3 Theoretical properties of "projected composites''

The Sedov (1959) model does not give a centrally-concentrated morphology due to geometrical properties of self-similar solutions. The solutions are 1-D and give a specific internal profile of the flow gas density: most of the mass is concentrated near the shock front. These factors and cumulation of the emission along the line of sight cause a shell-like morphology. If we consider a more complicated nonuniform ISM, we get beyond one dimension and need to consider additional parameters responsible for nonuniformity of the medium and orientation of a 3-D object.

Projection effects may essentially change the morphology of SNRs (Hnatyk & Petruk 1999). Densities over the surface of a nonspherical SNR may essentially differ in various regions. If the ambient density distribution provides a high density in one of the regions across the shell of SNR and is high enough to exceed the internal column density near the edge of the projection, we will see a centrally-filled projection of a really shell-like SNR. Such density distribution may be ensured e.g. by a molecular cloud located near the object.

What is a really shell-like 3-D SNR? We suggest that such a remnant has internal density profiles similar to those in the Sedov (1959) solutions. Thus, we separate a shell-like SNR (as an intrinsic property of a 3-D object) from its limb brightened projection (as a morphological property of the projection). Let us call shell-like SNRs with centrally-filled projections "projected composites''.

3.1 Hydrodynamic models

For simplicity, let us consider the case of a 2-D SNR and the characteristics of SNR and the surrounding medium which could be possible on smoothed boundaries of molecular clouds. Thus, SNR evolves in the ambient medium with hydrogen number density n distributed according to

$$n(\tilde{r})=n_{\rm o}+n_{\rm c}\exp(-\tilde{r}/h),$$ (1)

where $n_{\rm o}$ is the density of the intercloud medium, the second term represents the density distribution into the boundary region and the cloud, h is the scale-hight, $\tilde{r}$ is the distance. Let us take the explosion site to be at point $\tilde{r}_{\rm o}$ where $n(\tilde{r}_{\rm o})=2n_{\rm o}$. Other parameters are assumed to be $n_{\rm o}=0.1\ {\rm cm^{-3}}$, $n_{\rm c}=100\ {\rm cm^{-3}}$. The energy of the supernova explosion is $E_{\rm o}=1~10^{51}\ {\rm erg}$. We consider three basic evolutionary cases of SNR models which cover practically the whole adiabatic phase (models a-c, Table 1) and then we vary parameter h in intermediate model b (models d-f, Table 1), in order to see how the gradient of the ambient density affects X-ray characteristics of objects.

 

 

Table 1: Parameters of SNR models. h is the scale-height in the ambient medium density distribution (1). t is the age of the SNR, R and D are the radius and velocity of the shock front, $T_{\rm s}$ and $n_{\rm s}$ are the temperature and number density of the swept-up gas right behind the shock. $R_{\rm max}$, $R_{\rm min}$ ( $D_{\rm max}$, $D_{\rm min}$) are the maximum and minimum shock radii (shock velocities) of nonspherical SNR. Analogously, $T_{\rm s,\ \!max}$, $T_{\rm s,\ \!min}$ ( $n_{\rm s,\ \!max}$, $n_{\rm s,\ \!min}$) are maximum and minimum temperatures (number densities) of the gas flow right behind the shock. $L_{\rm x}^{>0.1\ {\rm keV}}$ is the thermal X-ray luminosity (for photon energy $>0.1\ {\rm keV}$) and $\alpha$ is the spectral index (at photon energy $5\ {\rm keV}$) of the thermal X-ray emission from the whole SNR. $T_{\rm ef}$ is the effective temperature of a nonspherical SNR defined by Hnatyk & Petruk (1999) as $T_{\rm ef}\propto M^{-1}$, where M is the swept-up mass. The contrasts in the distribution of X-ray surface brightness ${\rm log} (S_{\rm c}/S_{\rm max,\ \!2})$ and spectral index $\alpha _{0.95}/\alpha _{\rm c}$ are presented for the case of $\delta =90\hbox {$^\circ $ }$. Subscript "c" corresponds to the center of the projection, $\alpha _{0.95}$ is the value of the index at $0.95R_{\rm p}$, $R_{\rm p}$ is the radius of the projection

Parameter	Model
	a	b	c	d	e	f
$h,\ {\rm pc}$	2.5	2.5	2.5	5	10	40
$t,\ {\rm 10^3\ yrs}$	1.0	6.8	17.7	6.8	6.8	6.8
$\log T_{\rm ef},\ {\rm K}$	8	7	6.5	7	7	7
$M,\ {M_{\odot}}$	9.5	94	280	98	95	94
$R_{\rm max}/R_{\rm min}$	1.4	1.8	2.1	1.4	1.2	1.1
$D_{\rm max}/D_{\rm min}$	1.9	2.8	3.1	1.9	1.5	1.1
$T_{\rm s,\ \!max}/T_{\rm s,\ \!min}$	3.5	7.9	9.8	3.7	2.2	1.2
$n_{\rm s,\ \!max}/n_{\rm s,\ \!min}$	9.5	45	84	11	3.9	1.4
$\log L_{\rm x}^{>0.1\ {\rm keV}}$	34.1	36.7	37.3	36.4	36.2	36.1
$\alpha ^{5\ {\rm keV}}$	0.98	3.2	3.1	3.9	4.1	4.2
$\log (S_{\rm c}/S_{\rm max,\ \!2})$	0.43	2.1	2.3	0.72	-0.10	-0.54
$\alpha _{0.95}/\alpha _{\rm c}$	1.6	1.8	3.6	1.3	1.2	1.3

   
3.2 Emission models

Gas density n and temperature T distributions inside the volume of a nonspherical SNR are obtained with the method of Hnatyk & Petruk (1999).

The equilibrium thermal X-ray emissivities are taken from Raymond & Smith (1977).

We make simple estimations of the radio morphology of SNR in a nonuniform medium as described below, on the basis of a model for synchrotron emission from SNRs developed by Reynolds (1998, hereafter R98) and Reynolds & Chevalier (1981).

The volume emissivity in the radio band at some frequency $\nu$ is

$$S_\nu\propto KB_\perp^{(s+1)/2},$$ (2)

where K is the normalization of electron distribution $N(E){\rm d}E=KE^{-s}{\rm d}E$, $B_\perp$ is the tangential component of the magnetic field (perpendicular to the electron velocity or normal to the shock). Power index s is constant downstream because electrons lose energy proportionally to their energy (Eq. (6)) and remain essentially confined to the fluid element in which they were produced. Power index s is also constant over the surface of a nonspherical SNR because the shock is strong. Namely, in the first order Fermi acceleration mechanism, $s=(\sigma+2)/(\sigma-1)$, where shock compression ratio $\sigma$ does not depend on the ambient density distribution in the strong shock limit (Mach number $\gg$1) and $\sigma=4$ for $\gamma=5/3$.

The ambient field is assumed to be uniform (polarization observations support the assumption that the magnetic fields in molecular clouds may be ordered over large scales, e.g. Goodman et al. 1990; Messinger et al. 1997; Matthews & Wilson 2000 and others). Component $B_\parallel$ is not modified by the shock: $B_{\parallel,{\rm s}}/B_{\parallel,{\rm s}}^{\rm o}=1$ (indices "s" and "o" refer to the values at the shock and to the surrounding medium, respectively). Component $B_\perp$ rises everywhere at the shock by the factor of $\rho_{\rm s}/\rho^{\rm o}_{\rm s}=\sigma$. No further turbulent amplification of the magnetic field is assumed. Components $B_\parallel$ and $B_\perp$ evolve differently behind the shock front (R98; Reynolds & Chevalier 1981). Since the magnetic field is flux-frozen, $B_\perp(r)r{\rm d}r={\rm const}$, the tangential component in each 1-D sector of the remnant is

$$B_\perp(r)=B_\perp(a)\ {\rho(r)\over\rho(a)}\ {r\over a},$$ (3)

where a is the Lagrangian coordinate. Radial component

$$B_\parallel(r)=B_\parallel(a)\left({a\over r}\right)^2$$ (4)

due to the magnetic flux conservation, $B_\parallel {\rm d}\sigma={\rm const}$.

In each fluid element, energy density $\omega$ of relativistic particles is proportional to the energy density of the magnetic field

$$\omega\equiv\int\limits^{E_{\max}}_{E_{\min}}EN(E){\rm d}E= K\int\limits^{E_{\max}}_{E_{\min}}E^{1-s}{\rm d}E \propto B^2,$$ (5)

which yields $K\propto B^2\big(E_{\max}^{2-s}-E_{\min}^{2-s}\big)^{-1}$ for $s\neq 2$ and $K\propto B^2\ln \big(E_{\min}/E_{\max}\big)$ for s=2. We are to expect the variation of $E_{\max}$ and $E_{\min}$ over the surface of a nonspherical SNR and downstream. However, the maximum energy, to which particles can be accelerated, only varies few times during the whole adiabatic phase (R98). The ISM nonuniformity does not affect the evolution of SNR at the previous free expansion stage. Thus, we assume that possible variations of the minimum and maximum electron energies over the remnant's surface are likely to be minute and, in the first approach, we may neglect the surface variation of $E_{\max}$ and $E_{\min}$ caused by a nonuniform medium.

Figure 1: a-d) Logarithmic distributions of thermal X-ray surface brightness S in ${\rm erg\,s^{-1}\,cm^{-2}\,st^{-1}}$ for photon energy $\varepsilon >0.1\ {\rm keV}$ a,b), and radio surface brightness $S_\nu$ at some frequency in relative units c,d). The SNR model is b, power s=2. Angles $\delta$ and $\phi$ are shown in the figure. a,b):  $\log S_{\max}=-3.1$, $\Delta \log S=0.3$. c)  $\Delta \log S_\nu=0.3$. d)  $\Delta \log S_\nu=0.15$. The darker colour represents a higher intensity. The arrow indicates a magnetic field orientation

Each individual electron loses its energy due to the adiabatic expansion:

$$\dot{E}=E\ {\dot{\overline{\rho}}\over 3\overline{\rho}},$$ (6)

where $\overline{\rho}(r)=\rho(r)/\rho_{\rm s}$ (R98) and, therefore, $E(r)\propto\overline{\rho}(r)^{1/3}$ downstream. Thus,

$$K(r)\propto B(r)^2\overline{\rho}(r)^{(s-2)/3}.$$ (7)

Radio morphology depends on aspect angle $\phi$ between the line of sight and the ambient magnetic field (R98), also on inclination angle $\delta$ between the density gradient and the plane of the sky and, in a complex 3-D case, on the third angle between the density gradient and the magnetic field.

3.3 Results

Figures 1a,b demonstrate the influence of the projection on a thermal X-ray morphology of SNR. The X-ray brightness maximum located near the shock front in the shell-like projection ( $\delta=0\hbox{$^\circ$ }$) moves towards the centre of the projection with the increase of $\delta$ from $0\hbox{$^\circ$ }$ to $90\hbox{$^\circ$ }$. For clarity we only consider here the most emphatic limit case $\delta =90\hbox {$^\circ $ }$. Radio images (Figs. 1c,d) show that the radio limb-brightened morphology clearly appears at $\phi=0\hbox{$^\circ$ }$, i.e. if both the density gradient and the magnetic field are nearly aligned.

Variation of the magnetic field orientation changes the radio morphology from shell-like to barrel-like (Kesteven & Caswell 1987; Gaensler 1998). Contrast $\log (S_{\nu,\max}/S_{\nu,\min})$ in the radio surface brightness decreases with increasing $\phi$, from 2.7 ( $\phi=0\hbox{$^\circ$ }$) to 1.8 ( $\phi=90\hbox{$^\circ$ }$). Such behaviour of the radio morphology may be used for testing orientation of the magnetic field.

Thus, we found that the morphological properties of the projected composites match the basic features of the TXC class: centrally-peaked distribution of the thermal X-ray surface brightness is within the area of the radio shell; emission arises from the swept-up ISM material.

Figure 2: Distribution of density (solid curves) and temperature (dashed curves) along the line of sight inside SNR (model b, $\delta =90\hbox {$^\circ $ }$): 1 - at the center of the projection, 2 - at $0.95R_{\rm p}$. Emission measure $\int n_{\rm e}^2{\rm d}l=74\ {\rm cm^{-6}~pc}$ at the center and $\int n_{\rm e}^2{\rm d}l=1.1\ {\rm cm^{-6}~pc}$ at $0.95R_{\rm p}$

Let us consider physical properties of TXCs. a) The column number density increases from the edge towards the centre of the projection (e.g., for model b from $10^{18.9}\ {\rm cm^{-2}}$ to $10^{19.4}\ {\rm cm^{-2}}$). b) The diffuse optical nebulosity over the internal region of the projection may naturally take place in such a model. c) Emission measure $\int n_{\rm e}^2{\rm d}l$ ($n_{\rm e}$ is the electron number density, l is the length within SNR) is the highest in the X-ray peak because both $n_{\rm e}$ and l are maximum there (Fig. 2).

As Fig. 3 demonstrates, the distribution of X-ray surface brightness has strong maximum $S_{\rm c}$ around the centre and a weaker shell with second maximum $S_{\rm max,\ \!2}$ just behind the forward shock. It is essential that such a morphology takes place in different X-ray bands (lines b, 1, 2). The contrasts $S_{\rm c}/S_{\rm max,\ \!2}$ in X-ray surface brightness depend on the photon energy band and may lie within a wide range: in our models from 3 to 200 for $\varepsilon >0.1\ {\rm keV}$ (Table 1). The ratios of X-ray luminosity $\int S(r)2\pi r{\rm d}r$ of central region $R<0.2R_{\rm p}$ to the luminosity beyond $R>0.6R_{\rm p}$ are 0.16, 5.2 and 16 in models a, b and c, respectively. Thus, observational property d of TXCs takes place just at the adiabatic stage.

Figure 3: a and b. Evolution of the distribution of thermal X-ray surface brightness a) and spectral index b). Solid curves are labelled with the model codes according to Table 1; they represent $\log S$ in band $\varepsilon >0.1\ {\rm keV}$ and $\alpha$ at $5\ {\rm keV}$. Dashed lines represent model b in other bands: 1 - S in $\varepsilon =0.1-2.4\ {\rm keV}$, 2 - S in $\varepsilon >4.5\ {\rm keV}$ (multiplied by 102), 3 - $\alpha$ at $10\ {\rm keV}$. Radii are normalized to unity

Surface distribution of spectral index $\alpha=\partial\ln P_{\rm c}/$ $\partial\ln\varepsilon$, of the thermal X-ray emission where $P_{\rm c}$ is the continuum emissivity and $\varepsilon$ is the photon energy, gives us profiles of effective temperature T of the column of emitting gas ( $\alpha\propto T^{-1}$, if the Gaunt factor is assumed to be constant). Figure 3 shows that the temperature may either increase or decrease towards the centre. Decreasing takes place early in the adiabatic phase. Variation of the spectral index lies within factors 1.6 to 3.6 at the adiabatic stage (Table 1); the contrast in the spectral index distribution increases with age. Such values correspond to the possible range of temperature variation over the projection of thermal X-ray composites.

In order to reveal the dependence of the distributions of S and $\alpha$ on the ISM density gradient, a number of models with different h were calculated (Fig. 4 and Table 1). The surface brightness distribution has a stronger peak for a stronger gradient. With increasing h, the outer shell becomes more prominent in the projection. Only a scale-height of order $h<10\ {\rm pc}$ can cause projected composites. A less strong gradient of the ambient density makes effective temperature T more uniformly distributed in the internal part of the projection.