1. 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.

2. Over the course of a 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

4. Declination as a function of date
Summer Solstice
+23.45o
-23.45o
Winter Solstice

5. Meridian height is related to latitude and declination
90o+Decl-Meridian height
Due South

6. Day = number of days after 21 March
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?

7. Graphically – draw approximation to sine curve
and interpolate visually
Rule of 12ths: changes are 1:2:3:3:2:1 for sine curves

8. Draw a circle, and measure angles (Burch)
Could approximate angle
as days after 21 March
Radius=23.45

9. 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!!

10. Who are you going to trust? - Declination for
Oct. 16th from various sources
Source
Declination
(degrees)
Length of day
(hours)
Simple form.
Complex form
Random website
NOAA online
-9.25
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.

11. 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

12. 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

13. Shadow stick method

14. 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’

15. Steve Callahan – from Adrift

16. Example: Shadow stick work compared with
GPS on trip from Orlando to PA

17. 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
Winter Solstice
rises S of E
Equinox, rises
due east
Summer Solstice
rises N of E θ
Due East

18. Found in Greenland by the
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.

19. Interpretation of the scratches are hyperbolae
that give the position of the gnomon’s shadow
at different times of the day

20. 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

21. Terminator of earth – sunset in the summer over
Europe and Africa
Direction of sunset is
perpendicular to terminator

22. Length of day from latitude and declination
Side view
Top view
dec
you d
day
night
Terminator
Length of day = 24*d/360o

23. Example: Boston – constant lat, different dates

24. 16-Oct-2008, Decl. of sun = -10.28o hours of daylight

25. Time 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

26. Close up of daylight on Oct. 16 near latitude
of Boston
1o of Latitude =
0.04 hours = 2.4 min.

27. 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

28. 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!!

29. Special considerations for areas near the
North or South Pole

30. Midnight above the arctic circle – perfect for
determining latitude from horizon grazing!!

31. 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 days (minor for the primitive navigator)
Tidal forces from moon slows down the earth’s rotation ever so slightly (VERY minor)

32. Time zones are approx. 15o wide in longitude
centered on local noon (modulo political boundaries)

33. 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)

34. 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

35. 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

36. 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

37. 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)

38. A simultaneous plot of solar declination and time of the sun
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
analema

39. An “analemmatic” sundial corrects for the equation of time –
with different locations of the gnomon (shadow stick) depending
on date.

40. 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
Incoming

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

42. Polarization of the sky depends on the location of
the sun
Sun at zenith
Sun at horizon

43. Photo of the sky with a polarization filter

44. 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.

45. Calcite is “birefringent” – meaning that two different
polarization states of light have two different refractive
indices

46. 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.

47. 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:

48. 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

49. Oct. 14 Oct. 15 Oct. 16 Oct. 17 Oct. 18 Oct. 19 Oct. 20 Oct. 21
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