NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: lunars with and without altitudes
From: Dave Walden
Date: 2006 Nov 25, 22:19 -0800
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) --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To , send email to NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---