Untitled

A) Background

1) Parallaxes of the Sun (the horizontal parallaxes)

By definition the parallaxes of the Sun is the angle b shown below :

By trigonometry, sin b = R / r but the angle b is small and sin b can be approximated by b measured in radians. R is the radius of Earth and r is the distance from the observer to the object. We can get r using the relationship

r = R/ b

2) Observation from the Earth

Let's consider two observers on the Earth situated in points A and B on the same longitude (meridian), but at very different latitudes. The alignment of AB should be approximately perpendicular to the line of sight to the transit so as to keep the errors as small as possible. Venus is seen as a small disk on the face of the Sun at two different points A' and B'. This is because the lines of sight of A and B towards Venus are not identical.

Putting the two observations together using the reference stars it is possible to measure this parallax displacement.

3) A Geometrical Problem

Let's consider the plane defined by three points: the Earth's centre O, the Sun's centre C and the Venus centre V.

The triangles APV and BPC have the same external angles at P, hence

bv + b1 = bs + b2

bv - bs = b2 - b1 = Db

Where angle Db measures the distance between the different positions of Venus's trace on the face of the Sun. Rearranging the last equation gives

Db = bs (bv/ bs - 1)

Now Venus's parallax is bv = AB/(r_e- r_v) and the Sun's parallax bs = AB/r_e, hence the quotient bv/ bs = r_e/(r_e- r_v).
Substituting this into the equation above gives

Db = bs (r_e/(r_e- r_v) - 1) = bs r_v/(r_e- r_v)

In particular, we can get the solar parallax,

bs = Db (r_e/ r_v - 1)

Note that in order to measure Db it is necessary to superimpose the two Sun centres at C and then Db is the distance between the two traces of Venus observed at same time from A and B.

4) Kepler's Third Law

Let's take r_e as the Earth-Sun distance and r_v the Venus-Sun distance. We can calculate the ratio (r_v/ r_e)³ by using Kepler's Third Law as we know that the periods of revolution of Venus and of the Earth are 224.7 days and 365.25 days respectively.

(r_e/ r_v)³ = (365.25/224.7)²

therefore

r_e/ r_v = 1.38248

5) Final formulae for the Earth-Sun distance.

Using this result in the parallax formula from section 3, we get

bs = Db((r_e/ r_v) - 1)= Db (1.38248 - 1)

therefore

bs = 0.38248 Db

And finally using the parallax formula from section (1), the distance from the Earth to the Sun r_e is

r_e= AB/ bs

So we need to find AB from the location of the observers and to measure Db from observational data of the transit.