### The Astronomical Unit

October 3, 2012

The phrase, "room temperature," is often used in scientific papers and reports. Everyone knows what is meant by room temperature, but it's certainly not a definite temperature. I worked in a laboratory in which the room temperature varied wildly. In summer, that air conditioned building would be extremely cold first thing in the morning. I was happy to wear my thick lab coat to keep myself warm, and I would linger near the annealing furnaces.

After a few hours, more people would arrive to start their work day. Open fume hood doors would suck away all the cold air, and the HVAC system would have a hard time maintaining the temperature, so it would get hot. I would often run experiments overnight, and I always added a temperature channel to the data logger, so I could apply corrections when needed.

Astronomers have also had an imprecise measure that's similar to room temperature. That's the astronomical unit, usually abbreviated AU, which is the distance from the Earth to the Sun. It's a good distance measure for things in our Solar System, since it gives you an immediate sense of scale. Of course, since Kepler's time, we know that planetary orbits around the Sun, including Earth's own orbit, are elliptical, and not circular. The Earth can be as close to the Sun as 147,098,290 kilometers (the perihelion), or as far away as 152,098,232 km (the aphelion).

Since astronomers didn't want a variable astronomical unit that depended on the time of year, it was defined as the length of the semi-major axis of Earth's orbit, which averaged these two values to give 149,598,261 km. In 1976, in an effort to relate the astronomical unit to fundamental constants, it was defined in terms of the distance from the Sun at which the gravitational attraction was a certain value; or, equivalently, the distance from the Sun of an unperturbed circular Newtonian orbit of a particle having infinitesimal mass when its angular orbital velocity is 0.01720209895 radians per day.

There are some problems with that definition, the more severe of which is that the Sun is always losing mass. Furthermore, the gravitational constant, as I've written in a previous article (Big G, October 12, 2010), is the least precisely measured of all the fundamental constants. There is also a relativistic effect that now becomes important in the era of interplanetary spaceflight. Finally, at its August, 2012, meeting in Beijing, China, the International Astronomical Union redefined the astronomical unit as exactly 149,597,870,700 meters.[1-3]

Aristarchus of Samos (310 BC - c. 230 BC), who had an heliocentric model of the Solar System long before Copernicus, is quoted by Archimedes as estimating the astronomical unit to be 10,000 Earth radii, about half its actual value. Archimedes was interested in this distance, since he attempted to calculate the volume of the universe in his book, The Sand Reckoner.[4]

Archimedes wanted to calculate how many grains of sand were needed to fill the universe, so a sand grain was his voxel, or volume element. Knowing the value of the astronomical unit was one thing, but the distance to the outer shell of the universe, the sphere of the "fixed stars," was another. Archimedes assumed that the ratio of the diameter of the universe to the diameter of Earth's orbit was equal to the ratio of the diameter of Earth's orbit to the diameter of the Earth. In his age, I would have heartily endorsed such an estimate; otherwise, you would have no estimate at all.

About three centuries after Aristarchus, Ptolemy, author of the Almagest, made an estimate of the astronomical unit based his observations and a method used by Hipparchus (see figure). Astrometry was not that well developed then, and his method was very sensitive to small errors, so he calculated the astronomical unit as just 1,210 Earth radii.

Hipparchus construction for finding the distance to the Sun, as presented by Ptolemy and deciphered by Swerdlow (1969). The construction is based on the fact that the region of eclipse totality defines a conical shadow. Somehow, it's assumed that the apex of the cone is at the center of the Earth, but I've found no reference that comments on this. (Adapted from a Wikimedia Commons image.)

So, from antiquity, Aristarchus came close, and Ptolemy completely missed the mark with his estimate of the astronomical unit that's just 5% of the actual value of 23,481 Earth radii. The figures below show the modern estimates from 1635 onwards. It looks as if the new IAU value was selected to be slightly large to allow for the anticipated growth of Earth's orbit as the Sun loses mass.

The astronomical unit over the centuries, as expressed in Earth radii. (Data from Wikipedia, rendered using Gnumeric)

The astronomical unit over the centuries, as expressed in Earth radii (detail).

The present value is 23481.065877 Earth radii, 149,597,870.7 kilometers.

(Data from Wikipedia, rendered using Gnumeric)

Sergei Klioner of the Dresden University of Technology has championed the change since 2005. The new definition also has a practical advantage in his university teaching. Klioner is quoted in New Scientist as saying,
"I've been teaching celestial mechanics for 20 years and it was always a pain to explain the old definition. It was clear that it was unnecessarily complicated... I'm happy that I don't have to explain this any longer."[2]

### References:

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