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95 when the units of measurement are the astronomical units of length mass and time 7 15 16 equivalently by this definition one au is the radius of an unperturbed circular newtonian orbit about the sun of a particle having infinitesimal mass moving with an angular frequency of 0 017 202 098 95 radians per day 17 or alternatively that length for which the heliocentric gravitational constant the product g m is equal to 0 017 202 098 95 2 au 3 d 2 when the length is used to describe the positions of objects in the solar system subsequent explorations of the solar system by space probes made it possible to obtain precise measurements of the relative positions of the inner planets and other objects by means of radar and telemetry as with all radar measurements these rely on measuring the time taken for photons to be reflected from an object because all photons move at the speed of light in vacuum a fundamental constant of the universe the distance of an object from the probe is calculated as the product of the speed of light and the measured time however for precision the calculations require adjustment for things such as the motions of the probe and object while the photons are transiting in addition the measurement of the time itself must be translated to a standard scale that accounts for relativistic time dilation comparison of the ephemeris positions with time measurements expressed in barycentric dynamical time tdb leads to a value for the speed of light in astronomical units per day of 86 400 s by 2009 the iau had updated its standard measures to reflect improvements and calculated the speed of light at 173 144 632 6847 69 au d tdb 18 in 1983 the cipm modified the international system of units si to make the metre defined as the distance travelled in a vacuum by light in 1 299 792 458 second this replaced the previous definition valid between 1960 and 1983 which was that the metre equalled a certain number of wavelengths of a certain emission line of krypton 86 the reason for the change was an improved method of measuring the speed of light the speed of light could then be expressed exactly as c 0 299 792 458 m s a standard also adopted by the iers numerical standards 19 from this definition and the 2009 iau standard the time for light to traverse an astronomical unit is found to be τ a 499 004 783 8061 0 000 000 01 s which is slightly more than 8 minutes 19 seconds by multiplication the best iau 2009 estimate was a c 0 τ a 149 597 870 700 3 m 20 based on a comparison of jet propulsion laboratory and iaa ras ephemerides 21 22 23 in 2006 the bipm reported a value of the astronomical unit as 1 495 978 706 91 6 10 11 m 8 in the 2014 revision of the si brochure the bipm recognised the iau s 2012 redefinition of the astronomical unit as 149 597 870 700 m 10 this estimate was still derived from observation and measurements subject to error and based on techniques that did not yet standardize all relativistic effects and thus were not constant for all observers in 2012 finding that the equalization of relativity alone would make the definition overly complex the iau simply used the 2009 estimate to redefine the astronomical unit as a conventional unit of length directly tied to the metre exactly 149 597 870 700 m 20 24 the new definition also recognizes as a consequence that the astronomical unit is now to play a role of reduced importance limited in its use to that of a convenience in some applications 20 1 astronomical unit 149 597 870 700 metres by definition 149 597 870 700 kilometres exactly 92 955 807 273 miles 499 004 783 84 light seconds 8 316 746 3973 light minutes 1 581 250 740 98 10 5 light years 4 848 136 8111 10 6 parsecs this definition makes the speed of light defined as exactly 299 792 458 m s equal to exactly 299 792 458 86 400 149 597 870 700 or about 173 144 632 674 240 au d some 60 parts per trillion less than the 2009 estimate usage and significance edit with the definitions used before 2012 the astronomical unit was dependent on the heliocentric gravitational constant that is the product of the gravitational constant g and the solar mass m neither g nor m can be measured to high accuracy separately but the value of their product is known very precisely from observing the relative positions of planets kepler s third law expressed in terms of newtonian gravitation only the product is required to calculate planetary positions for an ephemeris so ephemerides are calculated in astronomical units and not in si units the calculation of ephemerides also requires a consideration of the effects of general relativity in particular time intervals measured on earth s surface terrestrial time tt are not constant when compared with the motions of the planets the terrestrial second tt appears to be longer near january and shorter near july when compared with the planetary second conventionally measured in tdb this is because the distance between earth and the sun is not fixed it varies between 0 983 289 8912 and 1 016 710 3335 au and when earth is closer to the sun perihelion the sun s gravitational field is stronger and earth is moving faster along its orbital path as the metre is defined in terms of the second and the speed of light is constant for all observers the terrestrial metre appears to change in length compared with the planetary metre on a periodic basis the metre is defined to be a unit of proper length but the si definition does not specify the metric tensor to be used in determining it indeed the international committee for weights and measures cipm notes that its definition applies only within a spatial extent sufficiently small that the effects of the non uniformity of the gravitational field can be ignored 25 as such the metre is undefined for the purposes of measuring distances within the solar system the 1976 definition of the astronomical unit was incomplete because it did not specify the frame of reference in which time is to be measured but proved practical for the calculation of ephemerides a fuller definition that is consistent with general relativity was proposed 26 and vigorous debate ensued 27 until august 2012 when the iau adopted the current definition of 1 astronomical unit 149 597 870 700 metres the astronomical unit is typically used for stellar system scale distances such as the size of a protostellar disk or the heliocentric distance of an asteroid whereas other units are used for other distances in astronomy the astronomical unit is too small to be convenient for interstellar distances where the parsec and light year are widely used the parsec parallax arcsecond is defined in terms of the astronomical unit being the distance of an object with a parallax of 1 the light year is often used in popular works but is not an approved non si unit and is rarely used by professional astronomers 28 when simulating a numerical model of the solar system the astronomical unit provides an appropriate scale that minimizes overflow underflow and truncation errors in floating point calculations history edit the book on the sizes and distances of the sun and moon which is ascribed to aristarchus says the distance to the sun is 18 to 20 times the distance to the moon whereas the true ratio is about 389 174 the latter estimate was based on the angle between the half moon and the sun which he estimated as 87 the true value being close to 89 853 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 380 and 1 520 earth radii 29 according to eusebius of caesarea in the praeparatio evangelica book xv chapter 53 eratosthenes found the distance to the sun to be σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας literally of stadia myriads 400 and 80 000 but with the additional note that in the greek text the grammatical agreement is between myriads not stadia on the one hand and both 400 and 80 000 on the other as in greek unlike english all three or all four if one were to include stadia words are inflected this has been translated either as 4 080 000 stadia 1903 translation by edwin hamilton gifford or as 804 000 000 stadia edition of édourad des places de dated 1974 1991 using the greek stadium of 185 to 190 metres 30 31 the former translation comes to 754 800 km to 775 200 km which is far too low whereas the second translation comes to 148 7 to 152 8 million kilometres accurate within 2 32 hipparchus also gave an estimate of the distance of earth from the sun quoted by pappus as equal to 490 earth radii according to the conjectural reconstructions of noel swerdlow and g j toomer this was derived from his assumption of a least perceptible solar parallax of 7 33 a chinese mathematical treatise the zhoubi suanjing c 1st century bce shows how the distance to the sun can be computed geometrically using the different lengths of the noontime shadows observed at three places 1 000 li apart and the assumption that earth is flat 34 distance to the sun estimated by estimate in au solar parallax earth radii aristarchus 3rd century bce in on sizes 13 24 7 12 256 5 477 8 0 011 0 020 archimedes 3rd century bce in the sand reckoner 21 10 000 0 426 hipparchus 2nd century bce 7 490 0 021 posidonius 1st century bce quoted by coeval cleomedes 21 10 000 0 426 ptolemy 2nd century 2 50 1 210 0 052 godefroy wendelin 1635 15 14 000 0 597 jeremiah horrocks 1639 15 14 000 0 597 christiaan huygens 1659 8 2 25 086 35 1 068 cassini richer 1672 9 5 21 700 0 925 flamsteed 1672 9 5 21 700 0 925 jérôme lalande 1771 8 6 24 000 1 023 simon newcomb 1895 8 80 23 440 0 9994 arthur hinks 1909 8 807 23 420 0 9985 h spencer jones 1941 8 790 23 466 1 0005 modern astronomy 8 794 143 23 455 1 0000 in the 2nd century ce ptolemy estimated the mean distance of the sun as 1 210 times earth s radius 36 37 to determine this value ptolemy started by measuring the moon s parallax finding what amounted to a horizontal lunar parallax of 1 26 which was much too large he then derived a maximum lunar distance of 64 1 6 earth radii because of cancelling errors in his parallax figure his theory of the moon s orbit and other factors this figure was approximately correct 38 39 he then measured the apparent sizes of the sun and the moon and concluded that the apparent diameter of the sun was equal to the apparent diameter of the moon at the moon s greatest distance and from records of lunar eclipses he estimated this apparent diameter as well as the apparent diameter of the shadow cone of earth traversed by the moon during a lunar eclipse given these data the distance of the sun from earth can be trigonometrically computed to be 1 210 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 38 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 for example in his introduction to ptolemaic astronomy al farghānī gave a mean solar distance of 1 170 earth radii whereas in his zij al battānī used a mean solar distance of 1 108 earth radii subsequent astronomers such as al bīrūnī used similar values 40 later in europe copernicus and tycho brahe also used comparable figures 1 142 and 1 150 earth radii and so ptolemy s approximate earth sun distance survived through the 16th century 41 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 1627 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 1635 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 42 by measuring the transit in two different locations one can accurately calculate the parallax of venus and from the relative distance of earth and venus from the sun the solar parallax α which cannot be measured directly due to the brightness of the sun 43 jeremiah horrocks had attempted to produce an estimate based on his observation of the 1639 transit published in 1662 giving a solar parallax of 15 similar to wendelin s figure the solar parallax is related to the earth sun distance as measured in earth radii by a cot α 1 radian α displaystyle a cot alpha approx 1 textrm radian alpha the smaller the solar parallax the greater the distance between the sun and earth a solar parallax of 15 is equivalent to an earth sun distance of 13 750 earth radii christiaan huygens believed that the distance was even greater by comparing the apparent sizes of venus and mars he estimated a value of about 24 000 earth radii 35 equivalent to a solar parallax of 8 6 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 transits of venus across the face of the sun were for a long time the best method of measuring the astronomical unit despite the difficulties here the so called black drop effect and the rarity of observations jean richer and giovanni domenico cassini measured the parallax of mars between paris and cayenne in french guiana when mars was at its closest to earth in 1672 they arrived at a figure for the solar parallax of 9 5 equivalent to an earth sun distance of about 22 000 earth radii 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 1669 as 3 269 000 toises this same year saw another estimate for the astronomical unit by john flamsteed which accomplished it alone by measuring the martian diurnal parallax 44 another colleague ole rømer discovered the finite speed of light in 1676 the speed was so great that it was usually quoted as the time required for light to travel from the sun to the earth or light time per unit distance a convention that is still followed by astronomers today a better method for observing venus transits was devised by james gregory and published in his optica promata 1663 it was strongly advocated by edmond halley 45 and was applied to the transits of v...