# 3.16 The Scale of the Solar System

One of the more interesting subplots of the story of astronomy has been the evolution of our understanding of the size of our Solar System. Initially, man saw the entire universe as quite small. It wasn't until Newton's death in 1727 that we could put reasonable limits on planetary orbits for the naked-eye planets, and even today we're revising our charts as we gain insights into the distributions of icy bodies at the edge of our Sun's planetary system. While it has literally taken millennia to map our solar system, the story is one that has undergone regular updates.

Before the story really began, mathematicians tried to use trigonometry to get at the distances to the Moon and even the Sun. Early measurements were limited by a lack of accurate equipment, but showed a desire by early scientists to use observations to get at answers. For instance, measurements by Aristarchus of Samos (310 B.C. - 230 B.C.) of the positions of the Sun and Moon when the Moon is at quarter phase (half lit), allowed him to use triginometery to estimate the ratio of distances to these two objects. His crude estimate of 1200 Earth radii (about 8 million kilometers), although far from the modern value of 150 million kilometers, was quite good considering their lack of modern observational tools like telescopes. His results were very wrong, but they were at least a start.

The first real progress came in accurately measuring the size of the planet we live on. Eratosthenes did this with a dramatic application of geometric reasoning around 200 B.C. He was a researcher and librarian at the great library in Alexandria, Egypt, where books containing much of the knowledge of the ancient world were stored. He completed a catalog of the 675 brightest stars, but his most famous achievement was measuring the size of planet Earth. Told that the Sun at summer solstice shone straight down a well to the south, near Aswan, he noted that on the same date in Alexandria, the Sun's direction was off vertical by about 6°, or 1/50 of a circle. Eratosthenes realized that this difference was due to the curvature of the Earth, and that the north-south distance from Alexandria to Aswan must be 1/50 of the circumference of the Earth. Measuring the distance in an ancient unit called stadia and multiplying by 50, he got an estimate of the Earth's size that was within (depending on the definition of stadia) 1-20% of the correct answer. At a time when few people traveled more than 50 miles in their lives, and when the first circumnavigation of the Earth was still more than 1000 years away, most educated Greeks knew the size of the planet they lived on! (And knew the Earth was round.)

For the next many hundred years, people would work on ordering the planets, and trying to build mathematical models that worked. It was correctly understood that the planets that move more slowly across the sky, like Saturn and Jupiter, are further away than the faster moving planets, like Mercury and Venus. It was also correctly understood that the stars live more distantly than the planets. What wasn't understood with exactly how faraway stars are. It was assumed that they were somewhat close, just beyond the orbit of Jupiter. It was actually this assumption, that stars are near, that led to the mistaken belief that the Earth is at the center of the solar system. With the Sun in the center, nearby stars would change in appearance, suffering severe parallax motion as the Earth orbited. Since this parallax isn't observed, the 1609 discovery by Galileo, which was originally theorized by Copernicus, that the Sun is at the center of our Solar System meant that we had to move the stars much more distant. The minimum distance to the stars became the distance required for the stars to not move as seen by the human eye as the Earth orbits the Sun. This single piece of understanding increased the size of the known universe at least 2 orders of magnitude. While this got us a bit closer to understanding the location of the stars, it still didn't help us understand the distances to the planets.

At the same time that Galileo was working on rearranging the solar system, Kepler was working to define the mathematics behind the planets' motions. Using Kepler's laws, astronomers can give the relative distances of all the planets from the Sun, based strictly on observing how long it takes to complete a single orbit. This doesn't give the absolute distances in units like kilometers, however. The best we could do, with the distance between the Earth and the Sun as a constant, was express the rest of the distances using the Earth-Sun distance.

Astronomers define the mean distance from the Earth to the Sun as one astronomical unit, or AU. Distances to other solar system bodies are quoted in multiples of AU. This is the most useful unit of measurement in solar system astronomy, outside the metric system. Using this distance as a yardstick, the distances to all the other planets can be calculated from their orbital periods and Kepler's third law. So the main challenge left in determining the scale of the Solar System was accurately measuring the absolute distance from the Earth to the Sun (or from the Earth to other planets, which could then be related to the Earth-Sun distance).

After the invention of the telescope, it was possible to measure the angular size of a planet. A modest-sized telescope shows the angular size of Mars to be about 30 arcseconds at its closest approach to Earth. If you assume that Mars is the same size as Earth, the small angle equation yields the distance, D = 206,265 × (d / a) = 80 million kilometers. This relies on a complete guess that Mars and Earth are the same size, so it's clearly not a very reliable number.

With the diameter of the Earth as the base, a parallax angle can also be used to measure the distance from Earth to Mars. This method uses a very long, skinny triangle — Mars is thousands of times further away than the size of the Earth. So the parallax angle is very small and very difficult to measure. Venus is a better subject for the experiment, because twice every 110 years it passes directly in front of the Sun, as seen from Earth. In astronomy, this is called a transit. The Sun provides a perfect backdrop for observing the small, dark disk of Venus. Using observations from two widely separated sites, the parallax angle can be measured, and the distance from Earth to Venus can be calculated. With that distance, plus a measurement of the relative distances between Earth, Venus, and the Sun, you can calculate the total distance from the Earth to the Sun. The relative distances between the orbits of Venus and Earth had been known since Nicholas Copernicus calculated them using the phases of Venus in the Mid 16th century.

There are two complications in using the parallax method to measure the distance to Venus. First, the Earth is spinning. The observations have to be made at exactly the same time, and an error in time measurement on a spinning planet leads to an error in distance, and thus an error in the measurement of parallax. Timekeeping is also important in navigation. The angle of stars can be used for navigation in latitude (for example, the use of Polaris). But accurate timekeeping is required for navigation in longitude. The Earth spins at 15° per hour, so a clock that is off by only half an hour after a long sea voyage would give a navigational error of 7.5° / 360° times the circumference of the Earth, or 1000 kilometers! Poor clocks were the reason for most shipwrecks and for the famously poor navigation of Christopher Columbus. This problem was solved in the middle of the 18th century, when the British government realized that an accurate clock was the key to mastery of the seas. In response to a competition for a prize of £20,000, John Harrison produced a clock accurate to 5 seconds over 80 days, which is an outstanding timekeeping device in any era.

Another problem with observing a transit of Venus was that you had to be lucky since pairs of Venus transits only occur twice every 110 years. Before Johannes Kepler died in 1630, he predicted that a transit of Venus would occur in 1631 — but unfortunately no one observed it because it happened during the night for European observers. A young astronomer and clergyman named Jeremiah Horrocks was able to predict and observe the next transit in 1639. He recorded a portion of the event, until sunset obscured the remainder. Using the relative sizes of the disk of Venus and the disk of the Sun, he calculated the Earth-Sun distance to be 14,700 Earth radii (98 million kilometers), which was much better, and much larger, than any previous measurement.

Knowing that a set of good observations of a Venus transit would fix the scale of the solar system, astronomers had to wait more than hundred years until the next pair of transits in 1761 and 1769. Edmond Halley predicted these transits in 1716. Aware that he would not live to see them, he urged future astronomers to make careful observations. His exhortations succeeded, and many scientific expeditions set out around the globe to get the necessary observations at well-separated latitudes. Even then, adventurers endured arduous sea voyages, only to be thwarted by pirates, typhoons, or clouds. Captain James Cook made a famous trip to observe the 1769 Venus transit in Tahiti. His observations, combined with those of other astronomers from around the world, established the distance from the Earth to the Sun to within 10% of its modern value of 150 million kilometers.

These transit measurements not only determined the correct scale of the solar system, they were also some of the first examples of a successful cooperative effort between international scientists. The recent Venus transit in 2004 provided an opportunity for amateur astronomers to participate in a similar cooperative experiment by reproducing these historical measurements themselves. The most recent transit of Venus occurred in 2012 — hopefully you observed it since you may not be alive for the next one!

The final definitive distances to the planets have only come since World War II. Since then, numerous measurements using radar have refined the sizes and eccentricities of the orbits of the other planets in our Solar System.

Eratosthenes' measurement of the Earth's circumference. Syene (S) is located on the Tropic of Cancer, so that at summer solstice the sun appears at the zenith, directly overhead. In Alexandria (A) the sun is???south of the zenith at the same time. So the circumference of earth can be calculated being 360?/???times the distance???between?A and?S. Erastothenes measured the angle???to be 1/50 of a circle and his access to knowledge of the size of Egypt gave a north/south distance??between Alexandria and Syene of 5000 stadia. His circumference of the Earth was therefore 250 000 stadia. Certain accepted values of the length of the stadia in use at the time give an error of less than 6% for the true value for the polar circumference. Click here for original source URL.