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
0
-10
-30
-23.45o
Date
-20
Winter Solstice
3/21/2009
2/21/2009
1/21/2009
12/21/2008
Summer Solstice
11/21/2008
10/21/2008
9/21/2008
8/21/2008
7/21/2008
6/21/2008
5/21/2008
30
4/21/2008
3/21/2008
Declination as a function of date
+23.45o
20
10
Series1
Meridian height is related to latitude and
declination
Meridian
height
Latitude =
90o+Decl-Meridian height
Due South
Calculating declination on a “desert island”
 360 
d  23.45 * sin( 
 * ( Day ))
 365 
o
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
30
Rule of 12ths: changes are 1:2:3:3:2:1 for sine
curves
20
Declination
10
0
Series1
-10
-20
-30
Date
Draw a circle, and measure angles (Burch)
 360 
 
 * Day
 365.24 
Could approximate angle
as days after 21 March
sin 

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
Source
Simple form.
Declination Length of day
(degrees)
(hours)
-10.3302
10.7663
Complex form
-9.5993
10.8588
Random website -8.6667
11.0666
NOAA online
-9.25
10.9028
Another NOAA
-9.02
11.000
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
instrument
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
changes
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
Equinox, rises
Summer Solstice
due east
rises N of E
θ
Due East
Winter Solstice
rises S of E
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)
Sun's amplitude v. Latitude
60
50
Amplitude
40
30
Series1
20
10
0
0
5
10
15
20
25
30
35
40
45
50
55
60
Latitude
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
dec
B
C
you
C
A
B
d
day
night
Terminator
A  sin( Lat )
B  sin( Lat ) * tan( dec)
C  cos(Lat )
Length of day = 24*d/360o
B
d  2 cos 1 

 C 
d  2 cos 1 ( tan( dec) tan( lat ))
Date
3/21/2009
2/21/2009
1/21/2009
12/21/2008
11/21/2008
10/21/2008
9/21/2008
8/21/2008
7/21/2008
6/21/2008
5/21/2008
4/21/2008
3/21/2008
Length of day (Hours)
Example: Boston – constant lat, different dates
Length of day
16
14
12
10
8
Length of day
6
4
2
0
16-Oct-2008, Decl. of sun = -10.28o hours of daylight
Daylight on Oct. 16
20
18
14
12
10
8
6
4
2
Latitude
66
53
40
27
14
1
-1
2
-2
5
-3
8
-5
1
-6
4
-7
7
0
Daylight hours
16
Daylight on Oct. 16
Time
• Need a watch – typical wristwatches are
pretty accurate
– Can calibrate – NIST time service:
– http://tf.nist.gov/service/its.htm
– 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
Daylight on Oct 16
11.1
11
Daylight (hours)
10.9
10.8
10.7
10.6
Daylight on Oct 16
1o of Latitude =
0.04 hours = 2.4 min.
10.5
10.4
10.3
35 36 37 38 39 40 41 42 43 44 45 46
Latitude
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
angles
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
( Sunrisetim e  Sunsettime)
LAN 
 /  Correction
2
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 simultaneous plot of solar
declination and time of the sun
(relative to mean solar time)
produces a figure 8 called an
analema
A photo of the sun at the same time
every day for a year
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
Incoming
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
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
indices
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:
– http://tycho.usno.navy.mil/vphase.html
B
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
Oct. 15
Oct. 16
Oct. 17
Oct. 18
Oct. 19
Oct. 20
Moon opposite Sun
180o to 270o
Oct. 21
Oct. 22
Oct. 23
Oct. 24
Oct. 25
Oct. 26
Oct. 27
270o to 0o
Oct. 28
Oct. 29
Oct. 30
Oct. 31
Nov 1
Nov. 2
Nov. 3
New Moon
0o to 90o
Nov. 4
Nov. 5
Nov. 6
Nov. 7
Nov. 8
Nov. 9
Nov. 10
90o to 180o