T-Stellar: Stellar Catalogs & Atmospheric Models

• T-Stellar is a comprehensive framework integrating stellar catalogs, analytic T(τ) relations, multidimensional abundance clustering via t-SNE, and energetic particle ionization models.
• It leverages empirical color–Tₑₓₜ calibrations and 3D radiative simulations to enhance stellar parameter derivation and improve outer boundary conditions in structure models.
• It employs advanced ionization modeling to interpret molecular ion emissions in proto-planetary disks, providing critical insights for asteroseismic and chemical tagging studies.

T-Stellar refers to a suite of technical concepts, datasets, and computational models concerning stellar properties, atmospheric thermal structures, chemical abundance-space visualization, and the role of stellar energetic particles in disk ionization. This entry integrates methodologies for cataloging fundamental stellar parameters (as in the K2-TESS catalog), analytic and grid-based atmospheric $T(\tau)$ relations crucial for stellar structure/evolution codes, and multidimensional abundance-space clustering via t-SNE, alongside physical modeling of proton-driven ionization in T Tauri disk environments.

1. Construction and Properties of T-Stellar Catalogs

A cornerstone for T-Stellar studies is the K2-TESS Stellar Properties Catalog, compiled by cross-matching K2 target lists with the Augmented TESS Target Catalog (ATTC), itself built from all 2MASS point sources with $J < 13$ mag and associated with multiple photometric, astrometric, and spectroscopic sources such as NOMAD, Tycho-2, Hipparcos, APASS, UCAC4, APOGEE, RAVE, and LAMOST (Stassun et al., 2014). The catalog covers over $117{,}521$ K2 targets with the following core attributes: coordinates, identifiers, photometry, proper motions, luminosity class flag (dwarf/subgiant vs. giant), effective temperature estimates via color–$T_\mathrm{eff}$ calibrations, Kepler-band magnitude, Guest Observer (GO) program IDs, and spectroscopic cross-matches. The reduced proper motion (RPM) method is applied to classify stars; dwarfs and subgiants are flagged using the empirical separation in the $(J-H)$ versus $RPM_J$ plane, based on Collier Cameron et al. (2007).

Effective temperature derivation employs empirical $V-K_S$ color–$T_\mathrm{eff}$ relations calibrated for dwarfs (Huang et al. 2015), cool dwarfs (Casagrande et al. 2008), and giants, with propagated uncertainties ($\sim2\%$ for dwarfs, $\sim1.7\%$ for giants, $19\,\mathrm{K}$ for M dwarfs). Extinction corrections with up to four de-reddened solutions are provided when applicable. Each target is traceable to original GO programs via string-separated investid lists; catalog access is via Vanderbilt’s Filtergraph portal, supporting interactive plotting, SQL-style filters, and programmatic download (Stassun et al., 2014).

2. Stellar Atmospheric $T(\tau)$ Relations: Formulation and Application

Stellar structure models rely on relations between temperature, $T$, and optical depth, $\tau$, as outer boundary conditions. The general form is:

$$\left(\frac{T(\tau)}{T_\mathrm{eff}}\right)^4 = \frac{3}{4}\,[\tau + q(\tau)],$$

where $q(\tau)$ is the Hopf function measuring deviations from the grey–Eddington solution ($q=\tfrac{2}{3}$) (Ball, 2021, Trampedach et al., 2014). Ball (2021) introduced a novel analytic $q(\tau)$ fit for the 3D RHD solar simulation grid of Trampedach et al., accurate to $<0.82\%$ in the solar case and to $<13\%$ across $37$ grid models with $$4400\,\mathrm{K}\lesssim T_\mathrm{eff}\lesssim6900\,\mathrm{K}$$ and $2.2\lesssim\log g\lesssim4.7$ ([Fe/H]$\sim0$) (Ball, 2021). The analytic gradient is:

$$\frac{dq}{dx} = \frac{c_1 + \exp[(x-a)/v]}{1 + \exp[(x-b)/w]},$$

with $x = \log_{10}\tau$ and physical meanings assigned to $(c_0, c_1, a, b, v, w)$ (offset, transition height, ramp location/width, logistic boundary/sharpness).

Implementation strategies in 1D models involve direct integration down to $\tau_{\rm eff}$ or modification of the radiative gradient, with proper handling of numerical issues (domain clamping, log-sum-exp stability, neglecting small hypergeometric terms for $x<0$). Compared to grey atmospheres, Trampedach–Ball fits yield cooler surface layers ($T/T_\mathrm{eff}\sim0.77$ vs. $(1/2)^{1/4}$), reflect the physical opacity transition, and reduce seismic mode frequency errors by tens of $\mu$Hz (Ball, 2021, Trampedach et al., 2014).

3. Three-Dimensional $T(\tau)$ Grids and Model Interpolation

Trampedach et al. (2014) supplied radiative $T(\tau)$ relations and opacities from a grid of 3D convection simulations, dramatically expanding the parameter dependence observed compared to 1D MARCS models (Trampedach et al., 2014). The grid spans $T_\mathrm{eff}=4200$–$6900$ K, $\log g=2.2$–$4.7$, solar metallicity. The Hopf function $q_\mathrm{rad}(\tau)$ and crucial derivatives (e.g., radiative temperature gradient)

$$\nabla_{\mathrm{rad}} = (3/16\sigma) \left(\kappa F_\mathrm{tot} p / g T^4\right) [dq/d\tau + 1]$$

are tabulated. Grid interpolation proceeds by Delaunay triangulation and barycentric weighting among the enclosing simplex, with code examples for Python and Fortran provided (Trampedach et al., 2014). Application requires transforming the optical depth via $d\hat{\tau} = f_\mathrm{rad}(\tau) d\tau$ (where $f_\mathrm{rad}$ is the fraction of the radiative flux to total flux), ensuring seamless interior-envelope boundary matching.

Key distinctions between the 3D and 1D $T(\tau)$: strong gravity and temperature dependence, smooth matching to diffusion limits, steeper atmospheric gradients due to overshoot-cooling, and proper treatment of turbulent pressure (up to $27\%$ in F dwarfs). This approach leads to improved asteroseismic agreement, more accurate convection zone depth determinations, and resistance to systematic errors from using scaled-solar profiles (Trampedach et al., 2014).

4. Chemical Abundance-Space Dissection via t-SNE

The t-distributed stochastic neighbor embedding (t-SNE) algorithm enables non-parametric, non-linear dimensionality reduction of high-dimensional stellar chemical abundance vectors ($x_i$, $i=1,\ldots,N$, $D=13$ abundance ratios plus [Fe/H]) (Anders et al., 2018). t-SNE works through the following sequence:

1. Conditional similarities in high-D—$p_{j|i}$ via isotropic Gaussian kernels with star-specific bandwidths $\sigma_i$ tuned to match a user-defined perplexity $p$ (effective local neighborhood size).
2. Symmetric joint probability—$p_{ij} = (p_{j|i} + p_{i|j}) / (2N)$, normalizing over all $i \ne j$.
3. Low-D embedding—Student-t kernel for the 2D vectors $y_i$, yielding $$q_{ij}=(1+\|y_i-y_j\|^2)^{-1}/[\sum_{k \neq l} (1+\|y_k-y_l\|^2)^{-1}]$$.
4. Objective—Kullback-Leibler divergence minimization: $$\mathrm{KL}(P\|Q)=\sum_{i\neq j}p_{ij}\log(p_{ij}/q_{ij})$$.

Robustness is validated through varying perplexity ($p=5$–$100$; fiducial $p=40$ for $N\simeq550$ stars) and Monte-Carlo error propagation. Perturbing abundances within twice their uncertainty, the major morphological features in chemical space persist, with both disc structure (thin, thick, inner/outer, super-metal-rich populations) and peculiar outliers retained. Input composition ([Fe/H], age, kinematics) only mildly modulates the clustering.

The t-SNE-derived 2D map distinguishes:

• Low-[$\alpha$/Fe] thin disc (black): [Fe/H] $\sim -0.3$ to $+0.25$, minimal scatter in [$\alpha$/Fe], ages $0$–$10$ Gyr.
• High-[$\alpha$/Fe] thick disc/inner disc (red/yellow): subpopulations split by Al, Mg, Ca, transitioning smoothly to super-metal-rich (orange).
• Super-metal-rich (orange): [Fe/H] $> +0.3$, enhanced [Y/Ba], [Cu/Fe], intermediate ages, cold orbits.
• Outer-disc (green): [Fe/H] $\sim -0.5$, elevated [$\alpha$/Fe], subsolar [Sr/Fe], [Y/Fe].
• Young local disc (grey): $\sim1$ Gyr, near-solar metallicity, low velocity dispersion, moderate s-process enrichment.

t-SNE also reliably identifies chemically peculiar groups, including s-process-enhanced old stars and a high-confidence pair (HD 91345/HD 126681), nearly identical in [X/Fe], age ($\sim12.8$ Gyr), and kinematics, consistent with a common birth origin (putatively a disrupted dwarf galaxy). Individual outliers are also isolated (e.g., s-enhanced HD 28701, debated Ti-rich candidate, spurious low-S/N odd-Z enhancements), demonstrating suitability for precision chemical tagging applications (Anders et al., 2018).

5. Stellar Energetic Particle Ionization in Proto-Planetary Disks

Young stellar objects like T Tauri stars exhibit particle fluences many orders of magnitude above the contemporary Sun. Their X-ray luminosity ($L_X \sim 10^{30}$–$10^{31}$ erg s$^{-1}$) scales the solar-proton spectrum upward, yielding $\sim1.5\times10^7$ protons cm$^{-2}$ s$^{-1}$ at $E>10$ MeV at 1 au (Rab et al., 2017). The SP ionization rate profile, modeled via Padovani et al. (2013) formulae, combines two power-law regimes with a cutoff at $N_E=2.5\times10^{25}$ cm$^{-2}$:

• $$\zeta_{\rm SP}(N_H) = \{[\zeta_L(N_H/10^{20})^a]^{-1} + [\zeta_H(N_H/10^{20})^b]^{-1}\}^{-1}$$ for $N_H\leq N_E$,
• exponential suppression for $N_H > N_E$.

SPs dominate ionization ($\zeta_{\rm SP} \sim 10^{-12}$–$10^{-8}$ s$^{-1}$) in upper disk layers ($z/r\gtrsim0.1$–$0.3$), but attenuate before reaching the disk midplane, contrasting with cosmic rays (uniform into midplane) and X-rays (surface/scattered penetration).

The principal chemical impact is on molecular ions:

• HCO$^+$ formation maximally enhanced in the SP-ionized layer ($N_{\rm H}\lesssim10^{24}$ cm$^{-2}$), J=3–2 line flux increased by $2$–$3\times$.
• N$_2$H$^+$ enhanced by up to $10\times$ in warm layers but less affected in the cold layer beyond the CO snow line unless CR flux is suppressed.
• Synthetic ALMA maps (beam 0.2") resolve snow line and vertical ionization stratification, enabling observational signatures of SP dominance through integrated intensity ratios, column profile inflections, and combined multi-transition analysis (e.g., H$^{13}$CO$^+$ vs. N$_2$H$^+$).

A plausible implication is that spatially resolved line diagnostics of molecular ions enable order-of-magnitude constraints on young star particle fluences, addressing the origin of meteoritic radionuclide anomalies (Rab et al., 2017).

6. Summary and Practical Recommendations

T-Stellar studies unify precise catalog-level parameterization (position, photometry, $T_\mathrm{eff}$, luminosity class, photometric/spectroscopic cross-matches, GO program links) (Stassun et al., 2014), advanced outer boundary physics ($T(\tau)$, Hopf function q$(\tau)$, grid interpolation, and gradient calibration) (Ball, 2021, Trampedach et al., 2014), nonlinear abundance-space clustering for chemical tagging (Anders et al., 2018), and first-principles modeling of energetic particle-driven disk chemistry (Rab et al., 2017). Recommended practices include:

• Utilize all-sky cross-matches and systematic RPM-based classification for robust T-Stellar samples;
• Employ analytic and grid-based $T(\tau)$ relations, avoiding scaled-solar proxies, and interpolate with barycentric weighting, ensuring seamless structure code integration and improved seismic predictions;
• Apply t-SNE dimensionality reduction for chemical tagging and substructure analysis, with careful hyperparameter and error assessment;
• Integrate SP ionization modeling in disk chemistry, interpreting molecular ion emission with ALMA-class observations coupled to full particle transport and thermal-chemical codes.

These cross-cutting methodologies enable precise characterization, evolutionary modeling, and chemical evolutionary analysis for diverse stellar populations and environments.