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In the spherical-collapse model, a bound overdense region initially expands with the background, but its additional Newtonian gravity slows that expansion. Its outer shell reaches a maximum turnaround radius, where its radial velocity is zero, and then falls inward. A perfectly spherical pressureless solution formally collapses to zero radius. In a realistic collisionless system, deviations from that idealization, shell crossing and violent relaxation convert coherent infall into random motions, producing a roughly stationary bound object.

Let the Newtonian gravitational potential energy of the homologous sphere be $U=-aG{M}^2/R$, where $a=3/5$ for a uniform sphere. At spherical-collapse turnaround, the kinetic energy of the bulk motion vanishes, so $E=U_{\mathrm{ta}}$. For the settled object, the virial theorem gives $2K_{\mathrm{vir}}+U_{\mathrm{vir}}=0$ and hence $E=U_{\mathrm{vir}}/2$. If energy and mass are conserved and the structure coefficient is unchanged,This is the virial radius from turnaround energy. It assumes negligible external pressure, mass loss and energy exchange. If the final structure coefficient differs, $R_{\mathrm{vir}}=(a_{\mathrm{vir}}/2a_{\mathrm{ta}})R_{\mathrm{ta}}$ instead. Half the turnaround radius is an idealized equilibrium size, not the radius at which each collisionless particle stops moving. The plotted settling time is schematic; the usual collapse-time convention is $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$.

Let $X$ be the hydrogen mass fraction, $f_g={\rho }_g/{\rho }_{\mathrm{tot}}$ the gravitating gas fraction, and $\chi =n_{\mathrm{tot}}/n_H$ the total-particle-to-hydrogen ratio. The optically thin gas cooling time and free-fall time of a uniform sphere arewhere ${\rho }_{\mathrm{tot}}=m_p{n}_H/(X{f}_g)$. Equating them gives the cooling-to-free-fall equality curve:At fixed composition, its shape is $n_{\mathrm{crit}}\propto T^2/{\Lambda }^2$. Gas above this curve cools faster than it falls; gas below it cools more slowly. Strong line cooling produces a deep low-density trough. In the high-temperature free-free regime, $\Lambda \propto T^{1/2}$ gives $n_{\mathrm{crit}}\propto T$.

The displayed uniform self-gravitating-cloud example uses $f_g=1$, $X=0.76$, $\mu \simeq 0.59$ and $\chi \simeq 2.24$. Holding these ionized-gas factors fixed makes the diagram transparent, but near the atomic threshold their actual temperature dependence changes numerical normalizations. With these conventions the supplied cooling values imply $n_{\mathrm{crit}}(1.5\times 10^4\mathrm{K})\simeq 9.6\times 10^{-10}\mathrm{c}{\mathrm{m}}^{-3}$ and $n_{\mathrm{crit}}(10^7\mathrm{K})\simeq 0.107\mathrm{c}{\mathrm{m}}^{-3}$. For gas supported in a dark-matter-dominated dark-matter halo, take $f_g<1$: at fixed $n_H$ the free-fall time of a uniform sphere is shorter and the critical curve shifts upward by $1/f_g$.

To add virial mass contours in a cooling diagram, use the uniform self-gravitating sphere consistently. Its Newtonian gravitational potential energy is $-3G{M}^2/(5R)$ and its thermal kinetic energy is $(3/2)M{k}_{B}T/(\mu m_p)$. The virial theorem gives $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(5R)$, while $M=(4\pi /3)R^3{\rho }_{\mathrm{tot}}$. Eliminating $R$ givesThus constant total gravitating mass gives slope three in the logarithmic density-temperature plane, with larger mass contours lying at smaller density. A different density profile changes the structural coefficient; the customary halo temperature scale $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(2R)$ shifts the mass contours but preserves their slopes. The horizontal illustrative formation-density lines are $n_{H,\mathrm{vir}}\simeq {\Delta }_v(2\times 10^{-7})(1+z{)}^3\mathrm{c}{\mathrm{m}}^{-3}$, using ${\Delta }_v=18{\pi }^2$. They show why increasing formation cosmological redshift helps gas reach the rapid-cooling region.

At the low-mass end, a dark-matter halo's virial temperature may not reach the atomic threshold. The atomic and molecular cooling thresholds for galaxy formation therefore favor an atomic-cooling mass of order $10^8{M}_{\odot }$ at early formation epochs, with its precise value varying as approximately $(1+z{)}^{-3/2}$. Smaller primordial systems require molecules; photoheating and stellar feedback further suppress their luminous output.

At the high-mass end, the virial temperature at a fixed assembly density scales as $M^{2/3}$. Beyond the line-cooling interval the gas becomes increasingly hot, while thermal bremsstrahlung improves only as $T^{1/2}$. Hence at fixed assembly density $t_{\mathrm{cool}}/t_{\mathrm{ff}}\propto T^{1/2}\propto M^{1/3}$ in that regime. Large systems can remain pressure-supported hot atmospheres for many dynamical times, instead of promptly fragmenting into one stellar object. This is the upper galaxy mass from gas cooling: it associates efficient condensation with roughly galactic, rather than group or cluster, masses and motivates the broad $10^8$–$10^{12}{M}_{\odot }$ interval.

The diagram gives characteristic cooling scales, not exact universal mass bounds. Density profiles, gas fraction, enrichment and formation epoch move the boundaries; the two supplied values alone do not determine a complete numerical cutoff. Some gas can cool in dense cores of larger dark-matter haloes, early smaller progenitors can form stars before merging, and subsequent galaxy mergers can assemble more massive galaxies. Long-lived hot atmospheres also require heating to prevent excessive late-time cooling. These qualifications explain why the cooling criterion for galaxy formation is useful without equating each dark dark-matter halo to one galaxy.

In the Lambda-CDM model, nearly Gaussian primordial density contrasts grow under gravity within an expanding universe containing cold dark matter, ordinary baryons and a cosmological constant. The cold dark matter is effectively collisionless and has negligible pressure on galactic scales. Before recombination, baryons are coupled to the photon fluid: radiation pressure and acoustic oscillations prevent their perturbations from behaving like pressureless matter. After recombination they can fall into the gravitational potentials already established by dark matter, subject to gas pressure and the Jeans mass.

For small density contrasts, evolution is linear. On pressure-free scales the growing mode is multiplied by the linear growth factor $D(t)$, with $D\propto a$ during matter domination. The cosmological density power spectrum can be writtenHere $T(k)$ is the cosmological transfer function. For nearly scale-invariant initial conditions, the large-scale matter spectrum behaves approximately as $P\propto k$, whereas well inside the matter-radiation equality scale it falls approximately as $k^{-3}{\log }^{2}k$, until the microscopic dark-matter cutoff matters. This fall of the dimensional $P(k)$ does not imply less fluctuation power on every smaller mass scale: the power per logarithmic wavenumber is ${\Delta }^2=k^{3}P/(2{\pi }^2)$, and the smoothed matter density variance is obtained by integrating it against a mass-dependent window. Over the relevant cold-dark-matter hierarchy, smaller mass windows generally have larger variance.

The hierarchical galaxy formation picture follows: fluctuations on small mass scales typically reach the nonlinear collapse threshold first, while larger objects assemble later through accretion and dark-matter halo mergers. It is a statistical ordering, not a claim that every small object precedes every large rare peak. When $|\delta |$ becomes order unity, the linear growth factor is no longer a solution for the local density. Collisionless dark matter develops multistream regions and bound dark-matter halos; phase mixing and violent relaxation redistribute orbital energies, and virialized structures approximately obey the virial theorem.

Baryons have an additional nonlinear route. Infall and shocks convert bulk kinetic energy into thermal energy, with characteristic virial temperatureUnlike collisionless dark matter, the gas can lose this energy through radiative cooling. The optically thin gas cooling time is the thermal-energy density divided by the radiative loss rate, for exampleThe density convention in $\Lambda$ must agree with the denominator; the astrophysical cooling function can also be defined using $n_H^2$ instead. The cooling criterion for galaxy formation compares this time with the collapse or supply time. Rapidly cooling gas loses pressure support, contracts and can form stars. Slowly cooling gas remains in a hot atmosphere. Stable virial shocks are not obligatory in every low-mass system: gas can also arrive in cold streams and cool while being accreted.

Angular momentum prevents indefinite radial contraction. Tidal torque theory supplies an initial halo spin, and later mergers change it. If gas radiates energy while retaining much of its specific angular momentum, it settles into a rotationally supported galactic disk rather than reaching the centre. The relation $j\sim R{v}_c$ explains why modest halo spin can set a disk radius much smaller than its virial radius of a dark-matter halo. Torques, bars and gravitational encounters can transport angular momentum outwards and feed central concentrations; radiative cooling alone does not remove it.

Galaxy mergers alter stellar structure as well as assembling mass. A major galaxy merger can strongly disturb or destroy an existing galactic disk, randomizing stellar orbits and creating a spheroid through violent relaxation. A gas-rich galaxy merger also permits dissipation, inflow and a burst of star formation; gas left over or accreted afterwards can rebuild a galactic disk. Minor galaxy mergers add stars to outer components, thicken disks and grow bulges. Dry galaxy mergers add stellar mass and can increase size without much new star formation. Halo merging therefore does not imply instantaneous merging of its galaxies: satellite orbital decay requires Chandrasekhar dynamical friction and can take a substantial time.

The atomic and molecular cooling thresholds for galaxy formation supply a lower characteristic scale. Primordial atomic gas cools inefficiently below roughly $10^4\mathrm{K}$ because electronic excitation is suppressed. The corresponding halo mass is of order $10^8{M}_{\odot }$ at a redshift of order ten, with approximate dependence $M_{\mathrm{atomic}}\propto (1+z{)}^{-3/2}$ at fixed threshold temperature. Molecular hydrogen can cool gas at hundreds of kelvin and permit smaller early objects, provided it forms and survives dissociating radiation. Metal-line cooling changes these thresholds after enrichment. Thus the atomic threshold is not an absolute minimum mass for all stellar systems.

At the other end, sufficiently massive dark-matter halos have high virial temperatures and low-density hot gas. Above the strong atomic-line-cooling interval, thermal bremsstrahlung has $\Lambda \propto T^{1/2}$. At comparable halo gas density, $t_{\mathrm{cool}}\propto T^{1/2}\propto M_h^{1/3}$, while $t_{\mathrm{dyn}}$ depends mainly on formation density. Cooling therefore becomes less able to condense all the gas within the available time. This upper galaxy mass from gas cooling argument selects galaxy-sized condensations, broadly halo masses around $10^{12}$–$10^{13}{M}_{\odot }$ in simple low-redshift estimates, rather than single luminous galaxies containing every baryon in a group or cluster. Its numerical scale depends on epoch, metallicity and gas profile. Subsequent galaxy mergers can build larger stellar systems; cooling is not an absolute upper bound on their final mass.

Finally, a Press-Schechter halo mass function has many low-mass objects and a steep high-mass cutoff, while the luminosity function of galaxies also has a faint component and a bright cutoff, often described by a Schechter function. The two shapes are related through the halo-to-galaxy luminosity mapping, not by identifying luminosity with total halo mass. In the idealized one-central-galaxy, no-scatter limit,If $d{n}_h/d{M}_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi (L)\propto L^{-1-(a-1)/p}$. A constant conversion gives similar shapes. Actual stellar feedback, inefficient low-temperature cooling and reionization suppress faint galaxies relative to small haloes, while long cooling times and active-galactic-nucleus feedback suppress luminosity at large halo masses. Satellites, scatter, stellar populations and dust further affect the correspondence. The halo hierarchy supplies the gravitational framework; cooling, angular momentum and feedback determine which parts become luminous galaxies.

Use the paper's positive binding-energy convention $U>0$; the physical Newtonian gravitational potential energy is $-U$. Assume positive masses, an isolated system, and no collision during the time interval under consideration. Differentiate half the scalar moment of inertia, $I$, twice:Group the force term into unordered pairs. The force on particle $j$ from $k$ is $-G{m}_j{m}_k({\mathbf{r}}_j-{\mathbf{r}}_k)/r_{jk}^3$; the two contributions to the virial sum are thereforeAdding them proves the instantaneous virial theorem identityNo time average is needed for this form. A vanishing average of $\ddot{I}$ would require additional boundedness assumptions.

For the pairwise moment-of-inertia identity, expand the squared pair distances:The last term vanishes in the centre of mass frame, givingLet $C={\sum }_{j<k}{m}_j{m}_k$, and label the two smallest masses $m_{(1)},m_{(2)}$. If $r=\min r_{jk}$, every term in $U$ is at most $G{m}_j{m}_k/r$, so $U\le GC/r$. One pair attains $r$, and its mass product is at least $m_{(1)}{m}_{(2)}$, so $U\ge G{m}_{(1)}{m}_{(2)}/r$. HenceThese constants show the minimum-separation binding-energy bounds: inverse binding energy is comparable to the nearest-pair distance, independently of configuration.

Similarly, every separation is at most $R=\max r_{jk}$ and at least one pair attains $R$. The pairwise identity gives the maximum-separation inertia boundsBoth lower bounds rely on actual extremal separations, rather than an arbitrary numerical lower or upper estimate for all the pair distances.

Negative energy does not imply a positive lower bound on the minimum separation. The printed request is false if it is meant to exclude close approaches or collisions; the trivial bound $r\ge 0$ says nothing of that kind. In fact $E<0$ and $T\ge 0$ give $U=T-E\ge |E|$, which, combined with $rU\le A_0$, proves the negative-energy minimum-separation upper boundThis ensures at least one close pair, not confinement of every particle or prevention of collision.

An explicit negative-energy gravitational collision is a pair initially at rest with separation $a$. Its total energy is $E=-G{m}_1{m}_2/a<0$. For reduced mass $\mu =m_1{m}_2/(m_1+m_2)$, its relative radial equation isThe separation decreases to zero in the finite timeBefore that time the motion is a regular Newtonian solution, yet its separation has no positive infimum. Even at fixed negative energy, bound Kepler orbits with eccentricity approaching one have arbitrarily small pericentre distance. Thus a missing angular momentum or collision-exclusion hypothesis cannot be supplied by the energy sign alone.

For $E>0$, the instantaneous virial theorem gives $\ddot{I}=T+E\ge E$. Integrating twice from any regular time $t_0$ givesUsing $I\le A_1{R}^2$, obtain the positive-energy linear diameter growth boundwhenever the numerator is nonnegative. Consequently, for a solution existing for arbitrarily large future times,This is the precise at-least-linear expansion statement. It does not assert that $R$ is monotone at every instant, nor that all individual particles escape. The large-time conclusion presupposes continued existence of the trajectory; the inequality itself holds on every nonsingular time interval.

The optically thin gas cooling time is thermal energy density divided by the radiative loss rate. Write $\chi =n_{\mathrm{tot}}/n_H$ for the number of thermal particles per hydrogen nucleus, $X$ for the hydrogen mass fraction, and $f_g={\rho }_g/{\rho }_{\mathrm{grav}}$ for the gas fraction of the gravitating mass. ThenThe last expression is the free-fall time of a uniform sphere. Equating the two times gives the cooling-to-free-fall equality curveAt fixed temperature, $t_{\mathrm{cool}}/t_{\mathrm{ff}}=\sqrt{n_{H,\mathrm{crit}}/n_H}$: above the curve, $t_{\mathrm{cool}}<t_{\mathrm{ff}}$; below it, cooling is slower. The temperature dependence is approximately $n_{H,\mathrm{crit}}\propto T^2/{\Lambda }^2$. Strong line cooling therefore creates low-density troughs, whereas the fully ionized bremsstrahlung tail gives $n_{H,\mathrm{crit}}\propto T$.

For the diagram choose a self-gravitating uniform gas cloud, $f_g=1$, with $X=0.76$, proton mass $m_p\simeq 1.67\times 10^{-24}\mathrm{g}$, and the fully ionized particle ratio $\chi =2+3(1-X)/(4X)\simeq 2.24$. Using this fixed ratio is a convenient sketch normalization; the varying low-temperature ion fraction changes the coefficient by a factor of order unity. The supplied cooling values then give $n_{H,\mathrm{crit}}\sim 10^{-9}\mathrm{c}{\mathrm{m}}^{-3}$ at $1.5\times 10^4\mathrm{K}$ and $n_{H,\mathrm{crit}}\sim 0.1\mathrm{c}{\mathrm{m}}^{-3}$ at $10^7\mathrm{K}$. A dark-matter-dominated potential has $f_g<1$, shortening the free-fall time and moving the equality curve upward by $1/f_g$.

To relate the plane to mass, use the virial theorem for a uniform self-gravitating sphere. Its gravitational potential energy is $-3G{M}^2/(5R)$, so $k_B{T}_{\mathrm{vir}}=GM\mu m_p/(5R)$, where $\mu =1/(X\chi )$. With $M=(4\pi /3)\rho R^3$, the virial mass contours in a cooling diagram satisfyThe coefficient $5$ is for this uniform-sphere convention; other halo profiles change it. The horizontal mean-density label is $n_{H,0}\simeq 2\times 10^{-7}\mathrm{c}{\mathrm{m}}^{-3}$. Real collapsing clouds are overdense, and their characteristic densities rise at earlier epochs.

The cooling criterion for galaxy formation requires a cloud to lose shock-generated virial heat rapidly enough to contract and fragment. At higher virial masses and temperatures, hydrogen-helium line cooling ceases to be efficient, so a growing bound aggregate can remain a hot, pressure-supported atmosphere rather than condense as one giant luminous galaxy. Combined with virial mass contours and the densities at which clouds assemble, the diagram gives a characteristic upper galaxy-scale cooling mass, conventionally of order $10^{12}{M}_{\odot }$ in the simple baryonic-cloud argument. Larger aggregates are predominantly groups or clusters containing smaller galaxies and hot gas.

This upper galaxy mass from gas cooling is a physical scale, not an exact universal mass cutoff derivable from the two cooling labels alone. Composition, formation density, geometry, metals and a dark-matter potential shift it. Cooling that is slower than free fall but faster than the available cosmic time can still yield gradual central condensation. Mergers of existing stellar galaxies can also assemble a larger stellar system without rapidly cooling the entire gas mass of its host halo.

Use a mass-normalized galactic distribution function $F$, so that $\rho =\int F{d}^{3}v$ and $\rho ⟨v_i{v}_j⟩=\int v_i{v}_{j}F{d}^{3}v$. Multiplying the stationary Collisionless Boltzmann equation by $v_j$ and integrating over velocity space givesAssume the galactic distribution function decays sufficiently rapidly for the velocity boundary term to vanish. Integration by parts then gives $\int v_j{\partial }_{v_i}F{d}^{3}v=-{\delta }_{ij}\rho$, and therefore the Jeans equations areThe second moment includes both ordered motion and velocity dispersion; no assumption of zero mean velocity was needed.

To obtain the tensor virial theorem, multiply the $j$th Jeans equation by $x_i$ and integrate over position. For an isolated, finite system with a vanishing spatial surface term,Since the left side is $-2K_{ij}$, $2K_{ij}+W_{ij}=0$. This convention puts all stellar second moments into $K_{ij}$; it does not separate ordered and random kinetic energies.

For a self-gravitating system, the trace $W$ is the Newtonian gravitational potential energy. Indeed, symmetrizing the pair integral givesThus the trace of the tensor virial theorem is $2K+W=0$, and the total energy obeysThese identities require self-gravity without an additional external potential or an omitted confining boundary pressure.

Let $⟨v_I^2⟩$ now denote a mass-weighted average over the entire initial system. Then $K_I=M_I⟨v_I^2⟩/2$. The gravitational radius is defined by $R_I=-G{M}_I^2/W_I$, so the virial theorem immediately yieldsThe gravitational radius measures total binding energy, rather than a particular geometric edge or half-mass radius.

For the accreted systems define the mass-weighted internal mean-square speed by $M_A⟨v_A^2⟩={\sum }_s{M}_s⟨v_s^2⟩$. If each satellite initially satisfies the virial theorem, its internal energy contributes to $E_A=-M_A⟨v_A^2⟩/2$. In parabolic dry-merger energy accounting, the orbital energy at large separation is zero. Assume a dry galaxy merger, no loss of mass or energy through escaping stars, no external work, and a final relaxed system satisfying the virial theorem. Then conservation of energy gives $E_F=E_I+E_A$ and $M_F=M_I+M_A$. Energy transferred by Chandrasekhar dynamical friction remains part of the total energy under these assumptions. Consequently,Applying the final virial theorem and dividing by the initial relation givesFor a mass doubling, $\eta =1$. A merger of two identical equilibrated galaxies has $ϵ=1$, hence $R_F/R_I=2$. For accretion through many minor galaxy mergers whose satellites have much smaller internal mean-square speeds, $ϵ\ll 1$, hence $R_F/R_I\simeq 4$, tending to $4$ in the cold-satellite limit. Equivalently, nearly fixed total energy makes the gravitational radius scale as $M^2$ during cold minor-merger size growth. Small satellite mass alone does not logically imply $ϵ\ll 1$: the factor four also requires this weak-binding assumption. Likewise, equal masses require comparable internal binding to give the factor two. These are the physical limits behind the two stated merger comparisons.

The spherical stellar structure equations apply to a planet as well, with its appropriate equation of state:The last equation is the local first law, including gravitational contraction through the specific entropy change. For an isolated non-burning planet, ${ϵ}_{\mathrm{nuc}}\simeq 0$, soIf the convective interior has nearly uniform specific entropy $s(t)$, positive outgoing luminosity gives $\dot{s}=-L_{\mathrm{int}}/\int Tdm<0$. This entropy loss of an isolated convective planet describes its secular cooling.

During assembly, the gravitational energy scale $G{M}^2/R$ provides a source of heat. Integrating hydrostatic equilibrium gives the virial theorem, $\Omega +3\int PdV=0$, neglecting the surface-pressure term. For a monatomic ideal gas this becomes $2U+\Omega =0$, hence $E=U+\Omega =\Omega /2$. Contraction makes $\Omega$ more negative, releases radiation, and heats part of the gas. A characteristic formation temperature has scale $k_{B}T\sim \mu m_{H}GM/R$ up to structure factors. The retained fraction depends on the accretion shock, producing hot and cold starts of a giant planet.

Formation releases heat, and an isolated planet subsequently loses specific entropy and intrinsic luminosity as it approaches a cooler, more degenerate state. The structure equations alone do not fix one initial temperature, and the early contraction of an ideal gas can increase its central temperature while its total energy decreases. “Cools” therefore need not mean that every depth becomes colder at every instant. Once degeneracy limits contraction, the thermal reservoir declines more directly; the emergent effective temperature usually falls along a fixed-mass cooling track.

This makes young giant planets attractive for exoplanet direct imaging: their thermal near/mid-infrared emission can be far brighter than that of old planets of the same mass. Imaging and spectroscopy measure projected separation, astrometry, luminosity, an exoplanet emission spectrum, temperatures and atmospheric molecular and exoplanet cloud signatures. Radius follows from luminosity and fitted temperature; mass and initial specific entropy generally require age/evolutionary information, or an independent dynamical mass measurement. A detected luminosity alone is not a model-independent planet mass.

A spherically symmetric stellar model has no preferred direction in the tangent plane at a fixed radius. Rotations about the radial axis exchange its two transverse velocity directions, so $⟨v_{\theta }^2⟩=⟨v_{\varphi }^2⟩$. This uses spherical symmetry of the velocity distribution, not just a spherical mass density.

The augmented density $\rho (r,\psi )$ treats radius and relative potential as independent variables, agreeing with the physical density only along $\psi =\psi (r)$. For any regular bound spherical galactic distribution function $F(E,L)$, write $P_r=\int v_r^{2}F{d}^{3}v$ and $P_t=\int v_{\theta }^{2}F{d}^{3}v$. At fixed $r$, $F_{\psi }=F_E$ and ${\partial }_{v_r}F=-v_r{F}_E$. Integration by parts therefore givesFor $v_t^2=v_{\theta }^2+v_{\varphi }^2$, the velocity measure is $2\pi v_{t}d{v}_{t}d{v}_r$, and $r{F}_L={\partial }_{v_t}F+v_t{F}_E$. Integrating the transverse derivative gives $-2P_r$, while integrating the radial derivative in $\int v_r^2{v}_t^2{F}_E{d}^{3}v$ gives $2P_t$. Consequentlywhere $\psi$ is fixed. These identities show that the augmented representation holds for any such spherical distribution, not only a monomial ansatz. With radial pressure vanishing at the zero-binding boundary, set Along the physical curve, the chain rule gives $dA/dr={\int }_0^{\psi }{\partial }_r\rho (r,q)dq+\rho (r,\psi ){\psi }^{\prime }(r)$. The Spherical Jeans equation then requiresThus the Jeans moments from an augmented density areThe partial derivative holds $q$ fixed. Substitution cancels the integral of ${\partial }_r\rho$ and leaves precisely $\rho {\psi }^{\prime }$ in the Spherical Jeans equation. This is a representation by a chosen augmented density, not a unique solution determined by the one-variable density: different extensions off the physical curve encode different anisotropies, and not every formal extension necessarily gives a nonnegative galactic distribution function.

For the hypervirial density-potential family, differentiate the potential and use the spherical shell theorem:Differentiating $M(<r)$ and dividing by $4\pi r^2$ givesThe enclosed mass tends to $M$ at infinity and to zero at the centre for every $p>0$. The same result follows from ${\nabla }^2\psi =-4\pi G\rho$, with the stated sign convention.

From now on set $G=M=a=1$. The separable augmented density is $\rho (r,q)=(p+1)r^{p-2}{q}^{2p+1}/(4\pi )$. Integrating it givesand henceWith $\psi =(1+r^p{)}^{-1/p}$ these are the requested radial expressions. The velocity-anisotropy parameter is $\beta =1-p/2$, and the total mean square speed is $⟨v^2⟩=\psi /2$.

The local kinetic-energy density is $t=\rho ⟨v^2⟩/2=\rho \psi /4$. With the potential zero at infinity, the gravitational energy density is $w=\rho \Phi /2=-\rho \psi /2$, where $\Phi =-\psi$. ThereforeThis is the local virial relation of the hypervirial model, a special property stronger than the global virial theorem. To compute the global energies as well, set $I=\int \rho \psi d^{3}x$. The substitution $u=r^p$ yields the global binding integral of the hypervirial modelThe beta function integral converges for all $p>0$. ThusRestoring dimensions multiplies $I,T,W$ by $G{M}^2/a$.

To derive the hypervirial distribution function, let $E=\psi -v^2/2$ be the positive relative energy and $L=rv\sin \vartheta$ the magnitude of the specific angular momentum. Seek $F=C{L}^{p-2}{E}^s$ for $E>0$ and zero otherwise. Its density isThe angular integral is $\sqrt{\pi }\Gamma (p/2)/\Gamma ((p+1)/2)$. Substitution $z=v^2/(2\psi )$ makes the radial integralMatching the augmented-density power ${\psi }^{2p+1}$ gives $s=(3p+1)/2$. Matching its coefficient gives the normalization of the hypervirial distribution functionAll integrals are finite in velocity for $p>0$, and $C>0$. Because $E$ and $L$ are orbital integrals, this is a steady solution by the Jeans theorem. As checks, $p=1$ gives $C=3/(4{\pi }^3)$ for the Hernquist model, and $p=2$ gives $C=24\sqrt{2}/(7{\pi }^3)$ for the isotropic Plummer model.

The density and potential do not uniquely fix the stellar distribution. The preceding coefficient is fixed within the chosen separable power-law model, but nonuniqueness of spherical galactic distribution functions remains without that extra restriction. An explicit positive counterexample uses the same $p=2$ density. Define the Osipkov-Merritt distribution function variable $Q=E-L^2/2$ and takeRescale the transverse velocity by $\sqrt{1+r^2}$. Then $Q=\psi -[v_r^2+(1+r^2)v_t^2]/2$ and the velocity Jacobian contributes $(1+r^2{)}^{-1}$. Direct integration givesexactly the same Plummer model density. This anisotropic Plummer distribution with unit anisotropy radius has $\beta =r^2/(1+r^2)$, so it is distinct from the isotropic $E^{7/2}$ distribution while remaining positive and spherical. Its second moments differ, so it does not share the extra augmented-density and local-virial restrictions of the chosen hypervirial model.

False. Stellar evaporation removes stars above the escape energy after two-body relaxation has redistributed their energies. The remaining self-gravitating cluster need not become less dense; losing mass is not the same as holding its radius fixed.

For a slowly evolving isolated cluster, apply the virial theorem at each approximate equilibrium: $E=T+W=W/2=-T$. Stars evaporating with nearly zero energy at infinity remove little total energy, so the bound cluster has nearly constant negative $E$. With gravitational radius $R_g$ defined by $W=-G{M}^2/R_g$,For homologous structure, its characteristic mean density therefore scales asIt increases as mass is lost. Escaping stars carrying positive energy make the bound energy more negative and strengthen the contraction tendency. This is virial contraction under stellar evaporation, related to the negative heat capacity of bound gravitational systems. A tidally limited globular cluster has additional radius constraints, so this scaling is not universal, but the claimed inevitable density decrease is false even in the simplest isolated evaporation model.