Disc Instability Model in Astrophysical Discs

• Disc Instability Model is a framework describing episodic, time-dependent accretion in discs around compact objects through mechanisms like thermal-viscous, gravo-magneto, and gravitational instabilities.
• It employs conservation laws with the Shakura-Sunyaev α-viscosity prescription to predict critical thresholds that trigger transitions between quiescent and outburst states.
• Observations in cataclysmic variables, X-ray binaries, and protoplanetary discs validate DIM predictions, highlighting its broad impact on stellar, planetary, and galactic astrophysics.

The Disc Instability Model (DIM) encompasses a family of theoretical frameworks responsible for episodic, time-dependent accretion phenomena in astrophysical discs around compact objects. Most notably, DIM staples include the thermal-viscous instability in hydrogen-rich discs, gravo-magneto instabilities in protoplanetary discs, and violent gravitational instabilities in high-redshift galaxies. This model is central to understanding dwarf nova outbursts, soft X-ray transient cycles, FU Orionis events, and fragmentation of protoplanetary discs, with predictions closely verified by detailed observations in stellar and extragalactic systems (Hameury, 2019, Dubus et al., 2018, Martin et al., 2013, Fenton et al., 2 Feb 2024, Cacciato et al., 2011).

1. Fundamental Physical Principles of DIM

The core mechanism of the DIM is the emergence of limit-cycle behavior driven by local instability criteria. In hydrogen-dominated accretion discs, this instability arises from rapid changes in opacity at temperatures where hydrogen ionizes or recombines. The governing equations are those of mass and angular momentum conservation, supplemented by the Shakura-Sunyaev $\alpha$-viscosity prescription: $$\frac{\partial\Sigma}{\partial t} = \frac{3}{R} \frac{\partial}{\partial R}\left[R^{1/2} \frac{\partial}{\partial R}\left(\nu \Sigma R^{1/2}\right)\right]$$ with $\nu = \alpha c_s H$, $c_s$ the sound speed, and $H$ the scale height. Viscous heating ($Q^+ = \frac{9}{8} \nu\Sigma\Omega_K^2$) is balanced against radiative cooling ($Q^- = \sigma T_{\mathrm{eff}}^4$), producing a characteristic “S-curve” in $\Sigma$–$T_{\mathrm{eff}}$ space. The stable branches correspond to hot (ionized) or cold (neutral) gas, while the intermediate segment is thermally and viscously unstable (Hameury, 2019).

2. Critical Thresholds and Outburst Cycle

At each radius, vertical disc structure calculations yield critical surface densities $\Sigma_{\min}$ and $\Sigma_{\max}$ (Hameury, 2019):

$$\Sigma_{\min} \simeq 39.9\,\alpha_{0.1}^{-0.80}\,r_{10}^{1.11}\,M_{1}^{-0.37}\ \mathrm{g\,cm}^{-2}$$

$$\Sigma_{\max} \simeq 74.6\,\alpha_{0.1}^{-0.83}\,r_{10}^{1.18}\,M_{1}^{-0.40}\ \mathrm{g\,cm}^{-2}$$

The corresponding critical accretion rates for stability are: $$\dot M_{\rm crit}^{-} \simeq 2.65 \times 10^{15}\, r_{10}^{2.58}\, M_1^{-0.85}\ \mathrm{g\,s}^{-1}$$

$$\dot M_{\rm crit}^{+} \simeq 8.07 \times 10^{15}\, r_{10}^{2.64}\, M_1^{-0.89}\ \mathrm{g\,s}^{-1}$$

A disc ring is unstable if $$\dot M_{\rm crit}^{-} < \dot M(R) < \dot M_{\rm crit}^{+}$$. The global DIM cycle proceeds through quiescence (mass accumulation, cold branch), fast-rise heating fronts ignited at $\Sigma_{\mathrm{max}}$, quasi-steady hot states, and return to quiescence via cooling fronts (Dubus et al., 2018, Bollimpalli et al., 2018). Recurrence times are set by cold-branch viscous timescales, rise durations by front-propagation speeds, and decay times by the hot-branch viscosities.

3. Advanced Instability Modes: Gravo-Magneto and Violent Gravitational Instabilities

Beyond the classic thermal-viscous DIM, discs may undergo gravo-magneto instabilities in regions where MRI turbulence alternates with gravitational transport. Layered disc models separate thermally ionized, MRI-active zones from shielded “dead zones” (Martin et al., 2013). Material piles up until self-gravity sufficiently heats the disc to activate the MRI, leading to outburst events that sweep mass inward and outward ("snow-plough" mechanism), draining up to ~25% of disc mass per event over $\sim10^3$ yr.

In high-redshift galaxies, violent gravitational instability in a gas-rich two-component disc regulates the Toomre $Q_{2c}$ parameter near unity. Sustained cosmological gas inflow and feedback maintain instability at $z\gtrsim1$, with stabilization occurring as stellar velocity dispersion dominates and gas turbulence cools below the sound speed limit (Cacciato et al., 2011).

4. Observational Validation and Testing

Large-sample tests employing Gaia distances and V-band lightcurves demonstrate that cataclysmic variables (CVs) and X-ray binaries cleanly partition into unstable (dwarf novae) and stable (nova-like) states in the $\langle\dot M\rangle$–$P_{\mathrm{orb}}$ plane, with the DIM’s predictions for critical stability lines robustly borne out (Dubus et al., 2018, Coriat et al., 2012). In the X-ray binary context, the inclusion of irradiation effects ($$\sigma T_{\rm irr}^4 = \dot M c^2 / (4\pi R^2)\mathcal{C}$$) refines stability boundaries, explaining the transient/persistent dichotomy and recurrence time trends when introducing the “transientness” parameter $$\tau = \dot M_{\rm ext} / \dot M_{\rm crit}(P_{\rm orb})$$ (Coriat et al., 2012).

5. Variants and Extensions: Irradiation, Winds, and Tidal Instabilities

The standard DIM is extended by irradiation of the disc and/or donor, inner disc truncation, and variable mass-transfer rates. Disc irradiation raises the minimum temperature of outer rings, suppressing or modifying instability thresholds (Bollimpalli et al., 2018). Mass-transfer fluctuations, often induced by irradiation feedback or donor variability, explain standstills (e.g., Z Cam stars) and superoutburst phenomena, which in SU UMa stars further require the operation of a tidal–thermal instability at the 3:1 resonance radius (Hameury, 2019). Magnetically driven winds extract angular momentum and mass, altering the decay rate and possibly dominating transport in quiescence.

6. Disc Instability in Protoplanetary Discs and Giant Planet Formation

The disc-instability model also governs fragmentation in massive protoplanetary discs when the Toomre $Q \lesssim 1$ and cooling time is sufficiently short ($$\tau_{\mathrm{cool}}\Omega \lesssim \beta_{\mathrm{crit}}$$). Simulations performed with SPH codes (e.g., PHANTOM) trace the evolution through the first and second hydrostatic core phases, revealing that resulting protoplanets are typically oblate spheroids accreting preferentially from polar directions, with morphologies and accretion rates closely tied to their angular momentum profiles and merger histories (Fenton et al., 2 Feb 2024). This non-spherical, anisotropic accretion morphology translates into observable properties, affecting spectral energy distributions and shock signatures.

7. Outstanding Problems and Future Directions

Major limitations of the current DIM framework include the poorly understood origin of angular-momentum transport ($\alpha$ parameter), especially in low ionization, quiescent discs; incomplete modeling of wind and outflow torques; and insufficient coupling of time-dependent disc spectra with detailed radiative transfer and multi-dimensional hydrodynamics (Hameury, 2019). The theoretical treatment of the low state and mass-transfer fluctuations in CVs remains incomplete. Progress hinges upon global MHD simulations, deeper observational constraints on quiescent discs, and improved treatment of disc irradiation and winds.

A plausible implication is that DIM remains the only model able to quantitatively explain the global phenomenology of accretion-driven outbursts, despite caveats in detailed lightcurve synthesis and transport physics. Its scope encompasses stellar mass transfer, planetary formation, and galactic-scale instability, underscoring its foundational role in contemporary astrophysics.