Venus Transit and the Astronomical Unit

1. Venus Transit and the Astronomical Unit BimanBasu bimanbasu@gmail.com

2. Measuring the Universe • Our idea of the structure of the Universe has changed drastically over the past centuries. • Earth-centred Universe has given way to a Sun-centred Universe. • Ingenious experiments led to the determination of the Earth's shape and size, the Earth-Moon distance, and further to the Sun-Earth distance.

3. Measuring the Earth • One of the earliest efforts to measure the Earth was by the Greek mathematician and astronomer Eratosthenes around 240 BC. • Using trigonometry and knowledge of the angle of elevation of the Sun at noon in Alexandria and Syene (now Aswan, Egypt) on summer solstice, he arrived at the conclusion that the Earth is spherical and has a circumference of about 40,000 km. • The calculation was based on the correct assumption that the Sun is so far away that its rays can be taken as parallel.

4. Eratosthenes's experiment

5. Measuring the distance of the Moon • In 129 BC, another Greek astronomer Hipparchus, used the coverage of the Sun by the Moon during a total solar eclipse to calculate the distance of the Moon from Earth, which came to about 90 times the radius of Earth. • Although the calculated distance was substantially larger than the actual distance, the method showed how parallax could be used to measure distances of astronomical objects.

6. Hipparchus's method From Hellespont (Lat. 4020N) the total eclipse could be seen, as the Moon fully covered the Sun. Viewed from Alexandria (Lat. 3120N), only a partial eclipse could be seen at the same moment due to parallax of the Moon.

7. Hipparchus calculations Since A and B differ 9 in latitude and the circumference of Earth is given by 2r, the distance AB is given by AB = (2r/360) × 9 where r is the radius of Earth. Similarly, since the distance CD is 1/5 the solar diameter and the Sun subtends an angle of 30' or 0.5 at Earth, the angle  is 0.1, which is the parallax of the Moon as seen from A and B.

8. Hipparchus calculations-II Here AB = (2R/360) × 0.1 where R is the Earth-Moon distance. Therefore, (2R/360) × 0.1 = (2r/360) × 9 or 0.1R = 9r, which gives R = 90r; that is, the Earth-Moon distance is 90 times the radius of Earth. The distance to the Moon comes out to be 5,73,300 km, which is about 50 per cent higher than the average value of 3,84,400 km.

9. What is parallax? • Parallax is the apparent displacement of a relatively close object compared to a more distant background as the location of the observer changes. • Parallax makes it possible for our eyes to perceive depth of field and see objects in 3-dimensions. • Astronomers use parallax to find distances of planets and stars. • Parsec is a unit of distance used in astronomy, equal to about 3.25 ly, which corresponds to the distance at which the mean radius of the Earth's orbit subtends an angle of one second of arc.

10. Parallax due to position of the eyes

11. Parallax of nearby stars When the stars are photographed six months apart, some of the nearby stars appear to shift slightly because of the shift in Earth's position.

12. Solar parallax Solar parallax is defined as the angle subtended by the radius of the Earth at the centre of the Sun. Its value has been computed to be about 8.8 arcseconds.

13. Astronomical transits An astronomical transit occurs whenever one celestial object, such as a planet or a moon, passes in front of another celestial object. When the Moon passes in front of the Sun it covers the Sun fully because both appear to be of the same size from Earth.

14. Planetary transits But when an inner planet like Mercury or Venus passes in front of the Sun it covers only a tiny portion of the Sun's disc. To viewers on Earth, during such a passage or transit the planet appears as a tiny black dot moving across the solar disc.

15. Transits of the inner planets • Only the inner planets Mercury and Venus show transits because their orbits are closer to the Sun than Earth’s and occasionally both come between the Sun and the Earth when they can be seen against the solar disc. • Transits of Venus are much fewer than transits of Mercury. • On the average, there are 13 transits of Mercury in a century compared to only two transits of Venus during the same period.

16. Transits of Mercury & Venus Being more distant and smaller, Mercury (left) appears much smaller than Venus (right) during transit.

17. Conditions for transit • Since both the inner planets occupy orbits between the Earth and the Sun, they would more than likely be seen to pass in front of the solar disk from time to time. • Mercury comes between the Sun and the Earth every 116 days on average, while Venus does so every 584 days on average.

18. Conditions for transit-II • But transits of the two planets are not as frequent because their orbits are tilted with respect to that of Earth: Mercury 7.0, Venus 3.23. • In order for a transit to occur, the planet, Sun, and Earth have to be in the same plane on the same side of the solar system. • This happens only when the planets are at any of the two nodes where their orbits cross the Earth's orbital plane.

19. Orbits of Mercury and Venus

20. Transits of Venus • Transits of Venus are much rarer than transits of Mercury because the orbital period of Venus is longer than that of Mercury. • Indeed, only seven such events have occurred since the invention of the telescope (1631, 1639, 1761, 1769, 1874, 1882, and 2004). • Transits of Venus show a clear pattern of recurrence at intervals of 8, 121.5, 8, and 105.5 years.

21. Transits of Venus-II • Transits of Venus are only possible during early December and June when the orbital nodes of Venus pass across the Sun. • The last transit of Venus was seen on 8 June 2004 and the next one is due on 6 June 2012. • After the 6 June transit the next transit of Venus will not be seen till the next century, on 11 December 2117; that is, after a gap of 105 years and six months.

22. Historical transits of Venus • The first person to predict a transit of Venus was the German astronomer Johannes Kepler, who calculated that one would take place on 6 December 1631. • Kepler died in 1630, and there is no record of anyone having seen the 1631 event. • Young English astronomer Jeremiah Horrocks and his friend William Crabtree were the first persons to observe a transit of Venus on 4 December 1639.

23. Jeremiah Horrocks (1618-1641) Jeremiah Horrocks observing the Venus transit of 4 December 1639.

24. Horrocks’ observation 1639 Horrocks had calculated that the transit was to begin at approximately 3:00 pm. He had about 35 minutes to observe the transit before sunset at 3.50 p.m. (Published in 1662 by Johannes Hevelius)

25. Venus transit cycles • Transits of Venus occur in a 'pair of pairs' pattern that repeats every 243 years. • First, two transits of a cycle take place in December, eight years apart. • There follows a wait of 121 years 6 months, after which two June transits occur, again eight years apart. • The pattern repeats after 105 years 6 months, beginning with two December transits, eight years apart.

26. Earth’s orbit around the Sun

27. The Astronomical Unit • The Astronomical Unit (AU) is the average distance between the Sun and the Earth. • It is a convenient unit to use when expressing distances within the solar system. • The AU, as defined in the International Astronomical Union (IAU) system of constants, is equal to 149,597,870 km.

28. Kepler's third law and the AU • According to Kepler's Third Law, the square of the orbital period of a planet (P) is proportional to the cube of the semi-major axis (a) of its orbit measured in AU; that is, P²a³. • If two planets have orbital periods P1 and P2, the ratio of their distances a1 and a2 from the Sun can be worked out. • Kepler's Third Law thus allows one to evaluate the dimensions of the solar system in relative units, e.g., in "astronomical units" (AUs), where 1 AU is the mean Sun-Earth distance.

29. The heliocentric model • The heliocentric model uniquely specifies the relative distances (from the Sun) to the planets according to Kepler’s Third Law, as follows: Mercury (0.38 AU), Venus (0.72 AU), Mars (1.5 AU), Jupiter (5.2 AU), and Saturn (9.5 AU). • The challenge then becomes determining the absolute distance to any one planet. • If a parallax measurement could be accurately obtained for any single planet, all the other distances including the AU would be trivial to calculate using Kepler’s laws.

30. Determining the Astronomical Unit • Astronomers have been trying to determine the value of the AU since Kepler’s time and various techniques have been used. • In 1716, the English astronomer Edmond Halley published a paper in the Philosophical Transactions of the Royal Society, describing exactly how the parallax of transits could be used to measure the Sun's distance, thereby establishing the absolute scale of the solar system from Kepler's Third Law.

31. Edmund Halley (1656-1742)

32. Transit of Venus and the A.U. • Halley’s method involved observing and timing a transit from widely spaced latitudes. • Although the method gave the first reasonable value for the Sun's distance from Earth, his method proved somewhat impractical since contact timings of the required accuracy are difficult to make.

33. Transit of Venus 1769 • Halley died in 1742, but the transits of 1761 and 1769 were observed from many places around the world. • The transit of 1769 was one of the most extensively observed transits of Venus of that time. • In 1771, French astronomer Jerome Lalande was able to use the combined measurements taken in 1761 and 1769 to determine the average Earth-Sun distance to be 153 ( 1) million km, as against the currently accepted value of 149.60 million km.

34. The effect of parallax on transit observation • Parallax causes a planetary transit to look slightly different for two observers at different latitudes on Earth. • Venus does not appear to enter or leave the Sun’s disc simultaneously from two widely different locations, and, observed at the same moment, Venus’ position on the disc of the Sun also differs slightly.

35. Determining the Sun’s distance • The parallax effect gives rise to two ways of obtaining the Sun’s distance from observations of the transit of Venus. i. Timing the start and end of the transit from two stations - Halley’s original 1716 method. ii. Photographing the Sun at the same moment from two stations and measure the northward or southward displacement of Venus due to parallax.

36. Determining the Sun’s distance From the time difference between T1 and T2 and the distance between A and B the distance between the Sun and Earth can be calculated.

37. Transit paths of 2004 and 2012 The transits of 1874 and 1882 occurred during ascending nodes, while the transits of 2004 and 2012 occur during the descending node.

38. Stages in Venus transit 2012

39. Timing the transit A transit of Venus across the Sun takes about 7 hours, but this time has to be measured to a precision of a few seconds to be of any use. To be useful, the most critical times are the first, second, third, and fourth contact. Unfortunately, an optical phenomenon called the ‘black drop’ effect makes it difficult to time the second and third contacts precisely.

40. Transit timings in India The transit of Venus on 6 June 2012 will start long before Sunrise in India and hence the timing of the 1st and 2nd contacts will not be visible. However, it may be possible to determine the apparent displacement of the position of Venus on the solar disc if observed from two distant latitudes.

41. View from India at sunrise

42. Recording position of Venus during transit The best way for amateur astronomers to record the position of Venus during transit would be to project the image of the Sun on a white card using a small telescope. CAUTION: NEVER LOOK AT THE SUN DIRECTLY!

43. Measuring Venus parallax • For measuring the parallax it is necessary to have at least two simultaneous observations at precisely the same instant from two locations in different latitudes along the same meridian. • Kanyakumari (8.06N, 77.30E) and New Delhi (28.40N, 77.12E) are good examples. • From Kanyakumari, Venus will appear a little northward on the solar disc than from New Delhi.

44. Locations for observing the transit of Venus

45. Recording Venus position on solar disc The positions can be recorded on identical Sun templates on cards with sketch pen, at 30-minute intervals, beginning at a predetermined time, from both locations.

46. Displacement of the tracks of Venus Path of Venus as recorded from Australia, India, and the Canary Islands during the 2004 Venus transit.

47. Geometry of Venus transit s = Δ ((re / rv ) - 1) [Can look up http://skolor.nacka.se/samskolan /eaae/summerschools/TOV2.html for details.]

48. Modern techniques of measuring A.U. • One of the modern methods for deriving the absolute value of the Astronomical Unit uses radar in combination with triangulation. • In this technique, the distance of Venus at its greatest elongation is measured using radar. • From the known velocity of radio waves of 300,000 km/s and the time taken for the signal to return, the distance of Venus can be determined with high accuracy.

49. Use of radar to measure A.U. Once the Earth-Venus distance is known accurately, the Earth-Sun distance or Astronomical Unit can be computed using simple triangulation.

50. Why the transit of Venus is important • Historically, the observation of the transit of Venus has been the most valuable technique for measuring the distance from the Earth to the Sun, or the Astronomical Unit. • Even today, the observation of a transit and using it to determine the value of the AU can be an enjoyable activity for anyone. • But it would need collaboration between groups located at widely separated places for getting reasonably accurate results.