Elliptical galaxies are categorized within the Hubble sequence based on their apparent ellipticity, denoted by the formula $\epsilon = 1 - \beta/\alpha$, where $\alpha$ and $\beta$ represent the major and minor axes of the projected image, respectively. These systems are traditionally labeled from E0 (spherical) to E7 (highly flattened), though it is physically observed that galaxies with ellipticities exceeding 0.7 do not exist. This morphological classification is heavily influenced by projection effects, as the orientation of a triaxial or spheroidal system relative to the observer’s line of sight determines the apparent shape; for instance, an oblate spheroid viewed pole-on always appears as an E0 regardless of its intrinsic flattening. Morphologically, ellipticals are subdivided into several distinct classes: cD galaxies are immense, luminous systems reaching diameters of up to 1 Mpc and are typically found at the centers of dense galaxy clusters; normal ellipticals range from giant (gE) to compact (cE) types; dwarf ellipticals (dE) possess lower surface brightnesses; and dwarf spheroidal (dSph) galaxies are extremely low-luminosity objects. The light distribution in these galaxies is typically modeled using the de Vaucouleurs $r^{1/4}$ law, which relates surface brightness to the effective radius ($r_e$), defined as the radius within which half of the total luminosity is emitted. A more general form is the Sérsic profile, where the exponent $1/n$ is adjusted to fit the distribution, with $n=4$ recovering the de Vaucouleurs law and $n=1$ describing the exponential profiles often seen in dwarf ellipticals. The physical parameters of these galaxies span a vast range, with absolute B magnitudes varying from −8 to −25, and total masses (including dark matter) extending from $10^7$ to $10^{14}$ solar masses. cD galaxies, the largest of the early-type systems, exhibit extended, diffuse envelopes and exceptionally high mass-to-light ratios, sometimes exceeding 750 solar units, indicating the presence of substantial dark matter halos. Despite their historical designation as “early-type” galaxies, this terminology does not imply an evolutionary sequence but rather reflects their position on the left side of the Hubble tuning-fork diagram.
The dynamics of elliptical galaxies are characterized by random stellar velocities (velocity dispersion, $\sigma$) rather than the systematic rotation dominant in spiral disks. The virial mass of these systems can be estimated using the dispersion of radial velocities through the relationship $M \approx 5R\sigma^2/G$, which accounts for the time-averaged kinetic and potential energy balance in equilibrium. A fundamental correlation exists between the mass of the supermassive black hole (SMBH) at a galaxy’s center and the velocity dispersion of the host’s spheroid, a relationship known as the M-Sigma relation. This suggests a deep physical link between the growth of central black holes and the formation of their parent galaxies. Historically, ellipticals were viewed as nearly devoid of gas and dust, but modern observations reveal a complex interstellar medium (ISM). Most normal ellipticals contain a high-temperature ($10^7$ K) X-ray emitting gas component with masses ranging from $10^8$ to $10^{10}$ solar masses, alongside smaller amounts of warm ionized gas and cold neutral hydrogen or molecular clouds. Their stellar populations are dominated by older, redder stars, and they exhibit distinct color and metallicity gradients; the central regions are typically more metal-rich and redder than the outer layers. This chemical enrichment is correlated with luminosity, as brighter, more massive galaxies tend to have higher overall metal abundances ([Fe/H]). Another identifying characteristic of elliptical galaxies is their high specific frequency ($S_N$) of globular clusters. While spiral galaxies usually have $S_N$ values around 0.5 to 1.2, elliptical galaxies show much higher concentrations, particularly the giant cD systems, which can host tens of thousands of clusters and have specific frequencies near 15. These old, metal-poor clusters within the stellar halo provide critical evidence for the early efficiency of star and cluster formation in the history of these galaxies. Ultimately, the complexity of these early-type systems, including their high-energy emission and evidence of supermassive central engines, challenges the notion of them being simple, “dead” stellar systems.
The Faber–Jackson relation is an empirical correlation between the intrinsic luminosity ($L$) of an elliptical galaxy and the velocity dispersion ($\sigma$) of the stars within it. It serves as the elliptical galaxy counterpart to the Tully–Fisher relation used for spiral galaxies, allowing astronomers to use internal stellar kinematics as a “standard candle” to determine cosmic distances.
Physical Basis and Required Equations: The relation is physically grounded in the virial theorem, which describes the equilibrium between kinetic and potential energy in gravitationally bound systems.
The Virial Mass Equation: The sources state that the mass ($M$) of an elliptical galaxy can be estimated using its radius ($R$) and its radial-velocity dispersion ($\sigma$) through the virial relationship:
$$M \approx \frac{5R\sigma^2}{G}$$
Luminosity and Surface Brightness: A galaxy’s luminosity is related to its effective radius ($r_e$) and its surface brightness ($I$) by the general proportionality:
$$L \propto I r_e^2$$
The Derived Power Law: By assuming that elliptical galaxies have roughly constant average surface brightnesses and mass-to-light ratios ($M/L \approx \text{constant}$), the virial equation can be rearranged. Substituting $R \propto \sqrt{L}$ into the mass-velocity relationship $M \propto R\sigma^2$ (and thus $L \propto R\sigma^2$) leads to the standard form of the relation:
$$L \propto \sigma^4$$
Expressing the luminosity in absolute magnitude, the empirical Faber-Jackson relation is:
$$log_{10}\sigma_0 = -0.1 M_B+ constant$$
As is readily apparent by inspecting observational data, there is considerable scatter in the Faber-Jackson relation. This is further reflected in the fact that the slope of the best-fit line through the data differs slightly from the idealized theoretical derivation. Apparently, the assumption that galaxies are a one-parameter family is not strictly true, which should not be surprising given the simplifications made in its development. Depending on the sample set used, the relation typically follows:
$$L \propto \sigma_0^\alpha, \quad \text{where } 3 < \alpha < 5$$
Multidimensional Scaling In an effort to find a tighter fit to the data, astronomers have introduced a second parameter into the expression: the effective radius ($r_e$). One representation of this empirical fit is:
$$L \propto \sigma_0^{2.65} r_e^{0.65}$$
In this framework, galaxies are visualized as residing on a two-dimensional “surface” within a three-dimensional “space” defined by the coordinates of luminosity ($L$), central velocity dispersion ($\sigma_0$), and effective radius ($r_e$).
Defining the Plane Known as the Fundamental Plane, this relationship combines the contributions of a galaxy’s gravitational potential well ($\sigma_0$) with its physical scale and light output. Alternatively, the fundamental plane can be expressed in terms of the effective surface brightness ($I_e$) of a galaxy at its effective radius:
$$r_e \propto \sigma_0^{1.24} I_e^{-0.82}$$
The fundamental plane appears to represent the whole family of elliptical galaxies and serves as a significant constraint on theories regarding the formation and virialization of these stellar systems.