Astrophys Space Sci (2006) 306:259–267 DOI 10.1007/s10509-006-9271-0
ORIGINAL ARTICLE
Stochastic Acceleration of Energetic Particles in the Magnetosphere of Saturn E. Mart´ınez-G´omez · H. J. Durand-Manterola · H. P´erez de Tejada
Received: 31 May 2005 / Accepted: 2 November 2006 C Springer Science + Business Media B.V. 2006
Abstract Voyager’s plasma probe observations suggest that there are at least three fundamentally different plasma regimes in Saturn: the hot outer magnetosphere, the extended plasma sheet, and the inner plasma torus. At the outer regions of the inner torus some ions have been accelerated to reach energies of the order of 43 keV. We develop a model that calculates the acceleration of charged particles in the Saturn’s magnetosphere. We propose that the stochastic electric field associated to the observed magnetic field fluctuations is responsible of such acceleration. A random electric field is derived from the fluctuating magnetic field – via a Monte Carlo simulation – which then is applied to the momentum equation of charged particles seeded in the magnetosphere. Taking different initial conditions, like the source of charged particles and the distribution function of their velocities, we find that particles injected with very low energies ranging from 0.129 eV to 5.659 keV can be strongly accelerated to reach much higher energies ranging from 22.220 eV to 9.711 keV as a result of 125,000 hitting events (the latter are used in the numerical code to produce the particle acceleration over a predetermined distance). Keywords Acceleration . Charged particles . Dione . Energetization . Monte Carlo method . Planetary magnetosphere . Plasma . Saturn . Stochastic mechanism . Titan
E. Mart´ınez-G´omez () · H. J. Durand-Manterola · H. P´erez de Tejada Department of Space Physics and Planetary Sciences, Institute of Geophysics, National Autonomous University of Mexico, C.P. 04510, Mexico D.F., Mexico e-mail:
[email protected]
1 Introduction The available observations in situ of the Saturn’s magnetosphere have been made by the Pioneer 11 (September 1979), Voyager 1 (November 1980), Voyager 2 (August 1981) and Cassini (July 2004) spacecraft. Both local and remote measurements provide us information on the structure and dynamics of this magnetosphere. While the Pioneer 11 and both Voyager encounters with Saturn supplied a wealth of definitive new knowledge, they have also introduced many fundamental questions that are still unanswered. The Pioneer and Voyager instruments found a planetary magnetic field aligned with the planet’s rotation axis, an extended high-density plasma sheet, the presence of hot plasma (∼35 keV) in the outer magnetosphere, and a population of trapped energetic particles which interacts strongly with the inner satellites and the rings. It was also determined that Titan has no intrinsic magnetic field and this satellite represents a significant plasma source for the outer magnetosphere. The vast hydrogen torus surrounding Titan that extends from 8 to 25 Saturn radii (R S ) plays an important role as a source of plasma and as a media interacting with ions that originate in other regions. However, the relative importance of other plasma sources (such as the solar wind, the Saturn atmosphere, the icy-covered surfaces of the inner satellites and the rings) has been studied as well by many authors (Eviatar and Richardson, 1986; Jurac et al., 2002). Saturn’s magnetosphere takes an intermediate place among Jupiter, Uranus and Neptune, with neutrals dominating the density and mass as at Uranus and Neptune but with plasma playing an essential role in the magnetospheric dynamics as at Jupiter (Richardson, 1998). The plasma probe on Pioneer 11 measured magnetospheric ions within the energy-per-unit charge range of 100 eV to 8 keV in the inbound pass between 16 R S and 4 R S . At radial Springer
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distances beyond the orbit of Rhea at 8.8 R S, the dominant ions are most likely protons and the corresponding typical densities and energies are 0.5 cm−3 and 1,000,000 degrees Kelvin (∼86 eV), respectively, with substantial fluctuations (Frank et al., 1980; Wolfe et al., 1980). The Voyager observations extended our knowledge of plasma distributions in Saturn’s magnetosphere. The plasma probe on each spacecraft provided simultaneous high-resolution measurements of electrons and ions over an extended energy/charge range (E/Q∼10 eV to 6 keV for both species), together with improved information on angular distribution. The Voyager electron measurements display large-scale radial gradients in electron temperature, increasing from 3 MeV per nucleon are present at distances larger than 6 R S ; (b) Bridge et al. (1981, 1982) confirm the existence of the Dione/Tethys torus of oxygen ions and add a wealth of data on the distribution of plasma in Saturn’s magnetosphere; (c) in the inner magnetosphere (≤10 R S ), the dust grains in the rings and the many large and small satellites reduce the population of particles E > 0.5 MeV to values of 103 times smaller than what otherwise be present (Van Allen et al., 1980; Baum et al., 1981). The sputtering and outgassing of their surfaces inject gas into the system; by some process not yet identified, particles of the resultant plasma are accelerated to energies of the order of tens of keV (Frank et al., 1980); (d) Krimigis et al. (1982) find that at specific energies ≥0.2 MeV per nucleon, H2 and H3 (molecular ions), as well as atomic hydrogen, helium, carbon and oxygen, are important constituents of the population of energetic particles in Saturn’s magnetosphere; (e) in the outer magnetosphere, the hydrogen and nitrogen torus produced by Titan maintains the plasma density between 10−2 and 0.5 ions cm−3 and a temperature of approximately 106 degrees Kelvin (Bridge et al., 1981; Gurnett et al., 1982). This plasma is, in general, convected in the corotation direction at a nearly rigid corotation speed. The energies of magnetospheric particles in this region extend to above 500 keV, in fact it is possible to find protons and ions above 2 MeV (Stone et al., 1981). Possible explanations to the observed high-energy population of particles involve the release of magnetic energy which heats the ion component of the plasma and then accelerates electrons to energies of about 2 MeV (Schardt et al., 1985). Another possibility is the Saturn’s magnetotail. In this region, as at Earth and Jupiter, changes in the tail configuration induced by interplanetary disturbances may lead to the acceleration of both ions and electrons to several hundred keV (Krimigis et al., 1981, 1982). Fujimoto and Nishida (1990) applied the recirculation model for particle acceleration in Springer
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order to explain the pitch angle and spectral characteristics of energetic electrons observed by Pioneer in the Jovian magnetosphere. In that model they combined the conventional radial and pitch angle diffusion processes with the essentially energy-conserving latitudinal diffusion in low altitudes and the pitch angle scattering in the plasma disk. Barbosa (1994) developed a model in which he combined the stochastic acceleration and the radial diffusion into an equation for energetic ions interacting with Magnetohydrodynamical (MHD) waves in Jupiter’s outer magnetosphere. His results provided a quantitative account of and an explanation for the high energy ion component observed in Jupiter’s magnetosphere. In order to explain the observations of energetic particles in the magnetospheric region, we propose a model in which the equation of motion includes the force due to the stochastic electric field originated from the magnetic field fluctuations measured by Voyager 1 and 2 spacecraft during their encounters with the Saturnian system, the magnetic force and the gravitational force. For this, we developed a numerical code in which different kinds of ions coming from Saturn’s upper atmosphere, Dione’s icy surface and Titan’s upper atmosphere are considered. In this work we use the SI units system.
2 Description of the model We use the magnetic field data given in the Saturn Solar Orbital Coordinate System (SSO) measured by the triaxial fluxgate magnetometer carried by the Voyager 1 and Voyager 2 spacecraft. We also use the trajectory position data with a resolution of 3 × 104 meters. The magnetic field data were averaged every 1.92 seconds during the Voyager 1 and Voyager 2 encounters with Saturn. The Voyager 1 moved through the magnetosphere from November 12 1980 at 01:54 Universal Time (UT) to November 14 1980 at 21:40 UT, while the Voyager 2 moved through the magnetosphere from August 25 1981 at 07:01UT to August 28 1981 at 20:37 UT. We will consider the first inbound magnetopause crossing and the fifth outbound magnetopause crossing as boundaries of the magnetosphere (Bridge et al., 1981; Ness et al., 1982). The SSO system is defined through the instantaneous vector that unites Saturn with the Sun. Some authors call this system of coordinates as “Sun-state”. In the time of the measurement, the direction x is the line that connects the center of mass of Saturn with the center of mass of the Sun, being positive towards the Sun. Direction z forms taking the cross product between the direction x and the vector speed of the Sun, z = x × v. Finally the direction y completes the east orthogonal system (y = z × x). Once we obtain the magnetic field fluctuations, it is important to determine the changes in the associated electric field. Knowing that ∇ · B = 0, we employ the Faraday’s and
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Table 1 Values of the parameters α, β for each spacecraft
Table 2 Mean free path for the most important ions in the magnetosphere of Saturn
a
The average radius for a dust grain is 2.5 × 10−7 meters
Component
α VOYAGER 1
α VOYAGER 2
β VOYAGER 1
β VOYAGER 2
E x E y E z
−3.13 × 10−7 −8.01 × 10−7 1.70 × 10−5
6.95 × 10−8 7.18 × 10−9 −6.16 × 10−7
3.00 × 10−4 2.12 × 10−4 1.89 × 10−3
2.55 × 10−5 1.29 × 10−6 7.36 × 10−4
Ion
Mean free path λ (m) (collisions with neutral C)
Mean free path λ (m) (Coulombian collisions)
Mean free path λ (m) (collisions with neutral dust in the E-ring)a
H He C N O
9.7972 × 1011 – 9.4019 × 1012 1.2939 × 1012 1.6002 × 1012
4.9895 × 1019 1.3447 × 1019 4.9294 × 1017 4.9082 × 1017 2.4037 × 1016
2.7599 × 1013 2.7599 × 1013 2.7599 × 1013 2.7599 × 1013 2.7599 × 1013
the Gauss’s Laws. In rectangular coordinates (x, y, z) we can write: ∂ Ey ∂ Ez ∂ Bx − =− ∂y ∂z ∂t ∂ By ∂ Ex ∂ Ez − =− ∂z ∂x ∂t ∂ Ey ∂ Ex ∂ Bz − =− ∂x ∂y ∂t
f (E) =
∂ Ey ∂ Ex ∂ Ez ρ + + = ∂x ∂y ∂z ε0
(1)
If we assume charge quasi-neutrality, i.e. ρ = 0, and that the changes of the magnetic field throughout the spacecraft trajectory are sufficiently small (∂ ≈), Equations (1) will solved and lead to:
Bz t
x y
+
xy (z)2
+
Bx y t z
1 1 + x (y) 2 + 2 (z) E x Bz E y = x − y t B y E x E z = x + z t E x =
for the normal, exponential, logistic, lognormal, Weibull and the extreme value probability distributions. The decision rule for this hypothesis test establishes that a smaller value of the Anderson-Darling statistic indicates that the chosen distribution fits the data better. We conclude that the Logistic probability distribution function is adequate:
1 x
(2)
Equations (2) show singularities when x = y = z = 0, in those cases, the value of the electric field was not considered. It is important to point out that does not affect the temporary distribution of the data because we applied the same t, i.e., every 1.92 seconds. Using the Voyager spacecraft data we require a function that better fits the field given in Equations (2). To find such function we do a goodness-of-fit hypothesis test through the value of the Anderson-Darling statistical criterion obtained
exp [−(E − α)/β] β(1 + exp[−(E − α)/β])2
(3)
In Equation (3) α is the expected value of this function whereas β is given in terms of the variance as: σ 2 = 3.289868β 2 . Using the adequate values of α and β for the Voyager 1 and Voyager 2 data shown in Table 1 we can obtain the electric field distribution function for each component. We need to simulate the stochastic behavior of the electric field given in Equation (3). For this, we employ a Monte Carlo Method in which the sampling rule can be chosen from the following techniques: direct inversion of the cumulative distribution function, rejection methods, transformations, sums, products, ratios, and composition (Hammersley and Handscomb, 1964). In this work we employ the direct inversion of the cumulative distribution function because it is most precise. An important consideration in the present study is to determine the nature of the magnetospheric plasma; that is, whether it is collisional or non-collisional. For this we need to estimate the mean free path, λ, whose calculated values in Table 2 indicate that 1011 ≤ λ ≤ 1019 meters, that is, greater than the size of the magnetosphere assumed to be 23 R S (∼109 meters); so that the plasma is in fact collisionless. The initial particle’s velocity distribution depends on the source they come from, for example, the Saturn’s and Titan’s upper atmosphere show a Maxwellian velocity distribution, v M , (Greiner, 1995). Due to the plasma anisotropy resulting from creation processes (pick-up), the distribution function of the particles coming from the icy-satellites is more exotic. Springer
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In our model it is possible to use different distribution functions because is independent of the chosen sampling method. In this work we use a Maxwellian distribution function for each source. In addition to this velocity it is necessary to consider the corotation velocity, v C , of the magnetospheric plasma deduced by MacLennan et al. (1982). In our model we will consider several particles to study the effects of the stochastic electric field on a velocity distribution, and then we will generate randomly the velocity of each particle. To simulate this situation, we use a Monte Carlo Method with the direct inversion of the cumulative distribution function as the sampling rule. In such case we do a modification of the Box-M¨uller Method (1958) for generating normal random numbers. Hence, we have: v M (χ1 , χ 2 ) =
k B TS cos χ 1 −2 log χ 2 mS
(4)
where χ 1 and χ 2 are random numbers in the interval [0,1], k B is the Boltzmann’s constant (k B = 1.3807 × 10−23 J/K), TS is the temperature of the plasma source, and m S is the mass of the ion. Thus, the initial speed for the particles is given by the following expression: v = vC + v M = r +
k B TS cos χ1 −2 log χ 2 mS
(5)
where = 1.638 × 10−4 rad/s, and r is the radial position vector of each particle. Once the particles are injected in the magnetosphere, they are under the action of several fields of force: the stochastic electric field (eE), the magnetic field (e(v × B)), and the gravitational field (G M S m s /r 2 ). Thus we have: F = eE + e | v × B | +
G MS m s r2
(6)
where e is the electric charge, E is the stochastic electric field, v is the velocity of the particle, B is the magnetic field, G is the gravitational constant, M S is the Saturn’s mass, m S is the particle’s mass and r is the radial distance between Saturn and the particle. For example in Equation (6) a proton with E∼10−5 V/m, B ∼10−6 T, v ∼107 m/s and r ∼108 m will be subject to FE ∼10−24 N, FM ∼10−18 N and FG ∼10−27 N. In order to calculate the gain (loss) of energy, Kr , for a particle after one interaction with the total force we have: K r = Fr r Springer
(7)
where Fr is the r -th component of the force shown in Equation (6) and r is the difference between two adjacent positions of the particle in each spatial direction. After each interaction with the total force, the kinetic energy is given by: K 2r = K 1r + K r = m 0 c
2
1 1−
2 v1r c2
− 1 + K r
(8)
where v1r is the initial component of the velocity in the r direction, c is the speed of light and m 0 is the particle’s rest mass. In Equation (8) we use the relativistic expression of the energy anticipating the fact that the particle could reach relativistic velocities. In addition to all the previous considerations, it is necessary to take into account the loss of energy due the electromagnetic radiation of the accelerated charged particles (Jackson, 1999) which is given by: Pr =
d Kr e2 ar2 = 2 2 dt 6ε0 π c3 1 − v2 c2
(9)
In Equation (9) e is the particle’s electric charge, ar is the acceleration, ε0 is the permittivity constant, c is the speed of light, and v is the final speed. Subtracting Equation (9) from Equation (8) and considering the relativistic expression for the final energy we obtain the final speed (v2 ) of the particle for each component r. v2r = c 1 −
K 1r
m 0 c2 + K r − P + m 0 c2
2 (10)
The procedure conducted between Equation (5) and Equation (10) is iterated several times according with a chosen number of interactions, the maximum number of interactions depends on the computing time. We take 23 R S as the upper limit of the size of the magnetosphere for computational purposes.
3 Results All the results referred here are from the Voyager 1 data. Similar results are obtained by the Voyager 2 data. Once the population of certain type of ions, for example C+ , H+ , He+ , O+ , OH+ , N+ and N2 + (see Table 3) interact with the stochastic electric field (Fig. 1), they win more energy in each interaction, as we can see in Fig. 2 and Fig. 3. However, in the case of Saturn’s upper atmosphere we can clearly distinguish between light and heavy ions. In such case, the lighter ion
Astrophys Space Sci (2006) 306:259–267 Table 3 Temperature of the source and the mass for the most important ions in the magnetosphere of Saturn
a
Atreya et al. (1984) Morrison (1986) c Soderblom and Johnson (1982) b
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Ion
Mass (in atomic mass unit) 1 amu = 1.66 × 10 −27 kg
C+
12.01
H+
1.01
He+ O+ OH+ N+
4.00 15.99 17.01 14.01
N+ 2
28.01
(H+ ) is being more accelerated than the others (like N+ ) by a factor of ∼10. In Fig. 4 and Fig. 5 a typical component of the final velocity is shown for a population of 10,000 particles of C+ coming from Titan’s upper atmosphere after 125,000 interactions with the stochastic electric field. We observe that the component perpendicular to the corotation velocity (z-component) experiments a separation into a bimodal distribution as seen in Fig. 4. It can be noticed that in the corotation y-direction the distribution does not show such separation into two different fluxes of particles (Fig. 5). When we start with particles located in the North Pole the final velocity distribution of all components are clearly bimodal. For example, in Fig. 6 we show the component in the y-direction because it is the corotation direction.
Source of particles
Temperature (K)
Saturn’s upper atmosphere Titan’s upper atmosphere Saturn’s upper atmosphere Titan’s upper atmosphere Saturn’s upper atmosphere Dione’s icy-surface Dione’s icy-surface Saturn’s upper atmosphere Titan’s upper atmosphere Titan’s upper atmosphere
1,000a 186b 1,000 186 1,000 86.59c 86.59 1,000 186 186
For particles coming from Dione’s icy-surface and Titan’s upper atmosphere the situation is very similar. In both cases the profile of the gain/loss of energy per particle plotted against the number of interactions does not depend on the type of ions. The natural tendency is to increase their energy as the number of interactions does. In fact, in the case of Titan’s particles the amount of gained energy is smaller than that obtained for particles that originated in Saturn’s upper atmosphere. The final velocity distributions show a typical Maxwellian profile. This is due to the fact that even though the particles do not collide with each other, the electric field fluctuations behave as dispersion centers of the energy of the particles and maintain the Maxwellian distribution. As a result, we can estimate the gain of energy associated from such distribution (see Table 4).
Fig. 1 Histogram of frequencies for the distribution of the stochastic electric field which interacts with the particles in the Saturn’s magnetosphere. It is derived from the magnetic field fluctuations obtained as the difference between the measured field and the calculated dipolar field within the Saturn’s magnetosphere. The magnetic field data were averaged every 1.92 seconds during the Voyager 1 Saturn encounter from November 12 1980 at 01:54 UT to November 14 1980 at 21:40 UT
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Fig. 2 Profile of the gain/loss of energy against the number of interactions with the stochastic electric field for several ions coming from the Saturn’s upper atmosphere (T = 1000 K). The particles are located in the Saturn’s equatorial plane at 1.1 Saturn radii. The results are based on the Voyager 1 data
Fig. 3 Profile of the gain/loss of energy against the number of interactions with the stochastic electric field for several ions coming from the Titan’s upper atmosphere (T = 186 K). The particles are located in the Titan’s equatorial plane at 20 Saturn radii. The results are based on the Voyager 1 data
4 Discussion When the particles are initially located in the Saturn’s equatorial plane at 1.1 R S (Fig. 2) we can see that the rate of gain/loss of energy is different for the heavy and light ions. For the particles located in the orbit of Dione (6.25 R S ) and the orbit of Titan (20 R S ) such difference between ions is less important due to the magnitude of the corotation velocSpringer
ity at their distances (∼60 km/s at Dione and ∼200 km/s at Titan). Also the amount of gained energy per particle is greater in Saturn than in Dione and Titan due to the same reason explained earlier. This fact can lead us to evaluate the efficiency of the stochastic mechanism. In this way, our model is more effective near the planet since the initial momentum of the particles is smaller than the impulse exerted by the magnetic fluctuations in the magnetosphere, as a whole
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Table 4 Expected gain of energy (in eV) for the population of some ions in the magnetosphere
a
Saturn He+
Saturn H+
Dione O+
Dione H+
Number of interactions
Equatorial plane (1.1 R S )a
North Pole (1.1 R S )
Equatorial plane (6.25 R S )
Equatorial plane (20 R S )
100 1,000 10,000 50,000 75,000 125,000
1.209 6.673 53.053 248.815 376.718 621.365
8.382 79.553 776.262 3,907.132 5,807.621 9,710.832
0.060 0.198 0.634 1.424 1.748 2.251
0.020 0.100 0.320 0.730 0.900 1.160
0.060 0.199 0.631 1.427 1.744 2.255
Titan C+
Titan N+
0.030 0.100 0.320 0.730 0.910 1.150
1 R S = 60,330 km
Fig. 4 Histogram of frequencies for the z-component of the final velocity reached by a population of 10,000 particles of C+ after 125,000 interactions. The ions come from Titan’s upper atmosphere (T = 186 K) and are initially located at 20 Saturn radii. The bimodal distribution appears as a result of the stochastic acceleration mechanism because the particles with low velocities have been accelerated to acquire higher velocities. The results are based on Voyager 1 data
Fig. 5 Histogram of frequencies for the y-component of the final velocity reached by a population of 10,000 particles of C+ after 125,000 interactions. The ions come from Titan’s upper atmosphere (T = 186 K) and are initially located at 20 Saturn radii. The velocity distribution slightly increases its value (
