# The Determination of the Distance to the Sun

## Basic Idea of the Contact Time Method

During a transit of Venus, four kinds of contact between Venus and the circular solar edge happen:

External contacts happen at points 1 and 4, internal contacts at points 2 and 3. Presumably, the internal contacts can be observed most easily. We call the corresponding moments contact times.

The exact contact times depend on our geographical position. The difference between the contact times measured at different locations offer the possibility of determining the distance to the Sun. In addition to the exactly measured contact times, only the geographical coordinates of the mutual observers must be known. Therefore, for the method proposed here only simple equipment is needed:

• a precise clock,
• a map which allows to read the coordinates of the own site and,
• for observing the Sun during the transit, a binocular or telescope with a magnification factor of at least 30. (Definitely, look at the security hints!!).

For the calculation of the distance to the Sun, measured contact times from at least two different sites are needed. However, for a satisfying result, the sites should be chosen so that the measured time difference is as large as possible. With respect to a certain direction, these sites must be widely seperated. Therefore, the participating schools are forced to international cooperation.

The cooperation platform which we use in this project offers the possibility of publishing and exchanging the observational results. You will find a suitable form for publishing your own results in the folder "Forum". The data in that form which you should send as soon as possible after your observation, will automatically be put into a list which will be accessible via the forum. In this way, the participating schools will find observation data suitable for comparison with the own measures. For the evaluation, we offer simple to be used programs.

## Foundation of the method

For the fact that different contact times will be measured at different sites, the geometric relations between Sun, Venus and Earth during the transit are deciding, because such a transit only occurs when Venus, on its path around the Sun, passes exactly through Sun and Earth. In this moment, the shadow of Venus strikes the Earth and, for an earthly observer, covers a certain part of the solar disc. To be more precise, it is the penumbra of Venus which, with increasing distance, becomes wider and less intensive. When this penumbra strikes the Earth it is already so weak that we register the corresponding "eclipse" only by observing the small disc of Venus on the solar disc. The little decrease of the sun's brightness caused by this passage, nowadays is one of the most successful method for the search for extrasolar planets.

Geometrically, the shadow of Venus is of conical shape. Therefore, in the distance of the Earth's path, a circular shadow emerges (in the above picture with a purple edge). In reality, its radius is more than 40-times as large as the earth's radius. Only observers within that circle can see Venus in front of the Sun. This cone of shadow takes, as Venus itself, 225 days to orbit the Sun. Because the Earth takes 365 days the shadow will overtake it during the transit.

Because of the inclination of Venus' orbit with respect to the plane of the earth's orbit the geometric circumstances become more complicated. For simplification reasons, we for the moment neglect that inclination and suppose Venus and Earth orbiting in the same plane. In this case, each time Venus is overtaking the Earth Venus can be observed moving over the solar disc on a central path. An observer will register the first external contact in the instant the edge of the circulating shadow is reaching him.

The first external contact of the transit will be firstly observed from that site at which the cone attains the Earth. For all other observers, the transit has not commenced at this moment.

While the shadow covers the Earth more and more its edge overtakes all observers on the daylight side of the Earth. At all sites which are striked by the shadow simultaneously the external contact is observed at the same instant. On the other side, observers which are seperated in the direction of the shadow's movement will measure different contact times.

The above picture illustrates that the same conclusions can be made for internal contacts: All observers which measure the internal contact at the same time are located on the surface of a cone whose apex coincides with that of Venus' umbra. Because of the simultaneous motion of both cones the same arguments can be applied for declaring different contact times.

According to the above picture, the first external contact will be observed at D later than at C. The following approximation shows how to calculate the distance to the Sun within this simplified model. It is the distance which has to be determined:

• Because of the large distance, the triangles and are similar.
• is the Earth's radius.
• is the difference between the contact times measured at C and D.
• The synodial period of Venus can be derived from the sideral periodes of the Earth and Venus .
• During the time interval Venus will move by .

From the premises made in this simplified model, a time difference of about 5.5 minutes follows between the points C and D.

## Real Conditions

In reality, the angles of interest and their differences are very small: At a certain location, an external contact differs from the corresponding internal contact by about 20 minutes while the whole transit takes about six hours.

As the above picture which is in proportion to the actual dimensions ilustrates the cone of Venus' shadow is as slim as to be be moved around the Sun like a pointer which always passes the same angle as Venus itself.

This pointer offers an excellent possibility to determine the distance to the Sun. For centuries, the distances in the solar system were known only in relation to each other because, at the sky, only angles can be measured but not the corresponding arcs. During the time interval which the edge of the cone takes to pass the Earth the arc is formed by the diameter of the Earth . However, the corresponding angle can be calculated because the (synodial) period of Venus is known. Therefore, the corresponding radius (that is the distance between Sun and Earth) can be calculated:

Unfortunately, the real conditions during the transit are not as simple as stated in the above simplified model.

1. Because of the inclination of Venus' orbit the distance between the solar centre and trail of Venus' passing the solar disc depends on the position of the orbital nodes of Venus.
2. Because of the inclination of Venus' orbit the motions of the shadow cone and the Earth do not move in parallel directions.
3. The Earth continues to rotate while the edge of the shadow is overtaking her.

In 2004, the transit will happen in the southern part of the solar disc. Accordingly, the Earth will cross the northern part of the shadow cone.

As a consequence, the cone's direction of movement and the direction of the shadow's edge moving across the Earth are not perpendicular. Therefore, it takes more time for the shadow's edge to cross the Earth.

The above picture shows the daylight side of the Earth on June 8th shortly before the shadow will reach it. The white lines connect sites from which the first (external or internal) contact will be observed simultaneously. Between two neighbored lines, the contact times change by one minute.

For comparison with own measured contact times for the ingress of Venus (points 1 and 2) reference data should be looked for differing from the own ones as much as possible. Therefore, you should look for sites whose position differs from your own by as many white lines as possible. The combination of measures from Australia and Europe should be optimal.

The same arguments can be applied to the observation of Venus' egress (points 3 and 4). The above picture shows the daylight side of the Earth on June 8th, 2004 shortly after the end of the transit. Obviously, contact times measured in Russia and Brasilia will form the best combination to derive the distance to the Sun.

written by Wilfried Suhr
last modification by Udo Backhaus: 02/09/2012