In Situ Heating of the 2007 May 19 CME Ejecta Detected by STEREO/PLASTIC and ACE
In situ measurements of ion charge states can provide unique insight into the heating and evolution of coronal mass ejections when tested against realistic non-equilibrium ionization modeling. In this work we investigate the representation of the CME magnetic field as an expanding spheromak configuration, where the plasma heating is prescribed by the choice of anomalous resistivity and the spheromak dynamics. We chose as a test case, the 19 May 2007 CME observed by STEREO and ACE. The spheromak is an appealing physical model, because the location and degree of heating is fixed by the choice of anomalous resistivity and the spheromak expansion rate which we constrain with observations. This model can provide the heating required between 1.1 and earth orbit to produce charge states observed in the CME flux rope.
However this source of heating in the spheromak alone has difficulty accounting for the rapid heating to Fe at lower heights, as observed in STEREO EUVI due to the rapid radiative cooling that occurs at the high densities involved. Episodes of heating and cooling clearly unrelated to spheromak expansion are observed prior to the eruption, and presumably still play a role during the eruption itself. Spheromak heating is also not capable of reproducing the high Fe charge states (Fe and higher) seen in situ exterior to the flux rope in this CME. Thus while the spheromak configuration may be a valid model for the magnetic topology, other means of energization are still required to provide much of the rapid heating observed.
The eruption mechanism of coronal mass ejections (CMEs) is currently an extremely active area of research. A variety of mechanisms have been discussed in the literature. Most of these require some form of magnetic reconnection during and after the eruption. The associated thermal heating should be an important diagnostic of the eruptive process, including the eruption dynamics and ejecta geometry, but until now has been relatively unexploited. Such heating gives rise to UV-X-ray emissions during the eruption, that can be remotely sensed, and also leaves its imprint in the charge states of ions detected in situ near 1 AU, as their ionization state responds to the newly heated plasma. These techniques give complementary insights into the heating and evolution. Spectroscopy is in some sense more direct, being a prompt signature of the eruption, but with existing instrumentation has only been possible relatively close to the solar surface where the signal is strong. Ion charge states, on the other hand, are routinely collected and CMEs, and their constituent parts (forward shock, flux rope, etc) can be easily identified. Further interpretation of such data require modeling of the non-equilibrium ionization (NEI) balance as the CME plasma expands outward into the ambient solar wind. In this paper we explore the consequences of a different source of thermal heating, that due to anomalous resistivity within a spheromak solution for the CME geometry as it expands into the solar wind. We compare the predictions of such a model to STEREO and ACE observations of the 19 May 2007 CME. The structure of the paper is as follows. We describe the status of our understanding of ion charge states observed in CMEs in more detail in section 2. Section 3 describes the 2007 May 19 CME, and some of our motivation for exploring a spheromak model. Section 4 gives the theory of spheromak resistive expansion, with some discussion of just how closely the spheromak model used here may represent a real CME. Section 5 outlines the NEI model incorporating the spheromak resistive heating, and we conclude in section 6 with a summary of our results in the context of previous research and discuss future modeling improvements.
2 NEI modeling of CME charge states
Interplanetary CMEs or ICMEs have many well-known signatures. Of these, in situ measurements of ion charge state distributions hold a unique potential for diagnosing the conditions throughout the eruption. The charge states evolve through ionizations and recombinations as a function of temperature and density up to heights of 3 to 6 solar radii heliocentric distance but are “frozen in” thereafter. Since low-Z ions “freeze-in” at higher densities (i.e. smaller heights or radii) than high-Z ions, they provide complimentary constraints on the temperature evolution.
A number of studies have examined the existing solar wind composition data and made inferences on freeze-in temperatures based on computed ionization distributions appropriate to coronal equilibrium (e.g. Zurbuchen and Richardson 2006, Zurbuchen 2004, Lepri and Zurbuchen 2004, Lepri et al. 2001, Henke et al. 2001, Gloeckler et al. 1998, Henke et al. 1998). However Rakowski, Laming & Lepri (2007) were the first to conduct time-dependent modeling of the ion charge state distributions of various elements detected in situ in order to draw quantitative conclusions regarding the thermal input and initial conditions in the corona during the CME eruption. Rakowski, Laming & Lepri (2007) modeled the charge states of 8 ICMEs from the event list of Lynch et al. (2003), chosen to cover a range of velocities.
Rakowski, Laming & Lepri (2007) considered a simplified geometry for the CME ejecta and parameterized the evolution. The model velocity and expansion were based on the observed phenomenology from plane-of-the-sky events showing that typically CME height-time evolution consists of three phases: initiation, acceleration and propagation (Zhang et al., 2001, 2004; Zhang & Dere, 2006; Sui & Holman, 2003; Lin et al., 2005; Sui et al., 2005). Heating of the CME core plasma during the acceleration phase was explored assuming a heating rate for the CME plasma proportional to the rate of kinetic energy increase, i.e. a constant fraction during the acceleration up to a final velocity . In 5 out of 8 sample CMEs studied, the dominant Fe charge states were neon-like (16+) or higher, indicating that high temperatures, comparable to flares (K), are involved. Starting the plasma from this temperature and allowing the ions to recombine as they expand could often account for the Fe ionization balance, with peaks around Fe and Fe (the Ne-like and Ar-like charge states, which have small recombination rates to the next charge states down, hence population “bottlenecks” here). However the lower-Z elements placed a limit on the maximum starting temperature (at least if assuming ionization equilibrium in the seed plasma). Above K, O would have been mainly O instead of O and O as observed and would not recombine significantly during the CME evolution. Evidently plasma must start out much cooler, and be further heated as the CME accelerates. Thermal energy inputs on the order of 2 to 10 times the kinetic energy were needed in 7 out of 8 events studied.
Similar conclusions to Rakowski, Laming & Lepri (2007) about the thermal energy input to CMEs have been reached from analysis of ultra-violet spectra taken by SOHO/UVCS. Akmal et al. (2001) studied a 480 km s CME observed on 1999 April 23, and find a thermal energy comparable to the bulk kinetic energy of the plasma. Ciaravella et al. (2001) give similar results for the 260 km s 1997 December 12 CME. More dramatically, Lee et al. (2009) studied the 2001 December 13 event, and using the combination of [O V] density diagnostics and non-equilibrium ionization modeling of O VI found that 75% of the magnetic energy must go into heat to match the UVCS observations. Rakowski, Laming & Lepri (2007) found a heat to kinetic energy ratio of at least 6 for the same event.
The motivations behind the current work are twofold. First, to test if our non-equilibrium ionization modeling of the charge state distributions can hold up to the scrutiny of additional constraints on the eruption dynamics coming from STEREO observations in the 2007 May 19 CME, as well as observations regarding the charge states closer to the sun. Secondly we explore whether a more physically and theoretically specified heating model when incorporated into our non-equilibrium modeling can reproduce the charge states of a real event with reasonable physical parameters.
3 The 2007 May 19 CME
The 2007 May 19 CME studied in this paper is qualitatively different from those CMEs discussed above. As one of the first major CME eruptions in the STEREO era the 19 May 2007 event has been intensely studied by multiple authors (Gopalswamy et al., 2009; Veronig et al., 2008; Li et al., 2008; Liu et al., 2008; Kilpua et al., 2009; Liewer et al., 2009; Bone et al., 2009; Kerdraon et al., 2010). It began as a filament eruption from the active region AR10956 and was detected as an ICME by STEREO B, ACE and possibly STEREO A on 2007 May 22. Multiple heating and cooling episodes were seen in the two days prior to eruption (Liewer et al., 2009; Bone et al., 2009; Li et al., 2008). The total unsigned flux declined 17% in the two days prior to the first eruption (Li et al., 2008). During which time at least 4 heating (and cooling) events happened which heated the filament to 1 MK or higher (Bone et al., 2009). According to Bone et al. (2009), the formation and merger of the filament which eventually erupted represents 6% of the total unsigned flux change during this two day period.
Our model of the heating during the CME eruption and evolution
through interplanetary space will need to match both the charge states
seen in the filament during eruption and detected in situ, as well as the density
and speed of the ejected material.
Liewer et al. (2009) present the STEREO EUV images showing the heating of
the filament in the hours and minutes immediately preceding the
eruption. Hot spots are seen in the EUVI 171 filter (mainly
Fe IX and XII) at heights near 1.07 R.
The magnetic cloud is determined
to have mainly interacted with STEREO B, which penetrated the center
during most of May 22 (DoY 142), while STEREO A only passed through
the magnetic cloud periphery (Kilpua et al., 2009).
The in situ Fe charge states seen with PLASTIC on STEREO B and A are shown
in Figure 1
The shock front around the 2007 May 19 CME launched at 958 km s, the average speed in transit to 1 AU was around 700 km s and the speed of the MC as it passed the STEREO B spacecraft was 482 km s (Kilpua et al., 2009). The filament that erupted from the active region at about the same time only rose at an average speed of 103 km s (Liewer et al., 2009), but it could have been dragged out by the CME and the fast solar wind to reach a similar coasting velocity as the ICME. It had no distinct initiation or acceleration phase, and clearly does not fit within the phenomenology described above and in Rakowski, Laming & Lepri (2007), which leads us to consider a spheromak model for the CME evolution.
4 Resistive Spheromak model of CMEs
4.1 Spheromak Field Configurations
Spheromaks are well known force-free configurations of plasma satisfying condition with spatially constant . They are solutions of the Grad-Shafranov equation in spherical coordinates with the poloidal current being a linear function of the flux (Chandrasekhar & Kendall, 1957). The basic spheromak solution is
where the spherical Bessel function can be expressed in terms of elementary functions, . The parameter is related to the size of the spheromak , defined by the surface where radial magnetic field is zero, given by the solution of ; with .
The spheromak model of CME topology is an attractive alternative to the usual flux rope model. It is self-contained with no unknown “length of flux rope” parameter to maintain connectivity to the sun and the heating within it is mathematically specified once the anomolous resistivity is chosen. The torus structure, seen on edge, may even resemble the filament shape in some CMEs (Kataoka et al., 2009). Previously, Lyutikov & Gourgouliatos (2010) found ideal self-similar solutions for expanding spheromak, with electric fields
(dot denotes differentiation with respect to time) and the corresponding non-radial velocity field
In this Section we discuss continuous heating of an expanding spheromak, generalizing the previous analysis to non-ideal self-similar expansion with .
Physically, the parallel component of the electric field should be related to the current density through Ohm’s law. Formally, the procedure described below breaks down the assumption of self-similarity, since the value of the parallel electric field component is not linearly proportional to the current density. Still, we assume that resistivity plays a subdominant role, so that the expansion remains approximately self-similar. In other words, we assume that resistivity leads to small deviation from the ideal self-similar expansion. The resulting dynamics remain self-similar, by assumption, but are a little different from the ideal case. We are interested not in the detailed properties of local resistive heating, but in general scaling relations. As we show below, the requirement that an expanding CME dissipates some of its initial energy in the form of heat, detectable by its effect on element charge states by the time it reaches Earth orbit, can be used to estimate the anomalous resistivity.
Expressing electric field in radiation gauge , where is vector potential, we find from the force-free condition,
This equations highlights two important points. First, it shows how the time-dependences of the poloidal components of the vector potential are related to the toroidal components, (this is related to the conservation of poloidal and toroidal fluxes). Second, we get one equation for the two functions . Since spheromak solutions are linear, a general solution is a linear combination of the two solutions. The two particular solutions, which in a static case correspond to two equivalent choices of the poloidal components of vector potential, in a time varying case correspond to different physical processes. Accordingly, there are two types of solutions, corresponding to two choices of vector potential in Eq. (4), and . We consider them in turn.
4.2 Resistive expansion of first type
Let us assume that scale with time as an arbitrary function of the expansion parameter, . The first type of solutions has and, from equation 1,
The first electromagnetic invariant . Thus, if the flow is ideal, consistent with the scaling chosen in Lyutikov & Gourgouliatos (2010). More generally, if we allow , we find the parallel electric field
where is a lengthy function not given explicitly here. The case of corresponds to ideal expansion. Since we expect that reconnection eliminates magnetic flux, physically realizable solutions correspond to . Magnetic flux decreases as .
4.3 Resistive expansion of second type
The second type of dissipative expansion corresponds to and
The parallel electric field in this case
For this solution the resulting electric field has a component along magnetic field for any . The resistive effects in these solutions are induced exclusively by spheromak expansion. Note, that this is different from resistive decay of a stationary spheromak, which proceeds homologously. Thus, expansion of the spheromak generally leads to appearances of parallel electric fields and associated dissipation, which can proceed much faster than resistive decay of stationary spheromaks.
4.4 Expansion-driven dissipation
Let us concentrate on the resistive expansion of first type, which produces equatorial dissipation concentrated in the flux rope, as in the CME observations we attempt to model. The dynamical models described above relate the resistivity to the overall dynamics. The volume integral over (see Eq. (6)) is
which leads to the dissipation rate
with the identification of the resistivity . This is justified by integrating the energy dissipation rate above between and to give a total dissipated energy
where the last step assumes . With total spheromak (magnetic) energy , the difference in energy between and is , of which a fraction is dissipated. Writing
and equating with in equation 11 yields . This is derived assuming an infinitesimal expansion. The generalization to arbitrary expansions gives
In the second case the dissipation rate
where , very similar to the first case when integrated over the spheromak volume. The distribution of the heating is different in the two cases, as illustrated in Figure 3. The amount of energy available for heating therefore depends on the value of the anomalous resistivity and varies with position in the spheromak about an average value. The distribution of volumetric heating rates for the first type of dissipative solutions (of most interest here; see below) is given in Figure 4.
To what extent can a spheromak approximate a real CME? Flux rope structures, similar to the equatorial portion of a spheromak are frequently observed in situ, through the rotation of the magnetic field as the spacecraft penetrates the CME plasma. The precise origin of the flux rope is not clear. Many authors assume that such a structure emerges fully formed from the photosphere (e.g., Fan & Gibson, 2004; Magara & Longcope, 2003). Alternatively in the breakout, (e.g. Antiochos et al., 1999; Lynch et al., 2004), tether-cutting (e.g. Moore et al., 2001) or flux cancelation (e.g. Linker et al., 2003) models the flux rope forms as the CME erupts by the reconnection along an arcade of magnetic loops. In such a case, the ends of the flux rope remain attached to the Sun during the eruption. A spheromak however is a self contained magnetic field configuration supported by internal currents, with no external attachment, and presumably if appropriate, must emerge fully formed from beneath the solar photosphere.
In general, the models in which a flux rope forms by reconnection predict an acceleration phase at the onset of flare reconnection. By contrast, the 2007 May 19 CME appears to have moved out from very low down at almost constant speed. There is no obvious feature in its height-time plot that would correspond to the onset of an acceleration phase coupled with reconnection to form a flux rope, suggesting that the observed flux rope must have formed much lower down. Thus its observed trajectory suggests that a spheromak might be a more appropriate description here, than it would have been for other CMEs considered previously in Rakowski, Laming & Lepri (2007). Nakagawa & Matsuoka (2010) find a similar trajectory for the magnetic cloud observed by ACE and Nozomi on 1999 April 16-18. They also conclude that the magnetic cloud is better fitted by a toroidal, as opposed to cylindrical, flux rope, though the model fitted is still approximate. Both flux ropes and spheromaks conserve helicity as they expand, and to do so must decrease their magnetic energy, a part of which can end up dissipated as heat. Kumar & Rust (1996) consider an expanding flux rope, and find temperatures reaching K. We apply similar ideas, derived within the context of a spheromak, to the 2007 May 19 CME. We assume that the expansion velocity of the spheromak entering the expression for the heating is given by the observed motion of the CME.
5 Spheromak non-equilibrium ionization model of the 19 May 2007 CME
As in Rakowski, Laming & Lepri (2007) we modeled the charge states within the CME ejecta for a variety of ions using an adaptation of the BLASPHEMER (BLASt Propagation in Highly EMitting EnviRonment) code (Laming & Grun, 2002, 2003; Laming & Hwang, 2003), which follows the time dependent ionization balance and temperatures of a Lagrangian plasma parcel as it expands in the solar wind. The fundamental equations are outlined below. Further details of the ionization and recombination calculation can be found in Rakowski, Laming & Lepri (2007) and references therein.
The density of ions of element with charge is given by
where are the rates for electron impact ionization, radiative recombination and dielectronic recombination respectively, out of the charge state . These rates are the same as those used in the recent ionization balance calculations of Bryans et al. (2006), with more recent updates given in Rakowski, Laming & Lepri (2007). The electron density is determined from the condition that the plasma be electrically neutral. The ion and electron temperatures, and are coupled by Coulomb collisions by
Here is the atomic mass of the ions of element and charge in the plasma, and is the Coulomb logarithm. The term in represents plasma energy losses due to ionization and radiation. Radiation losses can be taken from Summers & McWhirter (1979). In a CME from a filament eruption the densities and temperatures are such that radiative losses can be important, unlike most applications in the solar wind where they are generally negligible. The term gives the reduction in electron temperature when the electron density increases due to ionization. Recombinations, which reduce the electron density, do not result in an increase in the electron temperature in low density plasmas, since the energy of the recombined electron is radiated away (in either radiative or dielectronic recombination), rather than being shared with the other plasma electrons as would be the case for three-body recombination in dense plasmas. We include the last electron heating term to model the Ohmic dissipation given by equation 10 for the heating in the spheromak.
In addition to ionization and recombination there is also the geometry to consider in the density and temperature evolution. For lack of a better motivated profile we chose a uniform initial density and temperature, exploding from an initial radius of where is the starting height from the center of the sun in solar radii. Thus the underlying density evolution for adiabatic expansion goes as , which goes over to as and the CME expansion more closely follows that of the ambient solar wind.
There are two questions we are trying to answer with our simulations. First, how well can a spheromak solution explain the charge states seen in the 19 May 2007 event? To this end, two sets of models were explored. In model 1 we attempted to match both the filament heating and the ICME charge states using only the heating due to anomalous resistivity in the spheromak. In model 2 we allowed the plasma to already have been heated during the eruption to temperatures that would explain the EUVI observations, but then followed the subsequent heating in the spheromak. Second, given that fast CMEs, particularly at solar maximum, often show Fe charge states as high as Fe in situ, what are the highest Fe charge states obtainable by anomalous resistive heating in a spheromak model? In other words, how general could such a model be?
The baseline model 1 starts the plasma at a temperature of K, a density of cm, and a velocity of 10 km s accelerating to 500 km s. These were chosen as appropriate temperatures and densities for cool filament material, and the evolution is consistent with the final density seen in situ. We initially explored the effect on the final charge states of changing the anomalous resistivity, and thus the amount of magnetic flux dissipation, through the parameter . We assumed heating at the average rate for the spheromak volume. Values of of 0.01 and below leave the charge states essentially unaffected by the spheromak heating. (The charge states still evolve with the adiabatic expansion which changes the temperature and density as the CME erupts). At values between 1.5 and 2 one reaches the limit of diminishing returns whereby increasing does not result in higher charge states, due to the factor in equation 10. This corresponds to an anomalous resistivity between s and s, as much as times the Spitzer value, with a maximum Fe charge state of Fe. We note that increasing the final velocity or the acceleration rate does not substantially change this limit. A more dramatic spheromak expansion starting from a smaller sphere does heat the material faster, but it also reduces the density too quickly such that (a) the final densities are substantially lower than seen in situ and (b) ionization declines such that the highest charge states are not produced. Thus we conclude that the high charge states over Fe in fast CMEs cannot be reproduced purely with the spheromak solution and anomalous resistivity. This leaves the question of how well the spheromak can explain this particular 19 May 2007 CME.
We choose and compute ion charge states averaged over the whole spheromak assuming the distribution of heating rates in Figure 3. Since the 2007 May 22 CME flux rope does not exhibit Fe more ionized than Fe, its highest charge states can be explained with the baseline model 1 and an average . The overall weighted charge states of this model are shown in Figure 5. There are two obvious discrepancies between in situ observations and the Fe charge states modeled here. First, there are too many low charge states from the unheated portion of the spheromak. This is not a major concern as it is possible that that portion of the spheromak is simply not filled with plasma. Second, Carbon charge states seem to indicate that the plasma started hotter than in these simulations.
Additionally one can consider whether the spheromak heating could explain the EUV observations of the filament around launch. Figure 6 shows the early evolution of the charge states corresponding to the average heating rate near the sun in the baseline model 1 with . This started with the plasma at 1.1, above the heights of 1.07 where Liewer et al. (2009) first see Fe IX and Fe XII emission. Starting simulations below this height, with a density that evolves to become the observed density in situ at earth orbit, leads to strong radiative cooling that keeps the electron temperature below K until a radius of 1.1 is reached, whereupon the density is sufficiently reduced to lengthen the radiative cooling time. Hence the heating observed by Liewer et al. (2009), beginning before the actual eruption began, must have had a different origin.
Testing models that started at lower heights, we found that indeed Fe IX and even Fe XII could be produced within a short distance of launch. We did not use these for our in situ modeling because for reasonable starting filament densities the final density due to the rapid expansion was too low. Overall we conclude that while reproducing the EUV observations may be possible within a spheromak context, given the sequence of heating and cooling of the filament leading up the event as described by Bone et al. (2009), it is also plausible, and probably more likely, that whatever mechanism was heating the filament pre-eruption was responsible for the filament heating during the eruption.
The above discussion naturally leads us to our second model to determine if spheromak heating starting from temperatures seen in EUV observations can explain the in situ charge states. In Model 2 we started the plasma at K, cm, and a velocity of 100 km s similar to the velocity of the filament from (Liewer et al., 2009) allowing it to then accelerate to the coasting velocity of 500 km s. Starting from this high temperature, the additional heating of the spheromak has no noticeable effect on the charge states. While the temperature was chosen such that the charge states at low heights match observations, the final Fe charge states fall short of those detected in situ. We also experimented with starting at K but the higher density and/or at a slower velocity. However, radiative recombination dominated bringing the charge states and temperature rapidly back to their K levels. This has implications for the rapidity and amount of energy that had to be dumped into the filaments to produce their high charge states. A thorough examination of this phenomenology is warranted but we leave this to future work. Overall, starting from a higher initial temperature in a uniformly filled spheromak did not solve the problem of matching the highest Fe charge states.
6 Discussion and Conclusions
We have explored modeling this filament eruption as due to the heating provided in an expanding spheromak. Figure 6 shows the evolution of H, C, O, Si and Fe charge states obtained with an anomalous resistivity increased from the classical (i.e. Spitzer) value by a factor around , as the plasma temperature varies between K and K. Anomalous resistivity is thought to arise in conditions where plasma turbulence can scatter current carrying electrons, and can do so more effectively than the Coulomb collisions between electrons and ions. It may be estimated by replacing the isotropization frequency by Coulomb collisions with the relevant wave-particle interaction rate.
Solutions of the one dimensional Vlasov equation designed to model the Buneman instability in a reconnecting current sheet (Wu & Huang, 2009), and therefore model electrons in Langmuir turbulence, give quiescent values of the anomalous resistivity of the Spitzer value, with transient values as high as in a background plasma temperature of K. In our case, dropping the anomalous resistivity to the classical value, the dominant Fe charge state becomes Fe. The ratio of the electron plasma frequency to the collision frequency at in Figure 6 evaluates to , and probably represents the maximum enhancement in resistivity than one may reasonably expect. Lin et al. (2007) estimate a much higher resistivity from the observed width of current sheets trailing CMEs, as high as times the classical resistivity. Bemporad (2008) revisits this and argues that what is observed as a macroscopic current sheet is in fact an assembly of microscopic current sheets, each with width corresponding to anomalous resistivities similar to those given by (Wu & Huang, 2009), although possibly attributed to different modes of turbulence. We therefore conclude that magnetic energy dissipated in the expansion of a spheromak may account for the filament heating between K and K, but is unlikely to be the explanation for the high Fe charge states characteristic of temperatures of order K and higher. The nature of the heating specified in the spheromak solution is such that the heating rate due to anomalous resistivity cannot be increased without limit, even if an anomalous resistivity as high as allowed by the solution could be justified physically.
Higher charge states, beyond Fe must therefore result from other means of heating. Noting that in the 2007 May 19 CME, the highest Fe charge states are observed exterior to the flux ropes, we speculate that reconnection associated with the eruption is most likely. Type II and Type III radio bursts were observed during this eruption (Kerdraon et al., 2010) at times when the CME shock was between 1.2 and 1.4 heliocentric distance (Veronig et al., 2008). Bearing in mind that the plasma we observe and model will be at lower altitude that the CME shock, this corresponds very well to the epochs of heating shown in Figure 6. Recent results suggest that reconnection might be an efficient means of heating electrons (Drake et al., 2006; Oka et al., 2010).
In earlier work (Rakowski, Laming & Lepri, 2007), we expressed the heat input to the CME in terms of its kinetic energy input by relating the heating to the acceleration, tacitly assuming that the mechanism of acceleration inevitably also supplies heat to the CME plasma. In the case of the spheromak, the heating is related to the expansion, but not the acceleration, so the ratio of heat to kinetic energy does not have a natural interpretation. We estimate it, though, to facilitate comparison with Rakowski, Laming & Lepri (2007). From equations 11 and 12,
which close to the Sun evaluates to . This is similar to, but slightly lower than typical values found by Rakowski, Laming & Lepri (2007). The CMEs studied in that work frequently had charge states of Fe up to neon-like, rather higher than in the flux rope studied here which appear to be consistent with our estimate of thermal to kinetic energy in the CME.
- affiliation: Space Science Division, Naval Research Laboratory, Code 7671, Washington DC 20375
- affiliation: Space Science Division, Naval Research Laboratory, Code 7674L, Washington DC 20375
- affiliation: Department of Physics, Purdue University, West Lafayette, IN 47907
- Links to the level 2 data are available at http://fiji.sr.unh.edu/ .
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