The Venus Transit 2004... Extended InfoSheet B4
Approximated method for the calculation of the parallax

(1) 
Actually, the exact parallax are given by:

Then we have the following relation:
(2) 
and

or
(3) 
The measurement gives us the value D_{p} in terms of the solar diameter and we must also measure the diameter of the Sun, because if the distance EarthSun is unknown, this diameter may not be calculated.
To know the solar parallax, it is thus necessary to know the ratio of the SunEarth and SunVenus distances. However this ratio can be calculated thanks to Kepler's laws.
Kepler's first law says that the planets describe elliptic orbits around the Sun and that the Sun occupies one of the foci of these ellipses. At a given moment the radius vector R_{p} joining the centre of the Sun to a planet p is calculated using the following formula: r_{p} = a_{p} ( 1  e_{p} cos E) (4)
Here a_{p} is the semimajor axis of the ellipse, e_{p} is the eccentricity of the ellipse and E is an angle called eccentric anomaly which makes it possible to locate the planet on its orbit.
Kepler's third law provides a relation between the semimajor axes of the orbits and the periods of revolution of the planets, thus for the same central body all the orbits of the planets which revolve around it obey the following relation :
(5) 
Kepler's laws thus describe the orbits of the solar system apart from a scale factor. Knowledge of the periods of revolution of the planets gives us the ratios of the semimajor axes, thus the ratio of the semimajor axis of the orbits of Venus and of the Earth is equal to :
(6) 
and at any time T, the ratio of the radius vectors is equal to

Thus Kepler's laws allow the ratio of the radius vectors for any time T to be calculated.
Our measurement allows us to calculate the value πS, it is thus appropriate now to pass from this value to the value of the mean equatorial parallax of the Sun π_{0}.
The mean equatorial parallax of the Sun π_{0} is defined as the angle subtended at the centre of the Sun by the equatorial radius of the Earth when the Sun is at one astronomical unit from the Earth.
We have thus the following relation :
(8) 
R being the Earth's equatorial radius and a the value of the astronomical unit.
However, equation (1) gives us the value of the solar parallax π_{S} in terms of the R_{T} EarthSun distance and projection d of the distance between the observing points onto the plane normal to the EarthSun direction.
It is sufficient to express this distance D in terms of the Earth's radius and the EarthSun distance in astronomical units to have a relation between π_{S} and π_{0}.
(9) 
It only remains to calculate the ratio of D on R. The a/R_{T} ratio is provided Kepler's law (cf. Eq. 4). However if we take the vector product of the two vectors and we obtain :
(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 4).
Solving 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, so allowing the calculation of the cosine of the angle and then its sine using the relation : .
This calculation needs the Cartesian coordinates of the two points M_{1} and M_{2} and of the centre of the Sun C in an orthogonal reference frame (O, x, y, z) with origin at the centre of the Earth. We will use the apparent geocentric equatorial reference frame for this calculation.
ThE reference frame is defined by the plane of the Earth's equator at the time T of the observation (plane Oxy) and by the direction of the northern celestial pole of the Earth's rotation axis (Oz). In this reference frame we can define a Cartesian reference frame (x, y, z) and a polar reference frame (α, δ, R) where the two angles are called "right ascension" and "declination" (Figure 5). We pass from one system to the other using the following relations :
(12) 
and the converse relations:
(13) 
The direction of axis Ox at the time T is the direction of the vernal equinox on that date.
The ephemerides (i.e. Kepler's laws) give us the geocentric equatorial coordinates of the centre of the Sun (α, δ); the distance is not know but is not important because the vector may be replaced by its unit vector in equation 11.
A more complicated problem is the determination of the cartesian coordinates of the points M_{1} et M_{2} in this equatorial frame.
The position of a point on the surface of the Earth is given by its latitude and longitude (geographic); latitude is referred to the terrestrial equator, so it is like the ideclination angle. Longitude is referred to a zero meridian (the Greenwich meridian), so that it is similar to right ascension, but with a different origin from that of the celestial equatorial coordinates. Then we need to know for each date the angle between the direction of the Ox axis and the direction of the projection of the zero meridian on the equatoral plane (cf. Figure 5). This angle is related to the Earth's rotation: it is called the "sidereal time" of the Greenwich meridian and it increases by 360° during 23h 56m 4s (the sidereal revolution of the Earth).
Thus, it is sufficient to know the sidereal time of Greenwich T_{G} at 0h UTC on the day of the transit in order to know the sidereal time of Greenwich at any time t, and then the sidereal time of any point on Earth having a longitude λ.
(14) 
We pass from Greenwich sideral time to the sidereal time of the site M having longitude λ, by adding or substracting its longitude.
Beware! Sidereal time increases going eastwards from the Greenwich meridian; it is thus necessary to pay attention to the sign convention used for 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 λ and sidereal time with the Greenwich meridian is as follows: T_{λ} = T_{G}  λ (15)
Note that the two angles must be expressed in the same units (degrees or hours). Then the cartesian coordinates of a point M_{1} with geographical coordinates (φ_{1}, λ_{1}) at time t are given by:
(16) 
The length M_{1} M_{2} of the vector (its modulus) and its coordinates (X, Y, Z) are given by :
(17) 
The coordinates of the vector of the direction "centre of the Earth  Sun" is given by :
(18) 
The vector product and its modulus are then :
(19) 
and finally, using equation (11), we get :
(20) 
And the mean equatorial parallax is given (following Eq. (9)) by :
(21) 
Example 1: observation of the positions of the centres
We will take as an example the observation made at Antananarivo (Madagascar) and at Helsinki (Finland) at the time t=8h 30min on June 8, 2004.
The geographic coordinates of Antananarivo are :
The geographic coordinates of Helsinki are :
The geocentric equatorial coordinates of the Sun at 8h 30m UTC are provided by the ephemerides:
The sidereal time at Greenwich at a time t in UTC is provided by the following formula :
Then the sidereal time at Greenwich at 8h 30min is equal to :
It is necessary to convert to degrees before calculating the local sidereal time for the two cities.
From this, we deduce the local sidereal time at 8h 30m at Antananarivo :
And the sidereal local time at 8h 30m at Helsinki is :
We deduce the cartesian equatorial coordinates for these two places:
Antananarivo :

Helsinki :

The coordinates of the unit vector of the EarthSun direction are obtained through equation 18 :
The vector has coordinates :
Formula 20 allows us to calculate the value of d :
The ephemerides give us the ratio between the radii vectors and the ratio between the EarthSun distance and the semimajor axis of the orbit of the Earth on the given date : r_{T} / r_{V} = 1.397795 and r_{T} / a = 1.015087
Now, we just have to make an assumption about the measured values, i.e. about Δπ and about the solar diameter : we will suppose that Δπ= 0.015 φ and φ = 31.51'. That gives the value of Δπ : 28.359"
Equation 3 gives us the value of the solar parallax :
And equation 21 gives us the value of the mean equatorial parallax :
The value that we find is relatively close to reality, but it depends only on the separation of the apparent centres 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 separation of the apparent centres of Venus is not obvious. On a traditional photographic exposure where the apparent diameter is around 20mm, the distance between the centres is then about 0.3mm and a precision of one in a thousand corresponds to a measurement of 0.02mm.
In the preceding discussion we ignored a number of difficulties in order to simplify the problem. Below is a list of the complications which arise if we want to make a rigorous calculation:
We saw, in InfoSheet n°_{0}4b, that there are two simplified formulas allowing the direct calculation of the parallax by comparing the times of the same contact seen from two different places (Delisle method) or by comparing the duration of the transit observed from two different places (Halley method).
We will study simultaneouly these two aspects from the preceeding numerical example.
The mean equatorial solar parallax π_{0} is obtained by comparing two identical contacts using the following simplified formula (cf. formula 16 of the sheet n°_{0}4b) :
(22) 
If we neglect the uncertainties and the errors, then the formula becomes:
(23) 
Similarly, the mean equatorial solar parallax is obtained by comparing two identical durations using the following formula (cf. equation 21 of the Sheet n°_{0}4b) :
(24) 
i and j are the indices related to the same contacts: i = 1, j = 4 for the exterior contacts and i = 2, j = 3 for the interior contacts.
The coefficients A, B, C and the term dD/dt are calculated for each contact and are given by the following table :
Description of the contact  A  B  C 
dD/dt
"/min 
First exterior contact (index 1)  2.2606  0.0194  1.0110  3.0846 
First interior contact (index 2)  2.1970  0.2237  1.1206  2.9394 
Last interior contact (index 3)  1.0929  1.1376  1.9090  2.9391 
Last exterior contact (index 4)  0.9799  1.3390  1.8383  3.0842 
We will take again the example of the same two places with the observational hypothesis as follows :
In equations (22) and (23) the factors of the coefficients A, B, C are identical and may be calculated separately:
The differences between the dates of the first interior contacts is 3m 8s (3.1333m), and the use of the values of the coefficients A_{2}, B_{2}, C_{2} and dD/dt in the formula (22) will provide us the following equation :
That gives π_{0}=8.945'.
The difference in duration of the interior transits is 8m 52s (8.866m), and the use of the values of the coefficients A_{2}, B_{2}, C_{2}, A_{3}, B_{3}, C_{3} and dD/dt in equation (23) provides us the following equation :
Note that it is the value
and mainly its sign which may be used.
That gives π_{0}=8.822'.
Let us remember that these methods are not completely accurate and we should use more complete formulas for the reduction of the observations.
[1] Written by P. Rocher (IMCCE)
last modified: 2004/03/23 
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