Latitude and Longitude
• We now enter a realm where we worry
more about precision.
• Angular measuring devices
• Time measuring devices
• Corrections of one degree
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:
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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).
Altitude (height) measuring devices
over the years
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Kamal
Cross staff
Back staff
Astrolabe
Quadrant
Sextant
– Bubble
– Gyro-sextant
• Octant
Cross staff
Backstaff
Allows viewer to
look at shadow of
sun
Astrolabe
Sextant
Bubble sextant – artificial horizon for airplanes
Octant – note extra mirror
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
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)
Sighting tube for home-made quadrant
Estimated accuracy is maybe 20’
End of tube with alignment wires.
Image of sun’s shadow
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.
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
Refraction in the atmosphere always raises
the height of stars, sun, planets from true height
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
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
Standard Celestial Navigation
• Height is observed
• Corrections made
– Limb of sun – 16’
– Refraction
– Dip 1’ times height ( ft) if observing horizon,
no correction for plumb line or bubble
H c  H s  dip  refraction  semi  diameter
Use Hc as height of object
Latitude from meridian height of sun
Meridian
height
Latitude =
90o+Decl-Meridian height
Due South
Data from home-made quadrant on 27-Oct-08
Vertical angle
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
Fraction of day since midnight
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.
Declination is more reliable for
stars – it never changes
Have to use an artificial horizon
of some kind (plumb bob) at night.
Latitude =
90o+Decl-Meridian height
Meridian
height
Requires multiple sightings to find
maximum height – changes
slowly during passage.
Due South
Latitude from North Star
L  H c  Polaris  correction
Schedar
Cassiopeia
Polaris is offset toward
Cassiopeia by 41’
Subtract correction when
Cassiopeia is overhead
Polaris
Add correction when
Big Dipper is overhead
Dubhe
Big dipper/Ursa major
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.
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)
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).
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Columbus
Arab traders
Vikings
etc
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!!
d  2 cos 1 ( tan( dec) tan( lat ))
Length of day = 24*d/360o
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
Harrison chronometer
Developed in response to
prize offered by British Royal
Navy.
Many scientists in the day
looked at the “lunar” method.
Nautical chronometer circa 1900
Modern chronometer
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.
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
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
Add 4 hrs for UTC: 16:29:16
Add 16 min for EoT:
16:45:16
Vertical angle
This is 12:29:16
Time since noon at Greenwich:
4:45:16, at 15o/hour, this is
Long = 71.32 (Cambridge = 71.11)
Fraction of day since midnight
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.
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
Navigational Triangle
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.
Two equations for celestial observations
sin Hc  sin L sin d  cos L cos d cos t
sin d  sin L sin Hc
cos Z 
cos L cos Hc
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)