Activity of Comets Constrains the Chemistry and Structure of the Protoplanetary Disk

Abstract

Recent data of molecular clouds and protoplanetary disks constrain the composition and structure of the disk and planetesimals. Laboratory experiments suggest that dust accretion in disks stops at pebble sizes. Sublimation and recondensation of water ice at the disk water-snow line suggest that pebbles split into water-rich and water-poor ones. The same conclusion has been recently reached by models of cometary activity consistent with the structure of porous Interplanetary Dust Particles (IDPs) and of porous dust collected by the Stardust and Rosetta missions. The observation of crystalline water ice in protoplanetary disks by the Herschel satellite, the erosion of comets, and the seasonal evolution of the nucleus color require that the two pebble families have a water-ice mass fraction close to 33% and 2%, respectively. Here, we show that the diversity of comets is thus due to random mixtures with different area fractions $A_p$ and $A_r$ of water-poor and water-rich pebbles, predicting most of the data observed in comets: why the deuterium-to-hydrogen ratio in cometary water correlates to the ratio $A_p/A_r$, which pebbles dominate the activity of Dynamically New Comets (DNCs), what is the origin of cometary outbursts, why comets cannot be collisional products, and why the brightness evolution of DNCs during their first approach to the Sun is actually unpredictable.

1. Introduction

Comets survive passages inside the solar corona at temperatures of thousands Kelvin, and the evolution of their orbits are best explained by perturbations induced by rocket forces exerted by sublimation of ices exposed to sunlight [1]. These facts led to the definition of the dirty snow-ball model, where nuclei of comets are mainly composed of ices [2]. Meanwhile, the assumption that most carbon was stored into ices rather than in organic refractories led to conclude that ices were more abundant than refractories in the outer Solar System where comets were probably born [3]. Such a paradigm survived the discovery by the ESA Giotto mission that the nucleus of comet 1P/Halley is one of the darkest objects in the Solar System [4]. The non-uniform coma surrounding the nucleus was interpreted in terms of a refractory crust enveloping an ice-rich nucleus: fissures in the crust were actually the sources of the gas and dust jets observed in the coma [5,6].

However, thermophysical models of icy nuclei enveloped by a mantle or crust led to activity paradoxes which were actually never overcome. The first is that the pressure gradient either at the surface of sublimating ices or flowing through the insulating mantle/crust is always orders of magnitude lower than the tensile strength bonding sub-mm dust particles among them [7,8], whereas all observations of comets point out that the cross-section of dust in comae is dominated exactly by sub-mm dust [9,10,11]. This first paradox may be the source of the refractory mantle/crust itself which, however, becomes thicker and thicker as ices sublimate below it: when the mantle becomes thicker than the depth of the ice sublimation front, a comet should die forever, and this should happen after a few days of activity, opposite to observations [12,13].

These two fundamental paradoxes were ignored until today by models of comets enveloped in crusts/mantles, so that the fact they confirm that the nucleus mass is dominated by ices has little physical basis and is in practice a repetition of their main assumption [14,15]. In particular, crust/mantle models [14] cannot explain the measured erosion in Hapi’s smooth plains of comet 67P/Churyumov–Gerasimenko (67P hereafter) [16]; coma models based on a size and time-independent power law of the dust size distribution [15] are inconsistent with the measured 67P dust size distribution [10,17,18,19] and strongly underestimate the 67P dust fallout [20,21] which necessarily implies a low water mass fraction in 67P nucleus [20,22]. Moreover, these models [14,23,24] were unable to fit the steep water loss rate versus the heliocentric distance observed by the ESA Rosetta mission at 67P [25,26], unless a time and space-dependent crust thickness is assumed [27], making the crust models dependent on thousands of free parameters, i.e., unable to predict anything.

Following results of decades of dust tail models [9,17,18], the ESA Rosetta mission has confirmed that most mass of comets is in form of refractories [22] and that most volume of refractories is in form of hydrocarbons [28,29,30]. The structure of cometary dust [31] actually drives our comprehension of both the structure of nuclei and the activity of comets [32]. Cometary dust is porous, and ices, less abundant than refractories, necessarily are embedded inside the porous dust structuring every cometary nucleus [20,33]. This change of paradigm has provided a simple parameter-free analytical activity model which actually fits most available data regarding all comets [20,33,34,35,36,37], finding that nuclei of comets are structured in meter-sized blocks of water-enriched pebbles (WEBs hereafter) embedded in a matrix of water-poor pebbles [21], as discussed in detail in Section 7. Here, we review such a WEB-activity model and compare its outcomes to many available data collected at comets showing how they constrain the physics and chemistry of the solar protoplanetary disk.

2. How Much Ice in the Outer Solar System?

Estimates of the ice mass fraction characterizing the outer solar protoplanetary disk are provided by (i) the hydrogen-to-refractory mass ratio measured in the interstellar medium, $H/dust\approx 100$ [38], a value commonly adopted in all models of protoplanetary disks [39]; and (ii) by the hydrogen-to-water molar ratio measured in molecular clouds surrounding forming and young stars, $H/water$. Tens of measurements have provided the full ranges $1.2\times {10}^4\le H/water\le 8\times {10}^4$, to be compared to CO ($4\times {10}^4\le H/CO\le 2\times {10}^6$) and CO${}_2$ ($4\times {10}^4\le H/C{O}_2\le 5\times {10}^5$) [40]. The ratio between $H/water$ and $H/dust$ provides the refractory-to-water-ice mass ratio $6\le {\delta }_w\le 45$, i.e., a low ice fraction in the outer disk opposite to former assumptions discussed in the previous section.

Coupling the mineralogy provided by the dust samples collected by the NASA Stardust mission [41,42] and the relative abundances of minerals consistent with the dust bulk densities measured by the GIADA experiment of the ESA Rosetta mission [29,30,42], ${\delta }_w$ can be constrained following a completely independent approach. In fact, the average bulk density of Kuiper Belt Objects (KBOs hereafter) provides their ${\delta }_w$ by assuming [22] (i) that KBOs minerals are mainly sulfides, silicates, and hydrocarbons, sharing the same volume fractions observed in 67P [29,30]; (ii) which is the structure of water ice; and (iii) which is the actual abundance of carbon, oxygen, and nitrogen, i.e., depleted as in CI-Chondrites (a possibility supported by the results of Stardust at comet 81P/Wild 2 [41]) or enriched as in the end-case of solar composition (a possibility supported by the results of Rosetta at 67P [28]). Haumea’s star occultations have shown how unreliable estimates of KBO average density are [43] when based on KBO thermal data [44], so that KBO densities based on thermal data only are not considered here. The results shown in

Table 1. Refractory-to-water-ice mass ratios ${\delta }_w$ of Kuiper Belt Objects [22].

Kuiper Belt Object	Amorphous Ice CI-Chondritic	Crystalline Ice CI-Chondritic	Amorphous Ice Solar End-Case	Crystalline Ice Solar End-Case
Charon	3.6	4.1	6.8	9.4
Haumea	5.3	6.4	15.0	37.0
Pluto	6.1	7.6	24.0	190.0
Triton	16.0	30.0	∞	∞

perfectly match what is suggested by molecular cloud data, making the ${\delta }_w$ range in the outer disk even wider.

3. The Structure of Nuclei of Comets

A volume of comets dominated by refractories implies that ices are embedded inside the dust particles, either in form of ice grains or enveloping the refractory grains composing the particles [31,32,36]. Results of the missions Stardust and Rosetta and analyses of porous Interplanetary Dust Particles (IDPs hereafter) of probable cometary origin [31,42] show that dust particles cover any size from ≤1 $\mathsf{\mu }$m to ≈1 cm. Dust particles can be described as homogeneous porous aggregates of grains, which have all a size of about 0.1 $\mathsf{\mu }$m [31,45], are probably of interstellar origin [42,46], and are intermixed with sub-mm-sized rocks [29,30] which necessarily formed close to our protosun [41]. Interstellar grains probably accreted in the presolar cloud into fluffy particles of fractal dimension lower than two [47,48], which later were compacted into dust particles of much lower porosity and fractal dimension close to three by collisions in the disk [34,49]. According to laboratory experiments, accretion of dust stopped at the bouncing barrier, preventing the formation of dust clusters larger than 1 cm in the outer disk where comets probably formed [34]. These clusters are named pebbles and formed all planetesimals, possibly by the gravitational collapse of clouds of pebbles through streaming instabilities [34,39]. Therefore, pebbles are inhomogeneous porous clusters of dust particles [32] because dust particles have a wide size spectrum. Dust particles are homogeneous porous aggregates of grains, because all grains have similar sub-$\mathsf{\mu }$m sizes [31].

Nuclei of comets are thus characterized by three porosity levels: (i) from grains to dust particles, (ii) from dust particles to pebbles, and (iii) from pebbles to the nucleus, which imply a nucleus porosity ≥$1-{(1-0.35)}^3\approx 73\%$ (where 35% is the porosity of a close random packing of spheres [49]), consistent with the measured 67P nucleus porosity [50]. The voids among pebbles still preserve pristine fractals accreted in the presolar cloud [48,49]. However, their mass fraction and their role in cometary activity are negligible, so that they are not further considered here. The inhomogeneous structure of pebbles and the fact that the pebble volume is dominated by the largest dust particles [10,17,18] imply that cometary pebbles are characterized by cohesion strengths and gas permeability opposite than homogeneous aggregates of monomers all of the same size. In the latter, both strength and permeability depend on the monomer size, so that low gas permeabilities imply large cohesion strengths [7,8]. In realistic cometary pebbles, a low gas permeability coexists with a low cohesive strength, because the pebble diffusivity of gas originating from grains depends on the grain size, whereas dominating large particles have a few contact points among them, i.e., a low cohesive strength [32]. This is the key explaining cometary activity, i.e., dust ejection, because the sub-$\mathsf{\mu }$m pores inside the homogeneous particles imply gas pressures much larger than the cohesive strengths bonding an inhomogeneous pebble.

4. The WEB Activity Model

Diffusion of gas sublimating from ices stored inside a cluster of dust particles provides gas pressures strong enough to break the cluster itself. The diffusion timescale provides an upper limit of the cluster size, because cometary activity (i.e., dust ejection) at sunrise cannot onset before gas diffusion is complete within the cluster [39]. A dm-sized cluster has a diffusion timescale of half-an-hour, implying a similar onset delay, much longer than observed in 67P images taken by Rosetta. Thus, cometary activity confirms that the dust clusters are in fact cm-sized pebbles, which are necessarily the building-blocks of all active comets [34].

At the heliocentric distance $r_h$ au and solar zenithal angle $\theta$, the WEB-model [32] is defined by five analytical equations describing (i) the gas pressure $P\left(s\right)$ depending on the depth s from the nucleus surface, (ii) the gas flux Q from the nucleus surface, (iii) the temperature gradient $\nabla T$ at depths of a few cm below the nucleus surface, (iv) the heat conductivity ${\lambda }_s$ at depths of a few cm, and (v) the temperature $T_s$ of the nucleus surface

$$P\left(s\right)=P_{0}f\left(s\right)exp\left[-\frac{T_0}{T_s-s\nabla T}\right]$$

(1)

$$Q=\frac{14rP\left(R\right)}{3R}\sqrt{\frac{2m}{(T_s-R\nabla T)\pi k}}$$

(2)

$$\nabla T=\frac{\sqrt{\mathsf{\Lambda }Q/\sigma }}{8(T_s-R\nabla T)R}$$

(3)

$${\lambda }_s=\frac{32}{3}{(T_s-R\nabla T)}^3\sigma R$$

(4)

$$(1-A)I_{\odot }{r}_h^{-2}cos\theta =ϵ\sigma T_s^4+{\lambda }_s\nabla T+\mathsf{\Lambda }Q$$

(5)

where $r\approx 50$ nm and $R\approx 5$ mm are the radii of the grains of which cometary dust consists [31,42,45] and of the pebbles of which cometary nuclei consist [32,34], s is the depth from the nucleus surface, $f\left(s\right)=1-{(1-\frac{s}{R})}^4$ for $s\le R$, $f\left(s\right)=1$ elsewhere, $P_0$ and $T_0$ describe the saturation pressure versus temperature of the sublimating ice (

Table 2. Ice parameters [32,33].

Ice	${\mathit{P}}_0$	${\mathit{T}}_0$	$\mathsf{\Lambda }$	${\mathit{T}}_{\mathit{onset}}$	${\mathit{r}}_{\mathit{onset}}$
	Pa	K	J kg${}^{-1}$	K	au
H${}_2$O	$3.23\times {10}^{12}$	6135	$2.9\times {10}^6$	205	$3.8$
CO${}_2$	$2.89\times {10}^{12}$	3271	$5.7\times {10}^5$	111	13
C${}_2$H${}_6$	$1.44\times {10}^{12}$	2520	$7.5\times {10}^5$	89	18
CH${}_4$	$5.45\times {10}^9$	1182	$6.1\times {10}^5$	52	52
O${}_2$	$1.37\times {10}^{10}$	998	$2.9\times {10}^5$	44	60
CO	$1.73\times {10}^{10}$	942	$2.3\times {10}^5$	40	85

), $\mathsf{\Lambda }$ is the ice sublimation latent heat (Table 2), m is the mass of the gas molecule, k is the Boltzmann constant, $\sigma$ is the Stefan–Boltzmann constant, $A=1.2$% is the nucleus Bond albedo measured at 67P [51], $I_{\odot }$ is the solar flux at $r_h=1$ au and $ϵ\approx 0.9$ is the nucleus emissivity. The temperature $T_s-R\nabla T$ at $s=R$ provides directly Q, $\nabla T$ and ${\lambda }_s$, then Equation (5) provides $T_s$ and Equation (1) $P\left(s\right)$. Equation (1) provides $P=0$ at $s=0$ and $s\gg R$, and a maximum at $s\approx \frac{2}{3}R$.

A nucleus is active if the gas pressure overcomes the tensile strength $S=13s^{-2/3}$ mPa (with s in meters) bonding dust particles to the nucleus surface [8], thus fixing the activity onsets at the surface temperatures $T_{onset}$ and heliocentric distances $r_{onset}$ for $\theta =0$, neglecting any self-heating in nucleus concavities [24] (Table 2). As $T_s>T_{onset}$, $P\ge S$ for $s_m\le s\le s_M$, where $s_m$ and $s_M$ are the minimum and maximum ejected dust sizes, because dust is ejected from the nucleus surface, so that the depth s provides directly the dust size s.

5. Nucleus Ice-Depletion Versus Erosion

The linear rate at which the nucleus surface is depleted of the sublimating ice is [32]

$$D=\frac{(1+{\delta }_i)Q}{{\rho }_n}$$

(6)

where ${\delta }_i$ is the refractory-to-ice mass ratio (${\delta }_i={\delta }_w$ for water) and ${\rho }_n$ is the average nucleus bulk density [24].

The nucleus erosion is independent of Q and D, i.e., of ${\delta }_i$, because dust ejection depends on $T_s$ only, and not on how much ice is stored in the surface pebble, Equation (1). The linear erosion rate E is given by the size $s_M$ divided by the heat diffusion timescale, because the dust loss rate is dominated by the largest ejected dust [9,10,17,18], and because Equation (1) at depth $s_M>2R$ implies thermal equilibrium (gas transfer among pebbles is negligible [32]),

$$E=\frac{{\lambda }_s}{{\rho }_d{c}_P{s}_M}$$

(7)

where ${\rho }_d\approx 800$ kg m${}^{-3}$ is the average dust bulk density [30] and $c_P\approx {10}^3$ J kg${}^{-1}$ K${}^{-1}$ is the pebble heat capacity [34]. The complex structure and chemistry of cometary dust prevents realistic computations of the $c_P$ dependence on the temperature [33].

When $D<E$, cometary activity does never stop, because the gas diffusion erodes the surface pebbles before they are depleted of the sublimating ice, exposing the underlying ice-rich ones. On the contrary, when $D>E$, cometary activity stops, because the ice-depletion builds-up an insulating crust of dried pebbles before being eroded by the dust ejection, driven by the heat conduction. The condition $D<E$ provides an upper limit of ${\delta }_w$, which is ${\delta }_w<5$ at 67P perihelion, and ${\delta }_w<{10}^4$ at $r_h\ge 3$ au, Equations (6) and (7). Equation (7) predicts the erosion rate actually measured in 67P’s Hapi smooth plains at $r_h\approx 3$ au [16], and implies a dust ejection rate much larger than measured by Rosetta’s GIADA experiment [19] because most ejected dust falls back on the nucleus due to the low gas density in 67P coma [20,32].

The high gas density in 67P perihelion coma prevents any fallout of dust of size $s\le s_M$, so that the comparison between the computed erosion rate E, Equation (7), and the measured dust loss rate requires that, at 67P perihelion, dust is ejected from a few percent of the sunlit nucleus surface [32]. On the contrary, the measured water loss rate [25,26] requires that most sunlit nucleus ejects water to be consistent with the computed vapor flux Q, Equation (2). Therefore, at $r_h\approx 1$ au, the nucleus surface is divided in two completely different areas:

• A water-rich area $A_r$ of ${\delta }_w<5$ and depleted of other ices, where the activity is driven by water ejecting icy dust of size $s_m\le s\le s_M$, Equations (1)–(5), at the ejection rate

  $$Q_d=A_r{\rho }_{d}E$$

  (8)
• A water-poor area $A_p$ of ${\delta }_w\ge 50$, where the activity is driven by CO${}_2$ gas diffusing among the pebbles and ejecting chunks of size $s>s_M$, because at $r_h\approx 1$ au, the CO${}_2$ sublimation front is deeper than $s_M$ [37]. The chunk erosion exposes anyway pebbles rich in water ice, thus the water loss rate is given by $(A_p+A_r)Q$. Erosion of dust of sizes $s\le s_M$ during the dehydration of the water-poor superficial pebbles is inhibited by the high ${\delta }_w$ values in $A_p$ areas, preventing any thermal equilibrium in the surface pebbles [20].

This scenario is also consistent with:

• The negligible water distributed sources observed in 67P coma [52], because most refractory mass is ejected in forms of chunks which are water-inactive at 67P perihelion and fall back on the northern nucleus hemisphere in polar winter [20,22], where water ice is again distributed uniformly inside the chunks by frosting effects [53].
• The color of Hapi’s deposits consistent with ${\delta }_w\approx 100$ [21], i.e., with dm-sized chunks of ${\delta }_w\approx 50$ enveloped by a pebble-thick dry crust at the ejection because $D>E$ [20].
• The dominant cross-section of the dm-sized chunks composing all 67P dust deposits [54], fitting that predicted by the CO${}_2$-driven activity model [37].
• The inbound activity of the dust deposits during the northern polar summer, because chunks of ${\delta }_w<{10}^4$ become water-active at $r_h\ge 3$ au [32].
• The lack of water-rich chunks, which would be eroded into dust of size $s<s_M$ during their flight in the coma, inconsistent with the observed anisotropies and values of the dust flux [10] and the constraints on the water-distributed sources [52] (Section 8).
• The size of the smallest dust observed in the coma during steady activity, always consistent with the computed $s_m\approx 10\mathsf{\mu }$m [10,19,55].

6. The Heliocentric Dependence of the Water Loss Rate

At each heliocentric distance $r_h$, Equations (1)–(5) can be solved for each facet of area $dA$ composing the nucleus shape model and characterized by the angle $\theta$ providing $T_s>205$ K according to the nucleus season and rotation phase [35]. The integration of all $QdA$ values provides the $r_h$-dependence of the water loss rate to be compared to actual data [25,26]. Opposite to nucleus models based on dry crusts/mantles, the WEB-model fits all data within a factor two during years 2015 and 2016, despite it does not depend on any free parameter [35]. The steep $r_h$-dependence of the water loss rate is due to

• The steep exponential dependence on the temperature of Equation (1).
• The strong increase of the number of nucleus facets at $T_s>205$ K as $r_h$ decreases.

These facts further confirm that cometary activity is driven by dust ejection, and that the adopted gas diffusion parameters in Equation (2) have been correctly estimated.

The WEB-model fits all data within the measurement errors of the 67P water loss rates, excluded six months around perihelion and the inbound orbit during the year 2014. Around perihelion, the computed water loss rate is about a factor two larger than the measurements [35]. This disagreement is easily explained taking into account the balance between CO${}_2$-driven erosion and dehydration. If the CO${}_2$-driven erosion were a factor two slower than the dehydration of the pebbles exposed by the chunk erosion, the model would perfectly fit the water loss data also at perihelion. Alternatively, the CO${}_2$-driven activity model finds that only half of the 67P southern hemisphere may eject chunks [37].

The disagreement between data and model is much more severe during 2014, when the observed water loss rate decreases in time at values a factor twenty larger than predicted. A significant self-heating in the northern 67P hemisphere in polar summer [24] may explain the 67P water-driven activity onset at $r_h\approx 4.1$ au [18], but cannot increase much the water loss rate. The measured Hapi’s color [21] and the erosion $E_H$ measured in Hapi’s $A_H\approx 0.2$ km${}^2$ [16] provide anyway a direct explanation of the high water loss rates in 2014. In fact, water-distributed sources release the water loss rate $Q_s=A_H{\rho }_d{E}_H/({\delta }_d+1)$ [20], where ${\delta }_d\approx 100$ is the refractory-to-ice mass ratio of Hapi’s dust (previous section). It follows that, during 2014 only, 67P’s water-distributed sources were larger than the water loss rate $Q_n={\int }_{T_s>205K}QdA$ from the nucleus. Since during 2014 the $s_M$ value fastly increases [32], the WEB-model predicts both the values and the decreasing trend of the erosion $E_H$ actually measured [16]. Thus, the decreasing erosion rate $E_H$ into dust of ${\delta }_d\approx 100$ perfectly matches the measured values and trend of the water loss rate during 2014 too [25,26].

Regarding other comets, the ratio between the perihelion water loss rate and that at $r_h=3.8$ au of comet C/1995O1 Hale–Bopp is the same inbound and outbound [56], and similar to the 67P inbound one, thus suggesting significant water-distributed sources both inbound and outbound. In addition, this comet shows a water activity onset at $r_h\approx 4$ au, as well as all Jupiter Family Comets reaching $r_h<4$ au [57]. Some of these show an outbound activity up to $r_h\approx 6$ au [57], best explained by bound comae and fossil tails [33,58].

7. The Heliocentric Dependence of the Nucleus and Coma Colors

The Rosetta mission followed the evolution of the 67P coma color of equatorial and southern nucleus areas from $r_h\approx 3.5$ au inbound to $r_h\approx 3.5$ au outbound [21,59,60]. While the nucleus evolves from red far from the Sun to blue at perihelion [21,59,60], the dust coma shows the opposite trend, from blue far from the Sun to red at perihelion [59]. Color calibrations by means of water-ice absorption bands observed by the VIRTIS Rosetta instrument show that the bluer the color, the more abundant the exposed water ice [61]. Explanations of the nucleus color trend in terms of a crust becoming thinner as the comet approaches perihelion, i.e., transparent to the observation wavelength, thus revealing the icy interior [60], would require a crust thickness of a few microns at most, inconsistent with the dominant cross-section of dust collected by the Rosetta dust instruments [10,42] and with the independence of the nucleus color with respect to its surface temperature [21].

A quantitative explanation of the seasonal evolution of the nucleus color has been provided by the WEB-model only, i.e., by the seasonal evolution of water-poor and water rich areas [21,32], Figure 1. At $r_h>3$ au inbound, the equatorial and southern nucleus areas are covered of fallout of ${\delta }_w>{10}^3$, less thick than the depth of the CO${}_2$-ice sublimation front, composed of dm-sized chunks ejected during the previous perihelion [54], and dehydrated during many orbits around the nucleus [21]. Approaching to the Sun, the CO${}_2$-driven erosion exposes both water-poor and water-rich areas by the ejection of dm-sized chunks [37]. The exposed water-rich areas are eroded by water-driven activity into dust of size $s_m\le s\le s_M$ (Section 4). If the $r_h$-dependence of the CO${}_2$-driven erosion is steeper than the water-driven one, then the nucleus area fraction covered by water-rich areas increases inbound and decreases outbound [21]. Water-poor areas appear reddest because their thickness actually observed at visible wavelengths is dehydrated within a few seconds. Water-rich areas appear bluest as the blue patches of ${\delta }_w\approx 2$ observed on 67P nucleus [61,62]. Thus, the areal mixing of bluest water-rich and of reddest water-poor areas fits the measured water-ice abundance and its seasonal evolution if the water-rich areas are exposed water-enriched meter-sized blocks (WEBs) with a volumetric abundance of $(7.5\pm 2)$% and uniformly distributed inside the water-poor matrix [21]. WEBs are thus probably pristine, i.e., they accreted in the protoplanetary disk, so that all comets should be structured in WEBs. They are exposed by CO${}_2$-driven erosion into all the long-lasting blue patches inconsistent with the diurnal frost [53], and may be meters-sized when sampling the tail at large sizes of the WEB size distribution [21,61,63].

The WEB-model predicts that most outbound nucleus reddening is due to the fallout of dm-sized chunks occurring during most outbound orbits [21,22,58], consistent with bound comae at $r_h>4$ au [57]. Such a fallout is thinner and redder than that deposited on the northern hemisphere, because it is distributed over a larger nucleus surface and is dehydrated to ${\delta }_w>{10}^3$ during the orbits around the nucleus of the chunks composing it [21]. The fallout on northern Hapi can occur during perihelion only when Hapi is in polar night, because Hapi’s concavity prevents a long-lasting fallout, so that Hapi’s deposits maintain ${\delta }_w\approx 100$ as constrained by its color [21]. The northern fallout is thicker than the depth of the CO${}_2$ sublimation front [16], thus explaining because long-lasting blue patches have never been observed at nucleus latitudes $>+{30}^{\circ }$ [63]. The opposite color seasonal cycle of the dust coma [59] is due to the slow dehydration of optically-dominant mm-sized dust [19] of ${\delta }_d\approx 100$ ejected by Hapi at $r_h\approx 3$ au inbound versus the much faster dehydration of the dust ejected at perihelion [20], with optically-dominant sizes $s\le 0.1$ mm [10].

8. Water-Poor versus Water-Rich Pebbles

According to the WEB-model, nuclei of comets are composed of water-rich meter-sized WEBs embedded in a water-poor matrix. WEBs are composed of cm-sized pebbles of ${\delta }_r\approx 2$ (previous section), while the matrix is composed of cm-sized pebbles of ${\delta }_p\approx 50$ (Section 5). These ${\delta }_w$-values are outside the end-cases observed in molecular clouds (Section 2), suggesting that in the disk, some physical process transformed the wide unimodal ${\delta }_w$-distribution characterizing the molecular clouds (Section 2) into the bimodal ${\delta }_w$-distribution characterizing the disk where comets were born. This process is well known, namely sublimation and recondensation of water ice at the disk water-snow line, where dust acting as a cold finger accumulates frost on it [64], depleting other dust of its ice to maintain the initial ice budget in the disk [20]. This process has fundamental implications (

Table 3. Pebble parameters [20].

Physical Quantity	Water-Poor Pebbles	Water-Rich Pebbles
$\alpha$	$-3.0$	$-3.8$
${\delta }_w$	50	2
$D/H$	$(5.3\pm 0.7)\times {10}^{-4}$	$1.56\times {10}^{-4}$
Water Ice	amorphous	crystalline
Supervolatiles	rich	depleted

):

• All supervolatiles must sublimate at the disk water-snow line and cannot recondense on dust due to the dust temperature. This fact explains why all dm-sized chunks are water-poor and water-distributed sources in comae have sizes $s_m\le s\le s_M$ (Section 5).
• The amorphous water ice condensed on the dust grains in the molecular cloud becomes necessarily crystalline after condensation at the disk water-snow line, characterized by temperatures much larger than those making stable amorphous water ice [65].
• The deuterium-to-hydrogen ratio in the recondensed crystalline water ice must decrease from the high values typical of molecular clouds [66] to that of CI-chondrites, formed as well at the disk water-snow line, namely the Vienna Standard Mean Ocean Water (VSMOW) value $D/H=1.56\times {10}^{-4}$ [67].

The huge outward migration of dust from the inner Solar System, required to explain the sub-mm rocks observed in the dust of comets 81P [41] and 67P [30], thus involved also all the dust later accreted into water-rich pebbles. Both the large value of ${\delta }_p$ and the migration of dust carrying out crystalline water ice are consistent with the huge amount of crystalline water ice measured in outer disks at temperatures much lower than the crystallization one [20,68]. Water-rich pebbles either formed at the disk water-snow line and later migrated into the outer disk where comets formed, or water-rich dust migrated into the outer disk where later water-rich pebbles formed. WEBs possibly formed during the disk instabilities accreting protocomets thanks to the larger sticking of water-rich pebbles with respect to water-poor ones [21]. In both cases, all differences among comets cannot depend on where they formed, because all the water-rich pebbles accreted at the same disk water-snow line, whereas all the water-poor pebbles mix dust depleted of water and supervolatiles with dust accreted directly in the outer disk containing amorphous water ice and supervolatiles condensed in the molecular cloud. In other words, the chemistry of the disk is encoded in the two different families of pebbles, and the differences among comets do not depend on where they formed, but only on the different fractions $\frac{A_p}{A_r}$ in their nuclei [20].

Since dust ejected at $r_h\approx 1$ au is necessarily ejected by water-rich pebbles, whereas at large heliocentric distances, most dust comes from fallout deposits composed of water-poor pebbles, a different power index $\alpha$ of the size distribution of dust of size $s_m\le s\le s_M$ in the two families of pebbles directly explains the time-evolution of the dust size distribution observed in many comets [9,10,17,18,20]. The WEB structure of cometary nuclei constrains also the average refractory-to-water-ice mass ratio of a nucleus [20]

$${\delta }_n=\frac{A_p+A_r}{\frac{A_p}{1+{\delta }_p}+\frac{A_r}{1+{\delta }_r}}-1$$

(9)

where $\frac{A_r}{A_p+A_r}\approx \frac{2}{3}$ is the upper limit of the WEB fraction embedded in a water-poor matrix provided by the random packing of spheres [49], taking into account that the volumetric fraction of WEBs is the upper limit of the surface fraction of exposed water-rich areas [21]. Therefore, in all comets, $\frac{A_p}{A_r}\ge \frac{1}{2}$ and ${\delta }_n\ge 3.4$, perfectly matching independent estimates [22,30] and KBO data (Table 1).

9. Deuterium-To-Hydrogen Ratio in Water

Since all the water ejected by comet 67P during 2014 was surely coming from fallout composed of water-poor pebbles (Section 5), the $D/H$ measured then [69] provides the best estimate of $D/H_p$ (Table 3). Therefore, the average $D/H_n$ value in the nucleus is given by the linear combination of $D/H_p$ and $D/H_r$ weighed by the respective ice mass fractions

$$D/H_n=\frac{(1+{\delta }_p)A_{r}D/H_r+(1+{\delta }_r)A_{p}D/H_p}{(1+{\delta }_p)A_r+(1+{\delta }_r)A_p}$$

(10)

which is different from the average $D/H_c$ measured in the perihelion coma

$$D/H_c=\frac{(A_{p}D/H_p+A_{r}D/H_r)Q_n}{(A_p+A_r)(Q_n+Q_s)}+\frac{Q_{s}D/H_r}{Q_n+Q_s}$$

(11)

because $Q_n$ comes from both water-poor and water-rich areas, whereas $Q_s$ from water-rich areas only [20]. Equation (11) at 67P perihelion predicted [20] exactly $D/H_c=5.0\times {10}^{-4}$ measured at perihelion [70] according to $Q_s\ll Q_n$ (Section 5), and some fluctuations lower than the measurement error according to the Rosetta phase angle, because the dust flux measured by GIADA suggests negligible water-distributed sources at the nucleus terminator. However, the error affecting the $D/H_c$ data was evaluated assuming that $D/H_c$ is constant during all the Rosetta mission, and not dividing the data in two different sets, i.e., at low phase angles and at nucleus terminator [70]. In this case, the $D/H_c$ uncertainties become larger than the fluctuations predicted by Equation (11) according to 67P’s constraint $Q_s<\frac{1}{2}{Q}_n$ [20,52].

The WEB model applied to 67P perihelion with the dust loss rate $Q_d$ as input parameter fits all the 67 perihelion available data, and provides $D/H_n$ significantly lower than $D/H_c$ [20]. The WEB model applied to comet 103P/Hartley 2 with the perihelion $Q_n+Q_s$ as input fits as well all available 103P perihelion data [20], providing $D/H_c=1.{85}_{-0.3}^{+0.7}\times {10}^{-4}$, perfectly matching the measured $D/H_c=(1.61\pm 0.24){10}^{-4}$ [71]. 103P provides the lowest $\frac{Q_n}{Q_s}\ll 1$ and $D/H_c$ values observed in comets, i.e., the WEB volumetric fraction consistent with $\frac{A_p}{A_r}\approx \frac{1}{2}$ and $D/H_c$ consistent with the VSMOW value, opposite to 67P, providing the largest $\frac{Q_n}{Q_s}\gg 1$ and $D/H_c$ values ever observed. Therefore, Equation (11) predicts the so far unexplained correlation between $D/H_c$ and $\frac{Q_n}{Q_s}$ actually observed in all comets [20,72]. The larger the collapsing volume of the accreting comet, the less probable a close packing of WEBs in the whole nucleus, thus possibly explaining why only small comets have a VSMOW $D/H_c$ [72].

10. Activity of Comets beyond Jupiter

At $r_h>3.8$ au, the activity of comets is necessarily driven by supervolatiles (Table 2), i.e., depends on the $A_p$ value, opposite to the brightness at $r_h\approx 1$ au, depending on $Q_d$, i.e., on the $A_r$ value, Equation (8). This simple fact explains why the inbound evolution of the brightness of Dynamically New Comets (DNCs) is actually unpredictable, i.e., why DNCs of similar brightness at similar $r_h\gg 4$ au may show very different brightness at $r_h\approx 1$ au. Such a prediction depends on the $\frac{A_p}{A_r}$ value, which can be provided by $D/H_c$ measurements only (previous section), prevented by the very low water flux beyond Jupiter.

DNCs show activity beyond Uranus, e.g., C/2017K2 at $r_h=23.7$ au [73]. This evidence proves that supervolatiles cannot be trapped inside amorphous water ice, because the transition temperature from amorphous to crystalline water ice, making the impurities inside it free, occurs at $T_s\approx 140$ K [74]. Supervolatiles cannot be even trapped inside CO${}_2$ ice, because its activity onsets at $r_h=13$ au (Table 2). They can drive activity of comets if the heat transfer inside the superficial pebbles is faster than the ice depletion, providing an upper limit of the refractory-to-ice mass ratio ${\delta }_i$ of each supervolatile. Assuming that cometary activity beyond Jupiter is driven by at least the five ices listed in Table 2, most ${\delta }_i$ values become lower than ${\delta }_w=50$ in water-poor pebbles [33], another evidence that supervolatiles cannot be trapped inside a less abundant amorphous water ice. The ${\delta }_i$ values of CO and CO${}_2$ are consistent with the abundances measured in molecular clouds (Section 2). However, the fact that all the supervolatile ices sublimated at the disk water-snow line from water-rich dust (Section 8) suggests some recondensation of supervolatiles in the outer disk, again consistent with supervolatiles not trapped inside amorphous water ice.

The WEB model has been applied to models of dust tails, which are a powerful tool to infer physical parameters of comets observed from ground. They depend on the $\beta$-parameter proportional to ${\left({\rho }_{d}s\right)}^{-1}$ [9,17,18,33,75]. According to Equation (1), the WEB model predicts that the size distribution of the ejected dust can be well approximated by a lognormal distribution centered at $s\approx \frac{2}{3}R$ (Section 4). Since the ${\rho }_d$ distribution is well approximated by a lognormal distribution centered at ${\rho }_d\approx 800$ kg m${}^{-3}$ [30,75], it follows that also the $\beta$-distribution is a lognormal distribution centered at $\beta \approx 5\times {10}^{-4}$ [33,75]. The WEB model constrains the $r_h$-dependence of the nucleus erosion and of the dust ejection velocity [33], and the activity onset at $r_h\approx 85$ au (Table 2), consistent with observations [33,73]. Therefore, the probabilistic tail model depends on three free parameters only:

• The dust ejection velocity from the inner coma, constraining the tail width.
• The dispersion of the $\beta$ lognormal distribution, constraining the tail orientation.
• The $r_h$-dependence of the icy dust loss rate $Q_d$, constraining the tail length.

They are all correlated to the nucleus size [33]. The probabilistic tail model has been successfully applied to interstellar comet 2I/Borisov, showing that its nucleus and dust environment is very similar to the 67P one [75]. This evidence confirms that interpretations of dust polarimetry data still depend on the huge number of free parameters affecting all models of dust polarimetry [76,77,78], making their conclusions on the physical properties of comets much more uncertain than those based on well-constrained tail models [9,17,33,75].

The probabilistic tail model applied to five Oort comets shows that also DNCs are consistent with the WEB model, confirming that activity of comets begins in the outer Kuiper Belt [33,79] (Table 2). This evidence shows that comets are not collisional fragments, because if comets were born at $r_h<85$ au, their nucleus erosion started as soon as the disk became transparent to solar radiation at the computed average rate $E\ge 1$ m year${}^{-1}$, with erosion lifetimes orders of magnitudes faster than collisional lifetimes [80]. Close encounters with giant planets in fast inward migration [81] necessarily ejected protocomets to $r_h>85$ au, where collisions are negligible, well before the first catastrophic collision had enough time to occur [33]. Comets born at $r_h>85$ au cannot be collisional fragments too.

11. Outbursts

During outbursts, the brightness of comets suddenly increases, suggesting some kind of explosion in their nuclei [82]. The prototype showing a somewaht regular occurrence of outbursts is the Centaur 29P/Schwassmann-Wachmann 1, orbiting the Sun between Jupiter and Saturn [83]. Since its nucleus temperature is close to $T\approx 150$ K (Table 2), at which the exothermic transition of amorphous to crystalline water ice occurs [74], such an energy source was supposed to play a major role in cometary outbursts [65]. However, such a transition is exothermic only in case of pure amorphous water ice, otherwise it becomes an energy sink, probably because the released energy is adsorbed by the sublimation of the supervolatile ices trapped in the water ice itself [74]. Thus, the common assumption that all supervolatiles are trapped inside water ice [2,14], based on the observational bias of comets beyond Jupiter, excluded that this process is a possible source of outbursts.

Activity of comets well beyond Saturn is an evidence that supervolatiles are not stored inside amorphous water ice (previous section), so that it may be pure enough to ensure that its phase transition to crystalline is in fact exothermic, i.e., a working source of outbursts [74]. Taking into account that water-poor pebbles, where all amorphous water ice is stored, are characterized by ${\delta }_p\approx 50$ (Table 3), the energy supply by the phase transition becomes $Z\approx {10}^{19}exp-\frac{5370}{T}$ W m${}^{-3}$ [65]. This energy input is much lower than the solar energy input for $r_h>4$ au, and thus neglected in Section 4. During water-driven activity, the temperature at depths $s\le s_M$ is always $T\ge 200$ K [32], thus forcing the amorphous water ice to migrate to larger depths. However, if some landslide or fracture suddenly exposes amorphous water ice to, e.g., $T\approx 180$ K, then the phase transition supplies the energy input $Z\approx {10}^6$ W m${}^{-3}$, which provides $Q=2RZ/\mathsf{\Lambda }\approx 4\times {10}^{-3}$ kg m${}^{-2}$ s${}^{-1}$ and $P\approx 3\times {10}^4$ Pa, Equation (2), sufficient to detach sub-$\mathsf{\mu }$m-sized dust particles from the nucleus surface, i.e., dust much smaller than ejected during steady activity (Section 5). Both the water-flux measured in the most powerful outbursts observed in 67P [82] and the observation of sub-$\mathsf{\mu }$m-sized dust during 67P outbursts only [84] are consistent with this scenario.

The WEB activity model is also consistent with the two different kinds of outbursts observed by Rosetta on comet 67P [85]. In the first, both gas and dust are ejected: it has been described above by the phase transition of amorphous water ice to crystalline due to a sudden temperature increase of exposed water-poor pebbles, triggering the sublimation of supervolatile ices besides that of water ice. In the second, only the dust loss strongly increases mainly around the perihelion, without significantly affecting the gas loss rate. The ongoing exposition of a cluster of WEBs (Section 7) can locally enhance the dust loss without any appreciable increase of the total gas loss rate. The vapor flow from the nucleus is uniform, because it comes from both water-poor and water-rich areas, so that the dust flow coming from the exposed cluster of WEBs remains collimated in the coma, appearing as a much brighter jet [86]. The dusty outburst stops either when the WEBs in the cluster are eroded by their water-driven activity, or when a landslide suddenly buries them.

12. Conclusions

Activity of comets depends on the microstructure of the dust particles composing all nuclei and, in turn, constrains the structure of nuclei in form of cm-sized pebbles. Pebbles are inhomogeneous clusters of dust particles, which in turn are homogeneous porous aggregates of sub-$\mathsf{\mu }$m-sized grains. Pebbles are extremely fragile, so that the design of future cometary missions able to bring back unaltered samples proving the scenario proposed in this review will remain for long beyond our capabilities. The complex structure and chemistry of pebbles, which is the key to explain cometary activity, will probably remain hard to be simulated in our laboratories as well.

Nevertheless, the WEB activity model offers big improvements with respect to others: it fits most available cometary data without depending on free parameters. Moreover, many cometary processes constrain not only the structure, but also the chemistry of the pebbles formed in the protoplanetary disk (Table 3). The measured nucleus erosion, dust loss rates, seasonal evolution of the nucleus color, and the correlation between coma-distributed sources and $D/H$ in water require and converge to the same conclusion: sublimation and recondensation of water ice at the disk water-snow line was a fundamental process dividing all pebbles in two families of very different chemistry (Table 3), which may be the only source of the huge diversity observed among comets.

In the next years, new powerful telescopes will probe cometary activity in the outer Solar System, possibly checking at which distance it really onsets. New data of protoplanetary disks will probe whether the outward migration of dust and possibly pebbles is as ubiquitous as suggested by the pebble chemistry. Ongoing analyses of Rosetta data and new missions to comets will measure the actual WEB size distribution. New models of planetesimal accretion will probe how and when the formation of WEBs may occur. Additionally, as usual, unexpected discoveries will probe the effective robustness of the WEB model.