ISSA Explanatory Supplement
Zodiacal Dust Cloud Modeling Using IRAS Data
- Description of Model
- Fitting Procedure
Figure G.1 ZIP data (Murdock and Price 1985)
showing zodiacal brightness in the ecliptic plane as a function of
solar elongation. The dashed curve represents the predicted flux
(scaled by a factor of 1.5)
from the zodiacal dust cloud model.
The IRAS data are limited to solar elongation angles between 60° and 120° and consequently are not sensitive to material closer to the Sun than about 0.9 AU. However, comparison of the predicted model flux and Zodiacal Infrared Project (ZIP) data (Murdock and Price 1985) which looked to within 22° of the Sun at 10 and 20 µm shows excellent agreement in shape (Figure G.1), though there is a calibration scale discrepancy. It also implies that the r-1.8 power law is good to 0.4 AU. The inclination of the zodiacal dust cloud is 1.7° and its line of ascending nodes is at 69° ecliptic longitude, substantially different from the 3.4° and 87° deduced from the Helios measurements. However, since the Helios measurements were made between 0.3 and 1 AU and the IRAS data is primarily sensitive to material outside 0.9 AU, we attribute these differences to variation of the cloud symmetry plane with heliocentric distance.
The model presented here is based on the IRAS data as understood after the final calibration. Preliminary comparisons of IRAS data with the COBE-DIRBE data suggests that the IRAS gains and offsets require small change (§IV.D.3 and DIRBE Explanatory Supplement, 19 July 1993). Consequently, the physical parameters determined for the zodiacal dust cloud need reinterpretation. However, the purpose here is to represent accurately the variability of the infrared background and the current model does that quite well.
Figure G.2 The scanning geometry of the IRAS
satellite is illustrated. The scan coordinate system is defined
by the solar elongation angle , which
remains fixed for a scan, and the inclination angle i, which
changes at a constant rate.
The IRAS satellite was placed in a near-polar orbit of 99° inclination oriented with the orbital plane roughly normal to the Sun-Earth vector (Figure G.2). The orbit precessed through the year to maintain constant orientation to the Sun. Thus the nominal scan path during an orbit pointed directly away from the Earth and traced a line from ecliptic pole to ecliptic pole in a plane 90° away from the Sun. In practice, however, during any given half orbit (i.e., from one pole to the other) the satellite was tilted either toward or away from the Sun by varying amounts. It then swept out a cone on the sky at a constant angle from the Sun (the solar elongation angle) and with a constant azimuthal rate (3.84' s-1). The solar elongation varied by as much as ± 30° from normal but was usually within ± 10°.
The azimuth angle (referred to as the inclination angle) is arbitrarily defined to be -90° when the satellite passes the north ecliptic pole and increases in the direction of the scan. Since the descending part of the orbit occurred on the side of the Earth opposite the Earth's direction of motion (see Figure G.2), an inclination of 0° looks in the Earth's orbital plane back in the direction from which the Earth has come. Many of the coordinate angle references in this appendix (particularly on the plots) will be given in this solar elongation/inclination system.
Figure G.3 The IRAS bandpasses and a 200 K
blackbody, for reference.
The sky was observed through four wide bandpass filters, nominally centered at 12, 25, 60, and 100 µm (Figure G.3) (IRAS Explanatory Supplement 1988, §II.C). It is important for the modeling to use the exact bandpass shape, since temperature variations with heliocentric distance play an important role in the observed infrared flux. Consequently, fluxes will often be given in in-band Wm-2sr-1 and only converted to MJy sr-1 when appropriate.
Almost 6000 scans were made during the mission, about 1700 of which went from pole to pole. Of these, 200 scans were chosen which were representative of the range of solar elongations and which uniformly covered the time period of the mission. A typical scan is shown in Figure G.4.
The major portion of the flux seen at 12, 25, and 60 µm is due to the zodiacal emission we wish to model. The small local variations that are left are due to Galactic sources, which become dominant at 100 µm. It is important to note that these fluctuations are not noise (the noise is too small to show on these plots; the typical SNR is about 1,000). This proved to be a major problem in the fitting since it implied that the model should be fit to the local minima in a kind of "lower envelope" rather than the data as a whole. The method devised to handle this problem will be discussed in the section on fitting. At 100 µm there was almost no zodiacal emission on the sky that was not contaminated by a large amount of Galactic flux.
The variation of density out of the plane of the ecliptic is less clear. To fit Helios data, Leinert et al., (1978a) used a z-dependence of exp[-2.1(|z|/r)]. Collision models (Trulsen and Wikan 1980) indicate that exp[-r2] might be more appropriate. In this study the z-dependence is assumed to be of the form exp[-(|z|/r)] where and are free parameters.
In practice, we model not the density but the volumetric absorption cross-section (r,z). We will assume that the same functional form can be used, however, and complete our description of (r,z) with a reference value 0 =(R0 =1 AU,z=0).
The complete description of the volumetric absorption cross-section is then
The complete description of the temperature is
The complete description of our model for the emissivity is
= 0(/0)-1 > 0 (3)
To determine the cloud orientation parameters requires several scans spread out over the full time range of the mission (equivalently, the range of Earth orbital longitudes). In addition, to determine accurately the radial dependence of density and temperature requires the use of the full range of solar elongation angles. Finally, to constrain fully the temperature and to estimate the emissivity properties we must fit all four bands at once. Preliminary fitting with a subset of our model and of the data (Good, Hauser and Gautier 1986) gave credible values for those parameters fit but with much higher uncertainties than the present effort and with a poorer fit (further emphasizing the need for the full parameter set).
In addition to its inability to constrain all the model parameters, a single scan is contaminated by some unknown amount of Galactic light. It is impossible to separate out a smooth Galactic component in one scan, but if two scans covering the same position on the celestial sphere are observed at different times (i.e., through different amounts of the zodiacal dust cloud), fitting to both scans simultaneously will implicitly be sensitive to time variability, which can only be due to the cloud.
We are therefore forced to the conclusion that an accurate
derivation of the cloud parameters requires
simultaneous fitting to the full subset of scans described
above. The procedure used in the fitting is a) to
generate model estimates of all 200 scans using a given set
of model parameters, and b) to adjust the parameters
using the method of least squares
until the best fit is achieved.
The flux integral along any line-of-sight is given by
Fo (, ,t) = minmax Ro() 0linfinity (rc,zc)B(T(R)) dl d (4)
where and are the elongation and inclination of the observation, (r,z) and T(R) are as described above [with (rc, zc in cloud coordinates, not ecliptic)]. B(T) is the Planck function, Ro() is the spectral response of the detector/filter combination with nominal wavelengths 0= (12, 25, 60 or 100 µm) and l is the unit vector in the direction (, ) at time t. Positions in (rc, zc)-space are calculated from (l, , ) taking into account the orbital position of the Earth at time t (including eccentricity of the Earth's orbit) and the orientation of the dust cloud.
To generate the model scans, the flux from the model cloud was integrated along several lines of sight and over the bandpass. Thirteen reference points were used for each scan, spread out between inclination angles -90° and +90° but concentrated toward the ecliptic plane where the variation was most extreme. The resultant flux for the reference points was interpolated, using a cubic spline under tension, to give model fluxes for each of the real data points. The difference between this interpolated function and a full flux integration is typically less than 0.05%.
Each parameter in the model (the thirteen described above) was then perturbed slightly. The variations of the model with respect to these perturbations and the differences between the nominal model and the real data were then combined to generate updates to the estimated model parameters. This procedure was iterated until the parameters converged.
As mentioned previously, the fluctuations above the zodiacal background are not noise but Galactic structure. Consequently, those points that are strong positive excursions from the model (when the model has become reasonably accurate) should be given a very low weight to exclude the galaxy and the zodiacal dust bands. Such a weighting can be incorporated into the least-squares process since an uncertainty for each point is part of the scheme. Without this "lower envelope" approach the fit would be biased, especially at the longer wavelengths, by the Galactic emission.Table G.1 and the coefficients of correlation between the parameters are shown in Table G.2. Representative scans (for several elongations and times) are shown in Figures G.4, G.5 and G.6. Considering the number of parameters, even the highest of the correlations (0.91, between the power law exponents on and T) is quite small. We therefore conclude that the inclusion of all the free parameters in our model is justified and, moreover, that they are all required to fit the data properly.
|Class of Parameter||Values of Parameters||Units|
|Density|| 0=1.439±0.004 × 10-20
|Temperature||T0 = 266.20± 0.18
|Emissivity|| 0=37.75± 0.09
|Orientation|| O=68.61 ±0.03
|Offsets||12 µm=35.53 ±0.15× 10-8
25 µm=49.97± 0.14× 10-8
60 µm=2.19 ± 0.03× 10-8
100 µm=5.24± 0.07× 10-8
Figure G.4 A typical IRAS pole-to-pole scan
with the zodiacal dust cloud model fit (solid line).
This scan was at a solar elongation of 90°. The
zodiacal dust bands are visible as bumps at the ecliptic plane
(inclination 180°) and at ± ~ 10°. The
remaining structure is Galactic emission.
Figure G.5 Data and zodiacal dust cloud
model prediction for a scan
at solar elongation of 112°.
The profiles are broader and less intense than those in Figure
G.4 since this scan is looking through material farther from the Sun,
which is both cooler and has a larger z scale height.
Figure G.6 Data and zodiacal dust cloud model
prediction for a scan
at solar elongation of 67°.
The same discrepancy noted in Figure G.5 exists for these data.
These profiles are narrower and more intense since this scan is
observing material closer to the Sun.
The complete volumetric emissivity distribution description is given by
The emissivity properties of the dust material are extremely uncertain, although their overall emissivity is probably quite high. Models of the absorption/emission behavior for various materials (Roser and Staude 1978) show that several likely candidates for the dust (eg., olivine, obsidian) have fairly flat (though very uneven) emission properties between about 10 and 30 µm but then drop off as -n (n=1-3) out to beyond 100 µm. Our approximation is very crude, but the results of our fit require a decreasing emissivity at long wavelength and relatively flat emissivity between 10 and 30 µm. The large width of the IRAS filters precludes finer analysis of the composition of the dust.
ReferencesBriggs, R.E. 1962, Astron. J., 67, 710.
Misconi, N.Y. 1980, "The Symmetry Plane of the Zodiacal Cloud Near 1 AU", in Solid Particles in the Solar System, eds. I. Halliday and B.A. Mclntosh, (Reidel:Dordrecht), 49.
Roser, S., and H.J. Staude 1978, Astron. Astrophys., 87, 381.
Trulsen, J., and A. Wikan 1980, Astron. Astrophys., 91, 155.