**1AU = 2R _{E}/(ε_{V}-ε_{E}) = 2R_{E}/(ω_{V}Δt-ω_{E}Δt) (1)**

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 radius R
_{Shadow}of the shadow in the Earth's distance from the Sun (in AU), - the "impact parameter" p (in AU) and
- the speed v of the Earth relatively to the shadow (in AU/min).

Figure 1: Calculation of the relation between Δt_{ges}, the parameter p and the radii of the Earth and the shadow of Venus

**(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).

The time interval Δt_{ges} 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:

- We neglect the Earth's rotation during Δ t
_{ges}(≈15min). - We suppose the shadow's edge to be linear (R
_{Shadow}≈40R_{E}).

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.

We take the initial position of the Earth as its position at 18.00 UT on the first day of spring (March 20^{th}). The first rotations turns the Earth around its z-axis according to the time Δt_{tr}=jd-jd_{0} until the moment of transit.

This rotation can be written in the following form:

Here we use the following notation: **D**_{x}(α) (**D**_{y}(β), **D**_{z}(γ)) is the matrix for rotating around the x-, (y-, z-) axis by the angle α (β, γ).

jd and jd_{0} 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.

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

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 λ_{Sun} of the Sun:

Before the 2012 transit, the ecliptic latitude of Venus β_{V} is positive and its ecliptical longitude λ_{V}is smaller than that of the Earth λ_{E}. 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

First day of Spring | : | March 20^{th} |

Δ t_{tr} | = | 77.218056d |

ε | = | 23.4° |

λ_{Sun} | = | 75.3° |

α | = | 44.7° |

R_{Shadow} | = | 0.001850AU |

p | = | 0.5904R_{Shadow} |

v | = | 0.000471AU/h |

Δ t_{tr} | = | 77.482292d |

ε | = | 23.4° |

λ_{Sun} | = | 75.9° |

α | = | 27.7° |

The mathematical details are fully described in the German paper Ableitung der Sonnenentfernung durch Kontaktzeitmessungen beim Venustransit.

Editors: | Udo Backhaus |
last update: last update: 2020-03-06 | |||

Stephan Breil |