PROBLEM SET #3A. A = Ω 2r 2 2 Ω 1r 2 1 r2 2 r2 1

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PROBLEM SET #3A AST242 Figue 1. Two concentic co-axial cylindes each otating at a diffeent angula otation ate. A viscous fluid lies between the two cylindes. 1. Couette Flow A viscous fluid lies in the space between two infinitely long, concentic co-axial cylindes of adius 1 and 2 > 1.

1 PROBLEM SET #3A AST242 Figue 1. Two concentic co-axial cylindes each otating at a diffeent angula otation ate. A viscous fluid lies between the two cylindes. 1. Couette Flow A viscous fluid lies in the space between two infinitely long, concentic co-axial cylindes of adius 1 and 2 > 1. The cylindes otate with angula speeds of Ω 1 and Ω 2, espectively. We assume a high fluid viscosity. Such flow becomes unstable to fomation of Taylo votices if Ω 1 Ω 2. (a) Using the φ component of the Navie Stokes equation in cylindical coodinates, show that in steady state the fluid otates with tangential velocity (1) u φ = A + B whee constants (2) (3) A = Ω Ω B = (Ω 1 Ω 2 ) Assume that u = u z = 0 and that the flow is steady state and axisymmetic. (b) Assuming an incompessible flow find the diffeence in pessue between the inne and oute cylinde fo the limit 2 1 much smalle than 1 o 2. Use the adial 1

2 2 PROBLEM SET #3A AST242 component of the Navie Stokes equation. To do this expand A and B in tems of a whee a = 2 1 and a 1. Then P (the pessue diffeence) is P a P. In cylindical coodinates the components of the Navie Stokes equations ae u t + u u + u φ u φ + u u z z u2 φ = 1 [ p 1 ρ + ν ( ) φ z 2 1 ] 2 u u φ t + u u φ + u φ u φ φ + u u φ z z + u φu u z t + u u z + u φ u z φ + u u z z z ν 2 u φ 2 φ = 1 p ρ φ + ν +ν 2 2 u φ = 1 p ρ z + ν Note the non-tivial exta tems aising fom the coodinates. [ 1 [ 1 ( ) φ z ( ) φ z 2 ] u z ] u φ 2. Dynamic and kinematic viscosity The dynamic and kinematic shea viscosities, µ and ν ae elated by µ = ρν whee ρ is the density. Kinematic theoy allows us to estimate that ν vλ whee v is a themal velocity and λ is the mean fee path. It may be useful to define ρ = mn whee m is the mass of the paticles and n is the numbe of paticles pe unit volume. Show that the dynamic shea viscosity is not dependent on the paticle numbe density, n, if the tempeatue and collision coss section don t vay. 3. Accetion and Excetion Disks By combining the φ component of the Navie Stoke s equation and consevation of mass in cylindical coodinates it is possible to show that the basic equation that descibes viscous evolution of an accetion disk aound a point mass is Σ (4) t = 3 [ R 1/2 ] R R R (νσr1/2 ) whee Σ(R, t) is the disk suface density (mass pe unit aea) as a function of adius and time. Hee ν is the kinematic viscosity (units cm 2 /s). (a) Conside a steady state disk with Σ t = 0. In class we found Ṁ 3πΣν fo a steady state Kepleian disk. Show that when Σν is constant (and independent of adius) the equation above gives a steady state solution. Ae thee othe situations that would give a steady state solution?

3 PROBLEM SET #3A AST242 3 (5) The moe geneal equation fo angula momentum tanspot can be witten j t + 1 R R (ju R) = 1 R R (R3 νσ dω dr ) whee the angula momentum pe unit aea j = RΣu φ. The second tem on the left is the angula momentum flux caused by adial tanspot. Hee we have not assumed Kepleian otation (and consequently the equation contains a facto of dω/d). The tem on the ight hand side can be thought of as a toque pe unit aea j t. With vanishing viscosity, ν = 0, we expect that thee is no angula momentum tanspot o u R = 0. (b) Find a condition on the disk that would lead to excetion athe than accetion. This means the adial velocity u R is positive instead of negative. Recently excetion disks have been consideed fo cicumstella disks extenally tuncated by photo-evapoation (poplyds) and fo ou sola system (wok by Steve Desch). Bill Wad and Robin Canup have consideed them in the context of a pimodial cicum-jovian accetion disk. (c) Conside the adial component of the Navie Stokes equation in steady state. u R (6) u R R + u φ u R R φ u2 φ R = 1 Σ p R Φ R We can neglect the viscous tem because we expect the velocity shea only gives a stong viscous foce in the φ diection, not in the adial diection. We do not expect stong dependence on φ so deivatives with espect to φ can be dopped. Show that in the limit of u φ u R (7) u 2 φ = v2 c + c 2 R Σ s Σ R whee c s is the sound speed and v c is the velocity of a paticle in a cicula obit not affected by the gas (Kepleian velocity); v c R Φ R. The mean tangential velocity is slightly below that of the velocity of a paticle in a cicula obit if the density dops with inceasing adius. 4. Head winds in the minimum mass sola nebula Many papes efe to a suface density that is called the Minimum mass sola nebula. This is estimated fom the masses and spacing of the 4 giant planets in ou Sola

4 4 PROBLEM SET #3A AST242 system. The gas density (8) Σ gas = 2400 The dust density (9) Σ dust = 10 ( ) R 3/2 g cm 2 1 AU ( ) R 3/2 g cm 2 1 AU The density of solids (ices) is 3-4 times that of the dust. The above ae by Hyashi, C. 1981, Pog. Theo. Physics Supp. 70, 35. (a) Use equation 7 to estimate the diffeence between the cicula velocity of a paticle in a cicula obit about the Sun and the gas in the minimum mass sola nebula. Use a disk aspect atio of h/r = 0.1 and give you velocity diffeence in units of the cicula velocity. (b) Using an exponential scale height and h/r = 0.1, estimate the gas density, ρ gas, in the midplane at R = 1 AU. Figue 2. A planetesimal moving in a disk is moving faste than the ambient gas so it feels a headwind. The dag foce caused by this headwind exets a toque on the planetesimal causing it to spial inwads. The dag foce on a planetesimal that is embedded in a gas disk (10) F D = 1 2 C Dρ gas πs 2 v 2 whee C D is a dag coefficient, ρ gas is the gas density, v is the diffeence in velocity between the gas and the planetesimal and s is the adius of the planetesimal. (Note the dag foce depends on the aea of the object; hee A = πs 2 ). A planetesimal may be obiting at the Kepleian speed, howeve equation 7 implies that when the suface density of the gas dops with inceasing adius then the gas is moving slowe than the Kepleian speed. Consequently a planetesimal in a cicula obit

5 PROBLEM SET #3A AST242 5 would feel a headwind. This headwind would emove angula momentum fom the planetesimal causing it to spial inwads. The toque can be estimated fom the dag foce. The angula momentum of the planetesimal is mrv c whee m is the mass of the planetesimal, R is the adius fom the sta and v c the Kepleian velocity. (c) Show that a slowly in-spialing planetesimal in a nealy cicula obit at adius that is difting inwads at a speed Ṙ looses angula momentum at a ate v c (11) L m 2 Ṙ whee v c is the Kepleian velocity of a paticle in a cicula obit. Using the toque caused by the dag foce show that (12) t inspial = Ṙ R ( ρd ρ gas whee ρ d is the density of the planetesimal. ) svc C D v 2 (d) Assuming dag coefficient C D 1 and planetesimal density ρ d 1 g cm 3 estimate a timescale in yeas fo the inspial R/Ṙ of a mete sized planetesimal in the minimum mass sola nebula at 1AU. You timescale should be shote than a thousand yeas, pesenting a challenging poblem fo cuent planetesimal fomation models. 5. Radial tempeatue pofiles fo accetion disks Conside an accetion disk with an accetion ate sufficiently high that its themal stuctue is due to enegy viscously dissipated in the disk. (13) νσω 2 σ SB T 4 whee ν is the viscosity, (14) Ω = GM/R 3 the angula otation ate, M the mass the cental object, Σ the mass suface density, σ SB the Stefan-Boltzmann constant. The above tempeatue is that of the suface if the disk is optical thick, othewise it is appoximately the aveage tempeatue. The quantities Σ, ν, T, Ω can vay with adius R. (a) If the disk is optically thin how would its tempeatue scale with adius? In othe wods T is popotional to R to what powe? Assume a steady accetion ate independent of adius Ṁ 3πΣν and use equations 13, 14. (Do not assume a minimum mass Sola nebula.) (b) If the disk is optically thick but the disk opacity κ is independent of tempeatue and density how would its mid-plane tempeatue scale with adius? Assume viscous foces dissipate enegy in the disk mid-plane but the suface tempeatue is

6 6 PROBLEM SET #3A AST242 set by equation 13. Assume an α disk with α independent of adius and viscosity set by the popeties of the gas in the midplane. Assume Ṁ 3πΣν. The opacity of the disk elates the mid-plane and suface tempeatues. It may be useful to emembe that the sound speed c s T 1/2, how hydostatic equilibium elates c s, h, Ω whee h is a scale height and Ω is the angula otation ate (equation 14) and how viscosity ν is defined fo an α disk.

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