C) Results using the observations of 1769
The data below give us the times of the contacts observed at various locations. The drawing below was published in "A History of Astronomy" by A. Pannekoek and shows us the transits of 1761 and 1769. We will use the 1769 results of Vardö (straight line 3) and Tahiti (straight line 1) for the calculation. Measuring Db or A1A2 , B1B2 and D on the image of the Sun, we obtain A'B' and D on the same scale.
The times of the contacts of Venus with the Sun are used to calculate the duration of the transits: tA and tB.
We use the results C2 and C3 of Vardö and Tahiti for the calculation.
1) Distance between observers at points A and B
Vardö and Papeete (Tahiti) are in the same meridian and their latitudes are 70°21' N and 17°32' S. Note that Vardö observed the "midnight Sun".
Then the geometry of the problem changes and the new angle j to consider is
j = (90-j1) + 90 + j2 = 127º11'
and using the radius R=6378 km, we calculate,
AB = 2 R sin(j/2) =11425 km.
2) Distance Db between two observed Venus paths
a) Calculation of Db by direct measurement
Measuring directly the distance between the straight lines 1 and 3, we get A'B' = 1.5 mm when the diameter on the drawing is D=70 mm. Therefore
Db = (p / 360 ) (1.5 / 70 )= 0.00019 radians
2b) Calculation of Db by measurement of the chords
Measuring A1A2, B1B2 and D on the image of the Sun, we obtain A1A2= 52 mm (line 3), B1B2= 49 mm (line 1) and D=70mm. Then
A'B'/D = ½[
(1 -(49/70)2) - (1 - (52/70)2)] = 0.02235
And Db is given by
Db = ( 30 p / 360) 0.2235 = 0.00020 radians
c) Calculation of Db by measurement of the times of transit.
For Vardö, the location A, with observer Borgrewing, we calculate
tA = 15h 27min 28.6s - 9h 34min 32.6s = 21176s
For Tahiti, the location B, with observer Cook, we calculate
tB = 15h 14min 11s - 9h 44min 15s = 19796s
And using t0= 26775s
t'/t0 = ½[(1 -(19796/26775) 2) - (1 - (21176/26775) 2)] = 0.03076
And Db is given by
Db = ( p / 360 ) 0.03076 = 0.00027 radians
3) AND FINALLY the Earth-Sun distance re
Using the parallax formula from section (A5)we have
bs = 0.38248 Db
Using the solar parallax formula from section (A5), the distance from the Earth to the Sun re is
re = AB/ bs
Using the data from 1769 and AB = 11425 km we can calculate three different values of re as shown in the table.
| Db | re |
| 0.00019 | 157 106 km |
| 0.00020 | 149 106 km |
| 0.00027 | 111 106 km |
As you can see, it is difficult to produce consistent results, although the first two results are reasonable close to the currently accepted value.
