Propagation Radius in GM Discs

• Propagation radius is defined as the maximum outer radius reached by a disc outburst, extending beyond the locally unstable region in GM disc models.
• The outburst propagation is driven by a 'snow plough' mechanism that redistributes mass outward, triggering instability in initially stable regions.
• In fiducial models, the propagation radius is roughly twice the outer edge of the locally unstable interval, critically influencing global disc dynamics and accretion rates.

In the context of the Gravo-Magneto (GM) disc instability, the propagation radius is the maximum outer radius to which a disc outburst propagates during a global event triggered in a dead-zone-bearing circumstellar disc. In "Propagation of the Gravo-Magneto Disc Instability" (Martin et al., 2013), this quantity is distinguished from both the locally unstable radial interval, where no steady layered-disc solution exists, and the trigger radius, where the outburst is first initiated. The propagation radius characterizes the outer extent of the globally unstable region once the instability has been launched, and therefore measures how far a locally triggered transition between gravitationally driven turbulence and MRI turbulence can spread through the disc.

1. Definition and relation to local instability

The paper defines a locally unstable radial interval as the set of radii where, for the imposed infall accretion rate, no steady-state layered-disc solution exists. On the local state diagram at fixed radius, this is the gap between the upper thermally ionized MRI branch and the GM branch, where the midplane is self-gravitating and the surface layers are MRI active (Martin et al., 2013).

Within that framework, the trigger radius is the radius at which the global outburst is first initiated. Physically, it lies near the middle of the locally unstable interval, at the location where a local peak in midplane temperature and surface density reaches the MRI activation threshold $T=T_{\mathrm{crit}}$, so that the dead zone suddenly becomes MRI turbulent.

The propagation radius is a different quantity. It is the maximum outer radius reached by the global outburst. The paper emphasizes that the globally unstable region is larger than the locally unstable interval: it includes all radii interior to the trigger radius, all locally unstable radii exterior to the trigger radius, and some radii that were locally stable before the trigger but are driven unstable by mass redistribution during the outburst (Martin et al., 2013).

For the fiducial circumstellar disc model, the key radii are:

Quantity	Meaning	Fiducial value
$R_{\mathrm{inner,local}}$	Innermost locally unstable radius	$\approx 1.87$ au
$R_{\mathrm{trigger}}$	Radius where the outburst is first initiated	$\approx 2.6$ au
$R_{\mathrm{outer,local}}$	Outermost locally unstable radius	$\approx 4.51$ au
$R_{\mathrm{prop}}$	Maximum outer radius reached by the global outburst	$\approx 9$ au

The principal quantitative conclusion is that, in the fiducial case, the outer propagation radius is approximately twice the outer edge of the locally unstable interval: $$R_{\mathrm{prop}} \approx 2 \times R_{\mathrm{outer,local}}$$ (Martin et al., 2013).

2. Disc structure and the origin of the instability

The relevant disc structure contains three regimes. In the inner disc, MRI operates where $R_{\mathrm{inner,local}}$0. In the outermost disc, MRI can operate throughout the column if $R_{\mathrm{inner,local}}$1 because external ionization by cosmic rays or X-rays can penetrate the full column. Between these regions, a dead zone forms at the midplane: MRI is inactive there, while surface layers remain MRI active (Martin et al., 2013).

As mass piles up in the dead zone because transport is restricted, self-gravity may drive turbulence in the midplane if the Toomre parameter falls below a critical value. The GM instability occurs when the temperature in the dead zone rises to $R_{\mathrm{inner,local}}$2, producing a sudden transition from gravitationally driven turbulence to MRI turbulence.

The stability criteria used in the paper are explicit. The Toomre parameter is

$R_{\mathrm{inner,local}}$3

with $R_{\mathrm{inner,local}}$4 for Keplerian discs, and the disc is gravitationally unstable if $R_{\mathrm{inner,local}}$5 with $R_{\mathrm{inner,local}}$6 (Martin et al., 2013). MRI activation occurs either thermally, when $R_{\mathrm{inner,local}}$7, or by external ionization when $R_{\mathrm{inner,local}}$8.

For the fiducial model,

• $R_{\mathrm{inner,local}}$9,
• $\approx 1.87$0,
• $\approx 1.87$1,
• $\approx 1.87$2,
• $\approx 1.87$3 (Martin et al., 2013).

Under these parameters, the locally unstable interval is

$\approx 1.87$4

and the outburst is first triggered at

$\approx 1.87$5

3. Triggering and the “snow plough” mechanism

The paper’s global interpretation of propagation radius rests on the snow plough mechanism. When MRI is suddenly activated in high-surface-density dead-zone material, the effective viscosity jumps because $\approx 1.87$6 increases. This produces strong viscous torques and rapidly redistributes mass (Martin et al., 2013).

A snow plough in surface density forms at the trigger radius and launches waves that propagate both inward and outward. The inward branch drives all radii interior to $\approx 1.87$7 into outburst behavior, causing rapid accretion toward the star. The outward branch pushes mass into regions that were previously on the steady GM branch and locally stable in the pre-trigger steady state.

This redistribution changes the propagation problem from a local one to a global one. Outside the trigger radius, the increased surface density can push radii onto the upper MRI branch by raising temperatures above $\approx 1.87$8. It can also strengthen self-gravity by increasing $\approx 1.87$9 and reducing $R_{\mathrm{trigger}}$0, so that regions that were locally stable before the trigger become gravitationally unstable and then participate in the global outburst (Martin et al., 2013).

The resulting partition of the disc is explicit:

• all radii inside $R_{\mathrm{trigger}}$1 au become unstable and accrete inward;
• locally unstable radii outside $R_{\mathrm{trigger}}$2, between $R_{\mathrm{trigger}}$3 and $R_{\mathrm{trigger}}$4 au, also become unstable;
• locally stable GM radii beyond $R_{\mathrm{trigger}}$5 au can be pushed into instability by the outward snow plough, up to about $R_{\mathrm{trigger}}$6 au.

Beyond about $R_{\mathrm{trigger}}$7 au, outward mass redistribution still occurs, but it is insufficient to move the local state to the upper MRI branch; those radii remain near or move down the GM branch after the outburst (Martin et al., 2013).

A common misconception is to identify the propagation radius with the edge of the local instability interval. The paper’s central result is precisely that these are not the same quantity: the global outburst extends beyond the pre-existing locally unstable zone.

4. Quantitative structure of the fiducial outburst

The fiducial model gives a concrete chronology for the radial propagation of the instability. At $R_{\mathrm{trigger}}$8, the outburst is triggered at $R_{\mathrm{trigger}}$9 au. By $\approx 2.6$0, inward and outward waves are clearly propagating away from the trigger point. By $\approx 2.6$1, the inward-propagating wave has reached the inner boundary and the disc is in a high-accretion outburst state onto the star. At $\approx 2.6$2–$\approx 2.6$3, the dead zone begins to re-form, first at $\approx 2.6$4 au, and inward and outward re-formation fronts then propagate. By $\approx 2.6$5, the global outburst ends, although outward adjustments continue at larger radii (Martin et al., 2013).

The propagation character is asymmetric. Inward propagation is rapid, reaching the star within a few hundred years. Outward propagation continues to about $\approx 2.6$6 au during the outburst window and leaves the disc depleted and cooler compared to the pre-outburst steady GM solution.

The principal mass budget is also specified. Before outburst, the total disc mass is about $\approx 2.6$7; about $\approx 2.6$8 lies within the dead zone, and about $\approx 2.6$9 is non-self-gravitating. During the outburst, about $R_{\mathrm{outer,local}}$0 is accreted onto the star, which is about one quarter of the disc mass (Martin et al., 2013).

These numbers place the propagation radius in a broader dynamical context. It is not merely a geometric marker; it bounds the region over which the GM transition reorganizes transport, temperature, and mass distribution during a large-scale accretion event.

5. Governing equations and radius scalings

The local steady-state layered-disc analysis uses the mass-flux relation

$R_{\mathrm{outer,local}}$1

together with the radiative balance relation

$R_{\mathrm{outer,local}}$2

The MRI viscosity is

$R_{\mathrm{outer,local}}$3

and the gravitational viscosity is

$R_{\mathrm{outer,local}}$4

with

$R_{\mathrm{outer,local}}$5

for $R_{\mathrm{outer,local}}$6, and $R_{\mathrm{outer,local}}$7 otherwise (Martin et al., 2013).

The time-dependent viscous evolution is governed by

$R_{\mathrm{outer,local}}$8

with $R_{\mathrm{outer,local}}$9 switching between MRI-dominated and GM-dominated forms according to the local thermal, ionization, and gravitational state. The time-dependent model also solves an energy equation to evolve temperatures and determine activation (Martin et al., 2013).

The paper provides explicit fiducial radius formulae. The innermost locally unstable radius is

$\approx 4.51$0

which gives $\approx 4.51$1 au for the fiducial parameters (Martin et al., 2013).

The outer transition to full-column MRI activity, set by $\approx 4.51$2, is

$\approx 4.51$3

By contrast, the outermost locally unstable radius, $\approx 4.51$4 au, is obtained numerically for the fiducial model. The propagation radius, $\approx 4.51$5 au, is therefore a global dynamical output of the outburst calculation rather than a local steady-state boundary (Martin et al., 2013).

6. Interpretation, sensitivities, and limitations

The factor-of-two relation

$\approx 4.51$6

is reported for the specific circumstellar disc model considered. The paper explicitly states that this factor is physically tied to how much mass the sudden MRI activation can push outward before heating and self-gravity feedback cease to lift radii onto the upper MRI branch. It also states that the factor may vary with parameters such as $\approx 4.51$7, $\approx 4.51$8, opacity or cooling, $\approx 4.51$9, and $R_{\mathrm{prop}}$0 (Martin et al., 2013).

The dead-zone prescription is simplified. The paper assumes a constant $R_{\mathrm{prop}}$1 with radius for external ionization. It notes that a more realistic criterion might involve a critical magnetic Reynolds number, and that other non-ideal MHD effects, including ambipolar diffusion and dust or PAHs, can suppress ionization. These effects would alter the detailed shape of the limit cycle and the exact locally unstable interval, although the paper states that the snow plough mechanism and the qualitative propagation behavior are robust.

The viscosity prescriptions are likewise idealized. The model adopts $R_{\mathrm{prop}}$2, and the exact form of $R_{\mathrm{prop}}$3 is not regarded as crucial provided it declines steeply with $R_{\mathrm{prop}}$4. In the self-gravitating region, $R_{\mathrm{prop}}$5 and $R_{\mathrm{prop}}$6 implicitly scales with the cooling time (Martin et al., 2013).

The accretion history is also simplified: $R_{\mathrm{prop}}$7 is held constant, although the paper notes that in reality it likely declines. As $R_{\mathrm{prop}}$8 drops, the locally unstable interval shrinks and eventually disappears; for $R_{\mathrm{prop}}$9, a fully MRI solution exists everywhere.

A plausible implication is that propagation radius should be understood not as a universal disc scale, but as a model-dependent measure of how far a triggered GM outburst can extend under a particular combination of ionization, self-gravity, and viscous transport prescriptions. In the fiducial case, however, its meaning is precise: the outburst propagates to about $\approx 9$0 au, approximately twice the outer edge of the locally unstable interval, and thereby converts a local instability criterion into a global accretion event (Martin et al., 2013).