Astrophysical disk

An astrophysical disk is a flattened rotating distribution of gas, dust, stars, or other matter whose orbital motion strongly influences its structure and evolution.

In a thin constant-density disk with free surfaces $z=\pm H$, neglect slow radial gradients and let perturbations vary as $e^{ikr-i\omega t}$. With $h=p^{\prime }/\rho$, the local inviscid equations are $-i\omega u-2\Omega v=-ikh$, $-i\omega v+{\kappa }^{2}u/(2\Omega )=0$, $-i\omega w=-h_z$ and $iku+w_z=0$. Eliminating velocity gives $h_{zz}=k^2{\omega }^{2}h/({\omega }^2-{\kappa }^2)$. This local WKB approximation assumes $H/r\ll 1$ and $|kr|\gg 1$ and has both surface and inertial wave branches.

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For radial and vertical Lagrangian fluid displacements of a local axisymmetric mode, integrate the momentum equations and use incompressible flow plus the free-surface pressure condition of an incompressible disk. The result isFor nonnegative ${\kappa }^2$ and positive ${\Omega }_z^2$, every nontrivial oscillatory mode has real nonnegative ${\omega }^2$. The surface term is the gravitational restoring energy of the displaced vacuum interfaces.

For $0<{\omega }^2<{\kappa }^2$, set $S=kH\omega /\sqrt{{\kappa }^2-{\omega }^2}$. Even-pressure modes satisfy ${\omega }^2=-{\Omega }_z^{2}S\tan S$, while odd-pressure modes satisfy ${\omega }^2={\Omega }_z^{2}S\cot S$. In the case ${\Omega }_z>\kappa >0$, the branches approach $\kappa$ as $kH\to 0$. For each fixed vertical order $n\ge 1$ and large $kH$, $S\to n\pi /2$ and $\omega \sim \kappa n\pi /(2kH)$. Even orders have even pressure and odd orders odd pressure. The limiting boundary condition is zero normal derivative of pressure at both interfaces.

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For ${\omega }^2>{\kappa }^2$, set $Q=kH\omega /\sqrt{{\omega }^2-{\kappa }^2}$. Even-pressure and odd-pressure modes have $p^{\prime }\propto \cosh (Qz/H)$ and $\sinh (Qz/H)$ and obey ${\omega }^2={\Omega }_z^{2}Q\tanh Q$ and ${\omega }^2={\Omega }_z^{2}Q\coth Q$, respectively. At large $kH$, both become free-surface gravity waves withTheir opposite-parity splitting is exponentially small as the two interfaces decouple.

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Vertical hydrostatic equilibrium gives $p_0=\rho {\Omega }_z^2(H^2-z^2)/2$. At a vacuum interface, the Lagrangian pressure perturbation vanishes: $p^{\prime }+{\xi }_z{\partial }_z{p}_0=0$. The vertical equation gives ${\xi }_z=p_z^{\prime }/(\rho {\omega }^2)$, henceThe moving interface makes this different from setting the Eulerian perturbation of a fluid variable $p^{\prime }$ to zero on the original surface.

// Target: astrophysics.bigb

A Kuzmin disk is an infinitesimally thin axisymmetric astrophysical disk whose relative potential is $\psi =Gm/\sqrt{R^2+(|z|+b{)}^2}$, with $m,b>0$. Away from the disk this is a point-mass vacuum field. The jump of its vertical derivative is $-2Gmb/(R^2+b^2{)}^{3/2}$, so the Poisson equation for Newtonian gravity, written ${\nabla }^2\psi =-4\pi G\rho$, gives the displayed surface density. Integrating $2\pi R\Sigma$ gives total mass $m$.

Reflect a vacuum gravitational potential in the upper half-space evenly across $z=0$. For the convention in which gravitational acceleration is $\nabla \psi$, the derivative jump is $2{\partial }_z\psi (R,0^{+})=-4\pi G\Sigma (R)$. Thus a point mass $m$ below the plane produces a Kuzmin disk, and superposing point masses with weight $\mu (b)db$ produces $\Sigma (R)=(2\pi {)}^{-1}{\int }_0^{\infty }b\mu (b)(R^2+b^2{)}^{-3/2}db$. Positive weights give a nonnegative surface density wherever the integral converges.

For $B>0$ and $0<\beta <1$, the unreferenced relative potential of the Kuzmin reflection method diverges at large $b$. Referencing each component to the origin before integrating gives the finite potential differenceAt small $b$ the subtraction is integrable because $\beta >0$; at large $b$ the difference is $O(b^{-2})$, integrable because $\beta <1$. Its plane force is ${\partial }_R\psi =-C{R}^{\beta -1}$, where $C=(GB/2)B((1+\beta )/2,1-\beta /2)$. A circular speed $V=K{R}^{\beta /2}$ fixes $C=K^2$, givingThe beta function substitutions for the force and surface density have different exponents. Its central mass is locally finite, but its total mass diverges at large radius; a potential zero at infinity cannot be imposed. The force and density approach those of the Mestel disk as $\beta \downarrow 0$.

A disk warp is a radial change of the orbital plane of successive disk annuli. In the small-tilt approximation, a complex tilt $W$ records its magnitude and azimuth. Pressure and internal torques communicate the tilt through bending waves of a Keplerian disk in the inviscid thin-disk regime. A flat tilted disk has constant $W$ even though its absolute orientation differs from the reference equatorial plane.

Here $G$ is the complex internal horizontal torque, $P$ the vertically integrated pressure, and ${\omega }_1,{\omega }_2$ the approximate nodal precession and apsidal precession rates. In the exactly Keplerian case, local plane waves obey ${\omega }^2=k^{2}P/(4\Sigma )=k^2{c}_s^2/4$, so tilt disturbances propagate at $c_s/2$. The secular thin-disk approximation requires wavelengths large relative to the height while the local approximation requires them small relative to the orbital radius.

For $a>0$, $\Sigma \propto R^{-3/4}$, constant disk aspect ratio $ϵ=H/R$, and $r=R{c}^2/(GM)$, the inviscid steady disk warp equation reduces to $W_{xx}+W=0$ in the displayed coordinate. The vanishing inner torque gives $W_x(x_{\mathrm{in}})=0$; the outer tilt gives $W(0)=W_{\infty }$. The displayed solution holds when $\cos x_{\mathrm{in}}\ne 0$. If $x_{\mathrm{in}}=\pi$, it is $W=W_{\infty }\cos x$, with an aligned node at $r=2^{4/5}{r}_{\mathrm{in}}$ and opposite signed inner tilt. For negative spin the corresponding equation is hyperbolic, giving $W=W_{\infty }\cosh (x_{\mathrm{in}}-x)/\cosh x_{\mathrm{in}}$ with $x=4\sqrt{24|a|}{r}^{-5/4}/(5ϵ)$.

For an axisymmetric reflection-symmetric thin disk, the Laplace equation in cylindrical coordinates separates into regular radial Bessel functions and vertical decaying exponentials. The divergence theorem imposes ${\Phi }_z^{+}-{\Phi }_z^{-}=4\pi G\Sigma$. Hankel transform inversion gives $S(k)=-2\pi G{\int }_0^{\infty }\Sigma (R)J_0(kR)RdR$. Its radial derivative gives the galaxy rotation curve. Infinite scale-free models may require subtracting an additive divergent constant.

Let a disc contain fraction $f_d$ of total halo mass, take its edge gravity to be $v_d^2=C_{d}G{M}_d/R_d$, and equate its edge specific angular momentum to $q{R}_h{v}_{\mathrm{vir}}$. With $v_{\mathrm{vir}}^2=G{M}_h/R_h$, these conditions give $R_h/R_d=C_d{f}_d/q^2$. The structural coefficient $C_d$ is essential: disc mass alone does not uniquely determine its edge rotation curve.

A protoplanetary disk is a gas-and-solid astrophysical disk around a young star in which planet formation occurs. Its evolution combines accretion, radiation, gas dynamics and the growth of solid material. A minimum-mass solar nebula is one reconstruction of the material needed to form the Solar System.

A Mestel disk is a thin axisymmetric astrophysical disk with surface density inversely proportional to radius. Its galaxy rotation curve is exactly flat, $v_c=v_M$, and $v_c^2=GM(R)/R$ despite its nonspherical geometry. In the Legendre expansion of thin-disk gravity, all positive-degree interior and exterior terms cancel. Its total mass is infinite, so only potential differences, rather than a potential zero at infinity, are defined.

In the midplane Mestel disk potential $\Phi =v_c^2\log (R/R_0)$, take specific energy $E=(v_R^2+v_{\varphi }^2)/2+\Phi$ and $L_z=R{v}_{\varphi }$. A prograde galactic distribution function $f=A{L}_z^q{e}^{-E/s}$ for $L_z>0$, with $s>0$, has surface density proportional to $R^{q-v_c^2/s}$. Matching $\Sigma ={\Sigma }_0{R}_0/R$ gives $q=v_c^2/s-1>-1$ andThe denominator is the product of the full radial Gaussian integral and the positive-azimuthal half-line moment. Here $s$ is a squared-velocity scale, not a velocity dispersion itself. Changing the potential's additive constant changes $A$. With positive exponent $e^{\epsilon /s}$, $\epsilon$ must instead be the negative of the specific total energy.

The planar Mestel disk has $F(E)\propto e^{-E/v_0^2}$, giving independent centered Gaussian distributions for $v_R,v_{\varphi }$ with velocity dispersions ${\sigma }_R={\sigma }_{\varphi }=v_0$. Its streaming velocity is zero. A maximally prograde stellar distribution instead has mean $\sqrt{2/\pi }{v}_0$ and azimuthal variance $v_0^2(1-2/\pi )$, while all second raw moments remain unchanged.

The reflection-symmetric Newtonian gravitational potential $\Phi =v_0^2\log [(\sqrt{R^2+z^2}+|z|)/R_{\ast }]$ is harmonic off the plane. Its normal-derivative jump is $2v_0^2/R$, so the Poisson equation for Newtonian gravity gives surface density of a disk $S=v_0^2/(2\pi GR)$ and volume mass density $\rho =S\delta (z)$. The midplane circular speed is constant. The scale-free model has infinite total mass and does not use a finite-mass boundary condition at infinity.

The midplane galaxy rotation curve fixes the radial derivative of the Newtonian gravitational potential, not every exterior boundary condition. Adding $a|z|$ leaves that radial derivative unchanged but adds a uniform sheet of surface density of a disk $a/(2\pi G)$. The pure scale-free Mestel disk excludes such extra sources.

For a thin axisymmetric astrophysical disk, angular integration of the Newtonian kernel eliminates odd Legendre polynomials. Define $I_n={\int }_0^R\Sigma (s)s^{n+1}ds$ and $J_n={\int }_R^{\infty }\Sigma (s)s^{-n}ds$. The even coefficient is ${\alpha }_n=\pi [n!/(2^n((n/2)!{)}^2){]}^2$. In differentiating the paired inner and outer contributions, their moving-boundary terms cancel. The resulting galaxy rotation curve includes an inward inner-ring contribution and an outward exterior-ring contribution. For smooth surface density, the force at the disk is interpreted with a symmetric Cauchy principal value or a vanishing-thickness regularization.

A radiation-supported vertical disk has vertical hydrostatic equilibrium supplied mainly by radiation pressure. In a constant-opacity, optically thick layer in gravity $-{\Omega }^{2}z$, radiative diffusion gives $F=c{\Omega }^{2}z/\kappa$. This statement applies at a fixed radius and does not resolve the optically thin surface.

Combining $F=c{\Omega }^{2}z/\kappa$ with viscous heating $dF/dz=(9/4)\mu {\Omega }^2$ gives a height-independent dynamic viscosity $\mu =4c/(9\kappa )$ under constant opacity. This exact limiting relation differs from an approximate vertically averaged alpha disk closure.

A Keplerian disk rotates with angular frequency ${\Omega }_K=\sqrt{GM/r^3}$ set mainly by a central mass, so ${\Omega }_K\propto r^{-3/2}$.

The radius where Keplerian rotation matches the spin of the central star.

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.

A passive astrophysical disk receives most of its heat from external irradiation rather than internal viscous dissipation. For a stellar disk, a rough criterion is $GM\dot{M}/R_{\ast }\ll L_{\ast }$. The actual local requirement compares absorbed stellar flux with viscous flux; incidence angle, shadowing and absorption can invalidate a luminosity comparison alone.

For a thin illuminated surface $z=z_s(R)$ and a central point source, the radial light ray has slope $z_s/R$ whereas the surface tangent has slope $z_s^{\prime }$. Their angle difference is the grazing angle. The incident flux is $L_{\ast }\alpha /(4\pi R^2)$ to leading order. This requires a positive grazing angle and an unobstructed ray; a negative formal flux instead signals failure of direct illumination, not negative heating.

A thin circular disc presents projected area $\pi R^2|\cos i|$ to a sightline. Isotropic normals have a uniform distribution of $\cos i$ on $[-1,1]$, giving mean absolute cosine one half. The resulting interception cross-section is therefore half the face-on area; no extra orientation weighting is needed after this average is used in an incidence calculation.

At fixed cylindrical radius $R$ with $|z|\ll R$, the central gravitational potential is harmonic in height. The vertical epicyclic frequency equals ${\Omega }_K$ in this spherical potential.

The razor-thin disk approximation replaces a disk's vertical density profile by $\rho (x,y,z)=\Sigma (x,y)\delta (z)$. It is useful for horizontal dynamics on scales much longer than the disk thickness.

At an abruptly truncated razor-thin disk with nonzero edge surface density of a disk, the midplane radial force diverges logarithmically. Locally, mass occupies a half-plane and the inward force contains a nonzero angular factor multiplying $G{\Sigma }_{\mathrm{edge}}{\int }_{ϵ}^{\ell }ds/s$. The opposite exterior half-plane is absent, so its cancellation does not occur. Finite thickness or a taper taking the edge density to zero regularizes the force. Therefore a finite-edge galaxy rotation curve requires more than the ideal scale-free Mestel disk normalization.

A two-dimensional disk equation with pressure acceleration $-{\Sigma }^{-1}\nabla P$ requires an equation of state for vertically integrated pressure $P$ in terms of surface density of a disk $\Sigma$. A barotropic fluid closure is $P=P(\Sigma )$, with linear enthalpy perturbation $h^{\prime }=c_s^2{\Sigma }^{\prime }/{\Sigma }_0$. Merely giving $P=P(\rho )$ for a volume density does not close the two-dimensional equations unless a vertical conversion $\rho (\Sigma )$ is specified. Positive $c_s^2$ gives ordinary pressure restoration of an inertial-acoustic wave.

For a horizontal Fourier mode of wavenumber $k>0$, the midplane gravitational potential of a razor-thin disk is

For an isolated razor-thin disk, define the two-dimensional Fourier transform by $\tilde{g}(\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{x}}g(\mathbf{x})d^{2}x$. At $k>0$, the razor-thin disk Poisson kernel gives the displayed inversion. Since the transform of $1/|\mathbf{x}|$ is $2\pi /k$, inversion also gives $\Sigma =(4{\pi }^{2}G{)}^{-1}(1/|\mathbf{x}|)\ast {\Delta }_2\Phi$. The forward relation is $\Phi =-G(1/|\mathbf{x}|)\ast \Sigma$. An additive potential constant is irrelevant; a uniform infinite sheet and scale-free divergent potentials require separate zero-mode or potential-difference treatment rather than an arbitrary decaying boundary condition.

For $k>0$, horizontal Fourier transform gives $({\partial }_z^2-k^2)\tilde{\Phi }=4\pi G\tilde{\Sigma }\delta (z)$. Decay and continuity give an exponential on each side; its derivative jump fixes the coefficient. The uniform mode needs a separate linear vertical solution and a potential reference.

The vertical field of a disk has kernel $Gz/(|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^2+z^2{)}^{3/2}>0$. The triangle inequality therefore bounds a signed-surface density field by the field of its absolute surface density. If that surface density is everywhere smaller than a nonnegative comparison surface density, their vertical fields satisfy the displayed strict comparison, provided the integrals exist and the strict bound holds on positive measure. This is useful when comparing gravity-equivalent magnetic surface density with actual disk mass.

The surface density of a disk is its volume density integrated vertically, $\Sigma ={\int }_{-\infty }^{\infty }\rho dz$.

The vertical structure of a thin disk is set by the balance among vertical gravity, pressure, self-gravity, magnetic forces, and motions of gas and solids.

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.

For a thin disk in the harmonic vertical gravitational potential ${\Omega }_K^2{z}^2/2$, hydrostatic equilibrium and $p=c_s^2\rho$ with height-independent isothermal sound speed give the displayed Gaussian density. For perturbations with adiabatic index $\gamma$, the Brunt–Väisälä frequency is $N^2=(1-1/\gamma ){\Omega }_K^2{z}^2/H^2$. An isothermal background can therefore support buoyancy for adiabatic perturbations; isothermal perturbations instead have $\gamma =1$ and zero buoyancy frequency.

A locally isothermal disk in vertically harmonic external gravity has hydrostatic equilibrium $c_s^2{\rho }_z=-{\Omega }_z^{2}z\rho$, hence $\rho ={\rho }_0{e}^{-z^2/(2H^2)}$ with $H=c_s/{\Omega }_z$. Because $c_s$ is independent of height, integrating $p=c_s^2\rho$ gives the displayed pressure. The harmonic-gravity approximation has the accuracy of the thin disk expansion.

In an inviscid locally isothermal Keplerian disk, a radial normal mode of wavenumber $k$ has pressure variable $\xi$ proportional to a Probabilists' Hermite polynomial in $x=z/H$. Gaussian-weighted finite energy excludes the exponentially growing vertical branch. Polynomial termination gives $n=0,1,\dots$ and the local dispersion relation $({\hat{\omega }}^2-n{\Omega }_K^2)({\hat{\omega }}^2-{\Omega }_K^2)=k^2{c}_s^2{\hat{\omega }}^2$, where $\hat{\omega }=\omega -m{\Omega }_K$. At singular values introduced by eliminating velocity components, the original equations must also be checked. The $n=0$ propagating branch is an inertial-acoustic wave; $n=1$ has vertically uniform vertical velocity and odd pressure perturbation, permitting bending disturbances.

For the $n=1$, $m=1$ Hermite vertical mode of an isothermal disk, the local dispersion relation is $[(\omega -{\Omega }_K{)}^2-{\Omega }_K^2{]}^2=k^2{c}_s^2(\omega -{\Omega }_K{)}^2$. Its low inertial-frame frequency branches satisfy $\omega \simeq \pm k{c}_s/2$. These waves transport a changing tilt of successive annuli, or a warp, with group velocity $\pm c_s/2$. The vertical velocity is approximately independent of height. A rigid tilt is the zero-wavenumber, zero-pattern-frequency limit; the gas still executes vertical orbital motion at frequency ${\Omega }_K$ in its rotating frame.

For constant positive $\lambda =p_r/p_g$ and molecular gas constant $\mathcal{R}=k/({\mu }_m{m}_p)$, the gas–radiation equation of state gives $\rho =4\sigma T^3/(3c\lambda \mathcal{R})$. Vertical hydrostatic equilibrium in harmonic central gravity therefore gives $T=T_0(1-z^2/H^2)$ with $T_0={\Omega }^2{H}^2/[8(1+\lambda )\mathcal{R}]$. The mass density and pressure powers are three and four. Integrating both faces yields $\Sigma =(32/35){\rho }_{0}H$ and $\int pdz=\Sigma {\Omega }^2{H}^2/9$. With constant opacity, radiative diffusion produces a linear flux and height-independent dynamic viscosity $\mu =4c\lambda /[9\kappa (1+\lambda )]$. This is an interior formal profile; its low-density surface requires a photospheric treatment beyond diffusion.

The time for a vertical pressure or gravity response. A sound-crossing time and a small vertical oscillation period have this scale in a thin disk. It need not equal a damping time.

In harmonic stellar gravity, an ideal gas with a polytropic equation of state of index $n>0$ has a quadratic temperature profile and finite semi-thickness $H^2=2(n+1)\mathcal{R}{T}_0/{\Omega }^2$. This differs from the polytropic vertical structure of a self-gravitating disk.

For positive finite opacity and nonvacuum ideal gas satisfying radiative diffusion, zero flux implies $dT/dz=0$. Write $P=\mathcal{R}\rho T$, where $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units. If a polytropic equation of state $P=K{\rho }^{\gamma }$ with constant $K>0$ is also imposed, then $T=(K/\mathcal{R}){\rho }^{\gamma -1}$. For $\gamma \ne 1$, constant temperature would force constant density, contradicting vertical hydrostatic equilibrium $P^{\prime }=-\rho {\Omega }^{2}z$ on an open interval away from the midplane. The familiar finite polytropic slab for $\gamma >1$ is therefore a solution of the hydrostatic/adiabatic closure, not of this finite-opacity zero-radiative-flux system. For $\gamma =1$, the consistent isothermal profile is $\rho ={\rho }_m\exp [-{\Omega }^2{z}^2/(2\mathcal{R}{T}_m)]$, with no finite surface.

The disk scale height is the characteristic vertical thickness over which density or pressure decreases. Vertical hydrostatic balance in a thin Keplerian gas disk gives $H\simeq c_s/{\Omega }_K$.

The aspect ratio of an astrophysical disk is its vertical thickness scale divided by cylindrical radius. A thin disk has $H/r\ll 1$. In gas-supported vertical hydrostatic equilibrium with harmonic external gravity $-{\Omega }_z^{2}z$, $H/r\sim c_s/(r{\Omega }_z)$, where ${\Omega }_z$ is the vertical epicyclic frequency and $c_s$ is the sound speed. In a centrally dominated Keplerian disk, ${\Omega }_z=\Omega$, so this compares sound speed with orbital speed. If disk self-gravity supplies the vertical restoring force, the relation $H\sim c_s/\Omega$ requires replacement. A polytropic free-surface height and an isothermal scale height differ by model-dependent constants even though their aspect ratios have the same basic role.

A disk flares when its disk aspect ratio increases outward. This is stronger than the statement that its height increases: a disk with $H\propto r$ has constant aspect ratio and is not flaring by this definition. The radial temperature and surface density of a disk profiles determine flaring in a specified vertical equilibrium model.

Assume a passive disk has local isothermal sound speed squared $c_s^2=aT$, disk scale height $H=c_s/{\Omega }_K$, and irradiated height $z_s=fH$ with constant $f$. Its irradiation grazing angle of a disk and Stefan–Boltzmann law give $q^{\prime }=C{q}^8/R^3$ for $q=z_s/R$, where $C=4\pi \sigma (GM/a{)}^4/(L_{\ast }{f}^8)$. Integrating gives $q^{-7}=R_0^2/R^2+B$, with $R_0^2=7C/2$. Where the first term dominates and the thin disk assumptions hold, $q\simeq (R/R_0{)}^{2/7}$, $H\propto R^{9/7}$ and $T\propto R^{-3/7}$. The law cannot be extrapolated into a geometrically thick region using the same grazing-angle approximation.

The vertical epicyclic frequency is the frequency of a small vertical oscillation about a circular orbit. In an axisymmetric potential, ${\Omega }_z^2={\partial }_z^2\Phi$ at the midplane; in a spherical potential it equals the circular orbital frequency.

The radiation-to-gas pressure ratio compares radiation pressure with gas pressure. It is distinct from plasma beta, which compares gas pressure with magnetic pressure. Both are sometimes denoted by $\beta$, so the convention must be stated.

Assume a thin disk has height-independent opacity ${\kappa }_{\mathrm{op}}$ and radiation-to-gas pressure ratio $\beta$, with vertical gravity $-{\Omega }_z^{2}z$. Hydrostatic equilibrium and radiative diffusion implyBalancing its derivative against viscous heating $\rho \nu q^2{\Omega }^2$ fixes the dynamic viscosityIt is independent of height under these assumptions. For a spherically symmetric potential, ${\Omega }_z=\Omega$. Integrating $\rho \nu$ through the full surface-to-surface thickness $2H$ gives $\bar{\nu }\Sigma =2H\rho \nu$, which determines the thickness from the radial accretion rate solution.