A&A 609, A57 (2018) Astronomy DOI: 10.1051/0004-6361/201730951 & c ESO 2018 Astrophysics Dust arcs in the region of Jupiter’s Trojan asteroids Xiaodong Liu and Jürgen Schmidt Astronomy Research Unit, University of Oulu, 90014 Oulu, Finland e-mail: [email protected] Received 6 April 2017 / Accepted 26 October 2017 ABSTRACT Aims. The surfaces of the Trojan asteroids are steadily bombarded by interplanetary micrometeoroids, which releases ejecta of small dust particles. These particles form the faint dust arcs that are associated with asteroid clouds. Here we analyze the particle dynamics and structure of the arc in the region of the L4 Trojan asteroids. Methods. We calculate the total cross section of the L4 Trojan asteroids and the production rate of dust particles. The motion of the particles is perturbed by a variety of forces. We simulate the dynamical evolution of the dust particles, and explore the overall features of the Trojan dust arc. Results. The simulations show that the arc is mainly composed of grains in the size range 4−10 microns. Compared to the L4 Trojan asteroids, the dust arc is distributed more widely in the azimuthal direction, extending to a range of [30, 120] degrees relative to Jupiter. The peak number density does not develop at L4. There exist two peaks that are azimuthally displaced from L4. Key words. meteorites, meteors, meteoroids – planets and satellites: rings – minor planets, asteroids: general – zodiacal dust – celestial mechanics – solar wind D [m] 1. Introduction 1.e-05 0.001 0.1 1.e+01 1.e+03 1.e+05 1014 The study of the Trojan asteroids is a classical problem of ce- lestial mechanics and astronomy, but it is also a hot topic of re- ] 13 cent research and future space exploration (see review papers by 2 10 Emery et al. 2015; Robutel & Souchay 2010; Dotto et al. 2008; Jewitt et al. 2004, and references therein). In 2021, the NASA 1012 mission Lucy will explore the Trojan asteroids in a sequence of flybys. In this paper, we focus on the dynamical evolution of 11 10 dust expelled from the Trojans. Dust carries information about the composition of its source, which gives important constraints 1010 on the origin and formation of the Trojan group of asteroids. Cross Section (>D) [m Several papers studied the dynamics of dust in the Trojan region (Liou & Zook 1995; Liou et al. 1995; Zimmer & Grogan 2014; 109 de Elía & Brunini 2010). However, the configuration of the dust 10-8 10-6 10-4 10-2 100 102 104 distribution associated with the Trojan asteroids is still unclear D [km] to date. In this work, we answer this question through computer Fig. 1. Total cross section of the L4 Trojan asteroids (red line). The blue simulations of the long-term evolution of dust particles. The sim- dashed line denotes the upper limit of 6 × 1013 m2 for the cross section ulations are performed on a large computer cluster. of L5 Trojan asteroids that are larger than 10 µm in radius, as inferred by Kuchner et al.(2000). 2. Production rate of dust particles We use the size distribution of the L4 Trojans published by by Jewitt et al.(2000) and because the potential bias, if any, is Fernández et al.(2009) to estimate the total cross section of the small. The total asteroid cross section is obtained from the sec- Trojan asteroids. This is necessary to derive the production rate ond moment of the distribution (Fig.1). This second moment of dust particles. The distribution is modified from the distribu- has a logarithmic divergence toward small asteroid sizes, but for tion derived by Jewitt et al.(2000) by taking into account the practical purposes, this is not a problem because the change in systematic dependence of the albedo on asteroid size that has total cross section is very mild even when we vary the lower cut- been inferred by Fernández et al.(2009). Although this correla- off asteroid size from 10 µm to 100 km. For our modeling we use tion was not confirmed in the much larger data set from WISE a value of 1013 m2. (Grav et al. 2011, 2012), we continue to use the size distribu- To estimate the production rate of dust ejected from the sur- tion from Fernández et al.(2009, use their Fig. 6) because it is faces of the Trojan asteroids by impacts of interplanetary mi- calibrated to absolute numbers in the same way as the result crometeoroids, we follow the procedure by Krivov et al.(2003). Article published by EDP Sciences A57, page 1 of6 A&A 609, A57 (2018) By normalizing the differential distribution of ejecta particle 1018 + gsil=0.0 masses to the total mass production rate M , we obtain 16 10 gsil=0.5 gsil=1.0 1 − α 14 p(m) = M+ m−(1+α); (1) 10 1−α 1−α ] − -1 mmax mmin 1012 ) [s where mmax and mmin denote the highest and lowest mass of an min 10 ejected particle. The exponent α is the slope of the cumulative 10 p(>r production rate with a plausible range 0:5 < α < 1. Here we use 108 the value 106 α = 0:91; (2) 104 which was inferred from measurements on the Moon 0.1 1.0 10.0 100.0 r [µm] (Horányi et al. 2015) of the Lunar Dust Experiment dust detector min (LDEX) on board the Lunar Atmosphere and Dust Environment Fig. 2. Cumulative production rate of ejecta particles from Eq. (6). The Explorer (LADEE) mission. parameter gsil is the mixing ratio of silicate to ice of the target surface. It is practical to write Eq. (1) in the form The red, black, and blue lines denote the cumulative production rates for gsil = 0:0, gsil = 0:5, and gsil = 1:0, respectively. !−(1+α) 1 M+ 1 − α m p(m) = · (3) 1−α mmax mmax mmin mmax 1 − m For a given impact velocity vimp and projectile mass mimp, the max yield can be estimated from the empirical formula derived from We frequently use particle radii rg instead of mass m. From laboratory experiments by Koschny & Grün(2001a,b), p(m) dm = p(rg) drg, we obtain !−1 1 − g g −8 gsil sil sil + !−1−3α Y = 2:85 × 10 × 0:0149 × + 3 M 1 − α rg 927 2900 p(r ) = · (4) g 1−α r m mmin r !0:23 2:46 max max 1 − max mimp vimp mmax × × ; (9) kg m s−1 Here where g is the mixing ratio of silicate to ice of the target sur- !1=3 !1=3 sil 3 3 face. Equation (9) assumes a density of 927 kg=m3 for ice and rmin = mmin ; rmax = mmax ; (5) 3 4πρg 4πρg a density of 2900 kg=m for silicate (the original formula by Koschny & Grün(2001a,b) used a density of 2800 kg =m3 for sil- where ρg is the grain density. We note that p(m) and p(rg) are icate). For gsil = 1 we have a pure silicate surface and for gsil = 0 only defined for ejecta masses m with mmin < m < mmax. we have pure ice. For our modeling we use For the cumulative production rate, we obtain from Eq. (4) gsil = 1; (10) −3α Z r + rmin max 1 − α M r − 1 p(>r ) = dr p(r ) = max · (6) because the asteroid surfaces are most probably silicate rich min g g 3(1−α) α m rmin rmin max 1 − (even when the asteroid has a substantial interior ice fraction) rmax and because gsil = 1 leads to the smallest yields, which gives a To proceed, we need to quantify the total mass production lower bound for the particle number densities inferred from our rate M+. We again follow Krivov et al.(2003) and express it as model (Fig.2). For the typical projectile mass we use (Krivov et al. 2003) + M = YFimp S; (7) −8 mimp = 10 kg; (11) where Y is the yield, and S is the total cross section of the tar- get, that is, of all L4 Trojans (Fig.1). For the projectile mass because this particle mass (corresponding roughly to a radius of flux Fimp, we use the value 100 µm) dominates the mass flux at the distance of Jupiter. The largest ejecta have a mass on the order of the projectile. Thus, −15 −2 −1 Fimp = 10 kg m s ; (8) we use −8 which was derived by Krivov et al.(2003) from the Divine mmax = mimp = 10 kg: (12) model (Divine 1993) of the interplanetary meteoroid population for a spherical target with a unit surface that is on a heliocentric The precise choice of mmax has no strong effects on our results circular orbit at the distance of Jupiter. For the Trojans, the effect because Eq. (1) depends only weakly on mmax. of the gravitational focusing by a planet is not included. Newer The average impact velocity models have been published (Poppe 2016), but we continued to −1 use the number for the mass flux from the Divine model because vimp = 9 km s ; (13) this has led to a quantitative match with the measured ejecta clouds for the Galilean moons (Krüger et al. 2003), which are was also calculated by Krivov et al.(2003) from the Divine located at the same heliocentric distance as the Trojan asteroids. model evaluated at the distance of Jupiter. A57, page 2 of6 X. Liu and J. Schmidt: Dust arcs in the region of Jupiter’s Trojan asteroids 3. Dynamical model 1.5 The motion of dust particles from the Jupiter Trojans is influ- enced by a variety of forces.