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Planetary bow shocks: Asymptotic MHD Mach cones


A direct approach for determining the asymptotic MHD Mach cone is formulated and solved. An implicit analytical solution enables the calculation of the asymptotic downstream slope of MHD Mach cones at any clock angle for arbitrary M s , M a , and ϑ bv . The solution obtained includes all previously known symmetric cases. The elongation and shift of the asymptotic fast mode shock cross-section are studied for a wide range of upstream plasma parameters as well as its unusual ‘chopped’ shape under certain conditions. Our results may be useful for planetary shock modeling and MHD numerical codes verification.


  • Bennett, L., M. G. Kivelson, K. K. Khurana, L. A. Frank, and W. R. Paterson, A model of the Earth’s distant bow shock, J. Geophys. Res., 102(A12), 26927–26941, 1997.

    Article  Google Scholar 

  • Grad, H., Reducible problems in magneto-fluid dynamic steady flows, Rev. of Modern Phys., 32(4), 830–847, 1960.

    Article  Google Scholar 

  • Jeffrey, A., Magnetohydrodynamics, Oliver & Boyd ltd. publ., Edinburg, 1966.

    Google Scholar 

  • Kabin, K., A note on the compression ratio in MHD shocks, J. Plasma Physics, 66(4), 259–274, 2001.

    Article  Google Scholar 

  • Khurana, K. K. and M. G. Kivelson, A variable cross-section model of the bow shock of Venus, J. Geophys. Res., 99(A5), 8505–8512, 1994.

    Article  Google Scholar 

  • Landau, L. D. and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon publ., Oxford, 1984.

    Google Scholar 

  • Peredo, M., J. A. Slavin, E. Mazur, and S. A. Curtis, Three-dimensional position and shape of the bow shock and their variation with Alfvenic, sonic and magnetosonic Mach numbers and interplanetary magnetic field orientation, J. Geophys. Res., 100, 7907–7916, 1995.

    Article  Google Scholar 

  • Petrinec, S. M. and C. T. Russell, Hydrodynamic and MHD equations across the bow shock and along the surfaces of planetary obstacles, Space Sci. Rev., 79, 757–791, 1997.

    Article  Google Scholar 

  • Slavin, J. A., R. E. Holzer, J. R. Spreiter, and S. S. Stahara, Planetary Mach cones: Theory and observation, J. Geophys. Res., 89(A5), 2708–2714, 1984.

    Article  Google Scholar 

  • Spreiter, J. R. and S. S. Stahara, Magnetohydrodynamic and gasdynamic theories for planetary bow waves, in Collisionless Shocks in the Heliosphere: Reviews of Current Research, edited by by B. T. Tsurutany and R. G. Stone, pp. 85–107, Geophysical Monograph series No. 35, AGU publ., Washington, D.C., 1985.

    Chapter  Google Scholar 

  • Spreiter, J. R., A. L. Summers, and A. Y. Alksne, Hydromagnetic flow around the magnetosphere, Planet. Space Sci., 14, 223–253, 1966.

    Article  Google Scholar 

  • Verigin, M. I., G. A. Kotova, A. P. Remizov, V. A. Styazhkin, N. M. Shutte, T.-L. Zhang, W. Riedler, H. Rosenbauer, K. Szego, M. Tatrallyay, and K. Schwingenschuh, Shape and location of planetary bow shocks, Cosmic Research, 37(1), 34–39, 1999.

    Google Scholar 

  • Verigin, M., G. Kotova, A. Szabo, J. Slavin, T. Gombosi, K. Kabin, F. Shugaev, and A. Kalinchenko, Wind observations of the terrestrial bow shock: 3-D shape and motion, Earth Planets Space, 53(10), 1001–1009, 2001.

    Article  Google Scholar 

  • Whitham, G. B., Linear and nonlinear waves, A Wiley-Interscience publ., New York, 1974.

    Google Scholar 

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Verigin, M., Slavin, J., Szabo, A. et al. Planetary bow shocks: Asymptotic MHD Mach cones. Earth Planet Sp 55, 33–38 (2003).

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  • Slow Shock
  • Fast Shock
  • Mach Cone
  • Clock Angle
  • Asymptotic Slope