Introduction to solar motion geometry on the basis of a simple model This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 Phys. Educ. 45 641 (http://iopscience.iop.org/0031-9120/45/6/010) Download details: IP Address: 139.133.11.3 The article was downloaded on 12/08/2013 at 23:43 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Introduction to solar motion geometry on the basis of a simple model
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2010 Phys. Educ. 45 641
(http://iopscience.iop.org/0031-9120/45/6/010)
Download details:
IP Address: 139.133.11.3
The article was downloaded on 12/08/2013 at 23:43
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
AbstractBy means of a simple mathematical model developed by the authors, theapparent movement of the Sun can be studied for arbitrary latitudes. Usingthis model, it is easy to gain insight into various phenomena, such as thepassage of the seasons, dependences of position and time of sunrise or sunseton a specific day of year, day duration for different latitudes and seasons, andtime dependence of the solar altitude reckoned from the horizontal plane. Wepresent simulations of the Sun path alongside animated data. We show thatthe model adopted can also be used to explain the principle of a horizontalsundial and to determine shadow areas. We give recommendations as regardshow to build energy-effective houses and how to optimize solar cell positions.S Online supplementary data available from stacks.iop.org/physed/45/641/mmedia
Introduction
‘Why do winter and summer (seasons) exist onEarth?’ Following the Nobel prize winner VitalyGinzburg [1], one can ask a lot of people thisrelatively simple question and the majority ofstudents or even highly educated people will notprovide the correct answer [2]. Some of themargue that the Earth rotates around the Sun in anelliptical orbit and therefore the Earth is in theclosest position to the Sun in summer, whereas inwinter it is farther away. Only a small minorityknow that in fact the distance between the Earthand the Sun is minimal during winter in thenorthern hemisphere; this fact is unimportant inexplaining the passage of the seasons on the Earth.
Probably, the reasons for this lack ofunderstanding lie in the complicated explanationsusually given in modern physics and astronomy
textbooks which operate with three-dimensionalobjects and systems. The lack of clarity causeslow insight into the relative movement of thesimple system consisting of the Sun and theEarth. Here it is especially important to use thetopocentric observation from the Earth’s surfaceof the Sun path on the sky during differentseasons rather than the three-dimensional side-view in space. Understanding the movement ofthe Sun and the reason of the passage of theseasons can be also helpful to activate the broadharnessing of solar energy, which sometimes canbe done in a very simple manner, even in winter.The great potential of solar energy is substitutingthe exhausting fossil fuels and mitigating globalwarming [3–5]. Unfortunately, if one analyses thenumber of scientific publications including termssuch as ‘solar’, ‘photovoltaic’, or ‘climate∗’ itbecomes evident that their recent growth is slower
Figure 1. Histograms depicting the number of publications per year on various topics between 1999 and 2009 (data obtained from ISI Web of Knowledge, 9 February 2010). Top: absolute values; bottom: relative values with respect to the data for 1999. Note that the absolute number of publications including the key fragment ‘nano*’ was divided by 10 for convenience (top panel).
both in absolute and relative terms to ‘hot’ subjectsearches such as ‘nano∗’ (figure 1). Probably, oneof the reasons for such a situation is the scantpublic knowledge of the basic physical processesbehind the apparent solar movement through thesky.
We believe that the use of renewable energyand, especially, solar energy is very important forthe effective development of both small businessesand autonomous households consuming lowamounts of conventional energy sources likeelectricity, gas, coal, etc. Our interest in thistopic originates from our educational and civilactivities in Ukraine which we started togethera few years ago. Even after many years sincegraduating from secondary school we knew nextto nothing about solar energy and solar movementgeometry. After some studies it became clearto us that simple understandable models suitablefor the broad public audience do not exist in theliterature, while professional approaches are too
complicated. Therefore, we developed our ownsimple model suitable for explaining the basicprinciples of the visible Sun path on the celestialsphere, the revolution of the seasons, shifts in thedaily sunrise and sunset positions, etc.
Derivation of a simple parametrical modeldescribing the visible movement of the SunMain assumptions
The passage of the seasons on the Earth stemsfrom its revolution around the Sun and the tilt ofthe Earth’s axis relative to the revolution plane.We assumed the period of such a revolution tobe 365.25 days (the Julian year). The Earthrevolves around the Sun in an elliptical orbitwith low eccentricity (ε = 0.0167). The actuallow eccentricity of the Earth’s orbit as well asa sufficient external heating of our planet arerelated to the so-called anthropic principle, whichdescribes the conditions of life abundance in theUniverse [6, 7]. For the sake of simplicity wedecided to neglect the ellipticity and suppose thatthe Earth rotates around the Sun with constantspeed over an ideal circular orbit. The Earth’s axialtilt (or obliquity) is the angle between its rotationalaxis, and a perpendicular to the orbital plane (angleα in figure 2(a)). The tilt and direction of therotational axis with respect to the orbit change veryslowly (over thousands of years) [8]. We used thevalue of 23.45◦ as the actual obliquity α.
The Earth’s axial tilt is the main reason forthe passage of the seasons (figure 2(a)). Duringthe days close to the summer solstice the axisof the Earth is tilted towards the Sun and themaximal duration of a day is observed at anyplace in the northern hemisphere. Then the anglebetween the Sun’s rays and the ground surface(height of the Sun, or elevation angle, or altitude)is at a maximum at midday for latitudes betweenthe Northern tropic (the Tropic of Cancer) and theNorth Pole. After half a year the Earth appears atan opposite position corresponding to the wintersolstice with a minimum sunlight period and aminimal altitude at midday. To find the maximumaltitude, which can be observed from the Earth’ssurface at the extremal moments discussed above,let us consider a certain point A on the surfaceof an ideal spherical Earth at latitude ϕ. Thehorizontal surface for an observer at this pointcan be found as a tangent plane to the Earth’ssurface at point A. The projections of the indicated
642 P H Y S I C S E D U C A T I O N November 2010
Introduction to solar motion geometry on the basis of a simple model
summer solstice
A
ϕ
winter solstice
ϕ
A
P
NA
W
Sy
z
E
x
P
NA
W
S
E
B
C
D15°
–15°
Sun
(a)
(b) (c)
plane for summer and winter solstices are shownin figure 2(a) as green lines. It can be easilyestablished from figure 2(a) that at the summersolstice time the maximum altitude at the pointA is close to (90◦ − ϕ + α), whereas for thewinter solstice this quantity is about (90◦ −ϕ −α).Small deviations from those idealized relations aredue to the non-zero angular size of the Sun andthe fact that the solstice occurs at a particularmoment, which does not always correspond tonoon for a given point. To avoid complicationsconnected to the finite visible Sun diameter weshall consider the Sun in most cases as a point lightsource. Nevertheless, sometimes the finitenessof the Sun disc manifests itself conspicuouslyand will be discussed below. Within the sameapproach one can find that during the springor autumn equinoxes the observer on the Earth
surface can see the Sun at a maximum altitudeof (90◦ − ϕ). One can also easily conclude thatfor any latitude on the Earth the Sun rises fromthe East and sets in the West only during equinoxtimes.
Figure 2(b) shows a scheme of the visiblemovement of the Sun on the celestial sphereobserved from point A on the Earth surface inthe Northern hemisphere during the winter andsummer solstices, spring or autumn equinoxes.Due to the small visible diameter of the Sun,all circular arcs concerned lie in the parallelplanes and the circle centres are located on thepolar axis AP connecting the observer and theNorth celestial pole. For our purposes it isconvenient to introduce a plane of the celestialequator, which is tilted relative to the horizontalplane at an angle of (90◦ − ϕ) and intersects
November 2010 P H Y S I C S E D U C A T I O N 643
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the latter by the straight line West–East. Notethat the point A belongs to this line. Thus,for the terrestrial observer the Sun moves in thecelestial equator plane during the days of thespring and autumn equinoxes. In the temporalrange between the spring and autumn equinoxesthe Sun is always observed above the celestialequator, otherwise it moves below the equator. Toa good approximation all solar trajectories on thecelestial sphere lie in planes parallel to the celestialequator.
To determine (for a certain date) thecharacteristic deviation between the correspondingplane and that of the celestial equator one shouldintroduce a new parameter, declination θ , as theangle between the line AD (or AB) and thecelestial equator plane (see figure 2(c)). Thelines AD or AB indicate the direction betweenthe observer and the highest position of the Sunon the sky for the date concerned. Accordingto this definition, the summer solstice plane ischaracterized by the declination θ = α, thewinter solstice plane is declined by the angle (−α),whereas the declination for the spring or autumnequinoxes is equal to 0. To ensure a smoothtransition between the indicated points we usethe following interpolation function describing thedeclination at any day of the year
θ = α sin2πT
365.25, (1a)
where T is a date counted from the spring equinox:T = 0 for 20 March, T = 1 for 21 Marchetc. In the framework of the proposed modelsuch a dependence shows that at any particulardate the Sun rotates in a plane characterizedby a constant declination during the whole day,although actually the visible movement of the Sunis more complex.
We extracted the Naval Oceanography Por-tal [9] data containing the annual time depen-dence of the solar declination in the period from20 March 2010 till 20 March 2011 and com-pared those data with declination values derivedaccording to equation (1a). The results arepresented in figure 3(a). Comparison of theexact dependence with the interpolation curveshows that equation (1a) describes the observedbehaviour quite well. The few-days differencebetween the observed solstices and the autumnequinox, on the one hand, and the model data, on
Figure 3. Model properties. (a) Comparison of the Sun declination obtained from the equation (1a) (black points) and data given by the Naval Oceanography Portal (red points) in the period between 20 March 2010 and 20 March 2011 [9]. A maximal difference of 1.726° is observed at T = 202 (8 October 2010); (b) Duration of daylight over the course of a year derived from the adopted model for different places on the Earth: Singapore (red curve), Cairo (green curve) and Anchorage (blue curve). The black curve shows the observed duration of daylight in Anchorage obtained from the Naval Oceanography Portal [9].
20
10
0
–10
–20
0 50 100 150 200 250 300 350
23.5
24.5
23.588 90 92 94 96
3
0
–3
T (days)
decl
inat
ion
(°)
185 190180
20
16
0 50 100 150 200 250 300 350
13
12
11180 185 190
T (days)
dura
tion
of d
aylig
ht (
hour
s) 18
12
8
6
4
10
14
Anchorage (ϕ = 61° 13’) Cairo (ϕ = 30° 03’) City of Singapore (ϕ = 1° 14’)
(a)
(b)
the other hand, are mostly due to the Earth’s orbitellipticity.
Description of the model
Hereafter, a three-dimensional Cartesian coordi-nate system with the origin at point A will be used(see figure 2(b)). The AX , AY , and AZ axesare pointed towards the West, South, and zenith,respectively. The Sun always revolves on thesurface of the celestial sphere described in thesecoordinates as x2 + y2 + z2 = 1. The declinationθ for a particular day numbered by T is a constant,its value being determined by equation (1a). Theassumptions presented in main assumptions aresufficient to calculate the movement of the Sun
644 P H Y S I C S E D U C A T I O N November 2010
Introduction to solar motion geometry on the basis of a simple model
with respect to the fixed observer at the latitude ϕ:
x = − cos θ sin2π t
1440, (1b)
y = − cos ϕ sin θ − sin ϕ cos θ cos2π t
1440, (1c)
z = y
tan ϕ+ sin θ
sin ϕ. (1d)
Here t is a solar time in minutes reckoned frommidnight of the day numbered T , 0 � t < 1440.It is worthwhile to note that a season-dependentdifference between the local time and the solartime, accepted in our model, is non-zero for themajority of points on the Earth’s surface [10].The temporal difference can be estimated as thedifference between the highest point of the Sun onthe celestial sphere (the solar noon) at a particularday and the local noon of our civil time definedas 12:00 am. We emphasize that below only solartime will be used. The set of equations (1a)–(1d)is suitable to represent the visible Sun path bothin the Northern (0◦ � ϕ � 90◦) and Southern(−90◦ � ϕ � 0◦) Earth hemispheres. In whatfollows the results and conclusions are applicableonly for the Northern hemisphere, although theycan easily be fitted for the Southern hemisphere.
Daylight duration at different latitudes
The equations system (1) can be applied todetermine the daylight duration for a particularlatitude ϕ and declination θ , the latter beingdependent on T in agreement with equation (1a).It can be done by solving the equation z = 0,which gives the sunrise and sunset instants and, asa consequence, the duration of daylight in hours,according to the formula
D = 24
(1 − arccos(tan θ tan ϕ)
π
). (2)
Dependences D(T ) for different latitudescalculated on the basis of equation (2) arepresented in figure 3(b). Analysis of the presenteddata shows that at the equatorial points the daylightduration is constant over the whole year. Anincrease in latitude causes an enhancement ofthe difference between daylight durations at thesummer and winter solstices. If for certain θ
and ϕ one obtains the argument of the arccosinefunction larger than unity, equation (2) should be
augmented with a parameter-independent resultD = 24. This phenomenon called ‘polar days’(‘midnight sun’) happens in the summer timeclose to the solstice for latitudes ϕ � 90◦ − α,the equality determining the ideal position of theArctic polar circle.
A comparison between model and real datafor Anchorage presented in figure 3(b) shows thatequation (2) satisfactorily describes the daylightduration over the whole year, although minordifferences of 10–40 min can be found. Such adiscrepancy originates mostly from the non-zerosize of the Sun as a light source and properties ofthe Earth’s atmosphere. The Naval OceanographyPortal provides the following explanation: ‘theSun is not simply a geometric point. Sunrise isdefined as the instant when the leading edge ofthe Sun’s disc becomes visible on the horizon,whereas sunset is the instant when the trailing edgeof the disc disappears below the horizon. Theseare the moments of first and last direct sunlight.At these times the centre of the disc is belowthe horizon. Furthermore, atmospheric refractioncauses the Sun’s disc to appear higher in the skythan it would if the Earth had no atmosphere.Thus, in the morning the upper edge of the discis visible for several minutes before the geometricedge of the disc reaches the horizon. Similarly, inthe evening the upper edge of the disc disappearsseveral minutes after the geometric disc has passedbelow the horizon’ [9]. The latter phenomenonis especially pronounced at high latitudes wherethe Sun’s path near the horizon has a relativelymoderate slope (90◦ − ϕ). This value is equalto the angle between the celestial equator and thehorizontal plane (see also figure 2(b)).
Sun path diagramsAs we have already mentioned, the model (1a)–(1d) adequately describes the Sun trajectoryvisible to the fixed observer located at point A fora particular latitude ϕ. To better understand thegeometry involved it is convenient to imagine thatthis observer is positioned on the horizontal planecovered from the top by an imaginary topocentriccelestial hemisphere centred in point A. TheSun moves on the surface of this hemisphere,the corresponding path and coordinates at theparticular instant t of the day T being determinedby equations (1a)–(1d) with the control parameterϕ (0◦ � ϕ � 90◦). To make the whole picture
November 2010 P H Y S I C S E D U C A T I O N 645
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Figure 4. Three-dimensional Sun path diagram and its two-dimensional image for the data taken from figure 2b. Rainbow colours represent low (violet) and high (red) insolation of the horizontal plane in point A on the Earth’s surface.
–1
0
1 1
0
–1
0
1
xy
y
(west) (south)
(zenith)
as simple as possible and the most descriptivelet us pass from the full three-dimensional solartrajectory to its two-dimensional projection on thehorizontal plane. Path diagrams obtained in thismanner are limited by the skyline represented ingraphs 4, 5, and 6(a) as the circular border.
A transition from small angles betweenthe line connecting the observer with the Sunposition and the horizontal plane and largeangles approaching 90◦ (Zenith) is imitated bya transition of colours close to the violet partof the visible spectrum into those belonging tothe red end of the spectrum. The graphicalembodiment of the described approach usingthe Sun path data presented in figure 2(b) isdepicted in figure 4. Circular Sun paths onthe two-dimensional celestial sphere in the three-dimensional space are unambiguously imaged aselliptical arcs on the two-dimensional flat diagram.Those arcs in the framework of our model witha fixed declination during a particular day T arealways symmetrical with respect to the South–North line (see figure 4). Violet regions of the Sunpath diagram correspond to low insolation of thepoint A in the horizontal plane, i.e. to its weakheating, whereas red regions correspond to highinsolation. Numerical estimations of insolationvalues are presented in shadow path diagrams andhorizontal sundials.
Our diagrams demonstrate the differentcharacter of Sun paths at different latitudes. Forthis purpose, we selected a set of latitudes inthe Northern hemisphere with the separation ofabout 30◦ corresponding to well-known points:Singapore, Cairo (Egypt), Anchorage (USA), andthe North Pole. The comparison of their Sun pathdiagrams is presented in figure 5. They give aclear-cut answer to our starting question ‘What isthe origin of the passage of the seasons?’. Sucha comparison also gives the answer to anotherquestion ‘Why is it always hot at the equator andcold at the North Pole?’ The observed shift with ϕ
from the red area to the violet one establishes thesought explanation.
Shadow path diagrams and horizontalsundialsIt is natural that changes of the Sun path characterduring the year should lead to a drastic temporalevolution of shadow patterns on the Earth’ssurface. In this connection, let us consider theannual evolution of the shadow of a verticalstick. This phenomenon is of practical importance,e.g., for sundials still being a decoration of citycentres. The main principles of those primitiveclocks were well known from antiquity [11–13]but, unfortunately, have been partially forgotten inthe modern epoch. Their basic properties can beeasily derived using the Sun path diagram. Forexample, figure 6(a) shows the solar path schemefor the latitude of Dresden (Germany) showingSun positions at different instants of solar time fordays of solstices and equinoxes. Knowledge ofthe Sun coordinates makes it possible to predictthe temporal evolution of the shadow lengthsand directions. To be specific, we analysed thesimplest case of a shadow cast onto horizontalplane by a vertical stick of height L located atpoint A perpendicularly to the plane indicatedin figure 6(b) for instants between winter andsummer solstices. Hereafter, we label suchdependence as a ‘shadow path diagram’.
The data presented in figure 6(b) show thatshadow tips trace out branches of hyperbolas intime and space. The corresponding asymptotesare determined by sunrise and sunset points on thesolar map. The arms of the hyperbola branchesare oriented to the North for days between autumnand spring equinoxes, otherwise they are orientedto the South. The vertex of each hyperbola
646 P H Y S I C S E D U C A T I O N November 2010
Introduction to solar motion geometry on the basis of a simple model
Figure 5. Sun path diagrams for the winter solstice (black curve), spring or autumn equinoxes (dark yellow curve), and the summer solstice (red curve) at different latitudes of the Earth: (a) ϕ = 01º 14′ (Singapore); (b) ϕ = 30º 03′ (Cairo); (c) ϕ = 61º 13′ (Anchorage); and (d) ϕ = 90º (the North Pole; the curve for the winter solstice is absent because the Sun is invisible during the polar night. Note also that the concept of cardinal directions becomes meaningless in the area close to the North Pole).
(a)(b)
City of Singapore, ϕ = 01º 14´
NW NE
W
SW
S
SE
E
203040
50
60
70
80A
Cairo (Egypt), ϕ = 30º 03´
NW NE
W
SW
S
SE
E
203040
50
60
70
80A
N N
(c)(d)
Anchorage (USA), ϕ = 61º 13´
NW NE
W
SW
S
SE
E
203040
50
60
70
80A
North Pole, ϕ = 90º
203040
50
60
70
80A
N
(observed at the solar noon) is determined by thecurrent Sun declination as well as the latitude ofthe observation point, and therefore a distancebetween point A and a hyperbola vertex is givenby the formula
d(T ) = L tan(ϕ − θ(T )), (3)
where θ(T ) can be found from equation (1a).During equinox days the shadow path shapetransforms closely to a straight line, which isoriented from the West to the East with the
distance to point A equal to L tan ϕ. The analysisof shadow tip positions for a particular solar timeshows that all of them lie along a certain straightline. Bundle of lines corresponding to differentsolar times intersects at the focal point, F, locatedto the south of point A. The distance AF equalsL cot ϕ (see figure 6(c)). Parameters of linearequations describing those lines for a particularsolar time can be found as follows. First ofall, we take into account the above-indicated factthat the paths of shadow tips at equinoxes are
November 2010 P H Y S I C S E D U C A T I O N 647
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Figure 6. Explanation of the sundial idea on the basis of Sun path and shadow path diagrams. (a) The diagram for Dresden (latitude ϕ = 51° 03′) showing the Sun paths at different instants of the solar time t for the time T of the winter solstice, equinoxes, and the summer solstice. (b) Evolution of the shadow tip from the vertical stick located at point A of length L during the season transition from the winter to summer solstices. Black concentric circles show distances from point A which are equal to L, 2L, 3L, 4L and 5L. Marks indicating solar time are the same as in (a). (c) Demonstration of the fact that the shadow tips at any particular solar time (see (b)) lie on the straight lines (independent of the day T), which intersect at the focal point F. The distance AF equals L·cot(ϕ). The shadow paths are the same as in (b). (d) A sundial based on the data of (c) for practical use. A stick FB tilted by angle ϕ is placed at point F instead of the original vertical stick AB of length L located at point A.
B
F
87
9 10 11 12 13 14
17
Dresden, ϕ = 51° 03′
Dresden, ϕ = 51° 03′
Dresden, ϕ = 51° 03′
(a) (b)
sunset
19 Jun,T = 91
NW
SE
N2030
40
50
60
70
80
A
20 Mar,T = 0
19 Dec,T = 274
E
S
SW
W
NE
S
W E
N
19 Dec, T = 274
28 Jan, T = 314
27 Feb, T = 344
20 Mar, T = 0
19 Apr, T = 30
9 May, T = 50
19 Jun, T = 91
(c) (d)1519
8
7
16
17
S
EW
N10 141311
FA
WN
SE
1.0
0.5
0.0
02
40
–2–4
–6
A
15
16
straight lines. At the same time, only duringthese days is the celestial path of the Sun observedfrom point A uniform. Hence, we can easilyfind solar coordinates on the celestial sphere. For
example, the Sun rises at the equinox day at 6 amexactly in the East, an hour later it changes its
angular location on the celestial equator plane by180◦/12 = 15◦; at 8 a.m. the Sun can be found at
648 P H Y S I C S E D U C A T I O N November 2010
Introduction to solar motion geometry on the basis of a simple model
an angle of 30◦ with respect to its position atsunrise, etc. From this data we can calculate thecorresponding positions of the shadow tips at solartime instants equal to, say, N hours (N = 7–17).Having the latter points and bearing in mind thatall lines go through the point F, one finally findsthe equations for the shadow tip locations as
ys(t = N, ϕ, xs) = −L cot(ϕ)
+ xs
sin(ϕ) tan(15◦N). (4)
Here (xs, ys) are coordinates of shadow tips onthe horizontal plane for the stick concerned (seefigures 6(b) and (c)), the axis A–xs being orientedto the East and the axis A–ys being oriented to theNorth.
The shadow path diagrams can be used fordifferent practical purposes (see conclusions andoutlook). For instance, their analysis comprisesthe basis of the horizontal sundial theory. Notethat actually the non-zero diameter of the solardisc becomes significant. For example, a shadowcalculated according to the model presented abovein reality turns out to be shorter and its boundariestransform into ambiguously limited semi-shadows,especially for thin sticks and low altitudes of theSun. To reduce the ambiguity one can use a tiltedstick connecting points F and B instead of theoriginal vertical stick AB discussed above (seefigure 6(d)). The angle between the new stick FBand the horizontal plane is equal to the latitude ϕ,therefore the stick is parallel to the polar axis APpresented in figure 2(b). Such a trick results in thenew shadow lying completely on the proper timeline, which makes the time measurement mucheasier in practice than that for the original verticalstick where only the tip was significant. The newpattern overcomes the effect of the unavoidabledistortion by the finite size of the Sun disc.
Hunt for solar energy in practicalarchitecture as an application of theproposed modelAs we have seen, solar path diagrams explainqualitatively the passage of the seasons through theannual rotation of the Earth in its orbit. Moreover,our approach allows one to quantitatively estimatethe solar energy input (insolation) that can beharvested on the horizontal or any other plane.It is well known that the solar constant, i.e., theamount of radiant power falling onto a flat surface
unit area oriented perpendicularly to the directionof sunlight rays outside the Earth’s atmosphere,equals �max ≈ 1366 W m−2. After penetratingthe atmosphere and partial absorption the valueof the incident solar energy decreases but for thesake of simplicity we shall hereafter neglect thedifference between the actual weakened Sun flux�weak and the value �max indicated above. In thegeneral case where the angle between the surfaceconcerned and the direction of Sun rays is equal toδ the insolation can be found from the followingequation
�(δ) = �max sin δ. (5)
Of course the angle δ permanently changes withtime as a result of the Earth’s rotation around itsaxis as well as its annual path around the Sun.
Our model gives the coordinates of the Sunon the celestial sphere at any instant. Thus, we areable to estimate the maximum energy that can beharvested on special planes oriented to the North,South, East and West or is parallel to the horizontalplane. To compare incident energies for each planewe introduce a ‘solar cube’ with edges equal to1 m (figure 7(a)). Solar harvesting for differentlatitudes appropriate to Ukraine was chosen asan example. In figure 7(b) annual diagramsdepicting daily amounts of radiant energy reachingeach side of the solar cube are demonstratedin the ideal case. The calculated values werefound by integration of the flux in equation (5)over the insolation time for corresponding planes.In the framework of our model eastern andwestern sides always absorb equal amounts ofenergy although in reality a little difference existsdue to slow gradual variation of the declination.Analysis of the presented data for the cube topexplains quantitatively the observed revolution ofthe seasons. Namely, it is clear that in summerthe horizontal plane absorbs a few times moreenergy than in winter because the day is longerand, especially, because the Sun reaches higheraltitudes. The energy ratio summer/winter for thehorizontal plane severely depends on latitude andincreases for higher ϕ values.
The northern side of the cube makes asmall contribution only between the spring andautumn equinoxes, because in the winter time theSun never shines on its plane. But the mainpractically significant conclusion can be madefor the southern side. In the winter time whenadditional heating of our houses is required the
November 2010 P H Y S I C S E D U C A T I O N 649
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Figure 7. (a) ‘Solar cube’ and (b) contribution of its sides to the total daily radiant energy absorption, which can be harvested in the ideal case at different latitudes throughout the year. Example: at latitude ϕ = 52º on 19 June, the south, horizontal, east, west and north sides of the solar cube absorb 4.0, 11.9, 7.4, 7.4 and 3.2 kWh m–2 radiant energy, respectively. The contribution from the southern side is the most significant between autumn and spring equinoxes.
32
24
16
8
020.03 19.06 18.09 18.12 20.03
north sideϕ = 44°
32
24
16
8
020.03 19.06 18.09 18.12 20.03
north sideϕ = 48°
32
24
16
8
020.03 19.06 18.09 18.12 20.03
north side
ϕ = 52°en
ergy
(kW
h m
–2)
west side
east side
horizontal side
south side
west side
east side
horizontal side
south side
west side
east side
horizontal side
south side
y (south)
(a) (b)
x (west)
z
southern side absorbs about half of the energyharvested by the solar cube as a whole. Thus,vertical surfaces oriented to the South providethe most effective way to absorb solar energyand additionally heat our homes either throughwindows or special heating devices like Trombewalls [14, 15].
The ideas presented above concerning heatingof northern and southern sides of the solar cubewere intuitively understood and used by peasants,e.g., in old Ukraine, to construct country cottages.Their longer walls were oriented from the Eastto the West, most of their windows were locatedon southern walls, and windows on the northernwalls were absent or were rather small to decreaseheat losses. An additional paramount feature ofsuch houses was the existence of wide eaves on thesouthern roof edges, which protected the cottagesfrom overheating in summer but allowed forinsolation of internal rooms through the windowsin winter. The width of the eave S can be easilycalculated if you know the distance H between theupper edge of the window and the roof and the
local latitude ϕ (figure 8(a)). It is apparent that ahouse needs sunlight access between autumn andspring equinoxes when the Sun’s altitude does notexceed (90◦ − ϕ). Therefore, the eave width Sshould be selected in such a way as to allow thesunlight to pass through the southern windows ofthe designed house during the cold season and toprotect it against overheating during the day timein summer (see figure 8(a)):
S = H tan ϕ. (6)
Similar considerations can be used to proposea new kind of the greenhouse on the basis ofthe sector in the horizontal plane where the Sunappears in the winter time. As one sees fromthe solar path diagrams, in winter the Sun alwaysrises from a direction close to Southeast andsets towards Southwest. The exact directions ofsunrise and sunset are dependent on the locallatitude ϕ and the actual declination θ defined fromequation (1a). Obvious transformations usingequations (1a)–(1d) lead to the following angle β
between the sunrise (sunset) direction and the East
650 P H Y S I C S E D U C A T I O N November 2010
Introduction to solar motion geometry on the basis of a simple model
W
S N
E
A
E′
W′
19 June
20 March
18 December
S
H
northsouth
19 June
20 March
18 December
northsouth
glass
A↓ ↓A
insulationglass
A–A (top view)
(a)
(b)
(c)
(d)
ϕ
90º –ϕ
90º –ϕ
light-absorbingcoating
light-absorbingcoating
insulation
(the West)
β = arcsin(sin θ/ cos ϕ). (7)
On the winter solstice day the absolute valueof θ is equal to the Earth’s axis tilt: θ = −α =−23.45◦. In this case equation (7), e.g., for thelatitude of Dresden gives β ≈ −39◦ (figure 8(b)).The equation concerned can be used to construct agreenhouse, which takes into account the sunshinesector allowing maximum heating by a naturalenergy source in the cold season (figure 8(c)).
The greenhouse’s roof should be tilted at theangle (90◦−ϕ) with respect to the horizontal planeand two walls should be directed to the sunriseand sunset points calculated for the winter solstice.Access of the solar energy into the greenhouseshould be provided by a transparent glass pane;the heat supplied by the Sun is absorbed andaccumulated by the material of the walls (see side-and top-view images in figure 8(d)).
For the physical principles behind thegreenhouse effect, we refer to Besson et al, whogive a new teaching methodology of explaining
November 2010 P H Y S I C S E D U C A T I O N 651
V Khavrus and I Shelevytsky
thermal effects of interaction between radiationand matter [16]. In any case, to designgreenhouses or solar-harvest houses, one needsto consider the Sun path and ensure the optimalabsorption of solar energy by the building interiorin the coldest season.
Conclusions and outlookOur model can be applied to explain variousphenomena in one way or another dependingon the apparent Sun path on the celestialsphere. Furthermore, our approach can beuseful to quantitatively describe the Sun pathmechanically modelled, e.g., by the recentlypresented solar demonstrator [17]. Alternatingseasons, dependences of the sunrise and sunsetcharacteristics on the date, day durations fordifferent latitudes, temporal dependence of thesolar altitude for a fixed observer can bereproduced by rather simple computer calculationson the basis of the suggested model.
Of course, our simplified approach and cor-responding conclusions have certain limitations.Some of them can be easily overcome leadingto more accurate results, but to tackle othersubtleties more sophisticated insights and effortsare needed. In particular, we should indicatethat actually the set of equations (1a)–(1d) shouldbe replaced by a more accurate one including acontinuous time rather than integer ‘days’ (T ) andthe current time during this day (t), which wouldresult in a continuous changes of the declinationθ . Unfortunately, in that case the clarity ofour model based on the everyday experience ofstudents (24 h in one day and 365.25 days inone Julian year) will be lost. Furthermore, ifexact tabulated θ values were used to performmore precise calculations on the basis of the datapresented by the Naval Oceanography Portal [9]the convenient periodicity of equation (1a) wouldbe lost as well.
There are yet other areas of application of thesuggested approach. For instance, any variationof the Earth’s obliquity α might significantly alterabsorbed solar energy at high latitudes. Indeed,it is well known that the Earth’s obliquity slowlychanges with a period of about 40 000 years inthe range from 22.1◦ to 24.5◦ causing glacialcycles [8]. At the same time, other naturally drivenclimate changes (Milankovitch cycles) requiremore sophisticated explanations, in particular,
dealing with varying parameters of the Earth’selliptical orbit, explanations going far beyond thesimplicity of the proposed model. In any caseit should be noted that the Earth’s orbit changesvery slowly, specifically, at the scale of thousandsof years and cannot be responsible for the actualsharp climate changes observed during the last 30–40 years [3–5]. Prediction of the climate on otherplanets with orbits close to circular may be anotherpromising area of application of our model [7]. Inthis connection, the radiation flux constant (solarconstant in the case of Sun) can be easily foundfrom the power of the central star radiant energyand the orbital radius of a specific planet [5].
As for topical problems of everyday life, thepresented model, being the basis of the automaticor manual tracking system, would be of helpto control photovoltaic elements or solar thermalcollectors to maximize their output. It shouldbe taken into account that the solar constant�max (5) depends on the angle between rays andthe horizontal plane, since the light attenuation inthe atmosphere depends on the length of the pathand is stronger when the sunlight propagates atsmaller angles.
The model would also be useful to constructenergy-effective buildings [14, 15, 18]. Weemphasized above that in order to enhance theradiant energy absorption allowance should bemade for the sector where the Sun appears.However, during the cold season the shadow ofany construction covers a vast area, which canbe calculated using our model. Therefore, it isnecessary to bear in mind the nontrivial problemof the optimal arrangement of several buildingsor isolated constructions intended to avoid mutualshadowing. Doing this one should take intoaccount the actual latitude and monitor the time-dependent shadowed areas using the appropriatesoftware.
Similar programs might also be used toproduce animated cartoons where shadow areasof animated objects are of proper size. We alsowant to attract attention to another applicationof solar path and concomitant moving shadowdiagrams. Namely, Baker and Thornes used thesolar positions in Monet’s paintings of the Housesof Parliament in London to derive the dates andtimes of the depicted scenes, which allowed theauthors to extract from impressionist paintingsuseful information about London fogs and airquality more than hundred years ago [19].
652 P H Y S I C S E D U C A T I O N November 2010
Introduction to solar motion geometry on the basis of a simple model
Yet another important property of the modelis its ability to explain main features of horizontalsundials presented above. We are going to extendour analysis to the case of the gnomons positionedon the surfaces of vertical walls with their arbitraryorientation with respect to the East–West direction.The full sundial theory includes cumbersomemathematics hardly suitable for the high-schoollevel [11–13, 20–22]. On the other hand, thesimplicity of our mathematical model makes itappropriate to invite students to the challengingworld of solar geometry.
AcknowledgmentsThe authors cordially thank Professors AlexanderGabovich and Vladimir Fomin for useful discus-sions. VK acknowledges support from the BMBF(project 03X0076C).
Received 14 May 2010, in final form 31 July 2010doi:10.1088/0031-9120/45/6/010
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Vyacheslav Khavrus is a visitingscientist at the Leibniz Institute for SolidState and Materials Research Dresden,Germany. He completed a PhD inphysical chemistry at theL V Pisarzhevskii Institute of PhysicalChemistry of the National Academy ofSciences of the Ukraine in 1999. Hisscientific interests are nanotechnology,complex dynamic processes and solargeometry.
Ihor Shelevytsky is a professor anddeputy rector of the State PedagogicalUniversity in Kryvyi Rih, Ukraine.Besides teaching in information science,his research interests include real-timedigital signal processing using splinesand green energy solutions.
