NavList:

Lars Bergman writes:
>   2. Sometimes it can be very difficult, or actually impossible, to know
>   whether the upper or lower limb of the moon is the one to shoot. Then
>   you'll have to wait an hour or so until the moon has tilted suffiently
>   to show a clear limb but near full moon it is easy to make mistakes. A
>   false LL gives a too large altitude, a false UL gives a too small
>   altitude.

As one with some facility with math, and interest but no experience in
celestial navigation, I'll suggest a mathematical test for which limb
to shoot.  I hope you'll pardon non-standard nomenclature.  And the
length.

Let M be a unit vector from the observer toward the moon, S be a unit
vector toward the sun, and U be a unit vector up.  Then

   U cross M

is a horizontal vector toward the observer's left.  Consider the plane
containing that vector, the observer, and the moon.  If the sun is
above that plane, one should shoot the upper limb.  Otherwise, one
should shoot the lower limb.  Then

   (M cross (U cross M))

is a vector perpendicular to that test plane, and

   (M cross (U cross M)) dot S

is positive if the sun is above the plane.

If we use an East-North-Up cartesian coordinate system centered on the
observer, and

    Hm = altitude of the moon
    Am = azimuth of the moon
    Hs = altitude of the sun
    As = azimuth of the sun

then

    M = [ cos Hm sin Am, cos Hm cos Am, sin Hm ]'
    S = [ cos Hs sin As, cos Hs cos As, sin Hs ]'
    U = [ 0, 0, 1 ]'

where X' is the transpose of X (that is, a column vector).

Robert Eno didn't give the time and location of his recent
observations.  Suppose they were made from my home, at 2006-08-11 0600
UT.  Using Steve Moshier's program aa for ephimerides (see below) and
octave for calculation, we have:

  octave:2> Hm=38.597*pi/180
  Hm = 0.67364
  octave:3> Am=170.334*pi/180
  Am = 2.9729
  octave:4> Hs=-29.930*pi/180
  Hs = -0.52238
  octave:5> As=18.922*pi/180
  As = 0.33025
  octave:6> M=[cos(Hm)*sin(Am), cos(Hm)*cos(Am), sin(Hm)]'
  M =
 
     0.13123
    -0.77046
     0.62384
 
  octave:7> S=[cos(Hs)*sin(As), cos(Hs)*cos(As), sin(Hs)]'
  S =
 
     0.28103
     0.81980
    -0.49894
 
  octave:8> U=[0, 0, 1]'
  U =
 
    0
    0
    1
 
  octave:9> dot(cross(M, cross(U, M)),S)
  ans = 0.066259
 
which is positive, so the upper limb is the one to shoot.

(Is this overkill, or what? :-)

Links:
  aa      http://www.moshier.net/ 
  octave  http://www.gnu.org/software/octave/

          - Jim Van Zandt
 


vanzandt:/tmp$ aa

    Steve Moshier's Ephemeris Program v5.6

Planetary and lunar positions approximate DE404.
Terrestrial east longitude -71.4644 deg
geocentric latitude 42.5261 deg
Earth radius 0.99847
temperature 12.0 C
pressure 1010 mb
Input time is TDT.
Enter starting date of tabulation
Calendar date: Year (1986) ? 2006
Month (1-12) (1) ? 08
Day.fraction (1.000000) ? 11
Time: Hours (0) ? 06
Minutes (0) ?
Seconds (0.000000) ?
2006 August 11 Friday  5h 58m 52.396s  UT
2006 August 11 Friday  6h 00m 00.000s  TDT
Julian day 2453958.7500000
Enter interval between tabulations in days (1.000000) ?
Number of tabulations to display (1) ?
Planet number 0-9 or 88 to read star, 99 to read orbit (0) ? 3

                   The Moon

JD 2453958.75,  2006 August 11 Friday  5h 58m 52.396s  UT
2006 August 11 Friday  6h 00m 00.000s  TDT
nutation dRA 0.212s dDec -1.51"
Geometric lon -16.370 deg, lat -1.096 deg, rad 2.4061e-03 au
Apparent geocentric longitude 343.631 deg   latitude -1.096 deg
Distance 56.435 Earth-radii
Horizontal parallax    1d 00' 55.13"  Semidiameter    0d 16' 35.88" 
Elongation from sun 154.80 deg,  Illuminated fraction 0.95
Phase 1.9 days past Full Moon
    Apparent:  R.A. 23h 01m 22.074s  Declination -  7d 26' 55.48" 
Local apparent sidereal time  22h 31m 16.954s 
diurnal aberration dRA 0.016s dDec 0.00"
diurnal parallax dRA 23.982s dDec -2823.13"
atmospheric refraction 0.020 deg  dRA -0.607s dDec 71.73"
Topocentric:  Altitude 38.597 deg, Azimuth 170.334 deg
Topocentric: R.A. 23h 01m 45.465s   Dec. -  8d 12' 46.87" 
local meridian transit 2006 August 11 Friday  6h 30m 02.072s  UT
rises 2006 August 11 Friday  0h 52m 39.036s  UT
sets 2006 August 11 Friday 12h 20m 24.060s  UT
Visible hours 11.4625

...
Planet number 0-9 or 88 to read star, 99 to read orbit (3) ? 0

                   The Sun

JD 2453958.75,  2006 August 11 Friday  5h 58m 52.396s  UT
2006 August 11 Friday  6h 00m 00.000s  TDT
ecliptic long  138d 27' 05.24"   lat -  0d 00' 00.14"   rad 1.013504E+00
light time 8.4291m,  aberration dRA 1.329s dDec -6.23"
nutation dRA 0.273s dDec 5.18"
Leo Leonis.    Apparent:  R.A.   9h 23m 31.350s  Dec.    15d 17' 59.56" 
Apparent longitude 138.446 deg
Local apparent sidereal time  22h 31m 16.954s 
diurnal aberration dRA -0.016s dDec -0.02"
diurnal parallax dRA 0.129s dDec -7.26"
atmospheric refraction 0.000 deg  dRA 0.000s dDec 0.00"
Topocentric:  Altitude -29.930 deg, Azimuth 18.922 deg
Topocentric: R.A.  9h 23m 31.463s   Dec.   15d 17' 52.28" 
local meridian transit 2006 August 11 Friday 16h 51m 02.410s  UT
rises 2006 August 11 Friday  9h 47m 52.964s  UT
sets 2006 August 11 Friday 23h 53m 28.330s  UT
Visible hours 14.0932

...
Planet number 0-9 or 88 to read star, 99 to read orbit (0) ? 88
Name of star catalogue file:  (star.cat) ?
Line number (1) ? 49
2000 18 36 56.332  38 47  1.17   1.726   28.61 -14.0 0.1230   0.03 alLyr(Vega)         3238


               alpha Lyrae (Vega)

JD 2453958.75,  2006 August 11 Friday  5h 58m 52.396s  UT
2006 August 11 Friday  6h 00m 00.000s  TDT
approx. visual magnitude 0.0
Astrometric J2000.0:  R.A.  18h 36m 56.439s  Dec.    38d 47' 03.14" 
Astrometric B1950.0:  R.A.  18h 35m 15.706s  Dec.    38d 44' 25.91" 
Astrometric of date:  R.A.  18h 37m 09.756s  Dec.    38d 47' 24.46" 
elongation from sun 113.39 degrees, light defl. dRA 0.000s dDec -0.00"
annual aberration dRA 1.313s dDec 12.03"
nutation dRA 0.007s dDec -9.01"
    Apparent:  R.A.  18h 37m 11.075s  Dec.    38d 47' 27.49" 
Local apparent sidereal time  22h 31m 16.954s 
diurnal aberration dRA 0.010s dDec 0.13"
atmospheric refraction 0.015 deg  dRA 4.269s dDec 22.97"
Topocentric:  Altitude 46.401 deg, Azimuth 285.487 deg
Topocentric: R.A. 18h 37m 15.354s   Dec.   38d 47' 50.58" 
local meridian transit 2006 August 11 Friday  2h 05m 24.890s  UT
rises 2006 August 10 Thursday 16h 49m 16.999s  UT
sets 2006 August 11 Friday 11h 21m 32.762s  UT
Visible hours 18.5377


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