How Do Astronomers Weigh Planets and Stars? Watching What Orbits What

Kepler found in 1619 that for the planets, the square of the orbital period is proportional to the cube of the average distance from the sun. Newton then showed why, and in doing so turned the relationship into a measuring instrument, because the constant of proportionality in his version contains the mass of the central body. Rearranged, it says that if you can measure how far something orbits and how long it takes, you can calculate the mass it is orbiting. The masses of the orbiting object cancels out of the calculation when it is much smaller than the central one, which is why a moon of any size tells you the same thing about its planet. This single relationship, with adjustments, is responsible for almost every mass in astronomy, and the accuracy of the result depends almost entirely on how precisely the distance is known.

Each kind of object needs a slightly different application:

• •The sun: use the Earth's orbit, the period of one year and the distance of one astronomical unit, which is now measured directly by bouncing radar off Venus and timing the echo
• •Any planet with a moon: use the moon's orbit, which is why the masses of Jupiter, Saturn, Uranus, Neptune, Earth and Mars were all known well before spacecraft
• •Mercury and Venus, which have no moons: their masses were known only roughly from the tiny perturbations they cause in the orbits of other bodies, until spacecraft flew past and were themselves deflected, which gave precise values
• •Spacecraft generally: a probe passing a body is pulled off course by an amount that depends on the body's mass, and its radio signal is Doppler-shifted by its own acceleration, which can be measured to millimetres per second
• •Asteroids: mostly derived from how they nudge each other during close approaches, or from a moon where one exists, since many asteroids have them
• •The Earth itself: its mass required knowing the gravitational constant, which Henry Cavendish measured in 1798 with a torsion balance in a sealed shed, an experiment he described as weighing the world

A single isolated star cannot be weighed directly, which is why binary systems are so valuable. Roughly half of all stars have a companion, and a pair in orbit gives the sum of the two masses immediately from the period and separation; watching how each moves about the common centre of mass gives the ratio between them, and the two together give each mass separately. Eclipsing binaries, where the orbit is edge-on so that each star passes in front of the other, are the gold standard, because the eclipses give the orbital inclination, sizes and periods with great precision. From a few hundred such systems astronomers built the mass-luminosity relationship, the empirical finding that a main sequence star's brightness rises steeply with its mass, roughly as the mass to the power of three and a half. Once that relation is calibrated, the mass of an isolated star can be estimated from its brightness and spectrum alone, which is how masses are assigned to the vast majority of stars.

The same logic reaches far beyond the solar system. A planet does not orbit its star so much as both orbit their common centre of mass, so the star moves in a small circle, and that motion shows up as a periodic shift in the wavelengths of its spectrum. Measuring that wobble gives the planet's mass, with the caveat that an unknown orbital tilt means the result is a minimum unless the inclination is known from elsewhere. The transit method, in which a planet crossing the star's disc dims it slightly, gives the planet's radius instead, and a planet detected by both methods yields mass and radius together, and therefore density, which is how it is determined whether a body is rock, ice or gas. A third method, transit timing variations, works in systems with several planets, where each tugs the others and shifts their transit times by seconds or minutes, from which their masses can be solved.

At galactic scale the same technique produced the result that broke it. Measuring how fast stars and gas orbit the centre of a spiral galaxy at different radii should show the speed falling off with distance, as the planets do, since almost all the visible mass is concentrated in the middle. Vera Rubin and Kent Ford found in the 1970s that the curves stay flat out to the visible edge and beyond, meaning there is far more mass than can be seen, distributed in a halo, and the discrepancy is roughly a factor of five or six. The same excess shows up in the speeds of galaxies within clusters, first noticed by Fritz Zwicky in 1933, and in the way clusters bend the light of objects behind them. Gravitational lensing has since become a mass-measuring technique in its own right, requiring no orbits at all, since the deflection of light depends only on the mass in the way. What all these methods weigh is everything present, which is how the existence of dark matter was established: by weighing galaxies with gravity and then counting the light.

Almost every mass in astronomy comes from one relationship: an orbit's period and size give the mass of whatever is being orbited, with the orbiting object's own mass dropping out when it is much smaller. Planets are weighed by their moons or by the deflection of passing spacecraft, stars by their binary companions, and exoplanets by the wobble they impose on their star, with a transit adding the radius and therefore the density. Applied to galaxies the method revealed far more mass than light can account for.