Photometric properties of zodiacal light particles

ICARUS 62, 54--71 (1985)

Photometric Properties of Zodiacal Light Particles KARl L U M M E j Observatory and Astrophysics Laboratory University ~ff"Helsinki, Tfihtitorninmdki, 00130 Helsinki 13, Finland AN l)

EDWARD BOWELL Lowell Obxervatot3', P.O. Bor 1269, t;'lagst~t~i Arizona 86002 Received May 7, 1984: revised October 22, 1984 From published ground-based, spacecraft, and rocket photometry and polarimetry of the zodiacal light, a number of optical and physical parameters have been derived. It was assumed that the n u m b e r density, mean particle size, and albedo vary with heliocentric distance, and shown that average individual interplanetary particles have a small but definite opposition effect, a mean single-scattering albedo in the V band at I-AU heliocentric distance of (I.09 + 0.01, and a zerophase geometric albedo of 0.04. Modeled by a power law, both albedos decrease with increasing heliocentric distance as r .s4. The corresponding exponents for changes in mean particle size and number density are related in a simple way. The median orbital inclination of zodiacal light particles with respect to the ecliptic is 12° , close to the observed median value for faint asteroids and short-period comets. Furthermore, the color of dust particles and its variation with solar phase angle closely resemble those of C asteroids. These findings are, at least, consistent with the zodiacal cloud originating primarily from collisions among asteroids. Finally, a value of ~ I(P~p~ g was derived for the mass of the zodiacal cloud, where p~: is the mean particle radius (in micrometers) at I-AU-heliocentric distance. For extinction in the ecliptic, Am :- 10 s~: I~-"mag was obtained, where ~: is the solar elongation in degrees. , 198q Academic Press. Inc

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

Our goal in studying the photometric properties of the zodiacal cloud is twotbid. First, by evaluating the single-particle albedo and the spatial distribution of particles, we should be able to provide clues as to the composition, source, and perhaps dynamics of the cloud. Second, because zodiacal cloud particles are so distant from each other that they do not optically interact significantly, there is an opportunity to compute their single-scattering properties, which may be important in understanding the scattering of light by ensembles of particles in the regoliths of asteroids and satelAdjunct Scientist at Lowell Observatory, and Visiting Professor at the Department of Physics and As tronomy, University of Massachusetts.

lites. In these latter cases, the particles interact strongly with each other. At least two possibilities have been suggested for the origin of the zodiacal cloud. Whipple (1955, 1967) feels that short-period comets provide the main source of particles, with P/Encke being the major contributor over the past few thousand years. Delsemme (19761 and R6ser (1976), on the other hand, argue that short-period comets can account for only some 2 or 3% of the required mass. The main alternative, that the particles come from asteroids, has not been well explored because the dust production rate from asteroid collisions is unknown. Whatever the origin of the particles, the zodiacal cloud must be replenished on a time scale of about 10~ years, since Poynting-Robertson drag will cause depletion of small particles from the "~4

1tt119-11135/85 $3.00 ('opyright ~' 1985 by Academic Press, Inc. All rights of reproduction in any form reserved

ZODIACAL LIGHT PHOTOMETRY inner Solar System in that time (Wyatt and Whipple, 1950). Collection of airborne particles of supposed interplanetary origin has led to the conclusion that interplanetary particles consist predominantly of fluffy aggregates 10-100 /xm in diameter. There is general agreement that submicron-size particles do not play an important role in light scattering in the zodiacal cloud (Hanner et al., 1981). A good summary of these considerations, together with a presentation of much data on the zodiacal cloud, has been given by Leinert (1975). The single-scattering properties of particles in the Solar System are not well known. Saturn's rings have been well studied; but particles in that environment, although separate, interact considerably, particularly because of their high single-scattering albedo and because of mutual shadowing. Laboratory experiments carried out by a number of investigators appear to be the best way to understand light scattering by rough particles when Mie calculations are not appropriate (see, for example, Zeruil, 1976; Giese, 1980; Hanner et al., 1981). The scattering function of a rough particle may be described qualitatively as follows: At small scattering angles diffraction dominates, with a peak at scattering angle O = 0° whose width and height depend on the particle size. Outside the diffraction spike, the scattered intensity diminishes gently to a minimum, generally near 0 -- 90°. At large scattering angles there is a backscattered component, and for smooth spheres there is a strong increase toward 0 = 180° due to the glory. In Section 2, we derive empirically an expression for the mean single-particle phase function of zodiacal cloud particles. The dependence of brightness on ecliptic latitude is investigated in Section 3. In Section 4, by incorporating polarization data for both the zodiacal light and atmosphereless bodies, we derive the single-scattering albedo, the number density, and particle size, and determine their dependences on heliocentric distance. Then, in Section 5, we cal-

55

culate the total mass and extinction of the zodiacal cloud. Finally, in Section 6, we summarize our findings and their implications. 2. MEAN SINGLE-PARTICLEPHASE FUNCTION In the past, two methods have been used to derive the mean single-particle phase function P(O), where 0 is the scattering angle, of an interplanetary dust particle. One approach has been to fit a chosen Mie-scattering function (or matrix, if the polarization is also considered) to observations, and to evaluate the rather large number of free parameters (see, for example, Deirmendjian et al., 1961; Giese, 1961). However, it is well known that Mie scattering strictly pertains only to isolated, homogeneous, smooth spheres, so it is almost certainly inappropriately applied to the fluffy aggregates (or rough particles) that interplanetary particles are thought to comprise. A second approach has been to invert the brightness integral in order to obtain the single-particle phase function. This method was first introduced by Dumont (1973) and was later applied in a series of papers by Dumont and S~inchez (1975) and Dumont and Levasseur-Regourd (1980). Although mathematically rigorous, this inversion method is notoriously sensitive to the accuracy of the data, often leading to unphysical results such as negative values for P(O). In the following treatment we use a method that can be thought of as "direct." We make an "educated guess" for the particular form of P(O) and evaluate the appropriate free parameters by fitting the calculated surface brightness of the zodiacal cloud to the observational data. This method leads to an analytical expression for P(O) which, to our knowledge, has not previously been accomplished except by means of the Mie theory. It is usually assumed that the particle number density n(r) follows a power law as a function of heliocentric distance r and that all other parameters, such as the single-scattering albedo

56

LUMME AND BOWELL

iS0, the m e a n particle radius p, and the single-particle phase function P, are invariable (cf. Leinert, 1975). H e r e , we relax these restrictions and allow O3oand p (but not P) to vary with heliocentric distance. This separation of variables is of c o n s e q u e n c e only to the study of the polarization properties of interplanetary particles (Sect. 4). For computational reasons we choose to represent the d e p e n d e n c e on r by a p o w e r law: n(r) = n E ( ~ ) "'

~0(r) = ~E

(1) { r ~ ,',

p(r) = P E ~ )

,

where r0 = 1 A U , so that tiE, 60E, and OL pertain to the Earth-orbit environment. Of course, this p o w e r law breaks down as r 0 and r ~ ~. The o b s e r v e d surface brightness (see, for example, Leinert, 1975) is AF~ 1 l(e, fl; R) = (R/ro)l+ . sinJ+, e 47r ×

P(O)f(e:, [3, O) sin"OdO

(2)

A = ronETro~.~oE Id ~-- Pl q- P2 q-

2m,

where F~, is the incident solar flux at I AU (in erg cm -2 sec-l), R is the heliocentric distance of the d e t e c t o r (in AU), /3 is the ecliptic latitude, and f is the functional dependence of n(r) o n / 3 , normalized so that f ( e , 0, 0) = I. The surface brightness is usually given in units of S~°(V), or the number of 10th-mag solar-type stars (in the V band) per square degree (1 S~° unit = 6.37 x 10 -~2 times the solar irradiance per steradian). The p r e s e n c e of the factor 4n" in the d e n o m i n a t o r arises f r o m the way P(O) is normalized: in a c c o r d a n c e with the practices of m o d e r n radiative transfer studies (e.g., H a n s e n and Travis, 1974), we set P = 1 for isotropic scattering. The u p p e r limit of integration (or) implies that the zodiacal cloud extends to infinity. This m a y lead to

some uncertainty in the derived quantities if there is a sharp c u t o f f i n n(r) at 3.3 A U as o b s e r v e d by Pioneer 10 (Harmer et al., 1976). A check showed that the effect is quite small, in part because the quantity sin"0 in the integrand of Eq. (2) is small as 0 or. Any noticeable effect on I of the choice of the u p p e r limit of integration might also be apparent from the Helios I and 2 data (Leinert et al., 1981), for which 0.3 < r < 1.0 AU. Thus, the upper limit would be a function of r if n = 0 at r ~ 3.3 AU. H o w e v e r , the Helios m e a s u r e m e n t s indicated a remarkable constancy in the ratio l(r = 0.3 A U ) / I ( r = 1.0 AU) as a function of e. The g e o m e t r y of light scattered from a zodiacal cloud particle is depicted in Fig. I. It is clear from Eq. (2) that, on the ecliptic, P(O) is uniquely specified, once f is known from observations, if v is known (and, indeed, if the power-law assumption is valid). F r o m the Helios 1 and 2 data, Leinert et al. (1981) determined that v = 1.30 ± 0.05, a value we adopt here. As mentioned above, we choose to preselect the functional form of P(O) according to precepts that we find physically plausible: The function should include a c o m p o n e n t that represents isotropic scattering, the possible presence of a nonlinear brightening in the b a c k w a r d direction (here termed the "'opposition effect"), a c o m p o n e n t to describe the more gradual increases in P in

0

®

ro

FIG. I, Scattering geometry for a zodiacal light particle.

ZODIACAL LIGHT PHOTOMETRY the backward and forward directions (a sine function will suffice), and a c o m p o n e n t to represent the effect of a s y m m e t r y about 0 = 90 ° (cosine function). Of course, the number of ways in which the functional form of P can be expressed is infinite. However, any reasonable parameterization will lead to a very similar numerical description of P, because the integral Eq. (2) does have a unique solution. Thus we choose

0;

0 > 30 °,

(4)

which is derived from its original form ( L u m m e et al., 1980) by approximating sin a by 2 sin od2 and replacing ct by 7r - 0. Normalization of the phase function leads to al + 0.614az + ~ a3 = 1.

COMPACT

o

FLUFFY

--

BEST

PARTICLE ~--

FIT

0.3

0.2

(3)

where ~ s is the so-called " p h a s e function due to shadowing," derived by L u m m e and Bowell (1981a,b) to represent the opposition effect of atmosphereless bodies. Of course, one cannot anticipate that the shape and strength of the (possible) opposition effect of an average interplanetary particle is the same as that of the surface of an atmosphereless body. Nevertheless, because of functional simplicity, we assume here that I - c o s ½0 1 qbs(0) = exp 1.828 cos ½0 + 0.118 '

•

OA

P(O) = al + a2~s(0) + a3sin 0 + a4cos

p(o)

57

(5)

To establish how well our empirical phase function can represent laboratory measurements, we have fitted Eq. (3), assuming a4 = 0, to data by H a n n e r et al. (1981). Figure 2 shows that the fit is good for both a compact and a fluffy particle down to 0 = 60 °. Inserting Eq. (3) in Eq. (2) and using condition (5) leaves the four free parameters, a l , a 2 , a 4 , and A F o to be determined from the observational data. On the ecliptic, the main sources of data are Leinert (1975), who synthesized published observations in his Table III, D u m o n t and S~inchez (1975),

L

FIG. 2. C o m p a r i s o n b e t w e e n l a b o r a t o r y m e a s u r e m e n t s ( H a n n e r et al., 1981) a n d the e m p i r i c a l singlep a r t i c l e p h a s e f u n c t i o n , E q . (3), w i t h a4 = 0. C o m p a c t p a r t i c l e a = 33.6, m = 1.65 - 0.25i; fluffy p a r t i c l e a = 31, m = 1.65 - 0.25i.

and, at large solar elongations, Roosen (1970). In addition to these ground-based data, there are three data points at e = 15, 21, and 30 ° from rocket observations (Leinert et al., 1974, 1976). These agree well with the corresponding ground-based observations. Unfortunately, there are no data from the two Helios spacecrafts at/3 = 0 °. We have not considered the few published observations in the range 1° ~< e ~< 15° because of modeling difficulties arising from the assumption of a power-law variation of the brightness, the simplicity of the adopted phase function, Eq. (3), and the insensitivity of Eq. (2) at small elongations. This being the case, our method provides no information on the diffraction spike. Results in the U, B, and V bands are presented in Table I and Fig. 3. Data in U and B have been calculated from the V-band Helios data and B - V and U - B color indices given by Leinert et al. (1982). The corresponding fit to the V-band data is depicted in Fig. 4. Although these data refer to/3 :k 0 °, we suppose that the ratios Iv/IB and Iv/ Iu are insensitive to/3. We consider P(O) to be reliably determined in the range 30 ° ~< 0

58

LUMME AND BOWELL TABLE

1

WAVELENGTH DEPENDENCE OF PHASE UUNCTION PARAMETERSa i, EQ, (3). NORMALIZATION CONSTANT AF, IN S,~:~,~(V) UNITS, RMS ERRORS 1N COMPARISON TO DATA, A N D T H E H E N Y I : Y - - G R E E N S T E I N P A R A M E T E R S , EQ. (6)

Waveband

ttl

a:

Lt~

U B V

1.18 I.I I 1.02

0.90 0.85 0.83

-0.93 0.80 ~0.67

A f'

2,34 × 10~ 2.52 × 11)3 2.79 x I0 ~

~< 180 °. The uncertainty in P increases with decreasing 0 because integration from e to 7r b e c o m e s less and less sensitive to the form o f P. In all three w a v e b a n d s , the coefficient a4 o f the cosine term is very small (~<10 4), and the effect o f this term on P is correspondingly vanishingly small. We conclude, therefore, that the form of P is well represented by Eq. (3) with a4 = 0. in s o m e radiative transfer calculations it is c o n v e n i e n t to approximate P by the sum of two H e n y e y - G r e e n s t e i n phase functions PHG- Thus

rms

j,q

2.8 2.9 3.3

0.31 0.29 0.27

g2

b

g

0.30 0.28 0,26

0.51 0.50 0.50

-0.01 -0.01 -0.01

PHG(O, g) = bPHG(O, gl) + (I -

b)PHG(O, g 2 )

(6)

g = bg, + (I - b)g2.

Fitting the data described above, we obtain asymmetry factors g~, g2, and g and the relative contribution b as listed in Table i. The fit is good everywhere except as 0 ---, 180°, since a H e n y e y - G r e e n s t e i n function has zero derivative at 0--~ 180°; i.e., there is no opposition effect. io

401 r

l U,

f

BASED

o ROCKET --BEST FIT

35

20 U V

•OROUND

3 :o°

/

50o0

/ l / i

J

r

2o0o

//IB, 1ooo

$i.o

////

A;/

500 400 ~00 ~00

J r

__.~1100

too°

3'0.

6'0.

9~o•

o¢=7r-~

,to-

.... 4.

,0o.

single-particle phase functions. Dashed lines indicate the poorly know'n part o f the phase function. FI6. 3, D e r i v e d

UBV

i~0°

---~0 °

9~0° £

60 °

3~0°

0o

F I G . 4. Brightness in S-. . m ( V ) u n i t s o f the zodiacal light along the ecliptic as a function of solar e l o n g a l i o n . The dashed line shows the region where the model is insensitive to the observational dala.

ZODIACAL LIGHT PHOTOMETRY The Gegenschein region is particularly interesting because of the existence of an opposition effect in single particles. In Fig. 5 we compare the brightness of interplanetary particles observed near the antisolar point with the opposition effect seen in the integrated light of atmosphereless bodies. A convenient way of representing the latter data is by means of the Lumme and Bowell (1981a,b) theory in which a family of phase curves can be generated by varying the socalled multiple-scattering factor Q, the ratio of the intensity of multiply scattered light to the total intensity at zero phase angle. Evidently, it is not possible to obtain a close match between the two opposition effects. This in itself is not unexpected because one of the basic tenets of the Lumme-Bowell theory is that the opposition effect in regoliths is largely due to shadowing b e t w e e n particles. The color indices B-V and U-B of an average interplanetary dust particle can be obtained, as a function of ~, from ,

B-V = ( B - V ) o - 2.5 tog

A(B)PB(o0

00~ ~ X. t\k 0.I

zxm 02 X "

X

~

59

A(U)Pu(c0 U - B = (U-B)o - 2.5 log A(B)PB(c0

Using solar color indices (B-V)o = 0.63 and (U-B)o = 0.15 (Leinert et al., 1982), mean values of the color indices of interplanetary particles given in Table II result. We note the similarity both of the zerophase color indices and the amount of phase reddening at small phase angles with corresponding values for C asteroids. 3. D E P E N D E N C E ON E C L I P T I C L A T I T U D E

There exist both ground-based data [Dumont and Sfinchez (1976) and LevasseurRegourd and Dumont (1980) who treat a single data set in different ways] and Helios spacecraft data (Leinert et al., 1982) giving the dependence of the brightness of the zodiacal light on ecliptic latitude. All the data show a rapid decrease in brightness with increasing heliocentric latitude. It can be seen from Eq. (2) that the relative number density f ( e , / 3 , 0) can be determined if the single-particle phase function P(O), assumed to be independent of r, is known. A proper determination of f is crucial to questions bearing on the origin of the zodiacal cloud. In principle, there are two different ways to determine f from the data. The brightness integral, Eq. (2), which is an integral of Volterra's first kind, may be solved by inversion (a method given by Buitrago et al., 1980). No results for f have yet been pub-

o T A B L E I1 ~

;.

,~.

Ifl0*-E

O=O5O

"'~.

~0.

FIG. 5. Brightness of the G e g e n s c h e i n region of the zodiacal light. G r o u n d - b a s e d data are open and filled circles from D u m o n t and Sfmchez (1975), w h o give two possible sets o f data; pluses refer to the work of R o o s e n (1970); and triangles refer to L e i n e r t ' s Table I11 (1975). Brightness as a function of angular distance from the antisolar point (180 ° - e) is e x p r e s s e d in magnitudes relative to the brightness at e = 180 °. The opposition effect of the G e g e n s c h e i n derived in this paper (solid curve) is c o m p a r e d to that of a t m o s p h e r e l e s s bodies (dashed c u r v e s , see text).

(7)

MEAN COLORINDICESOF INTERPLANETARYDUST PARTICLESAS A FUNCTION OF SOLARPHASE ANGLE O~ c~ (deg.) 0

30 60 90 120 150

B-V

U-B

0.68 0.70 0.75 0.77 0.75 0.70

0.17 0.20 0.25 0.27 0.25 0.20

60

LUMME AND B O W E L L

lished using this m e t h o d . Alternatively, a " d i r e c t " m e t h o d m a y be e m p l o y e d : a functional f o r m for f is a s s u m e d , and the free parameters are determined by least squares. L e i n e r t et al. (1976) f o u n d that a " f a n m o d e l , " in which f d e p e n d s only on /3~> (the heliocentric latitude), explains the data m u c h b e t t e r than a " d i s k m o d e l , " w h e r e f d e p e n d s only on h (the height a b o v e the ecliptic; cf. Fig. I). T h e y suggested a f u n c t i o n

60 °

30"

DO°

n

i

120"

150° i

180"

r

i L H

,p, tax

• o -_-

I L2,

LRD ~, OS HELLOS PRESENT WORK 1 LEINERT ET AL.D976) 2 "(198t)

30C

I

20C

.#, (/3c.~) = e x p ( - 2 . 6 1 sin 13/3c, I) sin/3 sin(0 sin

fi<,~ =

sin

-

e)

L a t e r , L e i n e r t et al. (1981) c h o s e a s e c o n d function: f,(/3o) = e x p ( - 2 . 1 l s i n / 3 ~ 1 ) .

(9)

Inserting these f u n c t i o n s in Eq. (2), and using Eq. (3) for P , allows c o m p a r i s o n with

200

,-xe

Z \ \ NN\ ~\

150

\\ \

", \\

o- ~-

• o ----

\ x\

\\ \

~-\

GROUND B A S E D HELLOS P R E S E N T WORK 1 LEINERT ET AL(1976) 2 --. -(1981)

>-'.-..

100 o

~

•

--

,'~.

3'°-

25.

~'0"

\

(8)

&

;5"

•

•

9°-

5

Fro. 6. Brightness, in S~(V) units, of the zodiacal light along a plane perpendicular (A - ,k, 90°) to the meridian. Ground-based data are filled circles from Levasseur-Regourd and Dumont (1980) and pluses from Dumont and Sfinchez (1976). Note the systematic difference between the Helios and groundbased data. Predicted brightness, depending on the choice of the normalized latitude distribution, is also shown (see text).

J

X-X,., = 0 .

~.--

X k , , =180.

&-

~-

6;.

~0.

0o

Fro. 7. Same as Fig. 6, except in the ecliptic meridian: X X - 0 o r 180 °. the o b s e r v a t i o n a l data, as s h o w n in Figs. 6 and 7. F r o m the extensive data we have selected those in t w o planes: the meridian, X - X: -- 0 or 180 °, and a p e r p e n d i c u l a r plane, ,~ = 90 °. E x c e p t at the north ecliptic pole, there are no Helios data exactly in these planes. The Helios data presented in Figs. 6 and 7 have been extrapolated f r o m m e a s u r e m e n t s spanning the range 2.°8 to 169 ° in X - X~,; uncertainty due to e x t r a p o l a t i o n is p r e s u m a b l y small. A g r e e m e n t b e t w e e n the g r o u n d - b a s e d and Helios data can be c o n s i d e r e d no m o r e than satisfactory. The Helios data are systematically l o w e r by 17 _+ 10% at/3 = 16.°2, a n d b y 10 _+ 7% at 18 = 31 ° (a possible explanation is offered in Sect. 4). F r o m Figs. 6 and 7 it is clear that neither.f] nor f2 m a t c h e s the data v e r y well. We have refitted the o b s e r v a t i o n a l data, again a d o p t i n g a fan model. The function was c h o s e n empirically and has the f o r m J'(/3cc,) = e x p [ - ( & s i n / 3 < ~ + b2sin2/3~>)]. (10) W e f o u n d that the data can be well m a t c h e d

ZODIACAL LIGHT PHOTOMETRY with

61

ination of micrometeorite particles collected by Brownlee et al. (1980); these indib~ = 3.8 ___ 0.2 cated that p < 0.10 (Hanner, 1980). b2 = - 2 . 0 +-- 0.5. (11) We suggest here a new, semiempirical method of obtaining both the average geoIt is evident, particularly from Fig. 6, that metric albedo p and single-scattering albedo Eq. (I1) represents the observations better o30of interplanetary dust particles. Our idea thanfl or f2. A comparison amongf, fl, and f2 is shown in Fig. 8. The median latitude (at stems from the Helios observations f = 0.5) is/]o = 12 -+ 1°, rather than/3 ~ 20° (Leinert et al., 1981) that the polarization of that results fromf~ or f2. The uncertainty in the zodiacal cloud decreases by one-third our result reflects scatter in the data. The from 1 to 0.3-AU heliocentric distance, alvalue we derive for/3o is close to that for though the shape of the polarization curve both small asteroids and short-period com- remains remarkably unchanged. This obets, and therefore cannot be used as a dis- servation implies that the particle propercriminant for the origin of the zodiacal ties change in some way over the same range of distances. The elongation depencloud. dence of the polarization due to the zodiacal cloud is rather well known in the range 4. POLARIZATION AND ALBEDO The albedo of interplanetary particles is 30° ~< e _-< 180°. There has been some uncerone of the key indicators of their composi- tainty that light is negatively polarized in tion and, perhaps, origin. There exist two the Gegenschein region, 160° ~< e =< 180° published estimates of the geometric albedo (Wolstencroft and Rose, 1967; Frey et al., p of particles that differ by a factor of 40. 1974), although there is a measure of agreeCook (1978), by comparing estimates of in- ment that the observations are correct terplanetary-particle number densities with (Leinert, 1975). In any case, the average photometric data on the zodiacal light, de- polarization curve indicated by the ensemrived a geometric albedo p = 0.0058 at r = 1 ble of published observations (cf. Leinert, AU, with a steady decrease at increasing r. 1975, Fig. 10; Leinert et al., 1981) strongly Hanner (1980) used arguments similar to resembles those observed for atmosphereCook's, but different data, to obtain p = less bodies. For the Moon (Lyot, 1929) a 0.24. It is clear that this method can yield positive branch of polarization peaks at only a rough estimate of albedo, mainly be- phase angle c~ ~ 110°, the polarization cause of the large uncertainties in microcra- changes sign at the so-called inversion anter statistics used to infer the particle num- gle c~0 ~ 23°, and a negative branch of small ber density. Independent, and presumably amplitude extends to the opposition point, more direct, evidence comes from an exam- where the polarization disappears. Similar polarization curves have been observed for Mercury, Mars, some planetary satellites, 10 I - PRESENT WORK and many asteroids (e.g., Dollfus and ~, - - - - - - I LEINERT ET AL. 0976) Zellner, 1979), although the observations are more restricted in phase angle coverage. There is also extensive literature on 05 "~N the simulation of telescopic data using laboratory samples (e.g., Zellner, 1977), with the general conclusion that the surfaces of I i i i J atmosphereless bodies are rough and par15" 30" &5* ~ 75" 90* ticulate, most consisting of small, opaque grains. FIG. 8. Three different normalized latitude dependence functions (see text). In our opinion, the mechanism producing

62

LUMME AND BOWELL

this almost ubiquitous polarization signature remains unknown, but it is tempting to assume (and we do) that the mechanism is the same for all atmosphereless bodies, interplanetary particles, and particulate laboratory samples. We further assume that polarization is generated by light interacting with a single particle and is diluted in proportion to albedo by multiple scattering among particles (though the latter does not apply to interplanetary particles, of course). This assumption is supported by laboratory measurements (K. L u m m e , E. Bowell, and B. Zellner, unpublished) on rough basalt particles, where it was found that the functional form of the polarization was the same for a multilayer and a monolayer (in which multiple scattering between particles is absent). From studies of particulate samples in the laboratory, it is clear that albedo is the determinant o f the shape of the polarization curve (e.g., Dollfus et al., 1971). This finding leads us to seek, among the data for atmosphereless Solar System bodies, an empirical function of the polarization that is predominantly scaled by albedo. The polarized intensity lpol(O0 for a body of radius R0 can be written as a function of phase angle a rather than scattering angle 0 = "B" -- OH

where Dp is a single-particle phase function (not necessarily P), DR a function describing the probability of the emergence of light from a rough surface, and Ds a function describing the flux emerging in the face of mutual shadowing by other particles [though more generally here than in Eq. (4)]; all these phase functions are normalized to unity at a = 0 °. L u m m e and Bowell (1981b) found that, although (I)~ can be determined uniquely, the separation of Dp and DE is not unique, largely because of the unknown roughness parameter entering in DR. We assume here that the scattering properties of a regolith particle are the same as those of an interplanetary dust particle. This assumption is not critical to our results. Then Dp(a) -

/pol(a) =

-~--~j(a, dJt)zrR~Fc~DE(a),

(12)

w h e r e j is the polarized intensity for a single particle, F o (as before) is the incident solar flux, and ~E(a) the probability that a singlescattering event can o c c u r without extinction or shadowing due to surface roughness and the presence of other particles. The factor of 4¢r results from the way we have normalized the single-particle phase function P. According to L u m m e and Bowell (1981a), the probability function DE can be obtained from (I)I(OL) ~--- ( I ) p ( O g ) D E ( O ( ) OE(O/)

=

OR(Og)(l)S(Of),

(13)

(14)

where P(Tr - (~) is given by Eqs. (3)-(5). We obtain (in the V band) from Table I and Eq. (27) of L u m m e and Bowell (1981b) DE(a) exp[-- 3.34(tan ~)063] = 0.55 + 0.45Ds(Tr -- a) -- 0.36 sin ~' (15) From the definition of geometric albedo p (exactly at a = 0°), we obtain for the total flux I reflected by a body of radius R0 I(a) =

1

P(zr - c~) P(1r)

pRZF~:~Dobs(Ot),

(16)

where Dobs is the observed phase function (Dob~ = l at a = 0°). The percentage linear polarization ~ ( a ) is now given by lpol(O~)

~ ( a ) = 100 I(~)

= 100_J(a, o30) 7rDE(a) 4w PDobs(a)

(17) '

so that the polarized intensity j ( a , o5o) can be readily obtained from j ( a , O5o) -

p ~ (a) Dobs(a) 25 DE(C0

(18)

It follows that, providing our assumption that single particles can produce the observed polarization is correct, the po-

ZODIACAL LIGHT PHOTOMETRY larized intensity per particle j can be calculated for any object for which photometric and polarimetric data exist and for which the size, and therefore p, is known. Our data base for the polarimetry of atmosphereless bodies comprises I0 asteroids (six C type, two S type, and two others; Zellner and Gradie, 1976), the Moon (Lyot, 1929), Deimos (Zellner, 1972), and G a n y m e d e (Dollfus, 1975). F o r all these bodies there also exists reliable photometry (see, e.g., L u m m e and Bowell, 1981b). Moreover, the zero-phase geometric albedo p spans the range 0.042 to 0.47, a range we consider quite sufficient for any conclusions concerning zodiacal light particles. An evident deficiency in the data is the limited coverage in solar phase angle. Indeed, only for the Moon (amax ~ 160°) and Eros (O~max-~ 53 °) can we consider this coverage to be sufficient. F o r all the other objects O~max~< 30 °. This situation will improve when spacecraft data b e c o m e available. In spite of this limitation, we note that the resuiting j , Eq. (18), is remarkably consistent from one body to another, in the sense that it can be represented by just two components that depend separately on p and ct:

j(ot, p) = q(p)G(a),

(19)

where the functions q and G can be approximated by

q(p) = 0.87p°39;

63

• CASTEROD I o 5 ---+ OTHER -• DE M IOSMoON 0

a~ GANYMEDE

d,

o

d2

P

o,,

d,

FIG. 9. Albedo-dependent coefficient of linear polarization for some atmosphereless bodies compared to a power-law fit, Eq. (20). zodiacal cloud particles, we do not consider this scatter to be serious. Next, we wish to relate the geometric albedo p to the single-scattering albedo 030. To this end we have carried out radiativetransfer computations (unpublished) using the mean single-particle phase function P(O), given by Eqs. (3)-(5). As first implied by L u m m e and Irvine (1976), in a medium consisting of closely packed macroscopic particles which cast shadows on each other, the geometric albedo as calculated from the classical radiative-transfer theory PRT is related to the geometric albedo of a single particle pl by 030 pl = -~-P(Tr) + PRT;

o30 ~ 0.

(21)

p ~< 0.30

G(a) = e x p [ - 2 . 7 ( s i n ½00°-39] sin a sin(a - a0),

(20)

and where o~0 is the so-called inversion angle of the polarization curve. The data and q(p) are shown in Fig. 9; and fits for two objects, the Moon and the asteroid 56 Melete, are presented in Fig. 10. It was also found that, although G(ot) is the same for all the bodies considered, the factor q(p) does possess some real dispersion (cf. Fig. 9). In particular, the data for Ceres require a value o f q about 20% larger than the value derived from Eq. (20). H o w e v e r , since we are interested in the mean properties of the

This results from the way in which attenuation of light due to first-order scattering at a = 0 ° is calculated in the classical radiativetransfer theory. Our computations can be well represented by the relationship p = 0.46030L~3;

p <~ 0.30.

(22)

Inserting this equation into Eq. (20), we obtain q(030) = qo03~° q0 = 0.64 v0 = 0.44.

(23)

The phase function P(O) can now be de-

64

LUMME AND BOWELL

°]

:

8.O

.

6.0

/

°

l

•

/5

4.0

f /

0 qL

/ ®

V

-2.0 L

I

I

FIG. 10. Observed Linear polarization (in percent) of the Moon and the C asteroid 56 Melete compared to the semiempirical theory.

composed into two orthogonally polarized components Pj (0) and P2(0), perpendicular and parallel to the scattering plane, respectively. By definition

where the plus sign refers to the case where i = 1 and the minus sign to the case where i = 2. In Eq. (26), we have assumed an inversion angle a0 = 20 °, a mean value that applies with an accuracy of -+4° for all atmosphereless bodies hitherto observed. P, and P2 are given in Table III as a function of the single-scattering albedo 03o. We emphasize once more that diffraction is not included in P and that Eq. (26) refers to the average properties of a particle since the observed polarization of the zodiacal light results from particles of unknown size, and hence albedo distribution. The accuracy of P, and P2 decreases with decreasing scattering angle for reasons explained in Section 2, and because j is less accurately determined with increasing phase angle (i.e., decreasing scattering angle). We now return to the polarization of the zodiacal light. In accordance with Eq. (2) and using Eqs. (19) and (23) for the polarized intensity Ipoj(S, /3; r) of the zodiacal light: ApFQ 1 /poX(S, /3; R) = [(R/ro) sin s] '+~ • G(Tr - O)f(~,/3, O) sinuOdO A n =

½[Pt(O) + P2(O)] = P(O) l[pt(0) - P2(0)] = riO);

-~ _ u0 ronErrp~qo~oE

# = v, + v0u2 + 2u3. (24)

hence

The percentage linear polarization can be obtained from

P,(O) = P(O) + j(O) P2(0) = P(O) - j(O).

6~E

(25)

From Eqs. (3), (19), (20), and (23) we obtain in the V band, for a typical, relatively dark (O3o ~ 0.6) regolith or interplanetary dust particle

H.(s,/3) H ( s , ~) "

(28)

where

H,(s,/3) = ~ fl

G(rr - O)f(e, /3, O) sin~'OdO

0.64 - 0 . 6 7 sin 0 -+ ~ e x p [ - 2 . 7 ( c o s ½0)°.4] sin 0 sin(160 ° - 0),

},,. i

@(e,/3; R) = 100qo [(R/ro) sin s]~2

1

Pi = 1.02 + 0.83 ( - c o s ½0 ) exp 1.828 cos ½0 + 0.118

(27)

(26)

H ( s , /3) = ~

f~i P ( O ) f ( s , /3, O) sin~OdO.

(29)

ZODIACAL

TABLE

LIGHT

IV we calculate that

III

v2 = 0.54 -+ 0.03

ORTHOGONALLY POLARIZED COMPONENTS OF THE PHASE FUNCTION

P(O), EQ.

(26), AS F U N C T I O N S O F

/z = 1.00 -+ 0.02.

T H E S C A T T E R I N G A N G L E 0 AND S I N G L E - S C A T T E R I N G A L B E D O tO0 0 (deg.)

180 170 160 150 140 130 120 I10 100 90 80 70 60 50 40 30

030= 0.1

030= 0.2

t,~O= 0.4

t,5~= 0.6

PI

P2

Pt

P2

Pt

P2

Pt

P2

1.85 1.48 1.35 1.26 1.20 1.15 1.12 1.09 1.07 1.06 1.06 1.07 1.09 1.13 1.18 1.24

1.85 1.54 1.35 1.18 1.03 0.89 0.78 0.71 0.66 0.64 0.66 0,71 0.78 0.88 0.99 1.12

1.85 1.49 1.35 1.25 1.17 1.11 1.06 1.03 1.00 0.99 1.00 1.01 1.04 1.09 1.15 1.22

1.85 1.53 1.35 1.19 1.05 0.93 0.84 0.77 0.73 0.71 0.73 0.77 0.83 0.92 1.02 1.14

1.85 1.50 1.35 1.24 1.15 1.08 1.03 0.99 0.96 0.95 0.95 0.97 1.01 1.06 1.13 1.21

1.85 1.52 1.35 1.20 1.07 0.96 0.87 0.81 0.77 0.76 0.77 0.81 0.87 0.95 1.04 1.15

1.85 1.50 1.35 1.24 1.15 1.07 1.01 0.97 0.94 0.93 0.93 0.96 1.00 1.05 1.12 1.20

1.85 1.52 1.35 1.21 1.08 0.97 0.89 0.83 0.79 0.78 0.79 0.82 0.88 0.96 1.05 1.16

As mentioned earlier, the Helios spacecraft observed that the polarization diminished by about a third from a heliocentric distance of 0.3 to 1 AU. We denote this ratio by K(e,/3) and obtain from Eq. (28) K(e,/3) =

65

PHOTOMETRY

9~(e,/3; 0.3 AU) ~ ( e , / 3 ; 1 AU)

= (I),21~0-1) b-~ "

(30)

Using the data of Leinert et al. (1982), we have calculated K at b o t h / 3 = 16 and 31 °. Mean values are given in Table IV. That K does not vary with e and/3 can be judged from Figs. 11-13 and Table IV. From Table

TABLE

(31)

The near constancy of K is consistent with our assumptions about the " f a n - m o d e l " and the power-law d e p e n d e n c e of the parameters. F r o m Eqs. (28) and (30) we have (at 1 AU) q0sin°'3e Hp(e, t3) ~ ( e , / 3 ) = 100 5----------go3~ H(e,/3)

(32)

Next, we examine the observed polarization at three different ecliptic latitudes:/3 = 0 °, 1652, and 31.°0, and in three different wavebands. Ground-based observations pertain to the V band only. In his Table III, Leinert (1975) gives the polarization at/3 = 0 °. A second, independent ground-based data set comes from Dumont and Sfinchez (1976), who tabulated the degree of polarization at several values of e and /3. Although the shape of the polarization curve is very similar in both sets, the polarization in D u m o n t ' s and Sfinchez's data is everywhere less, being 0.83 - 0.07 that of Leinert, as can be judged from Fig. l l. No explanation for this disparity has been offered. There exist very homogeneous data at/3 = 1652 and 31 ° obtained by the Helios probes (Leinert et al., 1982). Only at/3 = 1652 are there data in U, B, and V (at/3 = 3150, both U and B data are missing). A comparison between the Helios and Dumont and Sfinchez data revealed that the latter are again reduced by almost the same

IV

W A V E L E N G T H AND L A T I T U D E D E P E N D E N C E OF K , T H E R A T I O OF POLARIZATION AT 0 . 3 A U TO T H A T AT 1.0 A U , AND OF T H E EXPONENTS v2 AND /.t, EQ. (31) Waveband

U B V

K

v2

/x

/3 = 1652

/3 = 3150

/3 = 1652

/3 = 3150

/3 = 1652

/3 = 3150

0 . 7 2 -+ 0 . 0 3 0.70 + 0.03 0 . 6 9 -+ 0.01

--0 . 6 8 -+ 0 . 0 3

0.49 0.53 0.55

--0.57

1.03 1.00 0.99

--0.98

66

LUMME AND BOWELL (7

30" i

60'

9~

i

i

120'

150"

PERCENTAGE rms DIFFERENCE BETWEEN PREDICTED AND OBSERVED POLARIZATION AS A FUNCTION OF THE V-BAND SINGLE-SCATTERING ALBEDO (oE AND ECLIPTIC LATITUDE /3

• •

2(

°o

15

•

/ •/ o

"~P(%)

TABLE V

~80"

i

~

B-0 °

0.07 0.08 0.09 0.10 0.11

1.8 1.2 1.3 1.8 2.3

~-

1672

B-3170

• • • 00

o DUMONT& SANCHEZ(1976) - - THEO~'

2.3 1.5 1.3 1.8 2.1

2.3 1.1 0.4 1.0 1.6

c a l c u l a t e d in t h e B a n d U w a v e b a n d s : A F o ( B ) oSE(V) ~3E(B) - A F t ( V ) ' 0

A F Q ( U ) o3z(V). O~E(U) -- AF~.~(V) "

(33)

T h e g e o m e t r i c a l b e d o PE o f a s i n g l e p a r t i -~ ~0.

,~"

,6.

~.

~

do.

FIG. 11. Observed linear polarization (in percent) of the zodiacal light at/3 = 0°i Two data sets are compared to theory. Note the systematic difference between the two sets. a m o u n t a s a t / 3 = 0 °. A t / 3 = 16?2, t h e f a c t o r is 0 . 8 4 + 0 . 0 5 ; a n d a t / 3 = 3170, t h e f a c t o r is 0.83 +- 0.06. W e a r e t e m p t e d , t h e r e f o r e , to r e j e c t D u m o n t a n d S ~ i n c h e z ' s d a t a in o u r analysis. F r o m E q . (32) it c a n b e s e e n t h a t t h e o n l y a d j u s t a b l e p a r a m e t e r is 6 E . A s b e f o r e , w e a s s u m e ~ = 20 ° ( o r 00 = 160°). O u r r e s u l t s a r e n o t s e n s i t i v e to t h e e x a c t v a l u e o f c~0 if c~0 = 20 + 4 ° , t h e r a n g e o b s e r v e d f o r all atmosphereless bodies. The observational d a t a at all t h r e e v a l u e s o f /3 c a n be e x p l a i n e d if O3E(V) = 0 . 0 9 --+ 0.01 w i t h n o e v i d e n c e t h a t O3E(V) d e p e n d s o n /3, as c a n be j u d g e d f r o m T a b l e V. T h e r m s e r r o r is about +2%, just within the quoted error of t h e H e l i o s d a t a ( L e i n e r t et al., 1982). A comparison between the observations and o u r s e m i e m p i r i c a l t h e o r y is d e p i c t e d in F i g s . 1 1 - 1 3 . F r o m E q . (2) a n d u s i n g t h e v a l u e s o f AF~v g i v e n in T a b l e I, OSE c a n be

o

HELIOS ro =I AU

:°e

~") ,o

s

0

-s FIG. 12. Observed linear polarization (in percent) of zodiacal light at /3 = 16.°2 from Helios data. Measurements at r = 0.3 AU have been divided by 0.7 fsee text).

ZODIACAL LIGHT PHOTOMETRY o

o •

67

Eqs. (19), (23), and (28), we can calculate that

li

Ipo~(V)

t o~(v) J

(35)

or '15

.ELIO_S ~-, Au

/

[ / o o l ( h ) ]2.27

\o

O~E(h) = Llpo~(V)J

lC

I

15(7

I

120"

I

I

90'

60'

3

FIG. 13. Same as Fig. 12, but at fl = 31 °.

cle in Earth-orbit environment is obtained from

O3E PE = - T e(Tr),

(34)

where the mean single-particle phase function at 0 = 7r can be calculated from the values of ai given in Table I. Values for O3E and PE are given in Table VI, and the resulting two components of the phase matrix in the V band are depicted in Fig. 14. An independent estimate of O3E(B) and O3E(U) can be obtained from the ratio of polarized intensities lpol(h)/lpol(V) (where X ~ B,U). From

O~E(V).

(36)

The polarized intensities in all three wavebands have been given by Leinert et al. (1982). We calculated the ratios and found that lpol(U)/Ipoj(V) = 1.09 -+ 0.03 and Ipol(B)/ lpol(V) = 1.09 + 0.02. Thus, Helios data show an increase in the observed polarized intensity with decreasing wavelength, in contrast to all other data on atmosphereless bodies in the Solar System, which exhibit the opposite behavior. This, together with two other items of evidence, leads us to speculate that there is a 15 + 5% error in the V-band calibration of the Helios data, in the sense that I(V) should be higher by 15%. Indeed, if the various discrepancies arise from just one cause, then this is the

20

L5

1.0

T A B L E VI W A V E L E N G T H D E P E N D E N C E OF THE S I N G L E - S C A T T E R I N G A L B E D O AND M E A N G E O M E T R I C A L B E D O OF D U S T PARTICLES AT I A U

Waveband U B V

O3E 0.08 ± 0.01 0.08 -+ 0.01 0.09 -+ 0.01

PE 0.04 ± 0.005 0.04 ± 0.005 0.04 ± 0.005

~o*

~*

FIG. 14. Single-particle phase matrix for linear polarization o f an average dust particle in Earth-orbit environment. Here P~ is the component polarized perpendicular to the scattering plane and Pz is the parallel component.

68

LUMME AND BOWELL

only possibility. First, as noted in Section be explained using this model, then k could 3, the Helios data are too low by 17 _+ 10% be estimated. at /3 = 1672 and by 10 _+ 7% at /3 = 31.°0 5. MASS AND EXTINCTION relative to the two ground-based data sets. Second, Leinert and Richter (1981) found To our knowledge, the only estimate of that, e v e n after correcting for differences in the total mass of the zodiacal cloud is that color, Pioneer 10 V-band m e a s u r e m e n t s of given by Whipple (1967), who derived a the brightness of the Milky W a y are higher value of 2.5 x 1019 g. In part, this paucity of by a factor of 1.20 -+ 0.14, whereas agree- published results arises from the depenment is m u c h better in B. All three pieces of dence of the normalization constant AFc.:, evidence point to the same conclusion: the Eq. (2), on three different variables: numI(V) values from the Helios probes are too ber density hE, particle cross section 7rp~:, low by 15 -+ 5%. This, h o w e v e r , remains and single-scattering albedo O3E (as before, only speculative as long as there is no direct subscript E refers to the Earth-orbit envievidence of an error in calibration. It ronment). Using the conversion factor of should be noted that such a calibration er- 6.37 x 10 t2 b e t w e e n F6;, and 1 S~(V) unit ror would not change the polarization, (Leinert, 1975) and the derived values of since each c o m p o n e n t would be affected AF.~ (Table I) and oSZ (Table VI), we obtain equally. If, indeed, we were to adopt a 15% n~:roTrp~-: = 2.0 x 10 7 (39) correction, Eq. (36) would yield values for O3E(U) and O3E(B) in very good agreement o r with those given in Table VI. nEp~: = 0.4 X 10 19 c m - i . (40) As has been shown above, the use of polarization data m a k e s it possible to obtain If, for instance, PE = 10 txm, then nE is o30 and its d e p e n d e n c e on the heliocentric equal to four particles per cubic kilometer. distance through the exponent u2 = 0.54. The total mass of the zodiacal dust cloud The other two e x p o n e n t s u~ and ~'3, Eq. (1), can be estimated from are still unknown. If it could be established that there exists a relation of the form f rr/2J'(/3o) cos/3od/3G M = 4~'D 0

dJo(p) = dJo(Po)(p/po)

~,

(37) (41)

between the single-scattering albedo and the m e a n particle radius such that k could be determined, we would have from Eqs. (2) and (27) 1.08 vl = 0.76 + - k 0.54 V3 ~

--

k

(38)

Use of any laboratory data would yield a very uncertain value of k b e c a u s e of its sensitivity to both the particle-size distribution and composition. We are currently making M o n t e Carlo c o m p u t a t i o n s on light scattering due to a rough particle. I f the polarized c o m p o n e n t s PI and P2, Eq. (26), can

where D is the density of a single particle ( - 3 g cm -3) and w h e r e f i s given by Eq. (10) and n(r) and p(r) are given by Eq. (1). We cannot extend our power-law approximation to infinity, because this would yield an infinite mass. Therefore, we assume that the outer b o u n d a r y of the zodiacal cloud is at rm = 3.3 AU, the heliocentric distance beyond which no zodiacal light was detected by Pioneer 10 (Hanner et al., 1976). We obtain from Eq. (41) 16re 2Dr 3 M - ~

nEp3LfZ~

L# = ( ' / 2 f ( B Q ) c o s Bod/3G = 0.3 JO

ZODIACAL LIGHT PHOTOMETRY (3.3) K

69

the single-scattering albedo thE, is needed to explain the observed polarization over a K large range of elongations and at three dif0.6 ferent ecliptic latitudes. These and other K = 2.1 +-(42) k similarities have convinced us to regard the Inserting numerical values into Eq. (42) and study of zodiacal light and atmosphereless using Eq. (40), we have as an order-of-mag- bodies as complementary. Light is scattered from a single large Sonitude estimate (taking K ~ 3) lar System body at essentially a single M = 1018pE (in grams), (43) phase angle, whereas for the zodiacal light, where PE is in micrometers. This agrees integration along the line of sight implies well with the value given by Whipple sampling over a large range of phase angles. (1967), assuming, as is commonly accepted, However, since interplanetary dust particles are well isolated, we are strictly obthat 10/xm
70

LUMME AND BOWELL

how involved. This inference is supported by the similarities of albedo, color indices, polarization, and the distribution of dust as a function of ecliptic latitude. Nevertheless, cometary origin of the dust cannot be ruled out, because at least some of the properties of cometary dust may resemble those of C asteroids. M. F. A'Hearn (1983, personal communication) has attempted to derive the single-scattering albedo for cometary dust particles. Preliminary results give very low values (O3o~< 1%), which, if correct, could provide a good discriminator between the two competing models of the origin of the zodiacal cloud. Also, the very recent results from the Infrared Astronomical Satellite (unpublished) show that there is an enhancement--a "ring"--in dust density at the heliocentric distance of the main asteroid belt. If interplanetary dust does indeed originate from collisions among asteroids, then one could compute collision rates and, in turn, the number of small (~
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