the Astronomical Unit calculation The Venus transit and the Astronomical Unit calculation William THUILLOT Institut de mécanique céleste et de calcul des éphémérides Brandys, May 2004 IMCCE/PARIS Observatory

The transit of June 8, 2004 On June 8, 2004, the planet Venus will pass in front of the Sun. Nobody alive today has seen such an event. Why does this event occur ? Why did it retain the attention of the astronomers in the past? What results can we expect? 5h40 UTC 11h05 UTC

The VT-2004 project Coordinated observations of a rare phenomenon Educational interest (wide public, schools) Measurements « easy » to make: timings Possibility to catch images (if experience…)

The VT-2004 project Educational interest Historical background closely related to the measurement of the Solar System (methods, distances, motions of the celestial bodies, exoplanets…) preparation of a scientific experiment and measurements with some scientific value Interest of exchanging information between participants , in particular: amateurs - schools amateurs – individuals succeeding in the measurement of the Earth-Sun distance (…and of the AU)

Mechanism Mini Solar eclipse Rare event Difficult to predict in the past (Kepler 1st) Rich historical background fundamental for : - Confirming the superiority of the Copernician model (Rudolphines Tables) - Measuring the Earth-Sun distance (and the AU)

Venus visibility West of the Sun Morning visibility Superior conjunction Gibbous phase Gibbous phase Sun West elongation East Elongation Crescent before The inferior conjunction Crescent after the inferior conjunction Inferior conjunction East of the Sun Evening visibility West of the Sun Morning visibility Fixed Earth

Motion of Venus / Earth… if Venus was in the ecliptic 6 2 t (days) 1 4 2 91 7 3 182 3 5 1 8 4 273 5 365 6 456 Earth 365.25 j Venus 224.70 j Synodic period 583.92 j 7 547 8 584

. More realistic… Orbital inclination (/ecliptic) : 3.4° Nœud descendant Orbital inclination (/ecliptic) : 3.4° Venus at Nodes : - 7 December (ascending node) 5 June (descending node) Conditions for a transit : conjunction Sun- Venus - Earth (584 d.) close to a node  Rare events Venus Earth Sun . Noeud ascendant

When transits of Venus can be observed ? Need of a close aligment of the Sun, Venus and the Earth (duration up to 8 hours) Very rare events (~ every 120 years, and 8 years after): Last events : 1874-1882 Following events: 2004 - 2012, then in 2117 The 2004 VT will be well observable from Europe

Short history of the Venus transits XVIIth, Dec. 1631, Dec Short history of the Venus transits XVIIth, Dec.1631, Dec.1639 XVIIIth, June 1761, June 1769 XIXth Dec. 1874, Dec. 1882

Kepler’s laws Each planet describes an ellipse of which the Sun is at one of the focus (1605) - area’s law – law related to the ratios of semi-major axis 1627 : Rudolphines Tables 1629: prediction of a transit of Mercury (november 1631) more…: prediction of a transit of Venus (december 1631) Kepler (1571-1630)

and the period of revolution T are linked Kepler’s third law The semi-major axis a and the period of revolution T are linked by a3/T2=constant for all the planets (1618).

Visibility of the Mercury transit of 1631

Gassendi in Paris 1631: Mercury transit Transit of Mercury on Nov 7, 1631 Calculation for Paris hour Sun (true solar time) 2e contact 5h 06 -21° 3e contact 10h28 +22° First observation of a transit Use of a darkroom ( and may be a lens ) Observation from Nov 5 (bad weather on 5 and 6) Starting from the sunrise on Nov 7, Gassendi saw a black spot Measured diameter of Mercury : 20" (true value : 10") Error of 5h from the Kepler’s predictions Three other observations in Europe "Le rusé Mercure voulait passer sans être aperçu, il était entré plus tôt qu'on ne s'y attendait, mais il n'a pu s'échapper sans être découvert " Mercurius in sole visus et venus invisa Parissiis anno 1631.

Gassendi in Paris 1631: Vénus transit Gassendi tried also to observe the 1631 Venus transit Main purpose: to check the Rudolphines Tables (Copernic system) Error of the Kepler’s predictions Unobservable : in Europe (during night) => America Unsuccessful observation of the 1631 Venus transit by Gassendi But in England… J. Horrocks understood that a second transit of Venus occurs 8 years later With W. Crabtree: organization of the 1639 observations

Visibility of the Venus transit of 1639

Observations of W. Crabtree 1639 Observations made at Manchester Cloudy until 3h35  10 min of observation possible only ! Amazed by the transit, he made no measure ! Painting of F. M. Brown, visible at the City Hall of Manchester

First observations of a transit of Venus: J. Horrocks Transit of Venus on Dec 4. 1639 local time Sun 2e contact 15h15 + 4° 3e contact 21h30 - 47° sunset 15h50 Horrocks: First observation of a transit of Venus Use of a darkroom with a refractor On Sunday 4 he observed from the morning, through clouds He stopped observing for religious obligations At 3h15 he continues his observations and the weather became fair

J. Horrocks (Venus in Sole Visa) 1639 He made three measures in a hurry before the sunset t distance (") 3h15 864 3h35 810 3h45 780 3h50 sunset Diameter of Venus: 1' 16“ (Kepler : 7’) Earth-Sun : 94 000 000 km

Transits during the XVIIIth century A fundamental question : the determination of the Solar parallax 1672 : Richer and Cassini (I) : Opposition of Mars 1677 : Halley observes a Mercury transit (St Helen Island) 1691: he presents a method to get the Solar parallax from the Venus transits 1716 : he call for observations for the next Venus transit  expeditions

Mean horizontal parallax The Sun-Earth distance cannot be directly measured Classical astronomy measures angles a R p Earth Mean horizontal parallax Measurement of p and R in order to compute a R = 6400 km and a ~ 150x106 km Then p ~ 10" ==> difficult to be measured A main problem in the past

Parallax of Mars (perihelic opposition in 1672) d D Paris R f Kepler: a 3 / T 2 = constant (aMars / a Earth)3 = (TMars / TEarth)2 aEarth = aMars - D (Mars-Earth) Cayenne Cassini et Richer ps = 9.5" ( a = 138x 106 km) Flamsteed ps = 10" ( a = 130x 106 km)

Transits during the XVIIIth century Halley died in 1742 but astronomers remember his call for observations Longitudes are not yet well known. Clocks are not good time keepers. Traveling is slow (sailing). Voyages are very expensive. Nobody has never observed a transit of Venus. Two methods for measuring the parallax : Method of Halley : The durations of the transits are compared => no problem with longitude. Method of Delisle : The times of contacts are compared => more observations but longitudes have to be known.

Method of E. Halley c b a a b c • a a • b • c The relative positions of the chords give the parallax Difficulty to get an accurate measurement No reference frame available But these positions are related to the duration of each transit Angular measurements are replaced by timing measurements accurate Requires observing sites far from each other  latitudes offset 1 s. of uncertainty ==> Parallax to 1/500 (Halley, 1716)

Use of the timing offset at the beginning or at the end of the event Method of J. Delisle Dt time t Use of the timing offset at the beginning or at the end of the event Topocentric observation (from the surface of the Earth) Geocentric view Advantages Less impact of the meteorological effects Increasing of the number of sites (partial observations usable) Disadvantages Timing measurement instead of a duration measurement  need to have absolute timing Comparaison between sites  need to accurately know the geographic position ! Requires maximum of timing differences -> longitudes offset

The transit of June 6, 1761 The French Expeditions for the observation: 2 of these voyages took place in countries allied of France. César-François Cassini de Thury (1714-1784) in Vienna (successful observation). the Abbot Jean-Baptiste Chappe d'Auteroche (1728-1769) to Tobolsk in Siberia (successful observation). Alexandre Guy Pingré : Rodrigues Island (north of Madagascar), Thanks to the compagnie des Indes (observation partially successful). Guillaume Joseph Hyacinthe Jean-Batiste Le Gentil de La Galaisière (1725-1792), left by sea in order to observe the transit in Indies at Pondichéry. Unfortunately the city of Pondichéry was taken by the English and he saw the transit from the ship, unable to make a measurement; he decided to wait until the next transit in 1769 Joseph-Jérôme Lefrançois de Lalande (1732-1807) observed from Luxembourg Palace in Paris.

The transit of June 6, 1761 The English two campaigns far from England to observe the event. Nevil Maskelyne (1732-1811) went to Sainte-Hélène where he was not able to observe because of clouds. Charles Mason (1728-1786), James Bradley and Jeremiah Dixon (1733-1779) was supposed to observe from Bencoolen (Sumatra). They were not able to make the observation because the French took the city. They observed then at Capetown. John Winthrop, professor in Harvard went to St-John (Terre-Neuve) where « surrounded by billions of insects " he succeeded to observe the last contact of the transit.

Le passage du 6 juin 1761 Projection de Hammer

The voyage of Chappe d’Auteroche The travel of Chappe d’Auteroche to Tobol’sk

Results from the transit of 1761 The number of observers was 120, on 62 sites (S. Newcomb, 1959). Note that some sites of observations were previously selected (Bencoolen, Pondichéry, Batavia) by Halley in 1716. 8.5" < P < 10.5" The large error is due to: - a bad knowledge of the longitudes of the sites of observation - the black drop effect which decreases the precision of the measurement of the time of the contacts. Disappointing results : no improvement of the measures from Mars.

Timing of the internal contacts: the black drop effect" Sun Before contact Internal contact Sun Expected Sun ~10 s after lcontact Sun Uncertainty of the contact measurement : 20s to 1 min.

The transit of Venus of June 3-4, 1769 The organization of the observations for 1769 were made by Lalande in France and Thomas Hornsby in England. They took benefit from the observations of the transit of 1761. 27 refractors were used, only 3 were used in 1761. General circonstances First contact with penumbra : le 3 à 19h 8m 31.2s First contact with shadow : le 3 à 19h 27m 6.7s Maximum of the transit : le 3 à 22h 25m 20.3s Last contact with shadow : le 4 à 1h 23m 35.7s Last contact with penumbra : le 4 à 1h 42m 11.2s

Visibility of the transit of 1769

The results from the transit of 1769 The English made 69 observations and the French 34. Finally 151 observations, were made from 77 sites. Four observations of the complete transit were made : Finland, Hudson Bay, California and Tahiti. Author(s) Values William Smith 8,6045" (1770) Thomas Hornsby 8,78" (1770) Pingré et Lalande 9,2" et 8,88" (1770) Pingré 8,80 (1772) Lalande 8,55"< P < 8,63" (1771) Planmann 8,43 (1772) Hell 8,70" (1773/1774) Lexell 8.68" (1771) et 8,63" (1772) The conclusion was that the parallax was from 8,43" to 8,80 " . This was a real improvement regarding the result of 1761 providing a parallax from 8,28 to 10,60".

The transits of the XIXth century The longitudes are now well determined The clocks are good time keepers. The travels are faster (steam, Suez channel). The travels are still expensive The photographs appeared (Daguerréotype) The experiences of the XVIIIth century are profitable.

An example: the observation at St-Paul The voyage of Commandant Mouchez at Saint-Paul. July 1874 : departure from Paris. August 9: Suez channel. August 30: arrival in Réunion Island September 22: arrival in Saint-Paul island in a tempest The probability of fair weather was only 8 to 10% In spite of tempest and bad weather, the observation was a success: 500 exposures of the transit were made

The observation at Saint-Paul

The transit of December 9, 1874

The transit of 1882 General circonstances Premier contact de la pénombre : 13h 49m 3.9s Premier contact de l'ombre : 14h 9m 1.3s Maximum du passage : 17h 5m 58.5s Dernier contact de l'ombre : 20h 2m 58.3s Dernier contact de la pénombre : 20h 22m 55.7s Les Français organisèrent dix missions : une mission à l'île d'Haïti (d'Abbadie), une au Mexique (Bouquet de la Grye), une à la Martinique (Tisserand, Bigourdan, Puiseux), une en Floride (Colonel Perrier), une à Santa-Cruz de Patagonie (Capitaine de Frégate Fleuriais), une au Chili (Lieutenant de vaisseau de Bernardières) , une à Chubut (Hatt), une au Rio-Negro (Perrotin, le directeur de l'observatoire de Nice), une au Cap Horn (Lieutenant de vaisseau Courcelle-Seneuil), une à Bragado (Lieutenant de vaisseau Perrin). Le Naval Observatory envoya huit expéditions à travers le monde pour observer le passage.

The transit of December 6, 1882

Reduction of photographs The measures on the plates were made through macro-micrometers with an accuracy of one micrometer. In France, 1019 plates were taken. All the measurements were made two times by two different persons. In fact more than 500 000 measurements were made.

8 June 2004 : How the Venus transit will appear ?

Description of a transit The duration of a Venus transit is from 5 to 8 hours t1 t1 : 1st contact t2 t2 : 2nd contact t4 t4 : 4th contact t3 t3 : 3rd contact t1, t4 : exterior contacts t2, t3 : interior contacts t1  t2 : ingress t3  t4 : egress Exterior contacts are not easily observable  Interior contacts will be more accurate

Geocentric circumstances Ecliptic Celestial pole On Tuesday 8 June Polar angle 8h 19m 43,5s UTC 5h 13m 33,2s UTC 5h 32m 49,8s UTC 11h 25m 53,8s UTC 11h 06m 37,1s UTC Duration of the general transit : 6h 12m 20,68s. Duration of the internal transit : 5h 33m 47,26s. Minimum of the geocentric angular distance : 10' 26,875".

Local circumstances East South Sun rise Meridian transit At 3h 50m UT POSITION OF THE SUN ON JUNE 8 (PARIS) at 11h 49.7 UT East South End of the transit at 11h 23m 34s UT Sun height : 63,5° Sun azimut : 346,4° Beginning of the transit at 5h 20m 6s UT Sun height : 12,4° Sun azimut : 249,3° Mid event at 8h 22m 53s UT Sun height : 41,9° Sun azimut :283,5° At Paris : T1 : first external contact at 5h 20m 06s UTC Z=159,8° P= 117,7° T2 : first internal contact at 5h 39m 48.s UTC Z= 164,2° P= 121,0° M : maximum at 8h 22m 53s UT center-center : 10’ 40,9” T3 : last internal contact at 11h 4m 20s UTC Z=228,9° P= 212,4° T4 : last external contact at 11h 23m 34sUTC Z=225,0° P=215,6°

Visibility of the Venus transit on 8 June 2004

Mercury transits Apparent diameter of Mercury 1/158 of the Solar diameter

Venus Transit in 1882

Equatorial mount / alt azimuth mount Venus trajectory on the solar disk as seen in an equatorial frame (for example in a refractor with an equatorial mount) North celestial pole Parallel to equator Venus trajectory on the solar disk as seen in an horizontal frame (for example in a refractor with a alt-azimuth mount) Zenith Direction of the celestial pole at T1 at T4 Parallel to horizon

How the Sun-Earth distance will be deduced from the observations ?

Calculation of the Sun-Earth distance in 2004 For the VT-2004 observations: Locations (longitudes, latitudes) well known Accurate timing (in Universal time) Pedagogic purpose (AU is well known…) Several calculations will be made: 1 connexion to the VT-2004 web server = 1 timing observation and 1 estimate of the individual measurement 2 partners: 2 timing observations from far sites Analysis of the whole campaign: a large number of timing observations

An approximation for two partners Sun A B rv re Venus Δβ D Earth βS Sheet « Calculating the Earth-Sun distance …» Assumptions: - Two observing locations, centers of the Earth, Venus, Sun are in the same plane - Circular orbits Measurement of the distance between two chords (re / rv )3 = (Te / Tv) 2 if eccentricities = 0 βS = Δβ (( re / rv) – 1) re = Δ / (Δβ . 0.38248) R 2l h dl = V dt Δβ = dl*l / h

AU online computation Sun f ( φ , X s , X v , π , t ) = Δ Venus R s R v Sun Venus f ( φ , X s , X v , π , t ) = Δ Relation between time t and parallax π Observer’s location φ Theory of Venus Theory of the Earth (Sun) Radii The registered users will send their own timing measurements to the vt2004 web server (same welcome page as registration) The server will compute the solution π of the equation : f (φ , X s , X v , π , to ) = R s +/- R v

AU determination: the global analysis Assuming geographical locations accurately known N equations of condition can be written (for N timing measurements) involving small corrections δX s , δ X v , δ π , δ R to be calculated O – C = offset of each timing O with respect to the theoretical calculated value C « Least square » method determination of correction δ π to the Solar parallax a .δXs + b .δ Xv + c .δ π + d .δ(Rs +/-Rv ) = O - C All along the data acquisition (starting from June 8), the server will compute the Solar mean horizontal parallax π + d π using all the data gathered Numerical values (t), statistics and graphs will be produced

1770’s parallax measurement Authors Values William Smith (1770) 8.6045" Thomas Hornsby (1770) 8.78" Pingré et Lalande (1770) 9.2" and 8.88" Pingré (1772) 8.80" Lalande (1771) between 8.55" and 8.63" Planmann (1772) 8.43" Hell (1773/1774) 8.70" Lexell (1771 / 1772) 8.68“ / 8.63"

Parallax measurements since the XVIIIth century Method / author Parallax Transits of 1761 and 1769 8.43" and 8.80" Transits of 1761 and 1769, Encke (1824) 8.5776" Transits of 1761 and 1769, (1835) 8.571 +/- 0.037" Parallax of Mars, Hall (1862) 8.841" Parallax of the asteroid Flora, Galle (1875) 8.873" Parallax of Mars, Gill (1881) 8.78" Transits of 1874 and 1882, Newcomb (1890) 8.79" Parallax of the asteroid Eros, Hinks (1900) 8.806" Parallax of the asteroid Eros, (1941) 8.790" Radar measurement, NASA (1990) 8.79415"

Small historic of the Sun-Earth distance measurement Method date parallax AU in " millions km Mars 1672 9.5 - 10 130 -140 Venus 1761 8.3 - 10.6 125 - 160 Venus 1769 8.5 - 8.9 145 - 155 Mars 1862 8.84 149 Flora 1875 8.87 148 Mars 1885 8.78 150 Venus 1874 - 82 8.790-8.880 148.1 - 149.7 Eros 1900 8.806 149.4 Eros 1930 8.790 149.7 radar 1970 8.79415 149.5978 Viking+radar 2000 149.597 870 691

The Astronomical Unit History of the International Astronomical Union (IAU) value of AU (106 km) De Sitter 1938 : 149.453 Clemence 1948 : 149.670 UAI 1964 : 149.600 UAI 1976 : 149.597 870 DE102 1977: 149.597 870 68 DE200 1982: 149.597 870 66 IERS 1992: 149.597 870 61 DE403 1995: 149.597 870 691

VT-2004 122 years later …VT-2004 Large number of observers Modern techniques (GPS, Internet, webcam images, …) What results will we get in 2004 ? Credits: aknowledgements to P. Rocher (IMCCE) and F. Mignard (OCA) for several frames

Acquisition and processing of the amateur observations Data Acquisition Acquisition and processing of the amateur observations W. Thuillot & J.E. Arlot Timings : - database and online processing - global analysis and results Images : - database and pipeline (Ondrejov) Access to the data base : - observational inputs - registered observers

Data acquistion Timings measurement Acquisition web page : same welcome page as « registration » 1 registration = 1 observation = t1, t2, t3 or t4 several instruments  several registrations check your profile (geographic coordinates !) AU and Solar parallax « observed » compared with the true values comparison with global results (individual /average, dispersion) global analysis  statistics page

Data acquistion Images data base Position of Venus with respect to the Solar limb can be used Field of vue must include the least distance to the limb …and the limb itself

VT-2004 AU calculation

VT-2004 : Geographic overview

Data acquisition and calculation Still in development, but new pages are in test for a week : try the AU calculation ! !