Question

Consider a region of a star with P=1.15e24 g s⁻² cm⁻³, T=3.5e9 K, rho=3.7e6 g cm⁻³. The equation of state is radiation dominated. We send through a shock at 20,000 km s⁻¹. What is the pressure, temperature, and density behind the shock? What burning processes can take place at this temperature? Assuming the material burns to completion, what elements will dominate the ejecta from this region of the star?

Answer 1

To solve this question, we will make use of the Rankine-Hugoniot conditions, which provide the laws of conservation of mass, momentum, and energy in the context of a shock wave.

Firstly, let's define our initial conditions. The initial density (rho_1) is 3.7e6 g cm⁻³, the initial pressure (P_1) is 1.15e24 g s⁻² cm⁻³, and the shock speed (v_1) is 20000 km s⁻¹, which is equivalent to 2e10 cm s⁻¹ after conversion.

Because we are dealing with a radiation-dominated gas, we take the adiabatic index (gamma) as 4/3.

Now, we can determine the post-shock conditions.

Applying the Rankine-Hugoniot conditions, we can find the post-shock density (rho_2) as rho_1 * ((gamma + 1) / (gamma - 1)), which calculates to approximately 2.59e7 g cm⁻³.

Next, the post-shock pressure (P_2) is determined through a formula derived from the momentum and energy conservation laws, P_1 * 2 * rho_1 * v_1^2 * ((gamma - 1) / (gamma + 1)). By inserting the given values, we find P_2 to be approximately 4.862e48 g s⁻² cm⁻³.

Finally, keeping in mind that in a radiation-dominated gas, the temperature is proportional to the pressure and inversely proportional to the density, we can use the ideal gas equation where P=nkT to find the post-shock temperature (T_2) as P_2 / rho_2. This gives T_2 a value of approximately 1.878e41 K.

Now, regarding possible burning processes and dominant elements in the ejecta, these would depend on multiple factors such as the post-shock temperature, density, the composition of the star, and additional complex physical processes within the star. A thorough evaluation of these factors would require the implementation of intricate stellar evolution models, necessitating insights deeper than can be provided in this context.

To get a more robust understanding, further astrophysical consultation or software dedicated to astrophysical calculations could be utilized. Some possible processes might include nuclear fusion, producing heavier elements like helium, carbon, or oxygen, but this largely depends on the specific conditions and star composition.