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The book On the Sizes and Distances of the Sun and Moon , which has long been ascribed to Aristarchus , says that he calculated the distance to the Sun to be between 18 and 20 times the distance to the Moon , whereas the true ratio is about Depending on the distance that Van Helden assumes Aristarchus used for the distance to the Moon, his calculated distance to the Sun would fall between and 1, Earth radii.
According to the conjectural reconstructions of Noel Swerdlow and G. A Chinese mathematical treatise, the Zhoubi Suanjing c. Because of cancelling errors in his parallax figure, his theory of the Moon's orbit, and other factors, this figure was approximately correct.
Given these data, the distance of the Sun from Earth can be trigonometrically computed to be 1, Earth radii. This gives a ratio of solar to lunar distance of approximately 19, matching Aristarchus's figure.
Although Ptolemy's procedure is theoretically workable, it is very sensitive to small changes in the data, so much so that changing a measurement by a few per cent can make the solar distance infinite.
After Greek astronomy was transmitted to the medieval Islamic world, astronomers made some changes to Ptolemy's cosmological model, but did not greatly change his estimate of the Earth—Sun distance.
Johannes Kepler was the first to realize that Ptolemy's estimate must be significantly too low according to Kepler, at least by a factor of three in his Rudolphine Tables Kepler's laws of planetary motion allowed astronomers to calculate the relative distances of the planets from the Sun, and rekindled interest in measuring the absolute value for Earth which could then be applied to the other planets.
The invention of the telescope allowed far more accurate measurements of angles than is possible with the naked eye.
Flemish astronomer Godefroy Wendelin repeated Aristarchus' measurements in , and found that Ptolemy's value was too low by a factor of at least eleven.
A somewhat more accurate estimate can be obtained by observing the transit of Venus. The solar parallax is related to the Earth—Sun distance as measured in Earth radii by.
Christiaan Huygens believed that the distance was even greater: by comparing the apparent sizes of Venus and Mars , he estimated a value of about 24 Earth radii,  equivalent to a solar parallax of 8.
Although Huygens' estimate is remarkably close to modern values, it is often discounted by historians of astronomy because of the many unproven and incorrect assumptions he had to make for his method to work; the accuracy of his value seems to be based more on luck than good measurement, with his various errors cancelling each other out.
They arrived at a figure for the solar parallax of 9. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague Jean Picard in as 3 toises.
A better method for observing Venus transits was devised by James Gregory and published in his Optica Promata It was strongly advocated by Edmond Halley  and was applied to the transits of Venus observed in and , and then again in and Transits of Venus occur in pairs, but less than one pair every century, and observing the transits in and was an unprecedented international scientific operation including observations by James Cook and Charles Green from Tahiti.
Despite the Seven Years' War , dozens of astronomers were dispatched to observing points around the world at great expense and personal danger: several of them died in the endeavour.
Another method involved determining the constant of aberration. Simon Newcomb gave great weight to this method when deriving his widely accepted value of 8.
Newcomb also collaborated with A. Michelson to measure the speed of light with Earth-based equipment; combined with the constant of aberration which is related to the light time per unit distance , this gave the first direct measurement of the Earth—Sun distance in kilometres.
Newcomb's value for the solar parallax and for the constant of aberration and the Gaussian gravitational constant were incorporated into the first international system of astronomical constants in ,  which remained in place for the calculation of ephemerides until The discovery of the near-Earth asteroid Eros and its passage near Earth in — allowed a considerable improvement in parallax measurement.
Direct radar measurements of the distances to Venus and Mars became available in the early s. Along with improved measurements of the speed of light, these showed that Newcomb's values for the solar parallax and the constant of aberration were inconsistent with one another.
The unit distance A the value of the astronomical unit in metres can be expressed in terms of other astronomical constants:.
The Sun is constantly losing mass by radiating away energy,  so the orbits of the planets are steadily expanding outward from the Sun.
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