Uncertainties on Asteroid Albedos Determined by Thermal Modeling

We present an analysis of the accuracy of geometric albedos determined for asteroids through the modeling of observed thermal infrared radiation. We show that albedo uncertainty is dominated by the uncertainty on the measured $H_V$ absolute magnitude, and that any analysis using albedos in a statistical application will also be dominated by this source of uncertainty. For all but the small fraction of asteroids with a large amount of characterization data, improved knowledge of the $H_V$ magnitude will be fundamentally limited by incomplete phase curve coverage, incomplete light curve knowledge, and the necessary conversion from the observed band to the $V$ band. Switching the absolute magnitude standard to a different band such a $r'$ would mitigate the uncertainty due to band conversion for many surveys, but this only represents a small component of the total uncertainty. Therefore, techniques making use of these albedos must ensure that their uncertainties are being properly accounted for.


Introduction
Thermal infrared sky surveys have produced infrared measurements of a large number of the known asteroids in the inner Solar system. Application of thermal models to these data have resulted in diameter and albedo constraints for over 100,000 asteroids from IRAS (Tedesco et al. 2004), AKARI (Usui et al. 2011), Spitzer (Trilling et al. 2010), and WISE/NEOWISE (Mainzer et al. 2011a) combined. Mainzer et al. (2015) present an overview of space-based studies of asteroids in the infrared, including a discussion of the techniques for thermal modeling.
Thermal infrared observations are primarily sensitive to the size of the asteroid observed. Once the orbit of the body is constrained, the thermal infrared flux is directly associated to the size via the thermal model used. Across a range of compositions and optical reflectivities, the emissivity of asteroids is consistently very close to a value of ∼ 0.9 (Lim et al. 2005;Vernazza et al. 2012). As shown by (Harris & Harris 1997), this means that changes to the measured absolute magnitude of an asteroid have only a minor effect on the calculated diameter. Masiero et al. (2018) presented an analysis of the accuracy of infrared diameters using the distribution of albedos observed for asteroid families. They found that the uncertainty on the H V absolute magnitude is a significant component of the overall albedo uncertainty, and dominates the albedo uncertainty for typical H V accuracies. This means that the knowledge of the absolute magnitude plays a critical role in our understanding of albedos. Pravec et al. (2012) performed an analysis of the H V values published in asteroid orbital catalogs (the most common source for these values) compared to objects tracked over long periods of time with photometrically-calibrated systems. They showed that while H V values found in orbit catalogs are generally good to a few-tenths of a magnitude for large and well-studied asteroids, smaller objects can show significant errors, both random and systematic, at the level of 0.5 mag. This is a result of a combination of effects, including using as assumed value for the phase curve G V parameter (Bowell et al. 1989), the absolute photometric calibration of the surveys providing photometry, some surveys using unfiltered observations for photometry, the accuracy of the estimation of mean brightness (to account for rotational light curve effects), changing viewing aspects resulting in different views of the asteroid's 3D shape, and the accuracy of the conversion from the observed band to the 'standard' V band used for H V calculation. For each of these uncertainty components, observations can reduce their individual contribution (such as was done by Pravec et al. 2012), however this requires large amounts of telescope time for each object to densely sample the rotational light curve, phase curve, and different apparitions to constrain the 3D shape. The majority of objects in Minor Planet Center's orbital catalog do not, and will not, have this level of knowledge without targeted densely-sampled followup covering a broad range of phase angles to constrain both the light curve and phase curve.
Here we investigate the effect that uncertainties on H V will have on the albedos derived when a survey provides diameter fits, as occurs with thermal infrared data. This is important to help us better understand the limitations of the derived albedo data sets, and how these uncertainties will affect our interpretations of the population as a whole and the sub-populations within the asteroids.

Relationship between diameter, absolute magnitude, and albedo
The empirical relationship between the size of a body, its geometric albedo, and the brightness is often described (e.g. Harris & Lagerros 2002) as: which can be rearranged as: where D is the size of the body in kilometers, H V is the phase-and distance-corrected magnitude (i.e. absolute magnitude) in the V band, and p V is the geometric albedo in the V band. The constant parameter C V is usually taken to be C V = 1329 km (for example, see the derivation in Pravec & Harris 2007).
The definition of geometric albedo is the ratio of the true scattering of light by the surface compared to an ideal scatterer, here a disk of area πr 2 = π D 2 4 that is 1 AU from the sun, 1 AU from the observer, and at phase of α = 0 • . Following (Jewitt et al. 2013), this relationship can be written: The constant of interest is a function of the stellar apparent magnitude of the Sun in the band of interest, here V band. Torres (2010) quote a value of V = −26.76 ± 0.03, which they derive by recomputing the calibrations of Bessell et al. (1998) using updated reference stars. From this, we then derive a constant of C V = 1330 ± 18 km. This implies that an albedo derived from a measured diameter and an H V magnitude will automatically have a ∼ 2.8% relative uncertainty from the uncertainty on the Solar V apparent magnitude, even before accounting for errors on D and H V . The previously-derived C V = 1329 value, based on a V = −26.762 ± 0.017 mag from Campins et al. (1985), is within measurement uncertainties of the value that is obtained with current Solar magnitude measurements.
An important point is that this constant is a function of the bandpass being used. The majority of current and planned sky surveys do not use the Bessell V filter, instead having moved to a filter set similar to that of the Sloan Digital Sky Survey (SDSS, Gunn et al. 1998;Smith et al. 2002). The conversion from the survey band to V will add an additional component of systematic uncertainty to any albedos determined. Further, this conversion will depend on the (unknown) composition of the object, as asteroids with different spectral curves have different colors, meaning that the systematic uncertainty will be different for different classes of object making comparisons between populations more difficult.
One option to reduce this conversion error is for the community to transition asteroid absolute magnitudes and albedos to a band that dominates the ongoing survey photometry. For example, the r band is typically the most sensitive to asteroids for ground-based surveys given their intrinsic brightness combined with filter responsivities. Using r = −27.05±0.03 (Vega mags, Willmer 2018) leads to a new constant value of C r = 1164 ± 16. This would, or course, require determination of the H r absolute magnitude and the G r phase parameter for all asteroids being studied, as well as a conversion technique to compare new albedos to literature p V values. However, as surveys such as the Legacy Survey of Space and Time at the Vera Rubin Observatory (LSST Science Collaboration et al. 2009) begin producing large quantities of asteroid photometry over many years, these measurements will become possible. Given that the majority of asteroid photometry over the next decade will be obtained in r or a closely calibrated band, it is worth careful consideration by the community whether now is the time to switch standards.
In counterpoint, there are arguments against switching standards as well. Foremost is the extensive amount of literature currently using H V and p V , as well as the numerous diagnostics that exist based on these parameters. In addition, the V band covers the peak of the distribution of reflected light from an asteroid. That makes it a closer analog to the true bolometric albedo, which is an important value needed for thermophysical modeling. Instead of switching standards, a concerted effort to provide accurate V −r indices for all surveys for a range of asteroid compositions might alleviate some of the problems created by the current system. Any change, of course, would require extensive community discussion and IAU approval.

Absolute magnitude uncertainty
Following Eq 2, we can see that the error on albedo will be a combination of the errors on diameter and absolute magnitude. While diameter error can be independently assessed based on comparisons between different determination methods (e.g. infrared modeling, radar modeling, or occultation chord fits), the true uncertainty on H V is more difficult to validate against an independent dataset.
Following Bowell et al. (1989), the absolute magnitude can be determined from fitting the phase-magnitude relationship of the asteroid using the equations: where α is the phase angle, R is the heliocentric distance, and ∆ is the geocentric distance at the time of observation. This is the simplified functional form adopted by the IAU (Marsden 1985), though a more precise calculation is presented by (Bowell et al. 1989) in their equation A4.
Other photometric phase functions have been developed, such as the H-G 1 -G 2 system and the H-G 12 system (Muinonen et al. 2010), however as these either have more parameters (in the case of the H-G 1 -G 2 system) or non-linear behavior (in the case of the H-G 12 system) they require more data to accurately fit the phase curve and thus will have comparable or lower accuracy for sparse data sets.
We note that in well-defined cases with extensive data and multi-parameter fits, like those presented in Muinonen et al. (2010), the uncertainty on H V can be of order 0.02 mag (1-σ). However as the authors of that paper note, a number of factors commonly encountered with photometric data can impair the determination of H V including changing geometry between apparitions, incomplete rotational coverage at each phase angle, coverage of only a narrow range of phase angles, and imperfect conversion of photometry from the observed band to V .
The work of Vereš et al. (2015) provides an ideal example of real world results from fitting absolute magnitudes to a large, photometrically-calibrated survey that is sparse in time. In that work, the vast majority of observations had individual photometric uncertainties < 0.1 mag, meaning that the individual observations did not place a fundamental limit on the accuracy of the H V determination. Through Monte Carlo simulations of different rotation states, those authors found that their statistical uncertainty on H V was 0.3 mag for objects with H < 18 mag (sizes larger than approximately 1 km) using the H-G relation from Bowell et al. (1989), or a slightly improved uncertainty of 0.25 mag under the H-G 12 system developed by (Muinonen et al. 2010). For cases where diameters are measured with an infrared survey with a nominal accuracy of 10%, the above uncertainty on H V would result in a relative uncertainty on albedo of 32−36%. In the converse case of an object with only optical observations and using an assumed albedo with perfect accuracy, this H V uncertainty alone would propagate to an uncertainty on diameter of 13 − 15%, with additional non-random uncertainty from the accuracy of the assumed albedo used. Vereš et al. (2015) note that the fits to the G V slope parameter are significantly worse than the H V fits in their work. The uncertainty on G V depends on the span of phase angles covered, with coverages > 20 • showing significantly smaller uncertainties than those with coverage < 10 • . In addition, objects that are only seen at high phase angles will have significant errors on H V due to the large lever-arm that the fitted value of G V has, as is often the case for near-Earth asteroids. Due to this uncertainty on G V , H V errors of ∼ 1 mag can be expected for objects on Earth-like orbits. This level of uncertainty for the H V value would correspond to an albedo uncertainty of ∼ 70% when using a infrared-determined diameter. In the case of a diameter calculated from H V and an assumed albedo, the uncertainty on this diameter will be ∼ 42%. As G V is only weakly correlated with taxonomy (see Table 6 from Vereš et al. 2015), asteroid color measurements cannot dramatically improve this.
An example of this situation is shown in Fig 1. Here, we assume an asteroid is detected with V = 21 mag (with negligible measurement uncertainty) at a phase of α = 60 • , 1 AU from the Earth and Sun. Based on (Vereš et al. 2015) we drew random G V parameters from a normal distribution following G V = 0.2 ± 0.2. This results in a median absolute magnitude of 19.00 +0.29 −0.54 (uncertainties shown are the 84 th and 16 th percentiles). For objects typically observed at smaller phase angles, like Main Belt asteroids, this situation will not occur and H V values will be better constrained. However, near-Earth objects are often observed at large phase angles, where this problem can result in significant uncertainties in their absolute magnitudes. For example, cadence simulations for the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) indicate that the twilight NEO survey, if carried out, would have approximately half of the survey fields at Solar elongations below 110 • , resulting in NEO detections predominantly at large phase angles (Jones et al. 2020).
As the range of phase angles covered increases, the constraint on the G V value for an object will improve, and thus the H V determination will also improve, however this is a strong function of the phase angles at which the object is observed. A Monte Carlo simulation of four V -band observations (assuming zero photometric error and perfect accuracy on the assumed albedo) in the range 10 • < α < 30 • yields a final uncertainty on H V of 0.23 mag and minimum calculated diameter uncertainty of 11%, while four observations drawn from 50 • < α < 70 • result in an uncertainty of 0.68 mag on H V and 30% on D. Increasing the number of observations to 64 further improves the H V uncertainty to 0.08 mag (4% on D) for 10 − 30 • and 0.31 mag (14% on D) for 50 − 70 • . Photometric measurement uncertainty will increase the true uncertainty on both H V and D, while the uncertainty on the assumed albedo will increase diameter uncertainty as well. It should be noted that these results depend on the assumption that the H-G or H-G 12 phase functions can adequately describe the opposition effect of all objects. Recent work by Mahlke et al. (2020) investigated the photometric phase curves of over 90,000 asteroids in two visible light bandpasses from the ATLAS survey. They show that even for objects with well-sampled phase curves, the difference in fitted absolute magnitude between the H-G 12 system and the H-G 1 -G 2 has systematic offsets of order 0.1 mag and random uncertainties of ∼ 0.2 mag. This will directly impact the accuracy of the albedos determined for asteroids from thermal modeling.

Discussion
The overall uncertainty on any given asteroid's albedo measurement is a combination of the uncertainties on the H V value and the diameter derived from thermal modeling. Comparisons of diameters determined from thermal modeling to those from different surveys as well as independent sources such as occultations and radar observations have shown that when multiple infrared measurements are available that sample the thermal emission portion of an asteroid's spectral energy distribution, the diameter uncertainty is approximately 10% (Mainzer et al. 2011b;Usui et al. 2014;Wright et al. 2018;Herald et al. 2020). This uncertainty is primarily caused by the deviation of the applied thermal model (usually a sphere with a simple temperature profile) from the asteroid's actual thermophysical properties. This propagates to an albedo uncertainty of ∼ 20%, which is comparable to the albedo uncertainty resulting from a typical H V uncertainty. Along with the uncertainty on the constant in Eq 2 the result is a top-level albedo uncertainty of ∼ 28% for typical values of well-studied objects of σ D = 10% and σ H V = 0.2 mag.
Using large samples of objects with assumed uniform properties, e.g. from asteroid families (Masiero et al. 2015) or selected by photometric colors , it is possible to obtain a mean albedo for the population that is known to higher precision than any single object's albedo. In this case, the uncertainty on a diameter inferred using this assumed albedo would be dominated by the accuracy with which the object has been assigned an albedo and the uncertainty on H V . For a case of σ H V = 0.2 mag and an albedo assumed to be known with arbitrary precision, the uncertainty on diameter will be ∼ 10%. As shown by Pravec et al. (2012), ∼ 0.2 mag is the smallest H V uncertainty that can be reasonably expected from current data for objects with D < 10 km, even after correcting for systematic errors in the orbital catalogs. For objects that are newly discovered, light curve properties and phase curve behavior will not be characterized to high precision without many years of observations, and the characterization accuracy will depend on the the observing cadence. These objects will thus have commensurately worse H V constraints, which would translate to larger uncertainties on inferred diameter even under the assumption of perfect albedo assignment. As discussed above, for newly discovered NEOs the diameter uncertainty from optical data alone can reach ∼ 50%, particularly for objects on Earth-like orbits.

Conclusions
As is noted by both Bowell et al. (1989) and Muinonen et al. (2010), the geometric albedo determined from the relation in Eq 1 might be more appropriately called a 'pseudo-albedo' as it is not a direct measurement, but rather inferred from models of models of measurements. This is not to say that it is not useful for population analysis or investigations of individual objects, as this is clearly demonstrated in the literature. Rather, as highlighted by Bowell et al. (1989), this relationship should be treated with caution as multiple assumptions go into a single derived value. As we have discussed here, the uncertainty on H V can have a large impact on our knowledge of p V , and it is nearly impossible to independently verify H V measurements against other, non-photometric data sources. In light of this, we urge caution in attempting to derive physical properties from albedos alone.