Example 1 for the

Evaluation of Twice Exposed Pictures

by means of simulated pictures of the 2004 transit of Venus

U. Backhaus, University of Duisburg-Essen

This example demonstrates the procedure of measuring and evaluating of transit photos in order to determine the distance to the Sun. The procedure is described in detail and explained with an example in a basic paper. While going through this example, that paper should be present.

Two "photos" are given which have been exposed twice with fixed camera simultaneously in Essen and Windhoek at 10.00 UT.

 

In order to derive the distance to the Sun from this pictures with the basic formaula

the following steps have to be done:

  1. Measuring the position of Venus' disc with respect to the Sun's disc

  2. Transforming these positions into rectangular coordinates in order to be able to determine the parallactic shift of Venus. First, this shift is calculated as a fraction of the apparent radius of the Sun on the pictures.

  3. In order to transform this fraction into the parallactic angle, the angular radius of the Sun ρS has to be determined. Therefore, the displacement between both pictures of the Sun must be measured.

  4. The distance to the Sun can be derived from the parallactic shift if the linear distance Δ between both sites is known. It can be calculated from the geographical coordinates of the sites.

  5. To be correct, not the linear distance itself is important but its projection parallel to the direction of the Sun. The determination of the projection angle w is the most difficult part of the procedure. It is done in three steps:

    1. Calculation of the equatorial coordinates of the sites (For this reason, the local sideral times of the exposures have to be determined for both sites.),

    2. Determination of the unit vectors between both sites and the direction to the Sun and

    3. Determination of the projection angle by means of the scalar product of these unit vectors.

  6. Derivation of the distance to the Sun by means of

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

  • geographical coordinates
    Essen φE=51.24° n.B. λE=7.0° östl.L.
    Windhoek φW=22.34° s.B. λW=17.05° östl.L.
  • equatorial coordinates of the Sun at the moment at which the pictures have been taken:
    αS=5h7m27s=76.86°
    δS=22°53'=22.88°
  • sideral time of Greenwich at 0.00UT:
    ΘG0=17h6m52s=256.72°
  • proportion between the solar distances of Venus and Earth
    rV/rE=0.715
  • the Earth's radius
    RE=6378 km
  • The time interval between both exposures is exactly 90 seconds.

(There are additional projects concerning the determination of these quantities by means of own measurements!)

Evaluation

  1. Determination of Venus' positions with respect to the solar disc

    The relative distances ρ' between Venus and the centre of the solar disc and the corresponding position angles can be determined by means of the computer program Bildauswertung.

    Results:

    Essen θ'1 = -73.15°  ρ'1 = 0.7832
    θ'2 = -72.54° ρ'2 = 0.7874
    Windhoek  θ'1 = -72.67° ρ'1 = 0.7601
    θ'2 = -71.88° ρ'2 = 0.7603

  2. Transforming to rectangular coordinates (x' = ρ'cosθ', y' = ρ'sinθ') and calculation of the parallactic shift

    Results (They are given by the program, too.):

    Essen x'1 = 0.2270  y'1 = -0.7496
    x'2 = 0.2363 y'2 = -0.7511
    Windhoek  x'1 = 0.2264 y'1 = -0.7256
    x'2 = 0.2365 y'2 = -0.7226

    From these results, the parallactic shifts can be calculated as multiples of the solar radius:

    f1 = 0.0240  f2 = 0.0285
  3. Determination of the angular radius of the Sun

    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 is smaller by the factor cosδS. Its angular velocity is then ω = ω0cosδS and, therefore, ω=13.82"/s on June 8th.

    On the photos, both pictures of the Sun are seperated by 279.3 pixels. During the corresponding time interval of 90s, the Sun has moved by 1243.8". Therefore, the scale of the pictures is 4.45"/pixels

    The solar radius on the pictures is 213 pixels. The angular radius of the Sun, therefore, is ρS=948.5"=15.81'.

    Therefore, we get the following parallactic shifts of Venus from the relative parallactic shift calculated above:

    β1 = ρSf1 = 22.8"  β2 = ρSf2 = 27.0"

  4. Calculation of the linear distance Δ/RE from the geographical coordinates of both sites

    First, the polar geographical coordinates are transformed to rectangular coordinates. The Earth's centre 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.

    Essen Windhoek
    x/RE = cosφcosλ   0.6214 0.8843
    y/RE = cosφsinλ 0.0763 0.2712
    x/RE = sinφ 0.7798   -0.3801

    Thus, the linear distance between both sites is Δ/RE = 1.21.

    Remark: Because of the difficulty of the following calculation of the projection angle, it is worthwhile to calculate approximations for the solar parallax by the assumption that this angle is nearly 90°:

    Approximations: πS1=7.5",   πS2=8.9"

  5. Determination of the projection angle w

    1. Calculation of the equatorial coordinates of the sites
      1. The declination δ of a site equals its geographical latitude. Therefore,

        δE = 51.24°, δW = -22.34°
      2. 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 in hours since 0.00 UT, the local sideral time of Greenwich is

        ΘG=ΘG0+1.0027379*t
        and, therefore,
        α = ΘG0+1.0027379*t+λ*4min/°
        At 10.00 UT, the local sideral time of Greenwich is
        ΘG=ΘG0+10h01m39s=3h08m31s
        The right ascension of Essen and Windhoek at 10.00 UT, therefore, are:

        αE=3h36m31s=54.13°, αW=4h16m43s=64.18°

    2. Calculation of the unit vectors in rectangular coordinates

      As in the case of geographical coordinates, the rectangular coordinates can be calculated by means of:

      x/RE = cosαcosδ
      y/RE = sinαcosδ
      z/RE = sinδ

      1. Unit vector pointing from Essen to Windhoek

        The rectangular equatorial coordinates of Essen and WEindhoek are:

        Essen Windhoek
        x/RE   0.3668 0.4029
        y/RE 0.5073 0.8326
        z/RE 0.7798   -0.3801

        The direction from Essen to Windhoek, therefore, is:

        eEW = (0.030, 0.270, -0.962)

      2. Direction to the Sun

        The unit vector follows directly from the given equatorial polar coordinates:

        eS = (0.209, 0.897, 0.389)

    3. Calculation of the projection angle w and of the projected distance between Essen and Windhoek

      The scalar product of both unit vectors equals the cosine of the angle between them:

      cosw = -0.1258 und damit w = 97.2°

      The projected distance, therefore, is

      Δsinw=1.20.

      (The above assumption, therefore, has been justified!)

    Now, all needed quantities have been determined and the solar parallax can be calculated:

    πS1 = 6.7", πS2 = 9.2"
  6. Calculation of the distance to the Sun

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

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

    We get then the final result:

    dS1 = 196 000 000 km, dS2 = 143 000 000 km.

    These results are not very satisfying because the errors in positioning Venus are quite large with respect to the parallactic effect. It is possible to get better results by combining the positions of many pictures and minimizing errors by statistical methods. This is the subject of an additional example.

Udo Backhaus
last modification:  March 28th, 2008