The Scale of the Solar System – Mechanical Engineering

Kepler’s laws agree with all observed planetary motions, and by the table in the previous section, they give the correct proportions of all planetary orbits. If the mean distance of Earth from the Sun is 1 AU (“Astronomical Unit”), then that of Venus is 0.723 AU, of Mercury 0.387 AU and that of Mars is 1.524 AU. But how much is that in kilometers, or miles? In other words–what are the actual dimensions, not just their ratios? Tycho still accepted the erroneous estimate by Aristarchus of the Sun’s distance, 20 times smaller than the actual one . Soon afterwards the telescope was discovered, and starting with Galileo, astronomers realized that Venus appeared as a round disk (or a crescent, when near Earth and presenting mostly its dark side). At its closest, Venus is nearly one minute of arc (1/60 degree) across. Assuming it was about as big as Earth, and using (for instance) Kepler’s laws, it was possible to estimate the distance of Venus, and from it, the distance of Earth from the Sun. That led to a much better estimate of about 15,000 Earth radii, more than 12 times the estimate by Aristarchus but still too small. If we know the proportions of all the orbits in the solar system, measuring just one actual distance in kilometers gives the scale of all orbits around the Sun. What one needs is a parallax, that is, a simultaneous observation of a planet from two widely separated points on Earth, providing a small difference in viewing angle. Remember how Hipparchus estimated the distance of the Moon? In a solar eclipse which was total in one location, at another location about 1000 kilometers away, only 80% of the Sun was covered. The body blocking the Sun–the Moon–was close enough that moving an observer by about 1000 kilometers shifted its apparent position in the sky by 1/5 the apparent size of the Sun, or about 0.1 degree.

Since the Sun’s distance sets the scale of the entire solar system, Tycho believed Mars was close enough for its apparent position in the sky to be shifted measurably as the Earth’s rotation carried an observer from one side of the globe to the other. Tycho (at least some of the time) believed he could see a difference with his pre-telescope equipment, but in fact, the solar system was much bigger than he had assumed, far beyond his abilities. A century later, at another close approach of Mars, Jean Richer (in 1672) used a telescope to get the first rough estimate of the distance of Mars, with an uncertainty around 30% Edmond Halley (1656-1742) suggested tracking the passage of the planet Venus in front of the Sun, in one of its infrequent “transits of Venus.” When this happens, a telescope observing the Sun (by projecting its image, or using a dark filter) sees the dark disk of Venus slowly creeping across the bright face of the Sun. By noting (1) where on the Sun’s disk is the crossing seen, (2) timing its duration at two far-apart points on Earth, and (3) comparing the times, one can calculate the distance to Venus and from it the scale of the solar system. “From Stargazers to Starships” presents a simplified calculation of the AU, based on the transit of 8 June, 2004, in sections #12c-e. Unfortunately, no “transits of Venus” happened in Halley’s own time. They occur in pairs, more than a century apart. One occurred in 1639–too early. The next ones did not take place until 1761 and 1769, and astronomers were prepared for them. One of the goals of the famous expedition by Captain James Cook to the Pacific Ocean was to observe the transit from a point far from other observers. Unfortunately, an unexpected observing effect, a “dark bridge” between the disk of Venus and the sky beyond (“black drop effect”), badly degraded the accuracy of the timing of the transit. No transits of Venus occurred in the 20th century, but one did occur on June 8, 2004, followed by another one in 2012. Observers in the US only saw the end of the 2004 event, though the world-wide web covered the phenomenon very well. Later astronomers realized that some asteroids passed quite close to Earth. Today we worry about any of them actually hitting Earth, but their discovery also made some astronomers happy. Because of their nearness, their viewing angles from separate locations were much larger, giving a bigger parallax and more accurate estimates of their distances. That gave a much better estimate of the AU. Still later the giant radio telescope whose (fixed) dish is nestled in a valley near Arecibo, Puerto Rico, was used to focus a radar signal whose beam was bounced off the planet Venus, and timing its “echo” gave an even more accurate estimate of the AU. Today, of course, one also can use the orbital mechanics of space probes, tracked by radio as they pass near major planets.

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