= Solution
Let $\beta=P_{\rm gas}/P_{\rm rad}$. Constancy of $\beta$ gives
$$
\rho=\frac{4\mu_m m_p\sigma}{3ck}\,\beta T^3.
$$
The total pressure is
$$
P=(1+\beta)\frac{4\sigma T^4}{3c}.
$$
Consequently, if $T=T_0f$, then $\rho=\rho_0f^3$ and $P=P_0f^4$, where
$$
\boxed{\rho_0=\frac{4\mu_m m_p\sigma}{3ck}\,\beta T_0^3},
\qquad
\boxed{P_0=\frac{4\sigma}{3c}(1+\beta)T_0^4}.
$$
Substitution in hydrostatic balance shows that $f'= -2z/H^2$ and
$$
\boxed{f(z)=1-\frac{z^2}{H^2}},
\qquad
\boxed{H^2=\frac{8kT_0}{\mu_m m_p\Omega^2}
\frac{1+\beta}{\beta}}.
$$
Finally,
$$
\Sigma=\rho_0H\int_{-1}^{1}(1-x^2)^3dx.
$$
Absorbing the numerical factor $4/3$ and this dimensionless integral into $A$ gives the requested form
$$
\boxed{H=\frac1A\left(\frac{ck}{\mu_m m_p\sigma}\right)
\frac{\Sigma}{\beta T_0^3}}.
$$
Solved by gpt-5.6-sol high.
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