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Zodiacal light gathered along the line of sight
003?~0633/83 $3.00 i-O.00 Pergamon Press Ltd.
ZODIACAL
LIGHT LATHERED
ALONG THE LINE OF SIGHT
THE VICINITY OF THE TERRESTRIAL ORBIT STUDIED WITH PHOTOPOLARIMETRY AND WITH DOPPLER SPECTROMETRY R. DUMONT
Observatoire
de Bordeaux, (Receiljd
F-33270 Fioirac,
15 February
France
1983)
Abstract-Theexpression for the zodiacal brightness integral is especiallysimpleiftheintegrand contains the ‘directional scattering coefficient’, 9,(a.u.- ‘), or equivalently the scattering cross-section per unit-volume. The two intersections of the terrestrial orbit with a line of sight lying in the ecliptic offer the possibility of isolating the contribution of the chord, with a conservative assumption of steadiness, but without the controversial assumption of a homogeneous zodiacal cloud. The zodiacal brightnesses between 60 and 120” elongation can beused toder~v~~~and~, thevalueof~anditshelio~ent~~radial derivative, bothat 1 a.u. andat ascattering angle of 90”. A polarimetric treatment leads to the local polarization degree, Pa, and to its heliocentric derivative, & Applied to all three available observational sources, this method invalidates the assumption of homogeneity, leading to a rather high relative gradient B/P0 near 1 a.u. (.- 12, - 16 or - 24%. according to the source, as the Sun’s distance decreases from 1.0 to 0.9 au.). The method is extended to Doppler spectrometry, taking advantage of the two equal projections on the line ofsight of the Earth’s velocity vector. The brightness 2, and the Dopplershift Ai,, observed at 90” elongation, together with the derivatives w.r.t. elongation e, of the brightness, i and of the Dopplershift, A’& can be used to retrievethemeanorbitalvelocity,~,oftheinterplanetaryscatterersin t~ere~onofthe terrestrialorbit.Thetwo most reliable observational sources lead, with fair agreement, to a relative excess fu - V)/V, over the terrestrial
velocity, of the order of + 257;.
1. INTRODUCTION
Observations of the zodiacal light (z.1.) provide information integrated along the whole line of sight (1.0s.). In the photometric case, this information is a brightness, Z (MTe3), generally expressed in S,, (equivalent to one G2V star of 10th magnitude V per square degree). In the polarimetric case, one has also access (at least) to the components of 2, Z, and Z!,, scattered by the Fresnel vectors respectively perpendicular to, and lying in, the scattering plane. In the spectrometric case, the wavelength dependence of the brightness, Z(1) = dZ/dl (ML-‘Ts3) is available. The integrated quantities are directly useful, as far as 2.1. is only considered a foreground source, against remotesources being theaim ofthestudy,and requiring to be corrected for it : diffuse galactic light (Morgan er al., 1976), cosmic light (Roach and Smith, 1968), faint targets for large instruments (Dumont and LevasseurRegourd, 1981). On the other hand, as far as the zodiacal cloud (z.c.) itselfis concerned, it is an enormous deficiency to ignore how the brightness (or its polarized components, or its derivative with 1) has been gathered along the 1.0,s. and this is one of the main origins of the ambiguities and the discrepancies between models of the Z.C. and of the scatterers. Several efforts have been made towards a
localization along the 1.0s. of the brightness contributions. Often these efforts con~ntrated on the last elemental section of the I.o.s., i.e. the optical behaviour of the scatterers in the immediate vicinity of the observer (Dumont, 1972, 1973 ; Dumont et al., 1979; Schuerman, 1979a, 1979b; Buitrago, 1979). For that single location on the 1.0.~.and for peculiar viewing directions, the local contribution by the end of the 1.0.~. can be determined free of assumptions. The knowledge, even partial and approximate, of what happens elsewhere on the 1.0.~. requires more or less arbitrary assumptions. If the Z.C. is assumed homogeneous, cylindrically symmetric and admitting a power law r - ” for the space density vs the heliocentric distance Y,there is a rigorous solution for inverting the brightness integral (Dumont, 1973). In the latter and much of subsequent work (Dumont and Sanchez, 1975; Dumont, 1976a, 1976b; Buitrago et al., 1981,1983), the assumptions have a rather high ““levelof arbitrariness”. In other cases (Leinert et al., 1976; Mujica et al., 1980; Dumont and Pelletanne, 198 1; Pelletanne, 1982) the dependence w.r.t. the major hypotheses is to a certain extent reduced. Recent space observations either have given support to these classical assumptions, or--more often-have invalidated them. The r-’ law for the space density is supported by the photometric results of Heiios probes
1381
1382
R. DUMONT
(Leinert et al., 1981), with v = 1.3. On the other hand, the polarimetric results of the same probes cannot be reconciled with a homogeneous Z.C. inside the Earth’s orbit. The data of Pioneer 10 led to the same conclusion outside the Earth’s orbit (Schuerman, 1980). Contrary to the cylindrical symmetry, which is strongly suggested, at least for an overall view, by the weakness of the seasonal oscillations of the z.1. features, the assumption of a macroscopically homogeneous cloud seems no longer satisfying. The present work is the first of a series, the purpose of which is to increase our ability to decipher the brightness distribution along the 1.0.~. and to decrease our recourse to physically arbitrary assumptions.
scattering cross-section of the unit-volume, or equivalently, the omnidirectional scattering coefficient. Both 9 and 0 will be expressed in a.u.-‘. In order to maintain the usual S,, unit of brightness, the solar intensity S has to be expressed in the corresponding unit. The irradiance unit will be that given by as many G2V stars of 10th magnitude V, as one steradian contains of square-degrees (3282). Adopting after Sparrow and Weinberg (1976) - 26.73 for the apparent magnitude V of the Sun, its irradiance at 1 a.u., i.e. its intensity, is S = dexC(26.73 + 10)/2.5]/3282 = 1.500 x 10” in the above unit. If the 1.0.~. is oriented positively like the propagation of light, our expression of the brightness integral is x0
2. THE
ZODIACAL
BRIGHTNESS:
z=s
AN INTEGRAL
rm29 dx.
OF WHAT?
Various expressions for the brightness integral have been published. Ordinarily the assumed models for grain size distribution and phase functions, as well as for their spatial distribution, appear in the integrand. A different, less model-dependent, way of expressing the integral is to concentrate on the intensity scattered by an elemental unit-volume, or an elemental section ofthe I.o.s., since we have no access to any microscopic knowledge inside these elements. If multiple scattering is admitted to be negligible, the contribution dZ by a section dx of the 1.0.~.near a point M is proportional to dx and to the solar irradiance at M. Denoting by S the intensity of the Sun (ML2Tm3), the irradiance (MTm3) at r astronomical units (a.u.) from thecenter ofthesunise = Sre2,exceptin thevery inner Z.C. (r = a few solar radii) where both the single scattering treatment and this expression of e may be insufficient. The proportionality factor is a scattering coefficient (L ‘). It depends not only on the scatterers at M, but on the scattering direction and it will be denoted 9 (directional scattering coefficient) : dZ = Srm29
dx.
In the polarimetric case, the local degree ofpolarization of the sunlight scattered by a small volume at M is B = (~l-~,i)/(~l+~,,), where gL and g,, correspond to Z, and Zll with equation (2).
3. RETRIEVAL
OF B AND 9J IN THE VICINITY
This paper concentrates on z.1. observations from the Earth, with the 1.0.~.in the ecliptic plane and at medium elongations E (solar and antisolar regions avoided). In the present section a method is introduced to obtain 9 and B at 1 a.u. and their heliocentric derivatives. Calling 0 a random position of the Earth and choosing the axes Ox, Oy as shown on Fig. 1, consider the Earth at A some time later and let Z(A) be the brightness
(1)
The product eg = .Y is the intensity (ML-rTm3) scattered towards the observer by the unit-volume surrounding M ; I has been extensively used in previous works, but the quantity 2 has the advantage of being independent of solar irradiance, therefore being an intrinsic property of the local matter. Since the energy subtracted from the incident beam by the volume do (per steradian and for the same scattering direction) is .Y dv = eg dv, an equivalent definition of 2 is the directional scattering cross-section of the unit-volume. The integral of 9 over the sphere, JJ& dw = 0, represents the total energy subtracted by scattering and will be the omnidirectional
OF THE
EARTH’S ORBIT
FIG. 1. GEOMETRY OF THE METHOD.
1383
Zodiacal light gathered along the line of sight observed from A at elongation e, along a 1.0s. parallel to Ox. Let B be the second intersection ofthe 1.0.~.with the orbit, which is admitted circular (a convenient, but not unavoidablesimplification).Thepossibilityofisolating the contribution ofthe chord BA, Z(ii)Z(B), would be an important step towards a localization of the information. At the time when the observation is made from A, the brightness that would be seen from B is unknown, but what was observed from B some time earlier is known. The assumption that Z(E) has remained unchanged while the Earth moved from B to A implies a steadiness of the Z.C.on the time-scale of a few weeks. Most of the observers, from Earth and from the apparent stability, space, have emphasized smoothness and cylindrical symmetry of the Z.C. (Sparrow and Ney, 1972; Weinberg and Sparrow, 1978 ; Levasseur-Regourd and Dumont, 1980; Richter et al., f 982; Leinert and Planck, 1982). Moreover, most of the ‘variations’ invoked in the literature do not conflict with our hypothesis ofsteadiness, at least those originating from the orbital motion of the Earth rather than from intrinsic local changes (Levasseur and Blamont, 1976 ; Dumont and Levasseur-Rego~lrd, 1978). As for the intrinsic changes reported near the ecliptic, they concern the solar cap (Banos and Koutchmy, 1973) or the antisolar cap (Bandermann and Wolstencroft, 1976; Robley, 1980) rather than the intermediate elongations. In summary, the invariance of Z(B) is a much less imprudent assumption than a homogeneous composition of the z.c., which is supported neither by theoretical nor observational arguments, and has been completely avoided in the present study. On this basis, the chord BA contributes to the brightness in A by +x0 Z(A) - Z(B) = s
r - ‘9(x, y) dx s -*o
(3)
where y = y0 is cst along the 1.0s. Introducing the development ofC8 with respect to x and toy, limited to the first order, g(x, y) = s$) c&,x where so follows :
is the central
Z(A)-Z(B)-S@!,
+ $,y
(4)
value (X = y = 0) of g%?,it
+~,y(&2/1
--y*)tg_ ‘(.x,/l -y())
or, with 56 and l-y,
= E = tg- ‘(x,/f -y,)
= cos E,
Z(E) - Z( - E) N S(2Ejcos
E) [go + ( 1 - cos E)L@,]. (5)
Within reasonable limits for E, say no more than 7c/6, the regression of the quantity (cos E/2E) [Z(E) - .Z( - E)] vs the quantity (1 -cos E) therefore gives access to SgO as the intercept and to S&, as the slope. This method can be extended to polarimetry, replacing Z and 3 by their components Z,, Z ,,, BL, 9 ,, The polarization degree of the sunlight scattered at 90” scattering angle by a small volume at 1 au. in the ecliptic will be B = (go1 -C@Oii)/(~oi +C@e,r). Its heliocentric gradient will be 9, = 9’o’[(l
-&J$,,-(l+Y&&].
If .@Yis significantly different from zero, something has to be heliocentric-dependent in the composition of the interplanetary medium in the region of the Earth’s orbit.
4. APPLICATION
TO AVAILABLE
DATA: THE RADIAL
PHOTOPOLARlMETRIC
INCREASE OF LOCAL
~LARIZATION
The above method has been applied to the three available observational sources which provide the polarized components of Zin the ecliptic as functions of elongation C, in the range 60” d 1~1< 120”: groundbased observations from Haleakala, Hawaii (Weinberg, 1964) and from Teide, Tenerife (Dumont and Sanchez, 1975); balloon observations by Gillett (1967). The first two sources give numbers, the latter only curves. The spectral domains are not very different (530 nm for the first ; 460 and 502 nm for the second; near Johnson’s B for the third). Giilett’s data are expressed in a S,,(B) unit and have been multiplied by 1.7 to be compared to the others, expressed in the classical S,, unit. In all three cases, values ofZ, and Z,, are available by steps of 5” in E. The couple E = 85 and 95” (E = 5’) has not been used, thedifference Z(S”)-Z( - 5”) being rather poorly known, due to its low value and to the fact that the figures are published in integral units of S, o, with lack ofsmoothness vs E.The regression of formula (5) has been made with five couples, 10,15,20,25 and 30”for E. Table 1 shows the results: values of gin,, g,,, .F and their heliocentric derivatives, with the r.m.s. uncertainties on all these quantities. Obviously, these CT errors represent the precision in the linearity of the regressions, not the accuracy of the measurements. The values at 1 a.u. can be compared to those already available in the literature, on a slightly different form (Dumont, 1972, 1973; Dumont and Sanchez, 1975; Leinert, 1975): the possibility of isolating the local contributions and polarization at the Earth and at a
1384
R. TABLE
DUMONT
1.LOCAL VALUESAND
HELIOCENTRICDERIVATIVESAT 1 a.u. OF THE DIRECTIONAL SCATTERINGCOEFFICIENT AND OF THE DEGREEOF POLARIZATION Weinberg (1964)
(a.u.-‘) ?O,, *B (ax-‘) ?yl *a (a.u.-‘) ayi, +o (a.u.-‘)
9ol*fs
To+” I yy + 0 (a.u. - ‘)
Dumont
110.2*0.5x lo-” 47.8*0.5x lo-” 63.6+6x lo-” 137.0&6x 10-l’ 0.395 k 0.006 -0.967+0.10
scattering angle of90” is well known. On the other hand, the heliocentric derivatives are new. They show, as understandable, more discrepancies than the values at 1 a.u. However, the three results are qualitatively the same : the local degree of polarization at right scattering angle, B, is rather rapidly decreasing as y increases, or increasing as r increases. When r goes from 1 to 0.9 a.u., the relative loss of 9, equal to @3,/10Y,,, would be 24% (Weinberg), 12% (Dumont and Sgnchez), 16% (Gillett). A similar treatment of the values compiled by Fechtig et al. (1981) from various observational materials gives a relative loss of 14%. By a different approach, Dumont and Pelletanne (1981) pointed out a similar effect. There is an evident connection between the sign of the gradient gpyand the regular decrease in the degree of polarization reported by the Helios group (Leinert et al., 1981) as these two probes approached the Sun. The same conclusion arises from their observations and from the present work : interplanetary scatterers polarize more and more, going away from the Sun. However, two main differences must be kept in mind : (i) Helios photopolarimeters did not point at the ecliptic, (ii) their polarizations are observed ones, i.e. integrated along the 1.0.~. There is another noteworthy agreement between Table 1 and previous observations, in a purely photometric domain. If we compute the relative gradient of the total directional scattering coefficient, g,/%l
= ($1+
Q/(%
+ %,I)
(6)
it has to fit the exponent v of the Y-” space density law commonly advocated : v = 1.2 (Dumont and Sgnchez, 1975);~ = l.s(Leinert et al., 1981). See thesurveypaper by Giese( 1980). According to Table 1, gY/gO is equal to 1.27 (Weinberg), to 1.22(Dumont and SBnchez), to 0.99 (Gillett). The significant values found for @Y imply the interplanetary matter to be heliocentric dependent at the level of the Earth orbit. This invalidates the assumption of a homogeneous cloud, that was
and Sanchez (1975)
74.9*0.4x 41.5*0.6x 70.1+4x 71.8+7x 0.288 k -0.365 f
10-l’ 10-l’ lo-” lo-” 0.006 0.05
Gillett (1967) 78.5*2x 10-l’ 43.5+ 1 x 10-l’ 50.3+ 17 x lo-” 70.8&9x lo-” 0.286+0.01 - 0.453 * 0.20
classically made, either explicitly or implicitly, in most previous studies.
5. EXTENSION
TO DOPPLER
SPECTROMETRY
Several attempts have been made to derive information about the kinematics of interplanetary scatterers from the Fraiinhofer lines shifted or even distorted by the Doppler effect. Apart from severe observational difficulties due to the weakness of the shifts (less than 1 A), a commonly emphasized snag (James, 1969 ; Fried, 1978) is that the radial velocities obtained are averaged along the whole 1.0.~. The above method can be extended in order to retrieve the radial velocity ofthe scatterers at 1 a.u. from the comparison of the spectra observed at B and at A (Fig. 1). However, the relative scarcity and lack of precision of the observations available up to now offer no chance of retrieving the heliocentric gradient in addition to the central value. Calling s, = dS/dl (MLTe3) the spectral distribution of the solar intensity, the brightnesses become wavelength-dependent and will be denoted i,(A)
= dZ(A)/d& i,(B)
= dZ(B)/dl
(ML- ‘T- 3).
Let 0, be the x-component of the mean velocity vector ofthe scatterers in the element (x, dx) of the 1.0.~.and let V be the velocity vector of the Earth. The apparent radial velocity of the element will be V cos E-v,, yielding a Dopplershift (positive towards the red) AI(x) = ic- ‘(V cos E-u,). At the wavelength di,
= s,(,I-Al)r-‘g
(7)
1 the element (x, dx) scatters dx
= s,[I+lc~‘(v,-
V cos E)]rm29
dx
(8)
an expression which is integrated from - co to -x,, for the observer in B, from - co to +x0 for the observer in A. The point is that, if the eccentricity of the terrestrial orbit is neglected, V cos E is the same at A and at B, so
Zodiacal
light gathered
that the subtraction is as legitimate as above, in order to isolate the contribution of the chord :
1385
along the line of sight
The difference in orbital velocity w.r.t. Earth is v-V
= -cl~‘A~,-(Z,/i,)cl~‘A’~,.
(14)
+x0 i,(A)-
i,(B) =
The photometric quantities Z, and 2, = -2, = Sg,, can be taken from the same authors as in Table 1 and from Table 1 itself, adding the first two lines (Table
di,
s -x0 +x0 =
S,[l+X’(u,-V
cos E)]r-‘9
dx.
(9)
s -x0 If the secant BA tends towards the tangent in 0, i.e. if E < 1 rad, r and cos E remain close to unity and v, practically keeps along the chord a constant value, which is the mean tangential velocity, v, ofthe scatterers near the terrestrial orbit. Then, .!?,may be taken outside of the integral :
2,(,4-i,(B)
= s,[r”+ic-‘(u-
+x0 9 dx
V)]
s -X0 = 2~~x,s,[~+Ic-‘(u-1/)].
(10)
In the very narrow spectral interval of an absorption line, 9 and B0 can be expected independent of i. Therefore, the difference i(A) -i(B) has the same spectrum as the Sun if u = V (i.e. if the scatterers are in circular keplerian orbits, practically co-rotating with the Earth); otherwise, the chord BA scatters what the Sun emits at a slightly different wavelength, 1-A&
= /z+icml(v-V)
(11)
(if u > V, Ai, < 0, the spectrum of the Sun is redder, the Dopplershift AL, is towards the blue). The observed derivatives with respect to x0 (or E,or E) of the Dopplershift and of the brightness, allow us to retrieve the orbital velocity. Calling Z, the brightness and A& the Dopplershift observed at x0 = E = 0 (or E = 90”), Z, and A’& their derivatives, let us write that the increment of brightness from 0 to A, i,E, has the Dopplershift A/1, and increases the observed Dopplershift by A’&& : Z,Ai,+i,EAi, or, restricting
= (Z,+i,E)(A&+A’i,E)
(12)
to first order, Ai., = AL, +(Z,/i,)A’&;.
TABLE ~.OBSEKVED
(13)
STUDY
Weinberg (1964)
rad-I)
QUANTITIES REQUIRED
OF THE MEAN ORBITAL
SCATTERERS
(S,,)
The photometric ratio required by equations (13) and (14) will be taken equal to 1.06 with 10% uncertainty. The spectrometric quantities Ai, and A’& can be expected from four observational sources : Daehler et al. (1968), Reay and Ring (1968), Hicks et al. (1974), Fried (1978). Since the measurements, in all cases, are rather scarce, all the dots between elongations 75 and 105” have been averaged to determine A& and all the dots in the elongation ranges 60-75” and 105-120” have been used to determine the slope w.r.t. E. Figure 1 and the present computations being valid for the case of the evening zodiacal cone, we have changed the sign of all Dopplershifts obtained in the morning zodiacal cone. The two series of observations by Hicks et al. (1974) have been mixed together. The high dispersion and the paucity of the dots in the quoted elongation intervals leads us to disregard the data of Daehler et ul. (1968). For the other three, Table 3 shows the results. Practically no information can be extracted, for our purpose, from the data of Hicks et a/.(1974), in which the r.m.s. dispersion ofthe dots is about three times that of the other two sources. Reay and Ring (1968) and Fried(1978), with r.m.s. dispersions less thanO.l A, both lead to a significant excess of the orbital mean velocity of the scatterers surrounding the Earth’s orbit, over the terrestrial velocity, of the order of + 25 or + 30%. Both these studies emphasized that the Dopplershifts were different from those expected for grains in circular Keplerian orbits and interpreted this in terms of prograde orbits with an excess of angular velocity. The present result confirms these interpretations, with the advantages of avoiding arbitrary models and of determining unambiguously which point of the 1.0.~. has which radial velocity.
VALUESOFTHEPHOTOMETKIC
FOR THE SPECTROMETRIC
Z.(I
2).
250 237
1.05
AT
VELOCITY
OF THE
1 a.u.
Dumont
and SBnchez (1975) 202 175 1.15
Gillett (1967) 181 183 0.99
1386
R. DUMONT TABLET. COMPUTATIONOFTHEDIFISRENCEOFMEANORBITALVELOCITYOFTHE SCATTERERSAT 1 a.".w.r.t. EARTH Reay and Ring (1968) Number 75” < ci-‘Al, Number 60”
of dots E < 105” (km s-l) of dots < E < 75”
9 -1.3k3.4
9
-16.1k9.7
u-T/
+9.1*5
-2.3+
= cX’Al,
(km s-l)
the
local
orbital
velocity,
will follow much
values
of
of the degree
the
is a matter
of good
the improvements
reduced
sensitivity
directional
ofpolarization
scattering
and of the mean observations
of their
accuracy,
to the uncertainties
and it with
ofmodels.
Acknowledgements--I am indebted to J. Delannoy, A. C. Levasseur-Regourd, J. C. Pecker, B. Pelletanne and M. Rapaport for helpful discussions, and to R. McCarroll and G. Mangenot for improvements in the manuscript. REFERENCES Bandermann, L. W. and Wolstencroft, R. D. (1976) Polarimetric observations of the zodiacal light in Hawaii from 1969 to 1974. Mem. Roy. astr. Sot. 81,37. Banos, C. and Koutchmy, S. (1973) Etude photombtrique d’un renforcement diffus observk dans la lumitre zodiacale, g une distance de 100 R, du sole& Icarus 20, 32. Buitrago, J. (1979) A three-dimensional equation for the inversion of the zodiacal light brightness integral. Planet. Space Sci. 21, 1043. Buitrago, J., Alvarez, P., Lopez, G., Mujica, A. and SBnchez, F. (1981) Method for determining dust particle density above the ecliptic plane. Planet. Space Sci. 29, 137. Buitrago, J., Gomez, R. and Sgnchez, F. (1983) The integral equation approach to the study of interplanetary dust. Planet. Space Sci. 31, 373. Daehler, M., Mack, J. E., Stoner, J. O., Clarke, D. and Ring, J. (1968) Measurements of the HP profile in the zodiacal light spectrum. Planet. Space Sci. 16, 795. Dumont, R. (1972) Inter&& et polarisation de la lumitre isol& de matike solaire diffuste par un volume interplanktaire. C. r. hehd. .%anc. Acad. Sci., Paris 2158, 765.
Fried (1978)
3 +2.1+3.0
13
-7.4k3.3
A great deal of the optical and kinematical properties of the interplanetary material in the region of the terrestrial orbit can be derived when associating the two intersections ofthat orbit with the line ofsight. This approach requires less questionable assumptions than usual, especially about the homogeneity of the zodiacal cloud, which is invalidated a posteriori. The knowledge of
12 + 19.4+ 12.2
105” < E < 120 cl~‘X1,(kms~‘rad-‘)
6. CONCLUSION
coefficient,
Hicks et al. (1974)
9 -9.1 16
k2.7
+7.6*5
Dumont, R. (1973) Phase function and polarization curve of interplanetary scatterers from zodiacal light photopolarimetry. Planet. Space Sci. 21, 2149. Dumont, R. (1976a) Ground-based observations of the zodiacal light. Lect. Notes Phys. 48, 85. Dumont, R. (1976b) Some formulae to interpret zodiacal light photopolarimetricdatain theeclipticfromgroundorspace. Lect. Notes Phys. 48, 115. Dumont, R. and Levasseur-Regourd, A. C. (1978) Zodiacal light photopolarimetry,IV. Annual variations ofbrightness and the symmetry plane of the zodiacal cloud. Absence of solar-cycle variations. Astron. Astrophys. 64, 9. Dumont, R. and Levasseur-Regourd, A. C. (1981) Zodiacal light and space observation offaint objects. Adu. Space Res. 1, 127. Dumont, R. and Pelletanne, B. (1981) Inversion de l’inttgrale de brillance le long de lignes de vi&e zodiacales et coronales. C.r. hehd. SPanc. Acad. Sci., Paris 273 II, 377. Dumont, R., Rapaport, M., Schuerman, D. W. and LevasseurRegourd, A. C. (1979) Inversion of the zodiacal brightness integral for an out-of-ecliptic photometer. Space Res. 19, 451. Dumont, R. and Sgnchez, F. (1975) Zodiacal light photopolarimetry, II. Gradients along the ecliptic and the phase functions ofinterplanetarymatter. Astron. Astrophys. 38, 405. Fechtig, H., Leinert, C. and Griin, E. (1981) Interplanetary Dust and Zodiacal Light (Landolt Biirnstein, VI, 2a, 228). Springer, Berlin. Fried, J. W. (1978) Doppler Shifts in the zodiacal light spectrum. Astron. Astrophys. 68, 259. Giese, R. H. (1980) Optical investigation of dust in the solar system. IAU Symp. 90, 1. Gillett, F. C. (1967) Measurement of the brightness and polarization of zodiacal light from balloons and satellites. NASA Sp-lSO,9. Hicks, T. R., May, B. H. and Reay, N. K. (1974) An investigation of the motion of zodiacal dust particles, I. Radial velocity measurements on Fraiinhofer line profiles. Mon. Not. R. astr. Sot. 166,439. James, J. F. (1969) Theoretical Fraiinhofer line profiles in the spectrum of the zodiacal light. Mon. Not. R. astr. Sot. 142, 45. Leinert, C. (1975) Zodiacal light-a measure of the interplanetary environment. Space Sci. Rev. 18,281. Leinert, C., Link, H., Pitz, E. and Giese, R. H. (1976)
Zodiacal
light gathered
Interpretation of a rocket photometry of the inner zodiacal light. Astron. Astrophys. 47, 221. Leinert, C. and Planck, B. (1982) Stability and symmetry of zodiacal light polarization in the antisolar hemisphere. Astron. Astrophys. 105,364. Leinert, C., Richter, I., Pitz, E. and Planck, B. (1981) The zodiacal light from 1.0 to 0.3 A.U. as observed by the Helios space probes. Astron. Astrophys. 103, 177. Levasseur, A. C. and Blamont, J. E. (1976) Evidence for scattering particles in meteor streams. Lecl. Notes Phys. 48, 58. Levasseur-Regourd, A. C. and Dumont, R. (1980) Absolute photometry of zodiacal light. Astron. Astrophys. 84, 277. Morgan, D. H., Nandy, K. and Thompson, G. I. (1976) The ultraviolet galactic background from TD-1 satellite observations. Mon. Not. R. as@. Sot. 177, 531. Mujica, A., Lopez, G. and SBnchez, F. (1980) Method for the determination of density and phase function of interplanetary dust. Planet. Space Sci. 28, 657. Pelletanne, B. (1982) Les intkgrales de brillance en photopolarimCtrie zodiacale; tentatives pour remonter aux contributions locales. Th&e, Universitk de Bordeaux I. Reay, N. K. and Ring, J. (1968) Radial velocity measurements on the zodiacal light spectrum. Nature, Lond. 219, 710. Richter, I., Leinert, C. and Planck, B. (1982) Search for short
along the line of sight
1387
term variations of zodiacal light and optical detection of interplanetary plasma clouds. Astron. Astrophys. 110, 115. Roach, F. E. and Smith, L. L. (1968) An observational search for cosmic light. Geophys. J. R. am. Sot. 15, 227. Robley, R. (1980) Change in the zodiacal light with solar activity. IAU Symp. 90, 33. Schuerman, D. W. (1979a) Inverting the zodiacal light brightness integral. Planet. Space Sci. 27, 551. Schuerman, D. W. (1979b) The brightness/unit volume of the zodiacal light as determined from Pioneer 10. Space Res. 19, 447. Schuerman, D. W. (1980) Evidence that the properties of interplanetary dust beyond 1 AU are not homogeneous. IAU Symp. 90,71. Sparrow, J. G. and Ney, E. P. (1972) Observations of the zodiacal light from the ecliptic to the poles. Astrophys. J. 174, 705. Sparrow, J. G. and Weinberg, J. L. (1976) The S,,(V) unit of surface brightness. Lect. Notes Phys. 48,41. Weinberg, J. L. (1964) The zodiacal light at 5300 A. Ann. Astrophys. 27, 718. Weinberg, J. L. and Sparrow, J. G. (1978) Zodiacal light as an indicator of interplanetary dust, in Cosmic Dust (Edited by McDonnell, J. A. M.). J. Wiley and Sons,
ZODIACAL
LIGHT LATHERED
ALONG THE LINE OF SIGHT
THE VICINITY OF THE TERRESTRIAL ORBIT STUDIED WITH PHOTOPOLARIMETRY AND WITH DOPPLER SPECTROMETRY R. DUMONT
Observatoire
de Bordeaux, (Receiljd
F-33270 Fioirac,
15 February
France
1983)
Abstract-Theexpression for the zodiacal brightness integral is especiallysimpleiftheintegrand contains the ‘directional scattering coefficient’, 9,(a.u.- ‘), or equivalently the scattering cross-section per unit-volume. The two intersections of the terrestrial orbit with a line of sight lying in the ecliptic offer the possibility of isolating the contribution of the chord, with a conservative assumption of steadiness, but without the controversial assumption of a homogeneous zodiacal cloud. The zodiacal brightnesses between 60 and 120” elongation can beused toder~v~~~and~, thevalueof~anditshelio~ent~~radial derivative, bothat 1 a.u. andat ascattering angle of 90”. A polarimetric treatment leads to the local polarization degree, Pa, and to its heliocentric derivative, & Applied to all three available observational sources, this method invalidates the assumption of homogeneity, leading to a rather high relative gradient B/P0 near 1 a.u. (.- 12, - 16 or - 24%. according to the source, as the Sun’s distance decreases from 1.0 to 0.9 au.). The method is extended to Doppler spectrometry, taking advantage of the two equal projections on the line ofsight of the Earth’s velocity vector. The brightness 2, and the Dopplershift Ai,, observed at 90” elongation, together with the derivatives w.r.t. elongation e, of the brightness, i and of the Dopplershift, A’& can be used to retrievethemeanorbitalvelocity,~,oftheinterplanetaryscatterersin t~ere~onofthe terrestrialorbit.Thetwo most reliable observational sources lead, with fair agreement, to a relative excess fu - V)/V, over the terrestrial
velocity, of the order of + 257;.
1. INTRODUCTION
Observations of the zodiacal light (z.1.) provide information integrated along the whole line of sight (1.0s.). In the photometric case, this information is a brightness, Z (MTe3), generally expressed in S,, (equivalent to one G2V star of 10th magnitude V per square degree). In the polarimetric case, one has also access (at least) to the components of 2, Z, and Z!,, scattered by the Fresnel vectors respectively perpendicular to, and lying in, the scattering plane. In the spectrometric case, the wavelength dependence of the brightness, Z(1) = dZ/dl (ML-‘Ts3) is available. The integrated quantities are directly useful, as far as 2.1. is only considered a foreground source, against remotesources being theaim ofthestudy,and requiring to be corrected for it : diffuse galactic light (Morgan er al., 1976), cosmic light (Roach and Smith, 1968), faint targets for large instruments (Dumont and LevasseurRegourd, 1981). On the other hand, as far as the zodiacal cloud (z.c.) itselfis concerned, it is an enormous deficiency to ignore how the brightness (or its polarized components, or its derivative with 1) has been gathered along the 1.0,s. and this is one of the main origins of the ambiguities and the discrepancies between models of the Z.C. and of the scatterers. Several efforts have been made towards a
localization along the 1.0s. of the brightness contributions. Often these efforts con~ntrated on the last elemental section of the I.o.s., i.e. the optical behaviour of the scatterers in the immediate vicinity of the observer (Dumont, 1972, 1973 ; Dumont et al., 1979; Schuerman, 1979a, 1979b; Buitrago, 1979). For that single location on the 1.0.~.and for peculiar viewing directions, the local contribution by the end of the 1.0.~. can be determined free of assumptions. The knowledge, even partial and approximate, of what happens elsewhere on the 1.0.~. requires more or less arbitrary assumptions. If the Z.C. is assumed homogeneous, cylindrically symmetric and admitting a power law r - ” for the space density vs the heliocentric distance Y,there is a rigorous solution for inverting the brightness integral (Dumont, 1973). In the latter and much of subsequent work (Dumont and Sanchez, 1975; Dumont, 1976a, 1976b; Buitrago et al., 1981,1983), the assumptions have a rather high ““levelof arbitrariness”. In other cases (Leinert et al., 1976; Mujica et al., 1980; Dumont and Pelletanne, 198 1; Pelletanne, 1982) the dependence w.r.t. the major hypotheses is to a certain extent reduced. Recent space observations either have given support to these classical assumptions, or--more often-have invalidated them. The r-’ law for the space density is supported by the photometric results of Heiios probes
1381
1382
R. DUMONT
(Leinert et al., 1981), with v = 1.3. On the other hand, the polarimetric results of the same probes cannot be reconciled with a homogeneous Z.C. inside the Earth’s orbit. The data of Pioneer 10 led to the same conclusion outside the Earth’s orbit (Schuerman, 1980). Contrary to the cylindrical symmetry, which is strongly suggested, at least for an overall view, by the weakness of the seasonal oscillations of the z.1. features, the assumption of a macroscopically homogeneous cloud seems no longer satisfying. The present work is the first of a series, the purpose of which is to increase our ability to decipher the brightness distribution along the 1.0.~. and to decrease our recourse to physically arbitrary assumptions.
scattering cross-section of the unit-volume, or equivalently, the omnidirectional scattering coefficient. Both 9 and 0 will be expressed in a.u.-‘. In order to maintain the usual S,, unit of brightness, the solar intensity S has to be expressed in the corresponding unit. The irradiance unit will be that given by as many G2V stars of 10th magnitude V, as one steradian contains of square-degrees (3282). Adopting after Sparrow and Weinberg (1976) - 26.73 for the apparent magnitude V of the Sun, its irradiance at 1 a.u., i.e. its intensity, is S = dexC(26.73 + 10)/2.5]/3282 = 1.500 x 10” in the above unit. If the 1.0.~. is oriented positively like the propagation of light, our expression of the brightness integral is x0
2. THE
ZODIACAL
BRIGHTNESS:
z=s
AN INTEGRAL
rm29 dx.
OF WHAT?
Various expressions for the brightness integral have been published. Ordinarily the assumed models for grain size distribution and phase functions, as well as for their spatial distribution, appear in the integrand. A different, less model-dependent, way of expressing the integral is to concentrate on the intensity scattered by an elemental unit-volume, or an elemental section ofthe I.o.s., since we have no access to any microscopic knowledge inside these elements. If multiple scattering is admitted to be negligible, the contribution dZ by a section dx of the 1.0.~.near a point M is proportional to dx and to the solar irradiance at M. Denoting by S the intensity of the Sun (ML2Tm3), the irradiance (MTm3) at r astronomical units (a.u.) from thecenter ofthesunise = Sre2,exceptin thevery inner Z.C. (r = a few solar radii) where both the single scattering treatment and this expression of e may be insufficient. The proportionality factor is a scattering coefficient (L ‘). It depends not only on the scatterers at M, but on the scattering direction and it will be denoted 9 (directional scattering coefficient) : dZ = Srm29
dx.
In the polarimetric case, the local degree ofpolarization of the sunlight scattered by a small volume at M is B = (~l-~,i)/(~l+~,,), where gL and g,, correspond to Z, and Zll with equation (2).
3. RETRIEVAL
OF B AND 9J IN THE VICINITY
This paper concentrates on z.1. observations from the Earth, with the 1.0.~.in the ecliptic plane and at medium elongations E (solar and antisolar regions avoided). In the present section a method is introduced to obtain 9 and B at 1 a.u. and their heliocentric derivatives. Calling 0 a random position of the Earth and choosing the axes Ox, Oy as shown on Fig. 1, consider the Earth at A some time later and let Z(A) be the brightness
(1)
The product eg = .Y is the intensity (ML-rTm3) scattered towards the observer by the unit-volume surrounding M ; I has been extensively used in previous works, but the quantity 2 has the advantage of being independent of solar irradiance, therefore being an intrinsic property of the local matter. Since the energy subtracted from the incident beam by the volume do (per steradian and for the same scattering direction) is .Y dv = eg dv, an equivalent definition of 2 is the directional scattering cross-section of the unit-volume. The integral of 9 over the sphere, JJ& dw = 0, represents the total energy subtracted by scattering and will be the omnidirectional
OF THE
EARTH’S ORBIT
FIG. 1. GEOMETRY OF THE METHOD.
1383
Zodiacal light gathered along the line of sight observed from A at elongation e, along a 1.0s. parallel to Ox. Let B be the second intersection ofthe 1.0.~.with the orbit, which is admitted circular (a convenient, but not unavoidablesimplification).Thepossibilityofisolating the contribution ofthe chord BA, Z(ii)Z(B), would be an important step towards a localization of the information. At the time when the observation is made from A, the brightness that would be seen from B is unknown, but what was observed from B some time earlier is known. The assumption that Z(E) has remained unchanged while the Earth moved from B to A implies a steadiness of the Z.C.on the time-scale of a few weeks. Most of the observers, from Earth and from the apparent stability, space, have emphasized smoothness and cylindrical symmetry of the Z.C. (Sparrow and Ney, 1972; Weinberg and Sparrow, 1978 ; Levasseur-Regourd and Dumont, 1980; Richter et al., f 982; Leinert and Planck, 1982). Moreover, most of the ‘variations’ invoked in the literature do not conflict with our hypothesis ofsteadiness, at least those originating from the orbital motion of the Earth rather than from intrinsic local changes (Levasseur and Blamont, 1976 ; Dumont and Levasseur-Rego~lrd, 1978). As for the intrinsic changes reported near the ecliptic, they concern the solar cap (Banos and Koutchmy, 1973) or the antisolar cap (Bandermann and Wolstencroft, 1976; Robley, 1980) rather than the intermediate elongations. In summary, the invariance of Z(B) is a much less imprudent assumption than a homogeneous composition of the z.c., which is supported neither by theoretical nor observational arguments, and has been completely avoided in the present study. On this basis, the chord BA contributes to the brightness in A by +x0 Z(A) - Z(B) = s
r - ‘9(x, y) dx s -*o
(3)
where y = y0 is cst along the 1.0s. Introducing the development ofC8 with respect to x and toy, limited to the first order, g(x, y) = s$) c&,x where so follows :
is the central
Z(A)-Z(B)-S@!,
+ $,y
(4)
value (X = y = 0) of g%?,it
+~,y(&2/1
--y*)tg_ ‘(.x,/l -y())
or, with 56 and l-y,
= E = tg- ‘(x,/f -y,)
= cos E,
Z(E) - Z( - E) N S(2Ejcos
E) [go + ( 1 - cos E)L@,]. (5)
Within reasonable limits for E, say no more than 7c/6, the regression of the quantity (cos E/2E) [Z(E) - .Z( - E)] vs the quantity (1 -cos E) therefore gives access to SgO as the intercept and to S&, as the slope. This method can be extended to polarimetry, replacing Z and 3 by their components Z,, Z ,,, BL, 9 ,, The polarization degree of the sunlight scattered at 90” scattering angle by a small volume at 1 au. in the ecliptic will be B = (go1 -C@Oii)/(~oi +C@e,r). Its heliocentric gradient will be 9, = 9’o’[(l
-&J$,,-(l+Y&&].
If .@Yis significantly different from zero, something has to be heliocentric-dependent in the composition of the interplanetary medium in the region of the Earth’s orbit.
4. APPLICATION
TO AVAILABLE
DATA: THE RADIAL
PHOTOPOLARlMETRIC
INCREASE OF LOCAL
~LARIZATION
The above method has been applied to the three available observational sources which provide the polarized components of Zin the ecliptic as functions of elongation C, in the range 60” d 1~1< 120”: groundbased observations from Haleakala, Hawaii (Weinberg, 1964) and from Teide, Tenerife (Dumont and Sanchez, 1975); balloon observations by Gillett (1967). The first two sources give numbers, the latter only curves. The spectral domains are not very different (530 nm for the first ; 460 and 502 nm for the second; near Johnson’s B for the third). Giilett’s data are expressed in a S,,(B) unit and have been multiplied by 1.7 to be compared to the others, expressed in the classical S,, unit. In all three cases, values ofZ, and Z,, are available by steps of 5” in E. The couple E = 85 and 95” (E = 5’) has not been used, thedifference Z(S”)-Z( - 5”) being rather poorly known, due to its low value and to the fact that the figures are published in integral units of S, o, with lack ofsmoothness vs E.The regression of formula (5) has been made with five couples, 10,15,20,25 and 30”for E. Table 1 shows the results: values of gin,, g,,, .F and their heliocentric derivatives, with the r.m.s. uncertainties on all these quantities. Obviously, these CT errors represent the precision in the linearity of the regressions, not the accuracy of the measurements. The values at 1 a.u. can be compared to those already available in the literature, on a slightly different form (Dumont, 1972, 1973; Dumont and Sanchez, 1975; Leinert, 1975): the possibility of isolating the local contributions and polarization at the Earth and at a
1384
R. TABLE
DUMONT
1.LOCAL VALUESAND
HELIOCENTRICDERIVATIVESAT 1 a.u. OF THE DIRECTIONAL SCATTERINGCOEFFICIENT AND OF THE DEGREEOF POLARIZATION Weinberg (1964)
(a.u.-‘) ?O,, *B (ax-‘) ?yl *a (a.u.-‘) ayi, +o (a.u.-‘)
9ol*fs
To+” I yy + 0 (a.u. - ‘)
Dumont
110.2*0.5x lo-” 47.8*0.5x lo-” 63.6+6x lo-” 137.0&6x 10-l’ 0.395 k 0.006 -0.967+0.10
scattering angle of90” is well known. On the other hand, the heliocentric derivatives are new. They show, as understandable, more discrepancies than the values at 1 a.u. However, the three results are qualitatively the same : the local degree of polarization at right scattering angle, B, is rather rapidly decreasing as y increases, or increasing as r increases. When r goes from 1 to 0.9 a.u., the relative loss of 9, equal to @3,/10Y,,, would be 24% (Weinberg), 12% (Dumont and Sgnchez), 16% (Gillett). A similar treatment of the values compiled by Fechtig et al. (1981) from various observational materials gives a relative loss of 14%. By a different approach, Dumont and Pelletanne (1981) pointed out a similar effect. There is an evident connection between the sign of the gradient gpyand the regular decrease in the degree of polarization reported by the Helios group (Leinert et al., 1981) as these two probes approached the Sun. The same conclusion arises from their observations and from the present work : interplanetary scatterers polarize more and more, going away from the Sun. However, two main differences must be kept in mind : (i) Helios photopolarimeters did not point at the ecliptic, (ii) their polarizations are observed ones, i.e. integrated along the 1.0.~. There is another noteworthy agreement between Table 1 and previous observations, in a purely photometric domain. If we compute the relative gradient of the total directional scattering coefficient, g,/%l
= ($1+
Q/(%
+ %,I)
(6)
it has to fit the exponent v of the Y-” space density law commonly advocated : v = 1.2 (Dumont and Sgnchez, 1975);~ = l.s(Leinert et al., 1981). See thesurveypaper by Giese( 1980). According to Table 1, gY/gO is equal to 1.27 (Weinberg), to 1.22(Dumont and SBnchez), to 0.99 (Gillett). The significant values found for @Y imply the interplanetary matter to be heliocentric dependent at the level of the Earth orbit. This invalidates the assumption of a homogeneous cloud, that was
and Sanchez (1975)
74.9*0.4x 41.5*0.6x 70.1+4x 71.8+7x 0.288 k -0.365 f
10-l’ 10-l’ lo-” lo-” 0.006 0.05
Gillett (1967) 78.5*2x 10-l’ 43.5+ 1 x 10-l’ 50.3+ 17 x lo-” 70.8&9x lo-” 0.286+0.01 - 0.453 * 0.20
classically made, either explicitly or implicitly, in most previous studies.
5. EXTENSION
TO DOPPLER
SPECTROMETRY
Several attempts have been made to derive information about the kinematics of interplanetary scatterers from the Fraiinhofer lines shifted or even distorted by the Doppler effect. Apart from severe observational difficulties due to the weakness of the shifts (less than 1 A), a commonly emphasized snag (James, 1969 ; Fried, 1978) is that the radial velocities obtained are averaged along the whole 1.0.~. The above method can be extended in order to retrieve the radial velocity ofthe scatterers at 1 a.u. from the comparison of the spectra observed at B and at A (Fig. 1). However, the relative scarcity and lack of precision of the observations available up to now offer no chance of retrieving the heliocentric gradient in addition to the central value. Calling s, = dS/dl (MLTe3) the spectral distribution of the solar intensity, the brightnesses become wavelength-dependent and will be denoted i,(A)
= dZ(A)/d& i,(B)
= dZ(B)/dl
(ML- ‘T- 3).
Let 0, be the x-component of the mean velocity vector ofthe scatterers in the element (x, dx) of the 1.0.~.and let V be the velocity vector of the Earth. The apparent radial velocity of the element will be V cos E-v,, yielding a Dopplershift (positive towards the red) AI(x) = ic- ‘(V cos E-u,). At the wavelength di,
= s,(,I-Al)r-‘g
(7)
1 the element (x, dx) scatters dx
= s,[I+lc~‘(v,-
V cos E)]rm29
dx
(8)
an expression which is integrated from - co to -x,, for the observer in B, from - co to +x0 for the observer in A. The point is that, if the eccentricity of the terrestrial orbit is neglected, V cos E is the same at A and at B, so
Zodiacal
light gathered
that the subtraction is as legitimate as above, in order to isolate the contribution of the chord :
1385
along the line of sight
The difference in orbital velocity w.r.t. Earth is v-V
= -cl~‘A~,-(Z,/i,)cl~‘A’~,.
(14)
+x0 i,(A)-
i,(B) =
The photometric quantities Z, and 2, = -2, = Sg,, can be taken from the same authors as in Table 1 and from Table 1 itself, adding the first two lines (Table
di,
s -x0 +x0 =
S,[l+X’(u,-V
cos E)]r-‘9
dx.
(9)
s -x0 If the secant BA tends towards the tangent in 0, i.e. if E < 1 rad, r and cos E remain close to unity and v, practically keeps along the chord a constant value, which is the mean tangential velocity, v, ofthe scatterers near the terrestrial orbit. Then, .!?,may be taken outside of the integral :
2,(,4-i,(B)
= s,[r”+ic-‘(u-
+x0 9 dx
V)]
s -X0 = 2~~x,s,[~+Ic-‘(u-1/)].
(10)
In the very narrow spectral interval of an absorption line, 9 and B0 can be expected independent of i. Therefore, the difference i(A) -i(B) has the same spectrum as the Sun if u = V (i.e. if the scatterers are in circular keplerian orbits, practically co-rotating with the Earth); otherwise, the chord BA scatters what the Sun emits at a slightly different wavelength, 1-A&
= /z+icml(v-V)
(11)
(if u > V, Ai, < 0, the spectrum of the Sun is redder, the Dopplershift AL, is towards the blue). The observed derivatives with respect to x0 (or E,or E) of the Dopplershift and of the brightness, allow us to retrieve the orbital velocity. Calling Z, the brightness and A& the Dopplershift observed at x0 = E = 0 (or E = 90”), Z, and A’& their derivatives, let us write that the increment of brightness from 0 to A, i,E, has the Dopplershift A/1, and increases the observed Dopplershift by A’&& : Z,Ai,+i,EAi, or, restricting
= (Z,+i,E)(A&+A’i,E)
(12)
to first order, Ai., = AL, +(Z,/i,)A’&;.
TABLE ~.OBSEKVED
(13)
STUDY
Weinberg (1964)
rad-I)
QUANTITIES REQUIRED
OF THE MEAN ORBITAL
SCATTERERS
(S,,)
The photometric ratio required by equations (13) and (14) will be taken equal to 1.06 with 10% uncertainty. The spectrometric quantities Ai, and A’& can be expected from four observational sources : Daehler et al. (1968), Reay and Ring (1968), Hicks et al. (1974), Fried (1978). Since the measurements, in all cases, are rather scarce, all the dots between elongations 75 and 105” have been averaged to determine A& and all the dots in the elongation ranges 60-75” and 105-120” have been used to determine the slope w.r.t. E. Figure 1 and the present computations being valid for the case of the evening zodiacal cone, we have changed the sign of all Dopplershifts obtained in the morning zodiacal cone. The two series of observations by Hicks et al. (1974) have been mixed together. The high dispersion and the paucity of the dots in the quoted elongation intervals leads us to disregard the data of Daehler et ul. (1968). For the other three, Table 3 shows the results. Practically no information can be extracted, for our purpose, from the data of Hicks et a/.(1974), in which the r.m.s. dispersion ofthe dots is about three times that of the other two sources. Reay and Ring (1968) and Fried(1978), with r.m.s. dispersions less thanO.l A, both lead to a significant excess of the orbital mean velocity of the scatterers surrounding the Earth’s orbit, over the terrestrial velocity, of the order of + 25 or + 30%. Both these studies emphasized that the Dopplershifts were different from those expected for grains in circular Keplerian orbits and interpreted this in terms of prograde orbits with an excess of angular velocity. The present result confirms these interpretations, with the advantages of avoiding arbitrary models and of determining unambiguously which point of the 1.0.~. has which radial velocity.
VALUESOFTHEPHOTOMETKIC
FOR THE SPECTROMETRIC
Z.(I
2).
250 237
1.05
AT
VELOCITY
OF THE
1 a.u.
Dumont
and SBnchez (1975) 202 175 1.15
Gillett (1967) 181 183 0.99
1386
R. DUMONT TABLET. COMPUTATIONOFTHEDIFISRENCEOFMEANORBITALVELOCITYOFTHE SCATTERERSAT 1 a.".w.r.t. EARTH Reay and Ring (1968) Number 75” < ci-‘Al, Number 60”
of dots E < 105” (km s-l) of dots < E < 75”
9 -1.3k3.4
9
-16.1k9.7
u-T/
+9.1*5
-2.3+
= cX’Al,
(km s-l)
the
local
orbital
velocity,
will follow much
values
of
of the degree
the
is a matter
of good
the improvements
reduced
sensitivity
directional
ofpolarization
scattering
and of the mean observations
of their
accuracy,
to the uncertainties
and it with
ofmodels.
Acknowledgements--I am indebted to J. Delannoy, A. C. Levasseur-Regourd, J. C. Pecker, B. Pelletanne and M. Rapaport for helpful discussions, and to R. McCarroll and G. Mangenot for improvements in the manuscript. REFERENCES Bandermann, L. W. and Wolstencroft, R. D. (1976) Polarimetric observations of the zodiacal light in Hawaii from 1969 to 1974. Mem. Roy. astr. Sot. 81,37. Banos, C. and Koutchmy, S. (1973) Etude photombtrique d’un renforcement diffus observk dans la lumitre zodiacale, g une distance de 100 R, du sole& Icarus 20, 32. Buitrago, J. (1979) A three-dimensional equation for the inversion of the zodiacal light brightness integral. Planet. Space Sci. 21, 1043. Buitrago, J., Alvarez, P., Lopez, G., Mujica, A. and SBnchez, F. (1981) Method for determining dust particle density above the ecliptic plane. Planet. Space Sci. 29, 137. Buitrago, J., Gomez, R. and Sgnchez, F. (1983) The integral equation approach to the study of interplanetary dust. Planet. Space Sci. 31, 373. Daehler, M., Mack, J. E., Stoner, J. O., Clarke, D. and Ring, J. (1968) Measurements of the HP profile in the zodiacal light spectrum. Planet. Space Sci. 16, 795. Dumont, R. (1972) Inter&& et polarisation de la lumitre isol& de matike solaire diffuste par un volume interplanktaire. C. r. hehd. .%anc. Acad. Sci., Paris 2158, 765.
Fried (1978)
3 +2.1+3.0
13
-7.4k3.3
A great deal of the optical and kinematical properties of the interplanetary material in the region of the terrestrial orbit can be derived when associating the two intersections ofthat orbit with the line ofsight. This approach requires less questionable assumptions than usual, especially about the homogeneity of the zodiacal cloud, which is invalidated a posteriori. The knowledge of
12 + 19.4+ 12.2
105” < E < 120 cl~‘X1,(kms~‘rad-‘)
6. CONCLUSION
coefficient,
Hicks et al. (1974)
9 -9.1 16
k2.7
+7.6*5
Dumont, R. (1973) Phase function and polarization curve of interplanetary scatterers from zodiacal light photopolarimetry. Planet. Space Sci. 21, 2149. Dumont, R. (1976a) Ground-based observations of the zodiacal light. Lect. Notes Phys. 48, 85. Dumont, R. (1976b) Some formulae to interpret zodiacal light photopolarimetricdatain theeclipticfromgroundorspace. Lect. Notes Phys. 48, 115. Dumont, R. and Levasseur-Regourd, A. C. (1978) Zodiacal light photopolarimetry,IV. Annual variations ofbrightness and the symmetry plane of the zodiacal cloud. Absence of solar-cycle variations. Astron. Astrophys. 64, 9. Dumont, R. and Levasseur-Regourd, A. C. (1981) Zodiacal light and space observation offaint objects. Adu. Space Res. 1, 127. Dumont, R. and Pelletanne, B. (1981) Inversion de l’inttgrale de brillance le long de lignes de vi&e zodiacales et coronales. C.r. hehd. SPanc. Acad. Sci., Paris 273 II, 377. Dumont, R., Rapaport, M., Schuerman, D. W. and LevasseurRegourd, A. C. (1979) Inversion of the zodiacal brightness integral for an out-of-ecliptic photometer. Space Res. 19, 451. Dumont, R. and Sgnchez, F. (1975) Zodiacal light photopolarimetry, II. Gradients along the ecliptic and the phase functions ofinterplanetarymatter. Astron. Astrophys. 38, 405. Fechtig, H., Leinert, C. and Griin, E. (1981) Interplanetary Dust and Zodiacal Light (Landolt Biirnstein, VI, 2a, 228). Springer, Berlin. Fried, J. W. (1978) Doppler Shifts in the zodiacal light spectrum. Astron. Astrophys. 68, 259. Giese, R. H. (1980) Optical investigation of dust in the solar system. IAU Symp. 90, 1. Gillett, F. C. (1967) Measurement of the brightness and polarization of zodiacal light from balloons and satellites. NASA Sp-lSO,9. Hicks, T. R., May, B. H. and Reay, N. K. (1974) An investigation of the motion of zodiacal dust particles, I. Radial velocity measurements on Fraiinhofer line profiles. Mon. Not. R. astr. Sot. 166,439. James, J. F. (1969) Theoretical Fraiinhofer line profiles in the spectrum of the zodiacal light. Mon. Not. R. astr. Sot. 142, 45. Leinert, C. (1975) Zodiacal light-a measure of the interplanetary environment. Space Sci. Rev. 18,281. Leinert, C., Link, H., Pitz, E. and Giese, R. H. (1976)
Zodiacal
light gathered
Interpretation of a rocket photometry of the inner zodiacal light. Astron. Astrophys. 47, 221. Leinert, C. and Planck, B. (1982) Stability and symmetry of zodiacal light polarization in the antisolar hemisphere. Astron. Astrophys. 105,364. Leinert, C., Richter, I., Pitz, E. and Planck, B. (1981) The zodiacal light from 1.0 to 0.3 A.U. as observed by the Helios space probes. Astron. Astrophys. 103, 177. Levasseur, A. C. and Blamont, J. E. (1976) Evidence for scattering particles in meteor streams. Lecl. Notes Phys. 48, 58. Levasseur-Regourd, A. C. and Dumont, R. (1980) Absolute photometry of zodiacal light. Astron. Astrophys. 84, 277. Morgan, D. H., Nandy, K. and Thompson, G. I. (1976) The ultraviolet galactic background from TD-1 satellite observations. Mon. Not. R. as@. Sot. 177, 531. Mujica, A., Lopez, G. and SBnchez, F. (1980) Method for the determination of density and phase function of interplanetary dust. Planet. Space Sci. 28, 657. Pelletanne, B. (1982) Les intkgrales de brillance en photopolarimCtrie zodiacale; tentatives pour remonter aux contributions locales. Th&e, Universitk de Bordeaux I. Reay, N. K. and Ring, J. (1968) Radial velocity measurements on the zodiacal light spectrum. Nature, Lond. 219, 710. Richter, I., Leinert, C. and Planck, B. (1982) Search for short
along the line of sight
1387
term variations of zodiacal light and optical detection of interplanetary plasma clouds. Astron. Astrophys. 110, 115. Roach, F. E. and Smith, L. L. (1968) An observational search for cosmic light. Geophys. J. R. am. Sot. 15, 227. Robley, R. (1980) Change in the zodiacal light with solar activity. IAU Symp. 90, 33. Schuerman, D. W. (1979a) Inverting the zodiacal light brightness integral. Planet. Space Sci. 27, 551. Schuerman, D. W. (1979b) The brightness/unit volume of the zodiacal light as determined from Pioneer 10. Space Res. 19, 447. Schuerman, D. W. (1980) Evidence that the properties of interplanetary dust beyond 1 AU are not homogeneous. IAU Symp. 90,71. Sparrow, J. G. and Ney, E. P. (1972) Observations of the zodiacal light from the ecliptic to the poles. Astrophys. J. 174, 705. Sparrow, J. G. and Weinberg, J. L. (1976) The S,,(V) unit of surface brightness. Lect. Notes Phys. 48,41. Weinberg, J. L. (1964) The zodiacal light at 5300 A. Ann. Astrophys. 27, 718. Weinberg, J. L. and Sparrow, J. G. (1978) Zodiacal light as an indicator of interplanetary dust, in Cosmic Dust (Edited by McDonnell, J. A. M.). J. Wiley and Sons,