= Solution
Corotation reaches the <speed of light> at $R_c=c/\Omega$. Extrapolating a dipole field gives $B_c\sim B_*(R_*/R_c)^3$ and <magnetic energy> density $\epsilon_c\sim B_c^2/\mu_0$, suppressing order-one factors. An outward <electromagnetic energy> flux of scale $\epsilon_c c$ through area of scale $R_c^2$ therefore gives
$$
\boxed{P\sim\frac{B_*^2R_*^6c}{\mu_0R_c^4}
=\frac{B_*^2R_*^6\Omega^4}{\mu_0c^3}.}
$$
This assumes a dipole-dominated field tied to the rotating star and an effective outgoing <Poynting flux> near the <light cylinder>. The dipole extrapolation, angular pattern and surface-field convention affect the coefficient. A strictly aligned vacuum dipole would not radiate, whereas an oblique vacuum dipole does. A plasma-filled magnetosphere opens field lines and can <spin> down even an aligned rotator, so the light-cylinder estimate is a scaling argument, not an exact vacuum emission calculation or literal rigid motion at $c$. Force-free calculations explicitly demonstrate this distinction.<Time-dependent force-free pulsar magnetospheres: axisymmetric and oblique rotators by Anatoly Spitkovsky>[abstract and spin-down luminosity versus inclination]{ref}
The <energy> comes primarily from <stellar rotation>. With constant <moment of inertia> $I$, $E_{\rm rot}=I\Omega^2/2$, and conservation of <energy> gives $I\Omega\dot\Omega=-P$. Thus
$$
\boxed{\dot\Omega\simeq-K\Omega^3,\qquad
K=\frac{B_*^2R_*^6}{I\mu_0c^3}.}
$$
For constant $K$, differentiating gives $\ddot\Omega=-3K\Omega^2\dot\Omega$, hence $n=\Omega\ddot\Omega/\dot\Omega^2=3$. Evolving field strength, inclination, <moment of inertia> or additional <torques> invalidate the constant-$K$ prediction; generally $n=3+(\Omega/\dot\Omega)(\dot K/K)$ in this parametrization.
For radial inflow onto <mass> $M$, take $v_{\rm ff}\sim(GM/r)^{1/2}$ and $\rho\sim\dot M/(4\pi r^2v_{\rm ff})$. <magnetic pressure> decreases as $B_*^2R_*^6/(\mu_0r^6)$, while <ram pressure> scales as $\rho v_{\rm ff}^2\propto\dot M(GM)^{1/2}r^{-5/2}$. Balancing them yields
$$
\boxed{R_M\sim\left(\frac{B_*^4R_*^{12}}{\mu_0^2GM\dot M^2}\right)^{1/7}
\propto B_*^{4/7}R_*^{12/7}\dot M^{-2/7}.}
$$
Numerical factors and disc geometry are not determined by this spherical estimate. The displayed proportionality holds $M$ fixed and requires a magnetospheric boundary outside the stellar surface.
In a disc, inflowing matter near $R_M$ has specific <angular momentum> approximately $\sqrt{GMR_M}$. If it is transferred to a slower star, the material <torque> is $N\sim\dot M\sqrt{GMR_M}$ and $I\dot\Omega\sim N>0$. The <corotation radius> is $R_{\rm co}=(GM/\Omega^2)^{1/3}$, distinct from the light-cylinder radius. As stellar <spin> approaches the inner disc's Keplerian <frequency>, net spin-up weakens. If $R_M$ lies outside corotation, the field rotates faster than the disc and can centrifugally expel matter in the propeller regime. A characteristic equilibrium is
$$
\Omega_{\rm eq}\sim\sqrt{GM/R_M^3},\qquad
T_{\rm eq}\sim2\pi\sqrt{R_M^3/(GM)}
\propto B_*^{6/7}R_*^{18/7}\dot M^{-3/7}
$$
for fixed <mass> and magnetic-coupling convention. A complete <magnetic torque> model can change the coefficient; a star also cannot rotate past its <mass-shedding limit>, of order $\sqrt{GM/R_*^3}$.
For <opacity> per unit <mass> $\kappa$, an isotropic radiative force balances gravity at $L_E=4\pi GMc/\kappa$. Efficient release of surface gravitational binding <energy> gives $L_{\rm acc}\sim GM\dot M/R_*$ and therefore the <surface accretion Eddington rate>
$$
\dot M_{\max}\sim\frac{4\pi cR_*}{\kappa}\propto R_*.
$$
The radius dependence matters: it uses efficiency $\eta\sim GM/(R_*c^2)$, not an arbitrarily fixed reference efficiency. Substituting this maximal rate into the equilibrium period gives
$$
\boxed{T_{\min}\propto B_*^{6/7}R_*^{18/7}(R_*)^{-3/7}
=B_*^{6/7}R_*^{15/7}.}
$$
<mass>, <opacity> and geometry factors are held fixed. This is the classical radiatively efficient Eddington-limited scaling, not a universal upper bound on every <mass> supply rate: beaming, advection, outflows and <opacity> changes can alter it. For sufficiently small field, $R_M\lesssim R_*$ and the magnetic scaling must be replaced by surface-disc and mass-shedding physics.
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