# Measuring the Angular Radius of the Sun

To get an own measure of the Astronomical Unit by means of the following formula for the Sun's parallax (see the related paper (in German or in English)

you need to determine, among other quantities, the angular radius ρS of the Sun.

By evaluating pictures taken during the transit of Venus (or Mercury, respectively) you get all results relative to the sun's angular radius ρS . For instance, by comparison of simultaneously taken pictures of different observers you get the relative parallax effect f. To be able to derive its absolute value Δβ you must know the angular radius of the Sun:

Δβ = f ρS

Of course, you know ρS roughly (about 15 arcminutes) and you can find its exact value - even for the day of interest. But:

We want to determine ρS by measurements of our own!

There are, at least, two simple possibilities for this measurement:

1. In nature, you often can find round or elliptical so called "sunspots". These spots are caused by the sunlight coming through little holes, for instance between the leaves of big trees:
Sunspots on the ground Light rays causing the spots on the ground
The spots are pinhole camara pictures of the sun! Therefore, the angular radius ρS of the Sun can be determined by measuring the linear radius r of a spot and its distance d to the hole of its origin:
ρS=arctan(r/d) ≈ r/d
2. A more exact method make use of the movement of the projection of the Sun due to the earth's dayly rotation. When you project the Sun on a sheet of paper with a simple optical lens, a binocular or a telescope, respectively, the picture is quite sharply limited.

Having the optical system fixed you will soon remark that the Sun's picture tends to wander over the sheet.

• Determine the speed of this movement. The most exact procedure for that is to draw a circle on the sheet, slightly larger than the projection picture, and to measure the time, which the picture needs to completely leave the circle (from "third contact" t1 to "fourth contact" t2 ):

Start "Third contact": t1 "Fourth contact": t2
• If you know the declination δS of the Sun (see Geographical coordinates) you can calculate the angular speed ω of the Sun:

Therefore, the angular radius of the Sun must be:

While the hour angle increases by Δτ the Sun is moving at the celestial sphere by rcosδSΔτ. The corresponding central angle is, therefore, cosδS Δτ.
The situation of the above picture, but now projected to the plane of meridian.

 Editors: Udo Backhaus last update: 01.04.2012 Stephan Breil