The first equation is vertical <hydrostatic equilibrium>. In a <thin disk>, stellar gravity has vertical component $-\Omega^2z$, giving $dP/dz=-\rho\Omega^2z$. The second is local energy conservation: Keplerian shear has $r\,d\Omega/dr=-3\Omega/2$, so viscous dissipation per unit volume is
$$
\mu(r\Omega')^2=\frac94\mu\Omega^2=\frac{dF}{dz}.
$$
The third equation is radiative diffusion. With radiation energy density $a_{\rm R}T^4=4\sigma T^4/c$, an optically thick medium carries
$$
F=-\frac{16\sigma T^3}{3\kappa\rho}\frac{dT}{dz},
$$
which rearranges to the stated gradient. Thomson scattering makes $\kappa$ approximately independent of density and temperature. The final equation adds perfect-gas pressure $k\rho T/(\mu_m m_p)$ and radiation pressure $a_{\rm R}T^4/3=4\sigma T^4/(3c)$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 321 / 2 / a / Solution

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A thin disk has vertical semi-thickness $H$ much smaller than cylindrical radius $r$. Vertical gravity is then approximately linear in height and orbital dynamics is nearly planar.


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 Incoming links: Thin disk