A Transit of Venus Turns the Sun's Parallax, Too Small as an Angle, into Seconds of Time Through Venus's Parallax ── misreading the time by 1 second shifts the parallax by 0.292107 percent, a fineness that would require 0.025688 arcseconds if read as an angle ── the separator is whether the parallax is read as an angle or as a time ── [Paper 806]

Abstract

The astronomical unit is fixed by the solar parallax, an angle of less than 9 arcseconds. Using the geometry of circular orbits, it is counted how a transit of Venus across the Sun magnifies that small angle and turns it into a time difference. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the relation between solar parallax and the astronomical unit, the method of finding the solar parallax from the times of a transit of Venus, and Venus's orbital elements are all known. The setting is idealized──the orbits of Earth and Venus are circles in the same plane. The two places are an Earth radius apart, and only the case where they lie along Venus's motion is considered. Observing difficulties are not treated──the blurring of the moment of contact (the black drop), atmospheric seeing, clock calibration and errors in the observers' positions are left out. The definition of the astronomical unit is not discussed──the present astronomical unit is fixed as a length, and the parallax of 8.794143 arcseconds here is a representative value for comparison. History is not treated──past expeditions and their resulting values are not given. Relation to earlier papers: Paper 297 showed that cosmic distance is a ladder of seven rungs and only the first (trigonometric parallax) is measured directly──this paper stands beside that seat: the first rung's parallax has the astronomical unit as its baseline, and here the mechanism for measuring that baseline's length by time rather than angle is counted. Paper 499 showed that not one of the four classical measurements of the speed of light measures one way, and counted Rømer's delay──turning that delay into a speed needs the length of the astronomical unit. This paper takes the same line in using idealized values and stands beside that seat. What is added is deriving how slight differences in solar parallax move the astronomical unit, deriving the magnification 2.614440 from the difference with Venus's parallax and the angular speed relative to the Sun, 241.777812 arcseconds per hour, by formula and by differences of positions, showing that a shift of 22.991763 arcseconds becomes a time difference of 342.340538 seconds, converting a 1-second misreading into an angular fineness of 0.025688 arcseconds, and placing the separator on reading as an angle or as a time. First, the solar parallax is small, and slight differences move the astronomical unit──149597884.970 km at 8.794143 arcseconds, and +0.047133 percent at 8.79 arcseconds (Section 2). Second, the difference from Venus's parallax magnifies the angle──with Venus at 0.723332 astronomical units, the magnification is 2.614440 (Section 3). Third, Venus moves slowly across the solar disk──241.777812 arcseconds per hour relative to the Sun, the same value from differences of positions on circular orbits (Section 3). Fourth, and this is the core. An angular shift becomes a time difference in seconds──at two places an Earth radius apart, a shift of 22.991763 arcseconds becomes a time difference of 342.340538 seconds (Section 3). Fifth, one second of time is a very fine angle──a 1-second misreading is 0.292107 percent of the parallax, a fineness that would need 0.025688 arcseconds if read as an angle (Section 4). Sixth, the separator is whether the parallax is read as an angle or as a time──"the solar parallax is too small to measure" is not decided true or false until one says whether it is read as an angle or as a time (Section 5). A transit of Venus turns the Sun's parallax, too small to measure as an angle, into a time difference measurable in seconds by magnifying it through Venus's parallax. From a solar parallax of 8.794143 arcseconds the astronomical unit is 149597884.970 km, and at 8.79 arcseconds it moves by +0.047133 percent──differences of 0.01 arcseconds must be told apart in an angle under 9 arcseconds. Venus, only 0.276668 astronomical units from Earth, spreads the parallax difference 2.614440 times, and moving only 241.777812 arcseconds per hour relative to the Sun, it turns a 22.991763-arcsecond shift between places an Earth radius apart into a 342.340538-second time difference. A 1-second misreading is 0.292107 percent of the parallax, which as an angle would be a fineness of 0.025688 arcseconds──nearness spreads the angle, and slowness stretches it into time. The separator is whether the parallax is read as an angle or as a time. Placed among the earlier papers──standing beside the seats of Paper 297's first rung and Paper 499's Rømer delay, the length of the astronomical unit that enters both was counted from the side of measuring by time. To be honest──the orbits were set as circles in one plane, and neither the black drop, the atmosphere, nor clock calibration is treated. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: transit of Venus, solar parallax, astronomical unit, Venus, time difference, angular speed. ----- 天文単位は、太陽の視差という 9 秒角に満たない角度で決まる。金星が太陽の前を横切る日面通過で、その小さな角度がどう拡大され、どう時刻の差に変わるかを、円軌道の幾何で数える。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──太陽の視差と天文単位の関係、日面通過の時刻から太陽の視差を出す方法、金星の軌道要素は、いずれも既知である。理想化する──地球と金星の軌道を円とし、同じ平面にあるとする。二地点のずれは地球の半径ぶんで、金星の進む向きに並んだ場合だけを見る。観測の困難を扱わない──接触の瞬間がにじむ現象(黒い雫)、大気のゆらぎ、時計の較正、観測地点の位置の誤差は入れない。天文単位の定義を論じない──現在の天文単位は長さとして定められており、本稿の視差 8.794143 秒角は比べるための代表値である。歴史を扱わない──過去の観測遠征とその結果の値は書かない。既刊との関係:論文297 は、宇宙の距離が七段のはしごで、直接測られているのは一段目(三角視差)だけであることを示した──本稿はその席の隣に置く。一段目の視差は天文単位を基線にしており、ここではその基線の長さを、角度ではなく時刻で測る仕組みを数える。論文499 は、光速の四つの古典測定に片道が一つも無いことを示し、レーマーの遅れを数えた──その遅れを光速に直すには天文単位の長さが要る。本稿も理想化した値を使う点で同じ線に立ち、その席の隣に置く。加えたのは、太陽の視差の小さな違いが天文単位を動かす量を出したこと、金星との視差の差による拡大 2.614440 と、太陽に対する角速度 241.777812 秒角/時を式と位置の差分で出したこと、22.991763 秒角のずれが 342.340538 秒の時刻の差になることを示したこと、時刻 1 秒の読み違いを角度の細かさ 0.025688 秒角に直したこと、分離子を「角度で読むか、時刻で読むか」に置いたことである。 第一に、太陽の視差は小さく、少しの違いが天文単位を動かす──8.794143 秒角で 149597884.970 km、8.79 秒角なら +0.047133 パーセント(第2節)。 第二に、金星との視差の差で、角度は拡大される──金星の軌道半径 0.723332 天文単位で、拡大は 2.614440 倍(第3節)。 第三に、金星は太陽面の上を、ゆっくり動く──太陽に対して 1 時間に 241.777812 秒角で、円軌道の位置の差分でも同じ値になる(第3節)。 第四に、これが本稿の芯である。角度のずれが、秒の時刻の差になる──地球の半径ぶん離れた二地点で、22.991763 秒角のずれが 342.340538 秒の時刻の差になる(第3節)。 第五に、時刻の 1 秒は、角度の細かさに直すと非常に細かい──時刻 1 秒の読み違いは視差の 0.292107 パーセントで、同じ細かさを角度で読むには 0.025688 秒角が要る(第4節)。 第六に、分離子は、視差を角度で読むか、時刻で読むか──「太陽の視差は小さすぎて測れない」は、角度で読むか時刻で読むかを言うまで正しいかどうかが決まらない(第5節)。 日面通過は、角度で測れない太陽の視差を、金星の視差との差で拡大して、秒で測れる時刻の差に変える。太陽の視差 8.794143 秒角から天文単位は 149597884.970 km で、8.79 秒角なら +0.047133 パーセント動く──9 秒角に満たない角度を、0.01 秒角の差まで読み分けねばならない。金星は地球から 0.276668 天文単位の近さにあって視差の差を 2.614440 倍に拡げ、太陽に対して 1 時間に 241.777812 秒角しか動かないので、地球の半径ぶん離れた二地点の 22.991763 秒角のずれは 342.340538 秒の時刻の差になる。時刻 1 秒の読み違いは視差の 0.292107 パーセントで、角度で読むなら 0.025688 秒角の細かさに当たる──近さが角度を拡げ、遅さが角度を時間に引き伸ばす。分離子は、視差を角度で読むか、時刻で読むかである。既刊との位置──論文297 のはしごの一段目と論文499 のレーマーの遅れに入っている天文単位の長さを、その席の隣に置いて、時刻で測る側から数えた。正直に言えば──軌道は円で同じ平面に置き、黒い雫も大気も時計の較正も扱っていない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:日面通過、太陽の視差、天文単位、金星、時刻の差、角速度。