NavList:

   

Reply

Antoine Couette wrote:
> Date : Feb 09 th, 2011
> observations took place in the [15:30 UT - 18:30 UT] time frame,
> Non moving Observer
> Height of eye : 17 ft above sea level, tide was high and (almost) steady.
> Temperature : 12�C / 53.6�F
> Pressure : 1017 Mb/hPa / 30.03 In. Hg
> All heights corrected for only Instrument error (Refraction, SD and 
> Parallax need to be performed)
> The horizon was exceptionally clear and sharp/well defined.


INITIAL ESTIMATES OF CLOCK ERROR, LATITUDE, LONGITUDE

Clock error estimate #1 = -15h45m.

Position estimate #1 = 47°N 002°W.


SUN ALTITUDE OBSERVATION

 > 1 - First set of Observations : 5 Sun Lower Limb shots :
 >
 > Watch Time t11=00h00m28.0s T11 = 15h45m28.0s H11 = 11�39.8'
 > Watch Time t12=00h01m43.0s T12 = 15h46m43.0s H12 = 11�29'4
 > Watch Time t13=00h02m57.0s T13 = 15h47m57.0s H13 = 11�19'2
 > Watch Time t14=00h04m05.0s T14 = 15h49m05.0s H14 = 11�10'0
 > Watch Time t15=00h05m28.0s T15 = 15h50m28.0S H15 = 10�58'6

Mean watch time = 00h02m56s, mean refracted altitude = 11°19.4' - 4.0' 
dip = 11°15.4'.

Estimated UTC = watch time - error = 00h02m56s - -15h45m = 15h47m56s.

Altitude residual (i.e., observed - computed) = -1°38.4' = -1.640.

Partial derivative of altitude with respect to north latitude = cos
azimuth = cos 231° = -.630° per degree of north lat. This is the effect
of a northerly change of latitude on computed altitude.

Partial derivative of altitude with respect to east longitude = sin az
cos lat = sin 231° cos 47° = -.530° per degree of east lon.

The partial derivative of altitude with respect to time is simply the
time equivalent of the above: -.530° per 4 minutes of time, or  -.133°
per minute. However, the sign must be reversed because my unknown is
actually clock error, not time. (An increase in clock error is 
equivalent to a decrease in time.)

With the values above, form an equation to represent the Sun altitude
observation. On the left is the altitude residual. On the right, the
partial derivatives are the coefficients and the variables are the
unknown corrections to latitude (∆φ), longitude (∆λ), and clock
error (∆CE).

-1.640 = -.630 ∆φ - .530 ∆λ + .133 ∆CE


MOON ALTITUDE OBSERVATION

 > 2 - Second set of Observations : 5 Moon Lower Limb shots :
 >
 > Watch Time t21=01h40m23.0s T21 = 17h25m23.0s H21 = 57�28'0
 > Watch Time t22=01h41m15.0s T22 = 17h26m15.0s H22 = 57�24'7
 > Watch Time t23=01h42m48.0s T23 = 17h27m48.0s H23 = 57�18'6
 > Watch Time t24=01h43m37.0s T24 = 17h28m37.0s H24 = 57�15'1
 > Watch Time t25=01h44m55.0s T25 = 17h29m55.0s H25 = 57�09'6

Mean watch time = 01h42m36s, mean refracted altitude 57°19.2' - 4.0' dip 
= 57°15.2'.

Estimated UTC = watch time - error = 01h42m36s - -15h45m = 17h27m36s.

Residual = -31.5' = -.525°. Partial derivatives of altitude with respect
to latitude, longitude, and time are -.945° / ° N lat, -.222° / ° E lon,
+.0555° / min CE. Form an equation as before:

-.525 = -.945 ∆φ - .222 ∆λ + .0555 ∆CE


JUPTER ALTITUDE OBSERVATION:

 > 3 - Third set of Observations : 5 Jupiter shots
 >
 > Watch Time t31=01h47m25.0s T31 = 17h32m25.0s H31 = 31�57'1
 > Watch Time t32=01h48m17.0s T32 = 17h33m17.0s H32 = 31�50'2
 > Watch Time t33=01h49m15.0s T33 = 17h34m15.0s H33 = 31�42'3
 > Watch Time t34=01h50m03.0s T34 = 17h35m03.0s H34 = 31�36'2
 > Watch Time t35=01h51m22.0s T35 = 17h36m22.0s H35 = 31�26'0

Mean watch time = 01h49m16s, Jupiter mean refracted altitude = 31°42.4'
- 4.0' dip = 31°38.4'. Estimated UTC = 17h34m16s. Put residual and
partial derivatives in an equation as before:

-1.523 = -.695 ∆φ - .490 ∆λ + .123 ∆CE


LUNAR DISTANCE TO JUPITER

 > 4 - Fourth set of Observations : 5 Jupiter Lunar Distances near limb
 >
 > Watch Time t41=02h05m07.0s T41 = 17h50m07.0s H41 = 29�37'5
 > Watch Time t42=02h06m14.0s T42 = 17h51m14.0s H42 = 29�38'0
 > Watch Time t43=02h07m25.0s T43 = 17h52m25.0s H43 = 29�38'4
 > Watch Time t44=02h08m17.0s T44 = 17h53m17.0s H44 = 29�38'8
 > Watch Time t45=02h09m24.0s T45 = 17h54m24.0s H45 = 29�39'1

Mean watch time = 02h07m17s, Moon near limb to Jupiter mean refracted
distance = 29°38.4'. Estimated UTC = 17h52m17s. Unlike altitudes, I know 
of no simple formulae for the partial derivatives. It's less trouble to 
get them from a program by observing the way lunar distance responds a 
change in latitude or longitude. Form an equation as before:

.092 = -.003 ∆φ - .010 ∆λ - .0057 ∆CE


SOLUTION #1

Here are the equations we created:
-1.640 = -.630 ∆φ - .530 ∆λ + .133 ∆CE
  -.525 = -.945 ∆φ - .222 ∆λ + .0555 ∆CE
-1.523 = -.695 ∆φ - .490 ∆λ + .123 ∆CE
   .092 = -.003 ∆φ - .010 ∆λ - .0057 ∆CE

Many scientific calculators can solve such a set of equations by the 
method of least squares. I used my old HP 49G, which has been out of 
production for years.

Solution:
∆φ = -.24°
∆λ = -.43°
∆CE = -15.3m

Add these corrections to the initial estimates of latitude, longitude, 
and clock error to obtain improved values. Then repeat the entire procedure.


ITERATION #2

Form new residuals, partial derivatives, and equations. The procedure is 
the same as before, so I'll show only the equations. This time the 
residuals (on the left) are much less than before. I.e., after applying 
the above corrections to latitude, longitude, and clock error, the 
predicted angles are now closer to the observed angles.

Sun altitude
0.048 = -.591 ∆φ - .553 ∆λ + .138 ∆CE

Moon altitude:
0.051 = -.909 ∆φ - .286 ∆λ + .072 ∆CE

Jupiter altitude:
0.050 = -.643 ∆φ - .525 ∆λ + .131 ∆CE

Lunar distance:
0.001 = -.003 ∆φ - .010 ∆λ - 0.0057 ∆CE

Solution:
∆φ = -.043°
∆λ = -.055°
∆CE = -.056m


RESIDUALS AFTER ITERATION #2

After applying the above corrections, we get these residuals:

Sun altitude 0.000°

Moon altitude -0.001°

Jupiter altitude 0.001°

Lunar distance 0.000°

All computed values are within .001° of the observations, so there is no
need for a third iteration.


FINAL RESULTS AND CHECK

clock time - UTC = -16h00m21s
Latitude = +46.717° = 46°43.0 N
Longitude = -2.485° = 002°29.1' W

For an independent check, use the above values and the USNO MICA program 
(which I did not use for the preceding computations). Compare its output 
to the observed angles. To compensate for MICA's error in ∆T (68.18 s 
vs. the correct value 66.35), and the UT1-UTC difference (-.17 s), 1) 
add 2.00 s to UTC and enter the sum as UT1 to MICA, 2) move the observer 
east by the (sidereal) arc equivalent of 1.83 s = 1.83 * 15 * 1.002738 
arc seconds.


Sun altitude
11°15.4' observed refracted LL
  + 11.6' refraction & SD (Nautical Almanac)
--------
11°27.0' unrefracted center
11 27.0' MICA


Moon altitude
57°15.2' observed refracted LL
  -   .6' refraction (Nautical Almanac)
  + 15.1  SD (MICA)
--------
57°29.7' unrefracted center
57 29.8' MICA


Jupiter altitude
31°38.4' observed refracted center
  -  1.6' refraction
--------
31°36.8' unrefracted center
31°36.8' MICA


MICA cannot compute the Moon to Jupiter distance, but it will give their 
topocentric azimuths and altitudes. After applying refraction to the 
altitudes, the separation angle may be found with a calculator.
29°38.4' observed, near limb to Jupiter center, refracted
  + 15.1' SD
--------
29°53.5' observed center to center, refracted
29°53.5' MICA


All computed angles agree with the observations within 0.1'.

-- 

   

Reply