Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 73 / 1

The first equation is vertical hydrostatic equilibrium: the upward pressure gradient balances the central mass's vertical gravitational force. For $|z|\ll r$, the point-mass acceleration is $-GMz/(r^2+z^2{)}^{3/2}\simeq -{\Omega }_K^{2}z$, where Keplerian rotation gives ${\Omega }_K^2=GM/r^3$. The disc's self-gravity and vertical inertia are neglected.

The second equation is local thermal balance. The divergence of the vertical radiative flux equals viscous dissipation. In the convention of this alpha disk, $\nu =\alpha p/(\rho \Omega )$, and the dominant Keplerian shear has $rd\Omega /dr=-3\Omega /2$. ThereforeThe radiative flux increases from zero at the symmetric midplane to a positive outgoing value at the upper surface. Radial heat transport, thermal advection and rapid storage of thermal energy are excluded from this local balance.

The third equation is radiative diffusion, equivalentlyIt applies to optically thick material, with $\kappa$ the opacity per unit mass and $\sigma$ the Stefan-Boltzmann constant. Here constant electron-scattering opacity represents Thomson scattering; the formula uses a diffusive, nearly thermal radiation field. Positive upward flux requires temperature to decrease upwards.

The final equation is the ideal gas pressure law, with mean molecular weight ${\mu }_m$ in proton-mass units. Writing $\mathcal{R}=k/({\mu }_m{m}_p)$ gives $p=\mathcal{R}\rho T$. The pressure is gas pressure because radiation pressure is assumed negligible. The four equations express vertical support, local heating balance, diffusive cooling and the gas equation of state.

Work at one radius and write $\mathcal{R}=k/({\mu }_m{m}_p)$. Use the surface density to fix the mass density normalization and introduce the height scaleChoose the dimensionless variablesThe pressure and temperature scales make the hydrostatic and ideal gas equations parameter-free. The flux scale fixes the heating coefficient at $9/4$, and the choice of $h$ makes the diffusion coefficient unity. Direct substitution gives the universal ordinary differential equationsFor example, the dimensionless coefficient in the temperature equation is $3\alpha \kappa {\Sigma }^2{\mathcal{R}}^4/(16\sigma {\Omega }^5{h}^6)=1$.

Let the upper free surface be at $\zeta ={\zeta }_s$, so its physical semithickness is $H=h{\zeta }_s$. Midplane symmetry and the radiative-zero disk surface conditions giveThe last condition is precisely $\Sigma ={\int }_{-H}^H\rho dz$. The unknown surface position and the two positive midplane values $P(0),\Theta (0)$ are determined as part of the normalized boundary value problem. Symmetry makes $P,D,\Theta$ even and $Q$ odd; $P^{\prime }(0)={\Theta }^{\prime }(0)=0$ already follow from the equations and are not additional independent boundary conditions.

The zero conditions do not mean zero surface flux. Integrating the heating equation gives $Q({\zeta }_s)=(9/4){\int }_0^{{\zeta }_s}Pd\zeta >0$. Near a regular surface with finite outgoing flux, dividing the hydrostatic equation by the temperature equation yields $dP/d\Theta \simeq {\zeta }_s{\Theta }^3/Q_s$. Thus $P\propto {\Theta }^4$, $D\propto {\Theta }^3$ and $\Theta \propto {\zeta }_s-\zeta$, so mass density also vanishes. The apparent singular quotient in the diffusion equation has a finite regular limit. All boundary conditions and equations are independent of the physical parameters: this is homologous vertical structure of a constant-opacity alpha disk.

The density-weighted mean kinematic viscosity isusing $\nu =\alpha p/(\rho \Omega )$. Under the preceding rescaling, letUniqueness of the universal dimensionless solution makes $I_P$ a positive numerical constant independent of $\alpha ,\Omega ,\kappa ,\Sigma ,\mathcal{R}$. HenceFor Keplerian rotation, ${\Omega }^{-2/3}=(GM{)}^{-1/3}r$. The constant-opacity gas-pressure disk viscosity law is thereforeThe particular scaling convention changes the representation of $I_P$ but not this physical law. Its nontrivial powers come from balancing viscous heating, diffusive cooling and vertical support together, rather than assuming a fixed temperature or thickness.

Estimate characteristic bulk terms. Put $ϵ=H/r\ll 1$ and assume smooth radial variation on scale $r$, secular evolution, and $\alpha \lesssim 1$. Vertical support gives $c_s^2\sim p/\rho \sim {\Omega }^2{H}^2$, while the alpha disk prescription and slow accretion implyThese velocity estimates follow from the viscous radial evolution scale and mass conservation. They are estimates for a slowly evolving thin disc, not a claim that geometric thinness excludes independently forced rapid vertical motion.

For vertical balance, the exact central force expands asAt $z\sim H$, the neglected correction has magnitude $\rho {\Omega }^{2}H{ϵ}^2$, compared with the retained force $\rho {\Omega }^{2}H$. Radial pressure support similarly changes the rotation from its Keplerian value byVertical meridional inertia scales as $u_r{u}_z/r$ or $u_z^2/H$, both $O({\alpha }^2{ϵ}^4{\Omega }^{2}H)$. Meridional viscous accelerations can scale as $\nu u_z/H^2$ or $\nu u_r/(rH)$, both $O({\alpha }^2{ϵ}^2{\Omega }^{2}H)$. Thus the gravity correction and possible viscous correction giveSome omitted terms are smaller than this leading order.

For the heating equation, the retained dissipation is $q_0^{+}\sim \rho \nu {\Omega }^2\sim \alpha p\Omega$. Radial pressure support and the finite-height correction change the dominant shear by a relative $O({ϵ}^2)$. Smooth vertical variation of the rotation gives ${\partial }_z(r\Omega )=O(ϵ\Omega )$, whose additional squared shear contributes $O({ϵ}^2{q}_0^{+})$. Meridional shear of order $u_r/H$ gives an additional relative $O({\alpha }^2{ϵ}^2)$ contribution.

Thermal advection and compressional work are of order $\rho u_r{c}_s^2/r$ and $p{u}_r/r$, respectively. Relative to $\alpha p\Omega$ they are $O({ϵ}^2)$. Thermal storage on the viscous evolution timescale $t_{\nu }\sim r^2/\nu$ has the same order, since $p/t_{\nu }\sim \alpha p\Omega {ϵ}^2$. For radiative diffusion with the same local diffusivity in different directions, smooth gradients give $F_r/F_z\sim H/r$, soConsequentlyThis is the thin-accretion-disk truncation error for the secular, radially smooth approximation. Neglect of disc self-gravity and radiation pressure, optical thickness, the opacity model and the idealized zero surface boundary have their own validity conditions; their errors do not become small solely because $H/r$ does. Near a sharp radial boundary layer or during thermal evolution on the heating timescale, the above estimates need not apply.