6 Dynamics

In the radius-velocity plane the flux from a thin shell of radius $r_{\rm s}$ expanding at velocity $v_{\rm s}$ is expected to form an ellipse which intersects the radius axis at $r_{\rm s}$and the velocity axis at $\pm$$v_{\rm s}$. We expect the velocity to increase with radius and for simplicity we can assume that the velocity field within the spherical volume is given by a linear form

$$v(r)=\frac{v_{\rm s}}{(r_{\rm s}-r_{\rm o})}(r-r_{\rm o})$$

where $v_{\rm s}$ is the postshock velocity of material at the shock radius $r_{\rm s}$and $r_{\rm o}$ is the radius within the remnant at which the velocity falls to zero. From the analysis of the Chandra data, (Gotthelf et al. 2001), we have an estimate of $r_{\rm s}=153$ arcsec. Adopting such a spherically symmetric velocity field precludes the possibility that the expansion is wildly asymmetric. In the plane of the sky the outer shock seen both in radio and X-ray images is remarkably circular (see for example the Chandra X-ray image scaled to highlight the faint outer filaments, Gotthelf et al. 2001). We therefore think it is unlikely that the velocity field or shock radius is very different along the line of sight. Assuming values for $v_{\rm s}$ and $r_{\rm o}$ we can calculate the radius of each pixel within the spherical volume from the observed radius on the plane of the sky and the Doppler velocity (along the line of sight). As the parameters are changed so the fractional rms scatter, $\Delta r/r$, of the flux about the mean radius varies. The top left panel of Fig. 10 shows the variation in $\Delta r/r$ as a function of $r_{\rm o}$ and $v_{\rm s}$.

Figure 9: Flux distribution of Si-K (red), S-K (green) and Fe-K (blue) in the radius-velocity plane. The solid line is the best fit shock radius (see text). The outer dotted line indicates the peak of the Fe-K flux distribution and the inner dotted line indicates the mean radius of the Si-K and S-K emission.

Figure 10: The velocity field which gives the minimum normalised shell thickness $\Delta r/r$. Each contour interval in the top left plot corresponds to a $\sim$2% increase in shell thickness. The lower panel shows the deprojected flux distribution of Si-K (red), S-K (green) and Fe-K (blue) as a function of radius from the centre of the remnant.

Figure 11: The left-hand panel is an image of Si-K (red), S-K (green) and Fe-K (blue). The small red circle indicates theposition of the Chandra point source. The white cross is the best fit centre from the fitting of the radial distribution. The right-hand panel is a reprojection of the same line fluxes onto a plane containing the line of sight, North up, observer to right. In both panels the outer solid circle is the shock radius $r_{\rm s}=153$ arcsec and the inner circle is the mean radius of the Si-K and S-K flux.

The minimum scatter of 0.16 is found at $r_{\rm o}=53$ arcsec and $v_{\rm s}=2600$ kms^-1. The velocity field which gives the minimum shell thickness is shown in the top right-hand panel. This result is very similar to the velocity field of an isothermal blast wave, Solinger et al. (1975), but we would like to stress this does not imply that Cas A has entered this phase of it's evolution. The remnant is almost certainly in between the ejecta-dominated and the Sedov-Taylor stages as modelled in detail by Truelove & McKee (1999). The lower panel of Fig. 10 shows the deprojected flux distributions as a function of radius in the spherical volume. The Si-K and S-K distribution match quite closely. The Fe-K distribution is somewhat broader and peaks at a larger radius. The deprojected Si-K and S-K flux distributions are very similar to the emissivity profiles derived from the Chandra data (Gotthelf et al. 2001) although their analysis was for a section of the remnant while the profiles in Fig. 10 are the average over the full azimuthal range. The solid semi-circular line in Fig. 9 is the best fit with $r_{\rm s}=153$ arcsec and $v_{\rm s}=2600$ kms^-1. The inner dotted line shows the mean radius of the combined Si-K and S-K flux and the outer dotted line shows the peak of the Fe-K distribution. The outer reaches of the Fe-K distribution straddle the shock radius derived from the Chandra image.

The MOS energy resolution cannot separate the red and blue components when they overlap. If we see both the distant red shifted shell and nearer blue shifted shell in the same beam the line profile is slightly broadened but the centroid shift is diminished. The observed Doppler velocities and the best fit value for $v_{\rm s}$ may be slightly biased by this ambiguity, however most beams appear to be dominated by either red or blue shifted knots and therefore this bias is expected to be small. It is fortuitous that the X-ray emission is distributed in clumps rather than a thin uniform shell since this enables us to measure the Doppler shift with a modest angular resolution without red and blue components in the same beam cancelling each other out.

The left-hand panel of Fig. 11 is a composite image of the remnant seen in the Si-K, S-K and Fe-K emission lines. The solid circle indicates the $r_{\rm s}=153$ arcsec and the dashed circle is the mean radius of the Si-K and S-K flux $r_{\rm m}=121$ arcsec. The X-ray image of the remnant provides coordinates x-yin the plane of the sky. Using the derived radial velocity field within the remnant we can use the measured Doppler velocities v_z to give us an estimate of the z coordinate position of the emitting material along the line of sight thus giving us an x-y-z coordinate for the emission line flux in each pixel. Using these coordinates we can reproject the flux into any plane we choose. The right-hand panel of Fig. 11 shows such a projection in a plane containing the line-of-sight, North upwards, observer to the right.

In this reprojection the line emission from Si-K, S-K and Fe-K are reasonably well aligned for the main ring of knots. The reprojection is not perfect because the MOS cameras are unable to resolve components which overlap along the line-of-sight and this produces some ghosting just North of the centre of the remnant. In the plane of the sky Fe-K emission (blue) is clearly visible to the East between the mean radius of the Si+S flux and the shock radius. Similarly in the reprojection Fe-K emission is seen outside the main ring in the North away from the observer. The Si+S knot in the South away from the observer in the reprojection is formed from low surface brighness emission in the South West quadrant of the sky image. The X-ray emitting material is very clumpy within the spherical volume and is indeed surprisingly well characterised by the doughnut shape suggested by Markert et al. (1983). However the distribution is distinctly different to that obtained in similar 3-D studies of the optical knots, Lawrence et al. (1995).

The expansion of Cas A has been measured in various ways; using the proper motion of optical knots (van den Bergh & Kamper 1983; Fesen et al. 1987; Fesen et al. 1988), from the proper motion of radio knots (Anderson & Rudnick 1995), using Doppler shifts of spectral lines from optical knots (Reed et al. 1995; Lawrence et al. 1995), Doppler shift of X-ray line complexes (Markert et al. 1983; Holt et al. 1994; Vink et al. 1996) and the proper motion of X-ray knots (Vink et al. 1999). These methods identify a number of distinct features with different dynamics; Quasi Stationary Flocculi (optical QSF), Slow Radio Knots in the South West (SRK), the main ring of radio knots, the main ring of X-ray knots (continuum + lines 1-2 keV), Fast Moving Knots (optical FMK) and Fast Moving Flocculi (optical FMF).

In proper motion studies it is conventional to express the motion as an effective expansion time $t_{\rm ex}=R/V$ (years) where R is the radius of the feature/knot from some chosen centre (arcsec) and V is the proper motion (arcsec/year). The deceleration parameter, the ratio of the true age over the expansion age, can be estimated as $m=t_{\rm age}/t_{\rm ex}$. There is no need to deproject the radius or velocity to estimate m. However if we then wish to estimate a true expansion velocity the R must be deprojected but still the ratio R/V will remain constant. Doppler measurements allow some form of deprojection and measured radii on the sky can be converted to actual radii within the volume of the remnant as described in the previous section. Given a radius in arc seconds $r_{\rm as}$ and velocity in kms^-1 $v_{\rm kms}$ we can calculate an expansion time in years $t_{\rm ex}$ assuming a distance in kpc $d_{\rm kpc}$, $t_{\rm ex}=4.63\times10^{3} r_{\rm as}d_{\rm kpc}/v_{\rm kms}$. Previous authors have used combinations of these measurements to refine estimates of the age and/or distance. Alternatively we can adopt some age and distance and compare the radii and expansion velocities of the various components. The original explosion probably occured in 1680 (Ashworth 1980) so the age in 2000 is $t_{\rm age}=320$ years. Distance estimates have varied over the years but recent studies (Reed et al. 1995) have settled on 3.4^+0.3_-0.1 kpc.

Table 3 gives estimates of the expansion parameters for the different components. Those marked with an asterisk are from proper motion studies which estimate the expansion time or the deceleration parameter directly. For these the $r_{\rm as}$ value has been estimated and the $v_{\rm s}$ calculated using the measured expansion time. From the Doppler measurements we get a measurement of $r_{\rm s}$ and $v_{\rm s}$ which are then used to estimate the expansion time or the deceleration parameter. Proper motion studies of X-ray emission track the movement of shock features in the plane of the sky while X-ray emission line Doppler measurements estimate the velocity of the postshock plasma $v_{\rm s}$ along the line of sight. The shock velocity $u_{\rm s}$ is related to the postshock plasma velocity, $v_{\rm s}=\alpha u_{\rm s}$. The factor $\alpha$ depends on the thermodynamics of the shocked gas but ranges between 0.58 for isothermal to 0.75 for adiabatic conditions, see for example Solinger et al. (1975). The present X-ray emission line (Xline) results in Table 3 have been calculated from the derived velocity field parameters, $r_{\rm s}=153$ and $v_{\rm s}=2600$using a mean value of $\alpha=0.65$. The error quoted reflects the uncertainty in this factor. The FMF are at large radii so it is likely that the deprojection correction is small and the value of 168 arcsec quoted is in fact the mean radius in the plane of the sky. For the SRK in the South West sector and the QSF the values quoted for $r_{\rm s}$ are just reasonable guesses.

 

 

Table 3: The expansion parameters for radio, optical and X-ray emissions. * Indicates proper motion studies. The results from this paper are given in the line labelled Xline (see text for details).

	$v_{\rm kms}$	$r_{\rm as}$	$t_{\rm ex}$	m
QSF*	$370\pm300$	$\sim$105	$4470\pm2300$	$0.07\pm0.06$
SRK*	$656\pm16$	$\sim$100	$2100\pm360$	$0.15\pm0.02$
Radio*	$2848\pm100$	$\sim$110	$604\pm12$	$0.51\pm0.02$
Xline	$4000\pm500$	$153\pm5$	$537\pm70$	$0.60\pm0.07$
1 keV*	$3456\pm105$	$\sim$110	$501\pm15$	$0.63\pm0.02$
FMK	$5290\pm90$	$105\pm1$	$312\pm9$	$0.98\pm0.03$
FMF*	$8816\pm28$	$168\pm6$	$300\pm9$	$0.99\pm0.03$

The expansion time and deceleration parameters derived from the observed Doppler shifts of the Si-K, S-K and Fe-K lines (Xline in Table 3) are in reasonable agreement with the radio proper motion observations (Anderson & Rudnick 1995) and in good agreement with the soft X-ray proper motion measurements of Vink et al. (1999) (1 keV in Table 3).