Post on 28-Nov-2014
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Times and hour anglesTimes and hour angles
References
Celestial bodies are reference for measuring passage of time Sundial
o 1500 – 1300 BCo Measure time of day by shadowo Hours shorter in winter
Water clocks / clepsydraso Water dripping at constant rate from small holeo Water filling up a container with markings on insideo Determine hours at night
Tower of Windso Octagonal structureo Sundials & mechanical hour indicators
Su Sung clock Mechanical clocks
o 14th centuryo Weight driveno Regulated by verge-and-foliot escapemento Period oscillation depended on driving force & friction => rate difficult to
regulate Spring-powered clocks
o Between 1500 & 1510o Smaller clocks & watches
Pendulum clocko 1656o Christiaan Huygenso Used natural period of oscillationo Error less than 1 minute a dayo Later refinements: less than 10 seconds a dayo William Clement
1671 Anchor / recoil escapement Less interferences with motion of pendulum
o George Graham 1721 Improvement accuracy to 1 second a day Compensating changes in pendulum’s length due to temperature
o John Harrison 1761 Refined temperature compensation techniques New methods to reduce friction Very accurate marine chronometer
Regional time zoneo 1850s
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o Established in New England 1905 => radio time signal transmitted to help ships find longitude Quartz clock
o Based on piezoelectric property of quartz crystalso Electric field => changes shapeo Squeezing / bending => electric fieldo In suitable electronic circuit => vibration & generation electric signal of
relatively constant frequencyo No gears / escapemento Frequency depends on crystal’s size, shape & temperature
Atomic clocko Atoms have stable resonanceso New international unit of time => 1 second = 9 192 631 770 oscillations of
cesiumo Accuracy 30 billionths of a second per year
World time scaleso Greenwich Mean Time evolved as time reference for the worldo Coordinated Universal Time
Runs at rate of atomic clocks Difference with Earth near 1 second => adjustment in UTC
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UT/GAT/time equation
Greenwich meridian time (GMT)o Former name for mean solar time at Royal Observatory in Greenwicho Now Universal Time (UT)
Mean solar timeo Related to mean sun
Running in plane of equator Constant velocity
o Begins at midnight Apparent (true) solar time
o Time interval between two successive lower transits of true sun for meridiano Begins at midnight
Time equation ()o Difference between mean time & apparent time (or passing mean sun and true
sun through meridian)o Given in Nautical Almanaco Causes for differences
Plane equator not same as plane Earth’s orbit around the sun (angle of obliquity = 23,5°)
Orbit of Earth around sun is ellipseo Remains practically constant from year to yearo No correction for 4 days in 1 year (16/04, 14/06, 02/09 & 25/12)o Maximum deviation on 03 or 04 / 11 = 16 min. 23 sec.
oo = C + Ro UT = GAT +/-
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LAT/LMT
Local meridian of observer = reference UT = LMT +/- gt GAT = LAT +/- gt gt
o Added when easto Subtracted when westo Calculation
15° of longitude = 1 hour 1° of longitude = 4 minutes Parts of 1° longitude = parts of 4 minutes
o Nautical Almanac (Conversion of arc into time)
ZT/ZD
World subdivided into zones of 15° longitude Time same throughout each zone Each zone has zone description or delay (ZD) indicating with UT ZT = UT +/- ZD ZD = (g + 7°30’)/15° (alles na de komma laten vallen)
ST/Legal time
Adopted in 19th century Zones subdivided or altered in shape for convenience of inhabitants Nautical Almanac/Admiralty List of Radio Signals Sometimes Daylight Saving Time Basis is UT, indicated times added or subtracted as listed Date line on 180° of longitude East of date line is one day ahead
WT
Watch time kept on board Shift takes place in 00-04h watch Main idea: next day watch time 12h00 as close as possible to LMT 12h00
Corrections to time
Chronometer indicate exact UT High precision watch but still an error Radio time signals for comparison
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Syntax times
GHA
Celestial bodies expressed by declination & Greenwich Hour Angle Angle between meridian of Greenwich & hour circle of body measured westward
from Greenwich meridian Groups of celestial bodies
o Suno Moono Planetso Stars
Add SHA to GHA of Aries to obtain GHA each celestial body Sun
o Sidereal hour angle changes 1° daily => GHA sun table insertedo GHA + I => GHA
Moono SHA changes more rapidly with average of 12,85° => separate tableo Extra correction due to irregular motiono GHA + I + v-correction => GHA
Planetso Irregular patterno Own table & v-correctiono GHA + I + v-correction => GHA
Starso Does not significantly change => NO separate tableo GHA Aries + I => GHA Aries => + SHA star => GHA star
LHA/P
Principle same, reference = local meridian GHA +/- g => LHA (+ if g = E, - if g = W) Polar angle (P)
o 000° < LHA < 180° => P = LHA & P = Wo 180° < LHA < 360° => P = 360° - LHA & P = E
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Relation between time and hour angle
Correlation between time & hour angleo Same referenceso Differ 180° or 12 hours
Basis time measurement = sun => sun at anti-meridian (180°) => 00h00
Syntax hour angles
GHA mean sun + 12h00 = UT LHA mean sun + 12h00 = LMT GHA mean sun = GHA apparent sun +/-
Traditional astro-positionTraditional astro-position
Basic principle of the position line
Vikings used sun & pole star to sail line of latitude No certain method of fixing ships position until mid 18th century 1837
o Captain Thomas Sumnero Establishing position lines from sightso Sextant altitude increases when sailing towards point where line between
celestial body and centre of earth intersects the earth’s surfaceo Theory used to elaborate a method to determine a line of position
Determine altitude & azimuth for estimated position for given time Difference between observed altitude & calculated altitude Convert minutes of difference in altitudes to miles Line perpendicular to azimuth towards or away from geographic
position of body Principle line of position
o Each celestial body for certain moment has geographical positiono All observers sighting celestial body with same declination => circle around
geographical positiono Geographical positions move on surface of the Eartho Earth & celestial equators in same plane =>
Latitude geographical position = declination of star Longitude geographical position = GHA
o Zenithal distance = distance between observer & geographical positiono = 90° - ho At least 3 circles for one positiono No circles drawn on chart but tangents
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The intercept
Difference between true altitude & calculated altitudeo True latitude = altitude by observationo Calculated altitude = expected altitude based upon assumed position
sin hc = +/-(sin * sin la) + cos * cos la * cos P
+ if & l same name & P < 90° - if & l same name & P > 90° - if & l not same name & P < 90°
hv – hc = h + => towards body - => away from body
Determination astronomical position
o Plot assumed positiono Line passing position in direction of Azimutho Measure intercept in right directiono Draw line of position perpendicular to Azimuth
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Azimuth
3 elements at the same timeo Altitude by sextanto Reading chronometero Compass bearing
Compass not precise enough => calculate Azimuth
tg la / tg P – tg / sin P = cotg Az / cos la A + B = C (Nories, see practical navigation) sin Az = sin P * cos * sec hv
Amplitudo
Special case of azimuth Celestial body on true horizon Amplitude = arc between prime vertical & body Quadrantal starting from E/W towards N/S sin A = sin * sec l Nories
Determination of compass errors
Calculated azimuth and amplitude => true bearing Comparing with compass bearing => total error
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Astro-position special casesAstro-position special cases
True latitude
Bearing due North or South Object crosses observer’s meridian (culmination) lv = * cu
cu = 90° - hcu
Exact momento Nautical Almanaco Calculation
Altitude and time of observation before estimated time of passage Note exact time when celestial body has same altitude tcu = (t1 + t2) / 2
True latitude with Polaris
lv = hv – 1° + a0 + a1 + a2 (Almanac) Polaris not exactly above true North Pole => corrections lv = hv + a0 + a1
o a0 = * cos Po a1 = ½ * ² * sin² P * sin 1’ * tg hv
Ex-meridian
Correction for meridian passage if not on true meridian lv = + cu
cu = - X X = ½ sec hv * sin 1’ * P² * cos l * cos X = A * P² A = in Nories
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True longitude
Longitude tempered by time Time aboard & time at another place with known longitude
Line of position
Celestial body’s bearing due East or West 2 elements needed
o Altitudeo LHA / Polar angle
cos P = tg * cotg lo Nautical Almanac to find LMTo LHA star = LHA Aries + SHA staro LMT star = LMT Aries + t (conversion of SHA to time)
sin hv = sin * csc l Bearing due east or west => take exact time Nautical Almanac => GHA & cos P = tg * cotg l LHA – GHA = lon (g)
At meridian passage
LHA = 360° LHA – GHA = lon (g)
Position fixPosition fix
Graphic solution
Intersection two or more lines of position => fix Lines of position separated in time => running fix
Noon position and its value in navigation
Navigational reasonso Indication position logbooko Calculation mean speed over ground since departureo Calculation mean speed over ground since previous noon positiono Calculation ETAo Plotting progress of voyage on large scale chart
Economical reasonso Charter party: current conditions + renewal or new chartero Economical speed orderedo Ship’s agent: ETAo Liner serviceso …
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Pagel method
Replaces graphical solution in case of true latitude and almost true longitude Find g to obtain true longitude g = cotg Az / cos lv * l g = C * l (Nories) gv = ge +/- g
Rise, set and twilightRise, set and twilight
Rise, set
Sunrise & sunset = times when upper limb of Sun is on visible horizon Moonrise & moonset = times when upper limb is on visible horizon Theoretical rise & set = relative to true horizon Visible rise & set = relative to visible horizon True horizon is 33,8’ above visible horizon Amplitude: when sun / moon with centre on true horizon
o Sun: lower limb semi-diameter above visible horizono Moon: 1/3 of surface visible above visible horizon (due to parallax)
Twilight
Before sunrise & after sunset Natural light provided by upper atmosphere (reflection of sunlight) Amount of light during twilight affected by state of atmosphere & local weather
conditions Limits applicable considering only position of sun below local horizon Civil twilight
o Sun geometrically 6° below horizono Terrestrial objects clearly distinguishedo Horizon clearly defined & brightest stars visible under good atmospheric
conditions & in absence of other illumination Nautical twilight
o Centre of sun 12° below horizono General outlines of ground objects distinguishableo Horizon = indistinct
Astronomical twilighto Centre of sun 18° below horizono Sky illumination very faint
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Special diagramsSpecial diagrams
Eclipses
General
Two kindso Solar eclipseo Lunar eclipse
General conditions: centres Earth, Moon & Sun one line in space Types
o Totalo Annular (shadowing disk covering complete, but off-centre)o Partial (centres almost one line in space, but shadowed disk not complete)
Eclipse of sun occurs during new moono Every month new moon => NOT every month eclipseo Moon passes to north or south of sun
Eclipse of moon occurs during full moono Moon passes north or south of earth’s shadowo Orbit around earth not same as earth’s orbit around sun (tilted 5°)
Point twice a year where moon appears neither north nor south => eclipseo Ascending node (moon moving from south to north)o Descending node (moon moving from north to south)
Regular pattern (= Saros)o Periodicity = 18 years 11 days 8 hourso Subsequent eclipses are visible from different parts of the globeo Same geographic region every 3 saroseso Nodes shift eastward => saros series doesn’t last indefinitelyo Exact duration & number of eclipses is not constanto Different saros series in progress at the same time
Solar eclipse
Moon between sun & earth Total or annular => declination & GHA of sun & moon must be the same 3 zones
o Eclipse zone => all observers can see total or annular eclipseo Penumbra zone => observers in this zone see partial eclipse & intensity of light
decreaseso Free zone => observers don’t remark anything
Lunar eclipse
Earth between sun & moon Total or annular => declination sun & moon opposite sign & same value & GHA
differs 180° Moon becomes dark red due to refraction
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Meridian passages of planets
In LMT In diagram
o Meridian passage sun in LMTo Meridian passage each planet in LMTo SHA sun & planets throughout yearo Possible conjunctions (verification if declination is same is necessary)