The Astronomical Unit from differential astrometry of the 2004 Transit of Venus?

By Daniel Fischer - 1st version of June 25, 2004
(with a 5% error in Earth's diameter corrected on July 5)

Based on a handful of medium-quality photographs of the full solar disk taken during the 2004 transit of Venus by P. Hombach in Germany and the author in South Africa, a first attempt to derive the AU by relative astrometry to two sunspots in AR 627 is made. Ways to improve the analysis are outlined.

Please point out to me any errors in the procedure you may find!

Most didactical projects during the June 8, 2004, transit of Venus concentrated on recreating the classical experiments performed in 1761, 1769, 1874 and 1882 in which expeditions sent all over the world to time the moments of ingress and egress of Venus' disk. Following Halley's famous recipe the hope was to deduce the distance of Venus from the Earth and - via Kepler's Third Law - the absolute value for the distance from the Earth to the Sun, the Astronomical Unit. As is well known the problem of determining the exact moments of the four contacts turned out to be so severe that the resulting value of the AU, even after four transits, was so unreliable that is was superseded almost immediately by other methods - one of the less glorious moments in the history of astronomy.

The idea behind the Halley method is, of course, to measure the parallax of Venus relative to the solar disk, caused by the different locations of the observers on the Earth: This causes their respective times of ingress and egress to differ a bit. But, I always wondered, wouldn't it be more straightforward to measure the parallax directly, by relative astrometry to features on the solar disk? B. Gährken had arrived at the same idea and wanted to use the short-lived and hard-to-photograph granules of the photosphere as a reference (scroll to the bottom of his website on the 2003 transit of Mercury) but my plan was always different: I wanted to measure the distance of Venus from sunspot umbrae throughout the transit and derive the parallax - and thus the AU - somehow from that dataset. (Still other attempts to get the parallax from photographs are described by Backhaus and Venus2004 and announced by the NSO).

Typical image used in this analysis (this one by P. Hombach, processed by D. Fischer)

The strong downward trend in solar activity since the last big outburst in October/November 2003 was a major concern: Would there be reasonable sunspots on the solar disk at all on June 8? Fortunately an old activity center, AR (10)627, just survived and still presented two pretty compact umbrae on that day - almost in the center of the disk - which I named »A« and »B«. There was another tiny umbra on the solar disk but it doesn't show up consistently on the images I'm working on here and so only the distances between Venus and A and B will be used. The activity region, belonging to the old solar cycle, sat close to the Sun's equator while Venus crossed the disk pretty deep in the South - thus a close appulse of the two would not happen. But for the method of data reduction used below this doesn't really matter. For this first attempt to derive the AU many approximations were made, hopefully reasonable ones. The strategy is as follows: Measuring the images was tedious but straightforward. Both observers had taken color slides; Hombach had supplied his in scanned format. I measured the latter with the cursor tool of Photoshop: The Sun's diameter was always measured twice (in x and y) to make sure the slide scanner had not introduced distortions, Venus and the spots were easy to »hit« with the tool's cross hairs to within a pixel or two. In the scans I had, the Sun had a diameter of 1056.5±2.2 pixels, without any noticeable distortions. My own slides, taken during a unique series of astronomical expeditions to Southern Africa in May and June of 2004 (described in detail in this report), were projected along a long hallway so that the Sun measured 550 mm on the screen; again x-vs.-y measurements showed that no distortions were introduced that way. The distances from Venus to spots A and B were then taken off the screen with a ruler, with a precision of about 1 mm. Thus in both cases the measurements are good to about 3 arc seconds.

8:00 UTC 258 270 12 251 265 14
8:30 UTC 245 262 17 245 261 16
9:00 UTC 250 266 16 258 274 16
9:30 UTC 269 285 16 285 297 12
10:00 UTC 302 314 12 318 331 13
10:30 UTC 336 354 18 358 374 16
11:00 UTC 385 398 13 404 419 15

In this table the projected distances between Venus and spots A and B (»V-A«, »V-B«) for South Africa and Germany (»RSA«, »GER«) as well as the differences are given in promille (1/1000s) of the solar disk diameter. Unfortunately these are all the image pairs between Fischer & Hombach that exist; nonetheless the moment of true parallax, when the measured value of German distance minus South African distance is equal to the true Venus parallax relative to the Sun should be included in the covered time interval (somewhere in the vicinity of 8:30 UTC, near the middle of the transit). As one sees immediately the distances were always significantly larger for the German observer, as is obvious since the transit took place over the far South of the solar disk: Being in South Africa shifted the chord more towards the center of the disk (and also made the transit longer by a few minutes as can be seen from F. Espenak's tables).

Why Delta should - in principle - reach a defined maximum

Determining the true parallax, i.e. the actual angle by which Venus shifts relative to the Sun's disk when one switches the observing location between South Africa and Germany, from this table of measurements is not easy at all: There simply are no clear global maxima of (V-(A,B))GER-(V-(A,B))RSA evident in the GER-RSA columns. Because Venus never came close to either spot, the actual increase and decrease of Delta that must have occured is by and large lost in the noise and cannot be retrieved easily due to the low number of data points. One can, however, conclude from the scatter of the difference values that the true parallax may have been around 15 promille of the solar disk diameter: that value corresponds to some P=28 arc seconds - which is almost exactly half of the diameter of Venus' disk that day. Clearly Venus is not infinitely far away!

Getting the physical distance between us two observers as projected into a plane perpendicular to the Venus-Earth vector (because that line in space is the origin of the parallax) from our geographical coordinates and planetary system ephemerides would involve an awful lot of spherical trigonometry (and possibilities for subtle errors :-), so it will be deferred to a later stage. Instead I used (a 1994 version of) the powerful planetarium software RedShift and had it plot the Earth as seen from the center of the solar system at the crucial times. The distance between the two observing sites was then simply measured off the CRT with a ruler and divided by the diameter of the Earth on the screen, then multiplied with the known true diameter of 12,750 km: Our sites were separated by a projected distance of roughly R=8000 km during these morning hours, as seen from the Sun and/or Venus.

How the parallax P is related to the AU

And here comes »my« AU ... With P and R determined, calculating the AU - with the help of Kepler's Third Law - is rather straightforward when one considers three triangles (see above): It is easy to see that S/V = tan(Q) = R/(E-V) and S/E = tan(P). Furthermore from Kepler we know that (Period of Venus/one Earth year)^(2/3) = V/E =: F. Thus S/(FE) = R/((1-F)E). Substituting S = E*tan(P) we see that tan(P)/F = R/((1-F)E) and thus the Astronomical Unit E can be calculated as E = R/tan(P)*(F/(1-F)). With P = 28 arc sec, R = 8000 km and the period of Venus being 224.7 days it follows that the AU is about 152 million kilometers, a value only 2% off (though the error bars must be substantial). Given the really moderate database - just a few pairs of moderate-quality photographs - and the numerous mathematical shortcuts taken here, one can be pretty happy. For now ...

Where do we go from here? The AU result is very sensitive to both P and R! E.g. if P were just one arc sec larger (remember that the measuring precision was only about 3 arc sec!) and R 200 km smaller, our AU would shrink by 10 million km. Thus in further steps more data sets from both Europe (where some have actually been promised to me) and the Southern hemisphere should be included to obtain better values for P. And for each pairing of locations R should be calculated rigidly, however tough the math may be. Plus the error budget must be handled in detail: Only then will we be in a position to say whether the differential astrometrical method is as good as or - applied with the utmost rigor - even better than the classical Halley technique. And whether it could have saved the day in the 19th century as my gut feeling continues to be ...