Quasi-Stars: Early SMBH Seed Models

• Quasi-stars are hypothetical astrophysical objects comprising a stellar-mass black hole embedded in a massive, radiation-pressure–dominated envelope that enables rapid black hole growth.
• They are modeled using an n=3 polytropic equation of state with inner boundary conditions at the Bondi radius, exhibiting mass limits akin to the Schönberg–Chandrasekhar limit.
• Their evolution has observable implications for early high-redshift SMBHs and compact infrared sources like JWST’s 'Little Red Dots', linking theory to cosmological surveys.

Quasi-stars are a class of hypothetical astrophysical objects central to the modeling of rapid supermassive black hole (SMBH) seed formation in the early universe. Conceptually, a quasi-star consists of a stellar-mass black hole embedded within a massive, hydrostatic, radiation-pressure–dominated envelope. The black hole accretes matter from the envelope at super-Eddington rates, and the liberated energy powers the envelope to radiate near the Eddington luminosity for the system’s total mass. This enables black hole growth far above rates possible for isolated stellar remnants and has significant implications for the observed population of high-redshift SMBHs and related compact infrared sources such as the "Little Red Dots" detected by JWST.

1. Quasi-star Structure and Governing Equations

Quasi-stars are modeled as composite objects: an accreting black hole at the center enveloped by a massive, extended, convective envelope dominated by radiation pressure. The envelope is well-described by an $n=3$ polytropic equation of state, yielding nearly constant ratios of gas pressure to radiation pressure throughout the bulk (0711.4078). For a total envelope mass $M_*$ far exceeding the black hole mass $M_{\mathrm{BH}}$, the central regions exhibit uniform temperature $T_c$ and density $\rho_c$.

Boundary conditions are imposed not at $r=0$ (as for standard stellar evolution models) but at an inner radius $r_0$, typically chosen to be the Bondi radius $B = (2GM_{\mathrm{BH}}) / c_s^2$, with $c_s$ the local sound speed (Ball et al., 2011). The inner mass boundary condition incorporates both the black hole and the cavity gas within $r_0$:

$$M_0 = M_{\mathrm{BH}} + M_{\mathrm{cav}}, \qquad M_{\mathrm{cav}} = \frac{8\pi}{3} \rho_0 B^3$$

The envelope must radiate away the accretion energy at its own Eddington luminosity (for $M_* + M_{\mathrm{BH}}$) rather than the Eddington luminosity for the BH alone. The photospheric effective temperature $T_{\mathrm{ph}}$ is a function of both $M_*$ and $M_{\mathrm{BH}}$ (0711.4078):

$$T_{\mathrm{ph}} = 1.0 \times 10^3\, \ell_{\mathrm{tr}}^{9/20} \tilde{\kappa}^{-9/20} \alpha^{-1/5} m_{\mathrm{BH}}^{-2/5} m_*^{7/20}\ \text{K}$$

where $\ell_{\mathrm{tr}}$ is an Eddington factor near unity, $\tilde{\kappa}$ the photospheric opacity normalized to electron–scattering, $\alpha$ a measure of accretion efficiency, and masses are in solar units.

2. Accretion Physics, Luminosity, and Mass Limits

Energy input to the envelope is governed by accretion onto the black hole. Due to angular momentum transport and the formation of thick disks, the actual accretion rate is limited versus naive Bondi estimates. The luminosity due to black hole accretion in models with inefficient accretion is given by (0711.4078):

$$L_{\mathrm{BH}} = 4 \pi G^2 \alpha M_{\mathrm{BH}}^2 \rho_c^{3/2} p_c^{-1/2}$$

Hydrodynamic energy transport is controlled by the envelope’s convective efficiency; the envelope channels the energy outward, enforcing the “global” Eddington limit:

$$L_{\mathrm{BH}} \simeq 1.4 \times 10^{38} \ell_{\mathrm{tr}}\, \tilde{\kappa}^{-1} m_*\ \mathrm{erg~s}^{-1}$$

The global Eddington constraint permits rapid black hole growth—far faster than possible for a naked black hole. However, the system develops mass limits connected to both microphysics and global structure. In Bondi-type quasi-star models, hydrostatic equilibrium breaks down when $M_{\mathrm{BH}}$ reaches roughly 10–12% of $M_*$, a result robust across codes and parametric choices (Ball et al., 2011, Ball, 2012). This limit is characterized by the “loaded polytrope” mass boundary condition:

$\phi_0 = \frac{1}{2n} \xi_0 + \frac{2}{3} \xi_0^3$

The appearance of a mass fraction limit is analogous to the Schönberg–Chandrasekhar (SC) limit in stellar evolution, which describes the maximum isothermal core mass fraction compatible with a polytropic envelope (Ball et al., 2012, Ball, 2012). In CDAF/ADAF-type models and recent saturated-convection frameworks, the black hole can accrete much higher fractions—up to $\sim$60% of $M_*$ in models where convection efficiently transports energy outward (Coughlin et al., 30 Apr 2024, Santarelli et al., 13 Oct 2025, Hassan et al., 21 Oct 2025).

3. Photospheric Properties and Hayashi-type Limits

As the quasi-star evolves, $T_{\mathrm{ph}}$ falls due to increasing $M_{\mathrm{BH}}$ and decreasing envelope mass. When $T_{\mathrm{ph}}$ drops below $\sim$10,000 K, the opacity in metal-free (Pop III) gas drops precipitously—reaching a floor near 4000–5000 K (0711.4078). This imposes a limiting temperature below which no hydrostatic envelope solutions are possible, analogous to the Hayashi track for protostars and red giants. At this limit, super-Eddington fluxes develop, the envelope disperses, and the quasi-star phase terminates.

In late-stage quasi-stars ($M_{\mathrm{BH}} \gtrsim 0.1 M_*$), the outer layers (photosphere) attain quasi-universal properties, largely insensitive to total mass or black hole mass (Begelman et al., 12 Jul 2025, Santarelli et al., 20 Oct 2025). Characteristic scaling relations describe this regime, e.g.,

$$T_{\text{eff}} \simeq 3000\, \beta_{-1}^{-1/5} M_6^{-1/20}\ \text{K}$$

and the radius,

$$R_* \simeq 6 \times 10^{16}\, \beta_{-1}^{2/5} M_6^{3/5}\ \text{cm}$$

with $M_6$ in $10^6\,M_\odot$ units and $\beta$ the convective efficiency parameter.

4. Numerical Modeling: Evolution Codes and Accretion-Wind Interplay

Quasi-star evolution has been treated using modified stellar evolution codes such as Cambridge STARS (Ball et al., 2011, Ball, 2012) and, more recently, MESA via the MESA-QUEST toolkit (Campbell et al., 16 Jul 2025, Santarelli et al., 13 Oct 2025, Hassan et al., 21 Oct 2025). In these frameworks, inner boundary conditions are imposed at $r_0$ above the Schwarzschild radius to avoid singularities, and models are validated via density profiles, envelope radii, temperature evolution, and surface luminosity comparisons.

Accretion is coupled to the envelope via prescriptions that either relate the convective energy transport, radiative efficiencies, and cavity mass, or scale accretion to envelope Eddington luminosity (Santarelli et al., 13 Oct 2025). Wind mass-loss is handled using various physical models:

• Reimers wind: $$\dot{M}_{\text{wind}} = 4 \times 10^{-13} \eta (L R / M)$$
• Dutch scheme: empirical, temperature-dependent prescriptions for outflows
• Super-Eddington wind: $\dot{M}_{\text{wind}} = 4\pi R_*^2 \rho_* \mu c_s$

Strong winds, especially in models with super-Eddington luminosities, can strip away the envelope and limit black hole growth (Santarelli et al., 13 Oct 2025, Hassan et al., 21 Oct 2025). Favorable (saturated-convection) conditions enable BH-to-total mass ratios up to $0.55$, while Bondi-limited configurations restrict this ratio to $\sim0.11$.

5. Connection to the Schönberg–Chandrasekhar Limit and Giant Evolution

The origin of mass-fraction limits in quasi-stars is intimately connected to the generalized Schönberg–Chandrasekhar limit (Ball et al., 2012, Ball, 2012). Mapping the Lane–Emden equation for polytropic models into the $(U,V)$ homology-invariant plane, solutions for the inner core (BH+cavity) only intersect fractional mass contours up to a maximum $q_{\max}$. When $q$ increases past $q_{\max}$, the envelope either cannot adjust via hydrostatic pressure or rapidly expands—mirroring the transition from post-main-sequence stars to red giants. This correspondence establishes quasi-star mass-fraction limits as a general property of composite stellar models.

Key relationships for discerning the existence and location of fractional mass limits include:

Variable	Definition	Context
$U$	$=d\log m / d\log r=3\rho/\bar{\rho}$	Local/mean density ratio
$V$	$=-d\log p / d\log r=Gm/r \cdot \rho/p$	Gravitational binding/pressure
$q$	$=\phi_0/\phi_1$	Fractional core mass

SC-like limits are thus predictive triggers for envelope expansion and quasi-star phase end.

6. Observational Manifestations and Cosmological Implications

Quasi-stars are proposed as progenitors of direct-collapse black hole seeds in the first galaxies, facilitating rapid formation of SMBHs at high redshift ($z \gtrsim 9$). Their radiative properties are predicted to resemble the "Little Red Dots" (LRDs) discovered with JWST (Begelman et al., 12 Jul 2025, Santarelli et al., 20 Oct 2025). In particular, late-stage quasi-stars radiate near the Eddington limit for the total mass, have effective temperatures (set by envelope physics) in the $\sim$3000–7000 K range, display significant Balmer breaks due to hydrogen n=2 populations, exhibit broad Balmer lines (from electron scatter broadening), and are X-ray suppressed due to huge electron column densities.

The short lifetimes of the quasi-star phase (tens of Myr) coupled with high LRD comoving density suggest that the formation pathway may have been nearly universal for SMBH precursors (Begelman et al., 12 Jul 2025, Santarelli et al., 20 Oct 2025). Scaling relations—for example,

$$\log(L/L_\odot) = 16\log(T_{\text{eff}}/\text{K}) - 50$$

and

$$M_{\text{star}} \simeq 10^6\, M_\odot (L/3.7 \times 10^{10}\,L_\odot)$$

—connect simulation outputs to observed properties.

Quasi-stars provide a cosmologically viable path for overcoming radiative, angular momentum, and accretion-rate barriers and assembling the SMBH population on timescales compatible with early-universe constraints. Their evolution, structure, and termination (via Hayashi-type floors and envelope ejection) are key to understanding the formation of massive black holes and the luminous infrared population at high redshift (0711.4078, Coughlin et al., 30 Apr 2024, Santarelli et al., 13 Oct 2025, Hassan et al., 21 Oct 2025).

7. Future Directions and Constraints

Continued modeling in MESA-QUEST and other frameworks will refine accretion physics (including rotation, magnetic fields, general relativistic corrections, and GR-RMHD), wind mass-loss prescriptions, and matching to observed high-redshift populations (Santarelli et al., 13 Oct 2025). A critical ongoing challenge is to balance envelope retention against wind-driven ejection, especially in the face of super-Eddington luminosities and feedback.

The self-similar nature of quasi-star evolution (i.e., mass ratios, envelope structures relatively independent of absolute scale) aids in integrating these results into cosmological simulations and predicting SMBH demographics. Connections between quasi-star termination, observational signatures (in LRDs), and large-scale galaxy evolution establish quasi-stars as a central object of paper in theoretical astrophysics and cosmology.