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Effect of radiation on dust particles in orbital resonances
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Journal of Quantitative Spectroscopy & Radiative Transfer 100 (2006) 187–198 www.elsevier.com/locate/jqsrt
Effect of radiation on dust particles in orbital resonances J. Klacˇkaa,, M. Kocifajb,1 a
Department of Astronomy, Physics of the Earth, and Meteorology, Faculty of Mathematics, Physics, and Informatics, Comenius University, Mlynska´ dolina, 842 48 Bratislava, Slovak Republic b Institute for Experimental Physics, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria
Abstract The effect of electromagnetic radiation on the dynamics of arbitrarily shaped cosmic dust particles is investigated. The paper concentrates on the motion of dust grains near commensurability resonances with a planet—mean-motion resonances—and possible capture of the grains in the resonances. A particle is in resonance with a planet when the ratio of the mean motions of the two objects is a ratio of two small integers. The most fundamental properties of the orbital evolution of spherical dust particles in the mean-motion resonances are shortly rederived: the solar wind effect is also included and the existing result is improved. The results for spherical particles are compared with the detailed numerical calculations for nonspherical particles. It is shown that the fundamental results valid for spherical grains do not hold, in general, for nonspherical particles. While spherical particles are always characterized by the secular decrease of the semi-major axes near mean-motion resonances, this may not be true for nonspherical particles. Nonspherical grains may exhibit an increase of the semi-major axes before capturing in the meanmotion resonances. This is caused by the effect of electromagnetic radiation on nonspherical dust grains. The eccentricities of spherical particles in the exterior resonances approach a limiting value, but nonspherical grains may not follow this behaviour. The interior resonances are characterized by a systematic decrease of eccentricity for spheres, but various behaviours exist in the case of irregularly shaped particles. The motion of a nonspherical dust particle under the action of electromagnetic radiation may be characterized by a small change of the semi-major axis during a long-time interval, but the particle is not captured in any mean-motion resonance. This kind of motion does not exist for spherical grains. r 2005 Elsevier Ltd. All rights reserved. Keywords: Electromagnetic radiation; Cosmic dust; Orbital motion
1. Introduction The existence of light pressure has been known since Maxwell. Maxwell used the equations for electromagnetic field, which were obtained by him in 1864 [1]. Later on Poynting [2] formulated the problem of finding the equation of motion for a perfectly absorbing spherical particle under the action of electromagnetic radiation. Robertson was the first who solved this problem [3]. His result has been applied to the astronomical Corresponding author. Tel.: +421 7 6029 5684. 1
E-mail addresses: [email protected] (J. Klacˇka), [email protected] (M. Kocifaj). On leave from Astronomical Institute, Slovak Academy of Sciences.
0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2005.11.037
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problems for several decades and is known as the Poynting–Robertson (P–R) effect. The Robertson’s case was relativistically generalized by Klacˇka [4], who showed that the generalized P–R effect holds only for the special case where the total momentum of the outgoing radiation per unit time is colinear with the incident radiation (in the proper/rest frame of reference of the particle). Since real particles interact with electromagnetic radiation in a complicated manner and particles of various optical properties exist (e.g., [5]), it is essential to have an equation of motion sufficiently general to cover a wide range of optical parameters, not just the limited cases previously investigated. This equation of motion within the accuracy to the first order in v=c can be found in [6]. The equation of motion for an arbitrarily shaped particle moving in a radiation field taking into account the radiation pressure caused by the anisotropy of thermal emission as well as scattering and absorption of light was presented by Klacˇka [7], in a relativistically covariant form. The experimental verification of the fact that real nonspherical dust particles under the action of radiation pressure forces move in a different way than spherical dust grains, may be found in [8]. This paper deals with one of the applications of the electromagnetic radiation action on the motion of cosmic dust grains. The aim is to treat the motion of dust grains near the orbital—mean-motion—resonances with planets (commensurability resonances). Several papers were published during the last two decades on this theme, e.g., [9–15]. However, all these papers consider dust particles as spheres and the Poynting–Robertson effect is used. The great advantage of such an approach is that some fundamental characteristics of the orbital motion can be obtained in an analytic way. However, real dust particles are nonspherical. The important question arises: Are the fundamental features, found for spherical particles, valid for arbitrarily shaped dust grains? As was shown by Klacˇka et al. [16], some qualitatively new results may exist when one admits that real particles are nonspherical—the results refer to the exterior mean-motion resonance. In this contribution we want to concentrate also on the interior mean-motion resonances.
2. Gravity and radiation—equation of motion to first order in v=c Let us consider an arbitrarily shaped dust grain under the action of solar gravitation and solar electromagnetic radiation. The equation of motion of the grain, is (see Eqs. (41)–(42) in [7]) 3 d~ v GM GM X ¼ 2 ~ bj e1 þ 2 dt r r j¼1
~ ~ v ~ ej v ~ e1 ~ v ~ þ 12 ej , c c c
(1)
where ~ v ~ e0j =cÞ~ e0j þ ~ v=c; j ¼ 1; 2; 3, ej ¼ ð1 ~ Z 1 2 pR B ðlÞfC 0ext ðlÞ C 0sca ðlÞg01 ðlÞg dl, b1 ¼ GM mc 0 Z 1 pR2 B ðlÞfC 0sca ðlÞg02 ðlÞg dl, b2 ¼ GM mc 0 Z 1 pR2 B ðlÞfC 0sca ðlÞg03 ðlÞg dl, b3 ¼ GM mc 0
ð2Þ
if thermal emission of the grain is neglected. R denotes the radius of the Sun and B ðlÞ is the solar radiance at a wavelength of l; G, M , and r are the gravitational constant, the mass of the Sun, and the distance of the particle fromR the center of the Sun, respectively. The asymmetry parameter vector ~ g0 is defined 0 0 0 0 0 0 0 by ~ g ¼ ð1=C sca Þ ~ n ðdC sca =dO Þ dO , where ~ n is the unit vector in the direction of scattering; ~ g0 ¼ g01~ e0j ¼ ð1 þ ~ v ~ ej =cÞ~ ej ~ v=c, ~ e0i ~ e0j ¼ dij ; dO0 is an elementary solid angle, and C 0sca is e01 þ g02~ e02 þ g03~ e03 , ~ 0 0 ¯ 0pr;2 =Q ¯ 0pr;1 , the cross-section for scattered radiation: C ext C abs ¼ C 0sca . We may mention that b1 b, b2 bQ 0 0 ¯ pr;3 =Q ¯ pr;1 correspond to the quantities used in [6]. b3 bQ
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In general, we have to take into account that the non-dimensional parameter for the Solar System b1 b ¼
r2 SA0 ¯ 0 L A0 ¯0 Qpr;1 Q GM mc 4pGM mc pr;1 0 2 ¯ 0pr;1 A ½m ¼ 7:6 104 Q m½kg
ð3Þ
(‘‘the ratio of the radiation pressure force to the gravitational force’’) may change during the motion. Here L is the rate of the energy outflow from the Sun, the solar luminosity. A quantity A0 is the geometrical cross section of a sphere of the volume equal to the volume of the particle. b may change and all three parameters b1 , b2 and b3 have to be numerically calculated at each point of the orbit (except for a very special cases, when the b parameters are constant during the motion). Since in the discussed applications b does not exhibit large changes during the particle motion, we will use (osculating) orbital elements for the case when central acceleration is given by GM ð1 bÞ~ e1 =r2 . This is a very good approximation to the reality. The central acceleration GM ð1 bÞ~ e1 =r2 defines the Keplerian orbits within this approximation. When dealing with the long-term orbital evolution, we may use the mean values of orbital elements—they may be defined as time mean values of orbital elements when the true anomaly changes in 2p radians: Z 1 T hgi gðtÞ dt. (4) T 0 As for simulations of nonspherical particles, the quantities fbj ; j ¼ 1; 2; 3g in Eq. (1) are calculated in two steps. The first step is defined by Eq. (2), and the second step is represented by the averaging (of the values obtained in the first step) over the azimuthal/rotational angle corresponding to the fixed axis of rotation of the particle. The fact that the rotational axis of the particle is fixed in our computational runs is consistent with the experimental results that the rotational axis exhibits only small changes with respect to some preferred orientation [17].
3. Secular evolution of orbital elements for spherical particle If we consider that the material within a dust grain is distributed in a spherically symmetric way, then b2 ¼ b3 ¼ 0 and b, defined by Eq. (3), is constant. This very special case is well-known as the Poynting–Robertson effect. In this very special case analytic calculations of the secular evolution of orbital ¯ 0pr;1 =ðR½g=cm3 s½cmÞ, for a homogeneous spherical elements are possible. Eq. (3) reduces to b ¼ 5:7 105 Q particle: R is mass density and s is the radius of the sphere. In order to obtain more general results, we will consider the evolution of a spherical dust particle under the action of solar gravity, solar electromagnetic radiation and solar wind. If we consider the most simple approximation of the solar wind [18], the equation of motion of the particle is ~ ~ ~ ~ d~ v GM GM b GM u v ~ e1 v v ~ e1 v ~ ~ ¼ 2 ~ 1 1 e1 þ b 2 þZ 0 , (5) e1 e 1 ¯ dt c u r r c r2 c u Q pr;1 where we neglect the decrease in the particle’s mass. The quantity u is the speed of the solar wind particles, Z 1=3 [7]. ¯ 0pr;1 ÞðGM =r2 Þðu=cÞ, we can rewrite Eq. (5) to the following Neglecting the solar wind pressure term Zðb=Q form: ! ~ d~ v GM ð1 bÞ Z GM ~ v ~ e1 v ~ ~ ¼ b 1 þ þ e , (6) e 1 1 ¯ 0pr;1 dt r2 c r2 c Q e1 =r2 as the where the second term causes the deceleration of the particle’s motion. We use GM ð1 bÞ~ central acceleration determining osculating orbital elements. The time averaging defined by Eq. (4) leads to
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secular evolution of orbital elements: ! da GM Z 2 þ 3e2 1þ 0 , ¼ b ¯ pr;1 að1 e2 Þ3=2 dt c Q ! de GM Z 5e=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi , 1þ 0 ¼ b ¯ pr;1 a2 1 e2 dt c Q di ¼ 0, dt
ð7Þ
where only the secular evolutions of the semi-major axis a, eccentricity e and inclination i of the particle are presented. T in Eq. (4) is a time interval between passages through two following pericenters. 4. Mean-motion resonances with planets—definitions According to the third Kepler’s law we have a3 n2 ¼ GM ð1 bÞ, a3P n2P ¼ GðM þ mP Þ,
ð8Þ
where a, aP are the semi-major axes of a particle characterized by parameter b (see Eq. (3)) and a planet with mass mP , respectively. n and nP are the mean motions characterizing the revolutions around the Sun of mass M . The first part of Eq. (8) uses the fact that the central Keplerian acceleration is given by the sum of the solar gravitational acceleration and radial component of the radiation pressure acceleration, as it was discussed in Section 2. Eq. (8) yields
2=3 mP 1=3 1=3 nP a ¼ ð1 bÞ 1þ aP . (9) n M If the dust particle is in a resonance with the planet, we can define the q-th order exterior resonance by the relation nP =n ¼ ðp þ qÞ=p, where p and q are integer numbers. Similarly, the qth order interior resonance is defined by the relation nP =n ¼ p=ðp þ qÞ, where p and q are integer numbers. In terms of orbital periods: T=T P ¼ ðp þ qÞ=p for exterior, T=T P ¼ p=ðp þ qÞ for interior resonances. Based on these definitions and Eq. (9), we can immediately write p þ q 2=3 mP 1=3 a ¼ ð1 bÞ1=3 1þ aP , (10) p M for the semi-major axis of the dust particle in the q-th order exterior resonance with the planet of mass mP . Similar relation can be obtained for the interior resonance. 5. Mean-motion resonances and spherical dust particles Let us consider a spherical dust grain under the action of the gravitational forces generated by the Sun and a planet moving around the Sun. Moreover, in reality the grain is evolving also under the action of the solar electromagnetic radiation and solar wind. The equation of motion of the particle is ! ~ ~ ~ ~ d~ v GM ð1 bÞ Z GM v ~ e1 v r ~ rP rP ~ ~ ¼ þ e1 b 1 þ 0 , (11) GmP e1 þ ¯ pr;1 dt r2 c r2 c j~ r ~ rP j3 j~ rP j 3 Q where ~ r and ~ rP are the position vectors of the particle and the planet with respect to the Sun, and, as a standard, m ¼ GM and G is the gravitational constant. Eq. (5) was used and the term generated by the gravitational force of the planet was added. If the particle is ejected from a parent body (comet, asteroid, planet, satellite of a planet), then its secular evolution is given by Eq. (7). The important property is that the secular evolution of the semi-major axis of the
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particle is a decreasing function of time. In this way, the particle approaches the Sun and sometimes it may reach a suitable condition to be captured in a mean-motion resonance. Being in the resonance, its secular evolution is characterized by a constant value of the semi-major axis defined by Eq. (9) or Eq. (10) (or, by analogous equation for the interior resonance). The particle can be captured for thousands or even of billions years. It depends on the size of the particle (note that b / size1 ), mass of the planet, type of the resonance (the smaller values of p and q, the longer capture times), and, also on the initial orbital conditions at the instant of the capture. Finally, when the capture terminates, Eq. (7) correctly describes the behaviour of the particle: the secular evolution of semi-major axis is, again, a decreasing function of time. We have already mentioned that the secular evolution of the semi-major axis is characterized by its constant value when a particle is in resonance with a planet. What can be said about the secular evolution of the eccentricity of a particle in such a situation? In order to show what really happens, we suppose that the planet is moving in a circular orbit around the Sun. In this case, we have a special gravitational problem of three bodies together with a presence of small nongravitational forces. In celestial mechanics this gravitational problem is called circular restricted problem of three bodies. There exists a quantity Tisserand’s parameter— see, e.g. [19], which does not change during the motion of the third body whose mass is negligible in comparison to the masses of the planet and the Sun. For our case, defined by Eq. (11), we can write the Tisserand’s parameter in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1b ð1 bÞað1 e2 Þ CT ¼ þ2 cos I, (12) a a3P where e is the eccentricity of the particle characterized by the parameter b and I is the inclination of the particle orbital plane with respect to the plane of the planetary orbit. The term 1 b in Eq. (12) corresponds to the reduction of the solar mass, as it is presented in Eq. (11), and, the considered gravitational problem is given by masses M ð1 bÞ, mP and mass of the dust particle. The velocity terms in Eq. (11) represent nongravitational terms in the following considerations. We have to stress that the Tisserand’s parameter C T stays constant only in a special case of the circular restricted problem of three bodies. However, Eq. (12) enables us to find the secular change of the eccentricity of a particle captured into a resonance. For this purpose, we will consider I ¼ 0 in Eq. (12), which is not in contradiction with the fact that the considered nongravitational effects do not change the inclination of the orbital plane (di=dt ¼ 0 in Eq. (7)). We can write dC T qC T da qC T de ¼ þ , (13) dt qa dt total qe dt total for the total time derivative of the Tisserand’s parameter C T defined by Eq. (12). However, according to Eq. (11), time derivatives of the semi-major axis and of eccentricity of the particle are caused by the gravitational perturbations of the planet (these terms will be denoted by the subscript G) and nongravitational perturbations caused by the solar electromagnetic radiation and solar wind (these terms will be denoted by the subscript NG and they correspond to the velocity terms in Eq. (11)): da da da ¼ þ , dt total dt G dt NG de de de ¼ þ . ð14Þ dt total dt G dt NG On the basis of the Eqs. (13)–(14) we can write dC T qC T da da qC T de de ¼ þ þ þ . dt G dt NG dt G dt NG dt qa qe According to Tisserand, gravitational terms alone do not change the value of C T : qC T da qC T de þ ¼ 0. qa dt G qe dt G
(15)
(16)
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Substituting from the Eq. (16) into the Eq. (15): dC T qC T da qC T de ¼ þ . dt qa dt NG qe dt NG
(17)
If we are interested in secular changes of the orbital elements a and e, then the particle stability in the resonance is characterized by the relation ðda=dtÞtotal ¼ 0 and Eq. (13) reduces to dC T qC T de ¼ . (18) dt qe dt total Eqs. (17)–(18) finally brings out the total secular change of the eccentricity of the particle de de qC T =qa da ¼ þ . dt total dt NG qC T =qe dt NG
(19)
Using the calculated partial derivatives of C T defined by Eq. (12) together with the Eq. (7) for the secular change of the eccentricity of a particle caused by nongravitational forces, one finally obtains ! pffiffiffiffiffiffiffiffiffiffiffiffiffi ( ) 1 e2 de GM Z n 1 þ 3e2 =2 ¼b 1þ 0 1 , (20) ¯ pr;1 dt nP ð1 e2 Þ3=2 c a2 Q if the definition represented by Eq. (9) has been used as well, and also the fact that mP 5M has been taken into account. Moreover, the symbol of angle brackets—denoting secular evolution—in Eq. (7) has been also omitted. Eq. (20) is an improved result of Liou and Zook [12]. Eq. (20) determines the secular evolution of the eccentricity of a particle characterized by the values b and ¯ 0pr;1 if the particle is captured into a mean-motion resonance. This equation enables us to find the detailed Q evolution. However, an important qualitative behaviour can be found very easily. If we take some special mean-motion resonance, we already know the value n=nP . If the initial (secular) eccentricity is small enough, then de=dt is always positive and the eccentricity of the particle is an increasing function of time during the stay of the particle in the mean-motion resonance. Eccentricity of the particle can only approach asympotically to a limiting value elim given by the condition that the term in the combined brackets equals zero: nP =n ¼ ð1 þ 3e2lim =2Þ=ð1 e2lim Þ3=2 . If the initial eccentricity is large enough, then de=dt is always negative and the eccentricity of the particle is a decreasing function of time during the stay of the particle in the meanmotion resonance. Again, eccentricity of the particle can only approach asympotically to the limiting value elim . 5.1. Mean-motion resonances and spherical dust particles—fundamental results In the zones of mean-motion resonances with a planet, the orbital evolution of a spherical dust grain in the circular restricted problem of three bodies, with a simultaneous action of the solar electromagnetic radiation (and the solar wind), is characterized by the following fundamental results: (i) secular evolution of the semi-major axis is a decreasing function of time outside the resonances, see Eq. (7), (ii) secular evolution of the eccentricity is a decreasing function of time outside the resonances, see Eq. (7), (iii) da=dt ¼ 0 in the resonance, (iv) secular evolution of the eccentricity in the exterior resonance is either an increasing function of time or a decreasing function of time and the values of the eccentricity cannot reach a limiting value elim , given by condition nP =n ¼ ð1 þ 3e2lim =2Þ=ð1 e2lim Þ3=2 , see Eq. (20), (v) secular evolution of the eccentricity in the interior resonance is a decreasing function of time, see Eq. (20), (vi) if the initial position vector (with respect to the Sun) and the velocity vector of the particle lie in the orbital plane of the planet, then the orbital plane of the spherical particle is conserved. The crucial question emerges: are the features, presented above, typical features for the real dust particles?
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6. Mean-motion resonances and nonspherical dust particles Real dust particles are nonspherical in shape. Thus, the total electromagnetic force represented by Eqs. (1)–(2) has to be used, instead of the Poynting–Robertson effect. Equation of motion for a particle in the problem of three bodies with the action of electromagnetic radiation and solar wind is 3 ~ ~ v ~ ej d~ v GM GM X v ~ e1 ~ v ~ ¼ 2 ~ þ bj 1 2 e1 þ 2 ej dt c r r c c j¼1 0 ~ ~ ~ Z Ainc GM ~ v ~ e1 v r ~ rP rP ~ b 0 þ þ , Gm e 1 P ¯ c A 0 r2 c j~ r ~ rP j3 j~ rP j 3 Q pr;1
ð21Þ
if the solar wind pressure term is neglected and A0inc is the cross section of the actually illuminated area of the particle. The following questions arise for the case of nonspherical particles: Can the particle be captured into a mean-motion resonance with a planet? How does the secular orbital evolution of the particle look in the zones of the resonances? We have done numerical calculations for a particle of a shape identical to that of the particle U2015 B10 [20]. The grain U2015 B10 (archived in NASA collection [21]) appears to be representative for cosmic dust. Its aspect ratio (the ratio of the largest and the smallest characteristic lengths of the particle) is equal to 1.4 and this value coincides well with the results of mid-infrared spectropolarimetry [22]. These results indicate that aspect ratios of cosmic dust grains lie between 4=3 and 3=2. We use the Discrete Dipole Approximation [23–25], to calculate the characteristics of the scattered radiation field because the method is applicable to the arbitrary targets. To accelerate the simulation of the particle motion in the solar radiation field we have precalculated a large database of scattering diagrams for particles of different sizes with a dense enough lattice Y B F, where the Euler’s angles Y, B, and F unambiguously identify the particle orientation (in local coordinate system) with respect to the direction of the incident radiation [26]. Choosing the initial slope of the particle rotation axis, the particle orientation is calculated at each point of its trajectory. An accurate interpolation of the scattering diagrams is applied to calculate orientationally averaged polychromatic efficiency factors for the radiation pressure. It has been supposed that the particle rapidly rotates around a rotational axis fixed in space (various rotational axes have been considered) and the orientation averaging over the corresponding azimuthal angle with respect to the Sun have been done; values of bj obtained this way were used in Eq. (21). The simulations have been done for the circular restricted problem of three bodies and solar wind is the only nongravitational effect besides the solar electromagnetic radiation. This enables comparison between the behaviour of the spherical dust grains (the results can be obtained even analytically—important from the point of view of the completeness of the possible solutions) and nonspherical grains. Moreover, it is evident that any difference between behaviour of spherical and nonspherical dust grains is due to the effect of nonsphericity of the grains and the terms b2 and b3 in Eq. (1) are relevant. As for the questions asked above, the answer is that the spherical dust particles do not yield any representative results for the real dust grains. Nonspherical particles are also captured in mean-motion resonances, but their orbital evolution is quite complex. It is documented in Figs. 1–5 which show possible evolution for outer and inner resonances with planet Neptune. In all cases the magnesium-rich silicate dust grains of the effective radius of 2 microns (shape U2015 B10) are considered, the relative change of the parameter b is less than 5% for the various orientations during the motion of the particles. Figs. 1 and 2 depict the exterior mean-motion resonances defined by the ratios T : T P ¼ 5 : 4 and 7 : 6. In both cases, the resonances are evident and no overlap with other resonances exists during the particle motion in the given commensurabilities. The secular evolution of the eccentricities in the resonances is not characterized by a monotonous function of time as it is in case of the spheres (Eq. (21) and summarization below Eq. (20)). The orbital inclinations are not conserved and they change due to the effect of the electromagnetic radiation, since other considered effects conserve the orbital plane. While the secular evolution of the semi-major axis is characterized by its increase at the moment of ejection from the resonance, in Fig. 1, the behaviour of the same orbital element decreases at the moment of ejection in Fig. 2 (however, the
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Fig. 1. Secular evolution of the orbital elements of the irregularly shaped dust grain in the zone of mean-motion resonance 5 : 4 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
decrease is more rapid than in the case of spheres). The secular evolutions of the eccentricities are increasing functions of time at the moments of the ejections from the resonances. The inner mean-motion resonances with Neptune (Figs. 3–5) are defined by the ratios T : T P ¼ 7 : 8, 15 : 16 and 6 : 7. The nonspherical particle in the Fig. 3 is characterized by the semi-major axis in the resonant zone 27:02 AUpap27:28 AU (theoretical value for b ¼ 0:036, and a ¼ 27:164 AU) and no overlap with other resonances during the capture exists. No such overlap was found neither for the resonance 6 : 7 (see Fig. 5). Here, the theoretical value of the resonant semi-major axis is 26.5 AU, and b 0:07. Fig. 4 probably shows only a mimic of a resonance. The corresponding theoretical resonance appears to be 15 : 16 but it is very close to other resonances. A complex ripple structure in the semi-major axis occurs due to the realistic motion of the nonspherical particle. This situation corresponds to some kind of hovering in semi-major axis. Such kind of behaviour does not exist in case of a spherical particle. Figs. 4, 5 show nonmonotonous behaviour of the secular evolution of the eccentricity and it is an increasing function of time at the moments of the ejections from the resonances, in both cases. Fig. 4 yields an interesting relation between the secular changes of eccentricity and inclination: e i constant; we do not know any theoretical explanation of this numerically found result. While the secular evolution of the orbital inclination is characterized by a constant value for the spherical particle in the considered model, nonspherical particles exhibit nonmonotonous character of the inclination evolution during the capture in the resonances. The secular evolutions of the inclinations may be increasing (Figs. 1 and 4) or decreasing (Fig. 2) functions of time at the moments of the ejections.
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Fig. 2. Secular evolution of orbital elements of irregularly shaped dust grain in the zone of mean-motion resonance 7 : 6 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
.
.
.
.
Fig. 3. Time evolution of the semi-major axis and the eccentricity for an irregularly shaped dust grain of the effective radius of 2 microns, b 0:036. The plots show interior resonance 7 : 8 of the magnesium-rich silicates (mass density 4 g=cm3 ) with Neptune.
The figures show that the nonspherical dust grains can be captured even in the case when the secular change of semi-major axis is an increasing function of time near the zone of the resonance, and, no limiting value of eccentricity exists in the resonance.
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Fig. 4. Secular evolution of orbital elements of irregularly shaped dust grain in the zone of mean-motion resonances near 15 : 16 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
6.1. Mean-motion resonances and arbitrarily shaped dust particles: fundamental results The resonant captures of arbitrarily shaped dust grains exist for exterior and interior mean-motion resonances with planets. An ejection of a grain from the exterior resonance is due to the grain’s close encounter with a planet, but also due to the strong effect of the electromagnetic radiation. While the semi-major axis and the eccentricity of a spherical grain are strictly decreasing functions of time shortly before the capture into the resonance (see Eq. (7)), they may be increasing or decreasing functions in case of nonspherical particles. Our results, depicted in figures, show that various evolutions of the semi-major axes and eccentricities for nonspherical dust grains can occur. Moreover, the orbital inclinations of the nonspheres can significantly change so particles do not move in the orbital plane of the planet. The most important physical result is that the resonant trapping, under action of (solar) electromagnetic radiation, is not normally expected for diverging orbits such as the orbits of nonspherical dust particles: as it was pointed out for exterior resonances [16], and completed for interior resonances in this paper, this is the first case when a physically justified force can generate such a behaviour. Nonspherical particles can move in a region of resonances, but the apparent capture in a resonance does not correspond to the real resonant motion. The effect of radiation can mimic the resonant motion.
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Fig. 5. Secular evolution of orbital elements of irregularly shaped dust grain in the zone of mean-motion resonance 6 : 7 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
7. Conclusions This contribution deals with the effect of the electromagnetic radiation on the dynamics of the cosmic dust particles. The paper presents the application of the equation of motion on mean-motion resonances with planets. In order to obtain significant results, we have used a model which can be treated analytically for the spherical particles. The completeness of the solution is guaranteed in this case. Thus, we consider the circular restricted problem of three bodies with the action of the solar electromagnetic radiation and the solar wind. The complete solutions of the motion of the spherical particles outside the resonances are presented in Eq. (7) and those inside the resonances in Eq. (20). Both sets of solutions are summarized in Eq. (20). Same gravitational and nongravitational effects are considered in the case of the nonspherical dust grains. This enables a simple comparison between the effects of the electromagnetic radiation on nonspherical and on spherical dust grains. Although some other nongravitational forces could be considered (Lorentz force, collisions among particles, better approximation of the solar wind, ...), the problem could not be analytically treated then. Simple understanding of the differences in action of the electromagnetic radiation on spherical and on nonspherical grains would not be possible. Moreover, the nongravitational effects may play differently relevant roles in various extra-solar planetary systems. However, the improvements mentioned above are to be done in the future work. Differences between the behaviour of the spherical and nonspherical particles are pointed out in this work. Although the spherical particles often enable analytical calculations, their orbital evolution in the zones of
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mean-motion resonances cannot be considered as a representative evolution for the real cosmic dust particles. The motion of a spherical grain is characterized by a decrease of the semi-major axis (and eccentricity) before the capture in the mean-motion resonance. Nonspherical grains may be captured even in case when the secular change of the semi-major axis has been an increasing function of time outside the resonance. This result is correct not only for the exterior mean-motion resonances as it was pointed out by Klacˇka et al. [16], but also for interior mean-motion resonances as shown in this paper. Resonant trapping is not normally expected for diverging orbits and the obtained results due to the effect of electromagnetic radiation on nonspherical dust particles represent the only known case when a physically justified force can generate such a behaviour. The effect of the electromagnetic radiation on the motion of the nonspherical dust particles can resemble the motion in a mean-motion resonance. The semi-major axis of the particle does not change for a long time but the motion does not correspond to any simple resonance. Acknowledgements This paper was supported by Project M772-N02 by the Fonds FWF and by VEGA grant No. 1/3074/06 and 2/3024/23. We specially thank Prof. Ch. Dellago for computer resources at the Institute of Experimental Physics. References [1] Maxwell JC. A Treatise on Electricity and Magnetism. Oxford: Oxford University Press; 1873. [2] Poynting JM. Radiation in the Solar System: its effect on temperature and its pressure on small bodies. Philos Trans R Soc London Ser A 1904;202:525–52. [3] Robertson HP. Dynamical effects of radiation in the Solar System. Mon Not R Astron Soc 1937;97:423–38. [4] Klacˇka J. Poynting–Robertson effect. I. Equation of motion. Earth, Moon and Planets 1992;59:41–59. [5] Mishchenko M, Travis LD, Lacis AA. Scattering absorption and emission of light by small particles. Cambridge, UK: Cambridge University Press; 2002 445pp. [6] Klacˇka J, Kocifaj M. Motion of nonspherical dust particle under the action of electromagnetic radiation. JQSRT 2001;70:595–610. [7] Klacˇka J. Electromagnetic radiation and motion of a particle. Cel Mech Dyn Astron 2004;89:1–61. [8] Krauss O, Wurm G. Radiation pressure forces on individual micron-size dust particles: a new experimental approach. JQSRT 2004;89:179–89. [9] Beauge´ C, Ferraz-Mello S. Capture in exterior mean-motion resonances due to Poynting–Robertson drag. Icarus 1994;110:239–60. [10] Jackson AA, Zook HA. A Solar System dust ring with the Earth as its shepherd. Nature 1989;337:629–31. [11] Liou J-Ch, Zook HA. An asteroidal dust ring of micron-sized particles trapped in the 1: 1 mean motion resonance with Jupiter. Icarus 1995;113:403–14. [12] Liou J-Ch, Zook HA. Evolution of interplanetary dust particles in mean motion resonances with planets. Icarus 1997;128:354–67. [13] Liou J-Ch, Zook HA, Jackson AA. Radiation pressure, Poynting–Robertson drag, and solar wind drag in the restricted three body problem. Icarus 1995;116:186–201. [14] Marzari F, Vanzani V. Dynamical evolution of interplanetary dust particles. Astron Astrophys 1994;283:275–86. [15] Sˇidlichovsky´ M, Nesvorny´ D. Temporary capture of grains in exterior resonances with Earth: Planar circular restricted three-body problem with Poynting–Robertson drag. Astron Astrophys 1994;289:972–82. [16] Klacˇka J, Kocifaj M, Pa´stor P. Motion of dust near exterior resonances with planets. J Phys 2005;6:126–31. [17] Krauss O. 2005. personal communication. + E. Interplanetary dust. In: Schwenn R, Marsch E, editors. Physics of the inner heliosphere I. Berlin, Heidelberg: [18] Leinert Ch, Grun Springer; 1990. p. 207–75. [19] Brouwer D, Clemence GM. Celestial mechanics. New York, London: Academic Press; 1961 598pp. [20] Kocifaj M, Kapisˇ insky´ I, Kundracı´ k F. Optical effects of irregular cosmic dust particle U2015 B10. JQSRT 1998;63:1–14. [21] Clanton VS, Gooding JL, Mckay DS, Robinson GA, Warren JL, Watts LA. Cosmic dust catalog (particles from collection flag U2015), NASA, Johnson Space Center (Houston, TX) 70/1 10, 1984. [22] Hildebrand RH, Dragovan M. The shapes and alignment properties of interstellar dust grains. Astrophys J 1995;450:663–6. [23] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186:705–14. [24] Draine BT. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys J 1988;333:848–72. [25] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am 1994;A11:1491–9. [26] Draine BT, Flatau PJ. User Guide for the Discrete Dipole Approximation Code DDSCAT.6.1’’, Freeware, hhttp://arxiv.org/abs/ astro-ph/0409262i2004.
Journal of Quantitative Spectroscopy & Radiative Transfer 100 (2006) 187–198 www.elsevier.com/locate/jqsrt
Effect of radiation on dust particles in orbital resonances J. Klacˇkaa,, M. Kocifajb,1 a
Department of Astronomy, Physics of the Earth, and Meteorology, Faculty of Mathematics, Physics, and Informatics, Comenius University, Mlynska´ dolina, 842 48 Bratislava, Slovak Republic b Institute for Experimental Physics, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria
Abstract The effect of electromagnetic radiation on the dynamics of arbitrarily shaped cosmic dust particles is investigated. The paper concentrates on the motion of dust grains near commensurability resonances with a planet—mean-motion resonances—and possible capture of the grains in the resonances. A particle is in resonance with a planet when the ratio of the mean motions of the two objects is a ratio of two small integers. The most fundamental properties of the orbital evolution of spherical dust particles in the mean-motion resonances are shortly rederived: the solar wind effect is also included and the existing result is improved. The results for spherical particles are compared with the detailed numerical calculations for nonspherical particles. It is shown that the fundamental results valid for spherical grains do not hold, in general, for nonspherical particles. While spherical particles are always characterized by the secular decrease of the semi-major axes near mean-motion resonances, this may not be true for nonspherical particles. Nonspherical grains may exhibit an increase of the semi-major axes before capturing in the meanmotion resonances. This is caused by the effect of electromagnetic radiation on nonspherical dust grains. The eccentricities of spherical particles in the exterior resonances approach a limiting value, but nonspherical grains may not follow this behaviour. The interior resonances are characterized by a systematic decrease of eccentricity for spheres, but various behaviours exist in the case of irregularly shaped particles. The motion of a nonspherical dust particle under the action of electromagnetic radiation may be characterized by a small change of the semi-major axis during a long-time interval, but the particle is not captured in any mean-motion resonance. This kind of motion does not exist for spherical grains. r 2005 Elsevier Ltd. All rights reserved. Keywords: Electromagnetic radiation; Cosmic dust; Orbital motion
1. Introduction The existence of light pressure has been known since Maxwell. Maxwell used the equations for electromagnetic field, which were obtained by him in 1864 [1]. Later on Poynting [2] formulated the problem of finding the equation of motion for a perfectly absorbing spherical particle under the action of electromagnetic radiation. Robertson was the first who solved this problem [3]. His result has been applied to the astronomical Corresponding author. Tel.: +421 7 6029 5684. 1
E-mail addresses: [email protected] (J. Klacˇka), [email protected] (M. Kocifaj). On leave from Astronomical Institute, Slovak Academy of Sciences.
0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2005.11.037
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problems for several decades and is known as the Poynting–Robertson (P–R) effect. The Robertson’s case was relativistically generalized by Klacˇka [4], who showed that the generalized P–R effect holds only for the special case where the total momentum of the outgoing radiation per unit time is colinear with the incident radiation (in the proper/rest frame of reference of the particle). Since real particles interact with electromagnetic radiation in a complicated manner and particles of various optical properties exist (e.g., [5]), it is essential to have an equation of motion sufficiently general to cover a wide range of optical parameters, not just the limited cases previously investigated. This equation of motion within the accuracy to the first order in v=c can be found in [6]. The equation of motion for an arbitrarily shaped particle moving in a radiation field taking into account the radiation pressure caused by the anisotropy of thermal emission as well as scattering and absorption of light was presented by Klacˇka [7], in a relativistically covariant form. The experimental verification of the fact that real nonspherical dust particles under the action of radiation pressure forces move in a different way than spherical dust grains, may be found in [8]. This paper deals with one of the applications of the electromagnetic radiation action on the motion of cosmic dust grains. The aim is to treat the motion of dust grains near the orbital—mean-motion—resonances with planets (commensurability resonances). Several papers were published during the last two decades on this theme, e.g., [9–15]. However, all these papers consider dust particles as spheres and the Poynting–Robertson effect is used. The great advantage of such an approach is that some fundamental characteristics of the orbital motion can be obtained in an analytic way. However, real dust particles are nonspherical. The important question arises: Are the fundamental features, found for spherical particles, valid for arbitrarily shaped dust grains? As was shown by Klacˇka et al. [16], some qualitatively new results may exist when one admits that real particles are nonspherical—the results refer to the exterior mean-motion resonance. In this contribution we want to concentrate also on the interior mean-motion resonances.
2. Gravity and radiation—equation of motion to first order in v=c Let us consider an arbitrarily shaped dust grain under the action of solar gravitation and solar electromagnetic radiation. The equation of motion of the grain, is (see Eqs. (41)–(42) in [7]) 3 d~ v GM GM X ¼ 2 ~ bj e1 þ 2 dt r r j¼1
~ ~ v ~ ej v ~ e1 ~ v ~ þ 12 ej , c c c
(1)
where ~ v ~ e0j =cÞ~ e0j þ ~ v=c; j ¼ 1; 2; 3, ej ¼ ð1 ~ Z 1 2 pR B ðlÞfC 0ext ðlÞ C 0sca ðlÞg01 ðlÞg dl, b1 ¼ GM mc 0 Z 1 pR2 B ðlÞfC 0sca ðlÞg02 ðlÞg dl, b2 ¼ GM mc 0 Z 1 pR2 B ðlÞfC 0sca ðlÞg03 ðlÞg dl, b3 ¼ GM mc 0
ð2Þ
if thermal emission of the grain is neglected. R denotes the radius of the Sun and B ðlÞ is the solar radiance at a wavelength of l; G, M , and r are the gravitational constant, the mass of the Sun, and the distance of the particle fromR the center of the Sun, respectively. The asymmetry parameter vector ~ g0 is defined 0 0 0 0 0 0 0 by ~ g ¼ ð1=C sca Þ ~ n ðdC sca =dO Þ dO , where ~ n is the unit vector in the direction of scattering; ~ g0 ¼ g01~ e0j ¼ ð1 þ ~ v ~ ej =cÞ~ ej ~ v=c, ~ e0i ~ e0j ¼ dij ; dO0 is an elementary solid angle, and C 0sca is e01 þ g02~ e02 þ g03~ e03 , ~ 0 0 ¯ 0pr;2 =Q ¯ 0pr;1 , the cross-section for scattered radiation: C ext C abs ¼ C 0sca . We may mention that b1 b, b2 bQ 0 0 ¯ pr;3 =Q ¯ pr;1 correspond to the quantities used in [6]. b3 bQ
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In general, we have to take into account that the non-dimensional parameter for the Solar System b1 b ¼
r2 SA0 ¯ 0 L A0 ¯0 Qpr;1 Q GM mc 4pGM mc pr;1 0 2 ¯ 0pr;1 A ½m ¼ 7:6 104 Q m½kg
ð3Þ
(‘‘the ratio of the radiation pressure force to the gravitational force’’) may change during the motion. Here L is the rate of the energy outflow from the Sun, the solar luminosity. A quantity A0 is the geometrical cross section of a sphere of the volume equal to the volume of the particle. b may change and all three parameters b1 , b2 and b3 have to be numerically calculated at each point of the orbit (except for a very special cases, when the b parameters are constant during the motion). Since in the discussed applications b does not exhibit large changes during the particle motion, we will use (osculating) orbital elements for the case when central acceleration is given by GM ð1 bÞ~ e1 =r2 . This is a very good approximation to the reality. The central acceleration GM ð1 bÞ~ e1 =r2 defines the Keplerian orbits within this approximation. When dealing with the long-term orbital evolution, we may use the mean values of orbital elements—they may be defined as time mean values of orbital elements when the true anomaly changes in 2p radians: Z 1 T hgi gðtÞ dt. (4) T 0 As for simulations of nonspherical particles, the quantities fbj ; j ¼ 1; 2; 3g in Eq. (1) are calculated in two steps. The first step is defined by Eq. (2), and the second step is represented by the averaging (of the values obtained in the first step) over the azimuthal/rotational angle corresponding to the fixed axis of rotation of the particle. The fact that the rotational axis of the particle is fixed in our computational runs is consistent with the experimental results that the rotational axis exhibits only small changes with respect to some preferred orientation [17].
3. Secular evolution of orbital elements for spherical particle If we consider that the material within a dust grain is distributed in a spherically symmetric way, then b2 ¼ b3 ¼ 0 and b, defined by Eq. (3), is constant. This very special case is well-known as the Poynting–Robertson effect. In this very special case analytic calculations of the secular evolution of orbital ¯ 0pr;1 =ðR½g=cm3 s½cmÞ, for a homogeneous spherical elements are possible. Eq. (3) reduces to b ¼ 5:7 105 Q particle: R is mass density and s is the radius of the sphere. In order to obtain more general results, we will consider the evolution of a spherical dust particle under the action of solar gravity, solar electromagnetic radiation and solar wind. If we consider the most simple approximation of the solar wind [18], the equation of motion of the particle is ~ ~ ~ ~ d~ v GM GM b GM u v ~ e1 v v ~ e1 v ~ ~ ¼ 2 ~ 1 1 e1 þ b 2 þZ 0 , (5) e1 e 1 ¯ dt c u r r c r2 c u Q pr;1 where we neglect the decrease in the particle’s mass. The quantity u is the speed of the solar wind particles, Z 1=3 [7]. ¯ 0pr;1 ÞðGM =r2 Þðu=cÞ, we can rewrite Eq. (5) to the following Neglecting the solar wind pressure term Zðb=Q form: ! ~ d~ v GM ð1 bÞ Z GM ~ v ~ e1 v ~ ~ ¼ b 1 þ þ e , (6) e 1 1 ¯ 0pr;1 dt r2 c r2 c Q e1 =r2 as the where the second term causes the deceleration of the particle’s motion. We use GM ð1 bÞ~ central acceleration determining osculating orbital elements. The time averaging defined by Eq. (4) leads to
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secular evolution of orbital elements: ! da GM Z 2 þ 3e2 1þ 0 , ¼ b ¯ pr;1 að1 e2 Þ3=2 dt c Q ! de GM Z 5e=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi , 1þ 0 ¼ b ¯ pr;1 a2 1 e2 dt c Q di ¼ 0, dt
ð7Þ
where only the secular evolutions of the semi-major axis a, eccentricity e and inclination i of the particle are presented. T in Eq. (4) is a time interval between passages through two following pericenters. 4. Mean-motion resonances with planets—definitions According to the third Kepler’s law we have a3 n2 ¼ GM ð1 bÞ, a3P n2P ¼ GðM þ mP Þ,
ð8Þ
where a, aP are the semi-major axes of a particle characterized by parameter b (see Eq. (3)) and a planet with mass mP , respectively. n and nP are the mean motions characterizing the revolutions around the Sun of mass M . The first part of Eq. (8) uses the fact that the central Keplerian acceleration is given by the sum of the solar gravitational acceleration and radial component of the radiation pressure acceleration, as it was discussed in Section 2. Eq. (8) yields
2=3 mP 1=3 1=3 nP a ¼ ð1 bÞ 1þ aP . (9) n M If the dust particle is in a resonance with the planet, we can define the q-th order exterior resonance by the relation nP =n ¼ ðp þ qÞ=p, where p and q are integer numbers. Similarly, the qth order interior resonance is defined by the relation nP =n ¼ p=ðp þ qÞ, where p and q are integer numbers. In terms of orbital periods: T=T P ¼ ðp þ qÞ=p for exterior, T=T P ¼ p=ðp þ qÞ for interior resonances. Based on these definitions and Eq. (9), we can immediately write p þ q 2=3 mP 1=3 a ¼ ð1 bÞ1=3 1þ aP , (10) p M for the semi-major axis of the dust particle in the q-th order exterior resonance with the planet of mass mP . Similar relation can be obtained for the interior resonance. 5. Mean-motion resonances and spherical dust particles Let us consider a spherical dust grain under the action of the gravitational forces generated by the Sun and a planet moving around the Sun. Moreover, in reality the grain is evolving also under the action of the solar electromagnetic radiation and solar wind. The equation of motion of the particle is ! ~ ~ ~ ~ d~ v GM ð1 bÞ Z GM v ~ e1 v r ~ rP rP ~ ~ ¼ þ e1 b 1 þ 0 , (11) GmP e1 þ ¯ pr;1 dt r2 c r2 c j~ r ~ rP j3 j~ rP j 3 Q where ~ r and ~ rP are the position vectors of the particle and the planet with respect to the Sun, and, as a standard, m ¼ GM and G is the gravitational constant. Eq. (5) was used and the term generated by the gravitational force of the planet was added. If the particle is ejected from a parent body (comet, asteroid, planet, satellite of a planet), then its secular evolution is given by Eq. (7). The important property is that the secular evolution of the semi-major axis of the
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particle is a decreasing function of time. In this way, the particle approaches the Sun and sometimes it may reach a suitable condition to be captured in a mean-motion resonance. Being in the resonance, its secular evolution is characterized by a constant value of the semi-major axis defined by Eq. (9) or Eq. (10) (or, by analogous equation for the interior resonance). The particle can be captured for thousands or even of billions years. It depends on the size of the particle (note that b / size1 ), mass of the planet, type of the resonance (the smaller values of p and q, the longer capture times), and, also on the initial orbital conditions at the instant of the capture. Finally, when the capture terminates, Eq. (7) correctly describes the behaviour of the particle: the secular evolution of semi-major axis is, again, a decreasing function of time. We have already mentioned that the secular evolution of the semi-major axis is characterized by its constant value when a particle is in resonance with a planet. What can be said about the secular evolution of the eccentricity of a particle in such a situation? In order to show what really happens, we suppose that the planet is moving in a circular orbit around the Sun. In this case, we have a special gravitational problem of three bodies together with a presence of small nongravitational forces. In celestial mechanics this gravitational problem is called circular restricted problem of three bodies. There exists a quantity Tisserand’s parameter— see, e.g. [19], which does not change during the motion of the third body whose mass is negligible in comparison to the masses of the planet and the Sun. For our case, defined by Eq. (11), we can write the Tisserand’s parameter in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1b ð1 bÞað1 e2 Þ CT ¼ þ2 cos I, (12) a a3P where e is the eccentricity of the particle characterized by the parameter b and I is the inclination of the particle orbital plane with respect to the plane of the planetary orbit. The term 1 b in Eq. (12) corresponds to the reduction of the solar mass, as it is presented in Eq. (11), and, the considered gravitational problem is given by masses M ð1 bÞ, mP and mass of the dust particle. The velocity terms in Eq. (11) represent nongravitational terms in the following considerations. We have to stress that the Tisserand’s parameter C T stays constant only in a special case of the circular restricted problem of three bodies. However, Eq. (12) enables us to find the secular change of the eccentricity of a particle captured into a resonance. For this purpose, we will consider I ¼ 0 in Eq. (12), which is not in contradiction with the fact that the considered nongravitational effects do not change the inclination of the orbital plane (di=dt ¼ 0 in Eq. (7)). We can write dC T qC T da qC T de ¼ þ , (13) dt qa dt total qe dt total for the total time derivative of the Tisserand’s parameter C T defined by Eq. (12). However, according to Eq. (11), time derivatives of the semi-major axis and of eccentricity of the particle are caused by the gravitational perturbations of the planet (these terms will be denoted by the subscript G) and nongravitational perturbations caused by the solar electromagnetic radiation and solar wind (these terms will be denoted by the subscript NG and they correspond to the velocity terms in Eq. (11)): da da da ¼ þ , dt total dt G dt NG de de de ¼ þ . ð14Þ dt total dt G dt NG On the basis of the Eqs. (13)–(14) we can write dC T qC T da da qC T de de ¼ þ þ þ . dt G dt NG dt G dt NG dt qa qe According to Tisserand, gravitational terms alone do not change the value of C T : qC T da qC T de þ ¼ 0. qa dt G qe dt G
(15)
(16)
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Substituting from the Eq. (16) into the Eq. (15): dC T qC T da qC T de ¼ þ . dt qa dt NG qe dt NG
(17)
If we are interested in secular changes of the orbital elements a and e, then the particle stability in the resonance is characterized by the relation ðda=dtÞtotal ¼ 0 and Eq. (13) reduces to dC T qC T de ¼ . (18) dt qe dt total Eqs. (17)–(18) finally brings out the total secular change of the eccentricity of the particle de de qC T =qa da ¼ þ . dt total dt NG qC T =qe dt NG
(19)
Using the calculated partial derivatives of C T defined by Eq. (12) together with the Eq. (7) for the secular change of the eccentricity of a particle caused by nongravitational forces, one finally obtains ! pffiffiffiffiffiffiffiffiffiffiffiffiffi ( ) 1 e2 de GM Z n 1 þ 3e2 =2 ¼b 1þ 0 1 , (20) ¯ pr;1 dt nP ð1 e2 Þ3=2 c a2 Q if the definition represented by Eq. (9) has been used as well, and also the fact that mP 5M has been taken into account. Moreover, the symbol of angle brackets—denoting secular evolution—in Eq. (7) has been also omitted. Eq. (20) is an improved result of Liou and Zook [12]. Eq. (20) determines the secular evolution of the eccentricity of a particle characterized by the values b and ¯ 0pr;1 if the particle is captured into a mean-motion resonance. This equation enables us to find the detailed Q evolution. However, an important qualitative behaviour can be found very easily. If we take some special mean-motion resonance, we already know the value n=nP . If the initial (secular) eccentricity is small enough, then de=dt is always positive and the eccentricity of the particle is an increasing function of time during the stay of the particle in the mean-motion resonance. Eccentricity of the particle can only approach asympotically to a limiting value elim given by the condition that the term in the combined brackets equals zero: nP =n ¼ ð1 þ 3e2lim =2Þ=ð1 e2lim Þ3=2 . If the initial eccentricity is large enough, then de=dt is always negative and the eccentricity of the particle is a decreasing function of time during the stay of the particle in the meanmotion resonance. Again, eccentricity of the particle can only approach asympotically to the limiting value elim . 5.1. Mean-motion resonances and spherical dust particles—fundamental results In the zones of mean-motion resonances with a planet, the orbital evolution of a spherical dust grain in the circular restricted problem of three bodies, with a simultaneous action of the solar electromagnetic radiation (and the solar wind), is characterized by the following fundamental results: (i) secular evolution of the semi-major axis is a decreasing function of time outside the resonances, see Eq. (7), (ii) secular evolution of the eccentricity is a decreasing function of time outside the resonances, see Eq. (7), (iii) da=dt ¼ 0 in the resonance, (iv) secular evolution of the eccentricity in the exterior resonance is either an increasing function of time or a decreasing function of time and the values of the eccentricity cannot reach a limiting value elim , given by condition nP =n ¼ ð1 þ 3e2lim =2Þ=ð1 e2lim Þ3=2 , see Eq. (20), (v) secular evolution of the eccentricity in the interior resonance is a decreasing function of time, see Eq. (20), (vi) if the initial position vector (with respect to the Sun) and the velocity vector of the particle lie in the orbital plane of the planet, then the orbital plane of the spherical particle is conserved. The crucial question emerges: are the features, presented above, typical features for the real dust particles?
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6. Mean-motion resonances and nonspherical dust particles Real dust particles are nonspherical in shape. Thus, the total electromagnetic force represented by Eqs. (1)–(2) has to be used, instead of the Poynting–Robertson effect. Equation of motion for a particle in the problem of three bodies with the action of electromagnetic radiation and solar wind is 3 ~ ~ v ~ ej d~ v GM GM X v ~ e1 ~ v ~ ¼ 2 ~ þ bj 1 2 e1 þ 2 ej dt c r r c c j¼1 0 ~ ~ ~ Z Ainc GM ~ v ~ e1 v r ~ rP rP ~ b 0 þ þ , Gm e 1 P ¯ c A 0 r2 c j~ r ~ rP j3 j~ rP j 3 Q pr;1
ð21Þ
if the solar wind pressure term is neglected and A0inc is the cross section of the actually illuminated area of the particle. The following questions arise for the case of nonspherical particles: Can the particle be captured into a mean-motion resonance with a planet? How does the secular orbital evolution of the particle look in the zones of the resonances? We have done numerical calculations for a particle of a shape identical to that of the particle U2015 B10 [20]. The grain U2015 B10 (archived in NASA collection [21]) appears to be representative for cosmic dust. Its aspect ratio (the ratio of the largest and the smallest characteristic lengths of the particle) is equal to 1.4 and this value coincides well with the results of mid-infrared spectropolarimetry [22]. These results indicate that aspect ratios of cosmic dust grains lie between 4=3 and 3=2. We use the Discrete Dipole Approximation [23–25], to calculate the characteristics of the scattered radiation field because the method is applicable to the arbitrary targets. To accelerate the simulation of the particle motion in the solar radiation field we have precalculated a large database of scattering diagrams for particles of different sizes with a dense enough lattice Y B F, where the Euler’s angles Y, B, and F unambiguously identify the particle orientation (in local coordinate system) with respect to the direction of the incident radiation [26]. Choosing the initial slope of the particle rotation axis, the particle orientation is calculated at each point of its trajectory. An accurate interpolation of the scattering diagrams is applied to calculate orientationally averaged polychromatic efficiency factors for the radiation pressure. It has been supposed that the particle rapidly rotates around a rotational axis fixed in space (various rotational axes have been considered) and the orientation averaging over the corresponding azimuthal angle with respect to the Sun have been done; values of bj obtained this way were used in Eq. (21). The simulations have been done for the circular restricted problem of three bodies and solar wind is the only nongravitational effect besides the solar electromagnetic radiation. This enables comparison between the behaviour of the spherical dust grains (the results can be obtained even analytically—important from the point of view of the completeness of the possible solutions) and nonspherical grains. Moreover, it is evident that any difference between behaviour of spherical and nonspherical dust grains is due to the effect of nonsphericity of the grains and the terms b2 and b3 in Eq. (1) are relevant. As for the questions asked above, the answer is that the spherical dust particles do not yield any representative results for the real dust grains. Nonspherical particles are also captured in mean-motion resonances, but their orbital evolution is quite complex. It is documented in Figs. 1–5 which show possible evolution for outer and inner resonances with planet Neptune. In all cases the magnesium-rich silicate dust grains of the effective radius of 2 microns (shape U2015 B10) are considered, the relative change of the parameter b is less than 5% for the various orientations during the motion of the particles. Figs. 1 and 2 depict the exterior mean-motion resonances defined by the ratios T : T P ¼ 5 : 4 and 7 : 6. In both cases, the resonances are evident and no overlap with other resonances exists during the particle motion in the given commensurabilities. The secular evolution of the eccentricities in the resonances is not characterized by a monotonous function of time as it is in case of the spheres (Eq. (21) and summarization below Eq. (20)). The orbital inclinations are not conserved and they change due to the effect of the electromagnetic radiation, since other considered effects conserve the orbital plane. While the secular evolution of the semi-major axis is characterized by its increase at the moment of ejection from the resonance, in Fig. 1, the behaviour of the same orbital element decreases at the moment of ejection in Fig. 2 (however, the
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Fig. 1. Secular evolution of the orbital elements of the irregularly shaped dust grain in the zone of mean-motion resonance 5 : 4 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
decrease is more rapid than in the case of spheres). The secular evolutions of the eccentricities are increasing functions of time at the moments of the ejections from the resonances. The inner mean-motion resonances with Neptune (Figs. 3–5) are defined by the ratios T : T P ¼ 7 : 8, 15 : 16 and 6 : 7. The nonspherical particle in the Fig. 3 is characterized by the semi-major axis in the resonant zone 27:02 AUpap27:28 AU (theoretical value for b ¼ 0:036, and a ¼ 27:164 AU) and no overlap with other resonances during the capture exists. No such overlap was found neither for the resonance 6 : 7 (see Fig. 5). Here, the theoretical value of the resonant semi-major axis is 26.5 AU, and b 0:07. Fig. 4 probably shows only a mimic of a resonance. The corresponding theoretical resonance appears to be 15 : 16 but it is very close to other resonances. A complex ripple structure in the semi-major axis occurs due to the realistic motion of the nonspherical particle. This situation corresponds to some kind of hovering in semi-major axis. Such kind of behaviour does not exist in case of a spherical particle. Figs. 4, 5 show nonmonotonous behaviour of the secular evolution of the eccentricity and it is an increasing function of time at the moments of the ejections from the resonances, in both cases. Fig. 4 yields an interesting relation between the secular changes of eccentricity and inclination: e i constant; we do not know any theoretical explanation of this numerically found result. While the secular evolution of the orbital inclination is characterized by a constant value for the spherical particle in the considered model, nonspherical particles exhibit nonmonotonous character of the inclination evolution during the capture in the resonances. The secular evolutions of the inclinations may be increasing (Figs. 1 and 4) or decreasing (Fig. 2) functions of time at the moments of the ejections.
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Fig. 2. Secular evolution of orbital elements of irregularly shaped dust grain in the zone of mean-motion resonance 7 : 6 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
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Fig. 3. Time evolution of the semi-major axis and the eccentricity for an irregularly shaped dust grain of the effective radius of 2 microns, b 0:036. The plots show interior resonance 7 : 8 of the magnesium-rich silicates (mass density 4 g=cm3 ) with Neptune.
The figures show that the nonspherical dust grains can be captured even in the case when the secular change of semi-major axis is an increasing function of time near the zone of the resonance, and, no limiting value of eccentricity exists in the resonance.
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Fig. 4. Secular evolution of orbital elements of irregularly shaped dust grain in the zone of mean-motion resonances near 15 : 16 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
6.1. Mean-motion resonances and arbitrarily shaped dust particles: fundamental results The resonant captures of arbitrarily shaped dust grains exist for exterior and interior mean-motion resonances with planets. An ejection of a grain from the exterior resonance is due to the grain’s close encounter with a planet, but also due to the strong effect of the electromagnetic radiation. While the semi-major axis and the eccentricity of a spherical grain are strictly decreasing functions of time shortly before the capture into the resonance (see Eq. (7)), they may be increasing or decreasing functions in case of nonspherical particles. Our results, depicted in figures, show that various evolutions of the semi-major axes and eccentricities for nonspherical dust grains can occur. Moreover, the orbital inclinations of the nonspheres can significantly change so particles do not move in the orbital plane of the planet. The most important physical result is that the resonant trapping, under action of (solar) electromagnetic radiation, is not normally expected for diverging orbits such as the orbits of nonspherical dust particles: as it was pointed out for exterior resonances [16], and completed for interior resonances in this paper, this is the first case when a physically justified force can generate such a behaviour. Nonspherical particles can move in a region of resonances, but the apparent capture in a resonance does not correspond to the real resonant motion. The effect of radiation can mimic the resonant motion.
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Fig. 5. Secular evolution of orbital elements of irregularly shaped dust grain in the zone of mean-motion resonance 6 : 7 with Neptune. Grain material: magnesium-rich silicate, effective radius : 2 mm, bulk density : 2 g cm3 , b 0:072.
7. Conclusions This contribution deals with the effect of the electromagnetic radiation on the dynamics of the cosmic dust particles. The paper presents the application of the equation of motion on mean-motion resonances with planets. In order to obtain significant results, we have used a model which can be treated analytically for the spherical particles. The completeness of the solution is guaranteed in this case. Thus, we consider the circular restricted problem of three bodies with the action of the solar electromagnetic radiation and the solar wind. The complete solutions of the motion of the spherical particles outside the resonances are presented in Eq. (7) and those inside the resonances in Eq. (20). Both sets of solutions are summarized in Eq. (20). Same gravitational and nongravitational effects are considered in the case of the nonspherical dust grains. This enables a simple comparison between the effects of the electromagnetic radiation on nonspherical and on spherical dust grains. Although some other nongravitational forces could be considered (Lorentz force, collisions among particles, better approximation of the solar wind, ...), the problem could not be analytically treated then. Simple understanding of the differences in action of the electromagnetic radiation on spherical and on nonspherical grains would not be possible. Moreover, the nongravitational effects may play differently relevant roles in various extra-solar planetary systems. However, the improvements mentioned above are to be done in the future work. Differences between the behaviour of the spherical and nonspherical particles are pointed out in this work. Although the spherical particles often enable analytical calculations, their orbital evolution in the zones of
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mean-motion resonances cannot be considered as a representative evolution for the real cosmic dust particles. The motion of a spherical grain is characterized by a decrease of the semi-major axis (and eccentricity) before the capture in the mean-motion resonance. Nonspherical grains may be captured even in case when the secular change of the semi-major axis has been an increasing function of time outside the resonance. This result is correct not only for the exterior mean-motion resonances as it was pointed out by Klacˇka et al. [16], but also for interior mean-motion resonances as shown in this paper. Resonant trapping is not normally expected for diverging orbits and the obtained results due to the effect of electromagnetic radiation on nonspherical dust particles represent the only known case when a physically justified force can generate such a behaviour. The effect of the electromagnetic radiation on the motion of the nonspherical dust particles can resemble the motion in a mean-motion resonance. The semi-major axis of the particle does not change for a long time but the motion does not correspond to any simple resonance. Acknowledgements This paper was supported by Project M772-N02 by the Fonds FWF and by VEGA grant No. 1/3074/06 and 2/3024/23. We specially thank Prof. Ch. Dellago for computer resources at the Institute of Experimental Physics. References [1] Maxwell JC. A Treatise on Electricity and Magnetism. Oxford: Oxford University Press; 1873. [2] Poynting JM. Radiation in the Solar System: its effect on temperature and its pressure on small bodies. Philos Trans R Soc London Ser A 1904;202:525–52. [3] Robertson HP. Dynamical effects of radiation in the Solar System. Mon Not R Astron Soc 1937;97:423–38. [4] Klacˇka J. Poynting–Robertson effect. I. Equation of motion. Earth, Moon and Planets 1992;59:41–59. [5] Mishchenko M, Travis LD, Lacis AA. Scattering absorption and emission of light by small particles. Cambridge, UK: Cambridge University Press; 2002 445pp. [6] Klacˇka J, Kocifaj M. Motion of nonspherical dust particle under the action of electromagnetic radiation. JQSRT 2001;70:595–610. [7] Klacˇka J. Electromagnetic radiation and motion of a particle. Cel Mech Dyn Astron 2004;89:1–61. [8] Krauss O, Wurm G. Radiation pressure forces on individual micron-size dust particles: a new experimental approach. JQSRT 2004;89:179–89. [9] Beauge´ C, Ferraz-Mello S. Capture in exterior mean-motion resonances due to Poynting–Robertson drag. Icarus 1994;110:239–60. [10] Jackson AA, Zook HA. A Solar System dust ring with the Earth as its shepherd. Nature 1989;337:629–31. [11] Liou J-Ch, Zook HA. An asteroidal dust ring of micron-sized particles trapped in the 1: 1 mean motion resonance with Jupiter. Icarus 1995;113:403–14. [12] Liou J-Ch, Zook HA. Evolution of interplanetary dust particles in mean motion resonances with planets. Icarus 1997;128:354–67. [13] Liou J-Ch, Zook HA, Jackson AA. Radiation pressure, Poynting–Robertson drag, and solar wind drag in the restricted three body problem. Icarus 1995;116:186–201. [14] Marzari F, Vanzani V. Dynamical evolution of interplanetary dust particles. Astron Astrophys 1994;283:275–86. [15] Sˇidlichovsky´ M, Nesvorny´ D. Temporary capture of grains in exterior resonances with Earth: Planar circular restricted three-body problem with Poynting–Robertson drag. Astron Astrophys 1994;289:972–82. [16] Klacˇka J, Kocifaj M, Pa´stor P. Motion of dust near exterior resonances with planets. J Phys 2005;6:126–31. [17] Krauss O. 2005. personal communication. + E. Interplanetary dust. In: Schwenn R, Marsch E, editors. Physics of the inner heliosphere I. Berlin, Heidelberg: [18] Leinert Ch, Grun Springer; 1990. p. 207–75. [19] Brouwer D, Clemence GM. Celestial mechanics. New York, London: Academic Press; 1961 598pp. [20] Kocifaj M, Kapisˇ insky´ I, Kundracı´ k F. Optical effects of irregular cosmic dust particle U2015 B10. JQSRT 1998;63:1–14. [21] Clanton VS, Gooding JL, Mckay DS, Robinson GA, Warren JL, Watts LA. Cosmic dust catalog (particles from collection flag U2015), NASA, Johnson Space Center (Houston, TX) 70/1 10, 1984. [22] Hildebrand RH, Dragovan M. The shapes and alignment properties of interstellar dust grains. Astrophys J 1995;450:663–6. [23] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186:705–14. [24] Draine BT. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys J 1988;333:848–72. [25] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am 1994;A11:1491–9. [26] Draine BT, Flatau PJ. User Guide for the Discrete Dipole Approximation Code DDSCAT.6.1’’, Freeware, hhttp://arxiv.org/abs/ astro-ph/0409262i2004.