Pad\'e approximant for the equation of motion of a supernova remnant

Abstract

In this paper we derive three equations of motion for a supernova remnant (SNR) in the framework of the thin layer approximation using the Pad\'e 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 Pad\'e approximated solution is always less than $10\%$. The theoretical decrease of the velocity over ten years for SNRs is evaluated.