If the albedo of an object is known, then the visible brightness of the object can provide an estimate of size. The main drawback of this method is the lack of knowledge of the physical properties of a newly discovered asteroid. While the orbit, and therefore distances from the Sun and Earth, at any time may already be well known, observational data on surface properties, such as albedo, have to await follow-up visible-infrared photometry and spectroscopy observations, which are more demanding in terms of target brightness and telescope time and may not be feasible until a later apparition—that is, many months or years after discovery (see Chapter 2). The 2010 National Space Policy of the United States specifically directed the NASA Administrator to “pursue capabilities, in cooperation with other departments, agencies, and commercial partners, to detect, track, catalog, and characterize near Earth objects (NEOs) to reduce the risk of harm to humans from an unexpected impact on our planet.” The most important physical parameter from the harm point of view is the size, which is the first thing the public would want to know, should a seriously hazardous object be detected in a survey. Accurate diameter measurements have the highest importance for planetary defense purposes and should therefore be made as soon as possible.
Visible reflectance spectra can provide taxonomic classification, which can be used to infer approximate albedo intervals for many asteroids. Some classes of asteroids, however, have relatively flat, featureless spectra, making taxonomic classification difficult without high-quality spectra.
Most solar radiation incident on an asteroid is absorbed, thereby heating it up and giving rise to the emission of thermal radiation in the mid-infrared region of the spectrum. The amount of solar energy incident on a surface element of an asteroid can be determined given knowledge of its orbit and the Sun’s radiation output. A simple model of the temperature distribution around the asteroid’s surface, which is normally assumed to be spherical, then suffices to enable a reliable prediction of the thermal-infrared brightness of the object as measured at an infrared telescope. In practice, an iterative procedure is used to match the model prediction to the measured brightness, resulting in a best-fit value of the diameter. A very simple asteroid thermal model might be a spherical black body, which absorbs all radiation incident on it and has a uniform equilibrium temperature around its surface. In practice, however, surface elements facing the Sun will be warmer than those on the night side of the object, and a telescope might be observing a side that is only partially illuminated by the Sun. In addition to the illumination and observing geometries, the rotation of the asteroid and its surface properties, such as cratering or roughness and whether the surface is dusty and porous (i.e., has low thermal inertia) or rocky (high thermal inertia), influence the thermal emission observed at the telescope. A significant number of small asteroids are either contact binaries or are highly irregular in shape. Averaging three to four thermal infrared measurements over a time span
of approximately 1 month or more should provide adequate sampling overview angles to allow accurate size estimation with a simple spherical model in many cases. Furthermore, repeated thermal-infrared measurements will reveal objects with large infrared lightcurve amplitudes, indicative of elongated and irregular shapes, of interest for further investigation.
A slowly spinning asteroid with low thermal inertia will have a prominent peak in surface temperature, and therefore thermal emission, on the side facing the Sun (see Figure 5.1). This situation is well described by the Standard Thermal Model (STM), which was successfully used to determine the sizes of main-belt asteroids and a few NEOs—for example, using ground-based telescopes such as the Infrared Telescope Facility (IRTF) and the orbiting Infrared Astronomical Satellite (IRAS) telescope.
The sizes of a significant number of asteroids, especially NEOs, derived using the STM were inconsistent with results derived using other techniques—for example, radar. In certain cases, the inconsistencies could be resolved by means of an alternative simple model, called the Fast Rotating Model (FRM; also referred to as the isothermal latitude model), which describes the surface temperature distribution in the case of fast rotation and high thermal inertia (see Figure 5.1).
A significant advantage of the STM and FRM is that they require observations in only one thermal-infrared wavelength band, typically in the range 5 to 20 μm. A very significant disadvantage is that, in the case of a newly discovered asteroid with unknown physical properties, there is no way of knowing which of the two alternative models should be applied. Selection of the wrong one can give rise to very large errors.
Over the past few decades, as more information on the physical properties of asteroids has been gathered, it has become clear that most objects have thermal properties somewhere between the extremes represented by the two simple models mentioned above. The rapid increase in computing power in recent years now allows detailed thermophysical models to be applied to observational data to provide very accurate sizes and reliable estimates of thermal inertia and surface roughness. Such models represent the current state of the art in the analysis of infrared and optical data of asteroids but require large amounts of observational data taken over wide ranges of observational geometry, and accurate information on the shape of the object in question. In the case of NEOs, such comprehensive data sets are normally built up over periods of years, using optical, infrared, and radar telescopes, during several apparitions of an object, and are currently available for just a few hundred asteroids. While thermophysical
modeling is a powerful and relatively accurate method, it cannot be applied reliably until the requisite amount of relevant data has been acquired.
For the estimation of sizes and albedos in the absence of information on physical properties, another option exists, the Near Earth Asteroid Thermal Model (NEATM), which represents a compromise between the STM and FRM and removes the problem of not knowing which of the two alternatives to apply.
The NEATM uses spherical geometry and is based on the STM, but with two important improvements. First, the NEATM incorporates a fitting parameter, normally called η, which in effect modifies the surface temperature distribution to allow the model thermal radiation to be fit more accurately to that observed at the telescope. Second, the observing geometry (Sun-asteroid-observer) is taken account of explicitly so that the radiation flux calculated from the model represents that part of the surface facing the telescope, which may be a combination of day- and night-side fractions of the asteroid. While the NEATM is still a relatively simple model based on spherical geometry, it has been used extensively by many groups since its publication,1 and its accuracy is well documented in the literature.
There are two ways of applying the NEATM, depending on the availability of thermal-infrared measurements in more than one band. If measurements in at least two well-separated thermal-infrared bands are available, then the model radiation fluxes can be fit to the observed fluxes by varying the asteroid size and the η parameter (see
1 A.W. Harris, 1998. A thermal model for near-Earth asteroids, Icarus 131(2):291-301.
Figure 5.2). The resulting best-fit value of η may contain useful bonus information on the thermal properties of the asteroid, such as thermal inertia. As shown in Appendix B, evidence from comparisons of NEATM results with results based on other techniques reported in the literature, such as detailed thermophysical modeling and radar observations, indicate that the NEATM often provides diameters accurate to ±10 percent or better. The NEATM is generally significantly more accurate than the STM or FRM.
If measurements in only one thermal-infrared band are available, then some assumption about the appropriate η value may be made. It has been shown from Near Earth Object Wide-Field Infrared Survey Explorer (NEOWISE) results, among others, that η depends on the observing geometry, increasing as the solar phase angle of the asteroid (i.e., the Sun-asteroid-observer angle) increases. In a number of studies, an appropriate value of η has been estimated on the basis of the phase angle at the time of the observations. While the accuracies of resulting diameters cannot match those of the fitted-η method described above, this technique may still give diameters to an accuracy of around ±20 percent (see Appendix B).
TABLE 5.1 Representative Error Estimates in Asteroid Diameter Determinations
|Method||Maximum Diameter Error||Typical Errora|
|Visible observations,b assuming albedo, pV, = 0.15 taking extreme range 0.01 < pV < 0.5||−80%, +100%|
|Visible observationsb assuming albedo, pV = 0.15 taking typical range 0.02 < pV < 0.35||−70%, +75%|
|α = 20°||−25%||−10%|
|α = 50°||−25%||−10%|
|α = 20°||+45%||+30%|
|α = 50°||+25%||+10%|
|IR/NEATM, fixed η = f(α), one purely thermal band||±40%||±20%|
|IR/NEATM, fitted η, ≥ two thermal bands|
|α = 20°||+5%||<5%|
|α = 50°||+15%||+5%|
NOTE: IR, infrared; FRM, Fast Rotating Model; NEATM, Near Earth Asteroid Thermal Model; NEO, near Earth object; STM, Standard Thermal Model.
b Diameters derived from visible photometry depend on measurements of the absolute visible brightness, H, of an object. Published values of H have typical errors of ~±0.3 magnitudes, which are included in the diameter error estimates given in the first two rows of the table. Values of H can be improved with follow-up ground-based observations although the faintness of most of the NEOs discovered in the next-generation surveys, and the need for multiple observations over at least 10° or so of solar phase angle, means this will be possible for only a subset of the most hazardous NEOs.
c For the IR/NEATM fixed-η mode, Harris et al. (2011) concluded that the RMS diameter uncertainty from Warm Spitzer observations is ±20%, based on a comparison with published NEO “ground-truth” results. In contrast, the estimated errors for the STM, FRM, and NEATM in fitted-η mode should be treated as best-case uncertainties. In the case of the FRM and NEATM, the error estimates are broadly consistent with the results of Mommert et al. (2018), which are based on a completely different and more comprehensive study. See A.W. Harris, M. Mommert, J.L. Hora, M. Mueller, D.E. Trilling, B. Bhattacharya, W.F. Bottke, et al., 2011, ExploreNEOs. II. The accuracy of the Warm Spitzer Near-Earth Object Survey, Astronomical Journal 141(3):75; and M. Mommert, R. Jedicke, and D.E. Trilling, 2018, An investigation of the ranges of validity of asteroid thermal models for near-Earth asteroid observations, Astronomical Journal 155(2):74.
d Error estimates given here are based on current knowledge of the NEO population. Errors in diameter determinations for unusual objects with physical properties outside of the observed normal ranges may be greater than the values given here.
Use of the NEATM in fitted-η mode requires measurements in two or more infrared bands in which reflected solar radiation is insignificant (wavelengths longer than about 4.5 μm for NEOs) or can be calculated and subtracted from the measured total fluxes. Corrections for the reflected component in wavelength bands below 5 μm often result in thermal flux values with relatively large uncertainties.2 To optimize the accuracy of diameter determinations, the wavelength bands of filters should be chosen so as to minimize contamination by reflected solar radiation (see Figure 5.3). For a typical NEO at 1 astronomical unit (AU) from the Sun, the reflected solar component is insignificant in filter bands beyond about 4.5 μm. The relative amount of reflected solar radiation increases with increasing albedo and heliocentric distance.
2 See, for example, N. Myhrvold, 2018, Asteroid thermal modeling in the presence of reflected sunlight, Icarus 303:91-113.
Apart from the infrared NEATM fixed-η case, the errors listed in Table 5.1 are based on modeling and do not account for real-world effects such as observational errors and irregular shapes, which must be included in a realistic overall error analysis. More irregularly-shaped NEOs may be found as improved surveys probe the smaller-size range of the NEO population (diameters of less than 150 meters). They may discover a greater proportion of small monolithic collisional fragments compared to more regularly shaped rubble piles found among the larger objects. Irregular shapes give rise to rotationally induced light curves with larger amplitudes in both reflected and emitted radiation, which may lead to increased errors in derived sizes, depending on how the light curves are sampled by the observations. In practice, the associated error can be minimized by taking measurements at several phases of the light curve, which is often done in thermal-infrared observations of asteroids. For example, the Wide-Field Infrared Survey Explorer (WISE) cryogenic survey made an average of 10 detections of a typical asteroid or comet spaced over ∼36 hours. Furthermore, several thermal-infrared measurements made over a number of weeks, thus sampling different aspect and phase angles, may be averaged to provide good size estimates with a simple spherical model even in the case of a contact binary or other highly irregular shape.
Finding: To optimize the accuracy of diameter determinations, a space-based infrared survey would have at least two filter bands at wavelengths longer than about 4.5 μm to minimize contamination by reflected solar radiation.
Modified versions of the NEATM (e.g., the Night Emission Stimulated Thermal Model)3 and the FRM (e.g., the Generalized FRM)4 have been proposed, which may perform better than the conventional NEATM for populations of NEOs with certain physical properties. Unfortunately, the physical properties of NEOs, especially small ones, are not well known. While these models have not been thoroughly tested in the way NEATM has been by many workers over many years, physical characterization of NEOs expedited by means of coordinated ground-based optical and radar observations, in addition to infrared measurements, would enable the applicability, scope, and accuracy of alternative modeling approaches to be tested. The well-documented archiving of all types of NEO observational data, enabling easy and flexible retrieval, is also important to allow new observations to be linked to existing data.
Thus, improved knowledge of the distribution of NEO physical properties will lead to improved models, which can be applied to existing and future infrared data sets to increase the accuracy of derived parameters, such as size. Furthermore, in the course of time, as more observational data become available for an object, thermophysical modeling based on an accurate shape model can significantly improve the initial results obtained from simpler models.
A number of groups are currently analyzing WISE data, in combination with other observations, to enhance the NEOWISE results published to date.
Studies (see Appendix B) indicate that an absolute size accuracy of ±10 percent or better is often achievable with current well-tested analysis procedures, given flux measurements in two or more thermal-infrared wavebands—that is, beyond about 4.5 μm. However, due to real-world effects, such as observational errors and irregular shapes, realistic expectations should allow for root-mean-square errors of ±20 percent. If flux measurements contain significant reflected solar radiation (i.e., at wavelengths below about 4.5 μm), the accuracies of derived sizes may be reduced.
3 S.D. Wolters and S.F. Green, 2009, Investigation of systematic bias in radiometric diameter determination of near-Earth asteroids: The night emission simulated thermal model (NESTM), Monthly Notices of the Royal Astronomical Society 400(1):204-218.
4 N. Myhrvold, 2016, Comparing NEO search telescopes, Publications of the Astronomical Society of the Pacific 128:045004.