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7 Chapter 2 Altitude Measurement Although altitudes and zenith distances are equally suitable for navigational calculations, most formulas are traditionally based upon altitudes since these are easily accessible using the visible sea horizon as a natural reference line. Direct measurement of the zenith distance, however, requires an instrument with an artificial horizon, e. g., a pendulum or spirit level indicating the local direction of gravity (perpendicular to the sensible horizon), since a suitable reference point in the sky does not exist. Instruments A marine sextant consists of a system of two mirrors and a telescope mounted on a metal frame (brass or aluminum). A schematic illustration (side view) is given in Fig The horizon glass is a half-silvered mirror whose plane is perpendicular to the plane of the frame. The index mirror, the plane of which is also perpendicular to the frame, is mounted on the so-called index arm rotatable on a pivot perpendicular to the frame. The optical axis of the telescope is parallel to the frame. When measuring an altitude, the instrument frame is held in an upright position, and the visible sea horizon is sighted through the telescope and horizon glass. A light ray coming from the observed body is first reflected by the index mirror and then by the back surface of the horizon glass before entering the telescope. By slowly rotating the index mirror on the pivot the superimposed image of the body is aligned with the image of the horizon line. The corresponding altitude, which is twice the angle formed by the planes of horizon glass and index mirror, can be read from the graduated limb, the lower, arc-shaped part of the triangular sextant frame (Fig. 2-2). Detailed information on design, usage, and maintenance of sextants is given in [3] (see appendix). Fig. 2-2 On land, where the horizon is too irregular to be used as a reference line, altitudes have to be measured by means of instruments with an artificial horizon. 2-1

8 A bubble attachment is a special sextant telescope containing an internal artificial horizon in the form of a small spirit level whose image, replacing the visible horizon, is superimposed with the image of the body. Bubble attachments are expensive (almost the price of a sextant) and not very accurate because they require the sextant to be held absolutely still during an observation, which is difficult to manage. A sextant equipped with a bubble attachment is referred to as a bubble sextant. Special bubble sextants were used for air navigation before electronic navigation systems became standard equipment. A pan filled with water or, preferably, a more viscous liquid, e. g., glycerol, can be utilized as an external artificial horizon. As a result of gravity, the surface of the liquid forms a perfectly horizontal mirror unless distorted by vibrations or wind. The vertical angular distance between a body and its mirror image, measured with a marine sextant, is twice the altitude. This very accurate method is the perfect choice for exercising celestial navigation in a backyard. Fig. 2-3 shows a professional form of an external artificial horizon for land navigation. It consists of a horizontal mirror (polished black glass) attached to a metal frame with three leg screws. Prior to an observation, the screws have to be adjusted with the aid of one or two detachable high-precision spirit levels until the mirror is exactly horizontal in every direction. Fig. 2-3 Fig. 2-4 A theodolite (Fig. 2-4) is basically a telescopic sight which can be rotated about a vertical and a horizontal axis. The angle of elevation (altitude) is read from the graduated vertical circle, the horizontal direction is read from the horizontal circle. The specimen shown above has vernier scales and is accurate to approx. 1'. 2-2

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41 sinh Lat coslatsindec sinlatcosdeccos t sinh Lat dlat sinh dt 0 t

42 dt tandec sin t tanlat tant dlat t tanlat 2 tandec 2 sint 2 tant 2 Dec tandec 2 tanlat 2 Lat Lon sint 2 tant 2 t 2 15 T 2 ht 1 h Lon 2 Lat' vkncosct 2 h T 1 h Lat 2 Lat 1 Lat Lon' vkn sinc coslat T 2 ht 1 h Lon 2 Lon 1 Lon 1knknot 1 nmh T 2 *h T 2 h t 15 T Transit T 1 T 2 * 2 dt dt tanlat tandec sin t tan t ddec dt

43 900'h tanlat sin t tandec tan t ddec' dth 900 tanlat tandec d Dec' tant dth [ ] [ ] t tanlat tandec d Dec' dth t' 3.82tanLat tandec ddec' dth + Hc arcsin sin80 sin1.445'' cos80 cos1.445''cos21.7' 10 0' 0.72'' Hc arcsin sin45 sin0.255'' cos45 cos0.255''cos3.82' 45 0' 0.13''

44 t arccos sinho sinlatsindec coslatcosdec Lon 1 t GHA Lon t GHA If Lon Lon If Lon Lon If Lon Lon dt cosho coslatcosdecsint dh

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46 cosd sindec M sindec B cosdec M cosdec B cosgha M GHA B cosd sindec M sindec B cosdec M cosdec B cos15ra M h RA B h

47 Az M Az B Az M Az Mapp Az B Az Bapp cosd app sinh Mapp sinh Bapp cosh Mapp cosh Bapp cos cos cosd app sinh Mapp sinh Bapp cosh Mapp cosh Bapp cosd sinh M sinh B cosh M cosh B cos cos cosdsinh M sinh B cosh M cosh B

48 cosd sinh M sinh B cosh M cosh B cosd app sinh Mapp sinh Bapp cosh Mapp cosh Bapp cosd sinh M sinh B cosh M cosh B 1 cosd app sinh MappsinH Bapp cosh Mapp cosh Bapp 1 cosd sinh M sinh B cosh M cosh B cosh cosh M B cosd sinh sinh app Mapp Bapp cosh cosh Mapp Bapp cosh M cosh B cosh Mapp cosh Bapp cosh Mapp cosh Bapp cosd sinh M sinh B cosh M cosh B cosh M cosh B cosd app sinh MappsinH Bapp cosh MappcosH Bapp cosh Mapp cosh Bapp cosd cosh M H B cosh M cosh B cosd app cosh Mapp H Bapp cosh Mapp cosh Bapp cosd cosh M cosh B cosh Mapp cosh Bapp cosd app cosh Mapp H Bapp cosh M H B cosd cosh M cosh B cosh Mapp cosh Bapp cosd app cosh Mapp H Bapp cosh M H B

49 H LMapp H1 LMapp H2 LMapp H1 LMapp WT D WT1 LMapp WT2 LMapp WT1 LMapp H Bapp H1 Bapp H2 Bapp H1 Bapp WT D WT1 Bapp WT2 Bapp WT1 Bapp tansd aug k 2 1 cosh sin 2 LMapp k HP sinh LMapp M k upper limb: cosh LMapp k lower limb: cosh LMapp k Lower limb: Upper limb: H Mapp H LMapp SD aug H Mapp H LMapp SD aug Limb of moon towards reference body: Limb of moon away from reference body: D app D Lapp SD aug D app D Lapp SD aug R i ' pmbar T C i Mapp,Bapp H tanh i tan 3 i 10 H i

50 sinp M sinhp M cosh Mapp R Mapp sinp B sinhp B cosh Bapp R Bapp H M H Mapp R Mapp P M H B H Bapp R Bapp P B T D T 1 T 2 T 1 D D 1 D 2 D 1 T D T 1 D D 2 D D 3 D 1 D 2 D 1 D 3 T 2 D D 1 D D 3 D 2 D 1 D 2 D 3 T 3 D D 1 D D 2 D 3 D 1 D 3 D 2 T 2 T 1 3h T 3 T 2 3h D 1 D 2 D 3 or D 1 D 2 D 3 T WT D T D UT WT T

51 P M f HP M sin 2LatcosAz M sinh Mapp sin 2 LatcosH Mapp f P M, improved P M P M H M H mapp R Mapp P M, improved Az M f HP M sin2latsinaz M cosh M cosd sinh M sinh B cosh M cosh B cos dcosd d cosh M cosh B sin dcosd sinddd sinddd cosh M cosh B sind dd cosh cosh sin M B d sind

52 d d Az M d D cosh McosH B sin daz M sind D cosh cosh sin M B Az sind M D improved D D D improved D f HP M cosh Bsin2Latsin Az M sinaz M Az B sind

53 Lat Dec 90 Lat Dec 90

54 sinh sinlatsindec coslatcosdeccos t 0 cos t sinlatsindec coslatcosdec t arccos tanlattandec Lon GHA t UT Sunrise,Sunset 12EoT Lon 15 t 15 Lon 1512 GMT Sunrise,Sunset EoTt

55 t Lon 1512GMT Sunrise,Sunset EoT cost Lat arctan tandec t arccos sinho sinlatsindec coslatcosdec Ho HP SD R H Dip Ho 0.15' 16' 34' Dip 50' Dip t arccos sinlatsindec coslatcosdec Az arccos sindec sinhsinlat coshcoslat Az arccos sindec cos Lat Az arccos sindec sinLat cosLat Az N Az if t Az if t 0

56 t arccos sinlatsindec coslatcosdec dsinh dt coslatcosdecsint dsinh coslatcosdecsintdt dt cosh coslatcosdecsint dh t 5.97 coslatcosdecsint Tm 24 coslatcosdecsint

57 Tm 4t 9 t 3

58 tanlat' 1f 2 tanlat

59 v'' sin2Lat1.163sin4Lat 0.026sin6Lat v Lat Lat'

60 cosz c l z c l cosz a l cosvsinz a l sinvcos 180 A a l arccos cosz a l cosvsinz a l sinvcosa a l

61 r r e 12e2 e 4 sin 2 Lat 1 e 2 sin 2 Lat p c arcsinsinhpsinz c l e 2 1 r 2 p 2 r e z c z c l p c A c arccos cosz l a cosz l c cosv sinz l c sinv z a arccoscosz c cosvsinz c sinvcosa c PA z a l z a p az arccos cosp cosz cosz l c a a l sinz a sinz a

62 Lat' arctan r p r e2 tanlat arctan 1f2 tanlat

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64 sin A 1 sin s 1 sin A 2 sin s 2 sin A 3 sin s 3 coss 1 coss 2 coss 3 sins 2 sins 3 cosa 1 coss 2 coss 1 coss 3 sins 1 sins 3 cosa 2 coss 3 coss 1 coss 2 sins 1 sins 2 cosa 3 cosa 1 cosa 2 cosa 3 sin A 2 sin A 3 coss 1 cosa 2 cosa 1 cosa 3 sin A 1 sina 3 coss 2 cosa 3 cosa 1 cosa 2 sin A 1 sina 2 coss 3

65 tan A A 1 2 tan A cos s 1 s 2 2 cos s 1 s 2 2 tan A A 1 2 tan A sin s 1 s 2 2 sin s 1 s 2 2 tan s 1 s 2 2 tan s 3 2 cos A 1 A 2 2 cos A 1 A 2 2 tan s 1 s 2 2 tan s 3 2 sin A 1 A 2 2 sin A 1 A 2 2 sin A 1 A 2 2 cos A 3 2 cos s 1 s 2 2 cos s 3 2 cos A 1 A 2 2 sin A 3 2 cos s 1 s 2 2 cos s 3 2 sin A 1 A 2 2 cos A 3 2 sin s 1 s 2 2 sin s 3 2 cos A 1 A 2 2 sin A 3 2 sin s 1 s 2 2 sin s 3 2

66 sins 1 tans 2 tan90 A 2 cos90 A 1 cos90 s 3 sins 2 tan90 A 1 tans 1 cos90 s 3 cos90 A 2 sin90 A 1 tan90 s 3 tans 2 cos90 A 2 cos s 1 sin90 s 3 tan90 A 2 tan90 A 1 cos s 1 cos s 2 sin90 A 2 tans 1 tan90 s 3 cos s 2 cos90 A 1 sin s 1 tan s 2 cota 2 sina 1 sin s 3 sin s 2 cot A 1 tan s 1 sin s 3 sina 2 cosa 1 cot s 3 tan s 2 sin A 2 cos s 1 cos s 3 cot A 2 cota 1 cos s 1 cos s 2 cosa 2 tan s 1 cot s 3 cos s 2 sina 1

67 cosz cos90 Lat AP cos90 Decsin90 Lat AP sin90 Deccos t cosz sinlat AP sindec coslat AP cosdeccos t sinh sinlat AP sindec coslat AP cosdeccos t H arcsinsinlat AP sindec coslat AP cosdeccos t

68 cos90 Dec cos90 Lat AP cosz sin90 Lat AP sinzcosaz sindec sinlat AP cosz coslat AP sinzcosaz sindec sinlat AP sinhc coslat AP coshccosaz Az arccos sindec sinlat APsinHc coslat AP coshc cosz cos90 Lat AP cos90 Dec sin90 Lat AP sin90 Deccos t sinh sinlat AP sindec coslat AP cosdeccos t cos t sinh sinlat AP sindec coslat AP cosdec t arccos sinh sinlat APsinDec coslat AP cosdec

69 sinr sintcosdec R arcsinsintcosdec sindec cosrsink sink sindec cosr K arcsin sindec cosr sinhc cosrcosk Lat AP Hc arcsincosrcosk Lat AP sinr coshcsinaz sinaz sinr coshc Az arcsin sinr coshc <

70 Az N Az if Lat AP 0N AND t LHA Az if Lat AP 0 N AND t 0 0 LHA Az if Lat AP 0 S cscr csctsecdec csck cscdec secr cschc secrseck Lat cscaz cscr sechc < log cscr log csct log secdec log csck log cscdec log secr log cschc log secr log seck Lat log cscaz log cscr log sechc

71 tanc d LoncosLat dlat d Lat cos Lat 1 d Lon tanc Lat B Lat A dlat cos Lat Lon B 1 tanc d Lon Lon A ln tan Lat B 2 4 ln tan Lat A 2 4 Lon Lon B A tanc

72 tanc ln Lon B Lon A tan Lat B 2 4 tan Lat A 2 4 C arctan ln Lon B Lon A tan Lat B 2 tan Lat A C C if Lat B Lat A AND Lon B Lon A 360 C if Lat B Lat A AND Lon B Lon A 180 C if Lat B Lat A dx d Lat cosc Lat B d 1 cosc d Lat Lat Lat B A cosc Lat A dkm Lat B Lat A 360 cosc dnm 60 Lat B Lat A cosc dkm Lon 360 B Lon A coslat dnm 60Lon B Lon A coslat

73 d AB arccos sinlat A sinlat B coslat A coslat B cos Lon AB Lon AB Lon B Lon A C A arccos sinlat B sinlat A cosd AB coslat A sind AB

74 coslat V sinc A coslat A Lat V arccos sinc A coslat A Lat V sgn cosc A arccos sinc A coslat A sgnx 1 if x0 0 if x0 1 if x0

75 coslon AV tanlat A tanlat V Lon AV Lon V Lon A Lon AV arccos tanlat A tanlat V Lon V Lon A sgn sinc Aarccos tanlat A tanlat V tanlat X coslon XV tanlat V Lat X arctan coslon XV tanlat V Lon XV Lon V Lon X cosc X sinlon XV sinlat V C X arccos sinlon XV sinlat V if sinc A 0 arccos sinlon XV sinlat V 180 if sinc A 0 C arctan coslat Lon B Lon A M Lat B Lat A Lat M Lat A Lat B 2 dkm Lat Lat B A 360 cosc dnm 60 Lat B Lat A cosc

76 dkm Lon 360 B Lon A coslat dnm 60Lon B Lon A coslat Lat B Lat A dkmcosc Lat Lat dnmcosc B A 60 Lon B Lon A dkmsinc coslat M Lon B Lon A dnmsinc 60cosLat M d i cosc i and d i sinc i

77 dx c'dloncoslat dy c'dlat dx cdlon dy c d Lat cos Lat Y Y 0 Lat P dy c 0 d Lat cos Lat cln tan Lat P 2 4 Y cln tan Lat P 2 45

78 X 0 Lon P dx clonp x ycoslat M Lat M Lat min Lat max 2

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81 MD B Az N

82 T 367yfloor 1.75 yfloor m 9 12 floor 275m UTh 9 d g T L M T L T L M 1.915sing 0.02sin2g T10 7 Dec arcsin sinl T sin cossinl RA 2arctan T cosdec cosl T

83 GHA Aries T 15UTh GHA GHA Aries RA x y floor x 360 GATh GHA 15 12h EoTh GAThUTh RAU cosg cos2g SD' 16.0 RAU ± ± ± ±

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85 dxnm 0.25cosLat AP dts dh' sin Az N dxnm dh' dts 0.25sin Az N coslat AP

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