The sun
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The sun. Due to the earth’s tilt and its orbit around the sun, the declination of the sun changes with time of year 23.45 o at summer solstice (21 June) 0 o at vernal and autumnal equniox (21 Mar, Sept) -23.45 o at winter solstice (21 Dec)

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The sun

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The sun

  • Due to the earth’s tilt and its orbit around the sun, the declination of the sun changes with time of year

    23.45o at summer solstice (21 June)

    0o at vernal and autumnal equniox (21 Mar, Sept)

    -23.45o at winter solstice (21 Dec)

    The sun looks like a “star” that changes declination, but also moves through the background of fixed stars over the course of the year.

In a given day

To a good approximation:

The sun moves like a star with

A fixed declination

A fixed Sidereal Hour Angle

Over the course of a year

  • To a good approximation:

    • The sun moves against the background of fixed stars

      • Declination changes like a sinusoid

      • SHA changes by approx. 1o per day

Declination as a function of date


Summer Solstice


Winter Solstice

Meridian height is related to latitude and declination



Latitude =

90o+Decl-Meridian height

Due South

Calculating declination on a “desert island”

d = declination

Day = number of days after 21 March

What if you don’t have a calculator or a table

of sines or MS Exel?

Graphically – draw approximation to sine curve

and interpolate visually

Rule of 12ths: changes are 1:2:3:3:2:1 for sine curves

Draw a circle, and measure angles (Burch)

Could approximate angle

as days after 21 March


Something you won’t remember on a desert island:

This varies from the other formula by as much as 2 degrees

Blame: orbital parameters of the earth, leap year differences, eccentricity

of orbit…..buyer beware!!

Who are you going to trust? - Declination for

Oct. 16th from various sources

NB – if you tried to estimate latitude from length of day,

you would see variations of 360 nautical miles!! Also- changes

most rapidly this time of year.

The sun moves across the sky at 15o per hour

With a compass can be used to tell local time

With a shadow stick can be used to find due

South (shortest shadow) or latitude

Watch method for finding South

  • Not talked about much because we all have digital watches these days.

  • Point the hour hand at the sun.

  • Due south is halfway between the hour hand and 12 on the watch

  • NB works only when the sun is below 45o in the horizon.

    • When sun is high, lines of azimuth converge - inaccurate

  • See tables in Gatty’s book for more accurate numbers using a compass

Shadow stick method

Precision of altitude measurements

These are limited by 1.) height of sun and 2.) measuring


With hands – only if very low in the sky (arctic) can one get

a degree or so.

With shadow stick – depends

on the geometry – maybe 1o

With quadrant – maybe 1o

Warning – do not look directly

at noon-day sun – use smoked

glass etc

With sextant – maybe 2-4’

Steve Callahan – from Adrift

Example: Shadow stick work compared with

GPS on trip from Orlando to PA

The sun will look like a star that has a declination

that varies with time of year – hence rising/setting azimuth


Rising and setting angles are (90o-Latitude) at the equinox

At any latitude – the maximum rising/setting angle north or south of

due east/wests called the sun’s amplitude

Winter Solstice

rises S of E

Equinox, rises

due east

Summer Solstice

rises N of E


Due East

Viking sun compass

Found in Greenland by the

archeologist C. L. Vebæk

Indicates 32 compass points.

Hole in the center is where

the gnomon (or pointer for

Sun-shadow) goes.

Markings on surface are

Interpreted as guides for

sunrise and sunset angles.

Interpretation of the scratches are hyperbolae

that give the position of the gnomon’s shadow

at different times of the day

From the azimuth/declination/latitude formula,

you can get the amplitude for any latitude

(use Rz formula from last week)

Note: it goes offscale at 66oN or S, at the artic circle

Terminator of earth – sunset in the summer over

Europe and Africa

Direction of sunset is

perpendicular to terminator

Length of day from latitude and declination

Side view

Top view







Length of day = 24*d/360o

Example: Boston – constant lat, different dates

16-Oct-2008, Decl. of sun = -10.28o hours of daylight


  • Need a watch – typical wristwatches are pretty accurate

    • Can calibrate – NIST time service:


    • Keep track of gain/loss

    • Precision of a few seconds possible (less than a nautical mile)

  • Limiting factors become accuracy of sightings

Close up of daylight on Oct. 16 near latitude

of Boston

1o of Latitude =

0.04 hours = 2.4 min.

Comments on latitude from length of day

  • Angular diameter of the sun is 32.5’ (2 minutes)

  • At horizon, must factor in refraction effects

  • Works best around the solstices

  • Almost impossible to use around equinox

    • Daylight the same at all latitudes

  • Accuracy of declination, other factors

Distortion of sunset

from refraction

Comparison – naïve approach (mine) to NOAA

calculation at solstice and Oct 16th

Oct 16th – 1o = 2.4 minutes of daylight

Solstice – 1o = 15 minutes of daylight

Oct 16th – 1o error in declination

Solstice – no error in declination

Oct 16th – 28 minutes difference in daylight

Solstice – 8.4 minutes difference in daylight

Oct 16th – difference in latitude = 6o

Solstice – difference in latitude = 0.56o

Huge difference!!

Special considerations for areas near the

North or South Pole

Midnight above the arctic circle – perfect for

determining latitude from horizon grazing!!

Mean Solar Time

  • The common meaning of “time” (how we set our watches) is mean solar time.

  • Places the highest point of the sun in the sky (solar meridian) roughly at noon.

    • Achieved by shifting time zones for every 15o of longitude

  • Greenwich Mean Time (GMT) is used as prime meridian

  • Variations caused by

    • Axial tilt of earth

    • Eccentricity of earth’s orbit (speeds up and slows down)

    • Position within time zone

    • Leap year effects – one year is not precisely 365.25 days (minor for the primitive navigator)

    • Tidal forces from moon slows down the earth’s rotation ever so slightly (VERY minor)

Time zones are approx. 15o wide in longitude

centered on local noon (modulo political boundaries)

Equation of time describes deviations of the sun’s

true position at noon from mean solar time

(negative means the sun is late relative to mean solar time)

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

Longitude from local noon

  • Variation of height of sun at meridian crossing is very slow – not accurate

  • Mid-point between a time of sunrise and sunset is most accurate

  • With a watch – measure identical height above horizon at sunrise and sunset using hands or kamal for accuracy

The kamal – used by ancient Arab sea traders

Based on same principle as use of hands to measure


Knotted string allows for range of angles

Board at end is calibrated in degrees

Hold knot in teeth and read off elevation from horizon

Local Area Noon

Correction from equation of time

Then – knowing watch time zone (from GMT) – caclulate

time difference from of LAN from GMT, and convert

to degrees from prime meridian

Accuracy of sighting of sunrise and sunset with a kamal

should be fairly accurate – much better than 1 degree.

Fraction of the sun’s diameter (10’ = 10 nautical miles)

The analema

A photo of the sun at the same time

every day for a year

A simultaneous plot of solar

declination and time of the sun

(relative to mean solar time)

produces a figure 8 called an


An “analemmatic” sundial corrects for the equation of time –

with different locations of the gnomon (shadow stick) depending

on date.

Why the sky is blue and polarized?

The light reaching your eyes from the sky is the result

of a single scatter off of air molecules. This scattering

is called “Rayleigh scattering”. It is larger at higher

Frequencies (shorter wavelengths) – so blue scatters best.

Also, light is polarized when scattered at 90o

Scattered light is polarized

at 90o scattering


At sunset, light has all the blue scattered out of it, and is red.

Polarization of the sky depends on the location of

the sun

Sun at zenith

Sun at horizon

Photo of the sky with a polarization filter

Polarization filter

The Viking sun stone

From Harafns Saga:

“the weather was thick and stormy…The king looked about and saw no blue sky…then the king took out the sunstone and held it up, and then he saw where the Sun beamed from the stone.”

Modern speculation is that the sun stone was Icelandic spar (calcite) that was used to get polarization information from the sky for the direction of the sun.

Calcite is “birefringent” – meaning that two different

polarization states of light have two different refractive


Speculation on the sun stone

  • The sun was often low on the horizon during the voyaging season

  • A lot of fog also was low in the sky and could obscure the sun

  • Sky polarization would be observable overhead, and could have been used

  • Large sources of calcite on east coast of Iceland.

The moon

  • Although the moon can be used for celestial navigation, it can’t really be used for “primitive” navigation, except for rough direction finding (i.e. not latitude and longitude – need sextant, clock and tables)

  • Tidal forces from the earth slowed down the moon’s rotation until it shows the same face to us.

  • The moon moves to the east in rotation by 12o per day (half a degree per hour).

  • Moves west like the sun, at 15o per hour

  • The lit side of the moon always faces the sun

    • Full moon rises opposite the setting sun around the time of the equinox

    • Website for moon phases:


Bright face of moon always faces the sun – phase of the moon tells you the angle to the sun


Direction of sun

14.5o per hour

Horns of moon

point south

Due South

Oct. 14

Nov. 4

Oct. 21

Oct. 28

Oct. 15

Nov. 5

Oct. 22

Oct. 29

Oct. 30

Oct. 16

Nov. 6

Oct. 23

Oct. 31

Nov. 7

Oct. 17

Oct. 24

Nov 1

Oct. 18

Nov. 8

Oct. 25

Oct. 26

Oct. 19

Nov. 2

Nov. 9

Oct. 27

Oct. 20

Nov. 3

Nov. 10

Moon opposite Sun

180o to 270o

270o to 0o

New Moon

0o to 90o

90o to 180o

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