Question

Use the Python ODE integrator solve_ivp to solve Equations (8) and (9) subject to the boundary conditions m(r=0) = 0 and rho(r=0) = rhoc. Your goal is to determine the radius and mass of model white dwarfs by starting the integration at rₛ = 0 and integrating outward to a distance rs where the density drops to zero, i.e. , rho(rₛ)= 0. Calculate solutions for 10 values of rhoc in the range 10⁻¹ to 2.5 · 10⁶ and take μe=2. [Hint: if starting your integration at exactly r=0 raises a ZeroDivisionError, can you start it at some tiny value of r instead?

Answer 1

To model white dwarfs using Python's solve_ivp, start the integration at a very small nonzero value of r and iterate over an array of central densities rhoc. Integrate until the density rho reaches zero to find the radius and mass for each case. Use numpy to create the array and solve_ivp for integration.

Step-by-step explanation:

To use Python's solve_ivp to solve Equations (8) and (9) with the given boundary conditions and calculate the radius and mass of model white dwarfs, integration must begin at some small value of r instead of zero to avoid ZeroDivisionError. Given the boundary conditions m(r=0) = 0 and rho(r=0) = rhoc, and integrating outward until the density rho drops to zero, solutions will be calculated for various values of central density rhoc within the range of 10⁻¹ to 2.5 · 10⁶, using μe=2.

Approaching the equations by starting at a nonzero, but very small r, such as rδ=10⁻¹⁰ m, allows for the simulation of white dwarf characteristics without mathematical singularity issues. Iterating through the range of central densities and solving for each case will enable the determination of white dwarf characteristics like mass and radius.

An array of central densities can be created using numpy functions like linspace or logspace to cover the specified range for rhoc. The integration process using solve_ivp would yield the radius and mass of the white dwarf for each case once the density reaches zero.