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.
Highlights
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
The three equations of motion are applied to four SNRs: Tycho, Cas A, Cygnus loop, and SN 1006
Examples of piece-wise solutions for an SNR can be found in [3]: A first energy conserving phase, r ∝ t0.4 followed by a second adiabatic phase where r ∝ t0.285
Summary
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 r ∝ t0.4 , see [1], and the momentum conservation in the framework of the thin layer approximation with CSM at constant density where R ∝ t0.25 , see [2]. Examples of piece-wise solutions for an SNR can be found in [3]: A first energy conserving phase, r ∝ t0.4 followed by a second adiabatic phase where r ∝ t0.285. The previous analysis allows posing a basic question: “Is it possible to find an analytical solution for SNRs given the three observable astronomical parameters, age, radius and velocity ?”.
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