Re: Orbit's pole

Frank Reed (f-reed@metabien.com)
Sat, 25 Jul 1998 19:34:35 -0700

Grzegorz Koralewski <caliban@kki.net.pl> said...
>I've tried to see the structure of Mir several times,
>but tracking it with my equatorially-mounted Newtonian
>has always been a barrier.
>Recently I thought, that actually hand-tracking would be
>much easier if I knew where the pole of the orbit
>was. What I'm looking for, is a computer program that
>will tell me RA and Dec of the orbit's pole. I would have to
>care only for one axis then.

I've recently been working on some Excel spreadsheet macros to do vector math
for satellite calculations, and this problem looked like a good exercise to
check out my work.  Perhaps the results might be of interest to some.

As has been pointed out by others, the satellite's path across the sky is
not a
circle on the celestial sphere, so it can't be tracked exactly by moving the
telescope about only one axis. However, the following example shows that a
polar axis can be chosen such that the variation about the declination axis is
only a fraction of a degree.

The proposed procedure for determining the polar axis alignment is as follows:

Azimuth and Elevation of the satellite as it crosses the sky.  You can then
choose three points along its path and calculate the center of the spherical
circle through these three points.  Align the polar axis of your telescope to
the center of this circle, and the declination settings for the satellite at
these three points will be equal for the three chosen points.  The declination
setting will vary somewhat at other points, but only a fraction of a degree
(at
least in the following example).  The three points should be chosen near the
beginning, middle, and end of the pass.

To calculate the center of the circle for the chosen points:

1.  Calculate the rectangular x, y, z components for the three points - for
each point the components are as follows:
 x = cos(El) * cos(Az)
 y = cos(El) * sin(Az)
 z = sin(El)

2.  Calculate the vector differences between the first and second point and
between the second and third point:

 V21 = V2 - V1
 V32 = V3 - V2

V21 is normal (perpendicular) to the plane of the great circle bisecting
points
1 and 2 on the celestial sphere, while V31, while V32 is normal to the
bisector
of points 2 and 3.

3.  Take the vector cross product VC of V21 and V32.

 VC = V21 cross V32 

The direction of the cross product is normal to the direction of both V21 and
V32, and thus aligns with the intersection of the planes of the two great
circle bisectors.  The intersection of this line of intersection with the
celestial sphere is the "center" of the spherical circle through the three
points.  The cross product is calculated as follows:

 VC.x = V21.y * V32.z - V21.z * V32.y
 VC.y = V21.z * V32.x - V21.x * V32.z
 VC.z = V21.x * V32.y - V21.y * V32.x

Conversion of VC to spherical coordinates will give the Azimuth and Elevation
of the center of the circle through the chosen points.

Following is a sample calculation for MIR's pass here in the Phoenix area this
evening.
20:50:00 to do the calulation.

The vector calculation results are as follows:
        Az      El     x          y          z
V1   255.00   12.40  -0.252781  -0.943393   0.214735
V2   320.70   34.80   0.635438  -0.520100   0.570714
V3    26.30   12.50   0.875236   0.432569   0.216440
V21                   0.888220   0.423293   0.355978
V32                   0.239798   0.952669  -0.354274
VC   140.72   49.69  -0.489091   0.400036   0.744674

Thus, the polar axis of the telescope should be aligned with Az = 140.72, El =
49.69 degrees.  The following table shows the satellite Az and El and the
position of the telescope setting circles aligned as stated above to center
the
satellite in the telescope's field of view.  Note that the declination setting
circle value is the same (-5.51 deg) at the three selected points and that for
satellite elevations above 10 degrees the variation about this value is less
than 0.4 degrees.

         Skytrace Ephemeris Data: MIR - Sat# 16609
         Date: 07/25/98 (MST) lon -111.898 late 33.484 el 1227 ft

   Satellite Position       Setting Circle
                            Angles - deg     
  Time      Az.    El.         RA    Dec 
------------------------    -------------

20:43:30    245.9    3.9    104.18  -6.73
20:44:00    248.2    6.3    107.49  -6.29
20:44:30    251.2    9.1    111.57  -5.91
20:45:00    255.0   12.4    116.57  -5.51
20:45:30    260.4   16.3    123.12  -5.36
20:46:00    268.1   21.2    131.94  -5.19
20:46:30    279.7   26.8    143.96  -5.27
20:47:00    297.3   32.3    160.28  -5.42
20:47:30    320.7   34.8    179.98  -5.51
20:48:00    344.1   32.3   -160.31  -5.42
20:48:30      1.6   26.9   -144.12  -5.24
20:49:00     13.2   21.3   -132.10  -5.17
20:49:30     21.0   16.4   -123.21  -5.30
20:50:00     26.3   12.5   -116.74  -5.51
20:50:30     30.2    9.2   -111.67  -5.85
20:51:00     33.2    6.5   -107.64  -6.15
20:51:30     35.5    4.1   -104.33  -6.60


--
Frank Reed
Metabien Software