Solar Geometry P L A N E O F S U N

Transcription

1 1 Slar Gemetry The Earth s daily rtatin abut the axis thrugh its tw celestial ples (Nrth and Suth) is perpendicular t the equatr, but it is nt perpendicular t the plane f the Earth s rbit. In fact, the measure f tilt r bliquity f the Earth s axis t a line perpendicular t the plane f its rbit is currently abut We call the plane parallel t the Earth s celestial equatr and thrugh the center f the sun the plane f the Sun. The Earth passes alternately abve and belw this plane making ne cmplete elliptic cycle every year. Winter Slstice (z Dec 21) Vernal Equinx (z March 21 ) SUN P L A N E O F S U N Autumnal Equinx (z Sept 23) Summer Slstice (z June 21 )

2 2 Summer Slstice On the ccasin f the summer slstice, the Sun shines dwn mst directly n the Trpic f Cancer in the nrthern hemisphere, making an angle δ = with the equatrial plane. We call δ the Sun declinatin angle. In general, the Sun declinatin angle, δ, is defined t be that angle made between a ray f the Sun, when extended t the center f the earth, O, and the equatrial plane. We take δ t be psitively riented whenever the Sun s rays reach O by passing thrugh the Nrthern hemisphere. On the day f the summer slstice, the sun is abve the hrizn fr the lngest perid f time in the nrthern hemisphere. Hence, it is the lngest day fr daylight there. Cnversely, the Sun remains belw the hrizn at all pints within the Antarctic Circle n this day. Winter Slstice On the day f the winter slstice, the smallest prtin f the nrthern hemisphere is expsed t the Sun and the Sun is abve the hrizn fr the shrtest perid f time there. In fact, the Sun remains belw the hrizn everywhere within the Arctic Circle n this day. The Sun shines dwn mst directly n the trpic f Capricrn in the suthern hemisphere n the ccasin f the winter slstice.

3 3 The Sun declinatin angle is δ = 23.5 n the winter slstice. The Sun declinatin angle, δ, has the range: 23.5 < δ < during its yearly cycle. Yearly Variatin in Declinatin [ Summer Slstice [ Vernal Equinx Autumnal Equinx c Day Number Winter Slstice/ Accurate knwledge f the declinatin angle is imprtant in navigatin and astrnmy. Fr mst slar design purpses, hwever, an apprximatin accurate t within abut 1 degree is adequate. One such apprximatin fr the declinatin angle is:

4 4 sin δ = cs [ ( N 173 ) ] ( 1 ) where the argument f the csine here is in degrees and N dentes the number f days since January 1. Vernal and Autumnal Equinx There are tw ccasins thrughut the year when the center f the Earth lies in the plane f the Sun. Since the Earth s Nrth Suth axis f rtatin is perpendicular t this plane, it fllws that n these tw days every lcatin n the Earth receives 12 hurs f sunshine. These tw events are knwn as the vernal and autumnal equinxes. The Sun declinatin angle has measure: δ = 0 n the days f the vernal and the autumnal equinx. Nte that The Earth is abve the plane f the Sun during its mtin frm the autumnal equinx t winter slstice t vernal equinx. Hence, δ < 0 during the fall and winter. The Earth is belw the plane f the sun as it mves frm vernal equinx t summer slstice and back t autumnal equinx (i.e. during spring and summer). S δ > 0 during these seasns. The latitude f a lcatin n the Earth is the angle between the line jining that lcatin t the center f the earth and the equatrial plane. Fr example, Chicag, Illinis is n the circle f latitude 41.8 N. All lcatins at the same latitude experience the same gemetric relatinship with the sun. The great semicircles alng the surface f the Earth jining the Nrth t the Suth ples are called lines f lngitude. Cities n the same circle f latitude Cities n the same line f lngitude The line f lngitude thrugh Greenwich, England is called the prime meridian. The lcatin P n the glbe has lngitude λ ο East and latitude ϕ ο Nrth.

5 5 Slar Nn Slar Nn is defined t be that time f day at which the Sun s rays are directed perpendicular t a given line f lngitude. Thus, slar nn ccurs at the same instant fr all lcatins alng any cmmn line f lngitude. Slar Nn will ccur ne hur earlier fr every 15 degrees f lngitude t the east f a given line and ne hur later fr every 15 degrees west. (This is because it takes the Earth 24 hurs t rtate 360.) Prblem: Determine the elevatin angles at which the sun s rays reach the surface f the Earth in Chicag, IL at slar nn at the times f an equinx r a slstice. At the Vernal and the Autumnal Equinx Thus, the mst direct angle at which the Sun s rays strike the grund in Chicag is at abut 48 n the days f a vernal and autumnal equinx.

6 6 At the Summer Slstice Thus, the Sun s rays strike the grund in Chicag mst directly at a vertical angle f abut 72 n the day f the summer slstice.

7 7 At the Winter Slstice On the day f the winter slstice, the Sun s rays strike the grund at an angle f abut 25 at slar nn in Chicag, IL.

8 It is evident that, n any day f the year, the mst direct angle f slar radiatin at a given lcatin ccurs at slar nn. Frm ur previus examples, it fllws that fr pints in the Nrthern hemisphere, the angle f slar elevatin at nn is given by: α = 90 ( ϕ δ ) ( 2 ) The symmetrical frmula fr slar elevatin at nn fr pints in the Suthern hemisphere is given by: α = 90 + ( ϕ δ ) ( 3 ) We have seen that at slar nn, the range f angles at which the sun strikes the grund at Chicag, Illinis varies frm a minimum f 24.7 at the winter slstice t a maximum f 71.7 at the time f the summer slstice. A sun path diagram fr Chicag appears belw. 8 In the abve diagram, the azimuth and slar elevatin angles are the crdinates f an bserver's lcal hrizn system in Chicag, Illinis. Yu may btain a sun path diagram fr any specified lcatin n Earth at any specified time f the year by entering the relevant infrmatin at the University f Oregn s Slar Radiatin Mnitring Labratry website:

9 The azimuth is the lcal angle Α between the directin f due Nrth and that f the perpendicular prjectin f the Sun dwn nt the hrizn line measured clckwise. Thus, we have the azimuth values: 0 = due Nrth, 90 = due East, 180 = due Suth, and 270 = due West. The angle f slar elevatin, α, is defined as befre, t be the angular measure f the Sun s rays abve the hrizn. Equivalently, it is the angle between the directin f the gemetric center f the Sun and the hrizn. 9 Α α S = = E W = 270 Sunrise Equatin N = 0 The sunrise equatin: cs ω = tan(ϕ ) tan(δ) (4) is used t determine the lcal time f sunrise and sunset at the latitude, ϕ, when the Sun declinatin angle is δ degrees. In this equatin, we have the quantities: ω = ω the hur angle expressed in degrees where 180 < ω < 0 at sunrise and 0 < ω < +180 at sunset. ϕ the latitude f a specified lcatin n Earth δ the Sun declinatin angle n a specified day. The hur angle, ω, is the angular distance between the meridian f the bserver and the meridian whse plane cntains the sun. Thus, the hur angle is zer at lcal nn (when the sun reaches its highest pint in the sky). At this time the sun is said t be due suth (r due nrth, in the Suthern Hemisphere) since the meridian plane f the bserver cntains the sun. The hur angle increases by 15 degrees every hur. ϕ ω Q δ Meridian perpendicular t sun rays Sun rays Declinatin angle Equatr Meridian f bserver at Q

10 10 It fllws that at lcal nn, the hur angle is zer: ω = 0. Using a 24 hur time scale fr the day, ( 0 < t < 24, then t = 12 at lcal nn, t = 6.5 at five and ne-halwe have that at time t, the measure f the hur angle hurs befre lcal nn and t = at six and ne-quarter hurs after lcal nn), is: ω = 15 ( t 12 ) (5) Upn slving (4) fr ω, setting that result equal t the right side f (5), and then slving it fr ( t 12 ), we find that (i) (ii) Nte that cases (a) and (b) nly relate t latitudes beynd +65.5, i.e. abve the Arctic Circle r belw the Antarctic Circle. Angle f Slar Elevatin fr Times f Day Other Than Lcal Nn It can be shwn that the angle f slar elevatin, α, may be expressed in terms f the Sun s declinatin angle, δ, n a given date, the current hur angle (i.e. the time f day) ω, and an bserver s latitude, ϕ, via the equatin: α = sin 1 ( sin δ sin ϕ + cs δ cs ω cs ϕ ). (7) Azimuth Equatin Sunrise ccurs at cs and Sunset ccurs at s 1 ( tanϕ tan δ ) 15 cs 1 ( tanϕ tanδ ) 15 Exceptinal cases: (a) If (tan δ tan ϕ) > 1, then n sunset that day. (b) If (tan δ tan ϕ) < 1, then n sunrise that day. hurs befre lcal nn hurs after lcal nn. One equatin which relates the Sun s azimuth angle, A, at a given lcatin, its angle f elevatin, α current hur angle ω at the bserver s latitude, ϕ, and the Sun s declinatinn angle, δ, n this date is: (6), the cs α cs A = sin δ cs ϕ cs δ cs ω sin ϕ (8) Slving fr the azimuth, A, we find: A = sinδ cs 1 csϕ csδ csω sinϕ csα where if ω < 0 then A = A. (9) Otherwise, ω > 0 and A = 360 A.

11 A knwledge f the angle f slar elevatin, α, and the azimuth, Α, allws us t calculate the length and the lcatin f a simple shadw. Example Describe the shadw due t a telephne pst that is h ft high at the time f day when the slar elevatin is α = α ο and the azimuth is Α = Α ο. 11 In the figure, we bserve that this shadw has length OP where in the right triangle, OPQ, we find that h tan α ο = OP h OP = ft. tan α Nte that the tip f the shadw is lcated at the pint P = ( x, y ) that is x ft East and y ft Nrth f the base f the pst. By analyzing the right triangle, ORP, we find that sin( Α ο 180 ) = x x = OP sin( Α ο 180 ) ft. OP Thus, x = h tan α sin( Α ο 180 ) = h sin tan α Α ο ft, and cs( Α ο 180 ) = y OP y = OP cs( Α ο 180 ) ft. S y = h tan α cs( Α ο 180 ) = h cs tan α Α ο ft. Therefre, the shadw s vertex is at P = h sin Α tan α ο, h cs Α tan α ο.

12 12 Example Describe the shadw psitining due t Crwn Hall at varius times f day n the ccasin f the summer slstice r autumnal equinx. 1. Shadw Psitin n Summer Slstice 2 Hurs After Slar Nn δ = 23.5 ο ϕ = 41.8 ο ω = 30 ο α = 59.1 ο Α = ο Α' = Α 180 ο = 63.1 ο Shadw Length = 14.4 ft Δ y = 6.5 ft nrthward Δ x = 12.8 ft eastward Building Dimensins: 220 ft x 120 ft x 24 ft 2. Shadw Psitin n Summer Slstice 4 Hurs After Slar Nn δ = 23.5 ο ϕ = 41.8 ο ω = 60 ο α = 37.4 ο Α = ο Α' = Α 180 ο = 89.4 ο Shadw Length = 31.4 ft Δ y = 0.33 ft nrthward Δ x = ft eastward

13 13 3. Shadw Psitin n Autumnal Equinx 2 Hurs After Slar Nn δ = 0 ο ϕ = 41.8 ο ω = 30 ο α = 40.2 ο Α = ο Α' = Α 180 ο = 40.9 ο Shadw Length = 28.4 ft Δ y = 21.5 ft nrthward Δ x = 18.6 ft eastward 4. Shadw Psitin n Autumnal Equinx 4 Hurs After Slar Nn δ = 0 ο ϕ = 41.8 ο ω = 60 ο α = 21.9 ο Α = ο Α' = Α 180 ο = 69.0 ο Shadw Length = 59.7 ft Δ y = 21.5 ft nrthward Δ x = 55.8 ft eastward