Re: Size of Earth's orbit

How is the size of the Earth’s orbit worked out?

This is a very old question.  Going back to the Greeks, Aristarchus of Samos (310-230 BCE) is the first in recorded history to propose a heliocentric model of the solar system where the Earth revolves around the Sun.  He had the following idea:  consider the right triangle formed when the Moon’s shadow covers exactly half of the Moon’s surface as seen from Earth.  Then Aristarchus measured the angle from Moon-Earth-Sun at 87 degrees.  Using trigonometry, this means that the Sun is approximately 1 / sin 3 degrees as far away from the Earth as the Moon.  Based on his estimates, Aristarchus put the Sun at 18 to 20 times the distance from the Moon, a figure much larger than any considered previously. 

On the other hand, there was good reason to believe that the Earth was the center of the Solar System, such as the lack of observable stellar parallax.  (Stellar parallax was finally observed with the aid of telescopes in 1838 by Bessel.)

That’s where matters stood until the 17^th century rolled around and Copernicus once again seriously put forth a heliocentric theory to explain the retrograde motion of the planets.    Using data from Brahe, Kepler modified this model so that the planets has elliptical orbits, and he formulated his Three Laws of Planetary Motion.  Unfortunately, none of these three laws can be used to determine how far away the Earth is from the Sun in absolute terms.  The third law roughly says that the square of the time needed for an orbit divided by the cube of the distance to the sun is a constant, but that only allows us to determine the ratios of the planetary distances, not the absolute number.  So calling the distance from the Earth to the Sun an Astronomical Unit, or A.U., Kepler’s third law can be used to show that Mars orbits at 1.52 A.U., but does not allow us to determine what 1 A.U. is in feet or meters.

 That’s where Venus comes into play.  Every so often, Venus passes between the Earth and the Sun (an event called a Transit of Venus).  If Venus orbited exactly in the ecliptic plane (the plane in which the Earth orbits the Sun), then transits would occur at periodic intervals (every 583.92 days in fact when Venus “laps” the Earth).  In fact the orbit of Venus is tilted 3.4 degrees from the orbit of Earth, and that makes Transits occur more rarely (there have been

only 6 since the first was predicted in 1631 by Kepler, with the next one coming in June 2004).

Now transits can only be observed via telescope, which fortunately for astronomy has just recently been developed by Galileo.  A young astronomer named Jeremiah Horrocks had such a telescope, and refined the data in astronomical tables to the point where he was able to predict a Transit of Venus missed by Kepler—in 1639.  (Of course, you never want to look at the Sun directly and especially not through a telescope!  Horrocks projected the image of the Sun onto a lined piece of paper to make his measurements.)

So here’s the data:  Horrocks measured the position of the shadow cast by Venus on the Sun at three different times.  From this (and Kepler’s Third Law) Horrocks could calculate the angle moved by Venus in the half hour he observed it (about 28 seconds, or 28/3600 of a degree).  Also, he could measure this angle in terms of the radius of Venus, that is, the distance traveled divided by the radius of the shadow cast by Venus.  Assuming that the Earth was roughly the same size as Venus (which turned out to be true) then the distance from Earth to the Sun could be measured in terms of Earth radii.  Horrocks found the distance from the Earth to the Sun to be about 59 million miles.  This was far closer to the true answer of 96 million miles than any previous estimate.

For comparison, Brahe has estimated 1 A.U. at 8 million miles, Kepler at 15 million.  After Horrocks, in 1672 Cassini estimated 1 A.U. at 87 million miles by measuring Mars at Paris while a colleague simultaneously made the same measurements in French Guinea in South America.  Later methods all use the same basic ideas, but with ever more sophisticated equipment.

Fast forward to today:  a mirror placed on the Moon by the Apollo astronauts allows us to determine very precisely the distance from the Earth to the Moon, and then Aristarchus’ original method can be used to give a precise estimate of the distance to the Sun.  Precise measurements of planetary orbits are still important today, not only for the clues they provide about unknown objects in the Solar System, but also for verification and refinement of General Relativity.  NASA lists 1 A.U. at 149,597,870.691 kilometers.

Mark Huber