IRAS Diameters and Albedos Revisited

Russell G. Walker

Monterey Institute for Research in Astronomy (MIRA)

200 Eighth Street Marina, CA 93933

Abstract.

A broadband variant of the Near Earth Asteroid Thermal Model proposed by A. Harris (1998) is applied to the observed IRAS asteroid 12 and 25 micron fluxes (IMPS database). The method allows separation of the three variables: the thermal beaming parameter, h, the wavelength independent emissivity, e, and the albedo, p_v, and permits direct estimation of these parameters. The key is that the sub-solar temperature on the asteroid surface is uniquely determined for any assumed surface temperature distribution from the ratio of the IRAS 12 and 25 micron fluxes.

Diameters derived from the model fluxes assuming e = 1.0 are compared to 54 occultation diameters and yield an average infrared emissivity e = 0.793 with a 1s uncertainty of 0.007. Using this value of e, the diameters, albedos, and thermal beaming parameters are derived for each asteroid using all 3336 observations of 654 asteroids with multiple observations at SNR > 10 in the 12 and 25 micron bands of IRAS. The mean value of h = 1.067 with a 1s uncertainty of 0.087 for asteroids of all classes represented in the dataset. There was no significant difference of h between classes C, S, F, M, K, and D. The resulting diameters are on average only about 1.4% larger than the IMPS diameters, however, the derived albedos average 11% smaller.

The paper presents the methods used to obtain the above results, as well as the new diameters, albedos, and beaming parameters for 654 asteroids. These data are available online at the MIRA website, http://www.mira.org.

Introduction

The standard thermal model (STM) (Lebofsky, et al 1986, Lebofsky and Spencer, 1989) was used to estimate diameters and albedos for 1884 asteroids detected in the IRAS sky survey (see IRAS Minor Planet Survey, IMPS, Tedesco, 1992). The STM calculates the flux from a non-rotating asteroid at zero phase angle and adopts a phase coefficient of 0.01 mag./deg. of phase. It assumes a beaming parameter, h = 0.756 and an infrared emissivity, e = 0.90.

Harris (1998) uses a near-Earth asteroid thermal model (NEATM) that integrates over the surface of the asteroid at the phase angle observed. This assumes a Lambertian emission model and was developed to analyze the thermal spectra of near-Earth asteroids (NEAs). Adopting e = 0.90, and treating h as a free parameter he finds that h = 1.2 is a good average value for NEAs.

Walker and Cohen (2002), attempting to fit the IRAS low resolution spectra (LRS) of main-belt asteroids with the IRAS STM, found that STM produced spectra that were too blue, that is, representative of a surface temperature distribution hotter than was consistent with the observed LRS. Thermal models, such as the NEATM with h in the range of 1.0 to 1.2 best represented the observed LRS spectral shapes.

This paper explores h and e in a larger subset of observations from IMPS, separating these variables through the constraint imposed on e by the applicable set of occultation diameters.

Model and Procedure

In keeping with the STM it is assumed the surface temperature decreases from a maximum T_SS at the subsolar point to zero at the terminator as

T(w) = T_SS cos^1/4 w (1)

where w is the angular distance from the subsolar point at a temperature T_SS , determined by the solar energy balance at the surface and given by

T_SS = [(1 – q p_v) S / (e h s)]^1/4 (2)

where S is the incident solar flux, p_v the visual albedo, q the phase integral, e the wavelength independent infrared emissivity, h the “beaming factor”, and s the Stephan-Boltzman constant.

With these assumptions the asteroid irradiance F(θ) at phase angle θ, received at the Earth in the IRAS bands is given by

F(θ) = (r/Δ)² e òòò R_λ B_λ(w) cos α sin α dα dφ dλ (3)

where r is the radius of the asteroid, Δ the Earth-asteroid distance, α the angle with respect to the normal to the surface element viewed, φ the azimuth angle, λ the wavelength of the emitted radiation, R_λ the normalized spectral response of the IRAS spectral band, and B_λ the surface element radiance given by the Planck function. The integrations are to be performed over the appropriate ranges of the variables. The relation between the solar illumination angle w at the surface element and the phase of the observation θ is given by

cos w = cos α cos θ + sin α sin θ cos φ (4)

It is clear from eq. (3) that the ratio of fluxes in any two spectral bands is independent of the radius and emissivity of the asteroid. It is also clear that for a given surface temperature distribution, such as that given by eq. (1), the flux ratio uniquely defines T_SS.

The procedure was to first compute tables of the model flux in the IRAS 12mm and 25mm bands from a 1 km diameter asteroid with unit emissivity at a geocentric distance of 1 AU. The tables were calculated for 180K £ T_SS £ 450K in steps of 1K, and for 0^o £ θ £ 90^o in steps of 1^o. A table of the ratio of the fluxes in the IRAS 12mm band to those in the 25mm band was then constructed.

Asteroid observations at 12mm and 25mm were selected from IMPS. To be included in our analysis set the asteroid was required to have multiple observations at a signal to noise ratio SNR ³ 10 in both bands. These criteria returned 3336 two-band observations of 654 asteroids. The flux ratio and its uncertainty were calculated, and T_SS with its uncertainty found by interpolation in the flux-ratio table for the phase of the observation. The diameter of the asteroid (with e = 1.0) was then found from eq. (3) and the model flux table for T_SS. The above steps were repeated for each observation of that asteroid, and its mean diameter was calculated using inverse variance weighting. Diameters (e = 1.0) were determined in this way for all 654 asteroids. A search of the resulting set found 55 asteroids in common with the lists of occultation diameters given by Millis and Dunham (1989), and Dunham et al. (2003). Figure 1 is a plot of the derived thermal diameters versus the occultation diameters. The slope of the straight line passing through zero is just e^1/2.

Thus the observations require e = (0.8907 ± 0.0036)² = 0.793 ± 0.007 to reconcile the thermal and occultation diameters. The 12mm and 25mm flux tables were then multiplied by 0.793, and the thermal diameters re-calculated for all the 654 asteroids in the sample by repeating the above procedure. The result is plotted in figure 2 for the occultation diameters.

The diameter, d, when combined with the asteroid absolute magnitude, H, yielded the visual albedo ( see Fowler and Chillemi, 1992)

p_v = [1329 x 10^-H/5 / d]^1/2 (5)

and the “beaming parameter”, h, was then derived from eq. (2). The IMPS database was the source of the absolute magnitudes and phase integrals.

Results

Figure 3 compares the NEATM diameters with the IMPS diameters for the 654 asteroids considered. The differences are small, the NEATM diameters being only 1.4% larger on average. This is not the case for the albedos shown in figure 4, where the NEATM albedos are, on average, 11% smaller than the IMPS albedos. The formal fractional uncertainties in the NEATM diameters and albedos (Table 1) are about a factor of two smaller than those from the IMPS reductions.

Table 1. Fractional uncertainties in the derived diameters and albedos
	NEATM	IMPS
Diameter	0.030 ± 0.014	0.057 ± 0.031
Albedo	0.061 ± 0.029	0.130 ± 0.082

Figure 5 shows the overall distribution of h derived using the NEATM procedure. The mean value for the entire sample is h = 1.067 ± 0.087. It is of interest to note that there are only 4 asteroids in the 654 asteroid sample with h < 0.80 and only one with h £ 0.756, the canonical STM value. The variation of h with taxonomic class was investigated for the 185 asteroids in our sample that had IMPS taxonomic classifications. The results are given in Table 2 and it is clear that there is little, if any, significant variation of h for the spectral classes represented.

Table 2. Variation of beaming parameter, h, with taxonomic class.
CLASS	h	Number of Asteroids
All classes combined	1.067 ± 0.087	654
C	1.076 ± 0.085	88
S	1.072 ± 0.085	63
F	1.056 ± 0.043	5
M	1.072 ± 0.082	22
K	1.097 ± 0.019	4
D	1.092 ± 0.144	3

Conclusions

The NEATM is a simple model that is, not only applicable, but preferable to the STM (yielding smaller uncertainties) to derive albedos and diameters for main-belt asteroids, as well as, for the near-Earth population, whenever our lack of the knowledge of their physical parameters prevents us from using more refined thermophysical models. In the case where observations of the thermal flux is available in two or more spectral bands, the subsolar point temperature may be determined from the flux ratio, and the diameter and albedo deduced without knowledge of the beaming parameter. All that is required is a model of the surface temperature distribution and the absolute magnitude of the asteroid. In the case of observations in a single spectral band, an assumption of the mean value of h = 1.067 will enable calculation of T_SS and an estimation of the diameter and albedo.

The wavelength independent infrared emissivity of 0.793 has been derived from comparison of the NEATM derived diameters with their occultation diameters. This value is significantly smaller than the 0.90, typical of many natural dielectric materials and which works well for the Lunar surface. This may be indicative of asteroid regoliths with higher metallic content that are thinner and less mature than the Moon’s due to the higher flux of impactors in the main asteroid belt coupled with the asteroid’s lower surface gravity.

A database of the 654 asteroids that have been processed using the two-color NEATM model has been created. A sample of this listing is given in Table 3. The full set is available on the MIRA website. The NEATM technique could be extended in the future to a reprocessing of the full IMPS database of 1884 asteroids.

It is of interest to note in figures 1, 2, and 3 that the solution for the diameter of 1Ceres appears anomalous. All authors attempting to model its thermal emission spectrum and reconcile it with its apparent diameter and albedo have noted this.

Table 3. This is a sample of the NEATM database of diameters, albedos, and beaming parameters. The full database is available on the MIRA website, www.mira.org.
	ID	Name	Diam.	sdiam	pv	spv	h	sh	nobs
			(km)	(km)
	1	Ceres	774.2	12.2	.091	.003	.909	.051	6
	2	Pallas	512.4	8.0	.126	.004	1.044	.051	7
	3	Juno	248.2	3.3	.191	.005	1.118	.054	8
	5	Astraea	126.7	2.9	.197	.009	1.161	.096	3
	6	Hebe	196.0	3.2	.236	.008	1.129	.062	6
	7	Iris	200.4	3.3	.269	.009	1.066	.061	6
	8	Flora	144.0	2.3	.208	.007	1.121	.065	7
	10	Hygiea	406.2	5.2	.068	.002	1.013	.045	9
	11	Parthenope	148.9	2.9	.190	.007	1.008