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 8^{th}, 2004.
Essen | IAS |
In addition to the pictures, the following informations are given:
Essen | φ_{E}=51.24° N | λ_{E}=7.0° S | |
IAS | φ_{W}=21.22° S | λ_{W}=14.86° E |
α_{S}=5h7m27s=76.86°
δ_{S}=22°53'=22.88°
Θ_{G0}=17h6m52s=256.72°
r_{V}/r_{E}=0.715
R_{E}=6378 km
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 |
Measuring the position of Venus on the solar disc
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):
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 | ρ_{S}=15.44' | |
IAS | x'_{1} = -0.2676 | y'_{1} = -0.6063 | |
x'_{2} = -0.2619 | y'_{2} = -0.6106 | ρ_{S}=15.80' |
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 ρ_{S}f, second the linear distance Δ between the sites and, finally, the solar parallaxe π_{S}. 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 ρ_{S}, r_{V}/r_{E}, (α_{S},δ_{S}) and Θ_{Gr0} must be changed in sheet "calculation"!) This has been done in comp2EssenNamibia.xls.
Derivation of the distance to the Sun
From the solar parallax π_{S}, the distance d_{S} 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:
d_{S} = 149 500 000 km
Perhaps, you will get an even more satisfying result by measuring the positions of Venus without downsizing the photos.
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:
Ascertain the positions (x_{Sun},y_{Sun}) of the Sun's centers and their radii r_{Sun} by using, for instance, the mouse pointer. Perhaps, you know a better method. We will use here the above values:
location | x_{Sun} | y_{Sun} | r_{Sun} | ||||
Essen | 442 | 1104 | 424 | ||||
1144 | 466 | 424 | |||||
IAS | 532 | 604 | 508 | ||||
1414 | 546 | 508 |
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 ω_{0} = 360°/24h = 1°/4m = 15"/s. However, if it is distant from the equator by its declination δ_{S}, the radius of its daily path, and therefore its angular velocity, is smaller by the factor cosδ_{S}. 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 ρ_{S} of the Sun:
We get the following results:
Δt | Δφ | Δ | scale | ρ_{S} | |||||||
Essen | 150 s | 2073″ | 948.6Px | 2.185″/Px | 15.44′ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
IAS | 120s | 1658″ | 883.9Px | 1.876″/Px | 15.88′ |
Detect the rectangular coordinates of Venus
x_{Venus}=x''cosα+y''sinα |
y_{Venus}=-x''sinα+y''cosα |
x_{V} | y_{V} | x' | y' | x'' | y'' | x_{Venus} | y_{Venus} | ||
---|---|---|---|---|---|---|---|---|---|
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 Δ_{V} and then in arcseconds Δβ:
Δ_{V} | Δβ | ||
---|---|---|---|
Essen-IAS | 0.02724 | 25.6'' |
Estimate now the angular distance of the observation sites as seen from the Sun β_{S}=(r_{E}/r_{V}-1)*Δβ:
β_{S} | |
---|---|
Essen-IAS | 10.18'' |
For simplicity, first make the following approximations:
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.
x'=R_{E}cosφcosλ | y'=R_{E}=cosφsinλ | z'=R_{E}=sinφ | Δ | ||||
---|---|---|---|---|---|---|---|
Essen | 0.6214R_{E} | 0.0763R_{E} | 0.7798R_{E} | ||||
IAS | 0.9010R_{E} | 0.2391R_{E} | -0.3691R_{E} | 1.19R_{E} |
In this case, the parallax of Sun is π_{S}≈8.6''.
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 | R_{E} | Δ/R_{E} | π_{S} | |
---|---|---|---|---|
Essen-IAS | 455Px | 383Px | 1.19 | 8.87'' |
Both directions span an angle w. In order to calculate this angle, the following steps have to be done:
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 Θ_{G} by means of
Θ = Θ_{G}+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
Θ_{G}=Θ_{G0}+1.0027379*t
and, therefore,
α =Θ_{G0}+1.0027379*t+λ*4min/°
At 08.00 UT, the local sideral time of Greenwich is
Θ_{G}=Θ_{G0}+8h01m39s=1h08m11s.
Therefore, we get following local sidereal times:
Θ | ||
---|---|---|
Essen | 1h36m11s | |
IAS | 2h07m37s |
With these facts we can get the equatorial coordinates:
α | δ | x | y | z | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Essen | 1h36m11s=24.0° | 51.24° | 0.572 | 0.255 | 0.780 | |||||
Essen | 2h07m37s=31.9° | -21.22° | 0.791 | 0.493 | -0.362 |
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.
x | y | z | w | Δ_{⊥} | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Essen-IAS | 0.184 | 0.201 | -0.962 | 81.1° | 1.17RE |
Hence we get the final result for the parallax of the Sun:
π_{s}=8.70''
From the solar parallax π_{S}, the distance d_{S} 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:
d_{S} = 151 200 000 km
This result depends sensitively from the pixel positions which you read from the pictures!
Essen | Namibia |
Editors: | Udo Backhaus |
last update: 22.05.2012 | |||
Stephan Breil |