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In this paper we derive three equations of motion for a supernova remnant (SNR) in the framework of the thin layer approximation using the Padé approximant. The circumstellar medium is assumed to follow a density profile of either an exponential type, a Gaussian type, or a Lane-Emden ( n = 5 ) type. The three equations of motion are applied to four SNRs: Tycho, Cas A, Cygnus loop, and SN 1006. The percentage error of the Padé approximated solution is always less than 10%. The theoretical decrease of the velocity over ten years for SNRs is evaluated.

The equation of motion for a supernova remnant (SNR) can be modeled by a single law of motion or multiple laws of motion when the appropriate boundary conditions are provided. Examples of a single law of motion are: The Sedov expansion in the presence of a circumstellar medium (CSM) with constant density where the radius, r, scales as

This section introduces three density profiles for the CSM: An exponential profile, a Gaussian profile, and a self-gravitating profile of Lane-Emden type.

This density is assumed to have the following exponential dependence on r in spherical coordinates:

where b represents the scale. The piece-wise density is

The total mass swept,

This density has the Gaussian dependence

and the piece-wise density is

The total mass swept,

where

The Lane-Emden profile when

The total mass swept,

The conservation of the momentum in spherical coordinates in the framework of the thin layer approximation states that

where

Assuming an exponential profile as given by Equation (2) the velocity is

where

and

In the above differential equation of the first order in r, the variables can be separated and integration gives the following non-linear equation:

In this case is not possible to find an analytical solution for the radius, r, as a function of time. We therefore apply the Padé rational polynomial approximation of degree 2 in the numerator and degree 1 in the denominator about the point

The resulting Padé approximant for the radius

and the velocity is

and

Assuming a Gaussian profile as given by Equation (4) the velocity is

where

and

The appropriate non-linear equation is

The Padé rational polynomial approximation of degree 2 in the numerator and degree 1 in the denominator about

The resulting Padé approximant for the radius

and the velocity is

and

Assuming a Lane-Emden profile,

where

and

The connected non-linear equation is

The Padé rational polynomial approximation of degree 2 in the numerator and degree 1 in the denominator for the left-hand side of the above equation gives

where

The Padé approximant for the radius is

where

and

and the velocity is

where

and

In the previous section, we derived three equations of motion in the form of non-linear equations and three Padé approximated equations of motion. We now check the reliability of the numerical and approximated solutions on four SNRs: Tycho, see [^{−1}, see

In order to reduce the unknown parameters from four to two, we fix

Name | Age (yr) | Radius (pc) | Velocity (km∙s^{−1}) | References |
---|---|---|---|---|

Tycho | 442 | 3.7 | 5300 | Williams et al. 2016 |

Cas A | 328 | 2.5 | 4700 | Patnaude and Fesen 2009 |

Cygnus loop | 17,000 | 24.25 | 250 | Chiad et al. 2015 |

SN 1006 | 1000 | 10.19 | 3100 | Uchida et al. 2013 |

Name | b(pc) | |||||
---|---|---|---|---|---|---|

Tycho | 0.1 | 1.203 | 8000 | 0.113 | 5.893 | −1.35 |

Cas A | 1 | 0.819 | 8000 | 0.1 | 6.668 | −3.29 |

Cygnus loop | 10 | 12.27 | 3000 | 45.79 | 6.12 | −0.155 |

SN 1006 | 1 | 5.49 | 3100 | 2.332 | 1.455 | −12.34 |

Name | b(pc) | |||||
---|---|---|---|---|---|---|

Tycho | 0.1 | 1.022 | 8000 | 0.561 | 8.517 | −10.469 |

Cas A | 1 | 0.741 | 7000 | 0.406 | 7.571 | −13.16 |

Cygnus loop | 10 | 11.92 | 3000 | 21.803 | 7.875 | −0.161 |

SN 1006 | 1 | 5.049 | 10000 | 4.311 | 4.568 | −18.58 |

Name | b(pc) | |||||
---|---|---|---|---|---|---|

Tycho | 0.1 | 0.971 | 8000 | 0.502 | 3.27 | −14.83 |

Cas A | 1 | 0.635 | 8000 | 0.35 | 4.769 | −23.454 |

Cygnus loop | 10 | 11.91 | 3000 | 27.203 | 7.731 | −0.162 |

SN 1006 | 1 | 5 | 10000 | 4.85 | 3.297 | −19.334 |

The goodness of the approximation is evaluated through the percentage error,

where

The expansion of an SNR can be modeled by the conservation of momentum in the presence of a decreasing density: here we analysed an exponential, a Gaussian and a Lane-Emden profile. The three equations of motion have complicated left-hand sides but simple left-hand sides, viz.,

Zaninetti, L. (2017) Padé Approximant for the Equation of Motion of a Supernova Remnant. Journal of High Energy Physics, Gravitation and Cosmology, 3, 78-86. http://dx.doi.org/10.4236/jhepgc.2017.31011