SGR 0418+5729 - How does a Young Neutron Star Spin Down to a 9 s Period with a Dipole Field less than 10^13 G?

The period derivative bound for SGR 0418+5729 (Rea et al. 2010) establishes the magnetic dipole moment to be distinctly lower than the magnetar range, placing the source beyond the regime of isolated pulsar activity in the P - dP/dt diagram and giving a characteristic age > 2 \times 10^{7} years, much older than the 10^5 year age range of SGRs and AXPs. So the spindown must be produced by a mechanism other than dipole radiation in vacuum. A fallback disk will spin down a neutron star with surface dipole magnetic field in the 10^{12} G range and initial rotation period P_0 ~ 100 ms to the 9.1 s period of SGR 0418+5729 in a few 10^4 to ~10^5 years. The current upper limit to the period derivative gives a lower limit of \sim 10^5 years to the age that is not sensitive to the neutron star’s initial conditions. The total magnetic field on the surface of SGR 0418+5729 could be significantly larger than its 10^{12} G dipole component.

💡 Research Summary

The paper addresses the puzzling timing properties of the recently discovered soft gamma repeater SGR 0418+5729. Observations show a spin period of 9.1 s and an extremely low period derivative upper limit of ˙P < 6 × 10⁻¹⁵ s s⁻¹. Interpreting this ˙P as pure magnetic dipole braking in vacuum would imply a surface dipole field B₀ < 1.5 × 10¹³ G and a characteristic age P/(2˙P) > 2.5 × 10⁷ yr, far older than the typical ∼10⁵ yr ages of other SGRs/AXPs and inconsistent with the source’s burst activity and X‑ray luminosity.

The authors propose that a fallback accretion disk, formed from supernova ejecta that failed to escape, provides the dominant torque during the early evolution. They model the disk’s viscous evolution (using the self‑similar Pringle solution) together with irradiation from the neutron star, which keeps the outer disk warm enough to stay active for a while. The inner edge of the disk is set by the Alfvén radius
r_A = 10⁹ cm µ₃₀^{4/7} Ṁ_in,15^{−2/7} (M/M_⊙)^{−1/7},
where µ₃₀ is the dipole moment in units of 10³⁰ G cm³ and Ṁ_in,15 the mass inflow rate at the inner edge in 10¹⁵ g s⁻¹.

When r_A lies inside the light‑cylinder radius r_LC = c/Ω, the neutron star is in the propeller regime (r_A > r_co). The torque is then
I Ω̇ = Ṁ_in (GM r_A)^{1/2} F(ω),
with the dimensionless factor F(ω) ≈ −ω² (ω = Ω/Ω_K(r_A)). Crucially, this torque is almost independent of the instantaneous mass inflow rate, so the period derivative remains roughly constant while the disk is active inside r_LC.

The disk mass flow declines as the disk cools; once the temperature drops below the critical T_p ≈ 100 K, the magnetorotational instability ceases, viscosity vanishes, and the disk enters a passive phase. Irradiation can delay this transition, but eventually Ṁ_in falls dramatically, causing r_A to expand faster than r_LC recedes. When r_A exceeds r_LC, the disk becomes disconnected from the magnetosphere, and the torque switches to pure dipole radiation, causing ˙P to drop sharply.

Numerical experiments explore a wide range of initial conditions. Acceptable models have an initial dipole field B₀ ≈ 1–2 × 10¹² G, an initial spin period P₀ ≈ 70–300 ms, and a fallback disk mass M_d ≈ 4 × 10⁻⁶–6 × 10⁻⁵ M_⊙. With these parameters, the disk spins the star down from the birth period to the observed 9 s within 10⁴–10⁵ yr. During the active phase the X‑ray luminosity is supplied by partial accretion (Ṁ_acc < Ṁ_in), yielding L_X ≈ 10³⁴–10³⁵ erg s⁻¹. As the disk fades, L_X falls to ≈ 2 × 10³¹ erg s⁻¹, consistent with the quiescent luminosity inferred for a distance of ~2 kpc.

The model naturally explains why SGR 0418+5729 appears “old” in the P–˙P diagram yet is actually young (∼10⁵ yr). The dipole component can be modest, while the total surface field (responsible for magnetar‑like bursts) may still be 10¹⁴–10¹⁵ G, decoupled from the dipole torque. The current upper limit on ˙P implies a lower age bound of ≳ 10⁵ yr if the disk torque has already ceased, or ≳ 2 × 10⁵ yr if the disk torque is still active. A future precise measurement of ˙P (e.g., ≈ 4 × 10⁻¹⁷ s s⁻¹) would discriminate between a pure dipole spin‑down scenario and ongoing disk torque.

In summary, the authors demonstrate that a fallback disk with realistic parameters can spin down a young neutron star with a relatively weak dipole field to the long 9‑second period of SGR 0418+5729 within the expected magnetar lifetime, while preserving the high total magnetic field needed for burst activity. This resolves the apparent age paradox and supports the broader idea that many SGRs/AXPs may be surrounded by long‑lived fallback disks that dominate their early rotational evolution.