The Earth orbits the Sun at 29.8 km/s, while Neptune, about 30 times farther out, orbits at 5.4 km/s[1]. Almost all of the Solar System's mass is in the Sun, so the only pull that matters for a planet is the Sun's gravity.
For a planet of mass to keep moving at speed on a circle of radius , it needs a force of toward the centre. That force is the gravity of the Sun, whose mass is : .
Setting the two forces equal gives . Cancelling and multiplying by leaves , and taking the square root gives .
is the same for every planet, so is inversely proportional to . Thirty times the radius means of the speed, and 29.8 km/s divided by 5.5 is 5.4 km/s, Neptune's value.
The left of the figure shows the orbits of three planets; the right plots against radius, with a dot for each planet.
In the disc of a spiral galaxy, stars and gas all circle the centre in the same direction. Seen at an angle, one side of the disc is coming toward us and the other is moving away.
Light from the approaching side is shifted to shorter wavelengths, and light from the receding side to longer ones. If the Doppler shift is and the original wavelength is , then , where is the speed along the line of sight and is the speed of light.
The two strips at the bottom of the figure are spectra of the light from the left and right ends of the disc, and the dashed line marks the original wavelength .
The disc is tilted to our line of sight, so the measured speed is corrected for the tilt angle to give the rotation speed[2].
Astronomers measure the line of hydrogen gas glowing around young stars in visible light, and the 21 cm radio waves from hydrogen atoms. The hydrogen seen at 21 cm reaches two to four times the radius of the stellar disc, so it gives speeds even where there are hardly any stars[3].
A rotation curve is a graph of rotation speed against distance from the centre. In most spiral galaxies, the speed rises near the centre and then stays almost constant out beyond the stellar disc[2]. In luminous spiral galaxies the curve is still flat 50 kpc from the centre (1 kpc is about 3260 light years)[2].
In the Milky Way, the speed at the Sun's position, 8.1 kpc from the centre, is 229 km/s. Between 5 and 25 kpc, it drops by only about 1.7 km/s for each kpc of radius[4].
The dots and the bright line in the figure are the measured rotation curve; the faint line is for mass concentrated at the centre.
A galaxy's starlight is strongest at the centre and fades quickly toward the edge. The rotation speed calculated from the distribution of stars and gas also falls in the outer parts[3].
If what shines were all the mass, the enclosed mass would almost stop growing beyond some radius. Beyond it is effectively constant, and the speed falls as , just as for the planets.
If all of the Milky Way's mass lay inside the Sun's orbit, the speed at 25 kpc would be km/s. The measured curve gives about 200 km/s there, some 70 km/s faster than this estimate[4].
The shaded area between the two lines in the figure is the part of the speed that shining matter alone cannot account for.
Write for the mass inside radius , and treat it as spread around the centre in a spherical shape. Solving for then gives , so if is constant, grows in proportion to .
At the Sun's position in the Milky Way, is 8.1 kpc, or m, and is 229 km/s, or m/s. Putting these into with gives kg, which is times the Sun's mass of kg.
Using 200 km/s at 25 kpc, the enclosed mass is times the Sun's mass[4]. The stars of the Milky Way together are estimated at solar masses, so there is about four times as much mass inside 25 kpc[5].
In 1970, Rubin and Ford measured the shifts of 67 ionised gas regions in the Andromeda galaxy and found that the enclosed mass grows roughly in proportion to radius between 4 and 14 kpc from the centre[6].
The two lines in the figure are enclosed masses: the straight one is , and the one that stops rising is the mass of the shining matter added up.
If is proportional to , a spherical shell of thickness holds the same mass, , at every radius.
The volume of the shell, , grows in proportion to , so the density is , inversely proportional to the square of the distance from the centre.
At twice the distance the density drops to a quarter, yet the mass still extends far beyond the stellar disc. This roughly spherical mass around the disc is called the dark matter halo; it gives off neither light nor radio waves, so it can be detected only through its gravity.
The figure shows the stellar disc edge-on and the halo around it, with the depth of the shading showing the density.
Including the halo, the Milky Way's total mass is estimated at solar masses, about 24 times the total mass of its stars[5].