RE: Satellite solar illluminati

Ted Molczan (molczan@fox.nstn.ca)
Sun, 26 Jan 1997 20:22:03 -0500

Rob Matson wrote to Greg Roberts on 22 Jan:

>Was following your thread with Ted regarding satellite shadow entry.  I =
spent
>a lot of time on the refractive aspects of this problem, and through a
>combination of shadow entry timings, runs of LOWTRAN 7 under various =
volcanic,
>cloud, and seasonal conditions, and adjustment for seasonal atmospheric
>variations, I feel that the algorithm I use in SkyMap is pretty good.

The apparent significant difference between my eclipse predictions and =
Rob's,
which started this thread, caused me to re-visit the issue over the past =
few=20
days. As a result, I believe I have confirmed the 15 km to 20 km =
atmosphere
allowance proposed by Rob, and I now have a much improved eclipse =
prediction
algorithm. However, for some reason, the difference between my =
predictions=20
and Rob's has not been resolved, and in fact it seems to have increased. =
That
is something that Rob and I can sort out later. Here is a general report =
on
my analysis and findings.

Background
----------
My approach to eclipse prediction has varied somewhat over the years, =
but I=20
was never completely satisfied with my state-of-the-art. I especially =
struggled
with atmospheric refraction and opacity, and largely avoided the Earth's
oblateness. Eventually, I decided to ignore these factors, and simply =
compute=20
the mid-point of the penumbra crossing, using the Earth's mean radius of =

6371 km, and a fixed apparent solar semi-diameter of 0.2666 deg. This =
often gave reasonable predictions, in my opinion because these factors =
often canceled
each other out. Of course, when they did not, then the predictions were =
poor.

Analysis
--------
I decided to try to compare one of my eclipse predictions with one of =
Rob's.
I recalled that Rob posted the following on 7 Jan:

>Based on Ted Molczan's latest predictive element set for the USA 129 =
payload,
>I took a look at the bright stars that the satellite should pass near =
for Rob
>McNaught on Jan. 7, 1997.  All times that follow are local times for =
Rob (UTC
>+ 10 hours):
>
>22:09:50   appulse w/ Alp Aps (mag 3.8)
>22:12:03   appulse w/ Alp Men (5.1)
>22:13:14   < 1-deg from Alp Dor (3.5)
>22:13:38   appulse w/ Del Cae (5.2)
>22:14:05   < 0.5-deg from 43 Eri (4.1)
>22:14:10   shadow entry begins
>22:14:22   payload completely in umbra

The search elements Rob referenced were:

USA 129 search  15.0  3.0  0.0  5.1 v
1 24680U 96072  A 97 05.34099537  .00000000  00000-0  00000-0 0    08
2 24680  97.8720  71.0700 0550000 112.3000 253.7500 14.72115144    09

Here is the result I obtained from my program, as it would have appeared
from Rob McNaught's vantage:

 7/ 1/97  10:00 - 15:00 UTC  J2000.0  EL > 15  Robert McNaught
99992A         99992A   99992   Bull =3D   0
SGP4  Age =3D    2.2 d  Unc =3D    0 s ( 50%)

  TIME      %I   Mv     AZ  EL    R.A.    DEC   FE   VANG  RANGE   ALT
--------    --  ----   ---  --   -----  ------  --   ----  -----  -----
12:12:50    49   5.3   185  57   04:51  -63:17   7   0.33   1109    960
12:13:13    54   5.1   192  64   04:41  -55:16   7   0.37   1038    954
12:13:33    59   4.9   204  71   04:34  -47:31   7   0.40    992    948
12:13:52    64   4.7   226  76   04:28  -39:36   8   0.42    964    942
12:14:10    68   4.6   262  78   04:24  -31:48   9   0.43    954    937
12:14:23 ES 71   4.6   287  76   04:21  -26:04  10   0.43    956    933

In this case, ES (enters shadow) at 12:14:23 refers to the mid-point of=20
the penumbra crossing, but this is one second after Rob predicts entry=20
into the umbra, i.e. total eclipse. It was difficult to make a full=20
comparison, because I computed only the mid-point, while Rob computed =
the=20
penumbra and the umbra entry. So I spent a couple of hours modifying my=20
program to do the same, with the following result:

  TIME      %I   Mv     AZ  EL    R.A.    DEC   FE   VANG  RANGE   ALT
--------    --  ----   ---  --   -----  ------  --   ----  -----  -----
12:12:50    49   5.3   185  57   04:51  -63:17   7   0.33   1109    960
12:13:13    54   5.1   192  64   04:41  -55:16   7   0.37   1038    954
12:13:33    59   4.9   204  71   04:34  -47:31   7   0.40    992    948
12:13:52    64   4.7   226  76   04:28  -39:36   8   0.42    964    942
12:14:10    68   4.6   262  78   04:24  -31:48   9   0.43    954    937
12:14:17 EP 70   4.6   276  77   04:22  -28:40  10   0.43    954    934
12:14:29 EU 72   4.6   295  74   04:20  -23:31  10   0.43    960    931

So both our predictions called for the eclipse to take 12 s, but mine =
started
7 s later. This difference was sufficient inspiration to experiment with =
some
of the ideas in Rob's discussion of atmospheric opacity and refraction, =
and to
generally improve my eclipse algorithm. I made the following changes:

- replaced the fixed Earth-radius model with an oblate-Earth model.
  This means the program now takes into account Earth's oblateness
  in determining the radius from the centre of the Earth at which the
  shadow grazes the Earth's surface. This required an additional =
iterative
  process in the program's eclipse routines. The difference between the
  Earth's equatorial and polar radius is about 21 km, so this cannot be
  ignored if one is going to experiment with the effects of the =
atmosphere,
  which is of the same order of magnitude.

- replaced the fixed apparent solar semi-diameter assumption with a=20
  proper calculation of the penumbral and umbral angles.

- added Rob's 15 km atmospheric opacity allowance and his corresponding
  0.2 deg refraction allowance (his winter setting).

The result was:

 7/ 1/97  10:00 - 15:00 UTC  J2000.0  EL > 15  Robert McNaught
99992A         99992A   99992   Bull =3D   0
SGP4  Age =3D    2.2 d  Unc =3D    0 s ( 50%)

  TIME      %I   Mv     AZ  EL    R.A.    DEC   FE   VANG  RANGE   ALT
--------    --  ----   ---  --   -----  ------  --   ----  -----  -----
12:12:50    49   5.3   185  57   04:51  -63:17   7   0.33   1109    960
12:13:13    54   5.1   192  64   04:41  -55:16   7   0.37   1038    954
12:13:33    59   4.9   204  71   04:34  -47:31   7   0.40    992    948
12:13:52    64   4.7   226  76   04:28  -39:36   8   0.42    964    942
12:14:10    68   4.6   262  78   04:24  -31:48   9   0.43    954    937
12:14:19 EP 70   4.6   280  77   04:22  -27:45  10   0.43    955    934
12:14:31 EU 73   4.6   298  74   04:19  -22:37  10   0.43    962    930

The difference between our predictions grew to 9 s! This is a =
significant
difference, because it is almost equal to the duration of the penumbral
crossing. This raised the question of which prediction was more =
accurate.=20

Fortunately, I received an observation of this very pass from one of my
correspondents, who wishes not to be identified. He stated that he =
tracked=20
the object into the shadow using 10x50 binoculars, until it vanished at=20
12:14:26 +/- 1 s. This would seem to be the approximate time of full =
passage=20
into the umbra. This was about 4 s later than Rob's prediction, and =
completely=20
outside his penumbral crossing period, but 5 s earlier than my =
prediction,=20
and in the middle of my penumbral crossing prediction.

However, the orbit upon which these predictions were based, had zero =
values
for orbital decay, so the pass, as well as the eclipse events, should =
have
occurred several seconds *prior* to the prediction. An accurate position =
and
time was measured by the same observer on this pass, which revealed that
the object was very near the predicted path, but 4.5 s earlier than =
predicted.
Subtracting this difference yields the following corrected prediction:

12:12:37    45   5.5   200  53   03:36  -62:41   7   0.32   1149    963
12:13:01    50   5.3   211  59   03:38  -54:24   8   0.36   1080    956
12:13:22    54   5.1   225  64   03:39  -46:25   8   0.39   1037    950
12:13:41    59   4.9   243  66   03:39  -38:45   9   0.41   1014    944
12:14:00    63   4.9   263  66   03:39  -30:53   9   0.41   1007    938
12:14:15 EP 66   4.8   278  65   03:39  -24:45  10   0.41   1014    934
12:14:27 EU 68   4.8   288  63   03:38  -19:56  10   0.40   1026    930

This reveals an even closer agreement between the observed vanishing of =
the=20
satellite at 12:14:26 and its predicted entry into the umbra, at =
12:14:27.

I made one additional test, this time using Rob's summer values of atm
opacity and refraction, 20 km and 0.1 deg, respectively:

12:12:37    45   5.5   200  53   03:36  -62:43   7   0.32   1148    962
12:13:01    50   5.3   211  59   03:38  -54:25   8   0.36   1079    955
12:13:22    54   5.1   225  64   03:39  -46:26   8   0.39   1037    949
12:13:41    59   4.9   243  66   03:39  -38:46   9   0.41   1014    944
12:14:00    63   4.9   263  66   03:39  -30:54   9   0.41   1007    938
12:14:11 EP 65   4.8   274  65   03:39  -26:25  10   0.41   1011    935
12:14:23 EU 67   4.8   284  63   03:38  -21:34  10   0.40   1021    931

In this case, the effect was to degrade the prediction, but it did seem
to confirm Rob's finding that the opaque part of the atmosphere extends
to 15 to 20 km altitude, and the corresponding refraction angles.

For some reason, Rob's prediction disagreed greatly with mine:

Predicted entries:
-----------------
Penumbra   Umbra
--------  --------
12:14:05  12:14:17   R. Matson, SeeSat-L message
12:14:15  12:14:27   Molczan,   15 km atmosphere
12:14:11  12:14:23   Molczan,   20 km atmosphere

Each of these predictions was adjusted for the observed 4.5 s early=20
satellite arrival. It was observed to enter umbra at 12:14:26 +/- 1 s.

I decided to perform some additional tests, using precise positional
observations of objects in mid-eclipse, provided by Russell Eberst.
Russell has confirmed that in each case, the object was observed to be
fading in mid-eclipse - he did not consult an ephemeris in deciding
whether or not the object was in mid-eclipse, so there is no bias.

The purpose of the test was to determine the frequency with which =
Russell's
observations fell within the penumbra; and the deviation of those that=20
fell outside. In each case, I obtained SGP/SGP4 orbital elements as =
close
in epoch to the observation as possible, typically no more than 3 or 4 =
days.
Even then, certain objects had experienced sufficient decay to cause=20
prediction errors of anywhere from several seconds to several minutes.
The predictions were adjusted to reduce these residuals to near-zero - =
an
essential step to ensure a proper comparison between the predicted and
observed eclipse events. Russell's precise positional observations were
absolutely vital to making these corrections.

I analyzed 28 of Russell's observations, made between Sep'95 and Dec'96,
plus the aforementioned 96072A obs by an anonymous correspondent.

Using Rob's 15 km atmosphere, and 0.2 deg refraction, I found that 5 obs =

were earlier than the predicted penumbra-entry, by an average of 6 s. =
One=20
point was later than the predicted umbra-entry, by 2 s.

Using a 20 km atmosphere, and 0.1 deg refraction, I found that 3 obs =
were
earlier than the predicted penumbra-entry, by an average of 1.3 seconds.
Six points were later than the predicted umbra-entry, by an average of =
3.2 s.

Finally, based on these two findings, I tried to "interpolate" to find
an atmosphere height that would more closely balance the number of early
and late events, and reduce their average deviation. So far, I have made =

and tested only one such interpolation, based on 18.33 km and 0.1261 deg
refraction.

I found that 5 points were early, by an average 1.8 s; and 4 points were
late, by an average of 2.8 s. This is a small improvement over both the=20
15 km and 20 km atmospheres, but Rob may be pleased to see that it falls
nicely between them.

I will provide more information on my findings, plus Russell's =
observations,=20
and my closest corresponding NORAD 2-line elements to anyone who is =
interested.

Thanks to Greg Roberts and Rob Matson for the inspiration and new ideas,
and to the outstanding observer Russell Eberst, without whom =
"in-sitting"=20
analysts would have nothing to do.

Ted Molczan