Past and present dynamics of the circumbinary moons in
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A&A 658, A99 (2022)
Astronomy
https://doi.org/10.1051/0004-6361/202141687
&
© ESO 2022
Astrophysics
Past and present dynamics of the circumbinary moons in the
Pluto-Charon system
Cristian A. Giuppone1 , Adrián Rodríguez2 , Tatiana A. Michtchenko3 , and Amaury A. de Almeida3
1
Universidad Nacional de Córdoba, Observatorio Astronómico - IATE. Laprida 854, 5000 Córdoba, Argentina
e-mail: cristian.giuppone@unc.edu.ar
2
Observatório do Valongo, Universidade Federal do Rio de Janeiro, Ladeira do Pedro Antônio 43, 20080-090, Rio de Janeiro,
Brazil
3
Instituto de Astronomia, Geofísica e Ciências Atmosféricas, USP, Rua do Matão 1226, São Paulo, SP 05508-090, Brazil
Received 30 June 2021 / Accepted 22 December 2021
ABSTRACT
Context. The Pluto-Charon (PC) pair is usually thought of as a binary in a dual synchronous state, which is the endpoint of its tidal
evolution. The discovery of the small circumbinary moons, Styx, Nix, Kerberos, and Hydra, placed close to the mean motion reso-
nances (MMRs) 3/1, 4/1, 5/1, and 6/1 with Charon, respectively, reveals a complex dynamical system architecture. Several formation
mechanisms for the PC system have been proposed.
Aims. Assuming the hypothesis of an in situ formation of the moons, our goal is to analyse the past and current orbital dynamics of
the satellite system. We plan to elucidate on in which scenario the small moons can survive a rapid tidal expansion of the PC binary.
Methods. We study the past and current dynamics of the PC system through a large set of numerical integrations of the exact equations
of motion, accounting for the gravitational interactions of the PC binary with the small moons and the tidal evolution, modelled by
the constant time lag approach. We construct stability maps in a pseudo-Jacobian coordinate system. In addition, considering a more
realistic model that accounts for the zonal harmonic, J2 , of Pluto’s oblateness and the ad hoc accreting mass of Charon, we investigate
the tidal evolution of the whole system.
Results. Our results show that, in the chosen reference frame, the current orbits of all satellites are nearly circular, nearly planar, and
nearly resonant with Charon, which can be seen as an indicator of the convergent dissipative migration experienced by the system in
the past. We verify that, under the assumption that Charon completes its formation during the tidal expansion, the moons can safely
cross the main MMRs without their motions being strongly excited and consequently ejected.
Conclusions. In the more realistic scenario proposed here, the small moons survive the tidal expansion of the PC binary without the
hypothesis of resonant transport having to be invoked. Our results indicate that the possibility of finding additional small moons in the
PC system cannot be ruled out.
Key words. celestial mechanics – planets and satellites: dynamical evolution and stability – planets and satellites: individual: Pluto
– methods: numerical
1. Introduction 2006; Kenyon & Bromley 2019c,a). Some have also considered
the moons’ masses to be one order of magnitude larger than those
The Pluto-Charon (PC) binary has a mass ratio of ∼0.122 and determined by observational data (Tholen et al. 2008; Pires Dos
is currently found in a dual synchronous state, which is the Santos et al. 2011; Youdin et al. 2012). The orbital periods of the
typical endpoint of tidal evolution, over 1–10 Myr for this par- small moons place them very close to or even inside (depending
ticular system (e.g. Farinella et al. 1979; Correia 2020). The on the uncertainties) the N/1 mean motion resonances (MMRs)
orbit of Charon is almost circular, as confirmed by a 1σ upper with Charon , namely, 3/1, 4/1, 5/1, and 6/1 for Styx, Nix, Ker-
limit of 7.5 × 10−5 (Buie et al. 2012). In the last 15 yr, the sys- beros, and Hydra, respectively. For example, Brozović et al.
tem has gained attention due to the discovery of its four small (2015) obtained the period ratios of a satellite and Charon of
moons (Styx, Nix, Kerberos, and Hydra) and their complex cir- 3.1565, 3.8913, 5.0363, and 5.9810 for Styx, Nix, Kerberos, and
cumbinary configurations. Several formation mechanisms for Hydra, respectively. While the double synchronous state of the
this satellite system have been proposed (Kenyon & Bromley PC binary is an indicator of the tidal evolution of the system, the
2021, and references therein). However, none of the proposed positions of the moons with respect to the MMRs could be an
scenarios of the past evolution of the whole system has pro- indicator of a smooth migration process.
vided strong conclusions on whether the four small moons could It is well accepted that Charon was formed as a result of a
survive the tidal expansion period of the PC binary. giant collision that most likely happened when the population of
The orbits of the small moons can be described with the Kuiper Belt was much denser than today (Canup 2005, 2011;
respect to the barycentre of the PC binary in the nearly circu- Ward & Canup 2006; Asphaug et al. 2006; McKinnon et al. 2017;
lar and coplanar orbital geometry. Several works have studied Walsh & Levison 2015). This giant impact could also have origi-
the dynamical stability of the small moons, considering the pos- nated the very small circumbinary satellites Styx, Nix, Kerberos,
sibility of the existence of putative satellites (e.g. Weaver et al. and Hydra. There is evidence that the satellites are the same age
Article published by EDP Sciences A99, page 1 of 18
A&A 658, A99 (2022)
as Pluto; indeed, the crater-counting data from New Horizons of several orders of magnitude in the eccentricities and incli-
imply that the surface ages of Nix and Hydra are at least 4 billion nations of the small moons. The solution from Showalter &
years (Weaver et al. 2016). Hamilton (2015) locked Pluto and Charon to the ephemerides of
Formation theories for the small moons in the PC system the Jet Propulsion Laboratory (JPL), PLU043. Our data are taken
include an ‘intact capture scenario’ and a ‘planetesimal cap- from JPL Horizons and correspond to the pre-computed solution
ture scenario’. In the scenario of intact capture (Canup 2005, PLU058/DE440, a fit to ground-based Hubble Space Telescope
2011), a proto-Charon grows rapidly, in about 30 h after the col- and New Horizons spacecraft encounter astrometry in the inter-
lision event, within a massive debris swarm produced during the val 1965–2018. To illustrate the uncertainties in orbital fits, we
collision and extending from 4 RP to 25 RP (in units of Pluto’s show in Table 1 the in-orbit errors (along the track) reported by
radius). The commonly assumed initial position of Charon in Brozović et al. (2015) and also include the amplitude observed
the disc is around 4 RP (e.g. Cheng et al. 2014a,b; Woo & Lee in the variation of orbital elements (see also Fig. A.1). The pre-
2018). Depending on the characteristics of the impact, the initial cise orbital dynamics of the small moons depends on the initial
orbit of Charon varies its form, from circular to highly eccen- osculating orbital elements and the masses of the moons.
tric (eC ∼ 0.50). The small moons belong to the debris swarm, With this in mind, we introduce an additional mechanism
and their initial positions are generally considered to be closer to that could provide a robust explanation for the existence of the
Pluto than today. four small circumbinary satellites at their current positions. For
To place the moons at their current locations, Ward & Canup this, we first give a global view of the dynamics of the cur-
(2006) proposed the mechanism of resonant transport. The main rent Pluto system of small satellites in Sect. 2. In Sect. 3 we
idea of the method is that, during the tidal expansion of the present the model, which describes the tidal interactions between
PC orbit, the small satellites can be captured into resonances Pluto and Charon, and discuss the choice of the parameter val-
with Charon and migrate outwards together with Charon. Several ues adopted in this paper. In the next section we analyse how
authors have tested this hypothesis and analysed the probability the tidal expansion of the PC binary, starting at different initial
of capture in the MMRs of the kind N/1 (e.g. Lithwick & Wu configurations, could affect the behaviour of the small moons
2008a; Cheng et al. 2014b; Woo & Lee 2018; Kenyon & Bromley located at their current positions (Sect. 4). In Sect. 5 we anal-
2021), which are considered to be strongest around binaries yse, in the frame of the in situ formation scenario, the effects of
(e.g. Cuello & Giuppone 2019; Gallardo et al. 2021). However, the tidal evolution of Charon’s orbit on a large grid of parame-
Lithwick & Wu (2008b) found some difficulties in adjusting the ter values and initial conditions. In that section we also study the
values of Charon’s eccentricity, eC : in order to safely transport impact of the zonal harmonic of Pluto’s oblateness on the moons’
Nix to the 4/1 MMR, it should be eC < 0.024, while in order dynamics. To overcome the problem of survival of the moons
to transport Hydra to the 6/1 MMR, it should be eC > 0.04. during the passages through the low-order MMRs with Charon,
Cheng et al. (2014b) have found stable solutions (that is, with- we investigate the effects of a mass-accreting Charon on the
out ejections from the system) for the test particles at the 5/1, moons’ behaviour in Sect. 6. Finally, we present our conclusions
6/1, and 7/1 MMRs but none at the 3/1 and 4/1 MMRs. More- in Sect. 7.
over, the orbits of the surviving particles were highly eccentric,
in contrast with the currently nearly circular orbits of the small
moons. In addition, the authors have found that, when the hydro- 2. Dynamical portrait of the current PC system
static value of Pluto’s zonal harmonic, J2 , was included in the
model, there was no stable transport at the regions near the It is generally accepted that the current orbital configuration of
N/1 MMRs. the PC binary and the four small moons is a product of the evo-
To overcome the problems of the resonant transport model, lutionary history of the whole system during its lifetime, since
the scenario of planetesimal capture gained more attention. This the collision event that gave rise to Pluto’s satellites. In this con-
scenario considers a ring of ejected material that ranges up to text, a detailed analysis of the current relative positions of the
60 RP , or even 200 RP ; this value is even higher in the intact Pluto system members may provide important constraints on the
capture scenario (e.g. Pires dos Santos et al. 2012; Desch 2015; dynamical past of the whole system.
Walsh & Levison 2015; Kenyon & Bromley 2019b). Smullen & A representative picture of the PC system is shown in Fig. 1;
Kratter (2017) studied the evolution of a debris disc resulting it is calculated with data from the JPL Horizons site (see
from the Charon-forming impact, established regions of stability Table 1). To construct it, we used a modified Jacobi reference
according to the tidal evolution of the PC binary, and charac- frame, in which orbital elements of the small moon are referred
terised the collisions onto Charon’s surface that might leave to the centre of mass of the PC system and the mutual interac-
visible craters. Woo & Lee (2018) studied several possibilities tions between the moons are neglected1 . In this reference frame,
for survival of the test particles in the regions of the known the motion of a particle of Styx’s mass is simulated through the
moons during the tidal expansion of the PC binary (in situ for- numerical integration over 200 yr (∼3500 Styx orbital periods)
mation scenario). Applying different tidal models, the authors and subsequently analysed using ∆e or ∆i indicators to better
found the most promising results when considered a constant- address the structure of the resonances. We calculate ∆e as the
∆t tidal model with a large dissipation coefficient (A ∼ 40) and amplitude of maximum variation in the orbital eccentricity of
an initially circular orbit for Charon (eC = 0.0); however, in the satellites during the integrations, ∆e = (emax − emin ). Sim-
their simulations they did not consider the impact of the zonal ilarly, ∆i = (imax − imin ) and ∆a = (amax − amin ). Regions with
harmonic of Pluto’s oblateness. higher ∆e lead to chaotic motion, and ∆e is a powerful indica-
The results obtained by either the tidal transport scenario tor of secular and resonant dynamics that also serves to identify
or the in situ formation scenario are still inconclusive due
to the poor knowledge of the precise orbital elements and 1 Pires Dos Santos et al. (2011) reported gravitational effects due to
masses of the small moons. The most precise orbital parame- Nix and Hydra on the test particles in the external region of the PC
ters of the PC system, reported in Brozović et al. (2015) and binary, but the masses of the moons used were one order larger than
Showalter & Hamilton (2015) (see Table 1), present a dispersion those reduced from the observations.
A99, page 2 of 18
C. A. Giuppone et al.: Past and present dynamics of the circumbinary moons in the Pluto-Charon system
Fig. 1. Dynamical maps on the a–e plane (left panel) and a–I plane (right panel) of the moon’s initial conditions calculated for the PC binary and
a small moon of Styx’s mass with initial Styx orbital elements from JPL in Table 1. The current positions of the moons are shown by red crosses.
The locations of the main MMRs are indicated, and the grey line in the right panel indicates the current inclination of Charon. The colour scale
varies with ∆e and ∆i, increasing from blue (regular motion) to red (chaotic motion), as shown in the colour bars on the top of the figure.
Table 1. Orbital elements and masses of the Pluto satellites.
Parameter Charon Styx Nix Kerberos Hydra
a/RP (a) 16.51 36.92 41.55 48.99 55.00
a/RP (b) 16.50 35.70 40.98 48.61 54.47
a/RP (c) – 35.91 40.99 48.64 54.49
e × 10−3 (a) 0.58 36.70 16.78 6.82 12.22
e × 10−3 (b) 0.05 0.01 0.00 0.00 5.54
e × 10−3 (c) – 5.8 2.0 3.3 5.9
inc (◦ ) (a) 112.90 112.86 112.92 112.57 113.09
inc (◦ ) (b) 0.0 0.0 0.0 0.4 0.3
inc (◦ ) (c) – 0.81 0.13 0.39 0.24
M (◦ ) (a) 356.76 18.79 49.94 326.33 32.45
ω(◦ ) (a) 293.25 348.42 293.25 293.25 293.25
Ω(◦ ) (a) 227.40 227.42 227.45 227.09 227.16
Mass (kg) (b) 1.59 × 1021 1.49 × 1015 4.49 × 1016 1.65 × 1016 4.79 × 1016
Errors (km) (b) 5 514 23 135 29
∆a (km) (a) 0.043 1761 1086 637 443
∆e (a) 7.13 × 10−5 0.034 0.025 0.018 0.016
∆i (◦ ) (a) 7.26 × 10−4 0.08 0.12 0.88 0.58
∆a (km) (d) 1. × 10−3 1792 1147 616 437
∆e (d) 1.8 × 10−7 0.041 0.031 0.015 0.015
∆i (◦ ) (d) ∼0.00 0.08 0.12 0.88 0.58
Notes. We use RP = 1188 km and mP = 1.303 × 1022 kg. M, ω, and Ω are the mean anomaly, argument of pericentre, and longitude of the node,
respectively. (a) Osculating orbital elements from the JPL Horizons site setting the ecliptic as reference plane, at epoch 1 January 2021. (https:
//ssd.jpl.nasa.gov/?horizons). (b) Averaged mean orbital elements from Brozović et al. (2015), derived based on 200 yr of orbital integration.
The epoch for the elements is JED 2451544.5. Errors correspond to in-orbit uncertainties; the mass of Styx has a solution with a null value, but we
adopted a 1σ value. (c) Orbital elements from Showalter & Hamilton (2015), obtained fitting a precessing ellipse. The epoch is Universal Coordinate
Time (UTC) on 1 July 2011. Orbital element variations (∆a, ∆e, and ∆i) correspond to those extracted from JPL ephemerides(a) and to N-body
integrations considering a three-body problem(d) in a 40 yr time span.
separatrices of MMRs (e.g. Dvorak et al. 2004; Ramos et al. where a, e, and I are the semi-major axis, the eccentricity, and
2015). the inclination of the particle, respectively. In the stability maps,
The instabilities in the motion of the particles are represented the initial conditions for Charon are given in Table 1, whereas
using a colour scale on the maps in Fig. 1, on the a–e plane the initial values of the angular variables of the circumbinary
(left panel) and a–I plane (right panel) of the initial conditions, moon are those given to Styx in Table 1. We note that all moons
A99, page 3 of 18
A&A 658, A99 (2022)
lie beyond the black continuous curve in the left panel, which deformations of the bodies are delayed by a constant ∆t, usually
delimits the domain of stable motion, according to the Holman– referred to as the tidal time lag (Mignard 1979). In this sce-
Wiegert criterion (HWC) derived in Holman & Wiegert (1999) nario, the PC tidal interaction is modelled following Cheng et al.
for circumbinary configurations and also noticed by Smullen & (2014a)2 , who assumed the collisional origin of Charon, when
Kratter (2017). Charon emerges with a non-zero orbital eccentricity, eC , and an
Figure 1 shows that the orbits of the moons are nearly circu- initial semi-major axis of aC = 4 RP (Ward & Canup 2006).
lar, nearly planar, and nearly resonant. Although these features Both Pluto and Charon would be spinning after the collision,
of the Pluto system have already been pointed out in previous with the spin axes oriented in some directions with respect to the
studies, it is still worth analysing them in more detail. Indeed, orbital plane of the binary. Due to the relatively small size and
as seen on the stability maps on the a–e and a–I planes, the mass of Charon, in comparison with Pluto, the contribution of
small satellites are confined to the very small eccentricity and its spin to the total angular momentum is small, and it evolves
inclination regions of regular motion (blue colour). According to quickly to the asymptotic tidal spin rate (in less than 10 yr). For
secular perturbation theories, very low eccentricities and incli- the sake of simplicity, in this work we assume zero obliquity for
nations are characteristic of the dynamical systems that have an both Pluto and Charon; the model of the PC binary, account-
angular momentum deficit (AMD) close to zero (Michtchenko ing for the arbitrary obliquities and the Sun’s perturbations, is
& Rodríguez 2011). Regarding initially excited dynamical sys- studied in Correia (2020).
tems (in this case, due to the collisional origin of Charon), the The total angular moment of the PC binary can be written,
low-AMD configurations are generally attained by the systems in the chosen reference frame and with the assumed approxima-
evolving under weak and slow dissipation. tions, as
The interpretation of the near resonant distribution of the q
small satellites requires the analysis of an additional dynamical L = CP ΩP + CC ΩC + MPC nC a2C 1 − e2C , (1)
mechanism, that is, a capture into MMRs. The stability maps in
Fig. 1 show the MMRs produced by Charon as vertical strips of where MPC = mP mC /(mP + mC ) and nC are the reduced mass
chaotic motion (red colour). The strong low-order MMRs, such of the binary and the mean motion of Charon, respectively.
as 3/2, 5/3, 2/1, 5/2, and 3/1, are located inside the domain of According to classical theories, L is conserved during the tidal
instability defined by the HWC (black curve in the left panel). In expansion of the PC binary. Hence, its amount for the PC sys-
the low-eccentricity region, the 3/1 MMR can be regarded as an tem can be obtained knowing the masses of Pluto and Charon,
inferior limit of stability of the motion, and Styx is stuck to it. mP and mC , and the current values of aC and nC . Indeed, at the
This moon lies so close to the 3/1 MMR that uncertainties in the dual synchronous state of the current PC binary, Charon’s orbit
determination of the angular variables of its orbit, such as the is circularised, that is, we can assume eC = 0 and ΩP = ΩC = nC .
mean longitude and the longitude of the pericentre, might put Then, by substitution of these conditions into Eq. (1), the L
it closer to the chaotic layer of the 3/1 MMR (e.g. in-orbit and value is obtained; in the following, this value is used to obtain the
radial errors reported in Brozović et al. 2015). The same is also angular velocity of Pluto, ΩP , resolving Eq. (1) for given start-
true for Nix with respect to the 4/1 MMR (see the Styx and Nix ing values of aC , eC , and ΩC , the angular velocity of Charon’s
panels in Fig. A.2) but is most noticeable for Kerberos and Hydra rotation.
with respect to the 5/1 and 6/1 MMRs, respectively (see the bot- Following Cheng et al. (2014a), we define the dissipation
tom panels of Fig. A.2). This fact suggests that the small moons parameter, A, as
may have been involved in one of the MMRs at some period of
their common history. Therefore, in this paper we analyse the k2C ∆tC mP 2 RC 5
interaction of the small moons with the main MMRs during the A= , (2)
k2P ∆tP mC RP
tidal expansion of Charon’s orbit.
It is worth emphasising that the uncertainties in the determi- where (k2C , k2P ) are the second-order Love numbers and (RP ,RC )
nation of the moons’ masses and the actual positions can provide are the radii for Pluto and Charon, respectively. We note that, for
a general view of the phase space of the PC system, but without the masses and the radii of Pluto and Charon (RC = 606 km),
specifying whether the objects are evolving near or inside the
MMRs. !2 !5
mP RC
' 2; (3)
mC RP
3. Tidal model thus, the constant A can be closely understood as the ratio of
time lags between Charon and Pluto. Cheng et al. (2014a) have
In this section we describe the tidal model, which we apply to also shown that, for an appropriate value of the ratio (A ∼ 10),
investigate the orbital expansion of Charon and the implications the eccentricity of Charon’s orbit, eC , remains roughly constant
for the orbital motion of the small moons. As described in the during most of the tidal evolution.
previous section, the orbital configuration of Pluto’s satellites Our current knowledge of the physical properties of the PC
is better represented in terms of the modified Jacobi elements. system does not allow us to constrain the range of A values
The initial Jacobi orbital elements are transformed to the Jacobi (see Eq. (2) and Lainey 2016; Quillen et al. 2017). However, it
rectangular coordinates and velocities in a three-dimensional ref- is easy to show through classical tidal theories that for A > 1 the orbital expansion and circularisation are
details, see Rodríguez et al. 2013). We considered the system dictated by the tides on Charon. Previous works adopted dif-
consisting of Pluto, Charon, and an external small moon, where ferent ranges for the possible values of A, namely, 1 < A < 18
two large bodies undergo mutual tidal interaction but the external
moon is not (directly) tidally affected, due to its small size and 2 We refer the reader to this work for a complete analysis of the PC
mass. We adopted a classical linear tidal model, in which the tidal evolution.
A99, page 4 of 18
C. A. Giuppone et al.: Past and present dynamics of the circumbinary moons in the Pluto-Charon system
(Cheng et al. 2014a,b); A = 10 or A = 40 (Woo & Lee 2018); and resonance (left-top panel), but its motion is still stable. It is worth
1 < A < 16 (Correia 2020). In this work, we initially explored noting here that the 3/1 MMR can be considered as an inferior
the wide range of 0.01 < A < 100; however, in order to place limit of the stability of the circumbinary motion, as shown in
test moons at the current Styx position, we adopted the range Fig. 1.
8 < A < 40. According to Fig. 2, the actual position of Styx has been
Cheng et al. (2014a) show the tidal evolution of the PC crossed by the 4/1 MMR at some moment in the past, as has
system for different initial eccentricities of Charon. Here, we that of Nix (right-top panel) at another period. Our simulations
summarise the common features observed in the numerical of the moons’ dynamics (described in the next section) show
integrations, which we refer to in our discussions below: (i) that the perturbations produced by this resonance increase with
Independently of the initial values of Charon’s eccentricity, eC , the increasing eccentricity of Charon. For Charon’s paths with
it presents a peak at some moment during the PC expansion; eC = 0.1 and 0.2, Styx and Nix spend some time evolving in
(ii) the time evolution of ΩP /n is correlated with the peak of the 4/1 MMR, which might be sufficient to strongly excite their
eC and can reach values of ∼10; (iii) the semi-major axis of eccentricities and even eject them from their current positions.
Charon attains its current value (16.5 RP ) and the orbit is cir- On the contrary, for Charon on a quasi-circular orbit (eC = 0.01),
cularised, whereas both rotations synchronise with the orbital the passages of Styx and Nix through the 4/1 MMR may occur
period after around 3 × 106 yr. It is worth emphasising that, quickly and without significant excitation of their motions.
according to tidal evolution theories, the double synchronous The width and, consequently, the dynamical effects of the
rotation is only attained when the orbit is completely circu- higher-order MMRs, such as 5/1, 6/1, and, 7/1, rapidly decrease
larised (see Cheng et al. 2014a). Also, at the end of the evolution, with decreasing order, as seen in Fig. 2. We expect that their pas-
the angular momentum exchange between Pluto and Charon sages through the current positions of the small moons would
ceases and both orbital and rotational components attain their not produce perturbations to destabilise their motion, particu-
equilibrium configurations in the system. larly for Charon evolving on a quasi-circular orbit during these
passages. Thus, we conclude that, in order to ensure the simulta-
neous orbital stability of the four small moons during the orbital
4. Dynamics of the moons in a tidally evolving PC expansion of the PC binary, Charon’s eccentricity should be con-
system strained to lower values. This fact points towards a dynamical
scenario in which the adequate choice of Charon’s eccentricity
In this section we investigate the dynamics of the small moons at the beginning of the tidal expansion is essential in order to
located at their current positions during the tidal expansion of guarantee the orbital stability of the system of moons. In the
Charon’s orbit. For this, we use the same techniques described next section we numerically simulate the impact of a migrat-
in Sect. 2. Considering that Charon’s orbital elements vary dur- ing Charon on the orbital motions of Styx, Nix, Kerberos, and
ing the migration, we mapped the neighbourhood of each moon, Hydra by taking simultaneous effects of the tidal evolution and
applying the conservative model for different values of the semi- gravitational perturbations into account.
major axis, aC , and the eccentricity, eC , of Charon’s orbit and
the fixed mass and initial conditions of the satellite (Table 1).
In this way, we can visualise the dynamical behaviour of each 5. Simulations of the moons’ behaviour under the
moon at different configurations of the PC binary acquired in the tidal expansion of Charon’s orbit
past. Figure 2 shows the dynamical maps on the aC –eC planes,
obtained for the current locations of Styx, Nix, Kerberos, and In this section we analyse the behaviour of Styx, Nix, Kerberos,
Hydra retrieved from JPL Horizons. The black curves show and Hydra during the orbital expansion of Charon’s orbit and
the positions of Charon on these planes as obtained through their passages through the MMRs. Here, we assumed the in
simulations of its tidal expansion in the past (described in the situ formation of the small moons, that is, our numerical sim-
previous section). Three values were used for Charon’s eccen- ulations start with the satellites at their current positions. We
tricity, namely 0.01, 0.1, and 0.2, while the other parameters were worked in the modified Jacobi reference frame, where small
fixed at aC /RP = 4, ΩC /nC = 2, A = 10, and ∆tP = 600 s. By moons are orbiting the barycentre of the PC binary. In this frame,
construction, the three paths converge to the current position of we neglected the mutual perturbation between the moons, due
Charon, at aC 16.5 RP and eC 0. to their small masses, which allowed us to solve their orbital
Now we can investigate the dynamical behaviour of each motions in a separate way. We solved the exact equations of
moon according to the position of Charon at any moment of motion for each satellite according to the tidal model described
its history. Indeed, for a chosen position of Charon in one of in Sect. 3 and following Rodríguez et al. (2013). We consid-
its paths on the aC –eC planes in Fig. 2, we can identify dynami- ered coplanar configurations and the initial mean anomalies, and
cal features that affect the behaviour of the moon at that instant. longitudes of pericentre were set to zero.
These features are mainly MMRs, which are identified by plot- We performed a set of numerical simulations, varying the
ting the moon’s dynamics with the colour scale. In this way, main parameters of the problem, namely, the initial semi-major
we can distinguish between the stable (blue-white) and unsta- axis and eccentricity of Charon’s orbit (aC , eC ), the initial rota-
ble (red) behaviour of satellites. The instabilities of the motion tion velocities of Pluto and Charon (ΩC and ΩP ), the time lag
are clearly related to the MMRs, whose locations are indicated of Pluto, ∆tP , and the constant of dissipation, A (see Table 2).
by the corresponding ratio in Fig. 2. The initial orbital angles of Charon were set to zero. We fixed
The strongest MMRs are of low order, 3/1 and 4/1, and a fiducial set of parameters such that aC /RP = 4, eC = 0.001,
we expect that the dynamical stability of the moons would be ΩC /nC = 2, A = 10, and ∆tP = 600s. Then, the simulations were
significantly affected by these resonances. Figure 2 shows that done varying the parameters one by one, choosing the values
Charon’s paths (black curves) never cross the 3/1 MMR, and, listed in Table 2. It is worth noting that the initial value of ΩP is
consequently, none of the four small moons suffer the destabil- not a free parameter but is determined by the conservation of the
ising effects of this resonance. Currently, Styx stays close to this angular momentum of the binary through Eq. (1).
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A&A 658, A99 (2022)
Fig. 2. Dynamical maps of Styx, Nix, Kerberos, and Hydra on the aC –eC plane of Charon’s initial conditions. On each map, black curves are
Charon’s tidal trajectories, obtained for initial eC values of 0.01, 0.1, and 0.2; the initial conditions of the corresponding moon are fixed at those
given in JPL from Table 1. The blue tones represent regular motions, while the brick tones correspond to increasing instabilities and chaotic motion;
the colour bar on the top shows ∆e. The locations of the main MMRs are indicated.
Table 2. Parameters adopted in the numerical simulations. the results obtained were discarded since Charon did not reach
its current semi-major axis after 100 Myr in these cases. We
integrated the equations of motions over 100 Myr for ∆tP = 6 s
Parameter Values
and ∆tP = 60 s; for ∆tP = 600 s, we considered 10 Myr as the
aC /RP 3, 4, 5, 6, 7 integration timescale.
eC 0, 0.001, 0.01, 0.1, 0.2 In the following we summarise the results obtained for each
ΩC /nC 0.5, 1.5, 2, 3, 4 of the four moons, emphasising the cases in which the final
A 0.01, 0.1, 1, 10, 100 orbital configuration of the small moon is similar to its current
∆tP (s) 6, 60, 600 one. We note that, despite the fact that the moons are currently in
nearly resonant configurations, the uncertainties in their masses
Notes. Here, nC is the mean motion of Charon. We note that the initial and orbital parameters could still place them inside the MMRs
value of ΩP is calculated taking the conservation of the total angu- (see our discussion in Sect. 1 and Fig. A.2). In Table 3 we include
lar momentum of the system into account (see Eq. (1)). The adopted the cases for which each of the moons reaches the end of the
parameters cover a broad range of initial conditions, as also suggested simulation in stable motion. Additionally, we identify the initial
in previous works (e.g. Canup 2005, 2011; Cheng et al. 2014a,b; Correia
conditions for which the final eccentricity of the moon is smaller
2020).
than 0.05, which is compatible with the current eccentricities of
the moons (see our Fig. A.1).
The masses and current semi-major axes of the external 5.1. Styx
moons were taken from Brozović et al. (2015) (see Table 1).
The initial orbital eccentricities of the moons were assumed to Styx is located at a mean distance of 42 650 km ('35.9 RP ) from
be almost circular (e = 10−5 ). We note that in this section we the barycentre of the PC binary and is close to the 3/1 MMR with
work in the frame of the planar problem, when the small moon Charon (see Figs. 1 and A.2). Other initial conditions different
evolves on the orbital plane of the PC binary. Physical parame- from those shown in Table 3 tested for this satellite lead to either
ters, such as the masses and radii of Pluto and Charon, are the ejection of the small moon from the system (Fig. 3, left panel)
same as described in Sect. 3. Since the tidal evolution of the PC or a final orbit that is different from the current one (Fig. 3, right
binary is not affected by the external small moon, we do not panel). In most cases, instabilities occur when Styx approaches
show the variations in their orbital and rotational components in the 4/1 or 5/1 MMRs, in this last case, only for a high initial eC .
this section (see Fig. 2 in Cheng et al. 2014a). In addition, we Figure 4 illustrates Styx’s orbital evolution obtained for
performed simulations for both ∆tP = 0.06 s and ∆tP = 0.6 s, but A = 100. We note that, despite an apparently slow increase,
A99, page 6 of 18
C. A. Giuppone et al.: Past and present dynamics of the circumbinary moons in the Pluto-Charon system
Table 3. Parameters and initial conditions under which the four small moons survive in the numerical simulations.
Parameter Styx Nix Kerberos Hydra
aC /RP 4 3, 4, 5, 6, 7 3, 4, 5, 6, 7 3, 4, 5, 6, 7
eC 0 0, 0.001, 0.01 0, 0.001, 0.01, 0.1, 0.2 0, 0.001, 0.01, 0.1, 0.2
ΩC /nC 1.5, 2 0.5, 1.5, 2, 3, 4 0.5, 1.5, 2, 3, 4 0.5, 1.5, 2, 3, 4
A 10, 100 10, 100 0.01, 0.1, 1, 10, 100 0.01, 0.1, 1, 10, 100
∆tP (s) 6, 60, 600 6, 60, 600 6, 60, 600 6, 60, 600
Notes. Values in bold correspond to the simulations resulting in a good agreement with the current orbital configuration of the small moons
(e < 0.05) according to the adopted criteria (see Sect. 5).
Fig. 3. Two typical examples of the Styx orbital evolution resulting
in either instability (left panel) or the final configuration, which is
significantly different from its current orbit (right panel). Left panel:
initial conditions used are aC /RP = 5, eC = 0.001, ΩC /nC = 2, A = 10,
and ∆tP = 600 s. Styx is ultimately ejected from the system when the
system crosses the 4/1 MMR between Charon and Styx (not shown),
resulting in a strong increase in Styx’s eccentricity. Right panel: initial
conditions used are aC /RP = 4, eC = 0.001, ΩC /nC = 2, A = 10, and
∆tP = 6 s. The orbital eccentricity is excited, due to the crossing of the
4/1 and 7/2 MMRs (not shown), to attain the mean value '0.1, which is
substantially larger than the current Styx eccentricity.
the semi-major axis of Styx remains close to the current value. Fig. 4. Examples of initial conditions for Styx. Top row: time evolution
The eccentricity acquires a mean value of 0.02, varying roughly of Styx’s semi-major axis (left) and eccentricity (right) during the tidal
between 0.007 and 0.04, which is also consistent with the oscil- expansion of the PC binary. The initial conditions used in the simula-
lation of Styx’s current eccentricity. The final value of the tions are aC /RP = 4, esC = 0.001, ΩC /nC = 2, A = 100, and ∆tP = 600 s.
mean-motion ratio is larger than the nominal location of the The final values of the orbital elements remain close to the current
ones. Bottom row: time evolution of the mean-motion ratio (left) and
3/1 MMR (see Fig. 1); the difference between the pericentre lon- ∆$ (right). We note that the mean value of nC /n is larger than 3. The
gitudes of Styx and Charon, ∆$, oscillates around 180◦ , which secular angle, ∆$, oscillates around 180◦ .
is an indicator of the dissipative migration (see Sect. 4). We have
verified that, for this particular simulation, all critical angles
associated with the 3/1 MMR circulate. configurations similar to the current orbit of Nix are shown
The results obtained for Styx point out that the scenario with in Table 3. Figure 5 shows examples of the orbital evolution
an initially quasi-circular orbit of Charon and a high dissipation obtained for aC /RP = 3 and aC /RP = 7. For aC /RP = 3 (orange
ratio (large A) favours the stability of Styx’s motion during the curves), the resulting orbit clearly deviates from the current
orbital expansion of the PC binary. However, we note in Table 3 orbital configuration of Nix. In this case, the trapping into the
that only very specific combinations of the parameters allow 4/1 MMR excites Nix’s eccentricity, up to e = 0.16. The final
Styx to remain on a stable orbit as the 4/1 MMR with Charon semi-major axis is larger than the current value, with a differ-
is crossed. For most of the explored initial conditions, Styx’s ence of '2 RP . Moreover, the critical angle associated with the
motion is strongly excited by the passage through this resonance, 4/1 MMR, φ4/1 = 4λ − λC − 3$C , librates around 180◦ , as do all
resulting in the escape from the system. In Sect. 6 we test the other angles associated with the same MMR, while ∆$ oscil-
orbital stability of Styx (and the other small moons) assuming lates around 0◦ (Fig. 5, bottom right). In this case, eC reaches
an on-time accretion of Charon’s mass. '4 × 10−5 at 107 yr in one of the simulations due to perturbations
from Nix (not shown here).
5.2. Nix On the other hand, for aC /RP = 7 (blue curves), the orbital
evolution of Nix occurs with the eccentricity and semi-major
Nix lies at a mean distance of 48 690 km ('41 RP ) from the axis close to their current values. All critical arguments asso-
PC barycentre and is currently close to the 4/1 MMR with ciated with the 4/1 MMR circulate in the case aC /RP = 7,
Charon. The values of the initial conditions resulting in final indicating that the moon is out of the resonance, although it is
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A&A 658, A99 (2022)
Fig. 5. Examples of initial conditions for Nix. Top row: time variations Fig. 6. Examples of initial conditions for Kerberos. Top row: orbital
in Nix’s semi-major axis (left) and eccentricity (right) for two different evolution of Kerberos for two initial conditions of eC , namely, 0.001
initial values of aC . The initial conditions are aC /RP = 3 and aC /RP = and 0.1. The other parameters are aC /R = 4, ΩC /nC = 2, A = 10, and
7, eC = 0.001, ΩC /nC = 2, A = 10, and ∆tP = 600 s. For aC /RP = 3, ∆tP = 600 s. These simulations are successful according to the adopted
the eccentricity of Nix is excited to a mean value close to 0.16. Bottom criteria (see Sect. 5). Bottom row: time variations in the mean-motion
row: time evolution of the mean-motion ratio (left) and two angles for ratio (left) and φ5/1 = 5λ − λC − $C − 3$ (right). We show the angle for
Nix (right), namely, the secular angle, ∆$, and the 4/1 MMR critical the specific case of eC = 0.001. The mean value of nC /n is larger than 5.
angle φ4/1 = 4λ − λC − 3$C . The mean value of nC /n is slightly smaller
than 4 for aC /RP = 7, whereas φ4/1 and ∆$ librate around 180◦ and 0◦ ,
respectively, for aC /RP = 3. 5.4. Hydra
Hydra is the most distant known satellite of the PC binary,
with a current barycentric semi-major axis of a ∼ 54.5 RP . This
still very close to it. In this case, the mean-motion ratio oscillates small satellite is close to the 6/1 MMR with Charon. Most of
around a mean value slightly smaller than the nominal position the simulations reproduce the current orbit of Hydra, as shown
of the 4/1 MMR (Fig. 5, bottom left), in accordance with the cur- in Table 3. Figure 7 shows the evolution for ∆tP = 6 s (brown
rent position of Nix given in Brozović et al. (2015) (see Figs. 1 curves) and ∆tP = 60 s (grey curves). We can note that, for
and 2). ∆tP = 60 s, Hydra is trapped in the 6/1 MMR, with the critical
The results shown in Table 3 indicate that, in comparison angle φ6/1 = 6λ − λC − $C − 4$ librating around 0◦ . Moreover,
with Styx, almost 40% of the tested initial conditions result in the ratio nC /n is slightly smaller than 6 for this specific trapping.
the final orbital configurations, which are similar to the current In the case with ∆tP = 6 s, all critical angles of the 6/1 MMR are
orbit of Nix. circulating.
As in the case of Kerberos, all initial conditions result in a
stable motion of the test moons, with e < 0.05, indicating good
5.3. Kerberos agreement with the current orbit of Hydra.
Kerberos is the third small satellite orbiting the PC binary, at a
mean distance of 48.6 RP , which places it close to the 5/1 MMR 5.5. Discussion of the moons’ behaviour and proximity to
with Charon. Table 3 shows that all numerical simulations result MMRs
in good agreement with the current orbit of this small moon.
Figure 6 shows the orbital evolution of Kerberos during the Analysing the parameter values in Table 3, we realise that it
tidal expansion of the PC binary for two values of the initial is unlikely that we can collect a specific set of parameters and
Charon eccentricity, eC = 0.001 (grey curves) and eC = 0.1 (red initial conditions that would result in a final configuration of
curves). For both initial conditions, the output of the simula- all four satellites that is consistent with the current one. This
tions satisfactory fits the current orbit of the moon. All critical fact is mainly due to the difficulty of finding Styx in a stable
angular combinations of the 5/1 MMR circulate, and the ratio configuration.
nC /n oscillates around a mean value slightly larger than 5, which Since the determination of the satellite’s current orbits is
places the moon outside, but still close to, the 5/1 MMR (see affected by uncertainties, it is not clear whether they are captured
Fig. 2). We show the angle φ5/1 = 5λ − λC − $C − 3$, obtained or not in the MMRs. Thus, for the sake of completeness, we show
for eC = 0.001, in the bottom-right panel of Fig. 6. At around in Table 4 the values of the parameters that provide the final
5 × 106 yr, this angle changes its behaviour from circulation to orbits of the moons trapped in the nearest MMR with Charon
libration around 0◦ . with at least one of the associated critical angles librating. It is
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C. A. Giuppone et al.: Past and present dynamics of the circumbinary moons in the Pluto-Charon system
5.6. Survival time of the small moons under J2 evolution
In order to refine our model, we introduced an additional force
due to the oblateness of Pluto. Pluto’s polar oblateness, param-
eterised by the zonal harmonic, J2 , is modelled as (Cheng et al.
2014b)
kfP R3P Ω2P
J2 = , (4)
3GmP
where k f P is the second-order fluid Love number of Pluto and
ΩP is the angular rotation velocity of Pluto. Recently, Correia
(2020) derived from the shape of Pluto a static value of J2 =
6 × 10−5 . In this work, we use both formulations, adding them to
the model described in Sect. 3.
We applied the model to study the behaviour of a ring
composed of 300 particles, each with a very small mass of
1.49 × 1015 kg (thus, we can neglect their mutual perturbations
and contributions in the total angular momentum of the system),
at different positions chosen from the interval of 10 RP < a <
55 RP of the barycentric semi-major axis. We used the constant
∆t model, A = 10, and the different initial values of Charon’s
eccentricity, eC . It is worth noting that the value of eC constrains
ΩP (and, equivalently, the initial rotation period of Charon). For
Fig. 7. Examples of initial conditions for Hydra. Top row: orbital vari- example, starting at aC = 4RP , with eC = 0.001 and ΩC /nC =2
ation in the semi-major axis (left) and eccentricity (right) of Hydra (i.e. PC = 0.38136 d) and using Eq. (1), we calculate the corre-
according to the adopted criteria. The initial conditions are aC /RP = 4, sponding initial rotation period of Pluto as being P0 = 0.13858 d
eC = 0.001, ΩC /nC = 2, A = 10, and ∆tP = 6 s. Bottom row: time varia- (P0 = 2π/ΩP ).
tion in the mean-motion ratio (left) and φ6/1 = 6λ − λC − $C − 4$ (right) We worked with two kinds of disc particles: one is coplanar
for two values of ∆tP , namely, 6 s and 60 s. We note that, for ∆tP = 60 s, with the orbital plane of the PC binary, and the other is inclined,
φ6/1 librates with a large amplitude around 0◦ . For ∆tP = 6 s, all critical with the angle randomly chosen from the interval −0.01◦ < i <
angles associated with the 6/1 MMR circulate. 0.01◦ . In the coplanar disc, the particles start on nearly circular
orbits, with eccentricities uniformly distributed in the interval
0.0 < e < 10−3 . For Charon’s initial configuration, we consid-
Table 4. Parameters resulting in captures in the MMRs with Charon. ered aC = 4 RP and three values of the eccentricity, eC , namely,
0.001, 0.1, and 0.2. In the case of the inclined disc, the particles
Parameter Nix Kerberos Hydra remain on nearly circular orbits, while aC = 4 RP and eC = 0.001.
In both cases, we set random initial conditions for the angular
aC /RP 3, 5, 6 3 3, 5, 6 variables of Charon’s and the particles’ orbits. The system com-
eC – 0, 0.001 0, 0.001 posed of the PC binary and the ring of particles was integrated
ΩC /nC 4 0.5, 1,5, 3, 4 0.5, 1,5, 3, 4 over 3 × 106 yr (approximately 150 × 106 periods of the binary)3 .
A – 0.01, 0.1, 1, 100 0.01, 0.1, 1, 100 The case of the initially inclined disc of particles was con-
∆tP (s) 60 60 60 sidered due to recently published results (Ćuk et al. 2020) that
demonstrate that the tidal evolution of particles at low incli-
Notes: Styx’s simulations did not produce any captures in the MMRs. nations can induce resonant trapping and the excitation of the
particles’ inclination, which is incompatible with the observed
low inclinations of the moons in the PC system. The tests with
an initially eccentric orbit for Charon were inspired by the results
remarkable that the motion of Styx in the 3/1 MMR is always
of the smooth particle hydrodynamic simulations reported in
unstable, which confirms our suggestion that the 3/1 MMR
Canup (2005, 2011).
delimits the domain of stable motion in the PC binary.
Figure 8 shows the survival times of the particles from the
For the sake of completeness, we compare our results with
different discs. The results were obtained considering the zonal
those previously reported by Woo & Lee (2018). Their model
harmonic, J2 , which evolves as the PC binary tidally expands,
considered A = 10 and A = 40 and an initial eC = 0 and eC = 0.2,
according to Eq. (4). We also investigated the behaviour of the
in the case of tidal model of constant ∆t. For a large A and
disc considering the cases of J2 = 0 and J2 = 6 × 10−5 , the lat-
an initial eC = 0, all test particles survive the entire simulation
ter value derived from Correia (2020); the results obtained are
(see their Table 3, last row). Moreover, the particles near the
discussed in the appendix. We note in Fig. 8 that the particles
MMRs with Charon have final eccentricity values larger than
can survive the tidal expansion of the PC binary (that is, remain
the current ones. On the other hand, for small and large A and
in the PC system after 3 × 106 yr) only when Charon starts on a
initial eC = 0.2, the final orbits do not match the current ones.
nearly circular orbit (eC = 0.001) and only in the region beyond
These results are in good agreement with ours; however, Woo
Nix’s orbit. Also, there are no significant differences between
& Lee (2018) only considered fixed values of ∆t = 600 s and
the coplanar and low inclined discs (blue crosses and orange
ΩP /n = 5.65 and an initial distance for Charon of aC = 4 RP ,
while our study presents a wider range of initial parameters and 3 According to Cheng et al. (2014a) and Correia (2020), the PC binary
tidal dissipation. acquires its current equilibrium position after (2−4) × 106 yr.
A99, page 9 of 18A&A 658, A99 (2022)
the disc of particles in the PC system in Sect. 6.4. In this section
we choose the plane of Charon’s orbit as a reference plane.
6.1. Tidal evolution of the PC binary with the accreting
Charon mass
We started by considering the evolution of the PC pair described
in Sect. 3, now including the accretion of Charon from the disc of
debris. The duration of the accretion process was arbitrarily fixed
at 104 yr after a giant impact, starting with Charon’s mass at 60%
of its current value, which is in accordance with the results in
Canup (2005) and Kokubo et al. (2000). Following Kokubo et al.
(2000), we adopted a logarithmic law for the mass accretion, but
we also considered an oligarchic growth defined as m ∝ t1/3 . The
evolution of the increasing mass of Charon is shown in the left
panel of Fig. 9 for both models. Assuming Charon to be a homo-
Fig. 8. Survival times of the small particles, for the different discs geneous sphere, we calculated its radius for a constant density of
extending from 10 RP to 55 RP , during the tidal expansion of the PC ρ = 1.70 g cm−3 and show its time evolution in the middle panel
binary, with the dissipation parameter A = 10, under the effect of the of Fig. 9. Finally, the right panel shows the time evolution of the
evolving zonal harmonic, J2 . The colour symbols are used to identify zonal harmonic, J2 , of Pluto’s oblateness during the tidal migra-
the particles from the different discs (see text). tion of a growing Charon. According to Eq. (4), J2 is a function
of the orbital spin of Pluto ΩP , whose starting value was chosen
dots, respectively). Contrarily, in the case of an initially eccentric as described below. For the chosen ΩP , J2 starts at ∼0.17 and
Charon orbit, all particles (green crosses and red dots) are ejected reaches 8 × 10−5 at the end of the migration, for both logarith-
from the system in less than 103 yr due to capture in the MMRs mic and oligarchic laws of the mass accretion. We note that both
and the subsequent excitation of the eccentricities, as described models represent almost the same evolution of J2 . We also show
in Cheng et al. (2014a) and Kenyon & Bromley (2019b). in the right panel of Fig. 9 the constant J2 of 6 × 10−5 derived in
For the quasi-circular Charon orbit, the low-order 3/1 and Correia (2020).
4/1 MMRs, at the positions of Styx and Nix, respectively, are It is worth emphasising that, in the growing mass scenario,
responsible for the rapid ejection of the particles (less than the total angular momentum of the PC binary given in Eq. (1) is
104 yr). In this case, the survivors are observed only beyond no longer conserved. Thus, we need to obtain the initial value of
Nix’s orbit. Moreover, even farther, in the regions around the Pluto’s spin, ΩP , and the dissipation ratio, A, in such a way that
current positions of Kerberos and Hydra (5/1 and 6/1 MMRs, both can guarantee the current double synchronous configuration
respectively), most of the fictitious moons do not survive more of the binary with orbital parameters compatible with those from
than 105 yr. It is clear that these times are incompatible with the Table 1.
hypothesis of primordial moons after the system settled into its Using a logarithmic law for Charon’s accretion and fixing
present configuration through an impact on Charon (Bromley & A = 10 and eC = 0.001, we varied the initial rotation period of
Kenyon 2020). Pluto, P0 = 2π/ΩP , in the range from 0.12 d to 0.21 d in order to
It is worth observing that, applying the simplified model, determine the value that assures, at the end of the migration, the
which considers the constant value of J2 (zero or J2 = 6 × 10−5 ), current configuration of the PC binary. We chose a low initial
we obtain that the particles starting on nearly circular and nearly eccentricity for Charon due to constraints already described in
planar orbits at the current positions of Styx and Nix can sur- Sect. 5.6. We simulated the tidal evolution of the PC binary and
vive the tidal expansion of the PC binary (for details, see the show the results in Fig. 10, where the time evolution of Charon’s
appendix). However, in the more realistic model, which accounts semi-major axis, aC (in units of RP ), is presented in panel (a). The
for the zonal harmonic, J2 , evolving during the expansion, the horizontal dashed line, which shows the final state of the current
additional perturbations destroy the stability of the particles in system, corresponds to the initial value P0 = 0.149 d. Smaller
the range 10 RP < a < 55 RP . Moreover, no particles remain at initial values of P0 lead to wider PC binaries, while larger values
the current positions of Styx and Nix. We repeated these exper- of P0 lead to more compact systems, more similar to the current
iments with a higher dissipation ratio, A = 20, and qualitatively PC configuration. The eccentricity of the binary is more effi-
obtain the same result. ciently damped for higher initial values of P0 , as shown in panel
b in Fig. 10. Panels c and d show the evolution of the angular
velocities of Pluto and Charon, ΩP and ΩC (in units of the mean
6. Accreting Charon’s mass motion nC ), respectively; it is clear that the system reaches the
double synchronous rotation more rapidly for higher values of
In this section we introduce a scenario that considers the mass P0 .
accretion of proto-Charon from the disc of debris left soon after The time evolution of the accreting system for the different
the giant impact. The growth of Charon occurs simultaneously values of the dissipation parameter, A, is presented in Fig. 11.
with the tidal expansion of its orbit around Pluto and could affect The values of A are chosen from the range [8 − 21], while the
the behaviour of the small circumbinary moons described in the initial rotation period of Pluto and the initial Charon eccentricity
previous sections. are fixed at P0 = 0.149 d and eC = 0.001, respectively. Charon’s
First, we analyse the evolution of the expanding binary with eccentricity, shown in the top panel of Fig. 11, is damped more
a growing Charon in Sect. 6.1. Then we introduce an additional efficiently as A increases. The evolution of ΩC (in units of the
small moon (either Styx or Nix) to assess the parameters of the mean motion, nC ) is shown in the bottom panel; we note that
tidal evolution in Sect. 6.2. Finally, we simulate the behaviour of the synchronous state of Charon is more quickly attained for
A99, page 10 of 18C. A. Giuppone et al.: Past and present dynamics of the circumbinary moons in the Pluto-Charon system
Fig. 9. Time evolution of Charon’s physical properties during the mass accretion: the mass, the radius, and the zonal harmonic, J2 , from left to
right. Colours represent the results obtained from the different accretion models. The dashed horizontal line (right panel) presents the constant J2
obtained in Correia (2020).
Fig. 10. Tidal evolution of the PC binary, for A = 10 and a logarithmic mass accretion of Charon during the first 104 yr, starting with the initial
eccentricity eC = 0.001. Panels show the orbital semi-major axis, aC , in units of RP (panel a), the orbital eccentricity, eC (panel b), and the spin
angular velocities of Pluto, ΩP (panel c) and Charon, ΩC (panel d) in units of the mean motion, nC . Each colour represents a different initial P0 , in
units of days.
higher values of the dissipation parameter, A. The variations in small moon. For the mass, we chose the extreme values of the
the semi-major axis of Charon and the rotation of Pluto, ΩP , with masses of Styx and Nix, given by the uncertainties reported in
the increasing A values are insignificant and are not shown in Brozović et al. (2015) and Showalter & Hamilton (2015). We set
Fig. 11. the initial eccentricity of the small moon at e = 0.001, the semi-
The tests described above allowed us to define the parameter major axis at a = 44413 km (a ∼ 37.4 RP ) and a = 50690 km
values of P0 = 0.149 d and A for the interval [8 − 21], which are (a ∼ 42.7 RP ) for the Styx and Nyx clones, respectively, and the
compatible with the current configuration of the PC binary. The angular elements equal to zero. As shown below, these initial
tests, done with the oligarchic mass accretion, indicate an initial values of a were chosen to obtain the current positions of the
value of P0 = 0.157 d; however, the corresponding simulations small moons. All simulations in this section were done with the
have shown results that are qualitatively similar to those obtained accretion time of 104 yr.
with the logarithmic mass growth model. We started analysing the behaviour of the Styx clones during
the tidal evolution of a growing Charon. We considered first a
6.2. PC binary and a small moon Styx mass of 1.49 × 1015 kg, which corresponds to the 1σ esti-
mate given in Brozović et al. (2015). Figure 12 shows the time
In this section the model is expanded with the additional of a evolution of the Styx orbital elements obtained for the values
small moon with the mass and orbital elements of either Styx or of the dissipation coefficient, A, ranging from 11 to 20; the ele-
Nix. As described in the previous section, the behaviour of these ments are the semi-major axis (top panel) and the eccentricity
two satellites is significantly affected by the passages through the (middle panel). We note that the test moon smoothly migrates
strong 4/1 and 5/1 MMRs. We explored the parameter space of inwards during the first ∼104 yr, a period during which Charon’s
the problem defined by A and mi , where mi is the mass of the mass is increasing. After the satellite reaches its current position
A99, page 11 of 18You can also read
