Astrophysical disk

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 is

ω^{2} \int_{- H}^{H} (∣ ξ_{r} ∣^{2} + ∣ ξ_{z} ∣^{2}) d z = κ^{2} \int_{- H}^{H} ∣ ξ_{r} ∣^{2} d z + Ω_{z}^{2} H (∣ ξ_{z} (H) ∣^{2} + ∣ ξ_{z} (- H) ∣^{2}) .

For nonnegative

κ^{2}

and positive

Ω_{z}^{2}

, every nontrivial oscillatory mode has real nonnegative

ω^{2}

. The surface term is the gravitational restoring energy of the displaced vacuum interfaces.

Inertial modes of an incompressible disk

For

0 < ω^{2} < κ^{2}

, set

S = k H ω / κ^{2} - ω^{2}

. Even-pressure modes satisfy

ω^{2} = - Ω_{z}^{2} S tan S

, while odd-pressure modes satisfy

ω^{2} = Ω_{z}^{2} S cot S

. In the case

Ω_{z} > κ > 0

, the branches approach

κ

k H \to 0

. For each fixed vertical order

n \geq 1

and large

k H

S \to nπ /2

and

ω \sim κnπ / (2 k H)

. Even orders have even pressure and odd orders odd pressure. The limiting boundary condition is zero normal derivative of pressure at both interfaces.

// Target: astrophysics.bigb

Surface modes of an incompressible disk

For

ω^{2} > κ^{2}

, set

Q = k H ω / ω^{2} - κ^{2}

. Even-pressure and odd-pressure modes have

p^{'} \propto cosh (Q z / H)

and

sinh (Q z / H)

and obey

ω^{2} = Ω_{z}^{2} Q tanh Q

and

ω^{2} = Ω_{z}^{2} Q coth Q

, respectively. At large

k H

, both become free-surface gravity waves with

ω^{2} \sim \frac{1}{2} [κ^{2} + κ^{4} + 4 Ω_{z}^{4} (k H)^{2}] = Ω_{z}^{2} k H + \frac{1}{2} κ^{2} + O ((k H)^{- 1}) .

Their opposite-parity splitting is exponentially small as the two interfaces decouple.

// Target: astrophysics.bigb

Free-surface pressure condition of an incompressible disk

Vertical hydrostatic equilibrium gives

p_{0} = ρ Ω_{z}^{2} (H^{2} - z^{2}) /2

. At a vacuum interface, the Lagrangian pressure perturbation vanishes:

p^{'} + ξ_{z} \partial_{z} p_{0} = 0

. The vertical equation gives

ξ_{z} = p_{z}^{'} / (ρ ω^{2})

, hence

p^{'} - \frac{Ω _{z}^{2}}{ω ^{2}} z p_{z}^{'} = 0 at z = \pm H .

The moving interface makes this different from setting the Eulerian perturbation of a fluid variable

p^{'}

to zero on the original surface.

// Target: astrophysics.bigb

Kuzmin disk ( $Σ (R) = mb / [2 π (R^{2} + b^{2})^{3/2}]$ )

A Kuzmin disk is an infinitesimally thin axisymmetric astrophysical disk whose relative potential is

ψ = G m / 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

- 2 G mb / (R^{2} + b^{2})^{3/2}

, so the Poisson equation for Newtonian gravity, written

\nabla^{2} ψ = - 4 π Gρ

, gives the displayed surface density. Integrating

2 π R Σ

gives total mass

m

Kuzmin reflection method ( $Σ (R) = - \partial_{z} ψ (R, 0^{+}) / (2 π G)$ )

Reflect a vacuum gravitational potential in the upper half-space evenly across

z = 0

. For the convention in which gravitational acceleration is

\nabla ψ

, the derivative jump is

2 \partial_{z} ψ (R, 0^{+}) = - 4 π G Σ (R)

. Thus a point mass

m

below the plane produces a Kuzmin disk, and superposing point masses with weight

μ (b) d b

produces

Σ (R) = (2 π)^{- 1} \int_{0}^{\infty} b μ (b) (R^{2} + b^{2})^{- 3/2} d b

. Positive weights give a nonnegative surface density wherever the integral converges.

Scale-free Kuzmin superposition ( $μ (b) = B b^{β}, 0 < β < 1$ )

For

B > 0

and

0 < β < 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 difference

ψ (R, z) = GB \int_{0}^{\infty} b^{β} [\frac{1}{R ^{2} + ( ∣ z ∣ + b ) ^{2}} - \frac{1}{b}] d b .

At small

b

the subtraction is integrable because

β > 0

; at large

b

the difference is

O (b^{- 2})

, integrable because

β < 1

. Its plane force is

\partial_{R} ψ = - C R^{β - 1}

, where

C = (GB /2) B ((1 + β) /2, 1 - β /2)

. A circular speed

V = K R^{β /2}

fixes

C = K^{2}

, giving

Σ (R) = \frac{K ^{2} R ^{β - 1}}{2 π G} \frac{Γ ( 1 + β /2 ) Γ (( 1 - β ) /2 )}{Γ (( 1 + β ) /2 ) Γ ( 1 - β /2 )} .

The 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

β ↓ 0

Disk warp ( $W (R) = β (R) e^{iγ (R)}$ )

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.

Inviscid warp equations of a nearly Keplerian disk ( $Σ R^{2} Ω (W_{t} - i ω_{1} W) = R^{- 1} G_{R}, G_{t} - i ω_{2} G = \frac{1}{4} P R^{3} Ω W_{R}$ )

Here

G

is the complex internal horizontal torque,

P

the vertically integrated pressure, and

ω_{1}, ω_{2}

the approximate nodal precession and apsidal precession rates. In the exactly Keplerian case, local plane waves obey

ω^{2} = k^{2} P / (4Σ) = 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.

Stationary bending warp of a Kerr disk ( $W = W_{\infty} cos (x_{in} - x) / cos x_{in}, x = 4 24 a r^{- 5/4} / (5 ϵ)$ )

For

a > 0

Σ \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_{in}) = 0

; the outer tilt gives

W (0) = W_{\infty}

. The displayed solution holds when

cos x_{in} \neq = 0

. If

x_{in} = π

, it is

W = W_{\infty} cos x

, with an aligned node at

r = 2^{4/5} r_{in}

and opposite signed inner tilt. For negative spin the corresponding equation is hyperbolic, giving

W = W_{\infty} cosh (x_{in} - x) / cosh x_{in}

with

x = 4 24∣ a ∣ r^{- 5/4} / (5 ϵ)

Hankel representation of thin-disk gravity ( $Φ = \int_{0}^{\infty} S (k) J_{0} (k R) e^{- k ∣ z ∣} d k$ )

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

Φ_{z}^{+} - Φ_{z}^{-} = 4 π G Σ

. Hankel transform inversion gives

S (k) = - 2 π G \int_{0}^{\infty} Σ (R) J_{0} (k R) R d R

. Its radial derivative gives the galaxy rotation curve. Infinite scale-free models may require subtracting an additive divergent constant.

Angular-momentum-conserving self-gravitating disc radius ( $R_{h} / R_{d} = C_{d} f_{d} / q^{2}$ )

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_{vir}

. With

v_{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.

Protoplanetary disk

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.

Protoplanetary disk dispersal

Removal of disk gas by accretion onto the host star, winds and photoevaporation. The finite availability of gas limits which solid cores can acquire massive envelopes. The disk's evolutionary state matters more than a single universal dispersal time.

Snow line

A transition beyond which a specified volatile can predominantly condense into solid grains under local temperature and pressure. Without qualification it often means the water-ice transition. Its location evolves with disk heating, cooling and composition; it is not a universal fixed orbital radius.

Mestel disk ( $Σ (R) = v_{M}^{2} / (2 π GR)$ )

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.

Rotating Mestel disk distribution function ( $f = A L_{z}^{q} e^{- E / s}, q = v_{c}^{2} / s - 1$ )

In the midplane Mestel disk potential

Φ = v_{c}^{2} lo g (R / R_{0})

, take specific energy

E = (v_{R}^{2} + v_{ϕ}^{2}) /2 + Φ

and

L_{z} = R v_{ϕ}

. 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

Σ = Σ_{0} R_{0} / R

gives

q = v_{c}^{2} / s - 1 > - 1

and

A = \frac{Σ _{0} R _{0}^{- q}}{π 2 ^{q /2} s ^{(q + 2) /2} Γ (( q + 1 ) /2 )} .

The 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^{ε / s}

ε

must instead be the negative of the specific total energy.

Velocity moments of an isotropic Mestel disk ( $σ_{R}^{2} = σ_{ϕ}^{2} = v_{0}^{2}$ )

The planar Mestel disk has

F (E) \propto e^{- E / v_{0}^{2}}

, giving independent centered Gaussian distributions for

v_{R}, v_{ϕ}

with velocity dispersions

σ_{R} = σ_{ϕ} = v_{0}

. Its streaming velocity is zero. A maximally prograde stellar distribution instead has mean

2/ π v_{0}

and azimuthal variance

v_{0}^{2} (1 - 2/ π)

, while all second raw moments remain unchanged.

Mestel disk potential-density pair ( $Φ = v_{0}^{2} lo g [(r + ∣ z ∣) / R_{*}]$ )

The reflection-symmetric Newtonian gravitational potential

Φ = v_{0}^{2} lo g [(R^{2} + z^{2} + ∣ z ∣) / R_{*}]

is harmonic off the plane. Its normal-derivative jump is

2 v_{0}^{2} / R

, so the Poisson equation for Newtonian gravity gives surface density of a disk

S = v_{0}^{2} / (2 π GR)

and volume mass density

ρ = S δ (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.

Boundary ambiguity of a flat disk rotation curve

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 π G)

. The pure scale-free Mestel disk excludes such extra sources.

Legendre expansion of thin-disk gravity ( $Φ = - 2 G \sum_{k \geq 0} α_{2 k} (R^{- 2 k - 1} I_{2 k} + R^{2 k} J_{2 k})$ )

For a thin axisymmetric astrophysical disk, angular integration of the Newtonian kernel eliminates odd Legendre polynomials. Define

I_{n} = \int_{0}^{R} Σ (s) s^{n + 1} d s

and

J_{n} = \int_{R}^{\infty} Σ (s) s^{- n} d s

. The even coefficient is

α_{n} = π [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.

Elliptic-integral circular-speed kernel of a thin disk ( $V_{d}^{2} (r) = 2 G PV \int s Σ (s) [K / (r + s) + E / (r - s)] d s$ )

Integrating a circular ring's Newtonian gravitational potential over azimuth gives

d Φ = - 4 G s Σ (s) K (k) d s / (r + s)

, with modulus

k = 2 rs / (r + s)

. Differentiating using the complete elliptic integral of the first kind derivative gives the displayed circular speed kernel, where

E

is the complete elliptic integral of the second kind. For an interior point of a smooth disk use a Cauchy principal value; beyond the outer radius the integral is ordinary. A discontinuous nonzero surface-density edge in an exactly zero-thickness disk has a logarithmically singular in-plane force at that edge.

Radiation-supported vertical disk

A radiation-supported vertical disk has vertical hydrostatic equilibrium supplied mainly by radiation pressure. In a constant-opacity, optically thick layer in gravity

- Ω^{2} z

, radiative diffusion gives

F = c Ω^{2} z / κ

. This statement applies at a fixed radius and does not resolve the optically thin surface.

Constant viscosity in radiation-supported vertical balance ( $μ = 4 c / (9 κ)$ )

Combining

F = c Ω^{2} z / κ

with viscous heating

d F / d z = (9/4) μ Ω^{2}

gives a height-independent dynamic viscosity

μ = 4 c / (9 κ)

under constant opacity. This exact limiting relation differs from an approximate vertically averaged alpha disk closure.

Keplerian disk

A Keplerian disk is an astrophysical disk whose orbital motion is governed mainly by a central point mass. Balancing radial gravitational acceleration with circular motion gives

r Ω_{K}^{2} = GM / r^{2}

, hence

Ω_{K} = GM / r^{3}

and orbital speed

v_{K} = GM / r

. Inner rings rotate faster than outer rings: this differential rotation supplies the shear on which many disk stresses act. The approximation requires small corrections from pressure gradients, disk self-gravity and relativistic gravity.

Corotation radius ( $r_{co} = (GM / Ω_{*}^{2})^{1/3}$ )

The corotation radius is where a central star's spin matches the local Keplerian rotation. Equating

Ω_{*} = GM / r_{co}^{3}

gives

r_{co} = (GM / Ω_{*}^{2})^{1/3}

. For example, doubling the stellar spin moves this radius inward by

2^{- 2/3}

. Its relevance is dynamical: magnetic coupling outside it tends to accelerate disk gas toward the faster stellar rotation, whereas coupling inside it can brake the gas. The propeller effect also depends on whether the magnetospheric truncation radius lies outside this reference radius.

Thin disk

 1  0

A thin disk has disk aspect ratio

H / r ≪ 1

, where

H

is its vertical thickness scale and

r

its cylindrical radius. The separation lets radial orbital motion and vertical hydrostatic equilibrium be treated at different orders. In a centrally dominated gas disk,

H / r \sim c_{s} / v_{K}

: a low sound speed compared with orbital speed naturally produces a flattened layer. Geometrical thinness alone does not imply large optical depth or efficient radiative cooling.

Passive disk

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_{*} ≪ L_{*}

. The actual local requirement compares absorbed stellar flux with viscous flux; incidence angle, shadowing and absorption can invalidate a luminosity comparison alone.

Irradiation grazing angle of a disk ( $α ≃ z_{s}^{'} - z_{s} / R = R d (z_{s} / R) / d R$ )

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}^{'}

. Their angle difference is the grazing angle. The incident flux is

L_{*} α / (4 π 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.

Mean projected area of a randomly oriented thin disc ( $⟨ A_{proj} ⟩ = π R^{2} /2$ )

A thin circular disc presents projected area

π R^{2} ∣ cos i ∣

to a sightline. Isotropic normals have a uniform distribution of

cos i

[- 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.

Thin-disk expansion of the stellar gravitational potential ( $Φ ≃ - GM / R + Ω_{K}^{2} z^{2} /2$ )

At fixed cylindrical radius

R

with

∣ z ∣ ≪ R

, the central gravitational potential is harmonic in height. The vertical epicyclic frequency equals

Ω_{K}

in this spherical potential.

Razor-thin disk approximation

The razor-thin disk approximation replaces a disk's vertical density profile by

ρ (x, y, z) = Σ (x, y) δ (z)

. It is useful for horizontal dynamics on scales much longer than the disk thickness.

Sharp-edge force singularity of a thin disk ( $g_{R} \sim G Σ_{edge} lo g (ℓ / ϵ)$ )

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 Σ_{edge} \int_{ϵ}^{ℓ} d s / 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.

Barotropic closure of a razor-thin disk ( $c_{s}^{2} = d P / d Σ$ )

A two-dimensional disk equation with pressure acceleration

- Σ^{- 1} \nabla P

requires an equation of state for vertically integrated pressure

P

in terms of surface density of a disk

Σ

. A barotropic fluid closure is

P = P (Σ)

, with linear enthalpy perturbation

h^{'} = c_{s}^{2} Σ^{'} / Σ_{0}

. Merely giving

P = P (ρ)

for a volume density does not close the two-dimensional equations unless a vertical conversion

ρ (Σ)

is specified. Positive

c_{s}^{2}

gives ordinary pressure restoration of an inertial-acoustic wave.

Razor-thin disk Poisson kernel

The razor-thin disk Poisson kernel converts a horizontal Fourier transform of surface density of a disk into its gravitational potential. Write the razor-thin disk as

ρ (x, z) = Σ (x) δ (z)

and use the Poisson equation for Newtonian gravity

\nabla^{2} Φ = 4 π Gρ

. For horizontal wavevector

k

of magnitude

k > 0

, the transformed equation is

(\partial_{z}^{2} - k^{2}) Φ (k, z) = 4 π G Σ (k) δ (z) .

A solution decaying away from the disk has the form

C (k) e^{- k ∣ z ∣}

. Integrating across

z = 0

gives the derivative jump

- 2 k C = 4 π G Σ

, hence the midplane relation

Φ_{m} (k) = - \frac{2 π G}{k} Σ (k) .

This multiplier turns the nonlocal horizontal gravity calculation into multiplication in Fourier space. It is useful in calculations of the self-gravity of an astrophysical disk, where longer wavelengths have a larger potential response at fixed density amplitude. The

k = 0

mode requires separate boundary information: an infinite uniform sheet has a gravitational potential proportional to

∣ z ∣

, which cannot decay at both vertical infinities.

Thin-disk potential-density inversion ( $Σ = - k Φ / (2 π G)$ )

For an isolated razor-thin disk, define the two-dimensional Fourier transform by

g (k) = \int e^{- i k \cdot x} g (x) d^{2} x

. At

k > 0

, the razor-thin disk Poisson kernel gives the displayed inversion. Since the transform of

1/∣ x ∣

2 π / k

, inversion also gives

Σ = (4 π^{2} G)^{- 1} (1/∣ x ∣) * Δ_{2} Φ

. The forward relation is

Φ = - G (1/∣ x ∣) * Σ

. 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.

Off-plane razor-thin Poisson kernel ( $Φ (k, z) = - (2 π G / k) Σ (k) e^{- k ∣ z ∣}$ )

For

k > 0

, horizontal Fourier transform gives

(\partial_{z}^{2} - k^{2}) Φ = 4 π G Σ δ (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.

Positive-kernel comparison of thin-disk fields ( $∣ Σ_{M} ∣ < Σ ⟹ ∣ \partial_{z} Φ_{M} ∣ < \partial_{z} Φ (z > 0)$ )

The vertical field of a disk has kernel

G z / (∣ R - R^{'} ∣^{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.

Surface density of a disk ( $Σ$ )

The surface density of a disk is the mass density integrated through its vertical column:

Σ (r) = \int_{- \infty}^{\infty} ρ (r, z) d z

. Its dimensions are mass per area. An axisymmetric annulus of width

d r

therefore contains

d M = 2 π r Σ (r) d r

. This reduces a three-dimensional mass inventory to a radial one, which is useful in viscous evolution of an accretion disk. Distinct vertical profiles can have the same column mass, so

Σ

alone does not determine the vertical structure of an astrophysical disk.

Vertical structure of an astrophysical disk

The vertical structure of an astrophysical disk describes how mass density, pressure and temperature vary away from its midplane. For a thin, centrally dominated gas layer, hydrostatic equilibrium gives

d p / d z = - ρ Ω_{z}^{2} z

. An isothermal ideal gas has

p = c_{s}^{2} ρ

, producing a Gaussian density profile with disk scale height

H = c_{s} / Ω_{z}

. Disk self-gravity or magnetic support changes that balance.^[ref]

Vertical mass coordinate of a disk ( $m (z) = \int_{0}^{z} ρ (z^{'}) d z^{'}$ )

In a reflection-symmetric astrophysical disk, the vertical mass coordinate measures mass per unit area above the midplane. It satisfies

d m = ρ d z

, so

\partial_{m} = ρ^{- 1} \partial_{z}

inside the material. If the total surface density is

Σ

, the upper face is

m = Σ/2

; the normalized coordinate

ζ = 2 m /Σ

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.

Isothermal vertical structure in stellar gravity ( $ρ (z) = ρ_{0} e^{- z^{2} / (2 H^{2})}, H = c_{s} / Ω_{K}$ )

For a thin disk in the harmonic vertical gravitational potential

Ω_{K}^{2} z^{2} /2

, hydrostatic equilibrium and

p = c_{s}^{2} ρ

with height-independent isothermal sound speed give the displayed Gaussian density. For perturbations with adiabatic index

γ

, the Brunt–Väisälä frequency is

N^{2} = (1 - 1/ γ) Ω_{K}^{2} z^{2} / H^{2}

. An isothermal background can therefore support buoyancy for adiabatic perturbations; isothermal perturbations instead have

γ = 1

and zero buoyancy frequency.

Vertically integrated pressure of an isothermal disk ( $P = c_{s}^{2} Σ = Ω_{z}^{2} Σ H^{2}$ )

A locally isothermal disk in vertically harmonic external gravity has hydrostatic equilibrium

c_{s}^{2} ρ_{z} = - Ω_{z}^{2} z ρ

, hence

ρ = ρ_{0} e^{- z^{2} / (2 H^{2})}

with

H = c_{s} / Ω_{z}

. Because

c_{s}

is independent of height, integrating

p = c_{s}^{2} ρ

gives the displayed pressure. The harmonic-gravity approximation has the accuracy of the thin disk expansion.

Hermite vertical mode of an isothermal disk ( $ξ_{xx} - x ξ_{x} + n ξ = 0$ )

In an inviscid locally isothermal Keplerian disk, a radial normal mode of wavenumber

k

has pressure variable

ξ

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

(ω^{2} - n Ω_{K}^{2}) (ω^{2} - Ω_{K}^{2}) = k^{2} c_{s}^{2} ω^{2}

, where

ω = ω - m Ω_{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.

Bending wave of a Keplerian disk ( $ω ≃ \pm k c_{s} /2 (m = 1, ∣ k ∣ H ≪ 1)$ )

For the

n = 1

m = 1

Hermite vertical mode of an isothermal disk, the local dispersion relation is

[(ω - Ω_{K})^{2} - Ω_{K}^{2}]^{2} = k^{2} c_{s}^{2} (ω - Ω_{K})^{2}

. Its low inertial-frame frequency branches satisfy

ω ≃ \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

Ω_{K}

in its rotating frame.

Constant gas-radiation ratio vertical disc structure ( $T \propto 1 - z^{2} / H^{2}, ρ \propto T^{3}, p \propto T^{4}$ )

For constant positive

λ = p_{r} / p_{g}

and molecular gas constant

R = k / (μ_{m} m_{p})

, the gas–radiation equation of state gives

ρ = 4 σ T^{3} / (3 c λ R)

. Vertical hydrostatic equilibrium in harmonic central gravity therefore gives

T = T_{0} (1 - z^{2} / H^{2})

with

T_{0} = Ω^{2} H^{2} / [8 (1 + λ) R]

. The mass density and pressure powers are three and four. Integrating both faces yields

Σ = (32/35) ρ_{0} H

and

\int p d z = Σ Ω^{2} H^{2} /9

. With constant opacity, radiative diffusion produces a linear flux and height-independent dynamic viscosity

μ = 4 c λ / [9 κ (1 + λ)]

. This is an interior formal profile; its low-density surface requires a photospheric treatment beyond diffusion.

Vertical dynamical timescale of a disk ( $t_{dyn} \sim Ω_{z}^{- 1}$ )

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.

Polytropic vertical structure in stellar gravity ( $T = T_{0} (1 - z^{2} / H^{2}), ρ \propto T^{n}$ )

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) R T_{0} / Ω^{2}

. This differs from the polytropic vertical structure of a self-gravitating disk.

Zero-radiative-flux obstruction for a polytropic ideal gas

For positive finite opacity and nonvacuum ideal gas satisfying radiative diffusion, zero flux implies

d T / d z = 0

. Write

P = R ρT

, where

R = k_{B} / (μ_{m} m_{p})

for constant mean molecular weight

μ_{m}

in proton-mass units. If a polytropic equation of state

P = K ρ^{γ}

with constant

K > 0

is also imposed, then

T = (K / R) ρ^{γ - 1}

. For

γ \neq = 1

, constant temperature would force constant density, contradicting vertical hydrostatic equilibrium

P^{'} = - ρ Ω^{2} z

on an open interval away from the midplane. The familiar finite polytropic slab for

γ > 1

is therefore a solution of the hydrostatic/adiabatic closure, not of this finite-opacity zero-radiative-flux system. For

γ = 1

, the consistent isothermal profile is

ρ = ρ_{m} exp [- Ω^{2} z^{2} / (2 R T_{m})]

, with no finite surface.

Disk scale height ( $H$ )

The disk scale height measures the vertical spread of matter or pressure. For an isothermal, non-self-gravitating thin disk,

ρ (z) = ρ_{0} exp [- z^{2} / (2 H^{2})]

and

H = c_{s} / Ω_{z}

. Stronger vertical gravity confines the same gas more tightly; a higher sound speed makes it thicker. Here

H

is a standard deviation, not a sharp surface; a polytropic surface uses a different convention.^[ref]

Disk aspect ratio ( $H / r$ )

The aspect ratio of an astrophysical disk is its vertical thickness scale divided by cylindrical radius. A thin disk has

H / r ≪ 1

. In gas-supported vertical hydrostatic equilibrium with harmonic external gravity

- Ω_{z}^{2} z

H / r \sim c_{s} / (r Ω_{z})

, where

Ω_{z}

is the vertical epicyclic frequency and

c_{s}

is the sound speed. In a centrally dominated Keplerian disk,

Ω_{z} = Ω

, so this compares sound speed with orbital speed. If disk self-gravity supplies the vertical restoring force, the relation

H \sim c_{s} /Ω

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.

Disk flaring ( $d (H / r) / d r > 0$ )

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.

Flaring passive disk ( $z_{s} / R \propto R^{2/7}, T \propto R^{- 3/7}$ )

Assume a passive disk has local isothermal sound speed squared

c_{s}^{2} = a T

, disk scale height

H = c_{s} / Ω_{K}

, and irradiated height

z_{s} = f H

with constant

f

. Its irradiation grazing angle of a disk and Stefan–Boltzmann law give

q^{'} = C q^{8} / R^{3}

for

q = z_{s} / R

, where

C = 4 πσ (GM / a)^{4} / (L_{*} f^{8})

. Integrating gives

q^{- 7} = R_{0}^{2} / R^{2} + B

, with

R_{0}^{2} = 7 C /2

. Where the first term dominates and the thin disk assumptions hold,

q ≃ (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.

Vertical epicyclic frequency ( $Ω_{z}$ )

The vertical epicyclic frequency is the frequency of a small vertical oscillation about a circular orbit. In an axisymmetric potential,

Ω_{z}^{2} = \partial_{z}^{2} Φ

at the midplane; in a spherical potential it equals the circular orbital frequency.

Radiation-to-gas pressure ratio ( $β_{rg} = p_{rad} / p_{gas}$ )

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

β

, so the convention must be stated.

Constant-pressure-ratio vertical disk model

Assume a thin disk has height-independent opacity

κ_{op}

and radiation-to-gas pressure ratio

β

, with vertical gravity

- Ω_{z}^{2} z

. Hydrostatic equilibrium and radiative diffusion imply

F_{z} = \frac{β}{1 + β} \frac{c Ω _{z}^{2}}{κ _{op}} z .

Balancing its derivative against viscous heating

ρ ν q^{2} Ω^{2}

fixes the dynamic viscosity

ρ ν = \frac{β}{1 + β} \frac{c}{q ^{2} κ _{op}} \frac{Ω _{z}^{2}}{Ω ^{2}} .

It is independent of height under these assumptions. For a spherically symmetric potential,

Ω_{z} = Ω

. Integrating

ρ ν

through the full surface-to-surface thickness

2 H

gives

\overset{ν}{ˉ} Σ = 2 H ρ ν

, which determines the thickness from the radial accretion rate solution.

Dust settling in an astrophysical disk

Dust settling is the vertical drift of solid particles toward a disk's midplane under gravity and drag force. In the strong-drag limit, a particle with aerodynamic stopping time

τ

settles through a linear vertical gravitational field at rate

Ω_{z}^{2} τ

Homologous vertical motion of an astrophysical disk

In homologous vertical motion, every height scales by one time-dependent thickness

H (t)

, so

z = H (t) ζ

and the vertical velocity is

u_{z} = \dot{H} ζ

. A fixed dimensionless density profile then expands and contracts without changing shape.

Slow cooling of an astrophysical disk ( $η = (1 + Ct)^{- 1/ (1 + 2 β)}$ )

A laminar column can contract through nearly hydrostatic equilibrium when its cooling time greatly exceeds its vertical dynamical timescale of a disk. For cooling

A T^{1 + β}

and fixed column mass, a similarity solution has

ρ \propto η^{- 1}

T \propto η^{2}

P \propto η

, and vertical velocity

u_{z} = \overset{η}{˙} z / η

. The coefficient

C

also depends on the initial mass density normalization.

Vertical breathing mode of an astrophysical disk ( $ω_{H}$ )

The vertical breathing mode is a coherent oscillation of a disk's thickness. For an adiabatic gas with pressure proportional to

H^{1 - γ}

in a harmonic vertical potential,

ω_{H} = γ + 1 Ω_{z}

Accretion disk

 1  0

An accretion disk is an orbiting flow in which stresses permit matter to move inward by transporting angular momentum to other matter or an outflow. Simply losing heat is insufficient for gas that retains too much angular momentum to reach the center. The stresses also dissipate orbital energy, potentially producing strong radiation. A spreading ring illustrates the exchange: some material can accrete while a smaller fraction moves outward carrying the required angular momentum.^[ref]

Accretion corona

A hot, comparatively optically thin plasma region associated with an accretion flow. Thermal or nonthermal electrons can transfer energy to softer seed photons through inverse Compton scattering, producing a harder radiation component. Its geometry, heating and coupling to the cooler disc are model dependent; a hard radiation spectrum alone does not uniquely determine these properties.

Accretion-disk tests of strong-field gravity

Radiating material close to a compact object probes relativistic orbital motion, gravitational redshift and light propagation. A thin disk's inferred inner edge can be compared with the predicted innermost stable circular orbit, while iron-line transfer and thermal-continuum fitting probe complementary aspects of the same geometry. Agreement tests a combined spacetime-and-disk model. A black-hole spin measurement made after imposing a Kerr metric is not alone an independent verification of that metric tensor; geometry tests must vary gravitational predictions and astrophysical nuisance parameters together.

Thermal-continuum fitting of an accretion disk

A geometrically thin, optically thick disk has a radially varying effective temperature. Relativistic transport and an atmosphere model convert its thermal spectrum into an inferred emitting area and inner radius. Black-hole mass, distance, inclination and spectral hardening are important external constraints. Selecting thermal-state observations with moderate luminosity limits contamination and the breakdown of the thin-disk approximation. Relating the fitted inner radius to the innermost stable circular orbit remains a physical modeling assumption.

Relativistic iron-line spectroscopy

An irradiated accretion disk can produce iron fluorescence with a known local energy scale. relativistic Doppler effect, gravitational redshift, beaming and light bending redistribute those photons into a broadened asymmetric line. The red wing is sensitive to the deepest emitting radii, while inclination and orbital motion influence the profile. Coronal illumination, ionization, absorption and the continuum model must be controlled before inferring an inner radius or testing a spacetime geometry.

Quadratic-polytrope disk with constant aspect ratio ( $Σ = \frac{GM}{K} [arsinh ϵ - ϵ / 1 + ϵ^{2}]$ )

For

p = K ρ^{2}

, a point-mass gravitational potential and a zero-density free surface at

z = \pm ϵr

, the barotropic cylindrical rotation theorem gives

Ω^{2} = GM / (r^{3} 1 + ϵ^{2})

. Vertical balance gives

ρ = GM [1/ r^{2} + z^{2} - 1/ (r 1 + ϵ^{2})] / (2 K)

. Integrating over the finite height produces the displayed constant surface density. This is a local radial equilibrium family; infinite radial extent would have infinite mass.

Thin-accretion-disk truncation error ( $O ((H / r)^{2})$ )

For smooth radial variation on scale

r

c_{s} \sim Ω H

ν \sim α Ω H^{2}

u_{r} \sim ν / r

and

u_{z} \sim H u_{r} / r

. The exact point-mass vertical gravity differs from

Ω_{K}^{2} z

by relative order

(H / r)^{2}

. Radial pressure support changes the rotation and dominant shear heating by the same order. Meridional viscous forces, radial radiative-flux divergence, thermal advection and compressional work are at most order

(H / r)^{2}

relative to their retained vertical terms for

α ≲ 1

. These estimates require secular disk evolution and exclude sharp radial boundary layers or independently rapid vertical motion.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

Wind-powered mass loss from a steady accretion disk ( $\dot{M}^{'} = 2 π RS, ν Σ = R \dot{M}^{'} / (9 π f)$ )

For a Keplerian disk, let

S

be the total loss through both faces per unit projected area and

\dot{M} > 0

the inward accretion rate. Annular mass conservation gives

\dot{M}^{'} = 2 π RS

. A circular orbit has specific orbital energy

- GM / (2 R)

, so a marginally escaping wind requires additional energy

GM / (2 R)

per unit lost mass. Equating this sink to a fraction

f

of the two-face viscous dissipation

(9/4) ν Σ Ω^{2}

gives the displayed relation. If the wind carries the local specific angular momentum

h = GMR

, conservation of angular momentum gives

(\dot{M} h - G)^{'} = h \dot{M}^{'}

, with viscous torque in an accretion disk

G = 3 π ν Σ h

. Eliminating the torque yields

R^{2} \dot{M}^{''} + \frac{3}{2} R \dot{M}^{'} = \frac{3}{2} f \dot{M}

Accretion-disk boundary layer ( $T_{BL}, L_{BL}$ )

An accretion-disk boundary layer connects nearly Keplerian rotation to a stellar surface rotating at a different angular speed. For a slowly rotating star, dissipation of the incoming orbital kinetic energy gives

L_{BL} ≃ GM \dot{M} / (2 r_{*})

, comparable to the standard thin-disk luminosity. If the emitting area scales as

r_{*} H

, its blackbody effective temperature scales as

T_{BL} \sim (r_{*} / H)^{1/4} T_{in}

. The coefficient depends on geometry, and stellar rotation reduces the available energy.

In magnetically channelled accretion, a strong magnetic field guides an accretion flow along flux tubes. The component of the Lorentz force density parallel to the magnetic field vanishes, so the longitudinal flow can be treated hydrodynamically while the field controls the transverse geometry. A magnetic dipole field near a neutron star channels matter toward small polar caps.

Accretion column

An accretion column carries an inward accretion flow onto a localized region of a star. In the strong-field slender approximation, the magnetic field controls the flux-tube area while longitudinal pressure and gravity determine acceleration. The signed outward-coordinate velocity is negative; the inward speed and mass accretion rate are usually defined positive.

Accreting-pulsar luminosity and material-torque consistency ( $N = \dot{M} GM r_{m}$ )

If both luminosity and spin-up torque are supplied by the same steady mass inflow,

L = GM \dot{M} / R

and

N = \dot{M} GM r_{m}

imply

r_{m} = N^{2} GM / (L^{2} R^{2})

. A model requiring

r_{m} < R

cannot describe an exterior magnetosphere. Beaming, opposing magnetic torque, outflows and a different radiative efficiency change this inference and must be made explicit.

Magnetic spin equilibrium ( $Ω_{*} \sim Ω_{K} (r_{m})$ )

The spin at which the net material and magnetic torque on an accreting star vanishes. The corotation radius is typically comparable to the magnetospheric truncation radius, but the precise torque balance depends on the coupling and any outflow.

Minimum magnetic-equilibrium spin period ( $T_{m i n} \propto B_{}^{6/7} R_{}^{15/7}$ )

With magnetic spin equilibrium at the magnetospheric truncation radius,

T \propto R_{M}^{3/2} M^{- 1/2} \propto B_{*}^{6/7} R_{*}^{18/7} \dot{M}^{- 3/7}

at fixed mass. Inserting the surface accretion Eddington rate changes the radius power to

15/7

. The relation assumes the boundary remains outside the star; very weak fields instead encounter surface-accretion and mass-shedding limits.

Propeller effect

A rapidly spinning magnetosphere adds angular momentum to inflowing matter outside the corotation radius, inhibiting accretion and potentially ejecting gas. The centrifugal onset condition does not by itself ensure that every parcel escapes to infinity.

Magnetospheric truncation radius ( $r_{m}$ )

The inner radius at which a stellar magnetic dipole field disrupts a surrounding accretion disk. In a free fall pressure-balance estimate with fixed aspect ratio

ϵ = H / r

r_{m} \sim (ϵ^{2} μ^{4} / (GM \dot{M}^{2}))^{1/7}

. Geometry and inflow structure change its numerical coefficient.

Accretion Alfvén radius ( $r_{Alf}$ )

The accretion Alfvén radius estimates where the magnetic stress of an accreting star competes with inflow stress and disrupts an approximately Keplerian disk. It is distinct from the Alfvén critical radius of an outgoing stellar wind. A dipole and a free-fall pressure estimate give the characteristic

B_{*}^{4/7} \dot{M}^{- 2/7}

scaling; the numerical coefficient depends on flow geometry, and an inferred radius inside the star is outside the model's domain.

Spherical magnetospheric pressure balance ( $R_{M} \sim [B_{}^{4} R_{}^{12} / (μ_{0}^{2} GM \dot{M}^{2})]^{1/7}$ )

A dipole's magnetic pressure scales as

B_{*}^{2} R_{*}^{6} / (μ_{0} r^{6})

. Spherical free fall with

ρ = \dot{M} / (4 π r^{2} v)

and

v \propto (GM / r)^{1/2}

has ram pressure proportional to

\dot{M} (GM)^{1/2} r^{- 5/2}

. Equating them gives the seventh-root scaling. Disc geometry changes the coefficient; the estimate ceases to describe an exterior magnetosphere if the resulting radius lies inside the star.

Transonic accretion in a dipolar flux tube

The dipolar flux-tube area has

A (r) = A_{⋆} (r / R_{⋆})^{3}

. Applying transonic accretion in a power-law tube with

n = 3

and a polytropic equation of state gives

c_{s}^{2} = \frac{2 c _{0}^{2}}{7 - 5 γ}, r_{s} = \frac{GM ( 7 - 5 γ )}{6 c _{0}^{2}}, ρ_{s} = ρ_{0} (\frac{2}{7 - 5 γ})^{1/ (γ - 1)} .

Here

c_{0}

and

ρ_{0}

are the reservoir sound speed and mass density, and

c_{s}

is the sound speed at the sonic point. The mass accretion rate is

\dot{M} = A_{⋆} (\frac{r _{s}}{R _{⋆}})^{3} ρ_{s} c_{s} .

For two equal polar caps of surface angular radius

θ_{⋆} ≪ 1

A_{⋆} ≃ 2 π R_{⋆}^{2} θ_{⋆}^{2}

. One loaded cap gives half the total rate. The finite-radius transonic branch exists for

1 < γ < 7/5

, must cross outside the star (

r_{s} > R_{⋆}

), and requires

θ_{⋆}^{2} r_{s} / R_{⋆} ≪ 1

for the sonic region to remain a narrow flux tube.

Isothermal dipolar accretion equation ( $(v - c_{s}^{2} / v) v' = 3 c_{s}^{2} / z - GM / z^{2}$ )

A narrow dipolar flux-tube area

A (z) \propto z^{3}

gives

A ρ v = \dot{M}

for inward speed

v > 0

. The isothermal magnetic Bernoulli integral in

Φ = - GM / z

then gives

(v - c_{s}^{2} / v) v' = 3 c_{s}^{2} / z - GM / z^{2}

. A smooth transonic branch crosses at

z_{s} = GM / (3 c_{s}^{2})

with

v_{s} = c_{s}

ρ_{s} = e^{5/2} ρ_{\infty}

and

\dot{M} = ρ_{s} c_{s} A (z_{s})

. The crossing must lie in the domain; subsonic branches need not attain it.

Critical adiabatic index for dipolar accretion ( $γ_{crit} = 7/5$ )

For transonic accretion in a dipolar flux tube, the Bernoulli equation at the sonic point contains

c_{s}^{2} [1/ (γ - 1) - 5/2]

. A positive reservoir energy requires

γ < 7/5

. Compression in the

r^{3}

dipolar flux-tube area raises the specific enthalpy more rapidly than in the

r^{2}

geometry of spherical Bondi accretion, whose corresponding upper index is

5/3

Accretion rate ( $\dot{M}$ )

The accretion rate is the mass crossing an inward-oriented cylindrical surface per unit time. A common convention takes

\dot{M} > 0

for inward flow, so the outward radial mass flux is

- \dot{M}

Hyperaccreting black-hole disk

A hyperaccreting black-hole disk has an accretion rate so large that photon cooling can be trapped and neutrino emission can dominate the cooling of its hot dense inner region. Such disks are models for the central engines of some gamma-ray bursts.

Neutrino-dominated accretion flow (NDAF)

A neutrino-dominated accretion flow is a hot, dense accretion disk in which weak-interaction emission removes much of the dissipated energy. At extreme accretion rates, escaping photons can become inefficient at cooling the inner gas, while reactions involving nucleons and leptons emit neutrinos. Hyperaccreting black holes have therefore been investigated as engines for gamma-ray bursts. Whether cooling wins over inward energy transport depends on the flow parameters; the name describes a regime, not every rapidly accreting disk.^[ref]

Viscous evolution of an accretion disk

Viscous evolution of an accretion disk follows the redistribution of surface density as internal stresses exchange angular momentum between rings. For an axisymmetric Keplerian disk without mass loss, combining mass and angular-momentum conservation gives

\partial_{t} Σ = (3/ r) \partial_{r} [r^{1/2} \partial_{r} (ν Σ r^{1/2})]

. Here

ν

is an effective kinematic viscosity. The equation describes both inward accretion and outward spreading; the sign of local mass flow follows from the radial gradient, rather than viscosity alone.^[ref]

Compact self-similar disk with density-linear viscosity ( $\overset{ν}{ˉ} = A r^{2} Σ, Σ = σ [(R / r)^{a} - 1]_{+}$ )

The Keplerian viscous diffusion equation with the displayed kinematic viscosity has two such nontrivial finite-support self-similar solutions:

a = 1

R = C t^{2/5}

, and

a = 5/4

R = C t^{1/2}

. Both have

σ = (15 A t)^{- 1}

for

A, C, t > 0

. Matching powers of radius eliminates the third power only when

(5/2 - 2 a) (2 - 2 a) = 0

. The first solution conserves disk angular momentum; the second conserves disk mass but requires an inner torque.

Nonaccreting similarity disk supplied by an inner torque ( $M = const, J \propto t^{1/4}, G (0) \propto t^{- 3/4}$ )

The

a = 5/4

r / (2 t)

but zero limiting mass flux at the origin. Its viscous torque in an accretion disk has a positive finite inner limit for

t > 0

, equal to

dJ / d t

. Thus disk mass conservation coexists with growth of angular momentum; an external central torque supplies that growth.

Accretion trajectories in a compact viscous similarity disk ( $r (t) = C t^{2/5} + K t$ )

For the

a = 1

\overset{r}{˙} = r / (2 t) - R / (10 t)

. An integrating factor gives the displayed trajectories. From

0 < r_{i} < R (t_{i})

, the arrival time at the origin is

t_{i} [1 - r_{i} / R (t_{i})]^{- 10}

, finite for every interior point. The outer-edge trajectory has zero surface density of a disk and carries no mass.

Wind-loaded Keplerian disk evolution ( $Σ_{t} = \frac{3}{r} \partial_{r} [r \partial_{r} (r \overset{ν}{ˉ} Σ)] + \frac{1}{r} \partial_{r} [2 (q - 1) r^{2} S] - S$ )

Subtract local specific angular momentum times mass conservation from conservation of angular momentum including wind losses S,T. With outward radial mass flux

F

and outward viscous torque in an accretion disk

G = - 2 π \overset{ν}{ˉ} Σ r^{3} Ω^{'}

, this gives

F h^{'} + G^{'} = 2 π r (S h - T)

. For Keplerian rotation and

T = qS h

, substitution in mass conservation yields the displayed equation. A wind with q=1 removes local angular momentum but supplies no additional drift term; q>1 can drive accretion even with zero viscosity.

Inward transport by an angular-momentum-enhanced disk wind ( $x^{'} = \int \frac{d r}{β r f}, g^{'} = f r^{2 - 1/ β} Σ, g_{t}^{'} = g_{x^{'}}^{'}$ )

For zero viscosity,

S = f (r) Σ

and constant

q > 1

, set

β = 2 (q - 1)

. On intervals where f>0, the wind-loaded Keplerian disk evolution equation becomes

Σ_{t} = β r f Σ_{r} + [β (2 f + r f^{'}) - f] Σ

. Multiplication by

f r^{2 - 1/ β}

and the displayed travel-time coordinate reduce it to constant-speed advection. Its profile is

g^{'} = F (x^{'} + t)

and its characteristics move inward with

d r / d t = - β r f

. The weight accounts for mass loss as well as radial compression.

Tracer diffusion in a power-law accretion disk ( $C_{t} - v C_{Y} = D C_{YY}, Y = ln R, v = 3 ν_{0} /2, D = ζ ν_{0}$ )

In a steady Keplerian accretion disk with

Σ = Σ_{0} R^{- 2}

ν = ν_{0} R^{2}

, and tracer diffusivity

k = ζ ν

, the trace passive scalar concentration obeys the displayed constant-coefficient advection-diffusion equation in logarithmic radius. Contaminant mass is proportional to

C d Y

, because

2 π R σ d R = 2 π Σ_{0} C d Y

. A radial impulse

C_{0} δ (R - R_{0})

therefore has logarithmic amplitude

C_{0} / R_{0}

and evolves by the Gaussian heat kernel centered at

ln R_{0} - v t

. The full-line solution idealizes radii far from the disk boundaries.

Compact self-similar disk with square-root density viscosity ( $S = τ^{- 1} (1 - x / (6 τ))_{+}^{2}$ )

For

ν = ν_{0} (Σ/ Σ_{0})^{1/2} (R / R_{0})^{9/4}

, the variables

x = R / R_{0}

S = (Σ/ Σ_{0}) x^{3}

and

τ = 3 ν_{0} t / (4 R_{0}^{2})

reduce the source-free Keplerian viscous diffusion equation to

S_{τ} = (x S^{3/2})_{xx}

. The displayed self-similar solution has an outer edge

R_{e} = 36 R_{0} τ

and a surface-density cusp proportional to

R^{- 3/2}

at the origin. The positive-part cutoff is essential: beyond the edge,

S^{3/2}

is the cube of the absolute value, so continuing the squared polynomial without truncation does not solve the same equation.

The disk mass is

8 π Σ_{0} R_{0}^{2} τ^{- 1/2}

and its angular momentum is

12 π Σ_{0} R_{0}^{2} GM R_{0}

, independent of time. It therefore describes accretion with outward spreading and conserved disk angular momentum, under the formal zero-torque condition at the origin. The inward mass flux changes sign at

R = 9 R_{0} τ /4

, so the outer portion moves outward while the inner portion accretes.

Viscous stability of an accretion disk ( $\partial_{Σ} (\overset{ν}{ˉ} Σ) > 0$ )