Chapter 8 Copyright Henning Umland All Rights Reserved

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1 Chaper 8 Copyrigh Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon as well as he imes of sunrise, sunse, and wiligh. A body can be always above he horizon, always below he horizon, or above he horizon during a par of he day, depending on he observer's laiude and he declinaion of he body. A body is circumpolar (always above he celesial horizon) if he zenih disance is smaller han 90 a he momen of lower meridian passage, i. e., when he body is on he lower branch of he local meridian (Fig 8-1a). This is he case under he following condiions: La Dec > 0 AND La + Dec > 90 A body is coninually below he celesial horizon if he zenih disance is greaer han 90 a he insan of upper meridian passage (Fig 8-1b). The corresponding rule is: La Dec < 0 AND La Dec > 90 A celesial body being on he same hemisphere as he observer is eiher someimes above he horizon or circumpolar. A body being on he opposie hemisphere is eiher someimes above he horizon or permanenly invisible, bu never circumpolar. The sun provides a good example of how he visibiliy of a body is affeced by laiude and declinaion. A he ime of he summer solsice (Dec ), he sun is circumpolar o an observer being norh of he arcic circle (La > ). A he same ime, he sun remains below he celesial horizon all day if he observer is souh of he anarcic circle (La < 66.5 ). A he imes of he equinoxes (Dec 0 ), he sun is circumpolar only a he poles. A he ime of he winer solsice (Dec 23.5 ), he sun is circumpolar souh of he anarcic circle and invisible norh of he arcic circle. If he observer is beween he arcic and he anarcic circle, he sun is visible during a par of he day all year round. Rise and Se The evens of rise and se can be used o deermine laiude, longiude, or ime. One should no expec very accurae resuls, however, since he amospheric refracion may be erraic if he body is on or near he horizon.

2 The geomeric rise or se of a body occurs when he cener of he body passes hrough he celesial horizon (H 0 ). Due o he influence of amospheric refracion, all bodies excep he moon appear above he visible and sensible horizon a his insan. The moon is no visible a he momen of her geomeric rise or se since he depressing effec of he horizonal parallax ( 1 ) is greaer han he elevaing effec of amospheric refracion. The approximae apparen aliudes (referring o he sensible horizon) a he momen of he asronomical rise or se are: Sun (lower limb): 15' Sars: 29' Planes: 29' HP When measuring hese aliudes wih reference o he sea horizon, we have o add he dip of horizon (chaper 2) o he above values. For example, he aliude of he lower limb of he rising or seing sun is approx. 20' if he heigh of eye is 8m. We begin wih he well-known aliude formula (see chaper 4). sin H 0 sin La sin Dec + cos La cos Dec cos cos sin La sin Dec cos La cos Dec Solving he equaion for he meridian angle,, we ge : ( an La an Dec) The equaion has no soluion if he argumen of he inverse cosine is smaller han 1 or greaer han 1. In he firs case, he body is circumpolar, in he laer case, he body remains coninuously below he horizon. Oherwise, he funcion reurns values in he range from 0 hrough 180. Due o he ambiguiy of he funcion, he equaion has wo soluions, one for rise and one for se. For he calculaions below, we have o observe he following rules: If he body is rising (body easward from he observer), is reaed as a negaive quaniy. If he body is seing (body wesward from he observer), is reaed as a posiive quaniy. If we know our laiude and he ime of rise or se, we can calculae our longiude: Lon ± GHA GHA is he Greenwich hour angle of he body a he momen of rise or se. The sign of has o be observed carefully (see above). If he resuling longiude is smaller han 180, we add 360. Knowing our posiion, we can calculae he imes of sunrise and sunse: GMT Surise / se 12 ± [ ] Lon[ ] EoT

3 The imes of sunrise and sunse obained wih he above formula are no quie accurae since Dec and EoT are variable. Since we do no know he exac ime of rise or se a he beginning, we have o use esimaed values for Dec and EoT iniially. The ime of rise or se can be improved by ieraion (repeaing he calculaions wih Dec and EoT a he calculaed ime of rise or se). Furher, he imes hus calculaed are influenced by he irregulariies of amospheric refracion near he horizon. Therefore, a ime error of ±2 minues is no unusual. Accordingly, we can calculae our longiude from he ime of sunrise or sunse if we know our laiude: Lon [ ] ± + 15 ( 12 GMT EoT ) Sunrise / se Again, his is no a very precise mehod, and an error of several arcminues in longiude is no unlikely. Knowing our longiude, we are able o deermine our approximae laiude from he ime of sunrise or sunse: [ ] Lon[ ] 15 ( 12 GMT EoT ) Sunrise / se La cos arcan an Dec In navigaion, rise and se are defined as he momens when he upper limb of a body is on he visible horizon. These evens can be observed wihou a sexan. Now, we have o ake ino accoun he effecs of refracion, horizonal parallax, dip, and semidiameer. These quaniies deermine he aliude (Ho) of a body wih respec o he celesial horizon a he insan of he visible rise or se. sin Ho sin La sin Dec cos La cos Dec Ho HP SD R H Dip According o he Nauical Almanac, he refracion for a body being on he sensible horizon, R H, is approximaely (!) 34'. When observing he upper limb of he sun, we ge: Ho 0.15' 16' 34' Dip 50' Dip Ho is negaive. If we refer o he upper limb of he sun and he sensible horizon (Dip0), he meridian angle a he ime of sunrise or sunse is: sin La sin Dec cos La cos Dec Azimuh and Ampliude The azimuh angle of a rising or seing body is calculaed wih he azimuh formula (see chaper 4): Az sin Dec sin H sin La cos H cos La

4 Wih H0, we ge: Az sin Dec cos La Az is +90 (rise) and 90 (se) if he declinaion of he body is zero, regardless of he observer's laiude. Accordingly, he sun rises in he eas and ses in he wes a he imes of he equinoxes (geomeric rise and se). Wih H cener 50' (upper limb of he sun on he sensible horizon), we have: Az sin Dec sin La cos La The rue azimuh of he rising or seing body is: Az N Az 360 Az if if < 0 > 0 The azimuh of a body a he momen of rise or se can be used o find he magneic declinaion a he observer's posiion (compare wih chaper 13). The horizonal angular disance of he rising/seing body from he eas/wes poin on he horizon is called ampliude and can be calculaed from he azimuh. An ampliude of E45 N, for insance, means ha he body rises 45 norh of he eas poin on he horizon. Twiligh A sea, wiligh is imporan for he observaion of sars and planes since i is he only ime when hese bodies and he horizon are visible. By definiion, here are hree kinds of wiligh. The aliude, H, refers o he cener of he sun and he celesial horizon and marks he beginning (morning) and he end (evening) of he respecive wiligh. Civil wiligh: H 6 Nauical wiligh: H 12 Asronomical wiligh: H 18 In general, an aliude of he sun beween 3 and 9 is recommended for asronomical observaions a sea (bes visibiliy of brigher sars and sea horizon). However, excepions o his rule are possible, depending on he acual weaher condiions. The meridian angle for he sun a 6 aliude (cener) is: sin La sin Dec cos La cos Dec Using his formula, we can find he approximae ime for our observaions (in analogy o sunrise and sunse).

5 As menioned above, he simulaneous observaion of sars and he horizon is possible during a limied ime inerval only. To calculae he lengh of his inerval, T, we use he aliude formula and differeniae sin H wih respec o he meridian angle, : d ( sin H ) d cos La cos Dec sin d ( sin H ) cos La cos Dec sin d Subsiuing cosh.dh for d(sinh) and solving for d, we ge he change in he meridian angle, d, as a funcion of a change in aliude, dh: d cos H cos La cos Dec sin d H Wih H 6 and dh 6 (H ), we ge: [ ] 5.97 cos La cos Dec sin Convering he change in he meridian angle o a ime span (measured in minues) and ignoring he sign, he equaion is saed as: T [ m] 24 cos La cos Dec sin The shores possible ime inerval for our observaions (La 0, Dec 0, 96 ) lass approx. 24 minues. As he observer moves norhward or souhward from he equaor, cos La and sin decrease (>90 ). Accordingly, he duraion of wiligh increases. When is 0 or 180, T is infinie. This is in accordance wih he well-known fac ha wiligh is shores in equaorial regions and longes in polar regions. We would obain he same resul when calculaing for H 3 and H 9, respecively: [ m] 4 ( [ ] [ ] ) T 9 3 The Nauical Almanac provides abulaed values for he imes of sunrise, sunse, civil wiligh and nauical wiligh for laiudes beween 60 and +72 (referring o an observer being a he Greenwich meridian). In addiion, imes of moonrise and moonse are given.