adsabs/AstroCLIMB huggingface dataset:
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: How a planet can tidally torque the host star into spin–orbit alignment by way of an axisymmetric m = 0 g-mode oscillation forced by the planet’s k = m orb = 2 potential. The star stretches and compresses along its spin axis (z-axis) twice for every orbit of the planet. The dark purple sphere marks the planet’s location, which is displaced ahead of the star’s tidal bulge due to tidal friction, while the light purple sphere displays the planet’s location in the absence of tidal friction. The top row shows different views of the same instant in time, when the star is maximally elongated but the planet is ahead at positive y. The bottom row shows a different time, when the star is maximally compressed and similarly lagged in phase relative to the planet. The tidal bulge at position vector  r  feels a force from the planet  F  and by extension a torque  T  =  r  ×  F  that is positive in the x-direction no matter the orbital phase (middle column). The torque T  x  > 0 acts to bring the stellar spin direction (﹩{\hat{{\boldsymbol{j}}}}_{\star }﹩) toward the x-axis, closer to the orbital angular momentum direction (﹩{\hat{{\boldsymbol{j}}}}_{\mathrm{orb}}﹩). Once spin and orbit are aligned, the torque vanishes (Figure 4, bottom left panel). Because the oscillation is symmetric about the z-axis (right column), there is no torque in the z-direction.
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Charting the limits of resonance locking. Computed curves are for an axisymmetric (m = 0) g-mode of a 1 M ⊙, 4 Gyr old main-sequence star (see Appendix B.2). The green bar (﹩{\psi }_{\mathrm{unlock}}^{\mathrm{lin}}﹩) marks a transition region dividing the space where resonance locking can operate (to the left) from where it cannot (to the right), computed under the assumption that tidal disturbances are linear and damp by radiative diffusion. In purple are plotted the analogous boundaries for ﹩{\psi }_{\mathrm{unlock}}^{\mathrm{nl}}﹩, computed assuming tidal disturbances are limited in amplitude by nonlinear mode–mode couplings, which spawn traveling waves that break and dissipate (see Appendix B). The nonlinear model expands the regime where resonance locking is possible, out to a/R ⋆ ≈ 8. Overlaid for context are observed projected obliquities λ (Albrecht et al. 2022; Rice et al. 2022a, 2022b; Espinoza-Retamal et al. 2023; Sedaghati et al. 2023; Siegel et al. 2023; Hu et al. 2024). See Section 5.2 for how we might interpret these data.
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Rate of Brunt–Väisälä frequency evolution (top panel, Equation (16)) and accumulated frequency change (middle panel, Equation (18)) vs. main-sequence age for stars of different masses as indicated. Although high-mass stars (red curves) increase their frequencies dramatically quickly toward the end of their hydrogen-burning lives (as their cores switch from being unstably to stably stratified), the speed-up phase is short-lived and consequently unlikely to be observed. Sampled over the entirety of their main-sequence lives, ΔN/〈N 0〉 is likely to be higher for low-mass stars, whose cores remain stably stratified throughout, than for high-mass stars (bottom panel showing median and ±1σ intervals, calculated by evaluating ΔN/N 0 at 1000 uniformly spaced times over the interval ﹩[{t}_{0},{t}_{\max }]﹩).
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: An axisymmetric m = 0 oscillation on the host star (top left) can be excited by the dominant k = m orb = 2 component of the planet’s tidal potential (top right) when ψ > 0 (bottom right showing that the radial tidal force integrated over the stellar equator is net outward). When ψ = 0, tidal forces cancel over stellar azimuth, and the m = 0 mode cannot be excited (bottom left).
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: A population synthesis based on resonance locking (open symbols) compared against observations (filled symbols). Data for projected obliquities λ (top) and obliquities ψ (bottom) are taken from Albrecht et al. (2022), Rice et al. (2022a, 2022b), Espinoza-Retamal et al. (2023), Sedaghati et al. (2023), Siegel et al. (2023), and Hu et al. (2024) for hot Jupiters (M p ≥ 0.3 M J, a/R ⋆ ≤ 10) above and below the Kraft break (T eff ≤ 6100 K in blue, T eff > 6100 K in red). In the model, obliquities are initially random (uniformly distributed in ﹩\cos {\psi }_{0}﹩ between −1 and 1) and subsequently damp by resonance locking over Gyr timescales. Damping is seen to be negligible for most hot stars (except those with ψ 0 ≲ 25°, which damp to 0) but significant for cool stars. Although roughly reproducing the break seen in the observations, the model fails to damp sufficiently cool star obliquities that start strongly retrograde; these evolve into the cloud of blue open symbols with obliquities between 10° and 70° sitting above the observations.
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Orbital evolution of hot Jupiters in resonance lock with a stellar gravity mode ({k, l, m} = {2, 2, 0}), integrated from t 0 = 1 Gyr through the stellar main-sequence lifetime or t = 12 Gyr, whichever comes first. Here planet orbits are assumed circular for all time. Obliquities damp more readily for cool stars (blue curves) than hot stars (red curves). For hot stars with ψ 0 = 145°, obliquities actually increase slightly, a consequence of their having J orb/J ⋆ < 1 (unlike for cool stars).
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Same as Figures 7 and 8 except for initial conditions {e 0, a 0} = {0.6, 0.07 au}. At these larger eccentricities, damping of ψ, e, and a takes longer, but the equilibrium fixed point of ψ, e = 0 can still be reached within main-sequence lifetimes for cool stars whose obliquities are not too high. Hot star obliquities remain near their initial values regardless.
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Same as Figure 7 except for initial eccentricities e 0 = 0.3. For such initial conditions, orbits can circularize around cool stars on the same timescale as the obliquity damps, but not hot stars. The obliquity evolution is similar to that for strictly circular orbits (compare with Figure 7).
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Plotting the magnitude of the Hansen coefficients (Equation (A2)) for the {k, ℓ} = {2, 2} component of the tidal potential. The ﹩| {X}_{2,2}^{2}| ﹩ coefficient dip at e ∼ 0.7 occurs because ﹩{X}_{2,2}^{2}﹩ changes sign. The ﹩| {X}_{2,2}^{2}| ﹩ coefficient is larger than ﹩| {X}_{2,0}^{2}| ﹩ at smaller eccentricities, until e ≳ 0.4−0.5. We always have ﹩| {X}_{2,-2}^{2}| \ll | {X}_{2,0}^{2}| ﹩.
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Same as Figure 13, except we replace γ  β  with ﹩{t}_{\mathrm{gr},\beta }^{-1}﹩ (Equation (B9)) once ﹩{E}_{\beta }\gt {E}_{\beta }^{\mathrm{brk}}﹩ and child modes become traveling waves. Left: energies of the parent (solid green line) and child (solid purple line) vs. time for Δ α  = 3.3 × 10−4 ω  α . For these parameters, the parent’s energy stays below the threshold ﹩{E}_{\alpha }^{\mathrm{thr},\mathrm{tw}}﹩ (Equation (B10)) for maintaining a traveling child wave. Thus, the child mode disappears as soon as it breaks, and the parent energy remains close to ﹩{E}_{\alpha }^{\mathrm{lin}}﹩. Middle: same as left but for Δ α  = 4.7 × 10−5 ω  α  (closer to exact resonance). Here the parent energy exceeds ﹩{E}_{\alpha }^{\mathrm{thr},\mathrm{tw}}﹩ and can sustain a traveling child mode. The parent and child energies asymptote to ﹩{E}_{\alpha }^{\mathrm{thr},\mathrm{tw}}﹩ and ﹩{E}_{\beta }^{\mathrm{tw}}﹩ (Equation (B11)), respectively. Right: equilibrium parent and child energies (evaluated at t > t end/2) vs. forcing frequency. Exact resonance is located at P orb/P  α  = 2. Away from exact resonance, child energies are too low to be plotted and leave parent energies near ﹩{E}_{\alpha }^{\mathrm{lin}}﹩. Closer to exact resonance, child energies grow to ﹩{E}_{\beta }^{\mathrm{tw}}﹩, causing the parent’s energy to flatline at ﹩{E}_{\alpha }^{\mathrm{thr},\mathrm{tw}}﹩ and preventing the parent from breaking.
Paper: Damping Obliquities of Hot Jupiter Hosts by Resonance Locking
Authors: J. J. Zanazzi, Janosz Dewberry, Eugene Chiang
Figure caption: Parent mode amplitudes, energy dissipation rates, and dissipation timescales using the same hot Jupiter + star model of Figure 14. Left: comparing the threshold energy ﹩{E}_{\alpha }^{\mathrm{thr},\mathrm{tw}}﹩ (Equation (B10)) needed for a parent to excite a traveling child mode to the parent breaking energy ﹩{E}_{\alpha }^{\mathrm{brk}}﹩. The parent may parametrically excite traveling wave child modes without itself breaking. Middle: average tidal dissipation rate (Equation (B3)) at times t > t end/2, either assuming the child modes behave as standing waves (crosses; see also Figure 13) or accounting for how child standing waves break to become traveling waves (filled circles; see also Figure 14). There is no resonant response when child modes are standing waves, but there is when child modes become traveling waves. Right: nonlinear dissipation timescale ﹩{t}_{\mathrm{diss}}^{\mathrm{nl}}﹩ (purple) compared to the stellar evolutionary timescale 3t ev/2 (brown bar; derived from the M ⋆ = 1 M ⊙ curve from Figure 6, top panel). Nonlinear damping may sustain resonance locks out to ﹩{(a/{R}_{\star })}_{\mathrm{crit}}\sim 8\mbox{--}10﹩. For comparison, the timescale ﹩{t}_{\mathrm{diss}}^{\mathrm{lin}}﹩ assuming linear damping is also shown.
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: JWST+HST imaging and JWST/NIRSpec spectroscopy of RX J2129–z8He II. Top: RX J2129–z8He II’s postage stamp images in multiple filters, cut from the HST CLASH (PI: M. Postman) and JWST/NIRCam (DD-2767; PI: P. Kelly) imaging mosaics. These stamps are oriented such that north is up and east is to the left, which is different from the orientation in Figure 1. The strong flux deficit in the ACS/F814W and NIRCam/F115W bands as compared to the other NIRCam filters indicates z ≳ 8. The rightmost panel in the top row zooms in on the [O III]–Hβ line complex taken from the full NIRSpec prism 1D spectrum shown in the bottom row. Middle and bottom: NIRSpec prism spectroscopy of RX J2129–z8He II. In the middle trace, we show the unsmoothed 2D prism spectra combined from the three individual slit-dithering sequences using the up-to-date reduction software MSAEXP (see Section 2.2). We detect prominent emission features of [O III] and Hβ at λ obs ∼ 4.5 μm and a strong continuum break blueward of Lyα at λ obs ∼ 1.1 μm. In the bottom row, we show the 1D spectrum optimally extracted from the 2D spectral trace in gray histograms with the 1σ uncertainty in cyan. We mark the locations of multiple rest-frame UV and optical emission lines using the magenta dotted lines, and the Lyman break is the blue dashed line. The strong detections of [O III]+Hβ and the Lyman break pinpoint a secure spectroscopic redshift of RX J2129–z8He II to be z spec = 8.1623 ± 0.0007. Bottom inset: a zoom-in view on the 2D spectrum around the rest-frame UV wavelength range, where the location of the He II line is marked by the red circle. The 2D spectrum is shown in units of S/N (i.e., ﹩{f}_{\nu }^{\mathrm{obs}}/{\sigma }_{\nu }^{\mathrm{obs}}﹩) per spatial pixel, color-coded in the range of [−1, 10].
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: Spectrophotometric and emission line analyzes of the full JWST–HST observations of RX J2129–z8He II. Top: the optimally extracted 1D NIRSpec prism spectrum is shown in blue histograms with a 1σ error spectrum in cyan bands. The broadband photometric results from NIRCam and WFC3+ACS are represented by the red circles and green diamonds, respectively. We perform full spectrum analyzes and fit Gaussian profiles to nebular emission features to estimate line fluxes and upper limits. The resulting best-fit continuum model and emission lines are represented by the red and orange curves. The top inset panel zooms in on the He II line, with significant detection. Bottom: the ratio between the continuum-subtracted f  λ  flux and the measured 1σ error spectrum. For the wavelength ranges without significant emission line flux contribution, this ratio follows a ﹩{ \mathcal N }(0,1)﹩ distribution, implying that the noise properties are well defined. As clearly shown here, the most pronounced emission features in the rest-frame UV and optical wavelengths are the He II and [O III]-Hβ lines, respectively. The gray shaded region marks the wavelength range affected by strong attenuation by the highly neutral IGM at z ∼ 8.
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: A color-composite NIRCam image of the RX J2129 galaxy cluster. We use the F115W+F150W, F200W+F277W, and F356W+F444W imaging as the blue, green, and red colors, respectively. The regions of formally infinite magnification at z = 8.16 predicted by the up-to-date GLAFIC lens model (Oguri 2010, 2021) are represented by the magenta curves. The location of RX J2129–z8He II is highlighted by the red circle. It is estimated to have a magnification factor of μ = 2.26 ± 0.14 (see Section 3). The inset shows a zoom-in view (3″ × 3″) of RX J2129–z8He II. Within this inset, we also show the orientation of the NIRSpec slit (0.″2 × 1.″4), constructed by three MSA slitlets, through which the prism spectroscopy of RX J2129–z8He II is taken by the JWST Director’s Discretionary program (DD-2767; PI: P. Kelly).
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: SFH of RX J2129–z8He II, obtained from three independent methods. The SFH given by our default BAGPIPES full spectrum fitting under the assumption of a double power-law model is represented by the red curve (best-fit value) with shaded regions (1σ confidence ranges). The SFHs given by the PROSPECTOR SED fitting assuming nonparametric and delayed-τ models are shown in black histograms and blue curves, respectively. All three SFHs returned by different assumptions and techniques reach excellent consensus that the SFH of RX J2129–z8He II is rapidly rising, in support of the existence of a very young stellar component likely associated with the Pop III signatures seen in the JWST/NIRSpec spectra.
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: The rest-frame UV and optical morphology of the entire RX J2129–z8He II galaxy, comprising two components: A and B. We use GALFIT to perform morphological modeling of both components in JWST/NIRcam imaging taken at filters F150W (upper panels) and F444W (lower panels). At z spec = 8.16, F150W and F444W correspond to rest-frame UV and optical wavelength ranges, respectively. In each row, we show the original observed image, GALFIT models of both components, and the residual removing their GALFIT models. On the observed images, we overlay the NIRSpec MSA slit, represented by the blue box. The size of all stamps is 3″ × 3″, with north up and east to the left. As indicated by the residual panels, we achieve reasonable morphological models of the entire galaxy.
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption:  EAZY fit to the broadband photometry of RX J2129–z8He II, performed on components A and B separately. The red points with error bars correspond to observed magnitudes with 1σ measurement uncertainties, whereas the triangles represent 2σ upper limits, all given in Table 2. The blue curve in each panel represents the best-fit SED model, with the open circles showing the model photometry in respective filters. The inset panel shows the probability distribution function of the photometric redshift, i.e., P(z). The photometric redshift constraints of the two components are both in good agreement with z spec = 8.1623 determined from the NIRSpec prism spectroscopy of component A. This strongly suggests that component B is also part of the galaxy RX J2129–z8He II.
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: We compare the photometry for both components of RX J2129–z8He II, normalized at their individual F200W flux. In each filter, the data points are slightly offset in wavelength for clearer visualization. The photometry suggests that component B has a comparably blue UV continuum yet possibly with substantially fainter oxygen lines in the rest-frame optical (probed by F444W), indicating that component B is more metal-poor than A. Also, component B’s SED shape might accommodate He II with similar EW, albeit at lower confidence.
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: The 1D posterior distributions of some key parameters fit from our spectrophotometric analyzes: spectroscopic redshift (z spec), stellar mass (M *; already corrected for lensing magnification), stellar dust attenuation (﹩{A}_{{\rm{V}}}^{{\rm{S}}}﹩), stellar phase metallicity (Z *), and UV spectral slope (β).
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: Emission line flux ratios from observations and models. Left: flux ratios of [O III]/[O II] vs. He II/Hβ predicted by the MAPPINGS V photoionization models (Sutherland & Dopita 1993) and measured in RX J2129–z8He II (as in Table 1) represented by the magenta star with 1σ error bars. The cyan, red, and blue symbols are calculated using the empirical spectra of metal-poor stars with atmospheric temperatures of 45 kK, 47.5 kK, and 55 kK, respectively. The diamonds, circles, and squares correspond to gas-phase metallicity (Z ISM) being half, one-tenth, and one-thousandth solar, respectively. It is obvious that “normal” O/B-type stellar populations cannot produce sufficient He II emission, in spite of their high [O III]/[O II] at low Z ISM. We construct physically calibrated phenomenological models of Pop III star spectra, with different assumptions of mass loss and the power-law index (α) in the helium-ionizing energy range (see text). Encouragingly, all these models successfully reproduce the observed value of He II/Hβ. The mass-loss models are better favored considering the observed value of [O III]/[O II]. Crucially, we find that high [O III]/[O II] alone is not necessary a good indication of the hard ionizing spectra, potentially from Pop III stars. Right: UV line ratios measured from galaxies that have reported He II detections (S/N ≳ 2.5). These measurements are color-coded in EWHe II  reported in the respective work. The arrows denote 2σ lower limits. RX J2129–z8He II shows greatly elevated He II/C III] and He II/C IV flux ratios, as compared to local galaxies (Berg et al. 2016; Senchyna et al. 2019), galaxies at the cosmic noon epoch (Nanayakkara et al. 2019), and CR7 at z = 6.6 (Sobral et al. 2018). We also overlay three sets of evolutionary tracks at different stellar metallicities given by the BPASS v2.1 stellar population synthesis models (Xiao et al. 2018). Among these three tracks, the measured flux ratio lower limits of RX J2129–z8He II clearly prefer that with extremely low metallicity. The gray diamonds represent the predictions of AGN narrow-line regions calculated using the CLOUDY code (Ferland et al. 1998), with larger symbol size corresponding to higher metallicity (i.e., 0.03, 0.1, 0.5 and 1× solar).
Paper: A Strong He II λ1640 Emitter with an Extremely Blue UV Spectral Slope at z = 8.16: Presence of Population III Stars?
Authors: Xin Wang, Cheng Cheng, Junqiang Ge, Xiao-Lei Meng, Emanuele Daddi, Haojing Yan, Zhiyuan Ji, Yifei Jin, Tucker Jones, Matthew A. Malkan, Pablo Arrabal Haro, Gabriel Brammer, Masamune Oguri, Meicun Hou, Shiwu Zhang
Figure caption: Realistic stellar spectra input to our photoionization modeling framework. The cyan, red, and blue curves correspond to the empirical spectra of metal-poor ([Fe/H] = −3) single stars with high atmospheric temperatures of 45 kK, 47.5 kK, and 55 kK, respectively, representing the typical populations of low-metallicity O/B stars. These empirical star model atmospheres are constructed using the non-LTE plane-parallel code TLUSTY (Hubeny & Lanz 2011; Hubeny et al. 2021). The black and green lines show our broken power-law models of Pop III stellar spectra, defined in Equation (7), with different assumptions of mass loss and the power-law index α (see text for more details). The ionizing sections of these Pop III spectra are calibrated by the time-averaged hardness ratios—described by Q(He0)/Q(H) and Q(He+)/Q(H) tabulated in Schaerer (2002)—with and without mass loss. The nonionizing sections are determined based upon our actual measurement of the UV spectral slope (β = −2.53) and the emission line EWs (i.e., EWHe II  = 21 Å). The three vertical dotted gray lines mark the energy levels of 13.6, 24.6, and 54.4 eV, corresponding to the ionization of H, He0, and He+ ions.