I N S T I T U T D ' A E R 0 l\l 0 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° 248 - 1982
The oblateness effect on the solar radiation incident at the top of the atmosphere of Mars
by
E. VAN HEMELRIJCK
B E L G I S C H I N S T I T U U T V O O R R U I M T E - A E R O N O M I E 3 - Ringlaan
F O R E W O R D
The paper entitled " T h e oblateness effect on the solar radiation incident at the top of the atmosphere of M a r s " has been presented at the symposium " T h e planet M a r s " held at the University of Leeds,
E n g l a n d , A u g u s t 27, 1982 as a part of the European Geophysical Society Meeting. It will be published in the Proceedings ( E S A Special Publica- tion SP-185) of the conference.
A V A N T - P R O P O S
L'article intitulé " T h e oblateness effect on the solar radiation inci- dent at the top of the atmosphère of M a r s " a été présenté au symposium
" T h e planet M a r s " qui s'est tenu à l'Université de Leeds, Angleterre, le 27 août 1982 d a n s le cadre de la réunion de la "Éuropean Geophysical Society". Il sera publié dans les comptes-rendus ( E S A Spécial Publica- tion SP-185) du symposium.
V O O R W O O R D
Het artikel getiteld " T h e oblateness effect on the solar radiation incident at the top of the atmosphere of M a r s " werd v o o r g e d r a g e n op het symposium " T h e planet M a r s " dat gehouden werd aan de Universiteit van Leeds, Engeland op 27 a u g u s t u s 1982 in het kader van de bijeen- komst van de "European Geophysical Society". Het zal gepubliceerd worden in de mededelingen ( E S A Special Publication SP-185) van het symposium.
V O R W O R T
Der T e x t " T h e oblateness effect on the solar radiation incident at the top of the atmosphere of M a r s " würde während des Symposiums
" T h e planet M a r s " der in Leeds im Rahmen der Sammlung des "European Geophysical Society" am 27 A u g u s t u s 1982 gehalten wurden, v o r g e t r a g e n worden. Dieses Artikel wird in die Proceedings ( E S A Special Publication SP-185) des Symposiums 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 S O L A R 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 OF M A R S
by
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 atmosphere of Mars (the outer planet with the smallest flattening
known), with and without the effect of the oblateness, are presented in a f i g u r e illustrating the seasonal and latitudinal variation of the ratio of both insolations. It is shown that for parts of the summer, the daily insolation of M a r s , assumed as an oblate planet, is slightly increased.
In winter, the flattening' effect results in a somewhat more extensive polar region, the solar e n e r g y input being always reduced ( b y more than 5 per cent near the poles) when compared to a spherical planet.
In addition, we also numerically studied the mean daily solar radiation.
It is found that the mean summer daily insolation is scarcely increased between the equator and the subsolar point, but decreased poleward of the above mentioned limit. In winter, however, the mean daily insolation is always reduced, the maximum loss of insolation attaining approximate- ly 2% in the 60-80° latitude interval. T h e partial gain of the mean summertime insolation being considerably smaller than the reduction d u r i n g winter season evidently yields a mean annual daily insolation which is decreased, maximally by about 0.5% near mid-latitudes, at all latitudes.
Résumé
L'insolation d i u r n e au sommet de l'atmosphère de Mars est calculée, d ' u n e p a r t en assimilant la planète 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 d u 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 une région symétrique 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'aplatissement implique une région polaire faiblement é t e n d u e , l'insolation d i u r n e d ' u n e planète Mars aplatie s ' a v è r e n t toujours petite (dans un r a p p o r t parfois s u p é r i e u r à 5% au voisinage des pôles) comparée à une planète 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 augmenté
légèrement pour des latitudes comprises e n t r e l ' é q u a t e u r 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 2% dans un i n t e r v a l l e de latitude allant de 60 à 8 0 ° . Le gain d'insolation d i u r n e moyenne en été étant beaucoup 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 0.5% au maximum).
Samenvatting
B e r e k e n i n g e n van de dagelijkse zonnestraling aan de rand van de atmosfeer van Mars ( d e uitwendige planeet met de kleinste a f p l a t t i n g ) , met en zonder a f p l a t t i n g s e f f e c t , worden voorgesteld 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 wordt aangetoond d a t , in de zomer, de dagelijkse zonnestraling van een afgeplatte planeet Mars lichtjes verhoogt in een gebied sym- metrisch gelegen ten opzichte van het zomersolstitium. In de winter 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 (met meer dan 5% in de nabijheid van de polen) v e r g e l e k e n met deze van een sferische planeet.
Er werd eveneens een numerieke studie gemaakt van de gemiddelde 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 wordt de gemiddelde dagelijkse zonnestraling g e r e d u c e e r d , maximaal met ongeveer 2% in het geocentrisch b r e e d t e i n t e r v a l 6 0 - 8 0 ° . De gedeeltelijke winst van de gemiddelde zomerse zonnestraling is echter heel wat kleiner dan het verlies in het winterseizoen zodat de gemiddelde dagelijkse zonnestraling genomen over een gans jaar kleiner is (maximaal met ongeveer 0.5% op gemiddelde b r e e d t e n ) op alle b r e e d t e n .
Zusammenfassung
Berechnungen der täglichen S o n n e n s t r a h l u n g am Rand der Atmos- p h ä r e des Planeten Mars ( d e r äusserer Planet mit d e r g e r i n g s t e n A b - p l a t t u n g ) , 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 g e f u n d e n dass, im Sommer, die tägliche Sonnenstrahlung eines abgeplattenen Planeten Mars ein wenig zunimmt in einem Gebiet, das symmetrisch liegt 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 es e r g i b t sich, dass die tägliche S o n n e n s t r a h l u n g auf einem abgeplattenen Planeten Mars immer weniger ist als diese auf einem sphärischen Planeten (bisweilen mit mehr dann 5%).
Es w u r d e auch eine numerische Analyse der mittleren täglichen Sonnenstrahlung d u r c h g e f ü h r t wobei wir g e f u n d e n h a b e n , 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. In Winter w u r d e die mittlere tägliche Sonnenstrahlung r e d u z i e r t mit einem Höchstwert von 2% in dem 6 0 - 8 0 ° B r e i t e n i n t e r v a l l . Die teilweise Gewinn der S o n n e n s t r a h l u n g im Sommer ist aber merklich kleiner als der V e r - lust w ä h r e n d der Wintersaison, so dass die mittlere tägliche Sonnen- s t r a h l u n g berechnet über ein ganzes Jahr als Funktion der ^Breite immer kleiner ist wobei der Höchstwert u n g e f ä h r 0.5% b e t r a g t .
1. I N T R O D U C T I O N
In studies i n v o l v i n g solar radiation problems, T h e instantaneous insolation, defined as t h e solar heat f l u x sensed at a g i v e n time by a horizontal u n i t area of t h e u p p e r b o u n d a r y of t h e atmosphere at a g i v e n point on t h e planet and per u n i t time, constitutes a v e r y important i n p u t d a t a . T h i s amount of solar e n e r y and its v a r i a b i l i t y with latitude and time is mainly g o v e r n e d by two factors : on one h a n d , the solar f l u x as a function of the orbital distance a n d , on t h e other h a n d , the cosine of t h e zenith angle of t h e incident solar r a d i a t i o n .
Considering more p a r t i c u l a r l y t h e l a t t e r p a r a m e t e r , it should be pointed out t h a t for a planet assumed to be s p h e r i c a l , t h e zenith distance may be expressed as a simple function of. l a t i t u d e , solar d e - clination and local hour angle of t h e S u n , t h e radius vector coinciding with the normal to t h e horizon plane. For an oblate, h o w e v e r , t h e r e is an angle ( v a n i s h i n g at t h e equator and t h e poles) between t h e two directions mentioned above. T h i s so-called angle of t h e v e r t i c a l is dependent upon t h e f l a t t e n i n g .
Although t h e oblateness effect on t h e solar radiation incident at t h e top of t h e outer planets J u p i t e r , S a t u r n , Uranus and Neptune ( V a n Hemelrijck, 1981a) and of t h e Earth ( V a n Hemelrijck, 1982b) has been t r e a t e d r e c e n t l y in d e t a i l , it should be emphasized t h a t t h e effect of the f l a t t e n i n g on the u p p e r - b o u n d a r y insolation of M a r s , the outer planet h a v i n g t h e smallest f l a t t e n i n g , has never been s t u d i e d . T h i s p a p e r , t h e r e f o r e , analyzes this e f f e c t .
O u r results are presented in t h e form of a contour map, i l l u s t r a t i n g t h e latitudinal and seasonal v a r i a t i o n of t h e ratio of the daily insolation with and without t h e influence of t h e oblateness and in a diagram g i v i n g t h e latitudinal variation of t h e numerical d i f f e r e n c e of both insolations corresponding to the southern and n o r t h e r n w i n t e r
hemisphere r e s p e c t i v e l y . For the sake of completeness we have included a f i g u r e showing the percentage d i f f e r e n c e of the mean (summer, w i n t e r and a n n u a l ) daily insolations as a function of l a t i t u d e .
2. 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
In this section we only b r i e f l y will mention the major expressions needed for the computation of the daily insolation with and without the oblateness e f f e c t . For more details see e . g . Ward ( 1 9 7 4 ) , Vorob'yev/ and Monin ( 1 9 7 5 ) , Levine et al. ( 1 9 7 7 ) , Brinkman and Mc. Gregor (1979),- Van Hemelrijck and Vercheval ( 1 9 8 1 ) and Van Hemelrijck ( 1 9 8 2 a , b , c ) .
T h e instantaneous insolation I at the u p p e r - b o u n d a r y of a planet can be expressed as :
I = S cos z (1) with : S = S / r j (2)
o 0and : r = a (1 - e ) / ( l + e cos W) (3)
© 0
where z is t h e . z e n i t h angle of the incident solar r a d i a t i o n , S is the solar f l u x at an heliocentric distance r_ and S is t h e solar constant at
w o
t h e mean S u n - E a r t h distance of 1AU taken at 1353 Wm" or 1 . 9 4 cal
- 2 - 1
cm ( m i n ) ( T h e k a e k a r a , 1973). F u r t h e r m o r e , in expression ( 3 ) , a , e and W are respectively t h e planet's semi-major a x i s , the eccentricity and t h e t r u e anomaly which is given by :
W = X - \ ( 4 )
0 p
where and A. are the planetocentric longitude of the Sun and the planetocentric longitude of the planet's perihelion. T h e numerical values of t h e parameters used for the calculations are listed in Table 1. in this table one can f i n d also t h e obliquity e , t h e equatorial radius ag, t h e polar radius a , the f l a t t e n i n g f = ( a - a ) / a „ and t h e sidereal
r p e p e
TABLE 1.- Elements of the planetary orbit and dimensions of Mars.
a ^ e K e a a f T
© p e p
(AU) (°) (°) (km) (km) (Earth days)
1.524 0.09339 248 25.20 3397.0 3379.5 0.00515 1.02
period of axial rotation T (sidereal d a y ) . Note t h a t t h e elements of the p l a n e t a r y o r b i t and the dimensions of Mars are t a k e n from the Handbook of the B r i t i s h Astronomical Association ( 1 9 8 0 ) and from Levine et al.
( 1 9 7 7 ) .
For a spherical p l a n e t , z may be expressed as :
cos z = s i n 0' s i n + cos <)>' cos cos h ( 5 )
where <f>1 i s t h e geocentric latitude ( w h i c h equals the geographic latitude d>), is the solar declination and h is the local hour angle of the S u n . Ci/
F u r t h e r m o r e , t h e solar declination can be calculated using the following expression :
s i n 8p = s i n e s i n ^ ( 6 )
T h e daily insolation lD can now be obtained by i n t e g r a t i n g expression ( 1 ) over daytime assumed to be equal to t h e time t h a t elapses between rising and setting of the Sun and is given by :
I = (ST/71) (h s i n <t>' s i n 6 + s i n h cos <t>' cos ( 7 )
D o Ci/ o • •*••
where h is the sunset ( o r s u n r i s e ) hour angle and may be determined o
from expression ( 5 ) by t h e condition t h a t at sunset ( o r s u n r i s e ) cos z = 0. Hence :
h = arc cos ( - t a n t a n <t>') ( 8 ) o G
i f
! < * / 2 - ' l6 0! •
In regions where the Sun does not rise (<t>' < - n / 2 + or (j)' > 7i/2 + 6Q) w e have hQ = 0; in regions where the Sun remains above the horizon all day (<t>' > n / 2 - 6Q or V < - n/2 - 60) we may p u t hQ = n.
I n t h e c a s e o f a n o b l a t e p l a n e t t h e r e is a n a n g l e v = <t>-<|>,/ t h e s o - c a l l e d a n g l e o f t h e v e r t i c a l , b e t w e e n t h e r a d i u s v e c t o r a n d t h e n o r m a l t o t h e h o r i z o n p l a n e ; i t v a n i s h e s a t t h e e q u a t o r a n d t h e p o l e s w h i l e e l s e w h e r e <f> > <t>' n u m e r i c a l l y . T h e a n g l e v is d e p e n d e n t u p o n t h e g e o c e n t r i c l a t i t u d e <|>' a n d t h e f l a t t e n i n g f b y t h e r e l a t i o n s h i p :
v = a r c t a n [ ( 1 - f ) "2 t a n <t>' ] - <K ( 9 )
D e n o t i n g t h e z e n i t h d i s t a n c e f o r a n o b l a t e p l a n e t b y Z , t h e f o l l o w i n g r e l a t i o n c a n e a s i l y b e o b t a i n e d b y a p p l y i n g t h e f o r m u l a s o f s p h e r i c a l t r i g o n o m e t r y :
cos Z = cos v cos z + s i n v ( - t a n <|>'cos z + s i n sec <t>* ) (10)
T h e d a i l y i n s o l a t i o n o f a n o b l a t e p l a n e t lD O c a n n o w b e f o u n d b y i n t e g r a t i n g e x p r e s s i o n ( 1 ) , w i t h i n t h e a p p r o p r i a t e t i m e l i m i t s , w h e r e c o s z h a s t o b e r e p l a c e d b y r e l a t i o n ( 1 0 ) y i e l d i n g :
ID 0 = *c o s v^ho o s i n s i n 6© + S i n ho o c o s <'>l C O S
+ s i n v f - t a n <b' ( h s i n ( b ' s i n S ^ + s i n h cos f cos 6 )
oo © oo w
+ h s i n sec ( b1] } ( 1 1 )
oo 0
w h e r e h , t h e local h o u r a n g l e a t s u n s e t ( o r s u n r i s e ) f o r a n o b l a t e oo
p l a n e t , is g e n e r a l l y s l i g h t l y d i f f e r e n t f r o m hQ. A s f o r a s p h e r i c a l p l a n e t , hQ o m a y b e d e r i v e d f r o m r e l a t i o n ( 1 0 ) b y p u t t i n g cos Z : 0 . A f t e r some r e a r r a n g e m e n t s , a n d as e x p e c t e d , t h e e x p r e s s i o n f o r hQ o i n t e r m s o f 8Q a n d <|> is f o u n d t o b e s i m i l a r t o f o r m u l a ( 8 ) .
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 ' q q / ' d a t
t h e t o p o f M a r s c a n n o w e a s i l y b e d e t e r m i n e d as a f u n c t i o n o f g e o - c e n t r i c l a t i t u d e <f>' a n d s o l a r l o n g i t u d e \Q.
T h e mean (summer, w i n t e r and a n n u a l ) daily insolations, h e r e a f t e r denoted as 0D)S/ a n d ^'d^A (sPh e r i c a l p l a n e t ) and C 'D O)S' ( T _ _ ) . „ and ( I ^ K (oblate p l a n e t ) r e s p e c t i v e l y , may be found by inte-
DO W DO A
g r a t i n g numerically relation ( 7 ) and ( 1 1 ) within the a p p r o p r i a t e time limits, yielding t h e total insolation over a season or a y e a r and by d i v i d i n g t h e obtained result by the corresponding length of the season
[ T _ = 3 8 1 . 3 (summer) and T.„ = 305.6 ( w i n t e r ) Earth days ] or by the s w
sidereal period of revolution or tropical y e a r ( T = 6 8 6 . 9 Earth d a y s ) . Note t h a t (TQ)a and (TQ Q )a c a n a , s o d i r e c t ,y b e computed from the knowledge of ( TD>S, 0D)W and ("i~D O)s/ ^ d o ^W l n d e e d' t a k i n9 into account t h e numerical values of T _ / T and T . . . / T , the average yearly
o O W O
insolations can, in a v e r y good approximation, be w r i t t e n u n d e r t h e following general form :
and ( I
D)
A= 0 . 5 6 ( I
d)
s +0 . 4 4 ( I
d)
w(12)
( W A = ° -5 6< W s + °-4 4 ( IDO>W ( 1 3 )
3. D I S C U S S I O N OF T H E R A T I O D I S T R I B U T I O N OF T H E D A I L Y I N S O L A - T I O N S
In two previous w o r k s , dealing with t h e oblateness effect on the solar radiation incident at t h e top of t h e atmospheres of the outer planets J u p i t e r , S a t u r n , Uranus and Neptune ( V a n Hemelrijck, 1982a) and of t h e Earth ( V a n Hemelrijck, 1 9 8 2 b ) , we studied q u a l i t a t i v e l y some characteristic f e a t u r e s of t h e ratio d i s t r i b u t i o n IQ Q/IQ i n t h e n o r t h e r n hemisphere both for summer ( 0 ° < A < 1 8 0 ° ) and w i n t e r ( 1 8 0 ° < <
3 6 0 ° ) p e r i o d . It should, h o w e v e r , be emphasized t h a t the results presented are also valid for the southern hemisphere and t h a t t h e y a r e e v i d e n t l y applicable to Mars. T h e following are the major conclusions t h a t were reached :
( 1 ) In summer and in t h e region of permanent s u n l i g h t ( h = hQ = 7T) t h e isocontours 'd c / 'd Pa r a , , e l t h e "nes of constant geocentric
latitude (J)1 and the daily solar radiation l pQ is greater than I p . T h e maximum value of the ratio ' q q / ' q can be e x p r e s s e d by the following relationship :
_2
( IDO/ ID)max = s i n *a r C t a n[ (1~f) cotan e ] } / c o s e (14)
( 2 ) In summer and in the region limited by the seasonal march of the S u n , the solar radiation of an oblate planet ( 'Dq ) is increased with respect to the insolation of a spherical one O q ) -
( 3 ) For latitudes between the subsolar point and the region where the S u n remains above the horizon all d a y , the ratio cos Z/cos z Is decreasing (with cos Z < cos z ) , whereas h 0 0/h 0 's increasing (with hQ Q > hQ) . Whether or not the regions mentioned in ( 1 ) and ( 2 ) are linked depends on the relative effect of those two ratios, both being function of f and e, and can only be evaluated by computation of the expression 'QQ^'D*
4) In winter, the effect of the flattening results in a more extensive polar region, the insolation is always reduced O q q < l p ) and the c u r v e s of constant ratio ' q q ^ ' d r o u9 'n' V parallel the boundary of the polar night except in the neighborhood of the equinoxes.
Application of expressions ( 7 ) and (11) leads to the isocontour map illustrated in F i g . 1, where values of constant ratio distribution ' q q / ' q are g i v e n on each c u r v e . Solar declination (lower p a r t ) and the region where the S u n does not set ( u p p e r p a r t ) are indicated by the dashed lines. T h e area of permanent d a r k n e s s is shaded and the region of enhanced solar radiation O q q > l p ) is dotted.
From F i g . 1 it can be seen that in the region of permanent s u n - light the incoming solar radiation O q q ) is increased when compared to
120 160 200 240 SOLAR LONGITUDE (degrees)
280 320 360
Fig. 1.- Seasonal and latitudinal variation of the ratio I ^ / I ^ of the daily insolation with (IJJQ) an<i without the oblateness effect at the top of the atmosphere of Mars. Solar declination (lower part) and the region where the Sun does not set (upper part) are indicated by the dashed lines. The area of permanent darkness is shaded, whereas the region of enhanced solar radiation (I^Q>Ip) is dotted. Values of constant ratio distribution a r e given on each curve.
I p . F u r t h e r m o r e , it follows f r o m e x p r e s s i o n ( 1 4 ) t h a t in t h e r e g i o n c o n s i d e r e d ( l _ _ / l _ ) = 1 . 0 0 1 9 ( = 0 . 2 % ) . T h i s e x t r e m e l y small g a i n of
DO D max
i n s o l a t i o n is p a r t i c u l a r l y d u e t o t h e e v e n small v a l u e of t h e f l a t t e n i n g . For c o m p a r a i s o n , i t is i n s t r u c t i v e to n o t e t h a t ( IQ Q/Ip ^ m a x 's e c'u a' t o
1 . 0 0 1 0 , 1 . 0 0 0 3 , 1 . 0 3 7 , 1 . 1 2 4 a n d 1 . 0 1 0 f o r t h e E a r t h , 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 r e s p e c t i v e l y . T h e e f f e c t of p a r a l l e l i s m b e t w e e n t h e i s o c o n t o u r s ' q q ^ ' d a n c' ' 'n e s c o n s t a n t g e o c e n t r i c l a t i t u d e <J>' is n o t i l l u s t r a t e d in F i g . 1 . T h e reason f o r t h i s n o n - r e p r e s e n t a t i o n lies in t h e f a c t t h a t t h e n u m e r i c a l v a l u e of 'Q Q/ 'D i s n e9 ' ' 9 'b , e small ( 1 . 0 0 1 2 a n d 1 . 0 0 0 3 a t <|>f = 70 a n d 8 0 ° r e s p e c t i v e l y ) .
I t can m a t h e m a t i c a l l y be p r o v e d t h a t , in s u m m e r , in t h e r e g i o n b o u n d e d b y t h e e q u a t o r a n d t h e solar d e c l i n a t i o n c u r v e , b o t h t h e l e n g t h of t h e d a y a n d t h e cosine of 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 of t h e f l a t t e n i n g . H e n c e , i t follows 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 a t t h e t o p of t h e a t m o s p h e r e of M a r s , 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 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 p l a n e t .
F i g . 1 also r e v e a l s t h a t t h e t w o zones w h e r e . I ^ ^ > l_. a r e l i n k e d
3 DO D
b y t w o c u r v e s c o i n c i d i n g r e m a r k a b l y well w i t h t h e two b r a n c h e s of an h y p e r b o l a . O u t s i d e t h e r e g i o n of i n c r e a s e d solar r a d i a t i o n t h e loss of i n s o l a t i o n d e p e n d s u p o n t h e solar l o n g i t u d e . For e x a m p l e , i f , a t
<j)' = 4 5 ° , \ i n c r e a s e s f r o m 0 to a p p r o x i m a t e l y 5 0 ° , t h e r a t i o 'Q Q^ 'D i n c r e a s e s f r o m 0 . 9 9 5 ( s 0 . 5 % ) to a b o u t 0 . 9 9 9 ( ^ 0 . 1 % ) . For t h e s a k e of c l e a r n e s s i t has to be p o i n t e d o u t t h a t , in s u m m e r , t h e i n c r e a s e o r d e c r e a s e in solar e n e r g y is s i g n i f i c a n t l y small.
As f o r all p l a n e t s ( V a n H e m e l r i j c k 1 9 8 2 a , b ) 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 e m p h a s i z e d t h a t t h e A r t i e c i r c l e s lD Q
= 0 a n d I p = 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?23 a t a solar l o n g i t u d e of 2 7 0 ° . T h i s v a l u e is h i g h e r t h a n t h e
one f o r t h e E a r t h ( 0 ? 1 4 ) b u t is c o n d e r a b l y lower t h a n those f o r t h e o t h e r planets v a r y i n g from about 0 . 4 ( J u p i t e r ) to a p p r o x i m a t e l y E>"
( S a t u r n ) . It is clear t h a t this f i n d i n g is a s c r i b e d to t h e e x t r e m e l y small f l a t t e n i n g .
F i g . 1 shows t h a t in w i n t e r , as stated e a r l i e r , t h e insolation iD Q
is always r e d u c e d when compared to l ^ . T h e incoming solar e n e r g y slowly decreases in passing from e q u a t o r l a t i t u d e s to m i d - l a t i t u d e s ; it is f o u n d t h a t at w i n t e r solstice and if 0' increases from 10 to about 55°
t h e ratio 'Q Q/ 'D d e c r e a s e s f r o m 0 . 9 9 9 (? 0.1%) to a p p r o x i m a t e l y 0 . 9 5 0 ( s 5%). À t h i g h e r l a t i t u d e s , p a r t i c u l a r l y near t h e region w h e r e t h e Sun does not r i s e , 'Q Q/ 'Q d r o p s v e r y r a p i d l y to z e r o . A l t h o u g h the loss of insolation is not v e r y s t r i k i n g it follows from Fig. 1 t h a t f o r p a r t s uT t h e w i n t e r t h e incident solar r a d i a t i o n is decreased by more t h a n t h r o u g h t h e f l a t t e n i n g .
A n o t h e r point about t h e c u r v e s is t h a t t h e y r o u g h l y p a r a l l e l t h e limit of t h e polar n i g h t e x e p t , of c o u r s é , in t h e v i c i n i t y of t h e e q u i n o x e s . F i n a l l y , it is also suitable to r e m a r k t h a t in summer, r e s p e c t i v e l y in w i n t e r , t h e c u r v e s of constant ratio 'Q Q/ 'D a r e Pe r f e c t"
ly symmetric w i t h r e s p e c t to t h e summer and w i n t e r solstices.
4. D I S C U S S I O N OF T H E N U M E R I C A L D I F F E R E N C E D I S T R I B U T I O N OF T H E D A I L Y I N S O L A T I O N S
In F i g . 2 we h a v e p l o t t e d t h e numerical d i f f e r e n c e [ in cal cm ( d a y ) "1] of t h e d a i l y solar r a d i a t i o n , w i t h o u t ( lD) and w i t h O Q Q) t h e oblateness e f f e c t , as a f u n c t i o n of l a t i t u d e c o r r e s p o n d i n g to t h e s o u t h e r n and n o r t h e r n w i n t e r hemisphere r e s p e c t i v e l y .
A l t h o u g h t h e isocontours r e p r e s e n t i n g t h e r a t i o d i s t r i b u t i o n of both insolations ( ' d ç / ' d ^ a r e Pe r f e c t |y identical in both h e m i s p h e r e s , it follows from t h e n o r t h - s o u t h seasonal a s y m m e t r y in t h e d a i l y insolation
SOLAR LONGITUDE (degrees)
Fig. 2.- Latitudinal variation of the numerical difference (1^ - I ) of the daily insolations corresponding to the southern (left) and northern (right) winter hemisphere. The areas of permanent darkness are shaded.
_ 2 Values of constant numerical difference distribution, in cal cm
produced by t h e e c c e n t r i c i t y of t h e o r b i t t h a t the absolute loss of solar radiation as a function of latitude and solar longitude is d i f f e r e n t in both hemispheres. Fig. 2 clearly indicates t h a t t h e decrease of solar e n e r g y in t h e n o r t h e r n w i n t e r hemisphere is obviously h i g h e r than in
- 2
t h e southern one reaching a maximum value as much as 2 cal cm ( d a y ) at midlatitudes; in the southern hemisphere t h e maximum loss of _ 1
- 2 - 1
insolation is about 1 . 5 cal cm ( d a y )
5. 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 F i g . 3 t h e influence of the f l a t t e n i n g is plotted in terms of the percentage d i f f e r e n c e [ 1 0 0 ( 7D Q - ' p V p ] a s a function of geocentric l a t i t u d e . As in section 2 , the bars over symbols s i g n i f y seasonal or annual a v e r a g e s .
Concerning more p a r t i c u l a r l y t h e mean summertime insolation, the influence of t h e f l a t t e n i n g , although v e r y small, is obviously e v i d e n t from Fig. 3. I n d e e d , in the latitude i n t e r v a l ( 5 0 - 6 0 ° ) it can be seen t h a t about 0.2% of the mean summertime insolation is lost t h r o u g h t h e oblateness e f f e c t . Outside this i n t e r v a l , t h e effect is of decreasing significance. Another i n t e r e s t i n g phenomenon is t h a t for latitudes between t h e equator and t h e subsolar p o i n t , the mean daily summer insolation of Mars, assumed as an oblate p l a n e t , is increased ( V a n Hemelrijck, 1 9 8 2 a , b ; Brinkman and M c G r e g o r , 1979). H o w e v e r , owing to t h e 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 , reaching only a maximum value of about 0.05% at a latitude of a p p r o x i - mately 1 0 ° .
D u r i n g w i n t e r season, as already stated p r e v i o u s l y , t h e daily insolation lD Q is always reduced with respect to ID; consequently ( 7 ). . < ( T O . , at any l a t i t u d e . Fig. 3 illustrates t h a t in w i n t e r the
v DOyW v D W 1
loss of insolation is of most importance between 45 and 85° where a percentage d i f f e r e n c e as much as 1% has been found with a maximum value near 2% in the 6 0 - 8 0 ° latitude i n t e r v a l .
F i g . 3.- Latitudinal v a r i a t i o n of the percentage difference [100(IpQ-Ip)/Ip] of the m e a n (summer, winter and a n n u a l ) daily i n s o l a t i o n s . The bars over symbols signify seasonal and annual a v e r a g e s .
T h e partial gain of the mean summertime insolation e q u a t o r w a r d of t h e subsolar point being considerably lower than . t h e corresponding loss of insolation in w i n t e r e v i d e n t l y results in a mean annual daily insolation which is reduced over t h e e n t i r e latitude i n t e r v a l as shown in Fig. 3. It can be seen t h a t the daily insolation averaged over a one year cycle is decreased by about 0.5% at m i d - l a t i t u d e s .
6. C O N C L U D I N G REMARKS
In the p r e s e n t p a p e r , we have investigated the influence of t h e f l a t t e n i n g on t h e solar e n e r g y input at t h e top of the atmosphere of Mars. As a result of this s t u d y we can d r a w n the general conclusion t h a t t h e e f f e c t of t h e oblateness causes n o n - n e g l i g i b l e , although r e l a t i - v e l y small, variations 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 t h e i n t e n s i t y of t h e daily solar r a d i a t i o n .
For p a r t s of the summer, t h e daily insolation of M a r s , assumed as an oblate p l a n e t , is slightly increased. T h e maximum increase of t h e incoming solar r a d i a t i o n , occuring at a geocentric latitude of about 65°
near summer solstice is approximately equal to 0.2%. Outside the dotted zone of Fig. 1, the d i r e c t solar radiation lD O is decreased when com- pared to I p . For example, in the neighborhood of t h e equinoxes t h e loss of insolation ranges from 0 . 1 to 0.5% ( e x c e p t at equator l a t i t u d e s ) , whereas elsewhere the oblateness effect is either u n i m p o r t a n t .
In w i n t e r , the influence of the f l a t t e n i n g causes the 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 , t h e rate of decrease d e - pending to a large e x t e n t on the geocentric l a t i t u d e . For comparison, at
\ = 2 7 0 ° , t h e loss of solar e n e r g y amounts to about 0 . 1 , 0 . 5 , 1 . 0 , 2 . 0 , 3 . 0 and 5.0% respectively at latitudes of approximately 5 , 20, 30, 45, 50 and 5 5 ° . M o r e o v e r , in the r e l a t i v e l y small area limited by t h e isocontour _ / l _ = 0 . 9 5 0 and the region of permanent d a r k n e s s t h e
effect of the flattening plays a more significant role and the decrease of incident solar radiation is c o r r e s p o n d i n g l y greater. Furthermore, it is particularly evident from Fig. 1 that the c u r v e s of constant ratio ' d o ^ ' d r o u9h lV ' parallel the Arctic Circle bounding the polar region in which there are d a y s without s u n r i s e .
Finally, we also have studied the latitudinal variation of the p e r - centage difference of the mean daily insolations. It is found that for latitudes equatorward of the subsolar point, the mean summer daily insolation of an oblate Mars is increased when compared to a spherical one, the maximum rise being extremely small (= 0.05%). At higher latitudes, there is a loss of insolation which is of most importance at mid-latitude regions (= 0.2%).
In winter, the horizon plane is always tilted away from the S u n causing both the cosine of the zenith angle and the length of the day to be reduced. Consequently, the daily insolation as well as the mean daily insolation are reduced, the latter decreasing maximally by about 2% at high latitudes.
Despite of the partial gain of the mean summertime insolation near the equator, the effect of the flattening can clearly be seen to reduce the mean daily insolation over the entire year.
In conclusion, we believe that the effect of the oblateness, although v e r y small except in the vicinity of the polar night, has to be taken into account in studies related to theoretical models for the calculation of solar global insolation on Mars.
A C K N O W L E D G E M E N T S
T h e author is grateful to J. Schmitz and F. V a n d r e c k for the realisation of the three illustrations.
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