Accretion Rate Fluctuations in Astrophysical Disks

• Accretion rate fluctuations are variations in the mass inflow of astrophysical disks caused by local instabilities and MRI-driven turbulence, modulated by viscous diffusion.
• Analytical Green’s function methods and MHD simulations reveal frequency-dependent propagation lags and broken power-law PSD features in systems from X-ray binaries to protoplanetary disks.
• Observational implications include the rms–flux relation, log-normal flux distributions, and coherent time lags that connect global disk structure with variability signatures.

Accretion rate fluctuations refer to stochastic, aperiodic, or quasi-periodic variability in the mass inflow rate, $\dot{M}(t)$, through astrophysical accretion flows. These fluctuations dominate the broadband timing properties of X-ray binaries, active galactic nuclei, accreting neutron stars, young stellar objects, and protoplanetary disks. The underlying mechanisms responsible for their origin, propagation, and observable imprints are traced to local instabilities, turbulence (principally MRI), and global flow geometry, resulting in complex variability phenomena across a vast range of spatial and temporal scales.

1. Physical Origin: Stochastic Fluctuation Generation

Accretion rate fluctuations originate from stochastic variability in the local viscosity, $\alpha(R,t)$, typically driven by the magnetorotational instability (MRI) on scales comparable to the disk thickness, $H$. These local fluctuations cause time-dependent surface density perturbations and mass inflow variability at each radius. In the standard formalism, the disk evolution is governed by the vertically averaged viscous diffusion equation,

$$\frac{\partial \Sigma}{\partial t} = \frac{3}{R}\,\frac{\partial}{\partial R}\Bigl[R^{1/2} \frac{\partial}{\partial R}(\nu\,\Sigma\,R^{1/2})\Bigr].$$

Small-scale random perturbations in $\alpha$ (e.g., OU processes on the local viscous timescale) induce local modulations in $\dot{M}(R,t)$ that propagate inward due to finite viscous transport rates (Rapisarda et al., 2017, Ahmad et al., 2017, Cowperthwaite et al., 2014, Hogg et al., 2015, He et al., 2 Aug 2025). In global MHD simulations, low-frequency Maxwell stress variations resulting from large-scale dynamo cycles are found to be the main source of coherent, slowly-evolving $\alpha$ fluctuations that seed observable accretion-rate propagation (Hogg et al., 2015).

2. Propagation and Diffusive Filtering

The spatial and temporal properties of $\dot{M}$ fluctuations are set by the propagation kernel—the Green’s function solution of the accretion disk’s diffusion equation. The viscous timescale at radius $R$, $t_{\rm visc}(R)=R^2/\nu(R)$, sets the local filter threshold: only perturbations with $f \lesssim 1/t_{\rm visc}(R)$ can propagate coherently inward. High-frequency perturbations generated at large radii are diffused away; only slow, long-wavelength modes survive to the inner disk (Mushtukov et al., 2017, Rapisarda et al., 2017, Ingram et al., 2013). The convolution of local fluctuation sources with the radial Green’s function yields strongly radius- and frequency-dependent propagation lags, generally described by

$$\dot{M}_{\rm in}(t) = \int_0^\infty \dot{M}_0(t-\tau)\,G(\tau)\,d\tau,$$

where $G(\tau)$ is the dimensionless propagation kernel, sharply peaked on timescales $\lesssim t_{\rm visc}$. Damped amplitudes and increasing phase lags are characteristic as fluctuations traverse significant radial extents, with outer-disk fluctuation amplitudes typically reduced at the inner edge due to diffusive damping (He et al., 2 Aug 2025, Mushtukov et al., 2017).

3. Timing Diagnostics: Power Spectra, Lags, and Variability Amplitudes

Accretion rate fluctuations underpin widely observed X-ray variability features:

• Power Spectral Density (PSD): The combined effect of local white or Lorentzian noise, filtered through viscous propagation, leads to broken power-law or "multi-hump" PSDs. In the canonical case, $P(f)\propto f^{-1}$ to $f^{-2}$ between breaks set by the outer and inner viscous timescales, and additional "bumps" correspond to distinct propagating regions (e.g., double-hump in hard-state black hole binaries, low-frequency break at truncated disk edge) (Rapisarda et al., 2017, Rapisarda et al., 2016, Mushtukov et al., 2017, Ingram, 2015). In neutron-star X-ray pulsars, high-frequency breaks are associated with dynamo fluctuation timescales near the Keplerian rate at the magnetospheric truncation radius, $f_{\rm br,high}\sim f_K(R_{\rm in})/k_d$ with $k_d\sim5$–$10$ (Mushtukov et al., 2019).
• RMS–Flux Relation and Log-normal Flux Distribution: Because inner (faster) fluctuations are multiplicatively modulated by slower (outer) fluctuations, segments of higher mean flux naturally carry higher variance, leading to a strictly linear $\sigma_{\rm rms} \propto \langle L\rangle$ relation and log-normal flux distributions (Hogg et al., 2015, Cowperthwaite et al., 2014, Turner et al., 2021). These features are robust signatures of multiplicative, propagating coupling.
• Time Lags and Coherence: Frequency-dependent hard lags between emission bands (hard photons lagging soft) arise naturally from finite viscous propagation times. Quantitatively, for propagating fluctuation models in $\alpha$-disks,

$t_{\rm lag}\propto f^{-p},$

with $p\simeq 0.54$–$0.7$; lags saturate and coherence breaks down above the local viscous frequency or over large radial separations (Ahmad et al., 2017, Hogg et al., 2015, Ingram, 2015).

• Amplitude and Timescale of Fluctuations: In protoplanetary disks, order-of-magnitude ($10^2$–$10^4$) variability in $\dot{M}$ and associated temperature and surface density cycles arise from coupled MRI-triggered heating/cooling wave propagation (Chambers, 25 Mar 2024). In YSO disks, stochastic variability saturates at $0.3$–$0.5$ dex over weeks–year timescales (Flaischlen et al., 2022, 0902.4235). In X-ray binaries, fractional rms values can be as high as $F_{\rm var}\sim 90\%$ (hard state, Cyg X-1) with propagating models reproducing empirical amplitudes and timing structures (Rapisarda et al., 2017). For neutron star and magnetic accretors, rms($\delta \dot M/\dot M$) spans $10^{-6}$–$10^{-2}$, with mean-reversion timescales of $10^6$–$10^8$ s corresponding to OU process fits (O'Leary et al., 3 Jun 2024, Melatos et al., 2022).

4. Multi-Zone Fluctuation Models and Application Across Accretion Regimes

State-of-the-art models—such as PROPFLUC and its double-hump generalization—decompose the flow radially (rings/zones), specifying local fluctuation spectra, radial viscosity prescriptions, and energy-dependent emissivity profiles. This formalism enables joint fitting of multi-band PSDs, cross-spectra, and phase/time lags, as demonstrated for both disc/hot-flow geometries and "corona+disk" sandwich flows (Rapisarda et al., 2017, Rapisarda et al., 2016, Ingram et al., 2013). In neutron star systems with strong magnetospheres, accretion rate fluctuations traced from the truncated disk to the star's surface dominate the observed power spectra and phase-resolved variability in X-ray pulsars, with magnetospheric boundary conditions setting the inner-edge timescales and break frequencies (Mushtukov et al., 2019, Mushtukov et al., 5 Apr 2024).

Propagating fluctuations also explain coherent, rapid state transitions, e.g., in type I X-ray bursts, where burst-induced enhancements to inner $\dot{M}$ (via Poynting–Robertson drag) are measured via time-variable persistent emission factors ($f_a$), rising on sub-second timescales and decaying in tens of seconds (Worpel et al., 2013). In protoplanetary contexts, coupled surface density and temperature waves, triggered at ionization fronts, result in recurrent $\dot{M}$ oscillations on $10^4$ yr periods, strongly affecting disk chemistry and planet formation (Chambers, 25 Mar 2024).

5. Statistical Inference, Simulation, and Population-Level Diagnostics

Reconstruction of $\dot{M}(t)$ fluctuations and their statistics is achieved via direct simulation (1D/3D viscous/MHD; see (Hogg et al., 2015, Cowperthwaite et al., 2014, Turner et al., 2021)), analytic Green’s-function approaches (Mushtukov et al., 2017, Rapisarda et al., 2017, Ingram et al., 2013), and data-driven inference tools such as Kalman and unscented Kalman filters (Melatos et al., 2022, O'Leary et al., 3 Jun 2024). Filtering simultaneous light curves and pulse periods in accreting pulsars yields time-resolved estimates of hidden state variables (mass inflow rate $Q$, Maxwell stress $S$, radiative efficiency $\eta$), with fluctuation statistics directly linked to viscous transport, magnetosphere-disk coupling, and MHD instability physics.

Table: Statistical Properties of Accretion Rate Fluctuations in Selected Systems

System/Model	RMS ($\delta\dot{M}/\dot{M}$)	Characteristic Timescale
BHXRB inner disk (PROPFLUC)	$0.9$ (hard); $0.42$ (soft)	$t_{\rm visc}\sim 0.1$–$10$ s
Protoplanetary disk	$10^3$–$10^4}$ (peak/trough)	$10^4$ yr
YSO disk (ONC, TTS)	$0.3$–$0.54$ dex (IQR)	days–year
XRP, $B\sim 10^{12}$ G	$10^{-6}$–$10^{-2}$	$10^6$–$10^8$ s

These statistics reflect underlying transport, geometry, and driver mechanism distinctions in each regime (Rapisarda et al., 2017, Chambers, 25 Mar 2024, 0902.4235, O'Leary et al., 3 Jun 2024).

6. Unified Implications and Observational Signatures

The propagating-fluctuations paradigm robustly accounts for:

• Observed, frequency-dependent PSD breaks, "bumps," and amplitude structure across all classes of disk accretors.
• The strict linearity of the rms–flux relation and emergence of lognormal flux distributions.
• Multiplicative coupling of variability leading to inter-band, frequency-dependent time lags and high inter-band coherence, consistent with observed X-ray light curves of black hole and neutron star binaries (Ingram, 2015, Hogg et al., 2015, Rapisarda et al., 2017).
• Disk-state dependence of variability morphology (single/double-hump PSDs; suppression of QPOs/enhancement in transition states).
• Connection of global flow/instability physics (e.g., heating/cooling fronts, MRI-dynamo cycles, magnetospheric torques) to observable fluctuation amplitudes, timescales, and lags.

Limitations remain in incorporating full non-linear, three-dimensional, relativistic, and radiative transfer effects, especially in the transition between flow regimes. Nevertheless, recent analytical, numerical, and data-driven advances demonstrate that accretion rate fluctuations, their propagation, and their observable spectral-timing signatures are now quantitatively understood as consequences of underlying stochastic transport physics modulated by global disk structure, local turbulence, and boundary conditions (Hogg et al., 2015, Mushtukov et al., 2017, Melatos et al., 2022, O'Leary et al., 3 Jun 2024, He et al., 2 Aug 2025, Chambers, 25 Mar 2024, Rapisarda et al., 2017).