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:– 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 devicesover the years

• Kamal• Cross staff• Back staff• Astrolabe• Quadrant• Sextant

– Bubble– Gyro-sextant

• Octant

Cross staff

Backstaff

Allows viewer tolook at shadow ofsun

Astrolabe

Sextant

Bubble sextant – artificial horizon for airplanes

Octant – note extra mirror

Finished quadrant Taking a sighting near sunset

Quadrant made from materials lying around the houseScrap wood, paper, pencil, metal tube, old banjo string,glue

Estimated accuracy is about ½ a degree (30’)

Quadrant markings made by successive halving of angles

Ab initio way of dividing up the angles, if degree markings are notavailable.

This one is divided into 64 angles = 1.40625o per angle.

Estimated accuracy of this is 21’ by interpolation (using triangles)

End of tube with alignment wires.Image of sun’s shadow

Sighting tube for home-made quadrantEstimated accuracy is maybe 20’

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 raisesthe height of stars, sun, planets from true height

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

At sunset, center of sun is actually 34.5’ below horizon

What you see

True position

34.5’

34.5’

True position

When sun just disappears, the center of the sun is 56’ belowthe horizon (almost one degree!).

What you see

Moment of sunset Height = 0

56’

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)( ftheight

diametersemirefractiondipHH sc

Use Hc as height of object

Due South

Latitude = 90o+Decl-Meridian height

Meridianheight

Latitude from meridian height of sun

Fraction of day since midnight

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Data from home-made quadrant on 27-Oct-08

Maximum height of sun = 24.9 units = 35.06o Naïve declination (from sines) = 14.06This gives latitude of 40.88Declination from table = 13.06Latitude 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

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.

Due South

Meridianheight

Latitude = 90o+Decl-Meridian heightDeclination is more reliable for

stars – it never changes

Have to use an artificial horizonof some kind (plumb bob) at night.

Requires multiple sightings to findmaximum height – changes slowly during passage.

Dubhe

Schedar

Cassiopeia

Big dipper/Ursa major

Polaris

Latitude from North Star

correctionPolarisHL c

Polaris is offset towardCassiopeia by 41’

Subtract correction whenCassiopeia is overhead

Add correction when Big Dipper is overhead

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 to measure latitude. Eg. Use a long stick with a rule atthe 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

Horizon (est)

Min. star height

Polar distance =(90o – Declination)

Latitude = (polar distance – minimum height)

Best to do during navigational twilight – 45 minutes after sunset

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

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!!

))tan()tan((cos2 1 latdecd

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 RoyalNavy.

Many scientists in the daylooked 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

Fraction of day since midnight

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Best information comes from period when sun is risingand setting – 3 hours after sunrise/before sunset. Height changesrapidly

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 isLong = 71.32 (Cambridge = 71.11)

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 oftwo LOP’s gives two possible locations.

Typical navigationalpractice is to assumea longitude and latitudeand, locally, the LOP’sare lines.

HcLHcLdZ

tdLdLHc

coscossinsinsincos

coscoscossinsinsin

WhereHc = height (after corrections for refraction)d = declination of objectL = latitudeZ=zenith anglet = hour angle (angle between meridian and star’s “longitude”)(LHA)

Two equations for celestial observations