Abstract
Dynamical evolution of a relativistic explosion resulting from a large amount of energy release in a homogenous medium is studied using the Khalatnikov equation describing relativistic, hydrodynamic, planar flow. The early phase of the explosion is idealized to two stages: a free expansion and a shock wave stage. By the hodograph transformation inverting the dependent and independent variables, the hydrodynamic equations for the relativistic flow are reduced to second-order linear equations in a velocity-enthalpy space and they are solved by the method of Laplace transformation. The propagation laws and flow structures of the relativistic expansion are obtained at each stage. In the free expansion stage, the flow with a sufficiently high sound velocity forms a thin shell of the energy density in the comoving frame at the front and accelerates the front. In the shock wave stage, the Lorentz factor of the shock front decreases logarithmically with time. The transition time from a free expansion to a shock wave stage suggests that the super-light expansion observed in extragalactic radio sources has no spherical geometry but must be confined to a narrow cone.
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References
Amai, S., Fukuda, H., Iso, C. and Sato, M.: 1957,Prog. Theor. Phys. 17, 241.
Beleńkii, S. Z. and Landau, L. D.: 1955,Usp. Fiz. Nauk 56, 309.
Blandford, R. D. and McKee, C. F.: 1976,Phys. Fluid 19, 1130.
Blandford, R. D. and McKee, C. F.: 1977,Monthly Notices Roy. Astron. Soc. 180, 343.
Blandford, R. D., McKee, C. F. and Rees, M. J.: 1977,Nature 267, 211.
Canuto, V. and Tsiang, E.: 1977,Astrophys. J. 213, 27.
Chevalier, R. A.: 1974,Astrophys. J. 188, 501.
Cohen, M. H., Kellermann, K. I., Shaffer, D. B., Linfield, R. P., Moffet, A. T., Romney, J. D., Seilstad, G. A., Pauliny-Toth, I. I. K., Preuss, E., Witzel, A., Shilizzi, R. T. and Geldzahler, B. J.: 1977,Nature 268, 405.
Cooper, F., Fry, G. and Schonberg, E.: 1975,Phys. Rev. D11, 192.
Courant, R. and Hilbert, D.: 1937,Methoden der Mathematischen Physik, II, Berlin.
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: 1954,Tables of Integral Transforms, McGraw-Hill, New York.
Harlow, F. H., Amsden, A. A. and Nix, J. R.: 1976,J. Comp. Phys. 20, 119.
Hoffman, F. de and Teller, E.: 1950,Phys. Rev. 80, 692.
Jones, J. W. and Tobin, W.: 1977,Astrophys. J. 215, 474.
Khalatnikov, I. M.: 1954,Z. Eks. Teor. Phys. 17, 241.
Landau, L. D. and Lifshitz, E. M.: 1959,Fluid Mechanics, Pergamon Press, London.
Liang, E. P. T.: 1977,Astrophys. J. 211, 361.
Marscher, A. P.: 1978,Astrophys. J. 219, 392.
Masuda, N. and Weiner, R. M.: 1978,Phys. Rev. D18, 1515.
Rees, M. J.: 1978,Eighth Texas Symp. on Relativistic Astrophysics.
Shapiro, P. R.: 1979 (preprint).
Shields, G. A. and Wheeler, J. C.: 1978,Astrophys. J. 222, 667.
Spitzer, L., Jr.: 1968,Diffuse Matter in Space, Interscience, New York.
Stanyukovich, K.: 1960,Unsteady Motion of Continous Media, Pergamon Press, London.
Synge, J. L.: 1957,The Relativistic Gas, North-Holland Publ., Amsterdam.
Takahara, F.: 1979 (preprint).
Vitello, P. and Salvati, M.: 1976,Phys. Fluids 19, 1523.
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Yokosawa, M., Sakashita, S. Relativistic hydrodynamics of a free expansion and a shock wave in one-dimension. Astrophys Space Sci 72, 447–475 (1980). https://doi.org/10.1007/BF00639150
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DOI: https://doi.org/10.1007/BF00639150