Dynamics and distribution of nano-dust particles in the inner solar system

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 40, , doi: /grl.50535, 2013 Dynamics and distribution of nano-dust particles in the inner solar system A. Juhász 1,2 and M. Horányi 2 Received 18 April 2013; revised 2 May 2013; accepted 3 May 2013; published 6 June [1] Dust particles in the approximate mass range of < m <10 20 kg produced near the Sun, due to collisions and breakup of larger interplanetary dust particles, have been shown to become entrained in the solar wind plasma flow. When these so-called nano-dust particles (NDPs) impact a spacecraft, they have been suggested to produce sufficiently large plasma clouds to cause a detectable signal in the onboard electric antennas. NDPs have been identified on the twin STEREO spacecraft, and the observed intermittent nature of their fluxes were suggested to represent the stochastic nature of their sources near the Sun. Here we show that even if the generation of NDPs remains a constant in time, their detectability near the ecliptic plane becomes intermittent due their interaction with the interplanetary magnetic fields. Citation: Juhász, A., and M. Horányi (2013), Dynamics and distribution of nano-dust particles in the inner solar system, Geophys. Res. Lett., 40, , doi: /grl Introduction [2] During the period of , the frequent spikes recorded by the antennas of the S/WAVES radio instrument [Bougeret et al., 2008] onboard the STEREO A and B spacecraft have been suggested to signal high-speed impacts of very small dust particles [Meyer-Vernet et al., 2009; Zaslavsky et al., 2012]. When a dust particle, with sufficiently large mass or speed, impacts a spacecraft, it could produce a large enough rapidly expanding plasma cloud to be detected by onboard electric antennas. The mass of the recorded dust particles has been suggested to be in range of < m < kg [Zaslavsky et al., 2012]. This is, however, just a putative estimate since the relation linking the measured signal to the mass and speed of the impacting dust grains has not been experimentally established for very small particles with very high velocities. These so-called nano-dust particles (NDPs), if fully entrained in the solar wind plasma flow, could be responsible for the observations [Meyer-Vernet et al., 2009]. [3] The observed NDP fluxes were highly nonuniform in time and space showing burst-like features that were not identical on the two spacecraft: STEREO B nano-dust flux measurements were more steady in time than seen by STEREO A [Zaslavsky et al., 2012]. Time-dependent 1 Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary. 2 Laboratory of Atmospheric and Space Physics and Department of Physics, University of Colorado, Boulder, Boulder, Colorado, USA. Corresponding author: M. Horányi, Laboratory of Atmospheric and Space Physics and Department of Physics, University of Colorado, Boulder, Boulder, CO , USA. (mihaly.horanyi@lasp.colorado.edu) American Geophysical Union. All Rights Reserved /13/ /grl NDP production in collisions between larger particles [Czechowski and Mann, 2012] and intermittent dust release from Sun-grazing comets [Ip and Yan, 2012] were suggested as possible causes for the reported variability. Here we propose a different mechanism that can naturally give rise to a highly variable nano-dust distribution near the ecliptic plane at 1 AU distance from the Sun. [4] The dynamics of nano-dust originating from the vicinity of the Sun has been investigated in detail, both numerically and analytically. Czechowski and Mann [2010, 2011, 2012] showed that nanometer-sized dust particles generated near the Sun (inside r =0.2AU) will remain trapped and subsequently lost due to evaporation. NDPs born outside r =0.2AU behave like ions and can be picked up by the magnetic fields of the solar wind plasma flow and accelerated to high velocities, v > 300 km/s, by the time they reach the orbit of the Earth. 2. Model Description [5] We follow the dynamics of a charged NDP in a heliocentric inertial (HCI) coordinate system centered on Sun, its x axis points to the solar ascending node in the plane of the ecliptic, and its z axis is along the rotation axis of the Sun. The equation of motion includes solar gravity, radiation pressure, and the Lorentz force dv = GMˇ(1 ˇ) Oe dt r 2 r + Q (v w) B, (1) m where v, m, andq are the velocity, mass, and the charge of the dust particle, Mˇ(= kg) is the solar mass, B is the interplanetary magnetic field (IMF), and w is the solar wind velocity. We set the particle density =210 3 kg/m 3, and the dust surface potential to be constant =+4 V, hence the charge Q =4 0 r g,wherer g is the radius of the dust particle. The magnetic field at r 0 =1AU is B 0 = 3.5 nt and that the solar wind flows strictly in the radial direction with a constant speed of w = 400 km/s. The ratio of the solar radiation pressure over the gravity of the Sun is ˇ = F rad /F gr. For NDPs, the value of ˇ ' 0.1 [Gustafson, 1994]. We have ignored the Poynting-Robertson drag since it is negligible compared to the radiation pressure force. [6] We use the Parker model of the IMF in the form of [Pei et al., 2012] B = pb 0 r 2 0 r 2 Oe r (r r s)ˇ sin Oe [1 2H( CS )], (2) w where (r,, ) are the conventional spherical coordinates, H is the Heaviside step function and CS specifies the latitude of the current sheet (CS), ˇ(= s 1 ) is the sidereal rotation rate (period 25.4 days), R S (= m) is the radius of the Sun, r s = 2.5R s is the radius of 2500

2 the solar wind source surface, and p is a parameter that determines the sign of the magnetic field. The value of p is 1, with the + sign corresponding to outgoing radial magnetic fields on the northern hemisphere of the Sun, so the polarized electric field E = wbpoints away from the CS. This situation is defocusing, while the opposite polarity, when E points toward the CS, is called a focusing period. The solar magnetic polarity reverses in 11 years, and , were defocusing, while and were focusing periods. [7] The three-dimensional current sheet is defined as the surface satisfying the equation tan 2 CS =tan (t, r) (3) sin 0 ˇt + ˇ(r r S ) w where (t) is the tilt angle of the CS at time t [Peietal., 2012]. [8] For the time dependence of the tilt angle, (t), we used the potential field source surface (PFSS) model (wso. stanford.edu/tilts.html) that gives the maximum latitude extent of the heliospheric current sheet as function of time (i.e., Carrington rotations) [Hoeksema, 1995]. [9] The direction of the magnetic field is opposite in the two regions separated by the CS surface. The constant 0 in equation (3) determines the initial orientation of the rotating CS and plays an important role in shaping the trajectories of the nano-dust particles (as described in section 4). [10] In a more realistic model for modeling the STEREO observation, we determined the magnetic field polarity (p) by using the source surface magnetic field model from the Wilcox Solar Observatory, based on photospheric observations of the Sun ( This method gives the position CS (r s ), CS (r s ) of the neutral line on the source surface at r s =2.5R S. Using the frozen in condition for the magnetic field, and assuming a strictly radial solar wind flow, the position of the footprint of the a magnetic field line s, s on the Sun that threads the position of a dust grain r d, d, d can be calculated. Hence, the parameter p at the position of the dust particle is set by the sign of the magnetic field at s, s. This method reproduces the real sectors structure of the IMF. For example, for 2007 January (CR2052) this method produces a four-sector structure, while the tilted CS method always gives rise to a two-sector IMF structure. However, the more realistic method cannot be used for making predictions., 3. Guiding Center Approximation [11] Before discussing numerical results, we start with a simplified model to gain analytical insight into the dynamics of charged nano-dust particles, using the guiding center approximation. The guiding center approximation to describe the motion of charged particles is typically valid for small r g <15nm grains in a unipolar IMF region, with gyroradii r L satisfying r L rb/b <0.1. These tiny dust grains are picked up by the solar wind just like ions [Luhmann, 2003] and start gyrating around the magnetic field lines while undergoing slow drift motions [Northrop, 1961, 1963; Winge and Coleman, 1968]. [12] In the guiding center approximation, the particle motion is decomposed into a gyration about its guiding center and the motion of the guiding center itself with a velocity (u g ). The guiding center velocity can be written as a sum of the parallel and perpendicular velocities with respect to the magnetic field line u g = u + u?. [13] The drift velocity u? = u E + u is [Northrop, 1961, 1963] u? =u E + b B Q rb + m Q g + u 2 (br)b (4) +u (u E r)b + u (br)u E +(u E r)u E, where g is the gravitational acceleration, u E = E b/b, b = B/B. Here = mv 2?/2B is the magnetic moment where v 0? = v? w? and w?, v? are the solar wind plasma and particle velocities perpendicular to B, respectively. In our HCI coordinate system, B =(B r,0,b ), w =(w,0,0),and E =(0,E,0). [14] The largest contribution to the dust velocity v comes from the so called E B drift (u E )(u E ' 300 km/s at r =1 AU) which is equal to the solar wind velocity component perpendicular to the magnetic field u E = w? (= w (wb)b). u E increases with increasing distance r and it has two components only u E = (u r,0,u ).Atr = 1 AU, the velocity of the particles is high, v ' u E = 300 km/s. We note, that u E is not proportional to the solar wind velocity. Choosing w = 800 km/s (high-speed streams), the value of u E at r =1 AU will be about u E ' 400 km/s only. This is due to the fact that for increasing value of w, the azimuthal component of the IMF B w 1 is decreasing, as the magnetic field lines are becoming more radial so that the normal component of w (which is equal to u E ) will be smaller too. [15] The acceleration (energy gain) of NDP comes from the total work done by the electric field during the drift motion which is W = Q R Evdt = Q R E u d dt. Since the electric field has only a component E =(0,E,0), only the vertical u component of the drift velocity is responsible for the acceleration of the nano-dust particle. Since u E has no component, it does not contribute to the dust acceleration. [16] It is lengthy, but straightforward, to calculate the analytical formulas for the vertical drifts (u ) in spherical coordinates and we do not list them here. The b rb and the b g drifts have opposite direction to E so they cause energy loss for a nano-dust grain. Only the so-called acceleration terms (the component of the last two terms in equation (4)) give the proper drift for particle acceleration. The largest contribution to u comes from the b (u E r)u E and b u (br)u E drifts. All of these drifts are proportional to the particle mass so larger NDPs drift (in the vertical () direction) faster than smaller ones. [17] The guiding center approximation is valid only up to a certain size only (r g <15 20 nm) since for larger particles the Larmor gyroradius, r L = mv 0?/(QB), becomes too large and the r L /L 1 condition is no longer satisfied, where L = B/rB. Also, for larger grains the gyroperiod is large, comparable to (or larger) than the particle s travel time from r =0.2to r =1AU ( 20 days). For small grains, the drift velocity is small (u ' km/s) so they can get to the near-earth orbit (NEO) region for at least some initial positions in periods of defocusing IMF. For larger (r g >10nm) particles, the drift is larger (u ' 100 km/s), hence they quickly move away from the ecliptic plane and could not be detected by spacecraft near the orbit of the Earth. We define 2501

3 [19] In focusing IMF configuration, some particles (depending on their size and initial condition) cross the CS, so an accelerating period is followed by deceleration due to the opposite direction of the E and v (and vice versa). Also, in certain cases (dominantly larger particles) particles can drift along the CS [Czechowski and Mann, 2012]. In this case, the final velocity at r =1AU will be smaller than u E. [20] In defocusing IMF, only the smallest grains (r g < 10 nm) can get to the NEO region, since larger particles drift quickly away from the ecliptic plane. In focusing IMF periods, larger dust fluxes in a broader size range can be expected to reach the NEO. Figure 1. The components of the drift velocity of a r g =7 nm (Q/m =10 5 e/m p ) radius dust grain that started from a circular orbit at r =0.2AU. The total drift from a numerical simulation (thick black line) and the total drift based on equation (4) (red thick line) are shown. The different drift terms (dashed lines) are labeled following their appearance in equation (4). the NEO region as r 1 < AU and z < z =0.05 AU. Figure 1 shows the different terms in equation (4) to the vertical drift velocity (u )forar g =7nm radius NDP. [18] The small particles that do not cross the CS follow a trajectory with monotonically increasing or decreasing latitude versus longitude, so only a fraction of them can get to NEO (Figure 2) depending on their starting positions. 4. Results From Numerical Simulations [21] One of the most interesting findings of the STEREO measurements is the highly variable nature of the observed NDP distribution along the Earth s orbit, where high-impact rates alternate with periods of no dust detection at all, with a periodicity of roughly 6 months [Zaslavsky et al., 2012]. It has been suggested that the temporal variability of NDP production rates in breakups and collisions [Czechowski and Mann, 2012], or the sporadic dust release from Sun-grazing comets [Ip and Yan, 2012] could be responsible for the intermittent nature of the nano-dust observations. Here we show that even in the case of a uniformly distributed, steady state nano-dust production near the Sun, the dust dynamics alone, due to the complex shape of the CS shape, naturally gives rise to a highly intermittent and azimuthally nonuniform nano-dust distribution along the orbit of the Earth. [22] We made trajectory calculations for two different sets of initial conditions for a large number of NDPs with fixed charge-to-mass ratios by integrating their equation of motion (equation (1)). Figure 2. The top row shows the trajectories of 360 particles with radii r g =3nm, started uniformly distributed along a circle in the ecliptic plane at r =0.2AU in the (left column) Heliocentric Aries Ecliptic (HAE) (x, y) plane projection, in the (middle column) HCI, coordinates, and the (right column) azimuthal dust density distribution in HAE. The orange lines represents the trajectories that can be detected near the ecliptic plane at 1 AU. In the (, ) plots, the continuous blue line represents the ecliptic plane and the dashed blue lines show the detectable limits set by z < z =0.05AU. The bottom row is for a N = 3600 particle simulation using initial conditions with random inclinations in the range 0<i <10 ı, random eccentricities in the range 0<e <0.01and random orbit orientations (0 <!, <2). 2502

4 JUHÁSZ AND HORÁNYI: NANO-DUST DYNAMICS Figure 3. Trajectories of (top) rg = 1 and (bottom) rg = 4 nm particles started uniformly distributed from a circular orbit in the ecliptic plane (thick blue line) at r = 0.2 AU in the HCI (, ) coordinate system. The detectable particles (orange lines) end in the strip determined by the two dashed blue lines set by z < z = 0.05 AU. [23] 1). NDPs were started uniformly distributed along a circular Kepler orbit in the ecliptic plane at r = 0.2 AU, with initial azimuthal velocities ' 66 km/s. The radial number density distribution of the dust is expected to follow n(r) r 1, hence we expect an increase in nano-dust production closer to the Sun due to dust-dust collisions and the breakup of larger particles [Mann et al., 2000]. However, nano-dust grains starting within this critical radius will remain trapped [Czechowski and Mann, 2010, 2011, 2012], and are not considered here. Hence, the dominant region of NDP production is expected to be just outside the trapping region. [24] 2). NDPs were started from a low inclination dust cloud with uniformly distributed random orbital elements 0ı < i < 10ı, 0 < e < 0.05, a = 0.2 AU, 0 <!, < 2 ) [25] The initial conditions were specified in the solar ecliptic coordinate system, with the z axis normal to the ecliptic plane, pointing north, the x axis extends in the direction toward vernal equinox, and the y axis completes the right-handed set. [26] The HCI latitude of very small grains barely changes since for them the u drift is very small. However their longitude will increase by an amount of = e i = 120ı while they get to r = 1 AU so the particle s trajectories will be long horizontal straight lines originating from the sinus curve (Figure 3). [27] The value of z at r = 1 AU determines a latitude region (which is about = 3ı for z = 0.05 AU). In Figures 2 and 3, the region between the two dashed (blue) sinusoidal lines represents the those positions where the z < z = 0.05 AU condition is satisfied. The choice of z is somewhat arbitrary as it represents a compromise between a vanishingly small value requiring an exceedingly large number of trajectories, or a z that is too large causing the longitudinal variability of the NDP trajectories to vanish near 1 AU. Figures 2 and 3 show that only selected initial positions lead to trajectories that satisfy this condition. Trajectories not ending in the strip are not detectable at NEO. Hence, the distribution at NEO is simply the result of the dynamics of the nano-dust grains that could result in a highly nonuniform density distribution near the ecliptic plane at 1 AU, even if the grains were initially uniformly distributed near the Sun. Without the 7.25ı tilt of the solar equatorial plane relative to the ecliptic, the smallest grains (rg 1 nm) would produce an approximately uniform dust density distribution at NEO, and no 180ı periodicity could be observed. [28] Larger grains drift faster so their latitude will increase/decrease in time from their initial latitudes (Figure 3). The longitude position ˆ0 where the directionality changes in the latitudes occurs is set by the initial orientation of the current sheet, 0 in equation (3). Also, the actual shape (tilt) of the CS introduces additional complications to the problem: If the particles cross the CS, the direction of their u drift velocities switches sign due to switch in direction of the electric field on the two sides of the CS. This can result in trajectories in the HCI (, ) plane to be more complicated, starting from the ecliptic plane (thick sinus curve), bending up or down and then at CS crossing changing direction again (Figure 3, bottom). [29] The requirement for a dust particle to be detectable at NEO ( z < z = 0.05 AU, or 3ı ) is a critical parameter in our argument. Choosing a much larger value results in almost all trajectories to remain detectable at the NEO region [Czechowski and Mann, 2012]. [30] The trajectories of the smallest grains (rg = 1 nm) are horizontal lines of lengths = e i = 120ı, due to their negligible vertical drifts. The equations describing the upper and lower latitude limits in HCI corresponding to z (Figure 3) are min = A sin ı, and max = A sin + ı, where A = 7.25ı and ı = arctan zau. The trajectories starting from the ecliptic plane ( Ec = A sin i ) and ending in the min < < max latitude range satisfy A sin i = A sin e ı, from which the values of e can easily be determined. For zau = 0.05 ı '= 3ı, we get two longitude regions at NEO: 33ı < HAE < 59ı and 213ı < HAE < 239ı, where we used the fact that HAE ' HCI + 76ı. These values agree with the numerical simulations very well. The above analytical treatment cannot be applied to larger grains since their vertical drift becomes fast for grains with rg > 1 nm and their trajectories bend up or down from the horizontal straight line, depending on the magnetic field polarity. [31] The largest dust particle that can reach the NEO environment is about amax = 50 nm for focusing and amax = 8 nm 2503

5 [34] A simple analytical model can be used to show that only a certain range of the initial dust longitudes results in detectable nano-dust at the NEO region. The detectable NDP distribution is nonuniform, with two peaks 180 ı apart that corresponds to an approximately 6 month periodicity, similar to the observations by STEREO. [35] The period will have a defocusing IMF field configuration in which the solar wind electric field (and particle vertical u drift velocity) points away from the current sheet. In this case, the nano-dust trajectories will rarely cross the current sheet, hence the larger grains (r g >10nm) will swiftly depart vertically away from the ecliptic plane, due to their large u drift velocities. In this period, only the smaller nanograins r g 10 nm can be expected at NEO. Figure 4. Azimuthal density distributions of nano-dust particles with radii r g = 3 (black line) and 10 nm (blue line) along the STEREO A and B orbits in 2007, calculated using the Wilcox B field model. The measurements by the STEREO A and B for 2007 are also shown (STEREO A red line, STEREO B green line). for defocusing periods. In the first case, the size limit is set by the increasing importance of the radiation pressure, while in the second case, particles with larger sizes will not cross the CS and quickly attain high-vertical drift velocities proportional to their mass and will be deflected above or below the ecliptic plane. [32] Figure 4 shows the calculated longitudinal nano-dust density along the STEREO A and B orbits for the year 2007, using the Wilcox B field model. N = 10, 000 dust grains were started at every Carrington rotation (CR = for the year 2007) and the longitudinal density distribution was calculated for every CR at the NEO region. From these distributions, we selected only those values that match the position of STEREO at that time. While the details of the STEREO measurements are not reproduced precisely, the highly variable nature of the measurements now can be explained by the effects of the IMF on the dynamics of the nano-dust particles. 5. Discussion [33] Our model simulations show that the longitudinal nano-dust density distribution near r = 1 AU is fluctuating in time and is spatially highly nonuniform (Figure 4). The expected nano-dust distribution at NEO also depends on the particle size, since the strength of the Lorentz force and the drift velocities are size dependent. We identified the drift mechanism that is responsible for the acceleration of the nano-dust particles to velocities ' 300 km/s. The intermittent nature of the nano-dust fluxes at NEO can be explained by the effects of the configuration of IMF alone. In reality, however, the temporal and spatial variability of the production rates and size distributions of the nano-dust particles near the Sun further complicates our ability to reproduce the STEREO measurements. [36] Acknowledgments. The authors acknowledge helpful discussions with E. Grün, and support from NASA s Solar and Heliospheric Physics Program. [37] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper. References Bougeret, J. L., et al. (2008), S/WAVES: The radio and plasma wave investigation on the STEREO mission, Space. Sci. Rev., 136, , doi: /s Czechowski, A., and I. Mann (2010), Formation and acceleration of nano dust in the inner heliosphere, Astrophys. J., 714, 89 99, doi: / x/714/1/89. Czechowski, A., and I. Mann (2011), Erratum: Formation and acceleration of nano dust in the inner heliosphere (2010, ApJ, 714, 89), Astrophys. J., 732, 127, doi: / x/732/2/127. Czechowski, A., and I. 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