KOMPARASI LAJU KONVERGENSI METODE EULER DAN RUNGE-KUTTA DALAM PENENTUAN MASSA DAN RADIUS TERSKALA WHITE DWARFS

Abstract

To determine structure of white dwarf stars, two quantities that are crucial are mass and radius of the stars. In order to determine these quantities, the analysis would be more effective computationally when we introduce scaled mass and scaled radius, i.e. to scale the quantities so that we have dimensionless mass and radius. In this article, we find numeric solutions to equation of state of white dwarfs using Euler and Runge-Kutta method to obtain scaled mass and radius of white dwarfs for a given value of central density. The solutions from the methods are also compared to see how fast the solutions converge. The equation of state used is the one used by Chandrasekhar and the electron to nucleon ratio used here is Ye = 1. First, Euler and Runge-Kutta method are applied to solve the coupled differential equation for scaled density and scaled mass. Solutions obtained are then compared for various step sizes. Scaled mass values obtained from Euler and Runge-Kutta method are 1.298014 and 1.298013 respectively for ℎ = 0.000001. Meanwhile, for the same h, scaled radius values are found to be 1.591624 and 1.591629 for the respective method. Comparisons of the two methods indicate that Runge-Kutta converges much faster than Euler method, just as expected. In addition, the convergence of Runge-Kutta is faster for determining scaled mass than for determining scaled radius.