1. Small particles in Pluto’s environment:
effects of the solar radiation pressure
P. M. Pires dos Santos, S. M. Giuliatti Winter, R. Sfair, & D. C. Mour˜o
a
arXiv:1108.0712v1 [astro-ph.EP] 2 Aug 2011
August 4, 2011
Abstract
Impacts of micrometeoroids on the surfaces of Nix and Hydra can produced dust particles and form a
ring around Pluto. However, dissipative forces, such as the solar radiation pressure, can lead the particles
into collisions in a very short period of time. In this work we investigate the orbital evolution of escaping
ejecta under the effects of the radiation pressure force combined with the gravitational effects of Pluto,
Charon, Nix and Hydra. The mass production rate from the surfaces of Nix and Hydra was obtained
from analytical models. By comparing the lifetime of the survived particles, derived from our numerical
simulations, and the mass of a putative ring mainly formed by the particles released from the surfaces of
Nix and Hydra we could estimate the ring normal optical depth. The released particles, encompassing
the orbits of Nix and Hydra, temporarily form a 16000 km wide ring. Collisions with the massive bodies,
mainly due to the effects of the radiation pressure force, remove about 50% of the 1µm particles in 1 year.
A tenuous ring with a normal optical depth of 6 × 10−11 can be maintained by the dust particles released
from the surfaces of Nix and Hydra.
Keywords: Kuiper belt: general - Planets and satellites: individual: Pluto
1 Introduction
Since the discovery of Nix and Hydra by Weaver et al. (2006) much effort has been made in order to find new
satellites and a tenuous ring located in the Pluto system. As proposed by many authors this ring would be
essentially composed by material produced from collisions between Pluto’s satellites and small Kuiper Belt
debris (Thiessenhusen et al. 2002; Stern et al. 2006; Steffl & Stern 2007).
Thiessenhusen et al. (2002) have showed that a dust cloud, mainly formed by ejecta produced by impacts
of micrometeoroids on the surface of Charon, can exist around Pluto and Charon. In their model the orbits
of the ring particles were disturbed only by the gravitational effects of the two massive bodies. A dissipative
force, such as the solar radiation pressure, was not taken into account. This dust cloud would be very tenuous
with a maximum optical depth of 3 × 10−11 .
A discussion on the collisional velocities of impactors from the Kuiper belt on the surfaces of Pluto’s
satellites is presented in Stern et al. (2006). By making some assumptions, such as the ring is composed by
ice particles with a mean lifetime of 105 yr, they could estimate a characteristic optical depth of τ = 5 × 10−6
for a ring located between the orbits of Nix and Hydra.
The first observational constraint on a Pluto’s ring was presented by Steffl & Stern (2007). Based on
the data obtained by the Hubble Space Telescope they argued that the Pluto system has no ring with
τ > 1.3 × 10−5 . If Pluto has a ring system it is either comparable to the Jupiter’s rings or it is a narrow ring
confined in less than 1500 km in width. They also estimated a lifetime of 900 yr for such a ring system based
on its optical depth constraint.
Stern (2009) discussed the dynamical evolution of the escaping ejecta from the surfaces of Pluto, Charon,
Nix and Hydra, in an attempt to propose an alternative hypothesis on the similarity found in the colors and
albedos of Pluto’s satellites. He found that the ejecta from Nix and Hydra can reach one another and also
the surface of Charon. In this analysis Stern (2009) considered the dissipative effect caused by the drag from
Pluto’s escaping atmosphere. He proposed that the similarities can be explained by the exchanging of the
1

2. Parameters Pluto
a (AU)a 39.482
ea 0.249
i (deg)a 17.140
D (km)b [2294]
M (kg)b 1.3 × 1022
Table 1: Parameters of the orbit of Pluto related to the Sun. a Semi-major axis, eccentricity and inclination
of Pluto are derived from Murray & Dermott 1999. b Diameter and mass of Pluto are given by Tholen et
al. 2008. Square bracket indicates assumed quantity.
ejecta material. However Canup (2011) argued that, given their current masses, similar albedos would imply
in high densities values for Nix, Hydra and Charon.
The motion of a dust grain in orbit around a planet can also be affected by the solar radiation force.
The Poynting-Robertson drag (PR drag hereafter) and the radiation pressure (RP component hereafter) are
components of the solar radiation force. The PR drag is mainly responsible for the collapse of the particle’s
orbit leading to a collision with the planet. The main effect of the RP component is the oscillation of the
particle’s orbital eccentricity as a function of the planet orbital period. If this oscillation is large enough the
particle can collide with the central body or it can be ejected from the system in a short period of time. A
detailed study of this dissipative force is presented in Burns et al. (1979).
In section 2 we firstly analysed the C parameter which is the ratio between the solar radiation pressure
and the gravity of Pluto (Hamilton & Krivov, 1996). This parameter allow us to verify the importance of the
solar radiation pressure in Pluto’s environment. We also numerically simulated a sample of ejecta particles
from Nix and Hydra’s surfaces under the effects of solar radiation force and the gravity of the four massive
bodies. In our model Pluto is an eccentric orbit and its tilted rotational axis was included. In section 3 we
estimated the mass production rates of dust ejecta from the surfaces of Nix and Hydra through an analytical
model. The combined results, analytical and numerical, can help us to constraint a normal optical depth of
a putative ring system. Our conclusions are presented in the last section.
2 Solar radiation force acting on dust particles
In this section we analysed the fate of the ejecta produced from the surfaces of Nix and Hydra under the
gravitational effects of the massive bodies and the RP component.
2.1 The strength of the perturbing force
First of all we calculated the dimensionless parameter C given by (Hamilton & Krivov, 1996)
9 n Fs r2
C= Qpr (1)
8 ns GM cρs
The C parameter gives the relative strength between the solar radiation force and the planetary gravity,
where n is the particle’s mean motion about the planet and ns is the mean motion of the planet around the
Sun, r is the position vector of the particle and r=|r|, Fs is the solar radiation flux density at the heliocentric
distance of Pluto, Qpr is the radiation pressure efficiency factor, c is the speed of light, G is the gravitational
constant, M is the mass of Pluto, and ρ and s are the density and the radius of the grain, respectively.
The variation of C as a function of the plutonian radius can be seen in Fig. 1 for two particles of sizes 1
and 10µm in radius. The orbital elements and physical parameters of Pluto are listed in Table 1, where a
is the semi major axis in Astronomical Unit (AU), e is the eccentricity, i is the inclination in degrees, D is
the diameter in km and M is the mass in kg (Murray & Dermott 1999, Tholen et al. 2008). The grains are
adopted to be spherical with an uniform density equals to 1g cm−3 and Qpr was assumed to be 1 (Burns et
al. 1979).
2

3. 3
10
2
10
Parameter value
1
10
0
10
-1
10
-2
10
-3
10 1 510 100 1000
Distance (plutonian radii)
Figure 1: Parameter C as a function of the distance of Pluto for a spherical grain with 1 (· · · ) and 10µm
(—) in radius and density equals to 1g cm−3 . The gray area represents the radial distance between the orbits
of Nix and Hydra.
As expected the smaller particle is more sensitive to the effects of the solar radiation pressure than the
larger particle. The strength of C is ∼ 10 for a 1µm sized particle and ∼ 1 for a 10µm sized particle placed
between the orbits of Nix and Hydra, about 45-58 plutonian radii (the gray area shown in Fig. 1).
We compared the results presented in Fig. 1 with a previous analysis by Sfair & Giuliatti Winter (2009)
on the orbital evolution of small particles located at the µ and ν uranian rings. Their results showed that C
is ∼ 1 for a 1µm sized particle (see their Fig. 2), smaller than the value shown in our Fig. 1. Their numerical
simulations did confirm that the solar radiation pressure has an important effect on the orbital evolution of
dust particles located in the µ and ν rings. Although Pluto is far from Sun, its small size, relative to the giant
planets, becomes the solar radiation pressure an important force to be taken into account in the analysis of
the orbital evolution of dust particles in Pluto’s environment (Pires dos Santos et al. 2010).
2.2 Numerical simulations
The equation of motion of a dust particle with cross section A, in an inertial reference frame, under the effects
of the solar radiation force can be written as
Fs AQpr r
˙ ˆ v
˙
mv = 1− S− (2)
c c c
ˆ
where r is the particle’s radial velocity, S is a unit vector in the direction of the incident radiation, and v is
˙
the velocity vector of the particle relative to the Sun (Burns et al. 1979). This force (eqn. 2) is composed by
a velocity-independent component (RP component) and a velocity-dependent component (PR drag).
The PR drag has long period effects while the RP component provokes, in a very short period of time, a
variation in the eccentricities of the dust particles which can lead to a collision or escape from the system.
Only for completeness we numerically simulated the effects of the PR drag in a sample of particles, with
radius ranging from 1 to 10µm, initially in circular orbits around Pluto. By assuming a constant decay rate
the 1µm sized particles will collide with Pluto in 106 years. Considering circumplanetary and equatorial
particles, this value is comparable to the characteristic orbital decay time derived by the analytic expression
3

4. Parameters Charon Nix Hydra
a (km) 19570.3 (a0 ) 49240. 65210.
e 0.0035 0.0119 0.0078
i (◦ ) 96.168 96.190 96.362
diameter (km) 1212 88 72
mass (kg) 1.5 × 1021 5.8 × 1017 3.2 × 1017
Table 2: Orbital and physical parameters derived by Tholen et al. (2008). Epoch JD 2452600.5. The
parameters of the orbit of Charon are relative to Pluto, the parameters of the orbits of Nix and Hydra are
relative to the center of mass of the Pluto-Charon system. The masses of Nix and Hydra were obtained
assuming the density equals to 1.63 g cm−3 .
given in Burns et al. (1979). By the other hand the RP causes an increase in the eccentricities of these dust
particles in a short period of time, less than 10 years. Therefore we do not take into account the long period
effects of the PR drag in our numerical simulations.
Stern (2009) has proposed that the ejecta escaped from the surfaces of Nix and Hydra can be responsible
for covering these two satellites and also the surface of Charon. Dust particles can escape from a parent
body when the ejecta velocity is larger than the escape velocity (vesc ) of that body. The range of the
characteristic ejecta velocity, produced from impacts of Kuiper belt objects in the Pluto system, is 0.01-
0.2 km s−1 (Stern, 2009). From the parameters derived from Tholen et al. (2008), the escape velocities of Nix
and Hydra are 0.042 km s−1 and 0.034 km s−1 , respectively, smaller than the characteristic ejecta velocity.
Pluto and Charon’s escape velocities are larger than 0.2 km s−1 and they will retain the ejecta produced
from the impacts on their surfaces.
Our initial sample of dust particles was perpendicularly ejected from the surfaces of Nix and Hydra
with initial velocity vi = 1.0vesc , large enough to escape from the parent satellite. Ejected particles with
vi > 1.0vesc can escape from the system in a very short period of time. The escaping ejecta, pure ice grains
with radii of 1, 5 and 10µm and scattering properties of an ideal material, are under the combined effects of
the gravity field of the four massive bodies and the RP. The initial conditions of the four massive bodies are
listed in Tables 1 and 2.
We take into account the variation of the solar flux during the orbital period of Pluto due to its large
eccentricity (Table 1). The planetary shadow and the light reflected from the planet were neglected in
first approximation, both effects are weaker than those caused by the RP (Hamilton & Krivov, 1996). The
Yarkovsky effect was also neglected since this effect is irrelevant for particles in the micrometer-sized range.
Figure 2 presents the values of the semimajor axis versus eccentricity after 1, 10 and 100 years. After
1 year (Fig. 2a) the ejected particles are distributed in a region which encompasses the orbits of Nix and
Hydra, located at 2.5a0 and 3.3a0 , respectively, where a0 = 19570.3 km (Table 2). The lifetime of the ejecta
from Nix and Hydra is determined by collisions with Pluto and by ejections from the system. After 100 years
only 7% of the total amount of particles is still in orbit around Pluto.
The oblateness of the central body can decrease the variation of the eccentricity of the dust particles
caused by the RP component (Sfair & Giuliatti Winter, 2009). By adopting the value of Pluto gravity
coefficient, as proposed by Beauvalet et al. (2010) to be O(−4), we verified that this effect can be neglected
near the orbits of Nix and Hydra.
Our results did not corroborate the results presented in Stern (2009). In our numerical simulations about
45% of the total amount of the dust particles collide with Pluto and Charon. Only a small fraction, less than
1%, of the ejecta produced by the small satellites, Nix and Hydra, can reach each other.
In the next section we analysed the mass and the normal optical depth of a ring generated by impacts of
interplanetary dust particles (IDPs), assumed to be interplanetary micrometeoroids, on the surfaces of Nix
and Hydra. The comparison of the numerical results and the mass production rate will help us to place an
upper limit to the ring normal optical depth generated by this mechanism.
4

5. 0.8
Eccentricity
0.6
0.4
0.2
0
0 1 2 3 4 5
Barycentric semimajor axis (a0)
0.8
Eccentricity
0.6
0.4
0.2
0
0 1 2 3 4 5
Barycentric semimajor axis (a0)
0.8
Eccentricity
0.6
0.4
0.2
0
0 1 2 3 4 5
Barycentric semimajor axis (a0)
Figure 2: Diagram of the instantaneous barycentric semimajor axis as function of the eccentricity for a set of
micrometer-sized particles (1µm, 5µm and 10µm) ejected from the surfaces of Nix and Hydra. The satellites
are located at 2.5a0 (Nix) and 3.3a0 (Hydra). Top: timespan of 1 year, middle: 10 years, bottom: 100 years.
5

6. 3 Ejecta Particles
Firstly the mass production of the ejected dust particles will be calculated by analysing the mass flux of
impactors at Pluto’s region and the yield parameter.
3.1 Mass production rate
Interplanetary meteoroid bombardment on the surfaces of planetary satellites was suggested as a possible
mechanism to produce and maintain dust rings around planets in the Solar System (see, e.g., Kruger et
al. 2000, Krivov et al. 2003). This section follows the approach summarized in Sfair & Giuliatti Winter (2011).
Although the dust fluxes beyond 18AU from the Sun have not been experimentally confirmed, estimatives
have been used to characterize the dust fluxes at all the giant planets distances and at the perihelion of Pluto
∞
(Krivov et al. 2003, Porter et al. 2010). Thus, we assume that the mass flux of impactors (Fimp ) at Pluto is
similar to the mass flux at Neptune, i.e., 1.0 × 10−16 kg m−2 s−1 (Porter et al. 2010), which corresponds to
the IDP flux at 30AU, close to the perihelion distance of Pluto. The superscript ∞ indicates that this value
was measured far from the central body and has to be corrected due to the gravitational focusing. However,
our results show that the diference between the mass flux of impactors far and close to Pluto is of O(−3).
Therefore, we assumed the mass flux of impactors at Pluto region to be Fimp = 1.0 × 10−16 kg m−2 s−1 .
The value of the mass production rate (M + ) of each satellite depends on the ejecta yield (Y ) defined
as the ratio of the total ejected mass to the mass of the impactors. Koschny and Gr¨n (2001) presented a
u
definition for Y which depends on the fraction of ice-silicate in the target, and on the mass (mimp ) and the
velocity (vimp ) of the impactor. Assuming that Nix and Hydra have pure ice surfaces and mimp ∼ 10−8 kg,
which corresponds to a small object of about 100µm in radius, with vimp = 2.6 km s−1 , gives Y ∼ 100.
The mass production rate produced by a satellite can be given by (Krivov et al. 2003)
M + = Fimp Y S (3)
where S is the cross section area of the satellite. This gives values of M + equal to approximately 6 × 10−5
kg s−1 and 4 × 10−5 kg s−1 for Nix and Hydra, respectively.
For a steady ring its mass is directly proportional to the lifetime (T ) of its particles. The mass of
the ring can be roughly estimate from the values of the mass production rate and the lifetime of the ring
particles obtained from our numerical simulations. We assumed the mass production rate M + to be equal to
10−4 kg s−1 , which corresponds to the sum of the mass production rates of Nix and Hydra.
The mass of a ring at distance R from the planet and width dR can also be estimated from its normal
optical depth given a size distribution of the grains. We assume that the size distribution of the dust follows
a power law of the form dN = n(s)ds = Ks−q , where dN is the number of particles in the interval [s, s + ds],
q is a power-law index and K is a normalization constant. We assumed the same distribution of the Uranus
internal ring where q = 3.5 (Colwell & Esposito 1990). This type of distribution is the most common for
very small dust (Burns et al. 2001).
A ring located between the orbits of Nix and Hydra at R ∼ 57000 km and dR ∼16000 km, with a particle
size distribution of 1-10µm and an optical depth of τ = 5 × 10−6 (Stern et al. 2006) has a mass equals
to 108 kg. To accumulate this amount of mass, only generated by interplanetary meteoroid bombardments
on the surfaces of the satellites, it would be necessary that the lifetime of the particles were 4 × 104 years.
However, as has been shown the RP component has a significantly effect on the dust particles and drives the
ejecta material to collisions or to escape from the system in a timescale much shorter than 104 years.
A putative ring with the same characteristics as described before but with a normal optical depth of 10−8
has a mass Mr = 2.4 × 105 kg. It will take about 80 years to form such ring. To accumulate a mass of 105
kg, only supplied by collisions between Nix and Hydra and the IDPs, the Fimp should be a hundred times
larger, if we considered the same properties of the impactors and the targets.
Since the solar radiation pressure has a significantly effect on these dust ring particles, leading most of
them to collisions or ejections from the system in a timescale of about 1 year, we ruled out a normal optical
depth of order 10−6 as has been proposed by Stern et al. (2006).
A ring with a normal optical depth of 6 × 10−11 takes less than 1 year to accumulate a mass of 103 kg. In
this timescale about 98% of the ejecta from Nix and Hydra are still in the system assuring an equilibrium
between the production and the loss of the dust particles.
6

7. The adopted size distribution of the dust population (Burns et al. 2001) has a large quantity of very small
particles, an equivalence of 3000 particles of 1µm for each particle of 10µm in radius. Although about 80%
of the total amount of the 10µm sized particles survive for almost 10 years we assumed the lifetime of the
ring to be the lifetime of the set formed by particles of 1µm in radius.
4 Discussion
Even in a distant region, such as the Pluto’s environment, the effects of the solar radiation pressure have
to be considered in order to better estimate the orbital evolution of dust particles. This dissipative force
can be divided into two components: the PR drag and the RP component. Although the PR drag is
mainly responsible for the decreasing in the semimajor axis of the particle, it is a long period effect. The
RP component can dictate the lifetime of a dust particle (Sfair & Giuliatti Winter, 2009) by increasing its
eccentricity and leading this particle to close encounters with the massives bodies in a very short period of
time.
We numerically simulated a set of dust particles perpendicularly ejected from the surfaces of Nix and
Hydra. These particles are under the gravitational effects of Pluto, Charon, Nix and Hydra and the RP com-
ponent. Our simplified model assumed an isotropic flux of impactors on the surfaces of Nix and Hydra at
the perihelion distance of Pluto.
The released particles, encompassing the orbits of Nix and Hydra, temporarily form a 16000 km wide
ring. However, collisions with the massive bodies, mainly due to the effects of the RP component, remove
about 50% of the 1µm particles in 1 year.
The mass production rate from the surfaces of Nix and Hydra was obtained from analytical models. We
assumed the mass flux of impactors at Pluto’s region to be similar to the mass of flux at Neptune.
By comparing the lifetime of the survived particles, derived from our numerical simulations, and the mass
of a putative ring mainly formed by the particles released from the surfaces of Nix and Hydra we could
estimate the ring normal optical depth. As a result we find that a tenuous ring with a normal optical depth
of O(10−11 ) can be maintained by the escaping dust.
It is worth to point out that the interplanetary environment in the outer Solar System is not well known.
Many assumptions have to be made in order to estimate a normal optical depth of a putative ring encom-
passing the orbits of Nix and Hydra.
The New Horizons mission will offer the best opportunity to obtain in situ measurements of the dust
fluxes during all the way and beyond Pluto. It has a dust counter onboard which can detect particles with
masses larger than 10−12 g (Hor´nyi et al. 2008). The spacecraft data will help to validate the numerical and
a
theorical models.
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