# Example: Evaluation of twice exposed transit pictures

 1. Evaluation of a single pair of simultaneously taken photos 2. Repetition of this evaluation by hand 3. Improve of this single result by combining all pairs of those pictures of two sites

Additionally, the evaluation of the measured positions of Venus is demonstrated on the basis of calculated positions during the coming transit.

This example demonstrates the analysis of two 2004 transit photos in order to determine the distance to the Sun. First, the evaluation will be done by using some programs. Second, we will demonstrate the underlying procedures by evaluating the pictures by hand.

It is possible to get better results by combining the positions of Venus in several pictures and minimizing errors by statistical methods. This is the subject of the third part of this example.

The elaboration will be done with two photos which have been simultaneously exposed twice with fixed camera in Germany (U. Backhaus, University of Duisburg-Essen) and Namibia (E. v. Grumbkow, Internationale Amateursternwarte IAS) at 8.00 UT on June 8th, 2004.

 Essen IAS

In addition to the pictures, the following informations are given:

• The geographical coordinates (φ,λ) of the observation sites:  Essen φE=51.24° N λE=7.0° S IAS φW=21.22° S λW=14.86° E
• The equatorial coordinates (αSS) of the Sun at the moment the pictures have been taken:

αS=5h7m27s=76.86°
δS=22°53'=22.88°

• The sideral time of Greenwich at 0.00UT on transit day:

ΘG0=17h6m52s=256.72°

• The proportion between the solar distances of Venus and the Earth on transit day:

rV/rE=0.715

RE=6378 km

• The time interval between the exposures was exactly 120 seconds in Namibia and 150 seconds in Germany.

### 1. Evaluation of a single pair of simultaneously taken photos

In this part of the example the evaluation will be done with the programs evalttransitpicts and calcparallax which have been written for this project. Therefore, only the results and the approach with these programs are listed.

In order to derive the distance to the Sun from the pictures with the basic formula

the following three steps have to be done:

 1. Measuring the position of Venus on the solar disc 2. Calculation the solar parallaxe 3. Derivation of the distance to the Sun
1. The positions of Venus are measured with the help of the program evalttransitpics.exe. After program start you must input the following data (example inputs in brackets):

1. the name of the file in which the program will write its results ("Essen0800.txt"),
2. the name of the picture to be evaluated ("Essen0800.jpg"),
3. the factor of size reduction, 1 means no reduction ("2", for a a more exact measurement choose "1"; the picture can be shifted so that the relevant part is visible on the monitor.),
4. the exposure time ("8.00" UT),
5. the declination of the Sun ("22.9" degrees),
6. the number of exposures of the picture ("2"),
7. the time difference between the exposures ("150" seconds),
8. the horizontal and vertical pixel position of the upper left edge of the picture ("0" and "0", negative values are possible) and
9. the initial pixel radius of the circle which will be fitted to the size of the solar disc ("400").

After these inputs the picture will be displayed und you will be able to fit four circles to the Sun and Venus, respectively:
1. to the first Sun (results: x=444, y=1104, r=424 pixels),
2. to the second Sun (results: x=1146, y=466, r=424 pixels),
3. to the first Venus (results: x=544, y=1374, r=14 pixels) and
4. to the second Venus (results: x=1250, y=732, r=14 pixels).

You may take a look to operation hints by pressing "F1".

After this fitting, the results will be displayed on the screen. Finally, the program will evaluate these pixel positions and derive
• the direction from celestial east to west,
• the scale of the picture (using the real movement of the Sun on the sky) and
• the positions of Venus on the both solar discs.
These results will be written into four result files Essen0800. Similarily, the picture from Namibia can be evaluated. Namibia0800 contains the results.

Results:

The edited pictures

 Essen IAS

and the rectangular coordinates of Venus in coordinate system of the Sun with the x-axis pointing from celestial east to west.

Essen x'1 = --0.2537 y'1 = -0.6299
x'2 = -0.2404 y'2 = -0.6292 ρS=15.44'
IAS x'1 = -0.2676 y'1 = -0.6063
x'2 = -0.2619 y'2 = -0.6106 ρS=15.80'

2. Calculation the solar parallaxe

Together with the geographical positions of the sites and the equatorial position of the Sun, the above results form the basis on which the programm calcparallax calculates first the parallactic shift ρSf, second the linear distance Δ between the sites and, finally, the solar parallaxe πS. On the basis of the above data the program calculates the following values and stores them into the file parallaxresults.txt:

 Parallactic displacement 25.6" Linear distance of the two observers Delta 1.19RE Projected distance of the two observers Delta*sin(w) 1.17RE Parallax of the Sun piS 8.8"

The same results can be derived by using the worksheet comp2Venuspositions.xls. (Attention: Because that sheet is prepared for 2012 the values for ρS, rV/rE, (αSS) and ΘGr0 must be changed in sheet "calculation"!) This has been done in comp2EssenNamibia.xls.

3. Derivation of the distance to the Sun

From the solar parallax πS, the distance dS to the Sun can be calculated by means of

Into this equation, the parallax must be inserted in radians, that means the value 8.8" has to be multiplied by the factor π/180/3600 = 4.848*10-6.

We get then the final result:

dS = 149 500 000 km

Perhaps, you will get an even more satisfying result by measuring the positions of Venus without downsizing the photos.

### 2. Repetition of this evaluation by hand

The above elaboration demonstrates the evaluation of two twice exposed pictures by using the programs offered here. For illustration reasons, in this part of the example the underlying procedure will been done without the help of these programs.

The following steps have to be done to get the result:

 1. Determination of scale and orientation of the pictures 2. Determination of parallactic displacement Δβ of Venus on the solar disc 3. Derivation of approximations for the Sun's parallax 4. Calculation of Sun's parallax πS by taking into account that the bonding vector is not rectangular on the of direction of Sun 5. Determination of the distance between the Earth and the Sun (1AU)

#### Evaluation

1. Scale and orientation of the pictures
• Ascertain the positions (xSun,ySun) of the Sun's centers and their radii rSun by using, for instance, the mouse pointer. Perhaps, you know a better method. We will use here the above values:

 location xSun ySun rSun Essen 442 1104 424 1144 466 424 IAS 532 604 508 1414 546 508
• Then, you can calculate the pixel movement of the Sun by applying Pythagoras' theorem and the angle α between the x-axis and the lower border of the pictures by using the arctan function:  location Δ α Essen 948.6Px 42.27° IAS 883.9Px 3.63°
• From the Sun's declination you can derive the real movement of the Sun from east to west between the both exposures:

At the equinoxes, when the Sun's position is exact on the celestial equator, it moves by exact 360° in 24 hours. Its angular velocity, therefore, is ω0 = 360°/24h = 1°/4m = 15"/s. However, if it is distant from the equator by its declination δS, the radius of its daily path, and therefore its angular velocity, is smaller by the factor cosδS. The solar angular velocity on transit day is, therefore, 13.82"/s. Finally, you can calculate (in this order) the real angular movement Δφ of the Sun, the scale of the picture and the angular radius ρS of the Sun:

We get the following results:

Essen 150 s 2073″ 948.6Px 2.185″/Px Δt Δφ Δ scale ρS
2. Determination of parallactic displacement Δβ of Venus on the solar disc
• Detect the rectangular coordinates of Venus

1. absolutely (xV,yV),
2. with respect to the center of the Sun (x',y'),
3. relatively to Sun's radius (x'',y'') and, finally,
4. relatively to the correctly oriented coordinate system (xVenus,yVenus):

The last coordinates can be calculated by rotating (x'',y'') clockwise by the above determined angle α:
 xVenus=x''cosα+y''sinα yVenus=-x''sinα+y''cosα

xV yV x' y' x'' y'' xVenus yVenus
Essen 544 1374 102 270 0.2406 -0.6368 -0.2503 -0.6331
IAS 416 920 -116 316 -0.2283 -0.6220 -0.2686 -0.6057
• With these coordinates of Venus you can affort the parallactic displacement of Venus between Essen and the IAS. At first relativ to the Sun's radius ΔV and then in arcseconds Δβ:

• Estimate now the angular distance of the observation sites as seen from the Sun βS=(rE/rV-1)*Δβ:

βS
Essen-IAS 10.18''
3. Derivation of approximations for the Sun's parallax
4. For simplicity, first make the following approximations:

1. The observation points are of maximum distance 2RE. In this case, the Sun's parallax is one half of the angular distance between the sites: πS≈5.1''.
2. The vector between the sites is perpendicular to the direction to the Sun:

In order to determine the linear distance between the sites the polar geographical coordinates (φ,λ) have to be transformed to rectangular coordinates. The Earth's center is the origin of the corresponding system of coordinates and the equatorial plane its x-y-plane. The x-axis points to the longitude of Greenwich.

In this case, the parallax of Sun is πS≈8.6''.

3. Take the linear distance between Essen and the IAS as be seen from the Sun from the following picture:

Measure the distance between Essen and Namibia of this copy and accumulate the proportion to the radius of the Earth. You will get the following results:

Δ RE Δ/RE πS
Essen-IAS 455Px 383Px 1.19 8.87''
5. Calculation of the Sun's parallax πS by taking into account that the bonding vector is not perpendicular to the direction to the Sun

Both directions span an angle w. In order to calculate this angle, the following steps have to be done:

• Calculation of the equatorial coordinates of the sites

The declination δ of a site equals its geographical latitude and the right ascension α of a site, at every moment, equals its local sideral time Θ. It can be derived from the sideral time of Greenwich ΘG by means of

Θ = ΘG+λ*4min/°

Here it is supposed that longitudes east of Greenwich are counted positive.

Sideral time runs faster than solar time by the factor 1.0027379. If one takes the time t in hours since 0.00 UT, the local sideral time of Greenwich is

ΘGG0+1.0027379*t

and, therefore,

α =ΘG0+1.0027379*t+λ*4min/°

At 08.00 UT, the local sideral time of Greenwich is

ΘGG0+8h01m39s=1h08m11s.

Therefore, we get following local sidereal times:

With these facts we can get the equatorial coordinates:

• With the help of the position of the Sun we can can get the angle w and, at least, the projected distance Δ. If you want to derive w, just convert the position in rectengular coordinates and develop the scalar product with the position´s unit vector and the direction vector to the Sun.

Hence we get the final result for the parallax of the Sun:

πs=8.70''

6. From the solar parallax πS, the distance dS to the Sun can be calculated by means of

Before applying this equation, the parallax in arcseconds has to be multiplied by the factor π/180/3600 = 4.848*10-6 in order to transform it into radiants.

We get then the final result:

dS = 151 200 000 km

This result depends sensitively from the pixel positions which you read from the pictures!

### 3. Improve of this single result by combining all pairs of those pictures of two sites

Because of the sensitivity of the evaluation better results can be expected if more than one measurement of both observers are combined.
The above observers observed and photographed the 2004 transit during several hours, see the result page of the 2004 transit project. The former results can be trasferred to two copies of our Excel sheet tableofVenuspositions.xls. (Attention: The "2012" in the title of "Introduction" must be changed into "2004" in order to set the correct reference time!) These sheets calculate the line fits to the measured positions of both observers. For further evaluation the sheets "Line" must be saved as tab-seperated text files.
The combination of these text files is done by the program comptransitpos.
1. It interpolates and extrapolates the data in order to calculate
1. the eventally missed positions at some appointed times,
2. the center of the transit and the according distance of Venus from the center of the Sun's disc,
3. the contact times resulting from the linear fit.
2. It visualizes the position results graphically:

 Essen Namibia

3. It compares the measurements of the observers and derives the solar parallax from the measured positions of two observers by three methods:
1. by comparison of the simultaneous measured positions (blue dots),
2. by comparison of the line fits (red line),
3. by taking the difference of the minimal distances to the center of the Sun as the parallactic displacement of Venus.
the results of these calculations are displayed in a graphic:

This evaluation of the complete series of position data allows to determine mean values and to estimate their statistical errors. It becomes clear now that the nearly exact result derived above from one pair of position data only was purely accidental.

 Editors: Udo Backhaus last update: last update: 2020-03-19 Stephan Breil