Star Stuff: Determining the Astronomical Unit from Mercury's Transit and Period

Acknowledgements to Eric and Daniel

We will determine the astronomical unit (the distance between the Earth and Sun) using Mercury's period with a time series of images of the transit of Mercury across the Sun taken by NASA's TRACE satellite.

In the image above, the blue circle is the Earth, the grey circle is Mercury, the yellow circle is the Sun, and the pink circle is TRACE. The red lines are drawn from TRACE (at the two different points shown), through the center of Mercury, to the surface of the sun. The green lines are drawn from TRACE to the apparent center at the surface of the sun. The white line divides the diagram symmetrically. Three angles are labelled (α, β, and ɵ), as well as the distance between Earth and Mercury, Δa.

Using basic geometry, we can state the following:
ɵ = α + β

Using trigonometry, this becomes
(Earth Radius)/Δa = α + (Earth Radius)/a (a being the astronomical unit)
assuming that the distance between TRACE and the surface of the Earth is negligible.

This can be simplified to the following:
*α=(Earth Radius)(1/Δa - 1/a)

Looking at the image of the transit above, we can see Mercury crossing with a sinusoidal path. This is the result of parallax, given TRACE's polar orbit. Geometrically, we can state that the ratio between this path's amplitude and the radius of the sun is equal to the ratio between our angle α and the angular diameter of the Sun (about half of a degree).

After gathering extremely precise measurements of the aforementioned amplitude relative to the radius, we can calculate α = 4e-5 radians.

In addition, using the relationship between the period of a planet and the distance from the Sun,