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A dynamical mass-to-light ratio must specify its aperture and whether the mass includes dark matter. A growing dark halo can increase ${\rm Y}(<R)$ outside the bright stellar body even when the stellar ratio is constant. Comparing a tidal-radius satellite mass with a large-aperture host mass is different from comparing both within their half-light radii. From $M\propto {\sigma }^{2}R$ and $L\propto {\sigma }^4$, the inferred ratio scales as $R/{\sigma }^2$, so equal ratios additionally require homologous radius scaling.

An empirical scaling between a disk galaxy's total baryonic mass and its characteristic flat rotation speed. The ideal isolated deep-MOND limit predicts $v_f^4=G{M}_b{a}_0$. Observational tests must specify the speed estimator, distances, gas masses and stellar mass-to-light ratios; the empirical relation alone does not determine a unique theory of gravity.

A dwarf spheroidal galaxy is a faint, diffuse, largely pressure-supported galaxy with little gas. A large dynamical mass-to-light ratio commonly indicates dark matter, subject to assumptions of equilibrium, member selection, and binary-star corrections. Its stellar light and total mass need not share one density profile.

The Faber-Jackson relation is an empirical correlation between luminosity and stellar velocity dispersion for elliptical galaxies, approximately $L\propto {\sigma }^4$ in its classical form. Slope, scatter, sample and band matter. A heuristic combines $M\propto {\sigma }^{2}R$, common mass-to-light ratio, and common characteristic surface brightness $L/R^2$. The luminosity-dispersion correlation alone does not imply those auxiliary assumptions, and extrapolation to a dwarf spheroidal galaxy is not guaranteed.

The fundamental plane relates effective radius, stellar velocity dispersion, and mean surface brightness of elliptical galaxies. Homologous virial equilibrium with constant mass-to-light ratio predicts $R_e\propto {\sigma }^2/I_e$; empirical exponents differ. The two-variable Faber-Jackson relation is a projection, not the entire plane.

For an isolated spherical stellar system in equilibrium with constant mass-to-light ratio, the virial theorem gives $2K+W=0$. Spherical averaging gives $K=3\pi {\rm Y}\int R\mu {\sigma }_p^{2}dR$, while $W=-4\pi G{{\rm Y}}^2\int r\lambda Ldr$. Dividing yields the displayed estimator, independent of the radial velocity-anisotropy parameter. It requires the full aperture and negligible boundary stress. A radially varying mass-to-light ratio or an independent dark matter halo prevents interpreting this expression as a general reconstruction of the mass profile.

The integrated halo star-formation efficiency is the fraction of a dark matter halo's reference baryonic mass converted into stars: ${ϵ}_{\ast }=M_{\ast }/(f_b{M}_h)$, with cosmic baryon fraction $f_b$. It is distinct from star-formation efficiency per free-fall time. Cooling, gas supply and retention, stellar feedback and active-galactic-nucleus feedback make it strongly dependent on dark matter halo mass. With stellar mass-to-light ratio ${{\rm Y}}_{\ast }$, a simple central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. Changing this efficiency changes both the mapping and the logarithmic Jacobian between the dark matter halo mass function and the luminosity function.

The mass-to-light ratio compares the mass assigned to an astronomical system with its luminosity in a specified band, often in solar units. A stellar value uses only stellar mass and depends on the stellar population, age, composition and band. A dynamical value can include gas and dark matter, so it need not equal the stellar value. In a halo-to-galaxy mapping, $L=M_{\ast }/{{\rm Y}}_{\ast }$ requires the stellar ratio and not the total dark matter halo ratio.

Write $x=m/M_{\odot }$, the initial mass function as $dN=A{x}^{-\alpha }dx$, and a main-sequence mass-luminosity relation as $L/L_{\odot }=x^{\beta }$. On $a<x<b$, the mass-to-light ratio in solar units isFor $\alpha =\beta =3$ this is exactly $1/(ab)$. The result concerns the specified unevolved stellar population; stellar remnants, evolved luminous stars and cooling substellar objects require different accounting.

The three probes constrain the same deep gravitational potential by different observations. A large member-galaxy velocity dispersion requires a large binding mass. Applying the virial theorem, with a gravitational scale radius $R$ and approximately isotropic orbits, gives a cluster velocity-dispersion mass estimate of order $M\sim R{\sigma }_v^2/G$, with a profile-dependent numerical coefficient. The inferred mass-to-light ratio is much larger than the stellar value. Membership, orbital anisotropy and equilibrium assumptions affect the precision.

The hot intracluster medium radiates through thermal bremsstrahlung and line emission. Its X-ray spectrum measures temperature, while its surface brightness constrains gas density. Confining this gas requires a potential with characteristic $GM/R\sim k_{B}T/(\mu m_p)$. Combining resolved profiles through the cluster hydrostatic mass estimator gives the total mass, not just the gas mass. The detected gas contributes substantial baryonic mass but does not normally account for the full gravitating mass. Nonthermal pressure or a merging cluster can invalidate a purely thermal hydrostatic equilibrium estimate.

Finally, gravitational lensing measures mass through light deflection. Strong gravitational lensing supplies multiple images and arcs; weak gravitational lensing measures coherent background-image distortion over larger radii. The lensing convergence is projected surface mass density divided by the critical surface density, irrespective of whether the material emits light. Cluster gravitational-lensing mass estimates agree broadly with the high masses implied by galaxy motions and hot gas, while having different equilibrium assumptions. Together these observations support a dominant nonluminous component in galaxy clusters, after accounting for their stars and observed gas.

For approximately circular motion in a spherically dominated Newtonian gravitational field, $v_c^2(r)=GM(<r)/r$. If most gravitating mass followed the concentrated luminous galaxies, then beyond their main light distribution the enclosed mass would nearly saturate, giving a declining Keplerian galaxy rotation curve, $v_c\propto r^{-1/2}$.

Instead, extended galaxy rotation curves commonly remain roughly flat. The spherical mass profile for a flat rotation curve then requiresThe inferred mass continues growing where the luminous contribution is small, so the mass-to-light ratio increases outward. Under the assumed gravitational dynamics this is evidence for an extended dark matter halo. Measurements of neutral-gas motion can probe radii beyond the bright stellar disk. A realistic disk contribution must be computed using its flattened geometry; the spherical formula describes a halo-dominated region and does not turn every disk rotation curve into a spherical mass profile.

A genuine nondegenerate bounce has ${\dot{a}}_{\ast }=0$ and ${\ddot{a}}_{\ast }>0$; using ${\ddot{a}}_{\ast }\ge 0$ also includes the limiting stationary case. Put $s=a_{\ast }^{-1}=1+z_{\ast }$, and define the present matter density parameter ${\Omega }_m=8\pi G{\rho }_0/(3H_0^2)$, ${\Omega }_{\Lambda }=\Lambda /(3H_0^2)$ and ${\Omega }_k=-k/H_0^2$. The dust-plus-cosmological-constant Friedmann equation and its acceleration equation becomeAt the bounce these giveFor a past bounce with $s>1$, eliminate ${\Omega }_k$ from the first equation:Subtracting ${\Omega }_m{s}^3/2$ and multiplying by the positive denominator givesThus the matter bound from a Friedmann bounce isStrict positive bounce acceleration makes this inequality strict. The zero of the Friedmann equation also requires positive curvature for positive matter and positive $\Lambda$.

An observed source at $z=6$ has $a_e=1/7$. Since the minimum scale factor cannot exceed the scale factor at emission, $z_{\ast }\ge 6$. The denominator of the bound increases with positive $z_{\ast }$, soThe observational lower bound is much larger. Cluster mass-to-light ratios, inferred independently from motions, gas temperatures and gravitational lensing, combined with the observed luminosity density imply substantial cosmic mass density. Measurements of the cosmic baryon fraction from representative clusters, combined with the baryon cosmological density parameter inferred from Big Bang nucleosynthesis, provide another estimate: ${\Omega }_m\simeq {\Omega }_b/f_b$. Large-scale clustering and peculiar velocities provide independent dynamical checks, although their interpretation must account for galaxy bias. These methods support matter densities of order a few tenths, rather than below one percent. The dust-plus-cosmological-constant bouncing model is incompatible with both a quasar at redshift six and the stated observational evidence for ${\Omega }_m\gtrsim 0.2$.

For a relaxed cluster, galaxy motions act as tracers of its gravitational potential. The virial theorem gives a dynamical mass of orderwith a profile-dependent coefficient and a projection correction because the observed velocity dispersion is usually only the line-of-sight component. For an isotropic uniform-sphere estimate using its one-dimensional dispersion, $T=3M{\sigma }_v^2/2$ and $W=-3G{M}^2/(5R)$ give $M=5R{\sigma }_v^2/G$. Measured dispersions require much more gravitating mass than the luminous galaxies provide, hence large cluster mass-to-light ratios. Interlopers, orbital anisotropy and departures from equilibrium are systematic uncertainties, so the independent gas and lensing measurements matter.

First velocity moment. Assume a stationary, spherical, nonstreaming collisionless stellar system with constant mass-to-light ratio ${\rm Y}$. Use a luminosity-weighted galactic distribution function, with luminosity density $\lambda$. Multiplying the Collisionless Boltzmann equation by $v_i$ and integrating over velocities, with its velocity-space boundary terms vanishing, givesSpherical symmetry makes the stress tensorHere ${\sigma }_t^2$ is one tangential component, as in the paper; the total tangential variance is $2{\sigma }_t^2$. Its radial divergence is $q^{\prime }+2(q-q_t)/r$, since the divergence of the radial unit vector is $2/r$. With the velocity-anisotropy parameter $\beta =1-{\sigma }_t^2/{\sigma }_r^2$, and enclosed mass $M(r)={\rm Y}L(r)$, this gives the Spherical Jeans equationThis uses a constant ratio of total gravitating mass to luminosity, rather than an arbitrary independent dark halo.

Projection and elimination of anisotropy. Write $R$ for projected radius and $r$ for spherical radius. The Abel transform of the luminosity density giveswhose inverse is the displayed density inversion in the paper. At a point on this line of sight, ${\sin }^2\theta =R^2/r^2$. The line-of-sight second stellar velocity moment is ${\sigma }_r^2{\cos }^2\theta +{\sigma }_t^2{\sin }^2\theta ={\sigma }_r^2(1-\beta R^2/r^2)$. Therefore the measured quantity $J(R)=\mu (R){\sigma }_p^2(R)$ satisfiesThe Spherical Jeans equation gives $\beta q=-r{q}^{\prime }/2-\gamma \lambda L/(2r)$. Substitution yieldsThe inner $\gamma L$ printed in the PDF's definition of $p$ must therefore be $\lambda L$: the literal expression has an extra gravitational factor and lacks the density. Both this derivation and dimensional consistency require the correction. Define the projected spherical Jeans residual byTo express it entirely in terms of the radial stress, putDifferentiating the nonsingular $z$ representation gives $A^{\prime }(R)=R{\int }_R^{\infty }{q}^{\prime }(r)/\sqrt{r^2-R^2}dr$. HenceIf $R^{2}A$ vanishes at zero and infinity, then ${\int }_0^{\infty }6\pi Rp(R)dR=6\pi [R^{2}A{]}_0^{\infty }=0$. These are the absence-of-boundary-stress conditions needed for the global result.

The virial relation, term by term. Interchange the order in the residual integral and useIt follows thatThe first term is $2K/{\rm Y}$: spherical averaging assigns one third of the integrated velocity-square to any fixed viewing direction, giving $K=3\pi {\rm Y}\int RJdR$. The gravitational potential energy isThus the integrated residual is exactly $(2K+W)/{\rm Y}$, and its vanishing is the virial theorem. Solving for $\gamma$ gives the global projected virial estimate of a mass-to-light ratio,It is independent of the radial anisotropy profile when the full aperture and the stated boundary conditions are used. A finite aperture retains a boundary term.

Let $p(r)=\rho (r){\sigma }_r^2(r)$ and use the velocity-anisotropy parameter $\beta =1-{\sigma }_{\theta }^2/{\sigma }_r^2$, with ${\sigma }_{\theta }^2={\sigma }_{\varphi }^2$. Here the tangential variance is per component; a definition using the sum of both tangential variances instead has an extra factor of two. The original PDF has this correct per-component ratio, which the TeX corrupts.

At a point along the line of sight, $r^2=R^2+z^2$. The radial direction makes an angle with the sightline satisfying ${\sin }^2\chi =R^2/r^2$. Spherical symmetry eliminates mixed moments, and projection givesUsing both sides of the galaxy and $dz=rdr/\sqrt{r^2-R^2}$,For this formula use $I$ as the projected density of the same tracer represented by $\rho$. If $\rho$ is a mass density and $I$ an observed surface brightness, a constant mass-to-light ratio rescales $I$ by that factor; otherwise replace $\rho$ by the luminosity density throughout the tracer equation. The gravitating $M(r)$ remains the total mass, not necessarily the tracer's mass.

The Spherical Jeans equation is $p^{\prime }+2\beta p/r=-\rho GM/r^2$. Eliminate $\beta p=-r[p^{\prime }+\rho GM/r^2]/2$ from the projected integrand:Rearranging provesNo constancy of $\beta$ was needed. Convergence of the projected moments and the steady spherical assumptions are required. This is the integral form of the projected spherical Jeans residual.

Although $\beta$ has disappeared explicitly, the unknown radial pressure $p$ still carries the orbital information. Thus the identity does not remove the mass--anisotropy--density degeneracy: projected brightness and one projected second moment do not uniquely determine both $M(r)$ and the radial/tangential partition. A central rise in velocity dispersion can reflect a compact central mass, radial orbital bias, or a changing mass-to-light ratio; flattening and departures from equilibrium add further ambiguities.

For example, within a point-mass-dominated region with tracer $\rho \propto r^{-k}$ and constant $\beta$, direct solution gives ${\sigma }_r^2=G{M}_{\bullet }/[(k+1-2\beta )r]$. For $k>1$ the projected velocity dispersion of a Keplerian cusp isThe beta function factors come from setting $r=R\sec \chi$ in the integrals along the line of sight. Different allowed $\beta$ generally change the mass needed to fit the same normalization; $k=2$ is a special cancellation, not the general case. Resolving the central black-hole sphere of influence and constraining orbital structure with higher velocity moments or additional kinematic directions strengthens a compact-mass inference. Brightness and velocity dispersion profiles alone do not establish a unique central black-hole mass, still less an event horizon.

Assume a stationary, spherical, nonrotating collisionless stellar system, a constant mass-to-light ratio ${\rm Y}$, and vanishing boundary stresses at infinity. Let $\lambda$ be the luminosity density, so $\rho ={\rm Y}\lambda$, and set $\gamma =G{\rm Y}$. Distinguish spatial radius $r$ from projected radius $R$. The enclosed luminosity and radial force areHere ${\sigma }_t^2$ is the velocity dispersion of either one of the two tangential components; thus $\beta =1-{\sigma }_t^2/{\sigma }_r^2$. If a combined tangential variance is used instead, its contribution must be divided by two in this definition.

To derive the Jeans equation, start with the steady Collisionless Boltzmann equation $v_j{\partial }_{j}f-({\partial }_j\Phi ){\partial }_{v_j}f=0$. Multiply by $v_i$ and integrate over all velocities. Integration by parts in velocity, with $f$ decaying fast enough to eliminate boundary terms, givesThe same equation holds with luminosity weighting because the tracer's luminosity is conserved along its orbit. Spherical symmetry makes the local stellar velocity moment tensor diagonal, with entries $q=\lambda {\sigma }_r^2$, $q_t=\lambda {\sigma }_t^2$, $q_t$. Its radial divergence is $q^{\prime }+(2q-2q_t)/r$: the second term accounts for the changing directions of the radial and tangential unit vectors. ThereforeThis is the Spherical Jeans equation; it follows from a stellar velocity moment rather than requiring a particular form of the galactic distribution function.

Integrating the luminosity density along a line of sight gives its surface brightness:The second identity is Spherical Abel deprojection. For the second stellar velocity moment, the angle between the radial direction and the line of sight has ${\sin }^2\theta =R^2/r^2$. The line-of-sight velocity dispersion is therefore ${\sigma }_r^2{\cos }^2\theta +{\sigma }_t^2{\sin }^2\theta ={\sigma }_r^2(1-\beta R^2/r^2)$, andFrom the Spherical Jeans equation, $\beta q=-r{q}^{\prime }/2-\gamma \lambda L/(2r)$. Substitution producesThe primary PDF has a genuine symbol error in its projected gravity term: it prints a second $\gamma$ multiplying $L$ inside the integral, where $\lambda$ is required. The corrected projected spherical Jeans residual isTo justify the derivative at the singular-looking lower limit, write $A(R)={\int }_0^{\infty }q(\sqrt{R^2+z^2})dz$. Differentiation then gives $A^{\prime }(R)=R{\int }_R^{\infty }{q}^{\prime }(r)(r^2-R^2{)}^{-1/2}dr$. In particular, $Rp=(R^{2}A{)}^{\prime }$. If $R^{2}A$ tends to zero both at zero and infinity, as for a finite isolated equilibrium with sufficiently decaying stresses,If a boundary stress survives, its contribution must be retained; the zero-integral statement is not a finite-aperture identity.

Now identify each integral with a term in the virial theorem. Let $K$ be the total kinetic energy, including all three random components, and $W$ the gravitational potential energy. Averaging over spatial directions in a spherical system gives one third of the total stellar velocity moment along any chosen line of sight, even if the velocities are locally anisotropic. ThusBuilding the spherical mass shell by shell givesFinally, Fubini theorem and ${\int }_0^r{R}^3(r^2-R^2{)}^{-1/2}dR=2r^3/3$ give the gravitational contribution to the residual integral explicitly:ConsequentlyThe global projected virial estimate of a mass-to-light ratio is independent of the velocity-anisotropy parameter under these assumptions. It does not remove the mass--anisotropy--density degeneracy for arbitrary radial mass profiles. In particular, if an independent dark matter halo makes mass fail to follow light, a single constant ${\rm Y}$ is no longer the physical enclosed mass-to-light ratio at every radius.

The host mass inside $D$ follows from the same singular isothermal sphere:Using the supplied host luminosity givesThis treats essentially all the stated optical light as contained in that aperture; a measured luminosity profile would refine that convention. Keeping consistent unrounded quantities,Thus the dwarf ratio is about twenty times the parent ratio; the rough equal-density tidal convention would give a factor thirty.

There is no logical contradiction in two objects sharing $L\propto {\sigma }^4$ but having different mass-to-light ratios. A virial theorem scaling gives $M\propto {\sigma }^{2}R$, so that $M/L\propto R/{\sigma }^2$. Equal ratios additionally require homologous relevant radii scaling as $R\propto {\sigma }^2$. Here the satellite's tidal radius scales as ${\sigma }_d$ in the fixed host field, not as ${\sigma }_d^2$. Equating the two ratios would require $r_t=D({\sigma }_d/{\sigma }_h{)}^2\simeq 56\mathrm{pc}$, far smaller than the derived tidal radius.

The heuristic derivation of the Faber-Jackson relation with common mass-to-light ratio and common surface brightness made auxiliary structural assumptions; the empirical relation alone cannot enforce them. Furthermore, the aperture dependence of mass-to-light ratio matters: a satellite's whole tidal mass and a host's $50\mathrm{kpc}$ halo mass are not homologous half-light-aperture masses. Dark matter need not follow stellar light, and a dwarf spheroidal galaxy is not automatically a scaled giant elliptical galaxy. The model calculates total gravitating mass: reading its density literally as stars alone would imply an implausibly large purely stellar ratio, exposing the limits of that interpretation.

Use the classical Faber-Jackson relation $L\propto {\sigma }^4$ as the explicitly requested extrapolation. The supplied host fixes its normalization:With the mass derived using circular-orbit Jacobi truncation of two singular isothermal spheres, the dynamical mass-to-light ratio isHence to one significant figureThe rough equal-density truncation convention gives ${{\rm Y}}_d\simeq 6\times 10^2{M}_{\odot }/L_{\odot }$ instead. Both conclusions indicate a very large dynamical mass-to-light ratio compared with a normal old stellar population.

Within their bright stellar bodies, typical giant elliptical galaxies have optical dynamical mass-to-light ratios of order a few to ten in solar units, with variations in band, initial mass function, and dark matter. Large outer apertures can give substantially larger ratios. Many dwarf spheroidal galaxies have much larger inferred ratios, from tens to hundreds and, for some very faint systems, still larger values. Thus the computed ratio is plausible as the scale of a very dark-matter-dominated faint dwarf, not as a purely stellar ratio. Dynamical inference depends on member identification, binary motions, equilibrium, and the assumed mass profile.

The stipulated Faber-Jackson relation is not a reliable universal extrapolation down to every dwarf spheroidal galaxy. Also, the fundamental plane of elliptical galaxies contains radius and surface brightness as well as velocity dispersion; the two-variable Faber-Jackson relation is a projection of that information, not an interchangeable definition of the full plane.

The result is in the broad range expected for an old, relatively faint stellar population, but is high compared with many observed stellar mass-to-light ratios of a few solar units in optical bands. The comparison must specify the band and whether the measured mass includes dark matter: for example, early-type galaxy dynamical measurements find an $I$-band normalization near $3.8$ solar units at velocity dispersion $200\mathrm{km}{\mathrm{s}}^{-1}$, including a contribution from dark matter. That is an illustrative observational comparison, not a direct equality with a bolometric stellar calculation.

For an old stellar population, an upper surviving main sequence mass near $M_{\odot }$ is reasonable because more massive stars have shorter lifetimes; low-mass main sequence stars also contain much of the surviving stellar mass. However, red giants and other evolved stars can contribute a large fraction of the light despite their small numbers, so omitting them generally makes this calculated mass-to-light ratio too large. Omitting stellar remnants has the opposite effect on the inferred mass, and a single cubic mass-luminosity relation is only approximate across the stated range. Finally, a single $m^{-3}$ initial mass function down to $0.1M_{\odot }$ is steeper and more low-mass-heavy than commonly adopted distributions, which flatten at low mass. Ten solar units is a useful toy-model estimate, not a precision stellar-population prediction.

The dark matter halo population follows gravitational hierarchical galaxy formation: many low-mass dark-matter haloes and an exponentially rare high-mass tail. The Press-Schechter halo mass function captures this broad form. A constant conversion from dark matter halo mass to luminosity would simply rescale that distribution and predict too much light from both very small and very large systems. The luminosity function instead has a shallower faint end and a sharp bright-end decline, often described by a Schechter function,Define the halo star-formation efficiency as ${ϵ}_{\ast }(M_h)=M_{\ast }/(f_b{M}_h)$, distinguishing this integrated baryon conversion from the star-formation efficiency per free-fall time. If the stellar mass-to-light ratio is ${{\rm Y}}_{\ast }$, a simple one-central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. For a monotone mapping without scatter,Thus a mass-dependent conversion efficiency and its Jacobian change the shape, not just the horizontal scale, of the luminosity function. Scatter, satellites and the stellar mass-to-light ratio add further differences.

In shallow potential wells, photoheating can prevent gas accretion, while stellar feedback from radiation, stellar winds and supernovae heats or ejects gas. These effects lower halo star-formation efficiency toward low mass, helping flatten the faint end. The need for atomic or molecular line cooling also affects the smallest systems. Intermediate-mass dark-matter haloes retain gas and cool efficiently, so the integrated conversion reaches a maximum around galactic dark matter halo masses.

In massive dark-matter haloes, long optically thin gas cooling times and stable hot atmospheres reduce fresh cold-gas supply. Active-galactic-nucleus feedback can replenish the thermal energy lost in cooling and suppress a cooling flow, reducing halo star-formation efficiency and sharpening the bright cutoff. Massive group or cluster dark-matter haloes contain many distinct galaxies rather than one object with luminosity proportional to their entire mass. Galaxy mergers redistribute stars and can grow already formed bright galaxies, but do not turn all hot dark matter halo gas into new stars. The luminosity function is the gravitational dark matter halo population filtered through strongly mass-dependent baryonic physics. The plotted abundance and efficiency curves are qualitative examples, not observational fits or calibrated mass-to-light relations.