Don’t torque like that - Measuring compact object magnetic fields with analytic torque models

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Table 2.

Resulting equilibrium fastness values, ω_eq.

Model	ω_eq
GL	0.349^(a)
W_I	0.949^(a)
W_II	7/8
W_III	$6 - \sqrt{3 \cdot 17 / 2} \approx 0.950$ $6-\sqrt {3\cdot 17/2} \approx 0.950$
R	N/A^(b)
M	${\begin{matrix} \frac{7 ω_{p}^{2} - ω_{p} \sqrt{(7 ω_{p})^{2} - 40 (2 ω_{p} - 1)}}{4 (2 ω_{p} - 1)} \\ 5 / 7 for γ_{c} \to \infty \end{matrix}$ $\left\{ \begin {array}{l} \frac {7\omega _{\textrm {p}}^2 - \omega _{\textrm {p}}\sqrt {(7\omega _{\textrm {p}})^2-40(2\omega _{\textrm {p}}-1)}}{4(2\omega _{\textrm {p}}-1)} \\ 5/7 \ {\textrm {for}}\ \gamma _{\textrm {c}}\rightarrow \infty \end {array}\right .$
D	$1 + c - \sqrt{1 / 2 + c^{2}}$ $1+c-\sqrt {1/2+c^2}$ , $c = \frac{9}{4 \sqrt{2}} \frac{ξ}{γ_{c}} + 1 / 2$ $c = \frac {9}{4\sqrt {2}}\frac {\xi }{\gamma _{\textrm {c}}} + 1/2$
K_I	20/21
G_I	5/7
G_II	$\frac{21 - \sqrt{3 \cdot 67}}{8} \approx 0.853$ $\frac {21 - \sqrt {3\cdot 67}}{8} \approx 0.853$

Notes.

^(a)

Equilibrium value is calculated numerically via root finding.

^(b)

This model never crosses over to spin-down.

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