Numerical investigations of potential systematic uncertainties in iron opacity measurements at solar interior temperatures

Iron opacity calculations presently disagree with measurements at an electron temperature of $\sim 180–195$ eV and an electron density of ($2–4)\times {10}^{22}{\mathrm{cm}}^{-3}$, conditions similar to those at the base of the solar convection zone. The measurements use x rays to volumetrically heat a thin iron sample that is tamped with low-$Z$ materials. The opacity is inferred from spectrally resolved x-ray transmission measurements. Plasma self-emission, tamper attenuation, and temporal and spatial gradients can all potentially cause systematic errors in the measured opacity spectra. In this article we quantitatively evaluate these potential errors with numerical investigations. The analysis exploits computer simulations that were previously found to reproduce the experimentally measured plasma conditions. The simulations, combined with a spectral synthesis model, enable evaluations of individual and combined potential errors in order to estimate their potential effects on the opacity measurement. The results show that the errors considered here do not account for the previously observed model-data discrepancies.

Figure 1

(a) Cross sections of the different target configurations, their labels, and the typical conditions. The tamper mass above the sample controls the sample conditions. (b) Experimental setup: $Z$ opacity science platform measures FeMg-attenuated and -unattenuated spectra in a single-shot experiment due to the finite backlight-to-sample distance, half-moon sample, and spectrometers fielded along $\pm 9^{\circ }$. (c) Comparison of measured and modeled Fe opacity at $T_e=182$ eV and $n_e=3.1\times {10}^{22}{\mathrm{cm}}^{-3}$ from Ref. [4]. The lower curve (blue) is the calculated opacity [7], which shows a lower bound-free opacity, narrower bound-bound lines, and more distinct opacity windows than the measured opacity (red, upper curve). We confirmed similar discrepancies from all models compared in [4].

Figure 2

Signal detected by the spectrometer at $\pm 9^{\circ }$. Backlight (red arrows) is attenuated by tamper at $-9^{\circ }$ and by FeMg foil and tamper at $+9^{\circ }$ (orange arrows). Self-emission (dashed green arrows) is attenuated by all plasmas between the emission source and the detector. $B$ indicates backlight radiation, and $T_X$ is the transmission of component $X$, where $X=a,b,c,$ and FeMg. Components $a$ and $b$ are the bottom and top tampers on the FeMg-embedded side, respectively. Component $c$ is the tamper on the tamper-only side. $I_X$ is the emergent self-emission at component $X$ where the self-absorption within the component is taken into account.

Figure 3

Simulated spectral image for $+9^{\circ }$ (top) and $-9^{\circ }$ (bottom) spectrometers. Due to the spatial resolution, finite source-to-sample distance, and $\pm 9^{\circ }$ line-of-sight difference, the backlight bright region appears on the FeMg-embedded side at $+9^{\circ }$, while it appears on the tamper-only side at $-9^{\circ }$. Dark vertical lines are FeMg bound-bound absorption features.

Figure 4

(a) Simulated spectra for the Thin CH tamper with the FeMg opacity sample (red curve) and the Thin CH tamper-only target (green curve). (b) The red curve is obtained by dividing the red curve in (a) by the green curve in (a). This is the imitation of our experimental data and takes into account (i) self-emission, (ii) the tamper-transmission difference, and (iii) axial- and temporal-gradient effects. The blue curve is the PrismSPECT FeMg transmission spectra computed at the single $T_e$ and $n_e$ value inferred from the Mg He-$\gamma$ and Ly-$\beta$ lines shown in (a). All spectra are convolved with instrumental broadening.

Figure 5

Comparison of the simulated (red curve) and modeled (blue curve) opacity spectra for the (a) Thin CH, (b) $CH+Be$, (c) and Thick CH cases. Instrumental broadening effects on the modeled spectra are taken into account in transmission spectra before converting them into opacity. Thick gray curves are the measured opacity from Ref. [4]. The discrepancies are not fully explained by any of the dynamic-gradient effects included in the simulation.

Figure 6

(a) Backlight radiation and (b) plasma self-emission observed by the detector through an aperture. The 50-$\mu \mathrm{m}$ slit further limits the detector view along the horizontal direction. The red rectangle illustrates the area that a point on the detector would observe through the aperture and slit. (c) Normalized time history of the simulated backlight radiation (solid red curve) and FeMg self-emission (dashed green curve). The FeMg self-emission peaks 1.5 ns earlier than the time of the backlight median. The FeMg temperature (dotted blue curve) peaks 8% higher than the inferred temperature, which would produce 36% more emission than that at the inferred temperature, 195 eV.

Figure 7

(a) Simulated FeMg-attenuated spectrum with (red curve) and without (blue curve) FeMg self-emission. The black curve (top) is the backlight radiation without FeMg attenuation. (b) The opacity spectra inferred from these simulations. The thick gray spectrum from 7 to 12.7 $\AA$ is the measured opacity published in Ref. [4]. The green curves at the bottom of each figure are the red curve minus the blue curve, indicating how much the self-emission would (a) increase the intensity spectra and (b) decrease the inferred opacity, respectively. The self-emission effect on the inferred opacity is always negative. The simulations rule out the hypothesis that discrepancies in the published range are produced by the self-emission.

Figure 8

Self-emission fraction computed for point calculation (dashed red curve) and detailed simulation (dashed green curve). The red curve is a simple estimate based on 195-eV FeMg self-emission divided by 350-eV Planckian backlight radiation. The detailed investigation (green curve) taking into account duration and area differences verifies that the simple self-emission investigation (dashed red curve) works reasonably well for our opacity platform.

Figure 9

(a) Simulated data for the FeMg-unattenuated spectrum (top, black curve) and the FeMg-attenuated spectra with (red curve) and without (blue curve) CH self-emission. (b) The opacity spectra inferred from these simulations. The thick gray spectrum from 7 to 12.7 $\AA$ is the measured opacity published in Ref. [4]. Green curves at the bottom of each panel are the red minus the blue curve, indicating how much the self-emission would (a) increase the intensity spectra and (b) decrease the inferred opacity. The tamper self-emission effect is significantly smaller than FeMg self-emission and negligible over the published range.

Figure 10

(a) $T_e\left(z\right)$ at the time of the backlight peak simulated for the Thick CH case with (blue curve) and without (red curve) FeMg sample. The shaded region indicates the FeMg region. (b) The resultant bottom-tamper (solid curves) and top-tamper (dotted curves) transmission spectra. While the bottom tamper transmission is slightly higher with FeMg, the top tamper transmission is significantly lower with FeMg due to extra attenuation of the heating radiation in the FeMg region.

Figure 11

(a) Simulated spectra for the FeMg attenuated spectrum (black curve) and the FeMg unattenuated spectra with (red curve) and without (blue curve) the tamper condition difference. (b) The opacity spectra inferred from these simulations. Green curves at the bottom of each panel show the simulated increase in measured opacity due to this effect. The thick gray spectrum from 7 to 12.7 $\AA$ represents the data published in Ref. [4]. The effect is notable for Thick CH data, but it is too small to explain the observed discrepancy.

Figure 12

(a) Simulated attenuated spectrum without sample or tamper self-emission (solid curve) and unattenuated spectrum without tamper self-emission and tamper-transmission differences (dashed curve). (b) The red curve is the inferred opacity spectrum from the simulation without the plasma self-emission and the tamper-transmission difference. It is compared with the opacity spectrum calculated under the inferred conditions (i.e., $T_e=195$ eV and $n_e=3.8\times {10}^{22}{\text{cm}}^{-3}$). Instrumental broadening effects on the modeled opacity (blue) are applied on its transmission. The difference (green spectrum at the bottom) confirms that the error associated with the integration over the predicted temporal and spatial gradients is negligible at $\lambda <12.5$ $\AA$.

Figure 13

Summary of numerically assessed systematic errors in the opacity measurement associated with (a) FeMg sample plasma emission, (b) CH/Be tamper self-emission, (c) FeMg-embedded-side and tamper-only-side tamper-transmission difference, and (d) time- and space-integration effects. The simulated inaccuracy is smaller (i.e., closer to 0) over the published range. The introduced systematic uncertainty is generally smaller for Thin CH (red curve) and larger for Thick CH (blue curve). The distinct gray curve shown from 7 to 12.7 $\AA$ is the discrepancy reported by Bailey et al. [4]. None of the investigated effects explain the reported discrepancy.