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. Mourao

August 4, 2011

Abstract

Impacts of micrometeoroids on the surfaces of Nix and Hydra can produced dust particles and form aring around Pluto. However, dissipative forces, such as the solar radiation pressure, can lead the particlesinto collisions in a very short period of time. In this work we investigate the orbital evolution of escapingejecta 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 obtainedfrom analytical models. By comparing the lifetime of the survived particles, derived from our numericalsimulations, and the mass of a putative ring mainly formed by the particles released from the surfaces ofNix and Hydra we could estimate the ring normal optical depth. The released particles, encompassingthe 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 releasedfrom 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 newsatellites and a tenuous ring located in the Pluto system. As proposed by many authors this ring would beessentially composed by material produced from collisions between Pluto’s satellites and small Kuiper Beltdebris (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 impactsof micrometeoroids on the surface of Charon, can exist around Pluto and Charon. In their model the orbitsof the ring particles were disturbed only by the gravitational effects of the two massive bodies. A dissipativeforce, such as the solar radiation pressure, was not taken into account. This dust cloud would be very tenuouswith 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’ssatellites is presented in Stern et al. (2006). By making some assumptions, such as the ring is composed byice 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 withτ > 1.3× 10−5. If Pluto has a ring system it is either comparable to the Jupiter’s rings or it is a narrow ringconfined in less than 1500 km in width. They also estimated a lifetime of 900 yr for such a ring system basedon 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 andalbedos of Pluto’s satellites. He found that the ejecta from Nix and Hydra can reach one another and alsothe surface of Charon. In this analysis Stern (2009) considered the dissipative effect caused by the drag fromPluto’s escaping atmosphere. He proposed that the similarities can be explained by the exchanging of the

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Parameters Plutoa (AU)a 39.482e a 0.249i (deg)a 17.140D (km)b [2294]M (kg)b 1.3 × 1022

Table 1: Parameters of the orbit of Pluto related to the Sun. aSemi-major axis, eccentricity and inclinationof Pluto are derived from Murray & Dermott 1999. bDiameter and mass of Pluto are given by Tholen etal. 2008. Square bracket indicates assumed quantity.

ejecta material. However Canup (2011) argued that, given their current masses, similar albedos would implyin 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) arecomponents of the solar radiation force. The PR drag is mainly responsible for the collapse of the particle’sorbit leading to a collision with the planet. The main effect of the RP component is the oscillation of theparticle’s orbital eccentricity as a function of the planet orbital period. If this oscillation is large enough theparticle can collide with the central body or it can be ejected from the system in a short period of time. Adetailed 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 pressureand the gravity of Pluto (Hamilton & Krivov, 1996). This parameter allow us to verify the importance of thesolar radiation pressure in Pluto’s environment. We also numerically simulated a sample of ejecta particlesfrom Nix and Hydra’s surfaces under the effects of solar radiation force and the gravity of the four massivebodies. In our model Pluto is an eccentric orbit and its tilted rotational axis was included. In section 3 weestimated the mass production rates of dust ejecta from the surfaces of Nix and Hydra through an analyticalmodel. The combined results, analytical and numerical, can help us to constraint a normal optical depth ofa 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 thegravitational 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)

C =9

8

n

nsQpr

Fsr2

GMcρs(1)

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 theSun, r is the position vector of the particle and r=|r|, Fs is the solar radiation flux density at the heliocentricdistance of Pluto, Qpr is the radiation pressure efficiency factor, c is the speed of light, G is the gravitationalconstant, 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 1and 10µm in radius. The orbital elements and physical parameters of Pluto are listed in Table 1, where ais the semi major axis in Astronomical Unit (AU), e is the eccentricity, i is the inclination in degrees, D isthe diameter in km and M is the mass in kg (Murray & Dermott 1999, Tholen et al. 2008). The grains areadopted to be spherical with an uniform density equals to 1g cm−3 and Qpr was assumed to be 1 (Burns etal. 1979).

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Distance (plutonian radii)1 510 100 1000

Para

met

er v

alu

e

10- 3

10- 2

10- 1

100

101

102

103

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 orbitsof Nix and Hydra.

As expected the smaller particle is more sensitive to the effects of the solar radiation pressure than thelarger particle. The strength of C is ∼ 10 for a 1µm sized particle and ∼ 1 for a 10µm sized particle placedbetween 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 Cis ∼ 1 for a 1µm sized particle (see their Fig. 2), smaller than the value shown in our Fig. 1. Their numericalsimulations did confirm that the solar radiation pressure has an important effect on the orbital evolution ofdust particles located in the µ and ν rings. Although Pluto is far from Sun, its small size, relative to the giantplanets, becomes the solar radiation pressure an important force to be taken into account in the analysis ofthe 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 effectsof the solar radiation force can be written as

mv =FsAQpr

c

[(1− r

c

)S− v

c

](2)

where r is the particle’s radial velocity, S is a unit vector in the direction of the incident radiation, and v isthe velocity vector of the particle relative to the Sun (Burns et al. 1979). This force (eqn. 2) is composed bya 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, avariation 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, withradius ranging from 1 to 10µm, initially in circular orbits around Pluto. By assuming a constant decay ratethe 1µm sized particles will collide with Pluto in 106 years. Considering circumplanetary and equatorialparticles, this value is comparable to the characteristic orbital decay time derived by the analytic expression

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Parameters Charon Nix Hydraa (km) 19570.3 (a0) 49240. 65210.e 0.0035 0.0119 0.0078i (◦) 96.168 96.190 96.362diameter (km) 1212 88 72mass (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. Theparameters of the orbit of Charon are relative to Pluto, the parameters of the orbits of Nix and Hydra arerelative to the center of mass of the Pluto-Charon system. The masses of Nix and Hydra were obtainedassuming 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 dustparticles in a short period of time, less than 10 years. Therefore we do not take into account the long periodeffects 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 responsiblefor covering these two satellites and also the surface of Charon. Dust particles can escape from a parentbody when the ejecta velocity is larger than the escape velocity (vesc) of that body. The range of thecharacteristic 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 Nixand 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 producedfrom the impacts on their surfaces.

Our initial sample of dust particles was perpendicularly ejected from the surfaces of Nix and Hydrawith initial velocity vi = 1.0vesc, large enough to escape from the parent satellite. Ejected particles withvi > 1.0vesc can escape from the system in a very short period of time. The escaping ejecta, pure ice grainswith radii of 1, 5 and 10µm and scattering properties of an ideal material, are under the combined effects ofthe gravity field of the four massive bodies and the RP. The initial conditions of the four massive bodies arelisted in Tables 1 and 2.

We take into account the variation of the solar flux during the orbital period of Pluto due to its largeeccentricity (Table 1). The planetary shadow and the light reflected from the planet were neglected infirst approximation, both effects are weaker than those caused by the RP (Hamilton & Krivov, 1996). TheYarkovsky 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. After1 year (Fig. 2a) the ejected particles are distributed in a region which encompasses the orbits of Nix andHydra, located at 2.5a0 and 3.3a0, respectively, where a0 = 19570.3 km (Table 2). The lifetime of the ejectafrom Nix and Hydra is determined by collisions with Pluto and by ejections from the system. After 100 yearsonly 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 particlescaused by the RP component (Sfair & Giuliatti Winter, 2009). By adopting the value of Pluto gravitycoefficient, as proposed by Beauvalet et al. (2010) to be O(−4), we verified that this effect can be neglectednear the orbits of Nix and Hydra.

Our results did not corroborate the results presented in Stern (2009). In our numerical simulations about45% of the total amount of the dust particles collide with Pluto and Charon. Only a small fraction, less than1%, 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 ofinterplanetary dust particles (IDPs), assumed to be interplanetary micrometeoroids, on the surfaces of Nixand Hydra. The comparison of the numerical results and the mass production rate will help us to place anupper limit to the ring normal optical depth generated by this mechanism.

4

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

Ecce

ntr

icity

Barycentric semimajor axis (a0)

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

Eccen

tric

ity

Barycentric semimajor axis (a0)

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

Eccentr

icity

Barycentric semimajor axis (a0)

Figure 2: Diagram of the instantaneous barycentric semimajor axis as function of the eccentricity for a set ofmicrometer-sized particles (1µm, 5µm and 10µm) ejected from the surfaces of Nix and Hydra. The satellitesare located at 2.5a0 (Nix) and 3.3a0 (Hydra). Top: timespan of 1 year, middle: 10 years, bottom: 100 years.

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3 Ejecta Particles

Firstly the mass production of the ejected dust particles will be calculated by analysing the mass flux ofimpactors 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 possiblemechanism to produce and maintain dust rings around planets in the Solar System (see, e.g., Kruger etal. 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, estimativeshave 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 (F∞imp) 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 tothe IDP flux at 30AU, close to the perihelion distance of Pluto. The superscript ∞ indicates that this valuewas 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 ) definedas the ratio of the total ejected mass to the mass of the impactors. Koschny and Grun (2001) presented adefinition for Y which depends on the fraction of ice-silicate in the target, and on the mass (mimp) and thevelocity (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 ringparticles obtained from our numerical simulations. We assumed the mass production rate M+ to be equal to10−4 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 normaloptical depth given a size distribution of the grains. We assume that the size distribution of the dust followsa 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 Uranusinternal ring where q = 3.5 (Colwell & Esposito 1990). This type of distribution is the most common forvery 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 particlesize distribution of 1-10µm and an optical depth of τ = 5 × 10−6 (Stern et al. 2006) has a mass equalsto 108 kg. To accumulate this amount of mass, only generated by interplanetary meteoroid bombardmentson 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 theejecta 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 timeslarger, 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 ofthem to collisions or ejections from the system in a timescale of about 1 year, we ruled out a normal opticaldepth 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 103kg. Inthis timescale about 98% of the ejecta from Nix and Hydra are still in the system assuring an equilibriumbetween the production and the loss of the dust particles.

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The adopted size distribution of the dust population (Burns et al. 2001) has a large quantity of very smallparticles, 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 thering 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 haveto be considered in order to better estimate the orbital evolution of dust particles. This dissipative forcecan be divided into two components: the PR drag and the RP component. Although the PR drag ismainly responsible for the decreasing in the semimajor axis of the particle, it is a long period effect. TheRP component can dictate the lifetime of a dust particle (Sfair & Giuliatti Winter, 2009) by increasing itseccentricity and leading this particle to close encounters with the massives bodies in a very short period oftime.

We numerically simulated a set of dust particles perpendicularly ejected from the surfaces of Nix andHydra. 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 atthe perihelion distance of Pluto.

The released particles, encompassing the orbits of Nix and Hydra, temporarily form a 16000 km widering. However, collisions with the massive bodies, mainly due to the effects of the RP component, removeabout 50% of the 1µm particles in 1 year.

The mass production rate from the surfaces of Nix and Hydra was obtained from analytical models. Weassumed 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 massof a putative ring mainly formed by the particles released from the surfaces of Nix and Hydra we couldestimate the ring normal optical depth. As a result we find that a tenuous ring with a normal optical depthof 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 dustfluxes during all the way and beyond Pluto. It has a dust counter onboard which can detect particles withmasses larger than 10−12 g (Horanyi et al. 2008). The spacecraft data will help to validate the numerical andtheorical models.

References

Beauvalet, L. et al. 2010, in 2010 Nix-Hydra Meeting, Baltimore, MD,( https://webcast.stsci.edu/webcast/detail.xhtml? talkid=1940&parent=1 ).Burns, J. A., Lamy, P. L., & Soter, S. 1979, Icarus, 40, 1.Burns, J. A., Hamilton, D. P., & Showalter, M. R. 2001, in Interplanetary Dust, ed. E. Grun, B. A. S.Gustafson, S. F. Dermott, H. & Fechtig, 641.Canup, R. M., AJ, 2010, 141, 35.Colwell, J. E., Esposito, L. W. 1990, Icarus, 86, 530.Hamilton, D. P., & Krivov, A. V. 1996, Icarus, 123, 503.Horani et al. 2008, Space Sci. Rev., 140, 387.Koschny, D., & Grun, E. 2001, Icarus, 154, 391.Krivov, A.V., Sremcevic, M., Spahn, F., Dikarev, V.V., & Kholshevnikov, K. V.. 2003, Planet. Space Sci.,51, 251.Kruger, H., Krivov, A.V., & Grun, E. 2000, Planet. Space Sci., 48, 1457.Murray, C. D. & S. Dermott, 1999, Solar System Dynamics.Pires dos Santos, P. M., Giuliatti Winter, S. M., & Sfair R. 2010, in 42nd DPS Meeting Bulletin, 42, AbstractNo. 40.03.

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Pires dos Santos, P. M, Giuliatti Winter, S. M., & Sfair, R. 2011, MNRAS, 410, 273.Porter, S. B., Desch, S. J., & Cook, J. C. 2010, Icarus, 208, 492.Sfair, R., & Giuliatti Winter, S. M. 2009, A&A, 505, 845.Sfair, R., & Giuliatti Winter, S. M. 2011, A&A, Submitted.Steffl, A. J., & Stern, S.A. 2007, AJ, 133, 1485.Stern, S. A. et al. 2006, Nature, 439, 946.Stern, S.A. 2009, Icarus, 199, 571.Thiessenhusen, K.-U., Krivov, A., Kruger, H., & Grun, E. et al. 2002. Planet Space Sci., 50, 79Tholen, D.J., Buie, M.W., Grundy, N.M. & G.T. Elliot, 2008, AJ, 135, 777.Weaver, H. A. et al. 2006, Nature, 439, 943.

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