The Astrophysical Journal, 633:L53–L56, 2005 November 1 䉷 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.
FINITE-TIME SHOCK ACCELERATION OF ENERGETIC STORM PARTICLES Chanruangrit Channok,1,2,3 David Ruffolo,2 Mihir I. Desai,4,5 and Glenn M. Mason4,6,7 Received 2005 June 23; accepted 2005 September 16; published 2005 October 12
ABSTRACT Energetic storm particles (ESPs) of various ion species have been shown to arise from suprathermal seed ions accelerated by traveling interplanetary (IP) shocks. The observed spectral rollovers at ∼0.1–10 MeV nucleon⫺1 can be attributed to the finite time available for shock acceleration. Using the locally measured shock strength parameters as inputs, the finite-time shock acceleration model can successfully fit the energy spectra of carbon, oxygen, and iron ions measured by the Ultra Low Energy Isotope Spectrometer (ULEIS) on board the Advanced Composition Explorer (ACE) during three ESP events. The inferred scattering mean free path in the acceleration region ranges from a typical IP value for the weakest ESP event down to 3.0 # 10⫺3 AU for the strongest event. This is consistent with the idea that proton-amplified waves result from the very intense particle fluxes in major events. Subject headings: acceleration of particles — interplanetary medium — shock waves — Sun: coronal mass ejections (CMEs) ambient spectrum as an input to the FTSA model. We then fit spectra for carbon, oxygen, and iron ions for three ESP events as measured by the ULEIS instrument on board the ACE spacecraft. We are able to infer a key parameter of shock acceleration, the scattering mean free path in the acceleration region, as well as its rigidity dependence, with evidence in favor of enhanced acceleration by proton-amplified waves for very intense ESP events.
1. INTRODUCTION
The solar system is pervaded by energetic particles, including Galactic cosmic rays and particles from various sources within and around the solar system. All of these particles, with the exception of those from impulsive solar flares, are believed to result from the Fermi acceleration mechanism (Fermi 1954), later developed into the theory of diffusive shock acceleration (e.g., Krymskii 1977; Axford et al. 1978; Bell 1978; Drury 1983). The basic theory predicts a power-law spectrum N( p) ∝ p⫺g, such as that observed for Galactic cosmic rays. However, in many cases the observed spectrum of particles from sources within the solar system rolls over at a certain energy, Ec, i.e., the power law does not persist to high energies. A spectrum that rolls over is often fit to a form suggested by Ellison & Ramaty (1985): j(E) ∝ p⫺g exp (⫺E/Ec ). The present work is motivated by observed spectra of particles of enhanced intensity in association with the passage of IP shocks, known as energetic storm particles (ESPs; Bryant et al. 1962). These typically exhibit spectral rollovers at 0.1–10 MeV nucleon⫺1 (Gosling et al. 1981; van Nes et al. 1985; Desai et al. 2004). We consider that the physical origin of such rollovers is the finite time available for shock acceleration, discussed by Klecker et al. (1981) and Lee (1983), and also evident in Monte Carlo simulations of particles injected at low energy (e.g., Decker & Vlahos 1986; Kang & Jones 1995; Giacalone 2005). The present work derives a simple model of finite-time shock acceleration (FTSA) and explores implications for the composition dependence of the spectrum. Given that recent composition measurements argue for a seed population at substantially higher energies than the solar wind (Desai et al. 2003), we use the
2. MODELING
Our approach is based on the probabilistic model of Bell (1978). The key quantities are the number of acceleration events, n, the rate of acceleration events, r, the rate of escape e, and the duration of the shock acceleration process, t. Our model aims to input observed seed spectra and use realistic formulae for r and e as a function of energy at an oblique shock. Since the energetic ions mostly travel along magnetic field lines, we consider a fixed set of field lines (a flux tube) and the evolution of N( p, t), the momentum distribution of ions in the acceleration region within that flux tube as the shock propagates outward. The particle speed, momentum, and energy refer to the plasma rest frame (solar wind frame), and we can use the resulting spectrum to fit observations in the spacecraft frame because the Compton-Getting transformation has a negligible effect on particle intensity at the energies of interest. We define an acceleration event as one cycle of diffusion back and forth across the shock. On average, after n acceleration events we have
(
pn p pn⫺1 1 ⫹
1
Department of Physics, Chulalongkorn University, Bangkok 10330, Thailand;
[email protected]. 2 Department of Physics, Mahidol University, Rama VI Road, Bangkok 10400, Thailand;
[email protected]. 3 Department of Physics, Ubonrajathanee University, Ubonrajathanee 34190, Thailand. 4 Department of Physics, University of Maryland, College Park, MD 20740. 5 Now at Southwest Research Institute, San Antonio, TX 78238; mdesai@ swri.edu. 6 Institute for Physical Science and Technology, University of Maryland, College Park, MD 20740. 7 Now at Applied Physics Laboratory, Johns Hopkins University, Laurel, MD 20723;
[email protected].
4 u1 ⫺ u 2 , 3 vn⫺1 cos v1
)
(1)
where vn and pn are the velocity and momentum after n times, u is the fluid speed along the shock normal, v is the magnetic field-shock angle, and subscripts 1 and 2 are for upstream and downstream, respectively (Drury 1983). The typical time for an acceleration event is Dtn p L53
4
vn
{k secu v ⫹ [1 ⫺ 冑1 ⫺ (B /B )] k secu v } , 1
1
2
1
1
2
2
2
(2)
L54
CHANNOK ET AL.
where k is the spatial diffusion coefficient along the shock normal and B is the magnetic field magnitude. The factor in square brackets is the fraction of upstream particles that is transmitted downstream. We then set rn p 1/Dtn. We use formulae from Jokipii (1987) in the limit that l p l 1, the upstream parallel mean free path, is much greater than the gyroradius: k1 p
vl 3
cos 2 v1 ,
k2 p
vl cos 3 v2
.
3 cos v1
(3)
Note that Jokipii (1987) assumes that l is proportional to the gyroradius, so l 1 /l 2 p B2 /B1. Finally, the escape rate e is slightly modified from Bell (1978) and Drury (1983) to account for the convective flux upstream: en downstream flux p rn ⫹ e n upstream flux p
en p
u2 , (1 ⫹ u1 /vn cos v1 ) 2 vn cos v1 /4
[
vn cos v1 4u 2
(
1⫹
u1 vn cos v1
⫺1
) ] 2
⫺1
rn .
(4)
The processes of inflow of the seed population, acceleration, and escape can be expressed in a Fokker-Planck equation as follows: ⭸N( p, t) ⭸ p I( p, t) ⫺ [R( p, t)N( p, t)] ⫺ e( p, t)N( p, t), ⭸t ⭸p (5) where N( p, t) is the momentum distribution of ESPs, I( p, t) is that of inflowing ions, and R( p, t) is the acceleration rate, equivalent to rn ( pn⫹1 ⫺ pn ). In practice, we discretize this as an initial-value system of ordinary differential equations: dNn (t) p In ⫺ (rn ⫹ e n )Nn (t) ⫹ rn⫺1 Nn⫺1 (t). dt
(6)
If there were no shock, we would have N p Nseed, the seed spectrum, and the inflow would balance escape from the region of interest with I p eNseed. Therefore, we use this inflow condition and set the initial condition to N p Nseed. The parameters rn, e n, and l represent average values experienced by the measured ESPs. We consider l, the upstream scattering mean free path along the magnetic field in the shock acceleration region, to be l p l0
(
P 1 MV
a
).
(7)
For the rigidity, P p pc/Qe, we use the mean ionization state Q as in Desai et al. (2004). We use the fourth-order RungeKutta method to solve the system of equation (6) and convert the result to a spectrum in energy per nucleon (E/A). Let us consider theoretical expectations for the composition (mass-to-charge ratio) dependence of the rollover energy. Observed seed spectra typically exhibit a power-law form at low energy (Desai et al. 2004), so seed particles are most abundant at the lowest momentum treated in our model, p 0, correspond-
Vol. 633
ing to a suprathermal velocity of v 0 p 200 km s⫺1 in the wind frame. The rollover momentum pc and rigidity Pc p pc/Qe can be estimated from d pc /dt p R( pc ) along with equations (1), (2), (3), and (7). For nonrelativistic ions, we expect a rollover in energy per nucleon at Ec Q2 e2 A p 2 P0a⫹1 ⫹ (a ⫹ 1)C t A A 2m 0 c 2 Q
[
2/(a⫹1)
]
,
(8)
where A and Q are the ion mass and charge numbers, respectively, m 0 is an atomic mass unit, e is an elementary charge, P0 is the minimum rigidity, and C is a constant independent of P, A, and Q. For a rollover well above the injection threshold, 2a/(a⫹1)
()
Ec Q ∝ A A
t 2/(a⫹1).
(9)
Above the rollover, the shock-accelerated spectrum is similar to the seed spectrum with a shift to higher energy. Rollovers due to FTSA in our simulations using observed seed spectra are consistent with equation (8), so that the rollover moves to higher energy with increasing t. Since the acceleration rate is inversely proportional to l 0, a higher value implies a lower rollover energy. The exponent a in equation (7) affects the shape of the spectrum, the rollover, and their composition dependence. We set the initial particle velocity in the wind frame, v 0 p 200 km s⫺1, to be somewhat higher than the value of u 2, according to evidence for a suprathermal seed population (Desai et al. 2003). The rollover energy is not sensitive to the choice of v 0. For a fixed duration t, when increasing the shock angles v1 and v2 according to the shock jump conditions (Ruffolo 1999), the rollover moves to higher energy, a well-established result in shock acceleration theory (Decker & Vlahos 1986). 3. FITS TO OBSERVED SPECTRA
We apply this model of FTSA to examine observed spectra for three ESP events typifying different compositional characteristics. These events correspond to events 13, 18, and 37 of Desai et al. (2003), respectively, and we consider the ambient and shock-accelerated spectra obtained by Desai et al. (2004). Some physical characteristics of these events are listed in the first eight columns of Table 1. Column (3) is the peak proton intensity at 1 MeV, I p, estimated as the geometric mean of the P6 (0.58–1.06 MeV) and P7 (1.06–1.90 MeV) LEMS30 ion fluxes from the Electron, Proton, and Alpha Monitor on ACE.8 Columns (4)–(8) are based on values from Desai et al. (2003). Also listed in the table is the duration t, which was obtained from the time of coronal mass ejection (CME) observation near the Sun and the shock arrival time at Earth from Cane & Richardson (2003). For each event, we vary the parameters l 0 and a to simultaneously fit the spectra for C, O, and Fe by minimizing x 2 p 冘 (D log N) 2, where D log N is the difference in log N(E) between the observations and our model results, summing over species and energy ranges. The derived values of l 0 and a are also listed in the table. During the fitting procedure, a best-fit absolute normalization of order unity was applied for each element and each set of parameters, for reasons as follows. The ambient spectrum was measured at 1 AU prior to the ESP event and is therefore only a proxy for the actual spectrum that was fed into the shock. This proxy seed spectrum has properties that correlate significantly with the shock-accelerated population, 8
See http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_EPAM.html.
No. 1, 2005
FINITE-TIME SHOCK ACCELERATION
L55
TABLE 1 Shock Acceleration Parameters for Three IP Shock Events Number (1)
Date of Shock Passage (2)
Ipa (3)
v1 (deg) (4)
v2 (deg) (5)
u1 (km s⫺1) (6)
u2 (km s⫺1) (7)
B2 /B1 (8)
t (hr) (9)
l0 (AU) (10)
a (11)
1 ....... 2 ....... 3 .......
1999 Jun 26 1999 Sep 22 2000 Oct 05
2.59 # 103 1.22 # 103 7.84 # 100
50 64 66
73 79 80
131 131 188
56.4 54.6 78.3
2.2 2.3 2.3
54.8 54.3 55.1
0.0030 0.039 0.100
0.05 0.08 0.08
a
Peak ion intensity (dominated by protons) at 1 MeV in units of cm⫺2 s⫺1 sr⫺1 MeV⫺1; see text for details.
apparently because the spectra in IP space have a correlation time of order a few days (Desai et al. 2004). However, because the actual seed spectrum is not measured, there is no information that would allow us to normalize absolute intensities. Thus, our goal is to fit the spectral shape. On the whole, the model is successful in explaining the differences between the seed spectra and ESP spectra (Fig. 1). First consider events 2 and 3, which were weak ESP events in the sense that the shock-accelerated intensity was not much greater than the ambient seed intensity. The observed shockaccelerated spectrum is steeper than the ambient spectrum, corresponding to the FTSA rollover (see also Decker 1983), and is similar to the ambient spectrum at higher energies. For the fixed duration t, the low rollover energy corresponds to a relatively long scattering mean free path, l (Table 1). All three events show only a weak dependence of l on energy. For event 3, l is consistent with typical values of l k p 0.08–0.3 AU for scattering along the magnetic field in the bulk IP medium (Palmer 1982), as derived from observed solar energetic particle time profiles. For event 2, l is somewhat lower. In contrast, event 1 was a major shock event that involved an intensity increase by roughly 2 orders of magnitude over the ambient seed spectrum. Our model spectra yield a hard power law at low energies, followed by a rollover and finally approaching the seed spectrum at high energies. Observed
shock-accelerated spectra have more gradual rollovers than our model spectra because the ions have actually experienced a variety of shock angles and plasma conditions, not the single values assumed in the present model. The inferred scattering mean free path l in the shock acceleration region is very low (l 0 p 0.003 AU). Figure 1 shows the Fe/O ratio as a function of energy per nucleon for all three events. The model can reproduce some aspects of the observed ratios for shock-accelerated particles, for example, that the Fe/O ratio is lower than that of the seed spectrum in each case. The model can explain the general trends of observed shock-accelerated Fe/O as a function of energy for events 1 and 3, but not for event 2. According to the model, the energy dependence of the Fe/O ratio in the weaker events 2 and 3 basically derives from the underlying seed spectrum, whereas that in event 1 is strongly affected by finite-time shock acceleration. The model Fe/O ratio for event 1 declines sharply at ∼0.5 MeV nucleon⫺1 because of the composition dependence of the rollover energy (eq. [9]), with the rollover for oxygen at a higher energy. Again, our FTSA model predicts a sharper decline than that seen in the data. Nevertheless, it does explain the deviation of the observed shock-accelerated Fe/O from the ratio found in the seed spectrum. At high energies, the model Fe/O ratio is nearly proportional to that of the seed spectrum for all three events.
Fig. 1.—Observed shock-accelerated spectra, observed seed spectra, and model shock-accelerated spectra for C, O, and Fe ions, in units of ions cm⫺2 s⫺1 sr⫺1 (MeV/nucleon)⫺1, and Fe/O ratio in three IP shock events.
L56
CHANNOK ET AL. 4. DISCUSSION
We have developed a model of finite-time shock acceleration (FTSA) and explore implications for the composition dependence of the spectral rollover energy Ec. Using the observed ambient spectrum as the input seed spectrum and using the measured shock geometry, the FTSA model provides a good fit to spectra of C, O, and Fe ions observed by the ACE ULEIS for three ESP events. The inferred values of the local scattering mean free path l in the acceleration region are important for qualitative understanding of shock acceleration in the IP medium. The concept that a major CME-driven shock generates proton-amplified waves has been developed through a series of papers (e.g., Ng & Reames 1994; Ng et al. 1999, 2003; Vainio 2003) and yields an impressive explanation of unusual elemental ratios as a function of time on 1998 April 20 (Tylka et al. 1999) with l ∼ 10⫺4 to 10⫺3 AU. Thus, we might expect to find l-values ranging from such low values, for strong events, to typical IP conditions with l k ∼ 0.08–0.3 AU (Palmer 1982) for weaker events. Our fits to ion spectra of event 1 (1999 June 26) indeed
Vol. 633
yield a much lower l for this major ESP event, with l p 0.003 AU at 1 MV. For event 3, l is similar to typical IP conditions. Vainio (2003) suggests that a key parameter is the peak proton intensity at 1 MeV, I p. As seen in Table 1, our results for event 3 are consistent with that work’s conclusion that there is no substantial wave growth when I p ! 10 cm⫺2 s⫺1 sr⫺1 MeV⫺1. We confirm that proton-amplified waves are apparently significant for ion acceleration in major ESP events but not for weaker ESP events. We are grateful to the members of the Space Physics Group, University of Maryland, and the Johns Hopkins University Applied Physics Laboratory for the construction of the ULEIS instrument. C. C. is grateful to Thailand’s Commission for Higher Education for supporting this work, including support for visiting the University of Maryland. Work in Thailand was also supported by the Rachadapisek Sompoj Fund of Chulalongkorn University and the Thailand Research Fund. Work at the University of Maryland was supported by NASA contract NAS5-30927 and NASA grant PC 251428.
REFERENCES Axford, I., Leer, E., & Skadron, G. 1978, Proc. 15th Int. Cosmic Ray Conf. (Budapest), 11, 132 Bell, A. R. 1978, MNRAS, 182, 147 Bryant, D. A., Cline, T. L., Desai, U. D., & McDonald, F. B. 1962, J. Geophys. Res., 67, 4983 Cane, H. V., & Richardson, I. G. 2003, J. Geophys. Res., 108, 1156 Decker, R. B. 1983, J. Geophys. Res., 88, 9959 Decker, R. B., & Vlahos, L. 1986, ApJ, 306, 710 Desai, M. I., Mason, G. M., Dwyer, J. R., Mazur, J. E., Gold, R. E., Krimigis, S. M., Smith, C. W., & Skoug, R. M. 2003, ApJ, 588, 1149 Desai, M. I., et al. 2004, ApJ, 611, 1156 Drury, L. O’C. 1983, Rep. Prog. Phys., 46, 973 Ellison, D. C., & Ramaty, R. 1985, ApJ, 298, 400 Fermi, E. 1954, ApJ, 119, 1 Giacalone, J. 2005, ApJ, 624, 765 Gosling, J. T., Asbridge, J. R., Bame, S. J., Feldman, W. C., Zwickl, R. D., Paschmann, G., Sckopke, N., & Hynds, R. J. 1981, J. Geophys. Res., 86, 547
Jokipii, J. R. 1987, ApJ, 313, 842 Kang, H., & Jones, T. W. 1995, ApJ, 447, 944 Klecker, B., Scholer, M., Hovestadt, D., Gloeckler, G., & Ipavich, F. M. 1981, ApJ, 251, 393 Krymskii, G. F. 1977, Soviet Phys. Dokl., 22, 327 Lee, M. A. 1983, J. Geophys. Res., 88, 6109 Ng, C. K., & Reames, D. V. 1994, ApJ, 424, 1032 Ng, C. K., Reames, D. V., & Tylka, A. J. 1999, Geophys. Res. Lett., 26, 2145 ———. 2003, ApJ, 591, 461 Palmer, I. D. 1982, Rev. Geophys. Space Phys., 20, 335 Ruffolo, D. 1999, ApJ, 515, 787 Tylka, A. J., Reames, D. V., & Ng, C. K. 1999, Geophys. Res. Lett., 26, 2141 Vainio, R. 2003, A&A, 406, 735 van Nes, P., Roelof, E. C., Reinhard, R., Sanderson, T. R., & Wenzel, K. P. 1985, J. Geophys. Res., 90, 3981
