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 devices over 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 =
o 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 6o mark.
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
34.5’
True position
What you see
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
90o+Decl-Meridian height
Due South

21. 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 (Cambridge = 42.38o )
Variance = 40’ Error in meridian height = 30’
Largest error was estimation of declination w/o tables – 1o
Fraction of day since midnight
Vertical 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 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
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 rising
and setting – 3 hours after sunrise/before sunset. Height changes
rapidly
Midpoint of parabolic fit is
(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 = (Cambridge = 71.11)
Fraction of day since midnight
Vertical angle

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)