Download Locating Geographic Coordinates Using Observations over the Sun

International School of Stavanger
Research Project
Locating Geographical Coordinates
Using Observations over the Sun
and Polaris Positions
Student Name:
Supervisor:
Mikhail Zakharov
David Tremblay, Cand. scient.
January - 2012
Table of Contents
Abstract.........................................................................................................................................
3
Introduction...................................................................................................................................
4
1. Observing the Daily and Seasonal Sun’s Motions Relative to Horizon………………………....
6
2. Solar Noon Time Variability………...........................................................................................
12
3. Locating the Geographical Coordinates from Observations over the Sun’s Motion..................
14
4. Alternative Method of Finding out the Latitude from the Height of Polaris………………………
19
5. Conclusion................................................................................................................................
25
6. Glossary….……………………..................................................................................................
26
7. Bibliography..............................................................................................................................
27
Appendix 1 – Solar Declination and Equation of Time.................................................................
28
Appendix 2 – Sun Positions Observed in Sandnes (Norway) from 24-Oct-2010 to 26-Mar-2011
30
Appendix 3 – Sun Positions Observed in Pula (Italy) on 24-Jun-2011………..............................
31
Appendix 4 – Sun Positions Observed in Sandnes (Norway) on 08-Oct-2011............................
32
Acknowledgements...…………….................................................................................................
33
2
Abstract
The research question of this study is the following:
How and with what accuracy the geographical coordinates of an observer on Earth can be found
from measured positions of the Sun and Polaris?
The study tested the following two hypotheses:
 The solar noon time and the maximum Sun’s height above horizon measured at any given date should
contain enough information for calculating the geographical coordinates (latitude and longitude) of the
observation site.
 The measured Polaris’ height above horizon allows finding out the latitude of any observation location in
the northern hemisphere.
The study is aimed at investigating the methods of determining the geographical coordinates through
observation over the two above mentioned stars’ positions.
Measured solar noon times when the Sun ascended to its highest point relative to the horizon were used to
determine the geographical longitudes of two observation locations: one in Sandnes (Norway) and another
one in Pula, Sardinia (Italy). Measured heights of the Sun above horizon at solar noon were used to
determine the geographical latitudes of the same observation locations.
Independently and as an alternative method, measured degrees above horizon of the Polaris star were also
used to determine the geographical latitudes.
A shadow technique was employed to observe the Sun’s motion and quantify its elevation above the horizon.
A direct pointing of the observation tool at Polaris was used in order to measure its position relative to the
horizon.
Calculation of geographical longitude from the measured solar noon time required application of the
tabulated equation of time correction in order to account for the Sun’s motion irregularity throughout the year.
Calculation of geographical latitude from the measured solar altitude required application of the tabulated
solar declination which accounted for seasonal variations of the Sun’s maximum heights above the horizon.
Geographical latitude of any observation location in the northern hemisphere is roughly equal to the height
of Polaris above horizon which is expressed in degrees.
The applied methods of locating the geographical latitudes and longitudes from observations over the Sun’s
motion provided accuracy within 0.2°. Locating the geographical latitudes from observations over Polaris
position provided accuracy within 0.5°.
Understanding of the methods tested by this study is important for the purposes of navigation.
3
Introduction
The primary objective of this project was to develop a better understanding of the Earth-Sun geometry and
find out how the knowledge about the Earth’s rotation, its revolution around the Sun, and the Earth’s position
relative to the Polaris star can be applied for the practical purpose of determining the coordinates of the
observer on Earth.
The main priority in the essay is given to discussion of the own experiments and interpretation of their
results. The related theoretical background information is not a focus area of this essay. However, the most
relevant astronomical facts are briefly mentioned in the text or referred to in the bibliography.
Staying at any location on the Earth, one can observe how the Sun moves relative to the horizon during the
day time, and how all visible stars move relative to the horizon at night time. This impression of the Sun’s
and stars’ movement is a result of the Earth’s own rotation. However, the trajectories of the above mentioned
celestial objects above the horizon do not stay constant every day. They are changing throughout a year due
to the Earth’s revolution around the Sun and also due to the fact that the Earth’s axis is tilted relative to its
orbit plane [2]. Accordingly, the day duration varies throughout a year (for any location beyond the Earth’s
equator).
The objective of my first experiment was to observe the seasonal variations of the Sun’s height at noon and
day duration. The first series of observations took place between 24 October 2010 and 26 March 2011.
Section 1 of this essay describes the experimental setup, summarizes the observed data and discusses its
interpretation.
Based on the experimental observations and as a result of the theoretical background study [2, 3], it became
apparent that the day duration and maximum height of the Sun above horizon depend not only on the
season and date, but also the latitude of the observation site. In fact, the following three parameters:
-
Date/time,
Sun’s height above the horizon at this date and time, and
Latitude
are interrelated with each other through equations. For any given latitude, date and time, the Sun’s height
above the horizon expressed in degrees can be calculated. This gave an opportunity to compare my
observed results with the theoretically calculated numbers [4] and in such a way to estimate the accuracy of
my experimental Sun height measurements. The accuracy analysis is also presented in Section 1.
During the first series of experiments, the Sun height measurements were made at the clock noon (i.e. at
12:00 winter time, or at 13:00 summer time). It soon became evident, that the Sun gets highest at certain
point in time during the day which appeared to be different from the clock noon. It gave motivation to explore
the notion of the so called solar noon and find out how much different it was from the clock noon.
It was experimentally observed that the solar noon time varied throughout a year. Discussion of this topic is
presented in Section 2 of the essay. The theoretical background study [5] revealed that the solar noon time
depended not only on the date, but also the longitude of the observation site. The following three
parameters:
-
Date,
Solar noon time at this date, and
Longitude
appeared to be interrelated with each other through equations. For any given date and longitude, the solar
noon time can be calculated.
4
All these findings gave an idea for my second series of experiments. Indeed, if for any given date, latitude
and longitude, the unknown solar noon time and the maximum Sun’s height above the horizon can be
calculated then the opposite should also be true. Namely, the solar noon time and the maximum Sun’s
height measured at any given date should contain enough information for calculating the geographical
coordinates (latitude and longitude) of the observation site. This hypothesis has been tested through my
experimental measurements and proved to be correct.
The variables for this experiment were geographical coordinates and dates. The above mentioned
measurements were made at two different dates in two different locations: one in Sandnes (Norway) and
another one in Pula, Sardinia (Italy). The objective of these exercises was to calculate the coordinates of the
two observation sites and estimate the accuracy of these calculations. The measurements in Sardinia were
made on 24 June 2011. Then observations in Sandnes were made on 8 October 2011. Section 3 of the
essay describes the experimental setup, summarizes the observed data, and discusses its interpretation and
accuracy.
Finally, the heights of Polaris above the horizon were measured in the same two locations in order to
determine the latitudes. This tests the second hypothesis. The experimental setup, observed data summary,
interpretation and accuracy analysis are presented in Section 4 of the essay.
5
1. Observing the Daily and Seasonal Sun’s Motions Relative to Horizon
The Sun’s daily motion – rising along the eastern horizon, tracing an arched path in the sky, and setting
along the western horizon – sets the most basic time cycle in our world: day and night. Midway between
sunrise and sunset the Sun ascends to its highest point relative to horizon. This event defines solar noon.
The interval from sunrise to sunset sets the length of the day [1].
As mentioned in the introduction, the objective of my first experiment was to observe the seasonal variations
of the noon Sun height. In addition, the day duration information over the same observation period was
collected from a local newspaper.
The simplest instrument to study the Sun’s daily and seasonal motion in the sky relative to the horizon is
called a gnomon. Basically, it is a short stick placed vertically in a flat place on the ground. The tip of the
shadow cast by a gnomon marks the end of a line that connects the shadow’s tip, the top of the stick, and
the Sun (Fig. 1). The shadow points opposite the Sun in the sky and the length of the shadow tells the height
of the Sun relative to the horizon. When the Sun hangs lower in the sky, the shadow is longer. At noon the
shadow has its shortest length of that day. In the northern latitudes, at noon the shadow points due north –
so the Sun is due south [1].
Figure 1. Experimental setup for measuring the Sun’s shadow at noon (the picture is copied from [1])
A device of this type was used for short term observations (within a few hours) in my second experiment
described in Section 3. However, for longer term observations within months, I needed a more stable,
stationary instrument which could not be affected by wind, birds, etc. Preferably, it would allow observations
to be made from inside the house.
A place had been found in the room where the Sun was clearly visible through a window during midday.
A cross was placed on the window which cast a shadow on a wall. A sheet of paper was permanently
attached to the wall (Fig. 2). Several times within the interval from about 11:00 to 14:30 and more importantly
during the midday (at 12:00 winter time or 13:00 summer time), the shadow of the cross was marked on the
paper by a pencil (Fig. 3).
A horizontal line was drawn on the paper at exactly the same level as the cross on the window.
This line marked the horizon (i.e. the shadow from the cross cast at sunrise would lay on this horizontal line).
Any elevation of the Sun above the horizon caused downwards movement of the shadow below the above
mentioned horizontal line.
6
Cross
Marked shadow from the cross
is one of many data points
()
Cross
Figure 2. Experimental setup for measuring the Sun’s height above horizon
7
Figure 3. Scanned sheet with marked points corresponding to the various Sun’s positions depending
on date and time
Figure 4. 3-Dimensional sketch of the experimental setup shown in Fig. 2
8
The height of the Sun above the horizon () could be calculated from a triangle ABC (Fig. 4) where A
corresponds to the cross on the window, C is a mark from the shadow of the cross, and B is the point on the
line of horizon just above C.
The angle of interest () was calculated from the right-angled triangle ABC. The BC length (l) divided by the
AB length (s) gave a ratio which was the tangent of the angle of the Sun’s height () above the horizon
(Fig. 4):
Tan = BC/AB = l/s
The trigonometric function ATAN() in Excel was used to find out the heights of the Sun expressed in radians.
Then the angles were converted from radians into degrees using the Excel function DEGREES().
The BC length was measured with a ruler. However, direct measurements of the AB length in 3D space
between the window and the wall with a tape measure appeared to be more difficult and less accurate.
Much easier and more accurate was to measure the BD length on a flat surface with the ruler and then
calculate the AB length from the right-angled triangle ABD:
|‫ = |ܤܣ‬ඥ|‫|ܦܣ‬ଶ + |‫|ܦܤ‬ଶ
where AD was the known horizontal distance from the cross on the window to the wall and the BD length
was measured with a ruler.
The series of observations over the Sun’s seasonal motion took place between 24 October 2010 and
26 March 2011. During the same period, records of the day durations were kept and calculated as a
difference between the times of sunset and sunrise. The sunrise and sunset information was collected
from a local newspaper (last page of the Stavanger Aftenblad).
The results are presented in Fig. 5 and in Table 1. The original measured data is presented in Appendix 2.
Figure 5. Sun Height at Noon & Day Duration for the period from 24-Oct-2010 to 26-Mar-2011
9
Table 1. Sun Height at Noon Sun & Day Duration for the period from 24-Oct-2010 to 26-Mar-2011
Date
(dd-mmm-yyyy)
Sunrise
(hh:mm)
Sunset
(hh:mm)
Day
Duration
(hh:mm)
Day
Duration
(hr)
Observed
(deg)
Sun Height at Noon
Reference Data [4]
(deg)
Deviation
(deg)
Comments
24-Oct-2010
25-Oct-2010
26-Oct-2010
04-Nov-2010
06-Nov-2010
07-Nov-2010
09-Nov-2010
17-Nov-2010
18-Nov-2010
19-Nov-2010
24-Nov-2010
25-Nov-2010
26-Nov-2010
27-Nov-2010
28-Nov-2010
03-Dec-2010
06-Dec-2010
20-Dec-2010
08:35
08:37
08:40
08:01
08:06
08:09
08:13
08:33
08:35
08:37
08:48
08:51
08:53
08:55
08:57
09:07
09:12
09:27
18:07
18:04
18:02
16:39
16:34
16:32
16:27
16:11
16:09
16:07
15:58
15:57
15:55
15:54
15:52
15:47
15:44
15:40
09:32
09:27
09:22
08:38
08:28
08:23
08:14
07:38
07:34
07:30
07:10
07:06
07:02
06:59
06:55
06:40
06:32
06:13
9.533
9.450
9.367
8.633
8.467
8.383
8.233
7.633
7.567
7.500
7.167
7.100
7.033
6.983
6.917
6.667
6.533
6.217
20.82
18.77
17.11
15.81
15.04
14.55
13.97
12.22
11.76
11.62
10.59
10.48
10.22
9.85
9.23
7.49
8.39
7.49
19.19
18.84
18.50
15.59
14.98
14.68
14.10
11.97
11.73
11.49
10.39
10.19
10.00
9.81
9.62
8.81
8.40
7.37
1.63
-0.07
-1.39
0.22
0.06
-0.13
-0.13
0.25
0.03
0.13
0.20
0.29
0.22
0.04
-0.39
-1.32
-0.01
0.12
Inaccurate measurement
22-Dec-2010
09:29
15:41
06:12
6.200
7.51
7.35
0.16
25-Dec-2010
26-Dec-2010
27-Dec-2010
02-Jan-2011
20-Jan-2011
23-Jan-2011
06-Feb-2011
12-Feb-2011
18-Feb-2011
23-Feb-2011
06-Mar-2011
26-Mar-2011
09:30
09:30
09:30
09:29
09:09
09:03
08:34
08:19
08:04
07:50
07:19
06:22
15:43
15:44
15:45
15:53
16:26
16:33
17:08
17:23
17:37
17:50
18:16
19:03
06:13
06:14
06:15
06:24
07:17
07:30
08:34
09:04
09:33
10:00
10:57
12:41
6.217
6.233
6.250
6.400
7.283
7.500
8.567
9.067
9.550
10.000
10.950
12.683
7.25
7.41
7.28
7.88
10.46
10.93
14.85
16.80
18.78
20.67
24.66
32.68
7.37
7.39
7.42
7.75
10.35
11.00
14.73
16.64
18.69
20.49
24.68
32.66
-0.12
0.02
-0.14
0.13
0.11
-0.07
0.12
0.16
0.09
0.18
-0.02
0.02
Inaccurate measurement
Inaccurate measurement
Fewest daylight hours,
lowest Sun height
The observation results demonstrated that the height of the Sun in the sky at clock noon and the day
duration varied with the date. The day with the fewest daylight hours was December 22. The lowest Sun
height at noon was observed on December 25. More precise reference data suggested that in 2010 the Sun
in Stavanger area was the lowest on December 22 (same as the shortest day).
This discrepancy can be attributed to the very slow daily changes of the noon Sun height around these dates
(<0.03º per day) and also to the limited accuracy of my instrument. As can be seen from Fig.5 and Table 1,
except for three obviously inaccurate measurements made on 24 & 26 October and 3 December 2010 with
an error exceeding 1.3º, all the other 28 recorded points corresponded very closely to the reference values.
Deviation for these 28 good quality points did not exceed 0.4º. For the relatively simple instrument, such
accuracy may be considered reasonably high.
The background theoretical research helped me with understanding a more comprehensive picture of the
Sun’s motion. Indeed, the height of the Sun in the sky at noon varies with the season [1]. During the
summer, the noon Sun hits its highest point in the sky for the year on the summer solstice (around June 21),
the day with the greatest number of daylight hours.
The noon Sun drops to its lowest point in the sky for the year in winter on the winter solstice (around
December 21), the day with the fewest daylight hours.
10
On the first days of astronomical spring and fall, the Sun’s height at noon is between its summer maximum
and winter minimum. These times are called equinoxes (ca. March 21 for spring and September 21 for fall).
Day and night durations become almost equal twice a year during these days.
This cycle of the Sun’s height defines the second basic unit of time: the year of seasons.
More theoretical information on this topic can be found in [2, 3]. In particular, it was important to learn that
the day duration and height of the Sun above the horizon depend not only on the season and date, but also
the latitude of the observation site. In fact, the following three parameters:
-
Date/time,
Sun height above the horizon at this date and time, and
Latitude
are interrelated with each other through equations. For any given latitude, date and time, the Sun height
above the horizon can be calculated, thus providing an opportunity to compare the observed results with the
theoretically calculated reference numbers and in such a way to estimate the accuracy of the experimental
Sun height measurements as mentioned above.
What is important for this study is that the maximum solar altitude (A) at any date and the latitude (L) are
related through the following equation:
Altitude (A) = 90º - Latitude (L) + Declination (D)
where (D) is a date/time dependent Solar Declination which is the angle between the rays of the Sun and
the plane of the Earth's equator. Solar declination values for any date of the year calculated using the
formula provided in [6] are presented in Fig. 6 and also provided in the table format in Appendix 1.
The table is used in calculations of the geographical coordinates as described in Section 3.
Solar Declination
25
20
15
5
0
-5
-10
-15
-20
-25
1-Jan-11
6-Jan-11
11-Jan-11
16-Jan-11
21-Jan-11
26-Jan-11
31-Jan-11
5-Feb-11
10-Feb-11
15-Feb-11
20-Feb-11
25-Feb-11
2-Mar-11
7-Mar-11
12-Mar-11
17-Mar-11
22-Mar-11
27-Mar-11
1-Apr-11
6-Apr-11
11-Apr-11
16-Apr-11
21-Apr-11
26-Apr-11
1-May-11
6-May-11
11-May-11
16-May-11
21-May-11
26-May-11
31-May-11
5-Jun-11
10-Jun-11
15-Jun-11
20-Jun-11
25-Jun-11
30-Jun-11
5-Jul-11
10-Jul-11
15-Jul-11
20-Jul-11
25-Jul-11
30-Jul-11
4-Aug-11
9-Aug-11
14-Aug-11
19-Aug-11
24-Aug-11
29-Aug-11
3-Sep-11
8-Sep-11
13-Sep-11
18-Sep-11
23-Sep-11
28-Sep-11
3-Oct-11
8-Oct-11
13-Oct-11
18-Oct-11
23-Oct-11
28-Oct-11
2-Nov-11
7-Nov-11
12-Nov-11
17-Nov-11
22-Nov-11
27-Nov-11
2-Dec-11
7-Dec-11
12-Dec-11
17-Dec-11
22-Dec-11
27-Dec-11
1-Jan-12
Declination (deg)
10
Date
Figure 6. Solar Declination
11
2. Solar Noon Time Variability
The main objective of the first series of experiments described above was to measure the height of the Sun
above the horizon every day at (when weather allowed doing so) exactly the same time at clock noon.
Occasionally at some dates, several measurements were also made prior to and after the clock noon.
These measurements suggested that the Sun got highest at certain point in time during the day which
appeared to be different from the clock noon. The time when the Sun gets highest is called solar noon time.
The observations suggested that the solar noon time varied throughout the observation period. Interpretation
of the Sun’s heights measured over a few hours in Sandnes (Norway) on the 24 October 2010 and on the 7
May 2011 (Fig. 7) gave two different solar noon time values: 13:21 and 13:36. This 15 minutes difference
could either be a result of insufficient accuracy of the angular measurements, or reflected a more
fundamental fact that the solar noon time varied. The “zigzag” pattern of the marks made at different dates,
but at the same time (11:00 or 12:00 or 13:30) which is obviously visible in Fig. 3 has also indirectly indicated
the variability of solar noon time and its cyclic nature.
24-Oct-2010
20
Sun Height (deg)
19
18
17
16
12:00
12:05
12:10
12:15
12:20
12:25
12:30
12:35
12:40
12:45
12:50
12:55
13:00
13:05
13:10
13:15
13:20
13:25
13:30
13:35
13:40
13:45
13:50
13:55
14:00
14:05
14:10
14:15
14:20
14:25
14:30
14:35
14:40
14:45
14:50
14:55
15:00
15
Local Time
07-May-2011
50
Sun Height (deg)
49
48
47
46
12:00
12:05
12:10
12:15
12:20
12:25
12:30
12:35
12:40
12:45
12:50
12:55
13:00
13:05
13:10
13:15
13:20
13:25
13:30
13:35
13:40
13:45
13:50
13:55
14:00
14:05
14:10
14:15
14:20
14:25
14:30
14:35
14:40
14:45
14:50
14:55
15:00
45
Local Time
Figure 7. Solar Noon Time measured in Sandnes (Norway) on 24-Oct-2010 and 07-May-2011
12
My background theoretical research not only confirmed that the solar noon time indeed varied throughout a
year, but also revealed its dependence on the longitude of the observation site [5, 11] which became a very
important learning for the purpose of this study. The following three parameters:
-
Date,
Solar noon time at this date, and
Longitude
appeared to be interrelated with each other. The time of solar noon depends on the Equation of Time (EOT)
and on the longitude of the observer’s location. The solar noon is slightly different for every day of the year.
This difference is accounted for by EOT.
For example, on the Greenwich meridian (0º longitude) the solar noon occurs at:
12:00 (clock noon) + 1 hour of Daylight Saving Time (if summer time is applied) - Equation of Time (EOT)
The Earth makes one full revolution (360º) in 24 hours (or 1440 minutes). One degree of the Earth’s
revolution corresponds to 4 minutes. Therefore, on any meridian east from the Greenwich meridian the solar
noon occurs earlier. This time difference can be calculated as:
4 minutes * Local Longitude
where the local longitude is positive to the east and negative to the west from the Greenwich meridian.
On the other hand, if one knows how sooner the Sun hits its highest point on any day in any observation
location as compared to the solar noon on the Greenwich meridian on the same day, then the observer’s
longitude can be calculated using the same formula. For example, if the solar noon occurred 40 minutes
prior to the same event in Greenwich, England, then the longitude of the observer is 10º to the east from the
Greenwich meridian. This will be used in calculations described in Section 3.
As mentioned above, Equation of Time is a date dependent correction. EOT gives the difference between
solar time and clock time for each day of the year. EOT values for any date of the year calculated using
the formula provided in [7] are presented in Fig. 8 and available in the table format in Appendix 1.
This parameter is also important for calculations of the geographical coordinates performed in Section 3.
Equation of Time (EOT)
20
15
10
0
-5
-10
-15
-20
1-Jan-11
6-Jan-11
11-Jan-11
16-Jan-11
21-Jan-11
26-Jan-11
31-Jan-11
5-Feb-11
10-Feb-11
15-Feb-11
20-Feb-11
25-Feb-11
2-Mar-11
7-Mar-11
12-Mar-11
17-Mar-11
22-Mar-11
27-Mar-11
1-Apr-11
6-Apr-11
11-Apr-11
16-Apr-11
21-Apr-11
26-Apr-11
1-May-11
6-May-11
11-May-11
16-May-11
21-May-11
26-May-11
31-May-11
5-Jun-11
10-Jun-11
15-Jun-11
20-Jun-11
25-Jun-11
30-Jun-11
5-Jul-11
10-Jul-11
15-Jul-11
20-Jul-11
25-Jul-11
30-Jul-11
4-Aug-11
9-Aug-11
14-Aug-11
19-Aug-11
24-Aug-11
29-Aug-11
3-Sep-11
8-Sep-11
13-Sep-11
18-Sep-11
23-Sep-11
28-Sep-11
3-Oct-11
8-Oct-11
13-Oct-11
18-Oct-11
23-Oct-11
28-Oct-11
2-Nov-11
7-Nov-11
12-Nov-11
17-Nov-11
22-Nov-11
27-Nov-11
2-Dec-11
7-Dec-11
12-Dec-11
17-Dec-11
22-Dec-11
27-Dec-11
1-Jan-12
EOT (minutes)
5
Date
Figure 8. Equation of Time
13
3. Locating the Geographical Coordinates from Observations over the Sun’s Motion
As demonstrated above, for any given date, latitude and longitude, the unknown values of solar noon time
and noon Sun height above the horizon could be calculated. The opposite was also assumed to be true.
My hypothesis was that the solar noon time and the maximum Sun’s height measured at any given date
should contain enough information for calculating the geographical coordinates (latitude and longitude) of the
observation site. This hypothesis has become a background for the next series of my experiments.
In addition to the couple of measurements mentioned in Section 2, more precise measurements of the solar
noon time and the maximum Sun’s height above the horizon were made on two different dates in two
different places: one in Sandnes (Norway) and another one in Pula, Sardinia (Italy) (Fig. 9).
Figure 9. Two observation locations (one in Norway and another one in Italy) marked on the map
The ultimate goal of these experiments was to calculate the coordinates of the above mentioned observation
sites and estimate the accuracy of these calculations. The measurements in Sardinia were made on 24 June
2011. Observations in Sandnes were made on 8 October 2011.
Experimental setup and procedure for the Sun’s height measurements in Sandnes remained unchanged.
However, for the experiment in Sardinia, I used a gnomon which was similar to the one described in
Chapter 1, but made from anything available during a summer trip (Fig. 10).
14
Figure 10. Gnomon assembled in Sardinia from improvised material
A height of the gnomon to the tip of the match over the horizontally placed cardboard was 27.5 cm.
The half-open book provided that its back was strictly vertical.
The first observation in Pula took place on 23 June 2011, but was interrupted by midday clouds. The more
successful observation on another day started approximately one hour before the expected solar noon and
finished about 2 hours after it. During the experiment, every 5-10 minutes I repeatedly marked on the
cardboard the position of the tip of the shadow thrown by the match and wrote the local time next to each
mark (Fig. 11). Then the shadow lengths were converted into the solar altitudes in the Excel spreadsheet
shown in Appendix 3 using the same formula and auxiliary Excel functions described in Section 1:
Tan = l/s
15
Figure 11. Scanned cardboard with marks tracking the Sun’s different positions around noon time
16
According to my interpretation (Fig. 12), the solar noon in Pula on that date occurred at 13:27 when the
shadow was the shortest. The highest solar altitude was 74.27° (or 74° 16’). At solar noon the shadow was
pointing in the north direction which was confirmed by compass.
The Sun heights measured in Sandnes on 8 October 2011 (Appendix 4) were interpreted similarly (Fig. 13).
The solar noon in Sandnes on that date occurred at 13:24. The highest solar altitude was 25.40° (or 25° 24’).
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
12:00
12:05
12:10
12:15
12:20
12:25
12:30
12:35
12:40
12:45
12:50
12:55
13:00
13:05
13:10
13:15
13:20
13:25
13:30
13:35
13:40
13:45
13:50
13:55
14:00
14:05
14:10
14:15
14:20
14:25
14:30
14:35
14:40
14:45
14:50
14:55
15:00
15:05
15:10
15:15
15:20
15:25
15:30
15:35
15:40
Sun Height (deg)
24-Jun-2011 (Pula, Italy)
Local Time
Figure 12. Solar Noon Time and the Highest Solar Altitude measured in Pula (Italy) on 24-Jun-2011
08-Oct-2011 (Sandnes, Norway)
Sun Height (deg)
26
25
24
12:00
12:05
12:10
12:15
12:20
12:25
12:30
12:35
12:40
12:45
12:50
12:55
13:00
13:05
13:10
13:15
13:20
13:25
13:30
13:35
13:40
13:45
13:50
13:55
14:00
14:05
14:10
14:15
14:20
14:25
14:30
14:35
14:40
14:45
14:50
14:55
15:00
23
Local Time
Figure 13. Solar Noon Time and the Highest Solar Altitude measured in Sandnes (Norway) on 08-Oct-2011
17
Then both of the measurements made in Italy and in Norway were interpreted in terms of latitudes and
longitudes using the formulas presented at the end of Section 1 and in Section 2. The tabulated values of
solar declination and equation of time for the corresponding dates were taken from Table A.1 (Appendix 1).
Observed data for the maximum solar altitudes, solar noon times and the geographical coordinates
calculated for Sandnes (Norway) and Pula (Italy) are presented in Table 2. The actual values for the same
parameters taken from [4,11] and from the Google Map [8] are also displayed in the table for reference.
As data comparison suggests, the accuracy of the maximum Sun altitude measurement was within 0.2º,
the accuracy of the solar noon time measurement was within 70 seconds.
The accuracy of this method for locating the geographical latitude and longitude both in Norway and Italy
was within 0.2º. It can be calculated that 1° of latitude or longitude roughly corresponds to 111 km
(40,000 km / 360° = 111 km). Therefore, in both cases the position of observer was located with accuracy
within about 31.5 km. In the global scale, such tolerance can be considered reasonably narrow given a
relative simplicity of the instruments and the method employed.
Table 2. Maximum Solar Altitudes, Solar Noon Times and Geographical Coordinates
for Sandnes (Norway) and Pula (Italy)
Pula, Sardinia (Italy)
Observation Location
Sandnes (Norway)
Observation Date
(dd-mmm-yyyy)
24-Jun-2011
08-Oct-2011
Observation Time
(hh:mm)
12:10 - 15:30
12:00 - 15:00
Solar Declination (from Table A.1)
(deg)
23.41
-5.88
Equation of Time (from Table A.1)
(minutes)
-2.30
13.19
Solar Noon on the Greenwich Meridian (*)
(hh:mm:ss)
13:02:18
12:46:49
Local Time Ahead of GMT
(hh:mm)
01:00
01:00
(*) calculated with accounting for Daylight Saving Time and Equation of Time
Observed Values
Actual Values [4,8]
Deviation
Observed Values
Actual Values [4,8]
Deviation
Maximum Solar Altitude
(deg)
74.27
74.42
-0.2
25.40
25.27
0.1
Geographical Latitude
(deg)
39.15
38.99
0.2
58.72
58.89
-0.2
Observed Values
Actual Values [8,11]
Deviation
Observed Values
Actual Values [8,11]
Deviation
13:26:22
00:00:38
13:23:33
13:24:43
00:01:10
8.99
-0.2
5.73
0.1
Solar Noon at the Observation Location
(hh:mm:ss)
13:27:00
Ahead of Solar Noon on the Greenwich Meridian
(hh:mm:ss)
00:35:18
Geographical Longitude
(deg)
8.82
00:23:16
5.81
18
4. Alternative Method of Finding out the Latitude from the Height of Polaris
Similarly to the Sun, at night time the stars move relative to the horizon – rising in the east, slowly travelling
in arcs against the sky, and setting in the west [1].
Some stars never dive below the horizon. Instead, they trace complete circles above it. As these stars swing
around, they draw concentric circles like the rings of a bull’s-eye (Fig. 14). The centre of these rings marks
the celestial pole, the point about which the stars seem to pivot. A modestly bright star called Polaris lies
close to the north celestial pole. Polaris is also known as the North Pole star.
Figure 14. Night sky picture taken in the Northern Hemisphere at long time exposure with the smallest arc
in the centre being traced by Polaris (the picture is copied from [12])
Ignoring a minor misalignment of Polaris with the north celestial pole, it can be assumed that for any
observation site in the Northern Hemisphere the height of Polaris above the horizon corresponds to the
geographical latitude of the observer regardless of the date, time and longitude. More theoretical information
on this topic is available in [9].
This astronomical fact gave an idea to one more experiment. The objective of this experiment was to
measure the height of Polaris relative to the horizon, calculate the latitude of the observation site and
estimate the accuracy of such latitude determination.
The measurements were made in the same two places in Norway (on the 12 October 2011) and in Italy (on
the 25 June 2011). In addition, an attempt of direct measurement of the Polaris’ height was made in Italy
using a telescope (on the 29 July 2011).
19
The instrument for measuring the height of Polaris above the horizon in Norway (Fig. 15) was assembled
from a straight 2.42 m long stick leaned on a stepladder and pointing at Polaris. The lower end of the stick
rested on the horizontal base formed by another similar straight stick. A carpenter’s level was used to check
the horizontality of the base. It was difficult to place an eye low enough to provide a visual alignment of the
main stick with Polaris. A mirror placed at the lower end of the stick helped with achieving the precise
alignment. A weight hanging on the thread attached to the upper end of the stick assured a verticality of the
thread which formed the third side of a right-angled triangle.
Figure 15. Instrument for measuring the height of Polaris above the horizon in Sandnes (Norway)
A tangent and then the angle corresponding to the height of Polaris above the horizon were calculated from
the lengths of the vertical and horizontal sides of the triangle measured by a tape. The formulas didn’t differ
from the ones provided in Section 1.
A very similar measurement of the Polaris’ height above horizon was made in Italy. However, since the
experiment was made during a summer trip to Sardinia, the instrument was assembled from improvised
material. I failed to find a carpenter’s level and long enough straight stick. Therefore, the experimental setup
was slightly modified (Fig. 16). A basin filled with water was used to make the base horizontal. A weight
hung on the thread from a tip of a short stick (the stick itself wasn’t meant to be aligned with Polaris).
A point on the horizontal base was marked so that its position was aligned with the tip of the stick and
Polaris. In other words, if Polaris were bright enough, a shadow from the tip of the stick would be thrown
exactly at the marked position. Similarly to the experiment in Norway, a mirror was used for visual alignment.
The lengths of the vertical and horizontal sides of the right-angled triangle measured by a tape gave the
height of Polaris above the horizon.
20
Figure 16. Instrument for measuring the height of Polaris above the horizon in Pula (Italy) assembled from
improvised material
21
Observed data for the Polaris height and the geographical latitudes calculated for Sandnes in Norway and
Pula in Italy are presented in Table 3. The actual values for the same parameters taken from [8, 10] are also
displayed in the table for reference. As data comparison suggests, the accuracy of the Polaris height
measurement was within 0.3º.
The accuracy of this method for finding out the geographical latitude both in Norway and Italy was within
0.5º.
Table 3. Heights of Polaris and Geographical Latitudes for Sandnes (Norway) and Pula (Italy)
Observation Location
Pula, Sardinia (Italy)
Sandnes (Norway)
Observation Date
(dd-mmm-yyyy)
25-Jun-2011
12-Oct-2011
Observation Time
(hh:mm)
23:00
22:00
Measured Horizontal Length ( s )
(cm)
109.6
125.8
Measured Vertical Height ( l )
(cm)
87.1
207.0
0.795
1.645
Tangent = l / s
Observed Values
Actual Values [8,10]
Deviation
Observed Values
Actual Values [8,10]
Deviation
Height of Polaris
(deg)
38.47
38.33
0.1
58.71
59.03
-0.3
Geographical Latitude
(deg)
38.47
38.99
-0.5
58.71
58.89
-0.2
Finally, an attempt of the direct Polaris height measurement was made in the same place in Italy using a
telescope. The telescope was levelled and aimed at Polaris. The angular scale on the instrument was used
to read the height of Polaris. The reading was 36.5º (Fig. 18).
However, the scale didn’t appear to be properly set to zero by manufacturer. It became obvious, when the
telescope was parked into the horizontal position. In this position, the reading on the scale was - 1.5º
(Fig. 19). Therefore, the 1.5º correction was to be added to the original 36.5º reading in order to arrive at
more accurate value of 38º. However, even with this correction, such direct measurement of the angle by
telescope appeared to be less accurate. Therefore, I preferred using the instrument shown in Fig. 16.
22
Figure 18. Direct measurement of the Polaris’ height above the horizon in Pula (Italy) using a telescope
23
Figure 19. Inaccurate zero setting on the scale when the telescope is parked into horizontal position
(+1.5º correction was applied to any current angle reading)
24
5. Conclusion
This study developed my skills required for planning and conducting scientific experiments. The project gave
me knowledge of how the geographical coordinates can be found from observations over the Sun’s motion.
It also confirmed that the altitude of the Polaris above the horizon is almost exactly the same as the
observer’s geographical latitude.
The applied simple instruments provided a surprisingly high accuracy of the angle measurements within 0.2°.
This translated into the accuracy of locating the geographical coordinates within ±0.2° which was equivalent
to ±31.5 km. As compared to the size of the Earth itself, such precision exceeded my expectations.
The study confirmed both of my hypotheses regarding the amount of information required for finding out the
longitude and latitude.
Understanding of the methods tested by this study has a practical importance. Such knowledge, for example,
is important for marine, aerial or onshore navigation during any long distance travel across the remote areas
without any visible landmarks (in oceans, in air above the clouds, in desert, tundra, taiga, Arctic or Antarctic,
etc.). Of course, the satellite Global Positioning System can provide more accurate information about the
coordinates when the GPS device is available. However, this doesn’t diminish the value of knowledge
acquired in course of work over this research project.
I presume the longitudes and latitudes can also be determined from observations over the positions of the
other stars visible at night time. Data interpretation method is expected to be more complicated, since it will
most probably require application of the star maps and tables with the time dependent corrections for star
positions. Alternatively, any star mapping software (such as Stellarium [10]) can help substitute the star
maps. Locating geographical coordinates using night observations over the star positions can become a
topic for the further research. Such knowledge can be of practical importance for navigation at night time.
25
6. Glossary
Geographical coordinates. Geographical coordinate system enables every location on the Earth to be
specified by a set of numbers. A common choice of coordinates is latitude, longitude and elevation, where
latitude and longitude represent horizontal position, and elevation represents vertical position.
Clock noon is a time of the day when the local clock time says 12 o’clock noon.
Solar noon is the time during the day when the Sun appears to have reached its highest point in the sky.
Usually, this is not the same time as “clock noon”. Solar noon always occurs half-way between local sunrise
and sunset.
Gnomon is the simplest instrument to study the Sun’s daily and seasonal motion in the sky relative to the
horizon.
Polaris (North Star or Pole Star or North Pole Star) is the brightest star in the constellation Ursa Minor. It is
very close to the north celestial pole, making it the current northern pole star.
North celestial pole is the imaginary point in the sky directly overhead to an observer at the Earth’s North
Pole.
Sun height, height angle, solar altitude angle or elevation is the angle between a line that points from the
site towards the centre of the Sun, and the horizon.
Polaris height, height angle, altitude angle or elevation is the angle between a line that points from the
site towards the Polaris, and the horizon.
Summer solstice in the Northern Hemisphere usually occurs some time between June 20 and June 21
each year, when the noon Sun reaches its highest position in the sky for the year. It is a day with the
greatest number of daylight hours.
Winter solstice in the Northern Hemisphere usually occurs between December 21 and 22 each year,
when the noon Sun reaches its lowest altitude above the horizon for the year. It is a day with the fewest
number of daylight hours.
Equinox occurs twice a year, when the Sun is in the same plane as the Earth’s equator. Around the
equinox, the night and day have approximately equal length. An equinox happens each year at two specific
moments in time around March 20/21 and September 22/23.
Astronomical spring and fall start from the dates of equinoxes on March 20/21 and September 22/23,
respectively.
Solar Declination is the angle between the rays of the Sun and the plane of the Earth’s equator.
Equation of Time (EOT) is the difference between solar time and clock time. The equation of time for each
day of the year can be found in tables or estimated with formulas.
26
7. Bibliography
th
1. Zeilik M. (1988). “Astronomy. The Evolving Universe”. 5 Edition. Published by John Wiley & Sons.
2. Pidwirny, M. (2006). “Earth-Sun Geometry”. Fundamentals of Physical Geography, 2
http://www.physicalgeography.net/fundamentals/6a.html
nd
Edition.
3. Schroeder D. V. (2011). “Understanding Astronomy. The Sun and the Seasons”. Department of Physics,
Weber State University.
http://physics.weber.edu/schroeder/ua/SunAndSeasons.html
4. Gronbeck C. (2009). “Sun Angle. The premiere tool for solar angle calculations”.
http://www.susdesign.com/sunangle/index.php
5. Print Your Solar Noon Calendar
http://www.spot-on-sundials.co.uk/calculator.html
6. Declination
http://en.wikipedia.org/wiki/Declination#cite_note-5
7. Equation of Time
http://en.wikipedia.org/wiki/Equation_of_time
8. Google Map
http://maps.google.com/maps?hl=en&tab=wl
9. Find Latitude by the Polaris
http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0203-1-12-celestial_navigation/polaris.htm
10. Stellarium Software
http://www.stellarium.org/
11. Local Solar Noon Calculator
http://viz.globe.gov/viz-bin/solarnoon.cgi
12. Using Circumpolar Stars to find the True North
http://www.world-mysteries.com/alignments/mpl_al3.htm
27
Appendix 1 – Solar Declination and Equation of Time
28
29
Appendix 2 – Sun Positions Observed in Sandnes (Norway) from 24-Oct-2010 to 26-Mar-2011
Table A.2. Measured Data Converted to Sun Height at Noon (Sandnes, Norway)
Calculated Measured
s = |AB|
l = |BC|
Calculated
Sun Height
at Noon
Date
Time
Known
|AD| (*)
Measured
|BD|
(dd-mmm-yyyy)
(hh:mm)
(mm)
(mm)
(mm)
(mm)
24-Oct-2010
13:00
180
109.0
210.4
80.0
0.380
20.82
25-Oct-2010
26-Oct-2010
13:00
13:00
180
180
142.5
139.0
229.6
227.4
78.0
70.0
0.340
0.308
18.77
17.11
04-Nov-2010
06-Nov-2010
12:00
12:00
180
180
139.5
141.5
227.7
229.0
64.5
61.5
0.283
0.269
15.81
15.04
07-Nov-2010
09-Nov-2010
12:00
12:00
180
180
142.0
145.0
229.3
231.1
59.5
57.5
0.260
0.249
14.55
13.97
17-Nov-2010
18-Nov-2010
12:00
12:00
180
180
144.5
145.5
230.8
231.5
50.0
48.2
0.217
0.208
12.22
11.76
19-Nov-2010
24-Nov-2010
12:00
12:00
180
180
148.5
143.2
233.4
230.0
48.0
43.0
0.206
0.187
11.62
10.59
25-Nov-2010
26-Nov-2010
12:00
12:00
180
180
145.6
148.0
231.5
233.0
42.8
42.0
0.185
0.180
10.48
10.22
27-Nov-2010
28-Nov-2010
12:00
12:00
180
180
139.3
139.5
227.6
227.7
39.5
37.0
0.174
0.162
9.85
9.23
03-Dec-2010
06-Dec-2010
12:00
12:00
180
180
134.0
135.3
224.4
225.2
29.5
33.2
0.131
0.147
7.49
8.39
20-Dec-2010
22-Dec-2010
12:00
12:00
180
180
123.5
126.2
218.3
219.8
28.7
29.0
0.131
0.132
7.49
7.51
25-Dec-2010
26-Dec-2010
12:00
12:00
180
180
126.6
125.0
220.1
219.1
28.0
28.5
0.127
0.130
7.25
7.41
27-Dec-2010
02-Jan-2011
12:00
12:00
180
180
131.8
121.0
223.1
216.9
28.5
30.0
0.128
0.138
7.28
7.88
20-Jan-2011
23-Jan-2011
12:00
12:00
180
180
115.5
110.5
213.9
211.2
39.5
40.8
0.185
0.193
10.46
10.93
06-Feb-2011
12-Feb-2011
12:00
12:00
180
180
117.5
107.5
215.0
209.7
57.0
63.3
0.265
0.302
14.85
16.80
18-Feb-2011
23-Feb-2011
12:00
12:00
180
180
100.0
114.5
205.9
213.3
70.0
80.5
0.340
0.377
18.78
20.67
06-Mar-2011
26-Mar-2011
12:00
12:00
180
180
108.5
107.5
210.2
209.7
96.5
134.5
0.459
0.642
24.66
32.68
l/s
(deg)
(*) 3-Dimensional sketch is shown in Fig. 4
30
Appendix 3 – Sun Positions Observed in Pula (Italy) on 24-Jun-2011
Table A.3. Measured Data Converted to Sun Height at Noon (Pula, Italy)
Date
Time
Known (*)
height of the
gnomon
(l)
Measured
length of
the shadow
(s)
(dd-mmm-yyyy)
(hh:mm)
(mm)
(mm)
24-Jun-2011
12:10
275.0
114.0
2.412
67.48
24-Jun-2011
24-Jun-2011
12:20
12:45
275.0
275.0
106.0
88.7
2.594
3.100
68.92
72.12
24-Jun-2011
24-Jun-2011
12:50
12:55
275.0
275.0
87.0
84.2
3.161
3.266
72.44
72.98
24-Jun-2011
24-Jun-2011
13:00
13:05
275.0
275.0
82.2
81.0
3.345
3.395
73.36
73.59
24-Jun-2011
24-Jun-2011
13:10
13:15
275.0
275.0
80.0
78.7
3.438
3.494
73.78
74.03
24-Jun-2011
24-Jun-2011
13:20
13:25
275.0
275.0
78.0
77.5
3.526
3.548
74.16
74.26
24-Jun-2011
24-Jun-2011
13:30
13:40
275.0
275.0
78.3
79.0
3.512
3.481
74.11
73.97
24-Jun-2011
24-Jun-2011
13:50
14:00
275.0
275.0
80.5
85.0
3.416
3.235
73.68
72.82
24-Jun-2011
24-Jun-2011
14:35
15:30
275.0
275.0
106.0
162.0
2.594
1.698
68.92
59.50
l/s
Calculated
Sun Height
at Noon
(deg)
(*) 3-Dimensional sketch is shown in Fig. 1
31
Appendix 4 – Sun Positions Observed in Sandnes (Norway) on 08-Oct-2011
Table A.4. Measured Data Converted to Sun Height at Noon (Sandnes, Norway)
Calculated Measured
s = |AB|
l = |BC|
Calculated
Sun Height
at Noon
(deg)
Date
Time
Known
|AD| (*)
Measured
|BD|
(dd-mmm-yyyy)
(hh:mm)
(mm)
(mm)
(mm)
(mm)
08-Oct-2011
08-Oct-2011
12:00
12:10
180
180
65.5
76.4
191.5
195.5
83.2
86.0
0.434
0.440
23.48
23.74
08-Oct-2011
08-Oct-2011
12:20
12:30
180
180
82.3
98.3
197.9
205.1
90.0
93.5
0.455
0.456
24.45
24.51
08-Oct-2011
08-Oct-2011
12:40
12:50
180
180
109.5
120.5
210.7
216.6
97.5
101.2
0.463
0.467
24.83
25.04
08-Oct-2011
08-Oct-2011
12:55
13:00
180
180
127.5
134.5
220.6
224.7
103.8
105.6
0.471
0.470
25.20
25.17
08-Oct-2011
08-Oct-2011
13:11
13:17
180
180
148.7
157.5
233.5
239.2
110.3
114.0
0.472
0.477
25.29
25.48
08-Oct-2011
08-Oct-2011
13:20
13:30
180
180
164.5
180.5
243.8
254.9
116.0
120.8
0.476
0.474
25.44
25.36
08-Oct-2011
08-Oct-2011
13:35
13:40
180
180
189.8
198.8
261.6
268.2
123.5
126.8
0.472
0.473
25.27
25.31
08-Oct-2011
08-Oct-2011
13:45
13:50
180
180
210.5
219.7
277.0
284.0
131.0
134.0
0.473
0.472
25.31
25.26
08-Oct-2011
08-Oct-2011
13:55
14:01
180
180
231.4
246.5
293.2
305.2
137.4
142.2
0.469
0.466
25.11
24.98
08-Oct-2011
08-Oct-2011
14:10
14:18
180
180
268.0
291.4
322.8
342.5
150.3
157.0
0.466
0.458
24.96
24.63
08-Oct-2011
08-Oct-2011
14:21
14:30
180
180
302.2
336.5
351.7
381.6
158.5
172.0
0.451
0.451
24.26
24.26
08-Oct-2011
14:48
180
415.5
452.8
195.0
0.431
23.30
l/s
(*) 3-Dimensional sketch is shown in Fig. 4
32
Acknowledgements
I am thankful to my science teacher in the International School of Stavanger and supervisor, Mr. David
Tremblay, whose guidance and support from the beginning of this school research project in 2010 enabled
me to develop an understanding of the subject. I am especially grateful to him for providing me with a
substantial and very informative scientific book on astronomy from his private library.
I am especially thankful to my mother Inna Zakharova who helped me with marking the Sun’s positions on
the working days when I was at school.
This project would not have been possible without the support of my father Mikhail Zakharov who pointed
out on a practical importance of my astronomical observations for the purposes of locating geographical
coordinates, helped me with correct application of trigonometric functions and encouraged the completion
of the project.
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