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Temperatures of Zodiacal dust
Planet. space Sci., Vol. 41, No. llj12, pp. 1099~1108, 1993 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/93 $7.00 +O.OO 0032-0633(93)E0044-D
Pergamon
Temperatures of zodiacal dust Peter Staubach,’ Neil Divine 1,2and Eberhard Griin ’ ‘pax-Planck-rnstitut fur Kernphysik? P.O. Box 103980,69029 Heidelberg, Germany *Jet Propulsion Laboratory, Pasadena, CA 91109, U.S.A. Received for publication
17 December 1993
1. Introduction Interplanetary dust is finely divided particulate matter that exists between the planets. Reflection of sunlight from the myriad of interplanetary dust particles which are concentrated in the plane of the solar system causes the zodiacal light [for a recent review see Leinert and Griin (1990)]. Illterplanetary dust has been studied by a number of methods: radar meteor observations provide orbits of sub-millimetre particles (Sekanina and Southworth,
Covvespondence to : P. Staubach
1975) ; from lunar impact craters the size dist~bution has been established ranging from sub-micron to mil~m~tre sizes (Grim et al., 198s); modern spacecraft detectors determined the micrometeoroid flux from 0.3 to 20 a.u. distance from the Sun (Humes, 1980; Griin et al., 1991) ; and from zodiacal light observations, both from the ground and with spacecraft, the spatial distribution has been established (Giese and KneiBel, 1989 ; KneiBel and Mann, 1991). Recently, both spacecraft [1&4S, Hauser et al. (1984)] and rocket-borne [ZIP, Murdock and Price (1985)] observations of the thermal emission of interplanetary dust in the IR have become available. These observations constrain the spatial distribution of interplanetary dust (Hanner, 1991) and in comparison with zodiacal light observations, provide means to determine local variations of optical and thermal properties of interplanetary dust (Levasseur-Regourd et al., 1991). In order to obtain the temperatures of inte~lanetary dust grains different approaches have been taken: Deul and Wolstencroft (1988) and Rowan-Robinson et al. (1990) assume the radial temperature distribution T(v) N rF6, with 6 = 0.5, while Good (1990) finds 6 = 0.36. Rose, and Staude (1978), Reach (1988) andTemi et al. (1989) compute T(v) from Mie theory. The inversion of the zodiacal emission integral gives 6 = 0.33 front and Levasseur-Regourd, 1988). In this study, dust concentrations in different regions of space are calculated from Divine’s (1993) model, temperatures according to particle sizes and heliocentric distance have been assumed and IR intensities of the zodiacal thermal emission are evaluated. These model-predicted intensities are compared with measured intensities and, in the case of differences between both, the particle temperatures are changed until conformity is reached. In Section 2 the Divine model is introduced and key characteristics of the five populations of meteoroids are demonstrated. Data sets representing the zodiacal thermal emission are described in Section 3. The method and assumptions for the calculation of particle temperatures are presented in Section 4 and, finally, contributions of
1100
1’. Staubach er al. : Temperatures of zodiacal dust
the different meteoroid populations to the thermat emission are detcrmincd and resulting particle temperatures arc discussed in the fifth and sixth sections, respectively.
2. The Divine model Concentrations of interplanetary particles are determined from the “Five Populations of Interplanetary Mctcoroids” model of Divine (1993) which classifies interplanetary meteoroids into five populations. Each population has characteristic distributions of orbital clcmcnts, i.e. perihelion distance N,(r). ecccntricityp,(e), inclination pi(i) and of particlc mass H&z). In order to define these populations Divine used different observational data sets. Data from radar meteors (Sekanina and Southworth, 1975) were taken as we11as in situ in~asurements of dust impacts onboard scvcral space probes, for example Ue!y~su.s,~x~~~r~~ 16 and 23, HEOS2 and Pioneer 8 and 9 for data near the Earth’s orbit. The meteoroid size distribution derived from lunar cratcring records is described by the Interplanetary Flux Model [Griin et al. (1985), set Fig. 11. Data from the Helios spacecraft for distances less than 1 a.u. (Griin et al.: 1980) and data from the Piuneer IO and 11 spacecraft obtained in the outer Solar System (Humes et al., 1980) were used. Also data sets originated from dust detectors currently operating onboard the Galileo and Ulysses spacecraft (Griin et al., 1991) were taken. Zodiacal light data sets from space probes and ground-based observations (Leinert et al., I981 ; Levasseur-Rcgourd and Dumont, 1980 ; Hanner et al., 1974) were also used. The five meteoroid populations are briefly described in order of decreasing particle mass : (1) the asteroidal populations with a logarithmic mean mass of 10 ’ g, has small ecc~nt~~ities and inclinations, its concentration increases outward from 1 a.u. and has its maximum in the asteroid belt ; (2) the core population, of mean mass lo-’ g, has particles also on orbits of small eccentricity and inclination. It makes the main contribution to the zodiacal light, as well as to most other data sets; (3) the halo population, of mean mass 10 ’ g, exits mostly beyond 2.5 a.u. and has random inclinations (including retrograde orbits). Tt is required to provide the nearly uniform lluxcs detected in the outer solar system by Pioneer f 0, Ykmrer I 1 and C~iy.~~.s ; (4) the inclined population, of mean mass IO- ’ gt has particles on near-circular, moderately inclined orbits inside 1 a.u.; and (5) the eccentric population, having mean mass below 10 I2 g, exists mostly inside 1 a.u., has highly eccentric orbits and is rcquircd primarily to match the angular distribution of individual impact events on He&x.
For the core and eccentric population a geometric albedo of 0.05 allows a good match to the measured zodiacal light intensities, while for all other populations a lower value of 0.02 is required in order for these populations not to contribute too large intensities to the zodiacal light. The combination of the five populations of
.l? 14
‘,~
I
\
I
,j\
\
/ . .._ ..~~~_-,_-_._ 10.” ic,”
10
I\
\
,-.,yy.-‘-;q .i‘.t*
‘\\
\
I% \ ‘.
. 10
.“(
‘0
;”
..,...
\\ \\
-$-pi xwl!~.
*
1
y’ .ncls,
‘y;> I ;
Q
Fig. 1. Cumulated fluxes calculated for a spinning flat plate at I a.~. whose spin axis points to North ecliptic pole. The sum of all conlributions corresponds to the measured flux of the rnterplanctary Dust Model (Griin et ul., 1985)
interplanetary meteoroids of different optical properties has the effect that the mean optical propertics vary with heiiocentric distance as required by the analysis of zodiacal light and thermal emission (Levasseur-Regourd rt uz., 1991). Figure 1 shows the contributions of the various populations to the measured flux at 1 a.~. (Grim ef al., 1985). It shows the mass characteristics of the diffcrcnt populations, e.g. the core population contributes to most of the mass range extending over about 10 orders of magnitude. The ccccntric population dominates the small particles while the asteroidal population contributes most to the larger ones. The contributions of the halo and the inclined populations are negligible in this case because the measurements took place near the Earth’s orbit, where thcsc two populations make only small contributions; they are dominant in the outer and inner solar system, rcspcctively. Figures 2-4 show a comparison of the Divine model with zodiacal light measurements, both at varying elongations, latitudes and heliocentric distances. The measured zodiacal light intensities and the sum of the intensities of the populations (the eccentric population is negligible for the zodiacal light and, therefore, not dis-
0
20
40
60
80
100
120
140
160
1
S&r tkmgatron ldeql Fig. 2. Zodiacal light measured with Helios at 1 a.~. (Leinert et al., 1981). Comparison between the populations and the measured intensities (X) at ecliptic latitude 16.2”
P. Staubach
et LII.: Temperatures
of zodiacal dust
1101
to the ecliptic plane is negligible with respect to the accuracy of the model and will not be taken into account.
Fig. 3. Zodiacal light measured from Earth (L~~asseur-Regourd and Dumont. 1980). CoI~l~~risoll between the populations and the modified (factor 0.913) intensities (X) at SObdr elongation 90
played) show the similarity of the model predicted values and the measured data. Outside about 3 a.~., measurements of zodiacal light are highly contaminated by background light, the effect of which is difficult to assess; therefore. the fit between the model predictions and the measurements is not satisfactory.
Intensities of the zodiacal emission were measured at four ditrerent wavelengths, at 12, 25.60 and 100 pm. The data were obtained from diagrams shown by Hauser cl ot. (1984). The ~incertainty of the ZRAS data consists of an error in the zero-point calibration and in not-calibrated low-frequency effects (25%). At the larger wavelength 60 and 100 /Lrn the influence of the interstellar emission is not negligible [see Dumont and Levasseur-Regourd (19X&)]. Therefore, the uncertainty is higher at these wavelengths. For these reasons the weights of those data (60 and 100 krm) in the model were reduced by a factor of IO compared with the weights of the other data. Although the factor of IO was used to reduce the influence of data points which have large uncertainties, the model should also match these data points. Data from Deul and Wolstencroft (1988) who used a special selection of the ZRAS zodiacal observation history file show a discrepancy with the other data of about a factor of IO. The reason for that disagreement is unknown and therefore this data set is not used in this study.
3. Measurements of the zodiacal thermal emission In order to determine particle temperatures. measurements of the zodiacal thermal emission are needed. Observations of this emission at infrared wavelengths have been made with balloons (e.g. Salama et ul.. 1987). rockets (e.g. Soifer CJ~ ul., 1973 ; Briotta et N/.. 1976 ; Murdock and Price, 1985) and satellites [e.g. Hauser er ul. (I 984)]. Because of their superior quality and quantity, we will use only the intensities measured with the IR satellite IRAS (Hauser et al., 1984) and with ZIP rocket flights (Murdock and Price, 1985). All intensities as a function of solar elongation, ecliptic latitude and wavelength were converted to the quantity nZ, (with unit W m ’ sr ‘). The fact that the symmetry plane of zodiacal emission is about I .5 inclined
3.2. The ZIP dutu set The Zodiacal Infrared Project ZIP consisted of two rocket flights, during which the intensities of the zodiacal emission were measured at different wavelengths between 2 and 30 pm. The intensity values were extracted from diagrams of Murdock and Price (1985). Because of the bigger display of the diagrams, some data from the ZIP cxperiment were taken from Temi et ul. (1989). Errors caused by reading the data from the diagrams are estimated to be smaller than the uncertainty in the actual measurement.
4. Method of temperature calculations c
For the evaluation of the intensity of the thermal emission the Planck function (Divine, 1992j.f; is needed :
with : c2
x = %T
Fig. 4. Zodiacal light at different heliocentric distances [Hanncr of al., 1974; Levasseur-Regourd and Dumont, 1980 (modified)] at ecliptic 0 and solar elongation 130’. Comparison between measured intensities (X) and the populations
and
c2=-.
hc k
I; di_ is the fraction of the energy of a blackbody radiated .in the interval of wavelength [&n+djL]. The intensity that reaches the observer at his position .s2 is an integral along the line-of-sight s of the sum over the contributions from all particle populations :
The factor I .‘n is a result of the fact that an obser\,er can only see hal’f of the surface of a spherical particle and the direction of radiation is pointed away normal to the surface. 71~’ is the geometrical cross-section in which (I is the radius of the particle. 0 if the Stc~in~Roltzmann constant. Here the dif-ferential number conccntrntion N,,, has to be used [cf Divine ( I993)]. The evaluation technique for this integral is shown in the Appendix. The thermal emission of ;I particle is ;I function of its heliocentric distance and depends on its composition. its structure and G/e. If the particle diameter is bigger than 10% of the wavclcngth, then the emissivity J: of the particles is set to one, otherwise I: =O. With this approximation WC took into account the wavelength depcndcncc of the absorption cfliciencies for dill‘erent particle si/es typical for absorbing materials. Particle concentrations along the line-of-sight for each measurement of the zodiacal emission were calculated. Temperatures ofparticles ofdiameter IO ‘. IO i and IO ” m were estimated and \~~lues for intermediate t~~asses were logarithmically interpolated. WC further evaluate these temperatures at two diffcrcnt heliocentric distances and LISC logarithmic interpolations in between. For IRAS the line-of-sight is dircctcd towards the outet solar system : therefore, the points of support are chosen at I and 3 ~1.~1. For the ZIP data. the heliocentric distances were wised because the solar elongation of 0.3 and 3 ‘I.LI. ’ the data is in the range between 4 and IX0 During the evaluation ofthc line-of-sight integral. contributions from intervals between assumed points along the line-of-sight are summed. The contribution from each interval must bc smaller than 10% of the total. otherwise this interval is split until the requirement is satisfied.
The iRAS and the ZIP data set were evaluated separately in order to compare potential systematic differences between the two data sets. The iteration of the particle temperatures was made with the help of some algorithms of the “Five Populations of Interplanetary Meteoroids” model. This model only has the ability to iterate one population at :I time. The core population is the dominant population for the /odiacal thermal emission. because it contributes most to the measured intcnsitics. Therefore. in the first run only the particle tcmpcraturcs of the core population arc itcrated. The agreement bctwccn the data and the model prcdicted values is described by mean and r.m.s. residuals [cf Divine (l993)]. The residuals arc diffcrcnccs of logarithmic quantities. For example ;I r.m.s. residual of0.23 mans that the discrepancy between data and model is 21 factor of 2. During an iteration the RMS residual is reduced. When a minimum is t-cached the intensities and the particle temperatures of the core population are stored. Then. another population is taken into account assuming the same temperatures. Contributions from this
population to the thermal emission are calculated and also stored. After performin, ~7these calculations for aII populations, the tcmpcraturcs of the cm-c population arc iterated again but this time by also taking into account the storod intcnsitics from the other populations which are added to those of the core population. These steps are repeated until the RMS residual for the sum of the intensities of al1 populations togcthcr is minimizcd. This pt-occd~~~-c ass~mus that thcrc arc no systcmatic diffcronccs in the thermal propcrtics of particles from diffcrcnt populations.
5. Contributions of the populations to the thermal emission
For the lRA.5’ data set the mean and the r.m.s. residuals arc + 0.02 I5 and 0.065 I. respectively. This is better than the residuals for the other data sets individually and indcates that there is a very good match between the model intensities and the IR.ASdata set. The residuals of the ZIP data set are not as good as for the IRAS data set. The mean and RMS residuals are + 0.0524 and 0.1934. Figure 5 shows the best fit of the complctc ZIP data st‘t. At small solar elongations the model predicted values arc systematically higher than the measured data. but at large solar elongations it is just the opposite. An omission ol the small solar elongations leads to better residuals. \vhcreas the residuals bccomc worst in the cast ofomitting large solar elongations. A similar effect is shown in the wavelength dependence. Here the omission of short wavclengths also leads to better residuals. Therefore. in addition to the complete ZIP data set ;I restricted data set is used with solar elongations greater than 63 and wavelength greater than IO /[tn. For this restricted data set the residuals are -0.0014 and 0.0X03. This indicates that the model predicted values match the restricted ZIP data almost as well as the IRAS data do. Additionally. the residuals of sin,@ data points now show no significant dependence on wavelength. solar elongation or ecliptic latitude. A possible explanation for the mismatch is the interferin g inlluence of the Sun on the
rnc~~sttrenl~~i~s themselves. because the e%zct is biggest at small solar elongation. Tile reason may bc either badly subtracted zodiacal tight or instrumental stray tight. Figures 6 to X show a representative selection of model fits to the rcslricted ZIP data set. It is rcmarkabie that for differ-
/
.-
cnt wavelengths, different solar elongations and ecliptic latitudes the agreement between the model predicted values and the data is excellent. However. it should hc mentioned that the tetnperatures resulting from the fit with the complete ZIP data set arc not much different from lhose obtained from the fit with the restricted ZIP duta set.
For the I/L-IS data set, solar ~lon~~t~ions of rhc lincs-ofsight are close to 90 Tables I and 2 show the results of the modci calculations for the IRAS data set. Both :I comparison between the measured and the model predicted values and Ihc relative contributions (‘%I) from each population as well as the intcnsitics for each population arc displayed together with wavelength, solar elongation and ecliptic latitude. The resulting temperatures will be discussed in Section 6. The dominant population contributing to the IRAS zodiacal emission is the core population (Fig. 9). The majority of particles in the region of space (close to the ecliptic) which contributes most to the thermal emission belongs to the core p~~pltl~t~i(~n.Two significant effects itrc observed :
( I) F:or the longer wovelengths. cspeciaily for 60 and 100 /ml. the relative contributions of the cot-c and the inclined populations are decreasing bccausc the emissivitics ofthc smailcr particles are tower and small particles arc nol ncgligibie in these populations. For the same reason, the eccentric population sho\vs up only al the shortest wavelength of the fR.45’ data set. The asteroidal and the halo population show opposite el-fects because the asleroidal population especially consists of bigger particles. For the halo popttl~ti~~n this cfl’cct is superposed by its dcpcndcncc on latitude. (2) Vuriations of the latitude at fixed wavelength and fixed solar ciongation (cf Table 2) have ;I strong inlluence on the contributions from the dilferent populations. This is an rfTcct of the latitude dependence of the populations in the “Five Populations of Interplanetary Meteoroids” model. The concentration of the halo population is constant over the whole range of Intilttdcs. but Ihc concentralions of the other populations are decreasing with increasing latitude below the level of the halo population. For this reason the contribution of the halo population increases with increasing latitude. At small ~~tveien~ths the ~OrltribLtti~~n ofthe core population is larger than 70%. Variations of the solar ciongation have no significant influence on the contributions of the various populations, because the examined range ol solar clonpation is too small in order to accoi~nt for special efrecls in the IRAS data set. Both for Ihe total ZIP data set and for the restricted data set in which only solar elongations larger than 63’ and wavelengths longer than 10 him arc uscd. the cot-cand the asteroidal population are showing similar contributions as the lRAS data set at the same solar eiongation of 90 (Table 3). The eccentric popui~~~i~~nis dccrcasing with increasing \vavelcngth bccausc of the
1104
P. Slaubach c’rtrl. : Temperatures of zodiacal dust
Table 1. /RAS data extracted from Hauscr c’t trl. (1984) : comparison
Latit 1Kk
(&g) 12.00 25.00 60.00 100.00
91.1 91.1 91.1 YI.1
0.0 0.0 0.0 0.0
12.00 25.00 60.00
Yl.1 91.1 9I I
12.00 12.00 12.00 12.00 12.00
68.X 81.5 90.5 98.4 103.3
15.00 25.00 25.00 25.00 25.00
68.8 81.5 90.5 98.4 103.3
13.00 25.00 60.00 100.00
91.1 Y I. I 9 I. I 91.1
Mcasurcd values log j. J, (Wm ‘sr ‘)
Sum
‘sr
-4.89 -4.94 -5.84 -- 6.30
-4.Y3 ~ 4.90 - 5.45 -5.95
xx.0
- 5.35
-5
X8.0 88.0
- 5.40 -6.44
-5.34 -5.94
0.0 0.0 0.0 0.0 0.0
-4.6’) -4.83 -4.91 -4.97 -4.YY
0.0 0.0 0.0 0.0 0.0 -71.0 ~71.0 -71.0 -71.0
CWC
log i J, (Wm
bct~~cen the model predicted and the measured intensities
‘)
log i J, (Wm ‘SI ~ ~ _ -
5.00 5.01 5.64 6.20
‘)
Inclined log iJ, (Wm ?sr
‘)
Asteroidal log iJ, (Wm ‘sr ‘)
Halo log ;I, (Wm ‘SI
Eccentric log 7.1, (Wm ‘sr ‘)
-6.32 -6.47 - 7.78 -8.07
-6.33 -5.76 - 6.08 -6.53
-6.57 -6.15 - 6.40 -6.73
-6.34 -8.65 -9.48 ~ 10.07
- 5.49 -6.19
~ 6.29 - 6.44 - 7.25
-6.96 -6.73 -7.34
-6.56 -6.15 -6.39
-6.95 -9.31 - IO.17
-4.68 -4.83 -4.93 -5.01 ~ 5.05
-4.76 -4.8Y -4.99 - 5.07 -5.1 I
-5.75 -6.1 I -6.31 -6.45 -6.53
~ 6.23 - 6.28 - 6.32 - 6.36 -6.39
~ 6.50 -6.54 -6.56 -6.58 ~ 6.60
-6.12 -6.26 -6.34 - 6.40 ~ 6.44
-4.77 - 4.90 -4.95 -5.01 -4.99
-4.71 m-4.83 --4.YO -4.95 -4.98
~~4.x0 -4.93 ~- 5.00 ~ 5.06 ~ 5.09
~ 5.94 ~ 6.21 -6.46 -6.60 -6.67
-5.73 -5.14 - 5.76 -5.78 -5.79
-6.13 -6.14 -6.15 -6.16 -6.16
-x.47 -8.58 -8.65 -8.71 -8.74
- 5.30 -5.37 -6.41 -6.52
_ 5.37 ~ 5.33 ~ 5.92 -6.41
- 5.36 ~ 5.46 -6.17 -6.74
-62Y -6.44 ~ 7.25 -7.99
-6.94 -6.70 ~ 7.30 ~ 7.x3
- 6.56 -6.15 ~ 6.3Y -6.73
3y I ._
-5.3x
small particle sizes. while the contributions of the halo and asteroidal population are increasing. For a fixed wavelength (10.94 /lm) in the ecliptic plane, the dependence of the solar elongation shows a significant increase of the contribution for the inclined population with a peak at about 39’ elongation (Table 4). This is the direction in which the concentrations of the core and the inclined population are almost of the same magnitude [Divine (1993). Fig. 91. The same effect appears at the wavelength of 20.87 Inn at a similar solar elongation. In addition, the influence of the asteroidal population at larger solar elongations is stronger.
6. Particle temperatures As a result of the fit of the model predicted intensities with the measured values from the ZIP and the JRAS data sets, the particle temperatures were obtained for three different particle sizes. Temperatures were interpolated for particle diameters between these values. For particles with a diato cooler temmeter larger than 10 ’ m no extrapolation peratures was made, because it is assumed that these particles show no size efrects (e.g. due to light scattering). Figure IO shows the particle temperatures as a function of diameter at I a.u. heliocentric distance. The blackbody temperature for an isothermal particle is 279K. One can
-6.X’) m-Y.25 - IO.1 I - IO.70
see that smaller particles are hotter and big particles are cooler than the blackbody temperature. The reason is that the model algorithm needs higher temperatures to fit the measured intensities at different wavelengths in a satisfactory way. The gradients of the fRAS and the ZIP curves at varying particle sizes are probably a hint that the material composition may be different for different particle sizes. Rtiser and Staude (197X) obtained temperatures for different materials (graphite. magnetite. andesite, olivine and obsidian) from Mie calculations. Compared with those temperatures, the results of our model are approximately between the two materials graphite and obsidian. The reasons why the model predicted temperatures are lower than those of the highly absorbing graphite can be threefold : (I) the material of the dust particles does not absorb visible radiation as well as graphite, i.c. the albedo is bigger than 0.05 ; (2) the concentrations and the distributions of the model are describing more particles in the interplanetary space (perhaps a factor of 1.5). so the model will reduce the temperatures of the particles to lit the model predicted intensities to the experimental data ; and (3) additionally, the uncertainty of the IRAS and the ZIP data (see Subsections 3. I and 3.2) can be responsible
Table 2. lRAS data set extracted from Hauser et d. (1984) : relative contributions
of the tive populations
SOIX
Wavclenpth
elongation
Latitude
VI.1
91.1 91.1 91.1
Core (“‘a) - . x5.7 78.1 64.1 Pi:!
Inclined (%I _ 4.1 2.7 1.5 0.x
Asteroidal (%)
Halo (%I
Eccentric (%)
4.0 13,7 97 -_ .I7 76.5
1.3 5.6 Il.2 16.5
3.9 0.0 0.0 0.0
5.3
15.7 35.1
2.2 0.0 0.0
~.
12.00
VI.1
25.00
60.00
VI.1 91.1
X8.0 X8.0 X8.0
80.4 12.2 56.0
10.0 8.0 3.9
2.1 4.1 4.0
12.00 I3.00 12.00 13.00 12.00
6X.X 81.5 90.5 V8.4 103.3
0.0 0.0 0.0 0.0 0.0
X3.3 85.6 85.7 x5.4 XL.3
8.6 5.2 4. I 3.6 3.3
2.9 3.5 3.0 4.4 4.6
25.00 25.00 ?S.OO 7-5.00 35 00
6X.X 81.5 90.5
0.0 0.0 0.0 0.0 0.0
80.8 79.5 75.2 16.9 76.2
5.9 3.6 1.7 77 _.2.0
9.6 12.1 13.6 14.7 15.1
3.8 4.x 5.5 6.2 6.5
0.0 0.0 0.0 0.0 0.0
80.8 73.0 57.2 46.0
9.5 7.7 4.8 2.6
2.2 4.7 4.2 3.8
5.1 15.1 33.8 47.6
2.4 0.0 0.0 0.0
I..
1‘.OO -. ‘5 .00
60.00 1~O.O(~
V8.4
103.3 91.1 VI.1
VI.1 91.1
-71.0 -71.0 -71.0 -.-7 I .I1
for the te~lper~~ttlre level, but this effect of c~libr;lti~~ll error is probably less important compared with the reasons above. The ~oInp~rison between the IFUS and the restricted ZIP data sets (Fig. 10) shows the elTect that, especially for big particles. the temperatures calculated with the IRAS data are significantly higher than the temperatures calculated with ZIP data. This is not sllrpris~~l~ because the intensities measured with IRAS are about 50% higher than those obtained from ZIP (see Temi cutCI/., 19XY). There are two dill’erent effects which cause ~IIlcert~~inties of the particle temperatures. Firstly. there is the uncer-
Fig. 9. Comparison bctwccn the four measured wavelengths of the lR.4.5’ data set (A’) and the populations at ecliptic latitude 0 and solar elongation 9 I. I
3.6 3.7 3.9 4.0 4.1
tainty of the intensities in the data sets. The problems could be in the calibration, but also in the subtraction of the direct radiation which originates from the Sun. The total Lillc~rt~~inty of the measured intensities can be estimated to a value of about 500/; (Murdock and Price, 19X5; Hauser cut~1.. 1984). Because of I _ T’, the relative uncertainty of the temperatures : AT T
z
IA1 4 f’
(4)
is about 12.5%. Secondly, the uncertainty of the model is caused by the uncertainty of the distributions describing the populations. The distribution with the biggest effect on the temper~~tLlrcs is the radial distribLltioii which controls the spatial concentration ofparticles. If. for example, the radial distributions of all (or only of some) populations contain too many part&s, then the model tries to compensate for this by decreasing the temperatures to get the same intensity. For example an increase of the logarithmic concentration of the radial distribution by 0.2 (this is a fairly realistic uncert~~inty, estimated by Divine) cxuses ;1 decrease of the temperatures (depending on particle size) of between I1 and 35% for diameters of 10 -I and 10 ’ m, respectively. An additional source of uncertainty is the cmissivity of the particles, which is a function of wavelength and particle size. We assume an emissivity of I for particles bigger than one-tenth of the considered wavelength and zero for smaller particles. A variation of the value [e.g. 0.5-0.7, Ney and Merril (1976)] has some influence on the results of this model. Further assumptions are the particle density and the shape of the particles
L:ttiiudc (kg)
(‘arc
(/1m1
SOI elongation (&g)
1o.w IO.94
4.x 12.1
0.0 0.0
X7.5
IO.94 IO.94
1X.5 24.2
0.0 0.0
lO.cf‘t IO.93
30.0 39.5
0.0 0.0
66.7 64.0 S7.6
IO.94 IO.94 IO.94
46.X 51.6 62.9
0.0 0.0 0.0
IO.% IO.93
X0.6
w~l~~l~ll~tll
10.v4 IO.94 IO.94 IO.94 IO.94
1nclincd ( “4 )
Asteroid
fklh
Eccentric
(74)
( ‘:b)
( %,)
II.0 71.8
0.0 0.0
0.0 0. I
1.4
0.0 0.0 0. I 0. I
0.1 0.2 0.3 03
2.4 2.5 3.0 3.7
60.0 66.X 75.X
30.7 33.3 39.0 41.8 34. I 26. I 14.0
0.2.
0.8
4.9
0.3 0.S 0.7 0.7 I.1 I.1 1.1 I.1 I.1
I .o 1.5 2.0 1.3 4.0 4.1 4.2 4.3 4.3
5.8 73 7.4 7.X X.7 X.7 x.7 X.7 X.7
( ‘:;I) 76.2
53.9
0.0
x3.1
5.9
X7.1 140.3
0.0 0.0
X3.4 82.0
5.X 3.3
I SO.0
0.0
x2.x
3.3
159.7 169.4 I79.0
0.0 0.0 0.0
81.7 82.7 X2.7
_i?.L 3.1 3.1
(spheres in the model). If the density (3.5 g cm ‘) of tho particles for example is lower. Ihc radius of the particle with the same mass is k~rger: therefore. the gcomctric cross-section also increases. So, the particle temperatures
I.9
decrease to compensate the resulting higher intensities. The total uncertainty of the particle temperatures is the ci’ft’ct of various ~ont~.iblitions, therefore 40% is a fairiy realistic estimation. Particle temperatures and their radial dependence ii for the IRAS and the restricted ZIP data sets at-c shown in Table 5. The temperature dependence ci is increasing with decreasing particle size. The values are between 0.35 and 0.56. Thcsc results are compatible with the values published in the literature (cf the Introduction).
7. Conclusions The determination of the temperature of interplanetary particles with the “Five Populations of Interplanetary Meteoroids” model and with IRAS and ZIP observations of the zodiacal emission was a first attempt to combine a large variety of diKerent independent data sets on interplnnctary dust. The temperatures determined thus can now be used to make inferences on the particle materials. Temperatures calculated with the IRAS data are signilicantly higher than those calculated with ZIP data. The un~ert~~ii~&~in the partick tenipe~-~tllres is about 40%.
P. Staubach (‘I (I/. : Tcmpcratures
II07
of’zodiacal dust
Table 5. Particle tcmpcraturcs and radial tcmpcraturc dcpcndcncc (5 ax ;I function of hclioccntric distance and particle size for the IRAS and the restricted ZIP data set Particle si/c IO -I m IO ( 111 IO “m
IRAS
I 3.11.
lX7K 75?K -I_
37YK
lRA.5
3 i1.u.
IOYK l42K 2llK
New data lnay be added in the I‘uture to reduce the tempcraturc uncertainties by using, for example, measurements of the zodiacal thermal emission from the COBE satellite. Also, new data from the impact ionization detectors onboard Galileo and ~l~~.sse.s will provide meteoroid fluxes as well as velocity and directional information in the heliocentric range LIP to 5 ~.LL. to improve the orbit distributions of some meteoroid populations. especially. of the halo population.
il~.lilzolc,/cc!yr~?~~,f~/. WC thank A. Salama for his constructive commcnts which helped to improve this paper.
References
h,,< ,\
0.4Y 0.52 0.53
ZIP
0.3 a.11. 117K 3XXK 74lK
ZIP 3 21.~. IIIK IhYK ?55K
h,,,, 0.35 0.44 0.56
Walker, R. G., IRAS observations or the dilrusc infrared background. ./ls/r.o~. .I. 278, L I 5 L 17. I9 X4. Humes, D. H., Results of Pior~cjc~,IO and I I mctcoroid cxpcrimcnts : intcrplanctary and near-Saturn. J. ~/c~J/I,I~.s.RM. 85, 5X41-5852. 1980. KneiRel, B. and Mann, I., Spatial distribution and orbital propcrtics of zodiacal dust. in OI?
Briotta, D. A., Pipher, J. L. and Houck, J. R., Report No. AFGL-TR-76-0736, 1976. Deul, E. R. and Wolstencroft, R. D., A physical model for thermal emission from the zodiacal dust cloud. A.srro/r. A.>~roph,~~.s. 196,277 786. 19X8. Divine, N., Five populations of inttqlanetary meteoroids. JPL IOM 5317-Y?- 14/I 5. Jet Propulsion Laboratory. Pasadena. CA. I’)‘)?. Divine, N., Five populations 01‘ intcrplanctary meteoroids. J. qlcq/~~~s.Rcs., in pas. I YY3. Dumont, R. and Levasseur-Regourd, A.-C., Properties of intcrplanetary dust from infrared and optical observations. .+lsrror~. .-l.sf/~q~l~~~.s. 191, I 54 160. I9 XX. Giese, R. H. and Kneillel, B., Three-dimensional model 01‘ the Lodiacal dust cloud : compatibility of’ proposed infrared models. Icrrrrrs 81, 369. I 989. Good, J. C., IR.AS constraints on ;L cold cloud around the Solar System. A.ctroph~x. .J.. 350. 408. 1990. Griin, E., Pailer, Iv., Fechtig, H. and Kissel, J., Orbital and physical characteristics ofmicromctcoroids in the inner solar system ;I:, obscrvcd bq IIc~/io.s I Phwt. Spctw SrY.28. 333. IYXO. Griin, E., Zook, H. A., Fechtig, H. and Giese, R. H., Collisional balance of the meteoritic complex. IUUI(.S 62, 244 172. 1985. Griin, E., Fechtig, H., Hanncr, M. S., Kissel, J., Linblad, B. A., Linker%, D., Morfill, G. and Zook, H. A., /)I .sit~ explorations of dust in the solar system and initial results from the Grrlilco Dust Dctcctor. in Oi%jir~ mid E~~oIutiot~ of Itztc~r.~~l~~t~~~t~~t~~~ i2rr.c t (&ted by A. C. Levasscur-Regourd and H. Hascgawu), pp. 21 2X. Kluwcr. Dordrccht. IYOI. Appendix : evaluation technique for the integrals Hanner, M. S., The infrared rodiacal light, in Ol-;qi/l crr~tl EW/LlfiWl of’ If~tc,~p/tr/Lc~trLr:1. Dust (edited by A. C. Levasscur-Rcgourd and H. Hasegawa). pp. I39 146. In order to evaluate the integrals over mass the ditTercntial Kluwcr. Dordrccht, IYYI. number concentration N,,, has to bc expressed by the differential Hanner. M. S., Weinberg, J. L., DcShields, L. M., II, Green, mass distribution H,,, which is available for the particles ol B. A. and Tollcr, G. Iv., Zodiacal light and asteroid belt : the each population characterized by mass, heliocentric distance and view from P~OIIC~UIO. J. qcq~/~~~.s. Rm. 79, 367 I 3675. 1974. ecliptic latitude. H\, is the cumulative mass distribution which Hauser, M. G., Gillett, F. C., Low, F. J., Gautier, T. N., Beichis the ratio of the number of particles having muss 111> M man, C. A., Neugcbauer, G., Aumann, H. H., Baud, B., to those having I~I > I g and N, is the cumulative number Boggcss, N., Emerson, J. P., Houck, J. R., Soifer, B. T. and N,,, = ~ iN,,;?n~ concentration of particles having mass 171> M
ct al. : Tempcraturcs
P. Staubach
1108 and at heliocentric latitude 1.
position
r spccitied
N,,,=
by distance
r and ecliptic
( 1.v\, HI,, HM
(5)
With the density p the radius (I can be expressed way :
of zodiacal
(WZ’H,,,)c (m’ff,,,),
dust
(17)
with :
in the following
3m ‘I
u = Intensity
!1 47ly
I of the thermal emission
from a value element is : 6
=
log Q,, -log Q., lo&;,
Q., zz $4
PI>=
rrT-'f; ,p7s,
i
(21)
1 log fnh . at
rn =
n7~,.
(22)
at
m =
liih.
(23)
h
Substituting
Q = nl., 1m and laler ’
I, = N,,m
(10)
Q,, = Qnl”:
(I 1)
dm(m’H,,)Q
In the cxx of I;!+ ij - 21 < 0.05 the approximation bad and WCwill LISC :
will bccomc
(12) For masses nl,, and q,Q
Choose I g:
a power
and Q,) become :
nT”f; Pa = .,., I-.
Q,> = $2.
(13)
Qc = QP&,
Q,, = Q,,4:.
(14)
law for the mass
distribution
with
no, =
(16) Approximation
of (m’H,,) in the interval
[HI,,< 0~ < nz,,] :
= Q,,(m’H,,,),,mi, The last mass interval way :
’ In
[uI~. x] can be evaluated
lllh IJI,,
(26)
in the following
Pergamon
Temperatures of zodiacal dust Peter Staubach,’ Neil Divine 1,2and Eberhard Griin ’ ‘pax-Planck-rnstitut fur Kernphysik? P.O. Box 103980,69029 Heidelberg, Germany *Jet Propulsion Laboratory, Pasadena, CA 91109, U.S.A. Received for publication
17 December 1993
1. Introduction Interplanetary dust is finely divided particulate matter that exists between the planets. Reflection of sunlight from the myriad of interplanetary dust particles which are concentrated in the plane of the solar system causes the zodiacal light [for a recent review see Leinert and Griin (1990)]. Illterplanetary dust has been studied by a number of methods: radar meteor observations provide orbits of sub-millimetre particles (Sekanina and Southworth,
Covvespondence to : P. Staubach
1975) ; from lunar impact craters the size dist~bution has been established ranging from sub-micron to mil~m~tre sizes (Grim et al., 198s); modern spacecraft detectors determined the micrometeoroid flux from 0.3 to 20 a.u. distance from the Sun (Humes, 1980; Griin et al., 1991) ; and from zodiacal light observations, both from the ground and with spacecraft, the spatial distribution has been established (Giese and KneiBel, 1989 ; KneiBel and Mann, 1991). Recently, both spacecraft [1&4S, Hauser et al. (1984)] and rocket-borne [ZIP, Murdock and Price (1985)] observations of the thermal emission of interplanetary dust in the IR have become available. These observations constrain the spatial distribution of interplanetary dust (Hanner, 1991) and in comparison with zodiacal light observations, provide means to determine local variations of optical and thermal properties of interplanetary dust (Levasseur-Regourd et al., 1991). In order to obtain the temperatures of inte~lanetary dust grains different approaches have been taken: Deul and Wolstencroft (1988) and Rowan-Robinson et al. (1990) assume the radial temperature distribution T(v) N rF6, with 6 = 0.5, while Good (1990) finds 6 = 0.36. Rose, and Staude (1978), Reach (1988) andTemi et al. (1989) compute T(v) from Mie theory. The inversion of the zodiacal emission integral gives 6 = 0.33 front and Levasseur-Regourd, 1988). In this study, dust concentrations in different regions of space are calculated from Divine’s (1993) model, temperatures according to particle sizes and heliocentric distance have been assumed and IR intensities of the zodiacal thermal emission are evaluated. These model-predicted intensities are compared with measured intensities and, in the case of differences between both, the particle temperatures are changed until conformity is reached. In Section 2 the Divine model is introduced and key characteristics of the five populations of meteoroids are demonstrated. Data sets representing the zodiacal thermal emission are described in Section 3. The method and assumptions for the calculation of particle temperatures are presented in Section 4 and, finally, contributions of
1100
1’. Staubach er al. : Temperatures of zodiacal dust
the different meteoroid populations to the thermat emission are detcrmincd and resulting particle temperatures arc discussed in the fifth and sixth sections, respectively.
2. The Divine model Concentrations of interplanetary particles are determined from the “Five Populations of Interplanetary Mctcoroids” model of Divine (1993) which classifies interplanetary meteoroids into five populations. Each population has characteristic distributions of orbital clcmcnts, i.e. perihelion distance N,(r). ecccntricityp,(e), inclination pi(i) and of particlc mass H&z). In order to define these populations Divine used different observational data sets. Data from radar meteors (Sekanina and Southworth, 1975) were taken as we11as in situ in~asurements of dust impacts onboard scvcral space probes, for example Ue!y~su.s,~x~~~r~~ 16 and 23, HEOS2 and Pioneer 8 and 9 for data near the Earth’s orbit. The meteoroid size distribution derived from lunar cratcring records is described by the Interplanetary Flux Model [Griin et al. (1985), set Fig. 11. Data from the Helios spacecraft for distances less than 1 a.u. (Griin et al.: 1980) and data from the Piuneer IO and 11 spacecraft obtained in the outer Solar System (Humes et al., 1980) were used. Also data sets originated from dust detectors currently operating onboard the Galileo and Ulysses spacecraft (Griin et al., 1991) were taken. Zodiacal light data sets from space probes and ground-based observations (Leinert et al., I981 ; Levasseur-Rcgourd and Dumont, 1980 ; Hanner et al., 1974) were also used. The five meteoroid populations are briefly described in order of decreasing particle mass : (1) the asteroidal populations with a logarithmic mean mass of 10 ’ g, has small ecc~nt~~ities and inclinations, its concentration increases outward from 1 a.u. and has its maximum in the asteroid belt ; (2) the core population, of mean mass lo-’ g, has particles also on orbits of small eccentricity and inclination. It makes the main contribution to the zodiacal light, as well as to most other data sets; (3) the halo population, of mean mass 10 ’ g, exits mostly beyond 2.5 a.u. and has random inclinations (including retrograde orbits). Tt is required to provide the nearly uniform lluxcs detected in the outer solar system by Pioneer f 0, Ykmrer I 1 and C~iy.~~.s ; (4) the inclined population, of mean mass IO- ’ gt has particles on near-circular, moderately inclined orbits inside 1 a.u.; and (5) the eccentric population, having mean mass below 10 I2 g, exists mostly inside 1 a.u., has highly eccentric orbits and is rcquircd primarily to match the angular distribution of individual impact events on He&x.
For the core and eccentric population a geometric albedo of 0.05 allows a good match to the measured zodiacal light intensities, while for all other populations a lower value of 0.02 is required in order for these populations not to contribute too large intensities to the zodiacal light. The combination of the five populations of
.l? 14
‘,~
I
\
I
,j\
\
/ . .._ ..~~~_-,_-_._ 10.” ic,”
10
I\
\
,-.,yy.-‘-;q .i‘.t*
‘\\
\
I% \ ‘.
. 10
.“(
‘0
;”
..,...
\\ \\
-$-pi xwl!~.
*
1
y’ .ncls,
‘y;> I ;
Q
Fig. 1. Cumulated fluxes calculated for a spinning flat plate at I a.~. whose spin axis points to North ecliptic pole. The sum of all conlributions corresponds to the measured flux of the rnterplanctary Dust Model (Griin et ul., 1985)
interplanetary meteoroids of different optical properties has the effect that the mean optical propertics vary with heiiocentric distance as required by the analysis of zodiacal light and thermal emission (Levasseur-Regourd rt uz., 1991). Figure 1 shows the contributions of the various populations to the measured flux at 1 a.~. (Grim ef al., 1985). It shows the mass characteristics of the diffcrcnt populations, e.g. the core population contributes to most of the mass range extending over about 10 orders of magnitude. The ccccntric population dominates the small particles while the asteroidal population contributes most to the larger ones. The contributions of the halo and the inclined populations are negligible in this case because the measurements took place near the Earth’s orbit, where thcsc two populations make only small contributions; they are dominant in the outer and inner solar system, rcspcctively. Figures 2-4 show a comparison of the Divine model with zodiacal light measurements, both at varying elongations, latitudes and heliocentric distances. The measured zodiacal light intensities and the sum of the intensities of the populations (the eccentric population is negligible for the zodiacal light and, therefore, not dis-
0
20
40
60
80
100
120
140
160
1
S&r tkmgatron ldeql Fig. 2. Zodiacal light measured with Helios at 1 a.~. (Leinert et al., 1981). Comparison between the populations and the measured intensities (X) at ecliptic latitude 16.2”
P. Staubach
et LII.: Temperatures
of zodiacal dust
1101
to the ecliptic plane is negligible with respect to the accuracy of the model and will not be taken into account.
Fig. 3. Zodiacal light measured from Earth (L~~asseur-Regourd and Dumont. 1980). CoI~l~~risoll between the populations and the modified (factor 0.913) intensities (X) at SObdr elongation 90
played) show the similarity of the model predicted values and the measured data. Outside about 3 a.~., measurements of zodiacal light are highly contaminated by background light, the effect of which is difficult to assess; therefore. the fit between the model predictions and the measurements is not satisfactory.
Intensities of the zodiacal emission were measured at four ditrerent wavelengths, at 12, 25.60 and 100 pm. The data were obtained from diagrams shown by Hauser cl ot. (1984). The ~incertainty of the ZRAS data consists of an error in the zero-point calibration and in not-calibrated low-frequency effects (25%). At the larger wavelength 60 and 100 /Lrn the influence of the interstellar emission is not negligible [see Dumont and Levasseur-Regourd (19X&)]. Therefore, the uncertainty is higher at these wavelengths. For these reasons the weights of those data (60 and 100 krm) in the model were reduced by a factor of IO compared with the weights of the other data. Although the factor of IO was used to reduce the influence of data points which have large uncertainties, the model should also match these data points. Data from Deul and Wolstencroft (1988) who used a special selection of the ZRAS zodiacal observation history file show a discrepancy with the other data of about a factor of IO. The reason for that disagreement is unknown and therefore this data set is not used in this study.
3. Measurements of the zodiacal thermal emission In order to determine particle temperatures. measurements of the zodiacal thermal emission are needed. Observations of this emission at infrared wavelengths have been made with balloons (e.g. Salama et ul.. 1987). rockets (e.g. Soifer CJ~ ul., 1973 ; Briotta et N/.. 1976 ; Murdock and Price, 1985) and satellites [e.g. Hauser er ul. (I 984)]. Because of their superior quality and quantity, we will use only the intensities measured with the IR satellite IRAS (Hauser et al., 1984) and with ZIP rocket flights (Murdock and Price, 1985). All intensities as a function of solar elongation, ecliptic latitude and wavelength were converted to the quantity nZ, (with unit W m ’ sr ‘). The fact that the symmetry plane of zodiacal emission is about I .5 inclined
3.2. The ZIP dutu set The Zodiacal Infrared Project ZIP consisted of two rocket flights, during which the intensities of the zodiacal emission were measured at different wavelengths between 2 and 30 pm. The intensity values were extracted from diagrams of Murdock and Price (1985). Because of the bigger display of the diagrams, some data from the ZIP cxperiment were taken from Temi et ul. (1989). Errors caused by reading the data from the diagrams are estimated to be smaller than the uncertainty in the actual measurement.
4. Method of temperature calculations c
For the evaluation of the intensity of the thermal emission the Planck function (Divine, 1992j.f; is needed :
with : c2
x = %T
Fig. 4. Zodiacal light at different heliocentric distances [Hanncr of al., 1974; Levasseur-Regourd and Dumont, 1980 (modified)] at ecliptic 0 and solar elongation 130’. Comparison between measured intensities (X) and the populations
and
c2=-.
hc k
I; di_ is the fraction of the energy of a blackbody radiated .in the interval of wavelength [&n+djL]. The intensity that reaches the observer at his position .s2 is an integral along the line-of-sight s of the sum over the contributions from all particle populations :
The factor I .‘n is a result of the fact that an obser\,er can only see hal’f of the surface of a spherical particle and the direction of radiation is pointed away normal to the surface. 71~’ is the geometrical cross-section in which (I is the radius of the particle. 0 if the Stc~in~Roltzmann constant. Here the dif-ferential number conccntrntion N,,, has to be used [cf Divine ( I993)]. The evaluation technique for this integral is shown in the Appendix. The thermal emission of ;I particle is ;I function of its heliocentric distance and depends on its composition. its structure and G/e. If the particle diameter is bigger than 10% of the wavclcngth, then the emissivity J: of the particles is set to one, otherwise I: =O. With this approximation WC took into account the wavelength depcndcncc of the absorption cfliciencies for dill‘erent particle si/es typical for absorbing materials. Particle concentrations along the line-of-sight for each measurement of the zodiacal emission were calculated. Temperatures ofparticles ofdiameter IO ‘. IO i and IO ” m were estimated and \~~lues for intermediate t~~asses were logarithmically interpolated. WC further evaluate these temperatures at two diffcrcnt heliocentric distances and LISC logarithmic interpolations in between. For IRAS the line-of-sight is dircctcd towards the outet solar system : therefore, the points of support are chosen at I and 3 ~1.~1. For the ZIP data. the heliocentric distances were wised because the solar elongation of 0.3 and 3 ‘I.LI. ’ the data is in the range between 4 and IX0 During the evaluation ofthc line-of-sight integral. contributions from intervals between assumed points along the line-of-sight are summed. The contribution from each interval must bc smaller than 10% of the total. otherwise this interval is split until the requirement is satisfied.
The iRAS and the ZIP data set were evaluated separately in order to compare potential systematic differences between the two data sets. The iteration of the particle temperatures was made with the help of some algorithms of the “Five Populations of Interplanetary Meteoroids” model. This model only has the ability to iterate one population at :I time. The core population is the dominant population for the /odiacal thermal emission. because it contributes most to the measured intcnsitics. Therefore. in the first run only the particle tcmpcraturcs of the core population arc itcrated. The agreement bctwccn the data and the model prcdicted values is described by mean and r.m.s. residuals [cf Divine (l993)]. The residuals arc diffcrcnccs of logarithmic quantities. For example ;I r.m.s. residual of0.23 mans that the discrepancy between data and model is 21 factor of 2. During an iteration the RMS residual is reduced. When a minimum is t-cached the intensities and the particle temperatures of the core population are stored. Then. another population is taken into account assuming the same temperatures. Contributions from this
population to the thermal emission are calculated and also stored. After performin, ~7these calculations for aII populations, the tcmpcraturcs of the cm-c population arc iterated again but this time by also taking into account the storod intcnsitics from the other populations which are added to those of the core population. These steps are repeated until the RMS residual for the sum of the intensities of al1 populations togcthcr is minimizcd. This pt-occd~~~-c ass~mus that thcrc arc no systcmatic diffcronccs in the thermal propcrtics of particles from diffcrcnt populations.
5. Contributions of the populations to the thermal emission
For the lRA.5’ data set the mean and the r.m.s. residuals arc + 0.02 I5 and 0.065 I. respectively. This is better than the residuals for the other data sets individually and indcates that there is a very good match between the model intensities and the IR.ASdata set. The residuals of the ZIP data set are not as good as for the IRAS data set. The mean and RMS residuals are + 0.0524 and 0.1934. Figure 5 shows the best fit of the complctc ZIP data st‘t. At small solar elongations the model predicted values arc systematically higher than the measured data. but at large solar elongations it is just the opposite. An omission ol the small solar elongations leads to better residuals. \vhcreas the residuals bccomc worst in the cast ofomitting large solar elongations. A similar effect is shown in the wavelength dependence. Here the omission of short wavclengths also leads to better residuals. Therefore. in addition to the complete ZIP data set ;I restricted data set is used with solar elongations greater than 63 and wavelength greater than IO /[tn. For this restricted data set the residuals are -0.0014 and 0.0X03. This indicates that the model predicted values match the restricted ZIP data almost as well as the IRAS data do. Additionally. the residuals of sin,@ data points now show no significant dependence on wavelength. solar elongation or ecliptic latitude. A possible explanation for the mismatch is the interferin g inlluence of the Sun on the
rnc~~sttrenl~~i~s themselves. because the e%zct is biggest at small solar elongation. Tile reason may bc either badly subtracted zodiacal tight or instrumental stray tight. Figures 6 to X show a representative selection of model fits to the rcslricted ZIP data set. It is rcmarkabie that for differ-
/
.-
cnt wavelengths, different solar elongations and ecliptic latitudes the agreement between the model predicted values and the data is excellent. However. it should hc mentioned that the tetnperatures resulting from the fit with the complete ZIP data set arc not much different from lhose obtained from the fit with the restricted ZIP duta set.
For the I/L-IS data set, solar ~lon~~t~ions of rhc lincs-ofsight are close to 90 Tables I and 2 show the results of the modci calculations for the IRAS data set. Both :I comparison between the measured and the model predicted values and Ihc relative contributions (‘%I) from each population as well as the intcnsitics for each population arc displayed together with wavelength, solar elongation and ecliptic latitude. The resulting temperatures will be discussed in Section 6. The dominant population contributing to the IRAS zodiacal emission is the core population (Fig. 9). The majority of particles in the region of space (close to the ecliptic) which contributes most to the thermal emission belongs to the core p~~pltl~t~i(~n.Two significant effects itrc observed :
( I) F:or the longer wovelengths. cspeciaily for 60 and 100 /ml. the relative contributions of the cot-c and the inclined populations are decreasing bccausc the emissivitics ofthc smailcr particles are tower and small particles arc nol ncgligibie in these populations. For the same reason, the eccentric population sho\vs up only al the shortest wavelength of the fR.45’ data set. The asteroidal and the halo population show opposite el-fects because the asleroidal population especially consists of bigger particles. For the halo popttl~ti~~n this cfl’cct is superposed by its dcpcndcncc on latitude. (2) Vuriations of the latitude at fixed wavelength and fixed solar ciongation (cf Table 2) have ;I strong inlluence on the contributions from the dilferent populations. This is an rfTcct of the latitude dependence of the populations in the “Five Populations of Interplanetary Meteoroids” model. The concentration of the halo population is constant over the whole range of Intilttdcs. but Ihc concentralions of the other populations are decreasing with increasing latitude below the level of the halo population. For this reason the contribution of the halo population increases with increasing latitude. At small ~~tveien~ths the ~OrltribLtti~~n ofthe core population is larger than 70%. Variations of the solar ciongation have no significant influence on the contributions of the various populations, because the examined range ol solar clonpation is too small in order to accoi~nt for special efrecls in the IRAS data set. Both for Ihe total ZIP data set and for the restricted data set in which only solar elongations larger than 63’ and wavelengths longer than 10 him arc uscd. the cot-cand the asteroidal population are showing similar contributions as the lRAS data set at the same solar eiongation of 90 (Table 3). The eccentric popui~~~i~~nis dccrcasing with increasing \vavelcngth bccausc of the
1104
P. Slaubach c’rtrl. : Temperatures of zodiacal dust
Table 1. /RAS data extracted from Hauscr c’t trl. (1984) : comparison
Latit 1Kk
(&g) 12.00 25.00 60.00 100.00
91.1 91.1 91.1 YI.1
0.0 0.0 0.0 0.0
12.00 25.00 60.00
Yl.1 91.1 9I I
12.00 12.00 12.00 12.00 12.00
68.X 81.5 90.5 98.4 103.3
15.00 25.00 25.00 25.00 25.00
68.8 81.5 90.5 98.4 103.3
13.00 25.00 60.00 100.00
91.1 Y I. I 9 I. I 91.1
Mcasurcd values log j. J, (Wm ‘sr ‘)
Sum
‘sr
-4.89 -4.94 -5.84 -- 6.30
-4.Y3 ~ 4.90 - 5.45 -5.95
xx.0
- 5.35
-5
X8.0 88.0
- 5.40 -6.44
-5.34 -5.94
0.0 0.0 0.0 0.0 0.0
-4.6’) -4.83 -4.91 -4.97 -4.YY
0.0 0.0 0.0 0.0 0.0 -71.0 ~71.0 -71.0 -71.0
CWC
log i J, (Wm
bct~~cen the model predicted and the measured intensities
‘)
log i J, (Wm ‘SI ~ ~ _ -
5.00 5.01 5.64 6.20
‘)
Inclined log iJ, (Wm ?sr
‘)
Asteroidal log iJ, (Wm ‘sr ‘)
Halo log ;I, (Wm ‘SI
Eccentric log 7.1, (Wm ‘sr ‘)
-6.32 -6.47 - 7.78 -8.07
-6.33 -5.76 - 6.08 -6.53
-6.57 -6.15 - 6.40 -6.73
-6.34 -8.65 -9.48 ~ 10.07
- 5.49 -6.19
~ 6.29 - 6.44 - 7.25
-6.96 -6.73 -7.34
-6.56 -6.15 -6.39
-6.95 -9.31 - IO.17
-4.68 -4.83 -4.93 -5.01 ~ 5.05
-4.76 -4.8Y -4.99 - 5.07 -5.1 I
-5.75 -6.1 I -6.31 -6.45 -6.53
~ 6.23 - 6.28 - 6.32 - 6.36 -6.39
~ 6.50 -6.54 -6.56 -6.58 ~ 6.60
-6.12 -6.26 -6.34 - 6.40 ~ 6.44
-4.77 - 4.90 -4.95 -5.01 -4.99
-4.71 m-4.83 --4.YO -4.95 -4.98
~~4.x0 -4.93 ~- 5.00 ~ 5.06 ~ 5.09
~ 5.94 ~ 6.21 -6.46 -6.60 -6.67
-5.73 -5.14 - 5.76 -5.78 -5.79
-6.13 -6.14 -6.15 -6.16 -6.16
-x.47 -8.58 -8.65 -8.71 -8.74
- 5.30 -5.37 -6.41 -6.52
_ 5.37 ~ 5.33 ~ 5.92 -6.41
- 5.36 ~ 5.46 -6.17 -6.74
-62Y -6.44 ~ 7.25 -7.99
-6.94 -6.70 ~ 7.30 ~ 7.x3
- 6.56 -6.15 ~ 6.3Y -6.73
3y I ._
-5.3x
small particle sizes. while the contributions of the halo and asteroidal population are increasing. For a fixed wavelength (10.94 /lm) in the ecliptic plane, the dependence of the solar elongation shows a significant increase of the contribution for the inclined population with a peak at about 39’ elongation (Table 4). This is the direction in which the concentrations of the core and the inclined population are almost of the same magnitude [Divine (1993). Fig. 91. The same effect appears at the wavelength of 20.87 Inn at a similar solar elongation. In addition, the influence of the asteroidal population at larger solar elongations is stronger.
6. Particle temperatures As a result of the fit of the model predicted intensities with the measured values from the ZIP and the JRAS data sets, the particle temperatures were obtained for three different particle sizes. Temperatures were interpolated for particle diameters between these values. For particles with a diato cooler temmeter larger than 10 ’ m no extrapolation peratures was made, because it is assumed that these particles show no size efrects (e.g. due to light scattering). Figure IO shows the particle temperatures as a function of diameter at I a.u. heliocentric distance. The blackbody temperature for an isothermal particle is 279K. One can
-6.X’) m-Y.25 - IO.1 I - IO.70
see that smaller particles are hotter and big particles are cooler than the blackbody temperature. The reason is that the model algorithm needs higher temperatures to fit the measured intensities at different wavelengths in a satisfactory way. The gradients of the fRAS and the ZIP curves at varying particle sizes are probably a hint that the material composition may be different for different particle sizes. Rtiser and Staude (197X) obtained temperatures for different materials (graphite. magnetite. andesite, olivine and obsidian) from Mie calculations. Compared with those temperatures, the results of our model are approximately between the two materials graphite and obsidian. The reasons why the model predicted temperatures are lower than those of the highly absorbing graphite can be threefold : (I) the material of the dust particles does not absorb visible radiation as well as graphite, i.c. the albedo is bigger than 0.05 ; (2) the concentrations and the distributions of the model are describing more particles in the interplanetary space (perhaps a factor of 1.5). so the model will reduce the temperatures of the particles to lit the model predicted intensities to the experimental data ; and (3) additionally, the uncertainty of the IRAS and the ZIP data (see Subsections 3. I and 3.2) can be responsible
Table 2. lRAS data set extracted from Hauser et d. (1984) : relative contributions
of the tive populations
SOIX
Wavclenpth
elongation
Latitude
VI.1
91.1 91.1 91.1
Core (“‘a) - . x5.7 78.1 64.1 Pi:!
Inclined (%I _ 4.1 2.7 1.5 0.x
Asteroidal (%)
Halo (%I
Eccentric (%)
4.0 13,7 97 -_ .I7 76.5
1.3 5.6 Il.2 16.5
3.9 0.0 0.0 0.0
5.3
15.7 35.1
2.2 0.0 0.0
~.
12.00
VI.1
25.00
60.00
VI.1 91.1
X8.0 X8.0 X8.0
80.4 12.2 56.0
10.0 8.0 3.9
2.1 4.1 4.0
12.00 I3.00 12.00 13.00 12.00
6X.X 81.5 90.5 V8.4 103.3
0.0 0.0 0.0 0.0 0.0
X3.3 85.6 85.7 x5.4 XL.3
8.6 5.2 4. I 3.6 3.3
2.9 3.5 3.0 4.4 4.6
25.00 25.00 ?S.OO 7-5.00 35 00
6X.X 81.5 90.5
0.0 0.0 0.0 0.0 0.0
80.8 79.5 75.2 16.9 76.2
5.9 3.6 1.7 77 _.2.0
9.6 12.1 13.6 14.7 15.1
3.8 4.x 5.5 6.2 6.5
0.0 0.0 0.0 0.0 0.0
80.8 73.0 57.2 46.0
9.5 7.7 4.8 2.6
2.2 4.7 4.2 3.8
5.1 15.1 33.8 47.6
2.4 0.0 0.0 0.0
I..
1‘.OO -. ‘5 .00
60.00 1~O.O(~
V8.4
103.3 91.1 VI.1
VI.1 91.1
-71.0 -71.0 -71.0 -.-7 I .I1
for the te~lper~~ttlre level, but this effect of c~libr;lti~~ll error is probably less important compared with the reasons above. The ~oInp~rison between the IFUS and the restricted ZIP data sets (Fig. 10) shows the elTect that, especially for big particles. the temperatures calculated with the IRAS data are significantly higher than the temperatures calculated with ZIP data. This is not sllrpris~~l~ because the intensities measured with IRAS are about 50% higher than those obtained from ZIP (see Temi cutCI/., 19XY). There are two dill’erent effects which cause ~IIlcert~~inties of the particle temperatures. Firstly. there is the uncer-
Fig. 9. Comparison bctwccn the four measured wavelengths of the lR.4.5’ data set (A’) and the populations at ecliptic latitude 0 and solar elongation 9 I. I
3.6 3.7 3.9 4.0 4.1
tainty of the intensities in the data sets. The problems could be in the calibration, but also in the subtraction of the direct radiation which originates from the Sun. The total Lillc~rt~~inty of the measured intensities can be estimated to a value of about 500/; (Murdock and Price, 19X5; Hauser cut~1.. 1984). Because of I _ T’, the relative uncertainty of the temperatures : AT T
z
IA1 4 f’
(4)
is about 12.5%. Secondly, the uncertainty of the model is caused by the uncertainty of the distributions describing the populations. The distribution with the biggest effect on the temper~~tLlrcs is the radial distribLltioii which controls the spatial concentration ofparticles. If. for example, the radial distributions of all (or only of some) populations contain too many part&s, then the model tries to compensate for this by decreasing the temperatures to get the same intensity. For example an increase of the logarithmic concentration of the radial distribution by 0.2 (this is a fairly realistic uncert~~inty, estimated by Divine) cxuses ;1 decrease of the temperatures (depending on particle size) of between I1 and 35% for diameters of 10 -I and 10 ’ m, respectively. An additional source of uncertainty is the cmissivity of the particles, which is a function of wavelength and particle size. We assume an emissivity of I for particles bigger than one-tenth of the considered wavelength and zero for smaller particles. A variation of the value [e.g. 0.5-0.7, Ney and Merril (1976)] has some influence on the results of this model. Further assumptions are the particle density and the shape of the particles
L:ttiiudc (kg)
(‘arc
(/1m1
SOI elongation (&g)
1o.w IO.94
4.x 12.1
0.0 0.0
X7.5
IO.94 IO.94
1X.5 24.2
0.0 0.0
lO.cf‘t IO.93
30.0 39.5
0.0 0.0
66.7 64.0 S7.6
IO.94 IO.94 IO.94
46.X 51.6 62.9
0.0 0.0 0.0
IO.% IO.93
X0.6
w~l~~l~ll~tll
10.v4 IO.94 IO.94 IO.94 IO.94
1nclincd ( “4 )
Asteroid
fklh
Eccentric
(74)
( ‘:b)
( %,)
II.0 71.8
0.0 0.0
0.0 0. I
1.4
0.0 0.0 0. I 0. I
0.1 0.2 0.3 03
2.4 2.5 3.0 3.7
60.0 66.X 75.X
30.7 33.3 39.0 41.8 34. I 26. I 14.0
0.2.
0.8
4.9
0.3 0.S 0.7 0.7 I.1 I.1 1.1 I.1 I.1
I .o 1.5 2.0 1.3 4.0 4.1 4.2 4.3 4.3
5.8 73 7.4 7.X X.7 X.7 x.7 X.7 X.7
( ‘:;I) 76.2
53.9
0.0
x3.1
5.9
X7.1 140.3
0.0 0.0
X3.4 82.0
5.X 3.3
I SO.0
0.0
x2.x
3.3
159.7 169.4 I79.0
0.0 0.0 0.0
81.7 82.7 X2.7
_i?.L 3.1 3.1
(spheres in the model). If the density (3.5 g cm ‘) of tho particles for example is lower. Ihc radius of the particle with the same mass is k~rger: therefore. the gcomctric cross-section also increases. So, the particle temperatures
I.9
decrease to compensate the resulting higher intensities. The total uncertainty of the particle temperatures is the ci’ft’ct of various ~ont~.iblitions, therefore 40% is a fairiy realistic estimation. Particle temperatures and their radial dependence ii for the IRAS and the restricted ZIP data sets at-c shown in Table 5. The temperature dependence ci is increasing with decreasing particle size. The values are between 0.35 and 0.56. Thcsc results are compatible with the values published in the literature (cf the Introduction).
7. Conclusions The determination of the temperature of interplanetary particles with the “Five Populations of Interplanetary Meteoroids” model and with IRAS and ZIP observations of the zodiacal emission was a first attempt to combine a large variety of diKerent independent data sets on interplnnctary dust. The temperatures determined thus can now be used to make inferences on the particle materials. Temperatures calculated with the IRAS data are signilicantly higher than those calculated with ZIP data. The un~ert~~ii~&~in the partick tenipe~-~tllres is about 40%.
P. Staubach (‘I (I/. : Tcmpcratures
II07
of’zodiacal dust
Table 5. Particle tcmpcraturcs and radial tcmpcraturc dcpcndcncc (5 ax ;I function of hclioccntric distance and particle size for the IRAS and the restricted ZIP data set Particle si/c IO -I m IO ( 111 IO “m
IRAS
I 3.11.
lX7K 75?K -I_
37YK
lRA.5
3 i1.u.
IOYK l42K 2llK
New data lnay be added in the I‘uture to reduce the tempcraturc uncertainties by using, for example, measurements of the zodiacal thermal emission from the COBE satellite. Also, new data from the impact ionization detectors onboard Galileo and ~l~~.sse.s will provide meteoroid fluxes as well as velocity and directional information in the heliocentric range LIP to 5 ~.LL. to improve the orbit distributions of some meteoroid populations. especially. of the halo population.
il~.lilzolc,/cc!yr~?~~,f~/. WC thank A. Salama for his constructive commcnts which helped to improve this paper.
References
h,,< ,\
0.4Y 0.52 0.53
ZIP
0.3 a.11. 117K 3XXK 74lK
ZIP 3 21.~. IIIK IhYK ?55K
h,,,, 0.35 0.44 0.56
Walker, R. G., IRAS observations or the dilrusc infrared background. ./ls/r.o~. .I. 278, L I 5 L 17. I9 X4. Humes, D. H., Results of Pior~cjc~,IO and I I mctcoroid cxpcrimcnts : intcrplanctary and near-Saturn. J. ~/c~J/I,I~.s.RM. 85, 5X41-5852. 1980. KneiRel, B. and Mann, I., Spatial distribution and orbital propcrtics of zodiacal dust. in OI?
Briotta, D. A., Pipher, J. L. and Houck, J. R., Report No. AFGL-TR-76-0736, 1976. Deul, E. R. and Wolstencroft, R. D., A physical model for thermal emission from the zodiacal dust cloud. A.srro/r. A.>~roph,~~.s. 196,277 786. 19X8. Divine, N., Five populations of inttqlanetary meteoroids. JPL IOM 5317-Y?- 14/I 5. Jet Propulsion Laboratory. Pasadena. CA. I’)‘)?. Divine, N., Five populations 01‘ intcrplanctary meteoroids. J. qlcq/~~~s.Rcs., in pas. I YY3. Dumont, R. and Levasseur-Regourd, A.-C., Properties of intcrplanetary dust from infrared and optical observations. .+lsrror~. .-l.sf/~q~l~~~.s. 191, I 54 160. I9 XX. Giese, R. H. and Kneillel, B., Three-dimensional model 01‘ the Lodiacal dust cloud : compatibility of’ proposed infrared models. Icrrrrrs 81, 369. I 989. Good, J. C., IR.AS constraints on ;L cold cloud around the Solar System. A.ctroph~x. .J.. 350. 408. 1990. Griin, E., Pailer, Iv., Fechtig, H. and Kissel, J., Orbital and physical characteristics ofmicromctcoroids in the inner solar system ;I:, obscrvcd bq IIc~/io.s I Phwt. Spctw SrY.28. 333. IYXO. Griin, E., Zook, H. A., Fechtig, H. and Giese, R. H., Collisional balance of the meteoritic complex. IUUI(.S 62, 244 172. 1985. Griin, E., Fechtig, H., Hanncr, M. S., Kissel, J., Linblad, B. A., Linker%, D., Morfill, G. and Zook, H. A., /)I .sit~ explorations of dust in the solar system and initial results from the Grrlilco Dust Dctcctor. in Oi%jir~ mid E~~oIutiot~ of Itztc~r.~~l~~t~~~t~~t~~~ i2rr.c t (&ted by A. C. Levasscur-Regourd and H. Hascgawu), pp. 21 2X. Kluwcr. Dordrccht. IYOI. Appendix : evaluation technique for the integrals Hanner, M. S., The infrared rodiacal light, in Ol-;qi/l crr~tl EW/LlfiWl of’ If~tc,~p/tr/Lc~trLr:1. Dust (edited by A. C. Levasscur-Rcgourd and H. Hasegawa). pp. I39 146. In order to evaluate the integrals over mass the ditTercntial Kluwcr. Dordrccht, IYYI. number concentration N,,, has to bc expressed by the differential Hanner. M. S., Weinberg, J. L., DcShields, L. M., II, Green, mass distribution H,,, which is available for the particles ol B. A. and Tollcr, G. Iv., Zodiacal light and asteroid belt : the each population characterized by mass, heliocentric distance and view from P~OIIC~UIO. J. qcq~/~~~.s. Rm. 79, 367 I 3675. 1974. ecliptic latitude. H\, is the cumulative mass distribution which Hauser, M. G., Gillett, F. C., Low, F. J., Gautier, T. N., Beichis the ratio of the number of particles having muss 111> M man, C. A., Neugcbauer, G., Aumann, H. H., Baud, B., to those having I~I > I g and N, is the cumulative number Boggcss, N., Emerson, J. P., Houck, J. R., Soifer, B. T. and N,,, = ~ iN,,;?n~ concentration of particles having mass 171> M
ct al. : Tempcraturcs
P. Staubach
1108 and at heliocentric latitude 1.
position
r spccitied
N,,,=
by distance
r and ecliptic
( 1.v\, HI,, HM
(5)
With the density p the radius (I can be expressed way :
of zodiacal
(WZ’H,,,)c (m’ff,,,),
dust
(17)
with :
in the following
3m ‘I
u = Intensity
!1 47ly
I of the thermal emission
from a value element is : 6
=
log Q,, -log Q., lo&;,
Q., zz $4
PI>=
rrT-'f; ,p7s,
i
(21)
1 log fnh . at
rn =
n7~,.
(22)
at
m =
liih.
(23)
h
Substituting
Q = nl., 1m and laler ’
I, = N,,m
(10)
Q,, = Qnl”:
(I 1)
dm(m’H,,)Q
In the cxx of I;!+ ij - 21 < 0.05 the approximation bad and WCwill LISC :
will bccomc
(12) For masses nl,, and q,Q
Choose I g:
a power
and Q,) become :
nT”f; Pa = .,., I-.
Q,> = $2.
(13)
Qc = QP&,
Q,, = Q,,4:.
(14)
law for the mass
distribution
with
no, =
(16) Approximation
of (m’H,,) in the interval
[HI,,< 0~ < nz,,] :
= Q,,(m’H,,,),,mi, The last mass interval way :
’ In
[uI~. x] can be evaluated
lllh IJI,,
(26)
in the following