DETECTING THE RAPIDLY EXPANDING OUTER SHELL OF THE CRAB NEBULA: WHERE TO LOOK

We will determine the gas density by combining the total mass with the density law and the inner and outer radii. The outer radius is unknown, but it must be specified to determine the gas density. Given our assumptions about the radius–velocity law, the outer radius will correspond to a particular highest expansion velocity. Chevalier (1977) gives a range of expansion velocities between 5000 km s⁻¹ and 10,000 km s⁻¹, Lundqvist et al. (1986) give a maximum expansion velocity of 5000 km s⁻¹, Sankrit & Hester (1997) assume a maximum velocity of 10,000 km s⁻¹, and Sollerman et al. (2000) quote 6370 km s⁻¹. We assume that the velocity at the outer radius is v_out = 6370 km s⁻¹, a velocity ∼3.8 times larger than the expansion of the observed nebula, and give results relative to this velocity. The v(r)∝r Hubble flow results in

$$\begin{equation} R_{{\rm out}} \approx 3.8\, R_{{\rm in}} = 1.9 \times 10^{19} \;({{\rm cm}}), \end{equation} \tag{ 1 }$$

while for the v(r)∝r² expansion law, the outer radius is

$$\begin{equation} R_{{\rm out}} \approx 1.9\, R_{{\rm in}} = 9.5 \times 10^{18} \;({{\rm cm}}). \end{equation} \tag{ 2 }$$

For the spectroscopic simulations we need to set the outer shell density n₀ at its inner edge R_in, the density law n(r)∝r^α, and the outer radius R_out. We investigate two density laws here, α = −3 and α = −4. The density law is determined by two quantities: how the expansion velocity varies with radius, v(r)∝r^γ, and how the mass flux varies with radius, MF∝r^β. We consider three cases, summarized in Table 1, as follows.

• 1.  
  The simplest case is a Hubble-law expansion, the sudden release of mass with a range of velocities so that γ = 1 and v∝r. For the mass flux, the simplest assumption is that the initial density distribution is constant, so that

  $$\begin{equation} {\rm MF} = 4\pi r^2 n(r)v(r) \propto 4\pi r^{2 + \alpha + \gamma } = 4\pi r^\beta \end{equation} \tag{ 3 }$$

  is constant. Since γ = 1, if α = −3, then β = 0, indicating mass flux conservation.
• 2.  
  As the second case, we still assume that the Hubble velocity law is maintained so that γ = 1 and v∝r. If α = −4, then β = −1, meaning that the mass flux decreases with increasing radius. This might happen if the outer layer of the star had a lower density.
• 3.  
  As a third case we also consider α = −4. Since we know that the expansion is accelerating (Trimble 1968), we will also investigate, as a sensitivity test, an arbitrarily different velocity law expansion with γ = 2 and v(r)∝r². In this case we also obtain β = 0, that is, the mass flux is conserved.

The density law for case I is

$$\begin{equation} n(r) = n_0 \left(\frac{r}{{R_{{\rm in}} }}\right)^\alpha = n_0 \left({\frac{r}{{R_{{\rm in}}}}} \right)^{ - 3}\, ({\rm cm}^{ - 3}) \end{equation} \tag{ 4 }$$

and for cases II and III is

$$\begin{equation} n(r) = n_0 \left({\frac{r}{{R_{{\rm in}} }}} \right)^{-4}\; ({\rm cm}^{-3}). \end{equation} \tag{ 5 }$$

We can calculate n₀ by mass conservation,

$$\begin{equation} M_{{\rm halo}} = 4\pi \int \nolimits_{R_{{\rm in}} }^{R_{{\rm out}} } mn_0 \left({\frac{r}{{R_{{\rm in}} }}} \right)^\alpha r^2 dr\;({{\rm gm}}). \end{equation} \tag{ 6 }$$

Here M_halo is the total mass of the outer shell and m is the mass per hydrogen for the assumed composition. Note that the composition of a supernova remnant is usually different in different parts; therefore we assume three different compositions for the outer shell: the abundances of some of the Crab filaments (Pequignot & Dennefeld 1983), solar abundances (recommended by Sollerman et al. 2000), and interstellar medium (ISM) abundances (which are basically solar abundances with grains). If μ is the mass of the proton then m = 3.8μ for the enhanced Crab abundances derived by Pequignot & Dennefeld (1983) and m = 1.4μ for solar and ISM abundances. A list of assumed abundances is given in Table 3(a) in Pequignot & Dennefeld (1983). We obtain the following expression for n₀ with the middle value m = 2.6μ and M_halo = 4M_☉

$$\begin{equation} \left. \begin{array}{c@{}} n_0 = \frac{{M_{{\rm halo}} }}{{4\pi mR_{{\rm in}}^3 \ln \frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}}} \\\ms\ms = 0.87\frac{{2.6\mu }}{m}\frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}\frac{{\ln 3.8}}{{\ln \frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}}}\quad ({\rm cm}^{ - 3}) \\ \end{array} \right\}\;{\rm case\; I;} \end{equation} \tag{ 7 }$$

$$\begin{equation} \left. \begin{array}{c@{}} n_0 = \frac{{M_{{\rm halo}} }}{{4\pi mR_{{\rm in}}^3 \left({1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)}} \\\ms\ms = 1.58\frac{{2.6\mu }}{m}\frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}\frac{{0.74}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}\quad ({\rm cm}^{-3}) \\ \end{array} \right\}\;{\rm case\; II;} \end{equation} \tag{ 8 }$$

$$\begin{equation} \left. \begin{array}{c@{}} n_0 = \frac{{M_{{\rm halo}} }}{{4\pi mR_{{\rm in}}^3 \left({1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)}} \\\ms\ms = 2.46\frac{{2.6\mu }}{m}\frac{{M_{{\rm halo}} }}{{4\,M_{{\odot}} }}\frac{{0.47}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}\quad ({\rm cm}^{-3}) \\ \end{array} \right\}\;{\rm case\; III}{\rm.} \end{equation} \tag{ 9 }$$

We see that the density depends on both the inner radius and the outer radius for the α = −3 law. This is important because the density determines the emission measure of the lines, and this depends on the uncertain outer radius. For the case of α = −4, the density depends only on the inner radius if the outer radius is much larger than the inner radius. Table 5 in Sollerman et al. (2000) also gave the densities in the inner edge for different density laws.

Before proceeding with the model we derived the kinetic energy for each of these hypotheses. The kinetic energy of the filaments is about 3 × 10⁴⁹ erg (Hester 2008), which is far less than the canonical 10⁵¹ erg seen in the ejecta of core collapse supernovae (Davidson & Fesen 1985). We calculate the kinetic energy in the outer shell to check if this makes up the missing energy. We obtain the kinetic energy of the outer shell using

$$\begin{equation} E_k = 6.86 \times 10^{50} \frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}\frac{{{\rm ln}3.8}}{{{\rm ln}\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}}}\frac{{\left({\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}} \right)^2 - 1}}{{13.44}}\;{\rm (erg),\; case\; I;} \end{equation} \tag{ 10 }$$

$$\begin{equation} E_k = 4.26 \times 10^{50} \frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}\frac{{0.74}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}\frac{{\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }} - 1}}{{2.8}}\;{\rm (erg),\; case\; II;} \end{equation} \tag{ 11 }$$

$$\begin{equation} E_k = 6.70 \times 10^{50} \frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}\frac{{0.47}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}\frac{{\left({\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}} \right)^3 - 1}}{{6.38}}\;{\rm (erg),\; case\; III}{\rm.} \end{equation} \tag{ 12 }$$

These provide about half of the missing energy, which is within the uncertainty in our assumed shell parameters. Table 5 in Sollerman et al. (2000) also gave the kinetic energies of the outer shell for different density laws.

The next step is to predict the full emission and absorption line spectra of the outer shell using photoionization models.

We obtain the luminosities of emission lines from the numerical calculations presented below. We use H i line emissivities given by Osterbrock & Ferland (2006, hereafter AGN3) and Ferland (1980). The luminosity of Hβ is

$$\begin{equation} L({{\rm H}\beta }) = \int \frac{{4\pi j_{{\rm H}\beta } }}{{n_e n_p }}n(r)^2 dV \end{equation} \tag{ 13 }$$

$$\begin{equation} \approx \frac{{4\pi j_{{\rm H}\beta } }}{{n_e n_p }} \times {\rm EM}\;({{\rm erg}\,{\rm s}^{ - 1} }), \end{equation} \tag{ 14 }$$

where 4πj_Hβ/n_en_p is the H i Case B recombination coefficient (AGN3). EM is the volume emission measure, defined as

$$\begin{equation} {\rm EM} = \int n(r)^2 dV \end{equation} \tag{ 15 }$$

$$\begin{equation} \approx \int \left[ {n_0 \left({\frac{r}{{R_{{\rm in}}}}} \right)^\alpha } \right]^2 dV\; ({\rm cm}^{-3}) \end{equation} \tag{ 16 }$$

corresponding to

$$\begin{equation} \left. \begin{array}{c@{}} {\rm EM} = \frac{4}{3}\pi n_0^2 R_{{\rm in}}^3 \left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 } \right] \\\ms\ms = 3.89 \times 10^{56} \left({\frac{{2.6\mu }}{m}} \right)^2 \left({\frac{{M_{{\rm halo}} }}{{4\,M_{{\odot}} }}} \right)^2\\ \ms\ms \quad\times\left({\frac{{ln3.8}}{{ln\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}}}} \right)^2 \frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 } \right]}}{{0.98}}\quad ({\rm cm}^{-3}) \\ \end{array} \right\}\;{\rm case\; I;} \end{equation} \tag{ 17 }$$

$$\begin{equation} \left. \begin{array}{c@{}} {\rm EM} = \frac{4}{5}\pi n_0^2 R_{{\rm in}}^3 \left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right] \\\ms\ms = 7.86 \times 10^{56} \left({\frac{{2.6\mu }}{m}} \right)^2 \left({\frac{{M_{{\rm halo}} }}{{4\,M_{{\odot}} }}} \right)^2\\ \ms\ms \quad\times\left({\frac{{0.74}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}} \right)^2 \frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right]}}{{0.99}}\quad ({\rm cm}^{-3}) \\ \end{array} \right\}\;{\rm case\; II;} \end{equation} \tag{ 18 }$$

$$\begin{equation} \left. \begin{array}{c@{}} {\rm EM} = \frac{4}{5}\pi n_0^2 R_{{\rm in}}^3 \left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right] \\\ms\ms = 1.82 \times 10^{57} \left({\frac{{2.6\mu }}{m}} \right)^2 \left({\frac{{M_{{\rm halo}} }}{{4\,M_{{\odot}} }}} \right)^2\\ \ms\ms \quad\times\left({\frac{{0.47}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}} \right)^2 \frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right]}}{{0.96}}\quad ({\rm cm}^{ - 3}) \\ \end{array} \right\}\;{\rm case\; III}{\rm.} \end{equation} \tag{ 19 }$$

Therefore we find the final expressions of the luminosity for Hβ

$$\begin{eqnarray} L({{\rm H}\beta }) &=& 1.43 \times 10^{31} \left({\frac{T}{{2.9 \times 10^4 }}} \right)^{ - 1.20} \left({\frac{{2.6\mu }}{m}} \right)^2 \left({\frac{{M_{{\rm halo}} }}{{4\,M_{{\odot}} }}} \right)^2 \nonumber\\ &&\times \left({\frac{{{\rm ln}3.8}}{{{\rm ln}\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}}}} \right)^2 \frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 } \right]}}{{0.98}}\;({\rm erg\; s}^{ - 1}),\;{\rm case\; I;} \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} L({{\rm H}\beta }) &=& 4.55 \times 10^{31} \left({\frac{T}{{2.3 \times 10^4 }}} \right)^{-0.833}\nonumber\\ &&\times\left({\frac{{2.6\, \mu }}{m}} \right)^2 \left({\frac{{M_{{\rm halo}} }}{{4\,M_{{\odot}} }}} \right)^2\left({\frac{{0.74}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}} \right)^2 \nonumber\\ &&\times\frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right]}}{{0.99}}\;({\rm erg\; s}^{ - 1}),\;{\rm case\; II;} \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} L({{\rm H}\beta }) &=& 1.20 \times 10^{32} \left({\frac{T}{{2 \times 10^4 }}} \right)^{-0.833}\nonumber\\ &&\times\left({\frac{{2.6\,\mu }}{m}} \right)^2 \left({\frac{{M_{\rm halo} }}{{4\,M_{\odot} }}} \right)^2\left({\frac{{0.47}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}} \right)^2 \nonumber\\ &&\times\frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right]}}{{0.96}}\;({\rm erg\; s}^{ - 1}),\;{\rm case\; III;} \end{eqnarray} \tag{ 22 }$$

where we suppose the temperature to be in the neighborhood of 2.9 × 10⁴ K for case I, 2.3 × 10⁴ K for case II, and 2 × 10⁴ K for case III as computed below, and use the temperature power-law fit to (4πj_Hβ)/(n_en_p) given by Ferland (1980). This is approximate due to the assumption of Case B H i emission. We show below that the Lyman lines are not optically thick and that continuum fluorescent excitation is important.

We can convert emission-line luminosities into surface brightness by dividing the luminosity by the area of emission on the sky. We assume that the lines form over a scale height determined by an effective radius, R_eff. The effective radius is defined as the position where half of the total line luminosity is formed. Emission line luminosities are determined by the emission measure, n²V (AGN3), so the inner highest-density regions are most important. We obtain the effective or "half luminosity" radius from

$$\begin{equation} \int \nolimits_{R_{{\rm in}} }^{R_{{\rm eff}} } \frac{{4\pi j}_{{\rm H}\beta}}{{n_e n_p }}n(r)^2 dV = L/2\;({{\rm erg\; s}^{ - 1} }). \end{equation} \tag{ 23 }$$

If we move (4πj_Hβ)/(n_en_p) out of the integral, equivalent to assuming that the temperature is constant, we find

$$\begin{equation} \left. \begin{array}{c@{}} R_{{\rm eff}} = R_{{\rm in}} \left({\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 }}} \right)^{{1}/{3}} \\\ms\ms = 6.26 \times 10^{18} \frac{{\left({\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 }}} \right)^{{1}/{3}} }}{{1.25}}\;({\rm cm}) \\\ms\ms = 1.25R_{{\rm in}} \\ \end{array} \right\}\;{\rm cases\; I\; and\; II;}\; \end{equation} \tag{ 24 }$$

$$\begin{equation} \left. \begin{array}{c@{}} R_{{\rm eff}} = R_{{\rm in}} \left({\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 }}} \right)^{{1}/{5}} \\\ms\ms = 5.67 \times 10^{18} \frac{{\left({\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 }}} \right)^{{1}/{5}} }}{{1.13}}({\rm cm}) = 1.15R_{{\rm in}} \\ \end{array} \right\}\;{\rm case\; III}. \end{equation} \tag{ 25 }$$

We convert the luminosities given above into surface brightness averaged over the full outer shell as it would be seen projected on the sky, in order to compare the results with observations. We obtain the surface brightness

$$\begin{equation} S({\rm H}\beta) = \frac{1}{{k^2 }}\frac{L}{{4\pi ^2 R_{eff}^2 }}\; ({{\rm erg\; s}^{-1}\; {\rm cm}^{-2}\; {\rm arcsec}^{-2}}) \end{equation} \tag{ 26 }$$

corresponding to

$$\begin{eqnarray} S({{\rm H}\beta }) &=& 2.76 \times 10^{- 19} \left({\frac{T}{{2.9 \times 10^4 }}} \right)^{ - 1.20} \!\left({\frac{{2.6\mu }}{m}} \right)^2\!\left({\frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}} \right)^2\nonumber\\ &&\times \left({\frac{{{\rm ln}3.8}}{{{\rm ln}\frac{{R_{{\rm out}} }}{{R_{{\rm in}} }}}}} \right)^2\frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 } \right]}}{{0.98}} \frac{{1.25}}{{\left[ {\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^3 }}} \right]^{{1}/{3}}}}\nonumber\\ &&\times({\rm erg\; s}^{-1} \;{\rm cm}^{-2}\; {\rm arcsec}^{-2}),\;{\rm case\; I;} \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} S({{\rm H}\beta }) &=& 9.42 \times 10^{ - 19} \left({\frac{T}{{2.3 \times 10^4 }}} \right)^{ - 0.833} \left({\frac{{2.6\mu }}{m}} \right)^2\nonumber\\ &&\times\left({\frac{{M_{\rm halo} }}{{4M_{\odot} }}} \right)^2 \left({\frac{{0.74}}{{1 - \frac{{R_{\rm in} }}{{R_{\rm out} }}}}} \right)^2\frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right]}}{{0.99}}\nonumber\\ &&\times\frac{{1.15}}{{\left[ {\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 }}} \right]^{{1}/{5}} }}\;({\rm erg\; s}^{-1} \;{\rm cm}^{-2}\; {\rm arcsec}^{-2}),\;{\rm case\; II;}\nonumber\\ \end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray} S({{\rm H}\beta }) &=& 2.17 \times 10^{ - 18} \left({\frac{T}{{2 \times 10^4 }}} \right)^{ - 0.833} \left({\frac{{2.6\mu }}{m}} \right)^2 \left({\frac{{M_{{\rm halo}} }}{{4M_{{\odot}} }}} \right)^2\nonumber\\ &&\times\left({\frac{{0.47}}{{1 - \frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}}}} \right)^2\frac{{\left[ {1 - \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 } \right]}}{{0.96}}\frac{{1.15}}{{\left[ {\frac{2}{{1 + \left({\frac{{R_{{\rm in}} }}{{R_{{\rm out}} }}} \right)^5 }}} \right]^{{1}/{5}} }}\nonumber\\ && ({\rm erg\; s}^{-1}\; {\rm cm}^{-2}\; {\rm arcsec}^{-2}),\;{\rm case\; III;} \end{eqnarray} \tag{ 29 }$$

where k = 206,265 converts luminosity into surface brightness. Table 5 in Sollerman et al. (2000) and Tziamtzis et al. (2009) also gave the surface brightness of the outer shell for different density laws.

The upper limit to the Hβ surface brightness corresponds to an upper limit to the mass in the shell for a given power-law index. The composition also affects S(Hβ) because, for Crab abundances, the heavy elements contribute to the total mass. This means that the hydrogen density and S(Hβ) are lower for the same mass but higher Z. The Hβ surface brightness is highest for models with solar abundances, where more of the 4 M_☉ is H so the density is higher. The coefficients in Equations (27)–(29) were evaluated for abundances intermediate between solar and Crab. The maximum expansion velocity also affects the surface brightness because it sets the outer radius that appears in the equations. A shell with a larger expansion velocity is more spread out and has lower density and lower S(Hβ).

With these assumptions the physical conditions in the outer shell, the ionization, and the temperature can be computed. The observations described below suggest that the upper limit to Hβ is about S(Hβ) < 4 × 10⁻¹⁸erg cm⁻² s⁻¹ arcsec⁻². If we apply different values of m, indicating different abundances, into Equations (27)–(29), we find that cases I and II have average surface brightness that are less than this observed limit for all abundances. Case III has a surface brightness that is under this observed limit for Crab abundances but above this observed limit for solar and ISM abundances. We consider all these models in the following to examine their predictions.

We use version 13 of the plasma simulation code Cloudy (Ferland et al. 2013) to predict the observed spectrum. We compute the luminosities of many emission lines and convert them to surface brightness by dividing the luminosities by the size derived above. We obtain different emission lines and surface brightness for the three different models. Figure 1 shows predicted spectra integrated over the full outer shell for case I with solar abundances. The upper panel shows the full range 0.1–100 μm. The lower panel shows the range 1–30 μm in greater detail. We focus on the UV–IR spectral region because this would be easiest to study with today's instrumentation.

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Tables 2–10 give the average surface brightnesses S for IR, optical, and UV emission lines, as defined by Equations (27)–(29), for models with different abundances. These can be compared to the best observational upper limit achieved to date for the outer shell, S < 1.2 × 10⁻¹⁷ erg cm⁻² s⁻¹ arcsec⁻² using long-slit spectra to search for Hα (Fesen et al. 1997, but with their value adjusted upward by a factor of 3.4 to correct for the observed extinction; Tziamtzis et al. 2009). All lines at or brighter than that limit are italicized in Tables 2–10.

Hα is brighter than the observational limit in case III with solar or ISM abundances, so at face value these models appear to be ruled out, at least for an outer shell containing the full amount of the missing mass. However, scattered light from the much brighter parts of the Crab is a major issue, as has been discussed by Tziamtzis et al. (2009). The Fesen et al. (1997) upper limit really corresponds to radii beyond about 03 from the bright edge of the main nebula (R_in), because their spectrum inside that radius is likely to be dominated by an unknown amount of scattered light. Figure 2 shows an example of the emissivity in three emission lines as a function of the depth into the shell from its inner edge at R_in for case I with solar abundances. Figure 3 shows the results of integrating these emissivities along the line of sight through the outer shell to find the predicted surface brightness as a function of R_proj/R_in for cases I, II, and III. We used solar abundances and assumed a distance of 2 kpc and a spherical shell. Note that Figure 3 is shown with linear scales for both radius and surface brightness, to make the wide range in surface brightness more obvious, and that each panel has been separately scaled in surface brightness. Each panel also shows the Fesen et al. (1997) Hα upper limit as a horizontal line beginning at a point 03 beyond R_in. The lack of an Hα detection does nothing to rule out cases I or II, nor does it firmly rule out case III. Further ground-based observations might be able to push these optical-passband limits slightly fainter, but observations in Hα or other lines that are also emitted by the main part of the nebula will require great attention to the scattered light issue.

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What is needed are unique spectroscopic tracers in the form of high-ionization lines not emitted by the filaments or (hopefully) by the thin [O iii]-emitting skin (Sankrit & Hester 1997) that surrounds the outer edge of the synchrotron bubble. Our models predict strong emission lines from high-ionization species of C, N, O, Ne, Si, Mg, and Fe, principally in the UV and IR parts of the spectrum. A surface brightness limit similar to that for Hα might be reachable in a few UV lines, notably C iv λλ1548, 1551, in about 3 hr of on-target exposure (plus an equal sky exposure) using Hubble Space Telescope/Advanced Camera for Surveys (HST/ACS) imaging with very heavy on-detector binning. But the most promising lines are in the mid-IR, particularly [Ne vi] 7.652 μm, which is also shown on Figure 3. Although these IR lines are somewhat fainter than the UV lines, they could be targeted with either SOFIA or (eventually) JWST mid-IR imagers and spectrographs. Archival Spitzer images and long-slit spectra also exist, and might be worth co-adding to search for these lines. Firm statements could be made about cases II or III if an IR measurement as deep as the Hα limit could be obtained.

An alternative to searching areas off to the side of the main part of the Crab would be to obtain spectra averaging over a fairly large area at the center of the Crab, where the projected expansion velocities are toward and away from us, and searching for these mid-IR lines with positive and negative velocity shifts corresponding to the shell structure. Lundqvist & Tziamtzis (2012) used this method in the optical passband to search for [O iii] and Ca ii lines. The [Ne vi] 7.652 μm line falls within the spectral range covered by the Spitzer/IRS, but Temim et al. (2012) do not report any strong feature at this wavelength in their IRS spectra of the Crab. The predicted spectral signature for such emission lines would be two broad peaks displaced symmetrically around the Crab's heliocentric systemic velocity of about 0 km s⁻¹ and separated by about 4000 km s⁻¹. We are in the process of carrying out the very careful reanalysis of the Spitzer spectra needed to search for faint features of this type. However, the low velocity resolution (4700 km s⁻¹) may prevent a clear distinction between any emission from an outer shell and emission from the ionized outer skin of the main part of the Crab (see Lundqvist & Tziamtzis 2012, their Figure 9).

Cloudy predicts the intensity of H i lines including line optical depths effect, collisional excitation and de-excitation, and continuum fluorescent excitation. The predicted H i intensities can be compared with Case B (pure recombination in which Lyman lines are optically thick) and Case A (Lyman lines are optically thin and there is no continuum fluorescent excitation).

Table 11 compares H i luminosities for the solar abundance of the case I Crab shell. It gives the computed luminosities with all processes included, along with the luminosities obtained from the computed density and temperature and assuming Cases A and B emission (Storey & Hummer 1995). The predicted lines are about 10%–140% brighter than Case B, an indication that continuum fluorescent excitation is important. The Lyman lines in the outer shell are not optically thick so continuum pumping is important, causing them to be brighter than would be found with pure recombination. The optical depth in Lyβ, for instance, is about 1, so neither Case A nor Case B formally apply. The predicted deviations are not large and Case B is, as is often the case, a fair approximation to the actual emission.

Figure 4 shows the gas kinetic temperature across the outer shell. It increases as the depth increases for all three models. This is because the Crab radiation field, which powers the outer shell, decreases at r⁻², because of the inverse square law. The gas density falls off faster, as r⁻³ or r⁻⁴. As a result the ionization parameter, the ratio of photon to hydrogen densities (AGN3), increases as r increases. Higher ionization parameter gas tends to be hotter.

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We compute optical depths for different assumptions about the expansion velocities. Tables 12 to 14 give the optical depths for models with Crab abundances, solar abundances, and ISM abundances, respectively. We continue to focus on the models with solar abundances. From Tables 12 to 14 we see that the optical depths for the C iv λλ1549 doublet are not much greater than 1. The lines mainly form over a small radius due to the density decline, so the wind acceleration should not be large over the line forming region. We make two assumptions to estimate the optical depth. First we assume a static shell, in which the lines are only thermally broadened. This would apply if there was no acceleration across the layer where the lines form. In this case there is sufficient opacity to produce the observed lines. In particular, the C iv λλ1549 doublet has an optical depth of 2.68, consistent with the Sollerman et al. (2000) tentative detection. We note that the optical depth of the O vi λλ1034 doublet is much larger than 1, which indicates strong absorption at that wavelength.

If the lines have a significant component of turbulence or if the expansion velocity changes across the line-forming region then the lines will be spread over a wider velocity range. Here the lines are optically thin.

Table 15 gives the computed line optical depths for both static and dynamic cases. We find the optical depth to be very small if we add a turbulence with velocity v = 1680 km s⁻¹ as the expansion velocity of the inner radius of the outer shell. The O vi λλ1034 doublet then becomes optically thin as well.

The truth will lie between these two limiting assumptions. We will consider dynamic models, in which the velocity is determined self consistently, in future papers.

3.4.1. Recombination Timescale

The recombination timescale is defined as (AGN3)

$$\begin{equation} t_{{\rm rec}} = \frac{1}{{n_e \alpha _{\rm B} ({T_e })}} \end{equation} \tag{ 30 }$$

where n_e is the electron density and α_B(T_e) is the Case B recombination coefficient at temperature T_e. The gas in the outer shell is photoionized by light from the visible Crab. Since Cloudy supposes that the gas atomic processes that are responsible for thermal and ionization equilibrium have reached steady state, we need to compare the age of the Crab with the recombination time to see if this is valid. We compute the recombination timescale for Ne⁺⁶ → Ne⁺⁵ for all three cases with solar abundances. We focus on this ion since it produces the strongest IR line. Since the temperature increases very slowly but the electron density decreases very quickly, we assume different radii have roughly the same recombination coefficient and evaluate it from the Badnell (2006), Badnell et al. (2003), and Badnell Web site (http://amdpp.phys.strath.ac.uk/tamoc/DR/). We find the recombination times are about 100 yr, 20 yr, and 10 yr in the inner edge of the shell for cases I, II, and III, respectively (Figure 5). All of these are much shorter than the age of the visible Crab, suggesting that the outer shell has reached photoionization equilibrium.

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3.4.2. Thermal Timescale

We calculate both the thermal energy (erg cm⁻³) and the cooling rate (erg cm⁻³ s⁻¹) as a function of the radius for all three cases with solar abundances. From the ratio we can find the cooling time. We also calculate the emission measure for different radii or different zones. The differential emission measure for each depth is then

$$\begin{equation} d{\rm EM} = 4\pi r^2 n(r)^2 dr. \end{equation} \tag{ 31 }$$

This gives an indication of which portions of the shell contribute most to the observed emission.

Figure 6 shows the cooling times and the differential emission measure across the Crab outer shell for all three cases with solar abundances. We find the cooling time for all cases to be much longer than the age of the visible Crab. Even for the inner edge, which produces much of the emission measure, the cooling times are still about 20, 10, and 6 times of the age of the visible Crab for cases I, II, and III, respectively. This indicates that the outer shell has not had time to reach thermal equilibrium, so it retains a memory of its temperature in the past.

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3.4.3. Effects on Predicted Spectrum