How stars orbit a galaxy
Stars go round the centre of a galaxy because of gravity. For a circular orbit, gravity must give just the right pull, so the speed is
v = √(G M ÷ r)
where M is the mass inside the orbit and r the distance from the centre. If almost all the mass is in the middle (like the Sun in the Solar System), then M stays the same and v goes down as r goes up. This is why Earth moves at 30 km/s and Neptune at about 5 km/s.
A graph of orbital speed against distance is called a rotation curve.
The flat rotation curve: the evidence
Astronomers measure speeds of gas and stars in galaxies using the Doppler shift: one side moves toward us (blue-shift), the other away (red-shift). Our own galaxy has a speed of about 220 km/s at the Sun's distance. Vera Rubin and others found, in the 1970s, that speeds in many galaxies stay flat or even rise far beyond the visible disc.
For a flat curve, v is constant, so M must grow in proportion to r. But the light fades away at the edge. So there must be a lot of mass that gives out no light: dark matter. Its extra mass is about 5 to 10 times the visible matter in a galaxy.
Other clues point the same way: the bending of light by clusters of galaxies (gravitational lensing), the fast motion of galaxies inside clusters, and the pattern in the early-universe microwave glow.
What dark matter is (and is not)
Dark matter is matter that does not give out, absorb or reflect light, so we can only know it through gravity. It forms a big, roughly round halo around each galaxy, much larger than the visible disc.
- It is not dark clouds of dust or gas (those we can detect by their effect on light).
- It is not mainly black holes or dim stars; surveys have ruled out enough of those.
- Most scientists think it is made of new, unknown particles. Detectors deep underground are searching for them. Nobody has caught one yet.
An alternative idea is that gravity itself changes at very small accelerations (modified gravity), but dark matter explains more observations together.
Dark energy and the budget of the cosmos
Galaxies are moving away from each other as space expands. In 1998, distant exploding stars showed that the expansion is speeding up, not slowing down. The unknown cause is called dark energy.
Putting everything together, the energy content of the cosmos is about:
- 5% ordinary matter (atoms: stars, planets, gas, us)
- 27% dark matter
- 68% dark energy
Dark matter pulls things together (it helped galaxies form). Dark energy pushes space apart. They are different things, even though both are 'dark'.
Key formulas and definitions
- Orbital speed v = √(G M ÷ r)
- Mass inside orbit M = v² r ÷ G
- Orbital period T = 2π r ÷ v
- Cosmic budget ≈ 5% ordinary matter + 27% dark matter + 68% dark energy
Worked examples
1. A star 8 kpc from the galaxy centre moves at 220 km/s. How long does one orbit take? (1 kpc = 3.09 × 10¹⁶ km, 1 year = 3.16 × 10⁷ s)
r = 8 × 3.09 × 10¹⁶ = 2.47 × 10¹⁷ km. T = 2πr ÷ v = 2π × 2.47 × 10¹⁷ ÷ 220 ≈ 7.05 × 10¹⁵ s. Divide by 3.16 × 10⁷ s per year: T ≈ 2.2 × 10⁸ years, about 220 million years.
2. If only the central mass pulled, a star at 10 kpc moves at 220 km/s. How fast should a star at 40 kpc move?
For a central mass, v ∝ 1 ÷ √r. The distance is 4 times, so speed is 1 ÷ √4 = half. Expected v = 110 km/s. Real galaxies show about 220 km/s: a big difference.
3. Find the mass inside 8 kpc if the speed there is 220 km/s. (G = 6.67 × 10⁻¹¹ N m² kg⁻², 1 kpc = 3.09 × 10¹⁹ m, M☉ = 2 × 10³⁰ kg)
r = 2.47 × 10²⁰ m, v = 2.2 × 10⁵ m/s. M = v² r ÷ G = (4.84 × 10¹⁰)(2.47 × 10²⁰) ÷ 6.67 × 10⁻¹¹ ≈ 1.8 × 10⁴¹ kg ≈ 9 × 10¹⁰ Suns, about 90 billion Suns.
Common mistakes
- Thinking dark matter is dark dust or gas. Dust and gas can be seen through their effect on light; dark matter cannot.
- Mixing up dark matter and dark energy. Dark matter pulls (gravity); dark energy pushes space to expand faster.
- Saying dark matter is proved to be a particle. It is the leading idea, but no particle has been found yet.
- Expecting stars at the edge to be slow. Measurements show they are about as fast as inner stars.