NSE with screening

SKYNET

2.4 Electron screening

2.4.5 NSE with screening

a 10% correction over the unscreened Boltzmann chemical potential. Intermediate screening is applicable between 7 and 9 GK. At 9 GK, ⁴He nuclei are broken up and free neutrons and protons start to dominate the composition. Thus ζ remains roughly constant at 1, and soΛandp(26)also stop changing rapidly becauseΛ₀does not depend strongly on the temperature. T ∼ 9 GK is also where weak screening sets in, because the thermal energy now overcomes the Coulomb interaction energy, which has been reduced by the smaller average charge per ion provided by the free protons. In that regime, the screening effect is at the 4% to 0.5% level.

Figure2.4 also shows that our screening corrections are indeed approximate. The weak and intermediate screening corrections never meet, so any scheme to transi- tion between them is necessarily approximate and arbitrary to some degree. But considering the fact that our transition scheme needs to work robustly for a wide range of compositions and ion charges, it seems to do reasonably well in interpo- lating between the different (somewhat disjoint) screening regimes. While progress has been made in improving screening calculations in various regimes, a unifying theory for screening across all regimes still eludes (e.g., Itoh et al., 1977; Shaviv and Shaviv,1996; Shaviv and Shaviv,2000; Chugunov et al.,2007).

using numerical derivatives to compute the Jacobian, with limited success. Another complication is that Equation (2.112) itself depends on allY_i on the right-hand side and thus has to be solved iteratively.

We find that it is much more robust to keep the screening corrections fixed during the NR iterations. This does not introduce any additional derivatives in the Jacobian.

NSE is computed exactly as shown in Appendix2.B, with the only difference that Equation (2.112) is used to computeY_i fromη_i, but the terms βµ_sc(Z_i)are constant throughout the NR iterations. In this case, to obtain an NSE composition that is self-consistent with the screening corrections it is based on, the NSE computation itself needs to be iterated.

We start by computing the NSE composition without screening (i.e., µ_sc(Z_i) = 0). We denote the resulting composition by Y⁽⁰⁾. Then we compute µ_sc(Z_i) based onY⁽⁰⁾and use these as the constant screening corrections for the next NSE computation that yieldsY⁽¹⁾. From this new composition we compute new screening corrections, which are used to computeY⁽²⁾ and so on. This iteration stops once max_i(|Y_i⁽ⁿ⁺¹⁾ −Y_i⁽ⁿ⁾|)< 10⁻¹²ornreaches 20 (both of these criteria can be changed by the user). This method of iteratively updating the screening corrections and computing NSE with them being fixed is not guaranteed to converge (but neither is the NR method itself). However, in practice we find that this method works very well and converges quite quickly in a large region of parameter space.

Figure2.5shows the NSE abundance distribution as a function of mass number A for three different temperatures. For all temperatures, ρ=10⁸g cm⁻³andY_e = 0.4, so this is the same composition as the one shown in Figure2.4. To show the impact of the screening corrections, the NSE compositions with and without screening are shown in the left panel. Screening is strongest forT = 3 GK and the effect of screening is to reduce the abundances belowA∼ 80 and slightly increase them above that mass number. Screening is weaker atT = 8 GK, but still enhances the high- mass abundances above A∼60. Finally, forT =13 GK, screening has virtually no effect. In the right panel of Figure2.5we show the ratio of the screening chemical potential µ_sc to the Boltzmann chemical potential µ_BZ(Equation2.128without the rest mass). µ_scdepends on the composition and the charge, hence it is the same for all isotopes of a given element. But the Boltzmann chemical potential also depends on the mass, partition function, and the abundance of a given isotope. Thus for a given charge numberZ, the ratio µ_sc/µ_BZ varies for the isotopes of that charge. In Figure2.5, we show the range of the chemical potential ratios as colored bands. The

66 0 10 20 30 40 50 60 70 80 90 100

10⁻¹⁰ 10⁻⁹ 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

Mass number A

RelativeabundanceY(A)

T = 3 GK T = 8 GK T = 13 GK

with screening without screening

1 10 100

0.01%

0.1%

1%

10%

Charge number Z µsc/µBZ

T = 3 GK T = 8 GK T = 13 GK

Figure 2.5: NSE compositions with and without screening at different temperatures. In all cases, ρ = 10⁸ g cm⁻³ and Y_e = 0.4.

Left: Abundances as a function of mass number Aof the compositions with and without screening. Screening pushes the abundance distribution to slightly higher masses. The effect atT = 3 GK is clearly visible. AtT = 8 GK the screening effect is quite small and at T = 13 GK it is practically absent. Right: The ratio of the screening chemical potential correction to the Boltzmann chemical potential (Equation2.128 without the rest mass). The bands show the range of this ratio for all isotopes with the same charge number Z. The screening correction can be as large as 70% in theT = 3 GK case, and as low as 0.004% forT = 13 GK. At high Z where the bands collapse, all isotopes of the same element have the same screening to Boltzmann chemical potential ratio, because their abundances are all extremely small.

bands collapse to a line at largeZ, because there the abundances of all isotopes are essentially zero. In the strongest screening case (T = 3 GK), the screening effect ranges from 0.1% to 70%. ForT = 8 GK, it ranges from about 0.02% to 10%. And forT =13 GK, the screening effect is much less than 1% except for very largeZ.