The laws of Kepler thus describe the orbits of the solar system except for a scale factor. The observation of the periods of revolution of planets gives us the ratio of the semi-major axis, thus the ratio of the semi-major axis of the orbits of Venus and of the Earth is equal to :

      ⁽⁶⁾

and at any date T, the ratio of the radius vectors is equal to

      ⁽⁷⁾

Thus the laws of Kepler make it possible to calculate the ratio of the radius vectors for any date T.

Our measurement allows us to calculate the value p_S, it is thus appropriate now to pass from this value to the value of the mean equatorial parallax of the Sun p₀ .

Calculation of the mean parallax of the Sun

The mean equatorial parallax of the Sun p₀ is by definition the angle under which one sees the equatorial radius of the Earth from the center of the Sun when the Sun is at one astronomical unit from the Earth.

We have thus the following relation :

      ⁽⁸⁾

R being the terrestrial equatorial radius and a the astronomical unit.

However the equation (1) gives us the value of the solar parallaxp_S according to the distance R _T Earth-Sun and of projection d of the distance between the points of observations on the normal plane to the direction Earth-Sun.

It is enough to express this distance D in terrestrial radius and the distance Earth-Sun in astronomical unit to have a relation between p_S and p₀.

      ⁽⁹⁾

It remains only to calculate the D on R ratio . The a/R_T ratio is provided to us by the first law of Kepler (cf formulae 4). However if one makes the vector product of the two vectors and one obtains :

      ⁽¹⁰⁾

However the product of the length of the first vector by the sine of the angle between the two vectors is equal to the distance d. In the same way the length of is equal to the distance R_T

Figure 5. - Solar parallax related to the points M₁ and M₂.

The resolution of equation 10 gives us the value of d.

      _⁽¹¹⁾

Note : if the concept of vector product is not known, one can use the scalar product of the same vectors, that allows to calculate the cosine of the angle, then his sine using the relation : .

Description of this calculation :

This calculation on the vectors requires to know the Cartesian co-ordinates of the two points M₁ and M₂ and of the center of the Sun C in a orthonormed reference frame (O, x, y, z) centered on the center of the Earth. We will use the geocentric apparent equatorial reference frame for this calculation.

This reference frame is defined by the plan of the terrestrial equator at the date T of the observation (plane Oxy) and by the direction of the northern celestial pole of the axis of rotation of Earth (Oz). In this reference frame one can define a Cartesian reference frame (x, y, z) and a polar reference frame (a, d, R) the two angles having the name of right ascension and declination. One passes from one system to the other by the following relations :

      ⁽¹²⁾

and the relations opposite

      _⁽¹³⁾

The direction of axis Ox at the date T is the direction of the vernal equinox at the same date.

The ephemerides (i.e. the Kepler's laws) provide us the equatorial geocentric coordinates of the center of the Sun (a, d); thedistance is not know but it is not important

because the vector may be replaced by its unit vector in equation 11.

The more complicated problem is the determination of the cartesian coordinates of the points M₁ et M₂ in this equatorial frame.

Figure 6. – Geocentric equatorial coordinates.

The position of a point at the surface of the Earth is given by its latitude and its longitude (geographic); the latitude is given referred to the terrestrial equator, so that it is an angle such as the declination. The longitude is given referred to a meridian origin (Greenwich meridian), so that it is an angle similar to the right ascension, but with an origin different from the one of the equatorial lestial coordinates. It is then necessary to know at each date the angle between the direction of Ox axis and the direction of the projection of the meridian origin in the equator plane (cf. figure 6). This angle is related to the rotation of the Earth: it is named the "sidereal time" of the Greenwich meridian and its increases by 360° during 23h 56m 4s (sidereal revolution of the Earth).

So, it is sufficient to know the sidereal time of Greenwich T_G at 0h UTC for the day of the transit to know the sidereal time of Greenwich at any time t , then the sidereal time of any point on Earth having a longitude l.

      ⁽¹⁴⁾

One will pass from the sideral Greenwich time to the sidereal time of the site M having the longitude l, by adding or substracting this longitude.

Attention, sidereal time increases when going towards east from of the meridian
of Greenwich; it is thus appropriate to pay attention to the convention of sign well used
to note longitudes.

If the longitudes are counted negatively towards east then
the relation linking local sidereal time to the meridian line of the site of longitude l and sidereal time with the meridianof Greenwich is as follows:

      _⁽¹⁵⁾

Attention les deux angles doivent être exprimés avec la même unité (degrés ou heures).

Alors les coordonnées cartésiennes d’un point M₁ de coordonnées géographiques (j₁, l₁) à l’instant t sont données par :

      ⁽¹⁶⁾

The length of the vector (its modulus) and its coordinates (X, Y, Z) are given by :

      ⁽¹⁷⁾

of the direction « center of the Earth - Sun » is given by :

The vector product and its modulus are then :

      ⁽¹⁹⁾

and finally, using formula (11), one will get :

      ⁽²⁰⁾

And the mean equatorial parallax is given (following (9) by :

      ⁽²¹⁾

Numerical application

Example 1: observation of the positions of the centers

We will take as an example the observation made at Antananarivo (Madagascar) and at Helsinki (Finland) at the date t=8h 30min on June 8, 2004.

The geographic coordinates of Antananarivo are :

Latitude :18° 52' south, longitude : 47° 30' east then j₁ = –18,866667° and l₁ = –47,5°.

The geographic coordinates of Helsinki are :

Latitude :60° 8' north, longitude : 25° 3' east, then j₂ = 60,133333° and l₂ = –25,05°.

The geocentric equatorial coordinates of the Sun at 8h 30m UTC are provided by the ephemerides :

Right ascension of the Sun a_s = 76°49' 36.493"

Declination of the Sun d_s = +22°53' 16.237"

The sidereal time at bGreenwich at a date t in UTC is provided by the following formula :

T_G (t UTC) = 17h 6m 51,31s + 1,002737908 t

Then the sidereal time at Greenwich at 8h 30min is equal to :

T_G = 17h 6m 51,31s + 8h 31m 23,78s = 25h 38m 15,09s = 1h 38m 15,09s

It is necessary to convert it in degrees before calculating the local sidereal time for the two cities.

T_G = 1h 38m 15,09s = 24,562875°.

From this, one will deduce the local sidereal time at 8h 30m at Antananarivo :

Tl₁ = 24,562875 – (–47,5°) = 72,062875°

And the sidereal local time at 8h 30m at Helsinki is :

Tl₂ = 24,562875 – (–25,05°) = 49,612875°

One will deduce from this the cartesian equatorial coordinates for these two cities :

Antananarivo :

Helsinki :

The coordinates of the unit vector of the direction Earth-Sun are obtained through formula 18 :

The vector has as coordinates :

Formula 20 allows us to calculate the value of d :

The ephemerides provide us the ratio between the radii vectors and the ratio between the distance Earth-Sun and the semi-major axis of the orbit of the Earth at the given date :

Now, we will just have to make an hypothesis on the measured values, i.e. on Dp  and on the solar diameter :

We will suppose that :

That gives the value of Dp : 28,359"

Formula 3 provides us the value of the solar parallax :

And formula 21 provides us the value of the mean equatorial parallax :

The value that we finds is relatively close to reality, but it depends only on the measure of the distance from the apparent centers of Venus on the solar disc and on the size of the solar diameter. The apparent size of the solar diameter can be measured with a good precision; on the other hand the measurement of the distance between the apparent centers of Venus is not obvious, on a traditional photographic stereotype, the apparent diameter being around 20mm, the distance from the centers is then of 0,3mm and a precision of one thousandth corresponds to a measurement to 0,02mm.

In the preceding forms, one occulted a certain number of difficulties to simplify the problem. Here the list of the complications which appear if one wants to make a rigorous calculation:
1. Because of the mutual disturbances, the orbits of the planets do not follow the laws of Kepler (valid only for two bodies) but more complex trajectories.
2. This is not the Earth which has a quasi-elliptic orbit around the Sun but the barycentre of the system Earth-Moon.
3. Following the movement of the axis of rotation of the Earth (precession and nutation), the origine Ox of the equatorial reference frame is not fixed relatively to time.
4. The light being propagated with a finished speed, the positions of the Sun and Venus at a given date t are not geometric positions but positions of two bodies at the date t – t_p, t_p being the time used by the light to go from each body and the Earth. Since these distances are not supposed to be known, it is necessary to iterate the calculations to take it into account.

Calculation of the parallax from the dates of the contacts or from the duration of the transit

We saw, in the sheet n°04b, that it exists two simplified formulae allowing to calculate directly the parallax from the comparison of the dates of the same contact seen from two different sites (Delisle method) or from the comparison of the duration of the transit observed from two different sites (Halley method).

We will study simultaneouly these two aspects from the preceeding numerical example.

The mean equatorial solar parallax is obtained by comparing two identical contacts using the following simplified formula (cf. formula 16 of the sheet n°04b) :

If one neglect the uncertainties and the errors, then, the formula becomes :

Same, the mean equatorial solar parallax is obtained by comparing the two identical duration, using the following formula (cf. formula 21 of the sheet n°04b) :

i and j are the index related to the same contacts  : i = 1, j = 4 for the external contacts and i = 2, j = 3 for the internal contacts.

The coefficients A, B, C and the term are calculated for each contact and are given by the following table :

Example 2: observation of the dates of the contacts

We will take again the example of the same two cities with the observational hypothesis as follows :

City n°1 : Antananarivo (j₁ = –18,866667° et l₁ = –47,5°)

Date of the first observed internal contact (index 2) : t2 = 5h 35m 30s UTC.
Date of the last observed internal contact (index 3) : t3 = 11h 8m 4s UTC  
Observed duration of the transit (internal): 5h 32m 34s.

City n°2 : Helsinki (j₂ = 60,133333° et l₂ = –25,05°)

Date of the first observed internal contact (index 2) : t2 = 5h 38m 38s UTC.
Date of the last observed internal contact (index 3) :t3 = 11h 2m 20s UTC
Observed duration of the transit (internal) : 5h 23m 42s.

In the formulae (22) and (23) the factors of the coefficients A, B, C are identical and may be calculated separately:

Calculation of the parallax using the first contacts :

The differences between the dates of the first internal contacts is –3m 8s (–3,1333m), and the use of the values of the coefficients A₂, B₂, C₂ and in the formula (22) will provide us the following equation :

That provides .

Calculation of the parallax using the duration of the internal transits

The difference in duration of the internal transits is 8m 52s (8,866m), and the use of the values of the coefficients A₂, B₂, C₂, A₃, B₃, C₃ and in the formula (23) provides us the following equation :

Attention, it is the value and mainly its sign which may be used.

That gives .

Let us remind that these methods are not completely accurate and one should use more complete formulae for the reduction of the observations.