THICK ACCRETION DISK

DOUWEI

2017/11/10GUIDEDBYBAIXUENING

CONTENT

• Basicmodel• Thickdisk• Properties• Discussion

PART I BASICMODELOFANACCRETIONDISK

WHAT IS ACCRETION DISK

•Central massive body• Protostars• Compact objects (white drawf, Neutron star and black holes)

• Surrounding material

HOW TO WORK?

ByNASA

SCALE (H/R)

Thindisk<<1Thickdisk~1

⇒ 𝜌 = 𝜌$ exp(−*+

,-+) and𝐻 = 01

2~ verticalscaleheight.

Ω5 =6789

�~Keplerianangularvelocity

𝐻𝑅 =

𝑐=𝛺𝑅~

𝑐=𝑣5~𝑣@A𝑣5

𝑐= isthevelocityofsound.

BASIC FUNCTIONS

• Continuity equation• Momentum equilibrium at radius• Angular momentum equilibrium

CONTINUITY EQUATION

• Massfluxconservation𝜕𝜌𝜕𝑡 + 𝛻 ⋅ 𝜌𝑣 = 0

CONT.

• Cylindricalcoordinate:𝜕𝛴𝜕𝑡 +

1𝑟𝜕𝜕𝑅 𝛴𝑣8𝑅 = 0

Where𝛴 = ∫ 𝜌𝑑𝑧 = 2𝜌𝐻 isdefinedassurfacedensity.

• Forastablesystem: OO@= O

OP= 0

• Forthesimplifiedequationisshownas:𝜕𝜕𝑅 𝜌𝑅𝐻𝑣 = 0

MOMENTUM EQUILIBRIUM AT R DIRECTION

• Cylindricalcoordinate• Thindisk:𝛺,𝑅 = 𝛺5,𝑅• Thickdisk:

𝑣d𝑣d𝑅 − 𝛺

,𝑅 = −𝛺5,𝑅 −1𝜌dd𝑅 𝜌𝑐R,

Accretion centrifugalforcegravitypressure• Advection-dominatedaccretiondisk(ADAF)

ANGULAR MOMENTUM EQUILIBRIUM

• Cylindricalwith𝜙 (AMbyviscosity)

Σ𝑣𝑅𝜕𝜕𝑅 (Ω𝑅

,) =𝜕𝜕𝑅 (Σ𝜈𝑅

𝑅𝜕Ω𝜕𝑅 𝑅)

Angularmomentumfluxchangedin2𝜋𝑅d𝑅 andviscosityWealsouseconstant𝛼 describingtheeffectofviscosity

𝜈 = 𝛼𝑐=𝐻 =𝛼𝑐=,

Ω5

PART II THICKDISK

ENERGY EQUATION

Σ𝑣𝑇d𝑠𝑑𝑅 = 𝑄f − 𝑄g

• Thindisk𝑄g = 4𝜋𝑅𝐹g

whenopticallythickandgeometrythinopticallythin?⇒ 𝜏 ≪ 1

ENERGY EQUATION

Opticallythin->• Free-freeemission(brakingradiation)• Emissivity~𝑁m,~𝜌,~𝛴,

Opticallythick->• Blackbodyradiation• Emissivity ~𝜎𝑇o~�̇�~𝛴

ENERGY EQUATION

Ifnocooling,justviscosity!

Σ𝑣𝑇d𝑠𝑑𝑅 = 𝑄f = 𝜈 𝑅

d𝛺d𝑅

,

Using𝑇d𝑠 = 𝐶|d𝑇 + 𝑝d𝑉3 + 3𝜖2 2𝜌𝐻𝑣

𝑑𝑐=,

𝑑𝑅 − 2𝑐=,𝐻𝑣𝑑𝜌𝑑𝑅 = 𝑄f

SOLVE : SELF-SIMILAR SOLUTION• O

O�𝜌𝑅𝐻𝑣 = 0

• 𝑣 ����− 𝛺,𝑅 = −𝛺5,𝑅 −

�����

𝜌𝑐R,

• Σ𝑣8𝑅OO8(𝛺,𝑅) = O

O8(Σ𝜈𝑅� O2

O8)

• 𝑇 �=��= 𝜈 𝑟 ��

�8

,

By wikiSolve the equations with varieties relative to 𝑅, we now have

𝜌 ∝ 𝑅g�,𝑣 ∝ 𝑅g

�,Ω ∝ 𝑅g

�,𝑐=, ∝ 𝑅g�

INTERESTING RESULTS

ThickDisk

• 𝑣 ≈ −���𝑣5

• Ω ≈ 0

• 01+

��+ ≈

,�

• �̇� ≈ −4𝜋𝑅𝐻𝑣𝜌 = 𝑐𝑜𝑛𝑠𝑡

ThinDisk

• 𝑣~𝛼 -�

,𝑣5

• Ω ≈ Ω5

• 01+

��+ =

-�

,≪ 1

INTERESTING RESULTS

Cooling does exist!• 𝑄f − 𝑄g = 𝑓𝑄f, 𝑓 = 0 representefficientcoolingand𝑓 = 1isnoradiativecooling.

• 𝐶| isrelativistic!Useafunctionof𝜖 = 0 representnorelativityand𝜖 = 0representrelativity.

• Using𝜖� = ��~ acriticalroleindeterminingthenatureofflow.

INTERESTING RESULTS

• 𝑣 ≈ − ���f,��

𝑣5

• Ω ≈ ,��

�f,��

�+ Ω5

• 𝑐=, ≈,��

�f,��𝑣5,

• -�~1 = 01

��

• Thindisk𝜖� → ∞• ADAF𝜖� < 1

PART III PROPERTIESOFTHICKDISK

BERNOULLI CONSTANT

• NormlizedBernoulliconstant(Energyconserved)

𝑏 =1𝑣5,

12 𝑣

, +12Ω

,𝑅, − Ω5,𝑅, +𝛾

𝛾 − 1 𝑐=, =

3𝜖 − 𝜖�

5 + 2𝜖′

𝑏 > 0 → gasreachinfinity(𝛺~𝑅g9+, 𝑣~𝑅g

�+)

𝑏 < 0 → cannotspontaneouslyescape

With𝑓 > 1/3 , 𝛾 < ��⇒ 𝑏 > 0!

• Winds?Jets?

• SeemoreinStone,Pringle(1999)

BRUNT -VÄISÄLÄ FREQUENCY

• 𝑁~Brunt- Väisälä frequency

𝜌$𝜕,𝑧�

𝜕𝑡, = −𝑔 𝜌 𝑧 − 𝜌 𝑧 + 𝑧� = −𝑔𝜕𝜌(𝑧)𝜕𝑧 𝑧′

⇒ 𝑧� = 𝑧$� 𝑒 g¥+� @ and𝑁 = − ¦�§

O�O*

�

O�O*> 0 →notstable!

INSTABILITY

• Dynamicalconvectiveinstability:

𝑁m��, = 𝑁, + 𝜅,

𝑁~Brunt- Väisälä frequency𝜅~ epicyclicfrequency

• Self-similar:𝑁m��, = �$��f©���g���(�f��)(�f,��)

⇒ 𝑓 > ,�+ ,

�𝜖

• Doublediffusiveinstability

PART IV DISCUSSION

�̇� > 𝑀ª̇ ⇒SLIM DISK

• 𝑃 = 𝑃¦¬= + 𝑃8¬�• 𝑇𝑑𝑠 = 𝑑𝐸�®=0¯=®@° − 𝑑𝐸8¬�®¬@®¯±

• Yuan(2003)

SUMMARY

• Non-standardmodel– ADAFisanopticallythingeometrythickmodelandcooledbyfree-freeemissionwhichisnotefficientcomparingtotheblackbodyradiation.

• Weuseaself-similarsolutionwhichisonlybasedontheradiustogetthefunctionofvelocity,density,angularvelocityandsoundspeed.

• Theremaybesomeinstabilityinthesimulationbuttheobservationareallstableuptonow.

THANKYOU!