I 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 - Avenue Circulaire B • 1180 BRUXELLES
A E R O N O M I C A A C T A
A - N° 236 - 1981
Some aspects of the solar radiation incident at the top of the atmospheres of Mercury and Venus
by
E. VAN HEMELRIJCK and J. VERCHEVAL
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 0 N O M I E 3 - Ringlaan
B • 1180 BRUSSEL
FOREWORD
The paper entitled "Some aspects of the solar radiation incident at the top of the atmospheres of Mercury and Venus" will be published in
" I c a r u s " , 1981.
A V A N T - P R O P O S
L'article intitulé "Some aspects of the solar radiation incident at the top of the atmospheres of Mercury and Venus" sera publié dans
" I c a r u s " , 1981.
VOORWOORD
De tekst "Some aspects of the solar radiation incident at the top of the atmospheres of Mercury and Venus" zal in het tijdschrift "Icarus"
1981 verschijnen.
VORWORT
Der Text "Some aspects of the solar radiation incident at the top of the atmospheres of Mercury and Venus" wird in " I c a r u s " , 1981 herausgegeben werden.
S O M E A S P E C T S OF 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 S OF M E R C U R Y A N D V E N U S
E. V A N H E M E L R I J C K and J. V E R C H E V A L
A b s t r a c t
A formalism has been developed for the calculation of the insolation on the planets Mercury and Venus neglecting any atmospheric absorption.
For M e r c u r y , the instantaneous insolation c u r v e s are repeated in a two-tropical year cycle, the distribution of the solar radiation being perfectly symmetric between both hemispheres. In addition to latitudinal variations, one observes a longitudinal effect expressed b y different instantaneous insolation distributions d u r i n g the course of the time; on the equator, the relative diurnal insolation variability may attain a factor of 3.
The small obliquity of Venus results in a nearly symmetric solar radiation distribution with respect to the equator except at the poles where an important seasonal effect has been found. It has to be noted that no longitudinal dependence exists. Finally, the insolation c u r v e s are repeated in a nearly half year cycle.
Résumé
Nous avons développé un formalisme pour le calcul de l'insolation des planètes Mercure et Vénus en négligeant toute absorption atmosphé- rique.
Pour la planète Mercure, les courbes d'insolation instantanée sont répétées au cours d'un cycle de 2 années tropiques avec une symétrie parfaite entre les deux hémisphères. Outre les variations en latitude, on observe un effet en longitude se traduisant par des distributions différentes de l'insolation instantanée au cours du temps; à l'équateur la variation relative de l'insolation diurne peut atteindre un facteur 3.
La faible obliquité de la planète Vénus donne lieu à une d i s t r i b u - tion de l'insolation quasi-symétrique par rapport à l'équateur, excepté aux pôles où un effet saisonnier important est observé. On note l'absence de tout effet sensible en longitude. Enfin, les courbes d'insolation se répètent avec un cycle voisin d'une demi-année.
- 2 -
Samenvatting
Een methode werd ontwikkeld voor het berekenen van de zonne- straling op de planeten Mercurius en Venus met verwaarlozing van eventuele atmosferische absorpties.
De kurven die de ogenblikkelijke zonnestraling op Mercurius weer- geven worden herhaald volgens een twee-jarige cyclus met een vol- maakte symmetrie tussen de beide hemisferen.
Behalve breedte-variaties neemt men ook een lengte-effect waar gekenmerkt door verschillende ogenblikkelijke zonnestralingsverdelingen in de loop van de tijd; aan de equator kan de relatieve variatie van de dagelijkse zonnestraling een factor 3 bereiken.
De kleine helling van de rotaties van Venus op zijn baanvlak resulteert in een bijna symmetrische verdeling van de zonnestraling ten opzichte van de equator uitgezonderd aan de polen waar een belangrijk seizoeneffect werd gevonden. Noteer dat geen enkele lengte-afhanke- lijkheid bestaat. Tenslotte worden de zonnestralingskurven herhaald over een periode van ongeveer een half tropisch jaar.
Zusammenfassung
Eine Methode wurde entwickelt für die Berechnung der Sonnen- strahlung auf den Planeten Merkur und Venus mit Verwahrlosung von eventuellen Atmosphärischen Absorptionen.
Die kurven der Augenblicklichen Sonnenstrahlung auf Merkur sind repetiert in einem zwei-järigen Zyklus mit einer perfekten Symmetrie zwischen den beiden Hemisphären.
Ausserhalb Variationen in Breite nehmt man auch eine Länge Wirkung wahr die ausgedrückt ist durch verschiedenen augenblicklichen Sonnenstrahlungsverteilungen als Funktion der Zeit mit besonders einer relativen Variation der täglichen Sonnenstrahlung die auf dem Äquator einen Faktor 3 erreichen kann.
Die geringe Neigung der Rotationsachse von Venus gegen die Bahnebene resultiert in einer ungefähr symmetrischen Verteilung der Sonnenstrahlung mit Rücksicht auf dem Äquator ausgenommen an den Polen wo ein wichtiger Saisoneffekt gefunden wurde. Eine Abhängigkeit der Länge wurde nicht konstatiert. Die Kurven der augenblicklichen Sonnenstrahlung auf Venus sind repetiert in einem ungefähr halb- järigen Z y k l u s .
1. I N T R O D U C T I O N
The knowledge of the distribution of the solar radiation incident at the top of the atmospheres of the planets of the solar system as a function of the coordinates of a surface element and time (or season) is indispensable to the s t u d y of its climatic history, energy b u d g e t and dynamical behaviour.
A l t h o u g h the u p p e r - b o u n d a r y insolation of the Earth's atmosphere and the radiation incident on its surface have been treated extensively, it must be pointed out that theoretical investigations relative to the other planets are scarce and often incomplete. Among the papers
published d u r i n g the last years we cite those of M u r r a y et al. (1973), Ward (1974), Vorob'yev and Monin (1975), Levine et al. (1977) and
Brinkman and McGregor (1979).
Concerning more particularly the inner planets Mercury and V e n u s , the distribution of incident solar radiation has only been studied partially (Soter and Ulrichs, 1967; H a n - S h o u Liu, 1968; V o r o b ' y e v arid Monin, 1975). A s an example, the diurnal insolation has never been analysed in detail. Moreover it should be emphasized that the insolation variation has generally been determined as a function of the latitude of a surface element neglecting any longitudinal effect. T a k i n g into account the large eccentricity of the heliocentric orbit of M e r c u r y and the slow rotation it is obvious that the longitude of a surface element is of prime importance for various radiation problems. Although this planet has practically no atmosphere, the analysis of its insolation may be of great interest to the study of the thermal budget on its surface.
I
The main objective of the present paper is to analyse in detail solar radiation problems of Mercury and Venus which are the nearest planets to the S u n . Another characteristic feature of those two slowly
rotating planets is t h a t t h e y have sidereal periods of axial rotation of t h e same o r d e r of magnitude as the sidereal periods of revolution in t h e i r heliocentric motions. F u r t h e r m o r e , the inclination of t h e equator with respect to t h e p l a n e t a r y o r b i t is v e r y small ( z e r o for M e r c u r y ) . However, two main differences have to be noted between t h e two planets. F i r s t , M e r c u r y rotates on its axis in the same direction in which it revolves around t h e Sun whereas Venus is the only planet in t h e solar system rotating in t h e opposite direction ( r e t r o g r a d e motion).
Secondly, the eccentricities of t h e i r orbits ( 0 . 0 0 6 7 8 for Venus and 0.20563 for M e r c u r y ) r e p r e s e n t the extreme values when listing this parameter of all planets of the solar system. Although t h e last two f i n d i n g s j u s t i f y a somewhat d i f f e r e n t theoretical approach to the insolation problem of t h e two planets we p r e s e n t , in a f i r s t section, a general formalism applicable to both planets. T h e n , t a k i n g into account t h e i r p r o p e r conditions we calculate the instantaneous as well as t h e diurnal u p p e r - b o u n d a r y insolation of the atmospheres of M e r c u r y and V e n u s .
2. GENERAL FORMALISM
T h e instantaneous insolation is defined as the solar heat f l u x sensed at a given time by a horizontal unit area of the u p p e r - b o u n d a r y of t h e atmosphere at a given point on t h e planet and per unit time.
In our calculations use will be made of the following parameters as illustrated in Fig. 1 (see also Ward, 1974).
( O ; x , y , z ) represents an equatorial coordinate system, whereas ( O ; x1, y ' , z1) defines an orbital coordinate system. I t should be noted t h a t in both systems the origin O is located in the center of the p l a n e t . T h e axes y and y' coincide and are directed toward the Sun at t h e epoch of the vernal equinox of the planets. T h e z axis contains the spin or rotation axis of the p l a n e t , r is the unit vector c h a r a c t e r i z i n g
Fig. 1.- Schematic representation of some parameters used in the equations for the instantaneous and diurnal insolations on Mercury and Venus.
- 7 -
the direction of a surface element and ^ is the unit vector directed toward the S u n . k and q> are respectively the longitude and the latitude of the surface element in the equatorial system, the y axis being taken as the origin of the longitudes (Note that the so-defined longitude is time dependent owing to the rotation of the planet). \Qa n d 6Qdefine the longitude and the latitude of the Sun in the same system. \Q and A' are the planetocentric longitude of the Sun and the planetocentric longitude of the planet's perihelion in the orbital system, and e is the 'obliquity or inclination of the equator with respect to the plane of the
orbit.
T h e instantaneous insolation I may now be written in the following form :
I = S(r.TQ) (1) with
I > 0 i f r . rQ > 0
I = 0 i f r . rQ ^ 0
(2)
(3)
S is the solar flux at an heliocentric distance r,~ and S is the solar u • o constant at the mean Sun - Earth distance of one astronomical unit. For the calculations presented in this paper we have adopted the most
- 2 - 1
reliable value of the solar constant of 1.94 cal cm min (or 2.79 x 103 cal cm"2 d a y "1) .
In expression ( 3 ) , a0, e and W are respectively the planet's semi- major axis, the eccentricity and the true anomaly. Table I represents, for the two planets under consideration, the numerical values of the parameters used for the computation of the instantaneous and diurnal and
S = S / ro' © 2
ro = ao (1 - e 2 ) / ( 1 + e c o s W )
- 8 -
T A B L E I . - Elements of the planetary orbits of Mercury and Venus
PLANET a e \'|p e T T o
( A . U . ) (°) (°) (Earth days) (Earth days)
M E R C U R Y 0.3871 0.20563 189.22 0 87.969 58.65 V E N U S 0.7233 0.00678 124.43 2 224.701 243.0
insolations. In this table one can find also the sidereal period of revolu- tion Tq (tropical year) and the sidereal period of axial rotation T (sidereal d a y ) . Finally, it has to be mentioned that the argument of perihelion is taken from Vorob'yev and Monin (1975).
The components of the unit vectors r and rQ are respectively :
r = ( c o s c p s i n A , c o s cp c o s A , s i n c p ) a n d
Tq = ( c o s 6q s i n A q, c o s 6q c o s Aq, s i n 6q )
With this in mind, we can write : S
I = — 2 ~ [ s i n c p s i n + c o s cp c o s S ^ c o s ( A . - A ^ ) ] ( 4 )
S
o r O i:
I = — ( 1 + e c o s W ) ^ [ s i n c p s i n 6 +
a0 '
c o s cp c o s c o s ( A . - A ^ ] ( 5 )
The solar radiation I has to be expressed as a function of time, the time dependence being a function of the arguments W, 60a n d (A - AQ). Indeed, taking t = 0 at the perihelion passage of the planet , the true anomaly W is given by :
W = n t + ( 2 e - f — ) s i n n t + j - e ^ s i n 2 n t + ^ e ^ s i n 3 n t ( 6 )
o 4 o 4 o l 2 o s
where nQ = designates the mean angular motion.
In relation (%), known in celestial mechanics as the equation of the center, we kept only terms up to the third degree in e. T h i s approxi- mation gives satisfactory results taking into account that the maximum value of the eccentrities considered here is approximately equal to 0.2 (Mercury).
- 1 0 -
Furthermore, the solar declination can be calculated using the following relations :
(7)
(8)
where |6Q| is always smaller than 90°.
If, on the other hand, one assumes that at t = 0 the longitude of the surface element is denoted by Aq, the time dependence of A. is described by the expression :
A = A ± nt o or
A = A + AA ± nt (9) P
2 7T
where n = is the rotational angular velocity of the planet and AA is the longitude difference between, the meridian of the surface element and the meridian crossing the line of apsides at the perihelion passage of the planet which is, as already mentioned, taken as t = 0. Note that the minus sign is adopted for Venus (retrograde rotation).
Finally, the following set of equations can also be obtained :
(10) (11) (12)
Hence, the angle (A - AQ) can be determined by combining ( 9 ) , (10), (11) and (12) yielding to :
s i n 6Q = s i n e s i n (A'^ + W) and
cos 6 = (1 - sin^ 6
A = arctg (cos e tg A ' )
AQ = arc cos (sec cos ^-'Q) P P A' = A' + W O P
- 1 1 -
A - AQ = arctg (cos e tg A^) + AA ± nt - arc cos [sec cos ( V + W) (13)
T a k i n g into account the expressions (5) to (8) and (13), the instantaneous insolation variation at the tops of the slowly rotating planets Mercury and Venus and for a surface element characterized by the coordinates (AQ, 9 ) or (AA, J) can now easily be obtained.
The diurnal insolation can be found by integrating numerically relation (5) over a period equal to the planet's solar day T , :
, T0
Id = [ I dt (14)
-1 t=o
It has to be pointed out that I = 0 when the condition r . i ^ i 0 is fulfilled. Furthermore the length of the solar day T0 , figuring in relation (14), is defined by :
fw rp - >Ti (15) 0 o
the plus sign being used for a planet rotating in the opposite direction in which it revolves around the Sun (of which Venus is the only case).
Practically, relationship (14) may also be written under the following form :
C1 To
Id = / I dt + / I dt (16)
0 2
where t^ and t^ correspond respectively to the time of setting and rising of the Sun; in this case I is always positive. The analytic expression of given by Ward (1974) and also by Levine et al.
(1977) and Vorob'yev and Monin (1975) is only applicable to rapidly rotating planets where rQ and 6 ^ are assumed to be constant in the integration of the instantaneous insolation I over a period equal to the planets solar day TQ. For slowly rotating planets this procedure is not correct when the eccentricity e or the obliquity e are not negligible.
- 1 2 -
3. SOLAR R A D I A T I O N I N C I D E N T ON MERCURY
3 . 1 . I n s t a n t a n e o u s i n s o l a t i o n
T h e o b l i q u i t y e of M e r c u r y b e i n g equal t o z e r o , t h e f o r m a l i s m d i s c u s s e d in t h e p r e v i o u s s e c t i o n may be s i m p l i f i e d . In t h i s case we h a v e :
6 o= 0
a n d
X - X Q = A X + n t - W ( 1 7 )
Hence, t h e amount of solar r a d i a t i o n i n c i d e n t on M e r c u r y 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 e q u a t i o n :
I ( A X , cp) = ~Y c o s cp c o s ( X - X ^ ) ( 1 8 )
s
r©
From r e l a t i o n ( 1 8 ) i t f o l l o w s also t h a t :
I ( A X,cp) / I ( A X , 0 ) = cos cp (19)
From ( 1 9 ) i t is o b v i o u s t h a t t h e i n s t a n t a n e o u s solar r a d i a t i o n at a g i v e n time a n d at a g i v e n l a t i t u d e c p can e a s i l y be d e t e r m i n e d b y m u l t i - p l y i n g b y t h e c o r r e c t i o n f a c t o r cos cp t h e i n s t a n t a n e o u s i n s o l a t i o n on t h e same m e r i d i a n , at t h e same moment and f o r a s u r f a c e element located on t h e e q u a t o r ( c p = 0 ) .
I n t r o d u c i n g t h e n u m e r i c a l v a l u e s f o r SQ, a0 a n d e ( T a b l e I ) e x p r e s s i o n ( 1 8 ) , a p p l i e d t o t h e e q u a t o r , f i n a l l y y i e l d s t o :
I ( A X , 0 ) = 2 . 0 3 3 x 1 0 4 ( 1 + 0 . 2 0 5 6 3 c o s W )2 c o s ( A X + n t - W )
(20)
- 1 3 -
Figure 2 illustrates the solar radiation incident at the top of the Mercurian atmosphere as a function of time and for three surface ele- ments characterized b y the following specific numerical values of AA : 0°, - 45° and - 90° (as already mentioned above, AA. designates a fixed meridian on the planet).
The time variation of the instantaneous insolation extends over three sidereal periods of revolution (or tropical y e a r s ) corresponding to approximately 264 Earth d a y s . However, it should be emphasized that one insolation cycle covers two periods of revolution, application of relation (15) (with the minus s i g n for M e r c u r y ) showing that Tq = 2 Tq
= 176 Earth d a y s .
A n analysis of figure 2 reveals that the insolation pattern as well as the maximum solar radiation incident s t r o n g l y v a r y in p a s s i n g from one meridian to another. To the best of our knowledge computations for AA. = 0 have been made by Vorob'yev and Monin (1975) whereas c u r v e s representing the insolation in a r b i t r a r y units on the meridians AA. = 0°, -75°, -90° and -105° have been reported by Soter and Ulrichs (1967).
It is also interesting to note that the calculations for AA. = - 90° result in a bizarre distribution of solar radiation in the neighbourhood of the perihelion passage of the planet showing a very short period of weak insolation preceded or followed by an even smaller period of complete d a r k n e s s [see also Soter and Ulrichs (1967) and H a n - S h o u Liu (1968)].
A n explanation for this phenomenon (not visible on Figure 2 due to the insufficiency of the scale adopted for the ordinate) will be given later ( F i g u r e 4).
The longitudinal dependence of the incident solar radiation as plotted in Figure 2 will be clarified by means of Figure 3. The dashed lines represent the theoretical insolation curves as a function of true anomaly W ( d u r i n g the course of one revolution) and for various values of the angle of incidence £ = A - A Q = A A + n t - W comprised between 0°
and 90°. The curve for £ = 90° coincides with the axis of abscissa. The
TIME (Earth d a y s )
Time variations of the instantaneous insolation at the top of the Mercurian atmosphere on fixed meridians characterized by the following specific numerical values of AX : 0°, - 45° and - 90°. The curves are related to the equator ( c p = 0). Perihelion and aphelion passages are also illustrated.
TIME ( E a r t h days)
o 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 88
I l i I I I I I I _ l 1 1 1 1 1 1 1 I 1 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
TRUE ANOMALY (Degrees)
Fig. 3.- Instantaneous insolation at the top of Mercury as a function of true anomaly (W), rotation angle difference (nt - W) or time (t). The dashed lines represent the theoretical insolation curves for various values of the angle of incidence £ = K - XQ = AA + nt - W. The solar radiation incident for surface elements characterized by longitude differences AX equal to 0°, - 45°, - 90° and - 135° is given by the full lines. The -I curves correspond to cp = 0.
f i g u r e shows t h e e f f e c t of £ f o r g i v e n v a l u e s o f W, w h e r e a s t h e i n - f l u e n c e o f t h e h e l i o c e n t r i c d i s t a n c e r is i l l u s t r a t e d b y t h e c u r v e s o f c o n s t a n t a n g l e o f i n c i d e n c e . As a r e s u l t , t h e i n s o l a t i o n v a r i e s b y a f a c t o r o f a p p r o x i m a t e l y 2 . 3 b e t w e e n p e r i h e l i o n a n d a p h e l i o n passage f o r a same z e n i t h a n g l e P l o t t e d a r e also t h e time scale t ( a t t h e t o p of t h e f i g u r e ) a n d t h e c o r r e s p o n d i n g v a l u e s of t h e r o t a t i o n a n g l e d i f - f e r e n c e ( n t - W) o b t a i n e d f r o m t h e f o l l o w i n g f o r m u l a e :
2 1/2 ( 1 - e ) s i n W and
s i n E = (21) (1+ecosW)
t = — (E - e s i n E) (22) n o
w h e r e E is t h e e c c e n t r i c a n o m a l y .
F i g u r e 3 can be a p p l i e d t o a n y m e r i d i a n AX t o o b t a i n t h e s o l a r h e a t f l u x a r r i v i n g at a g i v e n time on an h o r i z o n t a l u n i t area o f t h e u p p e r b o u n d a r y of t h e a t m o s p h e r e at a g i v e n e q u a t o r i a l p o i n t on t h e p l a n e t M e r c u r y . T h e f i r s t s t e p f o r d e t e r m i n i n g t h e c o r r e s p o n d i n g i n s o l a t i o n c o n s i s t s in a d d i n g a l g e b r a i c a l l y AX t o t h e n u m e r i c a l v a l u e o f t h e e x p r e s s i o n n t - W l e a d i n g t o t h e a n g l e o f i n c i d e n c e T h e n , t h e i n - s t a n t a n e o u s i n s o l a t i o n can be f o u n d b y i n t e r p o l a t i n g b e t w e e n t h e c u r v e s of c o n s t a n t T o i l l u s t r a t e t h e s e n s i t i v i t y of t h e i n s o l a t i o n t o c h a n g e s in AX we h a v e also p l o t t e d in F i g u r e 3 t h e solar r a d i a t i o n i n c i d e n t f o r AX = 0 ° , - 4 5 ° , - 90° a n d - 135°.
I t is i n t e r e s t i n g t o note t h a t t h e k n o w l e d g e o f t h e i n s o l a t i o n f o r AX r a n g i n g f r o m 0 ° t o - 90° allows also t h e d e t e r m i n a t i o n of I r e l a t i v e t o a n o t h e r m e r i d i a n o u t s i d e t h e above m e n t i o n e d i n t e r v a l ; in t h i s case, t h e f o l l o w i n g r e l a t i o n s h a v e t o be t a k e n i n t o a c c o u n t :
I ( A X + 180°, t + T ) = I ( A X , t ) (23) and
- 1 7 -
T T I(- 90° - a, 2 o
+ At) = I(- 90° + a, Y - At) • (24)
Expression (23) indicates that the curves l(AA) and l(AA + 180°) are identical but phase shifted over a sidereal period of revolution TQ; indeed, during the course of one revolution T (= 88 days), the planet Mercury spins exactly 1.5 times on its axis. This means that, for example, the position of the meridian AA at t = 0 will be occupied by the meridian (AA + 180°) at a time t equal to T .
On the other hand, during the first half of the revolution period
T 2 (t = = 44 days) the planet effectuates a ^ spin orbit coupling im-
plying that the meridians (- 90° + a) and (90° + a) occupy at t = 44 days, i.e. at aphelion, symmetrical positions with respect to the Sun- Mercury direction. During the second half of the revolution period the instantaneous insolation on the meridian (- 90° - a) at t = + At T
2 T0 equals the solar radiation falling on the meridian (- 90° + a) at t = g- - At; hence it follows that the two insolation curves are symmetric with
To
respect to the ordinate t = or W = 180°. These findings are illustrated in Figure 3 for the curves AA. = - 45° and AA = - 135° (a = 45°).
Figure 3 also reveals that in the vicinity of the perihelion (t = 0 and t = 88 respectively) the numerical value of the expression (nt - W) is slightly negative for 0° < W < 45° and scarcely above 180° for 315° <
W < 360°. The reason for this phenomenon is that in the neighbourhood of perihelion the Mercurian angular velocity of revolution (W) exceeds the rotational angular velocity (n) of the planet on its own axis (see also Soter and Ulrichs, 1967; Han-Shou Liu, 1968; Turner, 1978). This effect is illustrated in Figure 4, the curve representing the variability of W being obtained by Kepler's second law.
W = n (1 - e ) 2 x 1 / 2 ( a } 2
rO (25)
with
- 1 8 -
2
1 + + (2e + | e3) cos M + | e2 cos 2M + ^ e3 cos 3M (26)
where M is the mean anomaly.
Note that in expression (26) we kept only terms up to the third degree in e.
A s a consequence of this effect an interesting feature, already pointed out p r e v i o u s l y , r e g a r d s the meridian AA. = - 90°: at the peri- helion passage of the planet it is found that a v e r y short period of week insolation is preceded or followed by an even smaller period of complete d a r k n e s s . T h i s s u p r i s i n g phenomenon is clearly demonstrated in the upper part of Figure 4. Indeed, by analysing the instantaneous insolation curve it can be seen that a surface element is only illuminated from the eighth day (t = 8) with an insolation period extending to day 81 followed by a time interval of complete d a r k n e s s of approximately 7 d a y s . At the next perihelion passage of the planet, which occurs at day 88, the surface element considered here is insolated again for nearly 7 d a y s . Hence, the solar radiation is incident till day 95. At this moment the surface element considered returns into the nonilluminated hemi- sphere.
3.2. Latitudinal variation
It should be emphasized that in the theory of insolation presented above we only have considered a surface element located at the equator ( cp = 0). However, taking into account expression (19), the instan- taneous insolation received at any latitude can also easily be calculated as a function of time on a fixed meridian or as a function of any meridian at a given time.
A s an example, Figure 5 illustrates the distribution of the instan- taneous insolation on the meridian (AX = 0) crossing the line of apsides
£ )
r©
-19-
TIME (Earth days)
Fig. 4.- Mercurian angular velocity of revolution (W) and rotational angular velocity (n) of the planet on its own axis as a function of time (lower
/
part of the figure). The upper part shows the time variability of the instantaneous insolation at the top of tihe planet for cp = 0 (equator).
TIME ( E a r t h days)
Fig. 5.- Distribution of the instantaneous insolation at the top of Mercury on the meridian (AA. = 0) crossing the line cf apsides at the perihelion passage of the planet taken as t = 0. The incident solar radiation is given in contours of calories per square centimeter per day as a function of latitude ( c p ) and time (t) taken over 1.5 Mercurian solar days or approximately 264 Earth days. Areas of permanent darkness are dotted.
at t h e p e r i h e l i o n passage of t h e p l a n e t t a k e n as t = 0. T h e i n c i d e n t solar r a d i a t i o n is g i v e n in c o n t o u r s of c a l o r i e s p e r s q u a r e c e n t i m e t e r p e r d a y as a f u n c t i o n of p l a n e t a r y l a t i t u d e a n d time ( t a k e n o v e r 1 . 5 M e r c u r i a n solar d a y s o r 264 E a r t h d a y s ) . T h e r a t i o T / T _ b e i n g a b o u t 1 o G
2 i t is e a s i l y t o see t h a t t h e i n s o l a t i o n c u r v e s h o u l d be r e p e a t e d i n a t w o - m e r c u r i a n y e a r c y c l e .
From t h e f i g u r e , i t can be seen t h a t t h e maximum solar r a d i a t i o n is i n c i d e n t on t h e e q u a t o r at t h e f i r s t and t h e t h i r d p e r i h e l i o n passage o f
4 - 2 - 1
t h e p l a n e t w i t h a v a l u e of a b o u t 3 x 10 cal cm ( d a y ) . For e x a c t l y h a l f a s o l a r d a y (88 E a r t h d a y s ) all p a r t s o f t h e m e r i d i a n AX = 0 a r e in p e r p e t u a l d a r k n e s s , w h e r e a s a s u r f a c e element located at t h e poles n e v e r r e c e i v c c solar r a d i a t i o n . F u r t h e r m o r e , b c c o u c c of t h e z e r o o b l i q u i t y , t h e d i s t r i b u t i o n of t h e r a d i a t i o n i n c i d e n t on t h e M e r c u r i a n a t m o s p h e r e is p e r f e c t l y s y m m e t r i c w i t h r e s p e c t t o t h e e q u a t o r .
When c o m p a r i n g o u r r e s u l t s w i t h t h o s e of V o r o b ' y e v a n d Monin (1975) some a g r e e m e n t s as well as one s t r i k i n g d i f f e r e n c e a r e n o t i c e d . On one h a n d , t h e i n s o l a t i o n p a t t e r n is s i m i l a r a n d t h e c a l c u l a t e d maximum r a d i a t i o n i n c i d e n t is of t h e same o r d e r of m a g n i t u d e . On t h e o t h e r h a n d , t h e r e e x i s t s a d i s c r e p a n c y as t o t h e p o s i t i o n o f t h e m a x i m u m . O b v i o u s l y , t h i s maximum has t o o c c u r at t h e p e r i h e l i o n passage of M e r c u r y w h e n t h e S u n is on t h e m e r i d i a n p l a n e b u t t h i s is n o t t h e case a c c o r d i n g t o t h e p a p e r p u b l i s h e d b y t h e p r e v i o u s a u t h o r s . I t is d i f f i c u l t t o j u d g e w h e t h e r t h e c o m p u t i n g a l g o r i t h m o r a m i s t a k e in d r a w i n g t h e f i g u r e is r e s p o n s i b l e f o r t h e o b s e r v e d d i s a g r e e m e n t .
3 : 3 . D i u r n a l i n s o l a t i o n
For c o m p l e t e n e s s , we also h a v e i n v e s t i g a t e d t h e d i u r n a l i n s o l a t i o n lD o b t a i n e d b y i n t e g r a t i n g n u m e r i c a l l y e x p r e s s i o n ( 1 6 ) o v e r time d u r i n g t h e l i g h t time of t h e d a y . T h e i n t e g r a t i o n l i m i t s may be d e t e r m i n e d f r o m t h e f o l l o w i n g r e l a t i o n s :
- 2 2 -
n t 2 " W ( t2 ) = 2 7° ° " M
and
n tx - W ( t j ) = 90° - AA
( 2 7 )
(28)
In f i g u r e 6 we have plotted the v a r i a b i l i t y of the f u n c t i o n f ( t ) = n t - W ( t ) o v e r a time period equal to one M e r c u r i a n solar day and w h e r e W ( t ) is g i v e n b y formula ( 6 ) .
T h e p r o c e d u r e to obtain t^ and t^ is as follows. For a f i x e d meridian AX one calculates t h e c o r r e s p o n d i n g functions f ( t ^ ) and f ( t ^ ) g i v e n b y expressions ( 2 7 ) and ( 2 8 ) . T h e n , t^ and t^ can easily be f o u n d from f i g u r e 6 . T h e a c c u r a c y obtained with this g r a p h i c a l method to evaluate t h e d i u r n a l insolation is s u f f i c i e n t l y high c o n s i d e r i n g the small v a l u e of t h e solar radiation i n c i d e n t in the neighbourhood of t h e lower and u p p e r time limits.
T h e v a r i a b i l i t y of t h e d i u r n a l insolation on t h e e q u a t o r as a f u n c t i o n of t h e longitude d i f f e r e n c e between t h e meridian of a s u r f a c e element and t h e meridian crossing t h e line of apsides at t h e perihelion passage of t h e planet is g i v e n in F i g u r e 7. It can be seen t h a t t h e d i u r n a l insolation decreases by n e a r l y a f a c t o r of 3 as the longitudinal d i f f e r e n c e increases from 0 ° to 9 0 ° . T h i s e f f e c t has, to the best of our k n o w l e d g e , n e v e r been r e p o r t e d p r e v i o u s l y .
4. SOLAR R A D I A T I O N I N C I D E N T ON V E N U S
4 . 1 . Instantaneous insolation
T h e instantaneous insolation v a r i a t i o n at the top of t h e planet Venus and f o r a s u r f a c e element c h a r a c t e r i z e d b y the coordinates (Aq, cp) or (A\,cp) can also be determined from relations ( 5 ) to ( 8 ) and ( 1 3 ) .
- 2 3 -
TIME (Earth days)
F i g . 6 . - R o t a t i o n angle d i f f e r e n c e (nt-W) for Mercury as a function of time ( t ) .
-24-
LONGITUDE DIFFERENCE (degrees)
Fig. 7.- Diurnal insolation at the top of Mercury as a function of the longitude difference AA.. The curve corresponds to the equator.
I n t r o d u c i n g the numerical v a l u e s for SQ / A^ and e ( T a b l e I ) , e x p r e s s i o n ( 5 ) may be written u n d e r the following form :
I = 5 . 3 4 0 x 1 03 (1 + 0 . 0 0 6 7 8 cos W )2 [ s i n c p s i n
+ cos cp cos 6q cos (A - -^q)] (29)
It s h o u l d be pointed out that, t a k i n g into account the small value of the eccentricity of the orbit (e = 0.00678), formula ( 6 ) may be simplified. Keeping only terms up to the f i r s t degree in e, the t r u e anomaly W is now, in a v e r y good approximation, g i v e n by :
W = n t + 2 e sin n t (30)
o o
Following V o r o b ' y e v and Monin (1975) we f i r s t calculated the time variation, o v e r 3 V e n u s i a n solar d a y s ( = 350 Earth d a y s ) , of the i n s t a n - taneous insolation on the meridian (AX = 0) c r o s s i n g the line of a p s i d e s at the planet's perihelion p a s s a g e . Application of e x p r e s s i o n (29) leads to the isocontours s h o w n in f i g u r e 8.
From an a n a l y s i s of the solar radiation it may be concluded that the maximum is incident near the equator with a value of approximately
3 - 2 - 1
5.4 x 10 cal cm ( d a y ) . T h i s maximum o c c u r s at the f i r s t perihelion p a s s a g e (t=0) whereas the position of the other maxima is s l i g h t l y shifted with respect to the next aphelion or perihelion p a s s a g e s . It can be seen that for about half a solar d a y (= 58 Earth d a y s ) all p a r t s of the V e n u s i a n meridian AX = 0 are in permanent d a r k n e s s except for h i g h latitudes where the periods of non illumination ( o r illumination) increase in both hemispheres from half a solar day to approximately one solar d a y at the poles. O w i n g to the v e r y small obliquity the insolation d i s t r i - bution is q u a s i symmetric with respect to the planet's equator. A t the poles however this symmetry v a n i s h e s completely.
T
F u r t h e r m o r e , it is o b v i o u s that the ratio (= 0 . 9 2 ) cannot be e x p r e s s e d , as for M e r c u r y , b y a n y simple rational fraction a l t h o u g h the
- 2 6 -
TIME ( E a r t h days)
Fig. 8.- Distribution of the instantaneous insolation at the top of Venus on the meridian (AA = 0) crossing the line of apsides at the perihelion passage of the planet taken as t = 0. The incident solar radiation is given in contours of calories per square centimeter per day as a function of latitude (cp) and time (t) taken over 3 Venusian solar days or approximately 350 Earth days. Areas of permanent darkness are dotted.
numerical value is o n l y s l i g h t l y different from u n i t y . It follows, as a l r e a d y stated by V o r o b ' y e v and Monin (1975) that the i n s t a n t a n e o u s insolation pattern is o n l y q u a s i periodic in n a t u r e and that it s h o u l d r o u g h l y be repeated in a half y e a r cycle. T h e deviation from r i g o r o u s periodicity are however small since both the eccentricity and the o b l i q u i t y are small.
O u r r e s u l t s are compared with the data of V o r o b ' ye v and Monin ( 1 9 7 5 ) . T h e insolation c u r v e s are in reasonable agreement at all lati- t u d e s but o b v i o u s l y differ at the polar r e g i o n s . A l t h o u g h the maximum radiation is identical there does exist a difference relative to its position. A g a i n , it is difficult to elucidate the problem of p h a s e s h i f t .
For completeness, analogous calculations of the insolation as a function of time and latitude and for v a r i o u s v a l u e s of AA were made.
C o m p a r i s o n of a series of f i g u r e s , not p r e s e n t e d in t h i s p a p e r , reveals that the solar radiation difference is extremely small on p a s s a g e from one meridian to another. T h i s effect is a s c r i b e d to the small eccentricity (e = 0.00678) of V e n u s .
II 4.2. D i u r n a l insolation
Finally, the diurnal insolation I p as a function of latitude and for a f i x e d meridian (AX = 0) was studied b y numerical integration of re- lation ( 1 6 ) . It s h o u l d be noted that the u p p e r and lower time limits were taken from f i g u r e 8.
T h e diurnal insolation lD for the f i r s t solar d a y , a r b i t r a r y taken from t = 0 to t = TQ, is r e p r e s e n t e d in f i g u r e 9. From this f i g u r e it can be seen that the maximum diurnal insolation o c c u r s o v e r the equator
* 5 - 2 - 1
with a value of approximately 2 x 10 cal cm ( d a y ) . A s i g n i f i c a n t hemispheric asymmetry e x i s t s in that there is c o n s i d e r a b l y more i n s o - lation o v e r the s o u t h e r n polar r e g i o n s than o v e r the n o r t h e r n polar r e g i o n s d u r i n g t h i s solar d a y . For comparaison, the diurnal insolation
DIURNAL INSOLATION (cal cm2 day1)
F ig - 9- ~ Diurnal insolation at the top of Venus for the first solar day, arbitrary taken from t = 0 to t = TQ as a function of latitude (9).
received at the S o u t h and the N o r t h Pole amounts to about 1.1 x 10 3 - 2 - 1
a n d 2 . 9 x 10 cal cm ( d a y ) r e s p e c t i v e l y for the specific d a y mentioned above.
T h i s phenomenon is attributed to the fact that the total amount of the d i u r n a l insolation, especially at polar region latitudes, is s t r o n g l y d e p e n d e n t u p o n the a r b i t r a r y c h o s e n lower time limit of the so-called f i r s t solar d a y . indeed, integration of e x p r e s s i o n (16) o v e r a period equal to the p l a n e t ' s solar d a y T^ but for lower time limits t t 0 ( w h e r e t = 0 was taken at the planet's perihelion p a s s a g e ) yields numerical v a l u e s different from those obtained for t = 0. For example, if t = 0 is taken at the time of s u n r i s e at the S o u t h Pole, the S u n remains above its horizon o v e r a whole solar d a y , whereas the N o r t h Pole is in p e r - manent d a r k n e s s . O n the other h a n d , if t = 0 c o r r e s p o n d s to the time of maximum i n s t a n t a n e o u s insolation on one of the poles, the d i u r n a l insolation on both poles is approximately equal.
6. C O N C L U S I O N S
T h e solar radiation incident at the top of the atmospheres of the planets M e r c u r y and V e n u s d e p e n d s on a certain number of parameters related to the heliocentric o r b i t s of those two planets and also to the position of the s u r f a c e element s e n s e d b y this radiation. T h e intensity of the radiation at a g i v e n time is fixed by the heliocentric distance of the planet a n d b y the zenith angle of the S u n .
For M e r c u r y , the effect of the heliocentric distance c a u s e s a relative variation b y a factor of about 2.3 between perihelion and aphelion position; this result associated with the slow rotation period of M e r c u r y on its own a x i s implies a g r e a t d i v e r s i t y in the i n s t a n t a n e o u s insolation c u r v e s obtained at the equator on points of different l o n g i - t u d e s ; p e r t a i n i n g to the diurnal insolation, a similar conclusion can be d r a w n , the maximum relative variation attaining a factor of about 3.
M o r e o v e r , in the n e i g h b o u r h o o d of the perihelion p a s s a g e , the large eccentricity of M e r c u r y leads to an a n g u l a r velocity of revolution exceeding the rotational a n g u l a r velocity of the planet on its own a x i s . A s a c o n s e q u e n c e of this effect, a v e r y s h o r t period of weak insolation is preceded or followed b y an even smaller period of complete d a r k n e s s . T h e above mentioned phenomena were not found for the planet V e n u s , the orbital eccentricity being nearly equal to zero. O n the other h a n d , the small obliquity of its orbit g i v e rise to an important seasonal effect in the v i c i n i t y of the poles. O n M e r c u r y , this effect is n o n - e x i s t e n t , the d i s t r i b u t i o n of the incident solar radiation being perfectly symmetric with respect to the e q u a t o r . T h i s symmetrical b e h a v i o u r , which one practically o b s e r v e s also outside the polar r e g i o n s of V e n u s , yields to an identical rate of decrease of the insolation with i n c r e a s i n g latitude in both h e m i s p h e r e s .
T h e periodicity of the instantaneous insolation c u r v e s is fixed b y the length of the solar d a y of the planets : exactly two tropical y e a r s in the case of M e r c u r y and approximately half a tropical y e a r for V e n u s . It s h o u l d be emphasized that, in reality, the d i u r n a l insolation variations on V e n u s are only q u a s i periodic in n a t u r e when t a k i n g into account that both the. eccentricity of the orbit and the obliquity are n o n - z e r o a l t h o u g h v e r y small.
A C K N O W L E D G E M E N T S
T h e careful d r a w i n g s of J. S C H M I T Z are v e r y much appreciated.
We also wish to e x p r e s s o u r t h a n k s to the a n o n y m o u s referees for their helpful comments and c o n s t r u c t i v e criticisms.
REFERENCES
B R I N K M A N , A . W . and McGREGOR, J. ( 1 9 7 9 ) . T h e effect of the r i n g system on the solar radiation reaching the top of S a t u r n ' s atmo- sphere : D i r e c t radiation. I c a r u s . 38, 479-482.
H A N - S H O U LIU ( 1 9 6 8 ) . M e r c u r y has two permanent thermal bulges.
Science. 159, 306-307.
L E V I N E , J . S . , KRAEMER, D . R . and K U H N , W . R . ( 1 9 7 7 ) . Solar radia- tion incident on Mars and the outer planets : l a t i t u d i n a l , seasonal and atmospheric e f f e c t s . I c a r u s . 31^, 136-145.
M U R R A Y , B . C . , W A R D , W . R . and Y E U N G , S . C . ( 1 9 7 3 ) . Periodic inso- lation variations on Mars. Science. 180, 638-640.
S O T E R , S. and U L R I C H S , J. ( 1 9 6 7 ) . Rotation and heating of the planet M e r c u r y . N a t u r e . 214, 1315-1316.
T U R N E R , L . E . ( 1 9 7 8 ) . Diurnal motion of the Sun as seen from M e r c u r y . Am. J. Phys. 46, 475-479.
V O R O B ' Y E V , V . I . and M O N I N , A . S . ( 1 9 7 5 ) . U p p e r - b o u n d a r y insolation of the atmospheres of the planets of the solar systems. Atm. and Oceanic Phys. 1_1, 557-560.
WARD, W . R . ( 1 9 7 4 ) . Climatic variations on Mars. 1. Astronomical t h e o r y of insolation. J. Geophys. Res. 79, 3375-3386.
- 3 2 -
