AERO NOMICA ACTA

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 - 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

AERO N O M I C A A C T A

A - N° 2 5 5 - 1983

An estimate of the solar radiation incident at the top of Pluto's atmosphere

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 BRUSSEL

FOREWORD

The paper entitled "An estimate of the solar radiation incident at the top of Pluto's atmosphere" will be published in the "Bulletin de la Classe des Sciences de l'Académie Royale de Belgique". A more concise version of the text presented here is published as an article in

Icarus, 1982 under the title "The insolation at Pluto".

AVANT-PROPOS

Le travail intitulé "An estimate of the solar radiation incident at the top of Pluto's atmosphere" sera publié dans le "Bulletin de la Classe des Sciences de l'Académie Royale de Belgique". Une version plus concise du texte présenté ici est publiée comme article dans Icarus, 1982 sous le titre "The insolation at Pluto".

VOORWOORD

Het werk getiteld "An estimate of the solar radiation incident at the top of Pluto's atmosphere" zal gepubliceerd worden in het

"Bulletin de la Classe des Sciences de l'Académie Royale de Belgique".

Een verkorte versie van deze tekst wordt als artikel gepubliceerd in Icarus, 1982 onder de titel "The insolation at Pluto".

VORWORT

Der Text "An estimate of the solar radiation incident at the top of Pluto's atmosphere" wird in dem "Bulletin de la Classe des Sciences de l'Académie Royale de Belgique" heraugegeben werden. Ein kurzgefaster Version dieser Text ist in Icarus, 1982 publiziert unter dem Titel "The insolation at Pluto".

AN E S T I M A T E 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 TOP OF P L U T O ' S A T M O S P H E R E

by

E. V A N HEMELRIJCK

Abstract

Calculations of the daily solar radiation incident at the top of the atmosphere of Pluto are presented in a series of figures giving the seasonal and latitudinal variation for three fixed values of the obliquity (e = 60, 75 and 90°). It is shown that the maximum daily insolation is incident at the poles at solar longitudes near 110 and 250° with values ranging from 11 to 13 cal cm- 2 ( d a y )- 1 (North pole) and from 13.5 to 15.5 cal cm- 2 ( d a y )- 1 (South pole). At the equator, maxima of the order of 6 cal c m- 2 ( d a y )- 1 are found around autumnal equinox. The solar longitude intervals where the polar solar energy exceeds the equatorial one extend from 20 to 160° and from 200 to 340° respectively.

The steady increase of the polar insolation with increasing obliquity is accompanied by a corresponding loss of the equatorial solar radiation.

The large eccentricity of Pluto produces significant north-south seasonal asymmetries in the daily insolation, whereas the change in the obliquity causes mainly a global latitudinal redistribution, although the general pattern of the contour maps illustrating the variability of the daily solar radiation with latitude and season is nearly similar. It is also interesting to note that in the equatorial region summer and winter are, roughly speaking, repeated twice a year. In addition, we also numerically studied the latitudinal variation of the mean daily insolation.

It is found that in summer if e varies from 60 to 90°, the mean summer insolation increases by about 15% at the poles, remains approximately constant at a latitude of 20°, but decreases at the equator also by approximately 15%. In winter, an increase of the obliquity yields a mean daily insolation which is reduced at all latitudes (maximally by about 20%

near the equator), the influence of e being of decreasing significance at polar region latitudes. A s for Uranus, the equator receives less annual average energy than the poles; the ratio of both insolations amounts to about 0.9, 0.7 and 0.6 for an obliquity equal to 60, 75 and 90°

respectively. Our calculations also reveal that at a latitude close to 30°, the dependence of the mean annual daily insolation on the obliquity is either unimportant.

Résumé

L'insolation d i u r n e au sommet de l'atmosphère de Pluton est calculée et les résultats sont présentés dans une série de f i g u r e s i l l u s t r a n t les variations saisonnières et latitudinales pour t r o i s v a l e u r s données de l'obliquité (e = 60, 75 et 9 0 ° ) .

On montre que l'insolation d i u r n e maximale se manifeste aux pôles à des longitudes approximatives de 110 et 250° et avec des valeurs v a r i a n t de 11 à 13 cal c m^{- 2} ( j o u r )^{- 1} (Pôle N o r d ) et de 1 3 . 5 à 1 5 . 5 cal c m^{- 2} ( j o u r )^{- 1} (Pôle S u d ) . A l ' ë q u a t e u r , des v a l e u r s maximales de l ' o r d r e de 6 Cal c m^{- 2} ( j o u r ) "¹ ont été t r o u v é e s près de l'équinoxe d'automne. Les intervalles de longitude où l'énergie solaire reçue aux pôles est s u p é r i e u r e à celle à l'équateur s'étendent respectivement d ' e n v i r o n 20 à 160° et de 200 à 3 4 0 ° . L'augmentation p r o g r e s s i v e de l'insolation d i u r n e aux pôles en fonction de l'obliquité est accompagnée d ' u n e diminution correspondante de la radiation solaire à l ' é q u a t e u r . L ' e x c e n t r i c i t é élevée de Pluton donne lieu à une asymétrie N o r d - S u d importante dans l'insolation d i u r n e , tandis q u ' u n e variation de l'obliquité affecte la d i s t r i b u t i o n latitudinale globale. L'image générale des iso- contours r e p r é s e n t a n t les variations saisonnières et latitudinales de l'insolation en fonction des obliquités étudiées est qualitativement comparable. Notons que chaque année la région équatoriale connaît deux h i v e r s et deux étés.

Nous avons également étudié numériquement l'insolaton d i u r n e moyenne. Il en résulte que l'insolation d i u r n e moyenne en é t é , lorsque e v a r i e de 60 à 9 0 ° , augmente de 15% aux pôles, reste pratiquement constante a une latitude de 2 0 ° , mais diminue à l'équateur de 15%. En h i v e r , une augmentation de l'obliquité a pour e f f e t une p e r t e de l'insolation d i u r n e moyenne à toutes les latitudes ( d e 20% au maximum à l ' é q u a t e u r ) . L'influence de e est moins prononcée dans la région polaire. Comme pour U r a n u s , l'énergie solaire reçue a l'équateur s'avère toujours plus petite comparée à celle aux pôles; le r a p p o r t des deux insolations a t t e i n t des valeurs approximatives de 0 . 9 , 0 . 7 et 0 . 6 pour des obliquités respectives de 60, 75 et 9 0 ° . En o u t r e , les calculs indiquent qu'à une latitude de 3 0 ° , l'insolation d i u r n e moyenne annuelle est pratiquement insensible à une v a r i a t i o n de l'obliquité.

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Samenvatting

Berekeningen van de dagelijkse zonnestraling aan de rand van de atmosfeer van Pluto worden voorgesteld in een reeks van figuren die de seizoens- en breedteveranderingen weergeven voor drie gegeven waarden van de helling van de rotatieas op het baanvlak (e = 60, 75 en 9 0 ° ) .

Er wordt aangetoond dat de maximale dagelijkse zonnestraling zich voordoet aan de polen bij zonnelengten van ongeveer 110 en 250°

en met waarden die variëren van 11 tot 13 cal cm-2 ( d a g )^{- 1} (Noordpool) en van 13.5 tot 15.5 cal c m^{- 2} ( d a g )^{- 1} (Zuidpool). Aan de evenaar werden maximale waarden gevonden van de grootte-orde van 6 cal cm ( d a g ) rond de herfstequinox. De zonnelengteintervals waarbij de polaire zonneënergie groter is dan deze aan de evenaar strekken zich respectievelijk uit van ongeveer 20 tot 160° en van 200 tot 340°. De geleidelijke verhoging van de zonnestraling aan de polen met stijgende waarden van de helling van de rotatieas op het baanvlak gaat gepaard met een overeenkomstige vermindering van de zonnestraling aan de evenaar. De grote eccentriciteit van Pluto heeft een belangrijke noord- zuid asymmetrie in de dagelijkse zonnestraling tot gevolg, terwijl een verandering in de helling van de rotatieas t . o . v . het baanvlak voor- namelijk een globale breedteverdeling veroorzaakt. Het algemeen beeld van de isolijnen die de verandering weergeven van de dagelijkse zonne- straling in functie van de breedte en het seizoen is nagenoeg gelijk- vormig. Vermeldenswaardig is ook het feit dat in het evenaarsgebied de zomer en de w i n t e r , ruwweg gesproken, tweemaal herhaald worden in een jaar.

Bovendien werd ook de breedtevariatie van de gemiddelde dagelijkse zonnestraling numerisch berekend; in de zomer, en indien e verandert van 60 tot 90°, stijgt de gemiddelde dagelijkse zonnestraling met ongeveer 15% aan de polen, blijft nagenoeg constant rond een breedte van 20°, maar vermindert aan de evenaar eveneens met ongeveer 15%. In de winter gaat een stijging van de helling van de rotatieas gepaard met een verlies aan gemiddelde dagelijkse zonne- straling op alle breedten (maximaal met ongeveer 20% aan de evenaar).

De invloed van e is nochtans minder uitgesproken rond het poolgebied.

Zoals voor Uranus ontvangt de evenaar minder zonneënergie dan de polen; de verhouding van beide zonnestralingen bedraagt ongeveer 0 . 9 , 0 . 7 en 0 . 6 voor waarden van de helling van de rotatieas respectievelijk gelijk aan 60, 75 en 90°. De berekeningen wijzen ook uit dat de afhankelijkheid van de gemiddelde jaarlijkse zonnestraling aan een verandering in de helling van de rotatieas gering is bij een breedte van 30°.

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Zusammenfassung

B e r e c h n u n g e n der täglichen S o n n e n s t r a h l u n g an den Rand d e r Atmosphäre des Planeten Pluto sind v o r g e s t e l t in einer Reihe von A b b i l d u n g e n die die J a h r e s z e i t l i c h e n - und B r e i t e n v a r i a t i o n e n d a r s t e l l e n f ü r drei Werte d e r Schiefe der E k l i p t i k ( e = 60, 75 und 9 0 ° ) .

Es w ü r d e g e f u n d e n dass der Höchstwert der täglichen Sonnen- s t r a h l u n g sich manifestiert an den Polen f ü r Sonnenlängen von u n g e f ä h r 110 und 250° und mit Werte die wechseln von 11 bis 13 cal c m- 2 ( d a y )- 1

( N o r d p o l ) und von 13.5 bis 1 5 . 5 cal c m^{- 2} ( d a y )^{- 1} ( S ü d p o l ) . An dem Ä q u a t o r b e t r a g t d e r Höchstwert u n g e f ä h r 6 cal c m- 2 ( d a y )- 1 in d e r Umgebung des H e r b s t ä q u i n o k t i u m s . Die S o n n e n l ä n g e - i n t e r v a l l e wobei d e r polare Sonnenstrahlung grösser ist dann diese an dem Ä q u a t o r e r s t r e c k e n sich von u n g e f ä h r 20 bis 160° und von 200 bis 3 4 0 ° . Die systematische Erhöhung d e r täglichen Sonnenstrahlung an den Polen als Funktion d e r Schicfc der E k l i p t i k ist v e r b u n d e n mit einer- korrespondierenden V e r m i n d e r u n g d e r Sonnenstrahlung an dem Ä q u a t o r . Die grosse E x z e n t r i z i t ä t des Planeten Pluto r e s u l t i e r t in einer wichtigen N o r d - S ü d Asymmetrie der täglichen S o n n e n s t r a h l u n g , w ä h r e n d eine Variation in der Schiefe d e r E k l i p t i k vornehmlich eine globale Wiederverteiling als Funktion d e r B r e i t e v e r u r s a c h t . Das allgemeine Modell der K u r v e n gleicher S o n n e n s t r a h l u n g f ü r die d r e i Werte d e r Schiefe der E k l i p t i k ist u n g e f ä h r gleichförmig. Es ist auch interessant zu notieren dass in dem Ä q u a t o r g e b i e t der Sommer und der Winter sich u n g e f ä h r zweimal pro Jahr wiederholen.

Wir haben auch numerisch die B r e i t e n v a r i a t i o n e n der mittleren täglichen Sonnenstrahlung b e r e c h n e t . Es w ü r d e g e f u n d e n dass im Sommer und wann e wechseint von 60 bis 9 0 ° , die mittlere tägliche Sonnenstrahlung zunehmt mit u n g e f ä h r 15% an dem Polen, konstant bleibt auf einer B r e i t e von 20%, aber abnehmt an dem Ä q u a t o r ebenfalls mit u n g e f ä h r 15%. Im Winter ist eine Steigung der Schiefe d e r E k l i p t i k v e r b u n d e n mit einem V e r l u s t der täglichen Sonnenstrahlung auf allen Breiten wobei der Höchstwert etwa 20% b e t r a g t an dem Ä q u a t o r . Der Einfluss von e in der Umgebung des Polgebietes ist nicht so gross. Wie f ü r Uranus empfangt der Äquator weiniger Sonnenenergie als die Polen;

das V e r h ä l t n i s der beiden Sonnenstrahlungen b e t r a g t u n g e f ä h r 0 . 9 , 0 . 7 und 0 . 6 f ü r Werte der Schiefe d e r E k l i p t i k von 60, 75 und 9 0 ° . Unsere Berechnungen zeigen auch dass die A b h ä n g i g k e i t der mittleren täglichen Sonnenstrahlung an einer Variation der Schiefe der E k l i p t i k p r a k t i s c h u n b e d e u t e t ist auf einer B r e i t e von 3 0 ° .

- 4 -

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

T h e u p p e r - b o u n d a r y insolation of the atmospheres of the outer p l a n e t s , e x c l u d i n g Pluto, h a s been computed b y e . g . V o r o b ' y e v a n d Monin (1975) a n d Levine et al. ( 1 9 7 7 ) . T o the best of o u r k n o w - ledge, similar calculations for Pluto have n e v e r been p u b l i s h e d .

Pluto, the most d i s t a n t member of the solar s y s t e m so far k n o w n , was d i s c o v e r e d on F e b r u a r y 18, 1930 after 25 y e a r s of intense and systematic s e a r c h . T o date a n d t a k i n g into account the v e r y long orbital period of approximately 248 y e a r s , o b s e r v a t i o n s of the planet c o v e r only a little more t h a n 20% of its o r b i t . T h i s limited time period led to some v e r y d i v e r s e determinations of some orbital elements needed, for example, for the calculation of the solar radiation r e a c h i n g the top of the planet's atmosphere. A c t u a l l y , the orbital plane is relatively accurate determined, so all v a l u e s of the inclination ( i ) , the heliocentric longitudes of the planet's perihelion (TI^q) a n d the a s c e n d i n g node (fi^Q) are more or less compatible (see e . g . Duncombe et a l . , 1972;

N e w b u r n and G u l k i s , 1973; Nacozy and Diehl, 1974, 1978a,b; N a c o z y , 1980; Seidelman et a l . , 1980). Some other orbital parameters h a v e been more difficult to measure; p a r t i c u l a r l y the angle between the p l a n e t ' s s p i n a x i s and its orbit normal ( e ) is v e r y poorly determined ( D a v i e s et a l . , 1980). E v e n recent measurements v a r y s t r o n g l y a n d c o n s e q u e n t l y , it is not possible at this time to have sufficient confidence in a n y particular value. A l t h o u g h the d i s c o v e r y of the satellite of Pluto ( C h r i s t y and H a r r i n g t o n , 1978, 1980; H a r r i n g t o n a n d C h r i s t y , 1980, 1981) has o b v i o u s l y stimulated the o b s e r v a t i o n s , it s h o u l d be emphasized that a continued r e s e a r c h p r o g r a m is absolutely needed in o r d e r to improve the Pluto ephemerides.

In spite of the fact that the o b l i q u i t y ( e ) of Pluto is so questionable, we calculated the solar radiation incident on the planet for three fixed v a l u e s of e r a n g i n g from 60° ( A n d e r s o n a n d F i x , 1973) o v e r 75° ( G o l i t s y n , 1979) to 90° ( T h e H a n d b o o k of the B r i t i s h Astronomical

- 5 -

A s s o c i a t i o n , 1981). A l t h o u g h the two extreme v a l u e s of the o b l i q u i t y u s e d s p a n 30°, the calculations p r e s e n t e d in t h i s s t u d y enable u s to h a v e a quantitatively good estimate of the solar insolation. M o r e o v e r , t h e y illustrate f a i r l y well the s e n s i t i v i t y of the daily a n d the seasonal a v e r a g e insolation to c h a n g e s in the o b l i q u i t y in the above mentioned interval.

In a f i r s t section, we p r e s e n t e x p r e s s i o n s for the u p p e r - b o u n d a r y insolation of the outer planets. T h e n , t a k i n g into account the orbital a n d planetary data of Pluto, we calculate the planetocentric longitude of its perihelion ( ^ p ) a r ,d the length of the n o r t h e r n ( a n d s o u t h e r n ) summer a n d winter ( Tg a n d T ^ ) . T h e r e s u l t s of the daily insolation are p r e s e n t e d in the form of three c o n t o u r maps g i v i n g the incident solar radiation in calories per s q u a r e centimeter per planetary d a y as a function of latitude and solar longitude and in two f i g u r e s illustrating the seasonal variation of the equatorial a n d polar solar e n e r g y .

In addition, the latitudinal variation of the mean daily solar radiations are included in a series of three f i g u r e s .

2. S O L A R R A D I A T I O N I N C I D E N T O N A P L A N E T A R Y A T M O S P H E R E

T h e instantaneous insolation ( I ) at the u p p e r - b o u n d a r y of the

9

atmosphere of a planet of the solar s y s t e m can be e x p r e s s e d as (see e . g . Ward, 1974; V o r o b ' y e v and M o n i n , 1975; Levine et al., 1977; V a n Hemelrijck a n d V e r c h e v a l , 1981; V a n Hemelrijck, 1 9 8 2 a , b , c ) :

I; = S cos z (1)

with :

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S = So/ 4 (2)

a n d :

r = aQ (1 - e2) / ( l + e cos W) (3)

In e x p r e s s i o n s ( 1 ) to ( 3 ) , S , z, S , a & e a n d W are respectively the solar f l u x at an heliocentric distance rQ, the zenith angle of the incident solar radiation, the solar c o n s t a n t at the mean

- 2 - 1 3 S u n - E a r t h distance of 1 A U taken at 1.94 cal cm ( m i n ) or 2.79 x 10

cal c m "2 ( d a y ) "1 ( T h e k a e k a r a , 1973), the planet's semi-major a x i s , the eccentricity and the t r u e anomaly which is g i v e n b y :

W = Aq - Xp (4)

where \ and \p are the planetocentric longitude of the S u n ( o r solar l o n g i t u d e ) and the planetocentric longitude of the planet's perihelion (sometimes called the a r g u m e n t of p e r i h e l i o n ) . It has to be mentioned that the latter parameter for each planet in the solar s y s t e m ( e x c l u d i n g Pluto) has been g i v e n b y V o r o b ' y e v and Monin (1975) a n d Levine et al.

(1977).

For a spherical planet, the zenith angle ( z ) is g i v e n b y :

cós z = s i n <|> s i n 6 + cos <|> cos 6 cos h (5)

- 7 -

where <|> is the planetocentric (or planetographic) latitude, 6Q is the solar declination and h is the local hour angle of the S u n . Furthermore, the solar declination ( 6Q) can be calculated from the relation :

s i n = s i n £ s i n A. (6)

For a rapidly rotating planet, expression (1) can be integrated to yield the amount of solar radiation incident at the top of a planetary atmosphere over the planet's day C I q ) and is given b y :

I p = (ST/n) (hQ s i n <|> s i n 6Q + s i n hQ cos <|> cos 6Q) (7)

where T is the sidereal period of axial rotation (or sidereal d a y ) and hQ

is the local hour angle at sunset (or s u n r i s e ) and may be determined from expression (5) by the condition that at setting (or r i s i n g ) cos z = 0. Hence, it follows that :

h = arc cos ( - tan tan <|)) = o w

2 2 1/2 arc cos [ - tan <|> s i n e s i n ^q/ ( 1 - s i n e s i n K ^ ]

(8)

if

| + | < if/2 6q I

t

In regions where the Sun does not rise < - n/2 + 6

or <J> >

7t/2 + 6Q) we have h

= 0; in regions where the Sun remains above the horizon all day C<t> > n/2 - 6

or <J> < - n/2 - 6

) we may put h

= n.

hereafter denoted as ( l p )

/ O p )

and ( '

)

respectively, may be found by integrating relation (7) within the appropriate time limits, yielding the total insolation over a season or a year and by dividing the obtained result by the corresponding length of the season (~T"

or ~!~

) or by the sidereal period of revolution or tropical year ( T ). In our calculations and for the northern hemisphere, summer season is arbitrary defined as running from vernal equinox over summer solstice to autumnal equinox and spanning 180°; consequently, the planeto- centric longitudes of the Sun A

= 180° and \

= 360° mark the beginning and the end of the winter period. In the southern hemi- sphere, the solar longitude intervals (0, 180°) and (180, 360°) divide the year into astronomical winter and summer respectively.

As an example, the mean annual daily insolation may be written under the form :

Finally, the mean (summer, winter or annual) daily insolation,

T

o

(h sin d> sin + sin h cos <|) cos ô„) dt o

o • .vy

( 9 )

o

From equation (2) and by the aid of Kepler's second law

( ^ / r

)

dt = T

d \r/2n (1 - e

)

(10)

and after some rearrangements, expression (9) can be transformed into an integral over yielding :

2 n

S T sin (b sin e/[2no ^r ' i v ' © / o o © < 2 (1 - e2)1 / 2 a2] / (h - tan h ) sin d\

° (11)

where the dependence of hQ in terms of \Q is given by the second equality on the right hand side of relation (8) (see e.g. Vorob'yev and Monin, 1975). It should be pointed out that, considering the complexity of the integrand, equation (11) has generally to be integrated numerically. However, at the poles, the mean annual daily insolation can easily be obtained from (11) by putting <J> = ± n/2 and hQ = n (see e.g.

Murray et al., 1973; Ward, 1974; Vorob'yev and Monin, 1975).

Integration yields :

(ID)A = SQ T sin e/rt (1 - e2)1 / 2 a2 (12)

For the computations presented in this paper, the procedure to obtain the seasonal or annual average energy was as follows.

In a very good approximation, the total insolation over a season or a year ( ƒ lD dt) was obtained using a step-by-step summation U ( 'D) j (At).] where the solar longitude range (0-180°, 180-360°, 0-360°) has been divided into 18 (i = 1, . . . , 18) (summer and winter) or 36 (i = 1, . . . , 36) (year) intervals with a bandwith (A\Q)j of 10°. The adopted constant values 0D) j represent the calculated values in the center of each interval. Furthermore, (At)f or the length (in earth days) of the ith solar longitude interval can be

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determined by solving successively the following set of equations for the position of the S u n at the beginning and the end of each solar longitude interval (AAQ).. It has to be pointed out that all the relation- ships can be found in each text-book on spherical astronomy (see e . g . Smart, 1956).

The true anomaly (W) dependence of the eccentric anomaly ( E ) can be described by Lacaille's equation :

where W is given b y expression ( 4 ) .

Furthermore, the mean anomaly ( M ) in terms of E can be expressed by the Kepler equation :

E = 2 arc tan { [ ( 1 - e ) / ( l + e ) ]^{1 / 2} tan (W/2)} (13)

M = E - e sin E (14)

Finally, (At), can be written under the following form :

(At). = T AM/27T 1 o (15)

where

AM = ( M ) .+ 1 - (M).

- 1 1 -

T h e total amount of solar energy over a season or a year being calculated, the mean daily insolations can, as already mentioned previously, be determined by dividing the obtained result by the corresponding time interval. It is obvious that T

or T ^ can also easily be found by combining ( 4 ) , (13), (14) and (15). It is clear that the mean annual daily insolation 0 q )

may

derived from :

< V A

[ ( V S

S

V

V

0

3. P L A N E T O C F N T R I C L O N G I T U D E OF P L U T O ' S P E R I H E L I O N

In the preceeding section we noted that the position of perihelion ( \

) for the planets in the solar system is given by e . g . Vorob'yev and Monin (1975) and Levine et al. (1977). To the best of our knowledge, Pluto's argument of perihelion has never been published. With this in mind, an attempt is made to derive th above mentioned parameter based on a computing algorithm discussed by Vorob'yev and Monin (1975). According to their paper, the argument of perihelion may be written in terms of the heliocentric longitude of the planet's perihelion (n

) and the heliocentric longitude of the ascending node (Q ) as :

K^ = n - Q + A (17)

T o o

where A is the planetocentric longitude of the ascending node altered by 180°. A detailed description of the procedure to calculate A is beyond the scope of the present work. It should however be emphasized that the element A can be obtained from standard spherical trigonometric relationships. It is dependent upon several orbital and planetary data and can be expressed in the general form :

- 1 2 -

A = f ( i , fi , n , e, e , a , 6 )

' o' o' ' o' o' o (18)

where e^Q, a^Q arid 6^q are respectively the angle between the Earth's spin axis and its orbit normal (e^Q = 2 3 . ° 4 5 ) and the right ascension and declination of Pluto's north pole, the significance of the other quantities being already explained in the t e x t .

The adopted planetary data for the computation of \^p are given in Table I , whereas the results of the calculation are illustrated in Table I I .

It has to be pointed out that i, Q and n were taken from o o the paper by Seidelman et al. (1980) limited to two decimals. For the direction of the north pole, we adopted the recommended values for the equatorial coordinates ( a^Q, 6^q) recently published by the IAU Working Group on cartographic coordinates and rotational elements of the planets and satellites (Davies et a l . , 1980).

In Table II one can find also the length of the seasons ( T^g and T.„) for the three obliquities under consideration. From the table it

w

can be seen that \^p is only weakly dependent on e.

4. DISCUSSION OF CALCULATION

Table I I I represents the numerical values of the supplement- ary parameters used for the computation of l^D, 0^D)^S/ OQ^W ^{a n d} ^'D^A

- 1 3 -

TABLE I.- Adopted planetary data for Pluto.

i o n 0 a o 6 o

(°) (°) (°) (°)

17.14 109.51 222.50 305 5

-14-

TABLE II.- Computed planetary data for Pluto.

e X P T S T W

C ° ) ( ° ) (Earth days) (Earth days)

60 192.17 48455 42128

75 191.12 48187 42396

90 190.81 48108 42475

-15-

TABLE III.- Adopted supplementary planetary data for Pluto.

a 0 e T T

o

(AU) (Earth days) (Earth days)

(a) (a) (b) . (c)

39.72 0.2523 6.3867 90583

(a) From Seidelman et al. (1980) (b) From Davies et al. (1980) (c) From Golitsyn (1979)

- 1 6 -

" Y \

4 . 1 . D A I L Y I N S O L A T I O N

For t h e d a i l y i n s o l a t i o n , we h a v e f o l l o w e d t h e m e t h o d a d o p t e d b y V o r o b ' y e x / a n d Monin (1975) a n d L e v i n e e t a l . (1977) in p r e s e n t i n g o u r r e s u l t s in t h e f o r m o f a c o n t o u r map g i v i n g t h e seasonal d i s t r i b u - t i o n i n t e r m s o f t h e p l a n e t o c e n t r i c l o n g i t u d e o f t h e S u n t a k e n t o be 0 ° at t h e n o r t h e r n h e m i s p h e r e v e r n a l e q u i n o x . I n a d d i t i o n we h a v e i n c l u d e d t w o f i g u r e s s h o w i n g t h e l a t i t u d i n a l v a r i a t i o n o f t h e d a i l y i n s o l a t i o n at t h e e q u a t o r a n d t h e poles as a f u n c t i o n o f solar l o n g i t u d e .

A p p l i c a t i o n o f e x p r e s s i o n ( 7 ) leads t o t h e i s o p l e t h s i l l u s t r a t e d i n F i g . 1 ( s = 6 0 ° ) , F i g . 2 (e = 7 5 ° ) a n d F i g . 3 (e = 9 0 ° ) a n d t o t h e e q u a t o r i a l a n d p o l a r d i s t r i b u t i o n p l o t t e d in F i g s . 4 a n d 5. From t h e c o n t o u r maps a n d p a r t i c u l a r l y f r o m F i g s . 4 a n d 5 i t f o l l o w s 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 at t h e poles a r o u n d summer s o l s t i c e s

- 2 - 1

w i t h v a l u e s o f a b o u t 11 t o 13 cal cm ( d a y ) ( N o r t h p o l e ) a n d 1 3 . 5 t o

- 2 - 1

1 5 . 5 cal cm ( d a y ) ( S o u t h p o l e ) . C o m p a r i n g t h i s r e s u l t s w i t h t h e c o r r e s p o n d i n g i n t e n s i t i e s o f t h e o u t e r p l a n e t s r e p r e s e n t e d in T a b l e IV [ t h e o r b i t a l a n d p l a n e t a r y d a t a b e i n g t a k e n f r o m t h e H a n d b o o k o f t h e B r i t i s h A s t r o n o m i c a l A s s o c i a t i o n (1980) a n d f r o m V o r o b ' y e v a n d Monin ( 1 9 7 5 ) ] , w h e r e we also h a v e s u m m a r i z e d t h e s i d e r e a l d a y ( T ) , t h e o b l i q u i t y ( e ) , t h e s e m i - m a j o r a x i s (a ) , t h e e c c e n t r i c i t y ( e ) a n d t h e a r g u m e n t o f p e r i h e l i o n ( A^p) , i t m i g h t be somewhat s u p r i s i n g t h a t , t a k i n g i n t o a c c o u n t t h e f a r d i s t a n c e o f P l u t o , t h e maximum a m o u n t o f solar e n e r g y r e a c h i n g t h e t o p o f i t s a t m o s p h e r e is much h i g h e r t h a n t h e s o l a r r a d i a t i o n i n c i d e n t on J u p i t e r , S a t u r n , U r a n u s a n d N e p t u n e . T h i s p h e n o m e n o n , h o w e v e r , can e a s i l y be e x p l a i n e d b y a p p l y i n g , f o r i n s t a n c e , e x p r e s s i o n ( 7 ) t o t h e n o r t h pole a t summer s o l s t i c e . I n d e e d , b y p u t t i n g 0 = n / 2 , X = n/2 a n d h = n , r e l a t i o n ( 7 ) y i e l d s :

- 1 7 -

SOLAR LONGITUDE (degrees)

FIG. 1.- Seasonal and latitudinal variation of the daily insolation 0

) at the top of the atmosphere of Pluto for an obliquity e = 60°. Solar declination is represented by the dashed line. The areas of permanent

-o

_ 1

darkness are shaded. Values of l

, in cal cm (planetary day) ,are

given on each curve.

CD

120 160 200 240

SOLAR LONGITUDE (degrees)

280 320 360

FIG. 2 . - Seasonal and latitudinal variation of the daily insolation ( l

) at the

top of the atmosphere of Pluto for an obliquity e = 75°. See Fig. 1 for

full explanation.

ro

I

1

120 160 200 240

SOLAR LONGITUDE (degrees)

320 360

F I G . 3 . - Seasonal and latitudinal variation of the daily insolation Oq ) at the top of the atmosphere of Pluto for an obliquity e = 90°. See F i g . 1 for full explanation.

SOLAR LONGITUDE (degrees)

F I G . 4 . - Seasonal variation of the daily insolation ( lD) at the equator of Pluto for various values of the obliquity.

SOLAR LONGITUDE (degrees)

F I G . 5 . - Seasonal variation of the daily insolation ( lD> at the poles of Pluto for various values of the obliquity.

TABLE IV.- Some planetary data and maximum solar radiation incident (^jj)j^jj a t poles of the outer planets.

PLANET T

(Earth days)

£ (°)

a O (AU)

e (°)

XP

(°) [cal cm"

NORTH

D MAX

^ ( d a y ) "1] POLE

^ D ^ M A X [cal cm~z(day)~ 1

SOUTH POLE

EARTH 1 23.44 1 0.01672 282.05 1070 1150

MARS 1.02 25.20 1.5237 0.09339 248 440 630

JUPITER 0.41 3.12 5.2028 0.04847 58 2 .5 2.1

SATURN 0.44 26.73 9.539 0.05561 279.07 5.5 6.8

URANUS 0.45 82.14 19.18 0.04727 3.02 3.4 3.4

NEPTUNE 0.66 29.56 30.06 0.00859 5.23 1.0 1.0

( I^D)N p ( s s ) - (S^o T sin e / a²) [ ( 1 + e sin A p ) / ( 1 - e²) ]² (19)

\

The term in the second bracket, representing the dependence

^'D^NP(SS) ^{o n} ^ c e n t r i c i t y and the argument of perihelion, ranges from 0 . 9 ( S a t u r n ) over approximately unity ( U r a n u s , Neptune, Pluto) to 1.1 ( J u p i t e r ) . The above mentioned term being roughly equal for the five most remote planets it is evident that the polar solar intensity at summer solstice is mainly governed by the expression is the f i r s t bracket. Introducing the numerical values of S^Q and of the orbital elements for Pluto and the outer planets (Table I V ) in expression ( 1 9 ) reveals that 0^D)^{N P}(^{S S}) p | u t o exceeds considerably the energy supplied to the north poles of Jupiter, S a t u r n , Uranus and Neptune. It is obvious that this higher maximum results from the slow axial rotation ( T = 6.3867) and the v e r y large obliquity [Pluto rotates lying practically (e = 60, 75°) or really (e = 9 0 ° ) on its side in the plane of its orbit] which overcompensates for the extremely high value of the mean distance of Pluto from the Sun ( a^Q ~ 3 9 . 7 2 ) .

It should be pointed out that Table IV is somewhat misleading in that it might suggest t h a t , on one hand, the peak insolation at the north pole during summer solstice is equal to that of the south pole at its summer solstice for Uranus and Neptune and t h a t , on the other hand, the maxima in the solar radiation incident at Jupiter occur also over the poles.

As to the f i r s t remark, we note that they are slightly different with 0^D)^{N P}(^{s s )} > (^{| D )}S P ( s s ) - T h i s f i n d i n9 c a n e a s i lV ^{b e} evaluated by computing the maximum insolation at the north pole.

Indeed, from the conditions <f> = - n/2, \^Q= 3n/2 and h^Q = n it follows that expression ( 7 ) can be written as :

- 2 4 -

( ID)S P ( s s ) = ( So T S i n £ [ ( 1 " e S i n V / ( 1 ~ e 2 ) ] 2 ( 2 0 )

Dividing (19) by (20) yields :

( ID)N P ( s s )/ ( ID)S P ( s s ) = [ ( 1 + e s i n V / ( 1 " e s i n V ] 2 ( 2 1 )

Hence :

( ID^NP(ss) > ( ID)S P ( s s ) i f 0 < < 71 (J uPi t e r> Uranus, Neptune)

and

( ID)N P ( s s ) < ( ID)S P ( s s ) i f 71 < < 271 <E a r t h> M a r s> Saturn, Pluto

From (21) it is also easy to show that the ratio (I ^ >^Np ( s s

^ ' d ^ S P ( s s ) 's e q u a l t 0 u n' t y f °^r ^p = 0 or \p = 7i and has a minimum (or maximum) value at = n/2 or A^p = 3n/2. Uranus and Neptune, having arguments of perihelion roughly coinciding with their vernal equinoxes and their eccentricities being v e r y small, it follows from (21) that 0^D)^{N P}(^{S S}) = ( ' D ^ S P ( s s ) ' C o n c e r n i n9 ^{m o r e} particularly Pluto, A.^p being close to n, one would expect a similar conclusion. However, the maximum difference between the peak insolations attains approximately 20%; this effect is ascribed to the v e r y large eccentricity of its orbit.

-25-

T h e equatorial insolation d u r i n g summer solstice, hereafter denoted as ( 'D)E(s s) ' m aY be obtained from relationship ( 7 ) b y putting

<)> = 0, \Q = n/2 or 3n/2 and hQ = n/2 and is g i v e n b y :

( ID)E ( s s ) = CSo T cos e/n a2) [ ( l ± e s i n Xp)/(1 - e2) ]2

(22)

where the plus s i g n is for the n o r t h e r n summer solstice and the minus s i g n for the southern summer solstice.

D i v i d i n g (19) or (20) b y (22) yields :

< V P (S S )/ ( I D W ) = 71 t g £ ( 2 3 )

stating that the polar insolation at summer solstice is larger than that at the equator for e > 17°.7 (all planets except Jupiter) (see e . g . Ward, 1974; Levine et al., 1977). A s a consequence, Jupiter is the only planet in the solar system where the equator at summer solstice ( a n d even over the entire y e a r ) receives more daily insolation than the poles.

E x p r e s s i o n (23) clearly indicates that the value of (| D) p (s s) / (I ) „ , v is exclusively dependent on e. For Pluto, this ratio amounts to

v D E ( s s )

about 5.4, 11.7 and infinity (in this case the S u n does not rise at summer solstice) respectively for obliquities equal to 60,75 and 90°.

From F i g s . 1 to 3 and Fig. 5 it can be seen that (| D)N P(S S) and x increase with increasing e (Pluto's spin axis and

D S P ( s s )

consequently its pole tilts f u r t h e r away from the normal to the orbit plane so that the amount of polar e n e r g y in summertime is enhanced).

T h i s conclusion can also be derived from (19) and ( 2 0 ) , the obliquity

- 2 6 -

producing variations in the peak insolations through the sin e dependence of the solar f l u x . Furthermore, according to Figs. 4 and 5, the steady increase of 0

)

(

)

( ' d ^ S P ( s s )

P

V

corresponding loss of the equatorial solar radiation. Evidently, this systematic decrease of insolation can also be deduced from relation (22).

More generally, one can determine the solar longitude interval where 0

) p > ( '

)

- Indeed, from ( 7 ) it is easy to shown that :

v 2 2 1/2

( I

)

/ U

)

= ± 71 tan 6

= ± s i n e s i n \

/ ( 1 - s i n e s i n \

)

(2'0

the plus sign being used for the north pole, the minus sign for the south pole. Introducing the numerical values for e (60, 75 and 90°) in equation (24) it follows that, for the north pole, 0

)

> 0

)

if ranges from approximately 20 to about 160°. For the south pole, the polar daily insolation exceeds that of the equator in the approximate solar longitude interval (200-340°).

The contour maps and especially Fig. 5 reveal that the position of maximum solar radiation is shifted markedly, by about 20°, from the position of summer solstices. T h i s is due to the fact that thé perihelion position ( \

S 190°) is located approximately 80° to the left of the south summer solstice ( \ = 3n/2). It is worth pointing out that the solar longitude of maximum insolation can also mathematically be derived. However, taking into account the complexity of the computing algorithm, it is recommended to numerically calculate the polar insolation variability as a function of From the plotted curves ( F i g . 5) it can be seen that the maximum solar radiations occur near 110 and 250°.

It is well known that a large eccentricity (which is the case for Pluto) produces important north-south seasonal asymmetries in the

-27-

daily insolation ( F i g s . 1 to 3 and Fig. 5). On the other hand, a change in the obliquity causes mainly a global latitudinal redistribution ( F i g s . 4 and 5).

When comparing F i g s . 1, 2 and 3, it is also obvious that the general pattern of the three contour maps is only slightly different. In the solar longitude interval (n/2 - 3n/2) and especially at equatorial and midlatitudes the isocontours closely parallel the seasonal march of the S u n . In the region where the S u n does not set, the shape of the lines of constant daily insolation is r o u g h l y similar, although shifted with respect to the summer solstices, to the c u r v e limiting the area of permanent s u n l i g h t . Furthermore, the isopleths clearly illustrate that for nearly half the Pluto year (approximately 124 Earth y e a r s ) some parts of the planet are in permanent d a r k n e s s . It is evident that the zone where the S u n does not rise increases with increasing obliquity.

Another point of interest r e g a r d s the distribution of the daily solar radiation in the equatorial region. A t e < 45° (all planets except Uranus and Pluto), the Arctic circles bounding the polar region in which there are d a y s without sunset or s u n r i s e , lie outside the tropical zone (in which there are d a y s on which the S u n reaches the zenith) and because of the relatively small eccentricities the latitudinal insola- tion has one maximum and one minimum over the year if the polar night is considered as one maximum. Uranus (e = 82°14) and Pluto occupy a rather peculiar position in that they rotate lying practically or really on their sides in the orbital planes. In the polar regions the day and the night are approximately half a year long and the S u n is close to the zenith midway t h r o u g h the sunlit half of the year. Summer and winter are, roughly s p e a k i n g , repeated twice a year in the equatorial region, the two seasons being substantially more temperate than in the polar regions. T h i s interesting phenomenon is clearly demonstrated in Fig. 4 (contrast also this diagram with Fig. 5).

In two previous papers (Van Hemelrijck, 1982a,b) we discussed the oblateness effect (see also Brinkman and McGregor, 1979) on the solar radiation incident at the top of the atmospheres of the outer planets (all except Mars and Pluto). An investigation related to Pluto of the influence of the flattening cannot be made, the oblateness, defined as f = (a - a )/a where a and a signify respectively the e p e e p equatorial and the polar radius, being unknown (Newburn and Gulkis, 1973).

4.2. MEAN D A I L Y I N S O L A T I O N

The mean daily insolations, taken over a season or a year, are computed by the procedure discussed in section 2, the results of which appear in Figs. 6, 7 and 8.

Fig. 6, representing the latitudinal variation of the mean summer daily insolation for the three obliquities, shows that at the poles 0

) g increases with increasing obliquity ( e ) , whereas at the equator the opposite effect is found. For comparison, if e varies from

- 2 - 1

60 to 90°, ( 0

)

changes from 6.02 to 7.01 cal cm ( d a y ) at the north and south pole respectively; at the equator, however, the corresponding mean summer daily insolations amount to 2.68 and 2.23 cal cm ( d a y ) . All differences are of the order of 15%.

The sensitivity of the summertime polar insolation to changes in the obliquity can easily be illustrated by deriving the expression for ( ] _ ) _ . In fact, similar to relation (12) we obtain :

D S P

( ID)S = ( SQ T Tq

s i n

- e

)

(25)

stating that 0

)

monotonically increasing function of e. Note also that the summertime insolation changes only by about 1% between the two extremes of the length of the summer ( T _ ) (see table I I ) .

-29-

MEAN SUMMER DAILY INSOLATION [cal cm'^dayr']

FIG. 6 . - Latitudinal variation of the mean summer daiiy insolation O Q ) ^ at the top of the atmosphere of Pluto for various values of the obliquity.

On the other hand, the mean summer daily insolation for the equator can only be expressed by a complete elliptical integral of the second kind (Ward, 1974; Vorob'yev and Monin, 1975), but it can mathematically be proved that it is a monotonically decreasing function of sin e as illustrated in Fig. 6.

Finally, it should be pointed out that in both hemispheres the intersection of the c u r v e s representing the latitudinal distribution of the mean summer daily insolation as a function of e occurs at a latitude of approximately 20° and that 0 ^D)^S is only weakly dependent on e in the neighborhood of the above mentioned latitude.

Another point about the c u r v e s is that 0^D>^S > ^ ' d ^ ^>

( 7 ) . T h i s characteristic feature follows immediately from ?he theoretical analysis given in section 4.1 and evidently from Figs. 4 and 5. Moreover, the f i r s t inequality also clearly indicates that the c u r v e s are not symmetric with respect to the equator.

T h e mean winter daily insolations corresponding to the three obliquities under consideration are plotted in Fig. 7. Since in winter the Sun does not rise at the poles it is obvious that O Q ) w = A t a l t

latitudes, the effect of increasing obliquity can clearly be seen to reduce the insolation. As an example, the mean wintertime insolation at the equator varies by about 15% between the two extremes of the obliquity range and maximally by about 20% near the equator. At high

latitudes, particularly near the poles, the obliquity effect is of decreasing significance. Especially noteworthy is the fact that the loss of insolation when e varies from 60 to 75° is much higher than the reduction in the 75-90° obliquity interval. Furthermore, the north-south seasonal asymmetry in the distribution of the incident solar flux can also be seen from Fig. 7.

For the sake of completeness, the latitudinal variation of the mean annual daily insolation is given in Fig. 8. As already previously

-31-

M E A N W I N T E R DAILY I N S O L A T I O N [ c a l c m "2( d a y ) "1]

F I G . 7 . - L a t i t u d i n a l v a r i a t i o n o f t h e mean w i n t e r d a i l y i n s o l a t i o n (7,-A., at t h e t o p o f t h e a t m o s p h e r e o f

D W

P l u t o f o r v a r i o u s v a l u e s o f t h e o b l i q u i t y .

- 3 2 -

MEAN A N N U A L DAILY INSOLATION leal cm- 2 (day)"1]

F I G . 8 . - Latitudinal variation of the mean annual daily insolation 0^Q) ^A at the top of the atmosphere of Pluto for various values of the obliquity.

-33-

stated, ( i p ) ^ can be computed b y the general formalism d e s c r i b e d in section 2. H o w e v e r , it is expedient to calculate the a v e r a g e y e a r l y insolation b y e x p r e s s i o n ( 1 6 ) . T a k i n g into account the numerical v a l u e s of T _ , T.„ a n d T ( T a b l e s II a n d I I I ) , relation ( 1 6 ) can, in a v e r y s w o

good approximation, be written u n d e r the following general form :

( ÏD)A = 0.53 ( ÏD)S + 0.47 ( ÏD)W (Northern h e m i s p h e r e ) (26) or

( ID)A = 0.47 ( ID)S + 0.53 ( ID)W (Southern h e m i s p h e r e ) (27)

(Note that the ratios T _ / T and T.../T are only s l i g h t l y different when

v s o w o

e r a n g e s from 60 to 90°).

F i r s t , it s h o u l d be emphasized that, for the outer planets, there e x i s t s a critical obliquity (e = 54°) ( W a r d , 1974; V o r o b ' y e v and M o n i n , 1975; T o o n et al., 1980) past which the poles receive more annual a v e r a g e e n e r g y than the equator. Hence, this situation is not only realized b y U r a n u s (e = 82°14) but also b y Pluto when a s s u m i n g that s is equal or larger than 60°. T h i s interesting phenomenon is p a r t i c u l a r l y evident from Fig. 8 where it can also be seen that the steady increase in polar insolation [ ( i p ) ^ = 3.22, 3.60, 3.72] is accompanied b y a c o r r e s p o n d i n g decrease in the equatorial insolation [(7 ) = 2.87, 2.54, 2 . 3 7 ] . In other w o r d s , for £ = 60, 75 and 90°,

D A n

the decrease amounts respectively to about 11, 29 and 36%.

S e c o n d l y , from F i g s . 1, 2 and 3 a n d p a r t i c u l a r l y from F i g . 5, it follows that the daily insolation at the s o u t h pole d u r i n g summer solstice [ 0n)C D, \ = 12.5, 13.8 and 14.2 for £ = 60, 75 a n d 90° r e s p e c -

-34-

t i v e l y ] is a p p r e c i a b l y h i g h e r t h a n t h a t o f t h e n o r t h pole at i t s summer s o l s t i c e [ O r J » , r ^ \ = 1 0 . 0 , 1 1 . 3 , 1 1 . 7 ] . On t h e o t h e r h a n d ,

D N P ( s s )

a p p l i c a t i o n o f e x p r e s s i o n ( 1 2 ) y i e l d s a mean a n n u a l d a i l y i n s o l a t i o n w h i c h is t h e same. T h i s f i n d i n g can e a s i l y be e x p l a i n e d b y o b s e r v i n g t h a t t h e l e n g t h o f t h e n o r t h e r n summer (48455, 48187, 48108) is l o n g e r t h a n t h e l e n g t h o f t h e s o u t h e r n summer (42128, 42396, 42475) t h e r a t i o o f b o t h seasons b e i n g a p p r o x i m a t e l y e q u a l t o 1 . 1 5 . From F i g . 8 i t is o b v i o u s l y e v i d e n t t h a t t h e a b o v e m e n t i o n e d i n s o l a t i o n imbalance v a n i s h e s in t h e t o t a l t a k e n o v e r t h e y e a r , t h e c u r v e s b e i n g 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 l a t i t u d e <|> = 0 ° .

F i n a l l y , F i g . 8 c l e a r l y i l l u s t r a t e s t h a t a t a l a t i t u d e o f a p p r o x i m a t e l y 30° t h e mean a n n u a l d a i l y i n s o l a t i o n s a r e p r a c t i c a l l y e q u a l . For l a t i t u d e s b e t w e e n t h e e q u a t o r a n d t h e a b o v e m e n t i o n e d l i m i t i t is f o u n d t h a t ( 7^D)^A (e = 6 0 ° ) > 0^Q )^A (e = 7 5 ° ) > ( 7^D)^A (e = 9 0 ° ) ; b e y o n d 30° we h a v e : ( 7Q)A ( e - 9 0 ° ) > ( lQ)A (e = 7 5 ° ) > ( 7Q)A (e = 6 0 ° ) . T h i s c h a r a c t e r i s t i c b e h a v i o r r e s u l t s f r o m t h e c o m b i n e d i n f l u e n c e of t h e mean summertime a n d w i n t e r t i m e i n s o l a t i o n s ( F i g s . 6 a n d 7 ) .

I t is also i n t e r e s t i n g t o n o t e t h a t in t h e l a t i t u d e i n t e r v a l ( 0 - 4 0 ° ) a n d f o r e = 6 0 ° , t h e mean a n n u a l d a i l y i n s o l a t i o n is q u a s i c o n s t a n t , t h e maximum d i f f e r e n c e a t t a i n i n g o n l y a b o u t 2.5%.

5 . SUMMARY A N D C O N C L U S I O N S

I n t h e p r e s e n t p a p e r an a t t e m p t is made t o c a l c u l a t e t h e i n t e n s i t y a n d t h e p l a n e t a r y - w i d e d i s t r i b u t i o n o f t h e d a i l y s o l a r r a d i a t i o n on P l u t o as well as t h e l a t i t u d i n a l v a r i a t i o n o f t h e seasonal a n d a n n u a l a v e r a g e i n s o l a t i o n f o r t h r e e f i x e d v a l u e s o f t h e o b l i q u i t y (e = 60, 75 a n d 9 0 ° ) .

A n a n a l y s i s o f F i g s . 1 t o 5 r e v e a l s t h a t t h e maximum solar r a d i a t i o n o c c u r s - o v e r t h e poles a r o u n d summer s o l s t i c e s w i t h v a l u e s of a p p r o x i m a t e l y 11 t o 13 cal c m "² ( d a y ) "¹ ( N o r t h p o l e ) a n d 1 3 . 5 t o 1 5 . 5

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cal cm ( d a y ) ( S o u t h p o l e ) . T h i s maximum amount of solar e n e r g y exceeds considerably the corresponding solar radiation incident on J u p i t e r , S a t u r n , Uranus and N e p t u n e . O b v i o u s l y , this higer maximum results from t h e extremely slow axial rotation and t h e v e r y large obliquity of Pluto. Similarly to t h e E a r t h , Mars and S a t u r n , the absolute maximum is attained at t h e South pole. According to expression

( 2 1 ) it is t h e position of perihelion ( n < Ap < 2n) t h a t is responsible f o r this phenomenon.

A t summer solstice and if £ > 1 7 ° . 7 (all planets except J u p i t e r ) t h e poles receive more daily insolation than t h e e q u a t o r . More- o v e r , it can be proved that over practically the e n t i r e Pluto y e a r t h e polar insolation is g r e a t e r than t h a t of t h e equator where a maximum

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value of the o r d e r of 6 cal cm ( d a y ) is f o u n d .

When comparing Figs. 1, 2 and 3 and from Fig. 5 it can be seen t h a t the polar insolation increases with increasing o b l i q u i t y ; this g a i n , h o w e v e r , is accompanied by a corresponding decrease of the equatorial solar radiation.

T h e general p a t t e r n of the contour maps is only weakly dependent on e. A t equatorial and mid-latitudes and f o r 90° < \^Q < 2 7 0 ° , t h e isocontours closely parallel t h e solar declination c u r v e . A t polar latitudes, the shape of the lines of. constant daily insolation is r o u g h l y similar to the c u r v e limiting t h e zone of permanent s u n l i g h t .

It is also interesting to note t h a t in the equatorial region summer and w i n t e r are repeated twice a y e a r .

Finally, we also have studied the latitudinal v a r i a t i o n of the mean daily insolations. It is found t h a t in summer and if e v a r i e s from 60 to 9 0 ° , the mean summertime insolation increases with about 15% at the poles, b u t decreases at the equator with approximately t h e same q u a n t i t y . At a latitude near 20° the variation of the mean summer daily insolation with e is either unimportant.

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In winter, and at all latitudes, an increase of the obliquity causes the mean daily insolation to reduce, the influence of the obliquity being of decreasing significance at polar region latitudes.

The seasonal north-south asymmetries in the mean daily insolation are particularly evident from Figs. 6 and 7.

As stated previously, the ratio of the annual insolations at the equator and the poles is smaller than unity for e > 54°. For Pluto, this ratio amounts to about 0.9 (e = 60°), 0.7 (e = 75°) and 0.6 (e = 90°). For comparison, it is approximately equal to 2.4, 2.2, 18.4, 2.1, 0.7 and 1.9 for the Earth, Mars, Jupiter, Saturn, Uranus and Neptune respectively. Finally, Fig. 8 also clearly indicates that the hemispheric seasonal asymmetry in the solar radiation disappears when averaging the incoming solar energy over the year and that in the vicinity of <|> = 30°

the dependence of the mean annual daily insolation on e is rather small.

In conclusion, we believe the calculations presented in this work could help in studies of the climatic history, the energy budget and the dynamical behavior of Pluto. It is, however, evident that in the future observations have to be carried out systematically over an extended period of time in order to improve the accuracy of some orbital elements, especially the direction of the axis of rotation which is v e r y poorly determined.

ACKNOWLEDGEMENTS

We particularly thank Dr. V . K . Abalakin and Dr. M.E. Davies for bringing the Report of the IAU Working Group on cartographic coordinates and rotational elements of the planets and satellites to our attention. We are grateful to Dr. J . Vercheval and Dr. M. Scherer for clarifying discussions and to J . Schmitz and F. Vandreck for the realisation of the various illustrations.

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