Bound Orbit of Sub-micron Size Non-spherical Silicate Dust Particles in the Dust Band of the Asteroid Belt

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1 SUST Studies, Vol., No., ; P:-7 Bound Orbit of Sub-micron Size Non-spherical Silicate Dust Particles in the Dust Band of the Asteroid Belt M. A. Saklayen, Md. Mahmuduzzaman and M. H. Ahsan Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh. h.ahsan@sust.edu Abstract Infrared Astronomical Satellite (IRAS) discovered three dust bands in the asteroid belt. These dust bands are believed to arise from the collision activity of asteroids. Dynamics of spherical and non-spherical particles are quite different in these regions. In this study the β values (the ratio of radiation pressure force to the solar gravity) for cylindrical particles have been evaluated. The β value for cylindrical particle strongly depends on the incident angle of radiation. The dependence of β values on the incident angle of radiation has also been formulated. Orbits in the solar radiation field are computed numerically accounting for β that varies as a function incident angle of radiation or spin of the particle.. Introduction Bodies in the interplanetary space are not only attracted to the Sun by gravity but also repelled from it by radiation pressure due to the momentum carried by solar photons. Radiation pressure force is considerably important for very small particle (Sub-micron sized). It plays an important role in dynamical evolution on submicron size dust particles. The Pioneer and Helios spacecraft, as well as earlier in situ observations have directly measured dust impact counters of various types [,,]. It is aseptically evident that dust present in the solar system and also its existence in cloud throughout interstellar space is strongly implied by the extinction, reddening and polarization of star light [,5,6]. Dust can be produced in circumstellar space and can blow out in to interstellar space [7]. Low at el. [8] discovered the asteroid dust band associated with the Koronis and Themis families. It has proven that the debris from the asteroid becomes scattered through the interplanetary space. The laboratory experiments for collisions of solid bodies, e.g. Fujiwara and Stukamoto [9] suggested that the collisional debris has an irregular shape and is in the rotational state. The life time for a particle with a radius s to spiral from the Earth s orbit to the Sun, due to the Poynting- Robertson effect, is ~ /β yr, where β./s(µm) is the ratio of radiation pressure force to the gravity for the particle with density.5 g.cm - []. The Poynting-Robertson lifetime of such particle is of the order of 5 yr. Thus the inner solar system would be rapidly depleted of dust without continuing. Both asteroids and comets contribute solid materials to the interplanetary space. But it is not yet clear whether one source produces the bulk of the materials. Durda and Dermott [] found that about one third of the thermal flux detected in the IRAS 5µm wave band comes from the asteroidal dust. Yamamoto, Nakamura and Mukai [] estimated the production rate of collisional debris in the asteroidal belt. They found that the mass production rate by the catastrophic disruption of asteroids is higher than that by the impacts of interstellar dust. The discovery of the IRAS dust bands has proven that dust is produced in the asteroid belt. Asteroid families are the impressive remnants of energetic collisional events that were able to disrupt sizeable parent bodies and to disperse their fragments. Recently, the availability of a number (about ) of statistically reliable families [] have triggered a great deal of activity aimed at driving from their observable properties information on the composition of their parent bodies and on the physical processes of catastrophic disruption from which the families were originated. In particular, size and ejection velocity distributions of family members are very important, because they are directly related to the physics governing the outcomes of collisional break-up phenomena. Asteroids in a family common orbital elements lead to frequent collisions among the members. These collisions generate dust, which spirals towards the sun due to the Poynting-Robertson effect. Our ability to compute the radiation pressure force for solid particle is extremely limited. If the particle is spherical, homogenous and isotropic the Mie Scattering Theory is used to compute any desired optical

2 Bound Orbit of Sub-micron Size Non-spherical Silicate Dust Particles in the Dust Band of the Asteroid Belt properties of the grain. But our choice is really limited for non-spherical particles. The Discrete Dipole Approximation (DDA) developed by Purcell and Pennypacker [] is a very flexible and general technique for computing the optical properties of arbitrary shapes. The DDA replaces the solid particles by an array of number of dipoles, where the spacing between dipoles should be smaller compared to the wavelength of interest. In this study we computed the β value of a cylindrical silicate particle using DDA method, where the particles were ejected from the catastrophic disruption of parent bodies in the asteroid belt. We formulate the dependence of β on an incident angle of radiation for cylindrical particle and follow the trajectory of such particle ejected from the parent bodies as functions of shape of the particle and its spin rate.. Radiation Pressure Force The radiation pressure force on a target particle due to solar radiation is expressed as [] A R FR = B ( λ) Q pr ( s, m, λ) dλ... (a) c R where R is the distance of the particle from the Sun with a radius R and B * (λ) is the solar irradiance. A is the geometrical cross-section of the particle, c is the speed of light, λ is the wave length of incident radiation and Q pr is the efficiency factor of radiation pressure. The value of Q pr depends on the radius s, shape and the optical constant m* of the particle material. The values of solar spectrum B * (λ) are adapted from the data computed by Mukai[5]. The efficiency factor of radiation, Q pr can be written as Q = Q Q < cosθ >... (b) pr ext sca where, Q ext is the efficiency factor for extinction, Q pr is the efficiency factor for scattering and <cosθ> is the mean scattered radiation in the direction of propagation. The DDA produces Q ext, Q sca and <cosθ> as a function of wavelength of radiation as well as equivalent radius of the target particle. In the calculation complex refractive indices for wavelengths from.µm to 5. µm are used. The gravitational attraction force on the particle of mass m is GMm F G = R.() where, G is the gravitational constant and M is the mass of the Sun. A well known dimensionless parameter β is a ratio of the radiation pressure force to the gravitational attraction force is F β = F R G A = C m B ( λ) Q pr ( s, m, λ) dλ R where C is constant = of value.9 5 in cgs unit. cgm... (). Dependence of β of Cylindrical Particle on the Angle of Incident Radiation The β value for the cylindrical structure depends on the aspect ratio a/b (a is the length and b is radius of cylindrical particle) and incident angle of radiation. The β value of the circular cylindrical particle has been calculated, based on the DDA method [5] with a/b =.. Here in this study silicate and magnetite have been assumed to be the materials of cylindrical particle. The values of mass density are. g.cm - and 5. g.cm - for silicate and magnetite, respectively. The optical constant m* for silicate and magnetite are derived from the data complied in Mukai [5]. It is expected that in the derivation of β for the cylindrical particle, the incident angle of radiation Θ plays an important role, where Θ is defined as the angle between the solar radiation and the long symmetry axis of the cylinder. That is, Θ = º denotes that the solar radiation comes from the perpendicular direction to the circular end of the cylinder, while Θ = 9º means that the radiation comes perpendicularly to the curving side of the cylinder.

3 M. A. Saklayen, Md. Mahmuduzzaman and M. H. Ahsan Incident radiation Θ = º b Incident radiation Θ = 9º a Fig. : Geometry of a cylinder; where a is the length of curving side and b is the radius of its circular end. Arrows show the incident radiation Fig.a and b show a comparison of β of silicate and magnetite particle, respectively, as a function of an effective radius a eff for the sphere and the cylinder with an aspect ratio of a/b =. where a eff is the radius of the sphere with equivalent volume of the circular cylindrical particle. We computed the β value for the cylindrical particle with a eff from.µm to.µm with dipole number N = 6. The β values for the sphere were computed by the DDA, and have been checked that they agree very well with those computed by Mie theory..5 Sphere DDA C yl. Θ = C yl. Θ =9 β aeff (µm) Fig.a: A ratio of radiation pressure force to the solar gravity as function of effective radius for cylindrical silicate particle, two different incident angles and 9

4 Bound Orbit of Sub-micron Size Non-spherical Silicate Dust Particles in the Dust Band of the Asteroid Belt 6 5 S phere(dd A) Cyl. Θ = Cyl. Θ = 9 β aeff For the cylindrical particle, we found that the β value strongly depends on the incident angle of radiation Θ (see Figs.). Since in a/b =. the geometrical cross-section for Θ = 9º is larger than that for Θ = º and for the sphere with equivalent mass, the β value for Θ = 9º becomes always larger than other cases. In addition, in Θ = 9º it is found that the β value for silicate particle exceeds unity, whereas for the spherical particle and the cylinder with Θ = º, the β values never exceed unity. It is well known [,6] that for a spherical dielectric particle (silicate and water-ice) the β value is always less than unity. It is shown, however, that the irregularly shaped dielectric particle has the β value larger than unity in some cases mentioned above. On the other hand, it is also found that the β value for magnetite particle with Θ = º never exceeds unity. It is believed that the spherical particle consisting of the absorbing materials, such as magnetite, has β larger than unity in the range of particle radius less than µm. Recently Ripon Kumar Dey and Saklayen [7] computed the β for disc type absorbing and non-absorbing dust particles. In their study they also found that β values for non-spherical particle strongly depend on the angle of incidence of radiation. Therefore, the appearance of the absorbing particles with β < in all size range should be noted as a shape effect of the solar radiation pressure force on the particle. The Θ dependence of β is also examined for the cylinder with a eff =.5 µm and a/b =., for silicate and magnetite (see Fig.). From the curve fitting to the computed β values, we have derived the relation where, β is Fig.b: A ratio of radiation pressure force to the solar gravity as function of effective radius for cylindrical Magnetite particle, two different incident angles and 9 G β ( Θ) = β + k sin Θ..() FR at Θ = and k is a constant. Here, β =.8 and k =.55 for silicate, on the other hand, β = F.76 and k =.6 for magnetite.

5 M. A. Saklayen, Md. Mahmuduzzaman and M. H. Ahsan 6 5 magnetite β silicate Θ Fig. : Variation of β with incident angle of radiation Θ for cylindrical silicate and magnetite particles with an aspect ratio a/b =... Equation of Motion of Cylindrical Particle The equation of motion [8] of a particle in the solar system is given by d r r + µ =..(5) dt r where, µ = GM. If we consider the micron or submicron size particle, we should take into account the radiation pressure force. Direction of radiation pressure is radially outward from the Sun. Therefore, radiation modified two-body equation of motion for the particle is [8] d r r + µ ( β ) = (6) dt r It has been shown in sec. that for a cylindrical particle β varies with the incident angle of radiation. Therefore, eqn. (6) can be written as d r r + µ [ β ( Θ )] =.(7) dt r Radiation modified two-body method is used to find out the trajectory of cylindrical dust particle. The trajectory of the particle is followed by numerical integration of the equation of motion. The initial condition for the simulation is that the particle is set at AU, having the eccentricity and the orbital inclination is º are used. In the present case relative velocity of the particle released from the parent body is neglected. The RADAU numerical integrator [9] is used to solve eqn. (7). 5. Result and Discussions After releasing from parent body, the particle feels relative low gravitational attraction by a factor of (-β). Consequently, its orbit differs from those of the parent body. Fig. shows the orbit of a spherical particle with different β values. It is assumed that the cylindrical particle after release from a parent body has a spin axis aligned to the shortest cylindrical axis. Furthermore, it is assumed that the spin vector orbital angular momentum vectors are parallel. Therefore, for cylindrical particle varies as eqn.().

6 Bound Orbit of Sub-micron Size Non-spherical Silicate Dust Particles in the Dust Band of the Asteroid Belt 5 5 Y (AU) 9 β = β = X (AU) β =.7 β =.6 β =. Fig. : Trajectories of a spherical particle with different β values, released from the parent body with the circular orbit at AU, at X = AU and Y = AU toward the +Y direction. Heliocentric distance of such particle changes with spin. Fig. 5 shows the variation of heliocentric distances (X AU) with time during few days considering different spin of particle. Y (AU ) Sphere rotation/day rotation/day rotation/day.5 5 X (AU) Fig. 5: Heliocentric distance variation for spherical and cylindrical particle with different spin motion, where the similar calculation to Fig. is done.

7 6 M. A. Saklayen, Md. Mahmuduzzaman and M. H. Ahsan 8 Y (AU) X - (AU) Y (AU) X (AU) Y (AU) X (AU) Y (AU) X (AU) - Fig.6: Orbits of cylindrical silicate particle in X-Y plane with spin periods i.e. a) rotation/day, b) rotations/day, c) rotations/day and d) 6 rotations/day. It is found that the particle with slow spin changes the heliocentric distance more than that of the particles with fast spin. Saklayen and Mukai [] showed that for the slow spin particle, the predicted eccentricity varies from to larger than. This implies that the particle changes its position in a bound and unbound orbit, whereas the fast spin particle shows the eccentricity less than and it makes a bound orbit. Orbit of X-Y plane of cylindrical particles are shown in Fig. 6. We considered different spin periods, e.g., rotation/day, rotations/day, rotations/day and rotations/day. The particle with slow spin (e.g., rotation/day) changes the heliocentric distance much more than that of fast spin particle (e.g., rotations/day). As a result the slow spin particle makes a wide dust band with a bandwidth of.7 AU, which is wider than the main asteroid belt. It is found that the sub-micron size cylindrical particle with slow spin can easily diffuse from the asteroid belt. On the other hand, the fast spin particle makes narrow dust band with bandwidth of.55au, can easily stay in the asteroid belt. 6. Conclusions We have computed β values for finite cylindrical silicate particle with aspect ratio. for different angles of incident radiation. By using the resulting formula for a dependence of β on Θ, the trajectories of rotating cylindrical silicate dust particle in the asteroid belt are computed. It is found that the heliocentric distances of the orbit of particles fluctuate. So the collected dust from the dust detector on board space craft implies the instantaneous positions, eccentricities and inclinations of dust particles. References [] Fechting H., Gun E. and Kissel J. Laboratory Simulation in Cosmic Dust (J. A. M McDonnel Ed.) 978. [] Fechting H. In situ Records of Interplanetary Dust Particles-methods and results (H Esaesser and H Fechting Ed.) 976. [] Mcdonnel J. A. M. Micro particle Studies by Space Instrumention (J. A. M McDonnel Ed.) 978. [] Annested P. A. and Purcell E. M. Astron. Astrophs., 9,

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