, write $\rho =Mb/(4\pi R^3)$, $P=G{M}^{2}p/(4\pi R^4)$ and $T=\mu GMp/(\mathcal{R}Rb)$. The radiative diffusion in a star equation contains the dimensionless coefficient $D=3{\kappa }_0{\mathcal{R}}^{4}L/(64{\pi }^{2}ac{\mu }^4{G}^4{M}^3)$. A common fixed profile fixes $D$, so $L\propto M^3$ and the luminosity of a fixed-mass member is constant. Gravitational energy generation in a homologously contracting ideal-gas star gives $\dot{R}=-2EL{R}^2/(3G{M}^2)$ for a positive profile constant $E$. Thus $R^{-1}=R_0^{-1}+2ELt/(3G{M}^2)$. For CNO cycle heating $ϵ\propto \rho T^{16}$, the stellar nuclear fusion luminosity scales as $M^{18}{R}^{-19}$. Equating this with $M^3$ gives $R_{\mathrm{MS}}\propto M^{15/19}$. Neglecting $R_0^{-1}$, the arrival time is $t_{\mathrm{MS}}\propto M^{-34/19}$, hence a coeval cluster's newly arriving mass scales as $t^{-19/34}$. All comparisons hold stellar composition, opacity normalization and dimensionless profiles fixed.

A mass-preserving stellar homology with scale $x$ has mass density proportional to $x^{-3}$. For an adiabatic process with constant adiabatic index $\gamma$, its internal energy scales as $x^{-3(\gamma -1)}$ and its Newtonian gravitational potential energy as $x^{-1}$. Writing $a=3(\gamma -1)$, stationarity at $x=1$ requires $W_0=-a{U}_0$, and the second scale derivative is $a(a-1)U_0$. Thus this particular displacement is restoring for $\gamma >4/3$, destabilizing for $1<\gamma <4/3$, and marginal at $\gamma =4/3$. This is a test of homologous radial stability; it does not by itself test every stellar displacement.

For a monatomic ideal gas fully convective star, $P=K{\rho }^{5/3}$ and $K\propto R{M}^{1/3}$. Matching its adiabat to a photosphere with $P\sim g/\kappa$ and $\kappa \propto \rho T^{10}$ gives $T_{\mathrm{eff}}^{14}\propto M^{2}R$, and hence surface luminosity $L\propto M^{4/7}{R}^{16/7}$. Specific heating $ϵ\propto \rho T^5$ gives $L\propto M^7{R}^{-8}$. Equality yields the displayed laws and $T_{\mathrm{eff}}\propto M^{3/16}$, so the theoretical HR slope is $32/3$. Fixed composition, efficient convection and the surface matching approximation are essential.

For flux $F$ and effective dust opacity $\kappa$, radiative acceleration is $\kappa F/c$. Comparing it with the gravitational acceleration gives an Eddington luminosity condition appropriate to dusty gas. In an optically thick starburst galaxy, feedback can limit the star formation rate per area and drive a galactic outflow, while the gas supply limits the duration. A surface-density limit is not a universal ceiling on total star formation rate: the star-forming area matters. Radiation-supported starburst disk calculations give characteristic optically thick infrared fluxes near $10^{13}{L}_{\odot }{\mathrm{kpc}}^{-2}$.

A spherical mass density $\rho$ gives projected surface density $\Sigma (R)={\int }_{-\infty }^{\infty }\rho (\sqrt{R^2+z^2})dz$. On the positive half of the line of sight, substitute $r=\sqrt{R^2+z^2}$ to obtain the displayed Abel transform. Spherical symmetry is essential: a general three-dimensional density cannot be reconstructed uniquely from one projection.

A collisional self-gravitating stellar core can lose heat to its halo, contract and become hotter because it has negative heat capacity of a virialized gravitational system. The increased temperature contrast drives further outward heat flow and further contraction. The classical gravothermal catastrophe is the loss of a finite-density thermodynamic equilibrium at a finite total energy along the confined three-dimensional Newtonian isothermal sequence. It concerns slow two-body relaxation, not simply a collisionless instability. The dimensional form of the gravitational interaction and the thermodynamic ensemble matter: a two-dimensional logarithmic gravitational gas need not have the same finite-energy instability.

A spherical stellar model with total mass $M$, scale radius $a$, inner density proportional to $r^{-2}$ and outer density proportional to $r^{-4}$. Integrating its mass density gives $M(r)=Mr/(r+a)$, hence Newtonian gravitational potential $\Phi =(GM/a)\log [r/(r+a)]$, normalized to zero at infinity. Its circular speed satisfies $v_c^2=GM/(r+a)$: it tends to a constant inside the scale radius and becomes Keplerian outside. The central density cusp has vanishing enclosed mass as $r\to 0$ and does not imply a central point mass.

A stellar orbit with sensitive dependence on its initial phase-space point, often diagnosed by a positive Lyapunov exponent. It need not explore the whole energy hypersurface: regular islands and disconnected chaotic regions can coexist. If an orbit is dense in a connected energy-shell region, every continuous integral of motion is constant on its closure; that stronger condition excludes further independent continuous constraints within that region.