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Accretion Discs Trapped Near Corotation
Caroline R. D'Angelo, Hendrik C. Spruit
TL;DR
The paper addresses unexplained variability and torque behavior in magnetospherically accreting systems by modeling thin discs coupled to rotating stellar magnetospheres. It finds that discs can remain trapped near corotation at low and varying accretion rates, with steady or cyclic accretion and two distinct instability regions. The analysis also shows that cyclic accretion modifies the star’s average torque and may connect observed variability to disc-field interaction.
Problem
Observed spin reversals, hysteresis, recurrent outbursts, and stored disc mass are not naturally explained by standard magnetospheric-accretion interpretations.
Method
The paper explores trapped-disc evolution and cyclic instability across disc-field transition parameters, including cycle properties and effects on stellar spin.
Results
Trapped discs remain near corotation at low accretion rates; two cycle forms are identified, with RI increasing average spindown torque and the second reducing average torque.
Takeaways & Limitations
Observed cycles could provide clues to the physics of magnetospheric accretion and may relate trapped discs to unusual variability in X-ray binaries and EXors.
Takeaways & Limitations
The model neglects outflows and assumes interaction-region behavior that may vary on timescales longer than the modeled short field-line timescale.
Abstract
from arXiv · showhide
We show that discs accreting onto the magnetosphere of a rotating star can end up in a 'trapped' state, in which the inner edge of the disc stays near the corotation radius, even at low and varying accretion rates. The accretion in these trapped states can be steady or cyclic; we explore these states over wide range of parameter space. We find two distinct regions of instability, one related to the buildup and release of mass in the disk outside corotation, the other to mass storage within the transition region near corotation. With a set of calculations over long time scales we show how trapped states evolve from both nonaccreting and fully accreting initial conditions, and also calculate the effects of cyclic accretion on the spin evolution of the star. Observations of cycles such as found here would provide important clues on the physics of magnetospheric accretion. Recent observations of cyclic and other unusual variability in T Tauri stars (EXors) and X-ray binaries are discussed in this context.
1 INTRODUCTION
Magnetospheric accretion observations reveal spin, luminosity, and outburst behaviors that standard models do not naturally explain. This paper investigates trapped-disc states, in which material can remain near corotation and accrete steadily or cyclically without requiring propeller-driven mass loss.
- Observational motivation: Spin-up, spin-down, hysteresis, and nearly constant torque magnitudes across torque reversals challenge standard magnetospheric-accretion interpretations.These behaviors are reported in persistent X-ray pulsars and are not naturally explained when torque scales simply with accretion rate.
- Observational motivation: Short recurrent X-ray-pulsar outbursts imply that substantial disc mass can remain stored despite orders-of-magnitude drops in stellar accretion.The observed recurrence times are too short for a standard ionization-instability explanation, while the inferred storage conflicts with the usual inner-radius scaling.
- Physical picture: A centrifugal barrier need not expel transferred mass; instead, viscous discs can accumulate material outside corotation and alternate between storage and accretion.Mass loss through a magnetically driven wind is separate from the centrifugal-barrier effect considered here.
- Trapped states: Trapped discs keep their inner edge near corotation at low accretion rates, with either continuous or cyclic accretion.The study focuses on the cyclic case and its effects on instability properties, torque, luminosity, and observations.
- Paper objective: The paper expands the instability analysis across parameter space to characterize its cycles and connect them with observations.The authors also discuss persistent pulsars, recurrent X-ray-pulsar outbursts, and episodic bursts in young stars.
- Scope and boundary: Whether a disc remains trapped depends on disc-field interaction details and the ratio T_visc/T_SD; it can instead evolve into a dead disc.The model explicitly neglects outflows, so the disc does not enter the propeller regime for any accretion rate.
2 THE MODEL FOR MAGNETOSPHERIC ACCRETION
The model couples a thin-disc diffusion equation to magnetic torque and inner-edge evolution near corotation. Two transition length scales parameterize uncertain disc-field coupling, while the resulting equations describe accretion, suspended states, and dead discs.
- Disc-field interaction: The model represents magnetic coupling through a time-averaged toroidal-field strength and interaction width, with the interaction region treated as a free parameter.The assumed interaction width is smaller than the inner-edge radius and is explored over values suggested by simulations.
- Boundary conditions: Outside corotation, magnetic torque is transmitted outward by viscous stress and supplies the inner-edge surface-density boundary condition.Inside corotation, the viscous torque and surface density at the inner edge vanish in the standard accreting-disc limit.
- Torque prescription: The magnetic torque uses a dipole-field estimate, with a smooth transition function changing across the interaction width.The viscosity is fixed radially as ν = k_νr^1/2, corresponding to an α-disc with constant α and aspect ratio H/r.
- Accreting regime: For an inner edge inside corotation, the model relates edge position to accretion rate by balancing magnetic torque against the torque required to keep accreting matter corotating.The model uses the standard estimate associated with Spruit & Taam (1993).
- Transition region: The accretion transition is described using the comoving mass flux and a connecting function that changes from accreting inside corotation to non-accreting outside it.The separate transition scale Δr2 is varied independently from the magnetic-interaction width Δr.
- Evolution and scaling: The coupled equations evolve surface density and inner-edge radius together, with characteristic scales set by the initial corotation radius and viscous time.The characteristic accretion rate is defined as the rate that places the magnetospheric radius at corotation; the characteristic torque corresponds to angular momentum added at that rate.
- Dead-disc limit: A disc well outside corotation has vanishing accretion and becomes a dead disc whose surface-density profile is set by outward angular-momentum transport.A larger-distance sink, such as a companion’s orbital angular momentum or an effectively infinite disc, is required for a stationary outward flux.
3 SPIN EVOLUTION AND PHYSICAL PROPERTIES OF A TRAPPED DISC
A trapped disc differs from a standard accreting disc through mass accumulation near the centrifugal barrier, with its inner edge remaining close to corotation as accretion declines. The resulting torque changes from spin-up to spin-down across accretion regimes, while transition parameters control trapping and angular-momentum exchange.
- Disc structure: In a dead disc, zero accretion leaves the surface-density profile determined by angular-momentum transport from the star to the disc.
- Disc structure: Most energy is radiated near the inner edge because the surface dissipation varies as r^-7/2; with ν ∼ r^1/2, Ts ∼ r^-1.
- Disc structure: Near corotation, a dead disc has Ts ≈300 K for a T Tauri star and Ts ≈20000 K for a millisecond X-ray pulsar.
- Trapping: For ∆r2/rin = 0.05 or 0.1, rin remains close to rc even after ˙m decreases by several orders of magnitude; ∆r2/rin = 0.5 produces a more gradual transition.
- Angular-momentum exchange: At high ˙m the star spins up with ˙J ∝ ˙m^9/10, then spins down as rin moves outside rc, before spindown weakens in the effectively dead state.
- Angular-momentum exchange: When ˙J < 0, the spindown magnitude is proportional to ∆r/rin, while ∆r2/rin controls how far rin moves from rc as ˙m decreases.
4 CYCLIC ACCRETION
The parameter survey identifies two nearly separate instability regions with qualitatively different cyclic accretion behavior. RI produces long, strong outbursts over broad accretion rates, whereas RII is confined near the transition to steady accretion and maintains accretion throughout its low phase.
- 4.1 Parameter map of the instability: Two distinct instability regions, RI and RII, emerge from two-dimensional slices of the three-dimensional parameter space.The slices vary [∆r, ˙m] and [∆r2, ˙m] while holding the remaining interaction scale fixed.
- 4.1 Parameter map of the instability: RI spans ˙m/˙mc ≃ [10^-6, 10^-1] and broad ∆r/rin values, but is confined to ∆r2/rin = [0.002, 0.03].Its cycles feature large-amplitude outbursts followed by quiescent phases in which accretion onto the star drops to zero.
- 4.1 Parameter map of the instability: RII occurs near ˙m/˙mc ≃ 1, within ∆r/rin = [0.01, 0.07], and across a large range of ∆r2/rin.At ˙m = 1.05 ˙mc, sampled values of ∆r2/rin range from 0.028 to 0.15.
- 4.2 Period and Amplitude of Instability: Cycle period varies strongly with ∆r/rin and ∆r2/rin, while outburst amplitude is nearly independent of mean accretion rate.Larger ∆r/rin provides a larger mass reservoir, whereas smaller ∆r2/rin permits greater buildup during quiescence and longer outbursts.
- 4.2 Period and Amplitude of Instability: RII reaches amplitudes about 40% larger than the largest RI amplitude and periods up to about 60% shorter.Its amplitude remains roughly fixed near 10 ˙m/˙mc as the mean accretion rate decreases or ∆r2/rin increases, while the period lengthens.
- 4.3 Interpretation of the instability regions: A sufficiently leaky centrifugal barrier can prevent RI cycles by allowing accretion to match the incoming mass flow.The cyclic instability requires a sufficiently steep transition from accretion to pile-up outside corotation.
- 4.3 Interpretation of the instability regions: RII operates in a finite transition region near corotation, where small accretion-rate changes move the inner edge and substantially alter the torque.Its restricted accretion-rate range and short periods reflect the finite widths of the ∆r/rin and ∆r2/rin transition regions.
5 RELEVANCE FOR ASTROPHYSICAL SOURCES
The model connects trapped-disc instabilities and cyclic accretion with unusual variability in EXors and transient or persistent X-ray binaries. It also identifies observational signatures and limitations of these interpretations.
- EXors: EX Lupi’s 2008 outburst rose by 3 magnitudes in about 20 days and remained active for eight months.The profile resembles simulated rapid-rise, plateau, and rapid-decay cycles.
- EXors: The model predicts that a trapped EX Lupi disc has higher surface density and temperature than a standard accreting disc with the same inner radius.Its predicted quiescent surface temperature is 530–650 K.
- Transient X-ray binaries: Weak recurrent outbursts in IGR J00291+5934 and NGC 6440 X-2 suggest that substantial disc mass remains stored between accretion episodes.The observed recurrence times are about 27–31 days, with weak outbursts lasting several days.
- Interpretive limits: The interpretation remains uncertain because similar QPO frequencies challenge the proposed viscosity differences, and the model does not identify what directly triggers accretion outbursts.The disc-field interaction can halt accretion only on the shorter QPO timescale.
- Persistent X-ray pulsars: Persistent pulsars show spin-up rates lower than expected from X-ray luminosity, while spin-up and spin-down torques are often comparable within a factor of 2.The model attributes this behavior to a transition region near corotation that permits accretion and spin-down torque.
6 DISCUSSION
The discussion examines how magnetic-field interaction, accretion-rate regulation, and source properties affect trapped discs and instability cycles. It emphasizes model assumptions and unresolved physical and observational questions.
- Model assumptions: The model assumes that the disc-field interaction width remains fixed, although MHD simulations suggest field opening and reconnection can make it fluctuate.Small changes in this width can significantly alter outburst appearance and amplitude.
- Magnetic-field evolution: Magnetic-field diffusion may reduce the torque at the inner edge and increase the extent of the closed-field-line region.The diffusion timescale is assumed to be comparable to the viscous timescale.
- Model assumptions: The transition length between accreting and non-accreting solutions is less well understood and may vary with the disc-field interaction.An inclined dipole could permit simultaneous accretion and confinement of material.
- Accretion-rate regulation: The model treats the mean accretion rate as a free parameter and does not support disc-field interaction as the process that regulates that rate.Trapped-disc solutions become relevant only after the average rate falls below ṁcrit/ṁc ≃ 4∆r/rin.
- Open questions: It remains unresolved why some sources display trapped-disc behavior while most outbursting neutron stars show long quiescent intervals.The discussion links this difference to source properties such as viscosity, spin, and magnetic-field structure.
7 CONCLUSIONS
Magnetospheric accretion can remain trapped near corotation at low accretion rates, either steadily or through cycles. The cycle behavior depends on the transition width and can enhance stellar spin-down torque.
- Trapped states keep the disc’s inner edge near corotation even at very low accretion rates, with either steady or cyclic accretion.
- RI cycles arise when mass piles up outside corotation, then accretes and empties the pile before a new cycle begins.
- A narrow transition between accreting and suspended states produces RI cycles, whereas a broader transition produces continuous accretion.
- Cyclic accretion increases the average spin-down torque on the star.
- Observed EX Lup outbursts and cyclic activity in NGC 6440 X-2 and SAX J1808.4-3658 may be related to these accretion cycles.