EDP Sciences
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Volume 438, Number 1, July IV 2005
Page(s) 11 - 21
Section Astrophysical processes
DOI http://dx.doi.org/10.1051/0004-6361:20042260

A&A 438, 11-21 (2005)
DOI: 10.1051/0004-6361:20042260

The shocking properties of supersonic flows: Dependence of the thermal overstability on M, $\alpha$, and T $\mathsf{_{c}}\,$/$\,$T $\mathsf{_{0}}$

J. M. Pittard1, M. S. Dobson1, R. H. Durisen2, J. E. Dyson1, T. W. Hartquist1 and J. T. O'Brien1

1  School of Physics and Astronomy, The University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK
    e-mail: jmp@ast.leeds.ac.uk
2  Department of Astronomy, Indiana University, Swain Hall West 319, 727 East 3rd St., Bloomington 47405, USA

(Received 27 October 2004 / Accepted 8 April 2005 )

We present hydrodynamical calculations of radiative shocks with low Mach numbers and find that the well-known global overstability can occur if the temperature exponent ($\alpha$) of the cooling is sufficiently negative. We find that the stability of radiative shocks increases with decreasing Mach number, with the result that M=2 shocks require $\alpha \la -1.2$ in order to be overstable. Such values occur within a limited temperature range of many cooling curves. We observe that Mach numbers of order 100 are needed before the strong shock limit of $\alpha_{\rm cr} \approx 0.4$ is reached, and we discover that the frequency of oscillation of the fundamental mode also has a strong Mach number dependence. We find that feedback between the cooling region and the cold dense layer (CDL) further downstream is a function of Mach number, with stronger feedback and oscillation of the boundary between the CDL and the cooling region occuring at lower Mach numbers. This feedback can be quantified in terms of the reflection coefficient of sound waves, and in those cases where the cooling layer completely disappears at the end of each oscillation cycle, the initial velocity of the waves driven into the upstream pre-shock flow and into the downstream CDL, and the velocity of the the boundary between the CDL and the cooling layer, can be understood in terms of the solution to the Riemann problem. An interesting finding is that the stability properties of low Mach number shocks can be dramatically altered if the shocked gas is able to cool to temperatures less than the pre-shock value (i.e. when $\chi < 1$, where $\chi$ is the ratio of the temperature of the cold dense layer to the pre-shock temperature). In such circumstances, low Mach number shocks have values of $\alpha_{\rm cr}$ which are comparable to values obtained for higher Mach number shocks when $\chi
= 1$. For instance, $\alpha_{\rm cr}=-0.1$ when M=2 and $\chi=0.1$, comparable to that when M=10 and $\chi
= 1$. Thus, it is probable that low Mach number astrophysical shocks will be overstable in a variety of situations. We also explore the effect of different assumptions for the initial hydrodynamic set up and the type of boundary condition imposed downstream, and find that the properties of low Mach number shocks are relatively insensitive to these issues. The results of this work are relevant to astrophysical shocks with low Mach numbers, such as supernova remnants (SNRs) immersed in a hot interstellar medium (e.g., within a starburst region), and shocks in molecular clouds, where time-dependent chemistry can lead to overstability.

Key words: hydrodynamics -- shock waves -- instabilities -- ISM: kinematics and dynamics -- ISM: supernova remnants -- stars: winds, outflows

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