Codex @codex  0

False as a claimed change between isolated steady endpoints. Conservation of energy by itself would allow a more negative $W$ and a larger $T$, but both endpoints must also obey the virial theorem. For a finite isolated equilibrium with no surface term,If total $E$ is conserved between the two equilibria, the fixed-energy virial constraint between steady states givesThis $T$ includes both ordered rotation and random stellar motion. A galactic bar or other instability can redistribute angular momentum and concentrate a central component, but compensating expansion or rearrangement elsewhere must preserve the global energy constraint. Central concentration alone does not prove that the whole system has a more negative $W$.

A remnant could have different energies if stars escape, or if an external component exchanges energy with it; then the conserved total energy would include that other component. Those cases do not justify treating the bound disc simultaneously as a closed fixed-energy system and as a new equilibrium with increased $T$.

False. The singular isothermal sphere is infinite in extent and mass. The isolated finite-system virial theorem cannot be applied to its global divergent energies without boundary terms.

A direct Spherical Jeans equation counterexample is enough. For isotropic velocities with constant one-dimensional dispersion ${\sigma }^2$, its density slope givesThus ${\sigma }^2=v_0^2/2$ and the circular and isotropic speed comparison in a singular isothermal sphere isThis is realized by a positive isotropic distribution $F\propto \exp (E/{\sigma }^2)$, which gives a Maxwell-Boltzmann distribution of velocities and $\rho \propto \exp (\psi /{\sigma }^2)\propto r^{-2}$. Circular orbits have instead $⟨v_r^2⟩=0$ and $⟨v_{\theta }^2⟩=⟨v_{\varphi }^2⟩=v_0^2/2$, so their total mean square speed is $v_0^2$. The isotropic model contains eccentric orbits except for a measure-zero subset.

The boundary virial term of a singular isothermal sphere explains the different energy budgets of a finite spherical region. At radius $R$, its radial-pressure surface term is $4\pi R^3\rho (R){\sigma }_r^2$. It is nonzero in the isotropic model and zero in the circular-orbit model. Equality of the density therefore does not enforce equality of the velocity energy through an incorrectly omitted surface term.

The stellar relaxation time is the time on which accumulated discrete gravitational encounters change a typical star's velocity by an amount comparable to its original velocity. It concerns two-body relaxation, rather than the much faster orbital evolution in a smooth gravitational potential. A collisionless stellar system requires this time to greatly exceed the duration being studied.

In the straight-line impulse approximation, an encounter at relative speed $u$ has transverse acceleration $Gmb/(b^2+u^2{t}^2{)}^{3/2}$. ThusA small-angle gravitational encounter needs $\Delta v_{\perp }\ll u$. The transition to order-one deflections is therefore $b\sim Gm/u^2$, ignoring factors such as the equal-mass relative deflection. For the spherical estimate $u\sim v$,This is the lower cutoff of the weak-scattering estimate; closer encounters actually occur and require strong-scattering treatment.

The number of encounters in time $T$ with impact parameters in $[b,b+db]$ is $2\pi bdbnuT$. Independent random transverse directions make mean kicks cancel while their variances add. ConsequentlyEach logarithmic interval contributes equally, giving the Coulomb logarithm in stellar dynamics. Defining a deflection time by variance $v^2$ and setting $u=v$ gives $v^3/(8\pi G^2{m}^{2}n\log \Lambda )$. The paper instead uses an order-one normalization three quarters of this estimate:Both have the same physical scaling. The specified characteristic speed, the word “comparable”, the velocity-distribution average and strong-encounter cutoff do not fix that numerical coefficient uniquely. The crude equal-speed impulse calculation must not be claimed to determine $3/(32\pi )$ exactly.

For a self-gravitating, approximately virialized system, the virial theorem gives $v^2\sim GNm/R$. With $n=3N/(4\pi R^3)$, $\Lambda \sim N$ and the stellar crossing time $t_{\mathrm{dyn}}=R/v$, substitution into the paper's convention givesAn externally dominated gravitational potential or a different structural constant changes this substitution. For an N-body simulation lasting $T=K{t}_{\mathrm{dyn}}$, a tolerable fractional velocity-squared diffusion $\epsilon \ll 1$ requires $T/t_{\mathrm{rel}}\lesssim \epsilon$, henceThere is no unique smallest $N$ without $K$, the error tolerance and the force prescription. For scale, equality at $t_{\mathrm{rel}}/t_{\mathrm{dyn}}=100$ occurs near $N=7.1\times 10^3$; $N=10^5$ gives about $1.1\times 10^3$ crossing times. A calculation lasting 100 crossing times therefore needs substantially more than the first threshold to have negligible relaxation. Gravitational softening can raise the effective cutoff and reduce artificial scattering, but it also sets the spatial force resolution.

Put $u=\lambda v$ and let $n$ now mean number per area. During time $T$, the two strips of impact parameters between $b$ and $b+db$, on opposite sides of the trajectory, have combined area $2uTdb$. ThusThe same straight-line impulse approximation gives a kick $2Gm/(bu)$. Summing its squared size over these encounters givesassuming $b_{\max }\gg b_{\min }$. The relevant change is now the small random speed $u$, not the full circular speed $v$. Setting $D_v{t}_{\mathrm{rel}}\sim u^2$ yieldsThe razor-thin disk relaxation integral is dominated by nearby encounters, rather than equally by logarithmic intervals. Because kicks cease to be weak near the cutoff, its numerical coefficient is an estimate.

Here the weak-deflection cutoff must use the relative speed: $b_{\min }\sim Gm/u^2=Gm/({\lambda }^2{v}^2)$, which is larger than the spherical cutoff. Thus $t_{\mathrm{rel}}\sim u/(Gmn)$. Using $n\sim N/R^2$ and $v^2\sim GNm/R$ from the virial theorem givesFor example $n=N/(\pi R^2)$ gives the order-one coefficient $\pi /8$ in this last substitution; the paper's approximate equality suppresses such structural factors. The cancellation of $N$ is the central result.

An unsoftened point-particle system confined to an infinitesimally thin plane is therefore not collisionless over orbital times for $\lambda \sim 0.1$ in this encounter model. At fixed total mass, increasing $N$ decreases $m$ and the cutoff together; the enhanced importance of close planar encounters cancels the usual benefit of increasing particle number. This is a singular zero-thickness limit. A fixed nonzero gravitational softening length or physical thickness restores a different $N$ dependence, so the conclusion is not a prohibition on collisionless numerical models of disks.

With constant masses, differentiate the moment of inertia twice:For each pair, $F_{ij}=-G{m}_i{m}_j(r_i-r_j)/|r_i-r_j{|}^3$ and $F_{ji}=-F_{ij}$. Hence its contribution is $(r_i-r_j)\cdot F_{ij}=-G{m}_i{m}_j/|r_i-r_j|=V_{ij}$. Summing over unordered pairs gives $\sum r_i\cdot F_i=V$. Time-averaging over $[0,T]$ yieldsFor a bound system with $I^{\prime }(T)=o(T)$, or a stationary equilibrium with zero average left-hand derivative, the limit is the virial theoremBounded spatial extent together with bounded velocities suffices for the derivative condition; possible gravitational collisions must be excluded or otherwise regularized.

For the initially homogeneous sphere, $V_i=-3G{M}^2/(5R_i)$ and $K_i=0$. Conservation of energy and the virial theorem imply $E_f=V_f/2=V_i$, hence $V_f=2V_i$. With the usual homologous-sphere approximation, $V\propto -1/R$, soThese are the conventional collapse estimates. A final virial equilibrium is not, by itself, guaranteed to remain homogeneous: if $V_f=-{\alpha }_{f}G{M}^2/R_f$, the actual relation is $R_f/R_i={\alpha }_f/(2{\alpha }_i)$ with ${\alpha }_i=3/5$. Thus the factor of two applies exactly to the gravitational radius, and to the outer radius only under the stated structural approximation.

Here “non-interacting” means collisionless apart from gravity. Baryonic gas can collide, convert ordered motion to thermal energy, and radiate it. If radiative cooling is effective, the fixed-energy virial state loses pressure support and contracts further. Baryonic cooling can destabilize the collisionless virial state; baryons are not inevitably unstable if cooling is slow or heating and pressure support maintain equilibrium.

The shell equation is conservative because its radial acceleration is the negative derivative of the specific potential energyMultiplying the equation of motion by $\dot{r}$ givesHenceThis is energy per unit shell mass. The total energy of a uniform sphere requires integrating the energies of its shells.

For a sphere of physical radius $R$, the shell theorem gives $M(r)=M(r/R{)}^3$ and $dM=3M{r}^{2}dr/R^3$. Assemble the sphere shell by shell to obtain its Newtonian gravitational potential energy:The cosmological constant contributes a prescribed one-body potential $-\Lambda r^2/6$, rather than a pair interaction, so there is no additional factor $1/2$:Evaluating at the turnaround radius givesAs in the question, $\Lambda$ here includes the factor $c^2$ that would multiply the geometrical cosmological constant in SI units.

For the final equilibrium apply the scalar virial theorem. The gravitational virial is $W_G$, whereas the cosmological constant force contributesThus, neglecting a surface-pressure term,At fixed $R$, a positive cosmological constant reduces the kinetic energy needed for virial equilibrium because its outward acceleration partially offsets self-gravity.

For the uniform-sphere collapse with a cosmological constant, assume a uniform final sphere of the same mass, with no energy loss or escaping matter. PutThe sphere has no bulk kinetic energy at turnaround, so $E_{\mathrm{ta}}=-A-B$. The final virial theorem givesSince ${\rho }_{\mathrm{ta}}=3M/(4\pi r_{\mathrm{ta}}^3)$,Conservation of energy therefore givesSelect the collapsing branch continuous from $x=1/2$ at $\eta =0$, rather than an unrelated root of the cubic equation.

For $|\eta |\ll 1$, the root lies close to $1/2$. Linearize $x^3$ there:Substitution into the cubic equation gives $(\eta /2-2)x+1-\eta /2\simeq 0$, whenceThe rational expression is an approximation, not an exact solution of the cubic equation. In fact both it and the exact collapsing root agree through order ${\eta }^2$; their first difference is of order ${\eta }^3$.

With zero cosmological constant, spherical collapse gives $r_f=r_{\mathrm{ta}}/2$. For small positive $\Lambda$, the same sphere with the same turnaround radius reaches a smaller final radius. This does not mean that repulsion promotes collapse: as the sphere contracts, $W_{\Lambda }$ becomes less negative, so some released gravitational binding energy pays the energetic cost of moving inward against the repulsive force. Combined with the modified virial theorem, energy conservation requires a more compact endpoint. Positive $\Lambda$ can prevent turnaround altogether; an inward acceleration at $r_{\mathrm{ta}}$ requiresThe perturbative radius formula applies well inside this limit, for spheres that actually turn around. A small negative cosmological constant instead gives $r_f/r_{\mathrm{ta}}>1/2$ at fixed turnaround data.

Near the minimum, the Taylor expansion of the potential isNeglecting Hubble friction during one short oscillation reduces the Klein-Gordon equation to the harmonic oscillator equationThe virial theorem gives $⟨{\dot{\varphi }}^2/2⟩=⟨V⟩$, so $⟨P_{\varphi }⟩=0$. Restoring the slow cosmological damping in the averaged cosmological perfect-fluid continuity equation givesThe coherently oscillating inflaton therefore behaves as pressureless matter.

For $I=\frac{1}{2}\sum m_i|x_i{|}^2$, differentiation givesbecause the Newtonian potential is homogeneous of degree $-1$. Bounded positions and velocities imply $\overline{\ddot{I}}=0$, so the virial theorem gives $\overline{T}=-\overline{V}/2$.

Observed stellar velocities estimate $T$, while visible mass and size predict $V$. Galaxies and clusters often have far more kinetic energy than luminous matter can bind under the virial theorem. The additional gravitational potential is evidence for unseen dark matter.

For uniform density, Kelvin-Helmholtz contraction releases binding energy $|U|=3G{M}^2/(5R)$. If mass and luminosity are constant and stellar heating is negligible at 90 au, the contraction age isEnergy conservation, $L=-dU/dt$, givesThe virial theorem places roughly half of the released gravitational energy into internal heat. Including that effect gives $t_{\mathrm{KH}}\simeq 3G{M}^2/(10RL)$ and $\dot{R}\simeq -10L{R}^2/(3G{M}^2)$.

The density has the formInsert the ansatz ${\sigma }_r^2=k\Psi$ into the Spherical Jeans equationThe radial-power derivative cancels the anisotropy term because $p-2+2\beta =0$, leaving $k(2p+2)=1$. ThusTheir sum is independent of $p$:Therefore the local kinetic-energy density and gravitational potential-energy density areand they satisfy the local virial relation of the hypervirial modelat every radius. This pointwise identity is not generic. The ordinary virial theorem constrains suitable global integrals, with boundary terms when the system is truncated, but does not normally impose a virial balance shell by shell.

The shell feels only radial gravity and the radial force due to the cosmological constant, so its torque vanishes and its specific angular momentum $L=r^2\dot{\theta }$ is conserved. Multiplyingby $\dot{r}$ and integrating gives the conserved specific orbital energy

For a uniform sphere, assembling concentric shells gives its gravitational potential energyThe cosmological-constant potential per unit mass is $-\Lambda r^2/6$. Since $⟨r^2⟩=3r_{\mathrm{ta}}^2/5$ in a uniform sphere,

The scalar virial theorem weights a potential homogeneous of degree $n$ by $-nW$. Gravity has degree $-1$ and the $\Lambda$ potential degree $2$, so the final state obeysAt turnaround $T=0$, while the virial relation gives $E_f=W_{G,f}/2+2W_{\Lambda ,f}$. Conservation of energy, together with $M=4\pi {\rho }_{\mathrm{ta}}{r}_{\mathrm{ta}}^3/3$, then gives, for $x=r_f/r_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$,The root connected continuously to the $\Lambda =0$ solution has $x=1/2-\eta /8+O({\eta }^2)$, equivalentlyto first order in $\eta$. With $\Lambda =0$, virialization occurs at half the turnaround radius. At fixed turnaround state, positive $\Lambda$ makes this equilibrium root slightly smaller because its repulsive quadratic potential enters both energy conservation and the virial relation; sufficiently strong repulsion instead prevents a bound virialized state. Negative $\Lambda$ shifts the root in the opposite direction.

Direct evidence for stellar feedback includes expanding ionized shells and superbubbles around young associations, hot X-ray-emitting gas in supernova remnants, broad or split emission lines, and blueshifted absorption showing cool and warm outflows. P-Cygni profiles reveal massive-star winds, while extraplanar filaments and metal-enriched gas demonstrate transport away from star-forming disks. Indirect evidence includes the galaxy mass--metallicity relation, low baryon fractions and suppressed star formation in dwarf galaxies, chemically enriched circumgalactic gas, and correlations of outflow speed and mass loading with the star formation rate.

Rapid gas removal changes the gravitational potential before stellar and dark-matter orbits can respond adiabatically. Positions and velocities are initially unchanged, but orbital binding energies rise; orbits expand and become more eccentric, and some particles escape. Repeated burst--outflow--reaccretion cycles can irreversibly transfer energy to collisionless matter and turn a central dark-matter cusp into a core. This matters because dwarf-galaxy rotation curves are used to test dark-matter microphysics: a feedback-made core can mimic a non-cold or self-interacting dark-matter signature.

Write the initial potential energy as $W=-aG{M}^2/R$. The virial theorem gives $T=-W/2$. If a well-mixed fraction $ϵ$ remains after instantaneous mass loss, the immediate kinetic and potential energies are $T_a=ϵT$ and $W_a={ϵ}^{2}W$, soAfter revirialization at $R^{\prime }$, $E_f=W_f/2=-aG{ϵ}^2{M}^2/(2R^{\prime })$. Equating energies gives the impulsive mass-loss expansion lawThe remnant is bound only for $ϵ>1/2$; loss of half or more of the gravitating mass disrupts this idealized system.

For many infinitesimal, individually revirialized losses, put $ϵ=1+dM/M$ in the impulsive result. To first order, $dR/R=-dM/M$. Integration yields the adiabatic mass-loss expansion lawSlow loss causes finite expansion for every positive remaining mass fraction and has no sharp disruption threshold.

Finally consider an initially circular orbit of radius $R$ around a point mass $M$. Its speed and specific angular momentum obey $v^2=GM/R$ and $h^2=GMR$. Immediately after $M\to ϵM$, these remain unchanged, while the new specific energy isUsing the orbital eccentricity relation $e^2=1+2E^{\prime }{h}^2/(G^2{ϵ}^2{M}^2)$ givesIt is an ellipse for $ϵ>1/2$, parabolic at $ϵ=1/2$, and unbound for smaller $ϵ$, in agreement with the virial argument.

During a weak encounter, approximate one star's trajectory by a straight line with speed $v$ and impact parameter $b$. At longitudinal coordinate $z=vt$, the transverse acceleration isIntegrating from $t=-\infty$ to $\infty$ givesHere $Gm/b^2$ sets the gravitational acceleration, $b/v$ is the encounter duration, and the geometric projection produces the displayed transverse impulse.

In one crossing, the number of encounters with impact parameters in $(b,b+db)$ is, up to the system-geometry convention,Uncorrelated impulses add in mean square, soThe virial theorem gives $v^2\sim GNm/R$. Taking $b_{\max }\sim R$ and the strong-deflection scale $b_{\min }\sim Gm/v^2\sim R/N$ gives the Coulomb logarithm in stellar dynamics $\log \Lambda \sim \log N$ andVelocity memory is lost when the cumulative change reaches $v^2$, afterConsequently the two-body relaxation time iswith an order-unity prefactor depending on density profile and convention. The system is a collisional stellar system when $t_{\mathrm{rel}}$ is shorter than its age or the evolutionary timescale being studied.

For a confined system, $\mathcal{G}(t)$ stays bounded, so its long-time averaged derivative is zero. The preceding identities give the virial theorem

Estimate $T\sim M⟨v^2⟩/2$ and $V\sim -G{M}^2/⟨R⟩$. The virial theorem givesup to a profile-dependent factor. If this exceeds the stellar/gas luminous mass, the difference is attributed primarily to dark matter (with modelling uncertainties and non-equilibrium effects checked).

For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3G{M}^2/(5R)$ and $W_{\Lambda }=-\Lambda M{R}^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The virial theorem reads $2T+W_G=2W_{\Lambda }$. Defining $x=R_f/R_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$, conservation of energy gives $2\eta x^3-(2+\eta )x+1=0$. The collapsing branch has $x=1/2-\eta /8+O({\eta }^2)$. For small positive cosmological constant, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.