The observations of the last transit of Venus, made at different points of the earth, may be used to determine the sun’s mean equatorial parallax, a measure for its distance from the earth. The calculator below enables you to compute the mean equatorial solar parallax online from your own and others’ observations of the 2004 transit of Venus, employing either Halley’s or Delisle’s method.

The methods of Halley and Delisle

In his famous proposal submitted to the Royal Society in 1716, Edmond Halley explained how such a calculation could be performed. Because of the effect of parallax and the earth’s diurnal rotation, the duration of the transit of Venus, observed from two widely separated places, will differ from each other by a small amount of time. If this observed difference is found to be greater or less than the difference obtained theoretically from an assumed value of the solar parallax, then, according to Halley, the sun’s parallax will be greater or less in the same proportion.

Halley provided for an explanatory though inaccurate geometric construct to arrive at the transit’s duration from an assumed value of the solar parallax. The main principle of the calculation already set out and warmly recommended by Halley, the primarily work left to astronomers of a next generation therefore was to find the duration of the transit more accurately from theory.
On the occasion of the 1753 transit of Mercury, French astronomer Joseph-Nicolas Delisle noted that Halley’s reasoning with regard to the proportionality between the solar parallax and the duration of the transit could equally be applied to the moments of interior contact at either ingress or egress. The advantage of this alternative, which now bears Delisle’s name, was that observations from places where the transit would only be partially visible, could also be used to establish the sun’s distance, thus increasing the number of potential observing sites.
Enabled by the continuous development of new astronomical techniques, different and better ways came up in course of the nineteenth century to arrive at the sun’s distance using recordings of the transits of Venus. This web page, however, concentrates on the original methods proposed by Halley and Delisle, featuring the renowned online Parallax Calculator.

Learn more...
François Mignard, ‘The Solar parallax with the transit of Venus’. Observatoire de la Côte d’Azur (December 22, 2004)
Steven M. van Roode, ‘Halley’s method of durations – its history and appliance’. (August 23, 2005)
Robert Stawell Ball, ‘On the transit of a planet across the sun’. (1908)
NASA Connect Video, ‘The Venus Transit’. (May 19, 2004)

The Parallax Calculator

The calculator below enables you to compute the mean equatorial solar parallax online from your own and others’ observations of the 2004 transit of Venus, employing either Halley’s or Delisle’s method. The theoretical difference between the times of contact is computed using the iterative algorithm and Besselian elements provided by Jean Meeus in his book Transits (Willmann-Bell, 1989).

When you enter the longitude and latitude of each observer in degrees and decimals, please consider east longitudes and north latitudes as positive. The observed times of interior contact at ingress and egress, which you also must enter, should have the format hours.minutes.seconds, i.e. separated by dots. If desired, you may give the seconds any number of decimals, depending on the accuracy of your observation. If you are using Halley’s method, you may enter local times. Delisle’s method, however, requires that the times entered are in Universal Time. Also, if you check Delisle’s method, the times of contact at the not observed phases should be zero. Finally, clicking the compute button will yield the mean solar parallax expressed in seconds of arc and the associated mean distance of the sun from the earth expressed in kilometres.

For comparison, the values of the mean distance and parallax of the sun, adopted by the International Astronomical Union, are 149597870 km and 8″.794 respectively. The mean distance of the sun from the earth, being the yardstick for measuring other astronomical quantities, is aptly called the astronomical unit. Owing to the elliptical shape of the earth’s orbit, the momentary distance of the sun may be greater or smaller than one astronomical unit. On the day of the 2004 transit of Venus, the sun’s distance was 1.01507 AU. The indicated error is based upon an assumed uncertainty of about ten seconds in the recorded time differences, which is, to my own experience, a reasonable estimate. Other observers, however, found the sudden break between the black drop and a clear gap of sunlight very apparent, and were very certain that an error of ten seconds was too much.

This diagram shows the solar parallax obtained by applying Halley’s method to 42 separate pairs of submitted observations.

Results of the 2004 transit

From the contact timings submitted so far, which are listed in the calculator, a preliminary solar parallax may be computed by pairing complete observations of second and third contact. If only timings from widely separated locations are paired and combinations of neighbouring locations are accordingly disregarded, a total number of 42 matching observations is found. Subsequently, applying Halley’s method to these combinations and averaging the results, a solar parallax of 8''.538 is found, corresponding to a sun’s distance of 154084980 km. This compares to the true value of the mean solar parallax, the difference being only 2.9%.

It’s striking that three-quarters of the paired observations yield a solar parallax being significantly smaller than the true value. This indicates that somehow the observations are biased by the black drop effect. However, as the number of usable observations increases, I expect that the error will diminish and the resulting solar parallax will tend to the true value of 8″.794. Therefore, it’s important to submit your own observational data.

Links

The website of Chuck Bueter provides an extensive collection of links on different subjects related to the transit of Venus.
The website of Robert van Gent provides an extensive bibliography of original sources relating to transit of Venus, with links to many of the original publications.
David Sellers gives an explanation of the technique for measuring the sun’s distance.
Jürgen Giesen has a similar parallax calculator, which is based on a method by Patrick Rocher.

Acknowledgments

I wish to thank Bert Browne, David Sellers, Glenn Schneider and Jean Meeus for thinking along with me about the appliance of Halley’s method. In addition, I’m grateful to Andrew Norton for drawing my attention to a small (now fixed) bug in my parallax calculator, and to Paul Prideaux for pointing out a shortcoming in the calculator’s instructions. The Javascript code, forming the base of the parallax calculator, is by courtesy of Franco Martinelli. Finally, I wish to acknowledge the contributions of all observers who were willing to share their timings and thereby making this calculator of great value to all of us.