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    Re: lunars with and without altitudes
    From: Dave Walden
    Date: 2006 Nov 25, 22:19 -0800

    LOP via azimuth-intercept without horizon
    
    Since the lunar distance with GMT method still seems to be of interest,
    I'll mention a little calculation I've done for my own amusement.
    After carrying out quite a few calculations, (including a clearing of
    lunar distance via the fundamental equation with log tables---only
    takes 14 table look ups!) (thanks to Frank for bringing up the topic
    and for his calculator which I've used to confirm my results) and doing
    a few plots, I began to realize it would be possible to avoid the
    interpolations and iterations if one were willing to do some (well
    quite a lot actually) algebra.
    
    One sets up the equation for cleared lunar distance retaining the
    dependencies on latitude and longitude.  If one then takes the partial
    derivatives of this equation with respect to latitude and to latitude,
    the ratio of these partials gives the azimuth.  The difference between
    true and cleared lunar distances divided by the magnitude of a partial
    then gives the intercept.
    
    Below is a session using a CA (computer algebra) program called maxima.
     (Freeware, available for Linux and windows.)  It can be seen that
    I've used the example from Frank with Nautical Almanac data.  The
    Zn's are in degrees CW from North.  The intercepts are in degrees, so
    multiply by 60 to get minutes.
    
    batching #p/mnt/auto/floppy/lunarGEN.mac
    (%i2)                             diff : true
    (%o2)                                true
    (%i3)                             fpprec : 7
    (%o3)                                  7
                                                %pi x
    (%i4)                        sind(x) := sin(-----)
                                                 180
                                                %pi x
    (%o4)                        sind(x) := sin(-----)
                                                 180
                                                %pi x
    (%i5)                        cosd(x) := cos(-----)
                                                 180
                                                %pi x
    (%o5)                        cosd(x) := cos(-----)
                                                 180
    (%i6) LDtrue(gha1, dec1, gha2, dec2) :=
          180 acos(cosd(dec1) cosd(dec2) cosd(gha1 - gha2) + sind(dec1)
    sind(dec2))
    
    -------------------------------------------------------------------------
                                             %pi
    (%i7) ALTtrue(LAT, LONG, GHA, DEC) :=
               180 asin(cosd(LAT) cosd(DEC) cosd(GHA - LONG) + sind(LAT)
    sind(DEC))
    
    --------------------------------------------------------------------
                                               %pi
                                                  sind(HP) cosd(ALT)
                                       180 atan(----------------------)
                                                1 - sind(HP) sind(ALT)
    (%i8)      ALTPARALLAX(HP, ALT) := --------------------------------
                                                     %pi
                                                 %pi (90 - ALT)
                                      58.276 tan(--------------)
                                                      180
    (%i9)              REFRAC(ALT) := --------------------------
                                                  60
    (%i10) LDclear(LDobs, A, a, H, h) :=
                                   (cosd(LDobs) - sind(a) sind(h)) cosd(A)
    cosd(H)
        180 acos(sind(A) sind(H) +
    -----------------------------------------------)
                                                   cosd(a) cosd(h)
    
    ---------------------------------------------------------------------------
                                            %pi
    (%i11)                            LATAP : 38
    (%o11)                                38
    (%i12)                            LONGAP : 70
    (%o12)                                70
                                             6.0
    (%i13)                        GHAmoon1 : --- + 47
                                             60
    (%o13)                               47.1
                                            38.9
    (%i14)                       DECmoon1 : ---- + 27
                                             60
    (%o14)                             27.64833
                                              58.4
    (%i15)                          HPmoon1 : ----
                                               60
    (%o15)                              0.97333
                                             2.1
    (%i16)                        GHAstar1 : --- + 3
                                             60
    (%o16)                               3.035
                                             0.7
    (%i17)                        DECstar1 : --- + 28
                                             60
    (%o17)                             28.01167
                                           29.2
    (%i18)                        LDobs1 : ---- + 38
                                            60
    (%o18)                             38.48667
    (%i19) LDtrue1 : ev(LDtrue(GHAmoon1, DECmoon1, GHAstar1, DECstar1),
    numer)
    (%o19)                             38.75083
    (%i20)            A1 : ALTtrue(LAT, LONG, GHAmoon1, DECmoon1)
                                                     REFRAC(A1)
    (%i21)         a1 : - ALTPARALLAX(HPmoon1, A1) + ---------- + A1
                                                         60
    (%i22)            H1 : ALTtrue(LAT, LONG, GHAstar1, DECstar1)
                                      REFRAC(H1)
    (%i23)                       h1 : ---------- + H1
                                          60
    (%i24) LDclear1 : ev(LDclear(LDobs1, A1, a1, H1, h1), LAT : LATAP,
                                                              LONG :
    LONGAP, numer)
    (%o24)                             38.76112
    (%i25)       diffLAT : diff(LDclear(LDobs1, A1, a1, H1, h1), LAT)
    (%i26)   dLDcleardLAT : ev(diffLAT, LAT : LATAP, LONG : LONGAP, numer)
    (%o26)                             - 0.00742
    (%i27)      diffLONG : diff(LDclear(LDobs1, A1, a1, H1, h1), LONG)
    (%i28)  dLDcleardLONG : ev(diffLONG, LAT : LATAP, LONG : LONGAP, numer)
    (%o28)                              0.0119
                                         dLDcleardLONG
                                         --------------
                                         - dLDcleardLAT
                                180 atan(--------------)
                                          cosd(LATAP)
    (%i29)             Zn1 : ev(------------------------, numer)
                                          %pi
    (%o29)                             63.83353
                             (LDtrue1 - LDclear1) cosd(Zn1)
    (%i30)         Int1 : ev(------------------------------, numer)
                                      dLDcleardLAT
    (%o30)                              0.61094
                                            22.8
    (%i31)                       GHAmoon2 : ---- + 46
                                             60
    (%o31)                               46.38
                                            38.7
    (%i32)                       DECmoon2 : ---- + 27
                                             60
    (%o32)                              27.645
                                              58.4
    (%i33)                          HPmoon2 : ----
                                               60
    (%o33)                              0.97333
                                            38.1
    (%i34)                       GHAstar2 : ---- + 49
                                             60
    (%o34)                              49.635
                                            31.5
    (%i35)                       DECstar2 : ---- + 16
                                             60
    (%o35)                              16.525
                                           23.7
    (%i36)                        LDobs2 : ---- + 11
                                            60
    (%o36)                              11.395
    (%i37) LDtrue2 : ev(LDtrue(GHAmoon2, DECmoon2, GHAstar2, DECstar2),
    numer)
    (%o37)                             11.51992
    (%i38)            A2 : ALTtrue(LAT, LONG, GHAmoon2, DECmoon2)
                                                     REFRAC(A2)
    (%i39)         a2 : - ALTPARALLAX(HPmoon2, A2) + ---------- + A2
                                                         60
    (%i40)            H2 : ALTtrue(LAT, LONG, GHAstar2, DECstar2)
                                      REFRAC(H2)
    (%i41)                       h2 : ---------- + H2
                                          60
    (%i42) LDclear2 : ev(LDclear(LDobs2, A2, a2, H2, h2), LAT : LATAP,
                                                              LONG :
    LONGAP, numer)
    (%o42)                             11.50826
    (%i43)       diffLAT : diff(LDclear(LDobs2, A2, a2, H2, h2), LAT)
    (%i44)   dLDcleardLAT : ev(diffLAT, LAT : LATAP, LONG : LONGAP, numer)
    (%o44)                              0.01669
    (%i45)      diffLONG : diff(LDclear(LDobs2, A2, a2, H2, h2), LONG)
    (%i46)  dLDcleardLONG : ev(diffLONG, LAT : LATAP, LONG : LONGAP, numer)
    (%o46)                             - 0.00133
                                         dLDcleardLONG
                                         --------------
                                         - dLDcleardLAT
                                180 atan(--------------)
                                          cosd(LATAP)
    (%i47)             Zn2 : ev(------------------------, numer)
                                          %pi
    (%o47)                             5.789598
                             (LDtrue2 - LDclear2) cosd(Zn2)
    (%i48)         Int2 : ev(------------------------------, numer)
                                      dLDcleardLAT
    (%o48)                              0.69511
    (%i49)
    
    
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