© The Authors 2023

1 Introduction

Lunar swirls are high-albedo markings that are correlated with localized crustal magnetism with no discernable topography and appear unconnected to any geometrical feature such as craters, rilles, and domes (e.g., Hood & Williams 1989; Blewett et al. 2011). Although lunar swirls are well known, the processes that led to the formation of localized magnetic fields and lunar swirls remain contentious. It is not clear whether these two phenomena are interlinked. The magnetic anomalies with flux densities in the range of few tens to thousands of nanoteslas (Hemingway & Garrick-Bethell 2012; Hemingway & Tikoo 2018) on the lunar surface are considered to be effective in potentially blocking solar wind ions from reaching to the surface. The bright and dark markings of the swirl structure are likely to be correlated with the strength and direction of the magnetic field (e.g., Dyal et al. 1970; Coleman et al. 1972; Hood & Schubert 1980; Richmond et al. 2005; Vorburger et al. 2012). The partial surface shielding from the solar wind flux due to a magnetic anomaly has been considered as one of the most commonly accepted hypotheses of the formation of swirls (Bamford et al. 2016; Hemingway & Tikoo 2018). All swirl locations (except for a small structure near Mare Moscoviense; Hess et al. 2020) have been found to be coupled to a localized magnetic anomaly, but interestingly, not all magnetic anomalies have high-albedo swirl patterns (e.g., Hood & Williams 1989; Blewett et al. 2011; Denevi et al. 2016). The association of swirls with magnetic anomalies suggests that the swirl regolith is less affected by solar wind bombardment and thus offers a unique environment for studying the space-weather effects that mainly result from micrometeorite bombardments. Detailed spectroscopic studies of swirls suggest a difference in space-weather trends in comparison to the surroundings. Space-weathered normal mare surfaces exhibit low reflectance, a weak absorption band depth, and a steep positive spectral slope in the visible to near-infrared (NIR) range (Fischer & Pieters 1994). The swirls brighten across all wavelengths (Pieters & Noble 2016), and the surface hydration in the high albedo regions is low (Kramer et al. 2011; Wöhler et al. 2017; Hess et al. 2020). Recent results derived using hyperspectral imaging in the NIR wavelength domain suggest that magnetic shielding may lead to a longer preservation of swirl features, but this cannot be the only reason for swirl formation (Bhatt et al. 2018; Hess et al. 2020). Hess et al. (2020) examined spectral trends on and around swirls by considering the reflectance model of Hapke (2008) and the space-weather model of Wohlfarth et al. (2019). The observed spectral trends at six swirl locations are found to be consistent with a combination of effects induced by space weathering (Hood & Schubert 1980; Kramer et al. 2011; Blewett et al. 2011) and soil compaction. The soil compaction might be a result of interaction between the regolith and cometary material (Schultz & Srnka 1980; Shevchenko 1993; Pinet et al. 2000; Jeong et al. 2016). Another hypothesis of swirl formation is based on the effect of dust lofting, which allows the settlement of finely grained dust influenced by the magnetic fields and results in high albedo markings (Garrick-Bethell et al. 2011). Recent observations using the 70–80 cm pixel⁻¹ high-resolution digital elevation model of the Mare Ingenii swirls support dust transport as the swirl formation mechanism based on the topographic correlations with on-swirl locations (Domingue et al. 2022).

The Reiner Gamma swirl is one of the best-known swirl features. It is located in the western Oceanus Procellarum, which is the western part of the lunar nearside. This unique swirl has a fully evolved structure of high-albedo curve-shaped line markings that extend across an area of about 200 km². The Reiner Gamma swirl has been examined using observations at a variety of wavelengths with the aim to understand its spectral properties. This understanding might in turn lead to an understanding of its formation mechanisms (Thompson et al. 1970; Kramer et al. 2011; Blewett et al. 2011, 2021; Pieters & Noble 2016; Denevi et al. 2016; Hendrix et al. 2016; Hess et al. 2020). The spectral properties indicate a change in the physical regolith characteristics, which can be examined quantitatively using the polarimetric properties of the regolith obtained across a range of phase angles (Dollfus 1998, 1999). Imaging polarimetry is a powerful tool for a quantitative determination of the regolith structure parameters because the fine structure of the uppermost regolith layer and its opacity cause the polarization behavior (Dollfus 1998). Reiner Gamma is the only swirl feature that can be observed through Earth-based telescopes due to its position and spatial extent. Only a limited set of polarimetric studies have therefore been conducted to characterize the swirl (Pinet et al. 1995; Shkuratov et al. 2008, 2010; Jeong et al. 2016). These studies mainly focused on deriving the fraction P of linearly polarized flux, the total intensity I, and phase-ratio images in order to understand the regolith characteristics in a qualitative manner. The parameters P and I are correlated to the composition, grain size, and roughness (Dollfus 1998) and depend on the wavelength and phase angle. The composition of swirls is considered to be similar to their surroundings (Kramer et al. 2011; Blewett et al. 2011), and the polarization parameters for a swirl analysis therefore directly relate to the physical properties of the uppermost regolith layer. The albedo increases with a decrease in regolith grain size and reduced porosity, whereas P decreases with decreasing regolith grain size, but does not depend on the porosity (Hapke & van Horn 1963; Dollfus 1998). Additionally, the phase-ratio maps directly provide information on the surface roughness. However, the images obtained at different phase angles need to be adjusted to the same level of sharpness to avoid artifacts in the phase-ratio maps. The slope of the phaseangle dependent reflectance function increases with increasing surface roughness and increasing opposition effect strength (Hapke 2012), which directly influences the observed phaseratio values. Bhatt et al. (2021) derived phase-ratio maps by dividing intensity images at 98° by those at 84° and concluded that the phase-ratio anomalies of Reiner Gamma partially correlate with the soil compaction areas spectrally derived by Hess et al. (2020).

The objective of this study is to provide a quantitative assessment of the physical regolith properties at the Reiner Gamma swirl by deriving the opposition effect strength, surface roughness, and mean grain size information in order to shed light on the formation of the Reiner Gamma swirl. For this purpose, we conducted systematic telescopic observations at phase angles ranging from values below 10° to more than 110°, where the maximum polarization P_max is encountered at about 90°–100° and the minimum (negative) polarization P_min at about 10°. Systematic observations like this are important for differentiating between the surface roughness, the single-particle scattering behavior, and the opposition effect strength. We geo-referenced the telescopic polarimetric images to the global lunar coordinate system in order to obtain the single-scattering albedo, photometric roughness, and opposition effect strength using Hapke’s bidirectional reflectance model (Hapke 2012). The derived photometric roughness is used as an input along with the fraction of linearly polarized flux to derive the mean grain size for Reiner Gamma using the framework of Dollfus (1998, 1999). Because the relation between grain size and polarization is based on an empirical calibration that was established considering lunar and terrestrial samples, we also used the framework of Shkuratov & Opanasenko (1992), which is based on the correlation of the albedo with polarization around P_max.

2 Data and methods

We obtained the polarimetric observations using the 1.2 m telescope at the Mount Abu IR Observatory of the Physical Research Laboratory located in the northern Indian state Rajasthan. We used the electron multiplying CCD based optical imaging polarimeter (EMPOL) developed in-house (Ganesh et al. 2020) and acquired multiple sets of polarimetric images of the Reiner Gamma swirl and the ray systems of the craters Kepler and Aristarchus. The observations were collected with the two narrow-band continuum filters GC (green) and RC (red), corresponding to central wavelengths of 525.9 nm and 713.3 nm, respectively (Farnham et al. 2000). The number of steps per filter rotation is 48, corresponding to steps of the relative orientation angle of 7.5°. This provides tight constraints on the derivation of the polarization fraction and the angle of polarization. We observed the Reiner Gamma swirl, Kepler, and Aristarchus primarily at high phase angles, including observations around P_max (Appendix B) at a spatial resolution of ~1/3 arcsec/pix. The observations were conducted between January and March, 2021. The NASA JPL HORIZONS service¹ was used to generate ephemerides for the Moon and for the observing locations. The primary processing of observational data consist of (a) bias subtraction and flat-field correction, (b) image registration and subsequent averaging for each angle of the polarization filter, (c) derivation of polarimetric parameters, that is, intensity I, amplitude A, linear polarization fraction P = A/I, and angle of orientation W, (d) geo-referencing of the telescopic data of the Reiner Gamma swirl to the global lunar coordinate system after empirically correcting for the effect of variable atmospheric seeing, and (e) conversion of intensity images from digital number (DN) to reflectance. The flat-field frames were obtained during twilight for each observation night, corresponding to the RC and GC filters at every step of rotation of the half-wave plate, and the bias frames were acquired by setting the exposure time to zero. Subsequently, the intensity-based image registration procedure was carried out. We first constructed a reference image by coregistering and averaging all frames. Then we coregistered all images to the reference image only by introducing translation without any rotation, scaling, or shear. With this two-stage coregistration procedure, we minimized artifacts in the subsequent processing steps that otherwise might arise due to slight misalignments between individual frames. These misalignments are commonly caused by seeing-induced distortions and/or shifts due to guiding inaccuracies of the telescope. We used these individual sets of coregistered images (Appendix B) to derive the polarimetric parameters I, A, P, and W.

At large phase angles, P is inversely proportional to the albedo. This relation is known as the Umov effect (Dollfus 1998),

(1)

where A is the surface albedo and P_max is the maximum degree of polarization. The value of P_max is observed at phase angles around α = 100° for visible wavelengths (Shkuratov et al. 2015). C₁(d) is a grain-size-dependent parameter and can be considered constant for powdered material with a narrow grain size distribution. The exponent C₂ corresponds to 1 according to the original form of the Umov law, but it was found to be 1.36 for the lunar surface based on an analysis of returned samples (Dollfus 1998). Eq. (1) implies that the grain size is constant when the Umov law is valid. For relative considerations within one data set at a specific phase angle α, the parameter I(α) derived from imaging polarimetric observations can be seen as equivalent to the albedo. Figure 1 shows images of I(α), Ρ(α) and C(α) = P(α)_maxI(α)^1.36 for the Reiner Gamma swirl and the craters Kepler and Aristarchus. We find that the I(α) and P(α) maps of the craters Kepler and Aristarchus are strongly correlated in Fig. 1, but despite approximately identical I(α) values at the tail and the central oval part of Reiner Gamma, the tail has a higher degree of polarization. The variations seen in the images of the parameter C(α) of Kepler and Aristarchus mainly correspond to local topography, and the ray systems of these craters do not show significant variations from their surrounding mare regions. In contrast, strong variations occur at the Reiner Gamma swirl, which is also not associated with a discernible topography. The swirl structure is clearly distinguishable in the C(α) image of Fig. 1, indicating grain size variations with respect to the surrounding surface and also within the swirl structure (tail vs. central oval). The similar range of the variations in parameter C(α) in the region west of the swirl occurs in a rugged highland terrain and can be attributed to topography variation. From Fig. 1, we conclude that the polarization behavior at Kepler and Aristarchus can be explained by albedo variations and the known topography. However, the deviation from the Umov law in the case of Reiner Gamma indicates unique physical properties of the regolith and is further investigated by applying the Hapke model (Hapke 2012) and the framework of Dollfus (1998) and Shkuratov & Opanasenko (1992).

Phase-ratio images, that is, ratios of two I images taken at different phase angles, provide information on the surface roughness and porosity (Kaydash et al. 2009; Kreslavsky & Shkuratov 2003). The slope of the phase-angle-dependent reflectance function increases with increasing surface roughness (Hapke 2012). Figure 2 displays phase-ratio images of the craters Kepler and Aristarchus and the Reiner Gamma swirl between phase angles of 98° and 84°. The phase-ratio image of the Reiner Gamma swirl highlights photometric anomalies. The image sharpness was adjusted to construct the phase-ratio images (see below). The small craters in Fig. 2 and their ejecta have lower phaseratio values than Reiner Gamma. Kepler and Aristarchus show an inverse correlation between phase ratio and albedo (Fig. 1). In contrast, significant deviations from this inverse relation are apparent for the Reiner Gamma swirl, indicating anomalies in the regolith structure. The low phase-ratio values in Fig. 2 correspond to a steep slope of the reflectance function, suggesting a rough surface, whereas high ratio values correspond to a low slope of the reflectance function, indicating a smooth surface. The ray systems of Kepler and Aristarchus do not show these anomalies in the phase-ratio images, suggesting that their grain size and surface roughness do not differ significantly from the surrounding mare surface. The northern tail of Reiner Gamma is brighter in the phase-ratio image than in the central oval, suggesting grain size variations within the swirl structure. The Reiner Gamma swirl clearly has a positive phase-ratio anomaly compared to the surrounding surface. Figures 1 and 2 clearly show the peculiar polarimetric properties of Reiner Gamma. We therefore considered the swirl observations for a more detailed analysis.

Fig. 1 Comparison of Polarimetric parameters derived from the telescopic observations on 2021 February 5 at phase angle α = 106.41° using the RC filter for Kepler (first column), Aristarchus (second column), and Reiner Gamma (third column). The rows from top to bottom show the parameters I(α), (in DN), P(α), and C.

Fig. 2 Phase-ratio maps (98°/84°) of Kepler (left), Aristarchus (center), and Reiner Gamma (right), based on the observations obtained on 2021-January 5 and 2021-January 6 (Appendix B, GC filter). The ratio is shown in logarithmic scale for clarity.

2.1 Sharpness adjustment of the telescopic images

The sharpness level of the telescopic images combined for each polarizer angle is very similar because the images were acquired consecutively. However, the degree of sharpness may change significantly from one day to the next because the atmospheric seeing conditions are highly variable. To obtain consistent results and for an intercomparison of results from individual observations, all acquired images were adjusted to a common level of sharpness by introducing blurring or sharpening to all observational data sets. This is an important preprocessing step because the differences in sharpness between the individual images may cause strong spurious high-frequency structures in phase-ratio images. These unwanted high-frequency components can be minimized by blurring or deblurring to a common sharpness level. Here, blurring all data sets to the blurriest image is a more conservative approach than sharpening all images to the sharpest image, as it suppresses noise instead of possibly amplifying it. We derived both blurred and sharpened versions of the telescopic observations by filtering with appropriate Gaussian filters for blurring and Gaussian Wiener filters for deblurring. For this purpose, the amplitude spectra of small areas of I images were compared in a pairwise manner. The blurriest or sharpest image was set as a reference, and the remaining images were blurred or sharpened by estimating appropriate Gaussian filters. The filter strength was determined by analyzing the two-dimensional amplitude spectra of parts of the images. Appendix A provides the detailed steps conducted for sharpness adjustment.

2.2 Georeferencing of telescopic images

Hapke’s bidirectional reflectance model (Hapke 2012) allows us to estimate the physical characteristics of the surface. The model includes parameters that relate to the physical properties of the surface material. To apply the Hapke model to telescopic observations, we georeferenced the I images derived from the telescopic observations of Reiner Gamma to the lunar coordinate system. For this purpose, we constructed a hemispherical map with 64 pixels per degree of the Moon constructed from Lunar Reconnaissance Orbiter (LRO) Wide Angle Camera (WAC) image data (Speyerer et al. 2011) using orthographic projection, assuming the same observation conditions (subsolar and subobserver point) as during the telescopic observations. We applied a Gaussian blur to the WAC images of Reiner Gamma to bring it to the same level of sharpness as the telescopic observations. We used the high-pass filtered components of the observed images and the reference WAC global mosaic, with an adjusted standard deviation given by

(2)

where I_r is the WAC reference image, I_g is the reference image blurred with a Gaussian filter with a standard deviation σ = 2, and I_h is the resulting high-pass-filtered reference image. The standard deviation of I_h is given by σ(I_h), and I_adj is the final image with the adjusted standard deviation that was used for automatic registration. The geographical extent of the WAC mosaic was selected between (20° N, 70° W) and (0° N, 45° W), as all the telescopic observations are comprised by these limits. Two control points were manually selected in each observed image for an initial transformation, and I was then adjusted by minimizing the mean squared deviation between the high-pass-filtered image sets of the I image and the reference WAC mosaic in an automated way. We resampled our georeferenced telescopic observations to a uniform ground-projected sampling interval of 30 pixels degree⁻¹ by transforming them into a cylindrical projection, as shown in Fig. 3. The obtained resolution is sufficiently finer than any details resolved in the I images. A border of 10 pixels was discarded from each projected I image to reduce artifacts caused by the preceding filtering operations.

Fig. 3 Reiner Gamma telescopic observation in RC filter from 5th March, 2021, georeferenced to the LRO WAC mosaic (Speyerer et al. 2011). The homogeneous regions considered to convert DN into reflectance are marked in green.

2.3 Conversion into absolute reflectance

Sato et al. (2014) used 21 months of LRO WAC repeated observations and derived resolved Hapke parameter maps of the Moon in multiple wavelengths. Sato et al. (2016) compared the integrated set of Hapke parameters from Reiner Gamma with highlands and fresh crater ejecta in the vicinity and found an increase in single-scattering albedo and surface roughness for the central region of the Reiner Gamma swirl. However, the tail structure of Reiner Gamma was not examined in this analysis, and an integrated set of Hapke parameters was provided for the complete swirl structure. Here, our attempt is to derive the first set of pixel-resolved maps of the surface roughness and opposition effect strength from our telescopic data of varying phase angles using the Hapke model (Hapke 2012). A more detailed description of the Hapke model is given in Appendix C. We selected two reference regions around the Reiner Gamma swirl, which are covered in all telescopic images and belong to mare areas (Fig. 3). The selected reference regions do not include fresh craters with ejecta and are thus considered to be homogeneous. The highest and lowest quartile of the digital number (DN) values from the reference regions were removed in order to minimize outliers. We assumed a linear relation between DN and surface reflectance (Mei et al. 2016) and used the empirical line model (Smith & Milton 1999) to convert DN into reflectance,

(3)

where L_ri is the DN value corresponding to the pixels in an image i, k_i is a slope parameter calculated from the image i, and b is the offset term. We always neglected the offset term because we assumed that any DN offset has been removed by dark-frame subtraction. The average reflectance of the reference regions for each intensity image was then calculated by applying Hapke’s reflectance model, which has a set of free parameters (w, b, c, B_C0, h_C, B_S0, h_S, , and ϕ; see Appendix C).

Since we have a limited observational data set, we simplified the Hapke bidirectional reflectance model based on assumptions from previous works (Sato et al. 2014; Hapke 2012) for the reference regions. The assumed free parameters and their values are listed in Table 1. The single-scattering albedo w listed in Table 1 is wavelength dependent and was derived from the Moon Mineralogy Mapper (M³) level 1B data (Pieters et al. 2009) using the framework of Wöhler et al. (2017). The M³ derived w values of the reference regions were extracted from the Reiner Gamma albedo mosaic prepared by Hess et al. (2020) at the wavelengths corresponding to the central wavelengths of the GC and RC filters. The roughness parameter was set to 23.4° to avoid mathematical coupling of w and (Sato et al. 2014), and the porosity coefficient K was kept constant at 1. When we took the overall opposition effect into account, we did not differentiate between the individual contributions of the coherent backscatter opposition effect (CBOE) and the shadow-hiding opposition effect (SHOE). The CBOE theoretically dominates only at very low phase angles (g < 3°), (Shepard & Helfenstein 2007) and mainly depends on the transparency of the medium and the properties of subscatterers within larger particles (Hapke 2012). Thus, we set the CBOE strength to a constant value of 1 to avoid a competition between CBOE and SHOE. Because only a limited range of phase angles is available, the angular width h_s of the SHOE could not be determined and was kept constant to 0.05, as determined by Sato et al. (2014). The particle phase function was assumed to be constant over the telescopic coverage, and the parameter b that controls the shape of the forward-and backward-scattering lobes was kept constant at 0.235, corresponding to a typical mare setting (Sato et al. 2014). Based on the analysis of a laboratory sample, the Henyey-Greenstein double-lobed single-particle phase function (Eq. (C.2)) parameter c can be derived as a function of b (Hapke 2012; Sato et al. 2014),

(4)

This relation holds good for the equatorial lunar regions in which Reiner Gamma is located (Sato et al. 2014). The amplitude B_so of the SHOE is linked to the transparency of the grains and was approximated as

(5)

where the values α = 2.459 and β = 0.078 were obtained by an empirical regression to equatorial observations across the lunar surface at a wavelength of 643 nm (Sato et al. 2014). In order to convert the DN of the reference regions marked in Fig. 3 into reflectance, the slope k_i was derived from Eq. (3) such that the squared difference between pixel values in DN and corresponding reflectance value derived from the Hapke model was minimal. The total intensity (I) images were multiplied by their respective slope k_i factors to convert them into physical reflectance r_si (Eq. (3)).

Fig. 4 Flow chart of the telescopic data processing and the Hapke parameter fitting processes.

2.4 Hapke parameter analysis of Reiner Gamma

The georeferenced reflectance image sets in the RC and GC filters were used to derive the photometric roughness , the single-scattering albedo w, and the SHOE amplitude (opposition effect strength) B_so by following a multistep fitting process, resulting in an improved accuracy of the identified parameters. A factor was introduced to compare the B_S0 value obtained by model fitting to the empirical w-dependent estimate based on Eq. (5),

(6)

The value of represents the albedo-corrected amplitude of the SHOE.

All data processing steps performed on the imaging polari-metric observations of Reiner Gamma are summarized in Fig. 4. We reduced Hapke’s nine free parameters (w, b, c, B_C0, h_C, B_S0, h_s, , and ϕ) to a subset of three parameters , w, and B_S0 by keeping all other parameters constant, in accordance with a reference mare region during the fitting process (Table 1). The three free parameters were estimated per pixel by root mean square (RMS) minimization. Thus, we computed the mean squared error between the reflectance images using Eq. (7) as

(7)

with r_si as the reflectance of every image i, and r_model,i as the calculated model reflectance. The total number of images with valid data for a specific image pixel is given by n. This error function was used to determine the best-fitting single-scattering albedo Ω. The parameters and Ω are mathematically coupled and can have similar effects on the shape of the reflectance curve. This means that generally, an increase in enforces a high Ω (Eq. (C.1)). To decouple Ω and , we considered the reflectance-ratio images at two different phase angles (R_α1 /R_α1, α1 > α2). Figure 5 demonstrates that the influence of w is minimized by taking phase ratios between pairs of images.

Reiner Gamma is distinguishable in the 62°/9° phase-ratio image, whereas the structure nearly disappears in the 89°/62° phase-ratio image. The reflectance decreases more slowly with increasing phase angle of the swirl than on the surrounding mare surface, suggesting that the opposition effect is weaker on the swirl. As the ratios are largely independent of albedo and are highly sensitive to variations in the surface roughness and the opposition effect strength, we derived and from phaseratio images of all image pairs whose difference in phase angle exceeded 10°. The mean square error of the pairwise ratios we used to fit the Hapke parameters is

(8)

where e_r is the mean squared error of the pairwise ratios. The sum was calculated over m image pairs p, where r_s,p1 is the reflectance from the image with higher phase angle, and r_s,p2 is from the image with lower phase angle. Equivalently, r_model,p1 and r_model,p2 are the calculated model values with the same viewing geometry as presented in the images.

Equations 7 and 8 were used in an alternating manner to identify the free parameters using the multistep fitting process. Initial values were set to the mare reference values (Table 1). As the first step of the fitting process, the ratio pair error of Eq. (8) was minimized by optimizing Ω and and by setting to a constant value of 1, indicating no deviation from the empirically derived B_S0,e. The obtained value of Ω is not expected to match the true value very accurately in this step of the fitting process because Eq. (8) is only weakly sensitive to it. As the second step of the fitting process, the error function e_r of Eq. (7) was minimized to optimize the value of w while and were kept constant. In the last step of the fitting process, and were optimized while w was kept constant. The last two steps were repeated until the result converged. When no convergence was reached after 20 iterations, the process was aborted and the parameters were set to “invalid” for that specific pixel.

Fig. 5 Ratios of reflectances at different phase angles in the RC filter. The swirl structure is visible in the left ratio image, but is significantly reduced in the right ratio image.

3 Results

3.1 Reiner Gamma: Photometric parameter maps

We fit the Hapke model for a subsection of the data centered on Reiner Gamma in the extent 4.5°N to 11.3°N and 61.3°S to 54.5°S. This extent covers the central oval part and northern tail of the swirl and is covered in most of the observations. The three free parameters of the Hapke model Ω, , and B_S0 were fit, keeping all other free parameters set to those of a typical mare surface. The derived sets of resolved parameters are shown in Figs. 6 and 7 for the RC and GC filters, respectively, by processing the intensity images, considering images adjusted to the most blurred and the sharpest available intensity image from the covered range of phase angles. The spatial resolution of the resolved parameter maps is ~ 1 km, with comparable trends for the blurred and deblurred data sets. The lines visible across the tail of Reiner Gamma swirl in the parameter maps are due to the inclusion of multiple images with partial coverage of the tail area, so that the parameters were fit to a variable number of images. The image set processed considering the sharpest available intensity image as a reference provides more details without a significant increase in noise. Figures 8 and 9 visually compare and summarize the distribution of the Hapke parameters that were extracted from central oval, the northern tail, and the southern mare region (marked in Fig. 7) for the RC and GC central wavelengths, respectively. The statistics are based on the selection of regions of interest (ROIs) extracted from the central oval and northern tail of the swirl and the mare region south of the swirl. We merged the values of the blurred and deblurred data sets at each wavelength to generate this statistics. Interestingly, the central oval and tail of the swirl show distinct statistics for the RC and GC filters when compared to the surrounding mare.

The w (single-scattering albedo) maps of the Reiner Gamma swirl (Figs. 6 and 7) showed relatively higher Ω for the swirl than the mare region that surrounds it. More interestingly, within the swirl structure, we find that the albedo of the tail is about 25% lower than in the central oval for the two filters RC and GC (Figs. 8 and 9). The Reiner crater rim east of the swirl appears bright in the w map mainly due to topographic effects that are present and dominant at the crater rim. Topographic corrections were not taken into account while implementing the Hapke model because our objective was to derive the photometric parameters for the swirl, which do not have a significant topographic impression (Blewett et al. 2011). Sato et al. (2016) derived values of Ω in the range 0.25–0.3 for the central oval of Reiner Gamma, whereas in our case, the derived values for Reiner Gamma are in the range 0.12–0.16. This discrepancy is mainly due to the lower absolute average Ω value used for the reference mare region derived from M³ data during the conversion of DN into reflectance. The overall Ω variation trends from surrounding mare to Reiner Gamma swirl in Figs. 6 and 7 are comparable to the findings of Sato et al. (2016). According to Sato et al. (2016), the Ω values at the Reiner Gamma swirl are comparable to the highlands west of the Reiner Gamma swirl. However, this comparison in our study is limited to the coverage, which mainly includes the Reiner Gamma swirl and its surrounding mare regions. The Ω parameter is systematically lower for the GC filter than for RC filter values. This trend is expected with decreasing wavelengths. Interestingly, the relative increase when compared from the surrounding mare to the swirl remained the same in both cases.

By adapting the Hapke model to the phase-ratio images, we constructed the and maps. The derived values of and B_S0 are lowest in the oval in comparison to the tail and the surrounding mare region. The variations in and B_S0 are comparable for the RC and GC filter. The parameter expresses the albedo-corrected SHOE strength. For the surrounding mare, the median value of is ~0.95 (Figs. 8 and 9), suggesting a slightly weaker SHOE than predicted by Eq. (5). The central oval has a median value of that is still lower than the surrounding mare. In contrast, the median of exceeds 1 for the northern tail of Reiner Gamma (~1.2 in Figs. 8 and 9), suggesting a significantly stronger albedo-corrected SHOE than in the surrounding mare. The theoretical limit of B_S0 is 1.0, but the derived values are higher than the theoretical limit because of the simplification of the CBOE term of the Hapke model we employed. The B_S0 map follows the trends, and the derived values are comparable for both filters, suggesting that the effect of a variable phase function is not significant across the observed wavelengths. The and B_S0 maps show a clear difference between the central oval and the tail, suggesting that the opposition effect differs from the central oval to the northern tail, even though both regions have a higher albedo than the surrounding mare.

The photometric surface roughness is expressed by the slope angle of the surface irregularities integrated over all scales from the millimeter domain to the pixel size. The maps in the RC and GC wavelengths only show small variations, with a slight increase in roughness for on-swirl locations. However, the box plot of in Figs. 8 and 9 clearly shows that the median roughness of the northern tail is different from the central oval and the surrounding mare. Although the outliers are higher in the RC domain than in the GC domain, the trends are the same.

Considering the background mare as a reference, the tail is ~5% lower in surface roughness, whereas the central oval is ~5% higher in surface roughness. The significantly higher surface roughness values of small fresh craters and parts of the crater Reiner cannot be considered as a reference because we did not apply topographic corrections to the telescopic data based on the assumption that the swirl is a surficial feature (Blewett et al. 2011).

Fig. 6 Derived Hapke parameter maps. The telescopic images were processed by blurring to uniform sharpness. The top row shows the parameters for the RC filter, and the bottom row shows those for the GC filter.

Fig. 7 Derived Hapke parameter maps. The telescopic images were processed by sharpening to uniform sharpness. The top row shows the parameters for the RC filter, and the bottom row shows those for the GC filter. The Hapke parameters extracted from the regions marked in the RC albedo image are plotted in Figs. 8 and 9 for the RC and GC filters, respectively.

3.2 Grain size map of Reiner Gamma

The degree of polarization P is related to the microstructure of the surface. The phase angle P_max corresponding to the maximum value of P decreases as the grain size becomes smaller (Dollfus & Titulaer 1971). We extracted the polarization curve showing P as a function of the phase angle from four different locations on and around Reiner Gamma, as shown in Fig. 10. The profiles plotted here are average values of 10 to 20 pixels from four different regions: the eastern and western sides of central oval, the northern tail, and the surrounding mare. The P(α) values correspond to the positive branch of polarization, where the polarization direction is perpendicular to the scattering plane. In Fig. 10, P is variable across the swirl and also differs from its surrounding mare region. The eastern part of the central oval has the lowest polarization fraction, with a P trend similar to the northern tail structure of Reiner Gamma. The polarization fraction of the western central oval of Reiner Gamma is different from other on-swirl locations, but comparable to the surrounding mare. The maximum polarization P_max occurs at about 100° phase angle for the western part of the central oval and the surrounding mare, whereas we do not find a clear P_max value for the northern tail and eastern part of the central oval due to the lack of data for phase angles exceeding 120°.

At large phase angles, P follows the Umov effect described by Eq. (1). The parameter C₁(d) of Eq. (1) is grain size dependent and increases with increasing P. It is important to estimate the errors associated with P before deriving the grain size variation. We used high phase angle observations at phase angles in the range 70° to 100° and conducted an error analysis by fitting the pixel-wise polarization fraction P and angle of orientation Ω to a set of artificial observations with a realistic noise level, keeping the polarization fraction fixed at a specific phase angle. We found that the standard deviation of the polarization fraction P(α) is about 0.0026, independent of the true polarization fraction and similar for both filters RC and GC. In contrast, the standard deviation of Ω varies from 0.74° to 0.49° as P changes from 0.1 to 0.15, regardless of the phase angle. The measured value of P(a) is an integrated response of a mixture of small and large regolith grains, which is realistically derived based on laboratory experiments. Dollfus (1998) used measurements of P_grain/P_max of Apollo samples obtained as a function of phase angle to characterize the fine structure of the soil at submillimeter scales, represented by the median grain size M_d. Dollfus (1998) demonstrated that the observed P is a combination of the albedo A, the surface roughness , and the grain-size-dependent parameter C₁(d) produced by variations in the modulation ΓC₁(d) of C₁(d), where the contrast of a quantity x relative to a reference x₀ is given by Γx = (x − x₀) / (x + x₀). We applied the framework of Dollfus (1998) to derive ΓC₁(d) with respect to the reference mare region (green boxes in Fig. 3). We used the laboratory-based calibration given by Dollfus (1998) for a wide range of grain sizes and derived C₁(d) and M_d for the Reiner Gamma swirl region. For this purpose, we used as derived by the Hapke model fit and computed the albedo-independent parameter Q₀ by using the linearly polarized flux and intensity as

(9)

where P and I are measured from the telescopic data, C₂ = 1.36 (Eq. (1)), and P(α)/P_max derived for the reference region by applying a polynomial fit to Apollo samples (Dollfus 1998). The parameter C₁ (d) was obtained from the equation

(10)

given by Dollfus (1998), where Γ{C₁(d)} and Γ{Q₀} are the contrasts of C₁(d) and Q₀, respectively, with respect to the reference mare area. The quantity R(L, l, α)_(25°,0°) is a roughness-sensitive coefficient at the selenocentric optical longitude L and latitude l dependent on the phase angle α. We derived R(L, l, α)_(25°,0°) from our telescopic observations based on a polynomial fit to the laboratory measurements of Dollfus (1998), and Γ{θ, θ_ref} is given as [(θ − 25)/(θ + 25) − (θ_ref − 25)/(θ_ref + 25)], with θ_ref as the value for the reference region. Here we neglected the contribution due to multiple scattering because it is significant only for large Γ{Α} (Dollfus 1998), whereas the albedo variations within the swirl are small. The ratio P(α)/P_max is a phase-angle-dependent coefficient, where we inferred P_max = 0.138 based on a polynomial fit to the reference phase-angle-dependent polarization curve P(α), (Table I of Dollfus 1998). The coefficient F(α)/F_max of the sensitivity to the grain size is also tabulated by Dollfus (1998), considering Apollo 11 and 12 lunar samples and finely pulverized olivine basalts. A third-order polynomial fit was computed for the M_d and C₁ (d) values computed for lunar soils distinguishable by their maturity index (Dollfus 1998). The median grain size M_d was derived from the telescopic observations considering the lunar sample based the relation for C₁(d) from Eq. (10).

Figure 11 shows the maps of the parameters C₁(d) and M_d that were computed using the described framework of Dollfus (1998), taking the average of all observational data sets of a specific wavelength domain (Appendix B). The differences between the background mare and the swirl in (Fig. 11) indicate variations in surface texture, even though they share the same composition (Blewett et al. 2011). The distribution of C₁(d) and M_d is similar for the RC and GC filters. Within the swirl structure, the median grain size increases to up to 120 µm from the background mare level of ~45 µm, which is the typical value of mature regolith (Dollfus 1998). The western part of the central oval is in the same range as the background mare soil. In contrast, in the eastern part of the central oval, the maximum M_d value of ~120 µm is observed, while the northern tail exhibits grain sizes in the range 50–70 µm. The standard deviation of the median grain size is estimated by varying the standard deviation of the Hapke roughness and P(α). We found that the median grain size varies from 4 µm to 7 µm for P(α) changing from 0.1 to 0.15. However, increasing the Hapke roughness from near-zero to a high value only has negligible influence on the median grain size uncertainty values.

The derived C₁(d) and M_d maps are based on the correlations between the laboratory polarimetric measurements on a wide range of terrestrial samples, meteorites, and lunar samples and the structural characteristics of these samples, suggesting that this empirical relation should be used carefully. Therefore, we also applied the framework of deriving grain sizes proposed by Shkuratov & Opanasenko (1992) to our polarimetric observations of Reiner Gamma. Shkuratov & Opanasenko (1992) used the grain size parameter b, which is controlled by particle size and microstructural characteristics of the surface and is expressed by

(11)

where a and b are coefficients of a regression equation that is wavelength dependent and derived based on the linear correlation of the P_max and albedo A. Shkuratov & Opanasenko (1992) found that the structure of P_max and A relation may provide important information about the physical factors involved and established an empirical relation between the grain size d and the grain size parameter b based on measurements of lunar samples of different grain sizes,

(12)

where the grain size d is in micrometers and the relation holds only for phase angles around the polarization maximum. We applied the Shkuratov & Opanasenko (1992) framework to our telescopic data of Reiner Gamma obtained at a phase angle of 106.41° for the RC filter, as shown in Fig. 12. The GC filter data quality at the same phase angle is not adequate for grain size estimation. All other observations listed in Table B.1 around 90° phase angle only provide partial coverage of the Reiner Gamma swirl and could not be used to implement the framework of Shkuratov & Opanasenko (1992). The relative grain size variations from the central oval to the northern tail are compared in Figs. 11 and 12, and the trends are consistent between the two different filters used for polarimetric imaging. However, the absolute grain size values inferred using the two different calibrations based on laboratory measurements differ, suggesting that the empirical relations established between the physical parameters and polarimetric properties are dependent on the nature of the surface material. A broad range of lunar returned samples should be considered to establish an empirical relation between grain size and polarimetric properties. Extensive laboratory-based polarimetric measurements are required in this direction that use lunar returned samples and lunar analog materials.

Fig. 8 Box plot of the Hapke parameters extracted from central oval, the northern tail, and the mare region south of Reiner Gamma (area marked in Fig. 7) using the RC filter. The boxes and whiskers indicate the 50th and 95th percentiles in the distribution, respectively. Extreme outliers are shown with the plus symbol. The median value is shown by the horizontal line inside the box.

Fig. 9 Box plot of the Hapke parameters extracted from central oval, the northern tail, and the mare region south of Reiner Gamma (area marked in Fig. 7) using the GC filter. The boxes and whiskers indicate the 50th and 95th percentiles in the distribution, respectively. Extreme outliers are shown with the plus symbol. The median value is shown by the horizontal line inside the box.

Fig. 10 Polarimetric phase function in the GC domain for pixels extracted from (a) the northern tail, (b) the western part of the central oval, (c) the eastern part of the central oval, and (d) the off-swirl mare to the south. The dots represent the average value of P, and the solid lines are second-degree polynomial fits to the data points around the maximum polarization angle, excluding phase angles lower than 60°. The blue and red dots represent the observations in the negative and positive branch of the polarization, respectively. The error bars indicate the 1σ uncertainty of the polarization fraction, which is independent of the polarization fraction itself.

Fig. 11 Maps of the grain-size-dependent parameter C1(d) and the median grain size Md in µm of Reiner Gamma swirl using the framework of Dollfus (1998) for the RC (a and b) and GC (c and d) filters.

4 Discussion

The systematic telescopic observations conducted of the Reiner Gamma swirl using an imaging polarimeter enabled us to derive Hapke parameter maps and grain size maps of the bright part of the swirl. These newly derived maps provide important information for understanding the variations in regolith physical properties within the swirl. The derived Hapke parameter maps with a spatial resolution of ~1 km of Reiner Gamma shows that the microstructure of the northern tail varies with respect to the central oval of Reiner Gamma. These results are comparable to the recent outcomes presented by Wöhler et al. (2023) in the form of relative grain size maps. In general, the linear polarization flux and total intensity were used to characterize the lunar surface using polarimetric measurements (e.g., Dollfus 1998, 1999; Shkuratov et al. 2013, 2018; Jeong et al. 2016). The telescopic polarimetry of the Reiner Gamma swirl conducted by Jeong et al. (2016) focused on the distribution of these two parameters to understand relative changes in the regolith properties of the Reiner Gamma swirl with respect to maria and highlands. Jeong et al. (2016) concluded that the higher linear polarization flux of the swirl compared to the total intensity is mainly due to the collapse of regolith microstructures. Instead of favoring any of the three proposed formation mechanisms of lunar swirls (Sect. 1), Jeong et al. (2016) argued that the anomalous photometric properties of the swirl might also be due to unique compositional properties. The dust levitation model of swirl formation (Garrick-Bethell et al. 2011) suggests that on-swirl locations are enriched in feldspathic material. However, the M³ indicated spectral character of Reiner Gamma suggests variability in the absorption band depth rather than in the band center position, which is indicative of very similar mafic mineralogy across the swirl (Kramer et al. 2011; Bhatt et al. 2018; Hess et al. 2020). The band depth is mainly affected by grain size and/or soil maturity, although opaque minerals without absorption features may affect the composition as well. However, spectral studies of the same region by Hess et al. (2020) do not support an increase in opaque minerals on Reiner Gamma, which would otherwise darken and flatten the NIR reflectance spectra. Domingue et al. (2022) found topographic correlations for swirls in Mare Ingienii and suggested enhanced transport of intermediate-size (< 10 µm) dust particles in near-surface plasma sheaths introduced by a local magnetic field, resulting in destruction of the fairy castle structure of the regolith (Hapke & van Horn 1963) due to grain sorting. However, in this swirl formation scenario, we would not expect the bright part of the swirl with its largely uniform magnetic field strength to show variations in grain size (Figs. 11 and 12, eastern and western side of the central oval region). The spatial resolution of our telescopic polarimetry images prevents us from estimating the grain size distribution at craters within the on-swirl locations, which would otherwise help us to understand the dust transport mechanisms at on-swirl locations because electric fields at the crater edges and sharp shadow boundaries might influence the dust motion on local scales.

The detailed spectral analysis of the Reiner Gamma swirl conducted in the UV-VIS-NIR wavelength region suggested that the spectral characteristics of the on-swirl locations were similar to the surrounding fresh craters, supporting the hypothesis that the swirls were formed as a result of deflection of the solar wind by local magnetic fields (Blewett et al. 2011; Denevi et al. 2016; Kramer et al. 2011). The solar wind protons that are deflected away from on-swirl locations are deposited onto the off-swirl locations, which enhances the spectral darkening effect of space weathering (Blewett et al. 2011; Kramer et al. 2011). Hess et al. (2020) considered on-swirl and off-swirl spectra and modeled the effects of space weathering, but were unable to reproduce the off-swirl spectrum by artificially space weathering the on-swirl spectrum. Additionally, Hess et al. (2020) found that the swirl structure associated with a weak magnetic field in Mare Moscoviense does not show any difference in the 3 m absorption band with respect to the surrounding mare surface. Typically (i.e., in the presence of a magnetic field), it is weaker at swirl locations than in their surroundings, uncorrelated with the time of day, suggesting that the magnetic field shielding alone cannot explain on-swirl versus off-swirl spectral trends.

According to Blewett et al. (2011), the spatial extent of swirls is not always limited to the extent of the magnetic field. The swirl structure may extend outside the localized magnetic field, and the swirl patterns do not correlate with the strength of the magnetic anomaly. Hess et al. (2020) empirically derived a compaction significance spectral index dependent on the relative presence of reduced space weathering and soil compaction. This study found higher compaction for the northern tail and the western part of the central oval of Reiner Gamma in comparison to the eastern part. While comparing our grain size maps (Figs. 11 and 12) with the compaction map of Hess et al. (2020), we found a good correspondence between these two independently derived maps, suggesting that the spectrally derived information agrees well with the polarimetry-based characterization of the fine texture of the soil. The significant decrease in opposition effect strength and marginal reduction in surface roughness for on-swirl locations compared to the background mare suggests a smoothing and compaction of the uppermost regolith on the swirl. Integration of our newly derived Hapke parameter maps of Reiner Gamma with the spectral analysis of Hess et al. (2020) suggests that the magnetic anomaly is important to restrict the space weathering of swirls, but is not sufficient to form these features.

A cometary origin of swirls can explain the high albedo and the unique regolith texture found within the on-swirl locations (Schultz & Srnka 1980; Bruck Syal & Schultz 2015). Bruck Syal & Schultz (2015) modeled the impact of a comet nucleus and inferred plume-driven processing and extended zones of scouring in case of high-velocity cometary impactors. The hypervelocity collision of the dusty plasma environment of the inner coma provides a mechanism that generates magnetic anomalies associated with swirls. Swirls can be considered as the product of cometary impacts in which the microstructure of the regolith has been destroyed, leading to lower backscattering and high albedo with a slight change in roughness, as shown in Figs. 7 and 6.

An analogy of this behavior of the interaction between highvelocity cometary gas and the uppermost regolith layer is the regolith near a lander, which is affected by the jet of the landing rocket. Wu & Hapke (2018) studied the interaction between the jet of the Chang’e-3 landing rocket and the lunar surface based on a series of spectra acquired from the area affected by the Chang’e-3 lander in the VIS-NIR wavelength range. They concluded that the surface roughness does not change significantly between the disturbed and the undisturbed regolith. Our result of similar surface roughnesses on and off-swirl supports the findings of Wu & Hapke (2018) and Glotch et al. (2015). However, the variations in surface roughness, albedo, and amount of backscattering within the bright region of Reiner Gamma along with the significantly varying grain sizes in Figs. 11 and 12 cannot be attributed to a change in maturity alone. Phase-ratio images of high spatial resolution derived using the LRO Narrow Angle Camera data sets at several landing sites suggest that the regolith structure changes mainly due to the interaction between the jet of the landing rocket and the regolith, and the maturity degree should be considered as a secondary effect (Shkuratov et al. 2013, 2018).

Our results show that the high albedo of Reiner Gamma can be explained by its connection to the associated strong magnetic field, but the change in the amplitude B_S0 of the SHOE and the median grain size with a minor change in surface roughness is better explained by favoring the comet collision model (Bruck Syal & Schultz 2015) for the formation of Reiner Gamma. According to Shevchenko (1993) and Bruck Syal & Schultz (2015), the formation of the Reiner Gamma swirl is a recent event, a result of the high-velocity impact of a comet nucleus, which would produce large near-surface dynamic pressures, capable of creating extended zones of scouring that destroyed the fairy-castle structure of the regolith through soil compaction and reduction of shadowing effects, resulting in a high albedo and variable grain sizes. Our results suggest that the compacted soil of the Reiner Gamma swirl caused the unique forward-scattering nature of the swirls that was also observed in several previous studies.

Fig. 12 Maps of the grain size parameter b and the grain size d in m of the Reiner Gamma swirl using the framework of Shkuratov & Opanasenko (1992) for the RC filter. The observation was conducted at a phase angle of 106.41° (Table B.1).

5 Conclusion and outlook

Telescopic imaging polarimetry of the Reiner Gamma swirl was acquired at wavelengths of 526 nm and 713 nm from Mount Abu Observatory, India, primarily at moderately high phase angles (Appendix B) at a spatial resolution of ~1.2 arcsec pixel⁻¹. We derived the intensity, amplitude, and linear polarization fraction for each set of observations. We found an expected linear relation between the intensity and the linear polarization for the craters Kepler and Aristarchus, but a higher linear polarization at the tail and in the central oval part of Reiner Gamma, even though approximately the same intensity range holds at these locations. The grain-size-dependent parameter derived according to Dollfus (1998) and Shkuratov & Opanasenko (1992) clearly demonstrates unique regolith physical properties with respect to the surrounding surface and also within the swirl structure. In order to apply the Hapke model (Hapke 2012) to our telescopic data sets and for an intercomparison of results, we corrected for the effect of atmospheric seeing by applying appropriate Gaussian filters for blurring and Gaussian Wiener filters for deblurring in order to unify the sharpness level of all acquired data sets. The georeferenced telescopic data sets were converted into reflectance by applying the Hapke model to selected reference regions that were covered in all sets of telescopic observation. We used the M³-derived single-scattering albedo of selected reference areas and obtained the reflectance for every pixel by applying the empirical line model. From the set of Hapke free parameters, we modeled Ω, B_S0, and by following a multistep fitting process, resulting in an improved accuracy of the identified parameters. These three free parameters were estimated for each individual pixel by RMS minimization while all other free parameters were kept to constant values based on Sato et al. (2014). Additionally, we computed the albedo-corrected amplitude of the SHOE, defined as parameter . The photometric parameter maps of Reiner Gamma at 526 nm and 713 nm show relatively higher Ω for the swirl than in the background mare region and about 25% lower at the northern tail than in the central oval of Reinner Gamma. The B_S0 map follows the trends at both wavelengths and shows a clear difference between the central oval and the tail. The maps show only a slight increase for on-swirl locations when compared to background mare. However, within-swirl variations are more prominent with the tail, showing an ~5% lower surface roughness than the central oval.

We derived the grain size distribution of the Reiner Gamma swirl by adopting the framework proposed by Dollfus (1998) and applying it to the average of all observational data sets of a specific wavelength domain (Appendix B). Within the swirl structure, the median grain size increases to up to 120 µm in the eastern part of the central oval, in comparison to ~45 µm in the background mare. The northern tail exhibits grain sizes in the range 50–70 µm. The western part of the central oval exhibits grain sizes in the range of the background mare. We also used the framework proposed by Shkuratov & Opanasenko (1992) to estimate the grain size variability across the swirl. Both independent approaches provide consistent results in terms of relative changes from northern tail to the central oval. The Hapke parameter and median grain size analyses carried out in this work provide independent information about the lunar surface, suggesting variations in the microstructure of the regolith of Reiner Gamma within the on-swirl locations. Integrating our findings with the spectral analysis of Hess et al. (2020), we confirm the occurrence of surface alteration processes that might have disrupted the regolith microstructure on the Reiner Gamma swirl. Our results favor a cometary origin of swirls over other competing models, followed by preserved trends due to shielding by the crustal magnetic field that reduces solar wind interaction of the surface.

The Earth-based polarimetric observations of the swirls are limited to Reiner Gamma as all other pronounced swirls are located on the far side of the Moon. The first orbiter-based polarimetric measurements of the Moon are planned to be conducted by the wide-angle polarimetric camera (PolCam) on board the recently launched Korea Pathfinder Lunar Orbiter (KPLO) mission (Sim et al. 2020; Jeong et al. 2020). Polarimet-ric observations on global scales in UV-VIS wavelengths for a range of phase angles at a scale of fewer than 100 m pixel⁻¹ will be obtained for the first time. It will be interesting to extend our approach of the Hapke parameter study and grain size estimation to the global distribution of swirls using PolCam data to understand the differences in the regolith properties in more diffused swirl structures situated under varying strength of the magnetic field. The Hapke parameter maps of Reiner Gamma derived at a spatial resolution of ~ 1 km may serve as a ground truth for the data validation and correction for PolCam. Our telescopic observations may also be integrated with PolCam measurements to construct the polarimetric phase curve of Reiner Gamma.

The polarimetric phase curve of the lunar surface shows a negative minimum around 10° phase angle (Shkuratov et al. 2008). We will conduct a set of low phase angle measurements of Reiner Gamma at the Mount Abu Observatory. The behavior of Reiner Gamma at small phase angles and at high spatial resolution has rarely been studied over an extended range of phase angles. The addition of measurements at low phase angles will also be useful to constrain the opposition effect width and to improve the accuracy of the phase function model. Furthermore, we will relate our observations to laboratory-based hyperspec-tral polarimetric measurements of Apollo soils (Sun et al. 2022) to estimate the on-swirl and off-swirl grain size distributions of the regolith using measurements from the positive and negative polarization branch. Our near future observation plan includes the wavelength dependence of the Hapke parameters and its possible correlation with the physical properties of the regolith.

Acknowledgements

The authors thank the reviewer and the editor for their comments and suggestions. We downloaded the LRO WAC mosaic from the public domain. We acknowledge the technical staff of the Mount Abu IR Observatory of Physical Research Laboratory (PRL) for conducting the polarimetric observations with the 1.2 m telescope. M.B., K.A., S.G. and A.B. are funded by the Department of Space, Government of India. A.B. was J.C. Bose Fellow during the period of this work.

Appendix A Adjustment of the image sharpness

The sharpness of the intensity images for observations from different times obtained under different atmospheric seeing conditions was adjusted by computing the Fourier transform of a 121 × 121 pixel window and applying a Gaussian filter to it. The relative blur was determined by a least-squares fit of a quadratic function to the logarithm of the transfer function between the images according to

(A.1)

where t(u, v) is the decadal logarithm of the transfer function at pixel (u, v) in the transformed window, A(u, v) is the amplitude of the first transformed image at a certain pixel, and B(u, v) is the amplitude of the second transformed image at that pixel, after scaling to make the mean values equal. The value d(u, v) is the distance to the center of the transformed window (u_c, v_c), and a is the resulting relative blur parameter for the image pair. The fit of the quadratic function is limited to R, denoting the pixels (u, v) with d(u, v) ≤ 12, to reduce the effect of noise. If the fitted parameter a is negative, the transfer function has the shape of a Gaussian curve with a standard deviation of , so that A is more blurred than B. If a is positive, the transfer function is the reciprocal of a Gaussian curve, and therefore, B is more blurred than A. By pairwise comparison, the blurriest image in the set was determined. At each step, the blurrier image was kept for the next comparison, leading to the blurriest image as the reference frame. All other images were blurred by their transfer function relative to the reference frame. An example of the blur adjustment for the RC filter observations at multiple phase angles is shown in Fig. A.1.

An alternative approach is to sharpen all images to the level of the sharpest image considered as a reference, which we applied to all I images. The sharpening was achieved using a Wiener deconvolution (Petrou & Petrou 2010), given by

(A.2)

where M(u, v) is the Fourier transform of the Wiener filter that inverts the transfer function H(u, v) and the complex conjugate H^*(u, v). The spectral density of the noise is S_vv(u, v) and the spectral density of the original image is given by S_ff(u, v). These spectral densities can be expressed as the Fourier-transformed autocorrelation functions of the noise and the image, respectively. We assumed the noise to be constant white noise. In this case, only the spectral density of the original function has to be known. As the original image is not available, it can only be estimated. For a simplified equation, however, the ratio of the noise and the function can be set to a constant noise-to-signal ratio (NSR) Γ:

(A.3)

The parameter Γ was estimated from the power spectrum of the mage. Because the noise was assumed to be white noise, the corresponding power spectrum was considered to be constant, while the image power decreased towards higher spatial frequencies. We obtained the power spectrum of the image by binning the squared image MTF into spatial frequency bins with a size of 1 according to and taking their average values. The average spectrum values were selected between Ω = 11 and Ω = 50 as the approximated signal because outside this window, the noise-to-signal ratio estimate becomes too low. The noise approximation was selected between Ω = 50 and Ω = 100. The value of Γ in Eq. (A.3) was computed by the taking ratio of the average noise power and signal power.

This approach of determining a relative blur from the image transfer function between two images was found to cause the sharpness of the different images to appear visually similar when the images were blurred and sharpened, as shown in Fig. A.1.

Appendix B Observation set

We list the observation sets used in this study and the parameters derived for image calibration in Table B.1. The relative blur was computed from Equation A.1. The calibration factors k_b and k_s were computed from Equation 3 for the blurred and sharpened images, respectively. The noise to signal ratio (NSR) was estimated using Equation A.3.

Appendix C Hapke model: Reduction of free parameters using empirical approximations

The implementation of Hapke’s theoretical framework (Hapke 2012) is given here specifically to understand the microstructural properties of the Reiner Gamma swirl. The diffuse bidirectional reflectance function depending on the directions of incident and emitted light was summarized by Helfenstein & Shepard (2011) as

(C.1)

where Ω is the the volume-averaged single-scattering albedo, K is the porosity correction factor, g is the phase angle, and and µ_e are the cosines of the effective angles of incidence and emission angles, respectively. The original values of the angles are modified by the surface roughness to become effective values. The particle phase function is given by p(g). The shadow-hiding opposition effect (SHOE) is described by the function (1 + B_SH(g)), while B_CB(g) models the coherent backscatter opposition effect (CBOE). The shadowing function S (, µ_e, ψ) accounts for shadows occurring on a rough surface depending on the azimuth angle ψ between incident and emitted light. The function M(, µ_e/K, g) describes the multiple scattering of light between particles.

For the phase function p(g), we used the double-lobed Henyey–Greenstein single-particle phase function,

(C.2)

The parameters b and c are empirical values, where b ∈ [0, 1] controls the shape of the lobes, and c ∈ [−1, 1] specifies the relative strength of the forward and back scattering lobes. The first term of Eq. C.2 applies to the back-scattered lobe, and the second term applies to the forward-scattered lobe. The particle phase function describes the scattering behavior of one single particle. In the presence of multiple particles, however, light can be scattered multiple times. The modified isotropic multiple-scattering approximation, or MIMSA, attempts to include this effect (Hapke 2012). Anisotropic scattering by the particles is approximated by using average values of the phase functions and assuming that randomization after multiple scattering processes causes the increase in reflectance to become more isotropic. Hence, the reflectance due to single-particle scattering is increased by

(C.3)

where H(x) is the Ambartsumian-Chandrasekhar H function, which is approximated by

(C.4)

where r₀ is the diffusive reflectance, given by

(C.5)

Here, L₁(µ₀) is the average radiance scattered into the lower hemisphere from a single direction at the angle i, L₁(µ) is the average radiance scattered into the upper hemisphere from a single direction at the angle e, and L₂ is the average intensity scattered back into the lower hemisphere by a particle uniformly illuminated from the lower hemisphere (Hapke 2012).

Fig. A.1 Blur adjustment for the RC filter observations at multiple phase angles. The first, second, and third column shows images from 2021 February 5, 2021 March 4, and 2021 February 26, respectively. The difference is visibly reduced in the sharpened and blurred images.

The porosity coefficient K is given by

(C.6)

where ϕ is the filling factor of the medium, which is equivalent to 1-K. The SHOE used in Eq. C.1 is given by

(C.7)

where B_so and h_s are the SHOE amplitude and angular width parameters, respectively (Helfenstein & Shepard 2011). The CBOE function is given by

(C.8)

The parameters of the CBOE depend on the transparency of the medium and on the properties of subscatterers within larger particles (Hapke 2012).

The values i_e and e_e and the shadowing function S (µ_0e, µ_e, ψ) depend on the effective value of the photometric roughness, .

For i ≤ e, the shadowing function is

(C.9)

Where

When e ≤ i, it is

(C.10)

with

(C.11)

where y corresponds to i or e.

References

All Tables

All Figures

	Fig. 1 Comparison of Polarimetric parameters derived from the telescopic observations on 2021 February 5 at phase angle α = 106.41° using the RC filter for Kepler (first column), Aristarchus (second column), and Reiner Gamma (third column). The rows from top to bottom show the parameters I(α), (in DN), P(α), and C.
In the text

	Fig. 2 Phase-ratio maps (98°/84°) of Kepler (left), Aristarchus (center), and Reiner Gamma (right), based on the observations obtained on 2021-January 5 and 2021-January 6 (Appendix B, GC filter). The ratio is shown in logarithmic scale for clarity.
In the text

	Fig. 3 Reiner Gamma telescopic observation in RC filter from 5th March, 2021, georeferenced to the LRO WAC mosaic (Speyerer et al. 2011). The homogeneous regions considered to convert DN into reflectance are marked in green.
In the text

	Fig. 4 Flow chart of the telescopic data processing and the Hapke parameter fitting processes.
In the text

	Fig. 5 Ratios of reflectances at different phase angles in the RC filter. The swirl structure is visible in the left ratio image, but is significantly reduced in the right ratio image.
In the text

	Fig. 6 Derived Hapke parameter maps. The telescopic images were processed by blurring to uniform sharpness. The top row shows the parameters for the RC filter, and the bottom row shows those for the GC filter.
In the text

	Fig. 7 Derived Hapke parameter maps. The telescopic images were processed by sharpening to uniform sharpness. The top row shows the parameters for the RC filter, and the bottom row shows those for the GC filter. The Hapke parameters extracted from the regions marked in the RC albedo image are plotted in Figs. 8 and 9 for the RC and GC filters, respectively.
In the text

	Fig. 8 Box plot of the Hapke parameters extracted from central oval, the northern tail, and the mare region south of Reiner Gamma (area marked in Fig. 7) using the RC filter. The boxes and whiskers indicate the 50th and 95th percentiles in the distribution, respectively. Extreme outliers are shown with the plus symbol. The median value is shown by the horizontal line inside the box.
In the text

	Fig. 9 Box plot of the Hapke parameters extracted from central oval, the northern tail, and the mare region south of Reiner Gamma (area marked in Fig. 7) using the GC filter. The boxes and whiskers indicate the 50th and 95th percentiles in the distribution, respectively. Extreme outliers are shown with the plus symbol. The median value is shown by the horizontal line inside the box.
In the text

Fig. 10 Polarimetric phase function in the GC domain for pixels extracted from (a) the northern tail, (b) the western part of the central oval, (c) the eastern part of the central oval, and (d) the off-swirl mare to the south. The dots represent the average value of P, and the solid lines are second-degree polynomial fits to the data points around the maximum polarization angle, excluding phase angles lower than 60°. The blue and red dots represent the observations in the negative and positive branch of the polarization, respectively. The error bars indicate the 1σ uncertainty of the polarization fraction, which is independent of the polarization fraction itself.

In the text

	Fig. 11 Maps of the grain-size-dependent parameter C1(d) and the median grain size Md in µm of Reiner Gamma swirl using the framework of Dollfus (1998) for the RC (a and b) and GC (c and d) filters.
In the text

	Fig. 12 Maps of the grain size parameter b and the grain size d in m of the Reiner Gamma swirl using the framework of Shkuratov & Opanasenko (1992) for the RC filter. The observation was conducted at a phase angle of 106.41° (Table B.1).
In the text

	Fig. A.1 Blur adjustment for the RC filter observations at multiple phase angles. The first, second, and third column shows images from 2021 February 5, 2021 March 4, and 2021 February 26, respectively. The difference is visibly reduced in the sharpened and blurred images.
In the text