ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-1033-2017Nanodust dynamics during a coronal mass ejectionCzechowskiAndrzejace@cbk.waw.plKleimannJenshttps://orcid.org/0000-0001-6122-9376Space Research Center, Polish Academy of Sciences, Bartycka 18A, 00-716
Warsaw, PolandInstitut für Theoretische Physik IV, Ruhr-Universität
Bochum, 44780 Bochum, GermanyAndrzej Czechowski (ace@cbk.waw.pl)4September20173551033104910April201719July20171August2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/35/1033/2017/angeo-35-1033-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/1033/2017/angeo-35-1033-2017.pdf
The dynamics of nanometer-sized grains (nanodust) is
strongly affected by electromagnetic forces.
High-velocity nanodust was proposed as an explanation for
the voltage bursts observed by STEREO.
A study of nanodust dynamics based on a simple time-stationary
model has shown
that in the vicinity of the Sun the nanodust is trapped or,
outside the trapped region, accelerated to high velocities.
We investigate the nanodust dynamics for a time-dependent
solar wind and magnetic field configuration in order to find
out what happens to nanodust during a coronal mass ejection (CME).
The plasma flow and the magnetic field during a CME
are obtained by numerical simulations using a 3-D magnetohydrodynamic (MHD)
code. The equations of motion for the nanodust particles are
solved numerically, assuming that the particles are produced
from larger bodies moving in near-circular Keplerian orbits within
the circumsolar dust cloud. The charge-to-mass ratios for the
nanodust particles are taken to be constant in time. The simulation
is restricted to the region within 0.14 AU from the Sun.
We find that about 35 %
of nanodust particles escape
from the computational domain during the CME,
reaching very high speeds (up to 1000 km s-1).
After the end of the CME the escape continues,
but the particle velocities do not
exceed 300 km s-1. About 30 %
of all particles are trapped in bound non-Keplerian orbits
with time-dependent perihelium and aphelium distances.
Trapped particles are affected by plasma ion drag, which
causes contraction of their orbits.
Space plasma physics (charged particle motion and acceleration)Introduction
The vicinity of the Sun is a possible source region of nanometer-sized
dust grains (nanodust) produced by collisional fragmentation of
larger dust grains or released from comets
. Because of high charge-to-mass ratio, the effect of electromagnetic forces on nanodust is much
stronger than for larger grains. A dedicated study of the dust dynamics
near the Sun was restricted to
larger grains. studied
the nanodust
dynamics in a simplified model of solar wind and solar magnetic field by
assuming a purely radial, time- and distance-independent solar wind
velocity and a Parker spiral form of the magnetic field. It was found that,
depending on the initial position and velocity,
the nanodust particles can either be trapped near the Sun (in
non-Keplerian orbits strongly affected by electromagnetic forces)
or escape to large distances. The escaping particles can be accelerated
to high speeds comparable to those of the solar wind.
High-velocity submicron dust streams were discovered by Ulysses
within 1–2 AU from Jupiter . More recently,
high-velocity nanodust was proposed as an explanation for
voltage bursts observed by STEREO/WAVES at 1 AU from the Sun
. The
STEREO results were supported by the observations by the Radio and
Plasma Wave Science instrument on Cassini made during the time
when the spacecraft was close to the Earth's orbit
.
The results from STEREO/WAVES imply that the
flux of nanodust near the orbit of the Earth is variable in time by a
high factor, showing intermittent behavior.
have shown that this behavior
may arise during propagation of charged nanodust from the
source region in the vicinity of the Sun to the Earth's orbit. The
mechanism that they invoke relies solely on propagation and does not require
time dependence of the nanodust production rate. The results
of favor
the possibility that the nanodust particles observed by STEREO/WAVES
are coming from the vicinity of the Sun.
suggested that trapping conditions for nanodust may be affected by solar
wind and magnetic field perturbations such as those associated with
coronal mass ejections (CMEs). A sudden release of trapped nanodust would
lead to a temporary increase in the escaping particle flux. If
this increase were large enough to be observed, it could serve as a
signature of a trapped nanodust population.
Recently, investigated the relationship
between CMEs and the nanodust flux measured by STEREO. For a
subclass of CMEs, they found that the nanodust flux observed during
a CME includes an additional component. They interpret this component
as the nanodust particles accelerated within the CME.
In the present work, we study the effect of
a CME on nanodust dynamics in the vicinity of the Sun by numerical simulation.
In connection
with recent STEREO observations , we compare the
simulated velocity distribution of escaping nanodust
during a CME with the one corresponding to a time-stationary
situation.
We also discuss the effect of plasma ion drag ,
which was not considered in previous studies of nanodust
near the Sun
. Our additional
aim is to find out which results obtained in simple solar wind models
(e.g., existence of a trapped population, high velocity of escaping
nanodust) remain valid for a more realistic model including a CME.
As a model of
a CME we use the numerical solution of the MHD equations obtained
using the same method and parameters as in
. This solution, starting and ending
with time-stationary configurations, is restricted to the heliocentric
distances 0.005 AU <r< 0.14 AU and the time interval of 1.6 days.
We assume that the nanodust particles originate from the fragmentation of
larger bodies in the circumsolar dust cloud. The initial positions and
velocities of nanodust particles we take to correspond to circular
Keplerian orbits of different radii and inclinations situated within
the circumsolar dust disk or inside the spherical halo region
. Assuming
a simple form of the nanodust production rate as a function of distance
from the Sun, we can then determine the fractions of escaping and trapped
particles.
We found that, in the model CME, a fraction of ∼ 35 %
of the nanodust particles escapes away from the computational domain
within ∼ 1.3 days.
These particles have a broad velocity distribution extending to
∼ 1000 km s-1, far above the ∼ 300 km s-1 upper limit
for the remaining
particles.
This “rapidly escaping” population originates in the region of space
where the perturbation of the plasma flow and the magnetic field due to the
CME are the strongest.
For comparison, we also made calculations using the time-stationary
MHD solution (the initial or final configurations without the model CME).
In this case, only very few (fraction of ∼ 0.3 %)
particles escape within the same time interval.
In our particle simulations we included also the
“time-extended” plasma and magnetic field configurations. These consist
of the MHD model of the CME (with a time extent of 1.6 days) and
the subsequent time-stationary configuration (the final time frame of
the MHD model) assumed to continue without change for a longer
time period. In these time-extended models, we found that the
nanodust particles continue to escape from the computational
domain after the CME has left the computational volume.
The fraction of these “slowly escaping” particles reaches 13 %
∼ 1.4 days after the end of the CME and 21 %
after ∼ 4.4 days. Their velocity distribution
is different from the broad distribution of the rapidly
escaping population formed within the CME.
Perspective rendering of our model's magnetic field and flow
configuration. (a) Selected magnetic field lines (white) and absolute
velocity in the poloidal plane y=0 at t=20, i.e., immediately prior to
CME outbreak, showing a quiet Sun situation. The semitransparent disk marks
the ecliptic plane. (b) Magnified view of the streamer belt of closed
field lines (also at t=20) emanating from the Sun's surface (yellow sphere)
shown using a different set of lines. The cusps of the outermost field lines
extend to about 8.5R⊙. (c) Same situation as above at a
later time t=28, at which the CME has caused a considerable disturbance and
restructuring of magnetic topology. (d) Two isocontours of
r2ρ using the same color coding for Vr as above, again amended with
selected field lines. Note the CME's shell-like spatial structure with its
distinct ecliptic groove.
We also identified a subpopulation of particles moving in orbits similar to
the trapped orbits obtained in the simple model of
. We conclude that the trapping
mechanism is operating despite the difference between the
time-stationary,
constant-speed solar wind model used in
and the strongly time-dependent model
used in the present work. However, in the
MHD solution the trapped orbits are evolving in time, leading to some
losses of the trapped particle population. The losses increase
when the drag force is taken into account.
The paper is organized as follows. In Sect. 2 (and in the Appendix) the
numerical MHD model of the CME is described. Section 3 is an
introduction to
our numerical simulations of particle motion. Section 4 briefly presents
the simplified 2-D (heliocentric distance r and the radial
velocity v) phase space model derived from the guiding center approximation
, which we found helpful
to understand the trapping mechanism. The results of particle simulations
are presented and discussed in Sect. 5. The conclusions are summarized
in Sect. 6.
The MHD backgroundNumerical setup of the stationary background
We solve the equations of motion for the nanodust particles subject to a
solar wind plasma flow V=V(r,t) and a magnetic field
B=B(r,t), both of which are computed self-consistently
using the finite-volume MHD code CRONOS, which solves
the time-dependent equations of ideal MHD on a 3-D spherical grid.
It is based on the code that was already used by
but has been significantly improved in several ways since. Among other
things, it now allows for curvilinear grids, MPI-based parallelization, and
warrants magnetic solenoidality using a constrained transport method, thus
eliminating the need for the previously employed projection scheme.
For more details on the code, see and
.
Rather than using the full energy equation, we simply assume that pressure
P and number density n are related
via an isothermal equation of state P=c2nmp (mp
denoting the proton rest mass) with an isothermal sound
speed c=180 km s-1 that corresponds to a plasma temperature of
about
2.0 MK. This seems appropriate, given that we intend to compare
our findings to the much simpler case of a constant radial flow and a
magnetic field given analytically as a Parker spiral
.
Our equidistant numerical grid of size
(Nr,Nθ,Nϕ)=(290,72,180) covers the spherical
domain r∈[1,30]R⊙=[0.005,0.14] AU,
θ∈[0.1π,0.9π], and ϕ∈[0,2π], implying a cell
extension of Δr=0.1R⊙ in the radial direction and
2∘ in both azimuth and colatitude.
The region within angular distance θ0=0.1π from the polar axis
(with its cell sizes tending to zero in the ϕ direction) is deliberately
excluded to save computation time. This can be justified for a CME that is
launched in the equatorial plane and is unlikely to cause noticeable
distortions near the solar axis.
The fluid variables are initialized at t=0 as a radially expanding
flow:
n|t=0=n0(r/R⊙)-3,V|t=0=er(r/rc)c:r<2rc2c:r≥2rc,
with a base number density n0=1014 m-3 and the critical (sonic)
radius rc=GM⊙/(2c2), at which Vr=c. The velocity
profile of
Eq. () is chosen to be close to the classical solar wind
solution of in order to warrant sufficiently fast
convergence into a stationary state and also to conserve radial mass flux
(∼r2nVr) within r≤rc. The dipolar magnetic field used
by had to be modified in its θ component in
order to accommodate mirror boundary conditions at both
θmin=θ0=0.1π and θmax=π-θ0=0.9π,
and
it now reads
B|t=0=2cosθr3er+sinθr31-sinθ0sinθ2eθ
in units of 4cμ0n0mp≈83µT.
A derivation of Eq. () is provided in Appendix .
Within the inner radial boundary layer, quantities n, Br, Bθ,
and Vθ=0 are kept fixed at their initial values. We enforce
Vϕ=Ω⊙rsinθ (with Ω⊙=2.7µHz being the
Sun's angular rotation frequency) and extrapolate Vr linearly inwards
without fixing a specific boundary value (except that no backflow into the
Sun is permitted). Finally, Bϕ is fixed by keeping the field lines
straight across the boundary, i.e., the quantity (rBr/Bϕ) constant
along any radial direction. The respective boundary conditions at
rmax and ϕmin,max are outflowing and periodic.
This system is then self-consistently evolved until a sufficiently stationary
state is reached at t=t1=24 (in units of the sound crossing time
ts=R⊙/c≈ 1.5 h). This stationary state is characterized
by a helmet-streamer-like belt of closed magnetic lines at the equator and open,
almost radial field lines at higher latitudes. Along these open field lines, the
plasma flow forms a Parker-like wind, while a static region (“dead zone”) forms
under the closed equatorial field lines. The overall situation is thus reminiscent
of a quiet solar minimum configuration.
Modeling a CME eruption
CMEs are complex dynamical structures that come in a wide variety of morphologies
and exhibit an equally wide range of physical parameters. Despite the wealth of observational
data that have been accumulated and the modeling effort invested by the space-weather
community, key questions about their origin and the physical mechanisms that govern their
eruption and propagation remain unanswered to this day. Details on CME modeling may be
found in reviews by and and the
references given therein, while observational aspects have been summarized by,
e.g., and .
In the present context, our aim is merely to obtain a first assessment of how influential
the passage of a CME can be on a nanodust population near the Sun and an
indication of the type and expected magnitude of its effects. This justifies the use
of a rather simple density-driven CME model see, e.g.,, in which
the CME is launched at the
solar surface by a transient increase in the boundary value for n
(but not for V) in the
same manner and using the same parameters as already employed by
, except that the elaborate averaging method that was
used by these authors to implement spherical solar boundary conditions on a
Cartesian grid is no longer necessary in the present case.
Note in particular that this model CME has no internal magnetic flux rope whatsoever
but owes its magnetic structure entirely to that of the helmet streamer, which it
is able to open up and push outwards by virtue of its increased gas pressure.
The mass of the CME as
described by the model is about 5.7 × 1013 kg,
which puts the CME in the moderate to strong class.
The CME departs from a circular patch of radius 30∘ centered on the
θ=π/2, ϕ=0 direction, causing a temporary rupture of
field line connectivity
as it expands
and travels outwards, reaching peak speeds of about 5c≈900 km s-1.
The simulation is halted at tend,MHD=t2=50,
at which time the CME has completely left the computational volume, which
has again returned to a quiescent state. The numerical data for the entire
CME expansion process thus cover a time period of
26tc∼1.6 days.
The respective MHD configurations of the quiet Sun and during CME passage
are illustrated in the panels of Fig. , and velocity profiles for
both instants are shown in Fig. .
For numerical convenience, the plasma configuration {V,B}
is then interpolated linearly both on the spatial grid and between
consecutive time frames of separation Δt=1.
Plasma radial velocity profiles in the model
along the direction θ=67∘, ϕ=189∘
at the initial time of the CME (t=24) and
0.25 days later (t=28).
Particle simulations
Our calculations are similar to those in
but with the simple time-stationary
analytical model of the solar wind replaced by the time-dependent
numerical MHD solution corresponding to the model CME.
The electric charge of nanodust cannot be reliably estimated, so we
have to use an extrapolation from the results for the larger grains
.
In the calculations we use two sample values of the charge-to-mass
ratio: Q/m=10-4e/mp (for the grain radius s∼ 3 nm
with ∼ 8 V surface potential) and
Q/m=10-5e/mp (for s∼10 nm with ∼ 9 V
surface potential).
As in , we assume the nanodust
charge-to-mass ratio to be constant during the motion. The effect of
charge fluctuation is likely to be small because the charge jump
frequency is large compared to the frequency of Larmor rotation
. As a result, the particle
motion can be approximated by replacing the fluctuating electric charge
with the time-averaged value. We assume that this average
is approximately constant within the computational domain.
The equation of motion for a nanodust particle with mass m, electric
charge Q, and velocity v becomes
dvdt=Qmc(v-V)×B-GM⊙r2er+Fγ+Fion,
where v=dr/dt is the particle velocity and r the
particle position. The first term on the right-hand side of
Eq. ()
includes the electric field E=-(1/c)V×B induced by the
solar wind plasma flow, which is responsible for a large part of
acceleration
that a charged particle experiences in the simulation.
Fγ=GM⊙r2β1-vrcer-vc
is the force due to solar radiation,
the Poynting–Robertson force .
Since the radiation pressure-to-gravity ratio β for nanodust is
expected to be small (β∼0.1), in most of the calculations
presented here we set β=0, neglecting the Poynting–Robertson force.
In some calculations we also included Fion, the drag force caused by
solar wind proton impacts on a grain:
Fion=-FSW(r)CSW,pv-V|v-V|,
where FSW(r)=nSW(r)mp|v-V|2 is the solar wind proton
flux at r relative to the dust grain, and CSW,p is given by :
CSW,p=πa2232al(E)(2a≤l(E)),CSW,p=πa21-13l(E)2a2(2a>l(E)).
Here, a is the radius of the grain, and l(E) is the range of a proton of initial
energy E=(1/2)mp|v-V|2 passing through the material of the grain. Following
we assume l(E)∝E1/2 with l(1 keV) = 0.092 µm for
silicate grains. Similarly to , we simplify the
proton drag force by neglecting the thermal component of the proton velocity.
The drag force is weak compared to other forces. For the MHD solution used
in our simulations, its magnitude along a sample of nanodust trajectories
varies between ∼ 2
and ∼ 10 %
of the gravity force with approximately the same range for the CME and the
time-stationary configurations.
Equation () is solved numerically by the Runge–Kutta method with
linear interpolation used to incorporate the results (V and
B) from the MHD simulation.
The initial conditions for r and v are defined as
follows. We assume that the nanodust particles are released at zero
relative velocity from the parent bodies in circular Keplerian orbits.
The orbits are specified by the radius
r, the direction of the ascending node ψ, and the inclination
δ. As in , we disregard
the difference
between the solar equator and the ecliptic plane so that the
inclination δ is defined relative to the solar equatorial plane. The
initial position of the particle on the orbit is specified by the
azimuthal angle α. We choose nr=10 values of r (0.01 AU <r< 0.135 AU, logarithmic spacing), nψ=4 values of ψ
(22.5∘<ψ< 157.5∘), nδ=12 values of δ
(-66∘<δ< 66∘), and nα=10 values of α
(18∘<ψ< 342∘). One sample therefore consists of 4800 trajectories.
Most of our particle simulations stay within the time limits of the
MHD model for the CME (time from t1=24 to t2=50, total time
length ∼ 1.6 days).
We also use time-extended plasma configurations,
which consist of the CME solution in the (t1, t2) time interval and the
time-stationary configuration outside. We assume that the time-stationary
configuration for t<t1 can be approximated by the t=t1=24 time frame
and for t>t2 by the t=t2=50 time frame of the original MHD solution.
The initial and (maximum) final times for particle motion are taken to be
the same for all particles in the sample. All simulations
presented here (also including the time-extended calculations)
start from the same initial time equal to t1+Δt, where
Δt=2.6 in dimensionless MHD time units (∼ 0.16
days). The particle simulations restricted to the model CME end at
t2-Δt so that the time length of each of them is ∼1.32 days.
Nanodust creation rate as a function of the heliocentric distance
assumed in our calculations (Czechowski and Mann, 2010).
To calculate the averages over the ensemble of the initial conditions, we
have to assign probability weights to the initial points. For any quantity
A(r) dependent on the initial point r, we define the average
A as
A=∫d3rn(r)A(r)∫d3rn(r),
where n(r) is the nanodust creation rate at the point r, and
the integration runs over the region over which the initial points are
distributed. We want to approximate the volume integral by the weighted sum
over the set of our initial points. In spherical
coordinates (heliocentric distance r, heliographic colatitude θ,
and heliographic longitude ϕ), the region of space containing the
initial points (0.02 AU ≤r≤ 0.12 AU,
18∘≤θ≤162∘, 0∘≤ϕ≤360∘)
is divided into 800 bins centered at the points ri (i=1 to 10),
θj (j=1 to 10), and ϕk (k=1 to 8) with equal spacings
between logri, θj and ϕk. An initial point
rI (I=1 to 4800),
which belongs to the (i,j,k) bin, is assigned a weight wI equal to
the
volume of the bin
multiplied by n(ri)/N, where n(r) is the assumed nanodust creation rate
at the heliocentric distance r (see Fig. ) and
N is the number of initial points belonging to the bin. If the initial
point corresponds to the orbit with low inclination (|δ|≤13.5∘),
then n(r)=nd(r) (the rate for the dust disk); otherwise
n(r)=nh(r) (the rate for the dust halo).
The average A is then approximately given by
A≈∑wIA(rI)∑wI,
where the sum over all initial points is rI.
For comparison, we also repeated the calculations with a different
approximate expression for the sum over the initial states
used by . The results
for escape rates were higher by a factor of ∼ 1.3,
but our qualitative conclusions were not affected.
We use the same nanodust creation rates as .
The nanodust creation rates inside the circumsolar dust disk (nd(r))
and in the dust halo around the Sun (nh(r)) are shown in
Fig. .
These rates were found to underestimate the nanodust flux
derived by STEREO/WAVES . However,
as can be seen from Eq. (),
our results do not depend on the absolute values of the
nanodust production rates.
The dust–dust collision
and fragmentation model used for calculating the rates is described in
and based on
. The mass distributions of dust in the circumsolar
cloud are modified versions of
the interplanetary flux model by
. The spatial distributions are proportional to
∼r-1 in the disk and r-2 in the halo regions, both
flattening for r<10 R⊙. The velocities of collisions
between the dust grains (the parent bodies of the nanodust) are assumed
to scale as r-1/2.
Results
We present the results of simulations for different cases listed in
Table . The computational domain in space is limited to the
region described by the CME solution (see Sect. 2).
In each simulation, we follow the motion of
a sample of 4800 particles. The distribution over the initial
points is the same for all cases. In each sample, the initial time,
the values of
the charge-to-mass ratio Q/m, and the radiation pressure-to-gravity
ratio
β are the same for all particles. The
distributions shown in the figures (Figs. , ,
, , , and )
are obtained after weighing the
results with the assumed nanoparticle production rate (Fig. ),
as described in Sect. 3, Eq. (). The distributions over
final values mean the distributions over values reached by particles at
the end of the calculations (the moment of escape or reaching the final time
limit). The bins have equal size. The plotted fractions
sum to 1 when the non-escaping (solid lines)
and escaping (dashed or dotted lines) populations are combined.
In Table , “CME” denotes the calculation using the original
MHD solution for the CME (total time span 1.6 days), “CME + stationary”
is the time-extended configuration (the CME followed by the final time frame
of the CME), and “stationary” means that the final frame of the CME
was used as a time-independent distribution for the whole time period.
The “time length” column lists the differences between the final and initial
times used in particle calculations. These are taken a little shorter
than the total time length of the corresponding background plasma
configuration. The “(1-β)” notation
means that the gravity force in the equation of motion (Eq. )
is multiplied by (1-β) with β=0.1. “PR” means that the full
Poynting–Robertson force (Eq. ) with β=0.1 is included.
Apart from these two cases, the value of β was set to zero.
“Drag” denotes the results including the effect of the plasma ion drag force
(Eq. ).
“Trapped” means that only the trajectories in the trapped class (see
Sect. 5.3) are
included, and fesc denotes the (weighted) fractions of the
escaping particles (rapid or slow) calculated using Eq. ().
The (unweighted) numbers of escaping trajectories are given
in brackets below the corresponding fesc. The total number
of trajectories is 4800 for each case, except the last two.
The non-escaping fraction for each case is equal to 1 minus the sum of
fesc values for rapid and slow populations.
Escape fractions fesc for rapidly and slowly
escaping populations.
Distribution of the final velocity for
non-escaping and escaping particle populations obtained
in the CME model for a sample of 4800 nanodust
particles with Q/m=10-5e/mp.
As Fig. but for
the following
cases: (a)Q/m=10-4e/mp,
(b)Q/m=10-5e/mp with gravity modified
by the factor (1-β), β=0.1,
and (c)Q/m=10-5e/mp for the time-stationary
plasma configuration (initial time frame of the CME solution).
A significant fraction of particles (Table ) escapes from the
computational domain within the time extent (1.6 days) of the
original CME solution. Particles escape predominantly across the outer
boundary (r=0.14 AU) of the computational domain. Only a small fraction
(∼ 1.5 %, ∼ 200 trajectories) crosses the inner boundary
(r=0.005 AU), and an even smaller fraction (∼ 0.7 %, ∼ 10
trajectories) crosses the boundary in solar colatitude (θ=18∘
or 162∘). Inclusion of the ion drag force increases the fraction of
particles escaping through the inner boundary (∼3 % , ∼ 650
trajectories within 1.6 days).
The velocity distribution of this rapidly escaping population is different
from that of the remaining particles and extends to about 1000 km s-1
(Figs. and ). The results for different Q/m
(10-5e/mp, Fig. ; 10-4e/mp, Fig. a) are qualitatively
similar. Taking into account the effect of the radiation pressure
(the (1-β) factor in the gravity force or the full Poynting–Robertson force
term) does not greatly affect the results (Fig. b)
because of the small value of β (∼0.1) used for the nanodust
particles. The effect of the ion drag force on the velocity distribution of
rapidly escaping particles is also small. The ∼ 2 % increase in the
escaping fraction (Table ) is due to particles escaping through
the inner boundary. If the CME is replaced by the time-stationary model, the
escaping particles are essentially absent (Fig. c).
In heliographic longitude ϕ, the region of origin for the majority of the
rapidly escaping particles lies between 75 and 285∘
(Fig. ), which coincides with the region where the plasma flow
and magnetic field perturbations associated with the CME were the strongest.
The non-escaping particles originate mostly outside of this region.
The distribution of the final values of the colatitude θ of the
rapidly escaping particles is concentrated near 90∘ with the width
significantly less than the 27∘ width of the disk
(Fig. ). This suggests that the nanodust main acceleration
mechanism in the model CME may be related to the belt of closed magnetic
field lines and expanding arcs near the solar equator (also a magnetic
equator in the model). In fact, examination of the magnetic field structure
along the trajectories of the rapidly escaping particles shows that the
acceleration to very high speed (>500 km s-1) is associated with the
regions where the magnetic field acquires a large non-radial component
(Figs. and ). Strong acceleration of nanodust is
therefore a result of a strong electric field induced by the plasma flow. The
final velocity is within a factor of 2 from that of the local plasma. The
magnetic field lines have the form of expanding arcs (Fig. )
or, very close to the solar equator, of magnetic islands carried outwards
with the flow (Fig. ). In the latter case, the trajectories
often show multiple crossings of the solar equator plane at z=0
(Fig. ). This occurs for the negative grain charge, for which
the field polarity is “focusing”, but also for the positive charge
(“defocusing” polarity). Note that the peak in final θ
(Fig. ) is less prominent for the focusing configuration.
As Fig. but for the distribution in
initial heliographic longitude.
The distributions over final
values of solar colatitude θ for the nanodust particles
with (a)Q/m=10-5e/mp and (b)Q/m=-10-5e/mp.
For escaping particles (dashed line), there is a narrow peak
near the solar equator (θ=π/2) for both signs of Q.
(a) The particle and plasma speed
and (b) the magnetic field structure
along the trajectory of a rapidly escaping particle
with Q/m=10-5e/mp.
The thin lines in panel (b) are the selected magnetic field lines
encountered by the particle along its trajectory, which is shown as the
thick line. The
coordinate z is measured along the solar rotation axis with z=0 at
the solar equator.
As Fig. but the selected particle has
a negative charge (Q/m=-10-5e/mp) and is released closer
to the Sun.
Particles escaping after the end of the CME: the
slowly escaping population
The loss of particles from the computational domain continues after the end
of the CME. The final velocity distribution obtained for the time-extended
model (CME + the time-stationary period, total time length
∼ 6 days, Fig. ) includes three components: non-escaping
particles (solid lines), particles escaping during the CME (rapidly
escaping particles, dashed lines), and the particles escaping after the end
of the CME (slowly escaping particles, dotted lines). The slowly
escaping particles have a narrow velocity distribution (Fig. a) strongly suppressed beyond ∼ 250 km s-1.
Acceleration of nanodust particles to very high velocities
(∼ 1000 km s-1) is therefore restricted solely to the CME time
period.
The distribution of the initial heliographic longitude ϕ for slowly
escaping particles (Fig. c) is similar to that
for non-escaping particles. Unlike the rapidly escaping particles, the
slowly escaping particles (Fig. b) have no
sharp central peak in the distribution of final θ values.
Distributions of (a) final velocity, (b) final
colatitude, and (c) initial longitude for the time-extended case
(CME followed by the time-stationary configuration) lasting ∼ 6 days.
The distributions for non-escaping, rapidly escaping (escaping during the
CME), and slowly escaping particles (escaping after the end of the CME) are
shown.
As Fig. a but for the simulation including the
plasma ion drag.
Cumulative fraction of particles escaping from the computational
domain as a function of time for a time-extended (33 days long) simulation
including the ion drag. Rapidly escaping particles (39 %) are not
included in the figure. The result implies that only ∼ 6 % of all
particles remain inside at the end of the calculation: 100–39 % (rapidly
escaping), -55 % (present figure). A rough estimation of the
characteristic escape time gives ∼ 10 days.
The fraction of particles remaining inside the calculation region as
a function of time for the CME time-extended models with and without the ion
drag. The vertical dotted line marks the end of the CME. The dashed lines
shows the two parameter (B and τ) fits of the form
(A-B)exp(-t/τ)+B applied for the time period starting 1.32 days after
the initial time. The best fits give B=0.37 and τ=1.87 days without
drag and B=0.19, τ=2.9 days with drag.
The drag force increases the fraction of slowly escaping particles from 21 to
29 % (Table ) with 11 % contributed by the particles
escaping through the lower (rmin) boundary. The final velocity
distribution of slowly escaping particles (Fig. ) differs from
that for the no-drag case by having a second peak at
∼ 300 km s-1, representing particles escaping across the lower
boundary. The distributions in final θ and initial ϕ are similar
to those of the no-drag case.
Figure shows the fractions of nanodust surviving from the
sample of particles created at the same initial moment as a function of
time for the cases with and without ion drag. For the particles escaping
between the end of the CME and the end of the period covered by the
calculations, the exponential fit of the form (A-B)exp(-t/τ)+B gives
τ≈1.9 days and B=0.37 (without drag) and τ≈2.9 days
and B=0.19 (with drag). If the behavior of the surviving particle fraction
were described by the exponential formula, the value of B would be
equal to the asymptotic value of the surviving fraction.
We performed additional time-extended simulations to estimate the surviving
particle fraction after a longer time period. The results were the following: with drag
included, 6 % survived after 33 days (Fig. ); without drag,
25 % survived after 76 days. In the latter case, only trapped particles
were included in the simulation.
Trapped particlesTrapping and the guiding center approximation
Although the guiding center approximation is not used in our simulations,
it was found to be helpful in understanding the trapping mechanism
of nanodust in the vicinity of the Sun
. In this section we briefly summarize
the argument leading to the phase space model of
.
The equations for the parallel component v||G and the perpendicular
part vTG of the velocity of the guiding center are
dv||Gdt=g||-μ∂SB+VT⋅∂tb^+VT⋅(VT⋅∇)b^+v||G∂Sb^,vTG=VT,
where b^=B/B, g|| is the component of the gravity
force per unit mass parallel to b^, VT is the
perpendicular part of the plasma velocity, ∂t is the partial
derivative with respect to time,
∂S≡(b^⋅∇),
μ=(vT′)2/(2B) represents the adiabatic invariant, and
vT′=|vT-VT| is the perpendicular speed of the particle in the
plasma frame. In Eq. () the non-leading Q/m-dependent drift terms
are omitted. With this additional approximation, the particle (the guiding
center) slides along the magnetic field line convected with the plasma
flow.
In the study by
it is assumed that the plasma flow and the magnetic field are time stationary
with the plasma flow purely radial and the magnetic field in the
form of the Parker spiral:
b^=er-aeϕ(1+a2)1/2B=Cr2(1+a2)1/2,
where a=(Ω⊙r/V)sinθ, Ω⊙ is the angular
velocity
of solar
rotation, V is the solar wind speed, and θ is the heliographic
colatitude. C is constant along a magnetic field line and can be written as
C=B‾rr‾2, where B‾r is the radial component of B
at the reference distance r=r‾.
The sliding motion is determined by the force terms in Eq. (),
which can be explicitly calculated using Eq. (). We restrict
attention to the a≪1 region (corresponding to r≪ 1 AU). There is an
inward-directed gravity force
g||=-GM⊙(1-β)r2,
including the correction β due to radiation
pressure. The outward-directed forces are the “magnetic mirror” force
-μ∂SB=|vT0′|2r02/r3
and the “centrifugal” force
VT⋅(VT⋅∇)b^=Ω⊙2rsin2θ.
Here, vT0′ is the initial transverse velocity of the nanodust
particle in the plasma frame at the heliocentric distance r0, where
the particle is released from the parent body. The trapping occurs if the
gravity force stops the particle outward motion before the “centrifugal”
force becomes dominant.
A more detailed analysis
shows that, in the
region where a≪1, the
guiding center motion is described by the equations
dr/dt=v,dv/dt=W(r)-(a2/r)v2,
where r is the heliocentric distance of the guiding center,
v is the radial component of the guiding center velocity, and
W(r)=GM⊙(1-β)r2-1+r2r+rr13.
Here,
r2=|vT0′|2r02GM⊙(1-β),r1=GM⊙(1-β)Ω⊙2sin2θ1/3.
Note that the gravity force (Eq. ) for r>r2 dominates over the
magnetic mirror force (Eq. ) and for r<r1 over the
centrifugal force (Eq. ).
Equations () and () define the dynamical system in the
(r,v) phase plane. If W(r)=0 has real roots, the fixed points appear at
the root positions on the r axis in the (r,v) plane. If r2≪r1, the
fixed points are approximately at r≈r2 and r≈r1. The
trapped orbits encircle the fixed point r≈r2. The fixed point
r≈r1 is the outer boundary of the trapped region.
In the solar system, r1≈(0.16AU)(1-β)1/3sin-2/3θ. Since the CME model is limited to r< 0.14 AU, all
our simulations are restricted to the r<r1 region.
Trapped particles: simulation results
Trapped particle trajectories projected onto the (x, y) plane.
The top (Q/m=10-5e/mp) and bottom right (Q/m=10-4e/mp) trajectories follow from simulations for the case of the
time-extended (6.11 days long) CME model. For comparison, a trajectory
obtained in the model with constant plasma speed and Parker spiral magnetic
field is shown in the lower left part of the figure.
Distance versus time plots for a sample of trajectories of the nanodust
particles (Q/m=10-5e/mp) belonging to (a)
“rapidly escaping”, (b) “slowly escaping”, and (c)
“trapped” populations.
Distribution of the difference Δϕ between the final and
initial values of the heliographic longitude ϕ for the particles in the
“trapped” class for the case of the time-extended (6.11 days long) CME
model. The vertical dotted line shows the value of Δϕ corresponding
to solar rotation over 6.11 days.
Effect of the (photon) Poynting–Robertson force
and the plasma drag force on a trapped particle
trajectory. While the Poynting–Robertson force leaves the
minimum and maximum r values almost
unaffected, the plasma drag causes them to decrease with time.
The decrease in minimum r may lead to crossing of the
lower boundary of the calculational domain and/or
destruction of the nanodust particle by sublimation.
As explained in the previous subsection, the motion of a trapped particle in
the simple time-stationary model consists
of sliding between the two turning points along a rotating magnetic field
line. A class of particles with similar trajectories was identified in our
results. Two examples, projected onto the (x,y) plane, are shown in
Fig. (the upper curve and the curve on the right). A larger
sample is shown in Fig. c as distance versus time
plots. The trapped particles oscillate between the upper and lower
turning points in r, but the positions of the turning points and the time
interval between them may differ for subsequent cycles. Through inspection of the
distance versus time behavior, we found that the trajectories of at least
30 % of all particles in the time-extended (6.11 days long) CME model
and 51 % of particles in the 6.11-day time-stationary model can be
assigned to this class. We required that the oscillations in r persist for
the whole time span of the trajectory and that at least one full cycle of
radial oscillation (e.g., a particle starting from the upper turning point must
return to it at least once) is present.
Figure shows the distribution of the gain Δϕ in the
heliographic longitude (the difference between the final and initial values
of ϕ) achieved after 6.11 days by the particles in the trapped
class. Although some spread can be seen, there is a sharp maximum in the
distribution close to the angle (86.6∘) corresponding to the angle
of Carrington rotation over the period of 6.11 days. This is consistent with
trapped particles staying close to a rotating magnetic field line, as
suggested by the guiding center approximation.
The particles in the trapped class constitute the majority of particles
that remain inside the computational domain after the extended period of
∼ 6 days starting with the CME. Since the orbits of particles in the
trapped class evolve with time, some of these particles may ultimately
escape from the calculational volume. We attempted to estimate the
characteristic time for escape of trapped particles in the absence of
plasma ion drag force by extending the simulation time to about 76 days from
the beginning of the CME. We found the majority (70 %) of trapped
particles remaining inside the region.
Assuming that the nanodust particles are created continuously, the trapped
nanodust would accumulate in the vicinity of the Sun until some loss
mechanism balanced the creation rate. Our calculations suggest that the
main loss mechanism may be due to the plasma ion drag. The ion drag force
causes contraction of the orbits of trapped particles (Fig. ),
leading to a decrease in the aphelium and the perihelium distances and, in
many cases, to the particle crossing the inner boundary and falling into the
Sun. This process is responsible for the ∼ 11 % increase in the
slowly escaping particle fraction noted in the previous subsection. The
contraction rate of trapped orbits depends on their initial parameters and
occurs particularly fast for particles created near the lower boundary.
The result of the long-term (33 days) simulation (Fig. )
including the effect of drag permitted us to roughly estimate the average
characteristic escape time for nanodust (excluding the rapidly escaping
population) to be on the order of 10 days.
Loss through sublimation
The lifetime of small dust grains in the vicinity of the Sun, particularly
those belonging to the trapped class, can be limited by sublimation and
sputtering. An estimation (ignoring the effect of the ion drag) was made by
based on the results of for sublimation
and for sputtering. The survival probability against
sputtering after 100 orbits for 10 nm particles released from low-inclination orbit was found to be 0.5 for the initial heliocentric distance
0.12 AU, increasing to 0.8 for the initial distance within 0.06 AU from the
Sun.
In this subsection we present the estimation of the sublimation loss fraction
based on our simulations, taking into account the effect of the drag force on
particle trajectories. According to , the (fast) sublimation
for a compact silicate particle occurs within the heliocentric distance of
3 R⊙ with the sublimation time for a 100 nm particle equal to
0.01 years. Assuming that the lifetime of a spherical sublimating particle is
proportional to its radius, for the particular case of the time-extended
(6.11 days long) simulation (including the ion drag force) for Q/m=10-5e/mp particles with the radius 10 nm, we obtain the loss
fraction by sublimation after ∼ 6 days to be 15 % with 14 %
contributed by particles in the trapped class. The result without the ion
drag force would be 4.5 % with 4.1 % due to trapped particles.
Conclusions
We present the results of numerical simulations of the motion of nanodust
particles released from circular Keplerian orbits between 0.005 and 0.14 AU
from the Sun with the solar wind flow and the magnetic field approximately
corresponding to the MHD model of the CME by . The mass of
the CME as described by the model is about 5.7 × 1013 kg,
which puts the CME in the moderate to strong class. The MHD model includes a
simplifying assumption that the solar equator is the same as the solar
magnetic equator.
In part of our calculations we include the plasma ion drag
force, which was omitted in the earlier work on nanodust dynamics
near the Sun .
We follow the approach of . We find that
the ion drag force plays a very important role in nanodust dynamics.
In a simple time-stationary model of the solar wind with constant
radially directed velocity and the Parker spiral form of the magnetic field,
the nanodust released from low-inclination circular orbits within the region
r<0.14 AU would be trapped . Although the
MHD solution used in the present study is very different from this
simple model, we find that a similar trapping mechanism operates for a
significant fraction (∼ 35 %) of nanodust particles.
About 35 % of the nanodust particles released during the CME form the
rapidly escaping population with a broad velocity distribution extending
to 1000 km s-1. These particles come from the region where the
disturbance of plasma flow and magnetic field caused by the CME is strong.
The acceleration process is associated with the regions of closed magnetic
field lines: the expanding arcs in the CME region and the narrow belt in the
vicinity of the solar equator plane.
The remaining particles, which belong to neither the rapidly escaping nor
the trapped class, escape from the region after the time span
(∼ 1.6 days) of the model CME. These particles form the slowly
escaping population. To investigate their behavior, we extended the
calculations beyond the time span of the CME, assuming that the CME is
followed by a period described by a time-stationary MHD solution. We found
that the ion drag force becomes important for time-extended simulations.
The ion drag force differs from other forces (Lorentz, gravity, and
Poynting–Robertson) included in our calculations by its destructive effect on
trapped particle trajectories. The trapped orbits contract and ultimately
cross the inner boundary. The lifetime of trapped particles is consequently
limited by the ion drag force. From our simulations, we estimated the average
lifetime of nanodust to be ∼ 10 days.
The effect of the ion drag on particle trajectories increases the nanodust
destruction rate due to sublimation. Assuming that the results of
can be applied to nanodust, we estimated the fraction of
nanodust particles destroyed by sublimation over the period of ∼ 6 days
(1.6 days CME + time-stationary period) to be 15 % compared to
4.5 % if the ion drag is neglected. Almost all nanodust particles
destroyed by sublimation belong to the trapped population.
Since our computational domain was restricted to r<0.14 AU, we cannot
directly compare our results with the observations by STEREO/WAVES in the
vicinity of the Earth's orbit, as discussed by . To estimate
the flux of charged nanodust away from the source, it is necessary to have a
reliable model of nanodust propagation to the observation point (see
). For comparison with the observations, the simulations
would have to be extended to cover a wider region of space up to
∼ 1 AU from the Sun. Also, the model of the CME would have to be
modified to include the more realistic situation of the solar magnetic
equator inclined relative to the solar equator plane.
Calculated nanodust trajectories are available from
Andrzej Czechowski (ace@cbk.waw.pl). MHD simulation results are available from
Jens Kleimann (jk@tp4.rub.de).
Fixing the initial magnetic field
The usual magnetic field of a point dipole, which reads
B=2cosθr3,sinθr3,0
in spherical coordinates (r,θ,φ), derives from a vector
potential A=(sinθ/r2)eφ via
Br,Bθ,Bφ=B=∇×A=1rsinθ∂∂θsinθAφ,-1r∂∂rrAφ,0
and has radial field lines only at θ∈{0,π}. We now wish to
modify this field such that the field lines become radial at a colatitude
θ0 that may be freely specified, while maintaining the same magnetic
flux through the surface r=R⊙. For this purpose, we first note from
Eq. () that the transformation
sinθAφ→sinθAφ,mod+f(r)
leaves Br unchanged for any function f(r). This gauge freedom in the
choice of f(r) may be exploited to have Bθ vanish at
θ=θ0:
0=!Bθ|θ0=-1r∂∂rrAφ+f(r)sinθθ0=-1r∂∂rsinθ0r+rf(r)sinθ0,
from which we get
sinθ0r+rf(r)sinθ0=C⇒f(r)=C-sinθ0rsinθ0r
with C a constant of integration. The resulting Bθ component
becomes
Bθ,mod=-1r∂∂rrsinθr2+1sinθC-sinθ0rsinθ0r=1r3sinθ-sin2θ0sinθ=Bθ1-sin2θ0sin2θ.
Since Bθ,mod is obviously independent of C, we may choose
C=0 and finally obtain
Amod=sinθr21-sinθ0sinθ2eφBmod=2cosθr3er+sinθr31-sinθ0sinθ2eθ
as the desired initial condition ().
We note in passing that, alternatively, the same method could also be used to
keep |B|, rather than Br, dipolar at r=R⊙.
AC performed the nanodust simulations and analysis of the results.
JK designed the MHD model of the CME.
The authors declare that they have no conflict of
interest.
Acknowledgements
We thank the referee for the important suggestion to include the ion drag term in our
calculations.
We are grateful to Horst Fichtner for helpful comments,
discussions, and support, as well as to Ralf Kissmann for valuable technical
assistance with his CRONOS code. Furthermore, Jens Kleimann acknowledges financial support
from the Deutsche Forschungsgemeinschaft (DFG) via grant FI 706/8-2 and
from the Ruhr Astroparticle and Plasma Physics (RAPP) Center, funded as
MERCUR grant St-2014-040.
The topical editor, Elias Roussos, thanks two anonymous referees for help in evaluating this paper.
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