Uranium Abundances and Ages of r-process Enhanced Stars with Novel U ii Lines^*

We observed J0954+5246 with Keck/HIRESr (Vogt et al. 1994) on 2021 March 26 for a total of 6.6 hr. The observations were broken down as eight exposures of 1800 s, 1 exposure of 1400 s, and 1 exposure of 1350 s. The observations were taken with the red cross-disperser, using the 5.0 × 040 slit and with 1 × 1 binning, which yielded a resolving power of R ∼ 86,600. We used the blue and the green CCD chip data, which we reduced with MAKEE ¹⁶ using standard settings. The full wavelength range of the spectra was 3600–6800 Å. We corrected each spectrum for radial velocity by cross-correlating against a high-resolution spectrum of HD 122563. We normalized, coadded, and stitched the spectra to furnish the final spectrum with S/N per pixel of 185 at 4050 Å.

We observed J2038–0023 with Magellan/MIKE (Bernstein et al. 2003) on 2018 July 8, 2018 July 24, 2018 September 26, and 2018 November 11, for a total of 15.6 hr. We used the 035 slit and 2 × 1 binning, which yielded a resolving power of R ∼ 83,000. Data for this star already existed (Placco et al. 2017), but with 2 × 2 binning, which could have undersampled the profiles of the U ii lines, so we reobserved the star. We reduced the spectra from each night, together, using CarPy (Kelson et al. 2000; Kelson 2003) and corrected the radial velocity by cross-correlating against a high-resolution spectrum of HD 122563. We normalized, coadded, and stitched the spectra to furnish the final spectrum with S/N per pixel of 175 at 4050 Å.

We used the data from Frebel et al. (2007), who observed the star with VLT/UVES (Dekker et al. 2000) in 2005 and 2006, using image slicer No. 2 and 045 slit width to achieve a resolving power of R ∼ 75,000. The data are publicly available on the ESO raw data archive. ¹⁷ We used the BLUE 437 nm setting observations from 2006 April 22, 2006 April 23, and 2006 May 19, which amounted to a total exposure time of 3.0 hr. The wavelength range of the final spectrum was 3758–4990 Å, which included all of the lines of interest. We reduced the data using ESOReflex (Freudling et al. 2013), with order extraction set to the recommended linear method for data collected with an image slicer. We corrected the radial velocity of each exposure by cross-correlating with a high-quality rest-frame MIKE/Magellan spectrum of HE 1523–0901. We normalized, coadded, and stitched the spectra to furnish the final spectrum with S/N per pixel of 200 at 4050 Å.

We used data from Hill et al. (2002), who observed the star with VLT/UVES (Dekker et al. 2000) in 2000 using 045 slit width. The data are publicly available on the ESO raw data archive. We used the BLUE arm 380–510 nm setting observations from 2000 October 17 and 2000 October 19, which totaled 2 hr of exposure. The resulting resolving power was R ∼ 75,000. We reduced the data using ESOReflex (Freudling et al. 2013), with order extraction set to the recommended optimal method. We corrected the radial velocity of each exposure by cross-correlating to a high-quality MIKE/Magellan spectrum of HE 1523–0901. We normalized, coadded, and stitched the spectra to furnish the final spectrum with S/N of 125 per pixel at 4050 Å.

We took special care to constrain the abundances of elements that have transitions blended with the weak U ii lines. We identified that transitions of Fe i, CH, CN, and La ii are blended with the U ii lines investigated in this work. We obtained the Fe abundance using EW measurements of a subset of acceptable Fe i lines listed in Roederer et al. (2018), for each sample star. We estimated the uncertainty on the mean Fe abundance as the standard deviation in the abundances of the chosen Fe i lines. We determined the C abundance by fitting the λらむだ4313 Å G band of CH. Based on the quality of the data and the synthetic-spectrum fits, we set a fiducial uncertainty estimate of ±0.2 dex on the C abundance of all of the sample stars. With the C abundance fixed, we determined the isotopic ratio of ¹²C/¹³C by fitting the ¹³CH feature at λらむだ4217 Å. We determined the N abundance by fitting the λらむだ3876 Å CN molecular band for all of our sample stars, except for CS 31082–001, for which we could not derive a reliable N abundance. For the N abundance of J0954+5246, RAVE J203843.2–002333, and HE 1523–0901, we estimated an uncertainty of ±0.2 dex, based on the spectral synthesis fits. For each sample star, we determined the La abundance with the spectral synthesis of a subset of blend-free and acceptable La ii transitions listed in Roederer et al. (2018). We estimated the uncertainty on the mean La abundance as the standard deviation in the La abundances of the chosen La ii lines. We list the Fe, C, N, and La abundances and the ¹²C/¹³C isotopic ratio, determined for each star, in Table 3, along with their corresponding values from previous literature studies. The abundances determined in this work are in agreement with the values from the literature, within uncertainties. The exception to this case is our derived La abundance for J2038–0023, which disagrees with the abundance derived by Placco et al. (2017). However, this discrepancy is simply attributed to the difference in our adopted stellar parameters (see Section 3).

Table 3. Abundances and Isotopic Ratios of U ii Line Blends

	Source	J0954+5246 a	J2038–0023 b	HE 1523–0901 c	CS 31082–001 d
$\mathrm{log}\epsilon$(Fe)	This Work	4.55 ± 0.15	4.41 ± 0.12	4.53 ± 0.14	4.58 ± 0.11
	Other	4.51 ± 0.12	4.59 ± 0.12	4.50 ± 0.20	4.60 ± 0.13
$\mathrm{log}\epsilon$(C)	This Work	4.97 ± 0.20	5.11 ± 0.20	5.17 ± 0.20	5.75 ± 0.20
	Other	4.94 ± 0.20	5.08 ± 0.20	5.14	5.82 ± 0.05
12C/13C	This Work	4.0	4.6	3.5	19.0
	Other	⋯	⋯	∼3–4	>20.0
$\mathrm{log}\epsilon$(N)	This Work	5.82 ± 0.20	5.56 ± 0.20	5.88 ± 0.20	⋯
	Other	⋯	⋯	5.43	<5.22
$\mathrm{log}\epsilon$(La)	This Work	−1.06 ± 0.09	−1.06 ± 0.05	−0.47 ± 0.11	−0.65 ± 0.07
	Other	−1.15 ± 0.10	−0.76 ± 0.07	−0.63	−0.60 ± 0.04

Notes. Source of other work:

^a Holmbeck et al. (2018). ^b Placco et al. (2017). ^c Frebel et al. (2007). ^d Hill et al. (2002).

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So far in the literature, U abundances of RPE stars have been primarily determined using a single U ii line at λらむだ3859 Å. ²² Moreover, U abundance analyses have been carried out by different studies for individual stars, with each study varying in the employed method and atomic data.

In this study, we performed, for the first time, a homogeneous analysis to determine U abundances of four highly RPE stars using three U ii lines at λらむだ3859 Å, λらむだ4050 Å, and λらむだ4090 Å. We generated the linelist for the spectral synthesis of the U ii line-regions with linemake. We used the $\mathrm{log}{gf}$ measurements of the U ii lines from Nilsson et al. (2002a), who measured them with high accuracy by combining their branching fraction calculations with the radiative lifetime measurements of six U ii levels from Lundberg et al. (2001). We list the atomic parameters employed for the three U ii transitions in Table 4.

Table 4. Atomic Parameters of U ii Transition Lines

Species	λらむだ (Å)	χかい (eV)	$\mathrm{log}{gf}$	% Uncertainty
				in gf
U ii	3859.57	0.036	−0.067	12
U ii	4050.04	0.000	−0.706	7
U ii	4090.13	0.217	−0.184	13

Note. $\mathrm{log}{gf}$ values and percent uncertainty on gf values taken from Table 2 of Nilsson et al. (2002a). Excitation potential taken from linemake.

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We determined the final U abundance of each star as the weighted-average of the U abundances from the three U ii lines, i.e., $\mathrm{log}\epsilon$(U) = ${\sum }_{i}({w}_{i}\mathrm{log}{\epsilon }_{i})/{\sum }_{i}{w}_{i}$, where $\mathrm{log}{\epsilon }_{i}$ is the U abundance from line i and ${w}_{i}=1/{\rm{\Delta }}(\mathrm{stat}{)}_{i}^{2}$ for line i (McWilliam et al. 1995; Ji et al. 2019). We detail the method we used to estimate Δでるた(stat)_i , the statistical uncertainty on $\mathrm{log}{\epsilon }_{i}$, in Section 4.5. For the total uncertainty on the average-weighted U abundances, we accounted for systematic uncertainties (from stellar parameters and blends) and statistical uncertainties (from $\mathrm{log}{gf}$ measurement and continuum placement). We discuss this further in Section 4.5. We list the final weighted-average U abundance with the associated total uncertainty for all of the sample stars in Table 5.

Table 5. U, Th, and Eu Abundances and Nucleocosmochronometric Ages ^a

Source	$\mathrm{log}\epsilon$(X)	J0954+5246 b	J2038–0023 c	HE 1523–0901 d	CS 31082–001 e
Other Work	$\mathrm{log}\epsilon$(U)3859	−2.13 ± 0.20	−2.14 ± 0.20	−2.06 ± 0.12	−1.92 ± 0.17
This Work	$\mathrm{log}\epsilon$(U)3859	−2.45 ± 0.30	−2.50 ± 0.26	−1.93 ± 0.18	−2.00 ± 0.22
	$\mathrm{log}\epsilon$(U)4050	−2.50 ± 0.33	−2.34 ± 0.30	−2.00 ± 0.48	−1.60 ± 0.21
	$\mathrm{log}\epsilon$(U)4090	−2.60 ± 0.30	−2.50 ± 0.24	−2.15 ± 0.28	−2.00 ± 0.25
	$\mathrm{log}\epsilon$(U)	−2.50 ± 0.29	−2.47 ± 0.21	−1.96 ± 0.25	−1.87 ± 0.19
Other Work	$\mathrm{log}\epsilon$(Th)	−1.13 ± 0.10	−1.24 ± 0.10	−1.2 ± 0.05	−0.98 ± 0.13
This Work	$\mathrm{log}\epsilon$(Th)4019	−1.92 ± 0.09	− 1.70 ± 0.05	−1.22 ± 0.11	−1.18 ± 0.04
	$\mathrm{log}\epsilon$(Th)4086	−1.70 ± 0.09	− 1.63 ± 0.05	−0.95 ± 0.11	−1.09 ± 0.04
	$\mathrm{log}\epsilon$(Th)4095	−1.76 ± 0.09	− 1.57 ± 0.05	−1.01 ± 0.11	−1.10 ± 0.04
	$\mathrm{log}\epsilon$(Th)	−1.79 ± 0.18	−1.63 ± 0.21	−1.06 ± 0.19	−1.12 ± 0.16
Other Work	$\mathrm{log}\epsilon$(Eu)	−1.19 ± 0.10	−0.75 ± 0.10	−0.62 ± 0.05	−0.76 ± 0.13
This Work	$\mathrm{log}\epsilon$(Eu)	−1.16 ± 0.12	−1.16 ± 0.13	−0.53 ± 0.08	−0.81 ± 0.12
	Chronometer	J0954+5246	J2038–0023	HE 1523–0901	CS 31082–001
Age (Gyr)	U/Th	11.1 ± 6.4	13.5 ± 4.8	16.6 ± 5.1	11.1 ± 4.0
(±sys ± stat ± PR)		(±5.7 ± 1.9 ± 2.2)	(±3.8 ± 2.1 ± 2.2)	(±4.2 ± 2.0 ± 2.2)	(±2.8 ± 1.8 ± 2.2)
Age (Gyr)	U/Eu	12.0 ± 4.3	11.3 ± 3.1	14.2 ± 3.8	7.3 ± 2.6
(±sys ± stat ± PR)		(±3.9 ± 1.0 ± 1.6)	(±2.2 ± 1.4 ± 1.6)	(±3.3 ± 1.0 ± 1.6)	(±1.6 ± 1.3 ± 1.6)

Notes.

^a Nucleocosmochronometric ages listed are obtained in this work. See the text for details on uncertainty estimation for the elemental abundances and stellar ages. ^b Source of other work: Holmbeck et al. (2018). ^c Source of other work: Placco et al. (2017). ^d Source of other work: Frebel et al. (2007). ^e Source of other work: Hill et al. (2002).

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4.2.1. The λらむだ3859 Å U ii Line

While the λらむだ3859.57 Å line is the strongest U ii line discernible in the spectra of stars, blends from other transitions make its spectral synthesis quite difficult. This line is situated in the wing of a strong Fe i line at λらむだ3859.91 Å and is further blended with a CN feature at λらむだ3859.65 Å, which also resides in the wing of the Fe i line. Therefore, it is essential to constrain the wing of the Fe i line as well as the CN feature for a reliable U abundance determination.

To fit the wings of the strong Fe i line, we used the Unsöld approximation (Unsold 1955) multiplied by a factor of −8.77 for the Van der Waals hydrogen collision-damping coefficient of the Fe i line, for all of the sample stars. Furthermore, we adjusted the derived Fe i abundances of the stars by −0.15, −0.05, +0.02, and +0.03 dex for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. For most of the stars, the adjustment made to the derived Fe i abundance of the star is small and lies within the uncertainty on the Fe i abundance.

To fit the CN feature, we adjusted the derived N abundance of the stars by +0.0, +0.04, and +0.12 dex for J0954+5246, J2038–0023, and HE 1523–0901, respectively. We find these adjustments acceptable, since they are within the ±0.2 dex uncertainty on the derived N abundances. For CS 31082–001, we used $\mathrm{log}\epsilon$(N) = 5.22, which is the upper limit placed on the N abundance by Hill et al. (2002), as we could not derive a reliable N abundance for the star. We used the derived C abundance of the stars without any adjustments.

For the purpose of a better fit to the neighboring line features, we blueshifted the transition wavelength of the Nd ii line at 3859.42 Å and the Fe i line at 3859.21 Å by 0.07 Å. To fit the Nd ii line, we adjusted the derived Nd ii abundance of the stars by −0.12, +0.0, +0.04, and −0.05 dex for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. The adjustment is small for most stars and within the uncertainty of the Nd abundance.

We determined $\mathrm{log}\epsilon$(U)₃₈₅₉ = −2.45, −2.50, −1.93, and −2.0 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. In Figure 1, we show the resulting best-fit spectral synthesis model for the observed data of each star, along with the residuals between the model and the data. We also depict the ±0.2 dex abundance variation from the derived λらむだ3859 U abundance in the red-shaded region, for all of the sample stars.

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4.2.2. The 4050 Å U ii

The λらむだ4050.04 Å U ii line is blended with an La ii line at λらむだ4050.07 Å, which needs to be constrained well for U abundance determination. linemake obtains $\mathrm{log}{gf}$ = 0.11 for the La ii line from Corliss & Bozman (1962), but we used an updated $\mathrm{log}{gf}$ = 0.428, as measured by Bord et al. (1996). We substantiated this choice with spectral synthesis of the La ii line in RPE stars with minimal U contamination, which showed that the Bord et al. (1996) value provides a better fit to the observed spectra of these stars. Additionally, we blueshifted the transition wavelength of the La ii line by 0.02 Å to enable a better fit to the observed data. We also account for the hyperfine splitting (HFS) structure of this La ii line, as described in the Appendix. We applied the described prescription for the La ii line uniformly across all of the sample stars to determine their U abundances. We employed the derived La abundance of each star in the spectral synthesis without any adjustment.

We determined $\mathrm{log}\epsilon$(U)₄₀₅₀ = −2.50, −2.34, −2.00, and −1.60 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. We show the corresponding best-fit spectral synthesis models for all of the stars in Figure 2, along with the resulting residuals between the model and the observed data. We also depict ±0.2 dex variations in the λらむだ4050 U abundance with a red-shaded region for all of the stars. We generally find a good fit to the λらむだ4050 Å spectral region, as seen in Figure 2. We note an overestimation of the synthetic model flux around λらむだ4049.95 Å for J0954+5246 and CS 31082–001. This could possibly indicate an unidentified line between the Gd ii and U ii lines that has manifested itself more strongly in J0954+5246 and CS 31082–001, as compared to in J2038–0023 and HE 1523–0901. Alternatively, the abundance of the La ii HFS structure may not be well represented by the mean La abundances determined for the stars.

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4.2.3. The λらむだ4090 Å U ii Line

The λらむだ4090.13 Å U ii line is blended with one weak Fe i line at λらむだ4090.07 Å. We derived the λらむだ4090 U abundance of all of the sample stars with spectral synthesis, specifically, by fitting the U ii line to the redward wing of the Fe–U absorption feature. We determined $\mathrm{log}\epsilon$(U)₄₀₉₀ = −2.60, −2.50, −2.15, and −2.00 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. We show the corresponding best-fit spectral synthesis models for all of the stars in Figure 3, along with residuals between the models and the observed data. We also depict ±0.2 variation in the best-fit λらむだ4050 U abundance with the red-shaded region for all of the sample stars. We note that the two absorption features blueward and redward of the U ii line are currently unidentified in the linemake linelist. We tested the effect of these unidentified features by adding "fabricated" lines to mimic them and found that they have minimal-to-no effect on the U abundance determination. In Figure 4, we also show the best-fit spectral synthesis for a wider wavelength window of this line region, for all of the stars. This figure depicts that even though the immediately neighboring lines of the λらむだ4090.13 Å U ii line are unidentified, we found an optimal continuum placement for all of the stars using other spectral regions.

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We determined Th abundances for all of the sample stars using Th ii lines at λらむだ4019.13 Å, λらむだ4086.52 Å, and λらむだ4094.75 Å. We generated the linelists for spectral synthesis with linemake, using $\mathrm{log}{gf}$ values from Nilsson et al. (2002b).

The Th ii line at λらむだ4019.13 Å is the strongest Th line detectable in the optical spectra of stars. It is blended with a Ce ii line at λらむだ4019.06 Å, an Fe i line at λらむだ4019.04 Å, and ¹³CH lines at λらむだ4018.98 Å and λらむだ4019.15 Å (see Figure 5). For the spectral synthesis of this region, we employed the abundances of the blends without any adjustments. We determined $\mathrm{log}\epsilon$(Th)₄₀₁₉ = −1.92, −1.70, −1.22, and −1.18 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. The corresponding best-fit spectral synthesis model is shown in Figure 5 for each star, along with the residual between the mode and the observed data. We note that the synthetic spectrum is overestimated around the λらむだ4018.98 Å and λらむだ4019.25 Å regions for all of the sample stars. This suggests a need to revisit the atomic parameters of the lines in this spectral region and perhaps identify unknown transitions. Nevertheless, we expect a minimal effect of these wing-features on the Th abundance, which was robustly determined by constraining the fit of the synthetic spectrum to the core of the absorption feature.

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The Th ii line at λらむだ4086.52 Å is situated next to a La ii line and partly blended with a Ce ii line. With spectral synthesis of this line-region, we determined $\mathrm{log}\epsilon$(Th)₄₀₈₆ = −1.70, −1.63, −0.95, and −1.09 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001. For the spectral synthesis of CS 31082–001, we adjusted the derived Ce abundance of the star by −0.05 dex.

The Th ii line at λらむだ4094.75 Å is blended with a CH line and partly blended with an Er ii line. For the purpose of a good spectral synthesis fit to the region, we allowed an adjustment of the Er abundance within ±0.2 dex of the derived Er abundance for all of the sample stars. Since the Er ii line is blended with only a section of the blueward wing of the Th ii line, any adjustment of the Er abundance had a minimal effect on the synthetic-spectrum fit to the core of the Th ii line. Subsequently, we determined $\mathrm{log}\epsilon$(Th)₄₀₉₅ = −1.76, −1.57, −1.01, and −1.10 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively.

The final Th abundance of each sample star was obtained as the mean of the λらむだ4019.13, λらむだ4086.52, and λらむだ4094.75 Th abundances. We determined mean Th abundance as $\mathrm{log}\epsilon$(Th) = −1.79, −1.31, −1.06, and −1.12 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. We list the Th abundances obtained for each line and the corresponding mean Th abundance for each star in Table 5, along with the uncertainty estimates. Table 5 also lists the Th abundances determined in previous literature studies for comparison. For J0954+5246, HE 1523–0901, and CS 31082–001, we obtain good agreement with the Th abundances published in the literature. For J2038–0023, we note some discrepancy, which we attribute to the difference in the adopted stellar parameters (see Section 3).

We determined the Eu abundance of each sample star using Eu ii lines at λらむだ4219 Å, λらむだ4205 Å, and λらむだ4435 Å. We determined the mean $\mathrm{log}\epsilon$(Eu) = −1.16, −1.16, −0.53, and −0.81 for J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001, respectively. We list the mean Eu abundance, along with the uncertainty estimates and Eu abundance estimates from previous literature studies in Table 5. For J0954+5246, HE 1523–0901, and CS 31082–001, we find our derived Eu abundances to be in good agreement with the literature estimates within uncertainties. For J2038–0023, we note a discrepancy in the abundances, which we attribute to the difference in the adopted stellar parameters.

For the U abundances of the sample stars, we homogeneously accounted for various sources of systematic (Δでるた(sys)) and statistical (Δでるた(stat)) uncertainties. For Δでるた(sys), we considered the uncertainties on the stellar parameters (T_eff, $\mathrm{log}g$, and ξくしー) and the abundances of the blending elements. We list the individual systematic uncertainty components, ΔでるたT_eff, Δでるた$\mathrm{log}g$, Δでるたξくしー, and Δでるた(blend) in Table 6. For Δでるた(stat), we considered the uncertainties on the $\mathrm{log}{gf}$ values (Δでるた(log gf)) and the continuum placement of the synthetic spectra (Δでるた(cont)), which we also list in Table 6. For each U ii line, we estimated the individual uncertainty components, ΔでるたT_eff, Δでるた$\mathrm{log}g$, Δでるたξくしー, Δでるた(blend), Δでるた(log gf), and Δでるた(cont).

Table 6. Abundance Uncertainties

	ΔでるたTeff (K)	Δでるた $\mathrm{log}g$ (cgs)	Δでるたξくしー (km s−1)	Δでるた(blend)	Δでるた(sys)	Δでるた$\mathrm{log}{gf}$	Δでるた(cont)	Δでるた(stat)	Δでるた(total)
J0954+5246	+150	+0.30	+0.20	±1σしぐま			±0.5%
$\mathrm{log}\epsilon$(U)3859	+0.10	+0.10	+0.05	±0.23	±0.27	±0.05	±0.10	±0.11	±0.30
$\mathrm{log}\epsilon$(U)4050	+0.05	+0.05	+0.02	±0.30	±0.31	±0.03	±0.10	±0.10	±0.33
$\mathrm{log}\epsilon$(U)4090	+0.08	+0.09	+0.04	±0.23	±0.26	±0.06	±0.15	±0.15	±0.30
$\mathrm{log}\epsilon$(U)	+0.08	+0.08	+0.04	±0.25	±0.28	⋯	⋯	±0.07	±0.29
$\mathrm{log}\epsilon$(Th)	+0.15	+0.08	+0.00	⋯	±0.17	⋯	⋯	±0.05	±0.18
$\mathrm{log}\epsilon$(Eu)	+0.09	+0.07	−0.02	⋯	±0.12	⋯	⋯	±0.01	±0.12
$\mathrm{log}\epsilon$(U/Th)	−0.07	+0.00	−0.00	±0.25	±0.26	⋯	⋯	±0.09	±0.28
$\mathrm{log}\epsilon$(U/Eu)	−0.01	+0.01	+0.06	±0.25	±0.26	⋯	⋯	±0.07	±0.27
J2038–0023	+150	+0.30	+0.20	±1σしぐま			±0.5%
$\mathrm{log}\epsilon$(U)3859	+0.10	+0.05	+0.00	±0.20	±0.23	±0.06	±0.10	±0.12	±0.26
$\mathrm{log}\epsilon$(U)4050	+0.04	+0.17	+0.04	±0.14	±0.23	±0.04	±0.20	±0.20	±0.31
$\mathrm{log}\epsilon$(U)4090	+0.10	+0.08	+0.02	±0.08	±0.15	±0.05	±0.20	±0.21	±0.26
$\mathrm{log}\epsilon$(U)	+0.08	+0.10	+0.02	±0.14	±0.19	⋯	⋯	±0.09	±0.21
$\mathrm{log}\epsilon$(Th)	+0.18	+0.11	+0.01	⋯	±0.21	⋯	⋯	±0.03	±0.21
$\mathrm{log}\epsilon$(Eu)	+0.11	+0.07	+0.00	⋯	±0.13	⋯	⋯	±0.02	±0.13
$\mathrm{log}\epsilon$(U/Th)	−0.10	−0.01	+0.01	±0.14	±0.17	⋯	⋯	±0.10	±0.20
$\mathrm{log}\epsilon$(U/Eu)	−0.03	+0.03	+0.02	±0.14	±0.15	⋯	⋯	±0.09	±0.17
HE 1523–0901	+150	+0.30	+0.20	±1σしぐま			±0.5%
$\mathrm{log}\epsilon$(U)3859	+0.10	+0.10	+0.03	±0.08	±0.17	±0.04	±0.06	±0.07	±0.18
$\mathrm{log}\epsilon$(U)4050	+0.01	+0.25	+0.20	±0.25	±0.41	±0.03	±0.25	±0.25	±0.48
$\mathrm{log}\epsilon$(U)4090	+0.10	+0.10	+0.05	±0.11	±0.19	±0.05	±0.21	±0.22	±0.28
$\mathrm{log}\epsilon$(U)	+0.07	+0.15	+0.09	±0.15	±0.24	⋯	⋯	±0.07	±0.25
$\mathrm{log}\epsilon$(Th)	+0.14	+0.11	+0.0	⋯	±0.18	⋯	⋯	±0.06	±0.19
$\mathrm{log}\epsilon$(Eu)	+0.06	+0.04	−0.03	⋯	±0.08	⋯	⋯	±0.02	±0.08
$\mathrm{log}\epsilon$(U/Th)	−0.07	+0.04	+0.09	±0.15	±0.19	⋯	⋯	±0.09	±0.21
$\mathrm{log}\epsilon$(U/Eu)	+0.01	+0.11	+0.12	±0.15	±0.22	⋯	⋯	±0.07	±0.23
CS 31082–001	+150	+0.30	+0.20	±1σしぐま			±0.5%
$\mathrm{log}\epsilon$(U)3859	+0.10	+0.10	+0.00	±0.13	±0.19	±0.05	±0.10	±0.11	±0.22
$\mathrm{log}\epsilon$(U)4050	+0.05	+0.12	+0.03	±0.10	±0.17	±0.03	±0.13	±0.13	±0.21
$\mathrm{log}\epsilon$(U)4090	+0.10	+0.10	+0.00	±0.07	±0.16	±0.06	±0.18	±0.19	±0.25
$\mathrm{log}\epsilon$(U)	+0.08	+0.11	+0.01	±0.10	±0.17	⋯	⋯	±0.08	±0.19
$\mathrm{log}\epsilon$(Th)	+0.14	+0.06	−0.02	⋯	±0.15	⋯	⋯	±0.02	±0.16
$\mathrm{log}\epsilon$(Eu)	+0.07	+0.08	−0.01	⋯	±0.11	⋯	⋯	±0.05	±0.12
$\mathrm{log}\epsilon$(U/Th)	−0.06	+0.05	+0.03	±0.10	±0.13	⋯	⋯	±0.08	±0.15
$\mathrm{log}\epsilon$(U/Eu)	+0.01	+0.03	+0.02	±0.10	±0.11	⋯	⋯	±0.09	±0.14

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To estimate U abundance uncertainties from stellar parameters, we independently changed each stellar parameter by its uncertainty. We thus changed T_eff by +150 K, $\mathrm{log}g$ by +0.3 dex, and ξくしー by 0.2 km s⁻¹, for all of the stars. For every stellar parameter, we rederived the abundances of key elements and resynthesized the U ii lines. We report the resulting change in the U abundances as ΔでるたT_eff, Δでるた$\mathrm{log}g$, and Δでるたξくしー, for the respective stellar parameters.

We estimated Δでるた(blend) by changing the abundance of the blending element (e.g., Fe and La) by ±1σしぐま and rederiving the U abundance. Here σしぐま is the standard deviation in the abundance of the blending element, which we have also adopted as the uncertainty on the mean abundance of the respective blending element (see Section 4.1). We further limited the change in the abundance of the blending to ensure that the new synthetic-spectrum flux was within the S/N of the observed spectrum. We considered the following blending elements: Fe for the λらむだ3859 Å and λらむだ4090 Å U ii lines and La for the λらむだ4050 Å U ii line. While the λらむだ3859 Å U ii line is also blended with a CN feature, we find that the U abundance determination is most sensitive to the Fe abundance.

We estimated Δでるた($\mathrm{log}{gf}$) through varying the $\mathrm{log}{gf}$ values by the measurement uncertaintues listed in Nilsson et al. (2002a) and rederiving the U abundances. We estimated Δでるた(cont) for each U ii line by changing the local continuum placement in the spectral synthesis of the U ii line by ±0.5% and then rederiving the U abundance.

We determined the total uncertainty (Δでるた(total)) on the U abundance of each U ii line as the quadrature sum of the U ii line's systematic and statistical uncertainties. In turn, we determined Δでるた(sys) for each U ii line as the quadrature sum of Δでるた(T_eff), Δでるた($\mathrm{log}g$), Δでるた(ξくしー), and Δでるた(blend). Similarly, we determined Δでるた(stat) for each U ii line as the quadrature sum of the U ii line's Δでるた($\mathrm{log}{gf}$) and Δでるた(cont). For all of the stars, we list the resulting Δでるた(sys), Δでるた(stat), and Δでるた(total) for each U ii line in Table 6.

For the final U abundance of each sample star, we take the weighted-average of the U abundances from the three U ii lines. Therefore, $\mathrm{log}\epsilon$(U) = ${\sum }_{i}({w}_{i}\mathrm{log}{\epsilon }_{i})/{\sum }_{i}{w}_{i}$, where $\mathrm{log}{\epsilon }_{i}$ is the U abundance from line i and ${w}_{i}=1/{\rm{\Delta }}(\mathrm{stat}{)}_{i}^{2}$. Here Δでるた(stat)_i is the Δでるた(stat) uncertainty as estimated above for the U ii line i. We determined the Δでるた(total) on the weighted-average U abundance of each star as the quadrature sum of the corresponding Δでるた(sys) and Δでるた(stat). For the weighted-average U abundance, the systematic uncertainty components, Δでるた(T_eff), Δでるた($\mathrm{log}g$), Δでるた(ξくしー), and Δでるた(blend) are determined by taking the average of these components estimated for the three U ii lines. We then determined Δでるた(sys) as the quadrature sum of the averaged Δでるた(T_eff), Δでるた($\mathrm{log}g$), Δでるた(ξくしー), and Δでるた(blend). For the weighted-average U abundance of each sample star, we determined Δでるた(stat) by propagating the Δでるた(stat) estimates of each U ii for the weighted-average formula, i.e., 1/Δでるた(stat)² = ∑_i w_i , where ${w}_{i}=1/{\rm{\Delta }}(\mathrm{stat}{)}_{i}^{2}$. We list the final Δでるた(sys), Δでるた(stat), and Δでるた(total) for the weighted-average U abundance of each star in Table 6.

We also determined systematic and statistical uncertainties for the Th and Eu abundances of all of the stars. We determined Δでるた(sys) as the quadrature sum of ΔでるたT_eff, Δでるた$\mathrm{log}g$, and Δでるたξくしー. We determined Δでるた(stat) as the standard error of the mean Th and Eu abundances. Therefore, Δでるた(stat) = σしぐま/n, where σしぐま is the standard deviation of the abundances determined with different lines, and n is the total number of lines used. We then computed Δでるた(total) for the mean Th and Eu abundances of all of the sample stars as the quadrature sum of the corresponding Δでるた(sys) and Δでるた(sys).

Table 5 lists the final U abundance determined at each U ii line, along with its estimated uncertainty, for all of the sample stars. We note that the U abundances from the three U ii lines agree well, within uncertainties, for all of the sample stars. We also find that the λらむだ3859 U abundance determined in this work is consistent with previous literature estimates of the same within uncertainties for all of the stars; λらむだ3859 U abundance from the literature is also listed in Table 5.

Moreover, we find that in the case for all of the sample stars, the uncertainties on the U abundances of all three U ii lines are of the same order. This indicates that the new U ii lines provide similar precision for U abundance determination as the canonical λらむだ3859 Å U ii line. For all three U ii lines, we have homogeneously taken into account various sources of systematic and statistical uncertainties (see Section 4.5). Generally, we find the systematic uncertainties, specifically, from the blending elements to be the dominant source of the total uncertainty on the U ii line abundances. We also find that U abundance determination with all three U ii lines is sensitive to the continuum placement of the synthetic model. The sensitivity of the individual U ii lines to the various sources of uncertainties and the similar precision offered by the U ii lines impresses upon the advantage of using three U ii lines for U abundance determination, instead of just one.

We note, however, an exceptionally high uncertainty estimated for the λらむだ4050 U abundance of HE 1523–0901, on the order of ∼0.5 dex. This high estimate is driven by the La ii blend since the star has a relatively strong La feature relative to the other stars (see Figure 2). This highlights the fact that blends can have a significant impact on U abundance determination. However, we find that the spectral synthesis fit of the region is very good for the derived La abundance of this star.

As noted in Section 4.2, even though the spectral synthesis fits to the U ii lines are good—the goodness of the fits is further corroborated by the agreement between U abundances from the different U ii lines—there is indication of unidentified features in the λらむだ4090 Å and possibly λらむだ4050 Å spectral regions. Given the immense potential of these U ii lines on advancing nucleocosmochronometry and r-process studies, we recommend a detailed investigation into the atomic data, especially laboratory measurements of the transition lines in these spectral regions.

We further demonstrate the reliability of the new U ii lines with Figure 6, which shows (i) the residuals between the λらむだ4050 and λらむだ3859 U abundances in circular data points and (ii) the residuals between the λらむだ4090 and λらむだ3859 U abundances in square data points, for all of the sample stars. The residuals are plotted against the respective star's T_eff, $\mathrm{log}g$, [Fe/H], [C/Fe], [Eu/Fe], and spectrum S/N in different panels. The uncertainties on the residual data points are calculated by propagating the total uncertainties of the individual line abundances.

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First, we note that the residuals of the U abundances are all within ±0.2 dex (shown by the shaded gray region) for all of the stars. For CS 31082–001, while we find that the residual between the λらむだ4090 and λらむだ3859 U abundances is 0.0 dex, the residual between the λらむだ4050 and λらむだ3859 U abundances is +0.40 dex. This indicates that there might be an unidentified transition line in the λらむだ4050 Å U ii line region that is more prominent for CS 31082–001 than for the other stars. Alternatively, the abundance of the La ii HFS structure in this region may not be well represented by the mean La abundance determined for the star. While this relatively large residual may signify the need to better constrain the atomic data of this spectral region and/or the abundance of the La ii HFS structure, the uncertainty on the residual overlaps with the ±0.2 dex shaded region, subduing any serious concern.

Second, we note that the residuals of the U abundances show no discernible trend with respect to T_eff, $\mathrm{log}g$, [Fe/H], [C/Fe], [Eu/Fe], and spectrum S/N. This indicates that our current spectral synthesis models are of high fidelity and no identifiable systematic biases are being percolated to the U abundance determinations. Together, the ±0.2 dex range of the residuals between the U ii line abundances and the absence of any significant trend in the residuals with respect to key atmospheric and chemical properties as well as the data quality of the sample stars establish the reliability of the two new U ii new lines at λらむだ4050 Å and λらむだ4090 Å.

We provide revised U abundances for the RPE stars, J0954+5246, J2038–0023, HE 1523–0901, and CS 31082–001 via a homogeneous analysis of the three U ii lines at λらむだ3859 Å, λらむだ4050 Å, and λらむだ4090 Å. The resulting mean U abundance is computed as a weighted-average of the U abundances from the three U ii lines, with the weights assigned based on the statistical uncertainty on the U abundance of each line (see Section 4.2 for more details). We find that the total uncertainty of the weighted-average U abundances is on the order of ∼0.2 dex, for all sample stars. On the other hand, the total uncertainty of individual U ii line abundances is higher and typically in the range of ∼0.2–0.3 dex. This reiterates the advantage of using multiple U ii lines, which can potentially lend more precise U abundance than a single U ii line.

We compare the final weighted-average U abundances estimated in this work to previous literature estimates of λらむだ3859 U abundances in Figure 7, for all of the stars. We find that our U abundances agree well with the literature values within uncertainties, for all of the stars. In fact, the final U abundances agree with previous literature estimates within 0.05 dex for HE 1523–0901 and CS 31082–001, which we consider excellent agreement. This further establishes the reliability of the new U ii lines. For J0954+5246, we find that our U abundance is ∼0.4 dex lower than that determined by Holmbeck et al. (2018). We investigated the source of this discrepancy and suspect the cause to be differences in the adopted atomic data of the λらむだ3859.91 Å Fe i line. We also find that our U abundance for J2038–0023 is ∼0.3 dex lower than that determined by Placco et al. (2017). This discrepancy is attributed to the difference in the adopted stellar parameters, especially $\mathrm{log}g$ (see Section 3). Nevertheless, our U/Th and U/Eu nucleocosmochronometric age estimates compare well with those reported in Placco et al. (2017; see Section 5).

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We also discuss the uncertainty estimates on the final U abundances as determined in this work compared to previous literature estimates. In the case of J0954+5246 and HE 1523–0901, the total uncertainty estimated for the final weighted-average U abundance in this work is larger than the uncertainty quoted by the respective previous literature studies for their final λらむだ3859 U abundance (see Figure 7 and/or Table 5). For J0954+5246, Holmbeck et al. (2018) quoted a fiducial uncertainty of ±0.20 dex on their final U abundance, while we obtained an uncertainty of ±0.29 dex. For HE 1523–0901, Frebel et al. (2007) assigned an uncertainty of ±0.11 dex, solely arising from the effect of changing the Fe abundance by ±0.10 dex (although they considered additional sources of uncertainties in the age determinations). Having accounted for additional sources of uncertainties, we obtained a larger uncertainty of ±0.25 dex for the U abundance of HE 1523–0901. While Placco et al. (2017) also set a fiducial uncertainty of ±0.20 dex for J2038–0023, we find our detailed analysis renders a similar uncertainty of ±0.21 dex. Similarly for CS 31082–001, we find that our U abundance uncertainty of ±0.19 dex agrees with that of Hill et al. (2002), who also accounted for stellar parameters, oscillator strength, and observational fitting uncertainties and obtained an uncertainty of ±0.19 dex.

For every sample star, we determined two stellar ages, one using the U/Th chronometer and another using the U/Eu chronometer. We list the resulting ages in Table 5. There is good agreement between the U/Th and U/Eu ages of J0954+5246, J2038–0023, and HE 1523–0901. For CS 31082–001, the U/Eu age is ∼4.0 Gyr lower than the U/Th age due to its actinide-boost nature (Cayrel et al. 2001; Hill et al. 2002; Schatz et al. 2002). Even then, the U/Th and U/Eu ages of CS 31082–001 agree within uncertainties.

For stellar age uncertainties, we took into account systematic, statistical, and PR uncertainties as described in Section 5. These individual age uncertainty components are listed in Table 5, along with the total age uncertainties. For the U/Th and U/Eu stellar ages of all of the sample stars, the systematic uncertainties are the largest (and also the most dominant, in many cases), followed by the PR uncertainties and then the statistical uncertainties. Specifically, the systematic uncertainties of the stellar ages are driven by the large U abundance uncertainties from the blending elements.

The resulting total uncertainties on the ages of the sample stars are on the order of ∼4-6 Gyr for the U/Th ages and ∼3-4 Gyr for the the U/Eu ages. The U/Eu ages are more precise (although not necessarily more accurate) than the U/Th counterpart because of the shorter half-life of U, relative to Th. We estimate that to achieve a precision of 1 Gyr for the U/Th (U/Eu) ages, the systematic, statistical, and PR uncertainty components will each have to be driven down to less than 0.03 (0.04) dex, which might be challenging given the current limitations of unconstrained atomic data, LTE models, and uncertain r-process PRs. Improvements in these areas are thus highly encouraged.

 We note that we have not taken into account the systematic uncertainties associated with using just one set of theoretical PRs (from Schatz et al. 2002). Other r-process models have predicted slightly varying PRs for U/Th and U/Eu (e.g., Goriely & Arnould 2001; Farouqi et al. 2010). However, since the aim of this study was to investigate stellar ages estimated with revised U abundances from multiple U ii lines, we find using one set of PRs sufficient for the analysis.

We compare all of the stellar ages determined in this work to previous literature results in Figure 8, which shows the U/Th and U/Eu ages in the top and bottom panels, respectively. The dashed-black line in Figure 8 indicates the age of the universe, as determined by the Planck mission (Planck Collaboration et al. 2016). For the literature ages of J0954+5246 (Holmbeck et al. 2018), J2038–0023 (Placco et al. 2017), and HE 1523–0901 (Frebel et al. 2007), we display the ages determined by the respective studies using, specifically, the PRs of Schatz et al. (2002) to facilitate a consistent comparison to our age estimates. For CS 31082–001, Hill et al. (2002) determined U/Th age using the PR from Goriely & Arnould (2001), which we display. Also, Hill et al. (2002) did not determine the U/Eu stellar age of the CS 31082–001.

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We find that our stellar ages mostly agree well with previous literature results within uncertainties. The exception to this case is the U/Eu age of J0954+5246, which is much higher than previously determined in the literature. We attribute this discrepancy to a lower U abundance determined in this work relative to that determined in Holmbeck et al. (2018; see Section 6.3 more details). We also determined lower Th abundance relative to Holmbeck et al. (2018) so that in the U/Th ratio, the offset in the abundances is canceled.