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Centuries ago, people did not know how large the Sun was, nor how far away it was Estimates ranged from a few thousand miles (kilometers
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hadn t been invented yet) to a few million miles The distance to the Sun could not be measured by parallax relative to the background of stars because the Sun s brilliance obliterated the stars near it The distance to the Moon had been measured by parallax, as well as the distances to Mars and Venus at various times, but the Sun defied attempts to measure its distance until someone thought of finding it by logical deduction What follows is an example showing the sort of thought process that was used, and can still be used, to infer the distance to the Sun Let s update the measurement techniques from those of our forebears and suppose that we have access to a powerful radar telescope, with which we can measure interplanetary distances by bouncing radio beams off distant planets and measuring the time it takes for the signals to come back to us Given a central body having a known, constant mass, such as the Sun, all its satellites obey certain physical laws with respect to their orbits One of these principles, called Kepler s third law, states that the square of the orbital period of any satellite is proportional to the cube of its average distance from the central mass This is true no matter what the mass of the orbiting object; a small meteoroid obeys the rule just as does Earth, Venus, Mars, and Jupiter We know the length of Earth s year and the length of Venus s year; from this we can calculate the ratio (but not the actual values)
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of the two planets mean orbital radii Knowing this ratio is not enough, all by itself, to solve the riddle of Earth s mean distance from the Sun, but it solves half the problem The next step involves measuring the distance to Venus If we could do this when Venus is exactly in line with the Sun, then we could figure out our own distance by simple mathematics Unfortunately, the Sun produces powerful radio waves, and our radar telescope won t work when Venus is at inferior conjunction (between us and the Sun) because the Sun s radio noise drowns out the echoes However, when Venus is at its maximum elongation (its angular separation from the Sun is greatest either eastward or westward), the radar works because the Sun is out of the way At maximum elongation, note (Fig 4-8) that Venus, Earth, and the Sun lie at the vertices of a right triangle, with
Orbit of Venus
a Orbit of earth Earth
Venus at elongation
Figure 4-8 If we know the radius of Venus solar orbit and we can measure the distance to
Venus at its maximum elongation, then we can calculate our own distance from the Sun
The Moon and the Sun
the right angle at the vertex defined by Venus One of the oldest laws of geometry, credited to a Greek named Pythagoras, states that the square of the length of the longest side of a right triangle is equal to the sum of the squares of the other two sides In Fig 4-8 this means that a 2 + b 2 = x 2, where x is the elusive thing we seek, the average distance of Earth from the Sun Now that we know the value of a in the equation (by direct measurement) and also the ratio of b to x (by Kepler s third law), we can calculate the values of both b and x because we have a set of two equations in two variables Let s not drag ourselves through a detailed mathematical derivation here If you ve had high school algebra, you can do the derivation for yourself It should suffice to say that this scheme can give us a fairly good idea of Earth s mean distance from the Sun if the measurement is repeated at several maximum elongations and the results averaged However, even this will only give us an approximation because the orbits of Earth and Venus are not perfect circles In recent decades, astronomers have made increasingly accurate measurements of the distance from Earth to the Sun using a variety of techniques Once the distance from Earth to the Sun was known, the Sun s actual radius was determined by measuring the angular radius of its disk and employing surveyors triangulation in reverse (Fig 4-9)
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