The Determination of the Distance to the Sun

by using a Transit of Venus

Basic Idea for the Evaluation of twice exposed Photos

For two observers at different sites on Earth a transit lookes slightly different: Venus enters the Sun's disc at different times and doesn't leave it simultaneously. And, taken at the same moment, Venus' position in front of the Sun is not quite the same. This parallactic effect can be observed
  • if the duration of the transits as measured from different sites are compared,
  • if the exact moments of ingress or of egress are compared or
  • if two simultaneously taken photos are scaled to the same size and merged with exactly the same orientation.
This document shortly explanes the basic idea of the last method. The details are described in an additional paper.
  1. Determination of the distance to the Sun by measuring the parallactic displacement of Venus due to two sites on Earth
    1. The solar parallax πS is the angle by which the Earth's radius appears when observed from the Sun.

      Obviously, the following equation holds: sinπS = RE/dS or πS = RE/dS because πS is very small. Therefore, if the solar parallax is known the distance to the Sun dS can be calculated by means of the following equation:

      Precondition: The Earth's radius RE is known.

    2. The solar parallax can not be measured directly because nobody can observe the Sun from the Earth's centre. Instead, the Sun's position is measured with respect to the stars from two sites on Earth which are seperated as far as possible. The difference between both positions is the Sun's actual angle of parallax βS. That is the angle by which the distance between both sites appears from the Sun.

      From the parallactic angle, the solar parallax πS can be derived.

      Precondition: The linear distance Δ between the two sites is known.

    3. The Sun is very far away, its parallax, therefore, very small. For this reason, it is easier to measure the parallactic angle β of a nearer object. When passing the Sun's disc, Venus is much closer to the Earth than the Sun. Additionally, its position can easily be determined.

      The parallactic angle of the Sun can be derived from the actual parallax βV of Venus.

      Precondition: The ratio between the distances to Venus and to the Sun is known.

    4. In general, the line connecting both sites will not be perpendicular to the direction to the Sun.

      In that case, not the distance between the sites itself but its projection parallel to the direction to the Sun is of interest. The length of this projection can be calculated.

      Precondition: The angle of projection is known.

  2. Measurement of Venus' actual parallax by means of twice exposed photos

    The parallactic angle is to be measured by comparison of two photos which have been taken simultaneously at different sites on Earth. In order to do that the photos must be adjusted and compared:

    1. In general, the photos will have different scales and different orientations, as well.

    2. After having determined the Sun's radii on both pictures, it is possible to magnify (with a computer) one of the pictures so that the scales are the same:

    3. If then one of the pictures is rotated so that both have the same orientation (e.g. north above),
      then it will be possible

    4. to superpose the pictures so that the parallax of Venus becomes observable and measurable.

      However, it is difficult to determine the orientation of those pictures with sufficient accuracy!

    5. Exposing pictures twice with fixed camera is one possibility of solving that problem: If you expose the same negativ (or two digital pictures) twice within an interval of about two minutes the superposition will show two solar discs with nearly no intersection. This fact is due to the daily rotation of the Earth which causes the Sun apparently to move over the sky - exactly from east to west, parallel to the celestial equator!

      Therefore, the direction of the solar displacement shows us the exact direction from east to west!

    Proposals for multiple exposures of photos are made in another place.

    Udo Backhaus
    last modification:  March 28th, 2008