Structure and evolution of circumstellar disks during the early phase of accretion from a parent cloud


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1 Cente fo Tubulence Reseach Annual Reseach Biefs Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The pocess by which a cloud of gas and dust tuns into stas and planets is one of the most intiguing questions in astophysics. One of the steps in this pocess is the fomation of a disk suounding a young sta. The disk is impotant because planets may eventually fom within it and because it feeds mass to the sta. The cicumstella disk has two phases in its evolution. The fist is chaacteied by accetion fom the paent cloud diectly to the potosta and on to a adially gowing disk whose sie is compaable to the stella adius. In the second phase, accetion fom the cloud to the disk has ceased. The flow at this late phase of evolution is nealy Kepleian and thee is obsevational data at both mm and infaed wavelengths to povide guidance fo theoetical wok on the stuctue of the disk. In contast, vey little is known about the dynamics of the disk in the fist phase of evolution because the foming disk is heavily obscued by the paent cloud. Shu (1977) descibed the mechanism fo the spheically symmetic collapse of an initially isothemal cloud. He suggested that the collapse occus though the outwad popagation of a spheical aefaction wave. The expanding wave font educes the suppot of pessue gadient (against gavity) behind it. This allows gas paticles to fall adially inwad inceasing the mass of the cental potosta. Fo the fomation of a disk to take place it is necessay fo thee to be otation in the paent cloud. To descibe this pocess Cassen & Moosman (1981), and Teebey, Shu & Cassen (1984) assumed that the cloud collapses in an axisymmetic manne and each gas paticles falls with a specific angula momentum j = U φ whee is the adius in cylindical pola coodinates (Fig. 1) and U φ is the aimuthal velocity. The axis in Fig. 1 epesents the axis of a cylindical coodinate system. Sufaces of constant angula momentum ae epesented by nested cylindes. As the aefaction font popagates, a significant faction of the infalling gas will possess enough angula momentum so that it can miss the cental potosta and become incopoated to the gowing disk aound the cental sta. In the pesent wok we conside the evolution of the disk fom the time t 0 = ξt whee ξ is any positive numbe and t epesents the time when the cloud begins to miss the cental sta defined as ( ) 1/3 16 t = Ω 2, (1.1) 0 am3 0 is the initial stella adius, a is the speed at which the aefaction font popagates, Ω 0 is the otation ate of the paent cloud and m 0 is a dimensionless numbe equal to (Teebey, Shu & Cassen (1984)). If we assume that the total enegy in the infalling cloud is small compaed to gavitational potential enegy (negative), and kinetic enegy (positive) at the point of impact
2 210 Olusola C. Idowu Ω 0 A constant suface within the cloud j 1 j 2 Head of the aefaction wave Figue 1. Insideout collapse of a otating cloud showing the sufaces of constant angula momentum ing the disk, then the fluid element aives at a state of essentially fee fall. Theefoe the steamlines appoaching the disk ae well appoximated by eoenegy obits, i.e., paabolic tajectoies. Cassen & Moosman (1981), Ulich (1976), and Teebey, Shu & Cassen (1984) have used this basic fact to deive the velocity and density of infalling cloud onto the disk suounding the cental potosta. The velocity fo the infalling paticle in spheical pola coodinates (σ, θ, φ) is given by ( ) 1/2 ( GM U σ = 1+ cos θ ) 1/2 (1.2) σ ( ) 1/2 ( )( GM cos θ0 cos θ U θ = 1+ cos θ ) 1/2 (1.3) σ sin θ ( ) 1/2 ( GM sin θ 0 U φ = 1 cos θ ) 1/2 (1.4) σ sin θ whee θ is the colatitude and φ is the aimuthal angle. The density distibution is given by ( ) [ ( Ṁ ρ = 4πσ cos θ ) (1 2cot 2 θ )] 1 (1.5) U σ whee the angle θ 0 is the inclination of each obital plane elative to the otation axis; it is elated to the disk adius R d by ) cos 3 θ 0 (1 σrd σ cos θ = 0 (1.6) R d The disk adius R d is obtained fom Eq. (1.1) at a given t 0 and Ω 0. G is the gavitational constant, M is the mass of the cental sta, and Ṁ is the mass accetion ate onto the
3 sta given by Stuctue and evolution of a cicumstella disk 211 Ṁ = m 0a 3 (1.7) G (Teebey, Shu & Cassen (1984)). In cylindical pola coodinates (, φ, ), adial and axial velocities become ( ) 1/2 ( GM U = 1+ cos θ ) 1/2 ( ) 1 cos θcos θ0 (1.8) sin θ ( ) 1/2 ( GM U = 1+ cos θ ) 1/2 (1.9) while U φ emains the same. Using Eqs. (1.4), (1.5) and (1.8), Stahle et al (1994) consideed the motion of gas within a vetically mixed thin disk. The thin disk appoximation consists of neglecting any vaiation in the vetical diection. The goal of the pesent wok is to study via numeical simulation, the pocesses that occu within the disk as it accetes mateial. The motion esulting fom the infalling steams is assumed to be govened by the Eule equations fo compessible gas flow. The main finding so fa is that the gas initially within the disk aces towads the potosta at supesonic speed to fom an equatoial concentation of mass close to the inteio bounday of the disk. The mass concentation educes the adial velocity which in tun diminishes the ate of accetion of gas fom the disk to the cental potosta fom the inne bounday. The flow paamete scaling and the equations of motion used fo the simulation ae descibed in the next two sections. Some peliminay esults fom the flow simulation is discussed in the last section. 2. Nondimensionaliation Let us begin by denoting all dimensional quantities by tildes. The basic paametes of the poblem ae the otation ate of the paent cloud ( Ω 0 ), the initial cloud tempeatue ( T 0 ), the time since the beginning of the collapse ( t 0 ), the specific heat capacity of the gas at constant pessue ( c p ), the atio of specific heats (γ) and the gavitational constant ( G). Fom these basic paametes we can deive othe paamete. These ae the gas constant ( R = c p (γ 1)/γ), the initial speed of sound (ã 0 = γ R T 0 ), the mass accetion ate ( Ṁ) fom Eq. (1.7), and the mass of the cental sta ( M = Ṁ t 0 ). The fou paametes used to scale the flow ae G M, c p, Ṁ and the initial adius of the disk R d obtained fom Ω 0 via Eq. (1.1). The initial conditions of the flow is expessed in these fou paamete. Quantities ae nondimensionalied as follows ρ = 1/2 ρ( G M) R3/2 d, (u,u θ,u )= (ũ, ũ θ, ũ ) (2.1) Ṁ Ṽ k P = P ρṽ 2 k, T = T c p Ṽ 2 k, Φ= Φ Ṽ 2 k (2.2) = Rd, = Rd, t = ( t t 0 )Ṽk R d, (2.3)
4 212 Olusola C. Idowu whee G M Ṽ k =, (2.4) R d The initial Mach numbe of the paent cloud is given as G M Ma = (γ R) R (2.5) d T0 Values used fo the simulation discussed in this epot ae G = N.m 2 /kg 2, ξ =3, M = kg (0.1M ), R d = m(0.14au), T 0 =20K (initial cloud tempeatue), and γ R = m 2 /(s 2 K) (gas constant fo molecula Hydogen). This gives the a cloud otation ate Ω 0 = s 1, mass accetion ate Ṁ = M /y and Mach numbe Ma = Equations of motion and numeical methods of solution The flow of the gas within the disk is govened by the compessible Eule equations with a gavitational foce due to a cental point mass: Continuity equation ρ t + 1 Momentum equations t (ρu )+ 1 (ρu2 )+ 1 t (ρu θ)+ 1 (ρu u θ )+ 1 t (ρu )+ 1 Enegy equation whee t (e)+1 (ρu )+ 1 θ (ρu θ)+ (ρu ) = 0 (3.1) θ (ρu θu )+ (ρu u ) ρu2 θ (ρu u )+ 1 (u (e + P )) + 1 θ (ρu2 θ)+ (ρu θu )+ ρu u θ = P ρφ P θ = 1 θ (ρu θu )+ (ρu2 )= P ρφ θ (u θ(e + P )) + (u (e + P )) = 0 (3.2) (3.3) (3.4) (3.5) e = ρt γ + ρ 2 (u2 + u 2 θ + u 2 )+ρφ (3.6) is the total enegy. The gavitational potential tem is given as GM Φ= (3.7) ( ) 1/2 As a pelude to pope modeling of adiative cooling, we assumed the flow is isothemal i.e eplacing the enegy equation with T = T Numeical methods, bounday and initial conditions The numeical methods used fo solving the above equations ae simila to those used in studies of compessible jet flow by Feund (1997) and his code was used as a stating
5 Stuctue and evolution of a cicumstella disk 213 Figue 2. Initial velocity in a meidional () plane. The potosta is on the left of the diagam. ( min, min) =(0.1, 3.5), ( max, max) =(2.0, 3.5) point. Spatial discetiation in all diections was done using the sixthode Padélike scheme developed by Lele (1992). The details of the discetiation schemes ae discussed by Feund (1997). The fouthode RungeKutta method was used fo time advancement. To ensue stability we compute a time step based on the CouantFiedichsLewy (CFL) citeion. These numeical schemes wee tested fo diffeent steadystate compessible gas flows: solid body otation, adially conveging flow (nole flow), and adiabatic flow of gas in vetical hydostatic balance between themal pessue and the component of gavity. At the top, bottom, and oute adial boundaies of the computational domain, the flow may be locally supesonic o subsonic depending on the choice of initial conditions. To detemine the local flow at these boundaies, we apply the noneflecting bounday condition of Giles (1990) using the Cassen and Moosman flow (descibed ealie) as the efeence flow. To stat the simulation we need an initial condition which we expect will fom a steady o statistically stationay state. Fo the esults shown hee we assume that the initial condition is the same as the Cassen and Moosman flow except that to eliminate collisions of gas paticles at the midplane of the disk, we diminished the axial velocity Eq. (1.9) to eo at the midplane using the function f() = tanh(/ɛ max ) whee ɛ =0.3. We did not specify any bounday condition at the inne bounday because the initial Cassen and Moosman flow at this bounday is adially supesonic. We assumed that the gas is initially isothemal. The initial velocity field and density ae shown in Figs. 2 and 3 espectively. 4. Results We obseved that the gas initially inside the disk aces towads the sta to fom an equatoial concentation of mass close to the inne bounday of the disk (Fig. 4). The mass concentation significantly educes the adial velocity within the same egion (Figs. 5, 6 and 7). As the simulation pogesses, the flow at the inne bounday eventually becomes subsonic, making ou assumption of a supesonic outflow at the inne bounday invalid: at this point we stopped the simulation. It is possible that the equatoial concentation
6 214 Olusola C. Idowu Figue 3. Initial log of density contous. The maximum value is and the minimum is ( min, min) =(0.1, 3.5), ( max, max) =(2.0, 3.5) Figue 4. Log of density contous in meidional plane showing the equatoial concentation of density at the inne bounday. ( min, min) =(0.1, 3.5), ( max, max) =(2.0, 3.5) of mass is a tansient behavio and it will eventually be pushed into the potosta when thee is sufficient build up of mass within the disk. To answe this question popely we need to adopt a moe suitable bounday condition fo the inne bounday of the disk. This bounday condition should allow waves popagating fom inside the disk to leave the computational domain and disallow any incoming waves. The Giles bounday condition used fo the othe bounday cannot be used fo this because a efeence flow at the inne bounday is unknown. The mass flux at the inne bounday of the computational domain (Ṁout) was educed significantly (Fig. 8) because of the deceasing adial velocity at the inne bounday. The quantity plotted in Fig. 8 should be unity at steady state. Clealy, the calculation is vey fa fom such a state. Ou futue eseach effots will fist concentate on the pocesses involved in the appoach of a cicumstella disk to steady state and then study instabilities and tubulence evolving fom this state.
7 Stuctue and evolution of a cicumstella disk 215 Figue 5. Contou plot of the adial velocity at time t = Coodinates of the cones of the computational box ae ( min, min) =(0.1, 3.5), ( max, max) =(2.0, 3.5) 0 50 U/c Figue 6. Evolution of adial velocity at the inne bounday nomalied by the local speed of sound. : t =0.0000; : t =0.0004; : t =0.0009; : t =0.0014; : t = Acknowledgments The autho gatefully acknowledges D. Kaim Shaiff and D. Steven Stahle fo valuable discussions and contibutions duing the couse of this wok. REFERENCES Cassen P. & Moosman A On the fomation of potostella disks, ICARUS 48, Stahle S.W., Koycansky D.G., Bothes M.J., & Touma J 1994 The ealy evolution of potostella disks, Astophys. J. 431,
8 216 Olusola C. Idowu 0 50 U/c Figue 7. Nomalied adial velocity at the midplane at diffeent times. : t = ; : t =0.0004; : t =0.0009; : t =0.0014; : t = Ṁout Figue 8. Mass flux at the inne adial bounday nomalied by the total mass influx into the computational domain. t Teebey S., Shu F.H., & Cassen P The collapse of the coes of slowly otating isothemal clouds, Astophys. J. 286, Shu F.H Selfsimila collapse of isothemal sphees and sta fomation, Astophys. J. 214, Ulich, R.K An infall model fo the T Taui phenomenon, Astophys. J. 210, Feund, J.B., Moin,P.&Lele, S.K Compessibility effects in a tubulent annula mixing laye, Dept. of Mech. Eng., Stanfod Univesity Repot No. TF 72 Lele, S.K Compact finite diffeence schemes with spectallike esolution, J. Comp. Phys. 103,
9 Stuctue and evolution of a cicumstella disk 217 Giles, M.B Noneflecting bounday conditions fo Eule equation calculations, AIAA J. 18,
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