Magnetar Crust Seismology: Torsional Ringdown Modes

Torsional Mode

Crust Physics

Crust rigidity (shear modulus μ) 1.00 ×10³⁰ erg/cm³

Damping (quality factor Q) 30

Quake magnitude 1.00×

Readouts

Status: Quiescent crust

Mode frequency f_ℓ

— Hz

Time since quake

— s

Ring amplitude

Shear speed v_s

— km/s

How it works

A giant flare doesn't just eject a fireball of magnetically confined plasma — it also cracks the neutron star's solid crust. The crust then rings like a struck bell, executing global torsional (shear) oscillations in which surface material sloshes tangentially back and forth while the star's overall shape barely changes.

Shear wave speed:   v_s = √(μ / ρ)
Fundamental (n=0) toroidal mode:
  f_ℓ ≈ (v_s / 2πR) · √(ℓ(ℓ+1) − 2),  ℓ = 2,3,4,…
Ring-down:   A(t) = A₀ e^(−t/τ) sin(2π f_ℓ t),   τ = Q / (π f_ℓ)

Here μ is the crust's shear modulus, ρ its density, R the stellar radius (a canonical R = 10 km, ρ ≈ 10¹⁴ g/cm³ is used below), and ℓ the angular mode number — higher ℓ means more nodal rings and a higher pitch, exactly like higher overtones on a drum head.

ℓ buttons — pick the standing-wave pattern; the surface is colored by |P_ℓ(cos θ)|, the Legendre mode shape, and patches shift tangentially in the azimuthal direction with that amplitude.
Crust rigidity μ — a stiffer crust (higher μ, driven by the magnetic field's tension) carries shear waves faster, raising the frequency for every ℓ.
Damping Q — sets how many cycles the ringing survives before dying out; higher Q rings longer, exactly as in any damped oscillator.
Trigger Starquake — injects an impulsive kick and starts the exponentially-decaying ring-down.

Real-world relevance: this is the mechanism behind the quasi-periodic oscillations (QPOs) discovered in the X-ray afterglow of the 27 December 2004 giant flare from SGR 1806-20 — frequencies near 18, 30, 92 and 150 Hz were read as direct seismograms of a neutron star crust (Israel et al. 2005; Watts & Strohmayer 2006), the first asteroseismology ever performed on a magnetar.

High shear displacement Node (no displacement)

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