3D simulations of magnetospheric accretion in T Tauri stars: I. Disk truncation, stellar torques, and application to observations

Abstract: Young stars accrete material from their circumstellar disk through their magnetosphere while still contracting, two processes that impact their rotational evolution. We investigate stable and unstable accretion regimes (due to the interchange instability) and examine the associated stellar torques to assess the spin evolution of young stars. We perform 3D MHD simulations of disk accretion onto an inclined stellar dipole. We run 21 simulations with varying stellar stellar rotation rates, dipole field strengths and obliquities, and mass accretion rates. We find that stars with a ratio of truncation to corotation radius $R_t/R_{co} \gtrsim 0.80-0.85$ accrete via a stable regime, while accretion becomes unstable otherwise. Besides, our $R_t/R_{\ast}$ parametrization weakly depends on the mass accretion rate and the dipolar intensity, while strongly on the stellar rotation rate. We derive torque formulae for each flow component affecting the stellar rotation, i.e. accretion, magnetospheric ejections and stellar winds. Finally, we apply our results to a sample of young stars with measured magnetic fields, mass accretion rates, and rotational periods and find that most of them should currently accrete in an unstable regime and undergo spin-up torques. Our study comforts and expands upon previous results. Unstable accretion should lead to a net spin-up torque on the central star, while stable accretion can lead to stellar spin-down. When applying our truncation radius and torque prescriptions to observational data, we find that most young stars in our sample should be in a spin-up state. Thus, the angular momentum problem for young stars remains.

• The paper uses 21 three-dimensional resistive-viscous MHD simulations to show that accretion changes from stable funnels to unstable tongues near Rₜ/Rco ≈ 0.8–0.85, while deriving a truncation scaling dominated by stellar rotation rather than classical pressure balance.
• The study finds that star-disk interaction torques switch from spin-up to spin-down near Rₜ/Rco ≈ 0.836, with magnetospheric ejections and stellar winds sometimes braking stars, although winds cannot offset rapid spin-up in most unstable systems.
• Applying the model to 20 Classical T Tauri stars predicts that only about 20% accrete stably, roughly 38% occupy an ordered unstable regime, and only about 25% experience net spin-down including a 1% wind-efficiency assumption.

This paper presents a systematic study of magnetospheric accretion in Classical T Tauri stars (CTTs) based on 21 three-dimensional MHD simulations of an $\alpha$-viscous, resistive disk interacting with a tilted stellar dipole. The work pursues three objectives: characterizing stable versus unstable accretion regimes through the dynamics of disk truncation, deriving parametrizations of the stellar torques associated with accretion, magnetospheric ejections (MEs), and stellar winds, and applying these prescriptions to a sample of CTTs observed with spectropolarimetry and interferometry (2608.17869).

Numerical approach

The simulations solve the resistive-viscous MHD equations with the PLUTO code using a second-order Godunov scheme, HLLD Riemann solver, and divergence cleaning, on a spherical grid ($144 \times 128 \times 128$) extending to $34.28\,R_\star$, solved in the frame co-rotating with the star. A calorically imperfect equation of state allows nearly isothermal behavior for hot wind plasma and adiabatic behavior for cold disk material. The initial configuration comprises a hydrostatic polytropic disk, a Parker-like transonic stellar corona, and a dipolar field misaligned by $\Theta = 5^\circ$–$20^\circ$ from the rotation axis. Four parameters are varied across the 21 runs: fractional break-up rotation rate $f$ (0.038–0.15), polar field strength $B_\star/B_0$, obliquity $\Theta$, and disk-to-corona density contrast; transport coefficients are fixed at $\alpha_\mathrm{v}=\alpha_\mathrm{m}=0.2$. Each run is integrated for 30 stellar periods, with the first 10 discarded to remove transients.

Stable versus unstable accretion

The authors define the truncation radius geometrically as the equivalent-area radius of the $\beta_\mathrm{T}=1$ contour, where $144 \times 128 \times 128$0 compares total (thermal plus ram) disk pressure to magnetic pressure. An instability parameter $144 \times 128 \times 128$1, the ratio of the cavity perimeter to that of a circle of equal area, quantifies the amplitude of the interchange instability. The central result is a sharp transition at $144 \times 128 \times 128$2–$144 \times 128 \times 128$3: cases above this threshold show circular cavities and two ordered accretion funnels ($144 \times 128 \times 128$4), while below it the flow fragments into multiple tongues impacting the star at varying azimuths and latitudes. For $144 \times 128 \times 128$5 the instability parameter decreases again, interpreted as fewer but azimuthally wider tongues—an "ordered" unstable regime analogous to that found by Blinova et al., though at a slightly higher threshold than their $144 \times 128 \times 128$6. This threshold-based classification implies that the same quantity controlling spin evolution also controls accretion morphology, a correlation the paper exploits throughout.

Truncation radius scaling

A key quantitative finding departs strongly from the classical Ghosh & Lamb scaling. Fitting all cases yields

$144 \times 128 \times 128$7

with $144 \times 128 \times 128$8, $144 \times 128 \times 128$9, and $34.28\,R_\star$0: the truncation radius depends only weakly on the accretion parameter $34.28\,R_\star$1 and strongly on the stellar rotation rate, i.e., it tracks the corotation radius rather than the classical pressure-balance radius. In unstable cases, the disk begins fragmenting near corotation via the interchange instability before reaching the Ghosh & Lamb position, so the cavity size is set by the instability onset rather than by axisymmetric compression. The authors caution that this fit applies only within the explored range ($34.28\,R_\star$2, $34.28\,R_\star$3); for smaller ratios it likely provides only an upper limit, and the fixed $34.28\,R_\star$4 prescription could modify the scaling. Combining the scaling with the instability threshold yields a critical accretion rate above which accretion becomes unstable—roughly $34.28\,R_\star$5 for kG dipoles—implying that even moderately accreting CTTs should be interchange-unstable.

Stellar torques

The torque analysis separates contributions from closed-field accretion, MEs, and open-field stellar winds. Two results stand out. First, in the stable regime some cases exhibit null or even negative (spin-down) accretion torques, contrary to the usual assumption that accretion always spins the star up. The explanation is that when the disk rotates sub-Keplerian and slower than the star at the funnel base, Laplace forces transfer angular momentum from the star to the infalling gas; the excess angular momentum is then ejected through inflated field lines on the opposite hemisphere, effectively as MEs or conical winds. Second, the total star-disk interaction (SDI) torque transitions from spin-up to spin-down almost exactly at the instability boundary: the zero-torque condition evaluates to $34.28\,R_\star$6, consistent with the $34.28\,R_\star$7 found by Zhu in MRI-turbulent global simulations. In stable regimes the SDI spin-down timescale can be comparable to the Kelvin-Helmholtz contraction timescale, favoring spin equilibrium; in unstable regimes the spin-up timescale becomes much shorter than contraction, implying rapid spin-up unless another mechanism intervenes.

Stellar wind torque

The thermally driven winds in these models confirm the Réville et al. scaling of the Alfvén radius with the wind magnetization parameter, and show that the fractional open flux scales with the truncation radius as $34.28\,R_\star$8—closer disk truncation opens more flux and strengthens braking. Notably, high-density-contrast cases show the wind quenched by accretion, and those points are excluded from the fits. Winds with ejection efficiencies of 1–10% can provide spin-down torques comparable to or exceeding the SDI torque in stable regimes, potentially offsetting contraction; in unstable regimes they cannot balance the combined accretion and contraction spin-up—a wind efficiency exceeding unity would be required. The authors note that the unstable region is poorly sampled because of the quenched-wind exclusions, so wind efficiency there remains uncertain.

Application to observations

Applying the prescriptions to 20 CTTs (29 entries) observed with ESPaDOnS/SPIRou and ZDI, assuming a 1% wind ejection efficiency, the study finds that only about 20% of the sample should accrete stably ($34.28\,R_\star$9–$\Theta = 5^\circ$0), roughly 38% fall in the ordered unstable regime, and only ~25% experience a net spin-down torque including the wind. Only one star, AA Tau, has a total spin-down torque sufficient to overcome Kelvin-Helmholtz contraction. About ten stars have spin-up timescales longer than $\Theta = 5^\circ$1 yr, comparable to typical disk lifetimes, so their spin-up may be slow even if not fully arrested. These conclusions agree with independent estimates by Pittman et al. (~80% unstable/spin-up) despite different truncation-radius methods, though the median truncation radii differ (5.13 vs. 2.8 $\Theta = 5^\circ$2), which the authors attribute to their average-cavity definition versus H$\Theta = 5^\circ$3-based axisymmetric estimates. Comparison with VLTI/GRAVITY Br$\Theta = 5^\circ$4 sizes shows no clear correlation with $\Theta = 5^\circ$5 beyond agreement within a factor of two, consistent with Br$\Theta = 5^\circ$6 emission arising between 0.5 and 0.9 $\Theta = 5^\circ$7 plus wind contributions.

Limitations and open questions

Several caveats bear directly on the results. The torque and timescale estimates are snapshots assuming solid-body rotation; core-envelope decoupling could substantially alter the effective moment of inertia and hence the inferred spin evolution. The laminar $\Theta = 5^\circ$8-disk treatment fixes $\Theta = 5^\circ$9 and $20^\circ$0, and the resistivity prescription influences both the interaction extent and the instability criterion; no systematic diffusivity study is performed. All models neglect an intrinsic large-scale disk magnetic field, which could flatten the magnetospheric field decay and strengthen star-disk coupling. The truncation-radius fit degrades outside the explored parameter space, and the stabilizing effect of large obliquities ($20^\circ$1), relevant for several observed stars, is not modeled. Finally, whether unstable-regime spin-up can be prevented over Myr timescales requires coupling these torque laws to time-dependent stellar evolution models rather than snapshot estimates.

Conclusion

This work establishes, from a statistically meaningful ensemble of 3D MHD simulations, that the transition between stable and unstable magnetospheric accretion coincides closely with the transition between SDI spin-down and spin-up, both occurring at $20^\circ$2–$20^\circ$3. Its truncation-radius parametrization, dominated by the stellar rotation rate rather than the accretion parameter, revises the classical Ghosh & Lamb picture and provides observationally usable formulae for accretion regime and torque diagnostics. Applied to spectropolarimetric samples, the framework predicts that most actively accreting CTTs are currently in unstable, spin-up configurations, leaving the angular momentum problem of slowly rotating young stars unresolved and motivating time-dependent spin-evolution modeling with these new torque prescriptions.