Outbursts of comets at large heliocentric distances: Concise review and numerical simulations of brightness jumps

Planetary and Space Science 184 (2020) 104867

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Outbursts of comets at large heliocentric distances: Concise review and numerical simulations of brightness jumps M. Wesołowski *, P. Gronkowski, I. Tralle Faculty of Mathematics and Natural Science, University of Rzesz
ow, Pigonia 1 Street, 35-959, Rzesz
ow, Poland

A R T I C L E I N F O

A B S T R A C T

Keywords: Comets: General Cometary outburst Comets individual: 29P/SchwassmannWachmann 67P/Churyumov-Gerasimenko

The outburst of comet brightness is spectacular and well-known manifestation of the physical activity of these relatively small cosmic bodies. The outbursts of brightness are observed for periodic comets, as well as for the comets moving along parabolic trajectories. The most known comet representing such kind of activity is the comet 29P/Schwassmann-Wachmann. There is a number of hypotheses put forward so far in scientific literature, many of them are trying to establish ultimate cause of the phenomenon. However, careful analysis leads to the conclusion that there may be several different causes of cometary outbursts or even combination of them, which in favourable conditions may initiate the outbursts of brightness. The main goal of the work is not the searching for the new mechanisms of brightness outbursts, but rather, based on existing models to make new and more accurate calculations of the comet’s brightness jumps during their outbursts at large heliocentric distances. Numerical simulations of changes of brightness for three hypothetical comets orbiting in the solar system were carried out. The obtained results are consistent with observations of real comet outbursts.

1. Introduction In the Solar System there is a characteristic group of small celestial bodies called comets, which are orbiting around the Sun. The main component of a comet is its nucleus. It is an irregularly shaped body similar to an ellipsoid, cigar or peanut. The nucleus is the main carrier of the cometary mass. The main components of the nucleus are water ice, solidified carbon monoxide and carbon dioxide as well as dust, rock crumbs and admixture of organic materials. The shape of the comet’s orbits may be similar to an ellipse, parabola or hyperbola. When the comet approaches the Sun coming from the depths of cosmos, the comet’s appearance changes significantly. Far from the Sun, a comet does not demonstrate any physical activity, while gradually approaching the Sun, the comet nucleus begins to sublimate under the influence of solar radiation. In this way a comet’s head is formed that contains the nucleus; at the same time, ionic and dust tails also arise. The speed of sublimation increases until comet reaches its perihelion. Once passing perihelion, the comet shines most brightly and has the most spectacular form. Later, the sublimation rate decreases and away from the Sun it becomes a dark, inactive and finally undetectable object. During the next comet return to the Sun, the cycle of its activity described above can repeat. However, in addition to the described activity resulting from the heating of the

comet’s nucleus by solar radiation, sometimes a completely different kind of comet activity is observed. It is the outburst of comet brightness. It is worth mentioning that the majority of cometary outbursts takes place at the distances less than 5au from the Sun. This means that at the heliocentric distances where the sublimation rate of the dominant component of comets, that is water ice, is practically negligible, these cosmic bodies don’t manifest outburst activity. But there are some exceptions, eg. Comet Humason (1962 VIII) was flashed in 6au from the Sun, the famous comet 1P/Halley flashed at 14.3au heliocentric distance. The most famous example of comet outburst at large heliocentric distances, that is at distances where the sublimation of water ice is negligible, is comet 29P/Schwassmann-Wachmann (SW1, hereafter). The astronomers were interested in outbursts of comet brightness since the twenties of the last century. In November 1927, two German astronomers, Arnold Schwassmann and Arno Wachmann, discovered previously unknown comet, probably during one of its outbursts. The comet was just named after its discoverers as SW1-comet. The comet SW1 moves along the quasi-circular orbit going near the ecliptic plane between orbits of Jupiter and Saturn with the orbital period P
16 years. The semi-axis of its orbit is equal to a
6 au. A big surprise was that despite of relatively large distance from the sun, at which sublimation of water ice is practically negligible, this comet shows outburst activity even several dozens

* Corresponding author. E-mail address: [email protected] (M. Wesołowski). https://doi.org/10.1016/j.pss.2020.104867 Received 15 August 2019; Received in revised form 19 December 2019; Accepted 5 February 2020 Available online 18 February 2020 0032-0633/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/).

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brightness of comet 67P at the distances less than 5 au (Tubiana et al., 2015), but the detailed inquiry of changes of comet brightness at the heliocentric distances less than 5 au is beyond the scope of this article. During 2 years while the Rosetta spacecraft moved around the comet 67P, it was a unique opportunity to track the behaviour of this object when it was the most active. At that time, several dozens of outbursts of this comet were observed (Vincent et al., 2016b; Gicquel et al., 2017). Observations have shown that comet outbursts were caused by the emission of matter from its nucleus in the form of jets similar to cometary geysers. Three morphological types of plumes or emitted jets were distinguished during the outbursts (for more details see: Vincent et al., 2016b):

times per year. The cometary outburst is well-known phenomenon intensively discussed in scientific literature (for example, Cabot et al., 1996; Enzian et al., 1997; Groussin et al., 2004; Meech et al., 2009; Trigo-Rodriguez et al., 2008, 2010; Ipatov and A’Hearn, 2011; Zubko et al., 2011; Ipatov, 2012; Gronkowski and Wesołowski, 2015). The purpose of this work is not to search for the new mechanisms of cometary outbursts; instead, we deal only with a mathematical description of changes in the comet brightness during its outburst at a large heliocentric distances. The approach is based on numerical simulations of light scattering on cometary grains and make use of Grenfell and Warren (1999) method in its relation to the Lorenz-Mie theory. The most important ingredients of our approach are briefly discussed in Section 4. The Mathematica computation package was used intensively throughout; it is one of the modern packages used for symbolic and numerical calculations. Another advantage of this toolkit is that it enables to present the obtained results relatively easy and in transparent form by means of tables and graphs. Mathematica enables to calculate various special functions, for example such as Bessel, Neumann, Hankel and Ricatti-Bessel. This allows user to start programming immediately, without necessity to address to the source code of the additional library programs. It facilitates the work significantly and thus saves time when performing numerical simulations. The paper is organized as follows: the first part contains brief discussion of the cometary brightness outbursts, the second part presents the application of the Lorenz-Mie theory and Discrete Dipole Approximation theory to the description of light scattering by cometary dust-ice particles, while in the last part the results of numerical simulations of changes in luminosity of a comet during its outburst are presented.

(i) Type A: a significantly collimated stream of cometary matter. (ii) Type B: jets in the form of wide plumes or wide dust fans. (iii) Type C: complex events, often combining narrow and wide jets. Of course, the question how far the conclusions drawn from the observation of single comet can be generalized to other comets remains as yet open. Nevertheless, the next hypothesis can be safely put forward: the outburst of comet brightness might have different morphology and probably different sources. It is worth recalling that the greatest outburst of comet glow observed in the entire history of cometary research took place in October 24, 2007, when the comet 17P/Holmes increased its visible glow by about  14:5m . The comet was about 2.4 au from the Sun and increased the total visible brightness from 17m to about 2:5m in 2 days (see, e.g. Bockele’e-Morvan et al., 2008, Montalto et al., 2008; Moreno et al., 2008). So far, many authors tried to explain the outbursts of comets brightness by means of a single mechanism. In that respect one can distinguish three main approaches to the explanation of this phenomenon. The hypotheses of first type try to explain outburst of brightness by means of internal sources accumulated in the comet nucleus, among others, the transformation of water amorphous ice into a crystalline form (Smoluchowski, 1981; Prialnik and Bar-Nun, 1992; Cabot et al., 1996; Kossacki and Szutowicz, 2011), or the polymerization of HCN (Rettig et al., 1992). The second group suggests the collisions of comets with small interplanetary bodies as the reason for the outburst of brightness (Fernandez, 1990; Matese and Whitman, 1994; Gronkowski and Wesołowski, 2012), or the influence of solar wind on the comets (Intriligator and Dryer, 1991). At last, the third group relates this phenomenon to the internal structure and mechanical stresses within the comet nucleus, including hypothesis of destruction of comet’s nucleus subsurface layers (Gronkowski and Wesołowski, 2015), or the hypothesis of cometary ices melting (Miles, 2016). None of these hypotheses explains all the nuances and specific features of the phenomenon. Highly likely, outbursts of cometary brightness are caused by different, or several mechanisms occurring at the same time. However, without going into details of alleged primary cause of the brightness outbursts, the analysis of the rich observational material leads to the conclusion that the majority of outbursts are associated with the increase in the rate of cometary ices sublimation and ejection of the external surface layer from the part of the comet’s nucleus in the form of a cloud of grains and rock debris. In this way the amount of grains reflecting sunlight in the atmosphere of comet increases. At the same time, the exposure of the comet’s nucleus deeper layers rich in volatile substances to the Sun radiation, in its turn increases the rate of cometary matter sublimation. All this ultimately leads to a significant increase in the glow of comet, and as a consequence, one can finally observe the outburst of the cometary brightness. It is worth mentioning certain observations made during recent Rosetta’s space mission to comet 67P. According to them, comet outbursts may be caused by cometary avalanches and jets produced by cometary geysers (Grün et al., 2016; Leliwa-Kopystyn’ski, 2018; Lin et al., 2017; Pajola et al., 2016; Vincent et al., 2016b).

1.1. Outbursts of cometary brightness - the most prevalent contemporary hypotheses The outburst of the cometary brightness is sudden change in its luminosity by about -2m to -5m on average, lasting a few or a dozen or so days. It should be clearly emphasized that this sudden change of brightness should not be confused with a physical explosion of comet, similar to that of a mine or a bomb. As it was mentioned above, the most well-known comet exhibiting outburst activity is the comet SW1. Its outbursts of brightness have been observed since the twenties of the last century. The scenario of comet outburst is discussed briefly below (see also Hughes, 1991 and literature therein). Usually, in a quiet inactive phase, the comet looks like a diffuse, fuzzy oval cloud with barely noticeable central dense point. Suddenly, within a few hours or days, a ‘star-like’ bright nucleus is formed in its head expanding at a speed of 100 ms 1 - 400 ms1. Later, it transforms into a ‘planetary disk’ (see Hughes, 1990) and then into a specific cometary halo with the surface luminosity decreasing outside. During the outburst the cometary coma increases dramatically and reaches its maximum when the comet cloud has a diameter of several hundred thousand kilometers. During the ‘planetary disc stage’, the spectrum of comet is continuous and similar to the solar radiation spectrum. Therefore, one can conclude that comet glow is nothing else but solar radiation reflected and scattered by the ice and dust particles come from a comet. After some time, a few to several dozens of days, the comet glow decreases and its appearance returns to its original state. There are no changes in the orbital motion of the comet caused by the outbursts of its glow. In general, the majority of outbursts of other comets goes in a similar manner. At this point, it is worth recalling that July 4, 2005, at 5:52 UTC, Comet Tempel 1 collided with a shock probe launched by NASA Deep Impact Space Probe. The collision caused the crater on the surface of the comet’s nucleus and ejected large amount of dust into its atmosphere. Eventually, a significant outburst of brightness was observed for this comet (A’Hearn et al., 2005). It is worth mentioning however, that the results of ESA’s Rosetta’s mission to comet 67P/Churyumov-Gerasimenko (hereafter 67P) have disclosed the possibility for different morphology of cometary outbursts. Note that Rosetta space mission recorded several changes in

2. Cometary dust The source of the comet’s glow is mainly the reflection of the sunlight 2

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by dust grains in the coma. Studying the physical properties of cometary grains has a long history. Water ice in crystalline or amorphous form is the main component of cometary nuclei. On one hand therefore, it is assumed that ice grains with an admixture of dark material have to be present in the atmospheres of comets in significant amounts. This is confirmed both by the observations (Kawakita et al., 2004), as well as theoretical studies (Beer et al., 2006; Beer et al., 2008). On the other hand, both laboratory tests as well as the results of space missions, especially Rosetta mission to the comet 67P, have shown that cometary grains may have a complex, fragile structure of agglomerates composed of sub-micron monomers. The analysis of measurements carried out by the test instruments placed on the board of the Rosetta probe allows to distinguish the following families of cometary grains: BA - (Ballistic Aggregates), BAM1 - (Ballistic Aggregates Type 1), BAM2 - (Ballistic Aggregates Type 2), and (BCCA) - Ballistic Cluster-Cluster Aggregates (see: Skorov et al., 2016a and literature therein). In case of agglomerates composed of monomers containing primarily silicon, their density is estimated to be equal to ρag ¼ 2400kgm3 . For other types of admixtures, the density of such monomers can reach value of the order of ρag ¼

bn ¼

n  1 d sinðxÞ ; x dx x n 1 d cosðxÞ : x dx x

(7)

(8)

In order to calculate the effective scattering cross-section of the cometary particle given by Eq. (2) with sufficient accuracy, one should truncate the infinite series and include in the summation a sufficiently large number of terms. This number which is denoted as Lmax , is determined by the criterion introduced by (Eq. 9, Wiscombe, 1980): 1

Lmax ¼ x þ 4  x3 þ 1:

(9)

In this formula x stands for the particle size parameter defined by Eq. (1). This relationship is valid for any refractive index of cometary particles, because the convergence of the Mie series is determined entirely by the Bessel functions of x-argument. After carrying out these calculations, one finally gets the value of scattering factor or efficiency of scattering in the form given by Eq. (10): Q¼

σ scat : π a2eff

(10)

The calculations of effective scattering cross-section σ scat become more complicated for the non-spherical particles, because initially Lorenz-Mie theory was elaborated for spherical particles. However, when it comes to cometary matter, many authors who dealt with it, considered irregular shapes of dust particles and various levels of the porosity of them. In their papers numerical calculations were done using Discrete Dipole Approximation or T-Matrix theory (see e.g.: Purcell and Pennypacker, 1973; Mishchenko et al., 1996; Zubko et al., 2011). These two methods are very accurate but complicated and very time-consuming. In this work we propose another fast and relatively simple method for calculating the increase of the comet’s glow during its outburst. In order to determine the effective scattering cross-section for spherical of dirty water ice-grains, we use method based on the Mie-Lorentz theory. However, to determine the analogous characteristics for aggregates containing silicate monomers, we use the results of relevant calculations based on the Discrete Dipole Approximation theory and presented in the work by Zubko (2013).

(1)

4. The amplitude of increase in the brightness of a comet during its outburst

(2)

To calculate the changes of comet brightness during its outburst, one should accept some simplifying assumptions. For example, the comet nucleus is supposed to be a sphere of radius RN . We assume also the two-layer model of comet nuclei which was proposed by Skorov and Blum (2012). According to this model, the comet’s nucleus consists of the outer layer and the inner one. The first layer contains monomer agglomerates, while the second one is composed of both, monomer agglomerates and water ice grains. As it is stated in section 2, we assume that the direct cause of the cometary outbursts is the increase in

Scattering coefficients an and bn appearing in Eq. (2) can be written mψ n ðmxÞψ ’n ðxÞ  ψ n ðxÞψ ’n ðmxÞ ; mψ n ðmxÞξ’n ðxÞ  ξn ðxÞψ ’n ðmxÞ

(6)



as: an ¼

ξn ðxÞ ¼ x ðjn ðxÞ þ i  yn ðxÞÞ;

yn ðxÞ ¼  ðxÞn

where aeff is the effective radius of the particle, λ is the wavelength of sunlight incident on a given particle. The effective cross-section of scattering σ scat for spherical cometary grains is of the form (Bohren and Huffman, 1983): ∞    2π X ð2n þ 1Þ  an j2 þ bn j2 : k 2 n¼1

(5)

jn ðxÞ ¼ ðxÞn

As it is stated above, we consider two types of cometary grains, the first of them is of crumbs of water ice, while the second is of agglomerates of small sub-micron monomers. We assume for the sake of simplicity that the cometary grains of first type have a spherical shape. In order to calculate the efficiency of solar light scattering by these spherical grains of water ice, one can use the Lorenz-Mie theory. The scattering of sunlight by spherical particles were already intensively discussed in scientific literature (see e.g.: van de Hulst, 1981; Bohren and Huffman, 1983; Kitzmann and Heng, 2018). Therefore, we restrict ourselves only to quoting the most important formulae allowing to determine the scattering factor Q. To do this, we need to define at first some important parameter (Eq. 1), which we call the particle size parameter x:

σ scat ¼

ψ n ðxÞ ¼ x  jn ðxÞ;

where jn ðxÞ, yn ðxÞ are the spherical Bessel function (Eq. 7, ), which can be written as:

3. Light scattering by cometary dusty ice particles: application of Lorenz-Mie theory

2π aeff ; λ

(4)

Note that in Eqs. (3) and (4) m represents the relative refractive index of ice or dust grains, depending on the situation. In Eqs. (3) and (4) the following notations are used (see , ):

3500kgm3 (Laor and Draine, 1993). That is why in our work we accepted the value of monomers density equal to ρag ¼ 3000kgm3 . It should be expected that, on the one hand, a gentle sublimation from the comet could eject grains into a coma both in the form of ice crumbs and in the form of agglomerates. On the other hand, during the outburst of the comet’s brightness, some part of the nucleus surface layer can be ejected, probably in the form of debris of ice grains. Therefore, in the numerical calculations related to the determination of the amplitude of the comet’s brightness stroke during its outburst, we consider both the ice grains and the monomer agglomerates. We assume the sizes of monomers are of the order of 0.1 μm and the sizes of aggregates vary from microns to milimeters (Skorov et al., 2016a and literature therein).

x¼

ψ n ðmxÞψ ’n ðxÞ  mψ n ðxÞψ ’n ðmxÞ : ψ n ðmxÞξ’n ðxÞ  mξn ðxÞψ ’n ðmxÞ

(3)

3

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into account that in general, the comet at outburst stage can eject some part of the outer layer of its nucleus. So, we proceed distinguishing two cases:

the rate of cometary ices sublimation, accompanied (not necessarily, though) by the ejection of external surface layer from the part of nucleus. We also assume that ejected cometary material makes a cloud of spherical ice-dust grains as well as agglomerates of silicate monomers. Then, we can directly apply the Lorentz-Mie theory to the first type of particles, while for the particles of second type, - the results obtained by Zubko (2013) by means of Discrete Dipole Approximation. Then, to calculate changes in the brightness of comet Δm, one can make use of Pogson’s law: Δm ¼  2:5log

  Lðt2 Þ ; Lðt1 Þ

case (a) - the comet nucleus ejects a part of its external layer; this leads to uncovering surface layers of ice-rich CO and increasing the percentage of the surface of the comet nucleus with the maximum sublimation rate of this substance, case (b) - the comet nucleus does not eject part of its external layer, and the increase of its brightness is due to increasing the percentage of its surface revealing the massive rate of CO sublimation.

(11)

In case (a) symbol fCO ðt1 Þ stands for the percentage of comet nucleus active sublimation surface in a calm sublimation phase before the outburst. Parameter fCO ðt2 Þ means the percentage of comet nucleus active sublimation surface at a time t2 during the phase of strong sublimation. In this case one has to take into account the total scattering cross-section originating from the cometary grains raised to the comet atmosphere by rapid sublimation, as well as the total scattering cross-section of the cloud of cometary debris originating from the destroyed nucleus layer. Two parameters defined above, fCO ðt1 Þ and fCO ðt2 Þ, are obeyed the following relations (Eq. 12) and (Eq. 13):

Here Lðt1 Þ and Lðt2 Þ are the cometary luminosity observed from Earth, before and during the outburst, respectively. Note that in general, the contribution of the nucleus to the integrated brightness of comet is very small and can be neglected (Tancredi et al., 2000). Therefore, at a time t1 only total scattering cross-section of the icy and dust cometary grains raised to the atmosphere of comet by calm sublimation can be taken into account. Observe also that as it was discovered by Rosetta’s mission to the comet 67P, basically 100% of the illuminated surface of its nucleus was sublimationally active during the absence of outburst activity, but the rate of sublimation was small. There were recorded only slight changes in activity resulting from comet morphology. During the comet’s explosive activity, full sublimation activity is manifested by larger areas of nucleus than that of the calm phase, but their surface in relation to the total surface of the nucleus is still small. These areas become for some time more active than normally for various reasons, such as e.g. comet avalanches or sublimation, driven by more volatile substances than water ice. Sometimes these areas are called an active surfaces and are expressed as a percentage of the surface of comet nucleus and denoted as fH2 O . This is a means to measure and compare the actual activity of the comet with a virtual nucleus consisted of water ice only. Therefore, if a real comet has an active surface percentage of say, fH2 O ¼ 5%, then a virtual comet consisting of water ice only, having the same nucleus size and being at the same heliocentric distance, would sublimate 20 times more water than the real comet. This is because there is no water ice on the surface of the comet nucleus, but the sublimation of water occurs in the subsurface areas. The last one reduces the activity of comet nucleus, since most of the incident electromagnetic radiation coming from the Sun is absorbed by the dust on the surface. In other words, it can be assumed that fH2 O ¼ 5% of each square meter of that part of the surface of the comet’s nucleus, which is illuminated by the Sun, exhibits full sublimation activity of the water ice. In this paper, however, we deal with the comets which are at larger distances from the Sun than that of comet 67P. An average distance separating comet 67P from the Sun is a ¼ 3:46 au, and its sublimation activity is controlled by water ice. Therefore, not all conclusions drawn from the observations of comet 67P are valid for the hypothetical comets X1, X2, X3 considered in our paper. It is because these comets are at the distances 5, 10 and 15 au from the Sun, where the sublimation of water ice is negligible and a significant contribution to comets activity can be brought only by more volatile substances. For simplicity, the sublimation of our hypothetical comets at these heliocentric distances is assumed to be controlled only by CO ice. Therefore, in the absence of outburst, the total surface of nucleus illuminated by the Sun is supposed to be sublimationally active, but with the small rate fCO . Contrary, in the outburst phase the rate of sublimation from some part of nucleus increases significantly. In other words it means, just like in case of Rosetta mission, that if the percentage of the active real comet nucleus surface is say, fCO ¼ 1%, it is equivalent to that of a virtual comet consisting only of monoxide ice, of the same nucleus size as our hypothetical comets X1, or X2, or X3 and at the same heliocentric distance would sublimate during the outburst 100 times more monoxide ice compared to them. Of course, in the outburst phase the value of parameter fCO increases accordingly. Calculating the changes in comet brightness during its outburst, one should also take

fCO ðt2 Þ ¼ fCO ðt1 Þ þ ΔðfCO Þ; fCO ðti Þ ¼

(12)

Si ; 4π R2N

(13)

Sej : 4π R2N

(14)

ΔðfCO Þ ¼

Here ΔðfCO Þ (Eq. 14) means the increase in the percentage of the nucleus surface demonstrating sublimation due to the ejection of a certain damaged layer, Si is the surface of the part of nucleus exhibiting sublimation at the time ti , Sej means the size of surface of the comet’s nucleus ejected layer. The mass of the ejected layer Mej meets the following condition (see Eq. 15): Mej ¼ Sej  h  ρN ;

(15)

where h denotes the thickness of ejected layer, while ρN is cometary nucleus density. It should be emphasized that in carrying out calculations for the case (b), the same formulae as in case (a) are used, but also the following conditions are taken into account: Mej ¼ 0, Nej ¼ 0 and Sej ¼ 0. These conditions correspond to the absence of layer ejection from the comet nucleus. It also should be noted that the scattering efficiency Q is an integral characteristic accounts for the light scattering into an entire sphere circumscribed around the target particle. The comet however, is observed from the Earth along certain line characterized by some angle, called phase angle and denoted as α, i.e. the angle between two lines, the first one connecting the Earth and a comet, while another one connects the Sun and a comet (see: Appendix Fig.A1). The outbursts of comet brightness dealt with in the paper, are considered to occur at the heliocentric distances ranging from 5 au to 15 au and hence, they can be observed only at small phase angle α, smaller than 12∘ (see Appendix Fig.A2). The point is that an efficiency of scattering Q is governed mainly by forward scattering and as a result, in case of comets that we consider, the scattering angle θ approaches 180∘ . Hence, the comet glittering as it is seen from the Earth is the product of scattering cross-section of cometary particles and the phase function pðθÞ for a given scattering angle θ. Therefore, the sum of phase angle α and the scattering angle θ for the comets in question is equal to: α þ θ ¼ 180∘ . Phase function pðθÞ (Eq. 16) describes the probability of a photon to be scattered into an angle θ and is given by the following formula (Henyey and Greenstein, 1941): 4

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Planetary and Space Science 184 (2020) 104867

pðθÞ ¼

ð1  g2 Þ 4π ð1 þ g2  2gcosðθÞÞ3=2

;

(16)

where the parameter g describes the asymmetry of the phase function. The asymmetry parameter in Eq. (17) can be defined as (van de Hulst, 1981):

g¼

P 2 ∞ n¼1



   2nþ1  an anþ1 þ bn bnþ1 þ nðnþ1Þ Re an bn ;  2  2  P∞   n¼1 ð2n þ 1Þ an j þ bn j 

nðnþ2Þ Re nþ1

(17)

where the asterisk (superscript) denotes complex conjugate. The values of parameter g are calculated by means of Eq. (17) and they are presented, among others, in Table 1. As a result, Eq. (11) which expresses the increase in comet brightness during its outburst, now takes the form: Δm
 2:5log

  psi ðθÞ CðsiÞi þ CðsiÞej þ pice ðθÞCðiceÞej pice ðθÞCðsiÞi

:

(18)

Here psi ðθÞ and pice ðθÞ stand for the phase functions of cometary silicate grains and cometary ice grains, CðsiÞi are scattering cross-section of the silicate grains, respectively. Here we take into account only those grains that have been lifted from the surface of the comet nucleus to its head by the molecules of sublimating comet ices, CðiceÞej and CðsiÞej denote the scattering cross-section of ice grains and silicate grains coming from the destroyed and ejected layer of nucleus. In Eq. (18) the sub-subscript i ¼ 1 corresponds to a quiet, non-explosive phase of the comet, while i ¼ 2 corresponds to its outburst phase. Scattering cross-sections appearing in Eq. (18) are described by the following formulae: CðsiÞi ¼ π Ni

Z

rmax

Qsi r 2 gðrÞ dr;

(19)

rmin

CðsiÞej ¼ π bNej Fig. 1. The increase in comet X1 brightness Δm vs parameter fCO (t1); outburst is supposed to be at a heliocentric distance RH ¼ 5au. It was assumed at the calculations that the phase angle is equal to θ ¼ 11.54∘, and the values of phase functions are equal to: psi(θ) ¼ 1.20 and pice(θ) ¼ 0.71. It was also assumed that the percentage content of silicate grains and ice are equal to: i) b ¼ 0.9; c ¼ 0.1; ii) b ¼ 0.5; c ¼ 0.5; iii) b ¼ 0.1; c ¼ 0.9 . The calculations were carried out in the frame of model (a).

Z

rmax

Qsi r2 gðrÞ dr;

(20)

rmin

Table 1 Values of the physical cometary parameters for objects X1, X2, X3 used in numerical simulations. They are the same as in the works by Fanale and Salvail (1990); Schmitt (1991); Richardson et al. (2007); Reach et al. (2010); Kopp and Lean (2011); Kossacki and Szutowicz (2013); Brandt (2014); Encyclopedia of the Solar System - third edition (2014); Gronkowski and Wesołowski (2015), (2017); Wesołowski and Gronkowski (2018a), (2018b) and the literature therein.

Fig. 2. The same as in Fig.(1 i)), but parameter Δm is presented as the function of two variables Mej and ρag , while parameter f(t1)CO is chosen to be equal 0.1%. The calculations were carried out in the frame of model (b).

5

Parameter

Value(s)

Albedo () Heliocentric distance (au) Density of the comet nucleus (kg  m3 ) Density of the ice grains (kg  m3 ) Density of the silicate agglomerate (kg  m3 ) Radius of comet nucleus (m) Crystalline ice thermal conductivity (Wm1 K1 ) Dust conductivity (Wm1 K1 ) Initial temperature (K) Hertz factor () Porosity () Dust - gas mass ratio () Emissivity () Depth of cavity location (m) Constant ACO for carbon monoxide (Pa) Constant BCO for carbon monoxide (K) Latent heat of carbon monoxide sublimation (Jkg1) Radius of cometary head during the outburst (m) Radius of the cometary head during gentle sublimation (m) Minimum radius of cometary grains (m) Maximum radius of cometary grains (m) Solar constant (for d ¼ 1au) (W m2) Mean value of solar radiation wavelength (m) Asymmetry coefficient for water ice () Asymmetry coefficient for silicates materials ()

AN ¼ 0:04 RH ¼ 5; 10; 15 ρN ¼ 400 ρice ¼ 920 ρsi ¼ 3000 RN ¼ 1000 λcore (T) ¼ 567/T λdust (T)
2 T ¼ 51 h(ψ ) ¼ 0.01 ψ ¼ 0.7 κ¼1 ε ¼ 0.9 Δx ¼ 10 ACO ¼ 1.6624  109 BCO ¼ 764.16 H(T)CO ¼ 2.93  105 R(t2)h ¼ 109 R(t1)h ¼ 108 amin ¼ 106 amax ¼ 103 S ¼ 1360.80.5 λ ¼ 0.5015  106 gice ¼ 0.963 gsi ¼ 0.932

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CðiceÞej ¼ π cNej

Z

Planetary and Space Science 184 (2020) 104867

amax

Qice a2 f ðaÞ da;

subsurfaces layers of the comet nucleus. These equations (Eq. 28, Eq. 29) were formulated in the paper by Skorov Yu et al., 2017; they express energy conservation on the surface and the lower boundary of the external dust layer:

(21)

amin

where b is the fractional content (percentage) of ice grains in the total amount of grains contained in the comet’s head, c stands for the percentage of silicate grains in the total amount of grains in the comet’s head (of course, b þ c ¼ 1), Ni and Nej are the total number of cometary grains raised to the head of the comet by sublimation or destruction of the nucleus layer, respectively. In Eqs.(19)-(21) Qsi and Qice stand for the scattering efficiencies of silicates grains and water ice grains, respectively. They were calculated in the frame of Mie’s theory (see e.g.: van de Hulst, 1981; Bohren and Huffman, 1983) with taking into account the remarks made by Grenfell and Warren (1999) and Frisvad et al. (2007). Above all, in our calculations the wavelength λ is assumed to be determined by the Wien law applied to the Sun radiation. Note that in formulae (19)-(21), gðrÞ and f ðaÞ denote size distribution functions of cometary grains and a stands for their radius. The numbers of cometary grains NðiceÞi , NðsiÞi , NðiceÞej , NðsiÞej , are related to their total masses Mi , Mej and the total number of grains Ni , Nej through the following formulae (Eq. 22, Eq. 23): 4 Mi ¼ π Ni ρag 3

Z

ð1  AN ÞS dT ¼ εσ T 4 þ λdust ðTÞ ; dx R2H λdust ðTÞ

(22)

rmin

and 4 Mej ¼ π Nej ρgr 3

Z

amax

a3 f ðaÞ da;

(23)

Δm ¼  2:5log

amin

where ρag and ρgr stands for the density of agglomerates and cometary grains. At the same time, the mass Mi (Eq. 24) depends on the rate of sublimation from the comet surface and denoted by ZðTÞ according to the following expression (Gronkowski and Smela, 1998): Mi ¼ 4π R2N fCO ðti ÞκZðTÞmg

Rh ðti Þ : vth

Z X¼

Z Y¼

Qsi r 2 gðrÞ dr;

(31)

amax

Qice a2 f ðaÞ da:

(32)

amin

In Eq. (30), the parameters N1 , N2 and Nej are given the following relationships (see Eq. 33, 34 and 35):

(25)

N1 ¼

3fCO ðt1 ÞκR2N ZðTÞRh ðt1 Þmg R rmax ; vg ρag rmin r 3 gðrÞ dr

(33)

N2 ¼

3fCO ðt2 ÞκR2N ZðTÞRh ðt2 Þmg R rmax ; vg ρag rmin r 3 gðrÞ dr

(34)

3M R amax ej : a3 f ðaÞ da amin

(35)

where the pressure of cometary gases pðTÞ is the function of current temperature T given by (Eq. 26):

and

pðTÞ ¼ ACO  expð  BCO = TÞ;

Nej ¼

(26)

the values of coefficients ACO and BCO are given in Table 1 for carbon monoxide ices. We assume the mean thermal velocity of sublimating the cometary ices obeys the conditions:

πμ

rmax

and

(24)

mation rate of CO (expressed in kgm s units), and vth is the mean thermal velocity of gas molecules. The meanings of other symbols were already defined above. The sublimation rate of cometary water ice is given by the well-known Hertz-Knudsen formula:

sffiffiffiffiffiffiffiffiffiffiffi 8Rg T vth ðTÞ ¼ :

(30)

rmin

2 1

pðTÞ ; 0:5π vth

  psi ðθ2 Þ π N2 þ π bNðejÞsi X þ pice ðθ2 Þπ cNðejÞice Y ; psi ðθ1 Þπ N1 X

where symbols X and Y are given by Eq. (31), Eq. (32).

In this formula κ stands for the dust-gas mass ratio (κ ¼ 1), Rh ðti Þ is the radius of the cometary head corresponding to the phase denoted as i (see Table 1), mg the mass of cometary gas CO - molecule, ZðTÞ is the subli-

ZðTÞ ¼

(29)

Here we accept the following notation: S is the solar constant, RH means the heliocentric distance of a comet (expressed in au), AN and ε are the albedo and the infrared emissivity of the nucleus of a cometary nucleus, respectively, σ is the Stefan-Boltzmann constant, T denotes the nucleus surface temperature, and HðTÞ is the latent heat of CO sublimation. The parameters λdust (T) and λcore (T) mean the effective heat conductivity of the dust and core layer, respectively. In our numerical solution of these equations we assume, in accordance with the papers by: (Skorov et al., 2011; Skorov et al. 2016b; Skorov Yu et al., 2017 and see also the literature cited therein), that the comet’s nucleus is covered with a thin, dry layer of dust-agglomerates which are free of water ice that covers the interior layer containing dust and ice grains. Note that the permittivity of the ice-free porous dust layer, ΨðRag ; LÞ, is a function of the agglomerate radius Rag and the thickness of its layer L. Using Eqs.((11)–(27)), one can express the changes in comet brightness Δm as:

rmax

r3 gðrÞ dr;

  dT dT ¼ Ψ RAgg ; L ZðTÞHðTÞ þ λcore ðTÞ : dx dx

(28)

4πρgr

Now, one can calculate numerically by means of (30) the increase in comet brightness during its outburst. The calculations were carried out on the assumption that the ice grains are characterized by some distribution function f ðaÞ. The first distribution function which was used (see Newburn and Spinrad, 1985, 1989; Grün and Jessberger, 1990) is presented by following formula (Eq. 36):

(27)

 a0 M a0 N f ðaÞ ¼ k 1  ; a a

Here Rg stands for the universal gas constant, while μ means the molar mass of cometary ice. Notice, that in last formulae (25) and (27) two parameters, namely the rate of sublimation from the comet nucleus ZðTÞ and the mean thermal velocity vth , depend on the temperature of the comet nucleus surface T. The last one can be calculated by means of solving the equations of energy equilibrium at the surface and

(36)

where k is the normalization constant and a0 ¼ 1:59  106 m. The exponent M depends the heliocentric distance of a comet RH (expressed in au) according to the next formula (Eq. 37) which is given by de Freitas Pacheco et al., 1988:

6

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logðMÞ ¼ 1:13 þ 0:62  logðdÞ;

Planetary and Space Science 184 (2020) 104867

well as in the outburst phase fCO ðt1 Þ, were also considered as free parameters. Inspecting Figs. (1)-(8), it is easy to see that the obtained amplitudes of brightness jumps for the hypothetical comets X1, X2, X3 are comparable to the values observed for real comets exhibiting outburst activity at large distances from the Sun. Let us remind that according to the longterm observation of the SW1 comet carried out in the last century, increase of its outburst of brightness was of the order of -2m to -5m (Hughes, 1990, 1991). On the other hand, the observational campaign of this comet carried out in the years 2002–2010 recorded several dozens of outbursts, on average 7.4 times a year. The observed brightness changes were of the order of -1m to -4m (Trigo-Rodriguez et al., 2008, 2010). Comparison of the outburst of comet SW1 with the hypothetical comet (X/PC) are shown in Fig.(9).

(37)

The exponent N defines the slope of the size distribution at large a and it is of about
4. The second distribution function which was used at our calculations (Eq. 38) is Gaussian normal function:   ða  a0 Þ2 f ðaÞ ¼ A  exp 2σ 2

(38)

Here A stands for the normalization constant, a0 and σ are the average radius of cometary grains and dispersion, respectively. Our numerical tests have shown that two values of change in the comet’s brightness Δm obtained for these two different distribution functions with the same average values of radius a0 and dispersion σ are negligibly small. In the case of monomer agglomerates however, we assume after Lin et al. (2017) the following distribution function gðrÞ ¼ Cr 3:7 Þ, where r and C are the radius of the effective cross-section of fluffy aggregate and the normalization constant, respectively. Note that the average radius of agglomerates rav ¼ 14:2μm (see Appendix). It is worth emphasizing also, that the direct making use of Eq. (31) for numerical calculations is very time-consuming. One can avoid this by means of following approximation (Eq. 39): X
Qsi ðrav Þr 2av ;

6. Conclusions 6.1. General summary The paper presents a new method of calculating the brightness jump of comets during their outbursts of glow. The consideration is restricted only to the comets whose heliocentric distances are considerable, i.e. to those where the sublimation activity of the comet is not controlled by its most plentiful component, i.e. water ice, but by the CO ice which is the

(39)

Observe that the size parameter is equal to: x ¼ 177:732 () for rav ¼ 14:2μm, if one assumes the average wavelength of sunlight to be 0.502 μm. This justifies the approximation used, because for such large values of the size parameter x, the function Qsca ðxÞ becomes very slow monotonically decreasing. Therefore, taking the advantage of Zubko (2013) paper, we find that Qsca ðxÞ ¼ Qsi ðrav Þ
1. 5. Results As it was mentioned above, we assume the direct cause of comet outburst of brightness is the ejection of a part of the surface layer of nucleus into a void in the form of a cloud of cometary grains and dust. Depending on the distance separating comet from the Sun, the destruction of the surface layer can be caused by the pressure brought about by vapours, such as water, carbon dioxide or carbon monoxide which are in the deeper layers of the nucleus. Therefore we assume, that cometary grains may contain: a) water ice, b) silicate materials. This assumption is made because at the earlier stages of the comet’s passage through the perihelion, more volatile substances such as CO and CO2 did not survive in the surface layers of the nucleus. The calculations were carried out for three hypothetical comets X1, X2, X3, at three different distances separating them from the Sun. The physical characteristics of these hypothetical comets are presented in Table 1. Three heliocentric distances at which possible outbursts could occur, were chosen to be: RH ¼ 5au (outburst of X1), RH ¼ 10au (outburst of X2) and RH ¼ 15au (outburst of X3). Therefore, it is assumed the sublimation activity of comets X1, X2, X3 during their outbursts to be controlled by CO. Of course, sublimation of CO can occur out of the deeper layers of the nucleus exposed during the comet outburst. The calculations were made for two chemical compounds contained in the comet nucleus: water ice (the refractive index was taken to be nice ¼ 1.31 þ 0.05i), and silicates materials (the refractive index was supposed to be nsi ¼ 1.60 þ 0.05i). It should be noted here that ‘i’ is an imaginary unit that occurs in the second component of the sum defining the refractive index of cometary dust. The imaginary part of the refractive index is associated with its doping. It should be noted that the numerical values of the radii of the comet’s head during the quiet sublimation and during the outbursts of brightness, R(t1)h and R(t2)h was treated as free parameters. Specific numerical values were chosen in accordance with the observational data presented in: Trigo-Rodriguez et al., (2008), 2010; Hughes (1990), 1991; Encyclopedia of the Solar System - third edition (2014). The percentage of the sublimation active surface of the nucleus fCO ðt1 Þ in the calm phase, as

Fig. 3. The same as in Fig. 1, but the calculations are performed for the comet X2. Phase angle is chosen to be equal to θ ¼ 5.74∘, and the values of phase functions are: psi(θ) ¼ 6.33 and pice(θ) ¼ 4.99. 7

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Planetary and Space Science 184 (2020) 104867

Fig. 4. The same as in Fig.(3 i)), but parameter Δm is presented as the function of two variables Mej and ρag , while parameter f(t1)CO is chosen to be equal 0.1%.

Fig. 6. The same as in Fig.(5 i)), parameter Δm from is presented as the function of two variables Mej and ρag , while parameter f(t1)CO is chosen to be equal 0.1%.

Fig. 7. The same as in Fig. 1, but the calculations are performed for the model (b). In addition, it was assumed the large value of the parameter f(t2)CO.

Fig. 8. The same as in Fig. 7, but parameter Δm from is presented as the function of two variables Mej and f(t2)CO, while parameter f(t1)CO is chosen to be equal 0.1%.

crumbs can be determined by means of Mie-Lorenz theory. In turn, the scattering cross-section of aggregates was determined by means of Discrete Dipole Approximation theory, and the results, which were provided by Zubko (2013). At the calculations the geometric peculiarities resulting from the fact that we observe comets illuminated by the Sun from Earth, and hence, the so-called phase function have to be taken into account. The essence of the increase in comet brightness during its outburst is the rapid increase in the amount of comet dust in the comet’s atmosphere, which scatters sunlight. Two mechanisms lead to increasing the amount of matter surrounding the comet’s nucleus. The first is the rapid increase in the rate of sublimation from the comet’s nucleus. The second is the ejection of comet debris from the damaged part of the comet nucleus surface layer. The obtained results were compared with the

Fig. 5. The same as in Fig. 1, but the calculations are performed for comet the X3. Phase angle is chosedn to be equal to θ ¼ 3.82∘, and the values of phase functions are: psi(θ) ¼ 12.74 and pice(θ) ¼ 13.62.

second most abundant in comets. It has been established that cometary dust can take essentially two completely different forms. The first one is water ice crumbs contaminated with dust, and the second one are aggregates consisted of silicon monomers. The brightness jump of the comet during the outburst was calculated on the basis of Pogson’s law. Therefore, it was assumed that an effective scattering cross-section of ice 8

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different sets of appropriately selected values of these parameters can lead to the same brightness jump. Therefore, the problem opposite to the one considered in this paper is very complicated and requires many assumptions concerning whose correctness we are not sure of. Also, for similar reasons, paper does not deal with the state of comets evolution. 6.2. Remarks In this work numerous simulations and numerical tests were carried out and on this basis the following conclusions can be inferred. First, the values of Δm calculated by means of Eq. (30), are correct only for the comets whose spectrum is mostly continuous. In other words, at the calculation we assume the contribution of light flux coming from the scattering by ice and dust grains as most important, because only this flux determines the continuous spectrum of comet. Second, the outbursts of comet brightness last from several to several dozens of days. This is very short time in comparison with comet’s period of orbiting, and hence its phase angle changes very little. Therefore, it can be safely assumed that just before and during the outburst the angles α and θ are constants for the comets under consideration. As a result, we can simplify the numerator and denominator in the formula for Δm reducing the factor pðθ), because the changes of comet brightness are practically independent on the phase function pðθ), provided that the distribution functions f ðaÞ of particles emitted from the comet nucleus as a result of sublimation and recoil of the damaged nucleus layer, are the same or similar. Third, for the comets which are at large distances separating them from the Sun, in the second slightly more accurate approximation brightness jump Δm depends on the relation between scattering cross-sections of constituent elements of cometary matter in the coma before and after its outburst. Fourth, for comets whose perihelion is at the distance less than 1 au from the Sun, phase angle can changes significantly in relatively short period of time. As a result, simplified version of formula for Δm discussed above is not longer valid and one should use Eq. (30), taking into account all entries. Fifth, the analysis of obtained data concerning increase of comet brightness Δm has shown that this increase is greater when the parameter fCO (t1) is smaller. It results from the fact that the same amount of ejected material during the outburst is relatively larger for smaller values of fCO (t1), in comparison with material contained in the cometary coma in inactive phase. Sixth, for a comet whose sublimation activity is controlled by a single type of ice the increase in brightness Δm is an increasing function of comet’s heliocentric distance RH. This conclusion can be expected: it is due to the fact that the rate of sublimation is a decreasing function of the comet’s distance from the Sun. Therefore, the amount of grains emitted into the comet atmosphere during outburst, compared to the amount of grains which is already in the comet atmosphere in the inactive phase, is relatively larger for the larger distances separating comet from the Sun. It should be noted that some of these conclusions have been already formulated for the outbursts of other comets (see e.g.: Gronkowski and Wesołowski, 2015; Wesołowski and Gronkowski, 2018a; Wesołowski, 2019). At the end of the paper, three issues should be highlighted.

Fig. 9. Comparison of the actual brightness change for comet SW1 with the hypothetical comet (X/PC). The difference in brightness change in this comparison is a consequence of the fact that comet SW1 is a much larger body compared to comet X/PC. For comet SW1 it was assumed that its radius is 27 km, and in the case of comets X/PC 1 km. The outburst of comets took place at a heliocentric distance of RH ¼ 6au. The considerations include model (a) presented in this paper and parameter fCO(t1) ¼ 0.1%. It can be seen that for the same mass thrown out of the comet’s nucleus and the set value of the fCO(t1) parameter, the brightness changes are much larger for the smaller nucleus (comet X/PC) than for the larger (comet SW1). This is due to the fact that a fixed amount of ejected matter has a relatively larger scattering cross-section in relation to the matter already existing in the atmosphere of a smaller comet than in a larger one.

observations of comet SW1 outburst and one can asserts the satisfactory agreement between them (Fig. 9). Attention is drawn to the fact that the formula obtained in the paper and describing the change of comet brightness for some special situations becomes considerably simpler. In the paper some assumptions are made concerning the comet structure, as well as the values of several most important parameters, such as the chemical composition, the percentage of the surface nucleus which is sublimation-active both before and during the outburst, the radius of the nucleus, distance from the Sun and the amount of cometary matter ejected by the comet. It should be noted however, that the inverse problem of determining the comet composition based on a brightness jump is much more complicated. Determining the exact numerical values of the parameters mentioned above is difficult or even impossible, e.g. for a new comet that just recently has been observed in the solar system. A

 The results of our numerical calculations are qualitatively consistent with the observations of comets carried out during the outbursts (A’Hearn et al., 2005; Meech et al., 2009; Vincent et al., 2015; Vincent et al., 2016a). Thus, the method of calculating of increase in brightness during comet outburst presented in the paper, seems to be relatively effective and accurate.  Over the last decade, a great progress in numerical modeling of light scattering in the comet atmosphere has been achieved. The existence of highly irregular particles of cometary dust, called agglomerated debris particles has been postulated. The approach based on such observation was proposed in a number of papers (see, for example, Zubko et al., 2011, 2013; 2016; Picazzio et al., 2019). The authors 9

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nuclei of these celestial bodies and as consequence, lead to the outbursts of their brightness. This aspect of the comet evolution should be the target of future research, too.

plan to use method developed by Zubko and his collaborators in a future study of changes in comet brightness.  The analysis of results obtained during Rosetta mission to the comet 67P/Churyumov-Gerasimenko indicates that the comet dust can migrate over the surface of cometary nuclei (Wesołowski et al., 2019). This migration might bring about the significant changes in the surface topography of the comet’s nucleus (Fougere et al., 2014; Rubin et al., 2014; Pajola et al., 2015; Thomas et al., 2015a, 2015b). Note, the avalanches also has been observed for this comet (Pajola et al., 2017). All these can cause the emissions of cometary matter from the

Acknowledgements This work was done due to the support the authors gave gotten from the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzesz
ow.

Appendix Parameters characterizing comet’s orbit An important parameter describing the comet position relative to the Sun and Earth is the phase angle α. It is illustrated in Fig. A.1.

Fig. A.1. Relation between the phase angle α and the scattering angle Θ for the Sun light scattered in the atmosphere of the comet. The Sun, Earth and comet are denoted as S, E and C, respectively.

Fig. A2 illustrates the situation when the comet’s phase angle is of the greatest value. The distance separating a comet from the Sun is larger than 1au.

Fig. A.2. The phase angle α achieves its greatest value αmax when the the straight line EC is tangent to the Earth’s orbit. Then the angle SEC ¼ 90∘ .

Using Fig. A2, one can immediately conclude that the maximum phase angle αmax is described by the simple formula: sinðαmax Þ ¼

rE ; rc

(A.1)

illustrates the opposite relationship to Fig. 2 when the Earth is further from the Sun than the comet. The extreme situation when the comet is closer to the Sun than the Earth is illustrated by Fig. A4.

10

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Fig. A.3. The case when phase angle obeys the condition: 90∘ < α < 180∘ . Notations are analogous to those in Fig. A2.

Fig. A.4. The same as in Fig.A3, but the following condition is fulfilled: α ¼ 180∘ . Notations are analogous to those in Fig. A2.

The average radius of cometary agglomerates The average radius of cometary agglomerates rav was determined by means of data presented by Lin et al. (2017). The authors of this work monitored 45 short-lived outbursts of comet 67P occurred between July and September 2015. For each event, a dust cross-section Si and ejected mass Mi (where i is the number of outburst i ¼ 1, 2, …, 45) has been designated. Making use of these data, the following parameter can be easily calculated: Si Ni π r2i 3 ¼ ¼ ki ¼ ; 4ri ρN Mi Ni 4=3π r 3i ρN

(A.2)

where Ni stand for the number of ejected agglomerates in the i-th event, ri is their average radius and ρN means the density of comet’s nucleus. From Eq. (A.2) one gets: ri ¼

3 ; 4ki ρN

(A.3)

As the average radius of agglomerates rav , the arithmetic mean among 45 values of ri has been taken. As a result, rav ¼ 14:2μm.

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