Presentation on theme: "Latitude and Longitude"— Presentation transcript:
1 Latitude and Longitude We now enter a realm where we worry more about precision.Angular measuring devicesTime measuring devicesCorrections of one degree
2 Latitude and Longitude Modern technique:Observe height of two or more celestial bodies, where you know their declination and SHAKnow time of observationGives two lines of positionIntersection of two lines gives position.Process is called “sight reduction”In primitive navigation, don’t have access to:High precision measurement of height (angle above horizon)Precise declination and SHA tablesCalculators or trig tablesMaybe not a watch (e.g. Vikings, Polynesians)Must improvise height measurementTake advantage of tricksLongitude impossible without a watch (exception is “lunar method”, which requires tables and extensive calculations).
3 Altitude (height) measuring devices over the years KamalCross staffBack staffAstrolabeQuadrantSextantBubbleGyro-sextantOctant
10 Quadrant made from materials lying around the house Scrap wood, paper, pencil, metal tube, old banjo string,glueEstimated accuracy is about ½ a degree (30’)Finished quadrantTaking a sighting near sunset
11 Quadrant markings made by successive halving of angles Ab initio way of dividing up the angles, if degreemarkings are notavailable.This one is divided into 64 angles =o per angle.Estimated accuracy of this is 21’ byinterpolation (usingtriangles)
12 Sighting tube for home-made quadrant Estimated accuracy is maybe 20’ End of tube with alignment wires.Image of sun’s shadow
13 Modern versus “primitive tools” Tube was manufactured with a high precision drawing processBanjo wires, likewiseAngle calibration aided by compass, rulerUsing circle arcs works down to about 22.5oDivide chords visually at smaller anglesCut of board was preciseCan take advantage of local materials (e.g. metal broom-handle) which have more precision than primitive items.
14 Refraction and limb of sun When approaching the precision of a degree or fraction of a degree, refraction and the size of the sun becomes important.The diameter of the sun and moon are 32’, over half a degree.To get the highest precision – particularly near sunrise and sunset, must make corrections
15 Refraction in the atmosphere always raises the height of stars, sun, planets from true height
17 David Burch’s construction for refraction correction Prescription:Make a graph of refractionangle versus measured height.48’ vertical, 6o horizontal.Lay out compass from 48’, 6o to34.5’ and 0o degrees, swing arcdown to 6o mark.For higher angles, correction is60’ divided by measured height
18 At sunset, center of sun is actually 34.5’ below horizon When sun just disappears, the center of the sun is 56’ belowthe horizon (almost one degree!).What you seeTrue position34.5’True positionWhat you see34.5’56’Moment of sunsetHeight = 0
19 Standard Celestial Navigation Height is observedCorrections madeLimb of sun – 16’RefractionDip 1’ times if observing horizon, no correction for plumb line or bubbleUse Hc as height of object
20 Latitude from meridian height of sun 90o+Decl-Meridian heightDue South
21 Data from home-made quadrant on 27-Oct-08 Maximum height of sun = 24.9 units = 35.06oNaïve declination (from sines) = 14.06This gives latitude of 40.88Declination from table = 13.06Latitude using table declination is (Cambridge = 42.38o )Variance = 40’ Error in meridian height = 30’Largest error was estimation of declination w/o tables – 1oFraction of day since midnightVertical angle
22 Latitude from meridian passage of stars True for any star or planet where you know the declination.Basically the same as for the sun, but issues arise withfinding a horizon at night.Latitude =90o+Decl-Meridian heightDeclination is more reliable forstars – it never changesHave to use an artificial horizonof some kind (plumb bob) at night.Requires multiple sightings to findmaximum height – changesslowly during passage.MeridianheightDue South
23 Latitude from North Star SchedarCassiopeiaPolaris is offset towardCassiopeia by 41’Subtract correction whenCassiopeia is overheadAdd correction whenBig Dipper is overheadPolarisDubheBig dipper/Ursa major
24 Latitude from zenith stars A star at the zenith will have declination equal to yourlatitude.If you can get a good north-south axis, it is possible tomeasure latitude. Eg. Use a long stick with a rule atthe end and a plumb-bob to keep the stick verticalThis was used by the Polynesians to measure latitude –They would typically lie on their backs in their canoes andcompare the stars at the zenith to the tops of the masts.
25 Latitude from polar horizon grazing stars Best to do during navigational twilight – 45 minutes after sunsetLatitude = (polar distance – minimum height)Polar distance =(90o – Declination)Min. star heightHorizon (est)
26 Latitude sailingAccurate longitude determination only came when chronometers were availableBefore this, many voyages involved latitude sailing: sail along the coast until one reaches the latitude of the destination, then sail east or west along this latitude across the sea (checking position with astrolabe, etc).ColumbusArab tradersVikingsetc
27 Latitude from length of day Covered in “Sun and Moon” talkMost effective around solstices(+/-) a couple of months around solsticeUseless at equinoxCan also use this for stars – length of “stellar” day, if rising and setting are visible (eg. Antares)Star can’t be on equator, though!!Length of day = 24*d/360o
28 Time!!! Many navigational techniques require time Longitude for sureLatitude from length of dayLatitude from time of sunsetUntil the invention of the nautical chronometer, navigation was a combination of latitude observations and dead reckoning for longitude
29 Harrison chronometerDeveloped in response toprize offered by British RoyalNavy.Many scientists in the daylooked at the “lunar” method.
32 Longitude: Local Area Noon (LAN) Finding the time of maximum altitude of the sun (or a star) is difficult with any precision during meridianHeight is changing very slowlyMid-time between sunrise and sunset is local noon – but need to do correction for equation of time (EoT) to get longitude.Convert time of local noon to UTC (normally + 5 hours in eastern time zone, this week +4), add EoT if sun is early (like now), subtract EoT if sun is late (e.g. February 14th).Difference between this and 12:00 will give longitude in hours.Use 15o per hour to convert.
33 Memorization trick for E. o. T. – 14 minutes late on Feb Memorization trick for E.o.T. – 14 minutes late on Feb. 14th (Valentines day),4 days early three months later (May 15th), 16 minutes early on Halloween,6 minutes late 3 months earlier (June 26th)Approximate this – 2 weeks either side of points are flat, use trapezoids to connect
34 Finding LAN data from quadrant and watch Best information comes from period when sun is risingand setting – 3 hours after sunrise/before sunset. Height changesrapidlyMidpoint of parabolic fit is(fraction of day from midnightThis is 12:29:16Add 4 hrs for UTC: 16:29:16Add 16 min for EoT:16:45:16Time since noon at Greenwich:4:45:16, at 15o/hour, this isLong = (Cambridge = 71.11)Fraction of day since midnightVertical angle
35 LAN for StarsIf you can find stars that rise *and* set, you can find the meridian crossing time.Example – take the mid-point of rising and setting Antares.Again, need a good horizon, or artificial horizon.Stars low in the sky (southern stars) are best choices.Have to use the number of days after March 21st and SHA to use meridian crossing.
36 Full Treatment of Celestial Navigation Assume locations of all planets, sun, stars to arbitrary accuracyStandard is UTC (coordinated universal time)Assume a clock that is synchronized to UTCReturn to azimuth and celestial coordinates
38 For a given sighting, there is a circle of possible locations on the earth, called a “line of position”or LOP.The intersection oftwo LOP’s gives twopossible locations.Typical navigationalpractice is to assumea longitude and latitudeand, locally, the LOP’sare lines.
39 Two equations for celestial observations WhereHc = height (after corrections for refraction)d = declination of objectL = latitudeZ=zenith anglet = hour angle (angle between meridian and star’s “longitude”)(LHA)