DISTANCE AND EVOLUTIONARY STATE OF THE SUPERNOVA REMNANT 3C 397 (G41.1-0.3)

To interpret the radial velocities of absorption features in terms of distance, we adopt a kinematic model for the rotation of the Galaxy. We use a constant circular rotation speed model to relate radial velocities V_r to the Galacto-centric radius R. We adopt V₀ = 218 km s⁻¹ (218 ± 6 km s⁻¹ from Bovy et al. 2012) and R₀ = 8.33 kpc (8.33 ± 0.04 ± 0.14 kpc from Dékány et al. 2013). One can convert to distances for different choices of V₀ and R₀ and account for a circular rotation speed that varies with the Galacto-centric radius using the formula (e.g., see Sparke & Gallagher 2007):

$$\begin{eqnarray}&&{V}_{{\rm{r}}}={R}_{0}\;\mathrm{sin}(l)\;(V(R)/R-{V}_{0}/{R}_{0}).\end{eqnarray} \tag{ 1 }$$

Here, V(R) is the radius-dependent rotation curve. For constant or linear V(R), Equation (1) can be directly inverted to find R given V_r.

The tangent point is at R/R₀ = sin(l), which gives a tangent point radius for l = 411 of R_tan = 5.48 ± 0.10 kpc, and a tangent point distance d_tan = 6.28 ± 0.11 kpc. The adopted rotation curve model gives v_tan = 74.7 ± 2.5 km s⁻¹ at l = 411. However, the observed absorption extends to ∼90 km s⁻¹. Allowing for ∼5 km s⁻¹ nominal random velocities in the H i, the observed absorption from gas at the tangent point occurs over a range of velocities from v_tan − 5 km s⁻¹ to v_tan + 5 km s⁻¹. Because absorption is seen up to 90 km s⁻¹, we infer that v_tan is ≃85 km s⁻¹. This discrepancy between the calculated v_tan = 74.7 ± 2.5 km s⁻¹ and observed v_tan can be accounted for if V(R) is not constant. This requires V(R_tan) to be only slightly larger, e.g., with V(R_tan) = 230 km s⁻¹, the model v_tan is 86.7 km s⁻¹, which is consistent with the observed v_tan. This larger V(R_tan) = 230 km s⁻¹ is consistent with the published rotation curves determined using the tangent point method (such as given in Figure 2.21 of Sparke & Gallagher 2007). Other authors have suggested a higher rotation velocity for longitudes inside the solar circle in the range of 40°–55°: Reid et al. (2007) give 242 km s⁻¹; Levine et al. (2008) and Leahy et al. (2008) suggest similarly high rotation velocities. Below, we quote distances for two cases: (i) using a constant velocity rotation curve with V₀ = 218 km s⁻¹ (called "constant V(R)"), and (ii) using a linearly inward-increasing rotation curve with V₀ = 218 km s⁻¹ at R₀ and V(R_tan) = 230 km s⁻¹ (called "linear V(R)"). The latter is more complex, but is consistent with the observed tangent point velocity at l = 411, so we take it as the more realistic case. In reality, the fluctuations in measured rotation curves V(R) are probably complex and originate from velocity perturbations caused by the spiral arm structure of our Galaxy.

Figure 5 illustrates a top-down view of the Galactic plane showing the location of the Galactic center and the Sun, the solar circle, and the line of sight in the direction of 3C 397 (l = 41°). The positions and velocities are illustrated for the tangent point (radial velocity ∼85 km s⁻¹), farside gas moving at 50–60 km s⁻¹, and farside gas moving at 20–25 km s⁻¹.

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The main features of the absorption spectra for 3C 397 are as follows. There is absorption at least to the tangent point velocity, which implies that the SNR is beyond the tangent point distance d_tan. There is clearly no absorption at ≃20–25 km s⁻¹, yet there is strong H i emission at those velocities (see the background H i spectra in Figures 2 and 3). The emission implies that the H i at those velocities is behind the SNR and beyond the tangent point. For radial velocity ≃20–25 km s⁻¹, the H i is at a distance of 12.3–12.0 (±0.4) kpc (constant V(R)) or 11.3–11.1 (±0.3) kpc (linear V(R)). Thus, 3C 397 is in front of the H i at 25 km s⁻¹ (which, in turn, is in front of the 20 km s⁻¹ H i), i.e., the SNR is at a distance less than 12.0 ± 0.4 kpc (constant V(R)) or 11.1 ± 0.3 kpc (linear V(R)).

Weak (τ ≃ 0–0.2) or none H i absorption is observed for ≃50–60 km s⁻¹. However, the H i emission at velocities near ≃50–60 km s⁻¹ is very strong. In addition, the CO spectra in the bottom panels of Figures 2 and 3 show that there is molecular gas at 50–60 km s⁻¹, which should be surrounded by a significant amount of H i. This leads to the conclusion that there is significant H i at ≃50–60 km s⁻¹ and that most of this H i is behind the SNR because of the lack of strong absorption, i.e., the SNR is in front of the farside gas at 60 km s⁻¹. A small amount of nearside gas at ≃50–60 km s⁻¹, which is in front of the SNR, can account for the weak absorption. There is a secondary peak in the CO spectra at ≃65 km s⁻¹, which is associated with H i absorption (see the lower panels in Figures 2 and 3). This gas is in front of 3C 397 but could be at the nearside distance (2.3 kpc, constant V(R), or 3.1 kpc, linear V(R)) or farside distance (10.2 kpc, constant V(R), or 9.5 kpc, linear V(R)). The farside gas at 60 km s⁻¹ has a distance 10.4 ± 0.3 kpc (constant V(R)) or 9.7 ± 0.3 kpc (linear V(R)). This implies that the SNR is closer than 10.4 ± 0.3 kpc (constant V(R)) or 9.7 ± 0.3 kpc (linear V(R)). The SNR is beyond the tangent point (6.3 ± 0.1 kpc). If the 65 km s⁻¹ absorbing gas is at the farside distance, the lower distance limit for 3C 397 becomes 9.5 ± 0.3 kpc (linear V(R)).

In summary, the absorption to the tangent velocity but absence at ∼50–60 km s⁻¹ places 3C 397's distance between 6.3 ± 0.1 kpc and and 10.4 ± 0.3 kpc (constant V(R)) or 9.7 ± 0.3 kpc (linear V(R)). We use the result for the linear V(R) because it is consistent with the observed tangent point velocity.

The H ii region G41.1-0.2 has a recombination line velocity of 58 km s⁻¹ (Downes et al. 1980), placing it either at the near point (d = 2.0 kpc, constant V(R), or d = 2.8 kpc, linear V(R)) or the far point (d = 10.5 kpc, constant V(R), or d = 9.8 kpc, linear V(R)). The far point distances have uncertainties of the order of 3 kpc. The region A H i spectrum (see Figure 4) for G41.1-0.2 shows absorption to the tangent point velocity. This places G41.1-0.2 at the far point distance of 10.5 ± 0.3 kpc (constant V(R)) or 9.8 ± 0.3 kpc (linear V(R)). No absorption at ∼20–25 km s⁻¹ is consistent with the gas at ∼20–25 km s⁻¹ located at the farside distance of 11–12 kpc (see the paragraph above for details of the distance determination), behind the H ii region. Next we address the absence of (or weak) absorption at ≃60 km s⁻¹. The gas at ≃60 km s⁻¹ in this direction must be mostly at the far side distance to explain the lack of absorption by the nearside gas in this velocity range. This is supported by the CO spectra, which show two components of CO gas at 60 and 65 km s⁻¹. The CO gas at 65 km s⁻¹ is associated with H i absorption, so could be at the nearside or farside distance. The CO gas at 60 km s⁻¹ shows no absorption in the H i spectrum for region A, so it must be at the farside distance. Thus, the farside H i at ≃60 km s⁻¹ is consistent with the same velocity (allowing for a few km s⁻¹ velocity dispersion in the H i) and distance as the H ii region. Then the lack of absorption in the H ii region spectrum at ≃60 km s⁻¹ supports the conclusion that the farside ≃60 km s⁻¹ gas is just behind the H ii region. The distance of the H ii region then is the farside distance for 58 km s⁻¹, i.e., 10.5 ± 0.3 kpc (constant V(R)) or 9.8 ± 0.3 kpc (linear V(R)).

The above distance for the H ii region and the limits on distance for SNR 3C 397 do not allow us to determine whether or not they are physically close to each other: the SNR could be at the same distance as the H ii region or the SNR could be anywhere closer providing it is beyond the tangent point (d_tan = 6.3 ± 0.1 kpc).

From the H i absorption spectrum, the distance to 3C 397 is between 6.3 ± 0.1 and 9.7 ± 0.3 kpc (using the preferred linear V(R) model). For determining the physical state of the SNR, we use a nominal distance of 8.0 kpc, and write the distance as d₈ × 8 kpc. The uncertainty is ±1.7 kpc, dominated by the range of distances allowed by the H i absorption spectrum.

The angular size of SNR 3C 397, which is noticeably elongated in approximately the east–west direction, is estimated in different ways. The Chandra X-ray and VLA radio images (Safi-Harb et al. 2005) show 3C 397 to be roughly rectangular, with the long axis tilted approximately 30° counter-clockwise from the east–west direction. On the basis of the available images, the radio emission seems to extend slightly (∼10%) larger than the X-ray emission. The Chandra X-ray and VLA radio images yield angular diameters of 25 by 40 and 30 by 43. Yamaguchi et al. (2015) assume a prolate ellipsoid shape, with the unmeasured semi-axis along the line of sight assumed to be the same as the semiminor axis in the sky plane, i.e., they adopt semi-axes of 24, 13, and 13. We find better agreement with their quoted volume if the semi-axes are adjusted to 24, 14, and 14. These values give better agreement with the Chandra and VLA image sizes, so we adopt these values for the prolate spheroid case for 3C 397. This yields a volume for 3C 397 of 7.3 × 10⁵⁷${{d}_{8}}^{3}{\mathrm{cm}}^{3}$. It is also possible that 3C 397 is an oblate spheroid with semi-axes 24, 14, and 24. This would increase the volume to $1.25\quad \times {10}^{58}{{d}_{8}}^{3}$ cm³.

The density of the gas in 3C 397 is constrained from observations. The X-ray emission measures (EMs) constrain the density. The EMs for the hard and soft components were determined for the entire SNR from ASCA X-ray observations (Pshock + Pshock model; Table 1 of Safi-Harb et al. 2000), and for large regions of the SNR from Chandra X-ray observations (eastern and western lobes; Table 3 of Safi-Harb et al. 2005). We find that if we scale the EMs for the eastern and western lobe to the entirety of 3C 397, we obtain the same values as for the ASCA EMs: EM_soft = 32 and EM_hard = 0.05. The normalization of the EMs is ${10}^{-14}/(4\pi {d}^{2})\;\;\int {n}_{e}{n}_{H}{dV}$ cm⁻⁵.

The density is measured by Spitzer IR spectra. Yamaguchi et al. (2015) derive post-shock proton densities for two small regions on the rim of 3C 397 as 4.6 ± 0.4 and 8.5 ± 0.8 cm⁻³; thus, we use the average post-shock proton density of n_p = 6.5 cm⁻³. This yields an average post-shock electron density of 7.8 cm⁻³ for solar abundances. For a strong shock with a compression ratio of four, the pre-shock ambient density is n_p/4 ≃ 1.6 cm⁻³. A second estimate of the mean interstellar medium density around the SNR is found using the integrated IR flux. Yamaguchi et al. (2015) obtain a gas mass of $\sim 16{M}_{\odot }{d}_{8}^{2}$. This includes a factor of five increase to account for the efficiency of the detection of dust emission in the mid-IR. This mass corresponds to an ambient density of 2.1 ${d}_{8}^{-1}$ cm⁻³ or 1.2 ${d}_{8}^{-1}$ cm⁻³ for the prolate and oblate shapes of 3C 397, respectively. These agree well with the first estimate of ≃1.6 cm⁻³. Because the second estimate, which is global, agrees well with the first estimate, which is local, the two small regions measured by Spitzer seem to be representative of the mean post-shock density of 3C 397.

Next, we combine the measures of density and EM. There are two components seen in X-rays (the soft component and the hard component). The soft X-ray component has a proton density n_s and the hard X-ray component has a proton density n_h. The X-ray EMs depend on unknown filling factors: f_h for the hard X-ray component and f_s for the soft component. The expression for EM for the hard component including the filling factor is: ${\mathrm{EM}}_{\mathrm{hard}}={10}^{-14}/(4\pi {d}^{2})\;\;\int {f}_{h}\;\;{n}_{e}{n}_{h}{dV}$ cm⁻⁵, where the factor f_h accounts for the fact that the hard component does not fill the entire volume of the SNR. There is an analogous expression for EM_soft with f_s and n_s. We assume the two components fill the SNR volume: f_h + f_s = 1. The average interior proton density is given by ${n}_{{av}}={f}_{h}\quad {n}_{h}\quad +\quad {f}_{s}\quad {n}_{s}$. We take n_av = n_p, i.e., use the measured post-shock proton density.

For the distance 8 kpc, we find that the hard component has a density of n_h = 2.1 cm⁻³ and a filling factor of f_h = 0.993, and the soft component has n_s = 630 cm⁻³ and f_s = 0.007. For the distance 6.3 kpc, one has n_h = 2.4 cm⁻³, f_h = 0.995, n_s = 850 cm⁻³, and f_s = 0.005. For the distance 9.7 kpc, one has n_h = 1.9 cm⁻³, f_h = 0.991, n_s = 500 cm⁻³, and f_s = 0.009. If we use the oblate ellipsoid model for 3C 397 for the distance 8 kpc instead of prolate, we find n_h = 1.6 cm⁻³, f_h = 0.985, n_s = 330 cm⁻³, and f_s = 0.015. The values for the distances of 6.3 and 9.7 kpc change in a similar manner (decreased n_h, f_h, and n_s, and increased f_s).

For the above, we assumed the hard and soft X-ray components are each of uniform density and together fill the entire SNR volume. A more realistic model would take into account the expected hot interior region of the SNR. As a first approximation, we consider a Sedov remnant structure (e.g., Heiles 1964), which has a density less than 20% of the post-shock density and a temperature greater than twice the post-shock temperature for the inner 85% of the radius, or inner 61% of the volume. For this case, the sum of the filling factors is f_h + f_s ∼ 0.39. One finds n_h = 3.4 cm⁻³, f_h = 0.38, n_s = 530 cm⁻³, and f_s = 0.009 for the prolate volume model and the distance 8 kpc. These values are not much different than for the uniform interior case, except that the filling factor of the hard component is reduced and its density is increased. Values for distances of 6.3 and 9.7 kpc are similarly modified. In summary, the hard component fills most of the SNR with a density of ∼2–4 cm⁻³ and the soft component has a high density (∼300–800 cm⁻³) and a very small filling factor (∼0.005–0.015).

Next we consider the evolutionary state of the SNR 3C 397. Above, we found that the soft component has a very small filling factor ∼0.01. This is consistent with a very small volume of the total (pre-explosion) circumstellar medium consisting of dense clouds, whereas the bulk of the circumstellar medium is the material that is shocked to become the hard X-ray component. This means that the evolution of the SNR is determined by the density associated with the hard component. Secondary shocks propagating into the high-density, low filling factor clouds can account for the relatively bright soft X-ray component without significantly affecting the evolution of the SNR.

We use models that include the ejecta-dominated phase, the transition from the ejecta-dominated to the Sedov phase, and the generalized Sedov phase. These are described in Truelove & McKee (1999). Because the SNR is known to be from a Type Ia explosion, we take the circumstellar medium, except for the small dense clouds, to be of constant density, n₀. We use an ejecta power-law density profile with index n = 7 appropriate for an SN Ia (e.g., Truelove & McKee 1999).

Recent determinations of Type Ia explosion energies are as follows. Badenes et al. (2005) apply Type Ia models to several SN Ia and find (their Tables 3 and 4) E₀ = 0.42 − 1.36 × 10⁵¹; for Tycho's SNR, Badenes et al. (2006) find E₀ = 1.2 × 10⁵¹; and for the young Galactic SN Ia G1.9+0.3, Borkowski et al. (2013) find E₀ = 1.38 × 10⁵¹. Thus, for the explosion energy, E₀, we consider three cases: low (0.5 × 10⁵¹ erg), standard (1.0 × 10⁵¹ erg), and high (1.5 × 10⁵¹ erg).

For either the prolate model or the oblate model geometry of the SNR, the minimum shock radius is R_s = 4.0 d₈ pc and the maximum is R_s = 4.8 d₈ pc, i.e., the models need to produce shock radii, with the smaller radius corresponding to the narrow dimension of 3C 397 on the sky and the large radius corresponding to the long dimension. For the models here, we consider the nominal distance to 3C 397 of 8 kpc, as well as the lower and upper limits of 6.3 and 9.7 kpc. For 8 kpc, the minimum and maximum shock radii are 4.0 and 4.8 pc, for 6.3 kpc, they are are 3.2 and 3.8 pc, and for 9.7 kpc, they are 4.8 and 5.8 pc.

The average post-shock proton density for 3C 397 is n_p ∼ 6.5 cm⁻³ (Yamaguchi et al. 2015). Immediately behind the outer shock, the density of the shocked material (the hard component) is 4n₀. However, it decreases fairly rapidly inward, so the mean observed density of the hard component is $\sim 2{n}_{0}$. The exact relation between the pre-shock density n₀ and the observed mean density of the shocked gas is complex: it depends on the emissivity of the material and instrument sensitivity as well as the density and temperature structure of the SNR. Here we investigate n₀ values in the range ≃1.5–5 cm⁻³, which are consistent with the measured average post-shock proton density.

The observed post-shock temperatures are in the range 1.3–2.2 keV (Safi-Harb et al. 2005). We use the standard relation for an adiabatic shock between shock temperature T_s and shock velocity V_s:${V}_{s}=\sqrt{16{{kT}}_{s}/(3\mu {m}_{p})}$, where μ = 0.61 is the mean weight per particle and m_p is the proton mass. Thus, shock velocities corresponding to the observed temperatures are in the range 1050–1360 km s⁻¹.

A number of SNR models were calculated with E₀ and n₀ in the expected range by adjusting the age and phase of evolution to produce current shock radii corresponding to the long and narrow dimensions of 3C 397 for the three distances (nominal, lower limit, and upper limit). In addition, for the nominal distance of 8 kpc, we add one more model for an intermediate shock radius of 4.4 pc. This results in six values of shock radius to produce: 3.2, 3.8, 4.0, 4.4, 4.8, and 5.8 pc. Within the uncertainties, the long dimension of 3C 397 for the distance 6.3 kpc is the same as the short dimension for the distance 8 kpc, so we reduce the list of shock radii for modeling to five values: 3.2, 4.0, 4.4, 4.8, and 5.8 pc.

Table 1 presents a subset of the calculated models. Models with significantly too low or too high shock velocities are not shown. We find that the SNR is either in the transition from the ejecta-dominated to the generalized Sedov phase or early in the generalized Sedov phase. Thus, the use of the generalized Sedov phase and including the transition phase from the ejecta-dominated to the generalized Sedov phase is necessary. We find that none of the acceptable models have reached the stage where radiative losses affects the SNR. Typical timescales for radiative losses are ∼5000–10,000 years.

First we discuss the case for the distance of 8 kpc, which corresponds to 4.0, 4.4, and 4.8 pc shock radii. The low-energy models (E₀ = 0.5 × 10⁵¹ erg) are closest in evolutionary phase to the ejecta-dominated phase (t < t_RS): many of the low-energy models in Table 1 have an age t < t_RS, where t_RS is the time that the reverse shock reaches the center of the SNR. Among the low-energy models, the n₀ = 2 cm⁻³ model most closely yields the required range of shock velocities of 1050–1360 km s⁻¹ (derived from observed shock temperatures). However, a strong argument against most of the low-energy models is that there is no observed X-ray component corresponding to the high velocities required for the reverse shock. The standard energy models (E₀ = 1.0 × 10⁵¹ erg) produce shock velocities too high for n₀ < 3 cm⁻³ and shock velocities too low for n₀ > 5 cm⁻³. The high-energy models (E₀ = 1.5 × 10⁵¹ erg) produce  shock velocities too high for n₀ < ∼5 cm⁻³ and require densities inconsistent with the observed average post-shock proton density for n₀ > ∼5 cm⁻³. This allows only a narrow range of pre-shock density around n₀ ∼ 5 cm⁻³ to be consistent with the observed properties of 3C 397.

In practice, the pre-shock or ambient density can vary with position angle around the SNR. This means that for some directions n₀ is low and in other directions it is somewhat higher. Such a variation is consistent with the observed non-circular projection of 3C 397 on the sky. For a model with density varying with position angle, the age must be the same in all directions, e.g., for the 8 kpc distance, the E₀ = 1.0 × 10⁵¹ erg case (see Table 1), an age of ≃1350 years is obtained for: n₀ = 5 cm⁻³ (V_s = 1220 km s⁻¹) in a position angle with small radius (R_s = 4.0 pc); n₀ = 3 cm⁻³ (V_s = 1360 km s⁻¹) in a position angle with an intermediate radius (R_s = 4.4 pc); and n₀ ≃ 2 cm⁻³ (V_s ≃ 1510 km s⁻¹) in a position angle with a large radius (R_s = 4.8 pc). In addition, this range of model shock velocities is consistent with the range of observed shock temperatures in 3C 397. The E₀ = 1.0 × 10⁵¹ erg models are preferred over the E₀ = 0.5 × 10⁵¹ erg models because a high-mass/high-energy explosion is in better agreement with the the high abundances of nickel and manganese. Likewise, the E₀ = 1.0 × 10⁵¹ erg models are preferred over the E₀ = 1.5 × 10⁵¹ erg models because the higher energy models require too high an ambient density compared to the observed average post-shock proton density. In summary, the model which best fits the range of shock radii for the 8 kpc distance (4.0–4.8 pc) and the observed shock temperatures of 3C 397 is that with E₀ = 1.0 × 10⁵¹ erg, an age of 1300–1400 years, and pre-shock densities ranging from ≃2.0–5.0 cm⁻³.

Next we discuss the effects of taking the distance to 3C 397 as the lower and upper limit values. For the 6.3 kpc distance, the observed angular dimensions translate to shock radii of 3.2–4 pc. From Table 1, one sees that the observed shock velocities of 1050–1360 km s⁻¹ are produced only for n₀ ≃ 4 cm⁻³ if E₀ = 0.5 × 10⁵¹ erg, n₀ ≫ 5 cm⁻³ if E₀ = 1.0 × 10⁵¹ erg, and ${n}_{0}\gg 7$ cm⁻³ if E₀ = 1.5 × 10⁵¹ erg. This fact weakens the probability of a distance of 6.3 kpc because no model can be consistent with the observed density and at the same time consistent with observed X-ray temperatures (i.e., shock velocities). For the 9.7 kpc distance, the observed angular dimensions translate to shock radii of 4.8–5.8 pc. From Table 1, one sees that the observed shock velocities of 1050–1360 km s⁻¹ are produced only for n₀ ≃ 1–1.5 cm⁻³ if E₀ = 0.5 × 10⁵¹ erg, n₀ ≃ 2 cm⁻³ if E₀ = 1.0 × 10⁵¹ erg, and n₀ ≃ 3–4 cm⁻³ if E₀ = 1.5 × 10⁵¹ erg. Thus, the models applied with the larger distance of 9.7 kpc reproduce values compatible with both the density and the shock velocity constraints. In this case, a compatible age for 3C 397 (i.e., same age for shock radii of 4.8 and 5.8 pc) is ≃1750 years. This is obtained for a density of n₀ ≃ 2 cm⁻³ and a shock velocity of 1350 km s⁻¹, for a shock radius 5.8 pc and a density of n₀ ≃ 5 cm⁻³, and a shock velocity of 1130 km s⁻¹ for a shock radius 4.8 pc.

In summary, we find that the models predict compatible parameters for 3C 397 for distances from 8 to 9.7 kpc, matching both the observed post-shock density and the range of observed shock temperatures. The age and explosion energy of the SNR ranges from ≃1350 years and 1.0 × 10⁵¹ erg at 8 kpc to ≃1750 years and E₀ = 1.5 × 10⁵¹ erg at 9.7 kpc. The spatially variable pre-shock density is ≃2.0–5.0 cm⁻³ for either distance.