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Observations:
2010-10-26 14:01:25 from N40 35.1 W039 49.8 Sun alt 35 49.1
2010-10-26 20:04:48 from N42 09.5 W038 32.9 Jup alt 20 07.6
2010-10-27 00:36:33 from N43 08.1 W037 10.1 Moon to Jup 86 18.7

Times are UTC. Sun observation is near local noon and corrected for dip,
refraction, and semidiameter. Jupiter observation is at end of civil
twilight and corrected for dip and refraction. Azimuth is about 116°.
Moon (far limb) to Jupiter (center) angle includes refraction. All
observed angles are accurate, but the times and positions have
significant errors.

The object is to correct the errors. Since there are three measurements,
we can solve for only three unknowns. Therefore, assume all observations
have the same time, latitude, and longitude errors.

Begin with the first observation, the noon sight. At the estimated time
and position, computed altitude is 36 38.6, which is 49.5' greater than
the observation. One minute of time later changes computed altitude
+1.3'. One degree north changes it -59.6'. One degree west changes it -5.8'.

Jupiter computed altitude  = 20 38.4, which is 30.8' greater than the
observation. One minute later changes computed altitude +10.0'. One
degree north changes it -26.1'. One degree west changes it -40.2'.

The computed lunar distance = 86 18.6, which is .1' less than the
observation. One minute later changes the computed lunar distance +.5'.
One degree north changes it -.7'. One degree west changes it by +.2'.

We can now form an equation for each observation. On the left is the
required change in the computed angle to match the observation. On the
right are the unknown corrections in time (∆t), latitude (∆φ), and
longitude (∆λ) to make that happen. The coefficients are the partial
derivatives (e.g., change in altitude per minute of time) from the
paragraphs above.

-49.5 =  1.3∆t - 59.6∆φ -  5.8∆λ
-30.8 = 10.0∆t - 26.1∆φ - 40.2∆λ
   +.1 =   .5∆t -   .7∆φ +   .2∆λ

Solution:
∆t = +1.12 m = 1 m 7 s
∆φ = +.804° = N48.2'
∆λ = +.522° = W31.3'

Apply those corrections to the time and place of each observation. In
the table below I show the corrected values in the upper line of each
pair, as well as the computed altitude or lunar distance. In the lower
line are the true values for comparison.

2010-10-26 14:02:32 from N41 23.3 W040 21.1 Sun alt 35 49.5
2010-10-26 14:02:37 from N41 24.0 W040 18.0 Sun alt 35 49.1

2010-10-26 20:05:55 from N42 57.7 W039 04.2 Jup alt 20 07.8
2010-10-26 20:06:00 from N42 58.2 W039 05.5 Jup alt 20 07.6

2010-10-27 00:37:40 from N43 56.3 W037 41.4 Moon to Jup 86 18.7
2010-10-27 00:37:45 from N43 56.5 W037 30.0 Moon to Jup 86 18.7

Now the observations match the computed angles within a few tenths of a
minute. To practical accuracy we're finished, but let's repeat the 
correction process to try for a perfect match.

At the noon sight, computed altitude is .4' greater than the
observation. One second later changes computed altitude +.02'. One
minute north changes it -.99'. One minute west changes it -.09'.

At the Jupiter sight, computed altitude is .2' greater than the
observation. One second later changes it +.17'. One minute north changes
it -.43'. One minute west changes it -.66'.

At the lunar distance observation, computed lunar distance exactly
matches the observation. One minute later changes the computed lunar
distance +.01'. One minute north changes it -.01'. One minute west
changes it .00'.

The equations are:
-.4 = +.02∆t - .99∆φ - .09∆λ
-.2 = +.17∆t - .43∆φ - .66∆λ
0.0 = +.01∆t - .01∆φ + .00∆λ

Solution:
∆t = +.4 s
∆φ = +.4'
∆λ = +.1'

Apply corrections:

2010-10-26 14:02:32.4 from N41 23.7 W040 21.2 Sun alt 35 49.1
2010-10-26 14:02:37.0 from N41 24.0 W040 18.0 Sun alt 35 49.1

2010-10-26 20:05:55.4 from N42 58.1 W039 04.3 Jup alt 20 07.7
2010-10-26 20:06:00.0 from N42 58.2 W039 05.5 Jup alt 20 07.6

2010-10-27 00:37:40.4 from N43 56.7 W037 41.5 Moon to Jup 86 18.7
2010-10-27 00:37:45.0 from N43 56.5 W037 30.0 Moon to Jup 86 18.7

The computed angles and the observations are practically identical.
However, it's clear there's still a substantial error in longitude at 
the last observation. The main reason is that I had to assume all 
observations had an identical bias in time, latititude, and longitude. 
Most the error was of this nature. But on top of that, I injected a 3 NM 
error in the EP for the second observation and 6 NM error in the third 
EP (relative to the first observation EP). These are about 3% of the 
distance run. The directions of the errors were randomly chosen, but by 
happenstance they were at 86° and 275°, nearly reciprocal directions.

The results would have been better if both altitude observations had 
occurred at evening twilight, but I wanted to make the problem difficult.

Incidentally, the three points where the observations occurred do not
"line up" on a chart because there was a speed and course change between
observations.

I present this mostly as an interesting experiment rather than a 
practical method for reducing sights. Nevertheless, my lunar distance 
program uses a similar method to arrive at time and position. It's 
unable to do what I demonstrated here, though. The program is written 
for the classic lunar distance observation, where the altitudes and 
separation angle of the Moon and another body are observed at 
practically the same time and place.

-- 

   

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