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In the spherical approximation, equatorial centrifugal acceleration balances surface gravity when ${\Omega }_c^2{R}_c=GM/R_c^2$. For $I=\alpha M{R}^2$ and conserved angular momentum $J$, this gives the displayed critical radius. Real rotating gas can deform, lose matter and exchange torques before a spherical model remains accurate at this limit.

An attractive inverse-square force together with radial drag $-k\dot{r}/r^2$ preserves angular momentum $h=r^2\dot{\theta }$ but modifies the Binet equation for $u=1/r$ by a first derivative. For $k/|h|<2$ and $h\ne 0$, the reciprocal radius is a decaying oscillation about $p/h^2$. Forward angular distance $\mathrm{sgn}(h)(\theta -{\theta }_0)$ makes its damping sign explicit for either orbital orientation.

A dark-matter halo merger joins previously distinct dark-matter halos. Orbital angular momentum can become internal halo spin, while escaping material carries away some mass and angular momentum. Halo mergers and accretion are important reasons that early tidal torque theory is only an approximate predictor of the spin of a later virialized remnant.

A local odd mass density slope $\Sigma (x)-\Sigma (-x)=2{\Sigma }_0\beta x/r_0$ changes the cancelling one-sided torques into this net torque. A local outer cutoff $L$ adds the factor $1-(H/L{)}^2$. Positive slope transfers net angular momentum from satellite to disk. An infinite linear mass density law is only a formal local extension, since it cannot remain nonnegative on both sides.

An isotropic escaping stellar wind has the donor star's mean specific orbital angular momentum. With ${\dot{M}}_d<0$, retained fraction $f$, negligible spin and no additional wind torque, $\dot{J}=(1-f){\dot{M}}_d{a}_d^2\Omega$. The retained matter redistributes angular momentum internally. Unlike isotropic re-emission from a binary star, the escaping material originates at the donor rather than the accretor. With no retention this reduces to Jeans-mode mass loss.

For a singular isothermal sphere of dispersion $\sigma$ and gas fraction $f_g$, placing all gas inside radius $R$ onto a hole in time $R/\sigma$ gives ${\dot{M}}_{\mathrm{dyn}}\sim 2f_g{\sigma }^3/G$. This is an optimistic supply bound that omits angular momentum transport, star formation, and active-galactic-nucleus feedback. A numerical comparison with the Eddington accretion rate requires a gas fraction, reference radiative efficiency, opacity, and crossing-time convention.

The differential Newtonian gravitational field of a galaxy across a smaller bound system. Its tidal tensor can slowly change the angular momentum of distant comets, changing their periapses and their access to planetary encounters.

A galaxy merger combines two or more galaxies into a remnant. Their orbital angular momentum and energy are redistributed into internal stellar and halo motions, escaping matter, and, when present, dissipative gas. A bound pair produced by tidal capture of galaxies need not coalesce immediately.

In cylindrical coordinates, the Newtonian gravitational stress tensor gives radial angular-momentum flux $r{T}_{r\varphi }=r({\partial }_r\Phi )(r^{-1}{\partial }_{\varphi }\Phi )/(4\pi G)$. Nonaxisymmetric structure can correlate the field components and transport angular momentum. The isotropic negative-pressure part has zero off-diagonal component, so it gives no direct radial transport. This separates a shear flux from a contribution to normal force balance.

A potential $V(r,\varphi ,z)=f(r,\varphi -kz)$ is invariant under the screw transformation $(\varphi ,z)\mapsto (\varphi +ks,z+s)$. The Noether conserved quantity for a mechanical point symmetry isa linear combination of axial linear momentum and axial angular momentum.

During a spherical stellar homology contraction at fixed mass and angular momentum, a constant inertia coefficient gives $I=\alpha M{R}^2$ and solid-body rotation gives $\Omega =J/I\propto R^{-2}$. A fixed fluid label $s=r/R$ retains its enclosed mass and polar angle, so centrifugal acceleration scales as $R^{-3}$ and gravity as $R^{-2}$. Their ratio increases as $R^{-1}$. This assumes spin conservation, not continuous synchronization to an external orbit.

For two groups carrying angular momenta $L_a,L_b$ with mutual inclination $\Delta I$, resolving total angular momentum parallel and perpendicular to the second group's normal gives its orbital inclination relative to the invariable plane:For a positive denominator, $\tan I_b=\sin \Delta I/(L_b/L_a+\cos \Delta I)$. The larger-angular-momentum group lies closer to the invariable plane.

The inertia tensor about a point isIt maps angular velocity to angular momentum.

Constant areal velocity gives constant specific angular momentum $l=r^2\dot{\theta }$, hence zero transverse acceleration and a central force. For a focus-based ellipse $u=1/r=(1+\epsilon \cos \theta )/A$, one has $u_{\theta \theta }+u=1/A$. The Binet equation then gives the displayed radial acceleration, proving an attractive inverse-square law. An orbit's shape alone does not specify its acceleration without a time law.

The invariable plane is perpendicular to a system's total angular momentum. For an isolated planetary system, it is fixed while individual orbital planes precess.

For a Keplerian accretion disk with $\bar{\nu }\propto r^{5/2}$, dimensionless $g=\bar{\nu }\Sigma /({\bar{\nu }}_0{\Sigma }_0)$ satisfies $g_{\tau }=4x^{5/2}{g}_{xx}+6x^{3/2}{g}_x$. In $y=x^{-1/2}$ this is $g_{\tau }=y{g}_{yy}$. The ansatz $g=G(y/\tau )$ gives $G=A+B{e}^{-y/\tau }$; finite disk mass selects $A=0$. Its nonzero profile has logarithmically infinite disk angular momentum on an infinite radial domain.

The radius vector from the force centre to an orbiting body sweeps out equal areas in equal times. Its areal velocity is constant because a central force conserves angular momentum.

With zero cosmological constant and the component Hodge star operator convention $(\star j{)}_{abc}={ϵ}_{abcd}{j}^d$, the Einstein field equations and Killing equation give $d\star dm=-16\pi \star (T_{ab}{m}^b-\frac{1}{2}T{m}_a)$. If $m$ is tangent to a spacelike hypersurface, the trace term pulls back to zero. For $\partial \Sigma =S_{\infty }-{\bigcup }_i{S}_i$, Stokes theorem yieldsThe inner Komar charges cannot be dropped for a vacuum Kerr black hole, whose angular momentum is nonzero although its stress-energy tensor vanishes. A matter-only volume formula requires a regular filling with no inner boundaries.