How Do We Measure Distances in Space? Parallax, Cepheids and the Cosmic Ladder

Hold a finger at arm's length and close each eye in turn, and the finger jumps against the background. The jump is parallax, and its size depends on the distance to the finger. The Earth does the same thing on a larger scale: as it swings from one side of its orbit to the other, six months and 300 million kilometres apart, nearby stars shift very slightly against the more distant ones. The shift is tiny, less than one second of arc for even the nearest star, which is why it was not detected until 1838, when Friedrich Bessel measured it for the star 61 Cygni and found a distance of about ten light years.

Parallax is the only rung that involves no assumptions about the object; it is pure geometry. Its limitation is range. From the ground the shift is lost in the blur of the atmosphere beyond a few hundred light years. The Hipparcos satellite in the 1990s pushed it to about a thousand light years, and the Gaia satellite, launched in 2013, measures parallaxes for nearly two billion stars with a precision that reaches across much of the galaxy. The word parsec, the distance at which the parallax is one arcsecond, about 3.26 light years, comes from this method.

Beyond parallax, distance has to be inferred from brightness. A light bulb of known wattage looks fainter the further away it is, by a precise law, so if the true output of an object is known its apparent brightness gives its distance. Astronomers call such objects standard candles, and the problem is finding things whose true output can be known. The most important, discovered by Henrietta Leavitt at Harvard in 1912, are the Cepheid variables: giant stars that pulse in brightness over days or weeks, and whose period of pulsation depends strictly on their true luminosity. Time the pulse, and the true brightness follows; compare it with the apparent brightness, and the distance follows from that.

Cepheids are bright enough to be seen in other galaxies. Edwin Hubble found them in Andromeda in 1924 and settled a long argument by showing it was a separate galaxy far outside our own, and the relation between their distances and their speeds, published in 1929, was the first evidence that the universe is expanding. The Hubble Space Telescope was built partly to measure Cepheids in distant galaxies, and it did.

Cepheids give out at a hundred million light years or so. Beyond that the ladder relies on rarer, brighter events and on statistical relations:

• •Type Ia supernovae: exploding white dwarfs that all detonate at nearly the same mass and so reach nearly the same peak brightness, visible across billions of light years and calibrated on galaxies where Cepheids can also be seen
• •The Tully-Fisher relation: a spiral galaxy's rotation speed, measurable from its spectrum, tracks its true luminosity
• •Surface brightness fluctuations: the graininess of a galaxy's light smooths out with distance in a predictable way
• •Redshift: for the most distant objects, the stretching of their light by the expansion of the universe, converted to distance using the expansion rate the lower rungs established

Each rung inherits the errors of the one below, so a small mistake in the parallax of nearby Cepheids becomes a mistake in every supernova distance and in the expansion rate of the universe. That rate, the Hubble constant, has been the subject of a decade-long argument: the ladder, climbed from parallax through Cepheids to supernovae, gives about 73 kilometres per second per megaparsec, while measurements of the early universe from the cosmic microwave background give about 67, and the two groups have checked each other's work without finding the mistake. Either a rung of the ladder is miscalibrated or the universe contains something the standard model of cosmology does not.

Gaia's parallaxes have tightened the bottom of the ladder considerably, and the James Webb Space Telescope has re-measured the Cepheids in sharper detail and confirmed the ladder's figure. The disagreement, called the Hubble tension, is currently the most interesting problem the distance ladder has produced.

The ladder's rungs, and roughly how far each reaches: parallax, to a few thousand light years with Gaia; Cepheids, to about a hundred million; supernovae, to about ten billion; redshift, to the limit of the observable universe at around 46 billion light years in present-day distance. Every one of those numbers was reached by someone standing on the one before, which is why the discovery in 1838 that a faint star in Cygnus was ten light years away is still, in a sense, being used.

Parallax only gives a star's distance if you already know the size of the Earth's orbit, the baseline of the triangle. For centuries astronomers knew the relative distances of the planets very well, thanks to Kepler's laws, but not the actual scale in kilometres. One clever way to find it was the transit of Venus, when the planet crosses the face of the Sun. Observers in widely separated places see it cross along slightly different paths, and timing those differences gives the Sun's distance. The transits of 1761 and 1769 prompted expeditions worldwide, including James Cook's first voyage to Tahiti.

The results were good to within a few percent, limited by a blurring effect that made the moment of contact hard to judge. The modern answer is radar. From the 1960s astronomers bounced radio signals off Venus and timed the echoes, which pinned the scale of the solar system down with great precision. Lasers fired at reflectors left on the Moon by Apollo astronauts measure its distance to within centimetres. The astronomical unit, the average Earth to Sun distance, is now defined as exactly 149,597,870,700 metres, and every rung of the ladder above stands on it.

Since 2015 astronomers have had a way of measuring some cosmic distances that does not depend on the ladder at all. When two neutron stars or black holes spiral together, they send out gravitational waves, ripples in spacetime that detectors such as LIGO and Virgo can record. The way the signal's frequency rises as the objects close in reveals how powerful the source truly was, while the strength of the signal reaching Earth shows how much it has faded on the way. The comparison gives a distance directly, and by analogy with standard candles such events are called standard sirens.

The first clear test came in August 2017, when the merger of two neutron stars was detected and, within hours, telescopes found the flash of light from the same event in a galaxy about 130 million light years away. The gravitational waves gave the distance and the galaxy's light gave the speed of recession, and together they produced a value of the Hubble constant of about 70, with an uncertainty too wide to settle the tension between 67 and 73. Because sirens rely on none of the calibrations behind Cepheids or the microwave background, many more of them could act as an independent referee.

The arithmetic behind a Cepheid distance is simple enough to do by hand. Suppose a Cepheid in the Milky Way, 2,000 light years away by parallax, pulses with a period of ten days. Astronomers then find a Cepheid in another galaxy with the same ten-day period, so it must have the same true brightness, but it appears one million times fainter. Light spreads out with the square of distance, so a million times fainter means a thousand times further away: about two million light years. In practice the calculation must also allow for dust, which dims starlight and would otherwise make galaxies seem further away than they are.

Getting the calibration wrong once had dramatic consequences. In the 1940s the astronomer Walter Baade, working at Mount Wilson in California under skies darkened by the wartime blackout of Los Angeles, resolved individual stars in the Andromeda galaxy and recognised that stars come in distinct populations. In 1952 he announced that the period-luminosity relation had been calibrated partly on a fainter type of pulsating star, so the classical Cepheids in other galaxies were brighter, and therefore further away, than assumed. At a stroke the estimated size and age of the universe roughly doubled, easing an embarrassing conflict in which the universe had seemed younger than the Earth.

Astronomical distances are measured by a ladder of methods: parallax, the pure geometry of the Earth's orbit, for nearby stars; Cepheid variables, whose pulse period reveals their true brightness, for nearby galaxies; Type Ia supernovae and galaxy-scale relations for the distant universe; and redshift beyond that. Each rung is calibrated on the one below, so an error at the bottom propagates to the top, and a stubborn mismatch in the expansion rate is the ladder's current unsolved problem.