Efficiencies of Magnetic Field Amplification and Electron Acceleration in Young Supernova Remnants: Global Averages and Kepler's Supernova Remnant

This method, applicable whenever the process operates, relies on synchrotron self-absorption (SSA): the observation of the frequency at which a source becomes optically thin to synchrotron radiation, as well as the flux at that frequency, allows the determination of two out of the three quantities: source size, magnetic field, and electron energy density. However, the operation of SSA at observable radio frequencies requires conditions in a diffuse source that are fairly restrictive. Using the notation of Pacholczyk (1970), the absorption coefficient for a homogeneous synchrotron source with magnetic field B and electron energy spectrum N(E) = KE^−s electrons cm⁻³ erg⁻¹ can be written as

$$\begin{eqnarray}\begin{array}{rcl}{\kappa }_{\nu } & = & {c}_{6}(s){\left(1.25\times {10}^{19}\right)}^{(s+4)/2}{c}_{9}(s+1)\\ & & \times {{KB}}^{(s+2)/2}{\nu }^{-(s+4)/2}\,{\mathrm{cm}}^{-1},\end{array}\end{eqnarray} \tag{ 1 }$$

where the numerical constants are given in Pacholczyk (1970). For a typical electron energy index s = 2.5 (synchrotron spectral index α ≡ (s − 1)/2 = 0.75, with S_ν ∝ ν^−α ), this is

$$\begin{eqnarray}&&{\kappa }_{\nu }=6.13\times {10}^{21}{{KB}}^{9/4}{\nu }^{-13/4}\ \ {\mathrm{cm}}^{-1}.\end{eqnarray} \tag{ 2 }$$

If s > 2, the energy density in electrons u_e depends only on the lower energy limit to the spectrum E_l . Synchrotron emission basically requires E_l ≳ 10m_e c². An estimate of the required conditions for observable SSA can be made by characterizing both K and B in terms of energy densities. If u_e and u_B are both equal to some nonthermal energy density u_nonth, then a source of line-of-sight extent L will become opaque to SSA below a frequency of about

$$\begin{eqnarray}&&{\nu }_{1}\sim 2\times {10}^{6}{L}^{4/13}{u}_{\mathrm{nonth}}^{17/26}\ \ \mathrm{Hz}\end{eqnarray} \tag{ 3 }$$

(still for s = 2.5), and for that frequency to exceed 100 MHz, the source extent must satisfy

$$\begin{eqnarray}&&L\gtrsim 3\times {10}^{5}{u}_{\mathrm{nonth}}^{-17/8}\ \ \mathrm{cm}.\end{eqnarray} \tag{ 4 }$$

For a source extent less than 1 pc, the nonthermal energy density must exceed about 10⁻⁶ erg cm⁻³, or about 1 MeV cm⁻³. Thus, SSA is an important effect only for very high energy density circumstances, such as supernovae–but not SNRs.

Most radio supernovae show evolving spectra with a peak at some frequency that moves lower with time, attributed to synchrotron emission with some opacity setting in at lower frequencies. Some combination of free–free absorption, either coincident or foreground, and SSA is likely responsible (e.g., Chevalier 1984; Weiler et al. 2002). An extensive study by Chevalier (1998) attributes low-frequency absorption in 8 of 13 radio supernovae to SSA. He reduces the three required source parameters to two by assuming a fixed ratio (not necessarily one) of u_e to u_B and derives source sizes and magnetic fields on the assumption that ${u}_{B}\propto \rho {v}_{s}^{2}$, the postshock pressure. Magnetic field strengths depend fairly weakly on all parameters and are in the range 0.1–0.6 G at the time of the emission peak.

In one case, SN 1993J, the radio observations are sufficiently frequent and the frequency coverage so extensive as to allow a detailed determination of u_e and u_B independently and as a function of time (Fransson & Björnsson 1998), since the very long baseline interferometry observations of a more or less constant expansion velocity (for the first ∼100 days) of 2 × 10⁴ km s⁻¹ (Bartel et al. 1994) allow the radius to be inferred. They find that ${u}_{e}\propto \rho {v}_{s}^{2}$ describes the data well, and much better than u_e ∝ ρ alone (here ρ is the upstream, i.e., circumstellar medium (CSM), density). They determine _e ∼ 5 × 10⁻⁴. For magnetic field, the apparent deceleration beginning around day 100, with R ∝ t^m and m = 0.74, allows a discrimination between a model with ${u}_{B}\propto \rho {v}_{s}^{2}$ and one with B ∝ 1/R, with the latter description providing a better fit to data. Before day 100, they find ${u}_{B}/\rho {v}_{s}^{2}\sim 0.14$, independent of time. The suggestions of nonconstant _B and, worse, of a change in the very dependence of _B on supernova parameters are worrisome hints that the simple picture of constant epsilons is a poor description of the processes of shock acceleration and magnetic field amplification. While we can only observe SNRs evolve through a small fraction of their lifetimes, the results of SN 1993J should put us on notice that results for SNRs may fail to tell a complete picture.

The detailed supernova inferences rely on a simple one-zone emission model (though with spectral sophistication; Fransson & Björnsson 1998 evolve the electron distribution under both Coulomb and synchrotron losses) and on the operation of SSA as an absorption mechanism. They also apply to very high shock velocities and dense CSM. The importance of the processes of magnetic field amplification and particle acceleration is sufficiently great that testing assumptions such as the scaling of u_e and u_B with density and shock velocity, and possible evolution of efficiencies with time, in different regions of parameter space is a high priority. It is fortunate that methods exist to allow this in SNRs, especially to replace SSA for magnetic field determinations.

SNRs are far too diffuse for SSA to be an important mechanism, so inferences of magnetic field strengths rely on different techniques. Reynolds et al. (2012a) review these. Of particular interest is the "thin rims" argument, based on observations of X-ray synchrotron emission from young SNRs (Bamba et al. 2003; Vink & Laming 2003; Parizot et al. 2006), in particular, on the commonly observed morphology of thin tangential rims at the shock front (located, for instance, by Hα observations). This method relies on the assumption that the disappearance of emission a short distance downstream results from synchrotron losses on the emitting electrons as they are advected (or diffuse) downstream. A thorough treatment is given by Ressler et al. (2014), who also include a discussion of an alternative explanation for thin rims, decay of magnetic turbulence downstream (Pohl et al. 2005). Under the simplest assumptions (electron transport by pure advection, ignoring magnetic field damping, the delta-function approximation for the single-electron emitted spectrum), a straightforward relation can be derived between rim thickness and magnetic field strength (Parizot et al. 2006):

$$\begin{eqnarray}&&B\cong 210{\left(\frac{{v}_{s}}{1000\ \mathrm{km}\ {{\rm{s}}}^{-1}}\right)}^{2/3}{\left(\frac{w}{0.01\ \mathrm{pc}}\right)}^{-2/3}\ \mu {\rm{G}},\end{eqnarray} \tag{ 5 }$$

where w is the filament width in the radial direction. (We have assumed a small geometric correction factor ($4\bar{P}/{r}_{\mathrm{comp}}$) to be unity, where r_comp is the compression ratio and $\bar{P}=1$ for a perfect sphere. Since we will always be employing this relation in small regions at the very edge of the remnant, the locally spherical approximation should be quite good.) However, more elaborate treatments produce somewhat different values; Ressler et al. (2014) collect published values for the remnant SN 1006 ranging from 65 to 130 μG, with one report of 14 μG. All agree, however, in requiring amplification of magnetic field beyond simple shock compression. The situation is made more complicated by the presence in a few cases of thin radio rims, produced by electrons with energies far too low to be affected by synchrotron losses, and thus requiring magnetic damping, a process with few theoretical or observational constraints.

Given B, we then have the magnetic field energy density u_B ≡ B²/8π. We note that if B is determined in this way, a prediction results for ${\epsilon }_{B}\propto {B}^{2}/{v}_{s}^{2}\propto {v}_{s}^{-2/3}$, other things being equal. We shall apply Equation (5) in the analysis below.

To fix the efficiencies of shock energy deposition into magnetic field and relativistic electrons, one requires independent local measurements of density, shock velocity, magnetic field strength, and electron energy density. Such measurements have substantial uncertainties for an SNR. X-ray diagnostics of plasma density are fraught with uncertainties, requiring spectral modeling and assumptions about filling factors. Shock velocity determinations from X-ray spectra suffer from uncertainty about prompt electron heating in the shock. These difficulties require alternative methods for determining each quantity.

An excellent diagnostic for the density of a hot plasma is the temperature of grains embedded in it (e.g., Dwek & Arendt 1992), which are collisionally heated by the plasma. We have previously employed this method to obtain postshock densities of SNRs in the Large Magellanic Cloud (Borkowski et al. 2006; Williams et al. 2006) and of Kepler (Williams et al. 2012). The latter study involved analysis of a complete spectral mapping of Kepler with the Spitzer IRS and extraction of spectra between 7.5 and 38 μm at several locations. These spectra were fit with a dust-heating shock model in which postshock grains were heated as they were advected downstream in a constant-temperature, constant-density plasma (Borkowski et al. 2006). This model provided good descriptions of observed IR spectra, allowing extraction of postshock densities at various points in Kepler. These densities are listed in Table 3. We require ρ₀, which we infer by dividing those values by an assumed compression ratio r_comp of 4 and assuming cosmic abundances (mean mass per hydrogen atom μ = 1.4).

For shock velocities, we use proper motions of expansion, as measured from Chandra X-ray data (Katsuda et al. 2008), in 14 regions around the periphery. We then deduce magnetic field strengths using the simple thin-rim analysis of Equation (5), measured from the 741 ks Chandra image (Reynolds et al. 2007), the same image used for the second epoch of proper-motion measurements (Katsuda et al. 2008). Then, the electron energy density can be extracted from intensities measured from a radio synchrotron image (5 GHz VLA; DeLaney et al. 2002) using Equation (13).

Kepler's SNR is an excellent target with which to attempt this determination. The very strong N–S brightness gradient at all wavelengths indicates that the shock wave is expanding into highly asymmetric material; Blair et al. (2007) find that the density to the north is 4–9 times that to the south (see also Williams et al. 2012), and the shock velocities show up to a factor 3 of variation (Katsuda et al. 2008). These wide variations allow us to examine the dependence of efficiencies on several factors. Thin X-ray rims with nonthermal spectra can be seen in several locations around the periphery. Below we outline the quantitative results of our investigation.

The determinations of postshock density were based on observations of Kepler with the Spitzer IRS instruments: both orders of the low-resolution (LL) module (14–38 μm), and order 1 (7.5–14 μm) of the short-wavelength low-resolution (SL) module. The entire remnant was mapped with the LL instrument, and selected regions with the SL module. The observations and analysis are described in Williams et al. (2012). Figure 3 illustrates the sensitivity of model spectra to gas density. Models depend weakly on both ion temperature T_i and electron temperature T_e (see Blair et al. 2007; Williams et al. 2012, for details). Observed electron temperatures in young SNRs are typically a few keV (Reynolds et al. 2007); we assume kT_e = 1.5 keV but set uncertainties by allowing a range between 1 and 2 keV.

Broadly, variations of a factor of 30 were found in density between the faint southern rim and bright north, but values were obtained at many regions around the periphery. (Averaging over larger regions reduces the contrast to the range of 4–9 quoted in Blair et al. 2007). Figure 1 shows regions from which spectra were extracted, models fit, and densities obtained. Those densities are tabulated in Table 3. Statistical uncertainties of the fits are much smaller than the spread of values possible by changing the assumed electron temperature between 1 and 2 keV; that spread is shown in Figure 4.

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While we believe that the emission from each region is well characterized by the densities listed in Table 3, in a few regions the IR emission is so faint that it is not clear whether the densities listed there describe the immediate postshock density. Figure 1 shows that for Regions 4, 5, 8, and 12, detectable IR is only at the innermost edge. Both Figures 1 and 2 show the location of the blast wave as indicated by radio and nonthermal X-rays; the absence of immediate IR emission there suggests density variations along the line of sight, with IR appearing only once the density is somewhat larger (and shocks somewhat slower). For those regions, we have chosen to regard the densities of Figure 3 as upper limits. Higher spatial resolution IR observations, possible with JWST, will be required to improve our knowledge of the immediate postshock density in all locations.

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Proper motions of expansion were measured by Katsuda et al. (2008) comparing Chandra images obtained in 2000 and 2006, using images between 1.0 and 8.0 keV. We have converted them, and their uncertainties, into velocities for our nominal distance of 5 kpc. (Regions shown in Figure 1 correspond to whole Spitzer IRS pixels. They overlap, but are not identical with, the corresponding regions used by Katsuda et al. 2008.) Figure 4 shows both densities and shock velocities of various regions. As expected, these quantities anticorrelate. Densities vary fairly smoothly around the periphery of Kepler, with lowest values in the south and highest in the north, well correlated with the brightness in radio or X-rays, while the shock velocity (with larger errors) also varies fairly smoothly in the opposite sense. Postshock pressures ${P}_{2}\equiv \rho {v}_{s}^{2}$ (again, eliding the factor 2/(γ − 1)) are shown in Figure 4, where we have assumed ρ₀ = 1.4n_H m_H /r_comp, appropriate for neutral gas upstream. We take r_comp = 4.

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The pressure varies by about a factor of a few around the periphery of Kepler, not unexpected given the very strong gradient in external density, with highest pressure to the north, where the external density is much larger. This magnitude of pressure variation is comparable to the radial pressure gradient in the interior of a Sedov blast wave, where the dynamical timescale is comparable to the age.

For Regions 4, 5, 8, and 12, we regard our density determinations as upper limits, hence upper limits on the pressure. The variations in pressure we find, however, are relatively small (Figure 4), suggesting that the true densities in those regions are unlikely to be far below those in closely neighboring regions.

Regions shown in Figure 2 were selected for the presence of a "thin rim" of nonthermal X-rays, from which a magnetic field strength could be extracted, making the simplest assumptions about thin-rim physics as embodied in Equation (5). Radial profiles (averaged azimuthally over the region's azimuthal dimensions) were extracted; two examples are shown in Figure 5. Widths were measured at the intensity level halfway between the rim peak and the interior minimum. Uncertainties in this process were estimated to be 02, although, as we shall see below, the exact value turns out not to be critical. At 5 kpc, 10⁻² pc = 041, so for a radial width w of a rim in arcsec,

$$\begin{eqnarray}&&B=116{({v}_{\mathrm{sh}}/{10}^{8}\ \mathrm{cm}\ {{\rm{s}}}^{-1})}^{2/3}{(w^{\prime\prime} )}^{-2/3}\mu {\rm{G}}.\end{eqnarray} \tag{ 14 }$$

Note that since our shock velocities are obtained from angular proper motions, the ratio v_sh/w is independent of distance; we have also assumed 5 kpc to obtain the values for v_sh in Table 3. Values of magnetic field inferred from Equation (14) are listed in Table 4. Not all the rim locations of Table 3 showed distinct rims, so we restrict our analysis to the seven listed in Table 4. Uncertainties are difficult to gauge but, given the large spread shown in Table 1, seem to be of order a factor of 2. The relatively narrow range of measured filament widths results in a (noisy) trend of inferred B-values with increasing shock velocity, as expected from Equation (14).

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Table 4. Derived Properties of Regions

Region	B	ue	uB	ue /uB	P	e	B	ue /uBav a
	(μG)	(10−10 erg cm−3)	(10−10 erg cm−3)		(10−7 dyn cm−2)	(10−4)	(10−4)
3	105	53	4.36	12.2	7.9	68	56	7.1
4	157	0.74	9.84	0.076	< 11	> 0.69	> 9.1	0.20
5	161	2.44	10.3	0.24	< 12	> 2.0	> 8.3	0.68
7	121	15.9	5.8	2.75	3.1	52	19	2.7
8	148	2.54	8.7	0.29	< 2.1	> 12	> 42	0.61
10	80	55.6	2.5	22	2.6	220	9.9	4.7
13	60	189	1.4	133	9.0	210	1.6	9.7

Note.

^a Assuming the median value of B_av = 121 μG and u_Bav = 5.83 × 10⁻¹⁰ erg cm⁻¹ for all regions.

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We determine mean intensities over each small region assuming that it is homogeneous and that the line-of-sight depth L is equal to the transverse extent of the filament being sampled (so generally larger than the region's radial width). Flux densities in each region are measured from the 5 GHz radio image (DeLaney et al. 2002), then corrected for background taken from comparable or larger regions outside the remnant, divided by the solid angle ΔΩ, and extrapolated to 1 GHz assuming j_ν ∝ ν^−0.71 everywhere. The results are listed as 〈I₉〉 in Table 3, and the derived electron energy densities are given in Table 4. Quoted uncertainties result from the off-source rms fluctuation level of 0.14 mJy (DeLaney et al. 2002), scaled by the square root of the extraction area.

The magnitude of the variations among regions shown in Table 4 means that uncertainties of even factors of several cannot change the qualitative result of strong variations of efficiency. However, it is worth considering various possible sources of uncertainty.

7.1.1. Distance

7.1.2. Density Determinations

7.1.3. Shock Compression Ratio

7.1.4. Magnetic Fields

7.1.5. Radio Properties

The radio spectral index of Kepler between 1.4 and 5 GHz is observed to vary with location between about 0.65 and 0.8 over most of the remnant, including all of the regions we have measured (DeLaney et al. 2002). This corresponds to a range in electron energy index s of 2.3–2.6. In terms of α = (s − 1)/2, the spectral index dependencies of the electron energy density u_e are

$$\begin{eqnarray}&&{u}_{e}\propto {E}_{l}^{1-2\alpha }{(2\alpha -1)}^{-1}{c}_{j}(\alpha ){B}^{-(1+\alpha )}{\nu }^{\alpha }.\end{eqnarray} \tag{ 16 }$$

Evaluating this expression for the median magnetic field of 122 μG, E_l = 8.2 × 10⁻⁷ erg, and ν = 1 GHz, for α = 0.65 and 0.8, gives

$$\begin{eqnarray}&&\displaystyle \frac{{u}_{e}(\alpha =0.8)}{{u}_{e}(\alpha =0.65)}=2.4.\end{eqnarray} \tag{ 17 }$$

Very little of the scatter in the values of u_e at different locations in Kepler can be due to variations of α.

It is also unlikely that huge variations in the minimum energy of the electron distribution occur at different locations. Our fiducial value of 10m_e c² simply reflects the energy at which electrons are relativistic enough for the synchrotron formulae on which our analysis rests to apply.

The radio flux measurements for the different regions carry uncertainties, as listed in Table 3. Those were obtained from the off-source rms values of 4.8 GHz radio flux in the image of DeLaney et al. (2002), scaled by the square root of the extraction area. The signal-to-noise ratio for the regions ranges from 3 for region 3 to 31 for region 13. These cannot explain the orders-of-magnitude spread evident in Table 4 and exhibited in Figure 6.

Finally, we obtain emissivities by assuming a line-of-sight depth of our regions, which we take to be the longer dimension of our extraction region. Again, this can easily be in error by a factor of a few, but not by orders of magnitude.

Unfortunately, we cannot calculate formal uncertainties in u_e or u_B , or their ratio. While we report uncertainties in density and shock velocity, these cannot be propagated to functions of those quantities without knowledge of their distributions, quite unlikely to be Gaussian. We know even less about the uncertainties in magnetic field. The variations among authors illustrated in Table 1 are not statistical or systematic errors but results of applications of slightly different versions of the basic argument embodied in Equation (5). A better estimate is represented by the spread shown in Table 1 for the reference whose values we have used, Parizot et al. (2006)—of order 10%. But again, we do not know enough about the distribution of the uncertainties to be able to apply a simple error propagation formalism. Figures 6 and 7 illustrate what different levels of uncertainty would look like in the distributions; the values we have chosen represent our estimates of reasonable ranges based on the discussion above.

We are forced to conclude that the various sources of uncertainty are dwarfed by the enormous spread of values of u_e , u_B , _e , and _B at different locations on Kepler's periphery. The primary source of the spread in u_e /u_B results from the radio brightness variations (Equation (13)). We conjecture that the missing physics required to explain the spread has to do with acceleration of particles, electrons in particular, which evidently has a more complex dependence on parameters than we have included. No monotonic relation between u_e and shock velocity is apparent in the values in Tables 3 and 4; another likely possibility, the shock obliquity angle θ_Bn between the shock velocity and upstream magnetic field, may have a highly nonlinear effect on the ultimate population of ∼ GeV electrons producing the radio emission.

What has become of equipartition, the time-honored principle still often used to infer properties of synchrotron sources? First, in the SNR context, SNRs are very inefficient at producing either magnetic field energy or relativistic-particle energy. Compared, say, to extragalactic radio sources, relatively little of the total supernova energy ever winds up in nonthermal forms (that is, both epsilons are always small). It is easier to imagine wide variations in u_e /u_B when a much larger pool of thermal energy is available for any of various processes to produce electrons or magnetic field. In addition, absence of direct information on relativistic protons means that we simply have no idea what the total nonthermal energy density is (though very broad inferences from observed Galactic cosmic rays seem to require about 10% of total supernova energy winding up in relativistic baryons). So electron energy densities are a small fraction of a fairly small fraction, and we should not be astonished if there is no clear relation between the energy in relativistic electrons and other pools of energy. As for magnetic energy, when SNR magnetic fields are inferred using equipartition arguments, they essentially serve as proxies for the SNR mean surface brightness Σ, as can be seen from, for instance, Pacholczyk (1970), Equation (7.14), where B_equip ∝ (Σ/D)^2/7, D being the source diameter. There is no obvious mechanism that could operate to transfer energy to or from the magnetic field based on the energy in relativistic electrons. Figure 7 shows that the density-independent ratio u_e /u_B varies by orders of magnitude in regions of comparable shock velocity, emphasizing this point.

A related consequence of our results for Kepler is the problematic nature of the traditional Σ–D relation as a diagnostic tool. The large azimuthal variations we find suggest that global averages can be very misleading. While the average values of _e and _B we obtain from the global properties for Kepler (Table 2) do fall within the large range of values at particular locations, it is not clear that those global values represent any kind of mean of the physical properties. Great care should be taken in drawing conclusions based on such global values.

One might hope to extract from this exercise some clue as to what additional parameters might be required to explain the range of efficiencies. The absence of smooth trends of efficiencies with position angle around the remnant disfavors explanations such as a smoothly varying shock obliquity angle θ_Bn, as one might expect for a blast wave encountering a uniform upstream magnetic field. The large scatter for adjoining regions shown in Figure 6 requires relatively small-scale variations in properties important for particle acceleration and/or magnetic field amplification. Our attempt to extract information relatively independent of magnetic field inferences is shown in the right panel of Figure 7, where a noisy trend is visible as lower electron energy density with higher shock velocity. We do not claim the unambiguous existence of such a relation since the apparent trend is strongly dependent on one or two points, but the data might offer a clue toward identifying the additional physics evidently necessary to fully understand electron acceleration in strong shock waves.