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(I received your reply to my message, but I was already well into this
one, so I'll finish it tonight. If you're in France, there will be a
great waste of time if I hold this message until tomorrow. Here in
California we're 8 hours behind Greenwich. Also, since I'm in a hurry
to get this done, I have not worked the computations twice to verify
them. So if something "does not compute", it's probably my blunder.)

This compares your computation with HORIZONS for the case ∆T = 0.


antoine.m.couette@club-internet.fr wrote:
> -*-*-	Second, I am submitting here-after a fictitious example of an upcoming 
Lunar which shows that - if we were to ignore to-day the effects of delta-T 
on Lunars - we would significantly increase our error in UT determination, 
and therefore our Longitude uncertainty.
> 
> 23 Dec 2009   Height of Eye = 17 ' T = 59�F , P = 29.92 ' Estimated UT / 
Position :  16h58m53.0s / N3707.1W01253.7 - All observations assumed to occur 
at the same time - Distance SUN-MOON = 77�53'7 , SUNL = 5�50'1 , MOONL = 
50�05'0 . All sextant measures are corrected for Instrument error (and only 
instrument error), i.e. all other corrections (dip, refraction, SD, Parallax 
... ) need to be performed from the "raw data" hereabove.

Dip = .0667°, therefore lower limb apparent altitude = 5.7683° (Sun) and
50.0166° (Moon). Refraction (Nautical Almanac formula) = .1451° (Sun)
and .0138° (Moon). Lower limb observed altitude = 5.6232° (Sun) and
50.0029 (Moon).

> *** If we compute observer's position from the observed heights :
> 
> o	With delta-T = 0.0 second :
> 
> Lunar Distance UT  = 17h01m07.4s - i.e. TT = 17h01m07.4s -  hence UT error = 
-2m14.4s, cleared distance = 78�34.00', and
> We also get the following position at time of Lunar Distance :
> N3659.9W01317.1 (result of 2 body fix), and

As explained in my last message, I assume JPL HORIZONS uses ∆T = 66.06
seconds. To simulate ∆T = 0, I move your longitude east by 66.06 * 15 *
1.002738 arc seconds, or .2760°.

HORIZONS says Sun azimuth = 234.6730, unrefracted altitude = 5.8962,
semidiameter = .2710. Altitude - semidiameter = 5.6252 = unrefracted
lower limb altitude. The observed altitude was .0020° (.12') less.

HORIZONS says Moon azimuth = 155.5489, unrefracted altitude = 50.2545,
semidiameter = .2517. Altitude - semidiameter = 50.0028 = unrefracted
lower limb altitude. The observed altitude was .0003° (.02') less.

When we apply refraction to the HORIZONS altitudes, we get 6.0358 (Sun
center) and 50.2682 (Moon center). The refracted center to center
separation angle = 78.4162. Subtract the sum of the semidiameters
(.5227) to get the predicted limb to limb angle, 77.8935. The observed
angle was 77.8950, .0015° (.09') greater.

The predicted altitudes and lunar distance accurately match the
observations when I use your ∆T = 0 solution. Good work!


Now let's return to your original ∆T = 66.70. Here is the output from
my program. (The "target" body is the body other than the Moon; the Sun
in this case.)

Program Lunar2, by Paul S. Hirose.

Initial conditions.

estimated time:
2009-12-23T16:58:53.00 UT1
2009-12-23T16:59:59.68 Terrestrial Time
66.700 seconds delta T

estimated position:
+37.1183° - 12.8950° north lat, east lon
           - 13.1737° ephemeris east lon
5 meters above ellipsoid

atmosphere:
15° C (59° F) at observer
1013.3 mb (29.92" Hg) altimeter setting
1012.6 mb (29.90" Hg) actual pressure

Moon altitude observation:
  50.0167° observed lower limb altitude
   0.0133° refraction
   0.2517° unrefracted semidiameter
  50.2550° unrefracted altitude of center
  50.0853° predicted altitude
   0.1697° intercept
155.3688° predicted azimuth

Sun altitude observation:
   5.7683° observed lower limb altitude
   0.1428° refraction
   0.2710° unrefracted semidiameter
   5.8965° unrefracted altitude of center
   5.9377° predicted altitude
- 0.0412° intercept
234.5518° predicted azimuth

Moon to Sun predicted separation angle:
   78.5147° center to center, unrefracted
    0.1037° refraction
   78.4110° center to center, refracted
    0.2517° Moon near limb refracted semidiameter
    0.2680° target near limb refracted semidiameter
   77.8913° Moon near limb to Sun near limb
   77.8950° observed angle
    0.0037° observed - predicted

separation angle rate of change:
+0.00493° per minute (topocentric)
75% of total angular velocity

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

Solution, after 3 iterations.

corrected time:
2009-12-23T16:59:36.04 UT1
2009-12-23T17:00:42.76 Terrestrial Time
66.700 seconds delta T

corrected position:
+36.9973° - 12.9033° north lat, east lon
           - 13.1820° ephemeris east lon
79° LOP crossing angle

geocentric coordinates (true equator and equinox):
23.30103h + 0.4240°  Moon RA and dec.
0.2487° apparent semidiameter
18.14559h -23.4235°  Sun RA and dec.
0.2710° semidiameter

geocentric separation angle and rate:
   78.5632° center to center
+0.00767° per minute
84% of total angular velocity

illumination conditions:
234.7° 5.9° Sun unrefracted az, el
265° Moon to Sun position angle (0 = 12 o'clock)
101° Moon phase angle (0 = full, 180 = new)

position angles:
265° Moon to Sun
40° Sun to Moon

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

Moon altitude observation:
  50.0167° observed lower limb altitude
   0.0133° refraction
   0.2518° unrefracted semidiameter
  50.2550° unrefracted altitude of center
  50.2550° predicted altitude
   0.0000° intercept
155.5538° predicted azimuth

Sun altitude observation:
   5.7683° observed lower limb altitude
   0.1428° refraction
   0.2710° unrefracted semidiameter
   5.8965° unrefracted altitude of center
   5.8965° predicted altitude
   0.0000° intercept
234.6733° predicted azimuth

Moon to Sun predicted separation angle:
   78.5192° center to center, unrefracted
    0.1045° refraction
   78.4147° center to center, refracted
    0.2517° Moon near limb refracted semidiameter
    0.2680° target near limb refracted semidiameter
   77.8950° Moon near limb to Sun near limb
   77.8950° observed angle
    0.0000° observed - predicted

separation angle rate of change:
+0.00492° per minute (topocentric)
75% of total angular velocity


Now check this with HORIZONS. The longitude is moved west by .0027° to
correct for the discrepancy in ∆T (66.70 vs. 66.06).

HORIZONS says Sun azimuth = 234.6733, unrefracted altitude = 5.8965,
semidiameter = .2710. Altitude - semidiameter = 5.6255 = unrefracted
lower limb altitude. The observed altitude was .0023° (.14') less. This
is not a really bad result, but my program should be more accurate.

The problem is refraction. The Sun lower limb apparent altitude (i.e.,
with correction for dip) is 5.7683° The Nautical Almanac formula says
refraction is .1451° at this altitude. But my program uses the
Astronomical Almanac low altitude formula, which gives .1428°. A 1.6%
difference between refraction formulas seems reasonable at this low
altitude. In fact, I would avoid shooting a lunar with one body so near
the horizon.

If I use my program's refraction value to correct the observation, the
unrefracted lower limb altitude = 5.6255, a perfect match to the value
from HORIZONS.

HORIZONS says Moon azimuth = 155.5539, unrefracted altitude = 50.2550,
semidiameter = .2517. Altitude - semidiameter = 50.0033 = unrefracted
lower limb altitude. The observed altitude was .0004° (.02') less.

When we apply refraction to the HORIZONS altitudes, we get 6.0362 (Sun
center) and 50.2686 (Moon center). The refracted center to center
separation angle = 78.4129. Subtract the sum of the semidiameters
(.5227) to get the predicted limb to limb angle, 77.8902. The observed
angle was 77.8950, .0048° (.29') greater. Again, at this low altitude
the solution is sensitive to the refraction formula. If both bodies had
more altitude, I'm sure the results from my program would not be so
disappointing.

Of course, for these tests I could simply use the same refraction
formulas as my program. Then all the discrepancies would disappear. But
I think the real world refraction variations will be greater than the
differences between formulas, so this is a good lesson in avoiding low
altitude lunars.


I should have explained in my last message that the HORIZONS Web site is
http://ssd.jpl.nasa.gov/?horizons . There is a Web interface for the
system, but if you do many computations of the same type, I think it's
less trouble to use the email interface. You can save your old command
emails and simply modify the values as needed.


My lunar distance program is available at my Web site:
http://home.earthlink.net/~s543t-24dst/sofajplNet/LunarDist2.html

But it's not a "real program" because I'm not a "real lunars man". I
wrote it to demonstrate my positional astronomy DLL, and also so I could
run my own computations when lunars are discussed. There is no user
interface. To set the input data, you modify the values in the Basic
source code, re-compile the program, and run it! This is primitive, but
computers are so powerful now, I can barely stand up from my chair
before the recompile and execution are complete. Since I don't use the
program much, it's not worthwhile to write a proper user interface.
However, anyone is welcome to use this as a basis for a real program.
All the source code is free.


By the way, your first email has some strange format problems. There are
many • in the message and the degree symbol (°) doesn't appear.
When I view the message on the Web, • appears as a bullet (•), but
there's a little empty box where the degree symbols should appear.

There are many interesting points in your reply, but unfortunately I
don't have time to get to them before tomorrow.

-- 


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