Astronomical unit

• 1. Introduction

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  The astronomical unit is exactly 149597870700 metres, fixed by definition since 2012. For most of human history, nobody knew that number, or anything close to it. Around 280 BC, Aristarchus measured an angle between the Moon, the Earth and the Sun, and concluded the Sun was 18 to 20 times farther away than the Moon. The true ratio is about 389. He was wrong by a factor of twenty, and yet his method survived for two thousand years. How does a single distance go from a Greek geometry problem to a number nailed down to the metre? Why does the average gap between Earth and the Sun matter enough to send astronomers dying on expeditions across the globe? And what changed so completely in 2012 that the unit itself lost much of its old importance? Those questions sit between an ancient angle of 87 degrees and a constant the speed of light helped pin down.

• 2. A Number For Naming Things

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  One au is approximately equivalent to 499 light-seconds, which is roughly the time light takes to reach Earth. Light covers an astronomical unit in about 8 minutes 19 seconds. That small, human-scale gap is the yardstick for the Solar System. The astronomical unit is typically used for stellar system scale distances, like the size of a protostellar disk or the heliocentric distance of an asteroid. It is too small to be convenient for interstellar distances. There the parsec and the light-year take over, and the parsec itself is defined in terms of the astronomical unit. A parsec is the distance of an object with a parallax of one arcsecond, equal to about 206265 au and roughly 3.26 light-years. The light-year is popular in everyday writing but is not an approved non-SI unit, and professional astronomers rarely use it. There is one more, quieter job the unit does. When simulating a numerical model of the Solar System, the astronomical unit provides a scale that minimizes overflow, underflow and truncation errors in floating point calculations. The convenience of that scale is a thread that runs straight back to how the unit was first defined.

• 3. The Symbol Wars

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  In 1976, the International Astronomical Union used the single capital letter A to denote a length equal to the astronomical unit. That was only one entry in a long disagreement over how to write the thing down. In the astronomical literature, the symbol AU was common. In 2006, the International Bureau of Weights and Measures recommended ua, taken from the French unite astronomique. The non-normative Annex C to ISO 80000-3:2006, later withdrawn, also used ua. In 2012, the IAU noted that various symbols were presently in use and recommended au. The scientific journals published by the American Astronomical Society and the Royal Astronomical Society adopted it. In the 2014 revision and 2019 edition of the SI Brochure, the BIPM used au as well. ISO 80000-3:2019, which replaces the 2006 standard, does not mention the astronomical unit at all. The name itself is older than all this argument over letters. The name astronomical unit was first used in 1848.

• 4. From An Ellipse To A Metre

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  Earth's orbit around the Sun is an ellipse, and the semi-major axis is half the line joining perihelion and aphelion. The centre of the Sun lies on that line, but not at its midpoint. Because ellipses are well-understood shapes, measuring the extremes fixed the orbit mathematically and made predictions possible. That same line marks the largest straight-line distance Earth covers in a year, which sets the best chance to observe parallax, the apparent shift of nearby stars. Knowing Earth's shift and a star's shift lets you calculate the star's distance. Every uncertainty in the astronomical unit fed straight into uncertainty about the stars. Expected positions and distances are calculated in au from the laws of celestial mechanics and gathered into a table called an ephemeris. NASA's Jet Propulsion Laboratory HORIZONS System is one such ephemeris computation service. In 1976 the IAU adopted a definition built on celestial mechanics, tying the unit to the Gaussian gravitational constant taking the value 0.01720209895. Space probes then changed everything by measuring the time photons take to bounce back from a planet, multiplied by the speed of light.

• 5. Letting Light Set The Scale

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  In 1983, the BIPM redefined the metre as the distance light travels in a vacuum in 1/299792458 of a second. That replaced the 1960 to 1983 definition based on wavelengths of an emission line of krypton-86. The change came from a better method of measuring the speed of light, which could then be stated exactly as 299792458 metres per second. In 2009, the IAU reported the light-time for unit distance as 173.1446326847 in Barycentric Dynamical Time. From the metre's definition and that standard, the time for light to cross an astronomical unit works out to 499.0047838061 seconds, slightly more than 8 minutes 19 seconds. Multiply by the speed of light and the best IAU 2009 estimate was 149597870700 metres, drawn from a comparison of Jet Propulsion Laboratory and IAA-RAS ephemerides. The earlier 2006 BIPM value had been 1.49597870691 in scaled units. That long chain of estimates was about to be cut off and replaced by a flat declaration.

• 6. The 2012 Redefinition

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  In 2012, the IAU found that fully equalizing relativity would make the definition overly complex, so it simply took the 2009 estimate and made it law. The astronomical unit became a conventional unit of length tied directly to the metre, exactly 149597870700 metres. The reason for the change runs through general relativity. The metre is defined as a unit of proper length, and the CIPM notes its definition applies only within a region small enough to ignore the non-uniformity of the gravitational field. So a distance across the Solar System with no stated frame of reference is genuinely problematic. The 1976 definition never specified that frame, which made it incomplete, though it worked well for ephemerides. A fuller definition consistent with general relativity was proposed, and vigorous debate ensued until August 2012, when the current definition was adopted. The new definition openly recognizes that the astronomical unit now has reduced importance, useful as a convenience in some applications. One driver of that demotion is the Sun itself. The Sun is constantly losing mass by radiating away energy, so the planets' orbits are steadily expanding outward, which had led to calls to abandon the unit entirely.

• 7. A History Of Being Wrong

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  Many early measurements of the Earth-Sun distance are wildly incorrect, because they rely on the ratio of Earth's diameter to the distance of the Sun, a ratio of about 1/12000. Small errors in Earth's size blow up into huge errors in the distance. Aristarchus estimated the Moon-Earth-Sun angle at 87 degrees, when the true value is close to 89.853, and his result landed somewhere between 380 and 1520 Earth radii. Hipparchus, as quoted by Pappus, gave the distance as 490 Earth radii, derived from an assumed least perceptible solar parallax. The Chinese treatise Zhoubi Suanjing, from around the 1st century BCE, computed the distance geometrically from noontime shadows at three places 1000 li apart, assuming a flat Earth. In the 2nd century CE, Ptolemy estimated the mean solar distance as 1210 Earth radii, a figure roughly correct only because his errors cancelled. His procedure is so sensitive that changing a measurement by a few per cent can make the solar distance infinite. Islamic astronomers stayed close: al-Farghani gave 1170 Earth radii and al-Battani used 1108. In Europe, Copernicus and Tycho Brahe used 1142 and 1150, so Ptolemy's rough figure survived through the 16th century. Johannes Kepler was the first to see it had to be far too low, by at least a factor of three, in his Rudolphine Tables of 1627.

• 8. Transits And Expeditions

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  Godefroy Wendelin repeated Aristarchus's measurement in 1635 and found Ptolemy's value too low by a factor of at least eleven. The transit of Venus offered something better. By timing a transit from two locations, astronomers could calculate the parallax of Venus and from it the solar parallax, which cannot be measured directly because the Sun is too bright. Jeremiah Horrocks observed the 1639 transit and, in work published in 1662, gave a solar parallax of 15 arcseconds, equivalent to 13750 Earth radii. Christiaan Huygens compared the apparent sizes of Venus and Mars and estimated about 24000 Earth radii, remarkably close to modern values, though historians discount it because his errors cancelled by luck. Jean Richer and Giovanni Domenico Cassini measured the parallax of Mars between Paris and Cayenne in 1672, reaching a solar parallax of 9.5 and roughly 22000 Earth radii. They had an accurate radius of Earth from Jean Picard, measured in 1669 as 3269000 toises. That same year Ole R@mer discovered the finite speed of light in 1676, quoting it as the time light takes to travel from the Sun to Earth, a convention astronomers still follow. James Gregory's method, published in his Optica Promata of 1663 and championed by Edmond Halley, drove the great Venus transit campaigns. The transits of 1761 and 1769 became an unprecedented international operation, including observations by James Cook and Charles Green from Tahiti. Despite the Seven Years' War, dozens of astronomers were sent to dangerous outposts, and several died in the effort. Jerome Lalande collated the results into a solar parallax of 8.6. The near-Earth asteroid 433 Eros, passing close in 1900 to 1901, would push parallax measurement further than any planet had.