EDUCATIONAL ACTIVITY 2.
Calculating the Latitude of the observation site from images of the Midnight Sun.
Mr. Miguel Ángel Pío Jiménez. Astronomer Instituto de Astrofísica de Canarias, Tenerife.
Dr. Miquel Serra-Ricart. Astronomer Instituto de Astrofísica de Canarias, Tenerife.
Mr. Juan Carlos Casado. Astrophotographer tierrayestrellas.com, Barcelona.
Dr. Lorraine Hanlon. Astronomer University College Dublin, Irland.
Dr. Luciano Nicastro. Astronomer Istituto Nazionale di Astrofisica, IASF Bologna.
1 – Objectives of the activity.
In this activity we will learn to calculate the latitude of an observer from digital images
and calculating in them the height the Sun on the horizon.
The goals of this activity are to:
- Explain the phenomenon of the midnight Sun.
- Apply a methodology for the calculation of a physical parameter (Geographycal
Latitude) from an observable (digital images).
- Support the teaching and learning of mathematics and physics by applying
knowledge of mathematics (algebra and trigonometry) and basic physics
(kinematics) to solve a practical problem.
- Understand and apply basic techniques of image analysis (angular scale, distance
measurement, etc.).
- Work cooperatively as a team, valuing individual contributions and expressing
democratic attitudes.
2 – Instrumentation.
The activity will use digital images obtained from northern Europe during the transit of
Venus on June 6, 2012 (see sky-live.tv).
3 – Phenomenon.
3.1. Midnight Sun.
The Midnight Sun is a natural phenomenon that occurs in its fullness only north of the
Arctic Circle and south of the Antarctic Circle. Since in the southern hemisphere there are no
permanent settlements sufficiently close to the pole (except in the Antarctic bases which are
inhabited by a few scientists and military personnel), inhabited regions that can enjoy with this
phenomenon are all in the Northern Hemisphere: Alaska, Canada, Greenland, Norway, Sweden,
Finland, Russia and northern end of Iceland. Due to the inclination of the Earth's axis of rotation
to the ecliptic of about 23 degrees and 27 minutes, at these high latitudes the Sun does not set
during the summer.
Midnight Sun
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Figure 1: Render created to simulate a sequence of photos of the midnight Sun taken over 24 hours. The
Sun is at its maximum height at noon, and its lowest height at midnight. Credits: Anda Bereczky.
The midnight Sun remains visible above the horizon at these latitudes (see Figure 1). The
length of time for which the midnight Sun remains visible depends on the latitude. For example,
at the Artic circle (latitude 60 degrees) it is visible for only 20 hours around the summer solstice
(between June 22nd and 24th), while at the geographic poles it is visible for 6 months. At the
poles over the course of a year, sunrise and sunset are observed only once. During the six
months of day at the pole, the Sun moves continuously near the horizon, reaching its maximum
height in the sky at the summer solstice.
Due to refraction (bending of sunlight through our atmosphere), the midnight Sun can be
observed at latitudes slightly below the polar circle, but at most one degree below (depending on
local conditions). For example, the midnight Sun can be seen in Iceland, although most of the
country (the island of Grimsey is the notable exception) is located to the south of the Arctic
Circle. In the most northerly of the British Isles (and other places at similar latitudes) the sky
doesn’t get completely dark at this time of year.
As mentioned, the duration of the midnight Sun varies, depending on latitude. For
example, at the northernmost point of mainland Europe, the North Cape in Norway, the
midnight Sun can be enjoyed from May 14th until July 29th. However, slightly further south, at
the latitude of the Arctic Circle, in Rovaniemi, a place referred to as the reference of the circle,
it is only observable from June 12th to July 1st. A quarter of Finland's territory lies north of the
Arctic Circle and in the northernmost part of the country the Sun does not set for 73 days in
summer. In the Svalbard Islands, Norway, the northernmost inhabited region of Europe, there is
no sunset from around April 19th to August 23rd.
At the extremes are the poles themselves, where the Sun remains visible for half of the
year. After this time, the Sun remains above the horizon for several hours each day, and then
sets. The nights therefore get progressively longer, until the opposite time of year is reached,
when the Sun sets for months, leading to the long polar night.
In order to prepare to observe and enjoy this wonderful phenomenon, the best time will
be between about 23.00 and 01.00 hours from Norway. This is the time when we can see how
the Sun descends towards the horizon, acquiring the typical orange colour of a conventional
sunset, but then, rather than hiding, will reach a point where it will start to climb again,
transforming sunset into sunrise. For the rest of the “day”, the Sun is never seen high overhead
as we are used to seeing from other latitudes during summer. Therefore, during this period, and
just above the Arctic Circle, the Sun is not as intense as would be expected at this time of year.
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3.2. The White Nights.
Northern regions of Russia and other locations that are above 60 degrees latitude, but are
south of the Arctic Circle (or north of the Antarctic Circle in the southern hemisphere) do not
experience the midnight sun, but experience civil twilight at midnight. The Sun sets, but doesn’t
go more than 6 degrees below the horizon (which defines civil twilight), so daytime activities,
such as reading, are possible without artificial light, provided the sky is not cloudy. This natural
phenomenon is popularly known by the name of the White Nights.
Figure 2: Fireworks at the launch of the White Nights Festival in St. Petersburg, Russia. The
sailboat with scarlet sails is the frigate Standart (built by Peter I of Russia).
3.3. The Polar Night.
The polar night occurs during winter, when the Sun does not rise for a long period of time
and the days are in darkness. In places like the Svalbard islands, the polar night extends from
October 28th until February 14th.
In the regions within the polar circles (latitudes greater than 66 degrees north or south),
the length of time the Sun is below the horizon varies from 20 hours at the Arctic and Antarctic
Circles, to 179 days at the north and south poles. However, all this time is not classified as polar
night, as there may be some sunlight because of refraction. It is also worth noting that the Sun
stays above the horizon for 186 days (compared to 179 below). The difference is just due to the
fact that even if the Sun is only partially above the horizon, it is regarded as daytime.
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Figure 3: Night Pole at North of Europe. Longyerbyen, Svalbard, Norway.
4 – Methodology.
4.1. Sun apparent movement on the Celestial Sphere.
The position of the Sun appears to change during the course of the year, so that the points
where it rises and sets are constantly changing. In reality, of course, it is Earth’s position and
orientation with respect to the Sun, which changes (Figure 4). However, here we adopt the
convention in which the Sun is referred to as being in motion. On March 21st, at the vernal
equinox, the Sun rises in the east and sets in the west. As the days pass, these points move
northwards, first rapidly, then slowly, until June 21st, when at the summer solstice, the Sun
reaches its maximum height over the horizon for northern latitudes. Sunrise and sunset are then
at their most northerly positions.
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Figure 4: Representation of day-time and night-time as a function of the time of year and the orbit
of the earth around the Sun.
From June 21st, sunrise and sunset move gradually to their equinox positions, due east
and west, on September 22nd / 23rd, the autumnal equinox. They continue to move southwards
until December 22nd, the winter solstice. After that time, the sunrise and sunset positions begin
their northward migration once again, returning to their spring equinox positions after one year.
The Earth’s axis is tilted at an angle of 23°27' with respect to the line perpendicular to its
orbital plane (Figure 5). The trajectory of the Sun across the sky and its declination therefore
changes over the course of a year (Figure 6) where the declination of the Sun is the angle
between the rays of the Sun and the plane of the Earth's equator.
Figure 5: The Earth’s axis of rotation is tilted by 23o27' with respect
to the North Pole. Credits: Wikimedia commons.
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Figure 6: The declination of the Sun over the course of a year (www.astro.virginia.edu/class/oconnell/astr130).
Before the Sun crosses the equator on March 21st, its declination is negative (Figure 6),
while at the vernal equinox on March 21st its declination is zero. Then, the maximum height of
the Sun above the horizon is 90 degrees minus the latitude of the observer’s location, φ (i.e. 90
− φ). The length of the day is equal to the length of the night at the equinox. In the days
following, the declination of the Sun is positive, and continues to increase until it reaches +23°
27', at the summer solstice when it is overhead at noon for the Tropic of Cancer. In the northern
hemisphere this is the longest day and the shortest night of the year and the Sun reaches its
maximum height above the horizon (= 90o − φ = 23° 27'). From then on the declination of the
Sun begins to decrease until it is 0o again on September 21st, at which time the length of the day
again equals the length of the night. The declination continues to decrease, now with negative
values, until the winter solstice, when it is overhead at noon for the Tropic of Capricorn
(December 21st), reaching its minimum declination value of −23º 27'. The longest night and the
shortest day occur on this day and the Sun’s altitude above the horizon reaches its lowest annual
value for northern hemisphere observers. Figure 7 shows the latitudes at which the maximum
altitude of the Sun is greatest, for these key dates over the course of one year.
Figure 7: The latitudes for which the Sun is at its maximum declination over the course
of a year (www.geog.ucsb.edu/~joel/g110_w08/).
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4.2. Method to calculate the latitude.
We have seen in Section 4.1 that the maximum altitude of the Sun depends, not only on
the observer’s latitude, but also with the time of year, since the sun’s declination varies from –
23.5° at the winter solstice, to +23.5° at the summer solstice, and is zero at the equinoxes.
To calculate the height of the Sun at midnight, at a certain location on Earth, we use the
following expression:
hs = s – (90 – ),
where hs is the altitude of the Sun above the horizon at midnight, φ is the latitude of the observer
and δs is the solar declination for the day in question.
In our case,  is the quantity we wish to determine, therefore we rearrange the above
equation to get:
 = 90 – δs + hs
Equation [1]
keeping in mind that all values must be expressed in the same units, which in this case would be
degrees, to simplify calculations.
4.3. Calculating the height of the Sun above the horizon.
Measuring the Sun's altitude is relatively simple, and can be determined directly from the
images, at the time of passage of the Sun through the meridian midnight. That distance will be
measured in the image, so the simplest unit to use is the number of pixels. Then a scale factor is
applied which relates the size of a pixel to the angular scale in degrees and hs in units of degrees
can be determined.
5 – Measurement for June 6, 2012 from Tromsø, Norway.
5.1. Instrumental Description.
For the observation of the phenomenon and the recording of images in real time, a Canon
5D Mark II 21-Mpix will be used, in conjunction with a filter suitable for solar observation.
5.2. Example of the calculation for Tromsø.
We are interested in determining the latitude from which the observer is taking pictures of
the midnight Sun, for which, following the theoretical reasoning above, we need to know the
height of the Sun above the horizon at the moment, plus the value of the solar declination for the
observation day, so these values can be substituted into Equation [1].
Now let's do the calculation using astronomical software that lets us know, for June 6,
2012, the height at which the Sun will be above the horizon at the time of the midnight sun. In
this example, to calculate the scale factor we discussed, we use the size of the Sun in this image
(Figure 8), which is 42 pixels. The angular size of the Sun is a known quantity, 0.525º.
Applying the reasoning from the previous section, the scale value, ε, is:
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Figure 8: Schematic representation of the height hs to be used for the calculation.
𝑆𝑐𝑎𝑙𝑒 (ℇ) =
0.525°
= 0.0125 𝑑𝑒𝑔𝑟𝑒𝑒𝑠/𝑝𝑖𝑥𝑒𝑙
42 𝑝𝑖𝑥
Then, the height hs is measured to be 195 pixels, equivalent to saying that hs = 2°25'48".
From the ephemeris, we can get the value of the solar declination for a given day, which in this
case, is June 6, 2012. The value is:
δs = 22º 39' 39.42"
So, finally, substituting into Equation [1], we find the latitude of the observation to be:
ϕ = 90 – δs + hs = 69.77º = 69º 46' 8.58"
where δs, as mentioned, is the solar declination that day and hs is the height of the Sun above the
horizon, measured in degrees.
6 – Internet Directions.
 The celestial sphere.
http://csep10.phys.utk.edu/astr161/lect/celestial/celestial.html
 Introduction to astronomical coordinates systems.
http://www.astro.lsa.umich.edu/undergrad/labs/coords/index.html
 Transmission of the transit online by internet:
http://www.sky-live.tv
 Scientific expeditions of the Shelios group.
http://www.shelios.com
 Midnight Sun seen from Fjellheisen. 360º - Panorama
http://www.virtualtromso.no/en/panoramas-from-tromso/56-midnight-sun-seen-fromfjellheisen-cable-car.html
 Collection of some pictures.
http://www.fotosearch.com/photos-images/midnight-sun.html
Midnight Sun
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