Galactic Dynamics, Redshift, and Hubble’s Law

Galaxies are huge systems of stars, gas, dust and dark matter held together by gravity; the light of distant galaxies is redshifted, z = Δλ ÷ λ₀ (approximately v ÷ c for small z), and Hubble’s law, v = H₀d, relates the recession velocity of galaxies to their distance, showing that the universe is expanding, with H₀ ≈ 70 km/s per megaparsec. In brief: the rotation curves of spiral galaxies are flat, instead of falling off with distance as predicted from the visible matter, which is evidence for dark matter. The redshift of a galaxy is caused mainly by the expansion of space stretching the wavelength of light (cosmological redshift), not by motion through space. The Hubble time, 1/H₀ ≈ 14 billion years, is a first estimate of the age of the universe.

The discovery that galaxies move away from us, with more distant ones receding faster, changed our picture of the universe. This guide describes galaxies and how their rotation reveals dark matter, explains redshift, Hubble’s law and the expansion of the universe, compares the types of galaxies in a table, works through numerical examples and corrects common mistakes.

What are galaxies?

A galaxy is a gravitationally bound system of 10⁶ to 10¹² stars, with gas, dust and dark matter. The observable universe contains on the order of 10¹¹ to 10¹² galaxies. Types (Hubble’s classification): spiral galaxies (a flat rotating disc with spiral arms and a central bulge, such as the Milky Way and Andromeda; active star formation), elliptical galaxies (smooth ellipsoids of old stars with little gas, ranging from dwarfs to giants of 10¹³ M_☉) and irregular galaxies (such as the Magellanic Clouds). The Milky Way is a barred spiral with a diameter of about 100,000 light-years, roughly 2 × 10¹¹ stars, a supermassive black hole at its center, and a rotation period of about 230 million years at the Sun’s position (8 kpc from the center, orbiting at about 220 km/s). Galaxies are grouped in groups (the Local Group, containing the Milky Way and Andromeda), clusters (Virgo, with about 1,300 members) and superclusters, along filaments around large voids.

What do rotation curves show?

For a star at radius r in a circular orbit around the center of a galaxy, gravity provides the centripetal force: v² ÷ r = GM(r) ÷ r², where M(r) is the mass inside r, so v = √(GM(r) ÷ r). If most of the mass were in the visible bright central region, v would fall as 1/√r beyond it (as for the planets in the solar system, the Keplerian decline). Vera Rubin and Kent Ford measured in the 1970s the rotation speed of stars and gas in spiral galaxies from the Doppler shift of their spectral lines, and found that the curves stay flat at large radii, at about 200 to 250 km/s, far beyond the visible disc. A constant v implies M(r) ∝ r: the mass keeps growing linearly with radius. The visible matter accounts for only about 10 to 20 percent of the mass of a galaxy. The rest is dark matter, in an extended halo. Other evidence: the velocities of galaxies in clusters (Zwicky, 1933), gravitational lensing of light by clusters, and the pattern of the cosmic microwave background. Its nature is unknown (probably weakly interacting massive particles or other new particles); alternative theories modify gravity (MOND) but have difficulty explaining all the evidence.

What is redshift?

The spectral lines of distant galaxies are shifted to longer wavelengths. The redshift is defined as z = (λ_obs − λ_emit) ÷ λ_emit. For small z (up to about 0.1), the recession velocity is v ≈ cz. For large z, the relativistic formula must be used: 1 + z = √[(1 + v/c) ÷ (1 − v/c)] for a Doppler shift, but for cosmological redshifts the interpretation is that the wavelength is stretched by the expansion of space, with 1 + z = a_now/a_then (the ratio of the scale factor of the universe now to the time of emission): a galaxy at z = 1 emitted its light when the universe was half its present size. The most distant galaxies observed by the James Webb Space Telescope have z of about 10 to 14 (the universe was 300 to 500 million years old). A blueshift is seen for nearby objects moving toward us (Andromeda is approaching at about 110 km/s).

What is Hubble’s law?

Edwin Hubble found in 1929 that the recession velocities of galaxies are proportional to their distances: v = H₀ d. The Hubble constant H₀ is about 70 km/s/Mpc (measurements range from about 67 to 74 km/s/Mpc, a discrepancy known as the “Hubble tension”). A galaxy at 100 Mpc recedes at 7,000 km/s. The law is a consequence of the uniform expansion of space: all points move away from each other, so observers anywhere in the universe see the same pattern; there is no center. An analogy is the surface of a balloon with dots being inflated, or raisins in a rising loaf of dough. The expansion does not apply within bound systems (galaxies, the solar system, atoms). The inverse of H₀, the Hubble time t_H = 1/H₀ = 977.8 ÷ H₀ Gyr (with H₀ in km/s/Mpc) = 14.0 Gyr for 70 km/s/Mpc, is the age of the universe if the expansion rate were constant (the best present value, with the acceleration and deceleration history, is 13.8 billion years). Hubble’s law is used as the last step of the distance ladder: d = cz/H₀ for z ≪ 1.

Galaxy types compared

Worked example: redshift, distance and galaxy mass

1. Redshift from a line. The hydrogen-alpha line (rest wavelength 656.3 nm) in the spectrum of a galaxy is observed at 662.9 nm. z = (662.9 − 656.3) ÷ 656.3 = 0.01006.
2. Recession velocity: v = cz = 3.00 × 10⁵ km/s × 0.01006 = 3,017 km/s (the nonrelativistic approximation is valid since z ≪ 1).
3. Distance: d = v ÷ H₀ = 3,017 ÷ 70 = 43 Mpc = 140 million light-years.
4. Hubble time: 1/H₀: H₀ = 70 km/s/Mpc = 70 × 10³ ÷ 3.086 × 10²² = 2.27 × 10⁻¹⁸ s⁻¹, so 1/H₀ = 4.4 × 10¹⁷ s = 14.0 billion years.
5. Mass of the Milky Way from the rotation of the Sun. v = 220 km/s at r = 8.0 kpc (2.47 × 10²⁰ m): M(r) = v²r ÷ G = (2.2 × 10⁵)² × 2.47 × 10²⁰ ÷ 6.674 × 10⁻¹¹ = 1.79 × 10⁴¹ kg = 9 × 10¹⁰ M_☉ inside the orbit of the Sun.
6. Flat rotation curve. If v stays at 220 km/s out to 30 kpc, the mass inside that radius is M = v²r ÷ G = 3.4 × 10¹¹ M_☉, almost 4 times the mass inside the Sun’s orbit, while the visible matter ends at about 15 kpc: there is much unseen mass.
7. High redshift. A galaxy with z = 2 has light stretched by a factor of 3: the Lyman-alpha line (121.6 nm) is observed at 365 nm (near-UV). The universe was 1/3 of its present size when the light was emitted.

How was expansion discovered?

Vesto Slipher measured the redshifts of spiral nebulae from 1912. Edwin Hubble used Cepheids to show in 1923 that Andromeda is a galaxy far outside the Milky Way, and in 1929 published the velocity-distance relation (his original value of H₀ was about 500 km/s/Mpc, because of errors in the distance scale). Georges Lemaître had derived the expansion from general relativity in 1927, and Einstein’s static universe was abandoned. In 1998, observations of type Ia supernovae by two teams (Perlmutter, Schmidt, Riess) showed that the expansion is accelerating (Nobel Prize in Physics 2011), attributed to dark energy.

Which mistakes are common?

Mistake 1: Thinking that galaxies are moving through space away from us like an explosion

It is space itself that expands. There is no center of the expansion.

Mistake 2: Saying that the Hubble law applies to all objects

Gravitationally bound systems do not expand. Andromeda is approaching the Milky Way.

Mistake 3: Using v = cz for large redshifts

At z of order 1 or more, the relativistic or cosmological formulas must be used (v = cz would give a speed above c for z > 1).

Mistake 4: Assuming that the flat rotation curve is explained by the visible matter

The visible matter predicts a decline. The flat curve requires additional (dark) mass.