N S T I T U T D ' A E R O N O M I E S P A T I A L E D E B E L G I Q U E 3 • A v e n u e C i r c u l a i r e
B ' 1 1 8 0 B R U X E L L E S
A E R O N O M I C A A C T A
A - N° 249 - 1982
The oblateness effect on the extraterrestrial solar radiation
by
E. VAN HEMELRIJCK
B E L G I S C H I N S T I T U U T V 0 0 R R U . I M T E - A E R 0 I M 0 M I E
. 3 • R i n g l a a n B - 1 1 8 0 B R Ü S S E L
FOREWORD
The paper entitled "The oblateness effect on the extra- terrestrial solar radiation" will be published in Solar E n e r g y , 1982.
A V A N T - P R O P O S
L'article intitulé H'The oblateness effect on the extraterrestrial solar radiation" sera publiée dans Solar E n e r g y , 1982.
(
a )
VOORWOORD
De tekst "The oblateness effect on the extraterrestrial solar radiation" zal in het tijdschrift Solar Energy, 1982 verschijnen.
VORWORT
Der Text "The oblateness effect on the extraterrestrial solar radiation" wird in Solar Energy, 1982 herausgegeben werden.
T H E O B L A T E N E S S E F F E C T O N T H E E X T R A T E R R E S T R I A L S O L A R R A D I A T I O N
b y
E. V A N H E M E L R I J C K
A B S T R A C T
Calculations of the daily solar radiation incident at the top of the E a r t h ' s atmosphere, with and without the effect of the o b l a t e n e s s , are p r e s e n t e d in a f i g u r e illustrating the seasonal and latitudinal v a r i a - tion of the ratio of both insolations. It is s h o w n that, in summer, the daily insolation of an oblate Earth is s l i g h t l y increased in two r e g i o n s symmetric with respect to the summer solstice. In w i n t e r , the flattening effect r e s u l t s in a somewhat more e x t e n s i v e polar r e g i o n , the solar e n e r g y i n p u t being always r e d u c e d ( i n some cases b y more than 2 per c e n t ) when compared to a spherical one. In addition, we also numerical-
ly s t u d i e d the mean daily solar radiation. It is f o u n d that the mean summer daily insolation is s c a r c e l y increased between the equator a n d the s u b s o l a r point, b u t decreased poleward of the above mentioned
limit. In w i n t e r , h o w e v e r , the mean daily insolation is always r e d u c e d , the maximum loss of insolation attaining as much as 1 per cent in the 55-85° latitude interval. T h e partial g a i n of the mean summertime i n - solation being much smaller than the reduction d u r i n g winter season e v i d e n t l y yields a mean annual daily insolation which is d e c r e a s e d , maximally b y about 0 . 3 per cent, at all latitudes.
RESUME
L'insolation d i u r n e au sommet de l'atmosphère de la T e r r e est calculée, d ' u n e p a r t , en assimilant la T e r r e à une s p h è r e , d ' a u t r e p a r t , en t e n a n t compte de son aplatissement. Les résultats sont présentés dans une f i g u r e , i l l u s t r a n t les variations saisonnières et latitudinales du r a p p o r t des deux insolations. On montre qu'en é t é , l'aplatissement donne lieu à une insolation légèrement s u p é r i e u r e dans deux régions symétriques par r a p p o r t au solstice d ' é t é . En h i v e r , l ' e f f e t de l'apla- tissement implique une région polaire faiblement é t e n d u e , l'insolation d i u r n e d ' u n e T e r r e aplatie s ' a v é r a n t toujours plus petite ( d a n s un
r a p p o r t parfois s u p é r i e u r à 2 pour c e n t ) comparée au cas d ' u n e T e r r e s p h é r i q u e . Nous avons également étudié numériquement l'insolation d i u r n e moyenne. Il en résulte que l'insolation d i u r n e moyenne en été augmente légèrement pour des latitudes comprises e n t r e l'équateur et le point subsolaire, mais diminue pour des latitudes s u p é r i e u r e s . En h i v e r , l'insolation d i u r n e moyenne est toujours moindre, la réduction maximale a t t e i g n a n t une v a l e u r approximative de 1 pour cent dans un i n t e r v a l l e de latitude allant de 55 à 8 5 ° . Le gain d'insolation d i u r n e moyenne en été étant, beacoup plus petit que la p e r t e en h i v e r , il s'ensuit que l'insolation d i u r n e moyenne annuelle est r é d u i t e à toutes les latitudes ( d e l ' o r d r e de 0 . 3 pour cent au maximum).
S A M E N V A T T I N G
B e r e k e n i n g e n van de dagelijkse zonnestraling aan de r a n d van de aardse atmosfeer, met en zonder a f p l a t t i n g s e f f e c t , werden v o o r g e - steld in een f i g u u r die de seizoens- en b r e e d t e v e r a n d e r i n g e n van de v e r h o u d i n g van beide zonnestralingen w e e r g e v e n .
Er w e r d aangetoond d a t , in de zomer, de dagelijkse zonne- s t r a l i n g van een afgeplatte Aarde lichtjes verhoogt in twee gebieden symmetrisch gelegen ten opzichte van het zomersolstitium. In de w i n t e r . r e s u l t e e r t het a f p l a t t i n g s e f f e c t in een iets u i t g e s t r e k t e r poolgebied en de dagelijkse zonnestraling b l i j k t steeds kleiner te zijn ( i n sommige gevallen met meer dan 2%) v e r g e l e k e n met deze van een sferische A a r d e .
Er werd eveneens een numerieke studie gemaakt van de g e - middelde dagelijkse zonnestraling waarbij werd gevonden dat deze in de zomer een weinig verhoogt tussen de evenaar en het subsolaire p u n t , maar v e r m i n d e r t naar de pool toe. In de w i n t e r echter w o r d t de g e - middelde dagelijkse zonnestraling g e r e d u c e e r d , maximum met meer dan 1% in het geocentrische b r e e d t e - i n t e r v a l 5 5 - 8 5 ° . De gedeeltelijke winst van de gemiddelde zomerse zonnestraling is echter heel wat kleiner dan het verlies in het winterseizoen zodat de gemiddelde dagelijkse zonne- straling genomen over een gans jaar kleiner is (maximaal met ongeveer 0.3%) op alle b r e e d t e n .
- 3 -
ZUSAMMENFASSUNG
Berechnungen der täglichen Sonnenstrahlung am Rand d e r Erdatmosphäre mit und ohne E f f e k t der A b p l a t t u n g , sind v o r g e s t e l l t in einer A b b i l d u n g , die die Jahreszeitlichen- und B r e i t e n v a r i a t i o n e n des Verhältnisses der beiden Sonnenstrahlungen d a r s t e l l e n .
Es w u r d e gefunden dass, im Sommer, die tägliche Sonnen- s t r a h l u n g einer abgeplattenen Erde ein wenig zunimmt in zwei G e b i e t e n , die symmetrisch liegen mit Rücksicht auf dem Sommersolstitium. Im Winter r e s u l t i e r t das A b p l a t t u n g s e f f e k t in einem ausgedehnten Polgebiet und est e r g i b t sich dass die tägliche Sonnenstrahlung auf einer abge- plattenen Erde immer weniger ist als diese auf einer sphärischen Erde (bisweilen mit mehr dann 2%).
Es w u r d e auch eine numerische Analyse der M i t t l e r e n t ä g - lichen Sonnenstrahlung d u r c h g e f ü h r t wobei wir g e f u n d e n haben, dass im Sommer die mittlere tägliche Sonnenstrahlung e r h ö h t zwischen dem Ä q u a t o r und der S u b s o l a r p u n k t , aber abnimmt nach dem Pol. Im Winter w u r d e die mittlere tägliche Sonnestrahlung r e d u z i e r t mit einem Höchst- w e r t von 1% in dem 5 5 - 8 5 ° B r e i t e n i n t e r v a l l . Die teilweise Gewinn der Sonnestrahlung im Sommer ist aber merklich kleiner als der V e r l u s t w ä h r e n d der Wintersaison, so dass die mittlere tägliche Sonnenstrahlung berechnet über ein ganzes Jahr als Funktion der B r e i t e immer kleiner ist wobei der Höchstwert u n g e f ä h r 0.3% b e t r a g t .
I N T R O D U C T I O N
In studies involving the d i s t r i b u t i o n of solar radiation inci- d e n t at t h e Earth's surface and its v a r i a b i l i t y with latitude and season, t h e solar e n e r g y reaching t h e top of t h e atmosphere constitutes a v e r y important i n p u t d a t a . T h i s amount of solar r a d i a n t f l u x is governed b y two factors : on one h a n d , t h e solar f l u x as a function of t h e orbital distance a n d , on the other h a n d , t h e cosine of t h e solar zenith- angle.
Considering more p a r t i c u l a r l y t h e latter p a r a m e t e r , it should be emphasized t h a t f o r a planet assumed to be spherical, t h e zenith distance may be expressed as a simple function of t h e l a t i t u d e , t h e solar declination and the local hour angle of the S u n , t h e radius vector being normal to t h e horizon plane. For an oblate p l a n e t , h o w e v e r , t h i s is not t h e case since in general t h e r e is an angle between t h e two directions mentioned above. T h i s so-called angle of the vertical depends upon t h e f l a t t e n i n g b u t vanishes at t h e equator and t h e poles.
Although t h e u p p e r - b o u n d a r y insolation of t h e Earth's atmo- sphere [see e . g . 1 - 2 ] and the radiation incident on its surface [see e . g . 3 - 8 ] have been t r e a t e d e x t e n s i v e l y in recent y e a r s , it must be pointed out t h a t , to t h e best of our knowledge, the effect of t h e oblateness on the e x t r a t e r r e s t r i a l solar radiation has never been discussed at least on a q u a n t i t a t i v e basis. T h i s p a p e r , t h e r e f o r e , is an attempt to investigate in detail the influence of t h e f l a t t e n i n g on t h e radiation which would be incident in t h e absence of any atmosphere.
Our results are presented in the form of a contour map given t h e seasonal and latitudinal variation of the ratio of the daily insolation with and without t h e effect of t h e oblateness. For t h e sake of com- pleteness we have included a diagram showing the percentage d i f f e r e n c e of the mean (summer, winter and a n n u a l ) daily insolations as a function of l a t i t u d e .
- 5 -
1. D A I L Y I N S O L A T I O N W I T H A N D W I T H O U T T H E O B L A T E N E S S EFFECT
T h e instantaneous insolation I at the u p p e r - b o u n d a r y of t h e Earth can be expressed as : see f o r instance [ 9 - 1 1 ]
w i t h : and :
I = S cos 6 o _ z S = S / r
o s _
r = a ( 1 - e ) / ( 1 + e cos W)
(1) (2) ( 3 )
where 8 is t h e zenith angle of the incident r a d i a t i o n , S is t h e solar f l u x at an heliocentric distance r and S is the solar constant at the s o _„
mean S u n - E a r t h distance of 1 AU t a k e n at 1368 W m" [ 1 2 ] . F u r t h e r - more, in expression ( 3 ) , ag, e and W are respectively the Earth's semi-major a x i s , t h e eccentricity and t h e t r u e anomaly which is given by :
where A. and \p are the planetocentric longitude of the Sun and the planetocentric longitude of t h e Earth's perihelion. T h e numerical values of t h e parameters used for t h e computations are listed in T a b l e 1. In this table one can f i n d also the obliquity e , the equatorial radius ag, t h e polar radius ap, t h e f l a t t e n i n g f = ( ae - ap) / ae and t h e sidereal period of axial rotation T (sidereal d a y ) . Note t h a t the elements of t h e p l a n e t a r y o r b i t and t h e dimensions of the Earth are t a k e n from the Handbook of t h e B r i t i s h Astronomical Association [13] and from V o r o b ' y e v and Monin [ 1 ] .
w = \ - ( 4 )
For a spherical planet, t h e zenith angle %z may be expressed as :
TABLE 1.- Elements of the planetary orbit and dimensions of the Earth
a e e a a f T s P e p
(AU) (°) (°) (km) (km) (Earth days)
1 0.01672 282.05 2 3 . 4 5 6378 6357 0.00329 1
cos 0 = sin (b1 sin 6 + cos d>' cos 6 cos ID
z s T s s ( 5 )
where <J>' is the geocentric latitude (which in this case, equals the geographic latitude <J>), 6g is the solar declination and u)g is the hour angle of the S u n . Furthermore, 6g can be calculated using the following expression :
sin 6 = sin e sin A (6) s s
T h e daily insolation H , being defined as the amount of in- coming solar radiation at the top of the Earth's atmosphere over the planet's day, can now be obtained by integrating expression ( 1 ) over daytime assumed lu be equal to the time that elapses between rising and setting of the S u n ; we obtain :
H = (ST/n) (u) sin (J)o s s s s s s 1 sin 6 + sin UJ cos <t>' cos 6 ) (7)
where u)gs is the sunset hour angle and may be determined from expression ( 5 ) by the condition that at sunset cos 0z = 0. It follows that :
U) = arc cos (- tan 6 tan (J)') (8) ss s
if | <T| < n/2 - |5g ]
In regions where the Sun does not rise (<)>' < - n/2 + 6g or
<j>' > n/2 + 6s) we have u)gs = 0; in regions where the Sun remains above the horizon all day (<t>' > n/2 - 6g or <f>' < - n/2 - 6g) we may put u>ss = n.
In the case of an oblate planet ( F i g . 1 ) , there is an angle v , the so-called angle of the vertical, between the radius vector and the
CO
Normal to horizon plane
Fig. 1.- Representation of some parameters involved with the oblate calcu- lations.
normal to the horizon plane; as already mentioned in the introduction, it vanishes at the equator and the poles while elsewhere <|> > (j)1 numeri- cally. The angle v is dependent upon the geocentric latitude <|>* and the flattening f by the relationship :
v = arctan [ ( 1 - f ) ~2 tan <K] - <t>' (9)
Defininq 6 as the zenith distance for an oblate planet, the z,o following relation can easily be found by applying the formulas of spherical trigonometry :
cos 0 = cos v cos 6 + s i n v ( - tan d)* cos 8 + s i n 6 sec <)>') z , o z z s (10)
The daily insolation of an oblate planet Hq can now be obtained by integrating expression ( 1 ) , within the appropriate time limits, where cos ©z has to be replaced by relation (10) yielding :
H = (ST/71) {cos v (u) s i n (j)' s i n 6 + s i n u) cos <t>' cos 6 ) + o,o s s , o s SS,o s
s i n v [ - tan d>* (u) s i n <K s i n 6 + s i n u> cos <J>' cos 6 ) + s s , o s s s , o £>
U) s i n 6 sec <t>' ]} ( 1 1 ) ss ,o s
where uj , the local hour angle at sunset for an oblate planet, is ss,o generally slightly different form u)s s. As for a spherical planet, u> Q
may be derived from relation (10) by putting cos 0 = 0. After some rearrangements, and as aspected, the expression of u> Q in terms of 6 and <b is found to be similar to formula (8) and can be expressed* as
s T
U) = arc cos ( - tan 6 tan <|>) ( 1 2 ) s s , o s
T a k i n g i n t o a c c o u n t e x p r e s s i o n s ( 7 ) a n d ( 1 1 ) , t h e r a t i o H / H a t t h e t o p of t h e E a r t h can now e a s i l y be d e t e r m i n e d as a
0 , 0 o r
f u n c t i o n of g e o c e n t r i c l a t i t u d e <J>' a n d s o l a r l o n g i t u d e Ag.
2 . D I S C U S S I O N O F T H E R A T I O D I S T R I B U T I O N O F T H E D A I L Y I N S O - L A T I O N S
In a p r e v i o u s w o r k [ 1 4 ] , d e a l i n g w i t h t h e o b l a t e n e s s e f f e c t o n t h e solar r a d i a t i o n i n c i d e n t a t t h e t o p of t h e a t m o s p h e r e s of t h e o u t e r p l a n e t s J u p i t e r , S a t u r n , U r a n u s a n d N e p t u n e , we s t u d i e d q u a l i t a t i v e l y some c h a r a c t e r i s t i c f e a t u r e s of t h e r a t i o d i s t r i b u t i o n H / H in t h e 0 , 0 o n o r t h e r n h e m i s p h e r e b o t h f o r summer ( 0 < A. < 1 0 0 ° ) a n d w i n t e r ( 1 0 0 °
< A. < 3 6 0 ° ) p e r i o d . I t s h o u l d , h o w e v e r , be e m p h a s i z e d t h a t t h e r e s u l t s p r e s e n t e d a r e also v a l i d f o r t h e s o u t h e r n h e m i s p h e r e a n d t h a t t h e y a r e e v i d e n t l y a p p l i c a b l e to t h e E a r t h . T h e f o l l o w i n g a r e t h e major c o n c l u s i o n t h a t w e r e r e a c h e d :
( 1 ) In summer a n d in t h e r e g i o n of p e r m a n e n t s u n l i g h t ( ws s = uu = 7t) t h e i s o c o n t o u r s H / H p a r a l l e l t h e lines of c o n s t a n t g e o -
s s , o 0 , 0 o
c e n t r i c l a t i t u d e <t>' a n d t h e d a i l y solar r a d i a t i o n Hq is a l w a y s g r e a t e r t h a n H . T h e maximum v a l u e of t h e r a t i o H^ J H , o c c u r i n g a t <|>* =
o 0 , 0 0 n f 2 - e, can be e x p r e s s e d b y t h e f o l l o w i n g r e l a t i o n s h i p :
_ 2
(H /H ) = sin { arctan [(1 - f) cotan e]}/cos e (13) 0 , 0 o max
( 2 ) I n summer a n d in t h e e q u a t o r i a l r e g i o n limited b y t h e seasonal m a r c h of t h e S u n , t h e solar r a d i a t i o n of an o b l a t e p l a n e t Hq q
is i n c r e a s e d w i t h r e s p e c t to t h e i n s o l a t i o n of a s p h e r i c a l one H .
( 3 ) For l a t i t u d e s b e t w e e n t h e s u b s o l a r p o i n t a n d t h e r e g i o n w h e r e t h e S u n r e m a i n s a b o v e t h e h o r i z o n all d a y , t h e r a t i o cos
6 / c o s 0 is d e c r e a s i n g ( w i t h cos 0 , „ < cos 0 ) , w h e r e a s u> is z , o z z , o z s s , o ss
i n c r e a s i n g ( w i t h t h e c o n d i t i o n t h a t u) > uu ) . W h e t h e r o r how t h e SSIU bb
- 1 1 -
regions mentioned in ( 1 ) and ( 2 ) are linked depends on t h e r e l a t i v e effect of those two ratios, both being function of t h e f l a t t e n i n g and t h e o b l i q u i t y , and can only be determined by computation of t h e expression
H / H .
0 , 0 o
( 4 ) In w i n t e r , t h e e f f e c t of t h e f l a t t e n i n g results in a more extensive polar r e g i o n , the insolation is always reduced ( H Q < HQ) and the c u r v e s of constant ratio H / H r o u g h l y parallel t h e b o u n d a r y U ƒ u u of t h e polar n i g h t e x e p t in the neighborhood of t h e equinoxes.
Application of expressions ( 7 ) and ( 1 1 ) leads to the iso- contour map illustrated in Fig. 2 . , where values of constant ratio d i s t r i b u t i o n H / H are given on each c u r v e . Solar declination (lower
0 , 0 0
p a r t ) is indicated by the d o t t e d - d a s h e d line and t h e region where t h e Sun does not set ( u p p e r p a r t ) is r e p r e s e n t e d b y t h e dashed line. T h e area of permanent darkness is shaded. F i n a l l y , t h e two regions of enhanced solar radiation ( H > H ) are d o t t e d . O f o u
From Fig. 2 it can be seen t h a t in the region of permanent sunlight the incoming solar radiation Hq q is increased when compared to H . F u r t h e r m o r e , it follows from expression ( 1 3 ) t h a t in t h e region considered ( H / H ) = 1.00104 0.1%). T h i s extremely small gain
0 , 0 o max
of insolation is mainly due to the even small value of the Earth's f l a t - t e n i n g . For comparison, it is i n s t r u c t i v e to note t h a t (H 0 > c/H 0)m a x' i s equal to 1.00033, 1 . 0 3 7 , 1.124 and 1.010 f o r J u p i t e r , S a t u r n , U r a n u s and Neptune r e s p e c t i v e l y . T h e effect of parallelism between t h e iso- contours H / H and t h e lines of constant geocentric latitude <t>* in t h e
0 , 0 o
above mentioned zone and already pointed out previously is not illustrated in Fig. 2. T h e reason for this n o n - r e p r e s e n t a t i o n lies in t h e fact t h a t the numerical value of H / H is negligible small ( 1 . 0 0 0 8 and OfU u 1.0002 at <)>' = 70 and 80° r e s p e c t i v e l y ) .
It can mathematically be proved t h a t , in summer, in t h e
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
SOLAR LONGITUDE (degrees)
Fie. 2.- Seasonal and latitudinal variation of the ratio H /H of the daily
° o, o o
insolation with (H ) and without (H ) the oblateness effect at the o ,o o
top of the atmosphere of the Earth. Solar declination (lower p a r t ) is indicated by the dotted-dashed line and the region where the Sun does not set (upper p a r t ) is represented by ~he dashed line. The area of permanent darkness is shaded. The two regions of enhanced solar radia- tion (H > H ) are dotted. Values of constant ratio distribution
o,o o
H /H are given on each curve.
o,o o
t h e l e n g t h of t h e d a y a n d t h e cosine o f t h e z e n i t h d i s t a n c e a r e e n - h a n c e d b y t h e e f f e c t o f t h e f l a t t e n i n g . H e n c e , i t f o l l o w s t h a t in t h i s p a r t i c u l a r r e g i o n , t h e solar e n e r g y i n p u t at t h e t o p o f t h e a t m o s p h e r e o f t h e E a r t h assumed as an o b l a t e is i n c r e a s e d w i t h r e s p e c t t o t h e i n s o l a t i o n o f a s p h e r i c a l E a r t h .
I n c o n c l u s i o n , t h e r e e x i s t s t w o o b v i o u s l y d i s t i n g u i s h e d zones w h e r e t h e u p p e r - b o u n d a r y i n s o l a t i o n Hq q is g r e a t e r t h a n Hq. For t h e sake o f c l e a r n e s s i t has t o be e m p h a s i z e d t h a t t h e e n h a n c e m e n t is s i g n i f i c a n t l y s m a l l .
F i a . 2 also r e v e a l s t h a t t h e t w o zones w h e r e H > H a r e
a 0 , 0 0
n o t l i n k e d . I n d e e d , in t h e m i d - l a t i t u d e i n t e r v a l 4 5 - 6 0 ° , i t is o b v i o u s t h a t t h e e f f e c t of t h e o b l a t e n e s s causes t h e s o l a r e n e r g y o u t s i d e t h e E a r t h ' s a t m o s p h e r e t o be r e d u c e d , t h e loss of i n s o l a t i o n d e p e n d i n g m a r k e d l y u p o n t h e s o l a r l o n g i t u d e . For e x a m p l e , i f , at <J>' = 6 0 ° , A.g
i n c r e a s e s f r o m 0 t o a p p r o x i m a t e l y 5 0 ° , t h e r a t i o HQ O/ Hq i n c r e a s e s f r o m 0.995 0.5%) t o a b o u t 0.999 O 0.1%). T h e n o n - l i n k a g e of t h e a b o v e m e n t i o n e d areas is m a i n l y a s c r i b e d t o t h e e x p l i c i t e l y small v a l u e of t h e E a r t h ' s f l a t t e n i n g ( 0 . 0 0 3 2 9 ) .
As f o r all p l a n e t s [ 1 4 ] , t h e p o l a r r e g i o n is e x t e n d e d . I t s h o u l d , h o w e v e r , be emphasized t h a t t h e A r c t i c C i r c l e s H = 0 a n d H = 0 p r a c t i c a l l y c o i n c i d e , t h e maximum d i f f e r e n c e a t t a i n i n g s c a r c e l y
0 . 1 4 ° at a s o l a r l o n g i t u d e of 2 7 0 ° . T h i s v a l u e is c o n s i d e r a b l y l o w e r t h a n t h o s e f o r t h e o t h e r p l a n e t s v a r y i n g f r o m a b o u t 0 . 4 ( J u p i t e r ) t o a p p r o x - i m a t e l y 5 ° ( S a t u r n ) . I t is c l e a r t h a t t h i s f i n d i n g is also a s c r i b e d t o t h e small v a l u e o f f .
F i g . 2 r e v e a l s t h a t i n w i n t e r , as s t a t e d e a r l i e r , t h e i n s o l a t i o n of an o b l a t e E a r t h is a l w a y s r e d u c e d w h e n c o m p a r e d t o a s p h e r i c a l o n e . T h e solar r a d i a t i o n v e r y s l o w l y d e c r e a s e s in p a s s i n g f r o m e q u a t o r
l a t i t u d e s t o m i d l a t i t u d e s ; i t is f o u n d t h a t a t w i n t e r s o l s t i c e a n d i f (J)'
O 0.1%) to approximately 0.980 2%). At higher latitudes, especially near the region where the Sun does not rise, Hq 0/ Hq drops v e r y rapidly to zero. Although the loss of insolation is not v e r y striking it follows from Fig. 2 that for parts of the winter the incident solar radiation is decreased by more than 2%.
Another point about the curves is that they roughly parallel the limit of the polar night exept, of course, in the vicinity of the equinoxes. Finally, it is also suitable to remark that in summer, respectively in winter, the curves of constant ratio HQ 0/ Hq are per- fectly symmetric with respect to the summer and winter solstices.
3. D I S C U S S I O N OF T H E MEAN D A I L Y I N S O L A T I O N S
In the preceeding sections we investigated the oblateness effect on the extraterrestrial daily insolation. Here we discuss the influence of the flattening on the mean (summer, winter and annual) daily insolations.
In Fig. 3 this influence is plotted in terms of the percentage difference [100 (HQ - HO) / Hq] as a function of geocentric latitude, the bars over symbols signifying seasonal or annual averages.
Concerning more particularly the mean summertime insolation (the summer season being arbitrary defined as the period running from vernal equinox over summer solstice to autumnal equinox and spanning 180°), the influence of the oblateness, although very small, is obviously evident from Fig. 3. Indeed, in the latitude interval 5 0 - 6 0 ° it can be seen that about 0.15% of the mean summer daily insolation is lost through the flattening effect. Outside this interval, the effect is of decreasing significance. Another interesting phenomenon is that for latitudes between the equator and the subsolar point [ 1 4 - 1 5 ] , the mean daily summer insolation of an oblate Earth is increased. However, owing
-15-
30 AO 50 60 70 LATITUDE (degrees)
F i g . 3.- L a t i t u d i n a l v a r i a t i o n of the p e r c e n t a g e difference [100 ( H o o - Ho) / Hq] of the m e a n (summer, w i n t e r , a n n u a l ) daily i n s o l a t i o n s . The bars over
to the small f l a t t e n i n g , the rise of insolation is practically n e g l i g i b l e , r e a c h i n g only a maximum value of about 0.03% at a latitude of a p p r o x i - mately 10°.
D u r i n g winter season (180° < A. < 360°), as already stated p r e v i o u s l y , the daily insolation Hq is always r e d u c e d . C o n s e q u e n t l y , H < H at a n y latitude. F i g . 3 reveals that in winter the loss of o,o o 1 a
insolation is of most importance between 55 and 85° where a p e r c e n t a g e difference as much as 1% has been f o u n d with a maximum value of about 1.3% near 70°.
T h e partial gain of the mean summertime insolation e q u a t o r - ward uf the s u b s o l a r point being c o n s i d e r a b l y lower than the loss of insolation o v e r the same period in winter e v i d e n t l y r e s u l t s in a mean annual daily insolation which is r e d u c e d o v e r the entire latitude interval as illustrated in F i g . 3. It can be seen that the daily insolation a v e r a g e d o v e r a one y e a r period is decreased b y about 0.3% at latitudes from 45 to 65°.
4. C O N C L U D I N G R E M A R K S
In the p r e s e n t paper we have investigated the influence of the flattening on the solar e n e r g y i n p u t at the top of the E a r t h ' s atmo- s p h e r e . A s a result of this s t u d y one can d r a w n the conclusion that the effect of the oblateness c a u s e s n o n - n e g l i g i b l e , a l t h o u g h relatively small, v a r i a t i o n s in both the p l a n e t a r y - w i d e d i s t r i b u t i o n and the intensity of the extraterrestrial daily solar radiation.
In summer, the daily insolation is s l i g h t l y increased in two o b v i o u s l y d i s t i n g u i s h e d r e g i o n s : the f i r s t , near the poles, c o i n c i d i n g practically with the area of permanent s u n l i g h t , the s e c o n d , at lower latitudes, c o v e r i n g the region between the equator and the seasonal march of the S u n . T h e maximum increase of the incoming solar r a d i a -
-17-
t i o n , occuring at a geocentric latitude ( n / 2 - e ) , is approximately equal to 0.1%. Outside t h e two zones mentioned above, t h e d i r e c t solar r a d i a - tion H is decreased when compared to H . For example, in t h e
0 , 0 o neighborhood of t h e equinoxes the loss of insolation ranges from 0 . 1 to
0.5% ( e x e p t at equator l a t i t u d e s ) , whereas elsewhere the oblateness e f f e c t on the e x t r a t e r r e s t r i a l solar radiation is negligible small.
In w i n t e r , the influence of t h e f l a t t e n i n g causes t h e polar region to enlarge over an extremely small distance of about 15 km at w i n t e r solstice. T h e insolation is always r e d u c e d , the rate of decrease depending to a large e x t e n t on the geocentric l a t i t u d e . For comparison, at A = 2 7 0 ° , t h e loss of solar e n e r g y amounts to about 0 . 1 , 0 . 5 , 1 . 0 and 2.0% respectively at latitudes of approximately 10, 30, 45 and 5 0 ° . Moreover, in t h e r e l a t i v e l y small area limited by the isocontour H / H = 0.980 and the region of permanent darkness t h e effect of t h e
0 , 0 o
f l a t t e n i n g plays a more significant role and the decrease of incident solar radiation .is correspondingly g r e a t e r . F u r t h e r m o r e , it is p a r t i c u - larly e v i d e n t from Fig. 2 t h a t t h e c u r v e s of constant ratio HQ 0/ HQ
r o u g h l y parallel t h e A r c t i c Circle bounding the polar region in which t h e r e are days without s u n r i s e .
Finally, we also have studied the latitudinal variation of t h e percentage d i f f e r e n c e of the mean daily insolation. I t is found t h a t f o r latitudes between t h e equator and t h e subsolar p o i n t , t h e mean summer daily insolation of an oblate Earth is increased when compared to a spherical one, the maximum rise being extremely small 0.03%). A t higher latitudes, t h e r e is a loss of insolation which is of most impor- tance at midlatitude regions 0.15%).
In w i n t e r , the horizon plane is always tilled away from the Sun causing both t h e cosine of the zenith angle and t h e length of the day to be r e d u c e d . C o n s e q u e n t l y , t h e daily insolation as well as the mean daily insolation are r e d u c e d , the latter decreasing maximally b y
Despite of t h e partial gain of the mean summertime insolation near t h e e q u a t o r , t h e effect of t h e f l a t t e n i n g can clearly be seen to reduce t h e mean daily insolation over t h e e n t i r e y e a r .
In conclusion, as i n t e r e s t in applications of solar e n e r g y increases considerably over t h e last y e a r s , v e r y accurate values of t h e e n e r g y i n p u t at t h e top of t h e Earth's atmosphere are absolutely needed. We, t h e r e f o r e , believe t h a t the effect of the oblateness, although v e r y small, has to be taken into account in studies related to theoretical models f o r t h e calculation of solar global insolation.
-19-
ACKNOWLEDGEMENTS
I s h o u l d l i k e t o t h a n k J . S c h m i t z a n d F. V a n d r e c k f o r t h e r e a l i s a t i o n o f t h e t w o i l l u s t r a t i o n s .
N O M E N C L A T U R E
a e e q u a t o r i a l r a d i u s a p p o l a r r a d i u s a s s e m i - m a j o r a x i s e e c c e n t r i c i t y f f l a t t p n i n g
H o e x t r a t e r r e s t r i a l d a i l y i n s o l a t i o n o f a s p h e r i c a l E a r t h H o o e x t r a t e r r e s t r i a l d a i l y i n s o l a t i o n o f an o b l a t e E a r t h
H o e x t r a t e r r e s t r i a l mean d a i l y i n s o l a t i o n o f a s p h e r i c a l E a r t h H o , o e x t r a t e r r e s t r i a l mean d a i l y i n s o l a t i o n of an o b l a t e E a r t h
1 o e x t r a t e r r e s t r i a l i n s t a n t a n e o u s i n s o l a t i o n r s h e l i o c e n t r i c d i s t a n c e
T s i d e r e a l p e r i o d o f a x i a l r o t a t i o n
V a n g l e of t h e v e r t i c a l
S s o l a r f l u x at an h e l i o c e n t r i c d i s t a n c e rg S o solar c o n s t a n t (1368 Wm" )
W t r u e anomaly
G r e e k s y m b o l s
Ô s solar d e c l i n a t i o n e o b l i q u i t y
0 z z e n i t h a n g l e f o r a s p h e r i c a l E a r t h
0 Z - O z e n i t h a n g l e f o r an o b l a t e E a r t h
k s p l a n e t o c e n t r i c solar l o n g i t u d e
XP p l a n e t o c e n t r i c l o n g i t u d e o f t h e E a r t h ' s p e r i h e l i o n g e o g r a p h i c l a t i t u d e
geocentric latitude hour angle of the Sun
sunset hour angle (spherical Earth) sunset hour angle (oblate Earth)
- 2 1 -
REFERENCES
1. V . I . V O R O B ' Y E V a n d A . S . M O N I N , U p p e r - b o u n d a r y i n s o l a t i o n o f t h e a t m o s p h e r e s o f t h e p l a n e t s o f t h e s o l a r s y s t e m s . A t m . a n d Oceanic P h y s . 11, 557 ( 1 9 7 5 ) .
2 . N . R O B I N S O N , Solar R a d i a t i o n , E l s e v i e r , New Y o r k ( 1 9 6 6 ) .
3 . B . H . Y . L I U a n d R . C . J O R D A N , T h e i n t e r r e l a t i o n s h i p a n d c h a r a c t e r i s t i c d i s t r i b u t i o n o f d i r e c t , d i f f u s e a n d t o t a l s o l a r r a d i a - t i o n . Solar E n e r g y 4 , 1 ( 1 9 6 0 ) .
4 . D . V . H O Y T , A model f o r t h e c a l c u l a t i o n o f s o l a r g l o b a l i n s o l a t i o n . Solar E n e r g y 21, 27 ( 1 9 7 8 ) .
5 . J . SCHMETZ a n d E. R A S C H K E , A m e t h o d t o p a r a m e t e r i z e t h e d o w n w a r d solar r a d i a t i o n at g r o u n d . A r c h . M e t . G e o p h . B i o k l . B 2 6 , 143 ( 1 9 7 8 ) .
6 . S. B A R B A R O , S. C O P P O L I N O , C . LEONE a n d E. S I N A G R A , A n a t m o s p h e r i c model f o r c o m p u t i n g d i r e c t a n d d i f f u s e s o l a r r a d i a t i o n Solar E n e r g y 22, 225 ( 1 9 7 9 ) .
7 . M. C O L L A R E S - P E R E I R A a n d A . R A B L , T h e a v e r a g e d i s t r i b u t i o n o f s o l a r r a d i a t i o n - c o r r e l a t i o n s b e t w e e n d i f f u s e a n d h e m i s p h e r i c a l a n d b e t w e e n d a i l y a n d h o u r l y i n s o l a t i o n v a l u e s . Solar E n e r g y 22, 155, 1979.
8 . B . G O L D B E R G , W . H . K L E I N a n d R . D . M c C A R T N E Y , A c o m p a r i s o n o f some simple models u s e d t o p r e d i c t s o l a r i r r a d i a n c e on a h o r i - z o n t a l s u r f a c e . Solar E n e r g y 2 3 , 81 ( 1 9 7 9 ) .
9 . W . R . W A R D , C l i m a t i c v a r i a t i o n s on M a r s . 1. A s t r o n o m i c a l t h e o r y o f i n s o l a t i o n . J . G e o p h y s . Res. 79, 3375 ( 1 9 7 4 ) .
10. J . S . L E V I N E , D . R . KRAEMER a n d W . R . K U H N , Solar r a d i a t i o n i n c i d e n t on Mars a n d t h e o u t e r p l a n e t s : l a t i t u d i n a l , seasonal a n d a t m o s p h e r i c e f f e c t s . I c a r u s 31, 136 ( 1 9 7 7 ) .
11. E. V A N HEMELRIJCK a n d J . V E R C H E V A L , Some a s p e c t s o f t h e solar r a d i a t i o n i n c i d e n t at t h e t o p of t h e a t m o s p h e r e s o f M e r c u r y a n d V e n u s I c a r u s 48, 167 ( 1 9 8 1 ) .
12. R . C . WILLSON, Solar i r r a d i a n c e v a r i a t i o n s and solar a c t i v i t y . J . G e o p h y s . Res. 87, 4319 ( 1 9 8 2 ) .
13. Handbook of t h e B r i t i s h Astronomical A s s o c i a t i o n , Sumfield and Day L t d , p . 100-101, E a s t b o u r n e , East Sussex ( 1 9 8 0 ) .
14. E. V A N HEMELRIJCK, T h e oblateness e f f e c t on t h e solar r a d i a t i o n i n c i d e n t at t h e t o p of t h e atmospheres of t h e o u t e r p l a n e t s . I c a r u s , to be p u b l i s h e d (1982).
15. A . W . B R I N K M A N and J. McGREGOR, T h e e f f e c t of t h e r i n g system on t h e solar r a d i a t i o n r e a c h i n g t h e top of S a t u r n ' s atmosphere : d i r e c t r a d i a t i o n . Icarus 38, 479 (1979).
-23-
