DISTANCE TO SUN BY VENUS TRANSIT ------------------------------ John Pazmino NYSkies Astronomy Inc firstname.lastname@example.org www.nyskies.org 2012 June 12 Introduction ---------- After the transit of Venus in 2012 NYSkies was flooded with questions about the use of the transit to determine the Earth-Sun distance. There was ample discussion of the method in litterature issued for the transit but much of it was too technical or too simple. I here give a geometric walk-thru, but you do have to understand some trigonometry. Knock-off history --------- ---- Until the transits of the 1760s we did not know ell at all the earth-Sun distance for a linear size of the solar system. Cassini in the 1670s tried to capture a parallax from Mars at opposition but the instruments ere too crude for confident results. A better choice would be Venus, who approaches closer to the Earth than mars, yielding a larger and more easily measured parallax. Several attempts were made in the late 1600s during times when Venus was in dark sky as evening or morning star. Horrendous troubles broke out, explained below, that upset these efforts. Halley observed in the early 1700s a transit of Mercury and hit on the plan to observe the future Venus transits for getting a solid parallax. He passed on long before the transits but his mission was taken up by England and France. Both had colonies scattered across the globe from which to observe the events. The idea was to trace the exact path of Venus across the Sun's disc, then work out the shift as a function of separation of the observing sites. The effort was immense for its time, the first global science project in history. The results were far better than what we had until then. The project was repeated for the transits of the 1880s, with much improvement in the Earth-Sun distance.Even in the 20th century, parallax measurements were attempted on close-passing asteroids, like Icarus and Hermes. Since the Space Age, with interplanetary probes, transits of Venus are no longer needed. Distances thruout the solar system are measured by signal relaying with the spaceprobes. nearby objects, mainly Earth- threat asteroids, are measured by radar pinging. The transits of 2004 and 2012 were observed for information about Venus and to test instruments built for detecting extrasolar planets. Both transits were widely observed by NYSkies, other astronomers and the public. As a segment of human history, the stories of England and France work for the two 1700s Venus transits is full of adventure, danger, war, high science, travel, glory. For us in the United States, they prodded the colonies to agitate for a native American identity, leading in the late 1700s to the War of independence. Parallax from transits -------------------- Halley's plan was to trace the exact path of Venus across the Sun from as many dispersed places on Earth as possible. From building triangles from venus to paris of these observing sites the parallax of the planet could be worked out. The value so obtained was converted into the distance from Earth to Sun and by extension the distances to all planets in the solar system. The diagram is grossly exaggerated. A and B are the locations on Earth, V is Venus. The angle at V is really tiny. It turns out that for A and B one Earth radius apart. This angle is about the size of a US 25c coin, the quarter, seen from two short Manhattan blocks away, about 170 meters. A / +------ ------/ b | ------ ------ / | ------ V ------ | ------+------ | ------ ------ | ------ ------ / +------ ------/ a B / The lines of sight from A and B they V intersect the solar disc at a and b. At these points Venus is a round black dot against the Sun but at two different places on hum. The slanted lines a and b are the paths of venus across the Sun' disc, not shown,. The displacement of the paths from sites A and B go into the computation of parallax and then the Earth-Sun distance. Halley figured out that it was necessary only to time the ingress and egress of the Venus disc with the Sun, since the angular speed of the planet was known from solar system dynamics. The span of time it took Venus to cross the Sun was equivalent to the length and position of her path across the Sun. As simple as this sounds it was utterly heroic that these triangles were built from reports returned from the 1700s and 1800s transits. Apart from the problems of long-endurance travel, crossing war zOnes, setting up stations, and weather, there was the imperfect geography of the day. A team set up on a place whose exact lat-lon was not well established, making the base of the triangle loosely defined. It proved impossible to observe from the ideal stations one Earth radius apart face-on to the Sun. We had to scatter stations all over the world, even to places only discovered a few years earlier. The geometry of the triangles between arbitrary pairs of stations was far more difficult than for the A and B stations in the diagram above. The actual parallax was then fluffed up into the ideal situation of A and with many cross-checks in the network of station s. We cut thru these difficulties and consider the ideal situation of a parallax taken across a full radius of Earth, say from where the transit occurred in the local zenith and in the local horizon. This value is the horizontal parallax. Why use a transit? ---------------- Traditional parallax is measured against the stars, assumed permanently fixed on the celestial sphere. Altho by the mid 19th century stars were known to have proper motion against the celestial sphere, this motion was infinitesimal compared to a sensible parallax. It would be normal to acquire the parallax of Venus at nay time she was in dark sky against the stars. In the diagram above, without the slanted lines for a path on the Sun, points a and b are the positions of Venus relative to surrounding stars. Opportunities for dark-sky Venus parallax happen frequently enough, in both morning and evening, in the next couple months. The measured value would be applied to the triangle of Sun-Venus- Earth with sides fixed by the already known scale of the solar system. V + / \ / \ / \ +---------------------+ E S S-E is the Sun-Earth distance. S-V is the Sun-Venus distance. E-V, the distance from Earth to Venus that is determined from the parallax of Venus captured against the stars when Venus stands in a dark sky. All three sides are known ratios of each other from the application of known planetary motion. Once the linear length of the Earth-Venus arm of this triangle was determined, the other sides fall into line with their own linear dimensions. In the ideal case we should have obtained the Sun-Earth distance in the late 1600s. limited only by instrumental resolution for small angular displacement against the stars. When fixes on Venus were tried, we got discordant parallaxes. The uncertainties were due to the Venus phases and atmosphere. The phases made it tricky to fix on the center of the planet's disc. The atmosphere, discovered as such during the 1700s transits, distorted the shape of Venus, further confusing the location of her geometric center. A transit presents a nitid scene of just the black round dot of Venus and the bright disc of the Sun. Also, during a transit Venus is closest to Earth, at inferior conjunction, presenting the largest parallax she can offer. The Sun moves ----------- During a Venus transit we do NOT observe directly the full parallax of Venus. The Sun, against which we see Venus, is himself close enough to exhibit his own, unknown, parallax. The orbit radius of Venus is 0.723 that of Earth's. Accounting for excentricity and place of each planet relative to its own perihelion, the distance of Venus from Earth in the 2012 transit is 0.289 Earth orbit radius. It's larger than the mean distance mainly because Earth in June 2012 is nearing her apohelion, being a little father form Sun than 1.000 radius. This means that the parallax of Venus is 1/0.289 that of the Sun's or 3.460 times greater. The parallax angle is so tiny we can use a simple proportion via the small-angle principle. Against the celestial sphere Venus AND SUN are displaced by parallax such that Venus ON THE SUN shifts track by ((3.460)- (1))/(3.460) = 0.711 of its full value against the celestial sphere. ------------------------ geocentric position of Sun & Venus | | | | Sun parallax | \|/ Venus | ------------- horizontal position of Sun parallax | | | | Venus shift | | on Sun | | \|/ \|/ ------------------ horizontal position of Venus By missing the parallax shift of the background against which Venus is measured, erroneous values are calculated for the Earth-Sun distance. We must inflate the MEASURED displaced on the Sun by (3.460)/(2.460) = 1.407 times to remove the Sun's own parallax. From modern computations, like in the circumstances published for the 2012 transit, the Venus shift on the Sun is 21.8 arcseconds. This is about 1/3 of Venus's angular diameter. This must be inflate by 1.407 times to get 30.6 arcseconds. Earth-Sun distance ---------------- Now we can build the triangle. The base is 6,400km and apex is 30.6 arcsecond. By direct trigonometry we have that the long sides of this triangle, assumed equally long, is 6400km/tan(30.6 arcsec) = 43,740,000km (rounded). From the scale of the solar system this distance is 0.289 of the Earth-Sun distance. we have 46.740,000km/0.289 = 149,235,000km. This full distance is the Astronomical unit. Given some rounding here this is only a whisker off of the actual value of 149,600,000km. The discrepancy of 365,000km is less than the Earth-Moon distance. conclusion -------- The short explanation here of how we used the Venus transits to find the Earth-Sun distance, and then the size of the whole solar system, is an ideal situation. The actual work in the 1700s and 1800s took many years after the expeditions returned to their home countries. The final accepted size of the solar system was issued near the run of the 20th century . It stayed in force until radar and spaceprobes improved it in the 1960s. We now know the value to a couple kilometers, the diameter of a small town on Earth. It was an awesome thrill of this generation of astronomers to witness the two transits of the 21st century. The next pair come in the next century, long after all of us present today are passed on. With the separation between pairs being over a hundred years, it was possible that some astronomers lived and died without ever seeing one. In addition to the sight of a whole effing planet plodding its way across the Sun, there are spinoff lessons in history and global affairs, and, yes, maths and geometry.