ICARUS 48, 167--179 (1981)
Some Aspects of the Solar Radiation Incident at the Top of the Atmospheres of Mercury and Venus
E. VAN H E M E L R I J C K AND J. VERCHEVAL
Belgian Institute for Space Aeronomy, 3, Avenue Circulaire, B-II80 Brussels, Belgium
Received May 28, 1981; revised September 9, 1981
A formalism has been developed for the calculation of the insolation on the planets Mercury and Venus neglecting any atmospheric absorption. For Mercury, the instantaneous insolation curves are repeated in a 2-tropical-year cycle, the distribution of the solar radiation being perfectly symmetric between both hemispheres. In addition to latitudinal variations, one observes a longitu- dinal effect expressed by different instantaneous insolation distributions during 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 curves are repeated in a nearly half- year cycle.
1. INTRODUCTION
The knowledge of the distribution of the solar radiation incident at the top of the atmospheres of the planets of the solar sys- tem as a function of the coordinates of a surface element and time (or season) is in- dispensable to the study of its climatic his- tory, energy budget and dynamical behav- ior.
Although the upper-boundary insolation of the Earth's atmosphere and the radiation incident on its surface have been treated extensively, it must be pointed out that the- oretical investigations relative to the other planets are scarce and often incomplete.
Among the papers published during the last years we cite those of Murray et al. (1973), Ward (1974), Vorob'yev and Monin (1975), Levine et al. (1977), and Brinkman and Mc- Gregor (1979).
Concerning more particularly the inner planets Mercury and Venus, the distribu- tion of incident solar radiation has only been partially studied (Soter and Ulrichs, 1967; Han-Shou Liu, 1968; Vorob'yev and Monin, 1975). For example, the diurnal in- solation has never been analyzed in detail.
Moreover, it should be emphasized that the
insolation variation has generally been de- termined as a function of the latitude of a surface element, neglecting any longitudi- nal effect. Taking into account the large eccentricity of the heliocentric orbit of Mercury 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 practi- cally no atmosphere, the analysis of its in- solation may be of great interest to the study of the thermal budget on its surface.
The main objective of the present paper is to analyze in detail solar radiation prob- lems of Mercury and Venus, the nearest planets to the Sun. Another characteristic feature of those two slowly rotating planets is that they have sidereal periods of axial rotation of the same order of magnitude as the sidereal periods of revolution in their heliocentric motions. Furthermore, the in- clination of the equator with respect to the planetary orbit is very small (zero for Mer- cury). However, two main differences have to be noted between the two planets. First, Mercury rotates on its axis in the same direction in which it revolves around the Sun, whereas Venus is the only planet in the solar system rotating in the opposite
167
0019-1035/81/110167-13502.00/0 Copyright © 1981 by Academic Press, Inc.
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168 V A N H E M E L R I J C K A N D V E R C H E V A L direction (retrograde motion). S e c o n d l y ,
the eccentricities o f their orbits (0.00678 for Venus and 0.20563 for M e r c u r y ) r e p r e s e n t the e x t r e m e values w h e n listing this p a r a m - eter of all planets o f the solar s y s t e m . Al- though the last t w o findings j u s t i f y a some- w h a t different theoretical a p p r o a c h to the insolation p r o b l e m o f the two planets we present, in a first section, a general formal- ism applicable to b o t h planets. Then, taking into a c c o u n t their p r o p e r conditions, we calculate the i n s t a n t a n e o u s as well as the diurnal u p p e r - b o u n d a r y insolation o f the at- m o s p h e r e s o f M e r c u r y and Venus.
2. G E N E R A L FORMALISM
T h e i n s t a n t a n e o u s insolation is defined as the solar heat flux s e n s e d at a given time b y a horizontal unit a r e a of the u p p e r bound- ary o f the a t m o s p h e r e at a given point on the planet and per unit time.
In o u r calculations use will be made of the following p a r a m e t e r s as illustrated in Fig. 1 (see also Ward, 1974).
(O; x, y, z) r e p r e s e n t s an equatorial coor- dinate s y s t e m , w h e r e a s (O: x ' , y ' , z')
\. -, t,9. . - - - - "
j ;
EClUI'NOX t~RIHELION 0°~
FIG. 1. Schematic representation of some parame- ters used in the equations for the instantaneous and diurnal insolations on Mercury and Venus.
defines an orbital c o o r d i n a t e s y s t e m . It should be n o t e d that in b o t h s y s t e m s the origin O is located in the c e n t e r o f the planet. T h e a x e s y and y' coincide and are directed t o w a r d the Sun at the e p o c h o f the vernal equinox o f the planets. The z axis contains the spin or rotation axis o f the planet, r is the unit v e c t o r characterizing the direction o f a surface e l e m e n t and rc~ is the unit v e c t o r directed t o w a r d the Sun. h and ~0 are r e s p e c t i v e l y the longitude and the latitude o f the surface e l e m e n t in the equa- torial s y s t e m , the y axis being t a k e n as the origin o f the longitudes ( N o t e that the so- defined longitude is time d e p e n d e n t owing to the rotation o f the planet), he and 8~
define the longitude and the latitude o f the Sun in the s a m e s y s t e m , h~ and hp are the t
planetocentric longitude o f the Sun and the planetocentric longitude o f the p l a n e t ' s perihelion in the orbital s y s t e m , and E is the obliquity or inclination o f the e q u a t o r with r e s p e c t to the plane o f the orbit.
T h e instantaneous insolation I m a y now be written in the following form:
with
and
I = S(r • r,.>) (1)
I > 0 i f r .r~c > 0, / = 0 i f r . r ~ = < 0 ,
S = So~r> 2 (2)
r~ = a~(l - e2)/(1 + e cos W). (3) S is the solar flux at an heliocentric distance r o and So is the solar c o n s t a n t at the m e a n S u n - E a r t h distance o f I A U . F o r the calcu- lations p r e s e n t e d in this p a p e r we have a d o p t e d the m o s t reliable value of the solar c o n s t a n t o f 1.94 cal c m -2 min -1 (or 2.79 x 103 cal c m -2 day-a).
In e x p r e s s i o n (3), a~, e, and W are re-
spectively the p l a n e t ' s s e m i m a j o r axis, the
eccentricity, and the true a n o m a l y . Table I
r e p r e s e n t s , for the two planets under con-
sideration, the numerical values o f the pa-
r a m e t e r s used for the c o m p u t a t i o n o f the
SOLAR RADIATION ON MERCURY AND VENUS TABLE I
ELEMENTS OF THE PLANETARY ORBITS OF MERCURY AND VENUS
169
Planet a e h l e To T
(AU) (o) (°) ( E a C h d a y s ) ( E a c h d a y s )
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
i n s t a n t a n e o u s and diurnal insolations. In this table one c a n find also the sidereal pe- riod o f revolution T O (tropical year) a n d the sidereal period o f axial rotation T (sidereal day). Finally, it has to be m e n t i o n e d that the a r g u m e n t o f perihelion h~ is t a k e n f r o m V o r o b ' y e v and Monin (1975).
The c o m p o n e n t s o f the unit v e c t o r s r and r G are, respectively:
r = (cos ~0 sin h, cos ~ c o s h, sin ¢) and
r o = ( c o s 8 G s i n h o,
c o s 8 o cos h G, sin Be).
With this in mind, we c a n write:
I = ( S o / r o 2 ) [ s i n ~ sin 8 e
+ cos ~, c o s 8 e c o s (X - he)] (4) or
I = { S o / [ a e 2 ( 1 - e2)2]}
(1 + e cos W)2[sin ~ sin 8 o
+ cos ~¢ c o s 8® cos(X - he)]. (5) T h e solar radiation I has to be e x p r e s s e d as a function o f time, the time d e p e n d e n c e being a function o f the a r g u m e n t s W, 8 o, and (h - he). I n d e e d , taking t = 0 at the perihelion p a s s a g e o f the planet, the true a n o m a l y W is given by:
W = not
+ (2e - e3/4) sin not + (5/4)e 2 sin 2 not
+ (13/12)e a sin 3 not, (6) w h e r e n o = 27r/To designates the m e a n an- gular m o t i o n .
In relation (6), k n o w n in celestial m e - c h a n i c s as the e q u a t i o n o f the c e n t e r , we
kept only terms up to the third degree in e.
This a p p r o x i m a t i o n gives s a t i s f a c t o r y results taking into a c c o u n t that the maxi- m u m value o f the eccentrities c o n s i d e r e d here is a p p r o x i m a t e l y equal to 0.2 (Mer- cury).
F u r t h e r m o r e , the solar declination 8 o c a n be calculated using the following relations:
sin 8e = sin e sin (h~ + W) (7) and
c o s 8 o = (1 - sin* 8o) l/z, (8) w h e r e [Be[ is always smaller than 90 °.
If, on the o t h e r hand, one a s s u m e s that at t = 0 the longitude o f the surface e l e m e n t is d e n o t e d b y ho, the time d e p e n d e n c e o f h is d e s c r i b e d b y the e x p r e s s i o n :
h = ho + _ n t
o r
h = h p + A h +_ n t , (9) w h e r e n = 2 ~ r / T is the rotational angular velocity o f the planet and Ah is the longi- tude difference b e t w e e n the meridian o f the surface e l e m e n t and the meridian crossing the line o f apsides at the perihelion p a s s a g e of the planet which is, as a l r e a d y men- tioned, t a k e n as t = 0. N o t e that the minus sign is a d o p t e d for Venus (retrograde rota- tion).
Finally, the following set o f equations c a n also be obtained:
hp = arctan (cos ~ tan h~), (10) X G = a r c c o s (sec 8 e c o s X~), (11)
h o = h~, + W. t (12)
170 VAN HEMELRIJC K AND VERCHEVAL H e n c e , the angle (h - hQ) can be deter-
mined by combining (9), (10), (11), and (12), yielding:
h - h e
= arctan (cos ~ tan h'o) + Ah +_ nt - arccos [sec 8 e cos (h~ + W)]. (13) Taking into a c c o u n t the expressions (5)- (8) and (13), the instantaneous insolation variation at the tops o f the slowly rotating planets M e r c u r y and Venus and for a sur- face element c h a r a c t e r i z e d b y the coordi- nates (ho, ~o) or (Ah, ~o) can now easily be obtained.
The diurnal insolation, I m can be found by integrating relation (5) numerically o v e r a period equal to the planet's solar day Te:
ID = I - ~ I d t . (14)
J t ~ 0
It has to be pointed out that I = 0 when the condition r • r o ~ 0 is fulfilled. Further- more the length o f the solar day TQ, figuring in relation (14), is defined by:
1~To = 1/T-4- 1~To (15) the plus sign being used for a planet rotating in the opposite direction in which it re- volves around the Sun (of which Venus is the only case). Practically, relationship (14) may also be written under the following form:
ID = f~l l dt + ftre l dt, (16)
2
w h e r e tl and t2 c o r r e s p o n d respectively to the time o f setting and rising o f the Sun; in this case I is always positive. The analytic expression o f / o , given by Ward (1974) and also by L e v i n e et al. (1977) and V o r o b ' y e v and Monin (1975) is only applicable to rap- idly rotating planets where r e and 8 e are assumed to be constant in the integration o f the instantaneous insolation I o v e r a period equal to the planets solar day T e. For slowly rotating planets this p r o c e d u r e is not c o r r e c t w h e n the eccentricity e or the obliquity ~ are not negligible.
3. SOLAR RADIATION INCIDENT ON MERCURY
3.1. Instantaneous Insolation
The obliquity ~ o f M e r c u r y being equal to zero, the formalism discussed in the pre- vious section m a y be simplified. In this case we have:
and
8 e = O
h - h G = Ah + n t - W . (17) H e n c e , the a m o u n t o f solar radiation in- cident on M e r c u r y can be e x p r e s s e d by the following equation:
I(AX, ,p)
= (So/ro 2) cos ~o cos(h - ko). (18) F r o m relation (18) it follows also that:
HAM ~0)/l(Ak, 0) = cos ~p. (19) F r o m (19) it is obvious that the instanta- neous solar radiation at a given time and at a given latitude ~o can easily be determined by multiplying the insolation on the same meridian at the same m o m e n t by the cor- rection factor cos ~, and for a surface ele- ment located on the e q u a t o r ~ = 0.
Introducing the numerical values for So, a~, and e (Table I) e x p r e s s i o n (18), applied to the equator, finally yields:
I(Ak, 0) = 2.033 x 104(1 + 0.20563
cos W) 2 c o s ( A h + n t - W). (20) Figure 2 illustrates the solar radiation in- cident at the top o f the Mercurian atmo- sphere as a function o f time and for three surface elements characterized by the fol- lowing specific numerical values o f Ah: 0, - 4 5 , and - 9 0 ° (as already mentioned above, Ah designates a fixed meridian on the planet).
The time variation o f the instantaneous insolation e x t e n d s o v e r three sidereal pe- riods of revolution (or tropical years) corre- sponding to approximately 264 E a r t h days.
H o w e v e r , it should be e m p h a s i z e d that one
insolation cycle c o v e r s two periods o f revo-
SOLAR RADIATION ON MERCURY AND VENUS 171
30x 103
~'E u 26
~ 2 2
z 0
z
m 14
z z
~ 6
z
I;
I r l i i r ! ! i r i r !
Z Z Z
z o z o z
_c2
0 ~ 0 .~ 0
W ~ W W
\',t, "',
) e - ' . Ax=-9o* / \ ;.,,"-,. Ax:-9o°
v ,, -,, / ,7 ',,
/ \ " , / , i \ ',
i \ ', ", I ; # \ ', ',
, , ii \
/ \ , ~ " ~.,, ~'\
11 \ \ ', ,/ , ,
l \ i! ~ i 'l~l I \ \ I
/ j ill '~, • I I I / / # I
2 0 4 0 6 0 8 0 100 120 140 1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0
TIME (Earth days)
FIG. 2. Time variations of the instantaneous insolation at the top of the Mercurian atmosphere on fixed meridians characterized by the following specific numerical values of Ak: 0, -45, and -90 °. The curves are related to the equator (~o = 0). Perihelion and aphelion passages are also illustrated.
lution, application o f relation (15) (with the minus sign for M e r c u r y ) s h o w i n g that T o = 2To = 176 E a r t h days.
An analysis o f Fig. 2 r e v e a l s that the insolation p a t t e r n as well as the m a x i m u m incident solar radiation strongly varies in passing f r o m one meridian to a n o t h e r . To the b e s t o f o u r k n o w l e d g e c o m p u t a t i o n s for AX = 0 h a v e b e e n m a d e b y V o r o b ' y e v a n d Monin (1975) w h e r e a s c u r v e s r e p r e s e n t i n g the insolation in a r b r i t r a r y units on the me- ridians Ah = 0, - 7 5 , - 9 0 , and - 1 0 5 ° h a v e b e e n r e p o r t e d b y Soter and Ulrichs (1967).
I t is also interesting to note that the calcula- tions for AA = - 9 0 ° result in a bizarre dis- tribution o f solar radiation in the neighbor- hood o f the perihelion p a s s a g e o f the planet showing a v e r y short period o f w e a k insola- tion p r e c e d e d o r followed b y an e v e n smaller period o f c o m p l e t e d a r k n e s s [see also Soter and Ulrichs (1967) and H a n - S h o u Liu (1968)]. An e x p l a n a t i o n for this phe-
n o m e n o n (not visible on Fig. 2 due to the insufficiency o f the scale a d o p t e d for the ordinate) will be given later (Fig. 4).
T h e longitudinal d e p e n d e n c e o f the inci- d e n t solar radiation as plotted in Fig. 2 will be clarified b y m e a n s o f Fig. 3. The d a s h e d lines r e p r e s e n t the theoretical insolation c u r v e s as a function o f true a n o m a l y W (during the c o u r s e o f one revolution) and for various values o f the angle o f incidence
= k - h o = A h + n t - W c o m p r i s e d
b e t w e e n 0 and 90 °. The c u r v e for ~ = 90 °
coincides with the axis o f abscissa. The
figure shows the effect o f ~ for given values
o f W, w h e r e a s the influence o f the heliocen-
tric distance r o is illustrated b y the c u r v e s
o f c o n s t a n t angle o f incidence. As a result,
the insolation varies 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 perihelion and aphelion
p a s s a g e for a s a m e zenith angle ~. Plotted
are also the time scale t (at the top o f the
figure) and the c o r r e s p o n d i n g values o f the
172 VAN H E M E L R I J C K A N D V E R C H E V A L
TIME (Earth days)
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 B0 85 88
I~1= ' . . . . . . . . . . . . . . . . ]
30xl03 - 0 . . . . J
- - ~ - . 2 0 1
°
:-- /"--"i'-...
~E 22
- . . . . so " ' - . "'.'" .<.'. . - " - " - ' " . " " . . . . --
5 ____ 60 " ' - - . " ' - . " - . " ' " -~- .-" . . " " . - " " __
_z
z
~
6 0o . . . . . . . .
z z
I I I I
-0S -0.2 ~0 17 6.3 11.5 27.0 /*/*.A 65.9 90 114.1 135.7 153,0 165.5 173.7 178.3 180.2 180.5 180 ROTATION ANGLE DIFFERENCE (Degrees)
[ I I I I I I I I I 1 I I I I I i I I
0 20 /*0 60 80 100 ~20 140 160 180 200 220 240 260 280 300 320 3/*0 360
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 ~ = h - hQ - Ah + nt
-W. The solar radiation incident for surface elements characterized by longitude differences &h equal to 0, -45. -90, and - 135 ° is given by the full lines. The curves correspond to ~o = 0.
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 :
sin E = (I - e~)~/2sin W
/(1 + e c o s W) (21) a n d
t = ( 1 / n o ) ( E - e sin E), (22) 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 c a n b e a p p l i e d to a n y m e r i d i a n Ah to o b t a i n t h e s o l a r h e a t flux a r r i v i n g at a g i v e n t i m e o n a n h o r i z o n t a l u n i t a r e a o f t h e u p p e r b o u n d a r y o f t h e a t m o s p h e r e a t a g i v e n e q u a t o r i a l p o i n t o n t h e p l a n e t M e r - c u r y . T h e first 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 al- g e b r a i c a l l y Ah to 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 to 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 c a n b e 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 o f 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 o f the i n s o l a t i o n t o c h a n g e s in Ah w e h a v e a l s o p l o t t e d in F i g . 3 the s o l a r
r a d i a t i o n i n c i d e n t for Ah = 0, - 4 5 , - 9 0 , a n d - 135 °.
It is i n t e r e s t i n g to n o t e 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 Ah r a n g i n g f r o m 0 to - 9 0 ° a l l o w s a l s o t h e d e t e r m i n a t i o n o f I r e l a t i v e to a n o t h e r m e r i d i a n o u t s i d e the a b o v e m e n t i o n e d i n t e r v a l ; in this c a s e , 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 to b e t a k e n into a c c o u n t :
l ( A h + 180 °, t + To) = l ( A h , t) (23) a n d
1 ( - 9 0 ° - c~, T o ~ 2 + A t )
= 1 ( - - 9 0 ° + c~, T o ~ 2 - A t ) . (24)
E x p r e s s i o n (23) i n d i c a t e s t h a t t h e c u r v e s
I ( A h ) a n d I ( A h + 180 °) a r e i d e n t i c a l b u t
p h a s e s h i f t e d o v e r a s i d e r e a l p e r i o d o f r e v o -
l u t i o n To; i n d e e d , d u r i n g t h e c o u r s e o f o n e
r e v o l u t i o n T o ( ~ 8 8 d a y s ) , t h e p l a n e t M e r -
c u r y s p i n s e x a c t l y 1.5 t i m e s o n its a x i s .
T h i s m e a n s t h a t , f o r e x a m p l e , t h e p o s i t i o n
SOLAR RADIATION ON
of the meridian AA at t = 0 will be occupied by the meridian (AA + 1800) at a time t equal to T,.
On the other hand, during the first half of the revolution period (t = T,/2 = 44 days) the planet effectuates a $ spin-orbit cou- pling, implying that the meridians (-90” + ar) and (90’ + a) occupy at t = 44 days, i.e., at aphelion, symmetrical positions with re- spect to the Sun-Mercury direction. Dur- ing the second half of the revolution period the instantaneous insolation on the merid- ian (-90 - (Y) at t = TJ2 + At equals the solar radiation falling on the meridian (-90”
+ (u) at t = TJ2 - At; hence it follows that the two insolation curves are symmetric with respect to the ordinate t = T,/2 or W =
180”. These findings are illustrated in Fig. 3 for the curves AA = - 45” and Ah = - 135 (CY = 45”).
MERCURY AND VENUS 173 sion ( 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 neighborhood 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 illus- trated in Fig. 4, the curve representing the variability of W being obtained by Kepler’s second law.
with
w = n,( 1 - e2)l’2( a/r,)” (25)
(a/r,)” = 1 + e2/2
+ (2e + $e3) cos M + fez cos 2M + (13/4)e3 cos 3M, (26) Figure 3 also reveals that in the vicinity where M is the mean anomaly.
of the perihelion ( t = 0 and t = 88, respec- Note that in expression (26) we kept only tively) the numerical value of the expres- terms up to the third degree in e.
50 60 70 80 90 100 110 120 130
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 the planet for cp = 0 (equator).
1 7 4 V A N H E M E L R I J C K A N D V E R C H E V A L
As a c o n s e q u e n c e of this effect an inter- esting feature, already pointed out previ- ously, regards the meridian AX = - 9 0 ° : at the perihelion passage o f the planet it is found that a very short period o f weak inso- lation is preceded or followed by an e v e n smaller period of complete darkness. This suprising p h e n o m e n o n is clearly demon- strated in the upper part of Fig. 4. Indeed, by analyzing 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 o f com- plete darkness o f approximately 7 days. At the next perihelion passage o f the planet, which occurs at day 88, the surface element considered here is insolated again for nearly 7 days. H e n c e , the solar radiation is incident until day 95. At this moment the
surface element considered returns into the nonilluminated hemisphere.
3.2. Latitudinal Variation
It should be emphasized that in the the- ory o f insolation presented above we only have considered a surface element located at the equator (~ = 0). H o w e v e r , taking into account expression (19), the instanta- neous 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.
For example, Fig. 5 illustrates the distri- bution of the instantaneous insolation on the meridian (Ah = 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
90
60
3O
0
J - 30
-60
- 9 0 L 0
0 . . .
. . . M E r C U r Y . . .
...................... o .
...................... , °
. . . z . . .
~ . . . . .
o . . .- - . .......... ~ . . . O _
z . . . o . . .
t ~ . . . a . . . .
30 6 0 9 0 120 150 180 210 240
T I M E ( E a r t h d a y s )
FIG. 5. Distribution of the instantaneous insolation at the top of Mercury on the meridian (Ah = 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 (~0) and
time (t) taken o v e r 1.5 Mercurian solar days or approximately 264 Earth days. Areas of permanent
darkness are dotted.
SOLAR RADIATION ON MERCURY AND VENUS 175 planetary latitude and time (taken o v e r 1.5
Mercurian solar days or 264 E a r t h days).
The ratio T o / T G being a b o u t ½, it is easily to see that the insolation c u r v e should be re- peated in a 2-Mercurian year cycle.
F r o m the figure, it can be seen that the m a x i m u m solar radiation is incident on the e q u a t o r at the first and third perihelion pas- sage o f the planet with a value o f a b o u t 3 × 104 cal cm -2 (day) -1. F o r e x a c t l y half a solar day (88 E a r t h days) all parts o f the meridian Ah --- 0 are in perpetual darkness, whereas a surface element located at the poles n e v e r receives solar radiation. F u r t h e r m o r e , be- cause o f the zero obliquity, the distribution o f the radiation incident on the Mercurian a t m o s p h e r e is perfectly symmetric with re- spect to the equator.
When comparing o u r results with those o f V o r o b ' y e v and Monin (1975) some agreements as well as one striking differ- ence are noticed. On one hand, the insola- tion pattern is similar and the calculated m a x i m u m radiation incident is o f the same o r d e r o f magnitude. On the o t h e r hand, there exists a d i s c r e p a n c y as to the position o f the maximum. Obviously, this m a x i m u m has to o c c u r at the perihelion passage o f M e r c u r y w h e n the Sun is on the meridian plane but this is not the case according to the p a p e r published by the previous au- thors. It is difficult to j u d g e w h e t h e r the computing algorithm or a mistake in draw- ing the figure is responsible for the ob- s e r v e d disagreement.
3 . 3 . D i u r n a l I n s o l a t i o n
F o r c o m p l e t e n e s s , we also have investi- gated the diurnal insolation ID obtained by integrating numerically e x p r e s s i o n (16) o v e r time during the light time o f the day.
The integration limits m a y be d e t e r m i n e d from the following relations:
nt2
-W(t2) = 270 ° - Ah (27)
and
ntl - W ( t l ) = 9 0 ° - A)t. (28) In Fig. 6 we have plotted the variability
36C
~J 27q
a
k
5 0 100
TIME ( E Q r t h d o y s )
I I
l 150
FIG. 6. Rotation angle difference ( n t - W ) for Mer- c u r y as a f u n c t i o n o f time (t).
o f 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 Mercurian solar day and w h e r e W ( t ) is given b y formula (6).
The p r o c e d u r e to obtain tt and t~ is as follows. F o r a fixed meridian AA one calcu- lates the c o r r e s p o n d i n g functions f ( t ~ ) and f ( t ~ ) given by expressions (27) and (28).
T h e n , tl and t2 can easily be found from Fig.
6. The a c c u r a c y obtained with this graphi- cal m e t h o d to evaluate the diurnal insola- tion is sufficiently high considering the small value o f the solar radiation incident in the n e i g h b o r h o o d o f the l o w e r and upper time limits.
The variability o f the diurnal insolation on the e q u a t o r as a function o f the longitude difference b e t w e e n the meridian o f a sur- face e l e m e n t and the meridian crossing the line o f apsides at the perihelion passage o f the planet is given in Fig. 7. It can be seen that the diurnal insolation decreases by nearly a factor o f 3 as the longitudinal dif- ference increases from 0 to 90 °. This effect has, to the best o f our knowledge, n e v e r b e e n r e p o r t e d previously.
4. S O L A R R A D I A T I O N I N C I D E N T O N V E N U S 4 . 1 . I n s t a n t a n e o u s I n s o l a t i o n
The instantaneous insolation variation at
the top o f the planet Venus and for a sur-
176 VAN HEMELRIJC K AND VERCHEVAL
I [ I I I I I I
14
x 10 513
~E u
w u
I0
