AEROIMOMICA ACTA

Texte intégral

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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 O 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 I M O M I C A A C T A

A - N° 2 7 6 - 1 9 8 3

On the variations in the insolation at Mercury resulting from oscillations of the orbital eccentricity

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

B • 1180 B R U S S E L

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F O R E W O R D

T h e paper entitled " O n the v a r i a t i o n s in the insolation at M e r c u r y r e s u l t i n g from oscillations of the orbital e c c e n t r i c i t y " will be p u b l i s h e d in T h e Moon a n d the Planets, 1983.

A V A N T - P R O P O S

L'article intitulé " O n the v a r i a t i o n s in the insolation at M e r c u r y r e s u l t i n g from oscillations of the orbital e c c e n t r i c i t y " sera publié d a n s T h e Moon a n d the Planets, 1983.

V O O R W O O R D

De t e k s t " O n the v a r i a t i o n s in the insolation at M e r c u r y r e s u l t i n g from oscillations of the orbital e c c e n t r i c i t y " zal in het tijd- s c h r i f t T h e Moon and the Planets, 1983 v e r s c h i j n e n .

V O R W O R T

Der T e x t " O n the variations in the insolation at M e r c u r y

r e s u l t i n g from oscillations of the orbital e c c e n t r i c i t y " w i r d in T h e Moon

and the Planets, 1983 h e r a u s g e g e b e n w e r d e n .

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ON T H E V A R I A T I O N S IN T H E I N S O L A T I O N A T M E R C U R Y

R E S U L T I N G FROM O S C I L L A T I O N S OF T H E O R B I T A L E C C E N T R I C I T Y

by

E. V A N H E M E L R I J C K

Abstract

T h i s paper describes variations in the insolation on M e r c u r y resulting from fluctuations of the orbital eccentricity (0.11 ^ e ^ 0.24) of the planet. Equations for the instantaneous and the daily insolation are briefly discussed and several numerical examples are given illustrating the sensitivity of the solar radiation to changes in e.

Special attention is paid to the behavior of the solar radiation d i s t r i b u - tion c u r v e s near sunrise and sunset which at the warm pole of M e r c u r y (longitudes ± 90°) occur as the planet goes t h r o u g h perihelion. It has been found that for eccentricities larger than about 0.194 there exists two permanent thermal bulges on opposite sides of the Mercurian surface that alternately point to the S u n at every perihelion passage.

The critical value of e past which the S u n shortly sets after perihelion

is near 0.213.

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Résumé

Cet article décrit des variations dans l'insolation de Mercure qui proviennent des fluctuations de l'excentricité orbitale (0.11 S e â 0.24) de la planète. Les équations de l'insolation instantanée et diurne sont discutées brièvement et différents exemples numériques sont donnés, illustrant la sensibilité de l'insolation à des changements en e.

Beaucoup d'attention est prêtée au comportement des courbes de distribution de l'insolation près du lever et du coucher du soleil q u i , au pôle chaud de Mercure (longitudes ± 90°), se produisent lorsque la planète passe par le périhélie. Pour des excentricités supérieures à 0.194, il existe deux renflements thermiques permanents aux côtés opposés de la surface de Mercure qui, alternativement, pointent vers le Soleil à chaque passage du périhélie. La valeur critique de e au-dessus de laquelle le Soleil se couche durant une brève période après le passage au périhélie est de l'ordre de 0.213.

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Samenvatting

Dit artikel beschrijft variaties in de zonnestraling op Mercurius die afkomstig zijn van schommelingen van de baanexcentrici- teit (0.11 ^ e ^ 0 . 2 4 ) van de planeet. Vergelijkingen voor de ogenblikkelijke en dagelijkse zonnestraling worden bondig besproken en verscheidene numerieke voorbeelden, die de gevoelligheid van de zonne- straling aantonen t . o . v . veranderingen in e, worden gegeven. Veel aandacht wordt besteed aan het gedrag van de kurven die de verdeling van de zonnestraling weergeven bij zonsopgang en zonsondergang. Deze doen zich voor aan de warme pool van Mercurius (lengten ± 90°) wanneer de planeet doorheen het perihelium gaat. Voor excentriciteiten groter dan 0.194 bestaan er twee permanente thermische buiken aan de tegengestelde zijden van het Mercuriaans oppervlak die beurtelings naar de zon gericht zijn bij elke doorgang door het perihelium. De kritische waarde van e waarboven de zon een korte tijd ondergaat na de doorgang door het perihelium is ongeveer gelijk aan 0.213.

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Zusammenfassung

Dieses Artikel beschreibt Variationen in der Sonnenstrahlung auf dem Merkur, resultierend von Schwankungen der Bahnexzentrizität (0.11 ^ e ^ 0.24) des Planeten. Vergleichungen für die augenblickliche und tägliche Sonnenstrahlung werden bündig besprochen und verschiedene numerische Beispiele werden gegeben, die die Empfind- lichkeit von der Sonnenstrahlung illustrieren mit Rücksicht auf Veränderungen in e. Viele Aufmerksamkeit wird geschenkt am Benehmen der Kurven der die Verteilung der Sonnenstrahlung zeigen bei Sonnen- aufgang und Sonnenuntergang welche am warme Pol vom Merkur (Längen ± 90°) auftreten wenn der Planet hindurch das Perihelium geht. Für Exzentrizitäten grosser als 0.194 gibt eis zwei ständige thermische Bäuche an der entgegengesetzten Seiten von der merkurische Oberfläche der abwechselnd nach der Sonne gerichtet sind bei jedem Durchgang durch das Perihelium. Der kritische Wert von e worüber die Sonne während einer kurze Zeit sinkt nach dem Durchgang durch das Perihelium ist ungefähr 0.213.

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1. I N T R O D U C T I O N

Since the w o r k of B r o u w e r a n d v a n Woerkom (1950) on the t h e o r y of secular v a r i a t i o n s of the planetary elements, it has been possible to calculate the long-term periodic oscillation of the orbital eccentricity of M e r c u r y c a u s e d b y gravitational p e r t u r b a t i o n s from the S u n and the other planets. Now, we know ( C o h e n et a l . , 1973; Ward et al., 1976; V a n F l a n d e r n and H a r r i n g t o n , 1976) that the eccentricity ( e ) of M e r c u r y is c u r r e n t l y 0.20563, w h e r e a s it r a n g e s from a value near 0.11 to a theoretical maximum of approximately 0.24 d u r i n g the dynamical h i s t o r y of the planet. T h e oscillation has two s u p e r p o s e d p e r i o d s : one of 10 y r and another of 10 y r . 5 6

Some aspects of the solar radiation incident at the top of the atmosphere of M e r c u r y have been s t u d i e d b y e . g . Soter a n d U l r i c h s (1967), Liu ( 1 9 6 8 ) , V o r o b ' y e v and Monin (1975), V a n Hemelrijck and V e r c h e v a l (1981) and Landau (1982). In their computations the above mentioned a u t h o r s u s e d the p r e s e n t value of the eccentricity. H o w e v e r , the solar radiation being s t r o n g l y d e p e n d e n t u p o n this parameter, it is o b v i o u s that important periodic insolation v a r i a t i o n s on M e r c u r y are closely associated with fluctuations in e. In this p a p e r , t h e r e f o r e , we will s t u d y the effect of the time evolution of the eccentricity on the insolation at M e r c u r y .

In a f i r s t section, we briefly d i s c u s s some formulas needed

for the computation of the insolation. For more details on solar radiation

investigation we refer to Ward (1974), V o r o b ' y e v and Monin ( 1 9 7 5 ) ,

Levine et al. (1977), V a n Hemelrijck and V e r c h e v a l (1981, 1983),

L a n d a u (1982) and V a n Hemelrijck (1982a, b, c, d ; 1983a, b ) . T h e n ,

the influence of the oscillating orbital eccentricity on the instantaneous

as well as on the daily u p p e r - b o u n d a r y insolation is p r e s e n t e d .

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2. C A L C U L A T I O N O F T H E I N S O L A T I O N

T h e i n s t a n t a n e o u s insolation ( I ) is defined as the solar heat f l u x s e n s e d at a g i v e n time b y a horizontal unit area of the top of the atmosphere at a g i v e n point a n d per unit time.

A s s u m i n g the S u n as a point s o u r c e , I may be written u n d e r the following form

I = (S / r

2

) cos z i f z < 71/2 (1).

o' o '

1 = 0 i f z ^ n/2

with

r

O

= a

O

( 1

"

e 2 ) / ( 1 + e c o s W ) ( 2 )

and (for M e r c u r y ) ( V a n Hemelrijck and V e r c h e v a l , 1981)

z = AA. + nt - W (3)

where S is the solar constant at the mean S u n - E a r t h distance of 1 A U

o p

3

- 1

taken at 1.96 cal cm ( m i n ) or 2.82 x 10 cal cm ( p l a n e t a r y d a y )

(Wilson, 1982), r© is the heliocentric distance, a

Q

(0.3871 A U ) is the

semi-major a x i s , e is the eccentricity, W is the t r u e anomaly, z is the

zenith angle of the center of the S u n , AÀ is the longitude difference

between the meridian of the surface element c o n s i d e r e d and the meridian

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c r o s s i n g the line of apsides at the perihelion p a s s a g e of the planet which is taken as t = 0 and n (6.13811) is the rotational a n g u l a r velocity of the planet. Furthermore, keeping only terms up to the t h i r d degree in e (which is sufficiently accurate for o u r calculations), the true anomaly W is g i v e n b y

W = n t + (2e - e

3

/ 4 ) sin n t

o o

+ (5/4) e

2

sin 2 n t + (13/12) e

3

sin 3 n t (4) o o

where n^ (4.09235) designates the mean a n g u l a r motion.

A s already pointed out b y Landau (1975, 1982) a problem arises near s u n r i s e and s u n s e t when the horizon intersects the S u n ' s d i s k ( T h i s is particularly true at longitudes AA = ± 90° where e . g . for the c u r r e n t value of the eccentricity the S u n takes approximately 18 d a y s to rise or set). Indeed, for z > TT/2 some parts of the S u n may still be above the horizon and the instantaneous insolation will be different from zero. For z < 7t/2 the centroid of the visible portion of the S u n ' s d i s k will be smaller than z and I will again be greater than the insolation obtained b y e x p r e s s i o n ( 1 ) .

T a k i n g into account the finite a n g u l a r size of the solar d i s k at the intersection b y the horizon, the instantaneous insolation is, in a v e r y good approximation, g i v e n b y ( L a n d a u 1975, 1982)

I = (s /rh f cos z' (5)

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with

f = 1/2 + (u + sin u cos u)/rt (6)

z* = z - 2R c o s

3

u/3rt f (7)

where f is the fractional area of the Sun above the horizon, z

1

is the zenith angle of the centroid of this area and R is the angular radius of the Sun and depends upon the heliocentric distance r . Finally, the parameter u is defined as

u = arc sin [(rt/2-z)/R] (8)

T h e daily insolation ( l

D

) can be found by integrating relation ( 1 ) or ( 5 ) numerically over a period equal to the planet's solar day T (^ 176 Earth d a y s )

T

o

I

D

= ƒ I dt (9)

t = 0

Practically, relationship ( 9 ) may also be written as

T

0

I

D

= ƒ I dt + ƒ I dt (10)

o t

2

where t^ and 1

2

correspond respectively to the time of setting and

rising of the Sun [note that I is always positive using formula ( 1 0 ) ] .

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T h e integration limits (t^ and f °

r a

specific value of AA may be determined from the following relations ( V a n Hemelrijck and V e r c h e v a l , 1981)

n - w(t

2

) = 270° - AA (11)

and

n t j - W ( t

x

) = 90° - AA (12)

a n d from the g r a p h i c a l representation of the e x p r e s s i o n f ( t ) = nt - W ( t ) as a function of time. T h e a c c u r a c y obtained with this method is sufficiently h i g h c o n s i d e r i n g the small value of the solar radiation in the v i c i n i t y of the lower and u p p e r time limits.

3. 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

T h e d i s t r i b u t i o n of the equatorial i n s t a n t a n e o u s insolation on the meridians AA = 0 (the so-called hot pole) and AA = - 90°, (warm

pole) for three different values of the eccentricity (e = 0.11', 0 . 2 0 5 6 3

and 0 . 2 4 ) is respectively illustrated in F i g s . 1 a n d 2. It h a s to be

emphasized that the instantaneous solar radiation at a g i v e n time a n d at

a g i v e n latitude can easily be f o u n d b y multiplying the insolation on the

same meridian at the same moment b y the correction factor cos a n d

for a s u r f a c e element located on the equator ( = 0) ( V a n Hemelrijck

and V e r c h e v a l , 1981).

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From F i g . 1 (AA = 0 ) , it can be seen that the maximum solar radiation is incident at the f i r s t perihelion p a s s a g e of the planet with v a l u e s r a n g i n g from 3 . 1 x 10 (e = 0 . 2 4 ) to approximately 2 . 5 x 10 cal 4 4

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cm ( p l a n e t a r y d a y ) (e = 0 . 1 1 ) , the p e r c e n t difference of this two maxima being of the o r d e r of 25%. F u r t h e r m o r e , from d a y 10 until s u n s e t ( d a y 44) the difference between the insolation c u r v e s for e = 0.24 a n d e = 0 . 1 1 equals nearly 10%. For exactly half a solar d a y ( ^ 88 Earth d a y s ; not v i s i b l e on F i g . 1) or more p r e c i s e l y from the f i r s t aphelion p a s s a g e to the next one, all p a r t s of the meridian AA. = 0 are in permanent d a r k n e s s . Finally, another maximum is f o u n d on the t h i r d perihelion p a s s a g e of M e r c u r y .

In F i g . 2 we h a v e plotted the solar radiation incident at the top of the M e r c u r i a n atmosphere as a f u n c t i o n of time for AA. = - 90°

a n d for the three eccentricities u n d e r c o n s i d e r a t i o n . T h e time evolution of the instantaneous insolation e x t e n d s , as in F i g . 1, o v e r one sidereal period of revolution ( o r one tropical y e a r ) . F i g . 2 reveals that the insolation d i s t r i b u t i o n s are quasi-parallel a n d that the percentage differences are o b v i o u s l y h i g h e r than those obtained for AA = 0. T h e maximum insolation at aphelion amounts to about 1.6 x 10

4

(e = 0 . 1 1 ) and 1.2 x 10

4

cal c m "

2

( p l a n e t a r y d a y ) "

1

(e = 0 . 2 4 ) c o r r e s p o n d i n g to a percentage difference of the o r d e r of 35%. Before and after aphelion the percent differences are s i g n i f i c a n t l y h i g h e r .

T h e insolation c u r v e s r e p r e s e n t e d in F i g . 2 were computed b y a s s u m i n g the S u n as a point s o u r c e . H o w e v e r , near s u n r i s e and s u n s e t , which at longitudes ± 90° o c c u r as M e r c u r y goes t h r o u g h perihelion, more accurate calculations are needed ( L a n d a u 1975, 1982) p a r t i c u l a r l y due to the fact that the S u n takes mostly several d a y s to rise or set completely ( L a n d a u , 1982; V a n Hemelrijck and V e r c h e v a l , 1983).

T h e s e calculations are g i v e n in F i g . 3 for the three adopted v a l u e s of the eccentricity.

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INSTANTANEOUS INSOLATION [cal cm"

2

(planetary day)"

tn l/l 5 ° cn > cn o o m m o

a o

><

O CD O CD O

APHELION

.PERIHELION _L _L

Fig. 1.- Time variation of the instantaneous insolation at the top of the

Mercurian atmosphere on the meridian AA. = 0° and for the following

numerical values of the eccentricity e : 0.11, 0.20563 and 0.24. The

curves are related to the equator. Perihelion and aphelion passages are

also illustrated.

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Fig. 2.- Time variation of the instantaneous insolation at

" the top of the Mercurian atmosphere on the meridian hk = - 90°. See Fig. 1 for full explana- tion.

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F i g . 3.- Time variation of the instantaneous insolation at the top of the M e r c u r i a n atmosphere on the warm pole (AA. = - 90°) from 14 days b e f o r e to 14 days after the p e r i h e l i o n p a s s a g e .

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From t h i s F i g u r e , it can be seen that the solar radiation d i s t r i b u t i o n s l i g h t l y before the perihelion p a s s a g e of the planet r e s u l t s in a thermal bulge ( S o t e r a n d U l r i c h s , 1967; L i u , 1968; M o r r i s o n , 1970;

V a n Hemelrijck a n d V e r c h e v a l , 1981, 1983; L a n d a u , 1982) for eccentrici- ties equal to 0.20563 a n d 0 . 2 4 ; f o r e = 0.11 the temporal increase of the u p p e r - b o u n d a r y solar heat f l u x does not take place. F u r t h e r m o r e , it is o b v i o u s that the s h a p e of the c u r v e s is m a r k e d l y d i f f e r e n t , in p a s s i n g from the minimum to the maximum value of e.

T h e reason for the apparition of the thermal b u l g e s at large eccentricities is that in the n e i g h b o r h o o d of perihelion the M e r c u r i a n a n g u l a r velocity of revolution (W) exceeds the rotational a n g u l a r velo- city ( n ) of the planet on its own a x i s . T h i s phenomenon is illustrated in F i g . 4, the c u r v e s r e p r e s e n t i n g the variability of W being obtained b y K e p l e r ' s second law

W = n (1 - e

2

)

1 / 2

( a / r )

2

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o O O

with

(a /r )

2

= 1 + e

2

/2 + (2e + 3/4 e

3

) cos M + (5/2) e

2

cos 2M

v

O O

+ (13/4) e

3

cos 3M (14)

where M is the mean anomaly. Note that in e x p r e s s i o n ( 1 4 ) we kept only terms up to the t h i r d degree in e.

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F i g . 4 clearly demonstrates that W ^ n from 4 d a y s (e = 0 . 2 0 5 6 3 ) , r e s p e c t i v e l y 7 d a y s (e = 0 . 2 4 ) , before to 4 d a y s (e = 0 . 2 0 5 6 3 ) , respectively 7 d a y s (e = 0 . 2 4 ) , after the perihelion p a s s a g e . For the minimum value of the orbital eccentricity (e = 0 . 1 1 ) , W < n o v e r the entire time interval. O n the other h a n d it has been f o u n d ( V a n

Hemelrijck a n d V e r c h e v a l , 1983) that the minimum value of e past w h i c h W i n is approximately equal to 0.194. In other w o r d s , for eccentrici- ties l a r g e r t h a n the above mentioned limit, there exist two permanent thermal b u l g e s on opposite s i d e s of the M e r c u r i a n s u r f a c e that alternately point to the S u n at e v e r y perihelion p a s s a g e .

A s a l r e a d y stated p r e v i o u s l y the b e h a v i o r of the solar radiation d i s t r i b u t i o n c u r v e s ( F i g . 3) for e = 0.20563 a n d e = 0.24 is fundamentaly different. Indeed, for the actual value of e it can be mathematically demonstrated ( T u r n e r , 1978; L a n d a u , 1982; V a n Hemelrijck a n d V e r c h e v a l , 1983) that from 9 d a y s before to 9 d a y s after the perihelion p a s s a g e some fraction of the S u n is above the h o r i z o n ; from d a y + 9 the S u n is completely v i s i b l e . A s a c o n s e q u e n c e of the existence of the thermal bulge a n d owing to the fact that the S u n does not definitely set it follows that after r i s i n g a n d r e c e d i n g the S u n r i s e s again without d i s a p p e a r i n g completely ( F i g . 3 ) . T h e maximum value of the instantaneous insolation for e = 0.20563 amounts to about 250 cal

- 2 - 1

cm ( p l a n e t a r y d a y ) at d a y - 4 ; the minimum v a l u e , obtained symmetricaly with respect to the perihelion, reaches s l i g h t l y 15 cal cm

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_ I

( p l a n e t a r y d a y ) . It is also i n t e r e s t i n g to note that the 18 d a y s s p e n d s b y the S u n to rise ( o r s e t ) completely, c o r r e s p o n d r o u g h l y to 20% o f an orbital period ( T = ^ 88 Earth d a y s ) .

A s mentioned above, a s t r i k i n g difference e x i s t s for e = 0.24.

Fig. 3 reveals that, t a k i n g into account the finite a n g u l a r size of the

S u n ' s d i s k , the u p p e r limb of the S u n b r e a k s the horizon at d a y - 13,

whereas the lower limb d i s a p p e a r s v e r y s h o r t l y ( d a y + 1) after the

perihelion p a s s a g e . T h e S u n temporarily s e t s , then r i s e s a g a i n . For

this configuration it is shown that the relatively s h o r t period of weak

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, ' e = 0.24 %

3 in 0»

T3. 8*6

- n=6.13811 /

z o

3 - O >

LU (X O

>

B * 3 HI >

<r <

o z <

J L I I 1 L HI X

cc UJ

± CL i l I I I L -14 -12 -10 - 8 - 6 - 4 - 2 0 2 4 6 8 10 12 14

TIME SINCE PERIHELION PASSAGE (Earth days)

F i g . 4.- M e r c u r i a n angular v e l o c i t y of revolution (W) and rotational angular v e l o c i t y (n = 6 . 3 1 8 1 1 ) of the p l a n e t on its own axis as a function of t i m e .

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\

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insolation (14 d a y s ) is followed (or preceded on the next orbit) by a time interval of complete darkness of approximately 11 d a y s . Concerning more particularly the peak insolation of the thermal bulge, the sensibil- ity of I to changes in e is evident from Fig. 3. Indeed, it can be seen that the instantaneous insolation exhibits a sixfold [from 250 to 1500 cal

- 2 - 1

cm (planetary d a y ) ] increase in going from the c u r r e n t value of e to the theoretical maximum one. For completeness, it has to be said that the critical value of e past which the Sun shortly sets after perihelion is near 0.213 (Van Hemelrijck and Vercheval, 1983).

4. T H E D A I L Y INSOLATION

We also have investigated the daily insolation Ip as a function of the longitude difference AA. by application of expression ( 1 0 ) . T h e integration limits t^ and t^ have been determined from relations (11) and (12) and from Fig. 5 representing the variability of the function f ( t ) = nt - W(t) over a time period equal to one Mercurian solar day and where W(t) is given by formula ( 4 ) . It has to be noticed that t^

and t

2

corresponding to e = 0.20563 are taken from Van Hemelrijck and Vercheval (1981).

The distribution of the diurnal insolation, on the equator and for the three eccentricities studied in this paper, as a function of the longitude difference between the meridian of a surface element and the meridian crossing the line of apsides at the perihelion passage of Mercury is plotted in Fig. 6.

From an analysis of this Figure it may be concluded that the maximum daily insolation attained by a so-called hot pole (AA = 0) is about 1.25 x 10

6

(e = 0 . 1 1 ) , 1.40 x 10

6

(e = 0.20563) and 1.45 x 10

6

- ? - 1

(e = 0 . 2 4 ) cal cm (planetary d a y ) . T h e minimum values occur at a warm pole (AA = - 90°) and amount respectively to 7.95 x 10 , 5

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Fig. 5.- Rotation angle difference (nt - W) for Mercury as a function of time.

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F i g . 6.- D a i l y insolation at the top of M e r c u r y as a function of the longitude difference A \ . The curves correspond to the e q u a t o r .

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5.50 x 10

5

a n d 4.80 x 10

5

cal c m "

2

( p l a n e t a r y d a y ) "

1

. It follows that the d i u r n a l insolation d e c r e a s e s b y n e a r l y a factor of 1 . 6 , 2 . 5 a n d 3.0 as the longitude difference increases from 0 to 90°. F u r t h e r m o r e , it is p a r t i c u l a r l y e v i d e n t from F i g . 6 that the critical longitude difference past which the daily insolation at small eccentricities exceeds the one at large eccentricities is of the o r d e r of 55°.

5. S U M M A R Y A N D C O N C L U S I O N S

In the p r e c e d i n g sections emphasis is placed on the influence of orbital eccentricity v a r i a t i o n s on the instantaneous a n d daily insola- tion on the planet M e r c u r y .

T h e main r e s u l t s of t h i s s h o r t s t u d y may be summarized b y the following statements :

( a ) A t the hot pole of M e r c u r y , the maximum p e r c e n t difference in the i n s t a n t a n e o u s insolation in g o i n g from the minimum to the maximum value of the orbital eccentricity amounts to about 25% at the f i r s t and the t h i r d perihelion p a s s a g e of the planet.

( b ) A t the warm pole, the percentage differences in the i n s t a n t a n e o u s insolation are s i g n i f i c a n t l y h i g h e r with a minimum value of approximately 35% at aphelion.

( c ) T a k i n g into account the finite a n g u l a r size of the S u n , especially at the warm pole, it is f o u n d that there e x i s t s two permanent thermal b u l g e s on opposite sides of the M e r c u r i a n s u r f a c e that alternately face the S u n at e v e r y perihelion p a s s a g e for eccentricities l a r g e r t h a n about 0.194,

( d ) For 0.194 < e < 0.213 and at the warm pole the S u n r i s e s , recedes, then r i s e s again without d i s a p p e a r i n g completely.

( e ) For e > 0.213 the S u n r i s e s , temporarily s e t s , then r i s e s a g a i n .

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( f ) T h e daily insolation on the equator d e c r e a s e s b y n e a r l y a factor of 1.6 (e = 0 . 1 1 ) a n d 3.0 (e = 0 . 2 4 ) as the longitude difference r a n g e s from 0 to 90°.

( g ) A s eccentricity i n c r e a s e s , the daily insolation at the hot pole i n c r e a s e s but c o n v e r s e l y d e c r e a s e s at the warm pole. T h e longitude difference for which I p is practically i n d e p e n d e n t u p o n the orbital eccentricity is of the o r d e r of 55°.

In c o n c l u s i o n , we believe that l o n g - t e r m c h a n g e s in the i n s t a n t a n e o u s a n d daily insolations on M e r c u r y c a u s e d b y v a r i a t i o n s of the M e r c u r i a n o r b i t h a v e to be taken into account in o r d e r to better u n d e r s t a n d some aspects of the past and the p r e s e n t weather a n d climatc on M e r c u r y .

A C K N O W L E D G E M E N T S

We would like to t h a n k J. Schmitz and F. V a n d r e c k for the realisation of the illustrations. We are also grateful to L. Vastenaekel for t y p i n g the m a n u s c r i p t .

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