Transits of Venus – Halley’s Recommendation andThis Year’s Phenomenon
ハレーの推奨した金星の日面経過と今年の現象
Mitsuru SÔMA相馬 充（国立天文台）
National Astronomical Observatory of Japan,Mitaka, Tokyo 181-8588, Japan.
E-mail: mitsuru.soma@nao.ac.jp
Abstract: Edmond Halley (1716) recommended that the transit of Venus across the Sun’s disk in 1761be observed in order to measure the distance of an AU (astronomical unit), and assuming the solar
parallax to be 12.5 arcsec (its correct value is now known to be 8.794 arcsec) he estimated the differencein the duration of the transit on 1761 June 6 between the East Indies and Hudson Bay would be 17
minutes. He correctly calculated the effect of the diurnal motion of the observation stations due to theearth’s rotation but he missed the effect of the parallax due to the differences in the north-south directionof the observation stations, and therefore the transit in 1761 was not in fact a favorable event todetermine the distance of an AU as Halley considered. On the other hand the transit of Venus occurring
this year on 2012 June 6 is favorable to determine the distance of an AU. Although the distance of an
AU is known precisely as 149,597,870.700 km, it has still educational value to know how past
astronomers tried to obtain the distance of an AU from observations of transits of Venus.
Keywords: Transits of Venus, determination of an astronomical unit

1 INTRODUCTION
Transits of Venus across the Sun’s disk are rare events. A transit of Venus will occur on 2012 June 6.
The last transit of Venus occurred 8 years ago on 2004 June 8, but the next one will not occur for morethan 100 years until the year 2117. In fact the transits occur in cycles of 121.5, 8, 105.5, and 8 years
thfrom the 15 century to the 30 century. Fig. 1 shows the paths of Venus across the Sun’s disk for thetransits occurring between 1761 and 2255. We consider the reason of the rarity of the Venus’ transits
here.
In order for a transit of Venus tooccur, the Sun, Venus and the Earthhave to be lined up in this order. Since
Venus' orbit tilts by about 3°.4 withrespect to the Earth’s orbit, transits ofVenus occur only at the inferior
conjunctions of Venus which happen to
be near the days when the Earth passes
the ascending or descending node ofVenus’ orbit.
Because the Sun’sapparent diameter is about 32′, theheliocentric longitude of Venus at theinferior conjunction should be within
about 3°.5 in total around the longitudeof the ascending or descending node inorder for a transit to occur. The dates of
the Earth passing the ascending anddescending nodes are around Dec. 9
and June 7, respectively, and therefore
the transits occur only when an inferior
conjunction of Venus happens to bewithin about 3.5 days in total aroundthese dates.
1Since the sidereal revolution periods of Venus and theEarth around the Sun are 224.7008 days and 365.2564
days, respectively, the synodic period (the period from
one inferior conjunction to the next) of Venus is 583.9213
days. In Fig. 2 the orbit of the Earth is divided into 5 equalparts by the points A, C, E, B, and D where A is in thedirection of the descending node of Venus’ orbit. The
angle by which the Earth revolves around the Sun during
the synodic period of Venus is 583.9213d ×360°/365.2564d = 575°.52 = 360°+180°+ 35°.52. On
2004 July 8 the inferior conjunction of Venus occurred
when the Earth was near the point A, and therefore thetransit of Venus was seen on that day. The successive
inferior conjunctions occurred near the points of B, C, D,and E in this order, and the next inferior conjunction(which is the event on 2012 July 6) occurs after theinferior conjunction of 2004 by 583.9213d×5 = 2919.6d.
Since this interval is only 2.4d shorter than the length of 8years (365.25d×8 = 2922.0d), the inferior conjunction of2012 occurs near the point A but just before the Earth’s
reaching the point A.
The dates of these inferior
conjunctions are 2004 June 8 and 2012 June 6 andtherefore these dates fulfill the condition of seeing thetransits (within 3.5 days in total around the date of theEarth’s passing Venus’ descending node). Hence the transit of Venus occurs both on 2004 June 8 andon 2012 June 6. In 2020 the inferior conjunction of Venus will occur again near the point A, but its date is
2.4d further earlier than in 2012, and so the date will no longer fulfill the condition of seeing the transits.
The Earth’s positions at the dates when Venus’ inferior conjunctions of every 8 year occur movebackwards. The next transit of Venus will occur when the inferior conjunctions occurring near the point B
move backwards and reach the point near the direction of the ascending node of Venus’ orbit and its
date will be 2117 Dec 11, which is 105.5 years later from this year.
Note that the revolution period of the Earth with respect to the ascending or descending node ofVenus’ orbit is 365.2513 days, which is slightly different from the sidereal revolution period (365.2564
days) of the Earth around the Sun due to the movement of Venus’ orbit.

2 HALLEY’S RECOMMENDATION
Edmond Halley was an English astronomer who was born in 1656 and died in 1742. He recommended
in his paper written in 1716 that the transit of Venus across the Sun’s disk in 1761 be observed in order
to measure the distance of an AU (astronomical unit). His paper was written in Latin, but its English
translation was published by J. Ferguson (1761). Another English translation was published by Hutton etal. (1809) and is given on an NASA Eclipse Web Site or other Websites. Roode (2005) summarizes
Halley’s idea.
In Halley’s time relative distances between planets and the Sun were known from Kepler’s third law,
but their actual distances in kilometers were not known. The next transit of Venus was on 1761 June 6
and although Halley knew that he would be dead before the date of the transit, he recommended that itshould be observed in order to know the distance to the Sun. Assuming the solar parallax to be 12″.5 (itscorrect value is now known to be 8″.794) Halley found that for the transit of Venus in 1761 the greatest
difference to be expected was between locations in the East Indies, where Venus’ stay was shortened byeleven minutes, and Hudson’s Bay, where it was lengthened by six minutes, and therefore the differencein the duration between the East Indies and Hudson’s Bay would be 17 minutes. The difference wasproduced due to the fact that the two locations moved in the opposite directions by the diurnal motion
originated from the Earth’s rotation, as shown in Figs. 3, 4, and 5. In fact as seen from Hudson’s Bay
Venus would enter the sun's disk a little before the setting of Venus and the Sun, and would go off a little
after the rising of Venus and the Sun (i.e. in Hudson’s Bay most of the transit would occur below thehorizon).
He correctly calculated the effect of the diurnal motion of the two locations due to the Earth’s rotation
but he missed the effect of the parallax due to the differences in the north-south direction of theobservation stations, as is explained below.

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At the event in 1761 Venus passes over the southern part of the Sun (see Fig. 1), and therefore atthe northern stations on the Earth, like Hudson’s Bay, the apparent path of Venus over the Sun’s disk lies
south in comparison with that at the geocenter, and it makes the duration shorter, which cancels thelonger duration due to the rotation. As a result, as shown in Table 1, which gives the predictions for theEast Indies and Hudson’s Bay calculated with the current theory, the difference in the durations of thetwo locations is only 53 sec, which is not to be compared with 17 minutes predicted by Halley.
As shown above, the 1761 transit of Venus was not as favorable as Halley considered to determine
the distance of an AU.

3 TRANSIT OF VENUS IN 2012
At the transit of Venus in 2012 Venus passes over the northern part of the Sun (see Fig. 1). Therefore as
opposed to the event in 1761 the event in 2012 is favorable to determine the distance of an AU. In fact
3the effects of the Earth’s rotation and the parallax strengthen the difference in the durations for e.g. theSolomon islands and Iceland (see Fig. 6). In fact as shown in Table 2 the difference in the durations
becomes almost 21 minutes. Even within Japan, the difference in the durations can be about 2 minutes
as shown in Table 3.
As is discussed above, the transit in 2012 is favorable for determining the distance of an AU. Since
the distance of an AU is known precisely as 149,597,870.700 km (Luzum et al. 2011) from radarobservations to planets, transits of Venus has no scientific value now, but they still have educational
value about how past researchers tried to determine the distance of an AU from transits of Venus, andthe transit of Venus in 2012 is good for such educations.

4 JAPANESE TERMS FOR “TRANSITS”
There are two Japanese terms “経過” and “通過” forthe English word “transits”.
I will give somecomments about it below in Japanese.
金星が太陽の前面を通過する現象には，金星の「日面経過」，「日面通過」，「太陽面通過」などの用語が使用されている．混乱を避けるために，公開天文台ネットワーク (PAONET) では，一般の方に分かりやすいという理由で，2004 年 4 月に「太陽面通過」を使用するように提案した．しかし，理科年表等では「日面経過」が使用されている．ここではこの「日面経過」という用語について少し触れておく．
国立天文台暦計算室によると，「日面経過」という言葉が本暦 (当時の東京天文台が編集し神宮司庁が頒布していた暦書) に記載されたのは 1924 年 (大正 13 年) ５月８日の水星日面経過からで，理科年表(暦部) では理科年表が創刊された 1925 年 (大正 14 年) 版以後最初の水星日面経過が起こった 1927 年版(昭和２年版だが，昭和に改元される前に発行されたため大正 16 年版になっている) に「水星ガ太陽ノ面ヲ横切ル現象即チ水星ノ日面経過ガアル」と説明し，以後，この現象は「日面経過」で統一されている．
言葉を変えると別の現象を意味するという誤解を与える恐れがあるということで，国立天文台暦計算室
では今後も「日面経過」を使用する意向だということである．「経過」という用語は一般には「時の経過」のように使われるのが普通だと思われているだが，実は天文学では「掩蔽」に対する用語として「経過」が存在する．つまり，「掩蔽」とは一般に小さな天体の前面をより大きな天体が通過して小さな天体が隠される現象なのに対して，大きな天体の前面を小さな天体が通過する現象を「経過」という．
このことは大きな辞書にも説明されている．たとえば広辞苑で調べると「経過＝太陽面を内惑星・彗星などの天体が通り過ぎる現象。また、惑星面をその衛星が通過すること。」とある．いうまでもないことと思うが「通過」にはこのような特別な意味は載っていない．理科年表では 1950 年 (昭和 25 年) 版から 1970 年 (昭和 45 年) 版までの 21 年間，「木星の４大衛星の諸現象」というページがあったが，そこ4でも衛星が木星面を横切る現象を「経過」と言っていた．天文年鑑などでは現在でも木星の衛星現象で「経過」が使われている．また，前回の 1874 年 (明治７年) の金星日面経過を伝える当時の新聞記事でも「金星経過」なる語が使われていたことは，東京日日新聞・横浜毎日新聞・郵便報知新聞などの記事で確認できる (その一部は「星ナビ」2004 年６月号 p.79 の雨宮正実氏の投稿記事でも確認できる)．もっとも，当時は「金星過日」なる語も使われていた．
なお，文部省 (現在の文部科学省) と日本天文学会が著作権を所有する学術用語集の天文学編 (増訂版，1994 年刊行，発行所は日本学術振興会，発売所は丸善(株) ) には英語の transit の訳語に (「正中」以外に) 「太陽面通過」を当てているが，これは感心しない． transit の日本語は上で説明した「経過」である．つまり，太陽面だけでなく，木星面などの前面をその衛星などが通過するのも transit である．太陽の前面を通過することを英語で表すとすれば transit over the disk of the Sun のようになるだろう．学術
用語集の編者はこれらの事実を知らずに「太陽面通過」という訳語を transit に当ててしまったようだ．

5 REFERENCES
Ferguson, James, 1761: A plain method of determining the parallax of Venus, by her transit over the sun:and from thence, by analogy, the parallax and distance of the sun, and of all the rest of the planets,
London, Article VI on pp.15-26.
http://books.google.nl/books?id=jJ5bAAAAQAAJ&printsec=frontcover&hl=nl&source=gbs_ge_summar
y_r&cad=0#v=onepage&q&f=false
Halley, Edmond (Edm. Halleio), 1716: “Methodus singularis quâ Solis Parallaxis sive distantia à Terra,
ope Veneris intra Solem conspiciendae, tuto determinare poterit: proposita coram Regia Societate ab
Edm. Halleio J. U. D. ejusdem Societatis Secretario”, Philosophical Transactions of the Royal Society,Vol. 29, pp. 454-464 (in Latin).
http://rstl.royalsocietypublishing.org/content/29/338-350/454.full.pdf+html
Hutton, Charles; Shaw, George; & Pearson, Richard, 1809: “A new Method of determining the Parallax ofthe Sun, or his Distance from the Earth; by Dr. Halley, Sec. R.S. No 348, p.454. Translated from theLatin”, The Philosophical Transactions of the Royal Society of London, from their Commencement, in1665, to the Year 1800; Abridged, with Notes and Biographic Illustrations, Vol. 6, pp.243-249.
http://archive.org/details/philosophicaltra06royarich,
http://eclipse.gsfc.nasa.gov/transit/HalleyParallax.html,
http://en.wikisource.org/wiki/A_New_Method_of_Determining_the_Parallax_of_the_Sun
Luzum, Brian; Capitaine, Nicole; Fienga, Agnès; Folkner, William; Fukushima, Toshio; Hilton, James;
Hohenkerk, Catherine; Krasinsky, George; Petit, Gérard; Pitjeva, Elena; Soffel, Michael; Wallace,Patrick, 2011: “The IAU 2009 system of astronomical constants: the report of the IAU working group on
numerical standards for Fundamental Astronomy”, Celest. Mech. Dyn. Astr., Vol. 110, pp. 293-304.
The constants are also given at: http://maia.usno.navy.mil/NSFA/IAU2009_consts.html
Roode, Steven M. van, 2005: “Halley’s method of durations: Its history and appliance”.
http://www.transitofvenus.nl/halleysmethod.pdf

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