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The density-weighted mean kinematic viscosity of the universal vertical profile is $\bar{\nu }=\alpha \Omega h^2{\int }_{-{\zeta }_s}^{{\zeta }_s}Pd\zeta$. Substitution of the height scale gives $\bar{\nu }=C{\alpha }^{4/3}(\kappa /\sigma {)}^{1/3}{\mathcal{R}}^{4/3}{\Sigma }^{2/3}{\Omega }^{-2/3}$. For Keplerian rotation, ${\Omega }^{-2/3}=(GM{)}^{-1/3}r$. The positive numerical constant depends only on the dimensionless vertical profile and the convention for $\alpha$.

// Target: astrophysics.bigb

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.

Set $\mathcal{R}=k/({\mu }_m{m}_H)$, so $p_g=\mathcal{R}\rho T$ and the prescribed dynamic viscosity is $\mu =\alpha \mathcal{R}\rho T/\Omega$. Keplerian shear and radiative diffusion then giveDifferentiating the second expression, equating it to the first and dividing by $\rho$ givesThis reduction uses the gas-pressure viscosity closure, not an assumption that gas pressure dominates the hydrostatic support.

For $z\ge 0$, introduce the vertical mass coordinate of a diskLet $\Sigma ={\int }_{-z_s}^{z_s}\rho dz$ be the full surface density of a reflection-symmetric disk. The upper surface is at $m_s=\Sigma /2$. With $\zeta =2m/\Sigma$, the temperature equation becomesWrite $T=T_{\ast }t(\zeta )$ and chooseThe equation reduces exactly to $(t^4{)}^{\prime \prime }+t=0$. Midplane symmetry gives $t^{\prime }(0)=0$, and the zero surface temperature gives $t(1)=0$. The stipulated unique positive nontrivial dimensionless solution therefore applies at every radius and surface density, with all physical dependence contained in $T_{\ast }$ and the mass-coordinate scaling. A zero temperature at the ideal surface does not mean zero escaping flux: $F=-(4\sigma /(3\kappa )){\partial }_m{T}^4$ can have a finite nonzero surface value.

The required viscosity is the density-weighted mean kinematic viscosity, not an unweighted vertical mean. Since $\rho \nu =\mu$,Substituting the dimensionless solution gives, with $I_t={\int }_0^{1}t(\zeta )d\zeta$,For central mass $M_{\ast }$, $\Omega =(G{M}_{\ast }/r^3{)}^{1/2}$, so the requested explicit radius and surface-density dependence isThe universal integral need not be evaluated. No separate solution for $\rho (z)$ is needed to find this mean: the mass-coordinate transformation removes it from both the temperature equation and the viscosity integral. The result is the gas-pressure viscosity closure with mixed pressure support.

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.

In a reflection-symmetric astrophysical disk, the vertical mass coordinate measures mass per unit area above the midplane. It satisfies $dm=\rho dz$, so ${\partial }_m={\rho }^{-1}{\partial }_z$ inside the material. If the total surface density is $\Sigma$, the upper face is $m=\Sigma /2$; the normalized coordinate $\zeta =2m/\Sigma$ ranges from zero to one. Integrals weighted by mass density become ordinary integrals in $m$, which is especially useful for radiative diffusion and the density-weighted mean kinematic viscosity.