Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T05:33:11.651Z Has data issue: false hasContentIssue false
  • English
  • Français

Universality in the run-up of shock waves to the surface of a star

Published online by Cambridge University Press:  08 April 2011

C. GUNDLACH  and
R. J. LEVEQUE
C. GUNDLACH*
Affiliation:
School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK
R. J. LEVEQUE
Affiliation:
Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA 98195-2420, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the run-up of a shock wave from inside to the surface of a perfect fluid star in equilibrium and bounded by vacuum. Near the surface we approximate the fluid motion as plane-symmetric and the gravitational field as constant. We consider the ‘hot’ equation of state P = (Γ − 1)ρe and its ‘cold’ (fixed entropy, barotropic) form P = K0ρΓ (the latter does not allow for shock heating). We numerically find that the evolution of generic initial data approaches universal similarity solutions sufficiently near the surface, and we explicitly construct these similarity solutions. The two equations of state show very different behaviour because shock heating becomes the dominant effect when it is allowed. In the barotropic case, the fluid velocity behind the shock approaches a constant value, while the density behind the shock approaches a power law in space, as the shock approaches the surface. In the hot case with shock heating, the density jumps by a constant factor through the shock, while the sound speed and fluid velocity behind the shock diverge in a whiplash effect. We tabulate the similarity exponents as a function of the equation of state parameter Γ and the stratification index n∗.

JFM classification

Mathematical Foundations: Computational methods Compressible Flows: Gas dynamics Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 676 , 10 June 2011 , pp. 237 - 264
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Baiotti, L., Giacomazzo, B. & Rezzolla, L. 2008 Accurate evolutions of inspiralling neutron-star binaries: prompt and delayed collapse to black hole. Phys. Rev. D 78, 084033.CrossRefGoogle Scholar
Bale, D., LeVeque, R. J., Mitran, S. & Rossmanith, J. A. 2002 A wave-propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24, 955–978.Google Scholar
Barker, J. W. & Whitham, G. B. 1980 The similarity solution for a bore on a beach. Commun. Pure Appl. Math. 33, 447–460.Google Scholar
Bauswein, A., Janka, H.-T. & Oechslin, R. 2010 Testing approximations of thermal effects in neutron star merger simulations. Phys. Rev. D. (submitted)Google Scholar
Berger, M. & Kohn, R. 1988 A rescaling algorithm for the numerical calculation of blowing-up solutions. Commun. Pure Appl. Math. 41, 841–863.Google Scholar
Bouchut, F. 2004 Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser.Google Scholar
Duez, M. D. 2009 Numerical relativity confronts compact neutron star binaries: a review and status report. Class. Quant. Grav. 27, 114002.Google Scholar
Etienne, Z. B., Liu, Y. T, Shapiro, S. L. & Baumgarte, T. W. 2009 Relativistic simulations of black hole–neutron star mergers: effects of black-hole spin. Phys. Rev. D 79, 044024.Google Scholar
Font, J. A. 2008 Numerical hydrodynamics and magnetohydrodynamics in general relativity. Living Rev. Relativity 11, 7.Google Scholar
Gundlach, C. & LeVeque, R. J. 2010 Codes and simulations for this paper. Available at: www.clawpack.org/links/shockvacuum10.Google Scholar
Gundlach, C. & Murphy, J. 2011 Shock formation in stellar perturbations and tidal shock waves in binaries. Mon. Not. R. Astron. Soc. (submitted).Google Scholar
Gundlach, C. & Please, C. 2009 Generic behaviour of nonlinear sound waves near the surface of a star: smooth solutions. Phys. Rev. D 79, 067501. Erratum Phys. Rev. D 79, 089901.Google Scholar
Keller, H. B., Levine, D. A. & Whitham, G. B. 1959 Motion of a bore over a sloping beach. J. Fluid Mech. 7, 302–316.Google Scholar
Kiuchi, K., Sekiguchi, Y., Shibata, M. & Taniguchi, K. Longterm general relativistic simulation of binary neutron stars collapsing to a black hole. Phys. Rev. D 80, 064037.Google Scholar
Kyutoku, K., Shibata, M. & Taniguchi, K. 2010 Gravitational waves from nonspinning black hole-neutron star binaries: dependence on equations of state. Phys. Rev. D 82, 044049.CrossRefGoogle Scholar
LeVeque, R. J. 1997 Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 131, 327–353.Google Scholar
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
LeVeque, R. J. 2010 A well-balanced path-integral f-wave method for hyperbolic problems with source terms. J. Sci. Comput. (in press). Available at: www.clawpack.org/links/wbfwave10.Google Scholar
LeVeque, R. J. et al. 2010 clawpack software. Available at: www.clawpack.org.Google Scholar
Nakayama, K. & Shigeyama, T. 2005 Self-similar evolution of relativistic shock waves emerging from plane-parallel atmospheres. Astrophys. J. 627, 310–318.Google Scholar
Oechslin, R., Rosswog, S. & Thielemann, F.-K. 2002 Conformally flat smoothed particle hydrodynamics: application to neutron star mergers. Phys. Rev. D 65, 103005.CrossRefGoogle Scholar
Sakurai, A. 1960 On the problem of a shock wave arriving at the edge of a gas. Commun. Pure Appl. Math. 13, 353–370.Google Scholar
Toro, E. F. 1997 Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer.Google Scholar
Trangenstein, J. 2009 Numerical Solution of Hyperbolic Partial Differential Equations. Cambridge University Press.Google Scholar