Latitude and Longitude

1. Latitude and Longitude • We now enter a realm where we worry more about precision. • Angular measuring devices • Time measuring devices • Corrections of one degree

2. Latitude and Longitude • Modern technique: • Observe height of two or more celestial bodies, where you know their declination and SHA • Know time of observation • Gives two lines of position • Intersection 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 tables • Calculators or trig tables • Maybe not a watch (e.g. Vikings, Polynesians) • Must improvise height measurement • Take advantage of tricks • Longitude impossible without a watch (exception is “lunar method”, which requires tables and extensive calculations).

3. Altitude (height) measuring devicesover the years • Kamal • Cross staff • Back staff • Astrolabe • Quadrant • Sextant • Bubble • Gyro-sextant • Octant

4. Cross staff

5. Backstaff Allows viewer to look at shadow of sun

6. Astrolabe

7. Sextant

8. Bubble sextant – artificial horizon for airplanes

9. Octant – note extra mirror

10. Quadrant made from materials lying around the house Scrap wood, paper, pencil, metal tube, old banjo string, glue Estimated accuracy is about ½ a degree (30’) Finished quadrant Taking a sighting near sunset

11. Quadrant markings made by successive halving of angles Ab initio way of dividing up the angles, if degree markings are not available. This one is divided into 64 angles = 1.40625o per angle. Estimated accuracy of this is 21’ by interpolation (using triangles)

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 process • Banjo wires, likewise • Angle calibration aided by compass, ruler • Using circle arcs works down to about 22.5o • Divide chords visually at smaller angles • Cut of board was precise • Can 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 refraction angle versus measured height. 48’ vertical, 6o horizontal. Lay out compass from 48’, 6o to 34.5’ and 0o degrees, swing arc down to 6omark. For higher angles, correction is 60’ 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’ below the horizon (almost one degree!). What you see True position True position What you see 34.5’ 34.5’ 56’ Moment of sunset Height = 0

19. Standard Celestial Navigation • Height is observed • Corrections made • Limb of sun – 16’ • Refraction • Dip 1’ times if observing horizon, no correction for plumb line or bubble Use Hc as height of object

20. Latitude from meridian height of sun Meridian height Latitude = 90o+Decl-Meridian height Due South

21. Vertical angle Fraction of day since midnight Data from home-made quadrant on 27-Oct-08 Maximum height of sun = 24.9 units = 35.06o Naïve declination (from sines) = 14.06 This gives latitude of 40.88 Declination from table = 13.06 Latitude using table declination is 41.88 (Cambridge = 42.38o ) Variance = 40’ Error in meridian height = 30’ Largest error was estimation of declination w/o tables – 1o

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 with finding a horizon at night. Latitude = 90o+Decl-Meridian height Declination is more reliable for stars – it never changes Have to use an artificial horizon of some kind (plumb bob) at night. Requires multiple sightings to find maximum height – changes slowly during passage. Meridian height Due South

23. Latitude from North Star Schedar Cassiopeia Polaris is offset toward Cassiopeia by 41’ Subtract correction when Cassiopeia is overhead Add correction when Big Dipper is overhead Polaris Dubhe Big dipper/Ursa major

24. Latitude from zenith stars A star at the zenith will have declination equal to your latitude. If you can get a good north-south axis, it is possible to measure latitude. Eg. Use a long stick with a rule at the end and a plumb-bob to keep the stick vertical This was used by the Polynesians to measure latitude – They would typically lie on their backs in their canoes and compare 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 sunset Latitude = (polar distance – minimum height) Polar distance = (90o – Declination) Min. star height Horizon (est)

26. Latitude sailing • Accurate longitude determination only came when chronometers were available • Before 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). • Columbus • Arab traders • Vikings • etc

27. Latitude from length of day • Covered in “Sun and Moon” talk • Most effective around solstices • (+/-) a couple of months around solstice • Useless at equinox • Can 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 sure • Latitude from length of day • Latitude from time of sunset • Until the invention of the nautical chronometer, navigation was a combination of latitude observations and dead reckoning for longitude

29. Harrison chronometer Developed in response to prize offered by British Royal Navy. Many scientists in the day looked at the “lunar” method.

30. Nautical chronometer circa 1900

31. Modern chronometer

32. Longitude: Local Area Noon (LAN) • Finding the time of maximum altitude of the sun (or a star) is difficult with any precision during meridian • Height is changing very slowly • Mid-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. 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. Vertical angle Fraction of day since midnight Finding LAN data from quadrant and watch Best information comes from period when sun is rising and setting – 3 hours after sunrise/before sunset. Height changes rapidly Midpoint of parabolic fit is 0.5205 (fraction of day from midnight This is 12:29:16 Add 4 hrs for UTC: 16:29:16 Add 16 min for EoT: 16:45:16 Time since noon at Greenwich: 4:45:16, at 15o/hour, this is Long = 71.32 (Cambridge = 71.11)

35. LAN for Stars • If 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 accuracy • Standard is UTC (coordinated universal time) • Assume a clock that is synchronized to UTC • Return to azimuth and celestial coordinates

37. Navigational Triangle

38. For a given sighting, there is a circle of possible locations on the earth, called a “line of position” or LOP. The intersection of two LOP’s gives two possible locations. Typical navigational practice is to assume a longitude and latitude and, locally, the LOP’s are lines.

39. Two equations for celestial observations Where Hc = height (after corrections for refraction) d = declination of object L = latitude Z=zenith angle t = hour angle (angle between meridian and star’s “longitude”) (LHA)