The photometric functions of Phobos and Deimos I. Disc-integrated photometry

ICARUS 28, 405-414 (1976)

The Photometric Functions of Phobos and Deimos I. Disc-Integrated Photometry M. NOLAND AND J. V E V E R K A Laboratory/or Planetary Studies, Cornell University, Ithaca, New York 14853 R e c e i v e d M a y 30, 1975; r e v i s e d A u g u s t 4, 1975 We have used the integrated brightnesses from Mariner 9 high-resolution i m a g e s t o d e t e r m i n e t h e large p h a s e angle (20 ° t o 80 ° ) p h a s e c u r v e s o f P h o b o s a n d D e i m o s . T h e d e r i v e d p h a s e coefficients are fl = 0.032 + 0.001 m a g / d e g for P h o b o s a n d fl = 0.030 4- 0 . 0 0 1 m a g / d e g for D e i m o s , while t h e c o r r e s p o n d i n g p h a s e i n t e g r a l s are qpho~os= 0.52 and qDelmos= 0.57. The p r e d i c t e d i n t r i n s i c p h a s e coefficients o f t h e surface m a t e r i a l are fl~ = 0.019 m a g / d e g a n d fit -- 0.017 m a g / d e g for P h o b o s a n d D e i m o s , r e s p e c t i v e l y . T h e p h a s e curves, p h a s e coefficients a n d p h a s e i n t e g r a l s are t y p i c a l o f o b j e c t s w h o s e surface layers are d a r k a n d i n t r i c a t e in t e x t u r e , a n d are c o n s i s t e n t w i t h t h e p r e s e n c e o f a r e g o l i t h on b o t h satellites. The r e l a t i v e reflectance o f D e i m o s t o P h o b o s is 1.15 4- 0.10. T h e p r e s e n c e o f several b r i g h t p a t c h e s o n D e i m o s could a c c o u n t for t h i s slight difference in a v e r a g e reflectance.

1. INTRODUCTION The Mariner 9 high resolution camera (B-camera) obtained some 40 pictures of the Martian satellites Phobos and Deimos (Veverka et al., 1974), at an effective wavelength of 0.56~m. The large phase angle coverage, 18 ° ~ a ~ 83 °, is unobtainable from E a r t h and offers an opportunity to determine the phase functions of the two satellites in a region where the opposition effect is absent. Available Earth-based observations of Phobos and Deimos are limited to phase angles less than 20 ° . Photoelectric observations from Earth have the advantage of greater photometric accuracy t h a n spacecraft vidicon results, but, unlike spacecraft observations, they are plagued by intense scattered light from Mars. In addition Earth-based work is limited in aspect coverage. From Earth-based photographic measurements, Pascu (1973) infers linear phase coefficients for Phobos and Deimos in the range of 0.03 to 0.04mag/deg at small phase angles (a < 9°). Photoelectric observations of Deimos by Zellner and Capen (1974) yield a phase coefficient of Copyright © 1976by AcademicPress,Inc. All rights o f reproduction in any form rescrvexL Print~l in Great Britain

0.036mag/deg for 10 ° < a < 20 °. These results are in reasonable agreement with those obtained in this paper for larger phase angles.

2. TH~ B-CAMERA DATA

405

Seventeen pictures of Phobos, and six of Deimos were reduced photometrically, and the picture element by picture element (pixel by pixel) intensities were stored on digital tape. These pictures are known as the Reduced Data Record (RDR). The firstfew columns of Table I identify the pictures involved and give the relevant phase angles and sub-spacecraft longitudes and latitudes. Associated with each pixel of the reduced image is a number (in arbitrary "data number" units called D N ) which is nominally proportional'to the absolute brightness, or intensity at that point. Although the absolute photometric accuracy of the B-camera is known to be poor (Young, 1974), preliminary results published by Thorpe (1972, 1973a) indicate t h a t the relative photometry should be acceptable. Consequently, we have per-

406

NOLAND AND VEVERKA

TABLE I GEOMETRY OF PHOBOS--DEIMOS PICTURES a

Satellite/ spacecraft orbit I) D D ~D

Phase angle ~¢(deg)

Sub-spacecraft Apparent Apparent semisemiProjected Lat. W. Long. m a j o r minor area (deg) (deg) axis ( k i n ) a x i s (km) (km 2)

Measured illuminated fraction of area ~(1

73 111 159

22 31 51

--4 --1 --20

41 27 3

6.5 6.2 6.0

5.5 5.5 5.7

112.7 107.8 106.6

0.96 0.92 0.79

0.96 0.93 0.82

149 I ) 25 D 63

65 68 73

+27 --14 --14

355 20 7

6.0 6.1 6.0

5.8 5.6 5.6

109.1 107.4 105.3

0.62 0.74 0.66

0.71 0.69 0.65

P 129 P 129

18

--30

17

11.0

10.1

349.2

0.95

0.98

19 26 41 45 45 49 50 57 59 64 66 71 76 80 80 83

--27 --32 --41 --35 --39 --62 --62 --42 --71 +24 --19 +43 --27 --31 --48 --64

13 24 23 65 20 55 52 135 160 344 152 333 341 344 172 356

10.8 11.2 11.3 13.0 11.3 13.1 13.1 12.3 12.9 10.9 11.2 11.6 11.0 11.0 11.3 12.4

10.1 10.1 10.4 10.0 10.4 10.5 10.5 10.2 10.7 10.0 9.8 10.4 10.0 10.2 10.7 10.7

343.7 357.1 368.9 405.8 366.3 431.5 431.5 392.5 433.6 341.3 344.7 378.9 346.2 349.7 379.1 417.0

0.95 0.92 0.83 0.85 0.83 0.84 0.84 0.81 0.75 0.62 0.78 0.57 0.70 0.66 0.64 0.61"

0.97 0.95 0.88 0.85 0.86 0.83 0.83 0.76 0.76 0.72 0.71 0.67 0.62 0.59 0.59 0.56

P P P P P P P P P P P P P P P

53 221 131 161 48 48 73 34 150 43 171 31 77 89 80

a Assuming the axes of Phobos are 13.5 × 10.7 x 9.6km, and the axes of Deimos are 7.5 x 6 . 0 x 5 . 5 k m (Duxbury, 1974). b ½(1 + cos ~) = fraction of the projected area of a sphere illuminated at phase angle ~. formed our analysis without applying any corrections to the RDR. The results, not only in this paper, but especially in the t w o s u c c e e d i n g p a p e r s o f t h i s series ( N o l a n d a n d V e v e r k a , 1976a a n d 1976b), s u b s t a n t i a t e t h e a s s u m p t i o n t h a t for our purposes n o g r o s s e r r o r s e x i s t i n t h e relative photometry of the B-camera. On the basis of more recent studies T h o r p e (1975) feels t h a t i n f a c t i n a c curacies in the relative photometry of the B-camera may exist, especially at very low and at very high DN. Although the precise nature of the errors has not been determined, Thorpe suggests that a correction factor similar to that derived f o r t h e A - c a m e r a ( T h o r p e , 1973b) m a y

be appropriate. Accordingly, we have also analyzed the data using such a c o r r e c t i o n f a c t o r . T h e effect o n t h e d e r i v e d p h a s e coefficients is n e g l i g i b l e , w h i l e t h e s c a t t e r o f t h e d a t a p o i n t s is, i f a n y t h i n g , increased. In summary, errors in the relative photometry of the B-camera may exist, but if the errors are of the type suggested b y T h o r p e , t h e i r effect o n o u r r e s u l t s would be insignificant. Since the precise nature of these errors remains to be specified, w e c h o o s e t o p r e s e n t o n l y t h e r e s u l t s d e r i v e d f r o m t h e n o m i n a l l y reduced RDR data. Finally, we note that the upper lefth a n d c o r n e r o f t h e B - c a m e r a is k n o w n t o

PHOTOMETRIC FUNCTIONS OF PHOBOS AND DEIM0S

contain significant residual shading. Fort u n a t e l y , n o n e o f t h e s a t e l l i t e i m a g e s falls in this comer. 3. THE PHASE CURVES OF PHOBOS A N D DEMOS B y s u m m i n g t h e D N ' s for all p i c t u r e e l e m e n t s i n a g i v e n s a t e l l i t e i m a g e we o b t a i n ~ D N , w h i c h is p r o p o r t i o n a l t o t h e i n t e g r a t e d b r i g h t n e s s o r flux f r o m t h e satellite at that particular phase angle and viewing geometry. This quantity, listed in c o l u m n 3 of Table I I , depends on the s h u t t e r speed, t h e s a t e l l i t e - S u n d i s t a n c e and the satellite-spacecraft distance. We convert the ~DN values to a c o m m o n scale u s i n g t h e s h u t t e r s p e e d

407

factor given in c o l u m n 4 of T a b l e I I , a n d r e d u c e all o b s e r v a t i o n s t o c o m m o n s a t e l l i t e - S u n a n d satellite-~spacecraft distances. The reference distances chosen a r e : for P h o b o s , 1 . 4 3 A U a n d 5 7 1 0 k m ; for D e i m o s , 1 . 4 3 A U a n d 7 2 2 0 k m . T h e a p p r o p r i a t e c o r r e c t i o n f a c t o r s a n d corrected i n t e g r a t e d intensities are given in columns 4 t h r o u g h 8 of Table II. The corrected integrated intensities can now be reduced to relative magnitudes. I t is c o n v e n i e n t t o choose t h e zero o f t h e m a g n i t u d e scale a t t h e s m a l l e s t p h a s e a n g l e s o b s e r v e d (22 ° for D e i m o s ; 18 ° f o r Phobos). The resulting normalized relative m a g n i t u d e s are l i s t e d i n c o l u m n 3 o f T a b l e I I I , a n d are p l o t t e d a g a i n s t p h a s e a n g l e i n Figs. 1 a n d 2.

TABLE I I DERIVATION OF THE RELATIVE BRIGHTNESSOF EACH PIC'±"t):aE,IGNORINGGEOMETRICCORRECTION

Phase Satellite/ angle a revolution (deg)

RDR integrated intensity (~ DE)

Shutter speed factor a

Mars-Sun distance factor b

Satellitespacecraft distance factor c

D73 D lll D159 D149 D 25 D 63

22 31 51 65 68 73

186 194 549 016 147 728 310 414 74 080 151 063

1.0 0.5 0.5 0.5 1.0 0.5

1.028 1.057 1.100 1.100 0.986 1.014

1.945 1.000 1.941 0.578 1.496 1.158

P P P P P P P P P P P P P P P P P

18 19 26 41 45 45 49 50 57 59 64 66 71 76 80 80 83

355 334 325 463 451 466 562 576 823 693 108 303 103 55 301 678 455

1.0 1.0 1.0 0.5 0.5 0.5 1.0 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 0.5 0.5

1.071 1.071 1.014 1.160 1.071 1.100 1.014 1.014 1.028 1.000 1.100 1.000 1.115 1.000 1.028 1.042 1.028

4.792 4.916 4.329 2.946 3.324 3.067 1.572 1.581 1.280 1.000 6.458 1.671 5.665 6.475 2.027 1.018 1.577

129 129 53 221 131 161 48 48 73 34 150 43 171 31 77 89 80

461 551 394 370 800 754 506 703 590 707 800 095 800 581 010 420 556

Corrected integrated intensity 372 290 157 98 109 88

Relative integrated intensity

287 155 707 681 272 690

1.00 0.78 0.42 0.27 0.29 0.24

1 824 423 1 761 351 1 428 252 791 643 804 157 787 370 896 867 924 649 541 814 693 707 386 416 506 320 327 802 359 904 313 662 359 678 369 217

1.00 0.97 0.78 0.43 0.44 0.43 0.49 0.51 0.30 0.38 0.2I 0.28 0.18 0.20 0.17 0.20 0.20

a Shutter speed factor = 1.0 for 24msec exposure; 0.5 for 48msee exposure. Mars-Sun distance factor -- (Mars-Sun distance in AU/1.43) 2. c Satellite-spacecraft distance factor = (satellite-Mars distance in km/7220 or 5710) 2.

408

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Fzo. 1. Phase curve for Phobos, uncorrected for aspect variations. The phase angle is denoted by =; ¢ is the relative integrated brightness in magnitudes. The magnitude scale is normalized to zero at c¢= 18°. I

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FzG. 2. Phase curve for ]:)eimos, uncorrected for aspect variations. The magnitude scale is normalized to zero at c¢= 22 °. T h e p h a s e curves of P h o b o s a n d D e i m o s g i v e n in Figs. 1 a n d 2 a p p e a r to be linear, a n d a least-squares fit t o the points s h o w n yields a p h a s e coefficient ~ ___0.03 m a g / d e g for b o t h satellites. I f P h o b o s a n d D e i m o s were spheres, no f u r t h e r corrections would be necessary. B o t h objects are irregular, however, a n d t h e observations used in Figs. 1 a n d 2 r e p r e s e n t a v a r i e t y of aspects. I f we are to o b t a i n reliable p h a s e curves, some g e o m e t r i c corrections m u s t be applied.

Our first t a s k is to a d j u s t all i n t e g r a t e d brightnesses of a g i v e n satellite to t h e s a m e t o t a l p r o j e c t e d area. This is accomplished b y p r o p o r t i o n a t e l y increasing or decreasing t h e i n t e g r a t e d brightness, depending on w h e t h e r the picture h a s a p r o j e c t e d area smaller or g r e a t e r t h a n some chosen n o r m . W e a r b i t r a r i l y choose to normalize to the p r o j e c t e d area f o u n d a t the smallest o b s e r v e d p h a s e angle: = 1 8 ° for P h o b o s (P 129); a = 2 2 ° for D e i m o s (D 73).

400

P H O T O M E T R I C F U N C T I O N S OF P H O B O S A N D D E I M O S

I n order t o calculate the projected areas, we e m p l o y the triaxial ellipsoid models developed b y D u x b u r y (1974). Using these axes (Table IV) a n d the subspacecraft point (Table I), we can calculate the a p p a r e n t projected axes a n d area of each picture (Noland, 1975). These axes a n d areas are given in columns 5 t h r o u g h 7 of Table I. The projected area corrections a n d the resultant corrected m a g n i t u d e s are given in columns 4 a n d 5, respectively, in Table I I I . A l t h o u g h all i n t e g r a t e d brightnesses h a v e n o w been a d j u s t e d to the same t o t a l p r o j e c t e d area, t h e illuminated fractions which t h e y represent are still characteristic o f diverse viewing geometries. W e n e x t a d j u s t t h e i n t e g r a t e d b r i g h t n e s s of each

picture t o t h a t of a sphere of the same t o t a l projected area a n d average surface brightness at the same phase angle. I f f is the a c t u a l fraction of the surface illuminated, the correction factor is A M ~2.5 log [f/f (sphere)]. W e measure t h e illuminated fraction o f the projected area b y using a polarplanimeter a n d the longitude-latitude grids for each picture generated b y D u x b u r y (Veverka £t a/., 1974). This fraction is t a b u l a t e d in column 8 of Table I, while column 9 of the same table gives t h e corresponding fraction for a sphere. The correction factors a n d the " e q u i v a l e n t sphere" m a g n i t u d e s are given in columns 6 a n d 7 o f Table I I I . The resulting p h a s e curves are p l o t t e d in Figs. 3 a n d 4.

TABLE I I I EFFECT OF VARIOUS GEOMETRIC COP~RECTIONS ON THE RELATIVE BRIGHTNESS OF THE SATELLITES AS A FUNCTION OF PHASE ANGLE SaIne

Satellite/ revolution

Phase angle ~ (dog)

Normalized magnitude

Projected area correctiona

projected area magnitudes

Nonsphericity correctiona

Equivalen~ sphere magnitudes

I) D D D D D

73 111 159 149 25 63

22 31 51 65 68 73

0.00 0.27 0.93 1.44 1.33 1.56

0.00 --0.05 --0.06 --0.04 --0.05 --0.07

0.00 0.22 0.87 1.40 1.28 1.49

0.00 --0.01 --0.04 --0.14 +0.07 +0.02

0.0O 0.21 0.83 1.26 1.35 1.51

P P P P P

129 129 53 221 131 161 48 48 73 34 150 43 171 31 77 89 80

18 19 26 41 45 45 49 50 57 59 64 66 71 76 80 80 83

0.00 0.04 0.27 0.91 0.89 0.91 0.77 0.74 1.32 1.05 1.69 1.39 1.86 1.76 1.91 1.76 1.74

0.00 --0.02 +0.02 +0.06 +0.16 +0.05 +0.23 +0.23 +0.13 +0.24 --0.03 --0.01 +0.09 --0.01 0.00 +0.09 +0.19

0.00 0.02 0.29 0.97 1.05 0.96 1.00 0.97 1.45 1.29 1.66 1.38 1.95 1.75 1.91 1.85 1.93

--0.03 --0.01 --0.03 --0.06 0.00 --0.04 +0.02 +0.02 +0.08 --0.01 --0.17 +0.09 --0.17 +0.13 +0.14 +0.10 +0.09

--0.03 0.01 0.26 0.91 1.05 0.92 1.02 0.99 1.53 1.28 1.49 1.47 1.78 1.88 2.05 1.95 2.02

P

P P P P P P P

P P P P

° Area correction = 2.5log (projected area of satellite/projected area of either I) 73 or P 129). v Nonsphericity correction ----2.51og [measured illuminated fraction of surface/0.5 (I + cos ~)].

410

NOLAND AND VEVERKA I

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20

30

40

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60

70

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Fro. 3. Phase curve for Phobos corrected for departures from sphericity of the satellite surface. The magnitude scale is normalized to zero at = = 18°. I

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FxG. 4. Phase curve for I)eimos corrected for departures from sphericity of the satellite surface. The.'magnitude scale is normalized to zero at ~ = 22 °. I n principle, one f u r t h e r correction should be m a d e . T h e a v e r a g e surface b r i g h t n e s s a t a g i v e n p h a s e angle is also a function of aspect, since it depends on t h e d i s t r i b u t i o n of incident a n d e m e r g e n t angles o v e r t h e i l l u m i n a t e d area. Unless t h e a c t u a l surface s c a t t e r i n g law is known, w e can n o t c o m p e n s a t e for this effect. H o w e v e r , N o l a n d (1975) shows t h a t for a p r e s u m e d H a p k e - I r v i n e s c a t t e r i n g funct i o n , t h e additional correction to a n y one p o i n t is likely to be <0.1 m a g for p h a s e a~gies less t h a n 70 °, a n d t h a t t h e n e t

effect on the least-squares fit p h a s e coefficient is negligible. W e believe t h a t t h e p h a s e curves shown in Figs. 3 a n d 4 can be directly c o m p a r e d w i t h t h e p h a s e curves of spherical solar s y s t e m objects, such as the Moon. C o m p a r i n g Figs. 3 a n d 4 w i t h Figs. 1 a n d 2, we see t h a t t h e values of fl a n d M o h a v e n o t been significantly affected b y a p p l y i n g the series of corrections, b u t t h e deviations of individual points f r o m t h e straight-line fit h a v e decreased. T h e f o r m a l p h a s e coefficients for t h e p h a s e curves

PHOTOMETRIC FUNCTIONS OF PHOBOS AND DEIMOS

5. THE PHASE INTEGRALS OF PHOBOS AND DEIMOS

shown in Figs. 3 and 4 are: fl = 0.032 ± 0 . 0 0 l m a g / d e g for Phobos, f l = 0 . 0 3 0 ± 0.001mag/deg for Deimos. F o r comparison, L a n e a n d Irvine (1973) measure fl = 0.028mag/deg for the Moon at visible wavelengths.

Since our d a t a cover a large range of phase angles, we should be able to m a k e an adequate determination of the phase integrals of the two satellites

0 ~ 0

(1)

3~,

(4)

with fl ___0.03mag/deg. Earth-based observations (Section 1) show t h a t this value of the phase coefficient also applies reasonably well for a < 20 °. I f we assume t h a t t h e approximation to the real phase curve given b y (2) can be used in evaluating (1)--a procedure first discussed b y S t u m p f f (1948)--we have

(2)

q = 2 f : e-kB~sinado: •

Aft - flsphere -- fit = 0.013 mag/deg. Thus we would predict intrinsic phase coefficients of 0.019 and 0.017mag/deg for the surface materials of Phobos and I)eimos, respectively. These values are v e r y similar to the lunar fi~ = 0.018 mag/deg shown in Fig. 5 which we derived from the observations of Shorthill et al. (1969). I

°.

- 2 . 5 log ¢(a) =

Noland (1975) shows t h a t for phase angles between 20 ° a n d 80 °, there is a simple relationship between the phase coefficient of a uniform spherical object whose surface obeys (1) and the intrinsic phase coefficient of its surface material

i

(3)

I n Figs. 1 to 4 the observed phase curves of b o t h satellites are well represented b y straight lines

We define the intrinsic phase function for such surfaces to be

I

sin

where @(~) is the observed flux from a satellite at phase angle a relative to t h a t a t

The phase curves shown in Figs. 3 a n d 4 are typical of dark, t e x t u r a l l y complex surfaces for which I(i, ~, a) m a y be approxim a t e d b y (Veverka, 1971)

¢,(~) = - 2 . 5 log IV .f(o~)/C.f(O)].

2 j':

q=

4. THE INTRINSIC PHASE COEFFICIENTS OF THE SURFACES OF PHOBOS AND DEIMOS

I(i, E, ~) ---=C[cos//(cosi + cos E)]f(~).

411

= 2(1 +

e-~)/(1 + k2 f~),

(33

where k = (180/~) (In 10/2.5) = 52.77. Substituting the values of fl from Section 3 into (3') we have qPhobos 0.52 ± 0.03, qD©imos 0.57 -4- 0.03, =

=

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F]o. 5. The intrinsic phase curve for the lunar surface, based on observations by Shorthill e$ at. (1969). The magnitude scale is normalized to zero at ~ = 20°.

412

NOLAND AND VEVERKA

a t the effective wavelength of the Bcamera (A~0.56/~m). For comparison, th e value of the phase integral of the Moon, quoted b y Lane and Irvine (1973), is 0.61 for visible wavelengths. F o r objects such as Phobos and Deimos, whose surfaces are dark and intricate in texture, Stumpff's approximation should provide a good estimate of q, but one which is likely to be an upper limit (Veverka, 1976). The error of +0.03 quoted above is a lower limit since it results only from the +0.001mag/deg uncertainty in fl and not from any errors made in using approximation (3'). 6. RELATIVE BRIGHTNESS OF PHOBOS AND

DE~OS The ~ = O° intercepts of the corrected phase curves (Figs. 3 and 4) provide a good way of determining the relative magnitudes of Phobos and Deimos. Recall t h a t our Phobos results are normalized t o a spacecraft range of 5710km, whereas those for Deimos are normalized to a range of 7220km. In Fig. 3, zero magnitude corresponds to ~ D N = 1 824423 at a range of 5710km; in Fig. 4, zero magnitude corresponds to ~ DN = 372 287 at 7220km. Reducing the results to a commo n distance (7220km), and using the M0's given in Figs. 3 and 4, we find a magnitude difference of AM = 1.06 between Phobos and Deimos, assuming t h a t the opposition effects of the two satellites are identical. Corresponding values quoted by Earth-based observers (Pascu, 1973; Zeliner and Capen, 1974) range from A M = 1.00 to 1.05 at comparable wavelengths and at aspects TABLE

which result in similar relative projected areas. The next step is to use this magnitude difference to determine the relative reflectance of the two satellites. For this calculation we need to know the relative projected areas. Recalling t h a t all measurements have been adjusted to the projected areas of P 129 and D 73, we see from column 7 of Table I t h a t the relative projected area of P 129 to D 73 is about 3.10. This calculated ratio is in agreement with the value of 3.16 obtained by counting bright pixels on these pictures and extrapolating to zero phase. Adopting a value of / I M ~ 1, and 3.1 for the relative projected area of Phobos to Deimos, we find t h a t the relative reflectance of Deimos to Phobos is about 1.2 at ~ = 0% We can also use picture pairs at similar phase angles to determine the relative reflectance of the two satellites for ~ ~ 0% Since the phase coefficients of the two satellites are almost identical, such values should be applicable to ~ = 0 °. A comparison of D 159 (~ = 51 °) and P 48 (~ = 50°), for example, gives a ratio of about 1.1. Here again we use the relative number of bright pixels as a measure of relative areas. The possible errors in the reflectance ratio include photometric errors, phase coefficient errors and uncertainties in the relative areas. The latter error is potentially the largest: using the formal uncertainties in the axes quoted in Table IV yields an uncertainty in relative reflectance of 40%. At worst, then, the relative reflectance of Deimos to Phobos is ~1.2 i 0.5. However, the similarities of relative areas obtained b y counting pixels to relative areas determined from the IV

T R I A X I A L M O D E L OF P H O B O S AND D E I M O S a

Satellite

Largest axis (kin)

Intermediate axis (km)

Smallest axis (kin)

Phobos Deimos

13.5 ± 1 7.5+,3

10.7 ± 1 6.0 ± 1

9.6 ± 1 5.5 ± 1

a From Duxbury (1974).

PHOTOMETRIC FUNCTIONS OF PHOBOS AND DEIMOS

413

nominal axes would indicate t h a t the actual error is much less. Moreover, the similarity of the reflectance ratio obtained from P 48 and D 159 near a = 50 ° to the ratio determined b y extrapolating P 129 and D 73 to ~ = 0 ° indicates t h a t the combined uncertainty in phase coefficients and relative p h o t o m e t r y are probably only ~10%. We conclude t h a t the relative reflectance of Deimos to Phobos at 2 N 0.56#m is

in the Phobos data is due undoubtedly to the known presence of large topographic features (Veverka et al., 1974) seen under diverse lighting and viewing conditions. No a t t e m p t has been made to correct the phase curves presented here for changes with aspect or phase angle caused by large scale topographic roughness. The main conclusions of this paper can be summarized as follows:

R(Deimos)/R(Phobos) ~ 1.15 ± 0.10.

(1) I f P h o b o s and Deimos were spherical objects t h e y would have phase coefficients of about 0.03mag/deg between phase angles of 20° and 80 ° . The phase coefficient of Phobos m a y be slightly larger t han t h a t of Deimos: the formal values given in Table V are fl--0.032 ± 0.001 mag/deg for Phobos and fl = 0.030 q0.001mag/deg for Deimos. These values agree reasonably well with 0.036mag/deg found b y Zellner and Capen (1974) for Deimos at smaller phase angles, where there might be a slight increase due to the opposition effect. No trace of an opposition effect is expected in our phase curves, since the smallest phase angle reached b y our data is 18 ° .

Since the photometric function of both satellites is ver y lunar-like, no limbdarkening is expected at ~ = 0 °, and the term "reflectance" can be identified with either the normal reflectance or the geometric albedo (Veverka, 1976). 7. DISCUSSION AND CONCLUSIONS

The phase coefficients of Phobos and Deimos derived in this paper are summarized in Table V. Although the final results are largely independent of the nonsphericity corrections applied, the various corrections do reduce the scatter in the phase curves (cf. Figs. 1 and 2 with Figs. 3 and 4). The Deimos points scatter much less about the fitted phase curve t h a n do those for Phobos. At least two reasons for this tighter fit can be given: (1) all Deimos pictures were obtained at nearly identical aspects, (2) the face of Deimos viewed b y Mariner 9 is bland and devoid of large craters and other topographic features which produce large-scale shadows (Veverka et al., 1974). The greater scatter

(2) The predicted intrinsic phase coefficients for the surface materials of Phobos and Deimos are fit = 0.019 mag/deg and fit - 0.017mag/deg, respectively. These values are comparable to the lunar value of 0.018mag/deg (see Fig. 5). (3) For the phase integrals of the satellites we find q = 0.57 for Deimos and q = 0.52 for Phobos (at 56/~m), similar to the lunar value of 0.61.

TABLE

V

INTEGRATED-DISC PHASE COEFFICIENTS

Intrinsic phase function Corrected to equivalent sphere No correction for nonsphericity " Shorthill et al. (1969). Lane and Irvine (1973).

Phobos fl(mag/deg)

Deimos ~(mag/deg)

0.019 __0.001 0.032 _ 0.001 0.029 _+0.002

0.017 + 0.001 0.030 + 0.001 0.031 + 0.002

Moon 0.018° 0.028 b

414

NOLAND AND VEVERKA

(4) The phase curves, phase coefficients, and phase integrals are typical of objects whose surface layers are dark and intricate in texture, and are consistent with the presence of a regolith on both satellites (see also Noland et al., 1973; Gatley et al., 1974). The two succeeding papers in this series (Noland and Veverka, 1976a, 1976b) confirm that surface texture, and not large-scale topographic roughness, is responsible for the character of the observed phase curves. (5) The relative reflectance of Deimos to Phobos is 1.15±0.10 (at 0.56~m). Thus the surface material of Deimos is, on the average, slightly brighter than that of Phobos. The observed presence of several bright patches on Deimos (Pollack et a i , 1973; Veverka et al., 1974) whose albedo is about 30% higher than the mean disc-integrated value (Noland and Veverka, 1976a) could account for this slight difference in average reflectance. ACKNOWLEDGMENTS We are grateful to William Green and to the staff of the Image Processing Laboratory at J P L for reducing the Mariner 9 B-frame images to photometric form, to T. D u x b u r y for providing latitude/longitude grids for each satellite frame, and to J. B. Pollack, C. Sagan and two anonymous referees for helpful comments. We want to t h a n k especially T. Thorpe for helping us understand some of the mysteries of the Mariner 9 B-camera. This work was supported in part by Grants NGR 33-010-220 and NSG 7156 Planetology Program Office, NASA Headquarters. REFERENCES DUXB~RY, T. C. (1974). Phobos: Control network analysis. Icarus 23, 290-299. GATLEY, I., KIEFFER, H. H., MINER, E., AND NEUGEBAUER, G. (1974), Infrared observations of Phobos from Mariner 9. Astrophys. J. 19@, 497-504. HARRIS, D. L. (1961). Photometry and colorimerry of planets and satellites. I n Planets and Satellites (G. P. Kuiper and B. M. Middlehurst, Eds.), University of Chicago Press, Chicago, Illinois. LANE, A. P., AND IRVINE, W. M. (1973). Monochromatic phase curves and albedos for the lunar disc. Astron. J. 78, 267-277. NOLAND, M., VEVERKA, J., AND POLLACK,

J . B. (1973). Mariner 9 polarimetry of Phobos and Deimos. Icaru~ 20, 490-502. NOLAND, M. (1975). Photometric studies of Phobos, Deimos and the satellites of Saturn. Ph.D. Thesis, Cornell University. NOLAND, M., AND VEVERKA, J. (1976a). The photometric functions of Phobos and Deimos. II. Surface photometry of Deimos. Icarus, in press. NoI~ND, M., AND VEVERKA, J. (1976b). The photometric functions of Phobos and Deimos. I I I . Surface photometry of Phobos. Icarus, in press. PAscu, D. (1973). Photographic photometry of the Martian satellites. Astron. J. 78, 794-798. POLLACK, J. B., VEVERKA, J., NOLAI~I), M., SAGAN, C., DUXBURY, T. C., ACTON, C. H., JR., BORN, G. H., H~R~A~rN, W. K., AND SMITH, B. A. (1973). Mariner 9 television observations of Phobos and Deimos, 2. J. Geophys. Res. 78, 4313-4326. SHORTHILL, R. W., SAARI, J. M., AND BAIRD, F. E. (1969). Photometric properties of selected lunar regions. NASA Report CR-1429. STUMPFF, K. (1948). Concerning the albedos of planets and the photometric determination of the diameters of asteroids. (In German.) Astron. Nachr. 276, 118. THORPE, T. (1972). Mariner 9 television imaging performance evaluation (Mariner 9 TV subsystem calibration report). Project Report No. 610-237, Vol. II. Jet Propulsion Laboratory, Pasadena, California. THORPE, T. (1973a). Verification of performance of the Mariner 9 television cameras. Appl. Opt. 12, 1775-1784. THORPE, r . (1973b). Mariner 9 photometric observations of Mars from November 1971 through March 1972. lcarus 20, 482-489. THORPE, T, (1975). Private communication. VEVERKA, J., NOLAND, M., SAGAN,C., POLLACK, J. B., QUAM,~L., TUCKER, R. B., EROSS, B., DUXBURY, T. C., AND GREEN, W. (1974). Atlas of the moons of Mars. Icarus 23,206-289. VEVERKA, J. (1971). The physical meaning of phase coefficients. I n Physical Studies of Minor Planets (T. Gehrels, Ed.). NASA SP-267. VEVER~CA, J. (1976). Photometry and polarimerry of satellite surfaces. I. Photometry of satellite surfaces. I n Planetary Satellites (J. Barus, Ed.) Univ. of Arizona Press, Tucson, Ariz. YOUNG, A. T. (1974). Television photometry: The Mariner 9 experience. Icarus 21,262-282. ZELLNER, B. H., .~ND CAPEN, R. C. (1974). Photometric properties of the Martian satellites. Icarus 2S, 437-444.