Quasi-Star Instability Strip in HR Diagram

• Quasi-star instability strip is a defined region in the HR diagram, spanning effective temperatures of ~4000–5200 K, where massive, radiation-pressure dominated envelopes exhibit pulsational instabilities.
• The instability is driven by the κ-mechanism in helium and hydrogen ionization zones, with MESA and GYRE models revealing fundamental pulsation periods of 30–180 years and significant growth rates.
• This pulsation-induced variability regulates mass loss, shaping the final mass of black hole seeds and linking theoretical models to JWST observations of 'Little Red Dots'.

A quasi-star instability strip is a distinct region in the Hertzsprung–Russell (HR) diagram, occupied by massive, radiation-pressure dominated stellar envelopes surrounding growing black hole seeds, within which the envelopes are linearly unstable to radial pulsations. This instability strip, extending from effective temperatures $T_{\rm eff} \approx 5200$ K (the “blue edge”) down to $\sim4000$ K (the “red edge”), is a hallmark of the late evolutionary stages of direct-collapse black hole precursors. In these objects—denoted “quasi-stars”—the development of global oscillatory modes, primarily driven by the $\kappa$-mechanism in helium and hydrogen ionization zones, regulates both the observed variability of compact high-redshift sources such as JWST-discovered “Little Red Dots” and the feedback-limited growth of the nascent supermassive black holes they enshroud (Cantiello et al., 19 Dec 2025).

1. Physical Context and Definition

Quasi-stars, with envelope masses in the range $M_\star \sim 10^4$–$2\times10^5\,M_\odot$, represent a critical phase in the assembly of heavy seed black holes via direct collapse. These envelopes, initially in near-hydrostatic equilibrium, can become susceptible to global pulsational instabilities as they cool and expand. The “quasi-star instability strip” is defined by the loci in the HR diagram where the fundamental radial mode ($\ell=0$; $n_{\rm p}=1$) and, at lower temperatures, the first overtone ($n_{\rm p}=2$), transition from stability ($\Im(\omega)<0$) to linear over-stability ($\Im(\omega)>0$). This boundary is sharply delineated at a blue edge of $T_{\rm eff} \approx 5000$–$5200$ K and a red edge that tracks the Hayashi limit, near $T_{\rm eff} \sim 4000$ K, beyond which convective damping suppresses further pulsational driving.

2. Instability Mechanism: The $\kappa$-Mechanism in Ionization Zones

The underlying cause of the instability strip is the operation of the classical $\kappa$-mechanism, wherein opacity variations in partial ionization zones act as a heat engine. In quasi-stars, the dominant driver is the He II ionization layer at $T\sim4\times10^4$ K, responsible for broad destabilization due to its depth and the positive temperature derivative of the Rosseland mean opacity:

$$\left(\frac{\partial\kappa}{\partial T}\right)_\rho > 0.$$

Pulsational energy input is quantified with the non-adiabatic work integral formalism:

$$\frac{dW}{dr} = \frac{1}{\rho g}\,\Re[\delta p^*\,\delta\rho] \propto \Re\Bigl[\delta T^*\,\frac{\partial\kappa}{\partial T}\Bigr],$$

where zones with $dW/dr>0$ contribute to mode excitation. The hydrogen partial ionization zone, near $T\sim10^4$ K, provides secondary driving, characterized by a sharp but spatially confined peak in $dW/dr$. However, in the outermost layers, strong non-adiabatic cooling diminishes its net contribution.

3. Model Construction and Stability Analysis

Stability properties of quasi-star envelopes have been computed using the stellar evolution software MESA to generate hydrostatic structures, linked to the non-adiabatic oscillation solver GYRE. Linearized perturbation equations for radial displacement $\xi_r$ and pressure perturbation $\delta p$ are integrated from an inner acoustic cutoff $R_i$ (where convection becomes inefficient) to a free surface at $R_\star$:

$$\begin{align*} \frac{d\xi_r}{dr} &= \left(\frac{1}{r}-\frac{1}{\Gamma_1 p}\frac{dp}{dr}\right)\xi_r + \frac{1}{\rho r^2 \omega^2}\delta p,\ \delta p &= -\rho \omega^2 r \xi_r - \frac{dp}{dr} \xi_r, \end{align*}$$

augmented by a non-adiabatic energy equation coupling thermal and opacity perturbations. Growth rates are measured by

$$\eta = \frac{\Im(\omega)}{\Re(\omega)}, \quad \tau_{\rm growth} = \frac{1}{\Im(\omega)}\,\tau_{\rm dyn},$$

where $\tau_{\rm dyn} = \sqrt{R_\star^3/(G M_\star)}$ is the dynamical timescale. Unstable modes are identified by positive work integrals and the mode order ($n_{\rm p} = 1$ for fundamental, $n_{\rm p}=2$ for first overtone).

4. Periods, Growth Rates, and Evolution Across the Instability Strip

Within the instability strip, the fundamental mode periods ($P_1$) span $30$–$180$ yr, lengthening with increasing mass and as $T_{\rm eff}$ decreases. For example, for $M_\star=10^5\,M_\odot$ at $T_{\rm eff}=4680$ K, GYRE yields $\Re(\omega)\approx0.91$, $\Im(\omega)\approx+0.039$, giving $P_1\approx73$ yr, $\tau_{\rm growth}\approx273$ yr. The first overtone ($P_2\sim10$–$30$ yr) emerges unstable at lower $T_{\rm eff}$, although with lower growth rates and sometimes marginal damping (e.g., $\Re(\omega)\approx3.10$, $\Im(\omega)\approx-0.021$, $P_2\approx21.5$ yr). Nonlinear time-dependent MESA calculations confirm that $P_2$ can be successively amplified to produce supersonic motions at the stellar surface.

Table 1 summarizes the typical parameter regimes:

Mode	Period Range (yr)	Instability Onset
Fundamental	30–180	$T_{\rm eff} < 5000$–$5200$ K
First overtone	10–30	Lower $T_{\rm eff}$, overlapping red edge

5. Correspondence with Observed Little Red Dots

The quasi-star instability strip physically maps onto the observed properties of JWST “Little Red Dots.” The lensed source R2211-RX2, with $T_{\rm eff}\approx5000$ K and no variability, coincides with the predicted blue edge. Conversely, R2211-RX1, exhibiting a $\sim30$-year periodicity and hysteresis in its luminosity–temperature trajectory, resides deep within the unstable, pulsating domain. The observed overtone variability timescale of RX1 closely matches $P_2$ predicted by the models, while the longer $P_1$ is manifest as a secular drift.

6. Dynamical Consequences: Mass Loss and Quasi-Star Termination

As quasi-stars cool across the instability strip, nonlinear growth of radial pulsations is expected to induce substantial mass-loss analogous to super-AGB or Mira-star “super-winds.” MESA hydrodynamic calculations reveal supersonic photospheric velocities and shock generation, highlighting the onset of strong mass ejection. This pulsation-driven wind provides a negative feedback on accretion by dispersing the envelope: at the blue edge, the emergence of instability can trigger wind episodes that regulate or halt black hole growth, while at lower $T_{\rm eff}$ repeated mass-loss outbursts may occur if external accretion persists. Thus, the instability strip delineates a pivotal evolutionary bottleneck with direct implications for the final mass of the seed supermassive black hole.

7. Summary and Astrophysical Significance

The quasi-star instability strip, defined by $T_{\rm eff}\simeq4000$–$5200$ K and occupied for envelope masses $10^4$–$2\times10^5\,M_\odot$, is a robust and predictive feature of quasi-star evolution. Driven chiefly by the $\kappa$-mechanism in ionization zones, this instability strip controls the transition to violent radial pulsations, enhanced mass loss, and eventual envelope dispersal. Its boundaries and dynamical signatures are directly encoded in the observed properties of high-redshift compact red sources such as the Little Red Dots, furnishing a theoretical framework for interpreting their long-term photometric variability and for constraining the processes that set the initial mass function of supermassive black holes (Cantiello et al., 19 Dec 2025).