Zodiacal light

  

Figure: Annual variation of the brightness of the zodiacal light, as predicted by the DIRBE zodiacal light model for 12 $\mu$m wavelength. The three curves are for ecliptic latitudes 0$^\circ$ (upper curve), 30$^\circ$, and 60$^\circ$ (lower curve); the ecliptic longitude was 0$^\circ$ for all three curves. The part of the year during which each position is within the solar elongation range visible to Spitzer (80-120$^\circ$) is shown with solid curves; the rest of the year is dashed. An observer would use information like this in order to see the range of zodiacal light that will be present in his observation.

The absolute brightness of the zodiacal light will be taken from a three-dimensional model of the distribution of interplanetary dust and its scattering and thermal emission. The model was normalized to the time-variation of the sky brightness observed by the Cosmic Background Explorer (COBE) Diffuse Infrared Background Experiment (DIRBE) in 1989-90. The absolute calibration of DIRBE is very secure, and its gain stability was sufficient to allow it to monitor the temporal variation of the sky brightness in each 0.7-degree beamwidth. The temporal variation of the brightness of a fixed celestial position was due to the changing viewing angle through the Solar System dust cloud, which it is a unique signature of the Zodiacal Light. The range of solar elongation (angle between the line of sight and the Sun) was 64-124$^\circ$ for DIRBE, which is essentially the same as that planned for Spitzer. Therefore, the DIRBE observations sample the essentially the same parts of the Solar System that Spitzer will. The wavelength coverage of DIRBE was 1.25-240 $\mu$m, which spans the wavelength range of Spitzer. For comparison, the previous best estimator was the zodiacal light model fitted to the IRAS data (Good 1994). Differences between the COBE and IRAS zodiacal light models are certainly present, and they are largely attributed to the absolute calibration errors in the IRAS database. The IRAS absolute calibration for diffuse emission used offsets (as a function of time) were forced to match a particular model of the zodiacal light brightness of the ecliptic pole. The actual sky brightness measured by COBE is different, and it has an annual variation with a different phase than the IRAS calibration model. The calibration differences are always less than 20%, and for the purposes of background estimation for proposal planning, the IRAS data and model are generally adequate over their range of validity (12-100 $\mu$m).

The DIRBE zodiacal light model consists of three distinct components: a smooth dust cloud that extends over most of the Solar System, dust bands that extend from the asteroid belt to the Sun due to dust from the asteroid families, and a dust ring around the Sun at 1 AU due to particles in orbital resonance with the Earth. (Dust trails from short-period comets are bright but have a very low filling factor, so the probability of accidentally seeing one is small.) At each position, the particles are presumed to emit a modified blackbody spectrum, with free emissivities in each band (except for the normalization to unity at 25 $\mu$m). The scattering was modeled using a phase function and free albedos at 1.25-3.5 $\mu$m. Scattering is negligible at all Spitzer wavelengths except the shortest IRAC filters, where it will produce up to half the zodiacal brightness. The brightness for a time and celestial coordinates is calculated by integrating the model along the line of sight numerically. Preliminary versions of the model are described in some papers (Reach et al. 1995, 1996), and the final version is described by Kelsall et al. (1998).

  

Figure: Emissivity (solid curve, left axis) and albedo (dashed curve, right axis) in the zodiacal light model used for the Spitzer background estimator.

For the Spitzer/Spot zodiacal light estimator, we use the DIRBE model for the interplanetary cloud. We had to extend this model to deal with arbitraty wavelengths rather than the fixed 10 DIRBE wavelengths. The DIRBE model for thermal emission takes an assumed blackbody kernel (at each path length element along the line of sight), applies a color correction (again, at each path lenegth element), and then has a free parameter to scale the resulting brightness. This parameter is the "emissivity," and it was measured at 3.5, 4.9, 12, 25, 60, 100, 140, and 240 $\mu$m. We made a smooth fit of an analytic function for the emissivity, whcih is shown in Figure 2. The DIRBE model for scattered light takes the solar spectrum times the phase function times a color correction times a free parameter which is the "albedo." For our purposes, we removed the color correction, and we linearly interpolate between the albedos at 1.25, 2.2, and 3.5 microns. We set the albedo to zero at 4.9 microns and longer, and we interpolate between the 0.21 and 0 for wavelengths between 3.5 and 4.9 $\mu$m; the resulting values are shown in Figure 2. For the phase function, the DIRBE model has a functional form with three free parameters at three wavelengths (1.25, 2.2, and 3.5 $\mu$m). We found that using the same functional form but interpolating the parameter values as a function of wavelength yielded unacceptable results. Therefore, we evaluate the phase function appropriate for the specified scattering angle at each of the DIRBE wavelengths, then we interpolate the phase functions linearly as a function of wavelength. At wavelengths longer than 3.5 $\mu$m we adopt the 3.5 $\mu$m phase function. For completeness, we also included the visible-light phase function from Hong at 0.55 $\mu$m, and we interpolate the albedo and phase function as if it were another DIRBE wavelength. Due to these changes, the model in Spot will not match the Kelsall et al. (1998) model precisely, even when evaluated at the same wavelength, coordinates, and time.

One lien against the implementation of the zodiacal light model for Spot is that we have not taken into account the fact that Spitzer will be rather far from the Earth. Instead, we start integration through the cloud at the Earth. Because the Spitzer orbit is Earth-trailing with a semi-major axis near 1 AU, the distribution of zodiacal light brightness will be similar as seen from Spitzer to what we see from Earth. The most important difference is that the date that we give to the zodiacal light estimator should really be the "Spitzer date" which lags from the Earth date by approximately 6 days per year into the mission (using a 0.1 AU per year drift rate of Spitzer from the Earth). The worst errors will be incurred in the ecliptic, and for observations later into the mission, by which time we hope to upgrade the model. For the first year of the mission, the maximum error in the effective observing date is 6 days, which leads to an error in the background brightness of as much as 10% in the ecliptic plane (and less at higher latitudes). A second effect that will cause our background estimate to be inaccurate is that lines of sight from Spitzer will travel through significantly different parts of the Earth's resonant dust-ring. The maximum error due to this effect is less than 5%. In a future release, we will need to start the integration from the location of Spitzer at the time of observation.