Method of Contact Times: Mathematical Details

The equation

1AU = 2RE/(εVE) = 2RE/(ωVΔt-ωEΔt) (1)

of the Basic Idea is only valid when the shadow meets the Earth centrally. In the following we will show how to take into account the inclination of Venus' path.

Refinements

The inclination of Venus' movement

Changing the viewpoint one sees the Earth crossing the shadow of Venus decentrally (Fig. 1). Without knowing the AU the following values can be calculated by using Kepler's laws:

The relative sizes of Earth and Venus' shadow, however, are unknown until the AU has been determined! Therefore, figure 1 only serves as an illustration.


Figure 1: Calculation of the relation between Δtges, the parameter p and the radii of the Earth and the shadow of Venus

But the relative sizes can be determined when the time Δtges the Earth needs to move from position 1 to position 2 can be measured. In this time the Earth enters the shadow and contact 1 (or contact 2) is observed all over the Earth.

(2)

(3)

(4)

Because the Earth is much smaller than the shadow (this already was known!) the following relations hold approximately:

(5)

We therefore get

(5)

and we have derived the Earth's radius as a fraction of AU, that means the solar parallax has been determined:

(6)

This equation is the generalization of equation (1).

Linearisation

The time interval Δtges can not be measured directly because at the appropriate places the Sun just sets or rises. Additionally, the combination of two measurements only would not be exact enough. Therefore, the contact times have to be measured at as many sites as possible.

The relation between the geographical coordinates of the observers and the contact times measured by them is quite complicated. In order to linearize this relation we make the following simplifications:

  1. We neglect the Earth's rotation during Δ tges (≈15min).
  2. We suppose the shadow's edge to be linear (RShadow≈40RE).

Rotation of the coordinates

As first step we transform the geographical coordinates (φ,&lamba;) of the observers to rectangular coordinates (x,y,z):

The z-axis points to the noth pole of the Earth, the x-axis to the longitude of Greenwich. In the following figures the x-axis points to the right and the z-axis upwards. Figure 2 shows left the "initial" position of the Earth - its daylight side at 18.00 UT.

"Basic" position of the Earth In this orientation Venus' shadow moves from right to left.
Figure 2: Start and final position of the Earth

If we transform the coordinates so that the edge of the shadow moves exactly from right to left (fig. 2, right) then a linear relation between the (transformed) x-coordinates of the observers and the contact times measured by them will hold.

Rotation 1

We take the initial position of the Earth as its position at 18.00 UT on the first day of spring (March 20th). The first rotations turns the Earth around its z-axis according to the time Δttr=jd-jd0 until the moment of transit.

This rotation can be written in the following form:

Here we use the following notation: Dx(α) (Dy(β), Dz(γ)) is the matrix for rotating around the x-, (y-, z-) axis by the angle α (β, γ).

jd and jd0 are the Julian dates of the transit and the start time. The factor 1.002738 takes into account that the Earth's period of rotation is shorter than one day.

The result of this rotation is shown in figure 3.1.

Rotation 2

The second rotation around the y-axis turns the pole axis into its position on March 20th (as seen from the Sun).

(ε: obliquity of the ecliptic) The result of the first two rotations is shown in figure 3.2.

Rotation 3

Until transit day the north pole of the Earth turns - according to the Earth's movement arount the Sun - more and more into the direction to the Sun. That means the Earth must be rotated around the z-axis by the geocentric ecliptic length λSun of the Sun:

The result of these three rotations is shown in figure 3.3.

Rotation 4

Before the 2012 transit, the ecliptic latitude of Venus βV is positive and its ecliptical longitude λVis smaller than that of the Earth λE. According to these facts, in figure 3.3 the shadow would appear in the upper right edge of the picture. Therefore, the Earth must finally be rotated around the y-axis by the angle

As the result of all four rotations (fig. 3.4) the Earth looks like in figure 2 right!

1) D=D1=Dz((jd-jd0)*1.002738*2π) 2) D=D2D1=Dy(ε)D1
3) D=D3D2D1=Dz(-λSun)D2D1 4) D=D4D3D2D1=Dy(α)D3D2D1
Figure 3: Four rotations of the coordinates

In a summary the appropriate transformation can be written in the following form:

Concretisation

Parameters of contact 1/2

For June 5th, 22.14 UT, you can find or calculate the following values:

First day of Spring:March 20th
Δ ttr=77.218056d
ε=23.4°
λSun=75.3°
α=44.7°

With these values the matrix D can be calculated:

The transformation of the geographical coordinates, therefore, is:

For the determination of the solar parallax the following values had to be calculated, additionally:

RShadow=0.001850AU
p=0.5904RShadow
v=0.000471AU/h

Parameters of contact 3/4

For June 6th, 5.345 UT, you can find or calculate the following values:

Δ ttr=77.482292d
ε=23.4°
λSun=75.9°
α=27.7°

With these values the matrix D can be calculated:

The transformation of the geographical coordinates, therefore, is:

Realization with a worksheet

The algorithm described here has been realized in the worksheets evaluationofcontacttimes1+2.xls and evaluationofcontacttimes3+4.xls. The parameters of the rotation matrix are contained in cells B5-B7 of sheet "calculation", the other three parameters in cells E5-E7.

The calculation of the solar parallax and the distance to the Sun are done in rows 19-21. For the slope Δtges/(2RE) the quotient Δt/Δx built from two single observations or the slope of the fit line is taken.
Editors: Udo Backhaus
 last update: 04.09.2012
Stephan Breil