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2302.12141	Characterizing the nucleus of comet 162P/Siding Spring using ground-based photometry	Comet 162P/Siding Spring is a large Jupiter-family comet with extensive archival lightcurve data. We report new r-band nucleus lightcurves for this comet, acquired in 2018, 2021 and 2022. With the addition of these lightcurves, the phase angles at which the nucleus has been observed range from $0.39^\circ$ to $16.33^\circ$. We absolutely-calibrate the comet lightcurves to r-band Pan-STARRS 1 magnitudes, and use these lightcurves to create a convex shape model of the nucleus by convex lightcurve inversion. The best-fitting shape model for 162P has axis ratios $a/b = 1.56$ and $b/c = 2.33$, sidereal period $P = 32.864\pm0.001$ h, and a rotation pole oriented towards ecliptic longitude $\lambda_E = 118^\circ \pm 26^\circ$ and latitude $\beta_E=-50^\circ\pm21^\circ$. We constrain the possible nucleus elongation to lie within $1.4 < a/b < 2.0$ and discuss tentative evidence that 162P may have a bilobed structure. Using the shape model to correct the lightcurves for rotational effects, we derive a linear phase function with slope $\beta=0.051\pm0.002$ mag deg$^{-1}$ and intercept $H_r(1,1,0) = 13.86 \pm 0.02$ for 162P. We find no evidence that the nucleus exhibited an opposition surge at phase angles down to 0.39$^\circ$. The challenges associated with modelling the shapes of comet nuclei from lightcurves are highlighted, and we comment on the extent to which we anticipate that LSST will alleviate these challenges in the coming decade.	Abbie Donaldson, Rosita Kokotanekova, Agata Rożek, Colin Snodgrass, Daniel Gardener, Simon F. Green, Nafiseh Masoumzadeh, James Robinson	2023-02-23T16:27:58	http://arxiv.org/abs/2302.12141v1	# Characterizing the nucleus of comet 162P/Siding Spring using ground-based photometry ###### Abstract Comet 162P/Siding Spring is a large Jupiter-family comet with extensive archival lightcurve data. We report new $r$-band nucleus lightcurves for this comet, acquired in 2018, 2021 and 2022. With the addition of these lightcurves, the phase angles at which the nucleus has been observed range from $0.39^{\circ}$ to $16.33^{\circ}$. We absolutely-calibrate the comet lightcurves to $r$-band Pan-STARRS 1 magnitudes, and use these lightcurves to create a convex shape model of the nucleus by convex lightcurve inversion. The best-fitting shape model for 162P has axis ratios $a/b=1.56$ and $b/c=2.33$, sidereal period $P=32.864\pm 0.001$ h, and a rotation pole oriented towards ecliptic longitude $\lambda_{E}=118^{\circ}\pm 26^{\circ}$ and latitude $\beta_{E}=-50^{\circ}\pm 21^{\circ}$. We constrain the possible nucleus elongation to lie within $1.4<a/b<2.0$ and discuss tentative evidence that 162P may have a bilobed structure. Using the shape model to correct the lightcurves for rotational effects, we derive a linear phase function with slope $\beta=0.051\pm 0.002$ mag deg${}^{-1}$ and intercept $H_{r}(1,1,0)=13.86\pm 0.02$ for 162P. We find no evidence that the nucleus exhibited an opposition surge at phase angles down to $0.39^{\circ}$. The challenges associated with modelling the shapes of comet nuclei from lightcurves are highlighted, and we comment on the extent to which we anticipate that LSST will alleviate these challenges in the coming decade. keywords: comets:general comets - individual: 162P/Siding Spring ## 1 Introduction Photometric observations of comets can provide detailed insights into the nature and evolution of these small icy bodies, which are believed to have remained relatively unaltered since their formation 4.6 Gyr ago. Comets with short orbital periods ($<20$ yr) are particularly informative targets, as their properties can be monitored across multiple orbits. Jupiter-family comets (JFCs) are a dynamical subclass of short period comets with perihelia in the inner Solar System and whose orbits are dominated by Jupiter. The central nuclei of these comets are often obscured for observers on the ground by a coma of gas and dust, making it challenging to characterise their physical and surface properties remotely. To date, six short period comets have been imaged by spacecraft: comet 1P/Halley, and five JFCs (Snodgrass et al., 2022). These missions have provided the vast majority of the information we have on the shapes and surface features of comets thus far, demonstrating that their nuclei exhibit a variety of shapes and surface morphologies - from the highly elongated, _bilobed_ structure of 103P/Hartley 2 (Thomas et al., 2013) to the rounded, asymmetrical appearance of 81P/Wild 2 (Brownlee et al., 2004). Of these six comets imaged in situ, four are in highly elongated or bilobed configurations. Halley-type comet 8P/Itttle made a close passage to Earth in 2008, enabling radar imaging which indicated a bilobate nucleus (Harmon et al., 2010). With this addition, seven comet nuclei have reliably constrained shapes, and five of these are bilobed - we count 1P/Halley as a bilobed comet, although we note that from the spacecraft images (Keller et al., 1986) is shape can only be definitively described as elongated. This sample is too small to provide any statistically significant information on the shapes of the comet population; however, the high fraction observed to be bilobed may indicate that comets have physical characteristics or undergo formation and/or evolution processes that result preferentially in these shapes. Possibilities for the formation of bilobate nuclei have been explored at various dynamical stages: for example, slow growth by hierarchical agglomeration in the primordial disk (Davidsson et al., 2016); re-accretion following shape-changing collisions (Jutzi et al., 2017; Schwartz et al., 2018; Campo Bagatin et al., 2020); slow collisions between bodies of similar size (Jutzi and Asphaug, 2015); and sublimation-driven disruption in the Centaur region (Safrit et al., 2021). Bilobed shapes are also found in other small body populations. The New Horizons target 486958 Arrokoth (2014 MU${}_{69}$), a $\sim$30 km body in the Cold Classical population of trans-neptunian objects, appears to be comprised of two distinct, flattened lobes (Stern et al., 2019). In the near-Earth asteroid (NEA) populations, the contact binary fraction is estimated to be between 15 and 30 per cent (Virkki et al., 2022). While our current understanding of the shapes of comet nuclei is dominated by information returned from in situ missions, it is also possible to place constraints on shape information using ground-based observations. Radar facilities can be used when comets make close approaches to Earth, as was the case for comet 8P. When it is possible, delay-Doppler imaging is capable of providing nucleus size, rotation and shape information (Harmon et al., 2004). In extremely rare circumstances, a comet can pass in front of a star in such a way that its nucleus properties can be constrained by stellar occultation methods (discussed in e.g. Fernandez et al., 1999). This method relies on observers in multiple geographic locations and precise ephemerides for advance planning. However, due to the unpredictable nature of cometary orbits as they are affected by activity-driven torques, this technique has not yet successfully provided shape information for any comet nucleus. The most readily available means of acquiring shape information is from rotational lightcurves. The shape of the lightcurve is dependent primarily on the changing projected cross-section of the irregularly-shaped nucleus as it rotates. _Convex lightcurve inversion_ (CLI) is a technique that can be used to extract shape information from lightcurves (Kaasalainen et al., 2001; Kaasalainen et al., 2001), provided they cover a wide range of observing geometries. The CLI procedure generates a convex hull representing the object's global shape, 'inverting' the lightcurves to create a shape that is capable of reproducing the lightcurve at each input observing geometry and rotational phase. CLI is regularly employed to produce convex shape models of asteroids from their photometric disk-integrated lightcurves. The shape models for more than 3000 asteroids are stored in the Database of Asteroid Models from Inversion Techniques (DAMIT1; Durech et al., 2010), of which a large number have been produced using CLI. Footnote 1: [https://astro.troja.mff.cuni.cz/projects/DAMIT](https://astro.troja.mff.cuni.cz/projects/DAMIT) In contrast, only one comet nucleus has been modelled by CLI to date2. This was the target of ESA's Rosetta mission, comet 67P/Churyumov-Gerasimenko (hereafter 67P), prior to the re-dexvous phase of the spacecraft (Lowry et al., 2012). Using nucleus lightcurves collected around the 2006 aphelion passage (at heliocentric distances $r_{h}>$3 AU) and nucleus images from HST at $r_{h}\sim 2.5$ AU, the authors produced a convex model. The CLI model was later refined with the addition of lightcurves from the Rosetta approach (Motrola et al., 2014), which yielded a shape with properties similar to the original - consistent elongation and the presence of large planar surfaces. Both models produced pole orientation consistent with those found later by the spacecraft (Preusker et al., 2015), and the lightcurve data was used to detect a period change for 67P between the two perihelion passages. A direct comparison of the convex and Rosetta shape models can be found in Snodgrass et al. (2022). By definition, CLI cannot reproduce large-scale concave features. The substantial flat regions on the shape model of 67P masked what Rosetta observed to be a slim neck connecting two lobes of a rounded, bilobed shape. Large flat facets in convex models are strong indicators of large scale surface concavities (Devogele et al., 2015). Physical considerations regarding the criteria for the stability of any shape model may also be used to infer a bilobed shape, given that it is unlikely that single-lobed objects with extremely elongated axis ratios (a/b $>2.3$) would have formed naturally (McNeill et al., 2018; Jeans, 1919), though 67P has proven that bilobed shapes can be compact and rounded rather than elongated. Footnote 2: Two unusual objects which are related to JFCs have also had their shapes modelled by lightcurve inversion: (3552) Don Quiucte and 323P/SOHO In this work, we apply CLI to comet 162P/Siding Spring (hereafter 162P). The most recent physical properties measured for this comet are outlined in Table 1. 162P is a particularly large JFC, with effective radius $R=7.03^{+0.47}_{-0.48}$ km (Fernandez et al., 2013). Estimates of its geometric albedo ($\rho_{R}$) suggest it has one of the darkest surfaces of all studied comets. The comet has a relatively long synodic rotation period ($P_{SA}$) around 32.88 h, and has presented no evidence for significant period changes between orbits to date (Kokotanekova et al., 2018). Initially discovered as asteroid 2004 TU12, it was re-actrised as a comet after it was observed to display intermittent activity when closest to the Sun (Campins et al., 2006). These infrequent displays of weak activity may suggest that the comet is approaching dormancy, as defined by Kresak (1987) and Hartmann et al. (1987). The lack of a detectable coma over most of the comet's orbit has allowed for good lightcurve coverage of the nucleus at a range of viewing geometries, making 162P an appealing choice for this study. We obtained new lightcurves of the nucleus of 162P between 2018-2022, presented in Section 2 along with a description of the absolute photometric calibration process. The application of convex lightcurve inversion to our entire set of nucleus lightcurves is discussed in Section 3, including all results for 162P. The physical properties of the nucleus that we can derive from the shape model and a more general outlook on the possibilities of convex inversion for the lightcurves of comet nuclei are outlined in Section 4. ## 2 New observations and data reduction ### Observations In this work we have combined all published optical lightcurves of the nucleus of 162P/Siding Spring collected between 2007-2017 with new lightcurves obtained in 2018, 2021 and 2022. A complete summary of all the lightcurves is given in Table 2. Lightcurves collected prior to 2018 (ID 1-12 in Table 2) have been reported previously in Kokotanekova et al. (2017) and Kokotanekova et al. (2018). The new lightcurves were obtained in April 2018 (ID 13-15), July 2018 (ID 16-18), December 2021 (19-21), January 2022 (ID 22-28) and March 2022 (ID 29-33), and were collected and processed in this work as described below. For all observations, the field containing the comet was tracked at a sidereal rate, to enable the background stars and comet to be treated as point sources in the aperture photometry procedure described in \begin{table} \begin{tabular}{l c c} \hline \hline Property (symbol) & Value & Reference \\ \hline Orbital period & 5.46 y & Giorgini et al. (1996) \\ Inclination (_i_) & 27.5${}^{\circ}$ & Giorgini et al. (1996) \\ Rotation period ($P_{g}$) & 32.877${}^{+0.056}_{-0.065}$ h${}^{\star}$ & Kokotanekova et al. (2018) \\ Geometric albedo ($\rho_{R}$) & 0.022 $\pm$ 0.003 & Kokotanekova et al. (2017) \\ Effective radius ($R$) & 7.03${}^{+0.47}_{-0.48}$ km & Fernández et al. (2013) \\ \hline \hline \end{tabular} \end{table} Table 1: Summary of the physical properties of comet 162P. Section 2.2. The duration of each exposure was chosen such that the comet did not trail by more than the radius of the seeing disc. The April 2018 and March 2022 frames were obtained using the wide-field camera (WFC) on the 2.5-m Isaac Newton Telescope (INT) situated at Roque de los Muchachos Observatory (ORM), La Palma, Spain. The WFC is made up of four $2048\times 4100$ pixel CCDs, with a combined field of view (FoV) $34\times 34$ arcmin. The pixel scale of each CCD is 0.33 arcsec/pixel in $1\times 1$ binning mode. The target was kept in CCD4 throughout each epoch, and was imaged in the WFC SloanR filter in 2018, and in both WFC SloanR ($r^{\prime}$) and WFC SloanR ($g^{\prime}$) in 2022. The mean seeing varied nightly between $1.2-2$ for the 2018 observations, and $1.3-1.5$" in 2022. Bias frames were obtained nightly and used to subtract the bias level from all frames. Twilight flats in both filters were obtained when possible at the beginning of each night. These were used to flat-field correct frames taken on the same night (on nights when no twilight flats were obtained, frames were corrected using flats from the previous or next night). The July 2018 frames were collected with the European Southern Observatory (ESO) New Technology Telescope (NTT) at La Silla Observatory, Chile, using the ESO Faint Object Spectrograph and Camera (EFOSC). This is a multi-mode instrument with a $4.1\times 4.1$ arcmin FoV. The 162P frames were obtained using the Gunn-r ($r$) filter with the detector in $2\times 2$ binning mode, providing an effective pixel scale of 0.24 arcsec/pixel. The nightly seeing was around $1.4$". The bias subtraction and flat-field correction were applied to the frames by the same method as for the INT frames above. The December 2021/January 2022 photometry was acquired with the optical imaging component of the Infrared-Optical (IO:O) instrument on the fully robotic 2.0-m Liverpool Telescope (LT) at ORM. The IO:O is comprised of a $4096\times 4112$ pixel CCD, with an unbinned pixel scale of approximately 0.15 arcsec/pixel and effective FoV $10\times 10$ arcmin. The data were obtained in $2\times 2$ binning mode, using the SDSS-$r^{\prime}$ filter. Photometry was collected in short nightly observing blocks (OB) over a 2-week period ($\sim$5-25 exposures per OB) to probe as much of the object's long rotation period as possible. The average nightly FWHM of the seeing disc was $\sim$2". The frames were bias subtracted and flat-field corrected automatically in the IO:O data reduction pipeline. To minimize the possibility of the nucleus signal being contaminated by coma, the comet was observed at heliocentric distances $r_{h}>3.5$ au. We confirmed that 162P was inactive by aligning and stacking background-subtracted comet frames, and comparing the radial profile of the comet stack to the PSF of a scaled stacked comparison star for each epoch of observations. An example is shown for the LT observations in Fig. 1. The comet's radial profile is point source-like with no evidence of extended brightness at large distances from the centre, indicating that no detectable activity was present at the time of the observations. \begin{table} \begin{tabular}{l l l l l l l l l l} \hline ID & UT Start date & Instrument & Filters & Exposure time [s] & $r_{h}$ [au] & $\Delta$ [au] & $\alpha$ [deg]. & $\lambda_{E}$ [deg.] & $\beta_{E}$ [deg.] \\ \hline [MISSING_PAGE_POST] ### Absolute calibration procedure To combine the multi-epoch photometry obtained with multiple instruments, we followed the procedure outlined in Section 3.4 of Kokotanekova et al. (2017) with some alterations to the software used. This allowed us to transform differential photometry of the comet nucleus over the course of a night into absolutely-calibrated magnitudes using the $r$-band magnitudes of background stars from the Pan-STARRS 1 (PS1) catalogue (Chambers et al., 2016). We used aperture_photometry and related routines from the Astropy Photutils package (Bradley et al., 2020) to extract instrumental magnitudes, and calviacat(Kelley and Lister, 2019) to calibrate the photometry of the comet using comparison stars on the frames. For each night of observations, we identified background sources that appear on all frames. We then cross-checked each source with the PS1 catalogue using calviacat, and used only those with catalogue entries. We further filtered these sources to remove potential galaxies, sources that are too close together for reliable background subtraction, and sources with extreme colour indices compared to solar i.e $(g-r)<0$ and $(g-r)>1.5$. For the NTT frames we found a large number of background stars with $(g-r)>$$\sim$1.1, and so further limited the colour indices of the comparison stars to within $0<(g-r)<1.1$ to prevent biasing the colour towards more extreme values. We performed aperture photometry of the final background stars and the comet, using the suite of tools included in Photutils. For the photometry, we used a circular aperture with radius equal to the median FWHM of the background stars on each frame. This choice of aperture size was found to maximise the signal-to-noise ratio (SNR) of the comet. The number of comparison stars used for photometry was dependent on the field each night, and varied between $\sim$10-100 stars per night. For the INT frames, we corrected the background-subtracted aperture fluxes for the effects of distortion caused by distance from the optical axis of the wide-field instrument, as described in Gonzalez-Solares et al. (2008), before converting them to magnitudes. To absolutely calibrate the lightcurve points, it is necessary to account for the different colour responses of each instrument. We determined the colour term ($CT$) for each instrument following the iterative-gradient method detailed in Kokotanekova et al. (2017). The values obtained for the CTs are given in Table 3. For the LT $r^{\prime}$ frames and INT $g^{\prime}$ frames, we used the final gradient values as the CTs by which to correct our photometry. The CTs for the INT/WFC $r^{\prime}$ and NTT/EFOSC $r$ filters were previously derived using a much larger sample of comparison stars by the works stated in Table 3. We therefore ensured that their CTs provided suitable linear fits to our background stars, and used these values to correct for the $r$-band colour response of the INT/WFC and NTT/EFOSC. In order to determine the final $r_{PS1}$ magnitude for the nucleus, we required an estimate of its $(g-r)_{PS1}$ colour. We obtained this using the $g$ and $r$ frames from lightcurve ID 33. The observations on this night were taken in single-filter blocks of ten frames as $rgrg$, allowing us to infer the average $r$ magnitude at the time of the $g$ observations and vice versa. This gave an average $(g-r)_{PS1}$ nucleus colour of $0.48\pm 0.04$. This is consistent with value for $(B-V)$ of $0.76\pm 0.01$ determined by Lamy and Toth (2009) for 162P: converting this to $(g-r)_{PS1}$ using the expressions and coefficients given in Tonry et al. (2012) yields $0.51\pm 0.03$. The lightcurve IDs 1-12 reported in Kokotanekova et al. (2018) were calibrated using an average JFC surface colour index $(g-r)_{PS1}=0.58\pm 0.06$. These lightcurves were reprocessed following the procedures outlined in their work, but using the updated $(g-r)_{PS1}$ nucleus colour index found in this work to ensure consistent calibration. We corrected the calibrated lightcurves to account for the changing heliocentric ($r_{h}$) and geocentric ($\Delta$) distances and solar phase angle ($\alpha$) of the comet, to combine all lightcurve points to a consistent orbital configuration. The absolute magnitude H${}_{r}$, $r_{PS1}(1,\alpha=0)$ is given by Equation 1 and represents the magnitude of the comet at a theoretical position at distance 1 au from both the Sun and Earth and solar phase angle $\alpha=0$deg, assuming a linear phase function $\beta$. \[H_{r},r_{PS1}(1,\alpha=0)=m_{r}-5\log_{10}(R_{h}\Delta)-\beta\alpha \tag{1}\] Following the method of Kokotanekova et al. (2017), we fit the phase function $\beta$ by a Monte Carlo (MC) method which accounts for the photometric uncertainty of each lightcurve point. We replaced each lightcurve point with a value drawn randomly from a normal distribution with mean equal to the magnitude value and standard deviation equal to the uncertainty on that lightcurve point. We repeated this to produce 5000 randomised lightcurves, and used linear regression to determine the best fitting linear phase function for each random lightcurve. We plotted the resulting $\beta$ distribution and fit a Gaussian function to it, as shown in Fig. 2. We select the best fit value of $\beta$ and its corresponding uncertainty as the mean and standard deviation of this function, $\beta=0.0468\pm 0.0001$ mag deg${}^{-1}$. Fig. 3 shows the final absolute lightcurves from 2018 and 2021-2022, both phased to a period of 32.8638 h (the best fitting rotation period derived in this work, see Section 3). We measure a lightcurve amplitude $\Delta m$ for the 2021-2022 points of approximately 0.75 (not including the single outlying points at 2021-12-29 and 2022-03-05), though this is likely an overestimate due to the large amount of scatter in the lightcurves around the peak at rotational phase $\sim$0.2. Figure 1: Surface brightness profile of 162P in frames obtained on 01-01-2022. Black line shows radial profile of a field star, scaled to match the surface brightness of the comet within its smallest aperture. The comet profile and stellar profiles are indistinguishable within the uncertainties in the comet’s surface brightness. Inset shows the composite background-subtracted comet. \begin{table} \begin{tabular}{l c c l} \hline Instrument & Filter & CT & Reference \\ \hline INT/WFC & $g^{\prime}$ & $-0.044\pm 0.015$ & This work \\ INT/WFC & $r^{\prime}$ & $0.008\pm 0.004$ & Kokotanekova (2018) \\ NTT/EFOSC & $r$ & $-0.194\pm 0.005$ & Kokotanekova et al. (2017) \\ LT/IO:O & $r^{\prime}$ & $0.000\pm 0.005$ & This work \\ \hline \end{tabular} \end{table} Table 3: The colour terms (CT) used in the absolute-calibration process to account for the varying colour response of each instrument/filter. ## 3 Modelling the Nucleus Shape ### Convex lightcurve inversion With the absolutely-calibrated time series photometry at hand, we constructed the shape model of 162P by convex lightcurve inversion (CLI). The inversion procedure used in this work is the publicly available convexinv package by Durech et al. (2010), adapting the algorithms described by Kaasalainen & Torppa (2001) and Kaasalainen et al. (2001). The CLI software models the shape, sidereal period and rotation pole orientation that best fit the input lightcurves. The model is produced using a Levenberg-Marquardt algorithm (Press et al., 1992) to optimise the area and orientation each facet of a convex polyhedron by fitting a number of shape-related parameters. For a given combination of fitting parameters, the software computes a relative value of $\chi^{2}$ that quantifies how well the shape matches the observed lightcurve points (as defined in Kaasalainen & Torppa, 2001). The combination of parameters corresponding to the smallest value of $\chi^{2}$ are referred to as the 'best-fit' parameters. CLI is most effective when the object of interest is observed at a large number of viewing angles or _observing geometries_. This term refers to the object's position and orientation relative to both the observer on Earth and to the Sun. Changing the observing geometry changes the fraction of the object's surface that is illuminated at a given time, which in turn affects the reflected flux measured by the observer. This effect is compounded by the shape and rotation state of the object. If the aspect angle (the angle subtending the rotation pole and the observer's line of sight) changes significantly between observations, then the projected cross-section and brightness of the object will change, assuming it is non-spherical. For example, a triaxial ellipsoid rotating in a stable configuration about its shortest axis will have a maximum projected cross-sectional area when its aspect angle is equal to 0${}^{\circ}$ i.e. when the line of sight of the observer is perpendicular to the rotation axis. When lightcurves sample a wide range of different observing geometries, the projected cross-section of the object is viewed from many different orientations as the relative position of the object, observer, and Sun change. This allows for a well-constrained shape model. The solar phase angle $\alpha$ also plays a role in the quality of the final model: increased shadowing effects at large $\alpha$ can help constrain information about the shape that is not possible at small $\alpha$. For this reason, it is ideal for convex inversion to have access to lightcurves at large solar phase angles $\sim$20${}^{\circ}$ and above (Kaasalainen & Torppa, 2001). The CLI steps are described in detail below. A key aspect of the convexinv software is that input lightcurves can be treated as either _relative_ or _calibrated_. The software defines calibrated lightcurves as those that have been brought to the same magnitude scale at a unit distance of 1 au from the Earth and from the Sun. This corresponds to the form of the lightcurves produced in Section 2.2_without_ the phase function correction. Treating the lightcurves as calibrated effectively means that the shape optimisation procedure does not shift lightcurves from different epochs with respect to one another in the fitting process. In this case, to account for the effects of the changing solar phase angle between lightcurves, the software fits a light-scattering function during the shape optimisation procedure that includes a phase function. Rather than assuming an explicit scattering law, convexinv fits an empirical function, described in detail in Kaasalainen et al. (2001). The built-in phase function $f(\alpha)$ used by this scattering function to incorporate the effects of solar phase Figure 3: Lightcurves for 162P using photometry obtained in 2018 (upper) and 2021-2022 (lower). The comet magnitude at each point has been absolutely-calibrated to the Pan-STARRS 1 $r$ filter. The diamond-shaped lightcurve points were observed at the NTT, circular points with the INT, and triangular points with the LT. All points have been corrected for phase function using the best fit value for $\beta=0.0468$ mag deg${}^{-1}$ resulting from the Monte Carlo method described in the text. The lightcurves have been phased to a period of 32.8638 h, the best-fit sidereal period for the comet according to the convex inversion procedure described in Section 3. Figure 2: Distribution of phase function ($\beta$) values resulting from 5000 randomised lightcurve trials. The distribution has mean $\beta=0.0468\pm 0.0001$ mag deg${}^{-1}$. on the observed lightcurve points into the shape model is given by Equation 2: \[f(\alpha)=a_{s}\exp(-\frac{\alpha}{d_{s}})+k_{s}\alpha+1 \tag{2}\] The phase function is parametrized by $a_{s}$, $d_{s}$ and $k_{s}$, which are fitted in the convex inversion procedure, and $\alpha$ is the solar phase angle of the lightcurve points. It is important to note that $f(\alpha)$ and the linear phase function $\beta$ described in Section 2.2 are distinct. The convexinv software requires that the lightcurve points be heliocentric and geocentric-distance corrected and converted to intensity space. These intensities are not corrected for phase angle: determining the phase function correction is part of the shape optimisation procedure through fitting $a_{s}$, $d_{s}$ and $k_{s}$. To produce the shape model, we chose to implement and fit a phase function that is linearly dependent on phase angle, (i.e. fit a value solely for $k_{s}$ to convergence during the shape modelling procedure, fixing $a_{s}$ and $d_{s}$ at 0 and 1 respectively). We justify this choice below in Section 3.3.2. We stress however that no physical significance is placed on the final fitted values of these parameters, particularly $k_{s}$: we treat this as purely empirical. Identifying any relationship between the best-fit value for $k_{s}$, defined in intensity space and $\beta$, defined in magnitude space, is not necessary for this analysis. ### Period search To find the best shape, it is necessary to have a good measurement of the object's sidereal period. We used the periodscan procedure included in the convex inversion software package to find the best-fitting sidereal period for 162P. periodscan trials period values incrementally within a user-defined range. The range is typically based on prior knowledge of the object's rotation period because periodscan is computationally intensive, particularly for short rotation periods. Since the period of 162P was previously estimated to be $\sim$32.9 h, we used trial periods in the range 10-70 h. This ensured that periods around half and twice the literature estimate were also tested. For each trial period, periodscan starts from six initial pole locations and calls convexinv to simultaneously optimise the shape and scattering function. The relative $\chi^{2}$ value of the optimised shape at that trial period is determined and output - no other information about the shape or spin is stored at this stage. The resulting periodogram is shown in Fig. 4. The period corresponding to the minimum value of $\chi^{2}$ is $P=32.8638$ h, which is close to the literature synodic period. The next lowest $\chi^{2}$ value corresponds to $P=16.4319$h, half the value of the best fit period. We rule out this solution for the period as it is unlikely that the rotation of an irregularly-shaped object would produce a single-peaked lightcurve. There are several additional global peaks at $P=19.5079$ h, $P=29.2353$ h, $P=49.2190$ h and $P=52.1698$ h. We ruled these out as possibilities for the sidereal period by plotting the absolutely-calibrated lightcurve points shown in Fig. 3 phased to each of these periods in turn, none of which resulted in a coherent double-peaked lightcurve for either of the 2018 or 2021-2022 datasets. To associate a formal uncertainty with the best fit period value by the method described in Durech et al. (2012) leads to a period range defined by an increase of 5 per cent of the minimum $\chi^{2}$ value (assuming 800 degrees of freedom, from $\sim$900 individual lightcurve points and $\sim$100 model parameters). However, there are only minuscule differences in the models produced by period values within 10 per cent of the minimum $\chi^{2}$ value (Rozek et al., 2022), and as such we use this limit to express the uncertainty in the period. We therefore give the value of the best fit sidereal period as $P=32.8638\pm 0.0007$ h. For this analysis, we assume that the sidereal period is constant between observing epochs. This assumption is not unrealistic: 162P is a particularly large JFC and displays only very weak levels of outgassing, implying that it would experience negligible changes to its rotation period between orbits (Samarasinha and Mueller, 2013). The synodic period of 162P was previously estimated from the lightcurves acquired in 2007, 2012 and 2017 (ID 1-12) by Kokotanekova et al. (2017, 2018). The authors found that periods in the range $32.812-32.903$ h provided good fits to the combined lightcurves from all three epochs. By measuring the change in the phase angle bisector for each epoch, we determined synodic periods of 32.8856 h, 32.8794 h and 32.8866 h for the 2007, 2012 and 2017 epochs respectively, assuming a constant sidereal period $P=32.8638$ h. These values are all within the range of common synodic period values identified by Kokotanekova et al. (2018), implying that the sidereal period measured in this work is in agreement with these previously-estimated synodic periods. Additionally, the small deviations between the best individual synodic period fits for each of the 2007, 2012 and 2017 observing epochs in Kokotanekova et al. (2017, 2018) can be attributed to the comet's varying observing geometry. Figure 4: Periodograms demonstrating the quality of the sidereal period fit at each trial period using the periodscan routine over an interval of $10-70$ h (upper) and zooming in on interval $29-36$ h (lower). The dashed vertical line marks the period corresponding to the lowest value of $\chi^{2}$, $P=32.8638$ h, and the solid horizontal line shows the 10 per cent increase above the minimum $\chi^{2}$. ### Optimising shape and spin state To search for the optimal nucleus shape and rotation pole orientation, we created a coarse $5\times 5$ deg grid of ecliptic coordinates covering the entire celestial sphere (longitude $0^{\circ}\leq\lambda_{E}<360^{\circ}$ and latitude $-90^{\circ}\leq\beta_{E}\leq 90^{\circ}$). At each combination of pole coordinates, holding the sidereal period fixed, we ran convexinw with 50 iterations to find preliminary fits for each of the shape, spin state and phase function parameter $k_{s}$ (from Equation 2). The overall distribution of pole solutions does not depend on $k_{s}$ - the value of this parameter affects only the detailed features of the shape model at each pole orientation. Therefore we fix $k_{s}$ at the best-fit value from the initial pole search, and re-run the full sky grid search with 500 iterations to identify the pole orientations to search with higher resolution. The quality of the shape and spin state fit at each possible pole location are shown as a spherical projection of the $\chi^{2}$ plane in Fig. 5. The pole direction oversponding to the lowest value of $\chi^{2}$ at this stage lies at $(\lambda_{E},\beta_{E})=(120^{\circ},-50^{\circ})$. A second region of low $\chi^{2}$ values centred around $(\lambda_{E},\beta_{E})=(220^{\circ},-60^{\circ})$ is also identifiable from this distribution of pole solutions. We refined the solution by performing a $2\times 2$ deg grid search over a smaller region around this location, incorporating both areas of low $\chi^{2}$. At each grid point we fit both the shape and $k_{s}$ parameters with 500 iterations to ensure that $\chi^{2}$ was minimised. The best fit pole orientation was found to be at $(\lambda_{E},\beta_{E})=(118^{\circ},-50^{\circ})$. The uncertainty associated with these values was taken as the standard deviation in the $\lambda_{E}$ and $\beta_{E}$ values within 10 per cent of the $\chi^{2}$ minimum, giving final values $\lambda_{E}=(118^{\circ}\pm 26^{\circ})$ and $\beta_{E}=(-50^{\circ}\pm 21^{\circ})$. This solution implies an orbital obliquity of $167^{\circ}$ corresponding to retrograde nucleus rotation. The optimised facet areas and normals were transformed into a convex polyhedron by the minknowski procedure included in the software package. This shape was converted to a polyhedron with triangular facets by the standardfit procedure, to create the convex model shown in Fig. 6. The values of the best fit convexinw parameters and their uncertainties are given in column A of Table 4. This grid search method of identifying the shape and pole solution that minimises $\chi^{2}$ means that convexinv optimises an independent shape at each pole. We present the'most-likely' shape as that which minimises $\chi^{2}$, but note that statistically other shapes are only marginally less likely. To that end, we examine the shapes produced at pole orientations that result in slightly larger values of $\chi^{2}$. We find that for poles within a few degrees of $(118^{\circ},-50^{\circ})$, the shape is not appreciably altered from the reported best-fit shape. Around the second region of low $\chi^{2}$ values identified previously at $(\lambda_{E},\beta_{E})=(220^{\circ},-60^{\circ})$, the shape model found by the software is entirely unphysical. The model produced at this pole solution has a spin axis longer in length than its other dimensions, which implies an unstable rotation state. We therefore discard this as a possible solution, and continue this analysis using solely the best-fit model presented in Figure 6. To examine how well this shape model matched the observed lightcurve points, we generated synthetic lightcurves for each observing epoch and compared these to the observed lightcurve points as shown in Fig. 7. #### 3.3.1 Shape variability We employed a method similar to that of Lowry et al. (2012) to explore to what extent we could vary the shape shown in Fig. 6 and still obtain a statistically-significant fit to the lightcurves. We created a grid of factors in range $0.5-2.5$ by which to stretch the lengths of the principal axes $a,b$ and $c$ of the shape's equivalent-volume ellipsoid. This enabled us to stretch along two shape axes at once while holding the length of the third axis fixed. We generated synthetic lightcurves for all resulting stretched models, and determined a $\chi^{2}$ value according to how well each stretched model lightcurve matched the observed lightcurve points (it should be noted that this value of $\chi^{2}$ was calculated as standard, and is not the same as the relative $\chi^{2}$ metric used by convexinv). We show the resulting $\chi^{2}$ fits as contours at 1-$\sigma$ and 3-$\sigma$ confidence intervals for the three combinations of stretched axes in Fig. 8. The range of possible axes ratios that $a/b$ can have while providing a statistically-significant fit to the lightcurve points at the 1-$\sigma$ level is $1.4<a/b<2.0$. However, we are not able to constrain an upper limit for the $c$-axis in this way i.e. the length of the model's rotation axis. This is likely a direct result of the limited range of aspect angles at which the nucleus was observed - according to the best-fit pole orientation, the largest and smallest aspect angles at which 162P was observed vary by only $12^{\circ}$. We instead consider the rotational stability of the equivalent-volume ellipsoid that defines the axes $a,b,c$ along which the model is stretched. Since 162P is observed to be in a stable rotation state in every epoch, we assume that its axis of rotation, $c$, is the shortest in length. Should $c$ be stretched to be longer than $b$, then the model would no longer rotate about the axis with the largest moment of inertia, leading to unstable rotation. When $b$ is fixed, we can therefore constrain the maximum stretch factor applied to $c$ to be that Figure 5: The distribution of $\chi^{2}$ values corresponding to likelihood of the nucleus rotation pole orientation, measured at $5\times 5$ deg intervals over the entire ecliptic plane. These $\chi^{2}$ values have been projected onto a sphere of ecliptic longitude and latitude. The four spheres show the same solution from four viewing angles, along the cardinal direction labelled at the top left corner of each sphere. The darkest regions indicate the lowest values of $\chi^{2}$ where the pole is most likely oriented. The best-fit pole orientation is located at $(\lambda_{E},\beta_{E})=(120^{\circ},-50^{\circ})$ and is marked on the figure by a white cross in the E and S views. A second region of low $\chi^{2}$ is identifiable around $(\lambda_{E},\beta_{E})=(220^{\circ},-60^{\circ})$. which makes it equal in length to $b$. Imposing this upper limit means that the axis ratio $a/c$ can range from 1.5 to 5.5 and still provide a statistically significant fit to the lightcurve points at the 1-$\sigma$ level. For $b/c$, we impose that $b$ cannot exceed fixed $a$ in length, and that $c$ cannot exceed $b$. The resulting possible values for $b/c$ range from 1.5 to 4.0. The limits imposed from these stability-based arguments are shown as straight lines in the centre and right plots of Fig. 8. #### 3.3.2 Phase function considerations As described previously, we chose to force convexnv to fit a shape from the lightcurves using a purely linear phase function. The form of Equation 2 emulates empirically the typical behaviour of the surfaces of Solar System objects: increasing linearly in brightness with decreasing phase angle $\alpha$, and surging in brightness exponentially around $\alpha\sim 0^{\circ}$ in what is known as the opposition effect (OE). The phase functions of JFC nuclei acquired using ground-based observations have not yet revealed any evidence for a cometary OE. In every case, a purely-linear phase slope $\beta$ provides a suitable fit to the lightcurve points (reviewed in Snodgrass et al., 2011; Kokotanekova et al., 2017). Comet nucleus observations require negligible activity and therefore generally occur at large heliocentric distances, resulting in relatively small solar phase angles. However, nucleus observations at phase angles close to zero are extremely rare because they require an alignment of orbital nodes with the opposition direction. The 2018 lightcurves (ID 13-15) captured the nucleus of 162P close to phase angle zero for the first time, providing the opportunity to detect an OE if it occurred. To explore this possibility, we performed the shape modelling procedure twice: once exactly as described with a linear phase function; and the second time allowing convexnv to also fit the parameters $a_{s}$ and $d_{s}$ that characterise the exponential component of Equation 2, as well as $k_{s}$ in the phase function. The best-fitting shape model from this test was practically identical to the original shape model, as summarised in Table 4. With the addition of the 2018, 2021 and 2022 lightcurves, we expanded 162P's observed phase angle coverage to $0.4^{\circ}<\alpha<16.4^{\circ}$ (see Table 2). Using the shape model, it is possible to remove the effects of rotation from the lightcurve points, allowing the phase function to be determined with magnitudes at a consistent equatorial geometry in what is known as a reference phase curve, defined by Kaasalainen et al. (2001). The reference phase curve for 162P is shown in Fig. 9. We fit two phase functions to the points: a linear slope and an $H,G$ function (Bowell et al., 1989). The $H,G$ model provides a poor fit to the data points, particularly at small phase angles, while the linear function is in excellent agreement with the rotationally-corrected lightcurve points, indicating that 162P does not show evidence for an opposition surge in this dataset. From the linear fit, the phase function is characterised by a slope value of $\beta=0.051\pm 0.002$ mag deg${}^{-1}$ and intercept $H_{r}(1,1,0)=13.857\pm 0.020$. We use this value for $H_{r}$ to calculate 162P's $r$-band geometric albedo ($p_{T}$) using Equation 3. \[p_{T},p_{S1}=(k^{2}/R^{2})\times 10^{0.4(m_{\odot}-H_{r})} \tag{3}\] Here, $k=1.496\times 10^{8}$ km is the conversion factor between astronomical units and kilometres, m${}_{\odot}$ is the $r$-band PS1 magnitude of the Sun (-26.91 mag) and $R=7.03^{+0.47}_{-0.48}$ km is the effective radius of the comet nucleus. It should be noted that this value for $R$ was obtained using thermal IR measurements (Fernandez et al., 2013). Assuming that the nucleus size has remained unchanged, this yields a geometric albedo $p_{T}=0.022\pm 0.003$. To compare this with the existing literature values, we convert $H_{r}$ to $H_{R}$ and $H_{V}$ using the conversions described in Tonry et al. (2012), and derive values for $p_{R}=0.023\pm 0.003$ and $p_{V}=0.021\pm 0.002$. The value for $p_{R}$ is consistent with the values obtained by Kokotanekova et al. (2017, 2018) within the uncertainties, implying that 162P remains one of the darkest surfaces of all studied JFCs. ## 4 Discussion ### Nucleus properties from shape modelling #### 4.1.1 Shape The best-fitting shape model for 162P shown in Fig. 6 has an elongated axis ratio $a/b=1.56$, and appears relatively flat when viewed along the rotation pole. The stretching analysis performed in Section 3.3.1 revealed that the axis ratio $a/b$ can range from $1.4-2.0$ and still provide a suitable fit to the lightcurve points at the 1-$\sigma$ level. Typically for comets the lightcurve amplitude $\Delta m$ is used to place a lower limit on the axis ratio, since the orientation of the pole with respect to the observer is generally not known. For all known JFCs, the mean value for this lower limit $a/b=1.5$(Kokotanekova et al., 2017; Lamy et al., 2004). The axis ratios of the spacecraft-visited JFCs have been estimated more precisely from measurements of their shape dimensions. Using the literature values, we determine that they have a mean value $a/b=2.0$(Buratti et al., 2004; Thomas et al., 2013; Duxbury et al., 2004; Thomas et al., 2013; Jorda et al., 2016). When considering solely the bilobed JFCs (19P, 67P and 103P), this mean value becomes 2.4, and for 9P and 81P which are not bilobed, the average $a/b$ is 1.3. To determine these values we have used the ratio of the longest measured axis to the second-longest, contrary to the works stated above which quote $a/b$ for spacecraft targets as the ratio of longest to shortest axis length. To directly compare these values to the shape derived in this work, we calculate the relative lengths of the axis of the shape model to be: $a/b=1.6$; $b/c=2.2$; and $a/c=3.5$. The axis ratio $a/b$ indicates that the nucleus of 162P is more elongated than the non-bilobed JFCs, but not enough to match the average properties of the currently-known bilobed JFCs. It should be noted that we have not included 1P/Halley in this brief analysis. This is due in part to the fact that it is, by definition, not a JFC, and also to the contentious nature of describing it as bilobed. Adding the axis ratio for 1P, $a/b=1.96$, does not change the mean axis ratio for the spacecraft-visited comets, and yields $a/b=2.3$ for the mean elongation of the bilobed comets. For the five spacecraft-observed JFCs, the mean ratio of the \begin{table} \begin{tabular}{c c c} \hline Shape property & A & B \\ \hline $P$ [b] & $32.864\pm 0.001$ & $32.864\pm 0.002$ \\ $\lambda_{E}$ [$\arcmin$] & $118\pm 26$ & $118\pm 23$ \\ $\beta_{E}$ [$\arcmin$] & $-50\pm 21$ & $-50\pm 28$ \\ $a_{s}$ & - & 0.04 \\ $d_{s}$ & - & 0.01 \\ $k_{s}$ & -1.47 & -1.48 \\ $a/b$ & 1.56 & 1.58 \\ $b/c$ & 2.33 & 2.24 \\ \hline \end{tabular} \end{table} Table 4: A comparison of the shape model properties resulting from shape optimisation with a linear phase function (A) and with a linear-exponential phase function (B). $a_{s},d_{s}$ and $k_{s}$ are the best-fit parameters of the phase function described in Equation 2. The values $a/b$ and $b/c$ are the axis ratios calculated for an ellipsoid with the same volume as the shape model. The properties of the shape fit with a linear-exponential phase function are virtually identical to the shape produced with the linear phase function. longest-to-shortest axes $a/c$ is 2.2. This ratio becomes 2.7 when considering only the bilobed JFCs, and 1.5 for the non-bilobed. The value derived for 162P is 3.5, considerably higher than both of these estimates, and is similar to the dogbone-shaped JFC 103P. It is possible that the nucleus of 162P is long and relatively flattened in shape - objects in such configurations are known to exist in the Solar System, for example Arrokoth. A recent study of 3552 Don Quixote, a near-Earth object suspected to be of cometary origin, also resulted in a elongated and relatively flattened shape model using CLI (Mommert et al., 2020). The XY-plane of the 162P shape model is dominated by large regions of flat facets, which may be masking large-scale concave surface features that cannot be recreated by the lightcurve inversion procedure. The combination of these flat facet regions and the model axis ratios present tentative evidence for a bilobed shape: for example, the flat regions may be masking a sim neck connecting two distinct lobes. This interpretation should be treated with caution, as it is well-established that inferring concavities from lightcurves alone is not conclusive (Harris and Warner, 2020). Moreover, as demonstrated in Section 3.3.1, we were unable to constrain a maximum length for $c$ (model $Z$-axis) solely from fits to the observed lightcurve points at the 1-$\sigma$ level. It is therefore possible that the $Z$-axis has been underestimated somewhat by the convex inversion procedure, given the limited variation in viewing geometry covered by the lightcurves. While convexinv imposes no constraints on the possible shape dimensions and identified this model as the shape that best fits the input lightcurves when accounting for all variables, if the $Z$-axis (i.e. rotation axis) was continually oriented away from Earth then the true extent of this axis is almost impossible to quantify. However, the minimum value for both $a/c$ and $b/c$ was constrained to be 1.5. We therefore suggest that the nucleus of 162P is more elongated than the non-bilobed comets for which we have detailed shape information, and that its longest axis is at least 1.5 times greater in length than its axis of rotation. #### 4.1.2 Phase function We have used two different methods to measure the phase function of 162P in this work. We derived a value for $\beta=0.0468\pm 0.0001$ mag deg${}^{-1}$ using a Monte Carlo method on the entire lightcurve (without the removal of rotational effects). The uncertainty produced by this method is deceptively small, and accounts for the photometric and absolute-calibration uncertainties only. By correcting the lightcurves for the effects of rotation with the convex shape model, and fitting a line to the rotationally-averaged lightcurve points as a function of phase angle, we obtained a phase function value $\beta=0.051\pm 0.002$ mag deg${}^{-1}$. This value of $\beta$, derived using the shape model, is slightly steeper than the value obtained without correcting the lightcurve points for rotational effects. The newly derived 162P phase coefficient is larger than the $\beta=0.039\pm 0.002$ mag deg${}^{-1}$ determined by Kokotanekova et al. (2017) using the 2007-2017 datasets (IDs 1-12). The updated value of the phase function slope is derived for a broader range of phase angles (expanding the previous range of $4^{\circ}-12^{\circ}$ to $0.4^{\circ}-16^{\circ}$). Moreover, modeling the nucleus shape and correcting the photometry using the shape model accounts for the lightcurve shape effects which was not possible in Kokotanekova et al. (2017, 2018). We therefore adopt the Figure 6: Final shape model for 162P, viewed along three orthogonal directions. The model’s rotation axis is aligned with the Z-axis of the plot, and the longest shape axis is aligned with the X-axis. The model results from the best fit sidereal period $P=32.8638$h, and rotation pole orientation ($\Delta_{E},\beta_{E}$) = (118°, $-50^{\circ}$). The model is characterised by axis ratios $a/b$ = 1.56, $b/c$ = 2.33 and $a/c$ = 3.66 where $a,b,c$ are the three semi-axes of the equivalent-volume ellipsoid ($a>b>c$ in length). new PS1 $r$-band phase coefficient $\beta=0.051\pm 0.002$ mag deg${}^{-1}$ and the corresponding absolute magnitude $H_{r}=13.857\pm 0.020$, and conclude that the phase function of 162P is steeper than previously determined. With a minimum phase angle of $\alpha\sim 0.4^{\circ}$, and a total of four observing epochs at phase angles $\alpha<5^{\circ}$, the phase function of 162P is unique among other comet nuclei observed from the ground. Only two other JFCs have published nucleus lightcurves at phase angles less than 1${}^{\circ}$: 28P/Neujmin 1 (Delahodle et al., 2001) and 137P/Shoemaker-Levy 2 (Kokotanekova et al., 2017). The geometric albedos of these objects are $0.03\pm 0.01$(Jewitt & Meech, 1988; Campins et al., 1987) and $0.034\pm 0.006$(Kokotanekova et al., 2017) correspondingly. Like 162P, neither of these objects display any evidence for an opposition effect. To date, the only comet nucleus with a Figure 7: Fits to the model lightcurve for all the observational epochs in the dataset. Lightcurves taken on successive nights (i.e. across nights when differences in viewing geometry are presumed negligible) have been combined and displayed under the date of the earliest night in the group. The observed lightcurve points have been scaled to have a mean of zero. The value given for the phase angle on each lightcurve is an average for all the observed points from the combined epochs. clearly detected OE is 67P (Fornasier et al., 2015). However, this OE was _not_ detected from the ground - rather, it was observed in situ by the Rosetta spacecraft from the disk-integrated flux of the resolved nucleus. The geometric albedo of 67P is $p_{R}=0.065\pm 0.002$(Fornasier et al., 2015), which is more than double the measured albedos of 162P, 28P and 137P. It is therefore not entirely unexpected that 162P did not demonstrate evidence for an opposition effect in the phase angle range covered. Its surface is substantially darker than 67P, and darker still than 28P and 137P, both of which exhibited no OE. Moreover, other minor planet populations with low albedos such as Centaurs (Belskaya and Shevchenko, 2000) and some Jupiter Trojans (Shevchenko et al., 2012) are also known to display very narrow opposition effects with amplitudes less than 0.2 mag below phase angles $\sim$$0.1-0.2^{\circ}$. This work's finding that the phase function of 162P is steeper than previously estimated, while its albedo remains very small, brings into question whether 162P is in agreement with the potential relationship between geometric albedo and phase slope identified in Kokotanekova et al. (2018). That work compiled a database of 14 JFCs with well-constrained albedos and phase coefficients, and discovered a possible trend of increasing $\beta$ with increasing albedo. This behavior is opposite to the correlation between the linear phase coefficient and geometric albedo of asteroids (Belskaya and Shevchenko, 2000), and was interpreted as a possible evolutionary trend for comet nuclei in which the lowest-albedo comets have the most evolved surfaces. The new, steeper phase coefficient value of 162P places it closer to the asteroid correlation than the one found for JFCs. Since this is the first low-albedo comet phase function derived after accounting for the nucleus shape model, this finding challenges the possible correlation between comet phase functions and albedos. In addition, the phase function of 28P was initially reported by Delahode et al. (2001) as $\beta=0.025\pm 0.006$ mag deg${}^{-1}$ over a phase angle range of $\sim-0-15^{\circ}$. Schleicher et al. (2022) more recently determined a significantly steeper value for its phase function, $\beta\sim 0.05$ mag deg${}^{-1}$. This result makes 28P a potential second outlier for the phase function-albedo correlation hypothesis, further contesting this proposed evolutionary trend. Alternatively, if we continue to assume that the other comets in the sample have well-constrained phase functions, we must explain what makes 162P (and potentially 28P) the exception with a steep phase function and very small geometric albedo. One possible explanation for this discrepancy might come from the orbital analysis of the JFCs in the near-Earth space by Fernandez and Sosa (2015). In that work, the orbital integration of 162P revealed that it may originate from the outer Main Belt rather than from the Scattered Disk population currently believed to be the main source of JFCs. It is therefore constructive to compare the surface properties of 162P to those of very dark asteroids. However, very few low-albedo asteroids have well-constrained phase functions. Two notable examples are asteroids (101955) Bennu and (162173) Ryugu, studied in-situ by OSIRIS-REx and Hayabusa2 respectively. They have geometric albedos Figure 8: The 1-$\sigma$ and 3-$\sigma$$\chi^{2}$ confidence levels that result when the best-fit shape for 162P is stretched along principal axes $a$ and $b$ (left), $a$ and $c$ (centre) and $b$ and $c$ (right). The third axis is held fixed in every case. The black point on each plot marks the location of stretch factor 1.0 for both axes. From the left figure, the possible variation in $a$ and $b$ is constrained at the 1-$\sigma$ level to 40 percent along $a$ and 45 percent along $b$, giving a possible range $1.4<a/b<2.0$. From the contours in the centre and right plots, it is clear that we can place a lower limit on the possible length of the $c$ axis, but the upper limit for the $\chi^{2}$ contours are unconstrained in both cases. The vertical dashed line in the centre plot demonstrates the physical upper limit $c=b$ for this model from rotational stability arguments. The solid horizontal line in the right plot shows the additional limit imposed by not allowing $b$ to exceed $a$ in length, and the vertical dashed line indicates how this limits the possible length of $c$. Figure 9: The phase function for 162P. Every lightcurve (ID 1-33) has been corrected for rotational effects using a synthetic lightcurve generated from the shape model at that observing geometry, to obtain a rotationally-averaged magnitude for each night. The black solid line depicts the best linear fit to these data points, with a slope value of $\beta=0.051\pm 0.002$ mag deg${}^{-1}$ and intercept $H_{r}\,(1,1,0)=13.857\pm 0.020$. The dashed line shows the best $H$, $G$ fit to the points, which provides a poor fit to the data, particularly at low phase angles. The best-fit values for the $H$, $G$ function are $H=13.694\pm 0.022$ and $G=0.094\pm 0.025$. The original lightcurve magnitudes (with no rotational correction) are shown in grey. $0.044\pm 0.002$(Della Giustina et al., 2019) and $p_{V}=0.040\pm 0.005$(Tatsumi et al., 2020), larger than that estimated for 162P, and both have a small opposition surge detectable at $\sim 0-7^{\circ}$(Hergenrother et al., 2019; Tatsumi et al., 2020). To our knowledge, the only asteroid with comparable geometric albedo to 162P is the Jupiter Trojan (1173) Anchises with geometric albedo $p_{V}=0.027^{+0.006}_{-0.007}$(Horner et al., 2012). Interestingly, the phase function of Anchises at phase angles $0.3-2^{\circ}$ is remarkably shallow with $\beta=0.023\pm 0.008$ mag deg${}^{-1}$. Such a small sample of objects with both geometric albedo $<$0.05 and a well-constrained phase function offers limited possibilities to understand the unusual properties of 162P and 28P. However, future targeted campaigns or photometric data from surveys such as Rubin Observatory's Legacy Survey of Space and Time (LSST) provide the opportunity to significantly increase the number of objects with reliable phase functions over large phase angle ranges, and could enable us to build better statistics in order to test the ideas proposed to explain why 162P's phase function is steeper than those of other, similarly dark JFCs. ### Challenges associated with cometary convex inversion Comets are generally active within $\sim$3 au of the Sun, which is also where they are closest to Earth and most observable. As such, in most cases we cannot get reliable ground-based nucleus photometry over these parts of their orbits. We are therefore limited to observing when they are at heliocentric distances $\gtrsim 3$ au, yet still bright enough for sufficient signal-to-noise. Coupling this with the low orbital inclinations of the JFCs resulting in a limited range of body-centric latitudes visible to the observer, it is challenging to observe comet nuclei at a wide range of viewing geometries. The range of observing geometries obtained for 162P is sizeable for a short period comet nucleus, and is comparable to that of asteroids that have been previously modelled by convex inversion. This is likely due to 162P's large effective radius, which means that the comet remains sufficiently bright around aphelion. In addition, its low levels of activity mean that the nucleus signal is not typically contaminated by dust coma at heliocentric distance $\sim$3 au, making it an excellent candidate for targeted observations with limited telescope time available. The observer-centred ecliptic longitude and latitude of 162P in our dataset span a range of $\sim$48${}^{\circ}$ and $\sim$22${}^{\circ}$ respectively. In comparison, Lowry et al. (2012) obtained observations of the nucleus of 67P at observer-centred ecliptic longitude and latitudes over a range of 100${}^{\circ}$ and 15${}^{\circ}$ respectively to create its convex model. Their observations only varied in aspect angle by 17${}^{\circ}$ (between 53${}^{\circ}-70^{\circ}$) before the authors included images from HST at an aspect angle of 100${}^{\circ}$. For 162P, the best-fitting pole solution was oriented towards $(A_{\mathcal{E}},B_{\mathcal{E}})=(118^{\circ},-50^{\circ})$ (corresponding to R.A.= 290${}^{\circ}$, decl.= $-28^{\circ}$) implying that the lightcurves spanned a 12${}^{\circ}$ range of aspect angles from 86${}^{\circ}-98^{\circ}$. Rotation pole information is rare for low activity comets, because the most common means of determining pole orientation involves tracking or modelling the temporal evolution of morphological structures in the coma. The statistics for cometary pole distributions are therefore extremely limited. It is expected that the distribution of pole orientations for the population to be somewhat random, given that the torques exerted by the sublimation of surface volatiles are capable of dramatically altering the nucleus spin state (Samarasinha et al., 2004). Comparing the pole orientation derived for 162P to the limited existing orientations known for other comets is therefore unlikely to offer any particular insights into the properties of the population. Placing constraints on the pole orientation allows us to predict future geometries that may be available to further enhance the model and refine the z-axis. Prior to this work, it was possible to infer that 162P had been observed at a limited range of aspect angles due to the similar $\Delta m$ values of lightcurves obtained at different epochs. The upcoming LSST has the potential to extend the range of observing geometries for 162P and many other JFCs. To illustrate this, we used the pole solution to determine the aspect angle for 162P based on its current orbit, at 20-day intervals over a time frame which coincides approximately with LSST. The expected variation in aspect angle is shown in Fig. 10 as well as the comet's heliocentric distance at each timestamp. It should be noted that these points do not account for whether or not 162P will actually be observable by LSST - those that are not observable are plotted on a grey background. The figure shows that the nucleus exhibits a much greater range of aspect angles than those covered by our dataset, around 90${}^{\circ}$. However, the extremes in aspect range are all at low heliocentric distance $<2.5$ au, where observations are more likely to be affected by activity. This demonstrates the difficulty of implementing CLI on comet nuclei; unlike asteroids which are best observed close to Earth during which time they move quickly along the sky and offer a wide range of viewing angles, comets must be observed further out, considerably limiting the available viewing geometries. Nevertheless, LSST will produce an abundance of calibrated, temporally-sparse photometry for many comets, resulting in a much greater range of aspect angles than is realistic using present methods of targeted observations such as the lightcurves presented in this work. It has been shown to be possible to create reliable shape models of asteroids using sparse photometry from survey data combined with densely-sampled lightcurves (Durech et al., 2009). With a greater range of observing geometries available for a large number of known comets, we expect that it will prove possible to constrain the shapes and pole orientations of many more of these objects from ground-based observations alone. Figure 10: Variation in expected aspect angle of comet 162P over the approximate LSST operation period (early 2023-2033), extrapolated based on its present-day orbital elements and best-fit pole solution. The solid black line indicates the average aspect angle covered by the existing 2007-2022 observations, and the dashed lines show the maximum and minimum aspect angles covered by this dataset. The colour of each point corresponds to the heliocentric distance $r_{h}$ of the comet at that time, with the darkest coloured points being closest to the Sun. Grey background indicates where observations of 162P are not obtainable by LSST. ## 5 Summary This works presents photometric lightcurves of the nucleus of Jupiter-family comet 162P/Siding Spring obtained over 33 epochs between 2007-2022 when no cometary activity was detected. The dataset was analysed with the aim to derive a convex shape model of the comet's nucleus. * We collected new lightcurves for comet 162P in 2018 and 2021/2022, including in April 2018 when the comet was close to $0^{\circ}$ phase angle. With the addition of these observations, the total range of phase angles at which the nucleus has been observed was extended from $4^{\circ}-12^{\circ}$ to $0.4^{\circ}-16.3^{\circ}$. * Using photometry obtained in March 2022, we determined a $(g-r)$ colour estimate for the nucleus (in Pan-STARRS 1 filters) of $0.48\pm 0.04$ mag. This value is consistent with previous estimates of the nucleus colour. * We used this colour and the $r$-band magnitudes of background stars in the Pan-STARRS 1 catalogue to absolutely-calibrate the comet lightcurves, and update the calibration of existing photometry of the comet obtained in 2007-2017. * We used convex lightcurve inversion on the calibrated lightcurves to fit a shape model, pole orientation and sidereal period for the nucleus. Using a linear phase function to model the lightcurves, we obtain a shape with axis ratios $a/b=1.56$ and $b/c=2.33$ and pole orientation $(\lambda_{E},\beta_{E})=(118^{\circ}\pm 26^{\circ},-50^{\circ}\pm 21^{\circ})$. * To examine the extent to which this shape can vary while still fitting the lightcurves with statistical significance, we applied combinations of stretching factors to its principal axes, two at a time. We found that the $a/b$ axis ratio can vary between 1.4 and 2.0 and still fit the lightcurves at the 1-$\sigma$ level. We found that we could not place a statistical limit on the upper length of the rotation axis from the lightcurves alone, due to the limited range of observing geometries covered by the lightcurves. Using rotational stability arguments, the range of possible values for $b/c$ was constrained to $1.5<b/c<4.0$. * With the best-fit shape model we corrected the lightcurves for the effects of rotation, and fitted both a linear and $H$, $G$ phase function to the resulting distance-corrected magnitudes. The $H,G$ model provided a poor fit to the data, while the linear phase function $\beta=0.051\pm 0.002$ mag deg${}^{-1}$ matched the datapoints extremely well at all phase angles. We concluded that 162P did not display a detectable opposition surge in the 2018 lightcurves, and discussed the implications that a steep phase slope for an object with such a dark surface has on current hypotheses for comet surface evolution. * We suggested that 162P may be another example of a bilobed JFC based on the tentative evidence provided by the large planar regions in the model $XY$-plane, and its elongation. * We used the best-fit pole solution from the convex lightcurve inversion procedure to predict changes in the comet viewing geometry throughout the decade in which LSST will be operational which would allow us to further refine the shape. The aspect angle varies most extensively at heliocentric distances closest to the Sun, highlighting the challenges of obtaining reliable observations of comet nuclei at a range of viewing geometries. ## Acknowledgements The authors would like to thank the referee, David Schleicher, for insightful and helpful comments on this manuscript. We also thank Samuel Jackson for the useful discussions. This work was supported by the UK Science and Technology Facilities Council. This work was also facilitated by support from the International Space Science Institute in the framework of International Team 504 "The Life Cycle of Comets". RK acknowledges support by ESO through the ESO Fellowship. This work was supported in part by ESO's SSDF 21/22 (Student) Garching funding program. The new lightcurves presented were based on observations made with the Isaac Newton Telescope (UK PATT programmes I/2018A/07 and I/2022A/07), the Liverpool Telescope (programme XPL21B13) and the ESO New Technology Telescope (programme 0101.C-0709(A)). Previously reported lightcurves were based on observations at the European Southern Observatory under ESO programmes 089.C-0372(A) and 089.C-0372(B), the WHT and INT under UK PATT programmes W/2007A/20 and I/2017A/05, and the 2-m telescope at Rozhen Observatory, Bulgaria. The Pan-STARRS 1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. ## Data Availability The lightcurves used in this work can be accessed in an online Table 1 available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via [http://cdsarc.u-strasbg.fr/viz-bin/cat/J/MNRAS](http://cdsarc.u-strasbg.fr/viz-bin/cat/J/MNRAS). The shape model derived here will be shared on reasonable request to the corresponding author.	astro-ph.EP	[ "astro-ph.EP" ]
2301.08449	Revealing the supercritical dynamics of dusty plasmas and their liquid-like to gas-like dynamical crossover	Dusty plasmas represent a powerful playground to study the collective dynamics of strongly coupled systems with important interdisciplinary connections to condensed matter physics. Due to the pure Yukawa repulsive interaction between dust particles, dusty plasmas do not display a traditional liquid-vapor phase transition, perfectly matching the definition of a supercritical fluid. Using molecular dynamics simulations, we verify the supercritical nature of dusty plasmas and reveal the existence of a dynamical liquid-like to gas-like crossover which perfectly matches the salient features of the Frenkel line in classical supercritical fluids. We present several diagnostics to locate this dynamical crossover spanning from local atomic connectivity, shear relaxation dynamics, velocity autocorrelation function, heat capacity, and various transport properties. All these different criteria well agree with each other and are able to successfully locate the Frenkel line in both 2D and 3D dusty plasmas. In addition, we propose the unity ratio of the instantaneous transverse sound speed $C_T$ to the average particle speed $\bar{v}_{p}$, i.e., $C_T / \bar{v}_{p} = 1$, as a new diagnostic to identify this dynamical crossover. Finally, we observe an emergent degree of universality in the collective dynamics and transport properties of dusty plasmas as a function of the screening parameter and dimensionality of the system. Intriguingly, the temperature of the dynamical transition is independent of the dimensionality, and it is found to be always $20$ times of the corresponding melting point. Our results open a new path for the study of single particle and collective dynamics in plasmas and their interrelation with supercritical fluids in general.	Dong Huang, Matteo Baggioli, Shaoyu Lu, Zhuang Ma, Yan Feng	2023-01-20T07:04:42	http://arxiv.org/abs/2301.08449v1	Revealing the supercritical dynamics of dusty plasmas and their liquid-like to gas-like dynamical crossover ###### Abstract Dusty plasmas represent a powerful playground to study the collective dynamics of strongly coupled systems with important interdisciplinary connections to condensed matter physics. Due to the pure Yukawa repulsive interaction between dust particles, dusty plasmas do not display a traditional liquid-vapor phase transition, perfectly matching the definition of a supercritical fluid. Using molecular dynamics simulations, we verify the supercritical nature of dusty plasmas and reveal the existence of a dynamical liquid-like to gas-like crossover which perfectly matches the salient features of the Frenkel line in classical supercritical fluids. We present several diagnostics to locate this dynamical crossover spanning from local atomic connectivity, shear relaxation dynamics, velocity autocorrelation function, heat capacity, and various transport properties. All these different criteria well agree with each other and are able to successfully locate the Frenkel line in both 2D and 3D dusty plasmas. In addition, we propose the unity ratio of the instantaneous transverse sound speed $C_{T}$ to the average particle speed $\bar{v}_{p}$, i.e., $C_{T}/\bar{v}_{p}=1$, as a new diagnostic to identify this dynamical crossover. Finally, we observe an emergent degree of universality in the collective dynamics and transport properties of dusty plasmas as a function of the screening parameter and dimensionality of the system. Intriguingly, the temperature of the dynamical transition is independent of the dimensionality and it is found to be always 20 times of the corresponding melting point. Our results open a new path for the study of single particle and collective dynamics in plasmas and their interrelation with supercritical fluids in general. ## I Introduction A supercritical fluid [1; 2; 3] typically refers to a condensed state of matter in which the traditional liquid and gas phases cannot be separated anymore by a sharp first-order phase transition. For various substances, the supercritical fluid state can be achieved when the temperature and the pressure are above the corresponding critical point [1; 2; 3]. Supercritical fluids have been intensively investigated due to their widely applications in the nuclear waste, petrochemical, food, and pharmaceutical industries [4; 5; 6]. Although there is no traditional liquid or gas state within the supercritical regime, recently, several studies suggest that a liquid-like to gas-like dynamical transition can be identified in supercritical fluids using either the Frenkel line [7; 8; 9; 10; 11; 12; 13], the Widom line [14], or the Fisher-Widom line [15]. In the microscopic description of liquids proposed by Frenkel [16], atomic particle motion is a combination of quasiharmonic vibrations around potential minima and thermally induced jumps from an equilibrium position to a new one. These hopping processes give the ability to flow in liquids and happen at an average time $\tau$, also termed the liquid relaxation time [7; 8; 16]. This view, deeply inspired by the phenomenological ideas of Maxwell [17], implies that liquids behave effectively as solids for time scales shorter than $\tau$, or equivalently, for frequencies larger than the Frenkel frequency $\omega_{F}=2\pi/\tau$. Following this picture inspired by solid state theory, the minimal period of rigid-like vibrations is given by the Debye time $\tau_{D}$, which is around $0.1-1$ ps in classical liquids. When $\tau>\tau_{D}$, particles mainly vibrate at their equilibrium positions and hop rarely, so that the typical liquid behavior is exhibited, often termed as the "rigid liquid" state [7; 8]. In that regime, liquids are expected to support propagating shear waves at frequencies $\omega>\omega_{F}$. One can further derive a minimal cutoff wave-vector for their propagation [18], which implies a maximum propagation length approximately equal to the sound speed times the relaxation time $\tau$. Within this framework, the frequency of collective propagating shear waves in liquids, responsible for the emergent "rigidity", has to fall in the range $\omega_{F}<\omega<\omega_{D}$, with $\omega_{D}=2\pi/\tau_{D}$. The relaxation time $\tau$ decreases with the temperature. Physically, this is just reflected in a larger kinetic energy and therefore a stronger ability to rearrange. When $\tau<\tau_{D}$, the particles' hopping occurs more frequently than solid-like vibrations, collective shear waves disappear, and the "nonrigid" gas-like fluid state [7; 8] is approached. Following this logic, in supercritical liquids, the concept of Frenkel line [7; 8] has been introduced to discriminate the rigid liquid from the nonrigid gas-like fluid state and it has been formally defined by the condition $\tau\approx\tau_{D}$. In the past [7; 8; 9; 10; 11], several diagnostics, including the specific heat $c_{V}$, the velocity autocorrelation function (VACF), and the mean squared displacement have been used to determine the condition for the Frenkel line in various physical systems, as described in detail later. Furthermore, when $\tau\rightarrow\tau_{D}$, i.e., at the onset of the dynamical crossover, the propagation length of shear waves approaches the minimum available value approximately given by the interatomic distance. Therefore, the Frenkel line can be also defined as the disappearance of collective shear waves in fluids [18] and it is related to universal minimal values for different transport coefficients such as the shear viscosity and the thermal diffusivity [19; 20]. Dusty plasmas [21; 22; 23; 24; 25; 26; 27; 28], also termed complex plasmas, are partially ionized gases containing micron-sized dust particles, and they represent an excellent model system where the motion of individual dust particles can be directly tracked. In the laboratory conditions, these dusts have a typical charge of $-10^{5}e$ in the steady state [29; 30; 31], interacting with each other through the Yukawa repulsion [32], and leading to a much higher potential energy between neighboring dusts than their kinetic energy, i.e., these dusts are strongly coupled [24; 25; 26; 27; 28; 33] (cf. classical liquids). During experiments, these dusts can form into either a single layer two-dimensional (2D) suspension or a three-dimensional (3D) suspension, i.e., 2D [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50], exhibiting collective solid and liquid-like behaviors [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50], including the solid-liquid phase transition or melting [23; 29; 30]. Thus, dusty plasmas provide an incredibly powerful platform to explore collective dynamics of liquids and solids at the individual particle level [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50]. In the past thirty years, the solid-liquid phase transition of dusty plasmas has been systematically investigated, in both experiments and simulations, as well as theories [23; 29; 30]. Commonalities between fluid dusty plasmas and classical liquids, rooted in their shared strongly coupled nature, have been also explored, creating a beneficial exchange of ideas and results. As a concrete example, the dynamics of shear waves and the existence of a critical wave-vector, typical of liquids [18; 51], have been the subject of several theoretical and experimental studies in dusty plasmas [52; 53; 54]. However, so far, one important aspect of the collective dynamics of dusty plasmas remains elusive. In fact, due to the pure repulsive interaction between dust particles, there is no liquid-vapor phase transition [55] in dusty plasmas. Thus, a melted or fluid dusty plasma perfectly matches the definition of a supercritical fluid and should be thought as such. To the best of our knowledge, the supercritical nature of fluid dusty plasmas and the existence of a related liquid-like to gas-like dynamical crossover (Frenkel line) have been never disclosed before. In this work, we put forward this new interpretation and we further show that, as in classical liquids, the supercritical regime of 2D and 3D dusty plasmas can separated into a liquid-like and a gas-like phase by a dynamical crossover, the Frenkel line (see cartoon in Fig. 1). The rest of this paper is organized as follows. In Sec. II, we briefly introduce our simulation method to mimic 2D and 3D dusty plasmas. In Sec. III, we present the proposed supercritical nature of dusty plasmas and its features. We also provide various diagnostics to discriminate the liquid-like from the gas-like states, with the same resulting transition at 20 times of the melting point for both 2D and 3D dusty plasmas. In Sec. IV, we provide an interpretation of the agreement between our newly introduced diagnostics and the traditional Frenkel line criteria. In Sec. V, we give a summary of our findings. In Appendix A, we provide the details of our MD and Langevin simulations. In Appendix B, we present the calculated transport results from our simulation data (self-diffusion constant, shear viscosity, and thermal conductivity), as well as a new analysis of previous results from [67; 68; 69; 70; 71]. ## II Simulation method We perform equilibrium molecular dynamics (MD) simulations of 2D and 3D Yukawa liquids to mimic 2D and 3D fluid dusty plasmas as in [52; 56]. The dynamics of each particle $i$ is governed by the following equation of motion: \[m\vec{\mathbf{r}}_{i}=-\nabla\Sigma_{j}\phi_{ij} \tag{1}\] Figure 1: A putative phase diagram for dusty plasmas as a function of the coupling parameter $\Gamma$ and the screening parameter $\kappa$. The solid blue line indicates the first-order solid-liquid phase transition. The dashed red line refers to a possible dynamical crossover in the supercritical regime which is the subject of this work. where $\phi_{ij}=Q^{2}\frac{\exp(-r_{ij}/\lambda_{D})}{4\pi\epsilon_{0}r_{ij}}$ is the Yukawa repulsion between particles $i$ and $j$, separated by a distance $r_{ij}$. Here, $\lambda_{D}$ is the Debye length, $Q$ the charge, and $\epsilon_{0}$ the vacuum electric permittivity. To characterize dusty plasmas, we use the screening parameter $\kappa=a/\lambda_{D}$ and the coupling parameter $\Gamma=Q^{2}/\left(4\pi\epsilon_{0}ak_{B}T\right)$[24, 25, 26, 27, 28], where $a$ is the Wigner-Seitz radius of $(n\pi)^{-1/2}$[25] and $(4n\pi/3)^{-1/3}$[52, 57] as a function of the number density $n$ in 2D and 3D systems, respectively. Clearly, by increasing the screening parameter $\kappa$, the potential changes gradually from a long-range Coulomb-like potential to a hard-sphere-like repulsion. Timescales are normalized using the dusty plasma frequency $\omega_{pd}=\sqrt{Q^{2}/\left(2\pi\epsilon_{0}ma^{3}\right)}$[25] and $\omega_{pd}=\sqrt{3Q^{2}/\left(4\pi\epsilon_{0}ma^{3}\right)}$[52] for 2D and 3D dusty plasmas, respectively. For convenience, we will present all our results as a function of the reduced coupling strength $\Gamma/\Gamma_{m}$, where $\Gamma_{m}$ corresponds to the melting point determined from the static structure measured from simulations [58, 59]. In analogy to thermal systems, we can think of the coupling parameter $\Gamma$ as the effective inverse temperature for dusty plasmas, so that a higher $\Gamma$ value corresponds to a lower temperature. To simulate 2D and 3D dusty plasmas, we confine $N=4096$ and $8192$ particles in two simulation cells with the dimensions of $121.9a\times 105.6a$ and $32.5a\times 32.5a\times 32.5a$, respectively, using periodic boundary conditions. We always set the simulation conditions as melted dusty plasmas, i.e., for each chosen $\kappa$ value between 0.5 and 3, we specify various $\Gamma$ values lower than the corresponding melting point $\Gamma_{m}$ of 2D [58] and 3D dusty plasmas [59]. We also use the reduced coupling strength $\Gamma/\Gamma_{m}$[60, 61] to characterize the relative temperature of the studied systems. The integration time step is chosen to be $0.005$$\omega_{pd}^{-1}$, small enough as justified in [56], while the interparticle Yukawa repulsion at the radii beyond $22a$[56] and $8a$ is truncated directly for 2D and 3D systems, respectively. Other simulation details are the same as in [52, 56]. As the output of our simulations, the obtained time series of positions and velocities for all simulated particles are used to determine various physical quantities reported in the main text. We also perform Langevin dynamical simulations [62] of 2D and 3D dusty plasmas to confirm that our reported results are valid. More details can be found in Appendix A. ## III Results ### Heat capacity We start our discussion with the analysis of the heat capacity. In what follows, we will refer to the specific heat per particle, so that the total number of particles $N$ disappears from our expressions. Moreover, in dusty plasmas, the temperature is expressed in units of energy, so that $k_{B}=1$ as well [35]. At lower temperatures, or equivalently larger coupling parameters, the Frenkel frequency is much lower than the Debye one and collective shear waves display an almost gapless dispersion as in solids. In that regime, for 3D systems, we expect two transverse waves and one longitudinal wave there, each of which contributes $k_{B}T/2$ potential energy from the equipartition theorem [9, 63], and $k_{B}T/2$ kinetic energy. For large values of the coupling parameter, $\Gamma/\Gamma_{m}\gg 1$, the heat capacity approaches therefore the solid-like value, $c_{V}=3$. When transferring from the rigid liquid to the nonrigid gas-like state, the two transverse waves cannot be sustained anymore, and the heat capacity reduces to $c_{V}=2$[8, 9]. Qualitatively, this gradual decrease reflects the disappearance of the solid like oscillations into the diffusive gas-like motion. In other words, the number of transverse modes with frequency $\omega>\omega_{F}$ decreases towards the gas-like regime. As a result, the Frenkel line is determined by the simple condition $c_{V}^{\rm 3D}=2$ in 3D systems [9]. Following a similar argument, for 2D systems, there is only one transverse wave, so that $c_{V}$ decreases from 2 to 1.5 when the transverse wave cannot be sustained anymore. Thus, the Frenkel line for 2D systems can be determined from the condition $c_{V}^{\rm 2D}=1.5$. In the microcanonical ensemble, the specific heat $c_{V}$ can be derived from the fluctuations of the kinetic energy (KE) using [64, 65] \[\frac{2\left(\langle{\rm KE}^{2}\rangle-\langle{\rm KE}\rangle^{2}\right)}{Nd( k_{B}T)^{2}}=\frac{c_{V}-d/2}{c_{V}}, \tag{2}\] where $d$ is the dimensionality of the simulated system and $k_{B}$ the Boltzmann constant. Since the temperature is ex Figure 2: Specific heat $c_{V}$ for 2D and 3D dusty plasmas as a function of the reduced coupling strength $\Gamma/\Gamma_{m}$ for different values of the screening parameter $\kappa$. The vertical dashed line indicates the critical value $\Gamma/\Gamma_{m}=0.05$. The horizontal dotted line corresponds to the value 0.5. pressed in units of energy, the obtained specific heat $c_{V}$ is dimensionless, as in [35]. The numerical results from MD simulations are presented in Fig. 2 as a function of the reduced coupling strength $\Gamma/\Gamma_{m}$, for different values of the screening parameter $\kappa$ and for both 2D and 3D systems. Clearly, as the reduced coupling strength $\Gamma/\Gamma_{m}$ increases, the specific heat $c_{V}$ increases monotonically, for both 2D and 3D fluid dusty plasmas. Since the coupling strength has to be interpreted as an inverse temperature, this behavior is consistent with the expected heat capacity for a liquid, which contrary to that of solids, decreases monotonically with temperature. At $\Gamma\approx\Gamma_{m}$, the obtained values match the expectations from solid state theory, $c_{V}^{\rm 3D}=3$ and $c_{V}^{\rm 2D}=2$. In the opposite limit, $\Gamma/\Gamma_{m}\to 0$, the data approaches the ideal gas result, $c_{V}^{\rm 3D}=3/2$ and $c_{V}^{\rm 3D}=1$. From our simulation results in Fig. 2, $c_{V}^{\rm 2D}=1.5$ and $c_{V}^{\rm 3D}=2$ both occur at the same reduced coupling strength of $\Gamma/\Gamma_{m}=\gamma_{c}=0.05$, with $\gamma_{c}$ the dimensionless critical value. This clearly indicates that the Frenkel line for both 2D and 3D fluid dusty plasmas is located at the same $\gamma_{c}$, suggesting that the liquid-like-gas-like transition in dusty plasmas or Yukawa systems is independent of the dimensionality. Intuitively, this could be explained by considering the dynamics of transverse shear waves. In absence of shear excitations, the system is effectively isotropic and therefore insensitive to the number of dimensions. This is further proved by the universal collapse of the two curves in Fig. 2 for $\Gamma/\Gamma_{m}<\gamma_{c}$, where no collective shear waves can be sustained any more. On the contrary, in the rigid liquid phase, the heat capacity of the 3D system is consistently larger than the 2D counterpart simply because of the larger number of emerging propagating shear waves. Intriguingly, all the 3D and 2D curves collapse into a universal one for different values of the screening parameter $\kappa$. This hints towards a possible universality of our results with respect to the particle interaction potential chosen. As we will see, this universality pertains not only the thermodynamic properties but also the transport ones, such as shear viscosity and thermal conductivity. ### Velocity autocorrelation function and transport In order to verify further our results, we compute the normalized velocity autocorrelation function (VACF): \[C_{v}(t)=\frac{\left\langle{\bf v}\left(t\right)\cdot{\bf v}\left(0\right) \right\rangle}{\left\langle{\bf v}\left(0\right)\cdot{\bf v}\left(0\right) \right\rangle} \tag{3}\] for 2D and 3D fluid dusty plasmas of $\kappa=1$. The numerical results are shown in Fig. 3. Our obtained Frenkel line condition, $\gamma_{c}=0.05$ from the heat capacity $c_{V}$ is further confirmed by our results of the VACF in Fig. 3. In the rigid liquid state, the VACF contains significant oscillations due to the rigidity of the underlying phase. However, in the nonrigid gas-like state, the VACF does not contain oscillations any more, but rather a monotonic decrease typical of gas-like systems. Thus, in [8], the Frenkel line is proposed to be the critical temperature at which the oscillatory behavior in the VACF just disappears. In [8], it is also mentioned that, for some cases, such as systems with strong repulsive interactions, this second criterion from the VACF may be not accurate. In our systems, for 3D fluid dusty plasmas, as $\Gamma/\Gamma_{m}$ decreases, the oscillations of VACF gradually decay until they completely disappear at $\Gamma/\Gamma_{m}=0.05$, perfectly matching the Frenkel line extracted from the heat capacity criterion in Fig. 2. Besides the results for $\kappa=1$ shown in Fig. 3, we have also confirmed that, for all other simulated $\kappa$ values, the critical point at which the oscillations just disappear is always located at $\gamma_{c}=0.05$. Thus, the Frenkel line of 3D dusty plasmas determined by the VACF is always at $\gamma_{c}=0.05$, in agreement with the analysis of the heat capacity in Fig. 2. For 2D dusty plasmas, the dynamical transition of the VACF oscillations at $\Gamma/\Gamma_{m}=0.05$ is not as distinctive as for the 3D systems. This is consistent with the disclaimers mentioned in [8] and might be attributed to the anomalous diffusion in 2D dusty plasmas [66], which is absent in the 3D case. Besides the VACF results presented here, in Appendix B, we show that the various transport coefficients, such as the diffusion coefficient, the shear viscosity, and the thermal conductivity, calculated from our current simulation data and the previous results of Yukawa systems or one-component plasmas [67; 68; 69; 70; 71], are also optimal diagnostics for the dynamical crossover. Figure 3: Calculated normalized velocity autocorrelation function (VACF) $C_{v}(t)$ for 2D (a) and 3D (b) fluid dusty plasmas with $\kappa=1$ and different values of the reduced coupling strength $\Gamma/\Gamma_{m}$. The black lines correspond to the critical value $\gamma_{c}=0.05$. ### Local atomic connectivity and shear relaxation It is an important question to ask whether there exists any physical quantity linked to the microscopic dynamics which can identify the Frenkel line crossover. In the Frenkel/Maxwell theories, the re-arrangement time around local minima $\tau$ plays a fundamental role. A more microscopic characterization of such a process is given by the local atomic connectivity $\tau_{LC}$, which is defined as the time for one particle to maintain its surrounding neighbors, or equivalently the time for the atomic topological structure change [72, 73]. Within the Maxwell viscoelasticity theory, the shear stress relaxation time plays an equivalently important role, specially in the collective dynamics of gapped transverse waves. The shear stress relaxation time $\tau_{M}^{\rm ex}$, first introduced in [73] to quantify the viscoelastic response of dusty plasma liquids, controls the relaxation of the excess part (the particle interaction portion) of the shear stress autocorrelation function. Following [73], we calculate the shear stress relaxation time $\tau_{M}^{\rm ex}$ from the ratio of the excess parts of the shear viscosity $\eta^{\rm ex}$ to the infinite frequency shear modulus $G_{\infty}^{\rm ex}$ using \[\tau_{M}^{\rm ex}=\int_{0}^{\infty}G(t)^{\rm ex}dt/G_{\infty}^{\rm ex}\,, \tag{4}\] where $G(t)^{\rm ex}$ comes from the autocorrelation function of the particle interaction portion of the shear stress time series [73], while $G_{\infty}^{\rm ex}$ is just the initial value of $G(t)^{\rm ex}$[73]. The shear stress relaxation time $\tau_{M}^{\rm ex}$ is approximately equal to the Maxwell relaxation time $\tau_{M}$ when $\Gamma$ is large, in the so-called potential energy dominated regime [73]. From Fig. 4, we discover that the Frenkel line condition can be expressed as the product of the lifetime of the local atomic connectivity $\tau_{LC}$, or the shear stress relaxation time $\tau_{M}^{\rm ex}$, and the Einstein frequency $\omega_{E}$ to be unity, i.e., $\tau_{LC}\omega_{E}=1$ or $\tau_{M}^{\rm ex}\omega_{E}=1$, for 2D or 3D fluid dusty plasmas, respectively. Here, the Einstein frequency $\omega_{E}$ refers to the oscillation frequency of one particle in the environment where all other particles are assumed to be frozen stationary [25, 62]. As evident from the numerical data shown in Fig. 4, $\tau_{LC}\omega_{E}=1$ in 2D systems and $\tau_{M}^{\rm ex}\omega_{E}=1$ in 3D systems both occur at $\Gamma/\Gamma_{m}=0.05$, corresponding to the location of Frenkel line determined from $c_{V}$ in Fig. 2 and VACF in Fig. 3 above, as well as various transport coefficients presented in Appendix B. Thus, both $\tau_{LC}$ and $\tau_{M}^{\rm ex}$ can be used to quantify the relaxation time $\tau$ between two consecutive hops of a single particle [16]. Furthermore, the Einstein frequency $\omega_{E}$ is just proportional to the inverse of the minimum particle vibration period. As a result, the Frenkel's criterion [16] of $\tau/\tau_{D}\approx 1$ is qualitatively equivalent to $\tau_{LC}\omega_{E}\approx 1$ or $\tau_{M}^{\rm ex}\omega_{E}\approx 1$, consistent with the findings in Fig. 4. Thus, we propose the conditions $\tau_{LC}\omega_{E}=1$ and $\tau_{M}^{\rm ex}\omega_{E}=1$ as new diagnostics to determine the Frenkel line for supercritical fluids. To the best of our knowledge, these conditions have not been considered in classical liquids so far. In Fig. 4, it is clear that both $\tau_{LC}\omega_{E}$ in 2D and $\tau_{M}^{\rm ex}\omega_{E}$ in 3D dusty plasmas exhibit a universal collapse in the rigid liquid-like phase. On the contrary, this universal collapse seems to be less precise in the gas-like regime, $\Gamma/\Gamma_{m}<0.05$ where the trend of the data depends mildly on the screening parameter $\kappa$. We also notice that the product $\tau_{M}^{\rm ex}\omega_{E}$ in 3D dusty plasmas displays a discontinuous derivative at the Frenkel line which deserves further investigations. Finally, we also calculated $\tau_{LC}\omega_{E}$ in 3D and $\tau_{M}^{\rm ex}\omega_{E}$ in 2D fluid dusty plasmas, but in that case the universal collapse is less clear. Comparing with the other quantities presented in this work, which all show a universal collapse as a function of $\kappa$, we might speculate that this universality is an emergent property of the collective dynamics which fails, or at least fades away, for microscopic quantities. As a matter of fact, in the gas-like regime, the definition of $\tau_{LC}$ or $\tau_{M}$ is not able to reflect the collective dynamics any more. ### Instantaneous transverse sound speed Besides the aforementioned conditions $\tau_{LC}\omega_{E}=1$ and $\tau_{M}^{\rm ex}\omega_{E}=1$, we discover that the unity ratio of the instantaneous transverse sound speed $C_{T}$ to the average particle speed $\bar{v}_{p}$ can also discriminate the liquid-like and gas-like states and therefore be an optimal diagnostic for the location of the Frenkel line. Here, $C_{T}$ is the instantaneous speed of transverse sound which can be derived from the infinite frequency shear modulus $G_{\infty}$[74, 72, 55]. We observe that the unity ratio of $C_{T}/\bar{v}_{p}$ occurs at the same Figure 4: Local atomic connectivity time $\tau_{LC}$ and shear stress relaxation time $\tau_{M}^{\rm ex}$ multiplied with the Einstein frequency $\omega_{E}$ for various conditions of 2D (a) and 3D dusty plasmas (b). value of $\Gamma/\Gamma_{m}=0.05$ corresponding to the proposed Frenkel line. Some of these results for 2D dusty plasmas have already appeared in [74]. Here, new results for 3D dusty plasmas, as well as more results of 2D dusty plasmas under different conditions are presented. When $C_{T}/\bar{v}_{p}>1$, the transverse sound speed is faster than the motion of individual particles, corresponding to the "cooperative dynamics regime" [74], which is a defining property of rigid liquids. However, when $C_{T}/\bar{v}_{p}<1$, in the "individual dynamics regime" [74] belonging to the nonrigid gas-like state, the average speed of individual particles is larger than the transverse sound speed. Thus, the transition between the cooperative and individual dynamics regimes at $C_{T}/\bar{v}_{p}=1$ coincides exactly with the rigid liquid and nonrigid gas-like states separated by the Frenkel line. Our newly proposed criteria for the Frenkel line, $\tau_{LC}\omega_{E}=1$, $\tau_{M}^{\rm ex}\omega_{E}=1$ and $C_{T}/\bar{v}_{p}=1$, may be verified in the future in other physical systems, such as Lennard-Jones and soft-sphere systems [7, 8]. As compared with $\tau_{LC}\omega_{E}$, $\tau_{M}^{\rm ex}\omega_{E}$, $c_{V}$, and others, the concept of the speed ratio of collective to individual dynamics used in the diagnostic of $C_{T}/\bar{v}_{p}=1$ appears to be superior due to at least two reasons. First, from Fig. 5, the results of $C_{T}/\bar{v}_{p}$ for 2D and 3D fluid dusty plasmas collapse into one universal "master curve". This happens independently of the dimensionality of the system and the value of the screening parameter $\kappa$. Second, the concept of $C_{T}/\bar{v}_{p}$ can be generalized to different physical processes. For example, during compressional shocks in 2D dusty plasmas, a clear transition at the condition of $v_{left}/C_{l,preshock}=1$ has been observed in Figs. 4 and 5 of [75], where $v_{left}$ is the drift velocity of particles after shocks and $C_{l,preshock}$ is the longitudinal wave speed. Clearly, the main cooperative dynamics during compressional shocks [75] is represented by $C_{l,preshock}$, and not $C_{T}$ as above. At the same time, for compressional related dynamics, the average particle speed should be represented by the drift velocity $v_{left}$ in the postshock region along the shock propagation direction, and not the thermal velocity $\bar{v}_{p}$. Thus, in analogy to the $C_{T}/\bar{v}_{p}=1$ criterion, the predicted "phase" transition should be located at $C_{l,preshock}/v_{left}=1$, which is exactly what is observed in [75]. Therefore, when $C_{l,preshock}/v_{left}<1$, the compressed system exhibits a gas-like behavior in which many particles can penetrate the shock front to enter the preshock region, as confirmed in [76]. By extrapolation, we do expect that the speed ratio of the cooperative and individual dynamics may be regarded as a universal criterion valid for different physical processes. ## IV Discussion In this work, MD simulations are performed to investigate the collective and microscopic dynamics in 2D and 3D fluid dusty plasmas. We propose that, below the well-established solid-liquid phase transition (see Fig. 1), these systems should be regarded as supercritical liquids. To support this hypothesis, we reveal a dynamical transition between the rigid liquid and nonrigid gas-like states in the supercritical regime of dusty plasma, using the concept of Frenkel line. We probe this dynamical crossover with several microscopic and macroscopic thermodynamic and transport quantities such as the heat capacity, the VACF, the shear viscosity, and the thermal conductivity. Furthermore, we propose several new criteria to identify the Frenkel line, \[\tau_{LC}\,\omega_{E}=\tau_{M}^{\rm ex}\,\omega_{E}=C_{T}/\bar{v}_{p}=1\,, \tag{5}\] providing a new perspective into this old debate. We expect these measures to be generally valid beyond the dusty plasma system considered here. Starting from our conjecture, expressed as the cartoon in Fig. 1, we are now in the position to fundamentally redefine the phase diagram for 2D and 3D dusty plasmas, by adding a new dynamical crossover into it, as shown in Fig. 6. Importantly, from all our studied diagnostics, it is found that, for both 2D and 3D dusty plasmas, the transition point between the liquid-like and gas-like states is always at $\Gamma/\Gamma_{m}=0.05$. This value corresponds to a temperature of 20 times of the melting point and coincides with the recently proposed criterion to discriminate the strong and weak coupling regimes in dusty plasmas in [74], identified from the behavior of the shear viscosity. Thus, the proposed weak coupling regime of dusty plasma in [74] is just equivalent to the gas-like state, while the proposed strong coupling regime in [74] corresponds to the liquid-like state. From this point of view, the criterion of $\Gamma/\Gamma_{m}=0.05$ in [74] does contain fundamental Figure 5: The ratio of the instantaneous transverse sound speed $C_{T}$ to the average particle speed $\bar{v}_{p}$ for various conditions in 2D and 3D dusty plasmas. The vertical dashed line indicates the location of the Frenkel line as derived from the specific heat and the other diagnostics above. physical significance to discriminate the strong and weak coupling regimes and it may supersede the traditional criterion $\Gamma=1$[33]. The supercritical dynamics for 2D and 3D dusty plasmas revealed in this work presents an emergent degree of universality with respect to the dimensionality of the system and the value of the screening parameter $\kappa$, which hints towards the existence of a general fundamental origin. It would be fruitful to extend this analysis to different systems, potentials and conditions to ascertain, verify, and understand this conjectured universal character. ## V Summary In summary, we propose a fundamental re-definition of the phase diagram of 2D and 3D dusty plasmas by introducing a new dynamical separation between liquid-like and gas-like phases. Our results provide a strong evidence for the supercritical collective dynamics in strongly coupled plasmas and open the path towards a new interpretation of their fundamental nature which could be fruitful for the modern understanding of plasmas, classical liquids and supercritical phases of matter. ###### Acknowledgements. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12175159 and 11875199, the 1000 Youth Talents Plan, startup funds from Soochow University, and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. We thank Xiaqing Shi for helpful discussions. M.B. acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01) and the sponsorship from the Yangyang Development Fund. M.B. would like to thank Chulalongkorn University for the warm hospitality during the completion of this work.	physics.plasm-ph	[ "physics.plasm-ph", "cond-mat.soft", "cond-mat.stat-mech" ]
2302.11448	On properties described by terms in commutator relation	We investigate properties of varieties of algebras described by a novel concept of equation that we call \emph{commutator equation}. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra $\mathbf{A}$ satisfies the commutator equation $p \approx_{C} q$ if for each congruence theta in Con(\mathbf{A}) and for each substitution $p^{\mathbf{A}}, q^{\mathbf{A}}$ of elements in the same $\theta$-class, then $(p^{\mathbf{A}}, q^{\mathbf{A}}) \in [\theta, \theta]$. This notion of equation draws inspiration from the definition of \emph{weak difference term} and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid within the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold within the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a non-trivial idempotent Mal'cev condition then also the entire variety satisfies a non-trivial idempotent Mal'cev condition, statement that follows also form \cite[Theorem 3.13]{KK.TSOC}.	Stefano Fioravanti	2023-02-22T15:34:23	http://arxiv.org/abs/2302.11448v2	# On properties described by terms in commutator relation ###### Abstract. We investigate properties of varieties of algebras described by a new concept of equation that we call _commutator equation_. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra $\mathbf{A}$ satisfies the commutator equation $p\approx_{C}q$ if for each congruence $\theta$ in $\operatorname{Con}(\mathbf{A})$ and for each substitution $p^{\mathbf{A}},q^{\mathbf{A}}$ of elements in the same $\theta$-class, then $(p^{\mathbf{A}},q^{\mathbf{A}})\in[\theta,\theta]$. This concept of equation is inspired by the definition of _weak difference term_ and allows for further generalization of this notion. Furthermore, we prove that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a non-trivial idempotent Mal'cev condition then also the entire variety satisfies a non-trivial idempotent Mal'cev condition. This result represents an improvement of Taylor's characterization of the class of varieties satisfying a non-trivial idempotent Mal'cev condition obtained with a relaxation of the hypothesis. Moreover, we provide an algorithm to connect congruence equations that hold in the variety generated by the abelian algebras of the idempotent reduct of a variety and congruence equations that hold in the whole variety. ## 1. Introduction The genesis of this work is our interest in properties of classes of varieties described by equations. Starting from Mal'cev's description of congruence permutability as in [25], the problem of characterizing properties of classes of varieties as Mal'cev conditions has led to several results. In [27] A. Pixley found a strong Mal'cev condition defining the class of varieties with distributive and permuting congruences. In [15] B. Jonsson shows a Mal'cev condition characterizing congruence distributivity, in [8] A. Day shows a Mal'cev condition characterizing the class of varieties with modular congruence lattices. These results are examples of a more general theorem obtained independently by Pixley [28] and R. Wille [33] that can be considered as a foundational result in the field. They proved that if $p\leq q$ is a lattice identity, then the class of varieties whose congruence lattices satisfy $p\leq q$ is the intersection of countably many Mal'cev classes. [28] and [33] include an algorithm to generate Mal'cev conditions associated with congruence identities. These researches have also led to the problem of studying equations where the variables run not only over the congruence lattices but in possibly strictly larger sets as the lattices of all tolerances or of all compatible reflexive relations. Results about this problem can be found in [7, 12, 14, 19, 24, 32]. Furthermore, the study of Mal'cev conditions and more generally of properties of closed sets of operations has become more and more popular in the recent years because of its deep connection with CSPs and PCSPs problems [1, 2, 3, 9, 10, 11, 20, 21, 29, 34]. Following this branch of research, our aim is to introduce a new type of equations, that we will call _commutator equations_. Inspired by the definition and characterization of varieties with a _weak difference_ term (see [19] chapter 6.1) we model the new concept of equation to generalize this definition. A careful reader can easily see that the _weak difference_ is a weakening of a Mal'cev term obtained relaxing the characterizing equations to what we call commutator equations. This produce a property defining a Mal'cev condition strictly weaker than the one characterizing the class of varieties with a Mal'cev term. In Section 2 and 3 we introduce some of the basic concepts for our investigation. In Section 4 we prove an equivalent of the Taylor's result [31] for commutator equations. Namely, Theorem 4.4 shows that a variety $\mathcal{V}$ is Taylor if and only if the variety $\mathcal{V}^{\prime}$ generated by the abelian algebras of the idempotent reduct of $\mathcal{V}$ is Taylor. Furthermore, we provide a new characterization of this property in terms of congruence equations. This result establishes a connection between a variety $\mathcal{V}$ and the variety $\mathcal{V}^{\prime}$ generated by the abelian algebras of the idempotent reduct of $\mathcal{V}$ that we develop further in Section 5. In this last section our aim is to provide a Pixley-Wille type of algorithm for commutator equation. In the main result of the section, Theorem 5.3, we prove that, under mild assumptions on the lattice terms involved, weakening standard term equations produced by a congruence equation via the Pixley-Wille algorithm to commutator equations gives a property describing a Mal'cev class. ## 2. Preliminaries and Notations In this section we recall some of the basic definitions of universal algebra and we introduce the few new definitions we need. For other elementary concepts in general algebra (such as lattices, algebras, varieties, etc.) our textbook reference is [4]; for more advanced topics (such as abelian congruences and the commutator of congruences) we refer the reader to [19]. For the general theory of Mal'cev conditions and Mal'cev classes we refer to the classical treatment in [30] or the more modern approach in [19]. We recall the definition of _Centralizer_. Let $\mathbf{A}$ be an algebra. For $\alpha,\beta,\in\operatorname{Con}(\mathbf{A})$ we define $M(\alpha,\beta)$ as the set of matrices of the form \[\begin{pmatrix}f(\mathbf{a},\mathbf{u})&f(\mathbf{a},\mathbf{v})\\ f(\mathbf{b},\mathbf{u})&f(\mathbf{b},\mathbf{v})\end{pmatrix}\] where $f\in\operatorname{Pol}_{n+m}(\mathbf{A})$ and $\mathbf{a},\mathbf{b}\in A^{n}$, $\mathbf{u},\mathbf{v}\in A^{m}$ with $\mathbf{a}\equiv_{\alpha}\mathbf{b}$ and $\mathbf{u}\equiv_{\beta}\mathbf{v}$. For $\alpha,\beta,\delta\in\operatorname{Con}(\mathbf{A})$, we say that $\alpha$_centralizes $\beta$ modulo $\delta$_ if for all possible matrices $M\in M(\alpha,\beta)$ with $f\in\operatorname{Pol}(\mathbf{A})$ we have that: \[\text{if }f(\mathbf{a},\mathbf{u})\equiv_{\delta}f(\mathbf{a},\mathbf{v}), \text{ then }f(\mathbf{b},\mathbf{u})\equiv_{\delta}f(\mathbf{b},\mathbf{v}).\] In this case we write $C(\alpha,\beta;\delta)$ and we call the ternary relation $C$_centralizer_. We refer to [19][Section 2.5] for the various properties of the centralizer relation. We recall also the definition of the _two-term Centralizer_. Let $\mathbf{A}$ be an algebra. For $\alpha,\beta,\delta\in\operatorname{Con}(\mathbf{A})$ we say that $\alpha$_two-term centralizes $\beta$ modulo $\delta$_ if for all possible matrices \[\begin{pmatrix}f(\mathbf{a},\mathbf{u})&f(\mathbf{a},\mathbf{v})\\ f(\mathbf{b},\mathbf{u})&f(\mathbf{b},\mathbf{v})\end{pmatrix}\text{and} \begin{pmatrix}t(\mathbf{a},\mathbf{u})&t(\mathbf{a},\mathbf{v})\\ t(\mathbf{b},\mathbf{u})&t(\mathbf{b},\mathbf{v})\end{pmatrix}\in M(\alpha,\beta)\] where $f,t\in\operatorname{Pol}_{n+m}(\mathbf{A})$ and $\mathbf{a},\mathbf{b}\in A^{n}$, $\mathbf{u},\mathbf{v}\in A^{m}$ with $\mathbf{a}\equiv_{\alpha}\mathbf{b}$ and $\mathbf{u}\equiv_{\beta}\mathbf{v}$ we have that: \[\text{if }f(\mathbf{a},\mathbf{u})\equiv_{\delta}t(\mathbf{a}, \mathbf{u}),\,f(\mathbf{a},\mathbf{v})\equiv_{\delta}t(\mathbf{a},\mathbf{v}),\,\text{and }f(\mathbf{b},\mathbf{u})\equiv_{\delta}t(\mathbf{b},\mathbf{u})\text{ then}\] \[f(\mathbf{b},\mathbf{v})\equiv_{\delta}t(\mathbf{b},\mathbf{v}).\] We write $C_{2T}(\alpha,\beta;\delta)$ and we call the ternary relation $C_{2T}$_two-term centralizer_. We will use the TC-commutator as defined in [13] by Freese and McKenzie and we will refer to [19] for properties and basic results about the commutator. Moreover, in Theorem 5.3 we use the concepts of _linear commutator_ and _two terms commutator_, in symbols respectively $[\alpha,\beta]_{L}$ and $[\alpha,\beta]_{2T}$, as defined for example in [18] and [22]. Furthermore, a _weak difference term_ for a variety $\mathcal{V}$ is a ternary term $d(x,y,z)$ such that that for all $\mathbf{A}\in\mathcal{V}$, $a,b\in A$, and $\theta\in\operatorname{Con}(\mathbf{A})$ with $(a,b)\in\theta$, we have: \[d(b,b,a)\;[\theta,\theta]\;a\;[\theta,\theta]\;d(a,b,b),\] see [19, Chapter 6] for a deep treatment of the notion. We can observe that a weak difference term is Mal'cev on any block of an abelian congruence, thus in particular any abelian algebra in a variety which has a weak difference term is affine, see [19]. As a generalization of this definition we introduce the notion of _commutator equation_. Let $p$ and $q$ terms for a language $\mathcal{L}$. Then we denote by $p\approx q$ an equation involving the two terms. Through all the paper we will refer to this standard type of equations as _term equations_. As a relaxation of this concept we introduce the definition of commutator equation. Let $\mathbf{A}$ be an algebra and let $p$ and $q$ be $n$-ary terms of $\mathbf{A}$. We say that $\mathbf{A}$_satisfies_ the commutator equation $p\approx_{C}q$, in symbols $A\models p\approx_{C}q$, if for all $\theta\in\mathrm{Con}(\mathbf{A})$ and for all $a_{1},\ldots a_{n}\in A$ in the same $\theta$-class, we have: \[p(a_{1},\ldots a_{n})\;[\theta,\theta]\;q(a_{1},\ldots a_{n}).\] Clearly this concept of equation is weaker than the standard term equation. If an algebra satisfies the term equation $p\approx q$, then it satisfies also the commutator equation $p\approx_{C}q$ We recall the definition of _Taylor term_, introduced in [31] which will be widely mentioned through all the paper. An $n$-ary _Taylor term_ is a term satisfying the following term equations: \[f(x_{11},\ldots,x_{1n}) \approx f(y_{11},\ldots,y_{1n})\] \[\cdot\] \[\cdot\] \[\cdot\] \[f(x_{n1},\ldots,x_{nn}) \approx f(y_{n1},\ldots,y_{nn})\] with $x_{ij},y_{ij}\in\{x,y\}$ and $x_{ii}\neq y_{ii}$. These equations describe the class of varieties satisfying a non-trivial idempotent Mal'cev condition [31] and are called _Taylor varieties_. Before moving forward, we recall the definition of _Herringbone terms_ that can be found also in [19, Section 8.2] or in [22]. These terms are of interest since they are deeply connected with well behaved commutators in varieties as can be seen in [18] and [22]. Definition 2.1.: We call _Herringbone terms_ the two families of lattice terms $\{x^{i}\}_{i\in\mathbb{N}}$ and $\{y^{i}\}_{i\in\mathbb{N}}$ in the variables $\{x,y,z\}$ defined by: \[y^{0}(x,y,z) =y;z^{0}(x,y,z)=z;\] \[y^{n+1}(x,y,z) =y\vee(x\wedge z^{n}(x,y,z))\] \[z^{n+1}(x,y,z) =z\vee(x\wedge y^{n}(x,y,z)).\] We use the superscript instead of the standard subscript since the latter is in conflict to the usual subscrit notation for a sequence of variables. Note that this lattice terms form two possibly infinite ascending chains. Furthermore, let $p$ be a term for the language $\{\wedge,\vee,\circ\}$. Let $k\in\mathbb{N}$. We denote by $p^{(k)}$ the $\{\wedge,\circ\}$-term obtained from $p$ substituting any occurrence of $\vee$ with the $k$-fold relational product $\circ^{(k)}$. ## 3. Labelled graphs and regular terms In order to show the main results of Section 5 we recall the definition _labelled graph associated with a term_ as defined in [5], [6], [19]. We first give the definition of labelled graph. Definition 3.1.: Let $S$ be a set of labels. Then a _labelled graph_ is a directed graph $(V,E)$ with a labelling function $l:E\to S$. Following [19], let $p$ be a $\{\wedge,\circ\}$-term in the variables $\{x_{1},\ldots,x_{t}\}$. We introduce a construction producing a finite sequence of labelled graphs $\{\mathbf{G}_{i}(p)\}_{i\in I}$. The sequence starts with the labelled graph $\mathbf{G}_{1}(p)=(\{y_{1},y_{2}\}$, $\{(y_{1},y_{2}$ )\}\) with $l((y_{1},y_{2}))=p$ having an edge $(y_{1},y_{2})$ labelled with $p$ and connecting two vertices $y_{1}$ and $y_{2}$. For $s\geq 1$, from the labelled graph $\mathbf{G}_{s}(p)$ we continue selecting $w\not\in\{x_{1},\ldots,x_{t}\}$ such that $w$ is a label of an edge $(y_{i},y_{j})$ of $\mathbf{G}_{s}(p)$. Then we have two cases. * If $w=u\wedge v$ for some $u$ and $v$$\{\wedge,\circ\}$-terms. Then $\mathbf{G}_{s+1}(p)$ is obtained from $\mathbf{G}_{s}(p)$ by replacing the edge $(y_{i},y_{j})$ labelled $w$ with two edges connecting the same vertices $(y_{i},y_{j})$ and labelled $u$ and $v$ respectively; * if $w=u\circ v$ for some $u$ and $v$$\{\wedge,\circ\}$-terms. Then $\mathbf{G}_{s+1}(p)$ is obtained from $\mathbf{G}_{s}(p)$ by introducing a new vertex $y_{k}$ and replacing the edge $(y_{i},y_{j})$ labelled $w$ with two edges $(y_{i},y_{k})$ and $(y_{k},y_{j})$, labelled $u$ and $v$ respectively, connecting the same vertices in serial through $y_{k}$. The construction ends when for some $n\in\mathbb{N}$ none of the above steps can be performed for the labelled graph $\mathbf{G}_{n}(p)$. Thus $l(e)\in\{x_{1},\ldots,x_{t}\}$ for every $e$ edge in $\mathbf{G}_{n}(p)$. We note that the choice of $w$ can determine different sequences from a term $p$ but we can observe that the last graph of the sequence is always the same up to reorder of the vertices. We call the last graph of this sequence _labelled graph associated with_$p$, denoted by $\mathbf{G}(p)$. The main reason to introduce $\mathbf{G}(p)$ is stated in [6, Proposition 3.1] and in Claim 4.8 of [19]. The latter can be generalized to tolerances as in [23, Proposition 2.1] and also to relations in general [12][Proposition 3.3]. We include the most general version of this technical result in order to use it in Section 5. Proposition 3.2 (Proposition 3.3 of [12]).: _Let $\mathbf{A}$ be an algebra, let $R_{i}\subseteq A\times A$, for $1\leq i\leq n$, and let $p$ be a $\{\circ,\wedge\}$-term. Then:_ 1. _Let_ $Y\to A$_:_ $y_{s}\mapsto a_{s}$ _be an assignment such that for all edges_ $(y_{i},y_{j})$ _with label_ $X_{k}$ _of_ $\mathbf{G}(p)$_, we have_ $(a_{i},a_{j})\in R_{k}$_. Then_ $(a_{1},a_{2})\in p(R_{1},\ldots,R_{n})$_._ 2. _Conversely, given any_ $(a_{1},a_{2})\in p(R_{1},\ldots,R_{n})$_, there is an assignment_ $Y\to A$_:_ $y_{s}\mapsto a_{s}$ _extending_ $y_{1}\mapsto a_{1}$_,_ $y_{2}\mapsto a_{2}$ _such that_ $(a_{i},a_{j})\in R_{k}$ _whenever_ $(y_{i},y_{j})$ _is an edge labelled with_ $X_{k}$ _of_ $\mathbf{G}(p)$_, where_ $(y_{1},y_{2})$ _is the only edge of the graph_ $\mathbf{G}_{1}(p)$_._ ## 4. Taylor varieties and abelian algebras In this section our aim is to show a property describing Taylor varieties logically weaker then the standard presented in [31]. Taylor varieties are of interest for several reasons. Surprisingly the class of Taylor varieties forms a strong Mal'cev class [26], and this class of varieties has a deep connection with the Feder-Vardi conjecture that was proven to be true in [2] and [34] independently. Furthermore, modern results concerning the relationship between commutator and Taylor varieties can be found in [17]. Thus for many reasons Taylor varieties have always been a central theme of research in universal algebra. Before going deeply into the theory of commutator equation we make an easy observation. Remark 4.1.: Let $\mathcal{V}$ be a variety satisfying the commutator equation $x\approx_{C}p(x,\ldots,x)$. Then $p$ is idempotent. The remark follows from the fact that the commutator equation has to hold also for the $0$ congruence. In order to prove the main result of the section we present a lemma that can be seen in [22] in a slightly different version without a proof. From now on when not specified $\beta^{n}=y^{n}(\alpha,\beta,\gamma)$ and $\gamma^{n}=z^{n}(\alpha,\beta,\gamma)$ Lemma 4.2.: _Let $\mathbf{A}$ be an algebra and $\alpha,\beta,\gamma\in\operatorname{Con}(\mathbf{A})$. Let $\delta=\bigcup_{n\in\mathbb{N}}(\alpha\wedge\beta_{n})$. Then,_ \[[\alpha\wedge(\beta\vee\gamma),\alpha\wedge(\beta\vee\gamma)]\leq\delta.\] Proof.: We observe that Definition 2.1 yields that the chains of congruences $\{\alpha\wedge\beta^{n}\}_{n\in\mathbb{N}}$ and $\{\alpha\wedge\gamma^{n}\}_{n\in\mathbb{N}}$ are cofinal since: \[\alpha\wedge\beta^{n+1}\geq\alpha\wedge\gamma^{n}\geq\alpha\wedge\beta^{n-1}.\] Hence \[\delta=\bigcup_{n\in\mathbb{N}}(\alpha\wedge\beta^{n})=\bigcup_{n\in\mathbb{N }}(\alpha\wedge\gamma^{n}).\] This identity and the fact that $\wedge$ distributes over the union implies: \[\alpha\wedge(\beta\vee(\alpha\wedge\delta))=\alpha\wedge(\beta\vee(\alpha \wedge\bigcup_{n\in\mathbb{N}}(\alpha\wedge\gamma^{n}))=\bigcup_{n\in\mathbb{ N}}(\alpha\wedge\beta^{n})=\delta. \tag{4.1}\] Clearly a similar equality holds substituting $\beta$ with $\gamma$. From this two identities and the properties centralizer relation, see [19][Theorem 2.19 (8)], we obtain that $C(\beta,\alpha;\delta)$ and $C(\gamma,\alpha;\delta)$. The semidistributivity of the centralizer in the first component ([19][Theorem 2.19 (5)]) yields $C(\beta\vee\gamma,\alpha;\delta)$. From the definition of commutator we obtain $[\beta\vee\gamma,\alpha]\leq\delta$ and the claim follows from the monotonicity of the commutator. Lemma 4.2 in conjunction with the next theorem will be a key element to prove the main result of the section. The following theorem, connecting Olsak's result [26] to the equation characterizing Taylor varieties in [18], exploits [26] and the congruence equation in [18][Lemma 4.6] to get a simplified version of this equation. Indeed, the congruence equation characterizing Taylor varieties in [18] is produced using an arbitrary $n$-ary Taylor term and thus with a $6$-ary Olsak term can be refined to a more readable version involving a fixed number of congruences. Theorem 4.3.: _Let $\mathcal{V}$ be a variety. Then the following are equivalent:_ 1. $\mathcal{V}$ _is Taylor;_ 2. $\mathcal{V}$ _satisfies the term equations:_ (4.2) \[t(x,y,y,y,x,x)\approx t(y,x,y,x,y,x)\approx t(y,y,x,x,x,y);\] _for some idempotent term_ $t$_._ 3. $\mathcal{V}$ _satisfies the following congruence equation in the variables_ $\{\alpha_{1},\dots,\alpha_{6},\beta_{1},\dots,\beta_{6}\}$_:_ (4.3) \[\bigwedge_{i=1}^{6}(\alpha_{i}\circ\beta_{i})\leq(\bigvee_{i=1}^{6}\alpha_{i} \wedge\bigwedge_{i=1}^{2}(\gamma\vee\theta_{i}))\vee(\bigvee_{i=1}^{6}\beta_{ i}\wedge\bigwedge_{i=1}^{2}(\gamma\vee\theta_{i}));\] _where_ $\gamma=\bigwedge_{i=1}^{6}(\alpha_{i}\vee\beta_{i})$ _and_ \[\theta_{i}=(\bigvee_{j\in L_{i}}\alpha_{j}\vee\bigvee_{j\in L_{i}^{\prime}}\beta_ {j})\wedge(\bigvee_{j\in R_{i}}\alpha_{j}\vee\bigvee_{j\in R_{i}^{\prime}}\beta_ {j})\] _with_ \[L_{1} =L_{2}=\{1,5,6\} L_{1}^{\prime}=L_{2}^{\prime}=\{2,3,4\}\] \[R_{1} =\{2,4,6\} R_{1}^{\prime}=\{1,3,5\}\] \[R_{2} =\{3,4,5\} R_{2}^{\prime}=\{1,2,6\}\] Proof.: The equivalence of (1) and (2) follows from Olsak's result [26]. $(2)\Rightarrow(3)$ follows from [18][Lemma 4.6] adjusting the proof for the 6-ary Taylor term in (2). For sake of completeness we include the modified version of the proof. Let $\mathbf{A}\in\mathcal{V}$ and let $(a,b)\in\bigwedge_{i=1}^{6}(\alpha_{i}\circ\beta_{i})$. Thus there exist $u_{1},\ldots u_{6}\in A$ such that $a\;\alpha_{i}\;u_{i}\;\beta_{i}\;b$, for all $i\in\{1,\ldots,6\}$. Let $u=t(u_{1},\ldots,u_{6})$ and $v=t(a,b,b,b,a,a)=t(b,a,b,a,b,a)=t(b,b,a,a,a,b)$. We prove that: 1. $(a,u)\in\bigvee_{i=1}^{6}\alpha_{i}$; 2. $(u,b)\in\bigvee_{i=1}^{6}\beta_{i}$; 3. $(a,u),(u,b)\in\bigwedge_{i=1}^{2}(\gamma\circ\theta_{i})$. For claims (a) and (b) we observe that: \[a =t(a,\ldots,a)\;\alpha_{1}\circ\cdots\circ\alpha_{6}\;t(u_{1}, \ldots,u_{6})\] \[b =t(b,\ldots,b)\;\beta_{1}\circ\cdots\circ\beta_{6}\;t(u_{1}, \ldots,u_{6})\] For claim (c) we notice that $a$, $b$ and $v$ are in the same $\gamma$-class. Furthermore: \[u=t(u_{1},\ldots,u_{n})\;\theta_{i}\;v.\] Thus $a\;\gamma\;v\;\theta_{i}\;u$ and $b\;\gamma\;v\;\theta_{i}\;u$ and (c) holds. Putting (a), (b), and (c) together we obtain that $(a,b)$ is in the right side of the inequality (4.3) and (3) holds. For $(3)\Rightarrow(1)$ we can observe that the equation (4.3) is a spacial case of the equation in [18][proof of Lemma 4.6], hence (4.3) is non-trivial and thus it implies the satisfaction of a non-trivial idempotent Mal'cev condition. This can be seen considering an 8-element set $\{a,b,u_{1},\ldots,u_{n}\}$ with the partitions $\alpha_{i}$ generated by $(a,u_{i})$ and $\beta_{i}$ generated by $(b,u_{i})$, for all $i\in\{1,\ldots,6\}$. Hence (1), (2) and (3) are equivalent. We can see that equation (4.3) could be further simplified using the relation product and avoiding completely the use of $\vee$ and thus generating a strong Mal'cev condition through the Pixley-Wille algorithm [28, 33]. We are ready to prove the main result of the section which uses Theorem 4.3 to find a Mal'cev condition equivalent to be Taylor involving commutator equations. Theorem 4.4.: _Let $\mathcal{V}$ be a variety. Then the following are equivalent:_ 1. $\mathcal{V}$ _is Taylor;_ 2. _the variety_ $\mathcal{V}^{\prime}$ _generated by abelian algebras of the idempotent reduct of_ $\mathcal{V}$ _is Taylor;_ 3. $\mathcal{V}$ _satisfies the commutator equations:_ \[t(x,y,y,y,x,x)\approx_{C}t(y,x,y,x,y,x)\approx_{C}t(y,y,x,x,x,y)\] \[t(x,x,x,x,x,x)\approx_{C}x;\] 4. $\mathcal{V}$ _satisfies the following congruence equation:_ \[\bigwedge_{i=1}^{6}(\alpha_{i}\circ\beta_{i})\leq(\bigvee_{i=1}^{6}\alpha_{i} \wedge\bigwedge_{i=1}^{6}(\tau\vee\theta_{i}))\vee(\bigvee_{i=1}^{6}\beta_{i} \wedge\bigwedge_{i=1}^{6}(\tau\vee\theta_{i}));\] _where_ \[\theta_{i}=(\bigvee_{j\in L_{i}}\alpha_{j}\vee\bigvee_{j\in L_{i}^{\prime}} \beta_{j})\wedge([\tau,\tau]\circ\bigvee_{j\in R_{i}}\alpha_{j}\vee\bigvee_{j \in R_{i}^{\prime}}\beta_{j})\] _with_ $\tau=\bigwedge_{i=1}^{6}(\alpha_{i}\vee\beta_{i})$ _and_ \[L_{1} =L_{2}=\{1,5,6\} L_{1}^{\prime}=L_{2}^{\prime}=\{2,3,4\}\] \[R_{1} =\{2,4,6\} R_{1}^{\prime}=\{1,3,5\}\] \[R_{2} =\{3,4,5\} R_{2}^{\prime}=\{1,2,6\}\] 5. _there exists_ $n\in\mathbb{N}$ _such that_ $\mathcal{V}$ _satisfies the following congruence equation in the variables_ $\{\alpha_{1},\ldots,\alpha_{6},\beta_{1},\ldots,\beta_{6}\}$_:_ (4.4) \[\bigwedge_{i=1}^{6}(\alpha_{i}\circ\beta_{i})\leq(\bigvee_{i=1}^{6}\alpha_{i} \wedge\bigwedge_{i=1}^{2}(\tau\vee\theta_{i}))\vee(\bigvee_{i=1}^{6}\beta_{i} \wedge\bigwedge_{i=1}^{2}(\tau\vee\theta_{i}));\] _where_ $\tau=\bigwedge_{i=1}^{6}(\alpha_{i}\vee\beta_{i})$ _and_ \[\theta_{i}=(\bigvee_{j\in L_{i}}\alpha_{j}\vee\bigvee_{j\in L_{i}^{\prime}} \beta_{j})\wedge((\bigwedge_{i=1}^{5}(\alpha_{i}\vee\beta_{i})\wedge\beta_{6} ^{n})\circ\bigvee_{j\in R_{i}}\alpha_{j}\vee\bigvee_{j\in R_{i}^{\prime}} \beta_{j})\] _with_ $\beta_{6}^{n}=z^{n}(\bigwedge_{i=1}^{5}(\alpha_{i}\vee\beta_{i}),\alpha_{6}, \beta_{6})$ _and_ \[L_{1} =L_{2}=\{1,5,6\} L_{1}^{\prime}=L_{2}^{\prime}=\{2,3,4\}\] \[R_{1} =\{2,4,6\} R_{1}^{\prime}=\{1,3,5\}\] \[R_{2} =\{3,4,5\} R_{2}^{\prime}=\{1,2,6\}\] Proof.: $(1)\Rightarrow(2)$ is trivial since if $\mathcal{V}$ is Taylor then also the idempotent reduct of $\mathcal{V}$ is Taylor. For $(2)\Rightarrow(3)$, let us assume that the variety $\mathcal{V}^{\prime}$ generated by abelian algebras of the idempotent reduct of $\mathcal{V}$ is Taylor. Thus it has an Olsak term $t$ that satisfies the equations in (4.2). Let $\mathbf{A}\in\mathcal{V}$ and let $\theta\in\mathrm{Con}(\mathbf{A})$. Factoring $\mathbf{A}$ by $[\theta,\theta]$ we see that is sufficient to verify the equations in (3) for abelian congruences. Let $\mathbf{A}^{\prime}$ be the idempotent reduct of $\mathbf{A}$. We can observe that every $\theta$-class of $\mathbf{A}$ is an abelian subalgebra of $\mathbf{A}^{\prime}$ and thus it has a term $t$ satisfying (4.2) by (2). We claim that $t$ satisfies the equations in (3) for $\mathcal{V}$. Let $(a,b)\in\theta$, then $t(a,b,b,b,a,a)=t(b,a,b,a,b,a)=t(b,b,a,a,a,b)$ and $t(a,a,a,a,a,a)=a$. The commutator equations (4.2) follow directly remembering that a quotient by $[\theta,\theta]$ has been applied. For $(3)\Rightarrow(4)$, suppose that (3) holds. Then let $\mathbf{A}\in\mathcal{V}$ and let $\alpha_{1},\ldots,a_{6},\beta_{1},\ldots,\beta_{6}\in\mathrm{Con}(\mathbf{A})$ with $(a,b)\in\bigwedge_{i=1}^{6}(\alpha_{i}\circ\beta_{i})$. Then, with a similar argument of $(2)\Rightarrow(3)$ in the proof of Theorem 4.3 we have that there exist $u_{1},\ldots u_{6}\in A$ such that $a\ \alpha_{i}\ u_{i}\ \beta_{i}\ b$, for all $i\in\{1,\ldots,6\}$. Let $u=t(u_{1},\ldots,u_{6})$ and $v=t(a,b,b,b,a,a)$. Since $(a,b)\in\tau$ we have that $v\ [\tau,\tau]\ t(b,a,b,a,b,a)\ [\tau,\tau]\ t(b,b,a,a,a,b)$, by the equations in (3). We prove that: 1. $(a,u)\in\bigvee_{i=1}^{6}\alpha_{i}$; 2. $(u,b)\in\bigvee_{i=1}^{6}\beta_{i}$; 3. $(a,u),(u,b)\in\bigwedge_{i=1}^{2}(\tau\circ\theta_{i})$. For claims (a) and (b) we can observe that $t$ is idempotent, by Remark 4.1, and thus. \[a =t(a,\ldots,a)\ \alpha_{1}\circ\cdots\circ\alpha_{6}\ t(u_{1}, \ldots,u_{6})\] \[b =t(b,\ldots,b)\ \beta_{1}\circ\cdots\circ\beta_{6}\ t(u_{1}, \ldots,u_{6}).\] For claim (c) we notice that $a$, $b$ and $v$ are in the same $\tau$-class. Furthermore, we can prove that $v\ \theta_{i}\ u$, for $i=1,2$. \[t(u_{1},\ldots,u_{n})\ \bigvee_{j\in L_{1}}\alpha_{j}\vee\bigvee_{j \in L_{1}^{\prime}}\beta_{j}\ v\] \[t(u_{1},\ldots,u_{n})\ \bigvee_{j\in R_{1}}\alpha_{j}\vee\bigvee_{j \in R_{1}^{\prime}}\beta_{j}\ t(b,a,b,a,b,a)\ [\tau,\tau]\ v\] \[t(u_{1},\ldots,u_{n})\ \bigvee_{j\in R_{2}}\alpha_{j}\vee\bigvee_{j \in R_{2}^{\prime}}\beta_{j}\ t(b,b,a,a,a,b)\ [\tau,\tau]\ v.\] Hence, $a\ \tau\ v\ \theta_{i}\ u$ and $b\ \tau\ v\ \theta_{i}\ u$ thus (c) holds. Putting (a), (b), and (c) together we obtain that $(a,b)$ is in the right side of the inequality in (4). (4) $\Rightarrow$ (5) follows applying Lemma 4.2 to the equation in (4) with $\alpha=\bigwedge_{i=1}^{5}(\alpha_{i}\vee\beta_{i})$, $\beta=\alpha_{6}$, and $\gamma=\beta_{6}$. Thus $[\bigwedge_{i=1}^{6}(\alpha_{i}\vee\beta_{i}),\bigwedge_{i=1}^{6}(\alpha_{i} \vee\beta_{i})]\leq\bigcup_{n\in\mathbb{N}}(\alpha\wedge\beta^{n})$ and hence there exists $n\in\mathbb{N}$ such that: \[[\tau,\tau]\leq\bigwedge_{i=1}^{5}(\alpha_{i}\vee\beta_{i})\wedge\beta_{6}^{n}\] This yields the equation in (5). For $(5)\Rightarrow(1)$ we show that equation (4.4) is non-trivial and this implies (1) through the Pixley-Wille algorithm, see [28, 33]. Let $\mathbf{A}=\langle\{a,b,c_{1}\ldots c_{6}\},\pi\rangle$ be an algebra whose basic operations are only projections. Clearly every equivalence class is a congruence in $\operatorname{Con}(\mathbf{A})$. Let $\alpha_{i}$ be the partitions which identify only $\{a,c_{i}\}$ and let $\beta_{i}$ be the partitions which identify only $\{b,c_{i}\}$, for all $i\in\{1,\ldots,6\}$. Then $(a,b)$ is in the left side of (4.4). We prove that $(a,b)$ is not in the right side. First we can observe that $\bigwedge_{i=1}^{5}(\alpha_{i}\vee\beta_{i})\wedge\beta_{6}^{n}=0$ for all $n\in\mathbb{N}$, since $\beta_{6}^{n}=\beta_{6}$ for all $n\in\mathbb{N}$. Furthermore, $\bigvee_{i=1}^{6}\alpha_{i}$ has exactly the two classes $\{a,c_{1},\ldots,c_{6}\}$ and $\{b\}$ while the two classes of $\bigvee_{i=1}^{6}\beta_{i}$ are $\{c_{1},\ldots,c_{6},b\}$ and $\{a\}$. Moreover, we can observe that $(a,c_{j}),(c_{j},b)\notin\bigwedge_{i=1}^{2}\tau\vee\theta_{i}$ for all $j=1,\ldots,6$. Consequently $\bigwedge_{i=1}^{2}\tau\vee\theta_{i}$ has $\{a,b\}$ as the only non-trivial class. Hence we can conclude that $\{a\}$ is a class of $\bigvee_{i=1}^{6}\alpha_{i}\wedge\bigwedge_{i=1}^{2}(\tau\vee\theta_{i})$ and $\{b\}$ is a class of $\bigvee_{i=1}^{6}\beta_{i}\wedge\bigwedge_{i=1}^{2}(\tau\vee\theta_{i})$. Thus $(a,b)\notin(\bigvee_{i=1}^{6}\alpha_{i}\wedge\bigwedge_{i=1}^{2}(\tau\vee \theta_{i}))\vee(\bigvee_{i=1}^{6}\beta_{i}\wedge\bigwedge_{i=1}^{2}(\tau\vee \theta_{i}))$ as wanted. The previous theorem is an improvement of Taylor's characterization of the class of varieties satisfying a non-trivial idempotent Mal'cev condition obtained with a relaxation of the hypothesis. Note that there is no specific need of the Olsak term in the proof and it also works using a generic $n$-ary Taylor term. Nevertheless, the use the Olsak term allows to obtain simpler equations. Moreover, we can also see that the technique exploited in the previous theorem is clearly applicable in a wider context. In the next section we develop this intuition. ## 5. A Pixley-Wille type algorithm for commutator equations In this section our aim is to develop a Pixley-Wille type algorithm for commutator equations, something that can be partially seen in [22][Theorem 3.4] without a proof. Investigating this problem is of interest since a description of a systematic way of characterizing Mal'cev conditions described by commutator equations can produce idempotent Mal'cev conditions logically weaker than their congruence counterpart. Let $p\leq q$ be an equation for the language $\{\wedge,\circ\}$ in the variables $\{X_{s}\}_{s\in I}$. Let $\mathbf{G}(p)$ and $\mathbf{G}(q)$ be obtained from $p$ and $q$ through the procedure described in Section 3 and let $\{x_{1},\dots,x_{n}\}$ be the set of vertices of $\mathbf{G}(p)$. We define: \[T_{s}(p) :=\{(x_{i},x_{j})\mid(x_{i},x_{j})\text{ is an edge of }\mathbf{G}(p) \text{ with label }X_{s}\}\] \[Tt(q) :=\{(t_{i},t_{j})\mid(x_{i},x_{j})\text{ is an edge of }\mathbf{G}(q)\}\] \[Tt_{s}(q) :=\{(t_{i},t_{j})\mid(x_{i},x_{j})\text{ is an edge of }\mathbf{G}(q) \text{ with label }X_{s}\}, \tag{5.1}\] where the elements $\{t_{1},\dots,t_{l}\}$ of the pairs in $Tt(q)$ are $n$-ary terms of unspecified type with $t_{1}=x_{1}$ and $t_{2}=x_{n}$ in the variables $\{x_{1},\dots,x_{n}\}$, where $n$ is the number of vertices in $\mathbf{G}(p)$. Let $R\subseteq A\times A$ be a relation over the set $A$. We denote by $\operatorname{Eqv}(R)$ the equivalence relation generated by $R$. We define $\operatorname{Eq}(p\leq q)$ as the set of all equations of the form: \[t_{i}(x_{i_{1}},\dots,x_{i_{m}})\approx t_{j}(x_{i_{1}},\dots,x_{i_{n}}) \tag{5.2}\] such that $(t_{i},t_{j})\in Tt_{s}(q)$ and the vector of indices $(i_{1},\dots,i_{n})\subseteq\mathbb{N}^{n}$ satisfies $i_{d}=\min(\{i\mid(x_{i},x_{d})\in\operatorname{Eqv}(T_{s}(p))\})$ for all $d\in[m]$. This means that the variables that are in the equivalence relation generated by the pairs in $T_{s}(p)$ are collapsed. The equations (5.2) are produced by the Pixley-Wille algorithm [28, 33] that characterizes the Mal'cev condition described by the congruence equation $p\leq q$. In the previous definition we introduced terms of unspecified type. We consider those terms to be either of one letter $x_{1},x_{2},\dots$ or composed by a primitive operation symbols applied to letters, e.g $t(x_{1},\dots,$$x_{n})$. Namely, those terms are placeholders used to write equations that can be instantiated in the language of a given variety. Let $\operatorname{Eq}$ be a set terms equations of unspecified type and let $\mathcal{V}$ be a variety of type $\tau$. Then a $\tau$_-realization_ of $\operatorname{Eq}$ is a set of equations obtained from $\operatorname{Eq}$ by replacing each operation symbol, in all of its occurrences in $\operatorname{Eq}$, by some fixed term symbol of type $\tau$. Furthermore, a variety $\mathcal{V}$ satisfies a set of equations $\operatorname{Eq}$ of terms of unspecified type if there exists a $\tau$-realization of $\operatorname{Eq}$ such that the obtained equations hold in $\mathcal{V}$. As usual in the literature from now on we will use the same symbols for terms used as placeholders in some equations and their $\tau$-realization in a variety of type $\tau$. Let $p$ be a term for the language $\{\wedge,\circ\}$ and $q$ be a term for the language $\{\wedge,\circ,\vee\}$. Then we say that a variety $\mathcal{V}$ satisfies $\operatorname{Eq}(p\leq q)$ if there exists $k\in\mathbb{N}$ such that $\mathcal{V}$ satisfies $\operatorname{Eq}(p\leq q^{(k)})$. We modify the last part of the algorithm in [28, 33] in order to obtain a set of equations $\operatorname{Eq}_{C}(p\leq q)$ that characterizes the Mal'cev condition describing the class of varieties which satisfy $p\leq q$ over the algebras that belong to the subvariety generated by abelian algebras of the idempotent reduct. Algorithm 5.1.: _Let $p\leq q$ be an equation for the language $\{\wedge,\circ\}$. Let $\mathbf{G}(p)$ and $\mathbf{G}(q)$ be obtained from $p$ and $q$ with the procedure in Section 3. Let us consider $T_{s}(p)$ and $Tt_{s}(q)$ as in (5.1). We define $\mathrm{Eq}_{C}(p\leq q)$ as the set of all equations of the form:_ \[t_{i}(x_{i_{1}},\ldots,x_{i_{m}})\approx_{C}t_{j}(x_{i_{1}},\ldots,x_{i_{m}})\] _such that $(t_{i},t_{j})\in Tt_{s}(q)$ and the vector of indices $(i_{1},\ldots,i_{m})\subseteq\mathbb{N}^{n}$ satisfies $i_{d}=\min(\{i\mid(x_{i},x_{d})\in\mathrm{Eq}_{\mathcal{V}}(T_{s}(p))\})$ for all $d\in[m]$. This means that the variables that are in the equivalence relation generated by the pairs in $T_{s}(p)$ are collapsed._ In order to prove the main result of the section we need the following technical Lemma inspired by [22][Theorem 3.1]. Lemma 5.2.: _Let $\mathcal{V}$ be a variety, let $\mathbf{F}_{\mathcal{V}}(3)$ be the $3$-generated free algebra of $\mathcal{V}$, let $\alpha=\mathrm{Cg}(\{(x,z)\}),\beta=\mathrm{Cg}(\{(x,y)\}),\gamma=\mathrm{Cg} (\{(y,z)\})$, and let $t,s\in\mathbf{F}_{\mathcal{V}}(3)$ with $(t,s)\in\alpha\wedge y^{n}(\alpha,\beta,\gamma)$ for some $n\in\mathbb{N}$. Then, for all $\mathbf{A}\in\mathcal{V}$ and $\delta\in\mathrm{Con}(\mathbf{A})$ with $[\delta,\delta]_{2T}=0$, we have $t^{\mathbf{A}}(a,b,c)=s^{\mathbf{A}}(a,b,c)$ for all $a,b,c$ in the same $\delta$-class._ Proof.: First we prove that for all $\mathbf{A}\in\mathcal{V}$, $\delta\in\mathrm{Con}(\mathbf{A})$ with $[\delta,\delta]_{2T}=0$, and $u,v$ ternary terms such that $u^{\mathbf{A}}(a,b,a)=v^{\mathbf{A}}(a,b,a)$ and $u^{\mathbf{A}}(a,b,b)$$=v^{\mathbf{A}}(a,b,b)$ for all $a,b$ in the same $\delta$-class, then $u=v$ over the same $\delta$-block. This claim is included in a symmetric version in [22][Theorem 3.1]. In order to prove the claim we can observe that for all $a,b,c$ in the same $\delta$-class we have: \[\begin{pmatrix}u^{\mathbf{A}}(b,a,b)&u^{\mathbf{A}}(b,b,b)\\ u^{\mathbf{A}}(a,a,c)&u^{\mathbf{A}}(a,b,c)\end{pmatrix}\mathrm{and}\begin{pmatrix} v^{\mathbf{A}}(b,a,b)&v^{\mathbf{A}}(b,b,b)\\ v^{\mathbf{A}}(a,a,c)&v^{\mathbf{A}}(a,b,c)\end{pmatrix}\in M(\delta,\delta)\] and hence $u$$[\delta,\delta]_{2T}$$v$ over the same $\delta$-block and this yields $u=v$ over the same $\delta$-class. Moreover, suppose that $(t,s)\in\alpha\wedge y^{1}(\alpha,\beta,\gamma)=\alpha\wedge(\beta\vee(\alpha \wedge\gamma))$. Then there exists $k\in\mathbb{N}$ such that $(t,s)\in\alpha\wedge(\beta\circ^{(2k+1)}(\alpha\vee\gamma))$. Thus $t$$\alpha$$s$ and there exist $v_{1},w_{1},\ldots,v_{n},w_{n}\in\mathbf{F}_{\mathcal{V}}(3)$ such that \[s\ \beta\ v_{1}\ \alpha\wedge\gamma\ w_{1}\cdots v_{k}\ \alpha\wedge\gamma\ w_{k}\ \beta\ t.\] By a standard Mal'cev argument, $t(x,y,x)=s(x,y,x)$, $s(x,x,y)=v_{1}(x,x,y)$, $t(x,x,y)=w_{k}(x,x,y)$, $w_{j-1}(x,x,y)=v_{j}(x,x,y)$, $v_{i}(x,y,x)$$=w_{i}(x,y,x)$, and $v_{i}(x,y,y)=w_{i}(x,y,y)$ are identities of $\mathcal{V}$ for all $i\in\{1,\ldots,k\}$, $j\in\{2,\ldots,k\}$. Let $\mathbf{A}\in\mathcal{V}$ and let $\delta\in\mathrm{Con}(\mathbf{A})$ with $[\delta,\delta]_{2T}=0$. We can apply the previous claim to conclude that $v^{\mathbf{A}}_{i}(a,b,c)=w^{\mathbf{A}}_{i}(a,b,c)$ for $a,b,c$ in the same $\delta$-class. Thus, $s^{\mathbf{A}}(a,a,c)=t^{\mathbf{A}}(a,a,c)$ with $a\ \delta\ c$. Moreover, from $t(x,y,x)=s(x,y,x)$ and $s^{\mathbf{A}}(a,a,$$c)=t^{\mathbf{A}}(a,a,c)$ follows that $t^{\mathbf{A}}(a,b,c)=s^{\mathbf{A}}(a,b,c)$ for all $a,b,c$ in the same $\delta$-class, applying a symmetric version of the previous claim. Thus we proved the statement for $(t,s)\in\alpha\wedge y^{1}(\alpha,\beta,\gamma)$. If $(t,s)\in\alpha\wedge y^{n}(\alpha,\beta,\gamma)$ with $n>1$ the thesis follows applying inductively the previous strategy. Now we are ready to prove the main theorem of the section. The next result provides a bridge between congruence equations that hold in a variety $\mathcal{V}$ and congruence equations that hold in the variety $\mathcal{V}^{\prime}$ generated by abelian algebras of the idempotent reduct and thus is interesting for several reasons. Indeed, a systematic production of a Mal'cev condition by congruence equation valid in $\mathcal{V}^{\prime}$ is somehow surprising. This implies also that the commutator equations generated by the algorithm 5.1 under some mild assumptions produce a Mal'cev condition, result that justifies their study. Note that the proof of the next theorem makes deep use of a modified version of what is also called Mal'cev argument ([4, Lemma 12.1]). Theorem 5.3.: _Let $\mathcal{V}$ be a variety and suppose that $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$ is a congruence equation that fails on a $3$-element set with congruences with pair-wise empty intersection, where $p$ is a term for the language $\{\circ,\wedge,\vee\}$. Then the following are equivalent:_ 1. _the abelian algebras of the idempotent reduct of_ $\mathcal{V}$ _satisfy_ $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$_;_ 2. $\mathcal{V}$ _satisfies the commutator equations_ $\operatorname{Eq}_{C}(\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,$__$\gamma))$ _(see Algorithm_ 5.1_);_ 3. $\mathcal{V}$ _satisfies the equations:_ (5.3) \[\alpha\wedge(\beta\circ\gamma)\leq p(\alpha^{\prime},\beta^{\prime},\gamma^{ \prime});\] _where_ $\alpha^{\prime}=\alpha\circ G(\alpha,\beta,\gamma)\circ\alpha$_,_ $\beta^{\prime}=\beta\circ G(\beta,\alpha,\gamma)\circ\beta$_, and_ $\gamma^{\prime}=\alpha\circ G(\gamma,\alpha,\beta)\circ\alpha$ _with_ $G(\alpha,\beta,\gamma)\in\{[\beta\wedge(\alpha\vee\gamma),\beta\wedge(\alpha \vee\gamma)],[\gamma\wedge(\alpha\vee\beta),\gamma\wedge(\alpha\vee\beta)]\}$_._ 4. $\mathcal{V}$ _satisfies the equations:_ (5.4) \[\alpha\wedge(\beta\circ\gamma)\leq p(\alpha^{\prime},\beta^{\prime},\gamma^{ \prime});\] _where_ $\alpha^{\prime}=$__$\alpha\circ F(\alpha,\beta,\gamma)\circ\alpha$_,_ $\beta^{\prime}=$__$\beta\circ F(\beta,\alpha,\gamma)\circ\beta$_, and_ $\gamma^{\prime}=$__$\gamma\circ F(\gamma,\beta,\alpha)\circ\gamma$ _with_ $F(\alpha,\beta,\gamma)\in\{\beta\wedge y^{n}(\beta,\alpha,\gamma),\beta\wedge z ^{n}(\beta,\alpha,\gamma),\gamma\wedge y^{n}(\gamma,\beta,\alpha),\gamma\wedge z ^{n}(\gamma,\beta,\alpha)\}$_._ Proof.: Let us start from $(1)\Rightarrow(2)$. Let $\mathbf{A}\in\mathcal{V}$ and let $\theta\in\operatorname{Con}(\mathbf{A})$. Factoring $\mathbf{A}$ by $[\theta,\theta]$ we verify that $\operatorname{Eq}(\alpha\wedge(\beta\circ\gamma)\leq p)$ hold in every $\theta/[\theta,\theta]$-block of the quotient and thus $\operatorname{Eq}_{C}(\alpha\wedge(\beta\circ\gamma)\leq p)$ hold in $\mathbf{A}$. Let $\mathbf{A}^{\prime}$ be the idempotent reduct of $\mathbf{A}/[\theta,\theta]$. Since we factored by $[\theta,\theta]$ we can see that every $\theta/[\theta,\theta]$-class of $\mathbf{A}/[\theta,\theta]$ is an abelian subalgebra of $\mathbf{A}^{\prime}$ and thus it has terms satisfying $\operatorname{Eq}(\alpha\wedge(\beta\circ\gamma)\leq p)$ via the Pixley-Wille algorithm [28, 33]. We can observe that those terms witness the satisfaction of $\operatorname{Eq}_{C}(\alpha\wedge(\beta\circ\gamma)\leq p)$ in $\mathbf{A}$, since a quotient by $[\theta,\theta]$ has been applied. For $(2)\Rightarrow(3)$. From the hypothesis we have that there exists $k\in\mathbb{N}$ such that $\operatorname{Eq}_{C}(\alpha\wedge(\beta\circ\gamma)\leq p^{(k)})$ hold in $\mathcal{V}$. Then let $\mathbf{A}\in\mathcal{V}$, $a_{1},a_{2}\in A$, and let $\alpha,\beta,\gamma\in\operatorname{Con}(\mathbf{A})$ be such that: \[(a_{1},a_{2})\in\alpha\wedge(\beta\circ\gamma).\] Hence there exists $a_{3}\in\mathbf{A}$ such that $(a_{1},a_{3})\in\beta$ and $(a_{3},a_{2})\in\gamma$. We want to prove that \[(a_{1},a_{2})\in p^{(k)}(\alpha^{\prime},\beta^{\prime},\gamma^{\prime}). \tag{5.5}\] Let $Z=\{z_{1},\ldots,z_{u}\}$ be the set of vertices of $\mathbf{G}(p^{(k)})$ and let $X_{s}$ be a variable of $\mathbf{G}(p^{(k)})$. From the definition of $\operatorname{Eq}_{C}(\alpha\wedge(\beta\circ\gamma)\leq p^{(k)})$, we have that for all $(t_{i},t_{j})\in T_{s}(p^{(k)})$ there exist two $3$-ary terms $t_{i},t_{j}$ such that: \[t_{i}(x_{i_{1}},x_{i_{2}},x_{i_{3}})\approx_{C}t_{j}(x_{i_{1}},x_{i_{2}},x_{i_ {3}})\in\operatorname{Eq}_{C}(\alpha\wedge(\beta\circ\gamma)\leq p^{(k)})\] where the variables in $T_{s}(p)$-relation are collapsed. Let $\alpha$ be the variable substituted to $X_{s}$. Then, fixing this congruence, we have that $T_{s}(p)=\{(x_{1},x_{3})\}$ and \[t_{i}^{\mathbf{A}}(a_{1},a_{2},a_{3})\ \alpha\ t_{i}^{\mathbf{A}}(a_{1},a_{ 1},a_{3})\ [\beta\wedge(\alpha\vee\gamma),\beta\wedge(\alpha\vee\gamma)]\ t_{j}^{ \mathbf{A}}(a_{1},a_{1},a_{3})\] \[t_{j}^{\mathbf{A}}(a_{1},a_{1},a_{3})\ \alpha\ t_{j}^{\mathbf{A}}(a_{1},a_{ 2},a_{3}).\] With the same argument we can check that for all $\theta_{s}\in\{\alpha,\beta,\gamma\}$ and $\{\tau_{s},\delta_{s}\}=\{\alpha,\beta,\gamma\}\setminus\{\theta_{s}\}$: \[t_{i}^{\mathbf{A}}(a_{1},a_{2},a_{3})\ \theta_{s}\ t_{i}^{\mathbf{A}}(a_{i_{1}},a_{ i_{2}},a_{i_{3}})\] \[t_{i}^{\mathbf{A}}(a_{i_{1}},a_{i_{2}},a_{i_{3}})\ [\tau_{s}\wedge( \theta_{s}\vee\delta_{s}),\tau_{s}\wedge(\theta_{s}\vee\delta_{s})]\ t_{j}^{ \mathbf{A}}(a_{i_{1}},a_{i_{2}},a_{i_{3}})\] \[t_{j}^{\mathbf{A}}(a_{i_{1}},a_{i_{2}},a_{i_{3}})\ \theta_{s}\ t_{j}^{ \mathbf{A}}(a_{1},a_{2},a_{3})\] where $\theta_{s}$ is the congruence substituted to the variable $X_{s}$ of $\mathbf{G}(p^{(k)})$. Let $\rho:Z\to A$ be the assignment such that $z_{1}\mapsto a_{1}$,$z_{2}\mapsto a_{2}$ and $z_{i}\mapsto t_{i}^{\mathbf{A}}(a_{1},a_{2},a_{3})$ for all $3\leq i\leq u$. Thus we have that $(t_{i}^{\mathbf{A}}(a_{1},a_{2},a_{3}),$$t_{j}^{\mathbf{A}}(a_{1},a_{2},a_{3}))\in$$\theta_{s}\circ[\tau_{s}\wedge(\theta_{s}\vee\delta_{s}),\tau_{s}\wedge( \theta_{s}\vee\delta_{s})]\circ\theta_{s}$ whenever $(z_{i},z_{j})\in\mathbf{G}(p^{(k)})$. By (1) of Proposition 3.2, we have that $(a_{1},a_{2})\in p^{(k)}(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$$\subseteq p(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$. For $(3)\Rightarrow(4)$ follows applying Lemma 4.2. Indeed let ${\bf F}_{\mathcal{V}}(3)$ the free algebra of $\mathcal{V}$ over $3$ generators $\{x,y,z\}$. We see that $(5.3)$ hold in ${\bf F}_{\mathcal{V}}(3)$ by the hypothesis. Let $\alpha={\rm Cg}(\{(x,z)\})$, $\beta={\rm Cg}(\{(x,y)\})$, and $\gamma={\rm Cg}(\{(y,z)\})$. By Lemma 4.2 we have that $[\alpha\wedge(\beta\vee\gamma),\alpha\wedge(\beta\vee\gamma)]\leq\bigcup\alpha \wedge y^{n}(\alpha,\beta,\gamma)$, $\bigcup\alpha\wedge z^{n}(\alpha,\beta,\gamma)$. Thus we can substitute every occurrence in $(5.3)$ of $[\alpha\wedge(\beta\vee\gamma),\alpha\wedge(\beta\vee\gamma)]$ with $\bigcup\alpha\wedge y^{n}(\alpha,\beta,\gamma)$ or $\bigcup\alpha\wedge z^{n}(\alpha,\beta,\gamma)$ and the obtained equation holds in ${\bf F}_{\mathcal{V}}(3)$. Permuting the variables we can substitute every occurrence of $[\beta\wedge(\alpha\vee\gamma),\beta\wedge(\alpha\vee\gamma)]$ with $\bigcup\beta\wedge y^{n}(\beta,\alpha,\gamma)$ or $\bigcup\beta\wedge z^{n}(\beta,\alpha,\gamma)$ and of $[\gamma\wedge(\alpha\vee\beta),\gamma\wedge(\alpha\vee\beta)]$ with $\bigcup\gamma\wedge y^{n}(\gamma,\beta,\alpha)$ or $\bigcup\gamma\wedge z^{n}(\gamma,\beta,\alpha)$. Since we have done finitely many substitutions in the equation $(5.3)$ then there exists an $n\in\mathbb{N}$ such that the equations $(5.4)$ hold in ${\bf F}_{\mathcal{V}}(3)$. Hence, the equations $(5.4)$ hold in $\mathcal{V}$ via a standard Mal'cev argument. $(4)\Rightarrow(1)$. Let $\mathcal{V}^{\prime}$ be the variety generated by the abelian algebras of the idempotent reduct of $\mathcal{V}$. Since the equation $(5.4)$ is a congruence equation then, by the Pixley-Wille algorithm [28][33], it generates an idempotent Mal'cev condition that holds in $\mathcal{V}^{\prime}$ and we claim that this Mal'cev condition is non-trivial. In fact, $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$ is non-trivial and it fails on a $3$-element set choosing $\alpha,\beta,\gamma$ in a proper way. Furthermore, $\alpha,\beta,\gamma$ can be chosen with a pairwise empty intersection and thus with $\alpha\wedge\beta^{n}=\alpha\wedge\gamma^{n}=0$ and the same holds permuting $\alpha,\beta,\gamma$. Hence, we can see that if $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$ fails on a $3$-element set then also the equations $(5.4)$ fail choosing the same congruences. Furthermore, let ${\bf A}$ be an abelian algebra of the idempotent reduct of $\mathcal{V}$. By [18][Corollary 4.5], $[\phi,\phi]=0\Rightarrow[\phi,\phi]_{2T}=0$ for Taylor varieties since $[\phi,\phi]_{L}=0\Rightarrow[\phi,\phi]_{2T}=0$, where $[\phi,\phi]_{L}$ is the linear commutator [18] and thus $[\phi,\phi]_{2T}=0$ for all $\phi\in{\rm Con}({\bf A})$. From the hypothesis we have that ${\bf A}$ satisfies $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha^{\prime},\beta^{\prime},\gamma^{ \prime})$ and we prove that ${\bf A}$ satisfies $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$. Let $(a,b)\in\alpha\wedge(\beta\circ\gamma)$, then there exists $c\in A$ such that $(a,b)\in\alpha$, $(a,c)\in\beta$, and $(c,b)\in\gamma$. We show that $(a,b)\in p(\alpha,\beta,\gamma)$. Let ${\bf F}(\{x,y,z\})$ be the free algebra over three generators in the variety $\mathcal{W}$ generated by the abelian algebras of the idempotent reduct of $\mathcal{V}$. Furthermore, let $\overline{\alpha}={\rm Cg}(\{(x,z)\})$, $\overline{\beta}={\rm Cg}(\{(x,y)\})$, and $\overline{\gamma}={\rm Cg}(\{(y,z)\})$. Since ${\bf F}_{\mathcal{W}}(\{x,y,z\})$ satisfies $(5.4)$, by the Pixley-Wille algorithm [28, 33] we have that for all $(t_{i},t_{j})\in Tt_{s}(p(\overline{\alpha},\overline{\beta},\overline{ \gamma}))$ there exist $s_{i},s_{j}\in{\bf F}_{\mathcal{W}}(\{x,y,z\})$ such that $t_{i}\ \alpha_{1}\ s_{i}\ F(\alpha_{1},\alpha_{2},\alpha_{3})\ s_{j}\ \alpha_{1}\ t_{j}$ where $\alpha_{1}\circ F(\alpha_{1},\alpha_{2},\alpha_{3})\circ\alpha_{1}$ is the expression substituted to $X_{s}$ in $p$ and $\{\alpha_{1},\alpha_{2},\alpha_{3}\}=\{\overline{\alpha},\overline{\beta}, \overline{\gamma}\}$. Hence, $t_{i}(x_{i_{1}},x_{i_{2}},x_{i_{3}})=s_{i}(x_{i_{1}},x_{i_{2}},x_{i_{3}})$ and $t_{j}(x_{i_{1}},x_{i_{2}},x_{i_{3}})=s_{j}(x_{i_{1}},x_{i_{2}},x_{i_{3}})$ are identities of $\mathcal{V}$, where $x_{i_{1}},x_{i_{2}},x_{i_{3}}\in\{x,y,z\}$ and the variables in $\alpha_{1}$ relation are collapsed. Furthermore, $s_{i}\ F(\alpha_{1},\alpha_{2},\alpha_{3})\ s_{j}$ and thus we can apply Lemma 5.2 to these terms. Let $\psi:Y\to A$ be an assignment extending $y_{1}\mapsto a$,$y_{2}\mapsto b$ such that $y_{i}\mapsto t_{i}^{\mathbf{A}}(a,b,c)$, where $\{t_{i}\}_{i\in I}$ is the set of terms in appearing in $Tt(p(\alpha,\beta,\gamma))$. Let us consider the case $\alpha_{1}=\beta$, then, as a consequence of Lemma 5.2, $t_{i}(x_{1},x_{1},x_{3})=s_{i}(x_{1},x_{1},x_{3})$, and $t_{j}(x_{1},x_{1},x_{3})=s_{j}(x_{1},x_{1},x_{3})$, we have that \[t_{i}^{\mathbf{A}}(a,b,c)\ \beta\ t_{i}^{\mathbf{A}}(a,a,c)=s_{i}^{\mathbf{A}}(a,a,c)=s_{j}^{\mathbf{A}}(a,a,c)=t_{j}^{\mathbf{A}}(a,a,c)\ \beta\ t_{j}^{\mathbf{A}}(a,b,c).\] Thus $t_{i}(a,b,c)\ \beta\ t_{j}(a,b,c)$ and with the same strategy we can show similar relations for $\alpha_{1}\in\{\alpha,\gamma\}$ and hence, by Proposition 3.2, the assignment $\psi$ witness the satisfaction of $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$ in $\mathbf{A}$. Thus, the abelian algebras of idempotent reduct of $\mathcal{V}$ satisfy $\alpha\wedge(\beta\circ\gamma)\leq p(\alpha,\beta,\gamma)$ and (1) holds. We can see that the previous theorem requires to restrict the congruence equation $p\leq q$ that holds in the variety generated by the abelian algebras of the idempotent reduct of a variety to have $p=\alpha\wedge(\beta\circ\gamma)$. Although quite restrictive, this hypothesis is satisfied by the vast majority of the interesting known Mal'cev conditions characterized by congruence equations (modularity, distributivity, and many others). Moreover, the hypothesis $p=\alpha\wedge(\beta\circ\gamma)$ can be weaken further as shown in Theorem 4.4 but the difficulty in doing so comes from the constraints given by the commutator equations which run over a set of variables in the same $\theta$-class. This blocks the generalization of the proof of $(2)\Rightarrow(3)$ of Theorem 5.3. ## Conclusions In this work we have observed how some Mal'cev conditions can be characterized through commutator equations. This interesting connection that generalizes the Mal'cev condition produced by the weak difference term, might have many outcomes in the study of the relationship between Mal'cev conditions and commutator properties or TCT-types as shown in results such as [18, 19, 22]. Our aim with this work was to build the foundations of a new type of equations well-behaved in terms of Mal'cev conditions that could produce new insights about them. Further developments could be obtained using the procedure of Theorem 5.3 to known congruence equations in order to discover if the generated commutator equations furnish a known Mal'cev conditions or if they characterize new ones and if they are new what are the properties associated to them. Moreover, in commutator theory has been of great interest the study of the properties of commutators in varieties that satisfy certain Mal'cev conditions [18, 19]. Thus, a natural problem that could be worth to investigate is how various properties described by commutator equations change the behaviour of the commutator or of the solvability theory in varieties satisfying them. Examples of prototype results about commutator and solvability theory can be found in [19][Chapter 6] for the weak difference term. Finally, we want to highlight that this paper concerns properties satisfied by abelian algebras of the idempotent reduct of a variety. The fact that under mild assumptions those kind of properties generally produce Mal'cev conditions shows an unexpected deep connection between a variety and its abelian algebras of the idempotent reduct and, if needed, allows to check Mal'cev conditions in the more manageable setting of abelian algebras. ## Acknowledgement The author thanks Paolo Agliano, Erhard Aichinger, Sebastian Kreinecker, and Bernardo Rossi for many hours of fruitful discussions.	math.RA	[ "math.RA", "\"03C05", "08B05", "08B10\"" ]
2302.09207	RETVec: Resilient and Efficient Text Vectorizer	"This paper describes RETVec, an efficient, resilient, and multilingual text\nvectorizer designed fo(...TRUNCATED)	Elie Bursztein, Marina Zhang, Owen Vallis, Xinyu Jia, Alexey Kurakin	2023-02-18T02:06:52	http://arxiv.org/abs/2302.09207v3	"# RetVec: Resilient and Efficient Text Vectorizer\n\n###### Abstract\n\nThis paper describes RetVec(...TRUNCATED)	cs.CL	[ "cs.CL", "cs.AI" ]
2310.06325	"Long-time behavior for the Kirchhoff diffusion problem with magnetic\n fractional Laplace operator(...TRUNCATED)	"We consider a Kirchhoff-type diffusion problem driven by the magnetic\nfractional Laplace operator.(...TRUNCATED)	Jiabin Zuo, Juliana Honda Lopes, Vicentiu D. Rădulescu	2023-10-10T05:45:42	http://arxiv.org/abs/2310.06325v1	"# Long-time behavior for the Kirchhoff diffusion problem with magnetic fractional Laplace operator\(...TRUNCATED)	math.AP	[ "math.AP", "\"35R11", "35J20", "35J60\"" ]
2304.02976	"Unconstrained Parametrization of Dissipative and Contracting Neural\n Ordinary Differential Equati(...TRUNCATED)	"In this work, we introduce and study a class of Deep Neural Networks (DNNs)\nin continuous-time. Th(...TRUNCATED)	"Daniele Martinelli, Clara Lucía Galimberti, Ian R. Manchester, Luca Furieri, Giancarlo Ferrari-Tre(...TRUNCATED)	2023-04-06T10:02:54	http://arxiv.org/abs/2304.02976v2	"# Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equatio(...TRUNCATED)	eess.SY	[ "eess.SY", "cs.LG", "cs.SY" ]
2307.03692	Becoming self-instruct: introducing early stopping criteria for minimal instruct tuning	"In this paper, we introduce the Instruction Following Score (IFS), a metric\nthat detects language (...TRUNCATED)	"Waseem AlShikh, Manhal Daaboul, Kirk Goddard, Brock Imel, Kiran Kamble, Parikshith Kulkarni, Melisa(...TRUNCATED)	2023-07-05T09:42:25	http://arxiv.org/abs/2307.03692v1	"# Becoming self-instruct: introducing early stopping criteria for minimal instruct tuning\n\n######(...TRUNCATED)	cs.CL	[ "cs.CL", "cs.AI" ]
2305.15689	Zero-shot Approach to Overcome Perturbation Sensitivity of Prompts	"Recent studies have demonstrated that natural-language prompts can help to\nleverage the knowledge (...TRUNCATED)	Mohna Chakraborty, Adithya Kulkarni, Qi Li	2023-05-25T03:36:43	http://arxiv.org/abs/2305.15689v2	"# Zero-shot Approach to Overcome Perturbation Sensitivity of Prompts\n\n###### Abstract\n\nRecent s(...TRUNCATED)	cs.CL	[ "cs.CL", "cs.AI" ]
2303.01572	Transportability without positivity: a synthesis of statistical and simulation modeling	"When estimating an effect of an action with a randomized or observational\nstudy, that study is oft(...TRUNCATED)	Paul N Zivich, Jessie K Edwards, Eric T Lofgren, Stephen R Cole, Bonnie E Shook-Sa, Justin Lessler	2023-03-02T20:54:20	http://arxiv.org/abs/2303.01572v4	"# Transportability without positivity: a synthesis of statistical and simulation modeling\n\n######(...TRUNCATED)	stat.ME	[ "stat.ME" ]
2307.13481	Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems	"We provide an example for the generating matrix $A$ of a two-dimensional\nlattice $\\Gamma = A\\mat(...TRUNCATED)	Nicki Holighaus, Günther Koliander	2023-07-25T13:14:37	http://arxiv.org/abs/2307.13481v2	"# Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems\n\n###(...TRUNCATED)	math.FA	[ "math.FA", "\"42C40 (Primary)", "42C15", "11K38 (Secondary)\"" ]

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Arxiver Dataset with Category

This dataset is an enhanced version of the Arxiver dataset with additional category information. It contains 63,357 arXiv papers published between January 2023 and October 2023, with each paper enriched with its primary category and all associated categories.

Additional Features

Beyond the original dataset's fields (ID, title, abstract, authors, publication date, URL, and markdown content), this version adds:

primary_category: The main arXiv category of the paper (e.g., 'cs.AI', 'cs.CL')
categories: A list of all arXiv categories associated with the paper

These category fields were obtained through the arXiv API to provide more comprehensive paper classification information.

Dataset Structure

{
    'id': 'arxiv paper id',
    'title': 'paper title',
    'abstract': 'paper abstract',
    'authors': 'paper authors',
    'published_date': 'publication date',
    'link': 'arxiv url',
    'markdown': 'paper content in markdown format',
    'primary_category': 'main arxiv category',
    'categories': ['list', 'of', 'arxiv', 'categories']
}

Using Arxiver

You can easily download and use the arxiver dataset with Hugging Face's datasets library.

from datasets import load_dataset
dataset = load_dataset("real-jiakai/arxiver-with-category")

Alternatively, you can stream the dataset to save disk space or to partially download the dataset:

from datasets import load_dataset
dataset = load_dataset("real-jiakai/arxiver-with-category", streaming=True)

License & Citation

This dataset follows the same Creative Commons Attribution-Noncommercial-ShareAlike (CC BY-NC-SA 4.0) license as the original Arxiver dataset.

References

When using this dataset, please cite both this version and the original Arxiver dataset:

@misc{acar_arxiver2024,
  author = {Alican Acar, Alara Dirik, Muhammet Hatipoglu},
  title = {ArXiver},
  year = {2024},
  publisher = {Hugging Face},
  howpublished = {\url{https://huggingface.co./datasets/neuralwork/arxiver}}
}

@misc{jiakai_arxiver2025,
  author = {real-jiakai},
  title = {Arxiver with Category},
  year = {2025},
  publisher = {Hugging Face},
  howpublished = {\url{https://huggingface.co./datasets/real-jiakai/arxiver-with-category}}
}

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Size of downloaded dataset files:

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1.44 GB

Number of rows:

63,357