Accretion Disks: An Overview

Transcription

1 Accetion Disks: An Oveview Autho: Simon Konbeg Supeviso: Jügen Fuchs FYGB08 - Analytisk Mekanik Institutionen fö Ingenjösvetenskap och Fysik Abstact: This epot is intended as an oveview and intoduction to the fomation and popeties of accetion disks The cuent theoetical famewok suounding the fomation of stas is fist pesented followed by the specifics elating to disk fomation and the stuctue of thin disks This is supeseded by a bief discussion of the evolution of accetion disks and finally a modeately detailed oveview of angula momentum tanspot thoughout the disk Table of Contents I Intoduction 2 II Disk Fomation 2 III Thin Disks: Stuctue 3 A Steady Thin Disks 4 B Vetical Disk Stuctue 5 C Radial Disk Stuctue 6 D Tempeatue Pofile of Accetion Disk 6 IV Disk Evolution 7 V Angula Momentum Tanspot 7 A Tubulent Viscosity 8 B Shakua-Sunyaev Viscosity 9 C Magneto-Rotational Instabilities 9 D Dead Zones 9

2 2 I Intoduction The common consensus among scientists fo the oigin of stas is that they fom fom molecula clouds in a pocess which may be descibed thusly: The oiginally stable cloud becomes unstable due to any numbe of extenal factos which foces it to collapse unde its own gavitational attaction This collapse gives bith to a class 0 object a young stella object) which is a potosta that apidly accetes matte fom the suounding envionment This acceted matte patly adds to the gowing potosta but also foms an accetion disk Given time this will esult in a class I object which is simply a potosta mainly acceting matte fom its disk and citically suounded by a gas envelope which is less massive than the potosta itself This envelope will eventually get dissipated though jets winds and accetion leading to a class II object ie a sta suounded by an accetion disk Continuing this pocess of dissipation this time in the fom of accetion onto the sta fomation of potoplanets photoevapoation o othe dissipation methods the sta eventually tuns into a sta with a debis disk at which time the accetion stops and we have a class III object Accoding to Alecian [1] the potostella class 0 and I) phase lasts about yeas duing which the diffeence between low- and high-mass stas is thought to be negligible as it is believed that both high- and low-mass stas fom fom simila low-mass potostas with the diffeence being the amount of mass they eventually accete Duing the pe-main sequence phase class II and III) the sta adiates in the visible bands wheeas in the potostella phase the cental object is so obscued that it mainly adiates in the submillimetic to midinfaed spectum To make futhe discussions moe convenient we may want to use a Hetzspung-Russel HR) diagam Fig 1) which is a plot of the stella luminosity as a function of the effective tempeatue When the sta fist emeges fom the potostella phase it is pesent on the bithline It then follows a tack suitable to its mass until it eaches the zeo-age main-sequence ZAMS) at which point it stats its longest phase namely the main-sequence MS) The time until a sta eaches the ZAMS vaies widely depending on its mass accoding to Alecian [1] it may take as long as 100 My down to 015 My fo stas with masses of 1 M and 15 M espectively whee M is the mass of the sun Figue 1: PMS theoetical evolutionay tacks computed by Behend and Maede [2] and plotted in a Hetzspung-Russel HR) diagam The tacks stat on the bithline and end on the Zeo- Age Main-Sequence ZAMS) The tanspot of enegy inside the sta adiative o convective o both) is indicated with diffeent boken lines The zones suounded with blue bown) line epesent the egion whee the Hebig Ae/Be T Taui) stas ae situated figue fom [2] and caption fom [1] ) The familia enegy adiation fom the nuclea fusion of hydogen into helium in the coe of the sta will not stat until the end of the pe-ms phase Thus the enegy adiated duing the pe-ms phase is due to gavitational contaction An inteesting popety of vey massive stas > 20 M ) is that they neve undego a pe-ms phase and stat buning hydogen in its coe at the bithline II Disk Fomation As the molecula cloud aound a cental potosta contacts the mateial acceted onto the potosta makes it gow in a spheically symmetical fashion Due to the pesence of angula momentum and the action of gavitational and fictional foces a disk

3 3 will fom aound the potosta This is most easily undestood using a minimisation of enegy agument namely that in the pesence of otation the obit of least enegy is cicula Thee is also something known as an accetion shock: due to the incoming fluid elements fom the contacting molecula cloud will collide with matte coming fom the othe side The situation is depicted in Fig 2 Accoding to [3] an efficient dissipation of the heat fom the impact is sufficient to fom a thin disk stuctue III Thin Disks: Stuctue Conside a thin disk ie cool and nealy Kepleian which is axisymmetic We want to study the motion of an annulus of the disk ie a ing with a non-zeo height This will be geatly simplified using a vetically integated fom of the equations ie a suface density Σ instead of a density ρ Note that matte may flow though ou annulus with a speed v Clealy cylindical coodinates ϕ z) ae pefeable hee due to the symmety of the poblem Thus we define a suface density which is assumed to depend on and t: Σ t) := ż ρ z)dz Recall the continuity equation fo mass and the fact that mass flows though ou annulus along with speed v ): m t + mv ) = 0 t ρ) + ρv ) Figue 2: Schematic view of disk fomation duing the collapse of a otating spheical cloud adapted fom [3]) Because the disk is suppoted mainly by its otation but also by the gas pessue gadient we may appoximate the obital velocity as that of a Kepleian obit thus GM v ϕ = Ω K = 1) whee Ω K is the Keple fequency Ω K := GM 3 2) The vetical components of the velocity of the infalling mateial will get completely dissipated leaving only the paallel component v see Fig 2 This velocity is not necessaily equal to the Kepleian obital velocity at the point of infalling esulting in angula momentum tanspot as well as mass tanspot This continues until a nealy Kepleian obit has been established This leads to the consevation of mass continuity) equation Σ t + Σv ) = 0 3) by simply integating the continuity equation ove z Note that is a constant in time Due to the axisymmetical natue of the disk we see that the adial equation of motion is simply eg [4]) v 2 ϕ = GM The ϕ-equation of motion may be expessed as [4] v ϕ t v ϕ + v + v v ϕ = F ϕ whee F ϕ is the azimuthal component of the viscous foce If one wee to integate this ove z one would get accoding to [3]) 2 ΩΣ ) + t 2 ) 1 G Ω Σv = 2π 4) whee Ω = v ϕ and the tem on the ight hand side aises fom the viscous toques in the disk We would like to expess the angula momentum of a thin annulus of the disk

4 4 This is done in [3] and it is found to be 2π Σ 2 Ω whee is the width of the annulus Recall that toque is defined as the poduct of a foce F and the distance fom the mass cente thus G := F = F sinϑ) whee ϑ is the angle between F and We need to apply the toque equation to each annulus of the disk as it does not otate as a igid body Neighbouing annuli will exet foces onto each othe popotional to the obital velocity o the obital velocity gadient dω d This foce is known as shea o viscous) foce and is defined as A := dω dω whee d d is called shea Something which will be of consideable inteest late may now be noted namely that the total toque fom the annulus is G = 2π νσ dω d = 2π 3 νσ dω d 5) whee ν is the kinematic viscosity The fist expession is witten thus to show that it is the poduct of the cicumfeence the viscous foce pe unit length and the leve Using this we may ewite Eq 4) as 2 ΩΣ ) + 1 t 2 ) 1 Ω Σv = νσ 3 dω ) 6) d Conside the individual tems on the left hand side 1 2 ΩΣ ) = Ω Σ t t 3 ) ΩΣv = Ω 3Σv + Σ v + Σ c whee Ω is assumed to be time-independent If we etun to the continuity equation Eq 3) and evaluate the second tem we get Σv ) = Σv + Σ v + Σ v The left hand side of Eq 6) may now be witten as Ω Σ t + ) Σv ) + Σv 2 Ω + 2 Ω ) ) Hee the fist tem will vanish see Eq 3) and the second tem is ecognised as Σv 2 Ω ) by the poduct ule fo deivatives Finally Eq 6) may be witten as Σv 2 Ω ) = 1 νσ 3 dω ) d Solving this fo the adial speed v yields ) v = 1 νσ 3 dω d Σ 2 Ω) If we assume that Ω is the Keple fequency Ω K defined in Eq 2) we may evaluate this expession: 1 νσ 3 dω ) = 3 GM d 2 GM Σ 2 Ω ) = 2 v = 3 Σ νσ ) 1 νσ ) Insetion into the continuity equation Eq 3) yields Σ t = 3 [ νσ ) ] 7) A Steady Thin Disks A steady stuctue is one which emains unchanged in time This means that both adial and angula momentum is conseved and it is assumed that the vetical component of gavity fom the sta will pefectly cancel the vetical gas pessue gadient This is equivalent to saying that the system is in vetical hydostatic equilibium Conside a disk annulus at a distance fom the sta Matte can still flow though the annulus in adial diection with the velocity v Accoding to [3] we may wite the adial momentum consevation equation fo this steady-flow as v v v2 ϕ + 1 dp ρ дas d + GM 2 = 0 whee the fou tems aise fom adial mass flow centifugal foce gas pessue and gavity espectively

5 5 Recall the consevation of angula momentum equation Eq 6) The time deivative will clealy vanish steady state) esulting in 3 ) ΩΣv = νσ 3 dω ) d Integation yields 3 ΩΣv + C = νσ 3 dω d Because we would like to aive at a vey specific expession we will now ewite this expession as well as include facto of 1 into the integation constant 2π νσ dω d = Σv Ω + C 2π 3 8) Recall that dω d is called the shea If we evaluate the expession above at a point whee this vanishes we get C = 2π 3 Σv Ω = 2π Σv ) 2 Ω = M 2 Ω 9) whee the mass flow pe unit time is defined as M := 2π Σv Note that the eason this is negative is because the adial velocity is inwads and by the cylindical coodinates convention this esults in a negative sign To find a place whee the shea vanishes let us assume that the disk extends all the way down to the sta Then the shea would clealy vanish at the adius of the sta R At that adius C = MR 2 Ω = M GM R 10) whee again Ω = Ω K has been assumed We may hence be inclined to suggest a physical intepetation of this integation constant namely the influx of angula momentum though the disk We would now like to expess the suface density using this new constant To do this we note that fom Eq 9) we may expess the adial velocity as M v = 2π Σ Inseting this into Eq 8) and solving fo Σ yields Σ = MΩ 2πν dω d + C 2π 3 ν dω d Using the expession fo C in Eq 10) unde the assumption that Ω = Ω K yields the final expession Σ = M 1 3πν R B Vetical Disk Stuctue ) fo R 11) Recall that steady disks equie a vetical hydostatic equilibium Mathematically this may be expessed as [3]: 1 P ρ дas z = z GM 2 + z 2 ) 12) In ode to get an expession fo the vetical density stuctue we fist need to define some quantities The gas sound speed c s is defined as c 2 s := P ρ дas 13) whee P is the gas pessue and ρ дas is the gas density Fo an ideal gas eg [3]) kt д c s := 14) µm p whee k is the Boltzmann constant T д is the gas tempeatue µ is the mean molecula weight of the gas and m p is the poton mass If we pefom the deivative on the ight hand side of Eq 12) we get 1 P ρ дas z = GM 2 + z 2 ) 3/2z GM 3 z fo z Note that the condition z is equivalent to the statement thin disks We may now ewite Eq 12) as noting that Ω 2 K = GM 3 and that fom Eq 13) P = c 2 s ρ дas ) 1 cs 2 ρ дas ) ρ дas z = Ω 2 K z In ode to ewite this futhe we equie that the disk be vetically isothemal ie T z = 0 since that clealy implies that dc s dz = 0 cf Eq 14)) thus 1 ρ дas ρ дas z = Ω2 K cs 2 z

6 6 The solution to this diffeential equation is eadily found to be whee ρ дas z) = ρ c )e z2 /2H 2 дas H дas := c s Ω K 15) H дas is known as the gas pessue scale height and it is defined as the incease in altitude fo which the atmospheic pessue deceases by a facto of e [5] ρ c ) is the density at the midplane and H дas is also evaluated thee ie T д = T c in Eq 14) C Radial Disk Stuctue As we will see in a late section specifically Sect VB) afte a moe igoous teatment of viscosity we may wite the kinematic viscosity as ν = αc s h in what is known as α-paametisation and was fist intoduced by Shakua and Sunyaev in 1973 as a way of paametising ou ignoance of the angula momentum tanspot pocess [4] In the above equation h H дas and c s is the gas speed of sound as befoe Using this new elation togethe with Eq 15) in Eq 11) yields MΩ Σ = ) R 3παcs 2 1 whee we can now use Ω = Ω K and c s = kt c /µm p as well as the condition that R meaning that R / 0 to wite Σ = µm p GM M 3πk αt c fo R 3/2 An inteesting thing to note which also leads us nicely into the next section is that is that we have assumed the tempeatue of the midsection of the disk to follow a simple powe-law pofile ie T c q then the suface density Σ q 3/2 ie also a simple powe-law of adius Let us now examine whethe this is a sound assumption D Tempeatue Pofile of Accetion Disk To poceed we must again make some new definitions this time fo the viscous stess tenso T Fom expeiments we know that T visc appoximately v whee v is the velocity of the flow In wods: the magnitude of the shea stess in viscous flows is often popotional to the symmetic components of the velocity gadient [6] Explicitly T visc = ξθ g 2η σ whee thee is no dependence on because intuitively) thee would be no foce if the fluid is simply otating ie if v = 0 The fist coefficient ξ is called the bulk viscosity and it is often neglected fo astophysical flows The second coefficient η is sometimes called the shea viscosity and this is indeed the one that we will be using late To get tot his point we assumed the viscous stain popotional to the gadient of the velocity this is only tue fo Newtonian fluids In ode to compute anything we fist need to make some assumptions Recall that the shea A = dω d and that the geneated heat comes fom the diffeence in the obital velocity at diffeent adii fom the sta Thus the only component of the velocity of inteest fo ou pesent discussion is the gadient of the tangential velocity component in the adial diection! Thus we expect T viscϕ = νσ dω d whee the negative sign comes fom the conventions suounding the adial component of cylindical coodinates To get the enegy dissipated pe unit time and pe unit suface aea we take the viscous stess times the viscous stain whee the viscous stain is simply A Thus ) 2 dω E = νσ 2 d whee the change of sign simply comes fom the fact that we want positive enegy dissipation in the disk ie a puely physical agument)

7 7 Making the assumption that Ω = Ω K 3 2 Ω K yields E = 9 4 νσω2 K dω K d = We have yet to conside one aspect though the 3D natue of ou disk ie it adiates pesumably) equally well in positive z-diection as in the negative one Thus we must divide ou expession by 2 to get the coect enegy dissipation pe unit aea and unit time E = 9 8 νσω2 K Recall Eq 11) fom which we immediately see that M ν = 3π Σ since R Insetion into this equation yields E = 3 MΩ 8π K 2 when R Also note that unde the same assumptions and fom Eq 11) the mass accetion ate M = 3πνΣ Assuming the disk adiates like a black-body we may use the Stefan-Boltzmann law fo blackbody adiation E = σtdisk 4 whee σ is the Stefan- Boltzmann constant This yields the tempeatue pofile T disk = ie a 3/4 pofile 3 8πσ MΩ 2 K ) 1/4 IV Disk Evolution Recall Eq 7) epeated hee fo convenience: Σ t = 3 [ νσ ) ] 7) This descibes the tempoal evolution ie the evolution though time) of the suface density of the disk and is identified as a diffusion equation f t = D 2 f x 2 with a diffusion constant D As is seen in Fig 3 viscous foces dispese the matte which is oiginally located at a distance 0 fom the cente Figue 3: The viscous evolution of a ing of matte of mass m The suface density Σ is shown as a function of the dimensionless adius x = / 0 whee 0 is the initial adius of the ing and the dimensionless time τ = 12νt/ 0 2 whee ν is the viscosity caption and figue fom [7]) At this initial condition t = τ = 0 and = 0 ie all of the mass m is located at a distance = 0 fom the cental sta Thus the suface density is Σ t = 0) = m δ 0 ) 2π 0 whee δ 0 ) is the Diac delta function As is seen in the figue with time most of the mass will move inwad and accete onto the sta whilst a smalle potion of it will move outwad viscous speading of the disk) taking along the angula momentum V Angula Momentum Tanspot So fa we have mentioned the tanspot of angula momentum but we have neglected to explain its necessity Accoding to [3] the specific angula momentum o the angula momentum pe unit mass fo the case of a disk with mass 1 M with a size of 10 AU is cm 2 /s whilst a sta with the same mass otating at beak-up velocity has a specific angula momentum of cm 2 /s This shows the need fo a this mechanism Recall that angula momentum is conseved in ou system meaning such a pocess is viable The pocesses most likely to poduce such a mechanism ae a toque fom the extenal medium eg magnetic fields) viscosity inside the disk tanspoting angula momentum to the oute disk disk winds taking angula momentum away [3]

8 8 Let us assume fo a moment that the disk otates with Kepleian speed and conside but two paticles on the disk with massesm 1 andm 2 at a distance 1 and 2 fom the cente of the sta with mass M Then thei enegy and angula momentum may be expessed as E = T + V = = GM 2 2 L = i= m iv 2 ϕi i=1 m1 1 + m 2 2 ) i=1 GM m i i m i v ϕi i = GM m m 2 2 ) whee Eq 1) has been used in the last steps The consevation of angula momentum tells is that a small change in obit fo one of these masses must esult in a coesponding change fo the othe Mathematically this may expessed as L = 1 GM m L 1 = 1 L 1 = GM m L 1 = L 2 m = m Note that this clealy also applies to two neighbouing annuli in the disk Fo this type of behaviou we equie a coupling between the paticles o ings of mateial) Such a coupling may be attibuted to small tubulent andom motions because diffusion plays a ole in adial tanspotation not only inwad but also outwad We may also be tempted to conside the peviously discussed diffeence in otation speed o shea) but since we ae dealing with a gas the tubulent motions would lead to a adial mixing of mateial and thus cause the desied coupling between the annuli within the disk A Tubulent Viscosity Accoding to [3] thee is a simple appoximation to the molecula viscosity namely ν m λc s whee λ := 1 nσ m is the mean fee path of the molecules It is defined as the invese of the poduct between the gas paticle density n and the collisional coss section σ m between the molecules The quantity c s is of couse the gas sound speed defined in Eq 14) If we compute this numeically with c s = cm/s n = /cm 3 and σ m = cm 2 we obtain ν m = c s nσ m = cm 2 /s The viscous timescale may be expessed as eg [8]) t ν = 2 /ν m Thus t ν = 2 ν m yeas =10 AU when evaluated at a distance = 10 AU fom the cental sta In view of the fact that age of the univese is meely of the ode yeas we may safely ule out molecula viscosity as the main souce of the tubulent motions To gain some moe insight into the natue of these disks ecall the Reynolds numbe Re which is the atio of inetial foces to viscous foces ie the atio between the esistance to a change in motion and the cohesiveness of the gas It is useful to defining whethe the flow is lamina o tubulent the details of which ae beyond the scope of this text Suffice it to say we conside a flow to be tubulent when the Reynolds numbe is above Mathematically the Reynolds numbe is defined as eg [9]) Re = ρvl µ whee ρ is the density of the fluid v is the chaacteistic velocity of the fluid L is the chaacteistic dimension and µ is the dynamic viscosity of the fluid Fo ou puposes we may estimate it as [3] Re = V L ν m whee V is the chaacteistic velocity Thus V = c s = 05 km/s at 10 AU and L = h = 005 = 05 AU whee h is the scale height of the gas in the disk By diect computation we get that Re ie completely tubulent

9 9 B Shakua-Sunyaev Viscosity As we have seen the molecula viscosity is not nealy sufficient in poviding the amount of tubulence we equie Thus we tun ou attention to othe possible mechanisms Let us fist examine the wok done by Shakua & Sunyaev in 1973 whee they paametised the viscosity without identifying its souce This would then let us compae the viscosities of diffeent disks unde diffeent conditions This pocess of paametisation is also called α- paametisation fo easons which will become appaent pesently If we use the vetical scale height h as a epesentative scale and we use the gas sound speed c s as the chaacteistic velocity of the tubulent motions in the disk we may wite the viscosity ν as [3] ν = αc s h Using the same definitions of the viscous time scale as befoe ie t ν = 2 ν = 2 ) 2 h αc s h = 1 α Ω whee the definition of H дas = h ie Eq 15) has been used in the final step Following the estimations made in [3] ie the disk is vey thin and thus h/ 005 as well as the timescale being 1 My at a distance of 50 AU yields an α of 002 which fits well with obsevationally detemined values C Magneto-Rotational Instabilities Balbus & Hawley 1998) popose a model in which the pesence of magnetic fields poduces a coupling between diffeent annuli within the disk This inteaction may be viewed as a weak sping the schematic view of which is available in Fig 4 A equiement fo the instability of such a magnetised disk is that the obital velocity deceases with adius ie dω d < 0 Fo a Kepleian disk this is clealy valid as Ω = Ω K see Eq 2) fo the definition of which Figue 4: Two masses in obit connected by a weak sping The sping exets a tension focet esulting in a tansfe of angula momentum fom the inne mass m i to the oute mass m o If the sping is weak the tansfe esults in an instability as m i loses angula momentum dops though moe apidly otating inne obits and moves futhe ahead The oute mass m o gains angula momentum moves though slowe oute obits and dops futhe behind The sping tension inceases and the pocess uns away ie becomes unstable due to its acceleating pace figue and caption edited) fom Balbus & Howley [10]) This is not a sufficient peequisite howeve The disk also needs to be ionised since electically neutal gas does not inteact well with the magnetic field lines The citical ionization degee is examined by Sano & Stone 2002) and they found it to be n e /n tot [11] D Dead Zones Dead zones ae a esult of the local degee of ionisation dopping below the citical one The existence of these dead zones is vital to planetay fomation as the viscosity dops inside a dead zone meaning mateial flowing fom lage adii will accumulate thee The details of planetay fomation ae egettably outside of the scope of this text but suffice it to say this is not the only mechanism diving the fomation of planets

10 10 Refeences [1] E Alecian The envionments of the sun and the stas Spinge Lectue Notes in Physics ) / _7 [2] R Behend and A Maede Fomation of massive stas by gowing accetion ate Astonomy & Astophysics ) / : [3] Viscous accetion disks Pat of a couse the oiginal souce is not disclosed washingtonedu/couses/asto557/accetion_diskspdf [4] H Spuit Accetion disks Lectue in a couse given at Max-Planck-Institut fü Astophysik [5] Wikipedia Scale height [Online; accessed 21 Decembe 2016] php?title=scale_height&oldid= [6] A Quillen St242 lectue notes pat 3 Pat of the couse Astophysics II Astophysical Fluid Dynamics at Univesity of Rocheste pdf [7] J Pingle Accetion discs in astophysics Annual eview of astonomy and astophysics ) /annuevaa [8] F Casoli J Lequeux and F David Astonomie spatiale infaouge aujoud hui et demain infaed space astonomy today and tomoow 1st ed Vol 70 Les Houches - Ecole d Ete de Physique Theoique Spinge 2000) / [9] Wikipedia Reynolds numbe [Online; accessed 03 Januay 2017] indexphp?title=reynolds_numbe&oldid= [10] S A Balbus and J F Hawley Instability tubulence and enhanced tanspot in accetion disks Reviews of moden physics ) /RevModPhys701 [11] T Sano and J M Stone The effect of the hall tem on the nonlinea evolution of the magnetootational instability i local axisymmetic simulations The Astophysical Jounal ) /339504