Astron. Astrophys. 338, 340-352 (1998)

7. Discussion

The full thermophysical treatment allows us to understand the thermal behaviour and to explain the observations in our database. We tested the model also against additional preliminary IRTF and KAO data (C. Telesco, D. Osip, priv. communication) and ISOLWS ¹⁷ spectra (M. Burgdorf, priv. communication). All test cases confirmed the quality of the TPM on absolute and relative levels. We can now predict flux densities from mid-IR to submillimetre wavelengths for all 10 asteroids, including thermal and lightcurve effects. In that way we defined photometric standards for different photometric applications.

7.1. Accuracy of the individual standards

The best proof for the reliability of a theoretical model is the test against validated observations. We calculated the TPM flux for each data point in our database and compared observations and predictions one by one. We grouped afterwards the resulting ratios in certain wavelength bins (see Table 6) and analysed the weighted means and the weighted r.m.s. of . The results are summarized in Table 6. Our primary standards Ceres, Pallas, Vesta and Herculina show in the ISOPHOT far-IR range a r.m.s. of about . Since for the neighbouring regions similar r.m.s. are derived, we conclude that the TPM prediction for a given time will be better than . The scatter for the secondaries is larger, but in all cases, except Dione and Cybele, below . The r.m.s. values in table 6represent the expected uncertainties when using the asteroids for photometric calibration. The table shows also deficiencies in observations for individual objects (i.e. Dione) and some wavelength regions, mainly the submillimetre.

Table 6. The final weighted r.m.s. of for 6 wavelength bands in the range 7-. That is, calculated from Eq. (3).

As a final example, UKIRT Q-band data of Herculina are compared to model thermal light curves in Fig. 5. There are residual systematic differences in the absolute calibration of the two consecutive nights. Nevertheless, the model is able to predict the ligth curve variations without adding any offsets to the flux or shifting the time scale.

Fig. 5. Model and observed thermal light curves of 532 Herculina, at . Lines are for (solid), (dashed), and STM (dot-dashed). There is no shift added to the absolute level of the model flux or the rotational phase in order to fit with these specific data points.

7.2. General limitations

Our results are related to the 10 calibration asteroids. But there are many aspects which are also true for other asteroids. Depending on the needs for future projects we can extend the possibilities for new photometric standards.

Taxonomic type More than about 90 % of all asteroids (Zellner & Bowell 1979) belong to either of the taxonomic types C ("carbonaceous") or S ("stony") (Tholen 1984; Barucci et al. 1987). By neglecting any subgroups, the asteroids discussed here are either C- or S-types, except for Vesta which forms a group of its own. It is, therefore, not unreasonable to expect the results obtained here to be applicable to most other main-belt asteroids. In future investigations we hope to include M-type ("metallic") asteroids, since they could be high thermal inertia candidates.

Shape of asteroids One criterion for the preselection of the objects was a small amplitude in the lightcurve to allow for long integration times without significant flux changes. In general the TPM is valid also for more elongated objects, but this has not been extensively tested.

Wavelength range We aimed at the ISOPHOT far IR flux calibration, and optimised for a best agreement in that range. In principle TPM covers the total thermal emission from near IR to the radio region. Since we did not include reflected light, the valid range begins at around where the thermal emission dominates. The mm-range and beyond is also not well tested and needs further investigations.

Phase angle range TPM takes illumination geometry into account. In principle there are no limits for the phase angle range, but since all observations have been taken at phase angles smaller than , we had no possibilities to test the validity of the model at large phase angles, as they occur for instance for near-Earth asteroids (NEA).

Time range The model parameters can of course be improved with more observational data. The only quantity, however, that is "degrading" with time is the timing of the lightcurves. Except for Dione, the spin vectors given here should be good enough for decades. In most cases, it will only take a few additional observations to update the spin period and the timing of the lightcurves. The model results can therefore be used for IRAS data as well as for future projects.

7.3. Outlook

The thermophysical model proves to be an excellent tool to explain thermal data from the near IR to submillimetre wavelengths. Nevertheless our derived input parameters are only the starting point and can be improved by higher quality data sets and by filling the wavelength-phase angle space for each object. Some asteroids need also more lightcurve observations to clarify the ambiguities in the spin-vector solutions.

The strong need for far-IR photometric standards between the stars and the planets is not limited to ISOPHOT. Other ISO instruments (LWS ¹⁸ and SWS ¹⁹) have also large asteroid programmes for testing and improving their photometry (M. Burgdorf, A. Salama, priv. communications). First efforts to use the asteroids as photometric standards in the submillimeter (HHT ²⁰, JCMT ²¹) have started already. Future application for airborne (SOFIA ²²) and spaceborne (FIRST ²³, SIRTF ²⁴) experiments can be expected.

© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
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