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3. Coordinate transformations

  Object coordinates are usually given in the equatorial tex2html_wrap_inline10825, tex2html_wrap_inline11629 sytem.

The zodiacal light is given in terms of ecliptic coordinates tex2html_wrap_inline11631 with the zero point of tex2html_wrap_inline10929 in the Sun. Description of a line of sight by elongation tex2html_wrap_inline11635 and inclination i also is common. For the relation between these two sets of coordinates see Fig. 2 (click here) and Sect. 3.5 (click here) below.
Integrated starlight is naturally presented in galactic coordinates l,b.

For estimates of the diffuse background brightness at a given position, transformation between these three systems is necessary. Figures 3 (click here)-7 (click here) provide a simple way to do so graphically with about one-degree accuracy, which is sufficient for many applications. The underlying transformation equations are summarised below for ease of access.

Airglow, extinction and scattering are best described in the local horizontal system A, z (azimuth, zenith distance). The transformation to the other systems depends on time and on the observer's geographic coordinates. For the horizontal system, only the equations for the transformation to the equatorial system are given.

3.1. Ecliptic and equatorial coordinates

  The obliquity of the ecliptic is essentially constant, tex2html_wrap_inline11635 = 23.446tex2html_wrap_inline11647 for equinox 1950, respectively tex2html_wrap_inline11635 = 23.439tex2html_wrap_inline11647 for equinox 2000.

The precession of the vernal equinox along the ecliptic is p0 = 50.3 ''/year. Hence
 equation707
The north ecliptic pole is at tex2html_wrap_inline10825 = 18 h, tex2html_wrap_inline11657.
The north celestial pole is at tex2html_wrap_inline11659, tex2html_wrap_inline11661.

Both tex2html_wrap_inline10825 and tex2html_wrap_inline10929 are counted eastward from the vernal equinox. Apart from the minimal change in tex2html_wrap_inline11635, the transformation equations then are the same for 1950 and 2000:

3.1.1. Transformation tex2html_wrap_inline11669


 eqnarray717

3.1.2. Transformation tex2html_wrap_inline11671


 eqnarray721

3.2. Galactic and equatorial coordinates

  By IAU decision, for equinox 1950 the north galactic pole (NGP) is at tex2html_wrap_inline10825 = 12 h 49.0 m, tex2html_wrap_inline11629 = 27.4tex2html_wrap_inline11647 and the celestial pole at l = 123.0tex2html_wrap_inline11647, b = 27tex2html_wrap_inline11647 24.0'. Hence the ascending node of the galactic equator is at tex2html_wrap_inline11689 = 18 h 49.0 m = 282.25tex2html_wrap_inline11647, tex2html_wrap_inline11693 = 33.0tex2html_wrap_inline11647. For equinox 2000, the coordinates of the north galactic pole are tex2html_wrap_inline10825 = 12 h 51.42 m, tex2html_wrap_inline11629 = 27tex2html_wrap_inline11647 07.8', and we have tex2html_wrap_inline11689 = 282.86tex2html_wrap_inline11647, tex2html_wrap_inline11693 = 32.93tex2html_wrap_inline11647. The inclination of the galactic equator with respect to the ecliptic is tex2html_wrap_inline11713. As tex2html_wrap_inline10825 and tex2html_wrap_inline10929, l is also counted eastwards.

With these parameters, the transformations are as follows:

3.2.1. Transformation tex2html_wrap_inline11723


 eqnarray738

3.2.2. Transformation tex2html_wrap_inline11725


 eqnarray746

3.3. Galactic and ecliptic coordinates

  In ecliptic coordinates, for equinox 1950 the north galactic pole is at tex2html_wrap_inline10929 = 179.32tex2html_wrap_inline11647, tex2html_wrap_inline11731 = 29.81tex2html_wrap_inline11647, and the ascending node of the galactic equator at tex2html_wrap_inline11735 = 269.32tex2html_wrap_inline11647, l1 = 6.38tex2html_wrap_inline11647. For equinox 2000, the coordinates of the north galactic pole are tex2html_wrap_inline10929 = 180.02tex2html_wrap_inline11647, tex2html_wrap_inline11731 = 29.81tex2html_wrap_inline11647, and we have tex2html_wrap_inline11735 = 270.02tex2html_wrap_inline11647, l1 = 6.38tex2html_wrap_inline11647. The inclination of the galactic equator with respect to the ecliptic is tex2html_wrap_inline11759. As already mentioned, l is counted eastwards. With these parameters, the transformations are as follows:

3.3.1. Transformation tex2html_wrap_inline11761


 eqnarray768

3.3.2. Transformation tex2html_wrap_inline11763


 eqnarray777

3.4. Altazimuth and equatorial coordinates

  The transformation depends on local sidereal time tex2html_wrap_inline11765 and on geographical latitude tex2html_wrap_inline11767. Instead of elevation, zenith distance z = will be used. The zenith distance of the celestial pole is tex2html_wrap_inline11771. Both, azimuth A and hour angle tex2html_wrap_inline11775 are counted from the meridian through west.

3.4.1. Transformation tex2html_wrap_inline11777


 eqnarray791

3.4.2. Transformation tex2html_wrap_inline11779


 eqnarray796

3.5. Alternate ecliptic coordinates

  Instead of tex2html_wrap_inline11631 also a sun-centered polar coordinate system is used. Its coordinates are the angular distance from the sun, called elongation tex2html_wrap_inline11635, and a position angle i, counted from the ecliptic counterclockwise,called inclination. The relation between the two sets of coordinates, when describing the position of a field-of-view with respect to the sun, is shown in Fig. 2 (click here).
It is unfortunate that the obliquity of the ecliptic, used in Eqs. (3 (click here)) and (4 (click here)), and the angular distance from the sun, used in Eqs. (11 (click here)) and (12 (click here)), both are designated by the same letter tex2html_wrap_inline11635. However we did not want to change the commonly used notations. In practice this dual meaning rarely should lead to confusion.

  figure801
Figure 2: Relation between the coordinates used for presenting zodiacal light measurements. tex2html_wrap_inline10929 is counted positive towards east, i is counted positive counterclockwise from the ecliptic west of the Sun

3.5.1. Transformation tex2html_wrap_inline11825


 eqnarray848

3.5.2. Transformation tex2html_wrap_inline11849


 eqnarray904

The reader is cautioned that in some papers the differential helioecliptic longitude tex2html_wrap_inline11851 may be called "elongation" or may be designated as "tex2html_wrap_inline11635", contrary to our definition of elongation tex2html_wrap_inline11635 as the angular distance from the sun to the field-of-view.

  figure808
Figure 3: Relation between coordinates tex2html_wrap_inline11785 (lines) and tex2html_wrap_inline11787 (underlying dotted grid) for equinox 2000

  figure814
Figure 4: Relation between coordinates tex2html_wrap_inline11785 (lines) and l, b (underlying dotted grid) for equinox 2000

  figure820
Figure 5: Relation between coordinates l, b (lines) and tex2html_wrap_inline11785 (underlying dotted grid) for equinox 2000 - alternative projection

  figure826
Figure 6: Relation between coordinates tex2html_wrap_inline11787 (lines) and l, b (underlying dotted grid) for equinox 2000

  figure832
Figure 7: Relation between coordinates l, b (lines) and tex2html_wrap_inline11787 (underlying dotted grid) for equinox 2000 - alternative projection

 

Site IB(max) IB(min) IV(max) IV(min) Solar flux Corresponding dates ref.
ESO 22.74 21.69 164 78-02-05 1.
(22.20) (20.85) 168 80-12-06; 80-06-08; 88-12-05
22.97 161 78-02-08
(22.94) 116 78-02-08; 87-12-16; 87-12-19;
21.91 162 78-02-07
(22.02) 94 87-12-15; 87-12-16; 87-12-19;
Calar Alto 22.51(22.30) 61 95-05-26,27,28 2.
23.05(22.98) 176 89-05-06; 90-06-26; 93-06-21
(21.16) 206 89-05-04; 91-06-11; 91-06-16
21.79 61 95-05-27,28,29
San Benito Mt. 22.37 21.32 233 80-04-11; 81-07-28; 82-06-22 3.
23.08 78 76-04-30; 87-04-25; 87-06-29
22.07 76 76-04-30; 87-04-28; 87-06-29
Kitt Peak 22.65 114 88-01-21; 88-03-17; 88-06-14 4.
22.98 75 86-12-02; 86-12-30; 87-06-22
21.60 114 87-11-20; 88-03-17; 88-06-14
22.01 75 86-12-02; 86-12-31; 87-06-22
Crimea 21.91 21.10 122 68-04-28; 71-04-25; 70-08-09 5.
23.05 22.05 136 68-03-29; 68-04-06; 68-04-28
Hawaii 22.27 210 88-11-13; 89-03-28; 89-09-12 6.
23.03 142 87-08-26; 87-11-13; 89-06-10
21.21 166 85-12-13; 88-11-13; 89-03-28
22.05 102 86-06-02; 87-08-26; 88-07-18
McDonald 22.54 21.54 138 60-02-04; 72-12-30; 73-01-12 7.
Observatory 23.01 156 60-01-27; 72-01-11; 72-01-15
21.92 159 60-01-27; 72-01-15; 73-01-08
1. Mattila et al. (1996a).
2. Leinert et al. (1995), Leinert et al. (1996, unpublished).
3. Walker (1988).
4. Pilachowski et al. (1989).
5. Lyutyi & Sharov (1982).
6. Krisciunas (1990).
7. Kalinowski et al. (1975).
  Table 4: Comparison of B and V zenith sky brightnesses at different sites in units of mag/tex2html_wrap_inline11293''. The minimum/maximum values given are averages of the three smallest/largest sky brightness values (nightly averages) given for each site. In the case of ESO and Calar Alto, the numbers in boldface refer to actual B, V measurements, while the numbers in parantheses have been transformed from medium band filter measurements. The given solar 10.7 cm flux value (in units of 104 Jy) is the average of the three nights in question


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