Faye

New Astronomy 75 (2020) 101320

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BVR broadband photometry of comets 1P/Halley and 4P/Faye A.S. Betzler a b c

a,b,⁎

, O.F. de Sousa

T

c

Centro de Ciência e Tecnologia em Energia e Sustentabilidade, Universidade Federal do Recôncavo da Bahia, Feira de Santana, 44.085-132, Brazil Brazilian Meteor Observation Network (BRAMON), Nhandeara, 15.190-000, Brazil Programa de Astronomia do Recôncavo da Bahia, Centro de Formação de Professores, Universidade Federal do Recôncavo da Bahia, Amargosa, 45.300-000, Brazil

A R T I C LE I N FO

A B S T R A C T

Keywords: comets: individual [1P/Halley 4P/Faye] Techniques: Photometric Methods: Data analysis

We analyzed the visible broadband photometric data of comets 1P/Halley and 4P/Faye, obtained during their perihelion passages of 1986 and 1991, respectively at the Sanglokh Observatory (Tajikistan) and the European Southern Observatory (Chile). Applying the Lomb-Scargle periodogram in the V-band magnitudes and B-V color index, we find that the most probable periodicities are 79 ± 6 and 7.36 ± 0.04 days for 1P, and 6.1 ± 0.3 days for 4P. After comparing results of color and magnitude periodograms, we argue there is a systematic difference in the number of signals identified and the level of confidence of the same periodicity in the periodograms. Our results suggest the quest for periodicities in the color of the coma of active comets should be complementary to ones in magnitudes. We have verified that the distribution of the color B-V of Faye’s coma was invariable during and after the possible occurrence of a post-perihelion outburst. We verify a symmetry in the pre- and post-perihelion H0 photoelectric absolute magnitude of the comet Halley. The same issues were not observed in the B-V color index. We verify that the absolute magnitude H0 of the comet Halley differs from each other when calculated from the visual or photoelectric magnitudes, due to the section of the coma used to estimate these magnitudes. We also verified that this difference in the photometric aperture can compromise comparisons of B-V color distributions between active comets.

1. Introduction Nuclear activity in comets depends on their rotational state, as well as the shapes of jets, halos and other structures that eventually may be observed in the coma. The determination of the rotational state can be obtained by several methods, such as the analysis of the temporal evolution of the broadband filter magnitudes and the movement of structures (jets and fans) in the coma (Samarasinha et al., 2004). Alternatively, periodicities could be obtained from the temporal variation of the object’s color. Considering minor bodies in the solar system, various works suggest there is no detectable relationship between the V-band light curve phases with U-V and B-V color indexes for asteroids (e.g Schober and Surdej, 1979; Surdej and Schober, 1980). Studies searching for a similar relationship with comets are limited to only three objects and do not show a definitive answer. Leibowitz and Brosch (1986a,b) found periodicities of 52 and 9.5 h in the temporal variation of the color indexes M(514 nm)-M(700 nm) of the 1P/Halley and 21P/Giacobini-Zinner comets. This color index is the magnitude difference between two International Halley Watch (IHW) bandpass filters centered around emission lines of C2 and H2O +, respectively. Leibowitz and Brosch (1986a) also performed an equivalent

⁎

analysis on the distribution of the B-V color of the comet D’Arrest, but none significant periodicity was found at a confidence level lower than 80%.Considering that the study of color periodicity of active comets is still a barely explored question, our main objective in this study is to look for periodicities in the B-V color of the coma of active comets and compare the results obtained with magnitude V, with the aim of analyzing the similarities between the signals identified in the periodograms. The search for peridiocities in the B-V color can be advantageous in the study of rotational properties of active comets. Betzler et al. (2017) shows the B-V color is insensitive to large scale events, such as tail disconnection or outbursts, and geometrical events such as variations in the heliocentric distance or phase angle (Hartmann and Cruikshank, 1984; Jewitt and Meech, 1988). This makes applying geometrical corrections, as the ones applied in magnitudes, to be redundant. Furthermore, we consider problems in looking for periodical variations in the V-R or B-R colors given the possible sensibility that V-R color has with large scale events (see Betzler et al., 2017, Fig. 2). We chose the comets 1P/Halley and 4P/Faye as objects of study because they have a great amount of daily B-V measurements, from intervals varying from some months to two years. The magnitudes of these two comets were measured in the same photometric aperture

Corresponding author. E-mail address: [email protected] (A.S. Betzler).

https://doi.org/10.1016/j.newast.2019.101320 Received 14 May 2019; Received in revised form 23 August 2019; Accepted 23 September 2019 Available online 25 September 2019 1384-1076/ © 2019 Elsevier B.V. All rights reserved.

New Astronomy 75 (2020) 101320

A.S. Betzler and O.F. de Sousa

radius p during the entire observational period in question. Betzler et al. (2018) shows the use of a variable photometric aperture, associated to the variation of geocentric distance, introduces significant systematic errors in the apparent magnitudes of comets. During the pre-perihelion period, the comet Halley showed many outbursts and tail disconnection events (Brandt et al., 1999), with the detection of jets and other structures. However, only the outburst of December 12, 1985 UT (Watanabe et al., 1987) was registered in the data set analyzed in this study. On the other hand, the comet Faye is a low activity comet, not presenting any detectable structures in its coma (Farnham, 2009). This object presented a possible post-perihelion outburst in 1991 (Grothues, 1996). Assuming a connection between the temporal variation of the Vband magnitude and B-V color index with the nuclei rotation, we obtained new estimations for the rotational state of comets Halley and Faye, and we contribute to increment the sample of known cometary rotation periods. We consider these correlations to be reasonable given the similarity between 67P/Churyumov-Gerasimenko (hereafter, 67P) rotation period estimates through light curves obtained on Earth and in situ (Tubiana et al., 2011; Mottola et al., 2014). Specifically, it is suggested that the cometary activity has a diurnal variation in intensity from changing insolation conditions (De Sanctis et al., 2015; Tubiana et al., 2019). The cometary activity may come from discrete or surface areas or from the entire illuminated surface of the nucleus. (Kramer and Noack, 2016). Thus, the photometric variations observed in active comets may be connected to heterogeneous distributions of active areas on the surfaces of rotating nuclei of complex shape, such as that presented by 67P (Läuter et al., 2019). Additionally, we obtained photometric parameters such as the absolute magnitude H0 and the activity index n, and we analyzed the B-V color distribution of both comets. We studied the dependency of the photometric parameters and B-V color with the photometric aperture and we analyzed its influence as a possible source of error in cometary photometry.

Table 1 Observational circumstances and photometric parameters of 1P/Halley. ΔT is equal to Th − 2, 446, 298.44375.

2. Observations and data reduction 2.1. Observational data Comet Halley’s data are available in the archives of the International Halley Watch (IHW) (A’Hearn and Vanysek, 2006).The data were obtained between August 20, 1985 and May 25, 1987 at Mt. Sanglok observatory (Tadjikistan), with a 1-m f/13.3 telescope equipped with a photoelectric photometer and UBVR filters. Specifically, the BV filters can be considered as “near-standard” for data gathering (see Gerasimenko et al., 1986). An analysis of the central wavelength and wideness of bands B (λB = 436 nm; Δλ = 81 nm) and V (λV = 552 nm; Δλ = 84 nm) suggests similarities with the photometric system of Johnson and Morgan (1953). We analyse pre- and post-perihelion phase data of the comet Halley. The magnitudes are obtained with an aperture of 14.9”. This aperture corresponds to optocentric distances between 3.16 × 104 km (geocentric distance Δ= 5.835 AU) and 3.98 × 103 km (Δ= 0.735 AU). The cadence of the observations was one or two daily measurements, obtained with a few hours apart. The photometric calibration was done using standard stars in NGC-1807 and the Pleiades cluster (Gerasimenko et al., 1986). The photometric and observational parameters of this object are presented in Table 1. This dataset is the biggest provided by IHW obtained with the same instrumental apparatus (telescope, filter set and detector), photometric calibration procedure and aperture. This uniformity of observational parameters in this dataset could reduce systematic error in the magnitudes of the comets analyzed in this study. The quest of the use of a constant photometric aperture in cometary photometry is discussed in Section 2.3. We also analyzed another dataset corresponding to the pre- and post-perihelion passage of the comet Faye. The data were obtained by

ΔT(days)

V

B-V

Δ[AU]

r[AU]

α[∘]

V(1,1,0)

−0.01902 0.98074 2.97186 20.96865 21.97882 24.02268 24.95316 26.93812 28.02941 30.91305 32.91262 34.92839 35.98277 56.05449 57.02267 57.97505 58.95624 60.89680 64.07033 65.05421 73.06276 74.86957 75.85607 76.84652 77.06058 77.85072 81.82877 83.81231 83.90221 84.04111 84.79723 84.90182 84.99303 85.92764 85.97044 86.78003 86.90364 86.98694 88.72212 88.82252 91.82264 92.91308 93.73854 93.83643 100.78858 102.82451 110.79985 111.72295 112.77725 116.74704 117.73283 135.68866 138.66884 252.72735 257.72638 265.76190 290.75207 294.74313 295.74135 299.73402 501.06538 502.00172 532.92616 633.72840 640.72928 642.72348

14.14 ± 0.09 14.43 ± 0.05 14.36 ± 0.02 13.60 ± 0.02 13.63 ± 0.02 13.48 ± 0.02 13.64 ± 0.02 13.53 ± 0.02 13.47 ± 0.02 13.39 ± 0.03 13.28 ± 0.02 13.15 ± 0.02 13.15 ± 0.02 12.00 ± 0.01 11.96 ± 0.01 11.76 ± 0.02 11.76 ± 0.02 11.48 ± 0.01 11.41 ± 0.02 11.32 ± 0.01 10.51 ± 0.02 10.62 ± 0.02 10.49 ± 0.01 10.40 ± 0.01 10.36 ± 0.02 10.44 ± 0.02 10.21 ± 0.02 9.78 ± 0.02 9.74 ± 0.02 9.69 ± 0.02 9.62 ± 0.02 9.62 ± 0.01 9.65 ± 0.01 9.61 ± 0.01 9.61 ± 0.01 9.22 ± 0.01 9.19 ± 0.01 9.19 ± 0.01 9.20 ± 0.01 9.20 ± 0.01 9.13 ± 0.01 9.15 ± 0.01 9.26 ± 0.01 9.25 ± 0.01 9.14 ± 0.01 9.17 ± 0.01 8.69 ± 0.02 9.00 ± 0.01 8.69 ± 0.02 8.76 ± 0.02 8.63 ± 0.02 7.37 ± 0.04 7.34 ± 0.04 8.78 ± 0.01 8.99 ± 0.01 9.82 ± 0.02 10.82 ± 0.02 10.61 ± 0.02 10.81 ± 0.04 11.21 ± 0.03 14.10 ± 0.05 13.99 ± 0.02 14.89 ± 0.04 15.5 ± 0.1 15.6 ± 0.2 16.3 ± 0.1

0.8 ± 0.1 0.6 ± 0.1 0.72 ± 0.04 0.56 ± 0.04 0.66 ± 0.04 0.62 ± 0.04 0.63 ± 0.04 0.63 ± 0.05 0.6 ± 0.04 0.56 ± 0.06 0.69 ± 0.04 0.55 ± 0.04 0.60 ± 0.04 0.64 ± 0.02 0.71 ± 0.03 0.56 ± 0.04 0.68 ± 0.04 0.62 ± 0.02 0.68 ± 0.04 0.65 ± 0.02 0.66 ± 0.03 0.69 ± 0.04 0.63 ± 0.02 0.66 ± 0.02 0.66 ± 0.04 0.72 ± 0.04 0.68 ± 0.04 0.65 ± 0.04 0.65 ± 0.04 0.66 ± 0.04 0.69 ± 0.04 0.69 ± 0.02 0.70 ± 0.02 0.72 ± 0.02 0.70 ± 0.02 0.68 ± 0.02 0.68 ± 0.02 0.68 ± 0.02 0.72 ± 0.02 0.74 ± 0.02 0.65 ± 0.02 0.72 ± 0.02 0.69 ± 0.02 0.70 ± 0.02 0.69 ± 0.02 0.64 ± 0.02 0.62 ± 0.04 0.68 ± 0.02 0.59 ± 0.04 0.65 ± 0.04 0.62 ± 0.04 0.74 ± 0.08 0.74 ± 0.08 0.68 ± 0.02 0.68 ± 0.02 0.72 ± 0.04 0.63 ± 0.04 0.70 ± 0.04 0.71 ± 0.09 0.74 ± 0.06 0.8 ± 0.1 0.80 ± 0.05 0.6 ± 0.2 1.0 ± 0.3 1.1 ± 0.5 0.7 ± 0.3

3.291 3.264 3.207 2.674 2.644 2.581 2.550 2.488 2.456 2.361 2.297 2.234 2.202 1.559 1.527 1.495 1.464 1.402 1.309 1.279 1.045 0.991 0.991 0.964 0.938 0.913 0.818 0.776 0.776 0.776 0.756 0.756 0.756 0.737 0.737 0.720 0.720 0.720 0.688 0.688 0.651 0.642 0.634 0.634 0.626 0.638 0.735 0.751 0.768 0.841 0.860 1.217 1.272 0.803 0.955 1.211 2.029 2.157 2.189 2.315 4.243 4.236 4.090 5.608 5.784 5.835

2.863 2.851 2.826 2.602 2.589 2.564 2.551 2.525 2.513 2.474 2.448 2.422 2.409 2.142 2.129 2.115 2.102 2.074 2.033 2.019 1.907 1.878 1.878 1.864 1.850 1.836 1.778 1.749 1.749 1.749 1.735 1.735 1.735 1.72 1.72 1.705 1.705 1.705 1.676 1.676 1.632 1.617 1.603 1.603 1.498 1.468 1.346 1.331 1.315 1.254 1.238 0.961 0.916 1.633 1.707 1.822 2.171 2.224 2.238 2.291 4.560 4.570 4.870 5.788 5.848 5.866

17.2 17.4 17.9 21.9 22.1 22.5 22.7 23.1 23.3 23.8 24.1 24.4 24.6 25.5 25.5 25.3 25.2 24.9 24.2 23.9 20.1 18.7 18.7 17.9 17.0 16.1 11.5 8.8 8.8 8.8 7.3 7.3 7.3 5.7 5.7 4.0 4.0 4.0 1.4 1.4 6.4 8.7 10.9 10.9 27.5 31.9 45.7 46.9 48.0 51.6 52.2 52.1 50.3 28.9 30.6 31.4 27.7 26.7 26.5 25.5 12.1 12.1 7.7 10.0 10.0 9.9

7.19 7.51 7.50 7.32 7.39 7.31 7.51 7.48 7.46 7.51 7.49 7.45 7.49 7.50 7.53 7.41 7.48 7.35 7.53 7.53 7.50 7.82 7.69 7.71 7.79 7.98 8.23 8.04 8.00 7.95 8.00 8.00 8.03 8.11 8.11 7.84 7.81 7.81 8.04 8.04 8.09 8.12 8.24 8.23 7.89 7.77 6.36 6.53 6.07 7.27 6.92 7.04 6.65 6.29 6.45 6.71 7.01 6.90 7.82 7.27 7.32 7.92

Grothues (1996) from November 9, 1991 to January 22, 1992, using the Bochum 0.61-m telescope at La Silla (Chile) equipped with single channel photometer with a photomultiplier and a standard BV Johnson filter set. The data was calibrated using an appropriate number of Harvard E-regions UBVRI standard-stars (see Graham, 1982). The 2

New Astronomy 75 (2020) 101320

A.S. Betzler and O.F. de Sousa

observational cadence was similar to one used for the comet Halley observations. The photometric and observational parameters of this object are presented in Table 2. The photometric aperture used for the measurements was 45.4”, corresponding to optocentric distances between 1.07 × 104 km (Δ= 0.649 AU) and 1.92 × 104 km (Δ= 1.166 AU). Brazilian visual observers estimated the 1P/Halley’s coma angular diameters and degree of condensation (DC) between November 1985 and June 1986 and compiled at the Comet Section of the Observational Astronomy Network (“Rede de Astronomia Observacional”, REABrasil1). In order to verify the validity of the steady state hypothesis for the comas of these two objects, we used the fluxes in the R-band of comet Faye (Jorda et al., 1995) and magnitudes BC of the IHW photometric system of 1P/Halley (Svoren, 1990). These data were obtained in multiple apertures, which allowed the determination of the slope m of the coma through the use of Eqs. (2) and (9), as defined in Section 2.3. Betzler et al. (2017) showed that the slope has little variation according to the filter used, so that the establishment of the steady-state from magnitudes obtained through filters B or R can be extended to Vband filter.

Table 2 Observational circumstances and photometric parameters of 4P/Faye. ΔT is equal to Th − 2, 448, 576.72 .

2.2. Data processing The search for periodicities in the magnitudes V and color B-V was made with the use of a Python script, using routines provided by the Astropy Project (Collaboration et al., 2013). We searched for periodicities in the relationship between B-V color and V magnitude on time using the Lomb-Scargle periodogram (Lomb, 1976; Scargle, 1982). This algorithm is a traditional method of searching for periodic variations in data collected with irregular time intervals, such as our samples. We estimated the statistical significance of periodicities using the bootstrap method (see Efron, 1979; Kraft et al., 1991), generating the false alarm probabilities FAP (see Baluev, 2008). The FAP defines the probabilities that a measure is outside the confidence interval. The relationship between FAP and the confidence level CL is FAP = 1−CL (Wa̧s et al., 2014). The false alarm and CL were used to establish a ranking of the more probably periods found in the photometric data of the comets Faye and Halley. Thus, we estimated the uncertainty in the periods by means of the full width half maximum of the main peaks in the Lomb-Scargle periodogram. We apply the Kruskal-Wallis H-test (Kruskal and Wallis, 1952) to compare the B-V coma color distributions of both comets. This test is the non-parametric equivalent of the analysis of variance (ANOVA). It is used for comparing two or more independent samples of equal or different sizes. The Kruskal-Wallis test does not assume a normal distribution of the residuals, unlike the analogous one-way ANOVA.

A common procedure in cometary photometry is to couple the aperture radius p (in arcsec) to a constant optocentric distance measured in kilometers. Betzler et al. (2018) demonstrates that the use of this procedure could introduce systematic errors in the magnitudes of the comet C/2014 S2 (PANSTARRS). To avoid this problem, it was suggested to use a fixed photometric aperture radius (measured in arcsec) over the entire duration of the observational window. To demonstrate the need for this procedure, we will use an extension of the demonstration presented by Betzler et al. (2018). Initially, we consider that the azimuthally integrated intensity Ic(p) can be described by:

1

V

0.00000 0.02400 0.03600 0.85398 1.02498 1.83796 1.94596 7.82882 7.92282 8.03781 8.83179 8.91879 12.82867 12.91167 13.03767 13.84564 13.91264 14.82961 14.90861 15.84358 15.91357 16.92054 17.83451 17.92051 18.85148 18.89848 19.85544 19.89644 20.84640 20.88740 21.84537 21.88437 36.82572 37.86368 38.87962 39.87158 40.82153 40.87253 41.82148 41.86847 45.84027 45.87127 46.83821 46.88421 47.83816 47.89016 48.83710 48.88410 50.83700 66.84202

11.25 11.19 11.19 11.18 11.17 11.21 11.19 11.34 11.33 11.35 11.36 11.36 11.52 11.55 11.55 11.55 11.57 11.58 11.59 11.64 11.66 11.65 11.69 11.67 11.73 11.72 11.72 11.73 11.80 11.83 11.81 11.78 12.28 12.31 12.37 12.30 12.37 12.36 12.38 12.35 12.55 12.50 12.55 12.43 12.52 12.53 12.52 12.50 12.50 12.99

B-V

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.02 0.02 0.03 0.02 0.02 0.02 0.05 0.02 0.02 0.01 0.02 0.04 0.03 0.01 0.01 0.02 0.04 0.04 0.04 0.02 0.02 0.01 0.02 0.02 0.06 0.02 0.01 0.01 0.03 0.02 0.02 0.02 0.02 0.04 0.02 0.03 0.03 0.01 0.03 0.02 0.04 0.02 0.02 0.01 0.02 0.01 0.02 0.04 0.02 0.01

0.72 0.76 0.78 0.76 0.73 0.66 0.70 0.78 0.81 0.81 0.80 0.81 0.79 0.77 0.78 0.78 0.79 0.80 0.79 0.80 0.79 0.79 0.78 0.79 0.78 0.78 0.79 0.78 0.77 0.76 0.76 0.77 0.77 0.80 0.76 0.76 0.74 0.76 0.76 0.78 0.81 0.78 0.73 0.75 0.74 0.78 0.76 0.78 0.75 0.77

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.01 0.01 0.03 0.05 0.02 0.01 0.05 0.02 0.03 0.03 0.02 0.02 0.01 0.02 0.03 0.02 0.02 0.03 0.03 0.02 0.02 0.03 0.02 0.01 0.02 0.01 0.02 0.03 0.03 0.01 0.03 0.03 0.02 0.03 0.01 0.02 0.02 0.03 0.03 0.03 0.07 0.02 0.02 0.02 0.03 0.02 0.01 0.01 0.02 0.03

Δ[AU]

r[AU]

α[∘]

V(1,1,0)

0.649 0.649 0.649 0.652 0.653 0.656 0.656 0.681 0.681 0.682 0.685 0.686 0.705 0.706 0.707 0.711 0.711 0.716 0.717 0.722 0.723 0.728 0.734 0.734 0.740 0.740 0.746 0.747 0.753 0.753 0.759 0.759 0.871 0.879 0.888 0.896 0.905 0.905 0.913 0.914 0.950 0.950 0.959 0.959 0.968 0.969 0.978 0.978 0.997 1.166

1.593 1.593 1.593 1.593 1.593 1.594 1.594 1.595 1.596 1.596 1.596 1.596 1.599 1.599 1.599 1.600 1.600 1.601 1.601 1.602 1.602 1.603 1.604 1.604 1.605 1.605 1.607 1.607 1.608 1.608 1.609 1.609 1.638 1.641 1.643 1.646 1.648 1.648 1.651 1.651 1.662 1.662 1.665 1.665 1.668 1.668 1.671 1.671 1.677 1.734

16.6 16.61 16.62 17.05 17.14 17.57 17.63 20.51 20.55 20.60 20.97 21.01 22.69 22.73 22.78 23.11 23.13 23.50 23.53 23.89 23.92 24.29 24.63 24.66 24.99 25.01 25.34 25.35 25.68 25.69 26.00 26.02 29.84 30.04 30.22 30.40 30.56 30.57 30.72 30.73 31.30 31.31 31.43 31.44 31.55 31.56 31.67 31.67 31.88 32.82

8.37 8.32 8.33 8.32 8.31 8.33 8.32 8.20 8.25 8.22 8.22 8.19 8.17 8.20 8.16 8.15 8.08 8.06 8.03 8.04 7.97 7.96 7.93 7.92 7.93 7.92 7.96 7.95 7.95 7.95 7.94 8.02 7.89 7.89 7.87 7.87 7.86 7.87 7.89 7.91 7.85 7.87 7.84 7.88 7.86 7.83 7.84 7.89 7.91 7.82

where I0 is brightness intensity in the coma optocenter, m is a positive defined exponent, and p is the optocentric distance (arcsec). Considering a representation on the log10(Ic) × log10(p) scale, the negative value of exponent m is the slope of the linear adjustment. Based on photographic observations of three short-period comets, Miller (1961) suggested that this distribution presents m = 1 for distances p < 2.0 × 104 km, and being m ≈ 2 for longer distances. Specifically, the unitary value of exponent m implies that the coma is in steady state. Theoretically, in a steady state coma, the production of dust and its expansion would be constant, isotropic, and homogeneous. However, this exponent optocentric range differentiation is not necessarily uniform in all directions around the nucleus. There may be an asymmetry in the distribution of brightness in the solar and tail directions as a function of the solar radiation field (Festou and Barale, 2000) or cometary activity being concentrated in a few small regions on the surface of the nuclei. A good example is the great variation of the exponent observed in the coma and tail of the 67P comet (Rosenbush et al., 2017). However, Sen et al. (2019) suggests that the

2.3. Photometric aperture

Ic (p) = I0/ pm ,

ΔT (days)

(1)

http://rea-brasil.org/cometas/observ1p.htm. 3

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A.S. Betzler and O.F. de Sousa

67P coma was in a quasi-steady-state (m ≃ 1), in the range 3000–20,000 km from optocenter, in post-perihelic observations made in 2015 December 12, UT. Considering the action of solar radiation, Jewitt and Meech (1987) proposed that a steady-state coma can present exponent 1.0 ≤ m ≤ 1.5. According to Eq. (1), the flux F in a spectral band can be defined as:

F = c / pm − 2 .

(2)

The constant c is a function of I0, m and the photometric aperture radius p0 (see Betzler et al., 2017). The radius p0 can be interpreted as the optocentric distance corresponding to the beginning of the coma region modeled by Eq. (2). For photometric radius below p0, the action of seeing, and the telescope’s tracking errors become significant (Jewitt and Meech, 1987). The Eq. (2) makes it possible to obtain the ratio of the fluxes F measured with the same aperture p at different times T, obtained along the observational window. For the time T1:

F1 (p) = c1/ pm1− 2 ,

Fig. 1. Dependence between the pre and post-perihelion IHW’s BC-band (λcentral = 485.4 nm) magnitudes and the photometric aperture radius p (arcsec) for the comet Halley. Solid, dashed and dotted lines are best fits of a Eq. (9). The quadratic correlation coefficient R2 ∼ 1 for all the adjustments. Solid line: c0 = 13.70 ± 0.06 and m = 1.12 ± 0.04 . Dashed line: c0 = 12.79 ± 0.03 and m = 1.00 ± 0.02 . Dotted line: c0 = 10.37 ± 0.03 and m = 1.34 ± 0.02 . The vertical dash-dot line represents the photometric aperture radius p(=7.45”) used to obtain the data analyzed in the Section 3.1. Tp is the time of perihelion passage.

(3)

and for T2:

F2 (p) = c2/ pm2 − 2 .

(4)

With c1, c2 are constants and m1 and m2 exponents associated with time T1 and T2. considering the same aperture radius p, the ratio of the fluxes F1 and F2 to the times T1 and T2 is given by:

F1 (p) c = pm2 − m1 1 . F2 (p) c2

(5)

If m2 − m1 → 0, the flux ratio F1(p)/F2(p) is approximately constant and independent from photometric aperture radius p:

F1 (p) c = 1. F2 (p) c2

(6)

On the other hand, considering measurements made at different apertures radii (p1 and p2, with n = p1 / p2 ), for the times T1 and T2 is given by:

F1 (p1 ) c = n−m1 p2m2 − m1 1 , F2 (p2 ) c2

Fig. 2. Dependence between the pre and post-perihelion R-band flux F and the photometric aperture radius p (arcsec) for the comet Faye. Solid and dashed lines are best fits of a Eq. (2) for each observational season.The quadratic correlation coefficient R2 ∼ 1 for the adjustments. Solid line: c = (1.5 ± 0.2) × 10−17 Wm −2 Å −1”2 and m = 1.33 ± 0.05. Dashed line: c = (6 ± 1) × 10−18 Wm −2 Å −1”2 and m = 1.19 ± 0.08. The vertical dotted line represents the photometric aperture radius p(=22.7”) used to obtain the data which was analyzed in the Section 3.2. Tp is the time of perihelion passage.

(7)

and in the limit of m2 − m1 → 0 :

F1 (p1 ) c = n−m1 1 . F2 (p2 ) c2

(8)

Eqs. (7) and (8) suggests that the use of different photometric apertures can introduce significant variations in the measured fluxes and, consequently, in the magnitudes. The relationship between the flux F and the magnitude M of the coma is given by Betzler et al. (2017) (and previously by Li and Jewitt, 2015, with a different focus):

M = −2.5 log10 (F ) + Z0 = 2.5(m − 2)log10 (p) + c0,

value. For this task, we will use magnitudes of comet Halley measured in different apertures and times. The photometric data of the object were collected in two observational sessions by Svoren (1990), and presented in Fig. 1. At the 1985-11-09.2 UT observational session (T1), the comet Halley was 0.86 A.U from Earth and had an apparent magnitude Bc = 10.61 ± 0.02 for p1 = 24.50 ” ( p1 (km) = 1.5 × 10 4 km) and Bc = 11.24 ± 0.03 measured with p2 = 14.75” ( p2 (km) = 9.2 × 103 km). At the 1985-12-30.7 UT (T2) observational session, the object was at Δ = 1.1 A.U and had apparent magnitudes 8.06 ± 0.03 and Bc = 8.48 ± 0.01 respectively for the two previous photometric radii ( p1 (km) = 2.0 × 10 4 km and p2 (km) = 1.2 × 10 4 km). Applying Eq. (11) in the Halley’s comet data, we verified that the mean value of the F1(p)/F2(p), for each radius is equal to 0.14 ± 0.08 (1 σ). Using the same dataset, the value of F1 (p1 )/ F2 (p2 ) = (9 ± 1) × 10−2 . The ratio between Eqs. (6) and (8) is equal to n−m1. As n = 1.66 and m1 = 1.12 ± 0.02 (see Fig. 1), the value of n−m1 is equal to 0.566 ± 0.005. This value is compatible with what would be obtained if we directly divide the quantities F1(p1)/F2(p2) and F1(p)/F2(p), which is 0.62 ± 0.06. The previous results show that varying the photometric aperture

(9)

where Z0 is the zero point of the magnitude system and c0 is a constant. To calculate the ratio F1/F2, let’s use the following expressions. Using the Eq. (9), the flux can be defined as:

F = 10−0.4(M − Z0).

(10)

As the magnitudes were calibrated with the same catalog, we can assume that the zero points Z0 are similar. Thus, the ratio F1/F2 is given by:

F1 = 10−0.4(M1− M2), F2

(11)

where M1 and M2 are the magnitudes of the coma at time T1 and T2 respectively. To verify the validity of Eqs. (5) and (7), we must determine its 4

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introduces variations in the photometric flux of the Halley’s comet. Specifically, the systematic error introduced by varying the aperture radius from 24.50 to 14.75” is 0.42 ± 0.08 magnitudes. Assuming that there are no large variations of the exponent m during the total duration of the observational session of both comets, the difference m1 − m2 will tend to zero so that the ratio F1/F2 will be independent of the photometric aperture radius. This condition is reached when the part of the coma recorded within the photometric aperture is approximately steady state. This hypothesis appears to be valid for both comets: Applying Eqs. (2) and (9) in the pre- and postperihelion data of the comets Halley and Faye obtained by Svoren (1990) and Jorda et al. (1995) (Figs. 1 and 2), we find that both have exponent values within the range 1.0 ≤ m ≤ 1.5. Grothues (1996) obtained m ≃ 1 of the coma in solar and anti-solar during his observational run of the comet Faye which data was considered in this study. Analyzing the Fig. 2, we can concluded that the R-band flux of comet Faye increases linearly with the photometric aberture size, as predicted by Eq. (2) which implies a dust expansion at constant velocity up to a distance to the nucleus of at least 1.4 × 104 km, as observed in the 1P/Halley in similar heliocentric distances (see Danks et al., 1988). IR pre and post-perihelion observations of Halley’s comet done by Gehrz and Ney (1992) are generally consistent with the steady state model for nuclear ablation. According to previous results, we used photometric measurements obtained with a single aperture (in arcsec) in the V-band magnitudes of the comets Halley and Faye analyzed in sessions 3.1 and 3.2 of this work. Additionally, based in the Eq. (9), Betzler et al. (2017) demonstrates that the color index B − V is given by:

B − V = S log10 (p) + C ,

Fig. 3. Graphics of V − 5 log10 (Δ) × log10 (r ) for the comets 1P/Halley and 4P/ Faye.The lines are best linear fits. Solid line: H0 = 7.9 ± 0.2 and n = 3.8 ± 0.2 for pre-perihelion data of 1P, Dashed line: H0 = 7.6 ± 0.3 and n = 2.2 ± 0.3 for post-perihelion data of 1P, Dotted line: H0 = 9 ± 2 and n = 6 ± 4 for postperihelion data of 4P.The magnitude error bars have dimensions similar to the observational points.

periods. For this reason, this equation cannot be considered exclusive of comets with a specific dust-to-gas mass ratio, and probably represents a “mean” comet. This can explain why Schleicher (2010) considered Divine’s curve to be too shallow at smaller phase angles. In our observations, the phase angles were always lower than 50∘, allowing variations lower than 20∘ to comet Faye. This way, we consider that the phase corrections are not a significant source of error. The time of observation t, in Julian days, was corrected for the transit time of the light between the object and the Earth, using the equation (see Warner, 2016):

(12)

where S = 2.5(mB − mV ) and C = c0B − c0V being the difference between the exponents m and constants c0 in the filters B and V. Eq. (12) suggests that a systematic error can be present in the B-V colors measured with different photometric apertures. This situation occurs amongst the color distributions of the comets Halley and Faye, questioning the usefulness of its direct comparison. This question is addressed at Section 3.2.1. 2.4. Geometric corrections

th = t − 0.00576 Δ, Following Hughes (1990),the apparent or total magnitude V of a comet can be defined as:

V = H0 + 5 log10 Δ + 2.5n log10 (r ),

with the distance Δ being a function of the time t, and th is the observational heliocentric julian date.

(13)

where r and Δ are the heliocentric and geocentric distances, H0 is the absolute magnitude, with H0 = V (Δ = 1, r = 1), n is the activity index. The index n can be obtained by a linear fit of the relation between V − 5 log10 (Δ) and log10(r) from Eq. (13), as shown in Fig. 3. In this relationship, the angular and linear coefficients are respectively equal to 2.5n and the absolute magnitude H0. With the index n, the apparent V magnitudes of the comet were corrected to consider the effect caused by the heliocentric distance and phase angle α variation:

V (1, 1, 0) = V − 5 log10 Δ − 2.5n log10 (r ) + Φ(α ),

2.5. Rotational light curves For asteroids with an elipsoid shape and constant albedo, we expect their periodicity to be related to the second harmonic of the rotation period to be the most intense in a periodogram. However, this situation is not valid for all asteroids (Harris, 2012) and probably to active comets. In fact, well-defined peaks corresponding to twice the periodicity with the highest level of significance were not identified in the periodograms generated from magnitudes V(1,1,0) and B-V colors of comets Halley and Faye presented at Figs. 4 and 5. Thus, we use the periodicity best ranked in the periodogram associated with magnitude V to construct the light curves. For the composition of the light curve, we define the phase of each magnitude V(1,1,0) corresponding to periodicity P. After this, we calculate the moving average magnitude V (1,1,0) and phases, using three points with consecutive phases. Our intention with this procedure is to reduce the magnitude dispersion in similar rotational phases, which are common in light curves of active comets. We derived the light curve from a sixth-degree polynomial. The polynomial degree was chosen to minimize the residual mean square

(14)

where, V(1, 1, 0) is the magnitude that the comet would have been separated from the Earth and the Sun by 1 AU and phase angle equal to zero degree. The phase correction function Φ(α) was defined by Divine (1981) as:

Φ(α ) = 2.5 log10 (1 − 0.018α ),

(16)

(15)

∘

being valid for α ≤ 50 . Part of the parameters that define Divine’s curve (Divine, 1981) are based on the IR photometric data of six comets of short and long orbital 5

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deviation and to maximize the quadratic coefficient of determination (median R2 ∼ 0.6) for all light curves. 3. Results and discussion 3.1. 1P/Halley 3.1.1. The color of coma Using the H-test, we found that the pre- and post-perihelion coma’s B-V color distribution are different. We can attribute this result to the larger error of the B-V color index measured after the perihelion (see Table 1). The pre(2.863 ≤ r ≤ 1.315 AU) and post-perihelion (1.633 ≤ r ≤ 5.866 AU) mean colors are 0.66 ± 0.05 (1 σ) and 0.8 ± 0.1. Considering the standard deviation, the pre- and postperihelion B-V colors are similar to the ones estimated for the nucleus from post-perihelion observations with the Giotto probe, from March 1986 (0.72 ± 0.04, derived from Thomas and Keller (1989) by Lamy et al. (2004)), and all these colors are redder than solar ones (BV= 0.64, Holmberg et al. (2006)). However, Thomas and Keller (1989) concluded that the nucleus is redder than the dust in the coma. The data analyzed by Thomas and Keller (1989) is related to images obtained a few minutes before the minimal distance to the nucleus, implying that the data analyzed correspond to punctual observations. Similiarly, Cremonese et al. (2016) performed spectrophotometric analysis of 70 dust grain tracks of the coma and many regions of the nucleus of comet 67P observed by the Rosetta probe. This study showed that the majority of the observed grains had color similar to the nucleus. In this way, we consider that the short duration of the observations of comets Halley and 67P studied by Thomas and Keller (1989) and Cremonese et al. (2016) were insufficient to register significant alterations in the color of the coma, which time scale is of order of hours to weeks (see Betzler et al., 2017). This suggests that long duration observations are necessary, in order to better understand the process of grain size/composition population change of coma of comets with time.

Fig. 4. Lomb-Scargle periodograms from V(1,1,0)(=V) and B-V data of comet Halley. The confidence level above the horizontal dashed line is 95%. The letters labels correspond to the periods with larger confidence level in the periodograms: A (38.61 days), B (64.22 days), C (79.23 days), D (7.36 days), E (167.17 days), F (42.22 days), and G (3.75 days).

3.1.2. Photometric parameters In order to carry out the periodicity search in the V magnitudes, we had to correct them for the effects of distance and phase angle generating the V(1,1,0) magnitudes. For this, we use Eqs. (5) and 6 to estimate the H0 absolute magnitude and the activity index n. Since these parameters can vary before and after perihelion, the observations were separated into these two phases. The pre-perihelion phase had 52 measurements between August 20, 1985 and January 06, 1986. The post-perihelion phase had 10 magnitudes collected between April 30, 1986 and May 25, 1987. We obtained activity indexes n = 3.8 ± 0.2 and 2.2 ± 0.3 for the pre- and post-perihelion phase (Fig. 3). This asymmetry in activity around perihelion, with a tendency for post-perihelion reduction, was also detected in the visual magnitudes of Halley’s comet in 1910 (4.44 and 3.07) (Morris and Green, 1982) and 1986 (Ferrín, 2010). Hughes (1988) suggested that the activation and deactivation of regions in the nucleus could explain the asymmetry of preand post-perihelion values of the absolute magnitude H0 (= 4.53 ± 0.04 and 3.88 ± 0.04) identified in the visual magnitudes collected by observers of the British Astronomical Association in the last perihelion passage of comet Halley. Our results do not support this hypothesis. The absolute pre-and post-perihelion magnitudes are respectively H0 = 7.9 ± 0.2 and 7.6 ± 0.3, which does not suggest variation of this parameter. Analyzing the previous results, we can see that the pre- and postperihelion activity indices are similar, and a systematic difference occurs in the absolute visual and photoelectric magnitudes H0. We argue that the difference between these values and those estimated by Hughes (1988) can be attributed to the difference between the photometric apertures used for the determination of the magnitudes. The

Fig. 5. Lomb-Scargle periodograms from V(1,1,0)(=V) and B-V data of comet Faye. The confidence level in the horizontal solid line is 90%. The letter labels are A and C (59.01 days), B (6.01 days), D (9.18 days), and E (overlapping peaks of 23.13 and 24.63 days).

6

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16–18 weeks. We apply the Lomb-Scargle method to the combined sample of the magnitudes V(1,1,0) for both pre- (0.01902 ≤ ΔT ≤ 112.77725 days) and post-perihelion (252.72735 ≤ ΔT ≤ 642.72348 days). We found that the periods of 38.61, 64.22 e 79.23 days are the most significant with confidence levels greater than or equal to 99.87%. The fourth best ranking period is 7.36 days with FAP = 41.1% or CL = 58.9%. Not surprisingly, a period of 2.31 days was detected with FAP = 80.7% or CL = 19.3%. (Fig. 4). For the B-V color, the best ranked peaks of the periodogram were 167.17 days (FAP = 2.1% or CL = 97.9%) and 42.22 days (FAP = 11.9% or CL = 88.1%). Third, we have the periodicity of 3.75 days, with FAP equal to 20.7% and CL = 79.3%. The period of 38.61 days may be the second harmonic of the period of 79.23 days. However, the period of 64.22 days does not have an evident harmonic connection with the periodicities of 38.61 and 79.23 days. We do not rule out the possibility that the 64.22 day period is a high harmonic (10th) of the 642.74 days observational interval. We consider that the periods of 38.61 and 79.23 days may be linked to the rotational state of the nucleus. Given its greater intensity, we consider the signal of 79.23 days as the most significant. As in the case of magnitudes V (1,1,0), the periods of 42.22 days and 167.17 days appear to have a harmonic correlation with each other and with the period of 79.23 days. These relationships are almost identical to those occurring between the periodicities of 8–9 and 16–18 weeks suggested by Schleicher et al. (2015). The period of 3.75 days detected in the B-V color distribution could correspond to the second harmonic of the period of 7.36 days detected in our analyzes of the temporal variation of V-band magnitudes. Thus, taking into account the periodicities found in V(1,1,0) magnitudes and B-V colors, we suggested that the periodicities of 7.36 ± 0.04 days (Fig. 6) and 79 ± 6 days can be associated with the complex rotational state suggested for the Halley’s comet. The light curve corresponding to

Eq. (9) shows that a comet will appear brighter as the photometric aperture grows. Visual observers estimate the magnitudes considering the surface brightness of the coma integrated in a circle with a diameter of several minutes of arc. This value can be tens of times larger than the apertures that are commonly used in CCD or photoelectric photometry. This factor could justify the difference in observed values of the absolute visual and photoelectric magnitude H0. This hypothesis is corroborated by estimates of angular diameters and degree of condensation (DC) made by Brazilian visual observers in the REA-Brasil dataset. The DC is a scale that represents the visual impression of the coma’s surface brightness distribution. This scale is divided into nine classes in which DC 0 implies “Difuse coma of uniform brightness”, DC 2 is equal to “Difuse coma with definite brightening towards center” and DC 7 means “Condensation appears like a star that cannot be focused described as strongly condensed” (Schmude, 2010). Analyzing this dataset, we verified that the angular diameter of the coma ranged from 5.3 to 23.2’ and the DC ranged from 2 to 7, with a mode of 6, between November 1985 and June 1986. Thus, we concluded that the comet Halley presented a coma with a bright central condensation that had angular diameter with dimension superior to the photometric aperture used in the data analyzed in this study (2p = 14.9”). To study this hypothesis, we consider visual observations of comet Halley made by the experienced Brazilian observer V. F. de Assis Neto (1936–2004). In 1985-11-09.14 UT, he estimated that the comet Halley had angular diameter of 8.5’ and apparent visual magnitude 7.3. Independent of his observations, Svoren (1990) made a nearly simultaneous BC-band multiaperture photoelectric observation (1985-11-09.2 UT, Fig. 1), which made possible the estimative of the slope m = 1.12 ± 0.04 and c0 = 13.70 ± 0.06 with the Eq. (9). Applying this last equation, with p = 4.25’ (=8.5’ / 2 = 255”) and previous value of m and c0, the estimated B-band magnitude was 8.4 ± 0.3. Whereas, the pre-perihelion mean B-V color was 0.66 ± 0.05, the V-band magnitude is 7.7 ± 0.3 which is closer to Assis Neto’s estimates. These results consider that, according to Morris (1973), estimates of visual cometary magnitudes can present an error of 0.5 magnitudes. 3.1.3. Periodicities The determination of the rotational period of the comet Halley has been under several investigations, using data from 1910 and 1986, its latest perihelion passages. From the periodicities found, there is a multiplicity of solutions for the rotation period, position of the poles, the type of non-principal-axis (NPA) rotation state, among other traits. This denotes the complexity of determining the rotational state of a cometary nucleus. Using images collected by the space probes VeGa I and II, Sagdeev et al. (1986) proposed a rotation period of 53.5 ± 1.0 h (or 2.23 ± 0.04 days), with the spin vector directed toward ecliptic coordinates l = 22∘ and b = −63∘ . This periodicity is compatible with the values deduced by Grothues and Schmidt-Kaler (1996) and Voelzke et al. (1997) from photographic observations of dust tail and ion gas coma. A period of 7.4 days was inferred by Millis and Schleicher (1986) from the analysis of the temporal variation of the intensity of the lines OH, NH, CN, C3, CO +, and C2 bands and continuum emissions at 365 and 484.5 nm. Given the ambiguity of the previous values, Schleicher and Bus (1991) analyzed photographic plates obtained in 1910 at the Observatory of Cordoba (Argentina). Analyzing the gas and dust production, they found periodicity ranging from 7.35 to 7.60 days. Recently, Schleicher et al. (2015) reanalyzed IHW multi aperture photometric data, collected by 18 observatories in 1986. Using a rotation period of 7.35 days as reference, it was found that the comet’s light curve went from double-peak to triple-peaked every 8–9 weeks. The rotation period ranged from 7.2 to 7.6 days probably due to synodic effects and the interaction of solar radiation in isolated regions of the nucleus. Unexpectedly, they found that a phase shift of one-half cycle also took place during this interval, and therefore the actual beat frequency between the component periods is twice this interval or

Fig. 6. V-band phase-folded light curves of comets 1P/Halley (circles) and 4P/ Faye (diamonds) for the periods P. The V(1,1,0) magnitude is corrected for the effects caused by the variation of distance and phase angle. To minimize the noise in the light curve, three consecutive data points were binned to form a single point. The error bars are equal to the standard deviation of this three data points. The black dashed lines are the best fit to a polynomial function of 6th order. The 0.33 phase mark correspond to the average rotational phase which the 1985 December 12, UT Halley’s outburst that was probably registered in the data. 7

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colors, as modeled by Eq. (12).

the shortest period has a peak-to-peak amplitude of 0.9 ± 0.4 magnitudes, compatible with the value obtained by Neckel and Muench (1987) for a light curve phased to a period of 7.3 days. If the nucleus is modelled as a prolate ellipsoid with semi-axes a, b and c, where b = c and a > b, the axis ratio a/b and can be determined by (see Kokotanekova et al., 2017):

a = 100.4Δm, b

3.2.2. Photometric parameters To obtain the H0 and n parameters, we used 43 post-perihelion Vband magnitudes. The linear fit in Fig. 3 has correlation factor R2 ∼ 0.7 with n = 6 ± 4 and H0 = 9 ± 2 . The activity index and the absolute magnitude are close to the values proposed by Grothues (1996), using V’10 magnitude. this activity index value is almost three times superior to 1P/Halley pre and pos-perihelion indexes. However, this comparison is void of meaning given the great dispersion of the measurements. Taking this factor in consideration, the index n value is compatible with the interval between 2 and 8 defined by Meech and Svoren (2004).

(17)

where Δm is the light curve peak-to-peak amplitude. This definition appears to be correlated with the Halley‘s nuclear dimensions: using the Eq. (17) with Δm = 0.9 ± 0.4 magnitudes, we found a/ b = 2.3 ± 0.8. This is a reasonable value if when compared to a/ b = 2.1 ± 0.2 using in situ TV images obtained by the VeGa-1 spacecraft (Merenyl et al., 1990). The dispersion in some phases of the light curves in the V-band and in the B-V color are greater than the peak-to-peak amplitudes of the light curves. This trait can be attributed to impulsive variations in nuclear activity, probably generated by outbursts. In fact, a strong outburst with a southward jet was observed during 9h07m to 12h29m UT on December 12, 1985 by Watanabe et al. (1987), when they were observing the near nucleus region of comet P/Halley. In our dataset, this event’s probable beginning corresponds to the observation December 11.73, 1985 UT or day 112.77725 in the Table 1. In the V-band light curve, this outburst corresponds to the rotational phase 0.33 in the V-band light curve. The magnitude corresponding to this phase, presents larger error if compared with the adjacent measures. In a more recent paper, Vincent et al. (2016) analyzed a sequence of 34 outbursts observed by the Rosetta probe for the comet 67P, during its perihelion passage in 2015. It was possible to establish these events occur either in the early morning or shortly after the local noon. This appears to be the same situation as the outburst of December 12, 1985, assuming that the minimum of the light curve corresponds to the end of the solar illumination of an active region of the nucleus.

3.2.3. Periodicities Using the “halo method”, Whipple (1982) proposed rotation periods of 10.52 and 10.45 h to comet Faye using data collected in the perihelion passages of 1932 and 1969. Observations made with the Hubble Space Telescope by Lamy et al. (1996) suggested that the coma of this object is highly asymmetric. This feature may be explained either by an active source on the nucleus. Admitting this hypothesis, they found a pole orientation in the vicinity of α = 215∘ , δ = +10∘ (obliquity ∼ 70∘) with a source at +50∘ latitude. The asymmetry could also be explained by a projection effect of the dust tail. If this second interpretation is correct, then the temporal evolution of the coma is very slow, suggesting an extended source of dust on the nucleus. The latter hypothesis may indicate a longer rotation period for the 4P/Faye nucleus. In fact, Grothues (1996) suggested a rough estimative of nuclear rotation period of ∼ 10 days with a peak-to-peak nuclear magnitude amplitude of ∼ 0.5 magnitudes, by analyzing the temporal variation of the nuclear magnitude of the Faye’s comet nucleus obtained by Lamy and Toth (1995). The search for periodicities in the magnitudes V(1,1,0) suggests that the long periods of 23.13, 59.01, 24.63 days were best ranking with confidence levels respectively greater than or equal to 99.87%. The forth and fifth best ranking periods were 6.09 and 7.19 days with FAPs equal to 5.5% and 7.4% (Fig. 5). The possible harmonics or daily aliases (23.13 and 24.63 days) of the 59.01 days’ period are probably not associated with the rotational state of the nucleus. The periodicity of 6.01 days does not appear to be a harmonic or a daily alias of the observational period. Analyzing the B-V color periodogram, we verified that the periodicities of 9.18 and 59.01 days have higher confidence levels of 93.5% and 85.1%, respectively. The period of 59.01 days, corresponding to the observational interval of the analyzed data. Given the uncertainties of the period of 9.18 days, we cannot rule out the possibility that this is a harmonic of the period of 59.01 days. We do not find a clear harmonic relationship between best ranking periodicities in the V-band magnitudes and B-V color temporal variation. We could explain this fault to the small variation of the color with mean B-V value equal to 0.79 ± 0.02 (1 σ), or the nearly the half of the dispersion of the comet Halley’s coma color. This small dispersion may have made it difficult to identify a harmonic of the signal detected in magnitudes V. This characteristic can be considered similar to the difficulty encountered in the determination of rotational properties of asteroids with light curves of low amplitude ( ≤ 0.1 magnitudes) and long periods ( ≥ 24 h) (Warner and Harris, 2010). Based on the magnitudes V, we propose that the periodicity of 6.1 ± 0.3 days, with a light curve with peak-to-peak amplitude of 0.4 ± 0.1 magnitudes (Fig. 6), may be associated with the rotational state of the 4P/Faye. Using Eq. (17), the ratio between axis a/b in the nucleus is equal to 1.4 ± 0.1.

3.2. 4P/Faye 3.2.1. Outburst and comparison with 1P/Halley’s B-V color distribution Analyzing the temporal evolution of the V’10 geocentric magnitude, and linking with a fixed nuclear distance ( p = 16, 465 km),Grothues (1996) suggested that a small outburst occurred after the perihelion (Tp =1991-11-16.2 UT = JD 2,448,576.7) of comet Faye. The average V01 magnitude during the event (0 ≤ Δt ≤ 1.94596 days; see Table 2) and after (7.82882 ≤ Δt ≤ 8.91879 days) changed down from 11.67 ± 0.02 (1σ) to 11.77 ± 0.01. During the outburst, the water production rate may have raised by 10–15%. There is an observational gap between 1.94596 < Δt < 7.82882 days due the presence of the full moon in the sky. Using the H-test, we compare the B-V color indexes distribution during and after the outburst. The distributions are similar, considering a confidence level at 95% or higher. It corroborates the hypotheses of Betzler et al. (2017), in which the B-V color distributions are independent from the action of coma and phenomena as outbursts. Due to this information, we perform the photometric parameters and period searching using V-band magnitudes and B-V colors obtained at Δt ≥ 7.82882 days. The B-V colors of the comet Faye obtained from Δt ≥ 7.82882 days were compared with the H test with the colors pre- and post-perihelion of 1P/Halley. The comparison revealed that the pre-perihelion of 1P/ Halley are different from the color of comet Faye’s coma. A similarity between the post-perihelion color of comet Halley can be established with 4P/Faye’s data only at the level of confidence of 99%. (redundant with the next sentence) This can be caused by a real difference between the populations, originating from a difference of size/chemical composition between the grains, or a systematic error introduced by the difference between the photometric apertures used to obtain these

4. Summary and conclusions We performed a study on periodicities of the V-band magnitudes and B-V colors of the coma and the determination of photometric 8

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parameters n and H0 of the comets 1P/Halley and 4P/Faye. The main results of this study are:

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1. We verified that, in both comets, a same periodicity can have different confidence levels in the periodograms of magnitudes V and BV colors. Due to this reason, we do not recommend that the search for peridocities in the comas of the comets be exclusively based on B-V color. 2. From the V-band magnitudes and B-V color, we found that the periodicities 79 ± 6 and 7.36 ± 0.04 days are the most reasonable for 1P/Halley. For the latter periodicity, the light curve amplitude is 0.9 ± 0.4 magnitudes. 3. Considering the nucleus of the comet Halley as a prolate ellipsoid, the axis ratio is a/b= 2.3 ± 0.8 obtained from the light curve amplitude. This value is reasonable if compared with a/b= 2.1 ± 0.2 obtained from images of the nucleus by the space probe VeGa-1. 4. We have verified that the H0 absolute magnitudes of comet Halley, estimated from visual and photoelectric photometer magnitudes, can present great differences between them.This trend is not observed between the n photoelectric and visual activity indexes. The differences in the absolute magnitudes are possibly caused by the variation of the photometric apertures. 5. We did not detect an asymmetry of the photoelectric absolute magnitude H0 of comet 1P/Halley around the perihelion of 1986. 6. The B-V color distribution of the comet Halley, observed in 1986, was not equal in pre- and post-perihelion phases. 7. The pre- and post-perihelion mean B-V color index of the inner coma of comet Halley are 0.66 ± 0.05 (1 σ) and 0.8 ± 0.1 similar to the nucleus color. This result suggests the necessity of long duration observations to better understand the evolution of the grain population in a cometary coma. 8. The rotation period of 4P/Faye may be associated with a periodicity of 6.1 ± 0.3 days with a V-band light curve amplitude of 0.4 ± 0.1 magnitudes. The axis ratio is a/b= 1.4 ± 0.1. 9. We found the B-V color distribution during and after the possible occurrence of an outburst in the comet Faye is similar with a confidence level higher than 95%. Acknowledgements The authors thank REA-Brasil by kindly providing the visual data of the comet Halley, and to the anonymous reviewer for his valuable comments on our manuscript. This research has made use of NASA’s Astrophysics Data System. References A’Hearn, M., Vanysek, V., 2006. IHW Comet Halley Photometric Magnitudes, V2.0. NASA Planetary Data System. Baluev, R.V., 2008. Assessing the statistical significance of periodogram peaks. MNRAS 385, 1279–1285. https://doi.org/10.1111/j.1365-2966.2008.12689.x. Betzler, A.S., Almeida, R.S., Cerqueira, W.J., et al., 2017. An analysis of the BVRI colors of 22 active comets. Adv. Space Res. 60, 612–625. https://doi.org/10.1016/j.asr.2017. 04.021. Betzler, A.S., de Sousa, O.F., Betzler, L.B.S., 2018. Photometric study of comet C/2014 S2 (PANSTARRS) after the perihelion. Earth Moon Planets 122, 53–71. https://doi.org/ 10.1007/s11038-018-9521-5. Brandt, J.C., Caputo, F.M., Hoeksema, J.T., Niedner, M.B., Yi, Y., Snow, M., 1999. Disconnection events (DEs) in Halley’s comet 1985-1986. The correlation with crossings of the heliospheric current sheet (HCS). Icarus 137, 69–83. https://doi.org/ 10.1006/icar.1998.6030. Collaboration, A., Robitaille, T.P., Tollerud, E.J., et al., 2013. Astropy: a community python package for astronomy. A&A 558, A33. https://doi.org/10.1051/0004-6361/ 201322068. Cremonese, G., Simioni, E., Ragazzoni, R., Bertini, I., La Forgia, F., Pajola, M., Oklay, N., Fornasier, S., Lazzarin, M., Lucchetti, A., Sierks, H., Barbieri, C., Lamy, P., Rodrigo, R., Koschny, D., Rickman, H., Keller, H.U., A’Hearn, M.F., Agarwal, J., Barucci, M.A., Bertaux, J.-L., Da Deppo, V., Davidsson, B., De Cecco, M., Debei, S., Fulle, M., Groussin, O., Güttler, C., Gutierrez, P.J., Hviid, S.F., Ip, W.-H., Jorda, L., Knollenberg, J., Kramm, J.-R., Kueppers, M., Kürt, E., Lara, L.M., Magrin, S., Lopez Moreno, J.J., Marzari, F., Mottola, S., Naletto, G., Preusker, F., Scholten, F., Thomas, N., Tubiana,

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