Shadow entry of satellites

Greg Roberts (grr@da.saao.ac.za)
Thu, 23 Jan 1997 09:36:54 +0200

Copy to See-Sat:

Morning Rob,

Thanks for your message re the shadow procedure used in SKYMAP. Although
I have not used SKYMAP extensively - I only just recently revived my optical
tracking interest recently with the Dec 96 Titan launch on account of my
geographical location - I have had no problems with the question of shadow
entry. In all cases where I specifically choose a point for positional work
as high above the horizon as possible, and thus very close to earth shadow
for the 96072* objects, I have always seen the satellite and usually see it
going into shadow almost immediately afterwards so I have total confidence
in your shadow routine.

Ive dug the article on solar illumination out of our library at work-the
South African Astronomical Observatory where I am employed as an astronomer,
(same place as Willie Koorts except he is in our Electronics Department) and
the details are:

"Artificial Earth Satellites" Vol 39 No 2  1961 August MEMOIRS OF THE
BRITISH ASTRONOMICAL ASSOCIATION,Radio-Electronics section.
The article is by Gordon E.Taylor and starts on page 119,ends on page 138
of which approximately 13 pages are tables ( no pc's in 1961 !). Ill quote
the interesting bits from the article:
"The actual shadow entry or exit is clearly not an instantaneous phenomenon.
The width of the Earth's penumbral shadow increases as the distance from the
Earth increases. A further complication arises from the extent of the Earth's
atmosphere and its properties of refracting light and casting shadow.Thus
neither the outer edge of the penumbra or the division between the umbra
and the penumbra can be said to have a sharp,clearly-defined edge.
As the distance from the Earth increases,so the velocity of the satellite
decreases,both in actual km/sec and in angular motion as seen by the
observer.This fact,combined with the increasing width of the penumbra,will
cause the observed fading to occur more and more slowly. In addition the
rate of fading will also be affected by the angle as which the satellite 
enters the shadow,relative to the shadow's axis. For example Echo 1 at a
height of 1500km has,on occassions,been observed to gradually diminish in
brightness in the penumbral shadow for almost 20 seconds before final
extinction in the umbra..
The problem of determining whether or not a satellite is illuminated by
the sun may be solved in two stages. The first stage is to determine the
geocentric angular distance (Theta) between the Sun and the satellite.The
second stage is to determine,for a given height,the limiting geocentric
angular distance, ThetaP and ThetaE.
In view of the statements made in the first two paragraphs of this section,
the limits ThetaP and ThetaE have had to be chosen in a rather arbitrary
manner. However they have been selected to fit the observations of shadow
entry of Echo 1 so far available.
The limit ThetaP corresponds to the first appreciable fading of a satellite,
and is measured from the locus of a point on the axis of one side of the
penumbra.In other words,this limit is attained when the satellite crosses
the path of a ray of light from the centre of the solar disk and tangetial
to the Earth's surface, when the satellite is on the opposite side of the
Earth to the sun. Therefore, strictly speaking, ThetaP is not a precise
limit but merely an indication that the gradual transition from sunlight
to darkness is in progress.
The limit ThetaE indicates the point where the satellite has faded by about
10 magnitudes as compared with its brightness in direct sunlight.For
practical purposes this point could be described as the point of extinction."
Later:
"The formulae used for the formation of the limits ThetaP and ThetaE were:

                 cos x  =   6370/(6370 +H) = 6370/r
                 
                 ThetaP = 90.0 degrees + x
                 ThetaE = 90.7 degrees + x
                 
where 6370 is mean radius for the earth in kilometres, and H is height of
satellite above earths surface.
The value 90.7 degrees was chosen to fit observations of Echo 1. It may well
be that further observations,at different heights,will render some revision
of the values on which this table is based,but it would appear unlikely that
major alterations will be necessary"

Typical values are:

                 r(kms)    H(kms)       ThetaP      ThetaE
                -------------------------------------------
                 6570       200          104.2       104.9
                 6670       300          107.3       108.0
                 6770       400          109.8       110.5
                 6870       500          112.0       112.7
                 6970       600          114.0       114.7
                 7070       700          115.7       116.4
                 7370      1000          120.2       120.9
      
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Okay, rather a long quote but it might be of interest to others interested
in optical tracking so Ill post the above to See_Sat as well.

Cheers
Greg

End of bit for See-Sat