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antoine.m.couette@club-internet.fr wrote:
> The Lunar Distance example happened on May 22, 1901 in the evening . See 
> enclosure pages 1 to first part of page 7

The French gives me some trouble, but I believe "les hauteurs vraies"
are the unrefracted altitudes. On that basis, UT1 = 22:20:34 (10:29:55
mean astronomical time at Paris), lat = +47°49'33", lon = -79°37'23".

Arago's time nomenclature is unclear to me. Time scale A is a complete
mystery. I suspect M is chronometer time, and Tmp the mean time at
Paris. The PDF document says Tmp = 10:29:47, which is only 8 seconds
from my estimate.

Here's the output of my program. The initial altitude intercepts exceed
1°, but after only 4 iterations the solution converges to .1".

Initial conditions.

estimated time:
1901-05-22T22:30:00.00 UT1
1901-05-22T22:29:59.09 Terrestrial Time
-0.907 seconds delta T

estimated position:
+48°23'00.0" - 79°25'00.0" north lat, east lon
              - 79°24'46.4" ephemeris east lon
0 meters above ellipsoid

atmosphere:
26° C (79° F) at observer
1019.9 mb (30.12" Hg) altimeter setting
1019.9 mb (30.12" Hg) actual pressure

Moon altitude observation:
  54°58'23.9" observed center altitude
      0'38.9" refraction
  54°57'45.0" unrefracted altitude of center
  53°49'43.7" predicted altitude
   1°08'01.3" intercept
202°06'39.5" predicted azimuth

Sun altitude observation:
  23°59'34.9" observed center altitude
      2'04.9" refraction
  23°57'30.0" unrefracted altitude of center
  22°18'15.3" predicted altitude
   1°39'14.7" intercept
276°02'15.8" predicted azimuth

Moon to Sun predicted separation angle:
   62°46'14.1" center to center, unrefracted
       1'45.0" refraction
   62°44'29.1" center to center, refracted
      15'51.0" Moon near limb refracted semidiameter
      15'46.5" target near limb refracted semidiameter
   62°12'51.6" Moon near limb to Sun near limb
   62°09'26.0" observed angle
-  0°03'25.6" observed - predicted

separation angle rate of change:
+21.90" per minute (topocentric)
82% of total angular velocity

--------------------

Solution, after 4 iterations.

corrected time:
1901-05-22T22:20:33.83 UT1
1901-05-22T22:20:32.93 Terrestrial Time
-0.907 seconds delta T

corrected position:
+47°49'33.2" - 79°37'22.8" north lat, east lon
              - 80°36'17.6" ephemeris east lon
76° LOP crossing angle

geocentric coordinates (true equator and equinox):
  8h19m03.74s +14°28'19.1"  Moon RA and dec.
15'38.5" apparent semidiameter
  3h55m49.26s +20°23'15.4"  Sun RA and dec.
15'47.6" semidiameter

geocentric separation angle and rate:
   62°40'47.3" center to center
+30.80" per minute
87% of total angular velocity

illumination conditions:
274.4° 24.0° Sun unrefracted az, el
273° Moon to Sun position angle (0 = 12 o'clock)
117° Moon phase angle (0 = full, 180 = new)

position angles:
273° Moon to Sun
39° Sun to Moon

recommended limbs:
Use Moon upper limb.
Use Moon near limb.

Moon altitude observation:
  54°58'23.9" observed center altitude
      0'38.9" refraction
  54°57'45.0" unrefracted altitude of center
  54°57'45.0" predicted altitude
   0°00'00.0" intercept
198°19'29.2" predicted azimuth

Sun altitude observation:
  23°59'34.9" observed center altitude
      2'04.9" refraction
  23°57'30.0" unrefracted altitude of center
  23°57'30.0" predicted altitude
   0°00'00.0" intercept
274°25'51.6" predicted azimuth

Moon to Sun predicted separation angle:
   62°42'43.5" center to center, unrefracted
       1'39.6" refraction
   62°41'03.9" center to center, refracted
      15'51.3" Moon near limb refracted semidiameter
      15'46.6" target near limb refracted semidiameter
   62°09'26.0" Moon near limb to Sun near limb
   62°09'26.0" observed angle
-  0°00'00.0" observed - predicted

separation angle rate of change:
+21.69" per minute (topocentric)
82% of total angular velocity


> The Lunar Occultation example happened on Mar 02, 1901 - NOT the same 
> date as for the Lunar Distance - and you can find all relevant data from 
> the middle of page all the way down to the end.

I compute an occultation time 18 seconds different: 22:43:38.4 UT vs.
22:43:56.38 (Arago). At his time, my program says the star is -6.4"
inside the Moon's limb.

My program iteratively seeks the time, latitude, and longitude where all
three angles (separation angle and altitudes, the latter two perhaps
calculated) match the inputs. But in an occultation, the LOPs for the
Moon and star cross at a very small angle, so any error in altitude,
even roundoff error, has a large effect on position. Thus the latitude
and longitude move away from the correct values.

For that reason, the program's automatic solution is inaccurate for this
type of problem. I simply used its predicted separation angle and rate
of change to manually solve for time. Re-running the program with the
adjusted time gave this:

estimated time:
1901-03-02T22:43:38.40 UT1
1901-03-02T22:43:37.17 Terrestrial Time
-1.235 seconds delta T

estimated position:
+48°23'30.0" -  4°29'31.2" north lat, east lon
              -  4°29'12.6" ephemeris east lon
0 meters above ellipsoid

atmosphere:
15° C (59° F) at observer
1013.3 mb (29.92" Hg) altimeter setting
1013.3 mb (29.92" Hg) actual pressure

Moon altitude observation:
  52°51'07.7" observed center altitude
      0'43.4" refraction
  52°50'24.3" unrefracted altitude of center
  52°50'24.3" predicted altitude
- 0°00'00.0" intercept
181°39'12.9" predicted azimuth

kappa Cancri altitude observation:
  52°40'34.6" observed center altitude
      0'43.7" refraction
  52°39'50.9" unrefracted altitude of center
  52°39'50.9" predicted altitude
   0°00'00.0" intercept
181°20'46.3" predicted azimuth

Moon to kappa Cancri predicted separation angle:
    0°15'21.9" center to center, unrefracted
       0'00.3" refraction
    0°15'21.6" center to center, refracted
      15'21.6" Moon near limb refracted semidiameter
-  0°00'00.0" Moon near limb to kappa Cancri center
    0°00'00.0" observed angle
    0°00'00.0" observed - predicted

separation angle rate of change:
-21.31" per minute (topocentric)
94% of total angular velocity

geocentric coordinates (true equator and equinox):
  9h01m43.73s +11°47'53.5"  Moon RA and dec.
15'10.0" apparent semidiameter
  9h02m25.79s +11°03'44.2"  kappa Cancri RA and dec.
0'00.0" semidiameter

geocentric separation angle and rate:
    0°45'20.5" center to center
-15.87" per minute
51% of total angular velocity

illumination conditions:
322.5° -43.0° Sun unrefracted az, el
275° Moon to Sun position angle (0 = 12 o'clock)
28° Moon phase angle (0 = full, 180 = new)

position angles:
133° Moon to kappa Cancri
314° kappa Cancri to Moon

recommended limbs:
Use Moon upper limb.
Use Moon far limb.


The altitude intercepts are perfect because I used the altitudes
computed by the program as the "observations".

Star position, proper motion, and parallax from the Hipparcos catalog.
Delta T from the USNO MICA program. Sun and Moon position from the JPL
DE406 ephemeris. Precession and nutation with the IAU 2006 model.

Delta T was *negative* in 1901. That unusual circumstance revealed a bug
in my program, which in turn was due to a bug in my positional astronomy
DLL. It's in a feature which isn't used much, and fortunately there's an
easy workaround. I'll update the Web site with a description of the
workaround, and issue a fix soon.

-- 

   

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