MESA-QUEST: Quasi-star Simulation Toolkit

• MESA-QUEST is an open-source simulation toolkit that models quasi-stars—radiation-supported envelopes hosting accreting black holes—with precise 1D stellar evolution techniques.
• It incorporates advanced inner-boundary treatments, dual accretion schemes, and wind prescriptions to reproduce and extend previous quasi-star structural and evolutionary findings.
• The framework provides a flexible platform for exploring the viability of heavy black hole seed formation under direct-collapse conditions in the early universe.

Searching arXiv for the MESA-QUEST papers and related MESA context. MESA Quasi-star Evolutionary Simulation Toolkit (MESA-QUEST) is an open, MESA-based implementation of quasi-star models designed to study the formation and growth of heavy black hole seeds via direct collapse. It extends the 1D stellar evolution “star” module of the Modules for Experiments in Stellar Astrophysics (MESA) to represent a quasi-star as a hydrostatic, radiation-supported envelope hosting a central, accreting black hole, with the core goal of reproducing the structural and evolutionary behavior of quasi-stars found with the Cambridge STARS code and of providing a flexible platform for exploring additional physics and parameter sensitivity relevant to the viability of quasi-stars as heavy seeds of supermassive black holes (Santarelli et al., 13 Oct 2025, Campbell et al., 16 Jul 2025). The framework was introduced to address the rapid appearance of supermassive black holes at redshifts $z \geq 9$, for which direct-collapse black hole (DCBH) scenarios and quasi-star intermediaries remain a leading heavy-seed pathway (Santarelli et al., 13 Oct 2025).

1. Astrophysical context and scientific motivation

The origin of the first supermassive black holes (SMBHs) observed at $z \geq 9$ remains one of the most challenging open questions in astrophysics. Their rapid emergence suggests that massive “heavy seeds” must have formed early, possibly through the direct collapse of pristine gas clouds in the first galaxies (Santarelli et al., 13 Oct 2025). In the DCBH scenario, pristine gas in early protogalaxies collapses monolithically without fragmenting, aided by strong Lyman–Werner radiation fields that suppress $\mathrm{H}_2$ cooling, low metallicity, and angular-momentum transport by non-axisymmetric instabilities (Santarelli et al., 13 Oct 2025).

Within this framework, the end-state of such a collapse can be a quasi-star: a massive, radiation-supported envelope that harbors a central, accreting black hole. Quasi-stars are compelling because the embedded black hole can accrete at a rate set by the envelope’s Eddington limit rather than the black hole’s own Eddington limit, enabling rapid growth to intermediate-mass black holes of $\gtrsim 10^3\!-\!10^5\,M_\odot$ within the tight cosmic time budget implied by early SMBH detections (Santarelli et al., 13 Oct 2025). The earlier MESA-QUEST implementation described the same scientific objective in terms of porting and extending the methodology previously realized in the Cambridge STARS code into MESA’s modular framework, thereby enabling tighter numerical control, general-relativistic stability corrections, and straightforward inclusion of additional physics (Campbell et al., 16 Jul 2025).

The toolkit is therefore positioned at the intersection of DCBH formation theory, quasi-star structure theory, and numerical stellar evolution. Its scientific role is not merely to reproduce canonical quasi-star tracks, but to delineate the physical limits under which quasi-stars can remain stable, survive mass loss, and produce heavy seeds capable of evolving into the earliest SMBHs detected by JWST and Chandra (Santarelli et al., 13 Oct 2025).

2. Software architecture and integration with MESA

MESA-QUEST is built upon the Modules for Experiments in Stellar Astrophysics, specifically the 1D stellar evolution “star” module (Santarelli et al., 13 Oct 2025). In the initial implementation, all new quasi-star physics—inner boundary treatment, accretion source term, Tolman–Oppenheimer–Volkoff (TOV) correction, sound-speed regularization, and Bondi-radius limiter—are implemented in run_star_extras within the MESA-QUEST repository (Campbell et al., 16 Jul 2025). The broader MESA substrate contributes the EOS, opacity, convection, atmosphere, timestep, mesh-refinement, and solver infrastructure on which quasi-star extensions are layered [(Jermyn et al., 2022); (Paxton et al., 2010)].

The defining architectural modification is the replacement of the standard stellar center boundary conditions. Instead of the usual $m(r=0)=0$ and $l(r=0)=0$, MESA-QUEST imposes

$$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$

where $R_0$ is the inner boundary radius, $M_{\rm BH}$ is the black hole mass, $M_{\rm cav}$ is the cavity mass interior to $z \geq 9$0, and $z \geq 9$1 is the black-hole accretion luminosity (Santarelli et al., 13 Oct 2025). This converts a stellar-center problem into an envelope-on-core problem in which the unresolved central region is represented by a point mass plus a prescribed luminosity source.

Two inner boundary definitions are available. One is the Bondi radius $z \geq 9$2, which matches the quasi-hydrostatic envelope to a Bondi-like inflow. The other is the saturated-convection radius $z \geq 9$3, following Coughlin (2024), which places the hydrostatic boundary deeper, where outward convective energy transport saturates (Santarelli et al., 13 Oct 2025). The existence of these two boundary options is central to the framework because the boundary placement strongly affects the accessible black-hole growth regime.

The repository is public at http://www.github.com/andysantarelli/MESA-QUEST, and includes source modifications and example inlists illustrating how to enable quasi-star physics, choose inner boundary types, and select wind schemes (Santarelli et al., 13 Oct 2025). Standard MESA build procedures are followed; the customized star module is compiled and executed via the included inlists. Output includes black-hole mass, envelope mass, luminosities, radii, inner-boundary radius, and mass-loss rates (Santarelli et al., 13 Oct 2025). A typical 1D quasi-star track to $z \geq 9$4 black holes completes in hours on a modern workstation, with runtime depending on wind strength and whether the inner boundary approaches the photosphere (Santarelli et al., 13 Oct 2025).

3. Physical model, governing equations, and inner-boundary formalism

For $z \geq 9$5, MESA-QUEST solves the standard stellar structure equations with quasi-star-specific modifications (Santarelli et al., 13 Oct 2025). Hydrostatic equilibrium is written in MESA form with a GR correction,

$z \geq 9$6

where $z \geq 9$7 includes a Tolman–Oppenheimer–Volkov correction appropriate to the deep potential well and is recalculated each timestep (Santarelli et al., 13 Oct 2025). Mass conservation remains

$z \geq 9$8

and radiative transport is

$z \geq 9$9

Convection is treated with the Ledoux criterion and mixing-length theory, with $\mathrm{H}_2$0 in the fiducial runs (Santarelli et al., 13 Oct 2025).

The luminosity equation is modified so that the dominant energy source is black-hole accretion rather than nuclear burning: $\mathrm{H}_2$1 with $\mathrm{H}_2$2 in fiducial quasi-star models and the primary energy source supplied through the inner boundary condition $\mathrm{H}_2$3 (Santarelli et al., 13 Oct 2025). This formulation encodes the defining quasi-star assumption that photon transport and convection carry accretion power outward through the envelope.

The cavity mass $\mathrm{H}_2$4 accounts for gas within the inner boundary. For Bondi-type models, the unresolved density field is taken as

$\mathrm{H}_2$5

with $\mathrm{H}_2$6 or $\mathrm{H}_2$7 depending on whether angular momentum is transported inward or outward, and

$\mathrm{H}_2$8

where the Schwarzschild radius $\mathrm{H}_2$9 is typically negligible (Santarelli et al., 13 Oct 2025). In the earlier Bondi-only implementation, $\gtrsim 10^3\!-\!10^5\,M_\odot$0 was estimated by extending an inner $\gtrsim 10^3\!-\!10^5\,M_\odot$1 profile interior to the Bondi radius (Campbell et al., 16 Jul 2025).

The Bondi inner boundary is defined by

$\gtrsim 10^3\!-\!10^5\,M_\odot$2

with $\gtrsim 10^3\!-\!10^5\,M_\odot$3 evaluated at the innermost resolved cell (Santarelli et al., 13 Oct 2025). In the 2025 implementation, $\gtrsim 10^3\!-\!10^5\,M_\odot$4 is averaged over the innermost $\gtrsim 10^3\!-\!10^5\,M_\odot$5 zones and a safety limiter enforces $\gtrsim 10^3\!-\!10^5\,M_\odot$6 per timestep to stabilize the boundary (Campbell et al., 16 Jul 2025).

The saturated-convection boundary follows a deeper hydrostatic truncation based on dimensionless variables $\gtrsim 10^3\!-\!10^5\,M_\odot$7 and $\gtrsim 10^3\!-\!10^5\,M_\odot$8, where $\gtrsim 10^3\!-\!10^5\,M_\odot$9, with governing ODE

$m(r=0)=0$0

and

$m(r=0)=0$1

The associated convection-dominated accretion flow region satisfies $m(r=0)=0$2, and $m(r=0)=0$3 is adjusted accordingly (Santarelli et al., 13 Oct 2025).

4. Accretion prescriptions, luminosity coupling, and microphysics

MESA-QUEST implements two accretion prescriptions, each coupled directly to the inner luminosity boundary (Santarelli et al., 13 Oct 2025). The first is convection-limited Bondi accretion, following Ball et al. (2012), in which the black-hole growth rate is

$m(r=0)=0$4

where $m(r=0)=0$5 is the convective efficiency, $m(r=0)=0$6 the radiative efficiency, $m(r=0)=0$7 the adiabatic index, and the local $m(r=0)=0$8 and $m(r=0)=0$9 are evaluated at the inner boundary (Santarelli et al., 13 Oct 2025). The associated luminosity is

$l(r=0)=0$0

The second is envelope-Eddington-scaled accretion, motivated by quasi-star theory. In this formulation,

$l(r=0)=0$1

where $l(r=0)=0$2 is a control parameter, $l(r=0)=0$3 is the opacity evaluated at the deep core, and $l(r=0)=0$4 is the total quasi-star mass (Santarelli et al., 13 Oct 2025). The implied accretion rate is

$l(r=0)=0$5

The fiducial Eddington-scaled calculations span $l(r=0)=0$6 with $l(r=0)=0$7 (Santarelli et al., 13 Oct 2025).

The relevant radiative limits are the envelope Eddington luminosity,

$l(r=0)=0$8

and the black-hole Eddington limit,

$l(r=0)=0$9

Core opacities show that $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$0 is a critical boundary between compact, high-opacity envelopes and expanded, lower-opacity envelopes (Santarelli et al., 13 Oct 2025). This opacity sensitivity is significant because it controls whether the envelope enters runaway expansion or remains compact and stable.

The microphysics assumes primordial gas with fiducial composition $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$1 and $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$2 (Santarelli et al., 13 Oct 2025). Standard MESA EOS and opacity modules are used, with convection treated via the Ledoux criterion and mixing-length theory at $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$3 (Santarelli et al., 13 Oct 2025). The earlier implementation specifies MESA’s “ML1” option with $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$4 to match Ball et al. (2011), and emphasizes that radiation pressure is substantial to dominant in the interior (Campbell et al., 16 Jul 2025).

MESA itself contributes a more general infrastructure for time-dependent convection, improved energy conservation, automatic differentiation, and updated opacity and EOS handling (Jermyn et al., 2022). The MESA VI description notes that MESA does not include GR radiation hydrodynamics, multi-D radiation transport, or a native black-hole accretion flow model; in quasi-star applications these are represented as 1D source terms injected near the inner boundary (Jermyn et al., 2022). This contextualizes MESA-QUEST as a hydrostatic 1D quasi-star framework rather than a full GRRMHD treatment.

5. Winds, mass loss, parameter space, and practical workflow

A major extension of MESA-QUEST is the incorporation of multiple wind and mass-loss prescriptions (Santarelli et al., 13 Oct 2025). Three regimes are explored.

Regime	Prescription	Role in MESA-QUEST
Reimers	$$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$5 with $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$6	Lower bound
Dutch	van Loon et al. (2005) for $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$7; de Jager et al. (1988) for $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$8	Intermediate strength
Super-Eddington radiation-driven	Dotan & Shaviv (2011) form; Fiacconi et al. (2015) fit $$m(r=R_0)=M_{\rm BH}+M_{\rm cav}, \qquad l(r=R_0)=L_{\rm BH},$$9	Upper bound

The Reimers wind has minimal impact on quasi-star lifetimes in the reported models (Santarelli et al., 13 Oct 2025). The Dutch scheme is stronger than Reimers and curtails growth when envelopes expand and cool; the Nugis & Lamers WR component is not applied because it requires $R_0$0 (Santarelli et al., 13 Oct 2025). The quasi-star-specific radiation-driven winds provide the strongest mass loss and serve as an upper bound on envelope erosion (Santarelli et al., 13 Oct 2025).

The fiducial initialization procedure is explicit. An envelope is built by accreting at $R_0$1 onto a $R_0$2 protostar until $R_0$3, following supermassive-star construction approaches (Santarelli et al., 13 Oct 2025). A seed black hole of $R_0$4 is then embedded through the modified inner boundary conditions, using either a Bondi or saturated-convection inner boundary (Santarelli et al., 13 Oct 2025). The practical, reproducible setup specified for the repository includes primordial composition, Ledoux convection, $R_0$5, TOV correction, a selectable inner boundary type, one of the two accretion schemes, and a chosen wind prescription (Santarelli et al., 13 Oct 2025).

Representative controls listed in the documentation include:

• initial_mass = 20
• accumulate_mass_to = 1d4
• accretion_rate = 0.1
• use_ledoux_criterion = .true.
• mixing_length_alpha = 2.0
• use_TOV_correction = .true.
• bh_initial_mass = 10
• bh_inner_boundary_type = 'bondi' or 'saturated_convection'
• bh_accretion_scheme = 'bondi_convective' or 'alpha_edd'
• alpha_edd = 1.0
• radiative_efficiency = 0.1
• convective_efficiency = 0.1
• wind_scheme = 'none'/'reimers'/'dutch'/'radiation_driven'
• reimers_eta = 0.5 (Santarelli et al., 13 Oct 2025)

The recommended diagnostics are $R_0$6, $R_0$7, $R_0$8, $R_0$9, $M_{\rm BH}$0, $M_{\rm BH}$1, $M_{\rm BH}$2, $M_{\rm BH}$3, and $M_{\rm BH}$4 (Santarelli et al., 13 Oct 2025). Best-practice guidance in the repository notes that $M_{\rm BH}$5 is preferred for stability-focused studies, the saturated-convection boundary is preferred for maximizing black-hole growth, and the three wind prescriptions may be used to bracket uncertainties from lower to upper mass-loss limits (Santarelli et al., 13 Oct 2025).

6. Evolutionary outcomes, validation, and interpretive issues

The principal result is that quasi-stars can grow central black holes to $M_{\rm BH}$6 under favorable conditions (Santarelli et al., 13 Oct 2025). Without winds, saturated-convection inner boundaries yield substantially larger black-hole growth than Bondi boundaries. In saturated-convection models with $M_{\rm BH}$7, MESA-QUEST reaches $M_{\rm BH}$8, approximately five times higher than Bondi-limited cases, which yield $M_{\rm BH}$9 (Santarelli et al., 13 Oct 2025). This agrees with the trend found by Coughlin (2024), which suggested ratios up to $M_{\rm cav}$0, and exceeds the $M_{\rm cav}$1 ratio reported by Ball et al. (2012) under Bondi-like conditions (Santarelli et al., 13 Oct 2025).

A central interpretive claim of the framework concerns termination. In both boundary schemes, growth halts when the inner boundary radius $M_{\rm cav}$2 approaches the photosphere; the integration fails not because of a physical instability but because the hydrostatic region vanishes (Santarelli et al., 13 Oct 2025). On that basis, the reported black-hole-to-total-mass “limits” in earlier quasi-star studies are argued to be numerical or integration artifacts rather than physical caps (Santarelli et al., 13 Oct 2025). This is one of the major controversies addressed by the toolkit, and it is presented as a boundary-placement and stopping-criterion issue rather than as a new physical instability.

Envelope behavior depends strongly on $M_{\rm cav}$3. For $M_{\rm cav}$4, envelopes remain compact, hot, and dense; opacities remain high, preventing runaway expansion and maintaining stability over longer times (Santarelli et al., 13 Oct 2025). For $M_{\rm cav}$5, lower core opacities and higher luminosities drive envelope expansion; in Bondi models, the Bondi radius can show step-like increases when separated convective zones merge and the star becomes fully convective, lowering $M_{\rm cav}$6 and increasing $M_{\rm cav}$7 (Santarelli et al., 13 Oct 2025).

With winds included, the qualitative hierarchy is clear. Reimers winds reduce final $M_{\rm cav}$8 by $M_{\rm cav}$9 for $z \geq 9$00 in Bondi models and less in saturated-convection models (Santarelli et al., 13 Oct 2025). Dutch winds significantly curtail growth in models with envelope expansion; for $z \geq 9$01, the envelope is ejected promptly after expansion disrupts equilibrium, while $z \geq 9$02 survives marginally longer but still loses mass rapidly (Santarelli et al., 13 Oct 2025). Super-Eddington radiation-driven winds are most severe: quasi-stars survive only $z \geq 9$03 yr, and black holes grow by at most a factor of $z \geq 9$04 in mass across $z \geq 9$05 (Santarelli et al., 13 Oct 2025). This suggests that, unless envelope accretion replenishes the lost mass at comparable rates, heavy-seed formation can be strongly quenched (Santarelli et al., 13 Oct 2025).

Validation against earlier quasi-star work is explicit. The earlier Bondi-based implementation reproduces the fiducial quasi-star of Ball et al. (2011) to within $z \geq 9$06 in the final black-hole mass and closely tracks the same structural evolution (Campbell et al., 16 Jul 2025). Radial density profiles, inner and outer radii, luminosity evolution, and effective-temperature decline match the STARS results qualitatively, supporting the use of MESA as a validated quasi-star platform (Campbell et al., 16 Jul 2025). The later framework extends that validation by showing that Bondi inner-boundary models reproduce the qualitative behavior of Ball et al. (2012), while saturated-convection models recover the larger $z \geq 9$07 trend associated with deeper hydrostatic boundaries (Santarelli et al., 13 Oct 2025).

7. Relation to direct-collapse and supermassive-star studies, limitations, and future directions

MESA-QUEST is closely related to MESA studies of supermassive primordial stars, which provide envelope structures analogous to quasi-star progenitors (2208.00008). In those models, high accretion rates produce radiation-pressure-dominated, near-Eddington, cool red hypergiant envelopes with $z \geq 9$08, $z \geq 9$09, and collapse driven by general-relativistic instability during central hydrogen burning (2208.00008). MESA-QUEST adopts a comparable primordial-composition, Ledoux-convection, $z \geq 9$10, TOV-corrected numerical environment, but replaces the nuclear-burning core with a black-hole inner boundary and accretion-powered luminosity (Santarelli et al., 13 Oct 2025, 2208.00008).

The astrophysical implications are correspondingly specific. Quasi-stars can plausibly produce heavy seeds of $z \geq 9$11 under favorable conditions, especially with saturated convection near the black hole and modest winds (Santarelli et al., 13 Oct 2025). Environmental requirements are consistent with DCBH formation: pristine gas, strong Lyman–Werner backgrounds maintaining $z \geq 9$12, low angular momentum, and sustained inflows to maintain quasi-star envelopes against winds (Santarelli et al., 13 Oct 2025). A plausible implication is that the viability of the quasi-star pathway depends less on a unique hydrostatic ceiling to $z \geq 9$13 than on the mass-budget competition between black-hole feeding, envelope support, and wind-driven envelope loss.

Several limitations are explicit. Physics not yet included comprises rotation, magnetic fields, jets, photon trapping, GR-radiation hydrodynamics, and explicit envelope accretion during the quasi-star phase (Santarelli et al., 13 Oct 2025). The choice of inner boundary remains uncertain: neither Bondi nor saturated convection is definitively “correct,” and the true hydrostatic breakdown radius depends on complex multi-D physics (Santarelli et al., 13 Oct 2025). Likewise, the super-Eddington winds are implemented through semi-empirical fits rather than full radiation hydrodynamics, so mass-loss rates and critical thresholds may change in more complete treatments (Santarelli et al., 13 Oct 2025).

Future extensions identified by the framework are rotation, magnetic fields, GR-radiation hydrodynamics, and explicit envelope accretion (Santarelli et al., 13 Oct 2025). The earlier implementation had already identified wind mass loss, envelope accretion, and photon trapping as natural next additions made possible by MESA’s modularity (Campbell et al., 16 Jul 2025). In that sense, MESA-QUEST is both a research code for current quasi-star calculations and an extensible numerical framework for testing whether quasi-stars can remain a viable heavy-seed channel for the earliest SMBHs.