Lesson 3: Isothermal Hydrostatic Spheres. B68: a self-gravitating stable cloud. Hydrostatic self-gravitating spheres. P = "kt 2.

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1 Lesson 3: Isothermal Hydrostatic Spheres B68: a self-gravitating stable cloud Bok Globule Relatively isolated, hence not many external disturbances Though not main mode of star formation, their isolation makes them good test-laboratories for theories Hydrostatic equilibrium: Assumptions: isothermal cloud spherical symmetry molecular material (µ=.3) dp(r) GM (r)#(r) = " dr r Equation of state (ideal gas): Gravitational potential: P = "kt = "c s µm p dm (r) = 4"r #(r) dr 1

2 The boundary condition problem These equations can be directly integrated from the center, choosing ρ(0). But if we want to construct a cloud with given mass and given external pressure, we must try many ρ(0) to get a match. Example solutions are shown on the next slide. Numerical solutions: Numerical solutions: Plotted logarithmically (which we will usually do from now on) Bonnor-Ebert Sphere

3 Numerical solutions: Different starting ρ o : a family of solutions Numerical solutions: Singular isothermal sphere (limiting solution) Numerical solutions: Boundary condition: Pressure at outer edge = pressure of GMC 3

4 Numerical solutions: Another boundary condition: Mass of clump is given Must replace One boundary ρ c inner condition BC with one too of many outer BCs Summary of BC problem: For inside-out integration the paramters are ρ c and r o. However, the physical parameters are M and P o We need to reformulate the equations: Write everything dimensionless Consider the scaling symmetry of the solutions All solutions are scaled versions of each other 4

5 Scale-free formulation dp(r) GM (r)#(r) = " dr r d"(r) "(r)dr = d ln" GM (r) = # dr r c s " = " c e #$ 1/ % " = 4#G$ ( c ' * r & c s ) " = # /c s % 1 d ) M (r) = #(r) ' 4"r dr ' & *, $ 1 - M (r) = $ r dp(r) 4"r -r (' #(r)g dr + ' r #(r)g -P(r) = #(r) -r Scale-free formulation substituting ξ and Ψ yields the Lane-Emden equation 1 d [ " d" " ] d# d" = e$# "(# = 0) = 0; d" d# #=0 = 0 For M(r) we can write: r % 1 ( M (r) = $ 4"s #(s)ds = ' * & 4"# 0 ) 0 1/ 3/ cs % ( ' * & G ) + d, d+ = M (+) Scale-free formulation Now we consider models with mass of the cloud M(R) where R is at the edge of the cloud, corresponding to ξ=ξ 1. We get: $ 1 ' M (R) = & ) % 4"# 0 ( 1/ 3/ cs $ ' d+ $ & ) * 1 = c ' s d+ & ) R* 1 % G ( d* *1 % G ( d* *1 and therefore: GM (R) R = c s 1 " 1 d# d" "1 5

6 Scale-free formulation Similarly: 8 P(R) = "c c s = s 4#M (R)G $ & 4 d% ) 3 1 ( + ' d$ * " c 4#" c R 3 " 0 3M (R) = $ 1 3( d% /d$) $1 e,% 1 $ 1 Numerical calculations can be used to solve the Lane-Emden equation and give R, P(R) and ρ c /ρ 0 Solutions for Ψ and ρ/ρ c Scale-free formulation It turns out that P(R) is maximal for ξ 1 =6.5 # R m = 0.76 c & s % ( $ G" ' 1/ Compare with the expression for the Jeans length % # kt ( " J = ' * & µm H G$ 0 ) 1/ 6

7 Interpretation of maximum pressure When P(R) increases, the central density ρ 0 will increase and R will decrease (fixed M). Cloud mass becomes more and more concentrated toward the center At P(R) =P max gravity is about to overtake If we increase the pressure, cloud starts to collapse and a smaller P(R) would be required for stability Start of an inside-out collapse. Dimensionless mass Dimensionless mass: 7

8 Stability of BE spheres Many modes of instability One is if dp o /dr o > 0 Run-away collapse, or Run-away growth, followed by collapse Dimensionless equivalent: dm/d(ρ c /ρ o ) < 0 unstable unstable Stability of BE spheres Maximum density ratio =1 / 14.1 Bonnor-Ebert mass m 1 = 1.18 Ways to cause BE sphere to collapse: Increase external pressure until M BE <M Load matter onto BE sphere until M>M BE 8

9 A simple numerical model Temperature: 30 K Outer radius: 5000 AU Initial condition: BE sphere with ρ c = 1.x10-17 g/cm 3 ρ(r) A simple numerical model A more `realistic non-static model: Make perturbation, but keep mass the same. ρ(r) A simple numerical model ρ(r) Strong wobbles, but it remains stable 9

10 A simple numerical model Now add a little bit of mass (10%) to nudge it over the BE limit: ρ(r) Cloud collapses in a global way (not really inside-out) BE Sphere : Observations of B68 Alves, Lada, Lada 001 Magnetic field support / ambipolar diff. As mentioned in previous chapter, magnetic fields can partly support cloud and prevent collapse. Slow ambipolar diffusion moves fields out of cloud, which could trigger collapse. Models by Lizano & Shu (1989) show this elegantly: Magnetic support only in x-y plane, so cloud is flattened. Dashed vertical line is field in beginning, solid: after some time. Field moves inward geometrically, but outward w.r.t. the matter. 10

11 Magnetic field support In presence of B-field, the stability analysis changes. Magnetic fields can provide support against gravity. Replace Jeans mass with critical mass, defined as: M cr = 0.1 " $ B ' $ M R ' G 1/ #103 M sun & )& ) % 30µG( % pc( Magnetic field support Consider an initially stable cloud. We now compress it. The density thereby increases, but the mass of the cloud stays constant. Jeans mass decreases: M J " 1 # If no magnetic fields: there will come a time when M>M J and the cloud will collapse. But M cr stays constant (magnetic flux freezing) So if B-field is strong enough to support a cloud, no compression will cause it to collapse. Ambipolar diffusion Forces acting: f L = " 1 8# $B f drift = n n m n n i v% in v drift 11

12 Drag force: Collisions between ions and neutrals. The rate of collisions between ions and neutrals per neutral atom is: n i v" in σ in elastic scattering coefficient v relative velocity of ions as seen from neutral rest frame n I number density of ions <..> average over distribution function ions Momentum transfer: m i ( v r i " v r # m n ) n & % ( $ m n + m i ' Momentum transfer ions -> neutrals per unit volume " m f drag = n n m i % n $ '( v r i ( v r n )n i v) in # m n + m i & with v i -v n =v drift and m n +m i m i f drag = n n m n n i v" in v drift Example: infinite cylinder of uniform density f grav = "G#r# % & f drift = n i m n n n v$ in v drift ' v = drift "G# r n i n n m n v$ in drift timescale t drift = r v drift assume low ionization rate, neutrals are dominated by H, He $ " = ( n i m i + n n m n ) # n H m H 1+ 4 n ' He & ) % ( n H 1

13 Example: infinite cylinder of uniform density t drift = v" $ in & #Gm H % for <vσ in > cm 3 s -1 n i n H ' 1 ) ( 1+ 4n He /n H ( ) $ t drift " 5 #10 13 & % n i n H HI cloud: n i /n H Ambipolar diffusion not important Dense cloud: n i /n H t drift ~ 5 x 10 6 yr t drift >>t ff, so ambipolar diffusion not important during collapse ' ) years ( Ambipolar diffusion Magnetic pressure builds up during cloud contraction. Ambipolar diffusion acts to reduce magnetic pressure Contraction continues Once collapse set in, magnetic field remains frozen into the matter. 13

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