Chapter 6. Methods for Latitude and Longitude Measurement. Latitude by Polaris

Transcription

1 Chaper 6 Mehods for Laiude and Longiude Measuremen Laiude by Polaris The observed aliude of a sar being verically above he geographic norh pole would be numerically equal o he laiude of he observer (Fig This is nearly he case wih he pole sar (Polaris. However, since here is a measurable angular disance beween Polaris and he polar axis of he earh (presenly ca. 1, he aliude of Polaris is a funcion of LHA Aries. Nuaion, oo, influences he aliude of Polaris measurably. To obain he accurae laiude, several correcions have o be applied: La Ho 1 + a + a + a 0 1 The correcions a 0, a 1, and a depend on LHA Aries, he observer's esimaed laiude, and he number of he monh. They are given in he Polaris Tables of he Nauical Almanac [1]. To exrac he daa, he observer has o know his approximae posiion and he approximae ime. Noon Laiude (Laiude by Maximum Aliude This is a very simple mehod enabling he observer o deermine his laiude by measuring he maximum aliude of an objec, paricularly he sun. No accurae ime measuremen is required. The aliude of he sun passes hrough a fla maximum approximaely (see noon longiude a he momen of upper meridian passage (local apparen noon, LAN when he GP of he sun has he same longiude as he observer and is eiher norh or souh of him, depending on he observer s geographic laiude. The observer s laiude is easily calculaed by forming he algebraic sum or difference of declinaion and observed zenih disance z (90 -Ho of he sun. depending on wheher he sun is norh or souh of he observer (Fig. 6-.

2 1. Sun souh of observer (Fig. 6-a: La Dec + ( 90 Ho. Sun norh of observer (Fig. 6-b: La Dec ( 90 Ho Norhern declinaion is posiive, souhern negaive. Before saring he observaions, we need a rough esimae of our curren longiude o know he ime (GMT of LAN. We look up he ime of Greenwich meridian passage of he sun on he daily page of he Nauical Almanac and add 4 minues for each degree wesern longiude or subrac 4 minues for each degree easern longiude. To deermine he maximum aliude, we sar observing he sun approximaely 15 minues before LAN. We follow he increasing aliude of he sun wih he sexan, noe he maximum aliude when he sun sars descending again, and apply he usual correcions. We look up he declinaion of he sun a he approximae ime (GMT of local meridian passage on he daily page of he Nauical Almanac and apply he appropriae formula. Hisorically, noon laiude and laiude by Polaris are among he oldes mehods of celesial navigaion. Ex-Meridian Sigh Someimes, i may be impossible o measure he maximum aliude of he sun. For example, he sun may be obscured by a cloud a his momen. If we have a chance o measure he aliude of he sun a few minues before or afer meridian ransi, we are sill able o find our exac laiude by reducing he observed aliude o he meridian aliude, provided we know our exac longiude (see below and have an esimae of our laiude. Firs, we need he ime of local meridian ransi (easern longiude is posiive, wesern longiude negaive: T Transi [ GMT ] 1 EoT The meridian angle of he sun,, is calculaed from he ime of observaion: Lon ( T [ GMT] T [ GMT] Observaion Transi Saring wih our esimaed Laiude, La E, we calculae he aliude of he sun a he ime of observaion. We use he aliude formula from chaper 4: Hc arcsin ( sin La sin Dec + cos La cos Dec cos E E We furher calculae he aliude of he sun a meridian ransi, H MTC : H MTC 90 La E Dec The difference beween H MTC and Hc is called reducion, R: R H MTC Hc Adding R o he observed aliude, Ho, we ge approximaely he aliude we would observe a meridian ransi, H MTO : H MTO Ho + R

3 From H MTO, we can calculae our improved laiude, La improved : La Dec ± 90 ( improved H MTO (sun souh of observer: +, sun norh of observer: The exac laiude is obained by ieraion, i. e., we subsiue La improved for La E and repea he calculaions unil he obained laiude is virually consan. Usually, no more han one or wo ieraions are necessary. The mehod has a few limiaions and requires criical judgemen. The meridian angle should be smaller han abou one quarer of he expeced zenih disance a meridian ransi (z MT La E Dec, and he meridian zenih disance should be a leas four imes greaer han he esimaed error of La E. Oherwise, a greaer number of ieraions may be necessary. Dec mus no lie beween La E and he rue laiude because he mehod yields erraic resuls in such cases. If in doub, we can calculae wih differen esimaed laiudes and compare he resuls. For safey reasons, he sigh should be discarded if he meridian aliude exceeds approx. 85. If is no a small angle ( > 1, we may have o correc he laiude las found for he change in declinaion beween he ime of observaion and he ime of meridian ransi, depending on he curren rae of change of Dec. Noon Longiude (Longiude by Equal Aliudes, Longiude by Meridian Transi Since he earh roaes wih an angular velociy of 15 per hour wih respec o he mean sun, he ime of local meridian ransi (local apparen noon of he sun, T Transi, can be used o calculae he observer's longiude: Lon 15( 1 T EoT Transi Transi T Transi is measured as GMT (decimal forma. The correcion for EoT a he ime of meridian ransi, EoT Transi, has o be made because he apparen sun, no he mean sun, is observed (see chaper 3. Since he Nauical Almanac conains only values for EoT (see chaper 3 a 0:00 GMT and 1:00 GMT of each day, EoT Transi has o be found by inerpolaion. Since he aliude of he sun - like he aliude of any celesial body - passes hrough a raher fla maximum, he ime of peak aliude is difficul o measure. The exac ime of meridian ransi can be derived, however, from wo equal aliudes of he sun. Assuming ha he sun moves along a symmerical arc in he sky, T Transi is he mean of he imes corresponding wih a chosen pair of equal aliudes of he sun, one occurring before LAN (T 1, he oher pas LAN (T (Fig. 6-3: T 1 + T T Transi In pracice, he imes of wo equal aliudes of he sun are measured as follows: In he morning, he observer records he ime (T 1 corresponding wih a chosen aliude, H. In he afernoon, he ime (T is recorded when he descending sun passes hrough he same aliude again. Since only imes of equal aliudes are measured, no aliude correcion is required. The inerval T -T 1 should be greaer han 1 hour.

4 Unforunaely, he arc of he sun is only symmerical wih respec o T Transi if he sun's declinaion is fairly consan during he observaion inerval. This is approximaely he case around he imes of he solsices. During he res of he year, paricularly a he imes of he equinoxes, T Transi differs significanly from he mean of T 1 an due o he changing declinaion of he sun. Fig. 6-4 shows he aliude of he sun as a funcion of ime and illusraes how he changing declinaion affecs he apparen pah of he sun in he sky. The blue line shows he pah of he sun for a given, consan declinaion, Dec 1. The red line shows how he pah would look wih a differen declinaion, Dec. In boh cases, he apparen pah of he sun is symmerical wih respec o T Transi. However, if he sun's declinaion varies from Dec 1 a T 1 o Dec a T, he pah shown by he green line will resul. Now, he imes of equal aliudes are no longer symmerical o T Transi. The sun's meridian ransi occurs before (T +T 1 / if he sun's declinaion changes oward he observer's parallel of laiude, like shown in Fig Oherwise, he meridian ransi occurs afer (T +T 1 /. Since ime and local hour angle (or meridian angle are proporional o each oher, a sysemaic error in longiude resuls. The error in longiude is negligible around he imes of he solsices when Dec is almos consan, and is greaes (up o several arcminues a he imes of he equinoxes when he rae of change of Dec is greaes (approx. 1 arcminue per hour. Moreover, he error in longiude increases wih he observer's laiude and may be quie dramaic in polar regions. The obained longiude can be improved, if necessary, by applicaion of he equaion of equal aliudes [5]: an La an Dec sin an Dec 15 ( T T 1 is he meridian angle of he sun a T. is he change in which cancels he change in aliude resuling from he change in declinaion beween T 1 an, Dec. La is he observer's laiude, e. g., a noon laiude. If no accurae laiude is available, an esimaed laiude may be used. Dec is he declinaion of he sun a T. The correced second ime of equal aliude, T *, is: T * T T T 15 A T *, he sun would pass hrough he same aliude as measured a T 1 if Dec did no change during he inerval of observaion. Accordingly, he ime of meridian ransi is: * T 1 + T T Transi

5 The correcion is very accurae if he exac value for Dec is known. Calculaing Dec wih MICA yields a more reliable correcion han exracing Dec from he Nauical Almanac. If no precise compuer almanac is available, Dec should be calculaed from he daily change of declinaion o keep he rounding error as small as possible. Alhough he equaion of equal aliudes is sricly valid only for an infiniesimal change of Dec, ddec, i can be used for a measurable change, Dec, (up o several arcminues as well wihou sacrificing much accuracy. Accurae ime measuremen provided, he residual error in longiude should be smaller han ±0.1' in mos cases. The above formulas are no only suiable o deermine one's exac longiude bu can also be used o deermine he chronomeer error if one's exac posiion is known. This is done by comparing he ime of meridian ransi calculaed from one's longiude wih he ime of meridian ransi derived from he observaion of wo equal aliudes. Fig. 6-5 shows ha he maximum aliude of he sun is slighly differen from he aliude a he momen of meridian passage if he declinaion changes. Since he sun's hourly change of declinaion is never greaer han approx. 1' and since he maximum of aliude is raher fla, he resuling error of a noon laiude is no significan (see end of chaper. The equaion of equal aliudes is derived from he aliude formula (see chaper 4 using differenial calculus: sin H sin La sin Dec + cos La cos Dec cos Firs, we wan o know how a small change in declinaion would affec sin H. We differeniae sin H wih respec o Dec: ( sin H Dec sin La cos Dec cos La sin Dec cos Thus, he change in sin H caused by an infiniesimal change in declinaion,, is: ( sin H Dec ( sin La cos Dec cos La sin Dec cos Now we differeniae sin H wih respec o in order o find ou how a small change in he meridian angle would affec sin H: ( sin H cos La cos Dec sin The change in sin H caused by an infiniesimal change in he meridian angle, d, is: ( sin H d cos La cos Dec sin d Since we wan boh effecs o cancel each oher, he oal differenial has o be zero: ( sin H ( sin H Dec d Dec + d 0 ( sin H ( sin H d Dec d Dec

6 cos La cos Dec sin d ( sin La cos Dec cos La sin Dec cos d sin La cos Dec cos La sin Dec cos cos La cos Dec sin d an La sin an Dec an Longiude Measuremen on a Traveling Vessel On a raveling vessel, we have o ake ino accoun no only he influence of varying declinaion bu also he effecs of changing laiude and longiude on sin H during he observaion inerval. Again, he oal differenial has o be zero because we wan he combined effecs o cancel each oher wih respec o heir influence on sin H: ( sin H ( ( sin H ( sin H d + d Lon + La d La + Dec 0 ( sin H ( ( sin H ( sin H d + d Lon La d La + Dec Differeniaing sin H (aliude formula wih respec o La, we ge: ( sin H La cos La sin Dec sin La cos Dec cos Thus, he oal change in caused by he combined variaions in Dec, La, and Lon is: d an La sin an Dec an Dec + an sin an La d La d Lon an dla and dlon are he infiniesimal changes in laiude and longiude caused by he vessel's movemen during he observaion inerval. For pracical purposes, we can subsiue he measurable changes Dec, La and Lon for ddec, dla and dlon (resuling in he measurable change. La and Lon are calculaed from course, C, and velociy, v, over ground and he ime elapsed: [] ' v[ kn] cosc ( T T La 1 sin C Lon 1 cos La [] ' v[ kn] ( T T ( kno 1 nm h 1 kn / Again, he correced second ime of equal aliude is: T * T 15 The longiude hus calculaed refers o T1. The longiude a T is Lon+ Lon.

7 The longiude error caused by changing laiude can be dramaic and requires he navigaor's paricular aenion, even if he vessel moves a a moderae speed. The above consideraions clearly demonsrae ha deermining one's exac longiude by equal aliudes of he sun is no as simple as i seems o be a firs glance, paricularly on a raveling vessel. I is herefore undersandable ha wih he developmen of posiion line navigaion (including simple graphic soluions for a raveling vessel longiude by equal aliudes became less imporan. Time Sigh The process of deriving he longiude from a single aliude of a body (as well as he observaion made for ha purpose is called ime sigh. However, his mehod requires knowledge of he exac laiude, e. g., a noon laiude. Solving he aliude formula (chaper 4 for he meridian angle,, we ge: ± arccos sin Ho sin La sin Dec cos La cos Dec From and GHA, we can easily calculae our longiude (see Sumner's mehod, chaper 4. In fac, Sumner's mehod is based upon muliple soluions of a ime sigh. During a voyage in December 1837, Sumner had no been able o deermine he exac laiude for several days due o bad weaher. One morning, when he weaher finally permied a single observaion of he sun, he calculaed hypoheical longiudes for hree assumed laiudes. Observing ha he posiions hus obained lay on a sraigh line which accidenally coincided wih he bearing line of a erresrial objec, he realized ha he had found a celesial line of posiion. This discovery marked he beginning of a new era of celesial navigaion. A ime sigh can be used o derive a line of posiion from a single assumed laiude. Afer solving he ime sigh, we plo he assumed parallel of laiude and he calculaed meridian. Nex, we calculae he azimuh of he body wih respec o he posiion hus obained (azimuh formula, chaper 4 and plo he azimuh line. Our line of posiion is he perpendicular of he azimuh line going hrough he calculaed posiion (Fig The laer mehod is of hisorical ineres only. The modern navigaor will cerainly prefer he inercep mehod (chaper 4 which can be used wihou any resricions regarding meridian angle (local hour angle, laiude, and declinaion (see below. A ime sigh is no reliable when he body is close o he meridian. Using differenial calculus, we can demonsrae ha he error in he meridian angle, d, resuling from an aliude error, dh, varies in proporion wih 1/sin : d cos Ho cos La cos Dec sin dh Moreover, d varies in proporion wih 1/cos La and 1/cos Dec. Therefore, high laiudes and declinaions should be avoided as well. Of course, he same resricions apply o Sumner's mehod.

8 The Meridian Angle of he Sun a Maximum Aliude As menioned above, he momen of maximum aliude does no exacly coincide wih he upper meridian ransi of he sun (or any oher body if he declinaion is changing. A maximum aliude, he rae of change of aliude caused by he changing declinaion cancels he rae of change of aliude caused by he changing meridian angle. The equaion of equal aliude can be used o calculae he meridian angle of he sun a his momen. We divide each side of he equaion by he infiniesimal ime inerval dt: d an La sin an Dec an Measuring he rae of change of an in arcminues per hour we ge: an La an Dec 900 '/ h sin an [] ' Sine is very small, we can subsiue an for sin : Now, we can solve he equaion for an : an La an Dec 900 an an an La an Dec 900 Since a small angle (in radians is nearly equal o is angen, we ge: ['] [] ' π 180 an La an Dec 900 [] ' Measuring in arcminues, he equaion is saed as: [] ' 3.8 ( an La an Dec [] ' ddec/dt is he rae of change of declinaion measured in arcminues per hour. The maximum aliude occurs afer LAN if is posiive, and before LAN if is negaive. For example, a he ime of he spring equinox (Dec 0, ddec/dt +1'/h an observer being a +80 (N laiude would observe he maximum aliude of he sun a +1.7', i. e., 86.7 seconds afer meridian ransi (LAN. An observer a +45 laiude, however, would observe he maximum aliude a +3.8', i. e., only 15.3 seconds afer meridian ransi. We can use he las equaion o evaluae he sysemaic error of a noon laiude. The laer is known o be based upon he maximum aliude, no on he meridian aliude of he sun. Following he above example, he observer a 80 laiude would observe he maximum aliude 86.7 seconds afer meridian ransi. During his inerval, he declinaion of he sun would have changed from 0 o '' (assuming ha Dec is 0 a he ime of meridian ransi. Using he aliude formula (chaper 4, we ge: Hc ( sin 80 sin 1.445'' + cos 80 cos 1.445' ' cos 1.7' 10 0' 0.7' ' arcsin

9 In conras, he calculaed aliude a meridian ransi would be exacly 10. Thus, he error of he noon laiude would be -0.7''. In he same way, we can calculae he maximum aliude of he sun observed a 45 laiude: Hc ( sin 45 sin 0.55'' + cos 45 cos 0.55' ' cos 3.8' 45 0' 0.13' ' arcsin In his case, he error of he noon laiude would be only -0.13''. The above examples show ha even a he imes of he equinoxes, he sysemaic error of a noon laiude caused by he changing declinaion of he sun is much smaller han oher observaional errors, e. g., he errors in dip or refracion. A significan error in laiude can only occur if he observer is very close o one of he poles (an La!. Around he imes of he solsices, he error in laiude is pracically non-exisen.