---
title: Galaxy Morphology and Classification
draft: false
module: Galaxies and Dark Matter
moduleNumber: 10
lessonNumber: 2
order: 1002
summary: >
  Galaxies sort along the Hubble tuning fork from smooth ellipticals through
  lenticulars to grand-design and barred spirals, with irregulars off the end.
  The light of a spheroid follows the de Vaucouleurs quarter-power law while a
  disk fades exponentially, and the general Sérsic profile interpolates between
  them. Virial scaling relations — Tully–Fisher for disks, Faber–Jackson and the
  fundamental plane for spheroids — tie luminosity to internal motions, and the
  Schechter function fixes the abundance of galaxies as a function of luminosity.
topics: [Galaxies and Dark Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 25 — The Nature of Galaxies; §25.1 The Hubble Sequence; §25.2 Spiral and Elliptical Galaxies"
  - book: Maoz
    ref: "Ch. 6 — Galaxies"
---

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.

$$
% caption: The Hubble tuning fork: ellipticals of increasing flattening lead to the
% S0 junction, where the sequence splits into unbarred and barred spirals with bulges
% shrinking and arms opening from Sa to Sc.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% ellipticals handle
\foreach \x/\lab in {0/E0, 1.3/E3, 2.6/E6} {
  \draw[thick] (\x,0) ellipse (0.34 and 0.24);
  \node[black, anchor=north] at (\x,-0.35) {\lab};
}
\draw[black] (0.4,0) -- (2.3,0);
% S0 junction
\draw[thick] (3.9,0) ellipse (0.4 and 0.2);
\node[black, anchor=north] at (3.9,-0.32) {S0};
\draw[black] (2.95,0) -- (3.5,0);
% fork split
\draw[black] (4.3,0.05) -- (5.2,0.9);
\draw[black] (4.3,-0.05) -- (5.2,-0.9);
% upper prong: normal spirals
\foreach \x/\yy/\lab in {5.6/0.9/Sa, 7.0/1.15/Sb, 8.4/1.4/Sc} {
  \draw[thick] (\x,\yy) circle (0.3);
  \draw[thick] (\x,\yy) .. controls (\x+0.5,\yy+0.3) .. (\x+0.55,\yy-0.15);
  \draw[thick] (\x,\yy) .. controls (\x-0.5,\yy-0.3) .. (\x-0.55,\yy+0.15);
  \node[black, anchor=south] at (\x,\yy+0.45) {\lab};
}
\draw[black] (5.2,0.9) -- (8.4,1.4);
% lower prong: barred spirals
\foreach \x/\yy/\lab in {5.6/-0.9/SBa, 7.0/-1.15/SBb, 8.4/-1.4/SBc} {
  \draw[thick] (\x,\yy) circle (0.3);
  \draw[thick] (\x-0.32,\yy) -- (\x+0.32,\yy);
  \draw[thick] (\x+0.32,\yy) .. controls (\x+0.7,\yy+0.3) .. (\x+0.6,\yy-0.2);
  \draw[thick] (\x-0.32,\yy) .. controls (\x-0.7,\yy-0.3) .. (\x-0.6,\yy+0.2);
  \node[black, anchor=north] at (\x,\yy-0.45) {\lab};
}
\draw[black] (5.2,-0.9) -- (8.4,-1.4);
% irregular
\node[black, anchor=west] at (8.9,0) {Irr};
\end{tikzpicture}
$$

## 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,

$$
I(R) = I_0\, e^{-R/h_R},
$$

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,

$$
I(R) = I_e\,\exp\!\left\{-7.669\left[\left(\frac{R}{R_e}\right)^{1/4} - 1\right]\right\},
$$

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**

$$
I(R) = I_e\,\exp\!\left\{-b_n\left[\left(\frac{R}{R_e}\right)^{1/n} - 1\right]\right\},
$$

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 $n$.

$$
% caption: On a magnitude (log-surface-brightness) axis the exponential disk is a
% straight line while the de Vaucouleurs spheroid curves — steeper in the core and
% shallower in the wings — reflecting its stronger central concentration.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.0,0) node[right, black!70] {radius};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log surface brightness};
% exponential: straight decline
\draw[acc, very thick] (0.3,4.0) -- (8.2,0.5);
\node[acc, anchor=south west] at (5.4,2.5) {exponential disk, n=1};
% de Vaucouleurs: steep then shallow (convex)
\draw[black, very thick] (0.3,4.4) .. controls (0.9,2.6) and (1.8,1.9) .. (3.4,1.4)
  .. controls (5.4,0.9) and (7.0,0.7) .. (8.4,0.55);
\node[black, anchor=west] at (2.5,1.3) {de Vaucouleurs, n=4};
% R_e marker
\draw[black, densely dotted] (2.2,0) -- (2.2,3.2);
\node[black, anchor=north] at (2.2,0) {half-light radius};
\end{tikzpicture}
$$

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 $M = \Upsilon L$ and the luminosity through the
mean surface brightness, $L \sim I\,R^2$, so $R \sim (L/I)^{1/2}$. Then

$$
\sigma^2 \sim \frac{G\Upsilon L}{(L/I)^{1/2}} = G\Upsilon\, I^{1/2} L^{1/2}
\quad\Longrightarrow\quad
L \propto \frac{\sigma^4}{\Upsilon^2 I}.
$$

If $\Upsilon$ 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 $\Upsilon$ and $I$
are not strictly constant. Retaining them, ellipticals populate a thin **plane** in
the three-dimensional space of $(\log R_e,\ \log\sigma,\ \log I_e)$,

$$
R_e \propto \sigma^{1.24}\, I_e^{-0.82},
$$

whose tilt from the naive virial exponents ($R_e \propto \sigma^2 I_e^{-1}$) reflects
a systematic increase of $\Upsilon$ 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 $4$:

$$
L \propto v_{\max}^{4},
\qquad\text{or}\qquad
M_{\text{abs}} = a\,\log v_{\max} + b .
$$

The rotation speed comes from the 21-cm line width, independent of distance, so the
relation calibrates spiral-galaxy distances up the [distance
ladder](/astrophysics-cosmology/observational-foundations/the-cosmic-distance-ladder).

$$
% caption: The Tully–Fisher relation: disk luminosity rises as the fourth power of the
% flat rotation speed, so a distance-independent line width fixes the absolute
% magnitude and hence the distance.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.8,0) node[right, black!70] {log rotation speed};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log luminosity};
% slope-4 line
\draw[acc, very thick] (0.8,0.5) -- (7.8,4.2);
% scattered data points
\foreach \x/\y in {1.3/0.85, 2.1/1.2, 2.9/1.75, 3.5/2.0, 4.3/2.55, 5.1/2.95, 5.9/3.35, 6.7/3.75} {
  \fill[black] (\x,\y) circle (1.5pt);
}
\node[acc, anchor=north west] at (4.4,2.4) {L proportional to v to the fourth};
\end{tikzpicture}
$$

## 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**

$$
\Phi(L)\,\d L = \Phi^\ast\left(\frac{L}{L^\ast}\right)^{\alpha}
                e^{-L/L^\ast}\,\frac{\d L}{L^\ast}.
$$

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 $10^{-2}\ h^3\,\mathrm{Mpc^{-3}}$.

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 $\alpha > -2$.

$$
% caption: The Schechter luminosity function on log axes: a power law of slope alpha
% at the faint end bending to an exponential cutoff above the characteristic
% luminosity L-star, so galaxies much brighter than L-star are exponentially rare.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.0,0) node[right, black!70] {log luminosity};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log number density};
% faint-end power law (gentle rise to the left) then exponential drop
\draw[acc, very thick] (0.4,3.3) -- (5.4,2.35)
  .. controls (6.2,2.1) and (6.8,1.4) .. (7.2,0.5)
  .. controls (7.35,0.2) and (7.5,0.1) .. (7.7,0.05);
\draw[black, densely dotted] (5.4,0) -- (5.4,2.35);
\node[black, anchor=north] at (5.4,0) {L-star};
\node[acc, anchor=west] at (1.5,1.8) {faint-end slope alpha};
\node[black, anchor=west] at (6.5,1.7) {exponential decline};
\end{tikzpicture}
$$

## 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](/astrophysics-cosmology/galaxies/galaxy-clusters-and-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.[^co-morph][^maoz-gal]

[^co-morph]: Carroll & Ostlie, §25.1–25.2 — the Hubble sequence, surface-brightness profiles of spirals and ellipticals, and the fundamental-plane and Tully–Fisher scaling relations.
[^maoz-gal]: Maoz, Ch. 6 — galaxy classification, luminosity profiles, the Schechter luminosity function, and the morphology–density relation.