Galaxy Morphology and Classification

Galaxies span a factor of $10^6$ in luminosity and a wide range of shapes, yet their properties are not scattered at random. A morphological sequence organizes the shapes, radial light profiles distinguish the two structural families, and tight scaling relations connect a galaxy's luminosity to the internal motions of its stars. This lesson sets up the Hubble classification, derives the surface-brightness laws for disks and spheroids, obtains the Tully–Fisher, Faber–Jackson, and fundamental-plane relations from virial equilibrium, and states the Schechter luminosity function that counts galaxies as a function of brightness.

The Hubble sequence

Galaxies divide into two broad families and a bridge between them:

• Ellipticals (E0–E7). Smooth, featureless spheroids with little gas or ongoing star formation, supported by the random motions of old stars. The index $n$ in E$n$ encodes the projected flattening, $n=10(1-b/a)$, from round (E0) to lens-shaped (E7).
• Lenticulars (S0). A disk plus a prominent bulge but no spiral arms and little gas — a transition type between ellipticals and spirals.
• Spirals (Sa–Sc) and barred spirals (SBa–SBc). A rotating disk with spiral arms and a central bulge. Along the sequence Sa$\to$Sc the bulge shrinks, the arms open and become more clumpy, and the gas fraction and star formation rise. Barred and unbarred spirals form the two prongs of the fork.
• Irregulars (Irr). Disorganized, gas-rich, star-forming systems off the end of the sequence, common among low-mass and interacting galaxies.

The arrangement is descriptive, not evolutionary; Hubble's terms early (E, S0) and late (Sc, Irr) type persist as labels with no temporal meaning.

Surface-brightness profiles

The projected light of a galaxy, its surface brightness $I(R)$ in luminosity per unit area, distinguishes the two families more sharply than visual morphology.

Disks follow an exponential profile,

with central surface brightness $I_0$ and scale length $h_R$. In magnitudes per square arcsecond the profile is a straight line, $\mu (R)={\mu }_0+1.086(R/h_R)$.

Spheroids — elliptical galaxies and the bulges of spirals — follow the de Vaucouleurs quarter-power law,

where $R_e$ is the effective radius enclosing half the total light and $I_e$ the surface brightness there. The constant $7.669$ enforces the half-light definition. The quarter-power law is far more centrally concentrated and more extended in the outskirts than the exponential.

Both are special cases of the Sérsic profile

with $b_n\approx 2n-0.324$ chosen to keep $R_e$ the half-light radius. The Sérsic index $n$ measures concentration: $n=1$ recovers the exponential disk, $n=4$ the de Vaucouleurs spheroid, and elliptical galaxies span $n\approx 2$–$10$ with more luminous systems having larger $n$.

Integrating the profiles gives the total luminosity. For the exponential disk, $L=2\pi I_0{h}_R^2$; for the de Vaucouleurs law, $L=7.215\pi I_e{R}_e^2$. Both relate the total light to the central brightness and a single length scale.

Scaling relations from the virial theorem

Ellipticals and disks each obey a tight relation between luminosity and an internal velocity. Both descend from the virial theorem, $2K+U=0$, applied to a self-gravitating stellar system of mass $M$, radius $R$, and characteristic velocity $v$: $v^2\sim GM/R$.

Faber–Jackson (ellipticals). The internal velocity is the central velocity dispersion $\sigma$ of the random stellar motions, so ${\sigma }^2\sim GM/R$. Write the mass through a mass-to-light ratio $M={\rm Y}L$ and the luminosity through the mean surface brightness, $L\sim I{R}^2$, so $R\sim (L/I{)}^{1/2}$. Then

If ${\rm Y}$ and $I$ were exactly constant across ellipticals, this gives $L\propto {\sigma }^4$, the Faber–Jackson relation. The observed slope is close to $4$, and $\sigma$ measured from line widths yields distance-independent luminosities.

The fundamental plane. Faber–Jackson has real scatter because ${\rm Y}$ and $I$ are not strictly constant. Retaining them, ellipticals populate a thin plane in the three-dimensional space of $(\log R_e,\log \sigma ,\log I_e)$,

whose tilt from the naive virial exponents ($R_e\propto {\sigma }^2{I}_e^{-1}$) reflects a systematic increase of ${\rm Y}$ with mass. The plane is a sharper distance indicator than Faber–Jackson because it uses all three observables.

Tully–Fisher (spirals). For a rotationally supported disk the internal velocity is the flat rotation speed $v_{\max }$, and $v_{\max }^2\sim GM/R$. The same substitutions give $L\propto v_{\max }^4$, the Tully–Fisher relation. Empirically the slope in the infrared, where dust and young stars matter least, is close to $4$:

The rotation speed comes from the 21-cm line width, independent of distance, so the relation calibrates spiral-galaxy distances up the distance ladder.

The luminosity function

The number density of galaxies as a function of luminosity is the luminosity function $\Phi (L)$, well described by the Schechter function

Three parameters control it:

• $L^{\ast }$ — the characteristic luminosity where the function bends, marking the transition from the power-law regime to the exponential cutoff ($M^{\ast }\approx -20.5$ in the blue for field galaxies).
• $\alpha$ — the faint-end slope. For $\alpha \approx -1.1$ the number density rises gently toward faint galaxies, while the luminosity density $L\Phi (L)$ peaks near $L^{\ast }$.
• ${\Phi }^{\ast }$ — the normalization, of order $10^{-2}{h}^3\mathrm{Mp}{\mathrm{c}}^{-3}$.

The exponential factor suppresses galaxies far brighter than $L^{\ast }$: bright galaxies are exponentially rare. Integrating gives the total number and luminosity densities, $n={\Phi }^{\ast }\Gamma (\alpha +1)$ and $\mathcal{L}={\Phi }^{\ast }{L}^{\ast }\Gamma (\alpha +2)$, in terms of the gamma function, with the latter convergent for $\alpha >-2$.

Environment and the morphology–density relation

Morphology correlates with environment. In dense cluster cores the galaxy population is dominated by ellipticals and S0s, while spirals prevail in the low-density field — the morphology–density relation. The trend reflects environmental processing: in clusters, ram-pressure stripping by the hot intracluster gas removes a spiral's cold disk gas and quenches star formation, repeated high-speed encounters (galaxy harassment) heat the disk, and slow mergers build spheroids. The densest regions thus convert gas-rich disks into gas-poor spheroids, populating the early-type end of the Hubble sequence. This environmental sorting connects the morphology of individual galaxies to the large-scale structure in which they are embedded.

Summary

The Hubble tuning fork orders galaxies from ellipticals (E0–E7) through lenticulars (S0) to normal and barred spirals (Sa–Sc, SBa–SBc), with irregulars off the end; the sequence is descriptive, and bulge fraction, gas content, and arm structure vary along it. Disks follow an exponential surface-brightness profile $I=I_0{e}^{-R/h_R}$ and spheroids the de Vaucouleurs law $\propto \exp [-7.669(R/R_e{)}^{1/4}]$, both special cases of the Sérsic profile with index $n$ measuring concentration. The virial theorem yields the scaling relations: Faber–Jackson $L\propto {\sigma }^4$ and the tilted fundamental plane $R_e\propto {\sigma }^{1.24}{I}_e^{-0.82}$ for spheroids, and Tully–Fisher $L\propto v_{\max }^4$ for disks, each a distance-independent luminosity indicator. The Schechter function $\Phi \propto (L/L^{\ast }{)}^{\alpha }{e}^{-L/L^{\ast }}$ counts galaxies with a faint-end power law bending to an exponential cutoff at $L^{\ast }$. Morphology tracks environment, with early types concentrated in dense clusters through ram-pressure stripping, harassment, and mergers.¹²