The Geometric Albedo of (4179) Toutatis Estimated from KMTNet DEEP-South Observation
We aim to derive the geometric albedo of a near-Earth asteroid, (4179) Toutatis, to investigate the surface physical conditions. The asteroid has been studied rigorously not only via ground-based photometric, spectrometric, polarimetric, and radar observations but also via in situ observation by the Chinese Chang’e 2; however, the geometric albedo is less well understood.
We conducted V-band photometric observations when the asteroid was at opposition in April 2018 using three continental telescopes in the Southern Hemisphere that compose the Korean Microlensing Telescope Network (KMTNet). The observed time-variable cross section was corrected using the radar shape model.
We find that Toutatis has geometric albedo , which is typical of S-type asteroids. We compare the geometric albedo with archival polarimetric data and further find that the polarimetric slope–albedo law provides a reliable estimate for the albedo of this S-type asteroid. The thermal infrared observation also produced similar results if the size of the asteroid is updated to match the results from Chang’e 2. We conjecture that the surface of Toutatis is covered with grains smaller than that of the near-Sun asteroids including (1566) Icarus and (3200) Phaethon.
Key words: Minor planets, asteroids: general – Minor planets, asteroids: individual: (4179) Toutatis
Journal of the Korean Astronomical Society https://doi.org/10.5303/JKAS.2019.00.0.1
00: 1 99, 2019 May pISSN: 1225-4614 eISSN: 2288-890X
Published under Creative Commons license CC BY-SA 4.0 http://jkas.kas.org
\@footnotetextCorresponding author: M. Ishiguro
The geometric albedo is one of the most fundamental observed quantities for characterizing the physical properties and composition on asteroids. This parameter would provide clue as to hypotheses regarding the surface composition, degree of space weathering, and physical conditions such as the surface particle size if polarimetric data at large phase angles (Sun–asteroid–observer angle) are available (Shkuratov & Opanasenko 1992; Dollfus 1998). Because an absolute magnitude (at the opposite direction from the Sun viewed at unit heliocentric and observer distances) is related to the product of the geometric albedo and cross section of the asteroid, the accuracy of the size determination is a critical factor for deriving the geometric albedo in most cases. The sizes of asteroids have been investigated thoroughly via various techniques, namely, direct imaging of asteroids with cameras onboard spacecraft, delay-Doppler imaging (Ostro 1993), thermal–infrared observations (Tedesco et al. 2002; Mainzer et al. 2011; Usui et al. 2011), and occultation (Shevchenko & Tedesco 2006); nevertheless, the absolute magnitudes are less examined because of the scarcity of observational opportunities at opposition. In addition, magnitude variations due to an asteroid’s rotation have not been considered in many cases for the derivation, making the albedo values less certain. For these reasons, there are a limited number of reliable albedo estimates for asteroids.
Here, we aim to examine the geometric albedo of an Apollo-type near-Earth asteroid (NEA), (4179) Toutatis. The asteroid has been thoroughly studied by various observational techniques (i.e., photometry, spectroscopy, polarimetry, radar imaging, and in situ observation). It is classified into S-type (Lazzarin et al. 1994; Howell et al. 1994) or, more strictly, -type (Bus & Binzel 2002) and is possibly composed of undifferentiated L-chondrites (Reddy et al. 2012). The dimension is determined as by radar observation (Hudson & Ostro 1995). Toutatis is in an extremely slow, non-principal-axis rotational state (Ostro et al. 1995; Takahashi et al. 2013) having rotation and precession period of and , respectively (Zhao et al. 2015). The accurate shape model was established by the radar delay-Doppler imaging technique (Hudson et al. 2003).
More recently, in 2012 December, the Chinese spacecraft Chang’e-2 captured direct images of this asteroid with a maximum resolution better than 3 during the flyby (Huang et al. 2013). The study revealed, e.g., previously unknown basin at the end of the body, lineaments over the surface especially near the rim of it, and a sharp connection silhouette in the neck, as well as providing valuable information on the asteroid’s size estimation ( with nominal uncertainty of 10 ). Moreover, more than 50 craters larger than 36 and 30 boulders larger than 10 were identified and the chronology of craters and size distribution of boulders were investigated. From the cumulative boulder size frequency distribution analyses compared with (25143) Itokawa, it is suggested that Toutatis may have different preservation state or diverse formation history (Jiang et al. 2015). The shape is considered to have been produced from a low-speed impact between two components (Hu et al. 2018). An estimation of the hemispherical albedo was also made by Zhao et al. (2016) as 0.2083, 0.1269, and 0.1346 in R, G, and B bands of Chang’e-2 CMOS censor, respectively. Moreover, the close-orbit dynamics under Toutatis’ complex gravity field was explored. The asteroid is also a notable research target due to its extremely chaotic orbit in a 3:1 mean motion resonance with Jupiter and a weak 1:4 resonance with Earth (Whipple & Shelus 1993). In addition, the asteroid has a unique potential field in the close orbit (Scheeres et al. 1998). Thus, Toutatis has been central to the recent developments in NEA research since the 1990s.
Although Toutatis has been studied intensively, we found that the geometric albedo has a large ambiguity by a factor of , ranging from 0.13 (Lupishko et al. 1995) to 0.41 (Masiero et al. 2017). In this paper, we aim to establish a better estimate of the albedo value. Taking advantage of the continuous observation capability from KMTNet observatories in three different continents (Kim et al. 2016, see Sect. 2), we derived the geometric albedo of the slowly rotating asteroid. Together with the polarimetric properties given in Lupishko et al. (1995); Mukai et al. (1997); Ishiguro et al. (1997), we discuss the physical properties on the surface.
For future reference, Table 1 lists the some important symbols we used in Sections 2, 3, and Appendix A, their meanings, and their values and units. Most of the source codes we used for the data reduction and analysis are publicly available (see Appendix B).
|Category||Symbols||Description||Value and Unit|
|Magnitudes||Visual magnitude of the Sun|
|,||Visual magnitude and its uncertainty|
|Absolute magnitude ()|
|Phase angle||( or )|
|Time on the target (light-time corrected)||or|
|Ecliptic coordinate (longitude, latitude)||( or )|
|Physical||Total projected area viewed at|
|Geometric albedo in visual (V) band||-|
|Albedo at the phase angle of||-|
|The irradiance of the object of interest|
|of a Lambertian reflector at normal incidence|
|The radiance factor||-|
Variables with hiphen(-) in the last column are dimensionless. The units are basically given in SI format unless special units are dominantly used in this work. The units given in parentheses are dimensionless but are preferred to be explicitly written.
2 Observation and Data Reduction
The observation journal is summarized in Table 2. We conducted our observations using three 1.6-m telescopes in the Southern Hemisphere that compose the Korean Microlensing Telescope Network (or KMT-Net, Kim et al. 2016) as a part of DEEP–South project (Moon et al. 2016). Each telescope provides a field of view via identical mosaic CCD cameras with a pixel size of . The telescopes are located on three continents: at Cerro Tololo Inter–American Observatory (KMTNet-CTIO) in Chile, South African Astronomical Observatory (KMTNet-SAAO) in South Africa, and Siding Spring Observatory (KMTNet-SSO) in Australia. We employed the standard filter for deriving the geometric albedo, namely, the Johnson–Cousins -band filter. We obtained 10 frames per night at each observatory by setting the individual exposure time to 1 minute. The acquired images were preprocessed by a reduction pipeline for the KMTNet data at the KMTNet data center, which includes bias subtraction, flat-fielding, and cross-talk correction. A summary flow chart of the data reduction process is described below (see also Fig. 1).
|Date||Time in UT||Site||Seeing||Weather|
The average values of the UT date and time on the target (i.e., the light travel time is corrected) of the exposures made at same night, same observatory. The total span of the exposures in each row is less than 1 hour, which is significantly smaller than the rotational and precessional periods of Toutatis (–). The telescope sites: Cerro Tololo Inter–American Observatory (CTIO), South African Astronomical Observatory (SAAO), and Siding Spring Observatory (SSO). Number of exposures. The parameters are averaged for the ephemerides for each row (same night, same observatory group). Seeing and weather condition from the official daily report (see Notebook 6 of Appendix B). Weather code of 0 and 1 corresponds to clear and thin cloud, respectively.
The locations of Toutatis were specified via an initial analysis using the Moving Object Detection Program (MODP), which is developed internally by KASI. MODP updates the World Coordinate System (WCS) information in the header of each image first and specifies the positions of the asteroid in each image. For convenience of data reduction, we analyzed the clipped images with the dimension of pixels () centered on the Toutatis locations (which were calculated by the MODP). Cosmic-ray signals were removed by astroscrappy111https://github.com/astropy/astroscrappy version 1.0.5 using a separable median filter., which uses the L. A. Cosmic algorithm (van Dokkum 2001). We coadded all the exposures of the same night at the same observatory while shifting the individual images to increase the signal-to-noise (S/N) ratio. As a result, two composite images per night per observatory were generated: one for a co-added image with sidereal offset and the other by centering on Toutatis.
The detected signals in each composite image are compared with those of field stars from Pan-STARRS Data Release 1 (PS1, Flewelling et al. 2016). We first cone-searched PS1 objects within centered at Toutatis and with more than 5 observations in the and bands. We then used the criterion to reject non-stellar objects222https://outerspace.stsci.edu/display/PANSTARRS/How+to+separate+stars+and+galaxies, where iMeanPSFMag and iMeanKronMag are the mean i-band magnitudes using the point spread function and Kron radius photometry, respectively. If saturated stars resulted in blooming and/or contaminated nearby pixels, all the affected pixels were manually masked, and any stars near those masked pixels were rejected, with the minimum separation set to be , where is the full-width at half-maximum of the seeing disk. We fixed to be 4 pixels (). In addition, nearly colocated stars were rejected by using the DAOGROUP algorithm (Stetson 1987 implemented via photutils by Bradley et al. 2017) with a minimum separation distance of .
Next, we used the transformation formula
from Lupton (2005)333https://www.sdss.org/dr14/algorithms/sdssUBVRITransform/#Lupton2005 to obtain the Johnson–Cousins -band magnitude of the stars, where and are the mean aperture photometry magnitudes (MeanApMag) of the - and -band, respectively. We fit a linear line to and the instrumental magnitude from circular aperture photometry of the stars (aperture radius , inner and outer sky radii and , respectively) in the sidereally coadded images; only the stars with are used for the analysis. The regression line corrects the atmospheric extinction in relative photometry, and thus it was used to obtain the V-band magnitude of Toutatis by extrapolation in the Toutatis-centered images. The image with minimum number of stars used in the regression (April 7th from CTIO) had 9 stars. Due to the extrapolation, the uncertainty in , (obtained from the 1- confidence interval of the regression line), must be understood as a lower limit of the true uncertainty.
After the reduction, we compared the results with raw images, and it is found that an unexpected artifact pattern contaminated one of the datasets (2018-04-11 SAAO). We did manual photometry (selecting appropriate sky regions by visual inspection) to this single epoch; all other data were analyzed using the identical photometric reduction code described above. We summarized the observational quantities in Table 2 and plotted them as a function of time in Figure 2.
3 Derivation of the Geometric Albedo
3.1 Correction for Rotation and Distances
The obtained varies not only because of rotation and solar phase angle (, the Sun-asteroid-observer’s angle) effects but also because of the heliocentric and observer’s distances ( and , respectively). We first corrected the effect of the heliocentric and observer distances by deriving the reduced magnitude , which is defined as a magnitude when the asteroid is viewed at unit heliocentric and observers distances (i.e., ) but at arbitrary phase angles. This parameter is given by
The IAU two-parameter , magnitude system (Bowell et al. 1989) describes the -dependence of using the two parameters, , the absolute magnitude, and , the slope parameter:
Our observations were made at very small values. For a spherical object, the fraction of dark area (outside the terminator) is . Since our observations were made at , , and the non-spherical shape, such as that of Toutatis, may not significantly increase this value. Therefore, we assume that the total projected area of the target viewed from a direction will be the same as that of sunlit surface.
Furthermore, it has been known that the reflected intensity per projected area (which is closely related to the radiance factor; see Section 3.2) is nearly independent of the incidence and emission angle, but dependent only on the phase angle , when the object is viewed near opposition. That is, the reflected flux from a surface patch on Toutatis is proportional to its projected area, regardless of its orientation. This was first conjectured observationally by Markov & Barabashev (1926) from lunar observation, which was mentioned in the appendix of Dollfus & Bowell (1971). It was also confirmed by Lee & Ishiguro (2018) using a modern theoretical model: Figure 4 of theirs shows that the radiance factor (defined in Section 3.2) is reduced only or when the incidence angle is chagned from to or (the azimuthal angle and phase angle are fixed to 0). This means, if we correct the phase effect using Eq 3, the obtained will be linearly related to the logarithm of the geometrical cross-sectional area (Eq 5 below).
Figure 2 shows the and with respect to time, using following Spencer et al. (1995). The best rotational model using the high-resolution radar shape model (Hudson et al. 2004) to describe the observed light curve is shown as red solid line, and the other models that are within 1- confidence interval are shown as faint black dotted lines. The selection process of rotational models is described in Appendix A. We confirmed that any that is does not raise significant change in our results. Also the obtained reduced magnitudes and the amplitude of the light curve coincide well with the previously reported values (Spencer et al. 1995).
3.2 Radiance Factor and Geometric Albedo
The radiance factor is denoted as . Here, is the measured irradiance of the object, whereas is that of an imaginary object, namely a perfectly diffuse (Bond albedo of unity) Lambertian plate444Using Lambertian plate rather than Lambertian sphere has historical reason rather than mathematical or physical reason (see, e.g., Russell 1916). If it were Lambertian sphere, the value will be rather than . of the same projected area at the identical geometric configuration to that of the object, but oriented such that incidence angle of the sunlight is . Since the projected area is the same for both and , division of will be identical to the ratio of bidirectional reflectances of the object (which is unknown) and the Lambertian plate (which is ). Thus, at a perfect opposition (), coincides with by definition.
As described in Section 3.1, the value remains almost constant over the surface when the object is observed near opposition. Thus, the observed flux is is proportional to , where over the surface of Toutatis. The obtained reduced magnitude, , can then be converted into a logarithm of :
where is the V-band magnitude of the Sun at 1 au (Mann & von Braun 2015), and is the geometrical cross section in as a function of time. is a constant to adjust the length unit.
The calculated values are plotted against the phase angle in the upper panel of Fig. 3. As a simple approximation, we used a single power-law () to fit the points. From this plot, we considered four parameters of interest: the geometric albedo (), the albedo at phase angle of (), and two factors
which were used to check the validity of our results by comparing with previous works (explained below).
To investigate their uncertainties, we used a brute-force grid search algorithm (using fixed grid) to the parameters of power-law, and . The best-fit parameters were first found by simple chi-square minimization process where the reduced chi-square is defined as
Here, is the total number of the data points, is the degrees of freedom, and is the uncertainty in the for the -th observation. The contour in the 2-D parameter space (bottom panel of 3) is carefully investigated to assure that we searched large enough parameter space with fine enough grid size. The uncertainties of , , and are estimated based on the minimum and maximum values of each of them for the models within the 1- confidence interval (the gray region in top panel of Fig 3).
From the fit, the amplitude parameter is the geometric albedo by definition (). We obtained the geometric albedo in the V-band . This is a typical albedo value for an S-type asteroid.
The albedo at , , is important since this is used in experimental studies to compare with celestial bodies (see Sect. 4.4). It is defined as the ratio of the measured flux of the obeject of interest at to that of a perfectly diffuse Lambertian plate at the same geometrical configuration with the same projected area555The angle of is chosen by experimentalists, since the albedo measurements were found to give the most reproducible results with (see Geake & Dollfus 1986, and references therein). A smoked MgO screen illuminated at incidence angle of is used as an analog of Lambertian plate. The is then a ratio of the flux of the target of interest and the MgO screen, while both fluxes were measured at with zero incidence angle.. Thus,
as the black marker shown in Fig 3.
The two factors, and are
was investigated in Belskaya & Shevchenko (2000) and found to be for C-type and for S-type asteroids. From the V-filter result in the Figure 7 of Tatsumi et al. (2018), for the asteroid (25143) Itokawa, an S-type asteroid. was determined for asteroid Itokawa in Lee & Ishiguro (2018). For Itokawa, they found for and for (Note that the sign convention is opposite with their ). We employed and because the phase angle difference is larger than as Lee & Ishiguro (2018) and this is the interpolated region. Both and coincide with the expected values from previous works, but it is difficult to make further conclusion due to the uncertainties.
The obtained parameters are summarized in Table 3. In the last column of the table, we summarized the identical paramters derived by excluding one of the data points (and updating the degree of freedom accordingly), 2018-04-07 CTIO with , since this is the observation which deviated the largest from the shape model prediction (Fig 2). Comparing the two results, it can be seen that the derived parameters are robust against the exclusion of the deviating data value (parameters changed much less than the 1- confidence range).
The geometric albedos of asteroids have been derived by several techniques, including (1) infrared observations of thermal radiation, (2) time-delay and Doppler measurements of signals emitted from ground-based stations, (3) measurements of the time duration during occultation of background stars, (4) direct imaging with adaptive optics techniques, (5) in situ measurements of the bidirectional reflectance using spacecraft onboard cameras, and (6) an empirical method using polarimetric slopes. Since techniques (1)-(4) are contrived to derive the sizes (and shapes) rather than the albedo values of asteroids, it is essential to combine the visible magnitudes to derive the albedos using these four techniques (1)-(4). It is, however, true that the optical observations have not been conducted strenuously despite the recent comprehensive surveys, especially by means of (1) and (2). In general, the visible magnitudes have been obtained using uncalibrated observation data on the presumption that asteroids have a phase function (typically for an IAU , magnitude system). In fact, it is quite difficult to make optical observations of asteroids at the opposition point (or at least within a hypothetical circle having the equal apparent diameter of the Sun) because the orbital planes of almost all asteroids do not coincide with Earth’s orbital plane (i.e., ). Short time variations of the cross-sectional areas of asteroids also make it difficult to derive the albedo values even when the phase function and dimension of the asteroids are known. This effect is especially significant for elongated asteroids. In contrast, direct observations of asteroidal surfaces via space missions provide more reliable information about the albedos, eliminating the above uncertainties regarding the cross-sectional areas. However, note that the calibration processes of onboard data have some uncertainties, such as the standardization of magnitude systems and unavailability of flat-fielding data in space. Some in situ data were acquired at large phase angles (e.g., (433) Eros by the NEAR-Shoemaker mission). Thus, it is not straightforward to derive the accurate geometric albedo values.
4.1 Sources of Uncertainties
The parameters we obatained in this study, especially the albedos, may have suffered from sources that are difficult to be correctly quantified. For one of our observations (2018-04-07 CTIO with ), the magnitude is - fainter than the model prediction (Fig 2). Although a 1- of deviation is not necessarily significant, there is a possibility that the value actually was affected by unknown systematic shift (uncertainty), such as imperfect phase function modeling (Eq 3) and the weather effect (Table 2). Although it was almost invisible from this image, some artifact as in 2018-04-11 SAAO (see the last part of Sect 2) may have affected this image in a non-trivial way. This data, however, does not seem to critically affect the quantities derived in this work, as shown in Table 3.
In addition, as the radiance factor calculation requires the absolute physical value of the projected area (Eq 5), the uncertainty in the physical size affects the value. From Eq 5, ( to a first order approximation), if all other quantities including the observed values are fixed as constants. The radar model we used have the maximum size along each principal axes as (Hudson et al. 2003), which is slightly smaller but within the uncertainty ranges obtained by Chang’e-2, (Huang et al. 2013). Ignoring the uncertainty ranges, may have been underestimated systematically by in our estimation. Hence, our albedo estimations, and , could have systematically overestimated by and , respectively. These are negligible compared to the derived uncertainties (only of the error-bars; see Table 3).
4.2 Comparison with Flyby Observations
Chinese Chang’e-2 spacecraft succesfully obtained optical images of Toutatis during its December 2012 flyby at a distance of , and the hemispherical albedos were derived (Zhao et al. 2016). These results provide direct testing cases for the albedo estimations from different approaches including this work. It is, however, difficult to directly compare their results to ours, since theirs are estimated hemispherical albedo (assumed to be equal to the Bond albedo, ), thus different from bidirectional albedos derived in this work (e.g., and ).
Using the IAU H, G system (Bowell et al. 1989), the Bond albedo can be obtained from using numerical approximation of the phase integral:
where is the Bond albedo in V-band666 Conventionally suggested by Bowell et al. (1989) is widely being used. However, Myhrvold (2016) found that the best fitting function is . We confirmed the latter result and used it here.. Using the uncertainty in and , we obtain the extrema of . This is smaller than the values derived in Zhao et al. (2016): 0.2083, 0.1269, and 0.1346 in R, G, and B bands of Chang’e-2, respectively. Using Eq 10, these can be transformed into the geometric albedos of , , and assuming and identical phase functions, respectively.
The discrepancy may have caused by the assumptions used to compare the two physically different albedos (bidirectional and Bond). To directly measure the Bond albedo, one needs to observe the object at all the possible directions, which is impractical. Thus, we assumed a phase function (Eq 10) to convert the to , while Zhao et al. (2016) assumed illumination conditions (incidence angles) to convert the radiance () into the albedo, for example. The phase angle coverage of the observations in both works are also different.
Therefore, this discrepancy is a result of imperfect assumptions to indirectly infer the Bond albedo in the two works, while the uncertainties arise from such assumptions are not added as the error-bar. We emphasize that our and should be accurate within the derived uncertainty range, because these were obtained from observations made at , close to where these quantities are defined.
4.3 Comparison with Thermal Infrared Study
Admittedly, the sizes and albedos contained in infrared catalogs such as IRAS (Tedesco et al. 2002), AKARI (Usui et al. 2011), and NEOWISE (Masiero et al. 2017) are extremely helpful to gain the perspective of asteroids with different taxonomic types distributed across the solar system; nonetheless, it is also important to note that some asteroids with a small number of detections and elongated shapes have large uncertainties due to rotation.
Masiero et al. (2017)777The machine-readable table is available at http://iopscience.iop.org/1538-3881/154/4/168/suppdata/ajaa89ect1_mrt.txt obtained with the diameter for Toutatis, from infrared observation by NEOWISE the survey in combination with , , and the beaming parameter from 6 observations in both W1 and W2 bands at phase angle . The value can be transformed to , if we assume the posterior distribution of their estimation is truly, or at least similar to, Gaussian. It should be noted that the observations were made at a single phase angle, so the traditional NEATM may not give accurate estimation, but other advanced modelings may be more adequate, such as the advanced thermophysical model (ATPM; Rozitis & Green 2011, which is used for space mission related targets, e.g., Yu et al. 2014, 2015).
Meanwhile, Usui et al. (2014) noted a 22% deviation for the 1- confidence range among three radiometric survey catalogs by IRAS, AKARI, and WISE (Tedesco et al. 2002; Usui et al. 2011; Mainzer et al. 2011). The deviation is caused not only by the different types of thermal models but also by the observed rotational phases of the asteroids. It is worth noting that the size and albedo determination can be more uncertain for individual targets than the estimated 22% from the average trend.
Note also that there are currently ongoing discussions about the accuracy of the NEOWISE results (see, e.g., Myhrvold 2018a; Wright et al. 2018; Myhrvold 2018c). According to Wright et al. (2018), however, the results we used for Toutatis (Masiero et al. 2017) are not affected by any problem. Meanwhile, Myhrvold (2018b) discusses about a new perspective to the NEATM by introducing a new parametrization and physically valid yet simple generalization when the reflected sunlight is not negligible in the thermal spectra. This new approach may improve the NEOWISE results with better accuracy and consistency.
In case of Toutatis, one needs to pay close attention to the rotational phase because of the very elongated shape of the asteroid. In fact, the size given in NEOWISE catalog above is only 73 % of the mean diameter derived by radar observation (i.e., ). Using the relationship (derived in, e.g., the appendix of Pravec & Harris 2007), the value appears to be overestimated by a factor of . Despite this is not an accurate statement since the change in changes the shape of the thermal spectrum of the asteroid, we may crudely estimate that the updated albedo will be . This coincides with the expected albedo for an S-type asteroid.
4.4 Comparison with Polarimetric Studies
On the other hand, Lupishko et al. (1995) studied the polarimetric properties of this asteroid and derived using the empirical polarimetric slope–albedo law stated in Zellner et al. (1977). Subsequently, Mukai et al. (1997) also derived the identical value, , using the polarimetric slope–albedo law in Dollfus & Zellner (1979). The empirical relationship, first noted in Widorn (1977), It is written as
where denotes the polarimetric slope around the inversion angle in and where and are constant values. Note that the albedo in Zellner et al. (1977) and Dollfus & Zellner (1979) is different from the one that we currently apply. Thus, these authors derived a set of constants based on the laboratory polarimetric measurements, where the measurements defined the albedo at the normal () incidence and angle of emergence (i.e., , not ). It is challenging to measure the geometric albedo, which is defined at , because the light source is in the line of sight of the detector, thus hiding the sample. Although we acknowledge these early efforts, the polarimetric albedo in Lupishko et al. (1995) and Mukai et al. (1997) needs to be updated to match the definition of the geometric albedo.
There have been several efforts to derive the set of constants for obtaining the geometric albedo from the polarimetric slope (Masiero et al. 2012; Cellino et al. 2015; Lupishko 2018), involving better ways to calibrate the data from different sources. We estimated the geometric albedos using the polarimetric slope–albedo law using these sets of parameters, and the results are summarized in Table 4. To the first order, the Gaussian 1- confidence range of is estimated from Eq (11) and
where is the Gaussian 1- confidence range for the parameter . We summarize the geometric albedo derived with different observations and different sets of parameters (see Table 4). These polarimetric results are suitably consistent with the geometric albedo that we derived through the direct photometric observation at opposition.
|Lupishko et al. (1995)||Mukai et al. (1997)|
|Masiero et al. 2012|
|Cellino et al. 2015|
and are constant parameters in the polarimetric slope–albedo law given by . These parameters are determined by different authors in the left column of the above table. We used the slope parameters given in Lupishko et al. (1995) and Mukai et al. (1997).
Finally, we derived the grain size on the surface. The empirical relationships regarding the griain sizes (Geake & Dollfus 1986) include , not . In Section 3.2, we derived in the V-band. From previous polarimetric data, it is found that via the slope–albedo law (Lupishko et al. 1995; Mukai et al. 1997). By comparing with terrestrial rock powders (Fig 4), the grain size is estimated as .
The size is smaller than that of (3200) Phaethon (, Ito et al. 2018) and (1566) Icarus (, Ishiguro et al. 2017), that were calculated by using the same relationship. In addition, we compare the – relationship between these three asteroids and laboratory sample data (Figure 4). It is likely that (3200) Phaethon and (1566) Icarus are covered with grains larger than that of Toutatis, although the other factors such as the porosity of grains can affect the size estimate. These results may suggest that the near-Sun environment at less than approximately would affect the paucity of small grains on the surface.
Note that the observations or calculations of and for these asteroids were sometimes made in different wavelengths from laboratory samples in Geake & Dollfus (1986). Nevertheless, the estimated deviations of them due to the wavelength dependence are only about or smaller than the size of the markers in the figure.
In this work, we analyzed the observed data of an -type asteroid (4179) Toutatis at near-opposition configuration using KMTNet. We obtained a robust estimation of the geometric albedo from the observed radiance factors and the shape model. This value is typical for an S-complex asteroid, and thus shows the future potential of using KMTNet for the opposition monitoring. The other measures using the radiance factor ( and ), which were introduced in previous works, coincide with expected range, though uncertainties are large. The thermal modeling results and previous polarimetric measurements gave consistent results if we use appropriate diameter or empirical relationships, respectively. The albedo at is also determined from the radiance factor, and it was found that Toutatis is covered with particles smaller than the asteroids in the near-Sun environment using the empirical relationship about the grain size.
We thank to the two anonymous reviewers for their valuable comments which led to important improvements of the original draft. The observations on/around the opposition were made using telescopes as a framework of contract research between Seoul National University (SNU) and Korea Astronomy and Space Science Institute (KASI) for the DEEP–South project. This research was supported by the Korea Astronomy and Space Science Institute under the R&D program supervised by the Ministry of Science, ICT and Future Planning.
A Rotational Model
We calculated the projected geometrical cross-sectional area, of the asteroid viewed from observatories (i.e., on Earth) at the epoch of the observations (), assuming arbitrary rotational and precessional offsets, and . In the calculation, we employed the high-resolution shape model888https://sbn.psi.edu/pds/resource/rshape.html (Hudson et al. 2003, 2004). We considered the long-axis mode rotation and precession along in the ecliptic coordinate (Hudson & Ostro 1995). Although these rotation and precession periods are reported as and , respectively (Zhao et al. 2015), it is necessary to perform fine tuning by adding the arbitral offset and to fit the observed magnitudes to the modeled ones because the periods in the model are not determined so exactly to be extrapolated to the epoch of our observations (uncertainty in periods multiplied by the time difference is much larger than the periods themselves). We thus defined the rotational angle along the long-axis and the precession axis ( and , respectively) as follows:
for , where denotes the time of the reference epoch.
We used IDL POLYSHADE to calculate the total projected area for a given pair of with grid size of and and time step size of days. The projected area as a function of the direction to the observer is shown in Fig 5. The color and contour show the projected area viewed by an observer at an infinite distance to the direction specified the body-fixed frame, . As can be seen, an uncertainty of in the viewing direction cause small fractional uncertainty (normally ), which is much smaller than the photometric uncertainty of in Fig 2 and Table 2. Thus, the choice of the grid size of the offsets ( and ) are reasonable.
Each of the 12,960 models for pair is then spline interpolated when necessary. As mentioned in Section 3.1, the total reflected flux is proportional to the projected area when is small. Hence, for each rotational model, we fit such that , where and its uncertainty is (using Eq 3 with fixed ). Here, is the phase-corrected absolute magnitude, , calculated using Eq 3 at epoch . We used intensity unit rather than magnitude, since the uncertainties in magnitude is not Gaussian. The usual chi-square minimization process is not valid when the error-bar has non-Gaussianity. The reduced statistic is calculated for each case by using
where the denominator is the degree of freedom, is the number of data points, and is the -th observational time (Table 2). By changing and , we find such that . The minimum reached is then saved for each model.
The “good” models are selected such that the reduced -statistics is smaller than , where is the minimum of such reduced statistics of all the models. Because we have only one free parameter (), this gives the models with 1- confidence interval. The models are then converted into magnitude system using . The minimum model is the red solid line and other “good” models are black faint dotted lines in Fig 2. We confirmed any change of in the domain of does not significantly distort the results, except for a nearly parallel shift occured due to the nearly constant change (since all ’s are similar) in all values (Eq 3).
B On-Line Materials
This work was mostly carried out by open-source softwares and packages, and we decided to disclose the codes we used to obtain our results as well as this printed material. The original codes are available via GitHub platform999https://github.com/ysBach/KMTNet_Toutatis. It contains 5 notebooks:
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