# The Determination of the Distance to the Sun

## Basic idea for the evaluation of twice exposed photos

In the case of (nowadays only actual method of) digital photographing twice exposition can be replaced by combinating subsequently taken photos. Proposals for this method are made on an additional page.

 Determination of the distance to the Sun Measurement of Venus' actual parallax by means of twice exposed photos Exposing pictures twice with fixed camera Example for the superposition procedure

For two observers at different sites on Earth a transit looks 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 two simultaneously taken photos are scaled to the same size and merged with exactly the same orientation (this project),
• if the duration of the transits as measured from different sites are compared or
• if the exact moments of ingress or of egress are compared (see the contact times project).

This document shortly explanes the basic idea of the first method. The details are described in an additional paper.

### Determination of the distance to the Sun by measuring the parallactic displacement of Venus due to two distant 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 a very small angle. 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 separated 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 βS, 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, is 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 βS 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.

### 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. For this comparison, 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!

Exposing pictures twice with a camera fixed on a tripod is one possibility of solving that problem: If you expose the same negativ (or two digital pictures) twice within an interval of 120 seconds 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 indicates the exact direction from east to west!

Proposals for multiple exposures of photos are made on the photographing page.

### Example

The procedure of merging two simultaneously taken twice exposed pictures may be visualized by pictures of the 2004 transit which have been taken by E. v. Grumbkow in Namibia (right, Δt=120s) and by U. Backhaus in Essen, Germany (Δt=150s):

1. The pictures of the Sun are differently scaled because both optical systems had different focal length. Furthermore, the both cameras have been differently oriented.
2. In the first step of the procedure the pictures have been rotated so that the E-W-axis points exactly to the right.
3. After the Namibia picture has been rescaled in the second step and
4. the pictures have been cut so that the positions of the respective first images of the Sun coincide exactly
5. the pictures can be superposed.
The result of this procedure is a single picture of the Sun with two images of Venus whose displacement visualizes the parallactic effect.  Editors: Udo Backhaus last update: 22.05.2012 Stephan Breil 