Dynamics of astrophysical discs

Lecture 1

FDEPS, Kyoto

29.11.11

Gordon Ogilvie

gio10@cam.ac.uk

DAMTP, University of CambridgeNASA
2011年12月19日月曜日

Outline
Lecture 1: Introduction to astrophysical discs● Occurrence of discs, physical and observational properties● Orbital dynamics, mechanics of accretion● Equations of astrophysical fluid dynamics and MHD
Lecture 2: Evolution and structure of discs● Evolution of an accretion disc
● Vertical disc structure, timescales
Lecture 3: Local approximation and incompressible dynamics● Shearing sheet, shearing waves● Incompressible dynamics: hydrodynamic stability, vortices

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Outline

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Lecture 4: Compressible dynamics of astrophysical discs● Compressible dynamics: density waves, gravitational instability● Satellite-disc interaction
Lecture 5: Magnetohydrodynamics of astrophysical discs● Magnetorotational instability● Jet launching
Seminar: Astrophysical tides and planet–star interactions

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Astrophysical discs

Continuous medium in orbital motion around a massive central bodyorbital dynamics /celestial mechanics

fluid dynamics /continuum mechanics● Usually circular, coplanar and thin
● Usually Keplerian (dominated bygravity of central mass)

Ω=

�

GM
r3

�1/2

● Shearing, dissipative systems

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Astrophysical discs

Continuous medium in orbital motion around a massive central bodyorbital dynamics /celestial mechanics

fluid dynamics /continuum mechanics● Usually circular, coplanar and thinr
la m
gu ntuan eomm● Usually Keplerian (dominated bygravity of central mass)
GM
r3

�1/2

m

as

s

Ω=

�

● Shearing, dissipative systems● Accretion disc (angular momentum
out, mass in, energy liberated)
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Occurrence of discs● Spiral galaxies (different: dark matter, stars, time-scales)● Active galactic nuclei, quasars● Interacting binary stars● Protostellar / protoplanetary discs, solar nebula● Planetary rings, circumplanetary discs● Very rapidly rotating stars (Be stars)● Exotica : supernovae, gamma-ray bursts, ...

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NASA
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Hubble Space Terescope
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NASA
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Formation of discs● Collapse of rotating cloud (e.g. star formation)

rapidly rotating

slowly rotating

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centrifugally
supported2011年12月19日月曜日

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Formation of discs● Mass transfer / tidal disruption / merger

binary star
critical equipotential(Roche lobe)

satellite
Hill “sphere”● Other scenarios: captured stellar winds, stellar pulsations, ...
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disc or disk ?

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Physical composition● Weakly ionized H / H2 gas + solid particles (protoplanetary discs)
aspect ratio H/R � 0.1 , temperature 10 K � T � 103 K● Dense H / He plasma (interacting binary stars, AGN)
aspect ratio H/R � 0.03 , temperature 103 K � T � 107 K● Nuclear matter (exotica)● Metre-sized iceballs (dense planetary rings)
aspect ratio H/R ∼ 10−7 , random velocity ∼ mm s−1● Dilute plasma (some cases of black-hole accretion flows)
Relevant descriptions:● Gas dynamics● Magnetohydrodynamics● Kinetic theory
2011年12月19日月曜日[+ relativity+ radiation forceswhere needed]Characteristic length-scales and orbital periods● Planetary ring:r ∼ 105 km, t ∼ 10 hr● Protoplanetary disc:rout ∼ 100 AU, tout ∼ 1000 yr
rin ∼ 0.01 AU, tin ∼ 10 day● X-ray binary star:rout ∼ R⊙ , tout ∼ hr − dayrin ∼ 10 km, tin ∼ 10

−3

● AGN:srout ∼ 0.1 pc, tout ∼ 1000 yr

rin ∼ AU, tin ∼ hr

● Parsec● Astronomical unit
● Solar radius
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pc = 3.086 × 1018 cm

AU = 1.496 × 1013 cm

R⊙ = 6.960 × 1010 cm

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Configurations: binary stars

circumstellar disc(s)2011年12月19日月曜日

circumbinary disc

Configurations: protoplanetary systems

embedded planet

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gap-opening planet
circumplanetary discinterior + exterior discs

Observations: Saturn’s rings

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● Galileo (1610)
SMAISMRMILMEPOETALEUMIBUNENUGTTAUIRAS
ALTISSIMUM PLANETAM TERGEMINUM OBSERVAVI
I have observed the most distant planet to have a triple form

● Huygens (1659)

Galileo, 1610

AAAAAACCCCCDEEEEEHIIIIIIILLLLMMNNNNNNNNNOOOOPPQRRSTTTTTUUUUUANNULO CINGITUR TENUI PLANO NUSQUAM COHAERENTE AD ECLIPTICAM INCLINATO
It is surrounded by a thin flat ring, nowhere touching, and inclined to the ecliptic● Hooke, Cassini, ..., Laplace, Maxwell, ...
● Voyager 1 and 2 flybys (1980-1)● Cassini in orbit (2004-)
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Huygens,1659:System saturnium, pp.84

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Observations: Saturn’s rings● waves, wakes, gaps, ringlets, braids, shepherds, propellers, ...
● ciclops.org

Cuzzi, et al., 2010:Science, 327, 1470
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Observations: protoplanetary discs

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● Nebular hypothesis (solar nebula):
Swedenborg, Kant, Laplace (18th century)● Direct observations of protoplanetary discs (Hubble ST, 1995-)● Extrasolar planets around main-sequence stars (1995-)● Debris discs and transitional discs (Spitzer ST, infrared, 2003-)
Hubble Space Terescope
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Observations: cataclysmic variable stars

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SS Cygni
dwarf nova
V magnitude(range 12-8)
1900-2010
aavso.org
Also UV andsoft X-rays

http://aavso.org/
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Observations: X-ray binary stars (1960s-)
GRO J1655-40 (“Nova Sco 1994”)

1996 97
Cyg X-1

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98

99

2000

01

02

03

04

05

06

07

08

10 keV . . . 107 K● Unsteady accretion● Sources of variability ...

http://www.phys.lsu.edu/~rih/
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For an alternative point of view, try
http://web.archive.org/web/2011202230603/http://
www.accretiondisk.org/

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Orbital dynamics● Test particle in gravitational potential Φ● Cylindrical polar coordinates (r, φ, z)● Newtonian dynamics● Assume:● Special case:Φ = Φ(r, z)

axisymmetricΦ(r, −z) = Φ(r, z)

symmetric

Φ = −GM (r2 + z 2 )−1/2

point-mass potential → Keplerian orbits

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Orbital dynamics● Equation of motionr̈ = −∇Φ{r̈ − rφ̇2 = −Φ,rrφ̈ + 2ṙφ̇ = 0z̈ = −Φ,z● Specific energy

ε = 12 |ṙ|2 + Φ = const● Specific angular momentum

h = r2 φ̇ = const● Reduces to 2D problemr̈ = −Φeﬀ,rz̈ = −Φeﬀ,z● Effective potential

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2hΦeﬀ = Φ + 2
2r

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Orbital dynamics● Circular orbit in midplane (z = 0)
0 = Φeﬀ,z (r, 0)2h0 = Φeﬀ,r (r, 0) = Φ,r (r, 0) − 3rh2
ε = 2 + Φ(r, 0)
2r✓ by symmetry}h◦ (r)
definingε◦ (r)● Important relationdε◦h◦ dh◦h2◦= 2− 3 + Φ,r (r, 0)
dr
r dr
r
dε◦
h◦
= 2 = φ̇ = Ω◦dh◦r
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orbital angular velocity

Orbital dynamics● Keplerian caseGMΦ(r, 0) = −rh◦ = (GM r)1/2

GM
ε◦ = −
2r
�
�1/2
GM
Ω◦ =
r3

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Orbital dynamics● Reminder of general Keplerian orbits
GM rr̈ = −|r|3dhd= (r × ṙ) = ṙ × ṙ + r × r̈ = 0
dt
dt● Orbit is confined to plane ⊥ h , so introduce polar coordinates (r, φ) :GMr̈ − rφ̇ = − 2r2

h = r2 φ̇ = const

d
d
2 d
= φ̇
= hu
● Let r = 1/u and note that
:
dt
dφ
dφ
�
� ��
1
2 d
2 d
hu
− h2 u3 = −GM u2
hu
dφ
dφ u

d2 u
GM
+u= 2
2
dφ
h
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Orbital dynamics
d2 u
GM
+u= 2
2
dφ
h

● General solution (with two arbitrary constants)GMu = 2 [1 + e cos(φ − �)]h● Polar equation of conic section:e=0
circlebound(ε < 0)2011年12月19日月曜日

0<e<1ellipse
bound

λ
⇒ r=
1 + e cos(φ − �)e=1parabola
marginallyunbounde>1
hyperbola
unbound(ε > 0)1-23
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Orbital dynamics● Perturbations (δr, δz) of circular orbits in midplane ( h fixed)¨ = −Ω2 δr
δrr¨ = −Ω2 δzδzΩ2r = Φeﬀ
,rr (r, 0)Ω2z = Φeﬀ
,zz (r, 0)

z

[ Φeﬀ
,rz (r, 0) = 0 by symmetry]Ωrusually calledκΩz sometimes called µ(horizontal) epicyclic frequency
vertical (epicyclic) frequency● Orbit is stable if Ω2r > 0 (i.e. κ2 > 0 ) and Ω2z > 0i.e. if orbit is of minimum energy for given h

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Orbital dynamics
● Now
2
3h
κ2 = Φ,rr (r, 0) + 4◦
r
� 2�
d h◦
3h2◦
=
+ 4
3
dr r
r

1 dh2◦
= 3
r dr
dΩ◦
2
= 4Ω◦ + 2rΩ◦dr2Ωz = Φ,zz (r, 0)

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Orbital dynamics● Keplerian case
κ = Ωz = Ωdescending node
apocentre

pericentre
ascending nodeκ=Ωeccentric Keplerian orbit
2011年12月19日月曜日Ωz = Ω

inclined Keplerian orbit

Orbital dynamics

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● Precession● If κ ≈ Ω and/or Ωz ≈ Ω, describe as slowly precessing orbit
2π● Minimum r (pericentre) occurs at time intervals ∆t =
κ
2πΩ
∆φ =
κ
�
�
Ω
= 2π
− 1 + 2π
κ
�
�
Ω
= 2π
−1
mod 2πκ

http://en.wikipedia.org/wiki/File:Perihelion_precession.svg∆φ=Ω−κ● Apsidal precession rate∆t

● Similarly, nodal precession rate = Ω − Ωz2011年12月19日月曜日

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Orbital dynamics● Example 1: Kerr metric of rotating black hole
● Dimensionless spin parameter: −1 < a < 1(let a < 0 for retrograde orbit)● From general relativity:
�
�
3
r
c
1
x
=
Ω=
GM/c2
GM x3/2 + a
κ2
−1
−3/2
2 −2
=
1
−
6x
+
8ax
−
3a
x
2
Ω
Ω2z
−3/2
2 −2
=
1
−
4ax
+
3a
x
2
Ω

● Precession rates far from black hole ( x � 1 ):3c3
3(GM )3/2
Ω−κ≈
=
5/2
GM x
c2 r5/2

Einstein2ac32a(GM )2
Ω − Ωz ≈
=
3
GM x
c3 r3

Lense-Thirring

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Orbital dynamics● e.g. a = 0.5
event
horizonΩ2 · (r3 /GM )1.0Ω2z · (r3 /GM )

0.8
0.6
0.4

marginallystableorbit

κ2 · (r3 /GM )

0.2
0.0

5

�0.2
�0.4

unstable
circular orbits
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10

15

20

x

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Orbital dynamics● Example 2: Exterior of rotating planet or star (Newtonian)● Multipole expansion in spherical polar coordinates (R, θ, φ) :
�
�
�
�
∞
n
�
GM
Re
Pn (cos θ)
Φ=−
1−
Jn
R
R
n=2
multipolecoefficient

Legendre
polynomial
R

θ
M

Re

−2
−4
J
≈
1.63
×
10
,
J
≈
−9.4
×
10
● e.g. Saturn: 2
4

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Orbital dynamics
�

∞
�

GM
Φ=−
1−
Jn
R
n=2

● Find:

�

∞
�

�

Re
R

�

�n

Pn (cos θ)

�

�n

�

GM
Re
Pn (0)Ω = 3 1−(n + 1)Jn
r
r
n=2
�
�
�
�
∞
n
�
GM
Re
2
Pn (0)
κ = 3 1+(n + 1)(n − 1)Jn
r
r
n=2
�
�
�
�
∞
n
�
GM
Re
2
2
Pn (0)Ωz = 3 1 −(n + 1) Jn
r
r
n=2
2

● Related by κ2 + Ω2z = 2Ω2 (potential satisfies Laplace’s equation)

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Orbital dynamics● Precession rates for large r (using P2 (0) = −1/2 ):
� �2
3
Re
Ω − κ ≈ J2
Ω
2
r
� �2
3
Re
Ω − Ωz ≈ − J2Ω2r
● e.g. F ring of Saturn:Ω − κ ≈ 0.0045 Ω ≈ 2.6◦ /day

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Mechanics of accretion

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● Consider two particles in circular orbits in the midplane● Can energy be released by an exchange of angular momentum?
● Total energy and angular momentum:
E = E1 + E2 = m1 ε1 + m2 ε2

H = H1 + H2 = m1 h1 + m2 h2● In an infinitesimal exchange:dH

dE = dE1 + dE2 = m1 Ω1 dh1 + m2 Ω2 dh2
dH = dH1 + dH2 = m1 dh1 + m2 dh2● If dH = 0 thendE = (Ω1 − Ω2 ) dH1● In practice dΩ/dr < 0● Energy released by transferring angular momentum outwards
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Mechanics of accretion● Generalize argument to allow for exchange of mass:dM = dm1 + dm2 = 0
dH = dH1 + dH2 = 0

dHi = mi dhi + hi dmi

dEi = mi Ωi dhi + εi dmi= Ωi dHi + (εi − hi Ωi ) dmi
dE = (Ω1 − Ω2 ) dH1 + [(ε1 − h1 Ω1 ) − (ε2 − h2 Ω2 )]dm1● In practice d(ε − hΩ)/dr = −h dΩ/dr > 0● Energy released by transferring angularmomentum outwards and mass inwards● This is the physical basis of an accretion disc2011年12月19日月曜日

dH
dmEquations of astrophysical fluid dynamics

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● Astrophysical fluid dynamics (AFD):● Basic model: Newtonian gas dynamics:∂u1+ u · ∇u = −∇Φ − ∇p∂tρ∂ρ+ u · ∇ρ = −ρ∇ · u∂t∂p+ u · ∇p = −γp∇ · u∂t● Compressible● Ideal (inviscid, adiabatic)● Non-relativistic (Galilean-invariant)● Lagrangian (material) derivative:
D
∂
=
+u·∇
Dt
∂t
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u velocityΦ gravitational potential
ρ densityp pressure
γ adiabatic exponent

Equations of astrophysical fluid dynamics● Gravity:● Non-self-gravitating fluid:● Φ is prescribed (fixed / external potential)● Self-gravitating fluid:● Φ is determined (in part) from the density of the fluid:∇2 Φ = 4πGρ

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Equations of astrophysical fluid dynamics

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● Extensions of the basic model:● Viscosity:● Usually extremely small● May be needed to provide small-scale dissipation● May be introduced to model turbulent transport∂u1= ··· + ∇ · T∂tρ
T = 2µS + µb (∇ · u)I

� 1
1�
T
S=
∇u + (∇u) − (∇ · u)I2
3

2011年12月19日月曜日

T viscous stress tensorµ (shear) viscosityµb bulk viscosity
S shear tensor
I unit tensor

kinematic viscosity ν = µ/ρEquations of astrophysical fluid dynamics● Non-adiabatic effects:● Thermal energy equation:
Ds
ρT
=H−C
Dt

● Heating:T temperatures specific entropy
H heating / unit volume
C cooling / unit volume(non-adiabatic effects)● Viscous:
H = T : ∇u = 2µS2 + µb (∇ · u)2● Cooling:● Radiative: C = ∇ · F● Diffusion approximation:(optically thick regions)
16σT 3F =−∇T
3κρ
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σ Stefan-Boltzmann
constant
κ opacity(Rosseland mean)

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Equations of astrophysical fluid dynamics

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● Equation of state:p = p(ρ, T )● Ideal gas with radiation:
kρT
4σT 4
p = p g + pr =
+
µm mp
3c

● pr important at very high T

k Boltzmann constantµm mean molecular weight
mp proton massc speed of light● µm = 0.5 for fully ionized H, µm = 2 for molecular H, etc.
● Thermal energy equation in dynamical variables:
�
��
�
γ1 p
1
dp −
ρT ds =dργ3 − 1ρ����Dp γ1 p Dρ
1
=H−C
⇒
−
γ3 − 1Dt
ρ Dt● For ideal gas of constant ratio of specific heats, γ1 = γ2 = γ3 = γ
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Equations of astrophysical fluid dynamics● Extensions of the basic model:● Magnetohydrodynamics (MHD)● Radiation hydrodynamics (RHD)● Relativistic formulations● Kinetic theory / plasma physics● Simplifications of the basic model:● Incompressible fluid: ∇ · u = 0● Boussinesq / anelastic approximations● Barotropic fluid: p = p(ρ)

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Equations of astrophysical fluid dynamics

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● Magnetohydrodynamics (MHD):● Electrically conducting fluid (plasma, metal, weakly ionized gas)
● Pre-Maxwell equations (without displacement current):∂B= −∇ × E∂tsolenoidal∇·B =0constraint∇ × B = µ0 J

B magnetic field
E electric fieldJ electric current densityµ0 permeability of free space( ∇ · E equation not required)● Galilean invariance:x� = x − vtt� = tB� = BE� = E + v × B
J� = J2011年12月19日月曜日

Equations of astrophysical fluid dynamics

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● Ohm’s law:J � = σE �

⇒ J = σ(E + u × B)

in rest frame of conductor
σ : electrical conductivity
for conducting fluid with velocity u(x, t)● Combine with Maxwell:∂B= −∇ × E� �∂tJ= ∇ × (u × B) − ∇ ×σ= ∇ × (u × B) − ∇ × (η∇ × B)

= ∇ × (u × B) + η∇2 B

magnetic diffusivity
1
η=
µ0 σ
∝ resistivity

if η uniform● “Induction equation”: vector advection-diffusion equation∂ωcf. vorticity equation= ∇ × (u × ω) + ν∇2 ω for ω = ∇ × u∂t
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Equations of astrophysical fluid dynamics

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● Ideal MHD (perfect conductor: σ → ∞ , η → 0 ):∂B
= ∇ × (u × B)∂t
= B · ∇u − u · ∇B − B(∇ · u) + u(∇ · B)

● Magnetic field is “frozen in” to fluid:● Field lines behave as material lines● Magnetic flux through an open material surface is conserved● Valid for large magnetic Reynolds number
LU
Rm =
η

LU
cf. Re =ν(advection versus diffusion)● Much easier to achieve on astrophysical scales

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Equations of astrophysical fluid dynamics

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● Lorentz force per unit volume
Fm = J × B
1=(∇ × B) × B
µ0
�
�
2
1
|B|
=
B · ∇B − ∇µ02µ0curvature force:magnetic tension
|B|2
Tm =
µ0
Fm = ∇ · M

gradient ofmagnetic pressure
|B|2
pm =
(= magnetic energy density)
2µ0

2

BB
|B|
M=
−
I
µ0
2µ0

Maxwell stress tensor
2011年12月19日月曜日

If B = B ez ,

−pm
0
−pm
M= 0
0
0



0

0
T m − pm

Equations of astrophysical fluid dynamics● Lorentz force:● Magnetic tension + frozen-in field → Alfvén waves
�
�1/2
Tm
va =
cf. elastic stringρ
va = (µ0 ρ)−1/2 B

vector Alfvén velocity● Magnetic pressure → magnetoacoustic waves

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Equations of astrophysical fluid dynamics● Ideal MHD equations:∂u1
1+ u · ∇u = −∇Φ − ∇p +(∇ × B) × B
∂t
ρ
µ0 ρ
∂ρ
+ u · ∇ρ = −ρ∇ · u∂t∂p+ u · ∇p = −γp∇ · u∂t∂B= ∇ × (u × B)∂t∇·B =0● Or can expand out ××● E and J eliminated● Nonlinearities in equation of motion and induction equation

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Equations of astrophysical fluid dynamics● Total energy equation in ideal MHD:
�
�
�
�
2
∂
|B|
E×B
2
2
1
1
ρ( 2 |u| + Φ + e) ++ ∇ · ρu( 2 |u| + Φ + w) +
=0
∂t
2µ0
µ0
specificinternal
energy● For ideal gas of constant γ :pe=(γ − 1)ρ
pγpw =e+ =
ρ(γ − 1)ρ

specific Poynting
enthalpy
flux
E = −u × B● With self-gravity, Φ = Φint + Φext and only 12 Φint + Φext
contributes to the energy density
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Equations of astrophysical fluid dynamics

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● Forces as the divergences of a stress tensor:● Equation of motion can be writtenDuρ= −ρ∇Φ + ∇ · T
Dt● Related to conservative form for momentum:∂(ρu) + ∇ · (ρuu − T) = −ρ∇Φ∂t

● Contributions to stress tensor T :● pressure −p I

BB
|B|2
−
I
● magneticµ0
2µ0● viscous 2µS + µb (∇ · u)I
gg
|g|2
+
I (check using Poisson’s equation)● self-gravity −4πG 8πG∇2 Φ = 4πGρg = −∇Φ● also turbulent stresses from correlations of fluctuating fields
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Reference1

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● AAVSO,
http://www.aavso.org/● G. Galileo, 1610: Sidereus Nuncius,● C. Huygens, 1656: De Saturni Luna observatio nova
● C. Huygens, 1659: Systema Saturnium, pp84
● HubbleSite-Out of the ordinary...out of this world,
http://hubblesite.org/● J. N. Cuzzi, J. A. Burns, S. Charnoz, R. N. Clark, J. E. Colwell,
L. Dones, L. W. Esposito, G. Filacchione, R. G. French, M. M. Hedman,
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M. Sremcevic, M. S. Tiscareno, J. Weiss. 2010: An Evolving View ofSaturn’ s Dynamic Ring, Science, 327, 1470 --1475
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Reference2● NASA-Home,
http://www.nasa.gov/● Rob Hynes,
http://www.phys.lsu.edu/~rih/● www.accretiondisk.org,
http://web.archive.org/web/2011202230603/http://
www.accretiondisk.org/● Perihelion precession.svg,
http://en.wikipedia.org/wiki/File:Perihelion_precession.svg

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FDEPS, Kyoto© GIO 2011