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\def\rosat{{\it ROSAT}}

\begin{document}


\title{THE EXPANSION OF THE X-RAY REMNANT OF TYCHO'S SUPERNOVA
 (SN 1572)}

\author{John P.~HUGHES \\
{\it 	Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Cambridge, MA 02138 USA}}

\maketitle

\section*{Abstract}

The angular expansion rate of the X-ray remnant of the supernova (SN)
observed by Tycho Brahe in AD 1572 has been determined using
observations by \rosat. From two High Resolution Imager (HRI)
pointings separated in time by 4.5 yr, I determine a mean expansion
rate for the blast wave of $0.42 \pm 0.06$$^{\prime\prime}$
yr$^{-1}$, which is larger than published results from optical and radio
observations.  The X-ray expansion is consistent with the dynamical
evolution expected from a young ejecta-dominated remnant which has
experienced only modest deceleration. I find evidence for differential
expansion in that the interior of the remnant appears to be expanding
at a somewhat slower rate.  The expansion rate varies significantly
with azimuth as well. In particular, I find that two small knots in
the southeastern part of Tycho are moving more rapidly than any other
feature in the X-ray image.  Previous work has established that these
knots are clumps of highly enriched stellar ejecta with very different
abundance distributions.  This is the first evidence for ejecta
``bullets'' in the remnant of a Type Ia SN.

\section{Introduction}

The gaseous remnants of historical supernova explosions provide
astronomers with a valuable class of objects for investigating the
dynamical evolution of supernova remnants (SNRs). For these remnants
one knows the average expansion rate and thus a determination of the
current expansion rate yields an indication of the remnant's
deceleration. However the expected motions are small,
$<$1$^{\prime\prime}$ yr$^{-1}$, and thus significant measurements of
current expansion rates in SNRs require high angular resolution, long
time baselines, and deep observations.

\par

Tycho's SNR is an ideal target for such studies. Indeed, the expansion
of Tycho's SNR has been measured previously in both the optical
(Kamper \& van den Bergh 1978) and radio bands (Tan \& Gull 1985). The
observed proper motion of the optical filaments indicate expansion
rates ranging from 0.18$^{\prime\prime}$ yr$^{-1}$ to
0.28$^{\prime\prime}$ yr$^{-1}$. Since the amount of material probed
by the optical emission is a tiny fraction of the total mass of
Tycho's SNR and filaments appear only toward the north and east near
the remnant's periphery, observations in the optical band are unable
to provide a comprehensive picture of the expansion of the remnant.
Radio data reveal a nearly circular, limb-brightened, clumpy shell of
emission. The overall expansion of the entire radio remnant has been
determined to be $0.113 \pm 0.006$ \% yr$^{-1}$, although azimuthal
sectors of the remnant are observed to expand at rates varying from
$\sim$0.05 \% yr$^{-1}$ to $\sim$0.15 \% yr$^{-1}$ or more. Again, the
radio band probes material different from that in the X-ray or
optical: in this case regions of compressed magnetic field and
relativistic electron acceleration.

\section{X-ray Observations}

Figure 1 shows a 104 ks observation of Tycho taken by the HRI in
February 1995.  This image is about five times deeper than any
previous high resolution X-ray image and it reveals interesting new
features.  Most notable is the weak rim of emission seen clearly
around the entire remnant, which is believed to be the blast wave
propagating through the surrounding circumstellar medium. The time
baseline between this pointing and the first epoch one (done in July
1990) was about 4.5 years, roughly 1\% of the SNR's age.  A number of
techniques were developed to estimate background (made difficult by
the strong grain scattering halo from Tycho), generate exposure maps
(including telescope vignetting and spatial variations of the HRI
quantum efficiency), correct for aspect drift, and finally align the
images. The contours on the image show small-scale-size regions where
a significant change in X-ray flux has occurred in 4.5 years.  Notice
that the increased brightness regions (solid contours) are distributed
around the periphery of the remnant, while the regions of dimming
(dashed contours) appear further in.  Simulations were used to assess
the significance of these features and I find that most of the flux
changes are real and represent evidence for expansion of Tycho's SNR.

\begin{figure}[t]
%   \vspace{13pc}
   \centerline{\psfig{file=fig1.ps,height=4.0in}}
   \caption{Deep {\it ROSAT\/} HRI soft (0.2-- 2 keV band)
X-ray image of Tycho's SNR. Solid (dashed) contours show regions of 
increased (decreased) flux when this image is compared to one taken
4.5 years earlier. The contour levels are in units of $\sigma$ and 
the ones shown correspond to values of $-$4, $-$3, 3, 4, 5, and 6.}

\end{figure}

I determined the expansion rate by examining the count rates in
azimuthal sectors of a thin radial region at the edge of the remnant.
Quantitative results were obtained by shifting the first epoch image
along the direction of the center of each sector until the count rates
in the old and new images, from that azimuthal region, were the same.
The X-ray expansion rates as a function of azimuthal angle are shown
in Figure 2 along with radio and optical results for comparison.
Note that each azimuthal sector was $30^\circ$ wide, although we show
results for sectors with centers separated by $15^\circ$.

\begin{figure}[t]
%   \vspace{13pc}
   \centerline{\psfig{file=fig2.ps,height=4.0in}} 

   \caption{Expansion of Tycho's SNR in the X-ray, optical, and radio
   bands. The rate corresponding to the Sedov expansion law ($n=0.4$)
   is shown as the horizontal dashed line. The X-ray points at
   $105^\circ$ and $120^\circ$ include the location of the bright
   knots in the southeastern quadrant of Tycho. Note that these points
   are consistent with undecelerated motion (a value of 0.24\%
   yr$^{-1}$).}

\end{figure}

The azimuthally-averaged current expansion rate of the outermost rim
of the X-ray remnant is $0.42\pm0.06$$^{\prime\prime}$ yr$^{-1}$ (90\%
confidence level errors).  For a radial region that is slightly
interior to this but which still precedes the brightest ring of
emission, I find a slower expansion rate
$0.23\pm0.09$$^{\prime\prime}$ yr$^{-1}$.  When the age, $t$ of a 
remnant is known, it is possible to express its expansion rate in
terms of $n\equiv t\dot \theta/\theta$, where $\theta$ is the angular
size.  The mean radii of the selected radial regions in Tycho
are 248.9$^{\prime\prime}$ and 238.2$^{\prime\prime}$ so the expansion
rates are $n=0.72\pm0.10$ (outer edge) and $n=0.41\pm0.16$ (inner
edge).  The radio data, as analyzed by Tan \& Gull (1985), give an
average expansion rate of $n=0.462$. One possible explanation for the
difference with the X-ray results may be that the radio expansion was
obtained over the entire radial extent of the remnant.  This would
tend to ``dilute'' the effect of the fastest moving material at the
outermost rim of the remnant.

\section{Discussion}

The Sedov (1959) similarity solution for the dynamical evolution of
SNRs predicts an expansion rate of $n=0.4$. As is well known, this
model assumes that the ejecta are dynamically insignificant, so its
applicability to Tycho's SNR, where the ejecta are clearly dominant,
is questionable. Models where the SN ejecta are important have been
developed by Chevalier (1982) in which the ejecta and the ambient
medium are assumed to be in the form of power-law density profiles.
For parameters appropriate to Tycho (i.e., a uniform interstellar
medium and a steep power law, $\rho \propto r^{-7}$, for the exploded
white dwarf envelope), these similarity solutions predict a rate of
$n=0.57$.  Neither of these models accounts for differential
expansion.  Hamilton, Sarazin, \& Szymkowiak (1986) proposed a model
for Tycho wherein the blast wave was expanding as $n=0.70$ and the
ejecta (specifically the contact discontinuity) were expanding more
slowly: $n=0.60$. The azimuthally-averaged X-ray results presented
here are rather consistent with this latter model.

The expansion shows significant azimuthal variation as well.  The two
small X-ray bright knots in the southeast are particularly
interesting, since earlier work (Vancura, Gorenstein, \& Hughes 1995)
has revealed that these features are almost surely highly enriched
knots of SN ejecta with very different abundance distributions.  The
northern one is apparently rich in Si and S, while the southern one is
Fe rich.  These knots are moving considerably more rapidly than the
rest of the remnant (as shown clearly in Fig.~1), and, in addition,
the current expansion rate is consistent with their time-averaged
expansion rate, implying that these clumps of ejecta have not been
decelerated.

I propose that these are fragments or ``bullets'' of ejecta formed in
the SN explosion, which is believed to be Type Ia.  For an assumed
distance to Tycho of 2.5 kpc, the transverse velocity of these knots
is approximately 8300 km s$^{-1}$. The clump masses are of order
$0.002\,M_\odot$ (for the Si+S knot) and $0.0004\,M_\odot$ (for the Fe
knot) and the kinetic energies are $2\times 10^{48}$ ergs s$^{-1}$ and
$2\times 10^{47}$ ergs s$^{-1}$, respectively, again for purely
transverse motion.  These ejecta fragments in Tycho are similar in
many respects to the recently discovered ejecta fragments beyond the
apparent boundary of the Vela SN blast wave (Aschenbach, Egger, \&
Tr\"umper 1995).  Although the current velocities of the Vela ejecta
bullets are small (390--650 km s$^{-1}$), their angular distance from
the estimated site of the SN explosion require large mean transverse
velocities: 4000--5000 km s$^{-1}$ as well as significant
deceleration. The initial kinetic energy of the Vela knots is in the
range (0.3--5)$\times 10^{48}$ ergs s$^{-1}$, however, their masses,
0.01--0.5$\,M_\odot$, are quite a bit larger than the knots in
Tycho. 

In summary it appears likely that the rapidly moving, chemically
enriched features seen in Tycho's SNR are ejecta fragments being
observed at an early stage of evolution before much interaction with
the ambient medium has occurred.  Since their expansion rate is
greater than the rest of the remnant, at some time in the future they
will clearly precede the blast wave.  Their differing chemical
composition require widely different origination sites in the exploded
star: the Fe-rich knot must have originated from near the center where
explosive Si-burning led to only Fe-group nuclei and the Si+S-rich
knot must have come from further out where the flame temperature
dropped and explosive O-burning predominated.  Whether instabilities
of this type prove to be intrinsic to the nature of flame front
propagation in Type Ia SNe or arise from Rayleigh-Taylor and other
instabilities operating on the expanding ejecta at later times
remains to be studied. In any event the results presented here make
clear that Type Ia SN explosions are considerably more inhomogeneous
than previously thought.

\section{References}

%\vspace{1pc}

\def\apj{{\it Ap.~J.}}
\def\mnras{{\it M.N.R.A.S.}}
\def\nature{{\it Nature}}


\re
1.   Aschenbach, B., Egger, R., \& Tr\"umper, J., 1995, \nature, {\bf
373}, 587.
\re
2.   Chevalier, R.~A., 1982, \apj, {\bf 258}, 790.
\re
3.   Hamilton, A.~J.~S., Sarazin, C.~L., \& Szymkowiak, A.~E., 1986,
    \apj, {\bf 300}, 713.
\re
4.   Kamper, K.~W., \& van den Bergh, S., 1978, \apj, {\bf 224}, 851.
\re
5.   Sedov, L.~I., 1959, {\it Similarity and Dimensional Methods in
     Mechanics}, (New York: Academic).
\re
6.   Tan, S.~M., \& Gull, S.~F., 1985, \mnras, {\bf 216}, 949.
\re
7.   Vancura, O., Gorenstein, P., \& Hughes, J.~P., 1995, \apj, {\bf 441}, 680.

%\re
%1. 	Ikegami, H. et.al., 1985, {\it Publ.\ Astron.\ Soc.\ Japan} {\bf 39}, 941.
%\re
%2. 	Ikegami, H. and Shibata, T., 1987, {\it Phys.\ Rev.\ Lett.} {\bf 58}, 2722.
%\re
%3. 	Ikegami, H. and Shibata, T., 1987, {\it Phys.\ Rev.\ Lett.} {\bf 56}, 1566.

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\par\noindent
{\small This research was partially supported by NASA {\it ROSAT\/}
Grant NAG5-2156.}
\end{document}

