Exact critical Rayleigh relation for vertical-field magnetoconvection ($Q{\pi }^2=R_c-R_c^{2/3}{R}_0^{1/3}$)

At the minimum of the steady vertical-field magnetoconvection dispersion relation, stationarity gives $R_c=2(k_c^2+{\pi }^2{)}^3/{\pi }^2$. With nonmagnetic value $R_0=27{\pi }^4/4$, eliminating the critical wavenumber gives the stated relation and $k_c^2={\pi }^2[\frac{3}{2}(R_c/R_0{)}^{1/3}-1]$. These hold for every nonnegative Chandrasekhar number, including its zero-field limit.