Method of Contact Times: Basic Idea

Principle

The method of determining the distance to the Sun by measuring differing contact times at distant sites is based on the following idea: Since Kopernikus all angles and angular velocities in the solar system could be calculated. But to none of these angles the opposite length was known. If for one planet only the arc length due to the central angle covered by its motion during a certain time interval could be measured, its distance from the Sun could easily be calculated - and the size of our solar system would no longer be a secret. Figure 1: The edge of Venus' shadow overtakes the Earth.
(Shadow of Venus, a little program for visualizing the process)

During a transit the shadow of Venus encounters the Earth. While its edge overtakes the Earth (figures 1a and 1b) both planets cover certain central angles εE and εV on their orbits around the Sun. In figure 1c εV is marked with red color. If one can determine the time Δt the shadow's edge needs to overtake the Earth these angles can be calculated without knowing the absolute distances:

εVVΔt, εEEΔt

where ωV and ωE are the known angular velocities of Venus and Earth.

During that time interval, the diameter of the Earth is the arc length to the relative angle εVE. That means: The measurement of Δt will give us a known angle and the corresponding arc length and we will be able to calculate the distance between Earth and the Sun, the so called Astronomical Unit AU:

1AU = 2RE/(εVE) = 2RE/(ωVΔt-ωEΔt)

In this equation RE is the radius of the Earth which can be determined by own measurements (see the corresponding project).

Refinement

The above considerations must be refined for two reasons:

1. The entire time interval Δt can not be measured directly because the corresponding points on Earth will generally not be reachable. Additionally, observers at these places would see the Sun directly at the horizon!
2. Venus' plane of orbit is declined with respect to the ecliptic. Therefore, the shadow doesn't cover the Earth centrally (see figure 2; the corresponding pictures of the egress are shown in a little egress page) and the correlation between Δt and the angular velocities is more complicated.
Problem 2 is treated in the mathematical details. A simplified solution of the first problem may be outlined here (the details can be found in the mathematical details, too):

a) b) Figure 2: At the begin of the transit the shadow of Venus covers the Earth peripherally. The lines demonstrate the progress from minute to minute. A little movie of this run is also available.

We make the following idealisations:
• The shadow's movement over the Earth is uniform.
• The edge of the shadow is linear. (The diameter of the circular shadow is much larger than that of the Earth (by about the factor 40, but the exact value cannot be calculated without the knowledge of the AU.)
• We neglect the Earth's rotation while the shadow's edge is overtaking the Earth (Δt≈15min).
With these assumptions there exists a linear correlation between the coordinates of the sites on Earth in fig. 2b and the times they are hit by the shadow. In order to simplify this relation we rotate figure 2b so that the shadow's edge moves horizontally from right to left (figure 3). Figure 3: Figure 2b has been so rotated that Venus' shadow moves horizontally.

In the view of figure 3, only the (horizontal) x-coordinate of the shadow's edge changes linearily with time. Between the x-coordinates of the sites and the times they are hit by the shadow, therefore, the following equation holds:

x-RE=vsh(t-t1)

In this equation vsh isthe velocity of the shadow's horizontal movement>, t the time at which the contact is observed at a location with the coordinate x and t1 the (unknown!) time of the first contact between Earth and the shadow.

Evaluation

In order to determine Δ t=t2-t1 we need at least two time observations at distant sites (with different x-coordinates). With these data we can determine the parameters of the above equation and calculate Δ t by extrapolating it to x2=-RE and x1=RE.

To minimize observational errors it is, of course, better to measure the "eclipse times" at as many sites as possible and find the best linear fit to all of these data. The picture below (fig. 4) shows the accordant diagram with the observational data of the large 2004 transit project organized by the European Southern Observatory. Figure 4: ESO project observations of the second contact 2004 and their linearisation

On our data transfer pages we offer the possibility to upload own time measurements and to combine them with those of other participants for deriving an "own" measure for the Astronomical Unit. In order to simplify the necessary calculations we offer work sheets on our stuff page to fill in the observational data of contact 1+2 and contact 3+4 and to get measures of the AU.  Editors: Udo Backhaus last update: 01.06.2012 Stephan Breil 