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Writing an isophote's radial residual about its fitted ellipse as $\delta r=a_4\cos 4\theta +\cdots$, positive $a_4$ extends the major and minor axes and gives a pointed, disky shape; negative $a_4$ extends diagonal directions and gives a boxy shape. The sign stated here is for radial residuals, not an unconverted brightness residual. Massive slowly rotating elliptical galaxies commonly have boxy isophotes, while embedded rotating disks commonly make lower-mass elliptical galaxies disky.

Bright elliptical galaxies commonly have a shallow central surface-brightness cusp inside a break radius, whereas many less luminous elliptical galaxies have a steep cusp continuing inward. The latter can have many more luminous tracers at a fixed small radius, helping central black hole dynamical measurements. This empirical distinction is not a universal classification by mass alone. Measurements of central galaxy profiles quantify the contrast.

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.

The rotation-anisotropy diagram below uses the oblate-rotator flattening proxy for $\beta =0,0.2,0.4$. Each curve starts at $e=\beta$ and rises as $e$ increases. The plotted domains cover typical spheroid values, roughly $0\le e\lesssim 0.7$ and $0\le v/\sigma \lesssim 1.5$, with the upper axis extended to show the reference curve's rise. The shaded regions are schematic overlapping populations, not measurements or sharp classification boundaries.

Giant elliptical galaxies commonly have low ordered rotation for their observed flattening, lying below the oblate isotropic rotator reference. Anisotropic velocity dispersions and, in many cases, triaxial structure provide support. Lower-luminosity elliptical galaxies are more commonly rotating, disky systems near the isotropic reference. Classical galactic bulges of spiral galaxies often overlap this rotating-spheroid locus; disk-like bulges can be more strongly rotation supported. These are population trends with substantial overlap, not universal rules for individual galaxies.

Observations usually use projected ellipticity and a line-of-sight rotation measure such as peak speed divided by a central velocity dispersion. The intrinsic $e$ and global dispersions used in the derivation are different quantities. Inclination, apertures, and unresolved streaming can change both coordinates. Consequently a point below the reference suggests extra random-motion support, but does not uniquely measure $\beta$.

For a singular isothermal sphere, spherical integration of the mass density givesThe spherical shell theorem makes the radial attraction $GM(<R)/R^2$. Balancing it with circular centripetal acceleration yieldsThus the model has a flat galaxy rotation curve, resembling the outer galaxy rotation curves of many spiral galaxies. Central rotation curves usually rise, and not all real curves are flat throughout. A pressure-supported elliptical galaxy does not itself rotate at this test-particle circular speed.

The model has divergent central mass density, although finite enclosed mass at each finite radius, and an infinite total mass because $M(<R)\propto R$. Its potential is logarithmic, requiring an outer cutoff for a finite escape energy. A globally constant isotropic velocity dispersion, spherical symmetry, and one common profile for all gravitating components are further idealizations. Stellar light commonly has a quite different radial profile from the dark matter halo; real disks are flattened, and halos are neither exact isothermal spheres nor infinite. Tidal truncation also changes the distribution near the satellite boundary.

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.

Interpret the model comparison at the exam's 2008 epoch. A simulation's failure depends on its treatment of baryons, resolution, and adopted stellar feedback; it is not a theorem that the Lambda-CDM model cannot produce the observation under any physical implementation. The following were important tensions for then-standard realizations, with different implications for monolithic galaxy formation.

Old, alpha-enhanced massive spheroids and downsizing. The most massive quiescent elliptical galaxies tend to contain older stellar populations and elevated alpha element abundances relative to iron, indicating that much of their star formation occurred early and rapidly. Prompt core-collapse enrichment followed by shutdown before substantial delayed iron enrichment accounts for the abundance pattern. Models that spread a massive galaxy's star formation over many late progenitor-merger episodes can make its stars too young or insufficiently alpha enhanced. A rapid early dissipative collapse naturally gives a short formation interval and is therefore more immediately consistent with this particular observation. However, downsizing in galaxy formation is about stellar formation, not necessarily final assembly: late dry galaxy mergers can assemble already old alpha-enhanced stars. Early quenching and an appropriate chemical-enrichment model can consequently reconcile hierarchical assembly with the fossil record.

Thin, high-angular-momentum stellar discs. The existence of extended thin galactic disks, including systems with small galactic bulges, challenged simulations that produced compact central stellar concentrations or overly thick discs. Premature gas cooling into small progenitors, transfer of specific angular momentum to the halo, and repeated dynamical perturbations could cause the galactic angular-momentum catastrophe; satellite encounters also increase galactic disk heating. A comparatively quiet dissipative collapse with approximate conservation of baryonic specific angular momentum more readily produces an extended rotating disc, and avoiding subsequent strong perturbations helps retain thinness. Nevertheless, rapid collapse alone does not explain a disc's prolonged star formation and gas replenishment. Hierarchical histories with quieter late accretion, better feedback, delayed cooling, and dissipative disc regrowth can reduce these tensions; a problem in early implementations is not an unavoidable property of the cosmological framework.

Cores inferred in low-mass galaxies. Resolved inner galaxy rotation curves in a number of low-surface-brightness and dwarf systems favoured shallower central mass densities than the cusps of collisionless cold-dark-matter simulations. For a roughly constant-density dark-matter core, $M(<r)\propto r^3$ and $V_c\propto r$; an inner Navarro--Frenk--White profile has $\rho \propto r^{-1}$, hence $M(<r)\propto r^2$ and $V_c\propto r^{1/2}$. The differing slopes explain the cusp--core problem. Beam smearing, inclination, and noncircular motions complicate inference, but do not make the issue merely a normalization mismatch. A simple early dissipative collapse does not automatically solve it: baryonic contraction can deepen the central concentration. Potential fluctuations from gas loss and stellar feedback can instead redistribute dark matter and modify a cusp. Thus this observation tests the joint halo-and-baryon model rather than preferentially vindicating the old collapse picture.

Rapid formation of the stars and hierarchical assembly of the final galaxy are compatible histories. The comparison must keep those clocks separate, and the historical modelling tensions should not be presented as universal failures of all later simulations.

A K correction accounts for observing a redshifted spectrum through a fixed photometric passband, rather than the corresponding emitted band. In the same R-band astronomical magnitude convention,The luminosity distance already incorporates bolometric dimming; the additional correction depends on spectral shape and band response. At $z=3$, observed R-band light at $660\mathrm{nm}$ was emitted at $165\mathrm{nm}$, in the far ultraviolet. It is not a direct measurement of rest-frame R-band luminosity.

To fix the sign, let $L_{\nu }$ be emitted spectral luminosity and approximate R as narrow. Counting photon energy, arrival time and the transformed frequency interval gives $F_{{\nu }_o}=(1+z)L_{(1+z){\nu }_o}/(4\pi d_L^2)$. ThusFor an ordinary old elliptical galaxy, the far-ultraviolet spectrum is weak relative to the optical, so typically $L_{\nu }(165\mathrm{nm})/L_{\nu }(660\mathrm{nm})<1/4$. Then $K_R>0$: neglecting it overestimates the numerical absolute magnitude and underestimates the R-band luminosity. For a young, unobscured star-forming galaxy, hot stars make the far-ultraviolet strong, with an approximately flat $L_{\nu }$ spectrum often serving as a useful model. It gives $K_R=-2.5{\log }_{10}4\simeq -1.51$: neglecting it underestimates the numerical absolute magnitude and overestimates the R-band luminosity. Strong dust extinction or an unusual spectrum can reverse this latter sign; a galaxy label alone does not prove an exact correction. The band-integrated convention is given in Hogg and collaborators' definition of the K-correction.

Assume the stellar yield is common to the compared stellar populations and compare their mass-weighted stellar metallicities. Inverting the metallicity distribution in a leaky box result givesFor example, with $y=Z_{\odot }$, a solar-metallicity massive elliptical galaxy requires $\nu \simeq 0$, whereas $⟨Z{⟩}_{\ast }<0.01Z_{\odot }$ in a dwarf galaxy requires $\nu >99$. A supersolar stellar metallicity requires a correspondingly supersolar stellar yield: outflow alone cannot raise $⟨Z{⟩}_{\ast }$ above $y$. If $y=2Z_{\odot }$, a massive elliptical galaxy with $⟨Z{⟩}_{\ast }$ between $Z_{\odot }$ and $2Z_{\odot }$ has $0\le \nu \le 1$, while the same low-metallicity dwarf galaxy needs $\nu >199$.

For a yield of order solar, the required mass loading runs from order zero or unity in massive spheroids to order one hundred or more in faint dwarfs. Without a specified stellar yield, the robust comparison isThis explains the sense of the galaxy mass--metallicity relation; galactic gas inflow, incomplete depletion, and changes in the initial mass function can also change observed stellar metallicity.

A nonrotating elliptical galaxy need not have an isotropic velocity ellipsoid: unequal random-motion stresses in different directions can support an oblate or triaxial shape without substantial mean rotation. The tensor virial theorem, $2T_{ij}+W_{ij}=0$ for a stationary isolated system with appropriate boundary terms, balances this directional kinetic stress against the directional gravitational stress. Anisotropic random motions can supply the flattening, whereas an isotropic nonrotating equilibrium has much stronger restrictions on its shape.

Typical present elliptical galaxies are dominated by old stellar populations: their brightest short-lived stars have disappeared, leaving faint long-lived stars and stellar remnants, so luminosity per unit stellar mass is low. Ongoing star formation in many spiral galaxies and irregular galaxies supplies luminous young stars, giving a smaller stellar mass-to-light ratio in a fixed band; dark matter, the initial mass function and aperture must also be specified if the quoted ratio is dynamical.

Many lower-luminosity elliptical galaxies have dense central stellar cusps, giving far more photons and kinematic tracers near a central black hole than the shallow depleted cores common in massive elliptical galaxies, as measured in Central galaxy profiles. Thus central surface brightness can make the smaller black hole easier to measure, despite its smaller mass, whereas sparse central tracers and uncertain stellar orbital anisotropy weaken measurements in a large core. This is not a universal mass ordering: the black-hole sphere of influence $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$ must still be sufficiently resolved in angle, so distance, velocity dispersion and angular resolution also matter.

In the radial-residual convention $\delta r=a_4\cos 4\theta$, boxy isophotes have $a_4<0$ and disky isophotes have $a_4>0$, since negative $a_4$ extends the diagonal directions and positive $a_4$ extends the fitted axes. Massive elliptical galaxies tend to have slowly rotating, anisotropic stellar structures associated with dry galaxy mergers, whereas rotating embedded galactic disks, often associated with dissipation in gas-rich galaxy mergers, favour disky structure in lower-mass elliptical galaxies. These are population trends, as in Central structure measurements, rather than a universal mass rule, and brightness-residual Fourier coefficients must first be converted to this radial convention.