METUJE: Unified Wind Modeling Code

• METUJE is a unified modeling code that computes self-consistent atmospheres and wind structures for hot stars using NLTE CMF radiative transfer.
• It solves steady-state hydrodynamic equations without relying on parameterized CAK multipliers, enabling accurate predictions of mass-loss rates and velocity laws.
• Applications to hot subdwarfs and OB stars demonstrate its sensitivity to chemical composition and metallicity, providing key insights into wind diagnostics and stellar evolution.

METUJE (“Masaryk University Theoretical–unified Stellar Wind Equations”) is a one-dimensional, stationary, spherically symmetric global wind code for hot stars that computes a self-consistent atmosphere-and-wind solution from the quasi-hydrostatic photosphere to the supersonic outer flow. It is used for radiatively driven winds of hot stars, including hot subdwarfs and OB stars, and predicts the global wind parameters directly from stellar parameters and chemical composition, notably the mass-loss rate $\dot M$, velocity law $v(r)$, terminal velocity $v_\infty$, density $\rho(r)$, temperature $T(r)$, and NLTE level populations through the photosphere–wind transition (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

1. Concept and modeling domain

METUJE is described as a “global” or “unified” model because it treats the photosphere and stellar wind within a single framework rather than as separate domains. In the hot-subdwarf applications, the code was used with stellar parameters and abundances derived from TLUSTY/SYNSPEC atmosphere fits and from optical, ultraviolet, and photometric analyses; in the OB-star grid, it was used to generate metallicity-dependent wind models down to $0.01\,Z_\odot$ (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

The basic assumptions are stationary, one-dimensional, and spherically symmetric outflow. Published descriptions explicitly note the neglect of time dependence and multidimensional effects, including rotation, magnetic fields, and clumping (Krticka et al., 2024). Within those assumptions, METUJE attempts to solve the coupled atmosphere–wind problem self-consistently, so that the radiative force is not imposed externally but emerges from the NLTE CMF transfer solution.

The code’s intended domain is line-driven outflows from hot luminous objects. In subdwarfs, its role is to test whether a given abundance pattern permits a self-consistent wind and to quantify how evolutionary or diffusion-modified abundances affect $\dot M$ and wind diagnostics. In OB stars, its role is broader: to map the metallicity dependence of $\dot M$ and $v_\infty$, the disappearance of the bistability jump at low metallicity, and the onset of large scatter when only a small number of lines effectively accelerate the flow (Krticka et al., 2024, Krticka et al., 29 Aug 2025).

2. Physical and mathematical formulation

METUJE solves the standard steady-state hydrodynamic equations together with NLTE statistical equilibrium and CMF radiative transfer. In spherically symmetric form, the continuity and momentum equations are written as

$\dot M \;=\; 4\pi\,r^2\,\rho(r)\,v(r)$

and

$v(r)$0

The statistical-equilibrium equations for level populations are

$v(r)$1

These equations are solved concurrently with the CMF transfer problem, so the populations, opacities, emissivities, and radiative acceleration are mutually coupled (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

The total radiative acceleration is computed by direct frequency integration in the comoving frame:

$v(r)$2

or, in an equivalent published form,

$v(r)$3

A defining characteristic of METUJE is that this acceleration is obtained without Sobolev approximations and without parameterized CAK force multipliers; published summaries explicitly state that no parameterized CAK $v(r)$4, $v(r)$5, $v(r)$6 multipliers are used (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

The temperature structure is treated through radiative equilibrium in the photosphere and a consistent energy equation through the wind. One published description states that the code adjusts $v(r)$7 so that

$v(r)$8

while another writes the wind energy balance as

$v(r)$9

The 2025 OB-star summary gives the energy equation in the form

$v_\infty$0

This suggests that different publications emphasize different but compatible representations of the same thermal coupling problem (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

3. Numerical realization, boundary conditions, and atomic data

Published descriptions present METUJE as an iterative NLTE CMF hydrodynamics solver. The global iteration typically begins from a TLUSTY atmosphere extended by a simple $v_\infty$1-type wind law, after which the code alternates among CMF transfer, NLTE population updates, and hydrodynamic corrections until convergence in $v_\infty$2, $v_\infty$3, $v_\infty$4, and populations is reached. One summary gives convergence to better than $v_\infty$5; another reports successive-cycle changes below typically $v_\infty$6, or $v_\infty$7 for well-behaved O-star models (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

The hydrodynamic update is performed with finite-difference discretization and Newton–Raphson iteration, with explicit enforcement of smooth passage through the critical or sonic point (Krticka et al., 2024). The CMF transfer solver is described as using a short-characteristics or tangent-ray treatment for the angular dependence, while NLTE rate equations employ complete linearization with ALI-type acceleration. For iron-peak elements, a superlevel technique is used to reduce the number of levels (Krticka et al., 2024).

Published summaries place the inner boundary deep in the photosphere and the outer boundary near $v_\infty$8. The inner boundary is variously stated as Rosseland optical depth $v_\infty$9 or $\rho(r)$0, but in both formulations the base structure is matched to a TLUSTY hydrostatic model atmosphere, with a small subsonic velocity at the lower boundary (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025). The outer boundary is supersonic, with either zero incoming intensity from infinity, free-streaming extrapolation of the radiation field, or asymptotic approach to constant $\rho(r)$1.

The atomic-physics content is extensive. The subdwarf descriptions cite line data from the Opacity Project and Iron Project, bound-free cross-sections from TOPbase, electron scattering, and millions of transitions for heavy elements such as Fe and Ni; the helium-dominated study further lists NIST, VALD, and Kurucz line lists; the OB-star grid states that H, He, C, N, O, Ne, Mg, Si, S, and Fe-group elements from Sc through Ni are treated in NLTE (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025). In the helium-dominated subdwarf study, the code infrastructure is identified as Fortran 90/95 with OpenMP and MPI parallelization, a memory footprint of about $\rho(r)$2 when iron-group superlevels are active, and a wall-clock time of about $\rho(r)$3–$\rho(r)$4 hours for one converged model on 16 CPU cores at $\rho(r)$5 (Krticka et al., 2024).

4. Chemical composition, metallicity, and line driving

A central use of METUJE is to evaluate how composition modifies the CMF line force. In the helium-dominated subdwarf study, three model families are defined for fixed stellar parameters: the actual spectroscopic abundances, a model in which the H/He ratio is replaced by a scaled solar H+He mix at equal total mass fraction, and a fully solar composition following Asplund et al. 2009 (Krticka et al., 2024). This setup isolates the effect of heavy-element abundance anomalies from the effect of hydrogen and helium alone.

The code descriptions state that metals provide the bulk of the line driving, whereas hydrogen and helium contribute only a few percent to the radiative force. Replacing He by H+He changes $\rho(r)$6 only at the level of a few percent, while changes in C, N, O, Fe, and related metals produce corresponding changes in the computed mass-loss rate (Krticka et al., 2024). In one subdwarf summary, accurate abundances for HD 49798 do not significantly modify the mass-loss rate because increased Fe and Ni contribution to the radiative force is compensated by decreased force from other elements; in the helium-dominated merger products, modified abundances reduce the wind mass-loss rate by tens of percent; and in the OB-star grid the metallicity dependence steepens markedly for $\rho(r)$7 (Krticka et al., 2019, Krticka et al., 2024, Krticka et al., 29 Aug 2025).

The helium-dominated subdwarf paper quotes the approximate metallicity scaling

$\rho(r)$8

and also notes $\rho(r)$9–$T(r)$0 as a typical range (Krticka et al., 2024). The later OB-star grid refines this picture: $T(r)$1 for $T(r)$2, steepening to $T(r)$3 for $T(r)$4, with the physical explanation that iron-group lines dominate near the sonic point at solar and Magellanic abundances, whereas at $T(r)$5 most metal lines become optically thin and only a handful of strong UV resonance lines, including C III 977 Å, Si IV 1394/1403 Å, and O V 630 Å, drive the flow (Krticka et al., 29 Aug 2025).

A recurrent misconception in the subdwarf literature is that chemically peculiar surfaces must always imply dramatic changes in wind strength. The published METUJE applications do not support such a universal statement. Nucleosynthesis-modified abundances are reported not to significantly alter the strength of the wind of hot subdwarfs in general, although they can change $T(r)$6 by tens of percent and substantially improve agreement with UV wind line diagnostics (Krticka et al., 2024). This suggests that abundance effects are important, but not automatically dominant over the underlying stellar parameters.

5. Hot-subdwarf applications

The first dedicated hot-subdwarf application considered the hydrogen-dominated stars HD 49798 and BD+18$T(r)$7 2647, using abundances derived from UV and optical spectroscopy and then predicting wind structure with METUJE (Krticka et al., 2019). The reported stellar and wind parameters are:

Star/model	Stellar parameters	METUJE result
HD 49798, real abundances	$T(r)$8, $T(r)$9, $0.01\,Z_\odot$0	$0.01\,Z_\odot$1, $0.01\,Z_\odot$2
HD 49798, solar abundances	—	$0.01\,Z_\odot$3, $0.01\,Z_\odot$4
BD+18$0.01\,Z_\odot$5 2647, real abundances	$0.01\,Z_\odot$6, $0.01\,Z_\odot$7, $0.01\,Z_\odot$8	$0.01\,Z_\odot$9; no consistent solution
BD+18$\dot M$0 2647, solar abundances	—	$\dot M$1, $\dot M$2

The physical interpretation given in that study is sharply different for the two stars. HD 49798 has a strong wind that does not allow for chemical separation and therefore shows solar chemical composition modified by hydrogen burning. BD+18$\dot M$3 2647, by contrast, is reported to have no wind in the accurate-abundance model, and its abundances are therefore most likely affected by radiative diffusion. The star lies very close to the wind limit, and its peculiar chemical composition suppresses a self-consistent steady solution (Krticka et al., 2019).

The second subdwarf application addressed the helium-dominated merger products CD-46 8926 and CD-51 11879. Both have effective temperatures in excess of $\dot M$4; CD-46 8926 shows a strong overabundance of carbon and CD-51 11879 a strong overabundance of nitrogen. The modified abundances reduce the wind mass-loss rate by tens of percent, improve the predicted wind line profiles relative to observations, and lower the expected X-ray luminosities so that they agree with observations. The same study concludes that subdwarfs overabundant in helium are typically able to launch wind, and that such objects avoid the region in the Kiel diagram where winds are predicted to be absent. The proposed interpretation is suppression of gravitational settling of helium by the winds (Krticka et al., 2024).

6. Extension to OB stars and metallicities down to $\dot M$5

The 2025 OB-star grid extends METUJE beyond subdwarfs to line-driven winds of OB stars with metallicities down to $\dot M$6 (Krticka et al., 29 Aug 2025). In that regime, the code predicts that mass-loss rates decrease with decreasing metallicity and that the dependence steepens at very low $\dot M$7. The same study reports a disappearing bistability jump near $\dot M$8–$\dot M$9 for $\dot M$0, and gives terminal velocities that on average satisfy $\dot M$1 at $\dot M$2 and scale as $\dot M$3 at lower metallicity.

The low-metallicity behavior is presented as a line-statistics problem. At solar and Magellanic abundances, iron-group lines dominate near the critical point. As metallicity approaches $\dot M$4, most metal lines become optically thin, so the wind is accelerated by only a few strong lines. The resulting theoretical scatter is attributed to the low number of lines that effectively accelerate the wind at very low metallicities (Krticka et al., 29 Aug 2025).

The comparison to observations is qualified rather than absolute. For Galactic, LMC, and SMC stars, the predicted $\dot M$5 generally agrees with optical and UV diagnostics, corrected for clumping, to within a factor of about 2, and SMC B supergiants are described as well matched. At $\dot M$6–$\dot M$7, however, both theory and observations show large scatter. The observational part of that scatter is linked to inefficient shock cooling in weak winds, which can leave a considerable fraction of the wind at temperatures too high for standard UV and optical diagnostics. METUJE neglects shock heating and cooling, so in that regime its predicted $\dot M$8 may exceed values inferred from classical diagnostics, although the study notes that IR forbidden lines and X-ray diagnostics such as 10 Lac with JWST support the higher predicted rates (Krticka et al., 29 Aug 2025).

7. Validation, limitations, and projected developments

METUJE’s validation record in the supplied literature combines direct comparison to observed diagnostics with comparisons to other wind-modeling approaches. The helium-dominated subdwarf study reports reproduction of O-star wind models such as $\dot M$9 Pup and $v_\infty$0 Pup, with $v_\infty$1 and $v_\infty$2 in agreement with both CAK-style and Monte Carlo wind codes, and comparison of predicted UV P Cygni profiles of N V 1239/1243 Å and O VI 1032/1038 Å with IUE and FUSE data for canonical hot stars (Krticka et al., 2024). The same study also reports consistency checks of sonic-point location and critical-point integration against analytic CAK critical-point solutions in optically thin and optically thick line limits.

In the subdwarf context, the most specific validation uses HD 49798. There, the METUJE-predicted $v_\infty$3 and abundances were used to compute the X-ray eclipse light curve via

$v_\infty$4

The resulting ingress and egress profiles are reported to closely match the XMM–Newton measurements, whereas the solar-abundance model does not reproduce the observed eclipse shape (Krticka et al., 2019).

The limitations described in the literature are methodologically important. In the hot-subdwarf studies, some potentially wind-driving elements such as P and Cr could not be constrained spectroscopically and were assumed at solar ratios, introducing uncertainty at roughly the 30% level in $v_\infty$5 (Krticka et al., 2019). Solutions near the wind limit can fail to converge even when $v_\infty$6, which is identified as a necessary but not sufficient condition for a steady-state wind (Krticka et al., 2019). More generally, the published assumptions exclude rotation, magnetic fields, clumping, and time dependence (Krticka et al., 2024), while the OB-star metallicity study explicitly notes the neglect of shock heating and cooling (Krticka et al., 29 Aug 2025).

Planned developments listed in the OB-star work include incorporation of small-scale clumping, inclusion of X-ray emission from embedded shocks, multidimensional extensions for rotation and magnetic channeling, and a time-dependent treatment of the line-driven instability (Krticka et al., 29 Aug 2025). Taken together with the subdwarf and OB-star applications, these projected developments indicate a code base aimed at preserving the defining METUJE feature—a unified NLTE CMF hydrodynamic solution—while extending its physical realism in regimes where diagnostics and wind dynamics are presently limited by omitted microstructure or nonstationarity.