Accretion Disks As Probes of the Physics of Compact Objects

Home , Ring singularity

Accreon disks as probes of the physics of compact objects

Claus Lämmerzahl September 12, 2016

NewCompStar School 2017 Soﬁa, 11 - 15 September 2017 Inhalt

▶ Part I: General Introducon ▶ Part II: Nonrelavisc accreon disks ▶ Part III: Relavisc accreon disks

2/139 Part I: General Introducon Outline of Part I

Movaon - the exploraon of compact objects

Introducon - accreon

3/139 Part I: General Introducon Inhalt

▶ Part I: General Introducon ▶ Part II: Nonrelavisc accreon disks ▶ Part III: Relavisc accreon disks

4/139 Part II: Nonrelavisc accreon disks Outline of Part II

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

5/139 Part II: Nonrelavisc accreon disks Inhalt

▶ Part I: General Introducon ▶ Part II: Nonrelavisc accreon disks ▶ Part III: Relavisc accreon disks

6/139 Part III: Relavisc accreon disks Outline of Part III

Characterisc orbits General relavisc ﬂuid dynamics ▶ Congruences ▶ Fluids Thick disks ▶ Generalies ▶ Domain of existence ▶ The eﬀecve potenal ▶ Pressure Thin disks Slim disks Summary and outlook

7/139 Part III: Relavisc accreon disks Part I

General introducon

8/139 Inhalt

Movaon - the exploraon of compact objects

Introducon - accreon

9/139 gravity

orbits clocks

parcles lightpl ﬂuidspl

Movaon I

1 푅휇휈 − 2 푔휇휈푅 = 휅푇휇휈 Jürgen Ehlers 2006 For classifying soluons in general it is usual to focus primarily on properes of the metric and not on the maer variables (which may even be absent). But somemes it is of interest, not least since it is maer (including radiaon) that is observed, to characterize soluons in terms of the properes of maer.

10/139 gravity

orbits clocks

parcles lightpl ﬂuidspl

Movaon I

10/139 Movaon I

gravity

orbits clocks

parcles lightpl ﬂuidspl

10/139 Movaon II

Main quesons ▶ E. Ber et al.: Tesng general ▶ How to proof the existence of Black relavity with present and future Holes? astrophysical observaons (Topical ▶ What is a Black Hole? Review), Class. Quantum Grav. 32, Characteriscs of a Black Hole 243001 (2015) ▶ ▶ Event horizon L. Shao et al.: Advancing Astrophysics with the Square ▶ Singularity Kilometre Array, Proceedings of ▶ No hair / uniqueness Science, PoS(AASKA14)042 (2015) Black Hole foils ▶ Boson stars ▶ Planck stars ▶ gravastars ▶ Wormholes

11/139 How to explore Black Holes?

▶ parcle orbits ▶ point parcles — stars ▶ parcles with clocks — pulsars ▶ parcles with structure (spin, mass mulpoles) — rotang stars ▶ connua (gas, fluid, plasma - viscosity) — clouds, accreon disks, jets ▶ light effects ▶ light rays — deflecon, lensing, shadow ▶ polarized light ▶ waves ▶ merger, gravitaonal waves ▶ structure of merger ▶ structure of ring down

12/139 How to explore Black Holes?

12/139 no analyc soluon for gravitaonal ﬁeld of Neutron Stars are known → ﬁrst focus on Black Holes

How to explore Black Holes or Neutron Stars?

▶ parcle orbits ▶ point parcles — stars ▶ parcles with clocks — pulsars ▶ parcles with structure (spin, mass mulpoles) — rotang stars ▶ connua (gas, fluid, plasma - viscosity) — clouds, accreon disks, jets ▶ light effects ▶ light rays — deflecon, lensing ▶ polarized light ▶ waves ▶ merger, gravitaonal waves ▶ structure of merger ▶ oscillaons and their gravitaonal waves

13/139 How to explore Black Holes or Neutron Stars?

▶ parcle orbits ▶ point parcles — stars ▶ parcles with clocks — pulsars ▶ parcles with structure (spin, mass mulpoles) — rotang stars ▶ connua (gas, fluid, plasma - viscosity) — clouds, accreon disks, jets ▶ light effects ▶ light rays — deflecon, lensing ▶ polarized light ▶ waves ▶ merger, gravitaonal waves ▶ structure of merger ▶ oscillaons and their gravitaonal waves no analyc soluon for gravitaonal field of Neutron Stars are known → first focus on Black Holes

13/139 훽 훽

훽 deﬁcit angle

Plebański–Demiański Black Hole space-me staonary axially symmetric metric (Plebański & Demiański, AP 1976) 2 2 2 Δ푟 2 푝 2 Δ휗 2 2 푝 2 푑푠 = 2 (푑푡 − 퐴휗 푑휑) − 푑푟 − 2 sin 휗(푎푑푡 − 퐴푟 푑휑) − 푑휗 푝 Δ푟 푝 Δ휗 2 2 2 2 = 푔푡푡푑푡 + 푔푟푟푑푟 + 푔휗휗푑휗 + 푔휑휑푑휑 + 2푔푡휑푑푡푑휑 where 푝2 = 푟2 + (푛 − 푎 cos 휗)2 1 2 2 2 2 2 2 2 2 2 2 Δ푟 = (1 − 3 Λ푟 ) (푟 + 푎 ) − 2푀푟 − 푛 + 푄푒 + 푄푚 − Λ푛 (2푟 + 푎 − 푛 ) 1 2 2 4 Δ휗 = 1 + 3 푎 Λ cos 휗 − 3 Λ푎푛 cos 휗 2 퐴휗 = 푎 sin 휗 + 2푛(cos 휗 + 퐶) 2 2 2 퐴푟 = 푟 + 푎 + 푛 ▶ 푀 = mass, 푎 = Kerr parameter, Λ = cosmological constant, 푛 = NUT parameter, 푄푒 = electric charge, 푄푚 = magnec charge, ▶ this metric contains all standard black hole space–mes, Petrov Type D

14/139 Plebański–Demiański Black Hole space-me staonary axially symmetric metric (Plebański & Demiański, AP 1976) 2 2 2 Δ푟 2 푝 2 Δ휗 2 2 푝 2 푑푠 = 2 (푑푡 − 퐴휗훽푑휑) − 푑푟 − 2 sin 휗(푎푑푡 − 퐴푟훽푑휑) − 푑휗 푝 Δ푟 푝 Δ휗 2 2 2 2 = 푔푡푡푑푡 + 푔푟푟푑푟 + 푔휗휗푑휗 + 푔휑휑푑휑 + 2푔푡휑푑푡푑휑 where 푝2 = 푟2 + (푛 − 푎 cos 휗)2 1 2 2 2 2 2 2 2 2 2 2 Δ푟 = (1 − 3 Λ푟 ) (푟 + 푎 ) − 2푀푟 − 푛 + 푄푒 + 푄푚 − Λ푛 (2푟 + 푎 − 푛 ) 1 2 2 4 Δ휗 = 1 + 3 푎 Λ cos 휗 − 3 Λ푎푛 cos 휗 2 퐴휗 = 푎 sin 휗 + 2푛(cos 휗 + 퐶) 2 2 2 퐴푟 = 푟 + 푎 + 푛 ▶ 푀 = mass, 푎 = Kerr parameter, Λ = cosmological constant, 푛 = NUT parameter, 푄푒 = electric charge, 푄푚 = magnec charge, 훽 deﬁcit angle ▶ this metric contains all standard black hole space–mes, Petrov Type D

14/139 Structure of space-me metric singular for 푝 = 0, Δ푟 = 0, Δ휗 = 0 or sin 휗 = 0 ▶ 푝 = 0: ring singularity, curvature singularity (↔ 푟 = 0 in Schwarzschild) ▶ Δ푟 = 0: horizon, coordinate singularity (↔ Schwarzschild radius) ▶ Δ휗 = 0: for Λ ≠ 0 and 푛 ≠ 0 there is a horizon for a 휗 = 푐표푛푠푡. cone horizon. Is there a physical interpretaon? ▶ sin 휗 = 0: coordinate singularity along 푡-axis. For 푛 ≠ 0 this is a true singularity, which posion is related to the Manko-Riuz-parameter 퐶 ergoregion ▶ it is possible 푔푡푡 < 0: me-coordinate becomes space-like ⇒ observer cannot move along 푡. For 푎 ≠ 0 (and Λ = 0) the boundary 푔푡푡 = 0 lies outside the horizon ⇔ ergosphere, observer has to rotate with the Black Hole causality ▶ 푔휑휑 > 0 for 푎 ≠ 0 or 푛 ≠ 0. Then 휑-coordinate becomes melike. Then we have closed melike curves. hps://galileospendulum.org

15/139 Space-mes special cases of Plebański–Demiański Black Hole space-mes ▶ Schwarzschild ▶ Reissner-Nordström ▶ Kerr, Kerr-Newman ▶ Schwarzschild-de Sier, Kerr-de Sier space-mes with mulpoles ▶ Quevedo-Mashhoon ▶ Quevedo 푞 metric generalized Black Hole space-mes → Jua’s talk ▶ Gauss-Bonnet ▶ Axion-Dilaton ▶ Horava-Lifshitz ▶ ... wormholes neutron stars

16/139 Inhalt

Movaon - the exploraon of compact objects

Introducon - accreon

17/139 Accreon disk

18/139 Accreon disk; main features

▶ disk of some material ▶ spherical symmetric, radial accreon ▶ in general roughly axially symmetric ▶ mass transport in disks ▶ shape, inner orbits, thickness accreon disks ▶ jets are much closer to the event horizon than stars orbing SgrA*: accreon disks are a strong gravity probe ▶ accreon ▶ angular momentum transport ▶ energy dissipaon ▶ turbulence ▶ radiaon ▶ stability of disks ▶ oscillaons of disks also: accreon

19/139 First features material = fluid ▶ dust ▶ ideal fluid ▶ fluid with viscosity axial symmetry ▶ symmetry of space-me ▶ symmetry of maer distribuon restricon: circular orbits accreon, requires ▶ viscosity ▶ heat transfer and radiaon further issues ▶ self-gravitang disks (very complicated) ▶ not aligned disks (Bardeen-Peerson effect) ▶ observaon from radio to X-ray, spectrum and intensity to be calculated

20/139 Issues / Quesons models ▶ spin-fluid, ... ▶ parcles (ballisc model) ▶ granular media: Poynng-Robertson force, ▶ fluid Yarkovski force, ... ▶ plasma fundamental quesons ▶ spin fluid ▶ derivaon of the Navier-Stokes equaon (is ▶ granular medium an effecve equaon) equaons of moon ▶ ▶ parcles: geodesic equaon, with turbulence electromagnec force, spin forces ▶ viscosity (Mahison-Papapetrou-Dixon equaons) ▶ stability, causality ▶ fluid: Navier-Stokes equaon, plasma ▶ equaon structure formaon if equaon of moon is given, say Navier-Stokes equaon, then we want so solve this equaon ....

21/139 Method: from an analycal perspecve

22/139 Method: from a numerical perspecve

23/139 Method: from an analycal perspecve

we are more interested in the ▶ gravity / geometry aspects ▶ analyc approach ▶ needs analyc soluon for the gravitaonal ﬁeld ﬁrst task: accreon around Black Holes

24/139 Part II

Nonrelavisc accreon disks

25/139 Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

26/139 Connua 푡

푥1 only three types of relave moon 푥2 푗 푢푖 = 퐴푖푗훿푥 with 퐴푖푗 = 휕푖푢푗

퐴푖푗 can be decomposed into ▶ rotaon (preserves volume and internal orientaon, rigid) 푥01(푡2) 훿푥(푡2) ▶ 푥 (푡 ) expansion (changes volume and distances between 푡2 02 2 constuents) ▶ shear (changes distances between constuents, volume preserving) 푥 (푡 ) 01 1 훿푥(푡2) 푥 (푡 ) 푡1 02 1

rotaon shear expansion 27/139 Mass conservaon connuity equaon (휌 = mass density, ⃗푢= mean 3-velocity of ﬂuid) 휕휌 + ∇⃗ ⋅ (휌 ⃗푢)= 0 휕푡

28/139 Navier-Stokes equaon general equaon of moon for a ﬂuid 휕 ⃗푢 휌 ( + ( ⃗푢⋅ ∇)⃗ ⃗푢)= −∇푝⃗ + ∇휎⃗ − 휌∇푉⃗ 휕푡 휎 = viscosity tensor, proporonal to derivaves of the velocity, depends only on expansion and shear most general ansatz linear in derivaves of the velocity 휕 ⃗푢 휌 ( + ( ⃗푢⋅ ∇)⃗ ⃗푢)= −∇푝⃗ + 휇Δ ⃗푢+ (휇 + 휆∗)∇(⃗ ∇⃗ ⋅ ⃗푢)− 휌∇푉⃗ 휕푡 휇 = dynamical viscosity, shear viscosity, 휆∗ = bulk viscosity incompressible ﬂuid (compressibility plays no role in dust and plasma in astrophysics) 휕 ⃗푢 휌 ( + ( ⃗푢⋅ ∇)⃗ ⃗푢)= −∇푝⃗ + 휇Δ ⃗푢− 휌∇푉⃗ 휕푡

29/139 Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

30/139 Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

31/139 Basic equaons The model: thin disk

▶ all parcles move on circles task ▶ determine equaons of moon ▶ axial symetry, no 휑-dependence ▶ shape of thin disk ▶ no moon in 푧-direcon ▶ mass rate one takes cylinder coordinates ▶ disk evoluon

32/139 Basic equaons Basic equaons (lemmata) cylindrical coordinates 휕 1 휕 휕 ( ⃗푢⋅ ∇) ⃗푢= ((푢 ⃗푒 + 푢 ⃗푒 + 푢 ⃗푒 ) ⋅ ( ⃗푒 + ⃗푒 + ⃗푒 )) (푢 ⃗푒 + 푢 ⃗푒 + 푢 ⃗푒 ) 푟 푟 휑 휑 푧 푧 푟 휕푟 휑 푟 휕휑 푧 휕푧 푟 푟 휑 휑 푧 푧 휕푢 푢 휕푢 푢2 휕푢 휕푢 푢 푢 휕푢 휕푢 = (푢 푟 + 휑 푟 − 휑 + 푢 푟 ) ⃗푒 + (푢 휑 + 휑 푢 + 휑 휑 + 푢 휑 ) ⃗푒 푟 휕푟 푟 휕휑 푟 푧 휕푧 푟 푟 휕푟 푟 푟 푟 휕휑 푧 휕푧 휑 휕푢 푢 휕푢 휕푢 + (푢 푧 + 휑 푧 + 푢 푧 ) ⃗푒 푟 휕푟 푟 휕휑 푧 휕푧 푧 휕2 1 휕 1 휕2 휕2 ∆ ⃗푢= ( + + + ) (푢 ⃗푒 + 푢 ⃗푒 + 푢 ⃗푒 ) 휕푟2 푟 휕푟 푟2 휕휑2 휕푧2 푟 푟 휑 휑 푧 푧 휕2푢 1 휕푢 1 휕2푢 1 1 휕푢 휕2푢 휕2푢 1 휕푢 1 휕푢 = ( 푟 + 푟 + 푟 − 푢 − 2 휑 + 푟 ) ⃗푒 + ( 휑 + 휑 + 2 푟 휕푟2 푟 휕푟 푟2 휕휑2 푟2 푟 푟2 휕휑 휕푧2 푟 휕푟2 푟 휕푟 푟2 휕휑

1 휕2푢 1 휕2푢 휕2푢 1 휕푢 1 휕2푢 휕2푢 + 휑 − 푢 + 휑 ) ⃗푒 + ( 푧 + 푧 + 푧 + 푧 ) ⃗푒 푟2 휕휑2 푟2 휑 휕푧2 휑 휕푟2 푟 휕푟 푟2 휕휑2 휕푧2 푧 with 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 휕 ⃗푒 푟 = 0 , 푟 = ⃗푒 , 푟 = 0 , 휑 = 0 , 휑 = − ⃗푒 , 휑 = 0 , 푧 = 0 , 푧 = 0 , 푧 = 0 휕푟 휕휑 휑 휕푧 휕푟 휕휑 푟 휕푧 휕푟 휕휑 휕푧

33/139 Basic equaons Basic equaons (lemmata)

휕 1 휕 휕 ∇⃗ ⋅ 휎 = ( ⃗푒 + ⃗푒 + ⃗푒 ) ⋅ (휎 ⃗푒 ⊗ ⃗푒 + 휎 ⃗푒 ⊗ ⃗푒 + 휎 ⃗푒 ⊗ ⃗푒 + 휎 ⃗푒 ⊗ ⃗푒 + 휎 ⃗푒 ⊗ ⃗푒 푟 휕푟 휑 푟 휕휑 푧 휕푧 푟푟 푟 푟 푟휑 푟 휑 푟푧 푟 푧 휑푟 휑 푟 휑휑 휑 휑

+휎휑푧 휑⃗푒 ⊗푧 ⃗푒 + 휎푧푟 푧⃗푒 ⊗푟 ⃗푒 + 휎푧휑 푧⃗푒 ⊗휑 ⃗푒 + 휎푧푧 푧⃗푒 ⊗푧 ⃗푒 )

휕휎푟푟 휎푟푟 1 휕휎휑푟 휕휎푧푟 휎휑휑 1 휕휎휑휑 휕휎푟휑 휎푟휑 휎휑푟 휕휎푧휑 = ( + + + − )푟 ⃗푒 + ( + + + + )휑 ⃗푒 휕푟 푟 푟 휕휑 휕푧 푟 푟 휕휑 휕푟 푟 푟 휕푧 휕휎 휕휎 휎 1 휕휎 + ( 푧푧 + 푟푧 + 푟푧 + 휑푧 ) ⃗푒 휕푧 휕푟 푟 푟 휕휑 푧

34/139 Basic equaons Conservaon law conservaon law in cyindrical coordinates 휕휌 + ∇⃗ ⋅ (휌 ⃗푢)= 0 휕푡 in cylindrical coordinates

휕휌 1 휕 1 휕(휌푣 ) 휕(휌푢 ) 0 = + (푟휌푢 ) + 휑 + 푧 휕푡 푟 휕푟 푟 푟 휕휑 휕푧 symmetry and no 푧-moon 휕휌 1 휕 0 = + (푟휌푢 ) 휕푡 푟 휕푟 푟

35/139 Basic equaons Navier-Stokes equaon

Navier-Stokes in cylindrical coordinates

휕푢 휕푢 푢 휕푢 푢2 휕푢 휕푝 휌 ( 푟 + 푢 푟 + 휑 푟 − 휑 + 푢 푟 ) = − 휕푡 푟 휕푟 푟 휕휑 푟 푧 휕푧 휕푟 휕2푢 1 휕푢 1 휕2푢 1 1 휕푢 휕2푢 휕푉 +휇 ( 푟 + 푟 + 푟 − 푢 − 2 휑 + 푟 ) − 휌 휕푟2 푟 휕푟 푟2 휕휑2 푟2 푟 푟2 휕휑 휕푧2 휕푟 휕푢 휕푢 푢 푢 휕푢 휕푢 1 휕푝 휌 ( 휑 + 푢 휑 + 휑 푢 + 휑 휑 + 푢 휑 ) = − 휕푡 푟 휕푟 푟 푟 푟 휕휑 푧 휕푧 푟 휕휑 휕2푢 1 휕푢 1 휕푢 1 휕2푢 1 휕2푢 휌 휕푉 +휇 ( 휑 + 휑 + 2 푟 + 휑 − 푢 + 휑 ) − 휕푟2 푟 휕푟 푟2 휕휑 푟2 휕휑2 푟2 휑 휕푧2 푟 휕휑 휕푢 휕푢 푢 휕푢 휕푢 휕푝 휕2푢 1 휕푢 1 휕2푢 휕2푢 휕푉 휌 ( 푧 + 푢 푧 + 휑 푧 + 푢 푧 ) = − + 휇 ( 푧 + 푧 + 푧 + 푧 ) − 휌 휕푡 푟 휕푟 푟 휕휑 푧 휕푧 휕푧 휕푟2 푟 휕푟 푟2 휕휑2 휕푧2 휕푧

36/139 Basic equaons 휕푢 휕푢 푢2 휕푝 휕2푢 1 휕푢 1 휕푉 휌 ( 푟 + 푢 푟 − 휑 ) = − + 휇 ( 푟 + 푟 − 푢 ) − 휌 휕푡 푟 휕푟 푟 휕푟 휕푟2 푟 휕푟 푟2 푟 휕푟

휕푢 휕푢 푢 휕2푢 1 휕푢 1 휌 ( 휑 + 푢 휑 + 휑 푢 ) = 휇 ( 휑 + 휑 − 푢 ) 휕푡 푟 휕푟 푟 푟 휕푟2 푟 휕푟 푟2 휑 휕푝 휕푉 0 = − − 휌 휕푧 휕푧

centrifugal force ▶ thin gas ⇒ small viscosity 휇 ▶ small radial velocity

can neglect terms with products of 휇 and 푢푟

Navier-Stokes equaon

▶ axial symmetry: 휕휑 = 0 ▶ no moon in 푧-direcon: 푢푧 = 0

37/139 Basic equaons centrifugal force ▶ thin gas ⇒ small viscosity 휇 ▶ small radial velocity

can neglect terms with products of 휇 and 푢푟

Navier-Stokes equaon

▶ axial symmetry: 휕휑 = 0 ▶ no moon in 푧-direcon: 푢푧 = 0

휕푢 휕푢 푢2 휕푝 휕2푢 1 휕푢 1 휕푉 휌 ( 푟 + 푢 푟 − 휑 ) = − + 휇 ( 푟 + 푟 − 푢 ) − 휌 휕푡 푟 휕푟 푟 휕푟 휕푟2 푟 휕푟 푟2 푟 휕푟

휕푢 휕푢 푢 휕2푢 1 휕푢 1 휌 ( 휑 + 푢 휑 + 휑 푢 ) = 휇 ( 휑 + 휑 − 푢 ) 휕푡 푟 휕푟 푟 푟 휕푟2 푟 휕푟 푟2 휑 휕푝 휕푉 0 = − − 휌 휕푧 휕푧

37/139 Basic equaons ▶ thin gas ⇒ small viscosity 휇 ▶ small radial velocity

can neglect terms with products of 휇 and 푢푟

Navier-Stokes equaon

▶ axial symmetry: 휕휑 = 0 ▶ no moon in 푧-direcon: 푢푧 = 0

휕푢 휕푢 푢2 휕푝 휕2푢 1 휕푢 1 휕푉 휌 ( 푟 + 푢 푟 − 휑 ) = − + 휇 ( 푟 + 푟 − 푢 ) − 휌 휕푡 푟 휕푟 푟 휕푟 휕푟2 푟 휕푟 푟2 푟 휕푟

휕푢 휕푢 푢 휕2푢 1 휕푢 1 휌 ( 휑 + 푢 휑 + 휑 푢 ) = 휇 ( 휑 + 휑 − 푢 ) 휕푡 푟 휕푟 푟 푟 휕푟2 푟 휕푟 푟2 휑 휕푝 휕푉 0 = − − 휌 휕푧 휕푧 centrifugal force

37/139 Basic equaons Navier-Stokes equaon

▶ axial symmetry: 휕휑 = 0 ▶ no moon in 푧-direcon: 푢푧 = 0

휕푢 휕푢 푢2 휕푝 휕2푢 1 휕푢 1 휕푉 휌 ( 푟 + 푢 푟 − 휑 ) = − + 휇 ( 푟 + 푟 − 푢 ) − 휌 휕푡 푟 휕푟 푟 휕푟 휕푟2 푟 휕푟 푟2 푟 휕푟

휕푢 휕푢 푢 휕2푢 1 휕푢 1 휌 ( 휑 + 푢 휑 + 휑 푢 ) = 휇 ( 휑 + 휑 − 푢 ) 휕푡 푟 휕푟 푟 푟 휕푟2 푟 휕푟 푟2 휑 휕푝 휕푉 0 = − − 휌 휕푧 휕푧 centrifugal force ▶ thin gas ⇒ small viscosity 휇 ▶ small radial velocity can neglect terms with products of 휇 and 푢푟

37/139 Basic equaons Connuity equaon and Navier-Stokes equaon

휕휌 1 휕 0 = + (푟휌푢 ) 휕푡 푟 휕푟 푟 and

휕푢 휕푢 푢2 휕푝 휕푉 휌 ( 푟 + 푢 푟 − 휑 ) = − − 휕푡 푟 휕푟 푟 휕푟 휕푟

휕푢 휕푢 푢 휕2푢 1 휕푢 1 1 휕푉 휌 ( 휑 + 푢 휑 + 휑 푢 ) = 휇 ( 휑 + 휑 − 푢 ) − 휕푡 푟 휕푟 푟 푟 휕푟2 푟 휕푟 푟2 휑 푟 휕휑 휕푝 휕푉 0 = − − 휌 휕푧 휕푧

variables: 푢푟, 푢휑, 휌, 휇, 푝, (and 푉 with Newton ﬁeld equaon) ﬁnally required: equaon of state

38/139 Basic equaons New variables integraon of all equaons over 푧 ⇒ separaon of 푧-dependences 퐻 퐻 Σ(푡, 푟) = ∫ 휌(푡, 푟, 푧)푑푧 Π(푡, 푟) = ∫ 푝(푡, 푟, 푧)푑푧 −퐻 −퐻 separates the problem into (i) mathemacally think disk (휌, 휑, 푡) + (ii) height determinaon (푧; 휌, 휑) conservaon law gives 휕Σ 휕 퐻 퐻 휕휌 퐻 1 휕 1 휕 퐻 1 휕 = ∫ 휌푑푧 = ∫ 푑푧 = − ∫ (푟휌푢푟) 푑푧 = − (푟 ∫ 휌푑푧푢푟) = − (푟Σ푢푟) 휕푡 휕푡 −퐻 −퐻 휕푡 −퐻 푟 휕푟 푟 휕푟 −퐻 푟 휕푟 푟-component of Navier-Stokes equaon 퐻 2 퐻 퐻 휕푢푟 휕푢푟 푢휑 휕푝 휕푉 ∫ 휌 ( + 푢푟 − ) 푑푧 = − ∫ 푑푧 − ∫ 휌 푑푧 −퐻 휕푡 휕푟 푟 −퐻 휕푟 −퐻 휕푟 ⇒ 휕푢 휕푢 푢2 1 휕Π 휕푉 푟 + 푢 푟 − 휑 = − − 휕푡 푟 휕푟 푟 Σ 휕푟 휕푟

39/139 Basic equaons New variables

휑-component of Navier-Stokes equaon: use ℓ ∶= 푟푢휑 and 푢휑 = 푟Ω ⇒ 휕ℓ 휕ℓ 1 휕 휕Ω 휌 ( + 푢 ) = 휇 (푟3 ) 휕푡 푟 휕푟 푟 휕푟 휕푟 integraon over 푧 ⇒ 퐻 퐻 휕ℓ 휕ℓ 1 휕 3 휕Ω ∫ 휌 ( + 푢푟 ) 푑푧 = ∫ (휇푟 ) 푑푧 −퐻 휕푡 휕푟 −퐻 푟 휕푟 휕푟 dynamic viscosity 휇 depends on 푧, kinemac viscosity 휈 = 휇/휌 is constant ⇒ 휕ℓ 휕ℓ 휈 휕 휕Ω + 푢 = (Σ푟2 ) 휕푡 푟 휕푟 푟Σ 휕푟 휕푟 푧-component of Navier-Stokes equaon sll depends on 푧 1 휕푝 휕푉 = 휌 휕푧 휕푧 variables: Σ, Π, 푢푟, 푢휑, 휇 (and 푉 )

40/139 Basic equaons Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

41/139 The vercal structure of the disk Equaon of state

푧-component: analysis of pressure ↔ gravity, in 1D one needs an equaon of state 푝 = 푝(휌), assumpon polytropic

푛+1 푝 = 퐾휌 푛 mit 푛 > 0, 퐾 > 0 . 퐾 given by 휌 and 푝 at 푧 = 0 푛+1 − 푛 퐾 = 푝푐휌푐 with velocity of sound 푑푝 푛 + 1 1 푛 + 1 푝 푐2 = = 퐾휌 푛 = sound 푑휌 푛 푛 휌 we have 푛+1 1 1 − 푛 푝푐 − 푛 푛 2 − 푛 퐾 = 푝푐휌푐 = 휌푐 = 푐sound,푐휌푐 휌푐 푛 + 1 and for equaon of state 2 푛 푐sound,푐 푛+1 푛 푝 = 1 휌 . 푛 푛 + 1 휌푐

42/139 The vercal structure of the disk 1 푛 휕 2 휌 ⇒ 0 = (푛푐sound,푐 ( ) + 푉 ) , 휕푧 휌푐 ⇒ 1 푛 2 휌 푛푐sound,푐 ( ) + 푉 = 푐표푛푠푡 = 퐶 휌푐

퐶 can be related to 푉푐, then 푛 푉 − 푉푐 휌 = 휌푐 (1 − 2 ) . 푛푐sound,푐 consistent interpretaon: 휌 maximum at equatorial plane, falls oﬀ in 푧-direcon where 푉 becomes smaller one equaon for 휌 and 푉 , cannot be solved, needs further speciﬁcaon, ansatz

z-component of Navier Stokes

1 푛 1 휕푝 2 휕 휌 휕푉 = 푛푐sound,푐 ( ) = − 휌 휕푧 휕푧 휌푐 휕푧

43/139 The vercal structure of the disk ⇒ 1 푛 2 휌 푛푐sound,푐 ( ) + 푉 = 푐표푛푠푡 = 퐶 휌푐

z-component of Navier Stokes

1 1 푛 푛 1 휕푝 2 휕 휌 휕푉 휕 2 휌 = 푛푐sound,푐 ( ) = − ⇒ 0 = (푛푐sound,푐 ( ) + 푉 ) , 휌 휕푧 휕푧 휌푐 휕푧 휕푧 휌푐

43/139 The vercal structure of the disk 퐶 can be related to 푉푐, then 푛 푉 − 푉푐 휌 = 휌푐 (1 − 2 ) . 푛푐sound,푐 consistent interpretaon: 휌 maximum at equatorial plane, falls oﬀ in 푧-direcon where 푉 becomes smaller one equaon for 휌 and 푉 , cannot be solved, needs further speciﬁcaon, ansatz

z-component of Navier Stokes

1 1 푛 푛 1 휕푝 2 휕 휌 휕푉 휕 2 휌 = 푛푐sound,푐 ( ) = − ⇒ 0 = (푛푐sound,푐 ( ) + 푉 ) , 휌 휕푧 휕푧 휌푐 휕푧 휕푧 휌푐 ⇒ 1 푛 2 휌 푛푐sound,푐 ( ) + 푉 = 푐표푛푠푡 = 퐶 휌푐

43/139 The vercal structure of the disk one equaon for 휌 and 푉 , cannot be solved, needs further speciﬁcaon, ansatz

z-component of Navier Stokes

퐶 can be related to 푉푐, then 푛 푉 − 푉푐 휌 = 휌푐 (1 − 2 ) . 푛푐sound,푐 consistent interpretaon: 휌 maximum at equatorial plane, falls oﬀ in 푧-direcon where 푉 becomes smaller

43/139 The vercal structure of the disk z-component of Navier Stokes

43/139 The vercal structure of the disk Ansatz and soluon ansatz Taylor expansion 휕푉 1 휕2푉 1 휕2푉 푉 = 푉 + 푧 ∣ + 푧2 ∣ + 푂(푧3) = 푉 + 푧2 ∣ + 푂(푧3). 푐 휕푧 2 휕푧2 푐 2 휕푧2 푧=0 푧=0 푧=0 ⇒ 휕2푉 푛 ⎛ ∣ ⎞ ⎜ 휕푧2 ⎟ 2 푛 ⎜ 푧=0 ⎟ 1 푧 푐sound,푐 휌 = 휌 ⎜1 − 푧2⎟ = 휌 (1 − ( ) ) characterisc height ℎ = 푐 ⎜ 2 ⎟ 푐 2 ⎜ 2푛푐sound,푐 ⎟ 2푛 ℎ 휕 푉 √ ∣ ⎝ ⎠ 휕푧2 푧=0 with the true height 퐻 of the disk 푧 2 푛 √ 휌 = 휌 (1 − ( ) ) with 퐻 = 2푛ℎ . 푐 퐻

44/139 The vercal structure of the disk Disk form

Ρc 1.0

0.8

0.6

0.4 boundary at 푧 = 퐻 0.2 ▶ depends on 푛 z ▶ 푛 > 1 -3 -2 -1 0 1 2 3 smooth for ▶ linear growth for 푛 = 1 ▶ 1 ≤ 푛: large 푛 smoother boundary and thickness ▶ sharp boundary for 푛 < 1 increases ▶ for 푛 → ∞ we get a Gaussian curve ▶ dark blue line: 푛 = 1 ▶ 0 < 푛 < 1: smaller 푛 sharper boundary and thickness decreases 45/139 The vercal structure of the disk Further results one can integrate 퐻 √ Γ(푛 + 1) Σ = ∫ 휌푑푧 = 휌 퐻 휋 푐 3 −퐻 Γ(푛 + 2 ) wobei das 휌푐 wie auch das 퐻 i.a. von 푟 abhängen

퐻 √ (푛 + 1)Γ(푛 + 1) 푛 2 Π = ∫ 푝푑푧 = 푝푐퐻 휋 3 3 = 3 푐sound,푐Σ −퐻 (푛 + 2 )Γ(푛 + 2 ) 푛 + 2

46/139 The vercal structure of the disk Example 1: spherically symmetric central mass potenal 퐺푀 푉 = √ . 푀 푟2 + 푧2 ⇒ 2 2 2 휕 푉푀 3푧 1 휕 푉 퐺푀 2 2 = 5 − 3 ⇒ 2 ∣ = − 3 = ΩKepler 휕푧 2 2 2 2 2 2 휕푧 푟 (푟 + 푧 ) (푟 + 푧 ) 푧=0 ⇒ 푐 ℎ = sound,푐 ΩKepler

47/139 The vercal structure of the disk Example 2: inﬁnite self-gravitang disk

휕푉 Δ푉 = 푆 = 4휋퐺휌 푆 휕푧2 푐

(thereis only the 휌 in the equatorial plane, so that 휌 = 휌푐). Therefore 푐 ℎ = sound,푐 . √4휋퐺휌푐 translaonary invariant

48/139 The vercal structure of the disk Example 3: central mass with inﬁnite self-gravitang disk

Newton ⇒ superposion of potenals 푉 = 푉푀 + 푉퐷 ⇒ addion of second derivaves 휕2푉 휕2푉 휕2푉 ∣ = 푀 ∣ + 퐷 ∣ = Ω2 + 4휋퐺휌 , 휕푧2 휕푧2 휕푧2 Kepler 푐 푧=0 푧=0 푧=0 ⇒ thickness of disk 푐 푐 ℎ = sound,푐 = sound,푐 2 퐺푀 √ΩKepler + 4휋퐺휌푐 √ 푟3 + 4휋퐺휌푐

▶ small 푟: central mass dominant ▶ large 푟: disk dominant: constant thickness

49/139 The vercal structure of the disk Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

50/139 Slow accreon Slow acreon can eliminate Π from 푟 component of Navier-Stokes 2 푢2 3 휕 ln 푐2 푟 휕푢푟 푢푟 휕 ln 푢푟 휑 푟 휕푉 푛 2 휕 ln 푛 sound,푐 휕 ln Σ 2 + 2 = 2 − 2 − 3 ( 3 + + ) 푐sound,푐 휕푡 푐sound,푐 휕 ln 푟 푐sound,푐 푐sound,푐 휕푟 푛 + 2 푛 + 2 휕 ln 푟 휕 ln 푟 휕 ln 푟

푟 푟Ω 푢휑 thin disk ↔ 1 ≪ = = ↔ supersonic rotaon ⇒ ℎ 푐sound,푐 푐sound,푐 휕푢 휕푉 푟 푟 = 푢2 − 푟 휕푡 휑 휕푟 −1 slow accreon ↔ viscous me-scale ≫ dynamical me-scale = Ω = 푟/푢휑 ⇒

휕푢푟 푢푟 휏dynam 2 푟 ∼ 푟 = 푢푟푢휑 ≪ 푢휑 휕푡 휏viscous 휏viscous ⇒ 휕푉 푢2 = 푟 휑 휕푟 gravitaonal balance law

51/139 Slow accreon Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

52/139 Disk evoluon Mass accreon which mass determines the circular moon of a parcle at radius 푟? 퐺푀 푉 = 푉 + 푉 with 푉 = 푐 , 푉 = potental from disk mass tot 푀 퐷 푀 푟 퐷

푉퐷 can be split into (i) monopole term, (ii) mass within 푟, (iii) mass outside 푟 (ii) and (iii) complicated ⇒ neglect (ii) and (iii) ⇒ monopole approximaon 푟 ′ ′ ′ 푀(푟) = 푀푐 + 2휋 ∫ Σ(푟 )푑 푑푟 (1) 0 tricky calculaon 푢2 (푠) 1 1 퐺푀(푟) 2 1 푟 휑 1 − ℱ<(푘) 푟 ℱ>(푘) = 푢휑(푟) (1 − lim ( ) − ∫ ℋ(푟, 푘푟) 푑푘 + ∫ ℋ(푟, ) 푑푘) 푠→∞ 2 푘 푟 2 푠 푢휑(푟) 0 푘 0 푘 with 푠2Ω2(푠) 푑 ln Ω ℋ(푟, 푠) = 2 ( + 1) 푟2Ω2(푟) 푑 ln 푠

53/139 Disk evoluon Disk evoluon approximave soluon 휕푉 퐺푀(푟) 푟 = 푢2 =! 휕푟 휑 푟 diﬀerenaon with respect to 푟 휕(퐺푀(푟)) 휕 ln Ω = 푢2 (2 + 3) 휕푟 휑 휕 ln 푟 from the integral (1) 휕푀 = 2휋푟Σ (2) 휕푟 together ⇒ 휕 ln Ω 2휋퐺Σ = 푟Ω2 (2 + 3) (3) 휕 ln 푟 with this one can eliminate 휌푐

54/139 Disk evoluon Disk evoluon radial integraon of mass conservaon + mass monopole expression (1) ⇒ 휕푀 = −2휋푟푢 Σ 휕푡 푟 with (2) we eliminate Σ 휕푀 휕푀 + 푢 = 0 휕푡 푟 휕푟 휑-component of Navier-Stokes 휕ℓ 휕ℓ 휕Ω 휕 휕Ω 2휋푟Σ ( + 푢 ) = −푟푀 = (2휋Σ휈푟3 ) 휕푡 푟 휕푟 휕푡 휕푟 휕푟

55/139 Disk evoluon Disk evoluon finally (eliminang Σ using (3)) 휕 휕 휕 ln Ω 휕 ln Ω −푟4Ω2 = (휈푟3Ω3 (2 + 3)) 휕푡 휕푟 휕 ln 푟 휕 ln 푟 final disk evoluon equaon ▶ non-linear second order paral differenal equaon for Ω ▶ describes advecon and diffusion of angular momentum for viscous fluids and self-gravitang disks ▶ needs specificaon for kinemacal viscosity 휈

Illenseer & Duschl, MNRAS 2015

56/139 Disk evoluon Inhalt

Basic equaons

Thin disks ▶ Basic equaons ▶ The vercal structure of the disk ▶ Slow accreon ▶ Disk evoluon

Turbulence

57/139 Turbulence turbulent ﬂow is given by a mean part and a ﬂuctuang part ⃗푢= 푈⃗ + 훿 ⃗푢 with ⟨훿 ⃗푢⟩= 0 similar for 푝 and 휌 inseron into Navier-Stokes gives

휌휕푡(푈푖 + 훿푢푖) + 휌(푈푗 + 훿푢푗)휕푗(푈푖 + 훿푢푖) = −휕푖(푃 + 훿푝) + 휇Δ(푈푖 + 훿푢푖) − 휌휕푖푉 averaging: terms linear in 훿 ⃗푢 drop out, quadrac terms remain:

휌휕푡푢푖 + 휌푈푗휕푗푈푖 = −휕푖푃 + 휇Δ푈푖 − 휌휕푗⟨훿푢푗훿푢푖⟩ − 휌휕푖푉 푅 with Reynolds stress tensor 휎푖푗 ∶= 휌⟨훿푢푗훿푢푖⟩ ▶ Reynolds tensor undetermined, one needs a model ▶ one gets an equaon for the Reynold tensor through the third order Navier-Stokes equaon, but then a third order mean value remains undetermined – closure problem ▶ one way of approach: turbulence kinec theory ▶ problem remains unsolved ... people do numerical experiments

58/139 Part III

Relavisc disks

59/139 Inhalt

60/139 ⇒ two conservaon laws ̇ 퐸 ∶= 푔푡푡푡 + 푔푡휑 ̇휑 ̇ −퐿 ∶= 푔휑푡푡 + 푔휑휑 ̇휑 for Plebañski-Demiañski Δ Δ 퐸 = 푟 (푡̇ − 퐴 ̇휑)− 푎 휗 sin2 휗(푎푡̇ − 퐴 ̇휑) 푝2 휗 푝2 푟 Δ Δ 퐿 = 퐴 푟 (푡̇ − 퐴 ̇휑)− 퐴 휗 sin2 휗(푎푡̇ − 퐴 ̇휑), 휗 푝2 휗 푟 푝2 푟

Characterisc orbits

61/139 Characterisc orbits

▶ innermost stable circular orbit ▶ marginal stable orbit = boundary of set of all stable orbits ▶ marginal bound orbit = boundary of set of bound orbits geodec equaon 휈 휇 휇 휌 휎 0 = 퐷푢 ⇔ 0 = 푢 휕휈푢 + { 휌휎 } 푢 푢 staonarity and axial symmetry of space-me, Killing vectors 휉 = 휕푡 and 휂 = 휕휑 ⇒ two conservaon laws ̇ 퐸 ∶= 푔푡푡푡 + 푔푡휑 ̇휑 ̇ −퐿 ∶= 푔휑푡푡 + 푔휑휑 ̇휑 for Plebañski-Demiañski Δ Δ 퐸 = 푟 (푡̇ − 퐴 ̇휑)− 푎 휗 sin2 휗(푎푡̇ − 퐴 ̇휑) 푝2 휗 푝2 푟 Δ Δ 퐿 = 퐴 푟 (푡̇ − 퐴 ̇휑)− 퐴 휗 sin2 휗(푎푡̇ − 퐴 ̇휑), 휗 푝2 휗 푟 푝2 푟

61/139 ▶ inseron into Hamilton–Jacobi ▶ separaon of 푟 and 휗 equaons ▶ separaon constant = 푘 = Carter constant (Carter, PR 1968) ▶ introducon of Mino me 휏 through 푑휏 = 휌2푑푠 (Mino, PRD 2003) ▶ substuon 휉 = cos 휗 ▶ renormalizaon: all quanes in units of 푟푆

62/139 ▶ separaon of 푟 and 휗 equaons ▶ separaon constant = 푘 = Carter constant (Carter, PR 1968) ▶ introducon of Mino me 휏 through 푑휏 = 휌2푑푠 (Mino, PRD 2003) ▶ substuon 휉 = cos 휗 ▶ renormalizaon: all quanes in units of 푟푆

62/139 ▶ separaon constant = 푘 = Carter constant (Carter, PR 1968) ▶ introducon of Mino me 휏 through 푑휏 = 휌2푑푠 (Mino, PRD 2003) ▶ substuon 휉 = cos 휗 ▶ renormalizaon: all quanes in units of 푟푆

Soluon of geodesic equaon geodesic equaon 2 휇 휌 휎 푑 푥 휇 푑푥 푑푥 0 = + { } 푔 푢휇푢휈 = 휖 푑푠2 휌휎 푑푠 푑푠 휇휈 is equivalent to the Hamilton–Jacobi equaon 휕푆 휕푆 휕푆 2 = 푔휇휈 휕푠 휕푥휇 휕푥휇 separaon ansatz 1 푆 = 2 휖푠 − 퐸푡 + 퐿휑 + 푆푟(푟) + 푆휗(휗) ▶ inseron into Hamilton–Jacobi ▶ separaon of 푟 and 휗 equaons

62/139 ▶ introducon of Mino me 휏 through 푑휏 = 휌2푑푠 (Mino, PRD 2003) ▶ substuon 휉 = cos 휗 ▶ renormalizaon: all quanes in units of 푟푆

62/139 ▶ substuon 휉 = cos 휗 ▶ renormalizaon: all quanes in units of 푟푆

62/139 ▶ renormalizaon: all quanes in units of 푟푆

62/139 Soluon of geodesic equaon geodesic equaon 2 휇 휌 휎 푑 푥 휇 푑푥 푑푥 0 = + { } 푔 푢휇푢휈 = 휖 푑푠2 휌휎 푑푠 푑푠 휇휈 is equivalent to the Hamilton–Jacobi equaon 휕푆 휕푆 휕푆 2 = 푔휇휈 휕푠 휕푥휇 휕푥휇 separaon ansatz 1 푆 = 2 휖푠 − 퐸푡 + 퐿휑 + 푆푟(푟) + 푆휗(휗) ▶ inseron into Hamilton–Jacobi ▶ separaon of 푟 and 휗 equaons ▶ separaon constant = 푘 = Carter constant (Carter, PR 1968) ▶ introducon of Mino me 휏 through 푑휏 = 휌2푑푠 (Mino, PRD 2003) ▶ substuon 휉 = cos 휗 ▶ renormalizaon: all quanes in units of 푟푆

62/139 analyc soluon given by hyperellipc funcons 휎(푟)( ⃗푥) 휏 푟(휏) = ∓ 2 + 푟 with 휎(푟)( ⃗푥)= 0 , ⃗푥= ( 1) (푟) 0 휏 휎1 ( ⃗푥) 휎(휉)( ⃗푦) 휏 휉(휏) = ∓ 2 + 휉 with 휎(휉)( ⃗푦)=0, ⃗푥=( 1) (휉) 0 휏 휎1 ( ⃗푦)

Soluon of geodesic equaon

63/139 Soluon of geodesic equaon

푑푟 2 ( ) = ((푟2 + 푎2 + 푛2)퐸 − 푎퐿)2 − Δ (휖푟2 + 푘) =∶ 푅(푟) 푑휏 푟 푑휉 2 ( ) = Δ (1 − 휉2) (푘 − 휖(푛 − 푎휉)2) − (퐿 − 퐴 퐸)2 =∶ Θ(휉) 푑휏 휉 휉 2 2 2 푑휑 (푟 + 푎 + 푛 )퐸 − 푎퐿 퐿 − 퐴휉퐸 = 푎 + 2 =∶ 푓(푟) + 푔(휉) 푑휏 Δ푟 Δ휉(1 − 휉 ) 2 2 2 푑푡 (푟 + 푎 + 푛 )퐸 − 푎퐿 퐴휉 (퐿 − 퐴휉퐸) = 퐴푟 + 2 =∶ ℎ(푟) + 푗(휉) 푑휏 Δ푟 Δ휉(1 − 휉 ) analyc soluon given by hyperellipc funcons 휎(푟)( ⃗푥) 휏 푟(휏) = ∓ 2 + 푟 with 휎(푟)( ⃗푥)= 0 , ⃗푥= ( 1) (푟) 0 휏 휎1 ( ⃗푥) 휎(휉)( ⃗푦) 휏 휉(휏) = ∓ 2 + 휉 with 휎(휉)( ⃗푦)=0, ⃗푥=( 1) (휉) 0 휏 휎1 ( ⃗푦) 63/139 Soluon of geodesic equaon integraon of 휑 and 푡 moon

푟(휏) 푑푟 휉(휏) 푑휉 휑 − 휑 = ∫ 푓(푟)√ + ∫ 푔(휉) 0 푅 √Θ(휉) 푟0 휉0 푟(휏) 푑푟 휉(휏) 푑휉 푡 − 푡 = ∫ ℎ(푟)√ + ∫ 푗(휉) 0 푅 √Θ(휉) 푟0 휉0 푓, 푔, ℎ, and 푗 are raonal funcons (Hackmann, Kagramanova, Kunz, C.L., EPL 2009) soluons are characterized ▶ by zeros of 푅(푟): bound, escape, terminang orbit ▶ causal structure: spacelike and melike 푟- and 푡-coordinates

64/139 Characterisc orbits due to symmetry, one can represent the radial equaon of moon in terms of an eﬀecve potenal

1 푑푟 2 ( ) = −푉 (푟) = −푉 (퐸, 퐿; 푟) 2 푑푠 eﬀ eﬀ circular orbit characterized through

푑푟 푑2푟 = 0 and = 0 푑푠 푑푠2 in terms of the effecve potenal 푑푉 푉 (푟) = 0 and eff = 0 eff 푑푟

65/139 Characterisc circular orbits: stable orbits stable circular orbit 푑푉 (푟) 푑2푉 (푟) 푉 (푟) = 0 and eff = 0 and eff > 0 eff 푑푟 푑푟2

2 2 circular orbit is unstable if 푑 푉eﬀ(푟)/푑푟 < 0 푉eﬀ ⇒ boundary between stable and unstable orbits = boundary of the set of stable orbits = last stable orbit = ISCO = marginally stable orbit 푟 given by (recent general treatment Behesh & Gasperin, PRD 2016)

푑푉 (푟) 푑2푉 (푟) 푉 (푟) = 0 and eff = 0 and eff = 0 eff 푑푟 푑푟2 this are three equaon for 퐸, 퐿 and the radial coordinate 푟 in Schwarzschild-de Sier we also may have an OSCO

66/139 Characterisc circular orbits: bound orbits

▶ for asymptocally flat space-mes we have lim 푉 (푟) = 0 푟→∞ eff ▶ this corresponds to a certain energy 퐸 = 퐸0 bound circular orbits are characterized by 푑푉 (푟) 푉 (푟) = 0 and eff = 0 and 퐸 < 퐸 eff 푑푟 0 marginally bound circular orbits are characterized by 푑푉 (푟) 푉 (푟) = 0 and eff = 0 and 퐸 = 퐸 eff 푑푟 0

67/139 Rotaonal moon angular velocity 푑휑 푢휑 Ω = = 푑푡 푢푡 speciﬁc angular momentum or, in the case of geodesic moon, Keplerian angular momentum 푝 푢 푙 = 휑 = 휑 푝푡 푢푡 relaon 푔 + 푔 Ω 푔 푙 + 푔 푙 = − 휑푡 휑휑 ⇔ Ω = − 푡푡 푡휑 푔푡푡 + 푔푡휑Ω 푔푡휑푙 + 푔휑휑

68/139 Spherical symmetric space-me spherically symmetric metric 2 2 2 2 2 푑푠 = 푔푡푡푑푡 + 푔푟푟푑푟 + 푔휗휗푑휗 + 푔휑휑푑휑 2 conserved quanes 푑푡 푑휑 퐸 = 푔 , 퐿 = −푔 . 푡푡 푑푠 휑휑 푑푠 Keplerian angular momentum 휇 휑 푝휑 푔휑휇푢 푔휑휑푢 퐿 푙 = = 휈 = 푡 = − 푝푡 푔푡휈푢 푔푡푡푢 퐸 normalizaon (휖 = 1 for parcles and 휖 = 0 for light) 푑푡 2 푑푟 2 푑휗 2 푑휑 2 휖 = 푔 ( ) + 푔 ( ) + 푔 ( ) + 푔 ( ) 푡푡 푑푠 푟푟 푑푠 휗휗 푑푠 휑휑 푑푠 휋 spherical symmetry: can choose 휗 = 2 푑푡 푑휑 푑푡 2 푑푟 2 푑휑 2 퐸 = 푔 , 퐿 = −푔 , 휖 = 푔 ( ) + 푔 ( ) + 푔 ( ) 푡푡 푑푠 휑휑 푑푠 푡푡 푑푠 푟푟 푑푠 휑휑 푑푠

69/139 Spherical symmetric space-me eliminaon of 푡 and 휑

1 푑푟 2 퐸2 1 퐿2 ( ) = − + (휖 − ) =∶ −푉 (푟) 2 푑푠 2푔푡푡푔푟푟 2푔푟푟 푔휑휑 condions

2 2 퐿 푉 (푟) = 0 ⇒ 퐸 − 푔푡푡 (휖 − ) = 0 푔휑휑 푑푉 (푟) 푑 퐸2 1 퐿2 = 0 ⇒ (− + (휖 − )) = 0 푑푟 푑푟 푔푡푡푔푟푟 푔푟푟 푔휑휑 푑2푉 (푟) 푑2 퐸2 1 퐿2 2 = 0 ⇒ 2 (− + (휖 − )) = 0 푑푟 푑푟 푔푡푡푔푟푟 푔푟푟 푔휑휑

70/139 Schwarzschild condions (for 휖 = 1) 퐸2 − 1 푀 퐿2 퐿2푀 푉 (푟) = − − + − = 0 2 푟 2푟2 푟3 푑푉 (푟) 푀 퐿2 퐿2푀 = − + 3 = 0 푑푟 푟2 푟3 푟4 푑2푉 (푟) 푀 퐿2 퐿2푀 = −2 + 3 − 12 = 0 푑푟2 푟3 푟4 푟5 circular orbits (ﬁrst two equaons)

푀푟2 푟→∞ (푟 − 2푀)2 푟→∞ 퐿2 = ⟶ 푀푟 , 퐸2 = ⟶ 1 ⇒ 푟 ≥ 3푀 푟 − 3푀 푟(푟 − 3푀) marginal stable orbit 8 푟 = 6푀 = 3푟 , 퐿2 = 12 푀 2 , 퐸2 = ms S ms ms 9

71/139 Schwarzschild marginal bound orbit are given by 퐸2 = 1

2 2 2 푟mb = 4푀 = 2푟S , 퐿mb = 16 푀 , 퐸mb = 1

L2 12 L2 16 V V

x x 10 20 30 40 50 10 20 30 40 50

-0.05 -0.05

-0.10 -0.10

-0.15 -0.15

-0.20 -0.20

marginally stable orbit marginally bound orbit

72/139 Reissner-Nordström horizons 2 2 푟± = 푀 ± √푀 − 푄 condions 퐸2 − 1 푀 퐿2 + 푄2 푀퐿2 퐿2푄2 0 = − − + − + 2 푟 2푟2 푟3 2푟4 푀 퐿2 + 푄2 푀퐿2 퐿2푄2 0 = − + 3 − 2 푟2 푟3 푟4 푟5 푀 퐿2 + 푄2 푀퐿2 퐿2푄2 0 = −2 + 3 − 12 + 10 푟3 푟4 푟5 푟6 circular orbits (the ﬁrst two equaons)

푟2(푀푟 − 푄2) 푟→∞ (푄2 − 2푀푟 + 푟2)2 푟→∞ 퐿2 = ⟶ 푀푟 , 퐸2 = ⟶ 1 푟2 − 3푀푟 + 2푄2 푟2(푟2 − 3푀푟 + 2푄2)

73/139 Reissner-Nordström 푟 푀 posivity of 퐸2 and 퐿2 requires 3 푟2 − 3푀푟 + 2푄2 ≥ 0 and 푀푟 − 푄2 ≥ 0 2 3 1 √ 2 ﬁrst condion 2 ± 2 9 − 8(푄/푀) 2 1 2 1 ± √1 − (푄/푀) 3 2 1 2 2 (푄/푀) (푟 − 2 푀) ≥ 4 (9푀 − 8푄 ) right hand side automacally > 0 ⇒ 0 1 푄 푀 푟 3 1 푄2 ≥ + √9 − 8 forbidden domains are 푀 2 2 푀 2 ▶ below the blue parabola (second condion), 푟 3 1 푄2 ≤ − √9 − 8 ▶ inside the red parabolas (between the 푀 2 2 푀 2 horizons) ▶ inside the green parabola segments (ﬁrst condion)

74/139 Reissner-Nordström

L2 = 0, 10, 20, 30, 40, Q = 0.75 V radial coordinate of marginal stable bound orbits 2.0 given by cubic equaon 1.5

4 푄 1.0 0 = 푟3 − 6푀푟2 + 9푄2푟 − 4 푀 0.5 ⇔ r 5 10 15 20 25 30 푟 3 푟 2 푄 2 푟 푄 4 0 = ( ) −6 ( ) +9 ( ) −4 ( ) -0.5 푀 푀 푀 푀 푀

-1.0

75/139 Kerr Kerr metrics in Boyer-Lindquist coordinates 2푀푟 2푀푟푎 sin2 휗 푔 = 1 − 푔 = 푡푡 푟2 + 푎2 cos2 휗 푡휑 푟2 + 푎2 cos2 휗 푟2 + 푎2 cos2 휗 푔 = − 푔 = − (푟2 + 푎2 cos2 휗) 푟푟 푟2 + 푎2 − 2푀푟 휗휗 2푀푟푎2 푔 = − (푟2 + 푎2 + sin2 휗) sin2 휗 휑휑 푟2 + 푎2 cos2 휗 two Killing vectors and two constants of moon 푑푡 푑휑 푑푡 푑휑 퐸 = 푝 = 푔 + 푔 , −퐿 = 푝 = 푔 + 푔 푡 푡푡 푑푠 푡휑 푑푠 휑 휑푡 푑푠 휑휑 푑푠 ⇒ 푑푡 푔휑휑퐸 + 푔푡휑퐿 푑휑 푔푡휑퐸 + 푔푡푡퐿 = = − with Δ = 푔 푔 − 푔2 푑푠 Δ 푑푠 Δ 푡푡 휑휑 푡휑

76/139 Kerr normalizaon (in equatorial plane) 푑푡 2 푑푟 2 푑휑 2 푑푡 푑휑 1 = 푔 ( ) + 푔 ( ) + 푔 ( ) + 2푔 . 푡푡 푑푠 푟푟 푑푠 휑휑 푑푠 푡휑 푑푠 푑푠 radial equaon of moon

2 푑푟 1 1 2 2 ( ) = (1 − 2 (푔푡푡 (푔휑휑퐸 + 푔푡휑퐿) + 푔휑휑 (푔푡휑퐸 + 푔푡푡퐿) 푑푠 푔푟푟 Δ

− 2푔푡휑 (푔휑휑퐸 + 푔푡휑퐿) (푔푡휑퐸 + 푔푡푡퐿))) eﬀecve potenal (equatorial moon) 퐸2 − 1 푀 퐿2 + 푎2(1 + 퐸2) 퐿2 − 퐸2푎2 − 2퐸퐿푎 푉 (푟) = − − + − 푀 eﬀ 2 푟 2푟2 푟3

77/139 Kerr condions 퐸2 − 1 푀 퐿2 + 푎2(1 + 퐸2) 푀(퐿2 − 퐸2푎2 − 2퐸퐿푎) 0 = 푉 (푟) = − − + − 2 푟 2푟2 푟3 푑푉 (푟) 푀 퐿2 + 푎2(1 + 퐸2) 푀(퐿2 − 퐸2푎2 − 2퐸퐿푎) 0 = = − + 3 푑푟 푟2 푟3 푟4 푑2푉 (푟) 푀 퐿2 + 푎2(1 + 퐸2) 푀(퐿2 − 퐸2푎2 − 2퐸퐿푎) 0 = = −2 + 3 − 12 푑푟2 푟3 푟4 푟5 equatorial circular orbits (solving ﬁrst two equaons for 퐸 and 퐿) √ √ √ √ √ (푟 − 2푀) 푟 − 푎 푀 푀(푎2 + 푟2 + 2푎 푀 푟) √ 퐿 = ± √ √ 퐸 = √ √푟2(푟 − 3푀) − 2푎 푀푟 푟 푎 푀 − (푟 − 2푀) 푟 for 푟ms we get a forth order equaon ...

78/139 Kerr marginal bound orbit 1 푎 푎 푟 = 2푀 (1 − + √1 − ) mb 2 푀 푀 marginal stable orbit √ 푎2 √ 푎2 푟 = 푀 ⎜⎛3 + √3 + 푍2 − √(3 − 푍) (3 + 푍 + 2√3 + 푍2)⎟⎞ ms 푀 2 푀 2 ⎝ ⎷ ⎠ with 2 2 2 3 푎 3 푎 3 푎 푍 = 1 + √1 − (√1 + + √1 − ) 푀 2 푀 2 푀 2

(from Abramowicz & Fragile 2013)

79/139 푞-metric metric of a stac axially symmetric quadrupole −푞(2+푞) 1+푞 푀2 sin2 휗 2푀 (1 + 2 ) 푑푠2 = (1 − ) 푑푡2 − 푟 −2푀푟 푑푟2 푟 2푀 1+푞 (1 − 푟 ) −푞(2+푞) (1 + 푀2 sin2 휗 ) − 푟2−2푀푟 푟2푑휗2 − 1 푟2 sin2 휗푑휑2 . 2푀 푞 2푀 푞 (1 − 푟 ) (1 − 푟 ) conserved quanes 2푀 1+푞 푑푡 2 푑휑 퐸 = (1 − ) , 퐿 = 푟 2푀 푞 푟 푑푠 (1 − 푟 ) 푑푠 eﬀecve potenal (in equatorial plane, depends on 퐸) 푀 2 푞(2+푞) 퐸2 1 2푀 1+푞 2푀 푞 퐿2 푉 (푟) = (1 + ) (− + (1 − ) (휖 + (1 − ) )) eﬀ 푟2 − 2푀푟 2 2 푟 푟 푟2 looks like slightly deformed Schwarzschild

80/139 푞-metric for large 푟 1 − 퐸2 푀 퐿2 + 푞푀 2(4 + 3푞 − (2 + 푞)퐸2) 푉 (푟) = − (1 + 푞) + 2 푟 2푟2 (1 + 2푞)푀퐿2 + 푞푀 3 (− 2 + 2푞 + 5 푞2 + (2 + 푞)퐸2) − 3 3 + 풪(푟−4). 푟3 푑푉 푉 = 0 and eff = 0 can be solved eff 푑푟 2푀 푞 (푟 − 2푀)(푟 − (2 + 푞)푀) 퐸2 = (1 − ) 푟 푟(푟 − (3 + 2푞)푀) 2푀 −푞 (1 + 푞)푀푟2 퐿2 = (1 − ) 푟 푟 − (3 + 2푞)푀 2 2 inseron into 푑 푉eff/푑푟 = 0 gives marginal stable orbits 2 푟1,2 = 푀 (4 + 3푞 ± √4 + 10푞 + 5푞 ) 푟3 = (3 + 2푞)푀 no analyc soluon for marginal bound orbit 81/139 Inhalt

82/139 Inhalt

83/139 Congruences Congruences

Deﬁnion A congruence is a set of parcle 훿푥 푠2 trajectories 푠2 ▶ which do not intersect and 푢 which ﬁll all space-me 푢 ▶ 푢 and 훿푥 give closed 푠 푠 1 parallelograms [훿푥, 푢] = 0 1 훿푥

84/139 Congruences Congruences

Deﬁnions 휇 휇 ▶ 휇 projecon operator 푃푢 = 1 − 푢 ⊗ 푢, (푃푢)휈 = 훿휈 − 푢 푢휈 ▶ 휇 covariant derivave along 푢: 퐷푢 = 푢 퐷휇 Measurable quanes

▶ Relave distance 푟 = 푃푢훿푥 between two neighboring worldlines ▶ Relave velocity 푣 = 푃푢퐷푢푟 ▶ Relave acceleraon 푏 = 푃푢퐷푢푣 geodesic deviaon 휇 휇 휈 휌 휎 푏 = 푅 휈휌휎푢 푟 푢 one can relate 푣 to 푟 휇 휇 휈 휌 휎 푣 = 퐴 휈푟 with 퐴휇휈 = (푃푢)휇(푃푢)휈 퐷휌푢휎

85/139 Congruences Congruences

Kinemac descripon of a congruence

▶ rotaon 휔휇휈 ∶= 퐴[휇휈] ▶ 휇휈 휇 expansion 휃 ∶= 푔 퐴휇휈 = 퐷휇푢 ▶ 1 shear 휎휇휈 ∶= 퐴(휇휈) − 3 휃 complete analogy to 3-dimensional non-relavisc descripon of ﬂuids

86/139 Congruences Inhalt

87/139 Fluids Approaches

Approaches ▶ kinec theory ▶ energy momentum tensor Problems ▶ Coupling of viscosity to gravity

88/139 Fluids Energy momentum tensor

▶ 휈 maer can be described by an energy momentum tensor 푇휇 ▶ decomposion with respect to a hypersurface with normal 푛휇

휈 푇휇 푛휈 = 푝휇 = 4-momentum density

▶ decomposion with respect to a vector 푢휇

휇 휈 휈 푢 푇휇 = 푒 = energy 4-ﬂux

▶ 휇 휇휈 together with 푢 = 푔 푛휈

0 푖 휈 푇0 푇0 energy density energy ﬂux 푇휇 = ( 0 푖) = ( ) 푇푗 푇푗 momentum density momentum ﬂux = stress

▶ usually, 푇휇휈 is taken to be symmetric

89/139 Fluids Energy momentum tensor detailed decomposion with respect to a congruence

휈 휈 휈 휈 휈 휈 푇휇 = 휇푢휇푢 − 푝(푃푢)휇 + 푞휇푢 + 푞 푢휇 + 휋휇 휈 휈 휈 휈 휈 = (휇 + 푝)푢휇푢 − 푝훿휇 + 푞휇푢 + 푞 푢휇 + 휋휇 with ▶ 휈 휇 휇 ∶= 푇휇 푢 푢휈 rest energy density 휇 ▶ 휈 푝 ∶= (푃푢)휈 푇휇 pressure ▶ 휈 휈 휌 휇 푞 ∶= (푃푢)휌푇휇 푢 heat ﬂow 휌 ▶ 휈 휈 휎 1 휈 휎 휌 휋휇 ∶= (푃푢)휇(푃푢)휎푇휌 − 3 (푃푢)휇(푃푢)휌 푇휎 stress tensor

90/139 Fluids What is 푢? Eckart frame 푢휇 ∼ 퐽 휇 = maer current 퐽 휇 푢휇 = 푢휇 = . E √푔(퐽, 퐽) is used in low energy physics

Landau-Lifshitz frame 푢 ↔ energy current 푇 휇 푢휈 푢휇 = 푢휇 = 휈 . LL √푔(푇 (⋅, 푢), 푇 (⋅, 푢)) is used in high energy ion physics accreon disks: Eckart frame neutron stars: Landau-Lifshitz frame

91/139 Fluids Ideal ﬂuid energy momentum tensor for an ideal ﬂuid

휇 휇 휈 푇 휈 = (휇 + 푝)푢 푢휈 − 푝훿휇 equaon of moon 휈 퐷휈푇휇 = 0 projecon onto 푢휈: energy 퐷푢휇 = −(휇 + 푝)휃 projecon with 푃푢: Euler equaon

휇 휇휈 (휇 + 푝)퐷푢푢 = −(푃푢) 휕휈푝 no geodesic moon: pressure gradient gives force (dust: 푝 = 0 geodesics)

4 equaons for 푢, 휇, and 푝 ⇒ requires an equaon of state 푝 = 푝(휇)

92/139 Fluids Kinec theory

휇 is based on a distribuon funcon 푓(푥 , 푝휇) with properes ▶ number of parcles in a cell in phase space 푑푁 = 푓(푥, 푝)푑3푥푑3푝 ▶ normalizaon ∫ 푓(푥, 푝)푑3푥푑3푝 = 1

▶ ′ 휈 relavisc invariant: for 푝휇 → 푝휇 = 퐿휇 푝휈 we have

푑3푝 푑3푝′ = ′ 푝0 푝0

▶ 푓(푥, 푝) is a Lorentz scalar (that means that 푑3푥푑3푝 is a Lorentz scalar) moments 푑3푝 푀 휇휈1…휈푛 ∶= ∫ 푝휇휒휈1…휈푛 (푥, 푝)푓(푥, 푝) 푝0

93/139 Fluids Kinec theory: Moments

▶ 휒 = 1: number density 4-current 푑3푝 푁 휇(푥) = ∫ 푝휇푓(푥, 푝) 푝0 with number density and number density current 푝푖 푁 0 = ∫ 푓푑3푝 , 푁 푖 = ∫ 푓푑3푝 = ∫ 푣푖푓푑3푝 퐸 and rest mass density current 퐽 휇 = 푚푁 휇 ▶ 휒휇 = 푝휈: energy momentum tensor 푑3푝 푇 휇휈(푥) = ∫ 푝휇푝휈푓(푥, 푝) 푝0 ▶ 휒휈1휈2 = 푝휈1 푝휈2 : third moment 푑3푝 푆휇휈1휈2 (푥) = ∫ 푝휇푝휈1 푝휈2 푓(푥, 푝) 푝0

94/139 Fluids Equaons of moon total me derivave of distribuon funcon

휇 푑푓(푥, 푝) 푑푥 휕푓(푥, 푝) 푑푝휇 = 휕휇푓(푥, 푝) + = Π (= collision term) 푑푡 푑푡 휕푝휇 푑푡 for no collisions (푓휇 is an external force acng on the parcles)

휕푓 푑3푎 휈1…휈푛 휇 0 = ∫ 휒 (푝 휕휇푓 − 푚푓휇 ) 휕푝휇 푝0 paral integraon + boundary condions

푑3푎 휕휒휈1…휈푛 푑3푎 휈1…휈푛 휇 휇 휈1…휈푛 0 = 휕휇 ∫ 휒 푝 푓 − ∫ (푝 휕휇휒 + 푚푓휇 ) 푓 푝0 휕푝휇 푝0

95/139 Fluids Equaons of moon

▶ 휒 = 1: mass current conservaon 휇 휕휇퐽 = 0 ▶ 휒휇 = 푝휇: energy momentum conservaon

3 휇휈 휇 푑 푝 휕휇푇 = 푚 ∫ 푓 푓 푝0

휇휈 is there is no force: 휕휇푇 = 0

96/139 Fluids naive adding of a viscosity part to the energy momentum tensor violates causality: perturbaon of equilibrium and rest 휇 휇 휇 휇 = 휇0 + 훿휇, 푢 = 푢0 + 훿푢 in Navier Stokes to ﬁrst order in perturbaon, including viscosity

Viscosity energy momentum tensor with viscosity

휈 휈 휈 휈 휈 푇휇 = (휇 + 푝)푢휇푢 − 푝훿휇 + 휋휇 + (푃푢)휇Π viscosity ∗ ∗ 휋휇휈 = 휇휎휇휈 , Π = 휆 휃 , 휇, 휆 ≥ 0 Navier Stokes equaon

휈 휈 휌 휈휌 (휇 + 푝 + Π)푢 퐷휈푢휇 = −(푃푢) 휇휕휈(푝 + Π) + (푃푢)휇퐷휈휋

97/139 Fluids Viscosity energy momentum tensor with viscosity

휈 휈 휈 휈 휈 푇휇 = (휇 + 푝)푢휇푢 − 푝훿휇 + 휋휇 + (푃푢)휇Π viscosity ∗ ∗ 휋휇휈 = 휇휎휇휈 , Π = 휆 휃 , 휇, 휆 ≥ 0 Navier Stokes equaon

휈 휈 휌 휈휌 (휇 + 푝 + Π)푢 퐷휈푢휇 = −(푃푢) 휇휕휈(푝 + Π) + (푃푢)휇퐷휈휋 naive adding of a viscosity part to the energy momentum tensor violates causality: perturbaon of equilibrium and rest 휇 휇 휇 휇 = 휇0 + 훿휇, 푢 = 푢0 + 훿푢 in Navier Stokes to ﬁrst order in perturbaon, including viscosity

97/139 Fluids Viscosity

Navier Stokes for perturbaons (to first order) 휈 푖 푖푗 푖 휌휈 푖 (휇 + 푝 + Π)푢 퐷휈푢 + (푃푢) 휕푗(푝 + Π) − (푃푢)휈퐷휌Π = (휇0 + 푝0)휕푡훿푢 휇 Π = (퐷 푢 + 퐷 푢 ) + (휆∗ − 2 휇) 훿 휃 = −휇 휕 훿푢 푖푗 2 푖 푗 푗 푖 3 푖푗 0 푖 푗 this yields 푖 휇0 푖 휕푡훿푢 = Δ훿푢 휇0 + 푝0 diffusion equaon → instantaneous propagaon soluon: adding further terms (Müller 1967, Israel 1976, Romatschke, 2009) 휇 4 휅 휋 = (퐷 푢 + 퐷 푢 ) − 휏 ((푃 )휌 (푃 )휎푢휏퐷 휋 + 휋 휃) + (푅 + 2푅 푢휌푢휎) 휇휈 2 휇 휈 휈 휇 휋 푢 휇 푢 휈 휏 휌휎 3 휇휈 2 휇휈 휌(휇휈)휎 휆 휆 휆 − 1 휋 휋 휌 − 2 휋 휔 휌 − 3 휔 휔 휌 2휂2 휌(휇 휈) 2휂 휌(휇 휈) 2 휌(휇 휈) with transport coefficients 휏휋, 휅, 휆푖

98/139 Fluids Inhalt

99/139 Inhalt

100/139 Generalies The symmetries general model ▶ staonary axial symmetric space-me ▶ ﬂuid shares the same symmetries ▶ ﬂuid parcle move on circular orbits - no internal helical moon, for example Symmetries

▶ two Killing vectors 휉 = 휕푡 and 휂 = 휕휑 (me-like 휉 and rotaon 휂) ▶ 퐿휉푔휇휈 = 0 and 퐿휂푔휇휈 = 0 ▶ 퐿휉,휂푢 = 0, 퐿휉,휂휇 = 0 ▶ 휇 휈 휇 휈 휇 휈 in adapted coordinates 푔푡푡 = 푔휇휈휉 휉 , 푔휑휑 = 푔휇휈휂 휂 , 푔푡휑 = 푔휇휈휉 휂

ﬂuid parcles move on circular orbits 푢 = 퐴(휉 + Ω휂) normalizaon 2 1 퐴 = 2 푔푡푡 + 2Ω푔푡휑 + Ω 푔휑휑 we101/139 alsoGeneralies have from the corresponding deﬁnion 푔 + 푔 Ω 푔 푙 + 푔 푙 = − 휑푡 휑휑 ⇔ Ω = − 푡푡 푡휑 푔푡푡 + 푔푡휑Ω 푔푡휑푙 + 푔휑휑 Equaons of moon calculate 휈 휈 푎휇 = 푢 퐷휈푢휇 = 푢 퐷휈 (퐴(휉 + Ω휂)) some calculaon 푙 푎 = 휕 ln 푢푡 − 휕 Ω 휇 휇 1 − Ω푙 휇 can be reformulated as derivave on 푙 Ω 푎 = −휕 ln 푢 + 휕 푙 휇 휇 푡 1 − Ω푙 휇 equaon of moon Ω 1 −휕 ln 푢 + 휕 푙 = − 휕 푝 휇 푡 1 − Ω푙 휇 휇 + 푝 휇

102/139 Generalies von Zeipel theorem one further derivave of the equaon of moon, and ansymmetrizaon Ω 1 퐷 ( ) 휕 푙 = 휕 휇휕 푝 [휈 1 − Ω푙 휇] (휇 + 푝)2 [휈 휇] if there is an equaon of state, then 휕휇휇 ∼ 휕휇푝 and

Ω 퐷 ( ) 휕 푙 = 0 [휈 1 − Ω푙 휇] this is only possible provided Ω 퐷 ( ) ∼ 휕 푙 휈 1 − Ω푙 휈 this yields the von Zeipel theorem

Ω = Ω(푙)

103/139 Generalies von Zeipel cylinder from the Ω − 푙-relaon we get

푔푡푡푙 + 푔푡휑(1 + Ω푙) + 푔휑휑Ω = 0 we assign an 푙0 at a certain posion (푟0, 휗0) (what also implies a certain Ω0) and determine that surface on which 푙 = 푙0; inserng the 푙0 and Ω0 yields

2 0 = (푔푡푡(푟, 휗)푔푡휑(푟0, 휗0) − 푔푡휑(푟, 휗)푔푡푡(푟0, 휗0)) 푙0

+ (푔푡푡(푟, 휗)푔휑휑(푟0, 휗0) − 푔휑휑(푟, 휗)푔푡푡(푟0, 휗0)) 푙0

+ 푔푡휑(푟, 휗)푔휑휑(푟0, 휗0) − 푔휑휑(푟, 휗)푔푡휑(푟0, 휗0) this is a curve in (푟, 휗) and an axially symmetric surface in (푟, 휗, 휑) for spherically symmetric space-mes

0 = 푔푡푡(푟, 휗)푔휑휑(푟0, 휗0) − 푔휑휑(푟, 휗)푔푡푡(푟0, 휗0) this is pure geometry, no 푙0 appears here (see also Jefremov & Perlick CQG 2016)) 104/139 Generalies von Zeipel cylinder from the Ω − 푙-relaon we get

2 0 = (푔푡푡(푟, 휗)푔푡휑(푟0, 휗0) − 푔푡휑(푟, 휗)푔푡푡(푟0, 휗0)) 푙0

+ (푔푡푡(푟, 휗)푔휑휑(푟0, 휗0) − 푔휑휑(푟, 휗)푔푡푡(푟0, 휗0)) 푙0

0 = 푔푡푡(푟, 휗)푔휑휑(푟0, 휗0) − 푔휑휑(푟, 휗)푔푡푡(푟0, 휗0) this is pure geometry, no 푙0 appears here (see also Jefremov & Perlick CQG 2016)) 104/139 Generalies Examples

휋 we choose 휗0 = 2 ▶ Schwarzschild 1 − 2푀 1 − 2푀 푟 = 푟0 . 2 2 2 푟 sin 휗 푟0 ▶ Reisser-Nordström 2 2푀 푄2 1 − 2푀 + 푄 1 − + 2 푟0 푟2 푟 푟 = 0 . 2 2 2 2 푟 sin 휗 푟0 sin 휗0 ▶ Kerr ... looks complicated

105/139 Generalies Schwarzschild and Reissner-Nordström

6 6

4 4

2 2

z 0 z 0

-2 -2

-4 -4

-6 -6

0 2 4 6 8 10 0 2 4 6 8 10 x x

106/139 Generalies Inhalt

107/139 Domain of existence Gravitaonal potenal if the van Zeipel theorem holds, then there is a funcon 퐹 so that 1 Ω(푙) 휕 푝 = 휕 ln 푢 − 휕 푙 = 휕 ln 푢 − 휕 퐹 (푙) 휇 + 푝 휇 휇 푡 1 − Ω(푙)푙 휇 휇 푡 휇 with 퐹 (푙) given by 푙 Ω(푙′) 퐹 (푙) = ∫ 푑푙′ 1 − Ω(푙′)푙′ 푙0 1 therefore, 휇+푝 휕휇푝 can be integrated (“inner” = inner egde of the accreon disk where we have 푝 = 0)

푝 푑푝′ 푙 Ω(푙′) 푊 − 푊 = − ∫ = ln |푢 | − ln |푢 | − ∫ 푑푙′ . inner 휇(푝′) + 푝′ 푡 푡,inner 1 − Ω(푙′)푙′ 0 푙0 this is an eﬀecve potenal

108/139 Domain of existence 휇휈 from 1 = 푔 푢휇푢휈 we obtain 푔 푔 − 푔2 2 푡푡 휑휑 푡휑 푢푡 = 2 푔휑휑 − 2푔푡휑푙 + 푔푡푡푙 for stac space-mes 푔 푔 2 푡푡 휑휑 푢푡 = 2 푔휑휑 + 푔푡푡푙

푢푡 is deﬁned only in that space-me regions where the right hand side is posive ⇒ domains of existence of accreon disk

Gravitaonal potenal: special case for constant Keplerian angular momentum

휋 푙 = 푙(푟, 2 ) = 푙0 = 푐표푛푠푡. with 푙ms < 푙0 < 푙mb we have 퐹 = 0 and, thus, 푊 (푟, 휗) = ln 푢푡 + 푐표푛푠푡.

109/139 Domain of existence 푢푡 is deﬁned only in that space-me regions where the right hand side is posive ⇒ domains of existence of accreon disk

Gravitaonal potenal: special case for constant Keplerian angular momentum

휋 푙 = 푙(푟, 2 ) = 푙0 = 푐표푛푠푡. with 푙ms < 푙0 < 푙mb we have 퐹 = 0 and, thus, 푊 (푟, 휗) = ln 푢푡 + 푐표푛푠푡. 휇휈 from 1 = 푔 푢휇푢휈 we obtain 푔 푔 − 푔2 2 푡푡 휑휑 푡휑 푢푡 = 2 푔휑휑 − 2푔푡휑푙 + 푔푡푡푙 for stac space-mes 푔 푔 2 푡푡 휑휑 푢푡 = 2 푔휑휑 + 푔푡푡푙

109/139 Domain of existence Gravitaonal potenal: special case for constant Keplerian angular momentum

푢푡 is deﬁned only in that space-me regions where the right hand side is posive ⇒ domains of existence of accreon disk

109/139 Domain of existence Components of the accreon disk accreon disk may consist of more than one disconnected components - can be discussed using the condion for existence restricted to the equatorial plane accreon disk is not exisng if 휋 푢푡(푟, 휗 = 2 ) < 0. and splits at 푟 given by 휋 푢푡(푟, 휗 = 2 ) = 0.

110/139 Domain of existence ▶ in equatorial plane

(푟 − 2푀) 푟2 푢2 = 푡 푟3 − (푟 − 2푀) 푙2 ▶ 푢2 < 0 푟 < 2푀 푡 for becomes negave for 푟 > 2푀 if ▶ 2 3 2 for 푟 > 2푀 we have: 푢푡 < 0 if 퐶(푟) ∶= 푟 − (푟 − 2푀) 푙 = 0. This is solved for 푟3 sin2 휗 푙2 > √ 푟 − 2푀 푟 = 3푀 , 푙 = 푙disrupt = 3 3 푀 ⇒ border of the accreon disk the accreon√ disk disrupts up for 푙disrupt = 3 3 푀 at 푟 = 3푀 푟3 sin2 휗 − (푟 − 2푀) 푙2 = 0

Example: Schwarzschild

(푟 − 2푀) 푟2 sin2 휗 푢2 = 푡 푟3 sin2 휗 − (푟 − 2푀) 푙2

111/139 Domain of existence ▶ in equatorial plane

(푟 − 2푀) 푟2 푢2 = 푡 푟3 − (푟 − 2푀) 푙2

becomes negave for 푟 > 2푀 if ▶ 2 3 2 for 푟 > 2푀 we have: 푢푡 < 0 if 퐶(푟) ∶= 푟 − (푟 − 2푀) 푙 = 0. This is solved for 푟3 sin2 휗 푙2 > √ 푟 − 2푀 푟 = 3푀 , 푙 = 푙disrupt = 3 3 푀 ⇒ border of the accreon disk the accreon√ disk disrupts up for 푙disrupt = 3 3 푀 at 푟 = 3푀 푟3 sin2 휗 − (푟 − 2푀) 푙2 = 0

Example: Schwarzschild

(푟 − 2푀) 푟2 sin2 휗 푢2 = 푡 푟3 sin2 휗 − (푟 − 2푀) 푙2

▶ 2 푢푡 < 0 for 푟 < 2푀

111/139 Domain of existence ▶ in equatorial plane

(푟 − 2푀) 푟2 푢2 = 푡 푟3 − (푟 − 2푀) 푙2

becomes negave for 푟 > 2푀 if 퐶(푟) ∶= 푟3 − (푟 − 2푀) 푙2 = 0. This is solved for √ 푟 = 3푀 , 푙 = 푙disrupt = 3 3 푀

the accreon√ disk disrupts up for 푙disrupt = 3 3 푀 at 푟 = 3푀

Example: Schwarzschild

(푟 − 2푀) 푟2 sin2 휗 푢2 = 푡 푟3 sin2 휗 − (푟 − 2푀) 푙2

▶ 2 푢푡 < 0 for 푟 < 2푀 ▶ 2 for 푟 > 2푀 we have: 푢푡 < 0 if

푟3 sin2 휗 푙2 > 푟 − 2푀 ⇒ border of the accreon disk

푟3 sin2 휗 − (푟 − 2푀) 푙2 = 0

111/139 Domain of existence Example: Schwarzschild

▶ in equatorial plane 2 2 2 (푟 − 2푀) 푟 sin 휗 2 푢푡 = 2 (푟 − 2푀) 푟 푟3 sin 휗 − (푟 − 2푀) 푙2 푢2 = 푡 푟3 − (푟 − 2푀) 푙2 ▶ 푢2 < 0 푟 < 2푀 푡 for becomes negave for 푟 > 2푀 if ▶ 2 3 2 for 푟 > 2푀 we have: 푢푡 < 0 if 퐶(푟) ∶= 푟 − (푟 − 2푀) 푙 = 0. This is solved for 푟3 sin2 휗 푙2 > √ 푟 − 2푀 푟 = 3푀 , 푙 = 푙disrupt = 3 3 푀 ⇒ border of the accreon disk the accreon√ disk disrupts up for 푙disrupt = 3 3 푀 at 푟 = 3푀 푟3 sin2 휗 − (푟 − 2푀) 푙2 = 0

111/139 Domain of existence Example: Schwarzschild

112/139 Domain of existence ▶ 푟 < 푟−: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 < 0 for 푟 → 0 and 푃4 > 0 for 푟 → 푟− ⇒ 1 or 3 zeros, depends on 푙 ⇒ accreon disk may exist in [푟1, 푟−] or in [푟1, 푟2] ∪ [푟3, 푟−] with spling of accreon disk ▶ 2 푟− < 푟 < 푟+: here 푔푡푡 < 0 ⇒ 푢푡 < 0 ⇒ forbidden region ▶ 푟+ < 푟: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 > 0 for 푟 → ∞ and 푟 → 푟+, ⇒ 0, 2 or 4 zeros, depends on 푙 ⇒ spling of accreon disk

▶ space-me regions

▶ disrupon of disk

Example: Reissner-Nordström

113/139 Domain of existence ▶ 푟 < 푟−: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 < 0 for 푟 → 0 and 푃4 > 0 for 푟 → 푟− ⇒ 1 or 3 zeros, depends on 푙 ⇒ accreon disk may exist in [푟1, 푟−] or in [푟1, 푟2] ∪ [푟3, 푟−] with spling of accreon disk ▶ 2 푟− < 푟 < 푟+: here 푔푡푡 < 0 ⇒ 푢푡 < 0 ⇒ forbidden region ▶ 푟+ < 푟: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 > 0 for 푟 → ∞ and 푟 → 푟+, ⇒ 0, 2 or 4 zeros, depends on 푙 ⇒ spling of accreon disk ▶ disrupon of disk

Example: Reissner-Nordström

▶ 푟+ < 푟: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 > 0 for 푟 → ∞ and 푟 → 푟+, ⇒ 0, 2 or 4 zeros, depends on 푙 ⇒ spling of accreon disk ▶ disrupon of disk

Example: Reissner-Nordström

▶ 2 푟− < 푟 < 푟+: here 푔푡푡 < 0 ⇒ 푢푡 < 0 ⇒ forbidden region

▶ disrupon of disk

Example: Reissner-Nordström

▶ 2 푟− < 푟 < 푟+: here 푔푡푡 < 0 ⇒ 푢푡 < 0 ⇒ forbidden region ▶ 푟+ < 푟: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 > 0 for 푟 → ∞ and 푟 → 푟+, ⇒ 0, 2 or 4 zeros, depends on 푙 ⇒ spling of accreon disk

113/139 Domain of existence ▶ disrupon of disk

Example: Reissner-Nordström

(푟2 − 2푀푟 + 푄2) 푟2 sin2 휗 푢2 = . 푡 푟4 sin2 휗 − (푟2 − 2푀푟 + 푄2) 푙2 in equatorial plane 2 2 2 2 2 (푟 − 2푀푟 + 푄 ) 푟 (푟 − 푟−)(푟 − 푟+)푟 푢푡 = 4 2 2 2 = 4 2 푟 − (푟 − 2푀푟 + 푄 ) 푙 푟 − (푟 − 푟−)(푟 − 푟+)푙 complete discussion possible for all cases ▶ space-me regions ▶ 푟 < 푟−: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 < 0 for 푟 → 0 and 푃4 > 0 for 푟 → 푟− ⇒ 1 or 3 zeros, depends on 푙 ⇒ accreon disk may exist in [푟1, 푟−] or in [푟1, 푟2] ∪ [푟3, 푟−] with spling of accreon disk ▶ 2 푟− < 푟 < 푟+: here 푔푡푡 < 0 ⇒ 푢푡 < 0 ⇒ forbidden region ▶ 푟+ < 푟: nominator always posive, discussion of zeros of denominator 4 2 푃4 = 푟 − (푟 − 푟−)(푟 − 푟+)푙 = 0 푃4 > 0 for 푟 → ∞ and 푟 → 푟+, ⇒ 0, 2 or 4 zeros, depends on 푙 ⇒ spling of accreon disk

113/139 Domain of existence Example: Reissner-Nordström

l=3.7 l=4.0405 l=4.8

6 6 6

4 4 4

2 2 2

0 0 0

-2 -2 -2

-4 -4 -4

-6 -6 -6

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

114/139 Domain of existence Example: Kerr

푔 푔 − 푔2 2 푡푡 휑휑 푡휑 푢푡 = 2 푔휑휑 − 2푔푡휑푙 + 푔푡푡푙 2푀푟 2 2 2푀푟푎2 2 2 4푀푟푎 2 (1 − 2 2 2 )(푟 + 푎 + 2 2 2 sin 휗)sin 휗 + ( 2 2 2 ) = 푟 +푎 cos 휗 푟 +푎 cos 휗 푟 +푎 cos 휗 2 2 2푀푟푎2 2 2 4푀푟푎 2푀푟 2 (푟 + 푎 + 푟2+푎2 cos2 휗 sin 휗) sin 휗 + 2 푟2+푎2 cos2 휗 푙 − (1 − 푟2+푎2 cos2 휗 ) 푙 in equatorial plane

2푀 2 2 2푀푎2 4푀푎 2 (1 − 푟 )(푟 + 푎 + 푟 ) + ( 푟 ) 푢2 = 푡 2 2 2푀푎2 4푀푎 2푀 2 (푟 + 푎 + 푟 ) + 2 푟 푙 − (1 − 푟 ) 푙

115/139 Domain of existence Example: Kerr for 푎 = 0.75

l=5.5 l=5.5 l=5.5

10 10 10

5 5 5

0 0 0

-5 -5 -5

-10 -10 -10

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

116/139 Domain of existence Example: Kerr for 푎 = 0.95

l=5.5 l=5.5 l=5.5

4 4 4

2 2 2

0 0 0

-2 -2 -2

-4 -4 -4

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

117/139 Domain of existence Example: Kerr for 푎 = 1.75

l=6.0 l=8.20295 l=12.0

10 10 10

5 5 5

0 0 0

-5 -5 -5

-10 -10 -10

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

118/139 Domain of existence 푟 < 2푀 is forbidden; for 푟 > 2푀 denominator may be negave: crical angular momentum from vanishing of denominator and its ﬁrst derivave 0 = 푟3+2푞 − (푟 − 2푀)1+2푞푙2 , 0 = (3 + 2푞)푟2+2푞 − (1 + 2푞)(푟 − 2푀)2푞푙2 ⇒ 푟 = (3 + 2푞)푀 gives crical angular momentum 2+2푞 2+2푞 3+2푞 3 +푞 3 + 2푞 (3 + 2푞) 푀 (3 + 2푞) 푙 (3 + 2푞) 2 푙2 = = 푀 2 disrupt = disrupt 2푞 1+2푞 i.e. 1 +푞 1 + 2푞 ((3 + 2푞)푀 − 2푀) (1 + 2푞) 푀 (1 + 2푞) 2 disrupon of accreon disk depends on 푞 ↔ corresponding observaons can determine 푞

푞 metric

1+푞 2푀 푟2 sin2 휗 (1 − 푟 ) (− 푞 ) 2+푞 2 1+푞 푔 푔 (1− 2푀 ) 푟 sin 휗(푟 − 2푀) 푢2 = 푡푡 휑휑 = 푟 = 푡 2 2 1+푞 3+2푞 2 1+2푞 2 푔휑휑 + 푔푡푡푙 − 푟2 sin 휗 + 1 − 2푀 푙2 푟 sin 휗 − (푟 − 2푀) 푙 2푀 푞 ( 푟 ) (1− 푟 )

119/139 Domain of existence ⇒ 푟 = (3 + 2푞)푀 gives crical angular momentum 2+2푞 2+2푞 3+2푞 3 +푞 3 + 2푞 (3 + 2푞) 푀 (3 + 2푞) 푙 (3 + 2푞) 2 푙2 = = 푀 2 disrupt = disrupt 2푞 1+2푞 i.e. 1 +푞 1 + 2푞 ((3 + 2푞)푀 − 2푀) (1 + 2푞) 푀 (1 + 2푞) 2 disrupon of accreon disk depends on 푞 ↔ corresponding observaons can determine 푞

푞 metric

119/139 Domain of existence gives crical angular momentum 2+2푞 2+2푞 3+2푞 3 +푞 3 + 2푞 (3 + 2푞) 푀 (3 + 2푞) 푙 (3 + 2푞) 2 푙2 = = 푀 2 disrupt = disrupt 2푞 1+2푞 i.e. 1 +푞 1 + 2푞 ((3 + 2푞)푀 − 2푀) (1 + 2푞) 푀 (1 + 2푞) 2 disrupon of accreon disk depends on 푞 ↔ corresponding observaons can determine 푞

푞 metric

119/139 Domain of existence disrupon of accreon disk depends on 푞 ↔ corresponding observaons can determine 푞

푞 metric

1+푞 2푀 푟2 sin2 휗 (1 − 푟 ) (− 푞 ) 2+푞 2 1+푞 푔 푔 (1− 2푀 ) 푟 sin 휗(푟 − 2푀) 푢2 = 푡푡 휑휑 = 푟 = 푡 2 2 1+푞 3+2푞 2 1+2푞 2 푔휑휑 + 푔푡푡푙 − 푟2 sin 휗 + 1 − 2푀 푙2 푟 sin 휗 − (푟 − 2푀) 푙 2푀 푞 ( 푟 ) (1− 푟 ) 푟 < 2푀 is forbidden; for 푟 > 2푀 denominator may be negave: crical angular momentum from vanishing of denominator and its ﬁrst derivave 0 = 푟3+2푞 − (푟 − 2푀)1+2푞푙2 , 0 = (3 + 2푞)푟2+2푞 − (1 + 2푞)(푟 − 2푀)2푞푙2 ⇒ 푟 = (3 + 2푞)푀 gives crical angular momentum 2+2푞 2+2푞 3+2푞 3 +푞 3 + 2푞 (3 + 2푞) 푀 (3 + 2푞) 푙 (3 + 2푞) 2 푙2 = = 푀 2 disrupt = disrupt 2푞 1+2푞 i.e. 1 +푞 1 + 2푞 ((3 + 2푞)푀 − 2푀) (1 + 2푞) 푀 (1 + 2푞) 2

119/139 Domain of existence 푞 metric

119/139 Domain of existence Example: 푞-metric

푞 = −0.4999