Ablative Equilibrium Condition ππ , ππ , π£π Ablation area shell g z We define an Ablative RT equilibration problem by defining π, π, π£ at the ablation front (dotdash line in Fig. 1) and the gravity π. The hydro profiles are then calculated with different approximations in two regions below and above the ablation front. (1) In the shell above the ablation front, density is high, temperature is low, so the thermal conduction is neglected here. The shell is performing like a solid piston. So the hydrodynamic equations are approximated as: π£ = π£π π = π (π§) ππΎ οΏ½π (π + ππ£ 2 ) = βππ ππ§ where πΎ = 5/3,π£π is the fluid velocity at the ablation front and should have a negative value. π (π§) is a prescribed entropy function which is used to adjust the shape of the shell. Combining these 3 equations above we obtain: ππ π β² (π§)ππΎ + π π =β ππ§ π (π§) πΎ ππΎβ1 + π£π2 π(π§ = π§πππππ‘πππ ) = ππ This ODE can be readily solved to get the density profile and each quantity is then determined for the shell. (2) In the ablation area below the ablation front, density is low, temperature is very high. The profile in this region is in equilibrium, that is d/dt=0. Simply drop the d/dt terms in hydrodynamic equations: π π£ = ππ π£π β§ π (π + ππ£ 2 ) = βππ βͺ βͺ ππ§ 2 β¨ π οΏ½π£ οΏ½ πΎ π + π π£ οΏ½ β ΞΊ πποΏ½ = βπππ£ πΎβ1 2 ππ§ βͺ βͺππ§ β© π = ππ/π΄ π΄ is the average particle mass π΄ = ππ /(π + 1). For notational convenience, define a constant π β‘ ππ π£π . Combine the 4 equations above to 2 equations: π π 2 ππ β§ οΏ½ + οΏ½ = βππ βͺ π΄ ππ§ π πΎππ π3 ππ β¨π οΏ½ + β π οΏ½ = βππ βͺ 2 ππ§ β©ππ§ (πΎ β 1)π΄ 2π The second equation can be integrated analytically and yield: π3 ππ πΎππ + 2 β π οΏ½ = βπππ§ + ππππ π‘ οΏ½ (πΎ β 1)π΄ 2π ππ§ The integration constant is chosen as ππππ π‘ = πππ§πππππ‘πππ so the RHS terms cancel out at the ablation front. Now we need to solve the ODE system simultaneously: ππ πΎππ π3 β§ =οΏ½ + 2 + πππ§ β ππππ π‘οΏ½ /π (πΎ β 1)π΄ 2π ππ§ βͺ βͺ ππ = βππ ππ§ β¨ βͺ π 2 ππ βͺ π= + β© π π΄ This system need to be solved numerically. Also note that π is a function of T in Spitzerβs model. After π and π are advanced in each step, π can be found via π= π + οΏ½π 2 β 4π 2 π/π΄ 2π/π΄
© Copyright 2026 Paperzz