{'fraction_non_alphanumeric': 0.04624248745840163, 'fraction_numerical': 0.03357671484627229, 'mean_word_length': 4.256919060052219, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Rapid variations of dust colour in comet 41P/Tuttle-Giacobini-Kres\'{a}k

We monitor the inner coma of comet 41P/Tuttle-Giacobini-Kres\'{a}k searching for variations of its colour. Fast changes in colour of the comet 41P/Tuttle-Giacobini-Kres\'{a}k provide important clues for better understanding of the microphysical properties of its dust. Using the 61-cm and 70-cm telescopes we measured the apparent magnitude of the comet with the V and R Johnson-Cousins filters from January 29 until April 25 of 2017. The inner coma ({\sim} 2000 km) reveals fast and significant variations of colour. The most significant change was found between March 3 and 4 of 2017, when it changed from blue with a colour slope S {\approx}(-10.15 {\pm} 3.43){\%} per 0.1 {\mu}m to red with S {\approx} (16.48 {\pm} 4.27){\%} per 0.1 {\mu}m. This finding appears in good accordance with what was previously reported by Ivanova et al. (2017) for long-period comet C/2013 UQ4 (Catalina), suggesting that fast and significant variations of colour of dust could be a common feature of short- and long-period comets. We model observations of comet 41P/Tuttle-Giacobini-Kres\'{a}k using the agglomerated debris particles and conclude that its inner coma consists of a mixture of at least two types of particles made of Mg-rich silicates and organics or Mg-Fe silicates.


Introduction
Microphysical properties of dust particles in the vast majority of comets can be inferred only using remote-sensing techniques based on the interaction of dust with the solar radiation. Such interaction manifests itself in three groups of phenomena: elastic scattering of the sunlight in visible and near-IR bands, thermal emission of the absorbed solar radiation in mid-IR, and radiation pressure that affects the motion of dust particles and, thus, their spatial distribution in the coma and tail.
Photometric colour is defined as the ratio of the scattered-radiation fluxes measured at two different wavelengths. Its strength lies in the relative nature as it depends directly upon the microphysical properties of dust particles, and it is independent of their number within the field of view. This is important because it is known that the number of dust particles forming a cometary coma is a subject of rapid and significant variations. Photometric colour is quantified with either the colour index or colour slope (e.g., A'Hearn et al. 1984); the latter sometimes is referred to as the reflectivity gradient (Lamy et al. 2011). The colour slope is defined as follows: where Δm denotes the true colour index of the comet, i.e., reduced for the colour index of the Sun, and λ2 and λ1 are effective wavelengths of the utilized filters measured in μm (λ2 > λ1). As one can see in Eq. (1), S is normalized to the wavelength difference, making it, to some extent, independent of a specific choice of the filters.
Colour slope S also could be computed directly from the reflectivity of dust particles at different wavelength λ. In practice this could be done using the Afρ parameter that was investigated in numerous comets (e.g, A'Hearn et al. 1995;Moulane et al. 2018): geometric cross section of cometary dust particles to the total projected geometric cross section of the field of view (A'Hearn et al. 1995). We note that in the literature, the Afρ parameter is measured in cm. Obviously, by definition, f and ρ take on the same values at different wavelength ; whereas, the geometric albedo A is a function of . In addition, the geometric albedo A is defined at the exact backscattering with  = 0°; whereas, the vast majority of comets are observed at  ≠ 0°. The necessity to adjust the measured Afρ to the exact backscattering may cause some uncertainty in the resulting values.
Photometric colour is a key light-scattering characteristic of cometary dust that has been studied in numerous comets (e.g., Jewitt & Meech 1986;Lamy et al. 2011;Ivanova et al. 2015;Zubko et al. 2015;Picazzio et al. 2019). While the colour detected in a given comet was long considered to be a persistent feature of its dust (e.g., Jewitt & Meech 1986), it was later demonstrated that the colour can change over a considerable time period (e.g., Weiler et al. 2003;Li et al. 2014) and it is a function of the phase angle  (Zubko et al. 2014). Furthermore, Ivanova et al. (2017) have recently reported on outstanding variations of the photometric colour in comet C/2013 UQ4 (Catalina), whose dust turned from red to blue colour over only a two-day period.
Note, rapid variations of colour were previously found, for instance, in comet 17P/Holmes (Zubko et al. 2011), which was clearly related to its dramatic outburst activity at the end of October, 2007; whereas, comet C/2013 UQ4 (Catalina) did not reveal such activity during its observations. The findings by Ivanova et al. (2017) raise a legitimate question as to whether comet C/2013 UQ4 (Catalina) was an exceptional case in terms of fast and dramatic temporal variations of colour in its dust. In this paper we present results of our recent photometric survey of comet 41P/Tuttle-Giacobini-Kresák (hereafter 41P/T-G-K) over a three-month period in the first half of 2017. As it is shown in the next sections, photometric colour of dust in this shortperiod comet underwent dramatic short-term variations, very much resembling what was found in the long-period comet C/2013 UQ4 (Catalina) by Ivanova et al. (2017).

Observations and reductions
We observed comet 41P/T-G-K on 17 nights from January 29  However, this aperture size also nearly coincides with what was used previously in Ivanova et al. (2017). Furthermore, as shown below with modeling of dust-particle motion, the population of the dust within a 2,000-km aperture can be fully replaced during only a single day. Therefore, it is the best region for searching for fast changes of microphysical properties in the expelled dust.
Data overnight observations were averaged but values that deviated from the average by more than 3- were discarded. Photometry was conducted with the V and R broadband filters from the Johnson-Cousins photometric system that are centered at λ = 0.551 μm (FWHM = 0.088 μm) and 0.658 μm (FWHM = 0.138 μm), respectively. The colour index of the Sun with this pair of filters is equal to 0.368 (Burlov-Vasiljev et al. 1995, 1998. The CCD detector SBIG ST-10XME with 2 × 2 binning and resolution of 1.07 arcsec per pixel was used for observation at the 61-cm telescope at the Skalnaté Pleso. The CCD PL47-10FLI camera was used as a detector with full field of view of the CCD being 16 × 16 arcminutes, and the image scale was 0.95 arcsec per pixel for observations from the 70-cm AZT-8. Fig. 1 shows image of 41P/T-G-K obtained with the 70-cm AZT-8 telescope on observation station Lisnyky of the Astronomical Observatory of Taras Shevchenko National University of Kyiv (Ukraine) of 25 April 2017. The black circle marks part of the 41P/T-G-K coma whose colour was investigated in this study.
We followed the reduction procedure (bias subtraction, dark and flat field corrections, and cleaning cosmic-ray tracks) in the standard manner, using the IDL routines (e.g., Ivanova et al. 2016Ivanova et al. , 2017Picazzio et al. 2019). The morning sky was exposed to provide a flat-field correction for the non-uniform sensitivity of the CCD chip. All of the nights were photometric, the seeing value measured as the average FWHM of several sample stars ranged from 3.1 to 6.6 arcsec during our observations. Observations with a large seeing were not taken into account. The residual sky background was estimated with the use of an annular aperture. To perform an absolute flux calibration of the comet images, the field stars were used. The stellar magnitudes of the standard stars were taken from the catalogue UCAC4. UCAC4 is a compiled, all-sky star catalogue covering mainly the 8 to 16 magnitude range in a single bandpass between V and R.
Photometric uncertainty of the catalogue depends on star brightness and is estimated to be from 0.01 to 0.2 mag on average. Fig. 2 shows results of the three-month monitoring of the colour slope S of comet 41P/T-G-K (points). Measured values are given in Table 2. What immediately emerges from this figure is a significant dispersion of all 17 measurements. In particular, on 9 epochs, the comet was detected with confidence to be red, on 1 epoch its colour was unambiguously blue; whereas, on 7 other epochs the colour appeared to be neutral within the error bars. Such dispersion of the colour slope in comet 41P/T-G-K supports the conclusion previously drawn by Ivanova et al. (2017) that a sole observation hardly characterizes the colour of a comet.

Results and discussion
In addition to our measurements in Fig. 2, we also plot the colour slope S computed using Eq.
(2) from the Afρ parameter measured by Moulane el al. (2018) using a 10,000-km aperture (ρ = 5,000 km) in comet 41P/T-G-K (open circles). In particular, we adapt their data obtained with the HB narrow-band blue continuum (BC,  = 0.445 μm) and red continuum filters (RC,  = 0.715 μm) as they cover a relatively long time period, 14 epochs, which is consistent with our own observations. Note, they also exploited the HB green continuum filter (GC,  = 0.525 μm) that better corresponds to the V filter in our observations. Unfortunately, the GC filter was used only a few times. Nevertheless, we involve those data into a later analysis.
Although Moulane et al. (2018) draw attention to the colour variations in 41P/T-G-K, their analysis is largely qualitative as they do not provide the colour slope S. As one can see in Fig. 2, both sets of data were obtained at similar uncertainty, and within the error bars, they reveal a great deal of overlap. However, some of our data, such as the strong blue colour on March 3, 2017, appear to be unique.
In Fig. 3 we address the colour slope S obtained solely in our own observations. Here it is shown as a function of phase angle α. As one can see in Fig An important question with regard to the colour variations in the inner coma of comet 41P/T-G-K is whether such variations should be attributed to changes in the dust population or gaseous environment. In theory, gaseous emission potentially could affect photometric measurements conducted with the wideband filters. In practice, however, such a gaseous contamination does not appear to be a major factor. This was checked, for instance, by integrating high-resolution cometary spectra in visible with and without gaseous-emission lines taken into account. The difference between those two types of spectral integrations exceeds only a few percent (e.g., Jewitt et al. 1982;Picazzio et al. 2019), suggesting an extremely low contribution of the gaseous emission into the flux measured with wideband filters. Interestingly, this equally holds for longand short-period comets. One also should draw attention to comparative analysis of the color slope in Comet C/1995 O1 (Hale-Bopp) inferred from its high-resolution spectra and, simultaneously, from photometry with the wideband B and R filters. As reported in Weiler et al. appears in good accordance with C/1980 Y2 (Panther) and 38P/Stephan-Oterma. What is significant is that the comets C/1980 Y2 (Panther) and 38P/Stephan-Oterma were also investigated with the spectroscopic approach by Jewitt et al. (1982). Spectra of both comets in the visible reveal strong emission lines from various gases. However, integration of spectra over the wavelengths  = 0.5 -0.7 μm with and without those emission lines reveals only little difference, suggesting an extremely weak contribution of the gas emission, namely, a gaseous emission contribute only 1% of the total flux in comet C/1980 Y2 (Panther) and even less in 38P/Stephan-Oterma (Jewitt et al. 1982). Therefore, we conclude that our photometric measurements with the V and R filters correspond primarily to dust in comet 41P/T-G-K.
As can be seen in Fig Table 2. We compute the color slope S using values of the Afρ parameter presented in Table 2 of Moulane et al. (2018) for the GC and RC filters, and find S = (7.87  3.30)% per 0.1 μm for former date and S = (9.87  2.24)% per 0.1 μm for the later epoch.
These results match our finding within the error bars (see Table 2) despite that the measurements by Moulane et al. (2018) correspond to a bigger projected area of the coma, a radius of 5,000 km versus ~1,000 km in our measurements. Unfortunately, Moulane et al. (2018) does not report the continuum photometric data on the night between March 3 and 4, making our observations on those epochs unique.
We analyze the observations presented in Fig. 3  We note that the colour of cometary dust also depends on phase angle α. For example, in = 0.02). Therefore, we rule out all those fits to S on March 3. Furthermore, due to the small heliocentric distance of 41P/T-G-K, rh  1.18 au, the presence of micron-sized water-ice particles in its coma seems very unlikely. Thus, we conclude that on March 3, the coma consisted mainly of Mg-rich silicates with power index constrained to the range n  2.2 -3.
The colour slope measured on March 4 unambiguously suggests some wavelength dependence of the refractive index. In the middle and bottom plots we show modeling results obtained by varying the imaginary part of refractive index Im(m). We assume that at the shorter wavelength, Im(m) takes on a larger value than at the longer wavelength; whereas, the real part of refractive index Re(m) remains constant. Such a trend is common for cometary-dust species in visible (see, discussion in Zubko et al. 2015). One also needs to note that, unlike Im(m), a small variance of Re(m) does not significantly affect the light-scattering response of irregularly shaped particles (Zubko et al. 2014).
As one can see in the middle plot of Fig. 5, at ΔIm(m) = 0.01, all the modelling scenarios are capable of fitting the observation on March 4 at n < 2.3. What is significant is that some of these cases (i.e., Im(m) = 0.02 -0.03 at shorter wavelength λ(S) and Im(m) = 0.01 -0.02 at longer λ(L)) can be attributed to real materials like organics and Mg-rich silicates with some contamination from iron (Jenniskens 1993;Dorschner et al. 1995;Zubko et al. 2015).
The reddening further increases at ΔIm(m) = 0.02, see the bottom plot in Fig. 5. We discriminate here two solutions obtained with Im(m) = 0.04 -0.06 at shorter λ and Im(m) = 0.02 -0.04 at longer λ. They not only appear consistent with some organics and Mg-Fe silicates, but also fit the colour slope S on March 4 with the power index n  2 -2.9 that nearly coincides with what was inferred with Mg-rich silicates for March 3. Thus, within these scenarios both components of the 41P/T-G-K coma would obey the same size distribution.
As for a mechanism governing the replacement of pure Mg-rich silicates on March 3 with organics/Mg-Fe silicates on March 4 in the inner part of the 41P/T-G-K coma, we simulated the motion of dust particles within the model previously exploited in Zubko et al. (2015) and further expanded in Kochergin et al. (2018). The model is based on a time-domain approach of computation of dust-particle motion. At the initial time moment t0, a dust particle gets lifted off the nucleus surface at the place of its origin characterized with the radius-vector s0 and acquires initial velocity V0. We adapt R = 0.7 km for the radius of the 41P/T-G-K nucleus from Lamy et al. (2004). In general, the particle motion is affected by three forces, the solar gravity FSun, the cometary nucleus gravity Fnucleus, and the solar-radiation pressure Frad; the latter one acts in the direction opposite to the solar gravity (e.g., Fulle 2004). We infer the gravitational force of the 41P/T-G-K nucleus based on the assumption of its bulk material density of 0.5 g/cm 3 that is within the range detected in situ in cometary nuclei (Weissman & Lowry 2008).
The resultant of all three forces changes the dust-particle motion with the acceleration a0. We hold a0 constant during the first second that suggests a uniformly accelerated straight-line motion of the dust particle over the corresponding time period. At the end of the first time step, we update the particle velocity V1 and all three forces at its next location at s1, which yields a new value of their resultant acceleration a1. We repeat this whole computation procedure with time increment t = 1 s for a full day, i.e., 86,400 s. What emerges from this simulation is a sequence of dust-particle locations composing its trajectory. Note that we examine different values of the time increment and find that t = 1 s produces a well-converged modeling result. We also consider at every particle location whether it is shadowed by the nucleus from the direct solar radiation. If the particle is shadowed, we omit the solar-radiation pressure during the next increment of time t.
In the literature, the effect of the solar radiation on dust-particle motion is quantified with the β parameter that is a ratio of the solar-gravity force over the radiation-pressure force β = FSun / Frad (e.g., Fulle 2004). Frad is dependent on the radiation pressure efficiency Qpr whose computation requires a rigorous solution of the light-scattering for the given dust particle (e.g., Bohren & Huffman 1983). In application to dust particles with truly irregular morphology this can be accomplished only using a highly time-consuming numerical approach, such as, the discrete dipole approximation (DDA) (e.g., Zubko et al. 2015). This necessity is often ignored, and Qpr often gets preset to 1 instead. The reasoning behind this approximation is the asymptotic behavior Qpr  1 as the particle radius r >> 1 μm (e.g., Fulle 2004; Moreno et al. 2012), except for water-ice particles . This approximation was recently utilized, for instance, in application to comet 41P/T-G-K by Pozuelos et al. (2018), where the shape of its coma was modeled with particles whose radius r spans the range from 10 μm to 1 cm. However, the contribution of such large particles into the light-scattering response from a cometary coma is not established. While large particles with radius in excess of 10 μm do indeed exist in a cometary coma (e.g., Bentley et al. 2016), there also are present submicron-and micron-sized dust particles in much greater quantities (e.g., Mazets et al. 1986;Price et al. 2010), and these smaller particles are more efficient scatterers. As demonstrated by Zubko (2013), if the index in the power-law size distribution r -n ranges from n = 1.5 -3.5, the light-scattering at phase angles  ≤ 30° is predominantly governed by particles with radius r ≤ 1.5 μm. This apparently holds for comet 41P/T-G-K on March 3-4 of 2017, when its phase angle was   24.8-26°.
It is worth noting that when n > 3, which also includes the range n = 3.5 -3.75 suggested by Pozuelos et al. (2018), the total geometric cross section of the dust population is governed by the bottom limit of integration. However, in the analysis by Pozuelos et al. (2018) the bottom size of dust particles cannot be less than 10 μm in principle due to their approximation Qpr = 1. This contradiction makes their analysis difficult to reconcile with application to interpreting observations of 41P/T-G-K in the visible. In our modeling we exploit rigorous values of the β parameter in the Mg-rich silicate particles adapted from Zubko et al. (2015). It is significant that they are based on numerically exact DDA computations of Qpr in the agglomerated debris particles and integrated over the solar spectrum. In this work, we model the motion of Mg-rich silicate particles having radius r = 1 μm with β = 0.5.
In Fig. 6       whereas, direction to Earth is inclined left for some 24.8° that is shown with an arrow marked E.
The short side of the gray rectangular oriented perpendicular to line of sight has length of ~2000 km that corresponds to the aperture in our measurements. Length of trajectories correspond to 1-  April 25.01 10 V20, R20 3.1 1.05 585 * Date of start of observations ** V -V-filter; R -R-filter; number is the number of frames taken