Ultra-relativistic acceleration of electrons in planetary magnetospheres

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L405, doi:10.109/007gl036, 007 Ultra-relativistic acceleration of electrons in planetary magnetospheres Danny Summers 1 and Yoshiharu Omura Received 5 October 007; revised 10 November 007; accepted 6 November 007; published 5 December 007. [1] We present a ne particle acceleration mechanism called ultra-relativistic acceleration (URA). URA comprises electron energization due to a special form of nonlinear phase trapping by a coherent histler-mode ave for electrons ith an initial Lorentz factor g 0 satisfying g 0 > W /; is the ave frequency and W is the electron cyclotron frequency at the equator of an assumed dipole magnetic field. Radiation belt electrons that encounter a combination of relativistic turning acceleration (RTA) folloed by multiple URA interactions can undergo significant energy increase. Under ideal conditions, at Earth (L = 4) several-hundred-kev electrons can be energized to several MeV ithin a fe seconds, hile at Jupiter (L = 8), several-hundred-kev electrons can be energized by tens of MeV in a fe tens of seconds. URA can play a prominent role in generating the several-mev electrons observed in Earth s outer zone and the tens-of- MeV electrons observed in Jupiter s magnetosphere. More generally, e expect URA to be an effective electron energization mechanism in cosmic plasma environments that contain a magnetic mirror geometry and electromagnetic histler-mode emissions. Citation: Summers, D., and Y. Omura (007), Ultra-relativistic acceleration of electrons in planetary magnetospheres, Geophys. Res. Lett., 34, L405, doi:10.109/ 007GL Introduction [] A basic problem in the physics of planetary magnetospheres is to explain the formation of electron radiation belts. The main mechanism for generating an electron radiation belt is considered to be radial diffusion [Schulz and Lanzerotti, 1974] from an outer boundary. In this process the first and second adiabatic invariants are conserved and the third is violated. Typical electron energies in the Earth s outer radiation belt are 400 kev 10 MeV [Baker et al., 1986; Kress et al., 007] hile in Jupiter s inner magnetosphere electron energies are in the range MeV [Bolton et al., 004; Garrett et al., 005]. Radial diffusion alone cannot explain the observed relativistic electron fluxes at Earth [e.g., Miyoshi et al., 003; Green and Kivelson, 004] and Jupiter [e.g., depater and Goertz, 1990]. Possible models for MeV electron acceleration include recirculation processes [e.g., Fujimoto and Nishida, 1990] hich combine radial diffusion ith pitch-angle diffusion, though such models remain to be fully evaluated. The generation mechanism of the 0 50 MeV electrons 1 Department of Mathematics and Statistics, Memorial University of Nefoundland, St. John s, Nefoundland, Canada. Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Japan. Copyright 007 by the American Geophysical Union /07/007GL036 associated ith the observed synchrotron emissions at Jupiter is unresolved [e.g., Bolton et al., 004]. Local acceleration due to electron cyclotron resonance ith histler-mode chorus, associated ith violation of the first adiabatic invariant, can explain the peaks in relativistic electron phase space density observed near L = 4 at Earth [Summers et al., 1998, 00, 007]. Electron energization by the chorus acceleration mechanism should also be important in the radiation belts of Jupiter and Saturn here chorus aves have been observed [Coroniti et al., 1980; Kurth, 199; Hospodarsky et al., 007]. [3] We introduce a ne particle acceleration mechanism called ultra-relativistic acceleration (URA). URA consists of electron energization due to a special form of nonlinear phase trapping by a coherent histler-mode ave for electrons ith an initial Lorentz factor g 0 satisfying g 0 > W /, here is the ave frequency and W is the equatorial electron cyclotron frequency in an assumed dipole magnetic field. The URA process is complementary to, and distinct from, the relativistic turning acceleration (RTA) mechanism reported by Omura et al. [007]. RTA has been confirmed in the generation process of chorus emissions [Katoh and Omura, 007]. Electrons energized by RTA can subsequently undergo URA interaction. This is precisely because the turning point in the trajectory of an electron undergoing RTA corresponds to the Lorentz factor g = W /. Moreover, as e demonstrate belo, electrons accelerated by the RTA mechanism can then undergo multiple URA encounters and thereby be accelerated to very high energies. The present study is the first demonstration of nonlinear phase trapping at such high (multi- MeV) energies.. Theory [4] We study the relativistic motion of an electron in the presence of a coherent histler-mode ave ith constant frequency in an ideal dipole planetary magnetic field. The cyclotron resonance condition is kv k ¼ W e ðhþ=g; here W e (h) is the cyclotron frequency at distance h from the magnetic equator, k is the avenumber, g =[1 (v k + v? )/c ] 1/ here v k and v? are electron velocity components parallel and perpendicular to the dipole magnetic field, respectively, and c is the speed of light. From (1) the parallel resonance velocity V R can be ritten V R ¼ k ð1þ 1 W e ; ðþ g L405 1of6

2 L405 SUMMERS AND OMURA: ULTRA-RELATIVISTIC ACCELERATION OF ELECTRONS L405 here, near the equatorial plane, e approximate the dipole magnetic field by W e ¼ W 1 þ ah ; ð3þ ith a = 4.5/(LR p ) here R p is the radius of the planet; the cyclotron frequency W = eb 0 /m 0 here B 0 is the magnitude of the dipole field at the equator. [5] We adopt the cold-plasma dispersion relation for histler-mode aves, c k ¼ 1 þ pe ðw e Þ ; ð4þ here pe is the electron plasma frequency. [6] Equation (4) can be ritten ck ¼ dx here ð5þ x ¼ ðw e ðhþ Þ= pe ð6þ ith d = 1/(1 + x ). [7] Substituting (3) and (5) into (), using ah 1, and neglecting small variations in x, e obtain the approximate result V R ¼ cd 0 x 0 1 W g here x 0 = (W )/ pe and d 0 = 1/(1 + x 0 ). Note that equation (7) implies that V R is nearly constant if g W /. [8] We are interested in the dynamics of an ultrarelativistic electron ith an initial Lorentz factor g 0 such that g 0 > W /. From (7) such an electron has an initial parallel resonance velocity V R0 > 0. The phase trapping of a resonant electron by a coherent histler-mode ave is controlled by the value of the inhomogeneity ratio S, as given by Omura et al. [007; equation 11]. For the case of a constant-frequency ave, S is given by S ¼ 1 t d 0 þ d 0 W e g W e V R kgv? ; here t is the trapping frequency given by t = kv? W, and W = eb /m 0 ; B is the magnitude of the ave magnetic field, e is the electron charge, and m 0 is the electron rest mass. We assume that B is constant. Phase trapping occurs if jsj <1. [9] Using (), e rite (8) as S ¼ 1 V R d 0 kgv R t d W 0 e W : Assuming that jv R jv?, e neglect the first and second terms in the square brackets in equation (9). We hence obtain an approximate expression for S in the form, ð7þ ð8þ ð9þ here e have also used (3) and have applied the condition ah 1. [10] The time variation of the kinetic energy K of a stably-trapped electron is given by equation (13) of Omura et al. [007], namely, dk dt ¼ ee v? S; ð11þ here K = m 0 c (g 1), and E is the magnitude of the ave electric field. Substituting (10) in (11), putting v? = sc (here typically s 1), and using the result E /B = /k obtained from Maxell s induction equation, e find dk dt ¼ x 0 d 0 m 0 c 3 s agh: ð1þ We no assume that entrapping begins at t = 0 here h = h 0, and set or h ¼ h 0 þ V R t ð13þ h ¼ h 0 þ cd 0 x 0 1 W t; ð14þ g using (7). Substituting (14) into (1) e derive the equation, dg dt ¼ ðt þ bþg W t; ð15þ here ep have introduced the dimensionless time-variable t = x 0 cs ffiffiffi p a t and the dimensionless constant b = h0 s ffiffiffi a /d0. Equation (15) is valid from the time of entrapping at t =0,at hich g = g 0 and h = h 0, to the time of detrapping at, say, t = t d at hich g = g d and h = h d. Detrapping occurs ithin the range jsj 1. Hence, from (10), g d and h d satisfy the inequality g d h d d 0 W csa : ð16þ In general, the parameter b is small, namely b 1. For any given finite value of b, the exact solution of equation (15) is given by g ¼ W þ g 0 W exp t þ bt þ b W! p 1= ðb þ tþ exp t þ b b erf p ffiffiffi erf p ffiffi ð17þ S ¼ gv?ah d 0 W ; ð10þ of6

3 L405 SUMMERS AND OMURA: ULTRA-RELATIVISTIC ACCELERATION OF ELECTRONS L405 Figure 1. Trajectories of 3 resonant electrons ith initial energy 1.53 MeV, starting from the equator, interacting ith a coherent histler-mode ave. Of the 3 electrons, are phase-trapped and undergo URA thereby acquiring a significant energy increase. To aid clarity of presentation, trajectories of 4 groups of 8 electrons are plotted in red, magenta, blue, and green in descending order of the final kinetic energies attained. K is electron kinetic energy, v k is the parallel velocity, v? is the perpendicular velocity, h measures position relative to the equator, and t denotes time. (a) The kinetic energy profiles are shon from t = 0tot = 0.15 sec. The dashed curves denote approximate analytical solutions. here erf denotes the error function. In the useful special case that h 0 = 0, then b = 0 and equation (15) has the simple solution, exp t g ¼ W þ g 0 W : ð18þ Solutions (17) p and (18) are valid for 0 t t d here t d = x 0 cs ffiffi a td. By setting h d = V R t d in (16) and using (18), it follos that t d exp t d d p 0W ffiffiffi a c g 0 W 1 : ð19þ The detrapping time t d depends on the phase angle beteen the perpendicular velocity v? and the ave magnetic field B. If the trapped particle satisfies the second-order resonance condition exactly, then t d takes on the maximum value satisfying jsj =1. [11] Solutions (17) and (18) describe the systematic acceleration due to resonant phase trapping by a coherent histler-mode ave of an electron of initial Lorentz factor g 0 (>W /). We refer to this particular acceleration process as ultra-relativistic acceleration (URA). We expect that in a real dipole magnetosphere electrons can undergo multiple URA processes and, as a result, be accelerated to extremely high energies. We evaluate URA (in combination ith RTA) for Earth and Jupiter in the folloing section. 3. Test Particle Simulations [1] The relativistic equations of motion for electrons are solved assuming a dipole magnetic field and a coherent histler-mode ave ith constant amplitude B and constant frequency. We use the numerical scheme described by Omura and Summers [006]. The histlermode ave is assumed to be generated in the magnetic equatorial plane and to propagate aay from the equator in the positive h direction, here h is the distance along the dipole field measured from the equator. We further assume that the plasma density is constant along the magnetic field line, and e assume in all cases in the present simulations that pe /W =. [13] In Figure 1 e illustrate ho electrons can be energized by the URA mechanism in the Earth s magnetosphere at L = 4. For a typical value of the cyclotron frequency e take W /p = 14 khz. We set /W =0.4 and B /B 0 = 10 4 corresponding to the ave frequency 5.6 khz and ave amplitude 100 pt. The electrons are assumed to be located at the equator initially and to have initial Lorentz factor g 0 =[1 (v k0 + v?0 )/c ] 1/ =4, corresponding to the initial kinetic energy K 0 = 1.53 MeV, ith v k0 /c = and v?0 /c = , here v k0 and v?0 3of6

4 L405 SUMMERS AND OMURA: ULTRA-RELATIVISTIC ACCELERATION OF ELECTRONS L405 Figure. (a) The maximum detrapping time t d,max and (b) the maximum energy increase K d,max K 0 during trapping, as a function of initial energy K 0, for the URA process at Earth (L = 4), for the indicated ave amplitudes. are initial velocity components parallel and perpendicular to the dipole magnetic field respectively; the gyrophase angles are assumed to be equally spaced from 0 to p. The necessary condition for URA namely g 0 > W / is satisfied. Of the 3 electrons, are trapped and undergo URA. We sho the variation of the kinetic energy K ith h in Figure 1a from t =0tot = 0.15 sec, and in Figures 1b, 1c, and 1d e sho the respective variations of K, v k /c, and v? /c ith t from t =0tot = 0.1 sec. The acceleration (trapping) times are shorter than those for RTA, typically, and the energy gains are less. Hoever, the range of energy increase due to URA, here up to about 5%, is still substantial compared ith that for untrapped electrons, some of hich lose energy, as illustrated in Figures 1a and 1b. The range of trapping times is sec, the trapping time of a particular electron being determined by its initial gyrophase angle. Whether a given untrapped electron gains or loses energy also depends on its gyrophase angle. The dashed curve in Figure 1b represents the analytical solution (18) here K = m o c (g 1), hile the dashed curve in Figure 1a shos the analytical variation of K ith h, obtained from (18) by setting h = V R t (using (13) ith h 0 = 0). During the URA trapping process, the parallel velocity is nearly constant, as illustrated by the dashed line indicating the resonance velocity in Figure 1c. [14] In Figure e illustrate ho the maximum detrapping time t d,max for the URA process, and the associated energy gain K d,max K 0, depends on the initial energy K 0 for Earth at L = 4. We consider the ave amplitudes B = 50, 100, and 00 pt. For Figure e set h 0 = 0 for each value of g 0 (or K 0 = m 0 c (g 0 1)). To find t d,max at each g 0 e substitute t = t d into (18) to find the corresponding value g d. We then increase t d from 0 incrementally. We set the values of t d and g d satisfying the equality in (19) as the maximum values t d,max and g d,max. For each chosen ave amplitude, e then plot t d,max (or t d,max ) and g d,max g 0 (or K d,max K 0 ) as a function of g 0 (or K 0 ). The results in Figure sho that significant energy increases are possible over the course of a single URA interaction for a ide range of initial energies. We expect that the required detrapping times ill be satisfied over the course of electron interaction ith many ave packets. [15] Electrons may undergo multiple URA interactions, in addition to RTA, at a given L-shell in a planetary magnetosphere. In Figure 3a e illustrate ho electrons can be energized via the RTA and URA mechanisms in the Earth s magnetosphere at L = 4. The ave and background plasma parameters are the same as assumed in Figure 1. We assume electrons have the initial position h 0 W /c = For each electron, e set v k0 /c = , v?0 /c = corresponding to the initial Lorentz factor g 0 = and the initial kinetic energy K 0 = 511 kev. We then trace the trajectories of 3 particles ith gyrophase angles equally spaced from 0 to p. We sho the electron kinetic energy K as a function of position h in Figure 3a. At the loest energy 511 kev, some of the resonant electrons at the offequatorial position undergo RTA. We plot the trajectory of an electron that has been accelerated to the highest energy. After the RTA process, the electron is detrapped from the ave and undergoes adiabatic motion at the constant energy K = 1.14 MeV. The electron moves further in the positive h direction and is reflected at the mirror point. We assume that electrons that are detrapped after URA and RTA change their pitch-angles due to adiabatic motion and scattering through resonant interaction ith other histler-mode aves. Some electrons ill thereby satisfy the resonance conditions for subsequent URA processes. We then assume that the same electrons satisfy the cyclotron resonance condition at the equator and continue to be trapped. In each of the test particle simulations, e follo the trajectories of 3 resonant electrons ith different gyrophase angles starting from the initial energy after the previous detrapping. We plot only the most accelerated electrons. We repeat these processes ten times to reach an energy of MeV. The corresponding total trapping (acceleration) time is 1.7 sec. Electron energization by combined RTA and URA mechanisms can clearly be substantial at Earth. [16] In Figure 3b e illustrate composite RTA and URA interactions for Jupiter. The trajectories of trapped electrons in (h, K) space correspond to a single RTA interaction folloed by a series of URA interactions, just as assumed in Figure 3a. We assume that the Jovian radius R J =11R E, 4of6

5 L405 SUMMERS AND OMURA: ULTRA-RELATIVISTIC ACCELERATION OF ELECTRONS L405 processes over a large trapping time are fulfilled by many resonant interactions of seed electrons ith many ave packets. While e have assumed in Figure 3 that entrapping points for URA take place at the magnetic equator, the entrapping points may occur at any point beteen the equator and the detrapping points of the URA and RTA processes. [17] The efficiency of the URA mechanism increases as the electron energy increases. For a large value of g (1), the resonance velocity approaches the phase velocity of the ave, as is clear from (7). If the group velocity coincides ith the phase velocity, the electrons can then be in resonance ith the ave for a long time even if the length of the ave packet is short. Figure 3. Trajectories of resonant electrons in (h, K) space at (a) Earth (L = 4) and (b) Jupiter (L = 8), here K is kinetic energy and h is position relative to the equator. Electrons are assumed to undergo RTA folloed by a succession of URA interactions. The numbers at the end of acceleration processes are the maximum detrapping time t d,max in seconds. The total energy increase is significant in both cases. here R E is the Earth s radius, and e assume that the magnetic field intensity at the surface of Jupiter is 1.5 times that at Earth. At Jupiter e set L = 8 at hich the cyclotron frequency is about khz compared to 14 khz at L = 4 for Earth. The RTA and URA trajectories in Figure 3b are constructed using test particle simulations ith the ave amplitude B = 100 pt and ave frequency =0.4W.We assume that K 0 = 511 kev, and e find that electrons can be energized to 16 MeV after a single RTA interaction folloed by 10 cycles of URA, in a total acceleration time of 19 sec. As e found in Figure 3a for Earth, electron acceleration by composite RTA and URA processes at Jupiter can be significant. The total acceleration times over hich entrapping can occur should be limited by the duration of the histler-mode ave packets. We assume that the conditions for acceleration by combined URA and RTA 4. Conclusions [18] 1. We have introduced a ne particle acceleration mechanism hich e call ultra-relativistic acceleration (URA). URA consists of electron energization via a special form of nonlinear phase-trapping by a coherent histlermode ave for electrons ith an initial Lorentz factor g 0 satisfying g 0 > W /; is the ave frequency and W is the cyclotron frequency at the magnetic equator in an assumed dipole magnetic field. [19]. We have demonstrated URA by computer simulation, and e have obtained approximate analytical solutions for the electron kinetic energy as a function of time during the course of URA interaction. [0] 3. In a real planetary magnetosphere e expect that electrons ill encounter multiple URA interactions and undergo a corresponding substantial energy increase. Further, e expect that electrons ill encounter a combination of relativistic turning acceleration (RTA) and a succession of URA interactions. For such a combination of processes, e find that at Earth (L = 4) several-hundred-kev electrons can be energized to several MeV ithin a fe seconds, hile at Jupiter (L = 8), several-hundred-kev electrons can be energized by tens of MeV in a fe tens of seconds. [1] 4. URA, alone or in combination ith RTA, can play a significant role in generating the several-mev electrons observed in Earth s inner magnetosphere and the tens-of- MeV electrons in Jupiter s magnetosphere. More generally, e expect that URA and RTA can be effective electron energization mechanisms in the ider context of planetary, space, and cosmic plasmas. Necessary conditions for the URA and RTA mechanisms include a magnetic mirror geometry, an abundant supply of seed electrons and multiple histler-mode ave packets of sufficient duration. [] Acknoledgments. This ork as partially supported by Grantin-Aid , 17GS008 for Creative Scientific Research The Basic Study of Space Weather Prediction of the Ministry of Education, Science, Sports and Culture of Japan, and International Communication Foundation. D. S. acknoledges support from the Natural Sciences and Engineering Research Council of Canada under grant A-061. References Baker, D. N., et al. (1986), Highly relativistic electrons in the Earth s outer magnetosphere: 1. Lifetimes and temporal history , J. Geophys. Res., 91, 465. Bolton, S. J., et al., (004), Jupiter s inner radiation belts, in Jupiter: The Planet, Satellites and Magnetosphere, Cambridge Planet. Sci., vol. 1, edited by F. Bagenal, T. E. Doling, and W. B. McKinnon, p. 671, Cambridge Univ. Press, Ne York. 5of6

6 L405 SUMMERS AND OMURA: ULTRA-RELATIVISTIC ACCELERATION OF ELECTRONS L405 Coroniti, F. V., F. L. Scarf, C. F. Kennel, W. S. Kurth, and D. A. Gurnett (1980), Detection of Jovian histler mode chorus: Implications for the Io torus aurora, Geophys. Res. Lett., 7, 45. depater, I., and C. K. Goertz (1990), Radial diffusion models of energetic electrons and Jupiter s synchrotron radiation: 1. Steady state solution, J. Geophys. Res., 95, 39. Fujimoto, M., and A. Nishida (1990), Monte Carlo simulation of energization of Jovian trapped electrons by recirculation, J. Geophys. Res., 95, Garrett, H. B., S. M. Levin, S. J. Bolton, R. W. Evans, and B. Bhattacharya (005), A revised model of Jupiter s inner electron belts: Updating the Divine radiation model, Geophys. Res. Lett., 3, L04104, doi:10.109/ 004GL Green, J. C., and M. G. Kivelson (004), Relativistic electrons in the outer radiation belt: Differentiating beteen acceleration mechanisms, J. Geophys. Res., 109, A0313, doi:10.109/003ja Hospodarsky, G. B., et al. (007), Chorus observations at Saturn, paper presented at the European Planetary Science Congress, Eur. Planet. Netork, Potsdam, Germany, 0 4 Aug. Katoh, Y., and Y. Omura (007), Relativistic particle acceleration in the process of histler-mode chorus ave generation, Geophys. Res. Lett., 34, L1310, doi:10.109/007gl Kress, B. T., M. K. Hudson, M. D. Looper, J. Albert, J. G. Lyon, and C. C. Goodrich (007), Global MHD test particle simulations of >10 MeV radiation belt electrons during storm sudden commencement, J. Geophys. Res., 11, A0915, doi:10.109/006ja0118. Kurth, W. S. (199), Comparative observations of plasma aves at the outer planets, Adv. Space Res., 1(8), 83. Miyoshi, Y., A. Morioka, H. Misaa, T. Obara, T. Nagai, and Y. Kasahara (003), Rebuilding process of the outer radiation belt during the 3 November 1993 magnetic storm: NOAA and Exos-D observations, J. Geophys. Res., 108(A1), 1004, doi:10.109/001ja Omura, Y., and D. Summers (006), Dynamics of high-energy electrons interacting ith histler mode chorus emissions in the magnetosphere, J. Geophys. Res., 111, A09, doi:10.109/006ja Omura, Y., N. Furuya, and D. Summers (007), Relativistic turning acceleration of resonant electrons by coherent histler mode aves in a dipole magnetic field, J. Geophys. Res., 11, A0636, doi:10.109/ 006JA0143. Schulz, M., and L. Lanzerotti (1974), Particle Diffusion in the Radiation Belts, Springer, Ne York. Summers, D., R. M. Thorne, and F. Xiao (1998), Relativistic theory of ave-particle resonant diffusion ith application to electron acceleration in the magnetosphere, J. Geophys. Res., 103, 0,487. Summers, D., C. Ma, N. P. Meredith, R. B. Horne, R. M. Thorne, D. Heynderickx, and R. R. Anderson (00), Model of the energization of outer-zone electrons by histler-mode chorus during the October 9, 1990 geomagnetic storm, Geophys. Res. Lett., 9(4), 174, doi:10.109/00gl Summers, D., B. Ni, and N. P. Meredith (007), Timescales for radiation belt electron acceleration and loss due to resonant ave-particle interactions:. Evaluation for VLF chorus, ELF hiss, and electromagnetic ion cyclotron aves, J. Geophys. Res., 11, A0407, doi:10.109/ 006JA Y. Omura, Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Kyoto , Japan. (omura@rish.kyoto-u.ac.jp) D. Summers, Department of Mathematics and Statistics, Memorial University of Nefoundland, St. John s, NL, Canada A1C 5S7. (dsummers@math.mun.ca) 6of6