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Re: Captain Cook's Sep 07th, 1773 Lunar revisited
From: Paul Hirose
Date: 2012 Jul 20, 16:47 -0700

```Antoine Couëtte wrote:
> From Ulietea Harbour (S 16°45'33" W151°29'48") the Moon Upper Limb Height
was 43°29'0 at time UT = 17h02m55s0 and the Moon was in Azimuth 312°9. I am
computing this way with the Moon first since most likely the Moon Height is
the most reliable one for refraction correction and also for the reasons
given in my previous post on this subject.

I also computed a time sight with the Moon, at the same assumed position
(S16°45.55' W151°29.80' +5 meters), atmosphere conditions (78° F at
observer, altimeter setting 29.92), dip (3' 50"), and ∆T (+16.4 s).
Results (UT1):

17:07:19  Couëtte lunar distance
17:02:55  Couëtte time sight
17:03:18  Hirose time sight

At my time the observed altitude of the Sun (corrected for semidiameter,
refraction, etc.) is 17.1' more than the predicted altitude. The
observed lunar distance is .9' less than the predicted distance.

The Astronomical Almanac table in Section K says ∆T = +16 at 1773.0 and
also at 1774.0, so +16.4 s is reasonable.

My computation confirms that Antoine's assumed position, the lunar
distance, and the altitudes cannot all be correct:

-00.004' observed - predicted lunar distance
+41.95'  Moon altitude observed - predicted
-39.69'  Sun altitude observed - predicted

The above data are based on his assumed position and the time he
computed from the lunar distance (with extreme accuracy). For the Sun
and Moon observed altitudes I used the angles he gave, less 3' 50" for dip.

I also solved for time and position:

17:07:18.5 S16°45.55' W151°29.80'  initial values
17:05:54.8 S16°30.65' W151°53.78'  Hirose solution
17:06:04.9 S16°22.4'  W151°53.8'   Couëtte "Second Solution"

Kermit's solution makes the observed and computed lunar distances
virtually identical. That's good. But the observed Sun altitude is 3.8'
less than the computed value, and the Moon altitude 3.9' less.

Not that my solution is much help. The computed angles match the
observations to .01', but the position is tens of miles away from the
modern coordinates of the anchorage.

I have followed Alex's contributions with interest. His idea that
perhaps the date was wrong seemed good enough to check out. But I
discovered the lunar rate of 12°/day removes ambiguity. Given a lunar
distance and approximate date, but no information on the observer's
position, you can still compute a date and time accurate to about 2
hours. Simply assume the angle is geocentric. The parallax error will be
1° at most.

As for his theory that the altitude observations were not simultaneous
with the lunars, the Moon and Sun altitude rates were -10.7 and +14.1
'/min, respectively, and the observed - computed were +42.0' and -39.7'.
This implies the altitudes were shot 3 or 4 minutes before the mid time
of the lunars. But I wonder, were the observers that dumb? Isn't it more
logical, and just as easy, to shoot 5 lunars, 2 altitudes, then 5 more
lunars?

Here is the full output from my lunar program, where I start with
Kermit's values and let it solve for time and position. Ephemeris is the
JPL DE422, their latest long term ephemeris, successor to DE406. IAU
2006 precession and 2000A nutation models. GRS80 ellipsoid.

The "observed" altitudes are actually 3' 50" less than the observed
angles to correct for dip. My program does not automatically apply that
correction.

Program Lunar2, by Paul S. Hirose.

Initial conditions.

estimated time:
1773-09-07 17:07:18.52 UT1
1773-09-07 17:07:34.88 TT (Gregorian)
JD 2368884.5 + 0.713598 (TT)
16.400 seconds delta T

estimated position:
-16°45.55' -151°29.80' north lat, east lon
5 meters above ellipsoid

atmosphere:
25° C (78° F) at observer
1013.3 mb (29.92" Hg) altimeter setting
1012.7 mb (29.90" Hg) actual pressure

Moon altitude observation:
43°25.17' observed upper limb altitude
0.97' refraction
14.93' unrefracted semidiameter
43°09.26' unrefracted altitude of center
42°27.31' predicted altitude
0°41.95' intercept
311°56.18' predicted azimuth

Sun altitude observation:
12°34.57' observed lower limb altitude
4.02' refraction
15.89' unrefracted semidiameter
12°46.43' unrefracted altitude of center
13°26.13' predicted altitude
- 0°39.69' intercept
79°35.88' predicted azimuth

Moon to Sun predicted separation angle:
106°21.42' center to center, unrefracted
3.58' refraction
106°17.84' center to center, refracted
14.93' Moon near limb refracted semidiameter
15.84' target near limb refracted semidiameter
105°47.07' Moon near limb to Sun near limb
105°47.07' observed angle
-  0°00.004' observed - predicted

unrefacted separation angle rate:
-16" per minute (topocentric)
77% of total angular velocity

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

Solution, after 3 iterations.

corrected time:
1773-09-07T17:05:54.84 UT1
1773-09-07T17:06:11.24 Terrestrial Time
16.400 seconds delta T

corrected position:
-16°30.65' -151°53.78' north lat, east lon
53° LOP crossing angle

geocentric coordinates (true equator and equinox):
3h52m13.08s +15°39'46"  Moon RA and dec.
14.77' apparent semidiameter
11h05m49.32s + 5°48'19"  Sun RA and dec.
15.89' semidiameter

6h06m34.36s  local apparent sidereal time

geocentric separation angle and rate:
105°58.01' center to center
-27.1" per minute
85% of total angular velocity

illumination conditions:
79.9° 12.8° Sun unrefracted az, el
306.2° Moon to Sun position angle (0 = 12 o'clock)
73.5° Moon phase angle (0 = full, 180 = new)

position angles:
306.2° Moon to Sun
37.1° Sun to Moon

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

Moon altitude observation:
43°25.17' observed upper limb altitude
0.97' refraction
14.93' unrefracted semidiameter
43°09.26' unrefracted altitude of center
43°09.26' predicted altitude
0°00.00' intercept
312°25.26' predicted azimuth

Sun altitude observation:
12°34.57' observed lower limb altitude
4.02' refraction
15.89' unrefracted semidiameter
12°46.43' unrefracted altitude of center
12°46.43' predicted altitude
0°00.00' intercept
79°53.91' predicted azimuth

Moon to Sun predicted separation angle:
106°21.56' center to center, unrefracted
3.72' refraction
106°17.84' center to center, refracted
14.93' Moon near limb refracted semidiameter
15.84' target near limb refracted semidiameter
105°47.07' Moon near limb to Sun near limb
105°47.07' observed angle
0°00.000' observed - predicted

unrefacted separation angle rate:
-16" per minute (topocentric)
77% of total angular velocity

--
I filter out messages with attachments or HTML.

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