Light scattering in a spherical, exponential atmosphere, with applications to Venus

ICARUS 63, 354-373 (19851

Light Scattering in a Spherical, Exponential Atmosphere, with Applications to Venus DAVID WALLACH Department of Physics, Pennsylwmia State University, McKeesport, Pennsylvania 15132 AN D

BRUCE HAPKE Department of Geology and Planetary Science, University o["Pittsbur~,,h Pittsbur~,,h, Pennsylvania 15260 Received June 25, 1984: revised May 17, 1985 The problem of the reflection of light from an optically thick, spherical a t m o s p h e r e in which the scatterers are distributed exponentially with a scale height small compared to the radius of the planet is discussed. Exact formal solutions are obtained for the single scattered c o m p o n e n t . Useful a p p r o x i m a t e analytic solutions, which also include multiply scattered light, are given. The results are applied to the analysis of the Mariner 10 limb and terminator images of Venus. The altitude of the " ' d e t a c h e d " haze layer discovered by Mariner 10 is at 79-85 kin, but in places the haze exists above 100 km. This layer apparently is a stable, planetwide feature which forms at the top of the Pioneer V e n u s upper haze layer. It was similar in location, scale height, and thickness at the times of the two m i s s i o n s , in contrast to the lower, high-altitude haze which changed dramatically. We discuss two possibilities for the nature of the limb hazes. ( I ) The lower haze is probably the sulfuric acid cloud and the " d e t a c h e d " layer m a y be a separate water-ice haze. (2) The " ' d e t a c h e d " haze layer m a y not be separate at all. but part of the sulfuric acid haze, and the apparent "'gap" at 75-80 km m a y be the source region of a broadband absorber. The spatial distribution of the strong nearUV absorber, which m a y be elemental sulfur as first suggested by B. Hapke and R. Nelson ( 1975. J. Atmos. Sci. 32, 1212-1218). is e x a m i n e d in light of our results. Several a r g u m e n t s indicate that there is no nonabsorbing, overlying haze and that the UV absorber e x t e n d s to the top of the haze layer. ~ 1985 Academic Pres~. Inc

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

The availability of high-resolution spacecraft images of the limb and terminator regions of bodies with atmospheres makes it necessary to take the spherical geometry explicitly into account when considering the scattering of sunlight from planets. Light scattering in a spherical atmosphere was first examined by Chapman (1931) in connection with the brightness of the twilight sky as seen from the surface of the Earth, and this problem has been discussed many times since then (e.g., Rozenberg, 1966). Spherical geometry is considered briefly in Chandrasekhar (1960) and Sobelev (1975). It was taken into account by 354

0019-1(135/85 $3.0(1 Copyright c; 1985by AcademicPicss, Inc. All rights of reproduction in any folm rcser',cd

Young (1969) and Ajello and Hord (1973) in connection with the atmosphere of Mars and by O'Leary (1975) and Lane and Opstbaum (1983) for Venus. Numerical solutions have been given by Adams and Kattawar (1978) and Kattawar (1979). Most of the applications have dealt with the case where the scattering region is optically thin. In this paper we do not so restrict the discussion, but assume only that the scatterers are distributed exponentially with altitude with a scale height which is small compared to the radius of the planet and which is constant over the region in the atmosphere that controls the brightness. We also derive useful analytic approximate expressions for the reflected intensity. The results are ap-

VENUS SPHERICAL ATMOSPHERE

355

plied to the high-altitude hazes observed by the Mariner 10 spacecraft on the limb and terminator of Venus.

°

II. S I N G L E S C A T T E R I N G T H E O R Y

We assume that the atmosphere is optically thick and that the gas density at the altitude of the cloud tops is sufficiently low that refraction may be ignored. We begin by considering only those rays which have been scattered once. L a t e r we show how the multiply scattered intensity can be added in a relatively simple manner. The brightness of a portion of a planet in a typical spacecraft image is often interpreted for convenience as if the light came from a point, referred to in this paper as the "target p o i n t , " on a hypothetical, hard reference surface located on a sphere of radius R0 relative to the center of the planet. The light actually consists of photons scattered by particles located at various depths in the atmosphere along the line of sight viewed by the detector. L e t the density of scatterers at any radius R be given by n(R) = nl e x p [ - ( R - R1)/H], where H is the scale height, RI is the radius at which the optical depth is unity, and nl is the density of scatterers at RI. It is assumed that H ,~ Rl and R0. The optical depth r looking radially to level R is

~'(R) = f~ n(R)o-dR = nlo-He

tR RO/tI,

where tr is the extinction cross section of the particles in the atmosphere. Thus, n(R) can also be written

n(R) = e x p [ - ( R - RO/H]/crH.

(I)

The geometry of the spherical atmosphere problem is shown in Fig. 1. The line of sight of the detector intersects the spherical reference surface at the target point T, where it makes an angle e with the vector R0 which connects the planet's center C with T. Collimated sunlight of intensity J is incident on the top of the atmosphere and at T makes an angle i with the v e c t o r R0. As the sunlight travels through

I

/

/

FIG. 1. Scattering geometry: T is the target point, STD is the scattering plane, and C is the center of the planet; AC is parallel to ST, and CB is parallel to TD.

the atmosphere along the path L' it is exponentially attenuated, until at the volume element dV, located along the line of sight at radius R a distance L from T, its intensity is li,(L) = J e x p [ - ( G ( L ) ] , where G(L) is the slant-path optical depth to d V of the incident ray:

G(L) =

n(R ')o-dL' 1

H fo e IR'-R'~mdL'.

(2)

Let d V be a cylinder of unit cross-sectional area and length dL. The fraction of/in(L) scattered in d V per unit solid angle toward the detector is

P(g) dlout(L) = li.(L)n(R)O's ~ dV W

e -(R-RO/H

= li, T~ P(g)

H

dL,

(3)

where o-~ is the scattering cross section, w = o-~/o- is the single scattering albedo, g is the phase angle, and P(g) is the particle angular scattering function. The scattered radiance is further attenuated as it travels toward the detector, until it leaves the atmosphere with intensity

dl(L) = dlout(L) e x p [ - F ( L ) ] ,

356

WALLACH AND HAPKE

w h e r e F ( L ) is t h e s l a n t - p a t h o p t i c a l d e p t h to d V o f t h e e x i t r a y :

c o s i' = c o s e ' c o s g + ( c o s i - c o s e c o s ~,,) sin e ' / s i n e - [ ( R 0 c o s e + L) c o s g cos e cos g)Ro]/R

F ( L ) = f ~ n(R")o-dL" = 1 H

e Ik" U,~/HdL".

= (R0 c o s i + L c o s ~,,)/R.

(4)

(14)

N o w , f r o m (7) a n d ( I I),

T o find t h e t o t a l r e f l e c t e d i n t e n s i t y , d l is i n t e g r a t e d a l o n g t h e line o f sight L. C o m b i n i n g ( 2 ) - ( 4 ) , t h e r a d i a n c e r e a c h i n g the det e c t o r is l(i,e,g) = J ~

+ (cosi-

P(g)A(i,e,g),

Ro c o s e + L R

dR dL

cos e'.

(15)

Also, d L ....

(5)

H2

e ~'' R , V H d_ L dL",

w h e r e , f r o m (9) a n d (15)

where 1

A ( i , e , g ) = ~ l-~ e tr'l'

IGII.)I e

IR

dL

Ro/H alL.

~

(6) T h e r e l a t i o n s b e t w e e n the v a r i o u s q u a n t i ties in F i g . l a n d E q s . (1)-(6) a r e R = (R~ + 2RoL c o s e + L2) I/e,

(7)

R ' = (R 2 + 2 R L ' c o s i' + L'2) re,

(8)

+ L"

R" dR" dL""

= {R c o s e + L")/R" so that

dL

H2 "

e ~R" *¢,~m 1

H c

IR

RO/II

and R"

=

(R 2 +

2 R L " c o s e ' + L"2) ''2.

(9)

T h u s , Eq. (6) c a n be w r i t t e n in the a l t e r n a t e form

Also, R s i n e' = R0 s i n e ,

(10)

A(i,e,g) -

- fll

,

¢11. ,(;)d--{,dF d L

so that

-

R0 c o s e ' = (1 - ~ sin2 e 1/~

(Ro + 2RoL c o s e + L - ' - R~ sin e eP/e/R = (R0cos e + L)/R.

(ll)

Now, cos i - cos e cos + sin e sin ~ c o s 4),

(12)

w h e r e 4~ is t h e a n g l e b e t w e e n the p l a n e s ACB and CTD. Also, c o s i' = c o s e' c o s g + sine' singcos~b. U s i n g (11) a n d (12)

f

]'l

0

e iI-+~;~d F ,

(161

w h e r e Fl - F ( L = - ~ ) , a n d n o t i n g that F ( L - o~) _ O. T h e e x a c t s o l u t i o n for G ( L ) w a s o b t a i n e d b y C h a p m a n (1931) a n d c a n be e x p r e s s e d in t e r m s o f m o d i f i e d B e s s e l f u n c t i o n s o f the second kind. However, these equations m a y be put into m o r e m a t h e m a t i c a l l y t r a c t a b l e f o r m , using the a s s u m p t i o n that t t R0, as f o l l o w s . E x p a n d i n g Eq. (8) for R ' gives R ' - (R 2 + 2 R L ' c o s i' + L'2) v2 L' R l~l + ~ - c o s i '

L '2 + ~sin

2i'

(13)

+

)

VENUS SPHERICAL ATMOSPHERE

L

= R + L ' ( c o s i' + ~-R sin z i' + . . . .

357

.....-..\/~0/gH .

)

R,,- R, + -R0 - cot 2 e e

H

2H

sin e

N o w , most of the attenuation of the incoming ray will take place within an altitude difference of about a scale height of d V , or R ~< R' <~ R + H. For grazing incidence, i = 7r/2, it is trivial to show that the path length corresponding to an altitude difference of H is V ~ R H . Thus, the attenuation occurs primarily for L ' / R <~ ~ / 2 H / R = ~ v / 2 H / R o . Since H ~ R0, higher order terms in L ' / R may be dropped with negligible error. However, it is necessary to retain the first-order terms, since cos i' can be zero. Thus,

(~2~

erfc ,~ ~,, -/-z-ffor" ,, u

R e

G(L) -

cos e')

'

(22)

2-~ s--~T '

(23)

R, It

sin e

Ro cos2i' e 2 H sin 2 i

sin i

erfc \ and Ro

2X/~R~Ro/H

FI

-

sin e

e

.

RI

t- R 0 cot 2 e

2u

(24)

where erfc(x) is the complementary error function of argument x.

LI2 R' ~R

+ L'cosi'

+ ~ 0 sinzi" (17)

Similarly, R"=R

II1.A. General Expression for A

L"2

+L"cose'

+~sin

L2

R-~R0 + Lcos e + ~sin

2e,

(18)

2e,

(19)

L cos e' ~-- cos e + --if- sin 2 e, 1%

(20)

and L cos i' = cos i + -n-- (cos g - cos i cos e). /x0 H o w e v e r , cos i' will differ appreciably from cos i only when cos i or cos e or both are small, in which case the factor ( L / R o ) cos i cos e in the last equation will be of second order and may be ignored, so that to sufficient accuracy, L cos i' --~ cos i + w- cos g. 1%

(21)

With these approximations the integrations in (2) and (4) can be carried out to give V'~'Ro/2H F(L)

--

I11. APPROXIMATE ANALYTIC SOLUTIONS, INCLUDING MULTIPLE SCATTERING

sin e

R-R, e

n

dL

F+

G +

= 0.

(25)

Carrying out the differentiation gives _e-~R ROm + G ( - c o s e' + cos i'

COS g] sin 2 i/

_e_~R_R,Vn COS g + COS e' = 0,

(26)

sin ~ i from which the following transcendental equation for L* may be obtained:

R. . . . 2e,

e

The density of scatterers, which is proportional to e x p [ - ( R - R O / H ] , decreases exponentially with altitude, while the transmission of the atmosphere, described by e x p ( - F - G), increases exponentially with an exponential function of altitude. Thus, the integrand of A is sharply localized within a narrow region along the line of sight. This region from which virtually all the light received by the detector is scattered is called the " a c t i v e z o n e . " The point in the center of the active zone where the integrand is maximum is designated L* and can be found by setting the derivative of the exponent in A equal to zero:

2H sin2e

L*2 erfc

sin e

R ( L * ) = Ro - R i + L * c o s e + ~ s i n

2e

358

WALLACH AND HAPKE = HIn

COS g

1+ ~

sin- I

+

(

cos

- cos i* cos g] k/~-~Ro/2H__ _ sin 2 i/ sin i e r f c ( x /"- ~ c °' ssin ~ ) l2/ c ° s eH* }

e*

.020

e 2.1....... ~ ~i,,~i'i

(27A) .015

where L :~

c o s e * = cos e + ~ s m - e ,

<

(27B)

Q oc ,010 I,u

and

Z

t .5:

c o s i * -- c o s i +

~cosg.

(27C)

Equations (27) may be rapidly solved for L* by iteration as follows: A provisional value for R ( L * ) is obtained by setting L* 0 in the argument of the logarithm in (27A) and a first order estimate of L* calculated from R ( L * ) . Corrected values lbr R ( L * ) and L* are then found by inserting the provisional estimate of L* into the argument of the logarithm. One cycle of iteration is usually sufficient, except when both i and e are close to 90 ° , when several steps may be necessary for L* to converge. A useful analytic approximation lbr the error function is erf(x) = I + e

w'

(28)

.005

.05

.10

.15

L/R o

I-M;. 2. T h e i n t e g r a n d of A(i,e,g) for the c a s e i = 0. e = 90 ° , and g = 0, s h o w i n g the c e n t e r of the a c t i v e z o n e L* and the e q u i v a l e n t L o m m e I - S e e l i g e r point 1..

and G ( L ) and also is most sensitive to L: the other quantities are relatively slowly varying functions of L. If G ( L ) / F ( L ) is expanded in a Taylor series G(L) F(L)

G(L*) d /O'~ F(L*) + ~ ~ ,. ,IF(L) - F(L*)I,

where the minus sign is to be used for x -> 0 and the plus sign for x < 0. Equation (28) is asymptotically correct for large and small x and is accurate to within 1% everywhere. So far as we are aware, this expression has not been published before. An example of an active zone is shown in Fig. 2. Note that the length of the active zone is small fraction of the radius of the planet. The maximum length for all conditions considered in this paper is 6% of the radius. A first-order approximation to A can be obtained by noting that the factor e x p l - ( R - R t ) / H ] is c o m m o n to both F ( L )

it may readily be shown that the ratio of the second term to the first in the active zone is of order @ 0 or less. Thus, in the first approximation G ( L ) / F ( L ) is constant in the active zone and may be replaced with little error by its value at L*, where L* can be found as described above. Then (16) can be integrated to give the general approximate analytic solution (,'(L*)

A ( i , e , ~ ) ,~

e

• (;(L*I

=

G(L*) 1 + - -

F(L*)

(29)

VENUS SPHERICAL ATMOSPHERE

359

where FI is given b y (26) and G(L*)/F(L*) can be found b y substituting L = L*, e' = e*, and i' = i* f r o m (27) into (22) and (23). Equations (27)-(29) are c o n v e n i e n t for quickly calculating A(i, e, g) using a prog r a m m a b l e hand calculator or personal computer. Relative intensities calculated using Eq. (29) were c o m p a r e d with those f r o m numerically integrating Eq. (6) for the images used in this paper. The two solutions were found to be within 5% of each other in all cases.

which is the familiar L o m m e l - S e e l i g e r law of single scattering f r o m a plane-parallel atm o s p h e r e . H e n c e , except within about 10 ° from the limb or terminator, the curvature of the a t m o s p h e r e has negligible effect and the results for planar a t m o s p h e r e s m a y be used. N o t e that in this case A is independent of H ; in fact, it m a y readily be shown that plane-parallel a t m o s p h e r e s obey the L o m m e l - S e e l i g e r law independently of the f o r m of n(z) so long as n(~) = 0.

III.B. Disk Photometry

The following c o n v e n t i o n is adopted in analyzing limb images: when e < 7r/2 the radius of the nominal surface R0 is a s s u m e d to be equal to RI, the radius of the T = 1 level; w h e n the spherical surface tangent to the line of sight is a b o v e R~, e is taken to be equal to ~r/2 identically and R0 is defined as the radius of the sphere to which the line of sight is tangent. Using the a b o v e convention, cos e - 0 in the active zone. Since cos i is not small, G(L*) ~- e x p [ - ( R - RO/H] sec i; since cos e is small, expression (22) must be used for F(L*). N o w , the argument of the erfc in (22) is of the order of one, so that F(L*) is of the order of V'~-Ro/2H e x p [ - ( R - Rl)/ H]; hence, G(L*)/F(L*) ,~ 1. Thus, near the limb expression (29) for A b e c o m e s

The s e c o n d - o r d e r t e r m s in (17)-(21) m a y be neglected if cos i and cos e >> ~/-H~R0. The scale heights of scatterers in a planetary a t m o s p h e r e are usually of the order of the gas scale height, typically several kilometers to tens of kilometers, while the reference radius will be close to the optical radius of the planet, several thousand to several tens of thousands of kilometers, so that for all planets in the Solar S y s t e m this quantity will be of the order of 10 2. Specifically for Venus, the scale height of CO2 gas at a typical upper atmospheric t e m p e r a t u r e of 200 ° K is a p p r o x i m a t e l y 4 km, and the optical radius is 6120 km, so H~e/-H-~-R0 = 0.018. Thus, the second-order terms m a y be ignored except within, conservatively, 10° of the limb or terminator. When cos i and cos e are not small the arguments of the erfc in F(L) and G(L) are large, so that the a s y m p t o t i c expression erfc(x) = exp(-x2)/~/-~ x m a y be used. In this case,

a(i,e,g) -~ 1 - e x p ( - F i ) .

(31)

According to the convention we have adopted, when the line of sight does not intersect the r = 1 surface, e = 7r/2. In this case, inserting (24) into (31) gives A(Az) = 1

F(L) ~ e x p [ - ( R - Rl)/H]sec e, G(L) = e x p [ - ( R - R1)/H ]sec i, G(L)/F(L) = sec i/sec e,

- exp[ - ~¢/2~(R1 + Az)/H e-aZm],

F i -----~~ ,

and (23) b e c o m e s

A(i,e,g) ~ foe-('+ ~ ) V dF cos i = cosi+ cose'

III.C. Limb Photometry

(30)

(32)

where Az = R0 - R~ is the altitude of the surface tangent to the line of sight a b o v e the ~- = 1 level. N o t e that this a p p r o x i m a t e analytic e x p r e s s i o n for the relative limb brightness profile is independent of i. When the line of sight is exactly tangent to the z = 1 surface, A(Az = 0) -~ I, which is the same as the L o m m e l - S e e l i g e r value for

360

W A L L A C H AND H A P K E

e = ~-/2. T h e optical limb m a y be defined as the altitude AZL w h e r e the intensity drops to half its saturation limb value, or A(AzL) = 0.5. F r o m (32) this o c c u r s at

AZL/H = In ( k / - 2 ~ d H / l n 2).

(33)

F o r V e n u s , AZL ~ 5H. B e c a u s e (33) is so insensitive to R d H , AZL/H ~ 5 for all planets with e x p o n e n t i a l a t m o s p h e r e s . Slightly a b o v e the optical limb X/27rRdH e x p ( - A z / H ) <~ 1, so that (32) b e c o m e s A(Az) = 2N/~RwR~IHe x p ( - A z / H ) .

(34)

In this region a semilog plot o f brightness v e r s u s altitude is a straight line with slope - l / H . This b e h a v i o r o c c u r s w h e n the attenuation along b o t h the i n c o m i n g and outgoing paths is negligible, so that, f r o m (3), (5), and (6) J

1= ~

terers at the altitude o f unit slant-path optical depth for the i n c o m i n g ray.

III.E. Multiple Scattering T h e fact that the active zone is small c o m p a r e d with the radius o f the planet m e a n s that within this space the g e o m e t r y d e p a r t s f r o m being plane parallel by a very small a m o u n t . F o r all situations c o n s i d e r e d in this p a p e r the deviation was less than _+1.7 ° . This suggests that it is possible to apply the large b o d y o f solutions d e v e l o p e d for plane a t m o s p h e r e s to spherical atmos p h e r e s with little error. F o r any given value o f A(i,e,~) an equivalent L o m m e l Seeliger f a c t o r can a l w a y s be f o u n d that describes single scattering f r o m a plane-parallel a t m o s p h e r e with equivalent angles o f incidence i and e m e r g e n c e ¢; defined so that

A(i,e,,~) - 1/(I -~ cos d/cos 0. f ~ n(R)o-~P(g)dL.

05)

At these altitudes the intensity is proportional to the integral o f no-oP(g) along the line o f sight. This result was a s s u m e d without p r o o f by Y o u n g (1969) and O ' L e a r y (1975) in their earlier a n a l y s e s o f the a t m o s p h e r e s o f Mars and V e n u s , respectively.

(38)

A point, d e n o t e d by l~, along the line o f sight near the active z o n e with i' i and e' - ~ can thus be found f r o m (13) and I15): cos ~

-

cos

e

+ (LIRo) sin ~ e,

(39)

and cos i = cos i + (£1Ro) cos g.

(40)

T h e radius at this point is, f r o m (19).

III.D. Terminator Photometry In the vicinity o f the t e r m i n a t o r cos i is small but cos e is not. T h e n , since the altitude o f the active z o n e is well a b o v e the r 1 level, both e x p | - ( R - RI)/HI and cos i' are small c o m p a r e d to one. T h u s , in Eq. (26) for the c e n t e r o f the active zone the nonnegligible terms are

/~2

R -- R0 + l ] c o s e

+ 2R~ sin2e"

(41)

- A c o s e ] / [ A - (I - A) c o s g ]

(42)

Solving (38)-(41) gives /~ = R0[(l - A) cos i COS d

cos e +

sin ~ e[(l - A) cos i - A cos el/[A

- G cos e' +. cos e' - t),

- (I - A ) c o s ~ ] ,

or

G(L*) = 1.

(36)

(43)

and cos i = c o s i

Since Fl --> ~, F ( L * ) ~ l, and cos e is not small, Eq. (29) b e c o m e s A = 1/[I + G(L*)/F(L*)] = F(L*) -~ e x p [ - ( R ( L * ) - Rj)/H] sec e.

(37)

T h u s , the t e r m i n a t o r intensity is approximately p r o p o r t i o n a l to the density o f scat-

+ cos g[(1 - A) cos i - A cos e]/[A - (I - A) c o s g ] .

(44)

A n e x a m p l e o f the equivalent L o m m e l Seeliger point is s h o w n in Fig. 2. O v e r most o f the disk, i - i and d - e. H a p k e (1981) has derived the tbllowing

VENUS SPHERICAL ATMOSPHERE equation for the total intensity of light scattered from a plane-parallel atmosphere. w cos i Ir(i,e,g) = J 4~r cos i + cos e [P(g) + h(cos i)h(cos e) - I],

(45)

h(x) = (1 + 2x)/(1 + 2x~v/1 - w).

(46)

where

The part of (46) proportional to P(g) exactly describes singly scattered radiation, while that proportional to [h(cos i)h(cos e) - 1] approximately describes the multiply scattered component. Since the atmosphere is flat to - 1.7°, Eq. (45) may be used to describe scattering from a spherical atmosphere by replacing cos i and cos e, which apply to the hypothetical reference surface and have no special significance, by cos i and cos ~, respectively, to include the contribution of multiple scattering. Thus, W

Ir(i,e,g) -'~ J ~

A(i,e,g)

[P(g) + h(cos 0h(cos g) - 1].

(47)

Of course, any other solution for a plane parallel atmosphere may be used instead of (45), such as one of those developed by Chandrasekhar (1960). However, it should be noted that the changes in brightness at the limb or terminator take place across a small portion of the planet, over which i, e, and g are virtually constant. Thus, the only effect of multiple scattering and single-particle angular scattering functions is to cause A to be multiplied by a term which is constant for all practical purposes. IV. APPLICATIONSTO VENUS IV.A. General Remarks Even though both the Earth and Venus have nearly the same diameter and surface gravity the surface pressure of Venus is about 90 times greater than that on Earth. The atmosphere is primarily CO2, and the planet is completely covered by clouds. Russell (1899) was the first to propose that

361

the observed extensions of the cusps of the bright crescent of Venus were due to a haze layer. Dollfus (1966) estimated that the haze layer has a scale height of about 4 km. A similar result was obtained by Goody (1967). Sill (1972) and Young (1973) proposed that this haze layer is composed of sulfuric acid particles. The clouds contain a material which strongly absorbs near-UV light. Hapke and Nelson (1975) suggested that this material is elemental polymorphic sulfur; this hypothesis was also discussed later by Young (1977). A short-wavelength UV absorber has been identified as gaseous SO2 by Barker (1979). The present study was undertaken because Mariner l0 sent back several highresolution images of the limb and terminator of Venus. The limb images show a distinct dip in the distribution of brightness with altitude (Murray et al., 1974), which has been interpreted as implying a detached haze layer. O'Leary (1975) made a preliminary analysis of the limb pictures and estimated that the altitude of the dip was 80 to 83 km. Since the Mariner 10 flyby of Venus in 1974 the planetary limb hazes were also observed by Pioneer Venus at comparable spatial resolution and described by Lane and Opstbaum (1983). Because the characteristics of the limb hazes observed by the two spacecraft seem to be remarkably similar we restrict our discussion to the Mariner 10 data. Mariner I0 images of Venus were obtained from the Jet Propulsion Laboratory. They had been processed to remove vidicon distortions and were photometrically corrected (Danielson et al., 1975). For each picture element (pixel) the absolute radiance integrated between 200 and 700 nm incident on the front of the telescope was given. From knowledge of the position of the spacecraft relative to Venus, the positions of the four corners and the center point of each image were specified relative to a spherical reference surface centered on the planet with a radius of 6120 km, which

362

WALLACH AND HAPKE

is 69 km a b o v e the m e a n surface. Since the images consist of a three-dimensional curved surface projected onto a two-dimensional plane, a nontrivial part of the analysis involved calculating the locations and p h o t o m e t r i c p a r a m e t e r s for any given pixel. The details are given in Wallach (1979). The single scattering brightness factor A(i,e,g) was evaluated numerically using Eq. (6), (22), and (23) for several values of H and conditions appropriate to each image. The results were then c o m p a r e d with the o b s e r v e d values normalized at a common point. F o r increased a c c u r a c y the following a p p r o x i m a t i o n to the error function was used rather than (28):

2

[

e f t ( x ) = 7rtan J X / ~ x 4 X3

+5

I X5

+5 I

+ ~ x

I "r7

+~

+

~ "Vu

15 + 2 x 10 ~'.r ~l

(48)

This expression is accurate to within 0.1% for txl < 3. It has not been published before. Since only the slope of the logarithm of the relative brightness was used in the analyses, multiple scattering and single-particle phase functions will not affect the results and were not calculated, although we have shown in the previous section how this can easily be done.

1V.B. Terminator Photometry Only two high-resolution terminator images were available; both were of the evening terminator of Venus near the north pole. A p r o c e s s e d version of one of the pictures, n u m b e r 57779, is shown in Fig. 3. The shadings indicate increasing brightness levels from the u p p e r right corner to the lower left corner. The image has been contoured by clipping the five most significant bits f r o m the 8 bit n u m b e r representing the radiance of each pixel. Thus, the gray levels increase in brightness from I to 8 before the shading pattern repeats itself as level 9

through 16, and so on. Figure 4 shows the brightness along a line going from the lower left to the u p p e r right corner of the image. The unit of brightness is the DN, or data number, which can be related to the absolute radiance by a JPL-supplied conversion factor. Since the analysis of this paper uses only the relative brightness we used the DN unit. The equivalent surface distance on the reference sphere c o v e r e d by the line of Fig. 4 is about 230 km. The nominal angle of incidence changes from 89.66 to 91.36 ° . For each step of the plot the brightness factor, A was calculated for a variety of scale heights H. Multiple scattering was found to affect the slopes of the calculated brightness plots by less than 1% for any value of single scattering albedo and so was ignored. Figure 4 shows several of the calculated brightness profiles. The lines having the closest slopes to those of the observed data had H = 5 -+ I km. Brightness profiles were also calculated tbr several other lines drawn in different directions in Fig. 3 and were found to be consistent with H = 5 km. The center of the active zone corresponding to this portion of the terminator is 20 km a b o v e the reference sphere, or at an altitude of 89 km a b o v e the surface. Within the errors H is approximately the same as the gas scale height. The other terminator photograph, image n u m b e r 57777 (Fig. 5) was at larger nominal values of i, close to 94 °. It contains what appears to be the actual optical terminator, since in most of the picture the radiance is below the c a m e r a threshold sensitivity. Theoretical p h o t o m e t r i c profiles were fitted to DN values along the line from the lower left corner to the lower right corner, as well as to two lines parallel to the b o t t o m edge o n e - s e v e n t h and t w o - s e v e n t h of the full picture height a b o v e the bottom. The nominal angle of incidence along the bottom edge changes f r o m 92.37 to 93.45 ° . The observed radiances are plotted in Fig. 6, which also shows the calculations for various values of H. The best fits to the data were for curves with H = 9 _+ I km. The altitude of the

VENUS SPHERICAL ATMOSPHERE

363

FIc. 3. Bit-clipped image of an area near the terminator. Mariner 10 image No. 57779.

active zone was found to be 40 km a b o v e the reference level, or 109 km a b o v e the surface. Postflight engineering data indicated a c o n s e r v a t i v e uncertainty in the detector pointing angle of -+~°, which translates to an uncertainty of location of the target point of 150 k m on the reference surface. This contributed an additional error of -+ 1 km to H, so the scale height corresponding to the brightness distribution of image n u m b e r 57777 is H = 9 + 2 km, o v e r twice the gas scale height. A c a v e a t to the interpretation of the scale height inferred f r o m Fig. 5 is that m o s t of the brightness levels were just a b o v e the d e t e c t o r dark current. Although the data

were corrected for dark current, as well as other c a m e r a characteristics, the corrections w e r e b a s e d on preflight data which m a y h a v e changed s o m e w h a t during the trip to Venus. T h e s e low radiance levels m a y also possibly h a v e been affected by residual image or by scattered light from the brighter part of the planet, although there is no indication in any of the Mariner 10 data that these were significant problems. H o w ever, e v e n though the errors on H may be larger than estimated here, this would not alter the conclusion that in some places on Venus the scatterers, which are not gas because they have the wrong scale height, extend to altitudes of at least 1 i0 km.

364

WALLACH AND HAPKE

50

Z20 O

uJ 5 3.1 km

i

i

i

1

i

80 120 160 200 240 DISTANCE ON REFERENCE SURFACE, km

FIG. 4. Radiance along a line from the lower left corner to the upper right corner of Fig. 3. Heavy line: Mariner 10 data. Thin lines: theoretical profiles for several scale heights, normalized to the left end of the line. A profile calculated from the approximate analytic solution, Eq. (29), lbr tt 5 km lies between the lines for H = 4.6 and 5.1 km, but is not shown in order to avoid cluttering the figure.

IV.C. Limb Photometry Numerical calculations for conditions appropriate to the limb were carried out using Eqs. (6), (22), and (23). As predicted by the approximate analytic solution, Eq. (32), the relative radiance distributions were virtually independent of Sun angle. When the calculated brightness profiles were plotted against the altitude in units of scale height, the curves were approximately independent of H and only slightly dependent on

R1/H. The generalized theoretical limb profile is shown in Fig. 7. In this figure Az is the altitude of the spherical surface tangent to the line of sight above the r - 1 level. The generalized limb curve has the following properties. Changing the values of R~ or H shifts the curve laterally, but the amount of shift

is relatively insensitive to such changes. The brightness is almost constant from Az - 0, where r - 1, to about Az = 3H, where r = .05. Actually the brightness computed by exact numerical integration increases by about 5% between these two points. A b o v e the shoulder at Az = 3H the brightness drops and reaches the optical limb, defined as the altitude where the intensity has decreased by one-half, at about Az = 5H, at which level r = .007. At Az ~ 5H, slightly below the optical limb, the profile becomes a straight line on the semilog plot. The theoretical curves are compared with five of the observed limb profiles in Figs. 8-12. At high altitudes the curves are all reasonably straight on the semilog plots, consistent with exponential distributions of the scatterers at these altitudes. The scale heights are given on the figures and are between 2 and 3 km, except for image number 57840, for which H = 1.3 km. These values are less than the gas scale height, about 4 km. As the altitude decreases the brightness goes through a dip; each curve then increases abruptly to the shoulder, where it levels off, and in some of the images it decreases slightly, as predicted. In none of the cases investigated is it possible to match the portion of the curve below the dip with an exponential distribution of scatterers. A typical example of such an attempt is shown in Fig. 9. In that figure a theoretical curve with H = 0.8 km can be fitted to the portion of the data between the dip and just above the shoulder, but fails completely to match the data at the shoulder. H o w e v e r , in some of the cases the observed brightness below the dip approaches the theoretical curve for an exponential distribution extrapolated from the portion of the curve above the dip, almost as if the apparent " d e t a c h e d " haze layer were a continuation of the lower haze layer. Such a fit is excellent on image number 57839, reasonably good on 57856, 57855, and 57831, and less good on 57840. Using the theoretical fits to the brightness above the dip allows the position of the r

VENUS SPHERICAL ATMOSPHERE

365

FIG. 5. Bit-clipped image of the terminator. Mariner 10 image No. 57777. 1 level to be located on the curves, as shown. The theoretical analysis of Section IIl shows that above the optical limb the brightness is proportional to the integral of the scattering density [no'sP(g)] along the line of sight. Therefore, the brightness profiles can be inverted to give the relative scattering density as a function of altitude. This was done using the inversion technique described by Fjeldbo and Eshleman (1968). In performing the inversions it was assumed that the exponential distributions with the indicated scale heights could be extrapolated indefinitely with increasing altitude. The calculated relative scattering density distributions as a function of Az are shown in Figs. 8-12.

All the curves of scattering density versus altitude show a dip, including number 57831 which does not show it in the brightness distribution. Below the dips the scale heights change rapidly with altitude, but above the dips the distributions are exponential with constant scale heights between 1.3 and 2.9 km. The dip minima of the calculated scattering density profiles are at slightly higher altitudes than the minima of the brightness profiles. V. DISCUSSION

V.A. Altitude o f the Haze Features Fitting the theoretical curves for an exponential atmosphere to the limb brightness profiles, which appears to be empirically

366

WALLACH AND HAPKE 5O

2°i

BRIGHTNESS

~"'"":

H = 2.3 km

< <

•

k l = 9"2 km SCATTERING DENSITY

i J I 40 80 120 DISTANCE ON REFERENCE SURFACE, km

FIG. 6, Radiance along a line close to the bottom edge of Fig. 5. H e a v y line: Mariner 10 data. Thin lines: theoretical profiles for two scale heights, normalized to the left end of the line.

justified, at least, lor image numbers 57839, 57831, and 57856, allows the r -- I level to be located as shown in Figs. 8-12. The minima of the scattering density distributions o c c u r b e t w e e n 7 and 15 km a b o v e the r = I level. I f we adopt the Pollack et al. (1980) 1.00 .50 OPTICAL LIMB

=" .10 < .05

.01

I 4

6

8

J 10

Az/H

F[cJ. 7. The generalized limb brightness profile for an exponential a t m o s p h e r e ; z is the altitude of the spherical surface tangent to the line of sight above the r - I surface; c o n t i n u o u s line: exact numerical solution: d a s h e d line: approximate analytic solution, Eq. (32).

.1 __

_L 5

1'0

Az, km

~-1~

20

F[~. 8. Brightness profile perpendicular to the limb versus the altitude above the 7- = I level• Mariner 10 image No. 57839 (UV filter). C o n t i n u o u s line: data; dotted line: theoretical curve for the scale height given; d a s h e d line: scattering density no-oP(g) in relative units obtained from inversion of the brightness profile.

value, based on Pioneer Venus data, of 69 km for the altitude of this level the minima are at altitudes of 73-84 km, The associated m a x i m a are I - 3 km higher at 79-85 km. The optical thickness (looking radially) of the apparent detached layers are about 7 = 0.005. Thus, they are too optically thin to affect the photometric or polarimetric observations of Venus. The altitudes, optical thicknesses, and scale heights are consistent with the earlier independent estimate of O ' L e a r y (1975) from Mariner 10 data and are similar to those found by L a n e and O p s t b a u m (1983) from Pioneer Venus observations made 6 years later. It must be emphasized that the detached layers are not the "'thin upper h a z e " o b s e r v e d by Pioneer Venus (Kawabata et al., 1980), which are one to two orders of magnitude thicker, but a p p e a r to be a stable, planetwide feature which forms at

VENUS SPHERICAL ATMOSPHERE 100 ~

367

respond closely to the lower minimum in the temperature profile. The altitude of the terminator haze of image 57777 is close to that of the higher minimum in the temperature profile and implies local condensation there.

~ , . " . . . . . . • H = 0.8 km 3RIGHTNESS ~ . , . .

"-. H = 2.9 km Z 1

V.B. Limb Haze Models

t.u"

In this section we quantitatively examine two alternative explanations for the apparent detached haze layer visible on the limb images. V.B.1. The detached haze layer model. Turco et al. (1983) have calculated that at least three separate high-altitude hazes should be present on Venus. The lowest is the sulfuric acid haze which dominates the scattering at the r = 1 level near 69 kin. According to Fig. 2 of their paper, their model density in this haze decreases approximately exponentially with altitude with a scale height of about 2.6 km. The predicted mass loading at 69 km is M -- 500 /zg/m 3. These authors also predict a dis-

r,

[ I

NNN \

I

U

SCATTERING DENSITY

0

5l

lt0 ~,z, km

li5

20

FIG. 9, Same as Fig. 8 for image No. 57856 (orange filter). Theoretical curves for two different scale heights are shown.

100

the very top of the Pioneer Venus upper haze. The detached haze was observed by Mariner 10 over a latitude range from 3°S to 63°N (O'Leary, 1975) and by Pioneer Venus from 16°N to 43°N (Lane and Optsbaum, 1983). In contrast to the Pioneer Venus haze, which is highly episodic (Esposito, 1984), the detached haze and its associated " g a p " appear to be temporally and spatially stable. According to the Pioneer Venus measurements reported by Seiff et al. (1980) the temperature decreases from approximately 230°K at 79 km to a minimum near 160°K at 85 km, then rises to a maximum of 200°K at 110 km followed by another minimum at 115 kin. Thus, the scale height of the gas at these altitudes is between 3.5 and 5.0, which is approximately twice the particle scale heights deduced from the limb profiles. The altitudes of the limb profile detached haze maxima and the center of the active zone of terminator image 57779 cor-

BRIGHTN E S S - " ~ ' . . .. H = 2.5 km

SCATTERING DENSITY ~ /¢--\\ _

\

1

•10

I 5

ll0 ,~z, km

ll5

20

FIG. 10. Same as Fig. 8 for image No. 57855 (UV filter).

368

WALLACH AND HAPKE i - -

T

i

i

BR°HT\

10

Z £3 <

, t

OENS.TY

_J uJ cc 1

'10

\

\

5

10 ~z, km

_~

\

l i

1

--20

FrG. 11. Same as Fig. 8 for image No. 57831 filter).

~tJv

crete water-ice cloud at 85 km with maxim u m m a s s loading M = I 0 / ~ g / m ~, and an occasional m e s o p a u s a l ice haze at 115 km with m a x i m u m M = 0.03 p,g/m ~ The amount of material required to account for the hazes o b s e r v e d by Mariner 10 m a y be estimated as follows. For an exponential distribution the optical depth at any altitude is given by ~-(z) = n(z.)o-H. The average value of the scale heights derived from the limb profiles is 2.2 kin. Thus at the r = I level, noI / H = 4.5 x 10 ~'cm i Taking the particles as spheres of sulfuric acid with radius r = I /~m and density p = i.8 g / c m 3, o- = 2~'r 2, and M =- i~ ~ r ~ o n , the calculated mass loading of the cloud at 69 km is M = 5 0 0 / z g / m 3, in excellent agreement with the T u r c o et al. value. A b o v e the dip the m a x i m a in the observed profiles o c c u r at an average of 6. [ scale heights a b o v e the ~- = 1 level, where n o - = e x p ( - 6 . 1 ) / H = 1 x 10 Xcm ~. I f i t i s a s s u m e d that the haze particles are waterice spheres with average densities of I g/ cm 3 and radii of 1 p~m, the mass loading required to account for the observed bright-

ness is only M = 0.7/xg/m 3, well within the amount T u r c o et al. calculate could be present. The high-altitude haze o b s e r v e d on image 57777 is at the location predicted by Turco et al. (1983) for the mesopausal ice. Thus, the limb and terminator observations by M a r i n e r l 0 a p p e a r to be consistent with the T u r c o et al. (1983) model in which the cloud below the dip is sulfuric acid haze and the higher layers are water ice haze. V . B . 2 . The a b s o r b i n g l a y e r m o d e l . There are certain difficulties with the a b o v e interpretation which make it worthwhile to examine an alternate model. First, the brightness and scattering density profiles between the shoulder and the gap are inconsistent with an exponential distribution of scatterers. They decrease with increasing altitude nonexponentially much more rapidly than predicted by the Turco et al. (1983) model and would require an active sink of unknown nature at altitudes around

.... . °" "°'*".

100

- ' - ~ BRIGHTNESS~ \

7 - - -

'- H = 1.3 km ". ".

10

/ '+ \

_~,~ ~: SCATTERING DENSITY

0

_

_

i

5

110 3z, km

115

20

FiG. 12. Same as Fig. 8 for image No. 5784(I (UV filter).

VENUS SPHERICAL ATMOSPHERE 75-80 km. Second, it has already been noted in Section IV.C. that in several of the images the apparent detached haze layer seems to be an extension of the lower haze. This would be a remarkable coincidence if the two hazes were independent, or if they were separated by an active sink. Third, the temporal stability and planetwide nature of the apparent gap is easier to understand as resulting from a photochemical process than from condensation. Thus, we are led to suggest an alternate interpretation: the so-called detached haze layer is not a separate layer at all, but rather the apparent gap represents a local minimum in the scattering density. This minimum could be caused by a change in either n, o-s, or P ( g ) . A decrease in n would require an actual particle sink about one scale height thick and we have discussed the problems with this hypothesis in the preceding paragraph. A change in P ( g ) could be caused by a source of new type of particle and cannot be ruled out by present knowledge. Of the three possibilities it is the change in o-s which we find most intriguing: the minima may represent the formation of an absorber, probably by a photochemical process, at altitudes around 80 km. The absorber would have to be a broadband one since it is seen on both orange and UV images. There are a number of candidates for the possible high-altitude absorber, the most plausible being polymorphous elemental sulfur, chlorine gas, and disulfur monoxide. Hapke and Nelson (1975) first suggested that polymorphous sulfur was the longwavelength UV absorber in the Venus clouds. Although elemental S was not detected by the Pioneer Venus experiment, it still seems to be the best candidate (Toon et al., 1982). The ordinary cyclo-octal form of S, which is stable at room temperature, only absorbs in the UV. However, as Hapke and Nelson emphasized, newly formed S would probably have a large component of short-chain allotropes, which are known to absorb at longer wavelengths. As

369

the S particles grow and fall they would tend to polymerize, changing their absorbing properties with altitude and accounting for the apparent changing scale height below the minima. Unfortunately, the optical constants of polymorphous S are so poorly known that it is not possible to examine this hypothesis in any detail, although Hapke and Nelson (1975) showed that this substance could explain the spectral albedo of Venus. Pollack et al. (1980) have suggested that C12 may play a role in the absorption properties of the atmosphere of Venus. It is probably not the main near-UV absorber because chlorine also has absorption bands in the visible around 550 nm (Herzberg, 1950) that should be obvious in the Barker et al. (1975) spectrum of Venus, but are not. In order for an absorber of density n, and absorption cross section o-a to cause an appreciable drop in scattering density, we must have nao-~ ~ no'~2. The minima occur at about 5 scale heights above the ~" = I level, so no- ~ e x p ( - 5 ) / H = 3.1 x 10-8 cm -1. Taking o - a = 1.3 x 1 0 - 1 9 c m 2 in the UV (Hampson, 1980) gives n, = 1 x l0 I1 cm -3. This is two to three orders of magnitude above that predicted theoretically by Yung and Demore (1982) at this altitude. However, it corresponds to a mixing ratio of 1 ppm, which is well below the Pioneer Venus upper limit of 10 ppm (Hoffman et al., 1980), and thus cannot be excluded. The chlorine may be confined to a narrow range of altitudes because of rapid combination with H2 to form HC1. Because the layer is so thin it would only show up on limb images and not on disk photometry. Hapke and Graham (1984) have recently suggested that the long-wavelength absorber in the clouds may be solid SzO or the related polysulfur oxides, which can be formed from the products of photolysis of SO2, which absorbs strongly in the UV and blue. The evaluation of this hypothesis awaits the availability of quantitative data on absorption coefficients of these materials.

370

WALLACH AND HAPKE

Is there any other evidence besides the limb haze gap for a material which absorbs in the orange as well as in the UV in the Venus clouds'? In fact, some does exist. First, the Bond albedo of the planet appears to be around 92% at long wavelengths (lrvine et al.. 1968), rather than 100%,, although one of the co-authors of that reference informs us (A. Young, 1984, private communication) that the errors in those m e a s u r e m e n t s are large enough that an albedo of 100% cannot be excluded. Second, cloud contrasts as large as 5c~ occur in Mariner 10 images taken through the orange filter with an effective wavelength of 580 nm. This o b s e r v a t i o n is also consistent with an orange absorber, although A. Young (1984, private communication) points out that it might be possible to explain it by spatial variations in cloud optical thickness. Third, Pollack et al. (1980) and T o m a s k o et al. (1980) have analyzed the r a d i o m e t e r data on the Pioneer Venus sounder descent probe and shown that a long-wavelength a b s o r b e r is present a b o v e about 57 km. V.C. The L o c a t i o n o f the U V A b s o r b e r

In this section we examine the spatial distribution of the n e a r - U V absorber. The models which have been p r o p o s e d by various w o r k e r s to account for the photometric properties of the clouds of Venus can be grouped into a few general categories, depending on the degree of vertical and horizontal mixing of the absorber. M o d e l 1. The a b s o r b e r is well mixed vertically at all altitudes a b o v e a few optical depths: (a) The a b s o r b e r is horizontally confined to the darker areas. (b) The a b s o r b e r is present e v e r y w h e r e but is nonuniformly distributed horizontally, being m o r e concentrated in the dark areas.

M o d e l 2. A nonabsorbing haze overlies the absorber. (a) The a b s o r b e r is confined to the dark areas.

(b) The a b s o r b e r is present e v e r y w h e r e below the haze but is nonuniformly distributed horizontally. (c) The a b s o r b e r is uniformly distributed horizontally but the haze is thinner o v e r the dark areas. M o d e l 3. The a b s o r b e r is confined to a thin haze overlying nonabsorbing clouds. (a) The a b s o r b e r is confined to the dark areas. (b) The a b s o r b e r is present e v e r y w h e r e but is nonuniformly distributed horizontally. (c) The a s o r b e r is uniformly distributed horizontally but the haze is thicker o v e r the dark areas. All models of type 3 a p p e a r to be excluded by the Pioneer Venus probe radiometer data (Pollack et al., 1980) which shows that the a b s o r b e r extends down to altitudes around 57 km, corresponding to r =: 5. Models of type la, 2a, and 3a can be excluded by both ground-based telescopic observations (Barker et al., 1975; H a p k e and Nelson, 1975) and Mariner 10 data (Hapke. 1976) which clearly show that the UV absorber is present in both the light and dark regions. Esposito (1980) has asserted that model Ib is also incorrect. The basis for this assertion is the Pioneer Venus observation that the contrast between light and dark markings decreases as the phase angle increases and that this change can only be caused by an overlying, nonabsorbing haze. We examine this a s s u m p t i o n quantitatively. Esposito (1980) c o m p a r e d the UV reflectances r~ of bright areas, which tend to occur in the polar collar at latitudes around 45 °, with those rD of dark regions, which tend to be c o n c e n t r a t e d around 30 ° latitude. He calculated the contrast C - ( r B - - rD)/r~ and found that C d e c r e a s e d by a factor of 2 to 3 as the phase angle increased from a small value to around 120 °. He attributed this decrease to a nonabsorbing haze. However, he did not explicitly take into account the fact that the amount by which the reflectance of a cloud layer changes as the

VENUS SPHERICAL ATMOSPHERE angles of illumination i and viewing e vary depends on the cloud albedo, and this will also affect the contrast as the phase angle changes. T o illustrate quantitatively the expected magnitude of this effect we use the simple Eq. (45) for the bidirectional reflectance of a semi-infinite, uniform cloud of isotropic ( P ( g ) = 1) scatterers of single scattering albedo w, r

w cos i 4 cosi + cose h(cos i) h(cos e),

(49)

where r is the reflectance relative to a Lambert surface and h(x) is given by (46). The average UV albedo of Venus is about 0.5, which requires w = 0.95. A bright area at the subsolar point at zero phase with w = 0.97 will have rB = 0.60, and a dark one with w = 0.91 will have ro = 0.40, giving C = 33%. If the bright area is moved to latitude 45 ° and the dark area to 30 ° the zerophase contrast changes to C = 20%, which is typical o f the near UV contrasts (Hapke, 1976; Esposito, 1980). Keeping the latitudes constant and the two areas on the central meridian of the crescent, but increasing the phase angle to 120° , decreases the contrast to C = 11%. We conclude that most of Esposito's phase effect can be explained as due simply to the changing relative contribution of the multiply scattered c o m p o n e n t of the light and does not require a nonabsorbing haze layer. A second test, which can be made on high-resolution images of Venus at small phase angles, argues against models of type 2. N e a r both the limb and terminator the illuminating and reflected rays will lie along slant paths higher in the atmosphere than near the center of the disk. If a nonabsorbing layer were present we would expect to see the dark areas b e c o m e progressively brighter relative to the light areas until the contrast disappears close to the limb and terminator. Hapke (1976) specifically searched for such an effect in the Mariner

371

10 images and failed to find it. Neither does it seem to be present in the Pioneer Venus imaging data (e.g., Rossow et al., 1980). To be sure, the contrast decreases because of the effect described in the preceding paragraph, but the markings can easily be followed right up to the limb and terminator. A third argument that the UV absorber is present at high altitudes comes from the relative brightnesses of the Mariner 10 UV and orange limb pictures. When the images were photometrically corrected they were multiplied by factors which would give the radiance incident on the front of the telescope for a source with a Venus spectrum like that of Irvine et al. (1968) integrated between 200 and 700 nm. Thus, the correction implicitly assumes that the UV reflectances are about half the orange ones. If there were no absorber in the haze the brightnesses of the limb profiles above the shoulder would be expected to be systematically higher in the UV than in the orange. Several Mariner I0 limb radiance profiles are published here and additional ones are in O ' L e a r y (1975). In fact, no such effect is present. A fourth argument that the UV absorber is present in the high-altitude hazes is seen in the limb profiles (Figs. 8-12). According to Kawabata et al. (1980), the optical thickness of the haze layer at low latitudes is of the order of 0.05-0.10, so that A z / H ~ 2-3. The points on the limb profile (Fig. 7) corresponding to these values are located on the fiat part of the curve just below the shoulder. If an nonabsorbing layer were present the curves would be expected to fall below the predicted brightness as the altitude decreases below this level and the line of sight penetrates more deeply into the absorbing clouds. N o n e of the limb profiles observed by Mariner 10 or Pioneer Venus (Lane and Opstbaum, 1983) display this behavior. Except for the gaps the greatest discrepancies o c c u r just above the shoulders, where the brightnesses are too low, implying, if anything, a larger, rather than smaller, concentration of absorber at higher altitudes.

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Thus, we conclude that all models of type 2 are inconsistent with the available data. By elimination, the Venus clouds are best described by models of type I b, in which the absorber is well mixed vertically to the top of the haze. VI. CONCLUSIONS W e h a v e d i s c u s s e d the reflection of light f r o m a s p h e r i c a l p l a n e t w h e n the scale height o f the s c a t t e r e r s is small c o m p a r e d with the r a d i u s o f the p l a n e t a n d h a v e derived an approximate analytic expression for the i n t e n s i t y . T h i s t h e o r y was u s e d to p h o t o m e t r i c a l l y a n a l y z e the h i g h - r e s o l u t i o n M a r i n e r 10 l i m b a n d t e r m i n a t o r images. T h e r e s u l t s o f t h e s e a n a l y s e s m a y be s u m m a r i z e d as follows. T h e h a z e exists a b o v e 100 km in places a n d m a y b e the m e s o p a u s a l w a t e r - i c e haze p r e d i c t e d b y T u r c o e t al. (1983). T h e a p p a r e n t d e t a c h e d haze l a y e r is at a l t i t u d e s a r o u n d 7 9 - 8 5 k m , a n d m a y r e p r e s e n t the w a t e r - i c e h a z e also p r e d i c t e d by T u r c o et al. (1983). A n a l t e r n a t i v e m o d e l w h i c h is c o n s i s t e n t with the a v a i l a b l e data is that the a p p a r e n t gap b e t w e e n the m a i n a n d det a c h e d h a z e s is c a u s e d b y a b r o a d b a n d abs o r b e r at 7 5 - 8 0 k m , a n d the d e t a c h e d haze is a c o n t i n u a t i o n o f the m a i n sulfuric acid haze layer. T h i s q u e s t i o n c o u l d be r e s o l v e d b y a n o c c u l t a t i o n e x p e r i m e n t from a Ven u s - o r b i t i n g s p a c e c r a f t . T h e a v a i l a b l e evid e n c e i n d i c a t e s that the n e a r - U V a b s o r b e r , w h a t e v e r its c o m p o s i t i o n , is p r e s e n t at all a l t i t u d e s , e v e n in the h i g h - a l t i t u d e haze layers. T h e a p p a r e n t gap m a y be the s o u r c e r e g i o n of this a b s o r b e r . ACKNOWLEDGMENTS We thank A. Young for constructive comments on a preliminary version of this paper. This work was partially supported by grants from the National Aeronautics and Space Administration Planetary Geology Program and a private educational trust. During 1982-1983 B. Hapke was a National Research Council, National Academy of Sciences, fellow at the Space Science Division, NASA-Ames Research Center, and a portion of this work was done there.

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