Star and Planet Formation Sommer term 2007 Henrik Beuther & Sebastian Wolf 16.4 Introduction (H.B. & S.W.) 23.4 Physical processes, heating and cooling, radiation transfer (H.B.) 30.4 Gravitational collapse & early protostellar evolution I (H.B.) 07.5 Gravitational collapse & early protostellar evolution II (H.B.) 14.5 Outflows and jets (H.B.) 21.5 Pre-main sequence evolution, stellar birthline (H.B.) 28.5 Pfingsten (no lecture) 04.6 11.6
18.6 25.6 02.7 09.7 16.7 23.7 Clusters, the initial mass function (IMF), massive star formation (H.B Protoplanetary disks: Observations + models I (S.W.) Gas in disks, molecules, chemistry, keplerian motions (H.B.) Protoplanetary disks: Observations + models II (S.W.) Accretion, transport processes, local structure and stability (S.W.) Planet formation scenarios (S.W.) Extrasolar planets: Searching for other worlds (S.W.) Summary and open questions (H.B. & S.W.)
ore Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss07.html and http://www.mpia.de/homes/swolf/vorlesung/sommer2007.htm Emails: [email protected], [email protected] Last week Molecular line emission, line formation Dust continuum emission Heating and cooling processes Radiation transfer Column density calculations Applications I: Molecular gas structure of the Galaxy based on CO observations
Applications II: Molecular outflows Applications III: Temperature estimates from NH3 Tkin = Trot (11 22T11T22) Star Formation Paradigm Isothermal sphere, Hydrostatic equilibrium, Bonnor-Ebert spheres Jeans analysis Virial equilibrium Rotational support of molecular cloud stability Magnetic field support of molecular cloud stability
Isothermal Sphere I Isothermal Sphere II --> Numerical integration Boundary conditions: (0) = 0 (0) = 0 Gravitational potential and force are 0 at the center. Isothermal Sphere III
Density and pressure (P=a2) drop monotonically away from the center --> important to offset inward pull from gravity for grav. collapse. After numerical integration of the Lane-Emden equation, one finds that the density /c approaches asymtotically 2/2. Hence the dimensional density profile of the isothermal sphere is: (r) = a2/(2Gr2) Isothermal Sphere IV With: r = (at2/(4GGc))* = c exp() Isothermal Sphere V The beginning is for a radius 0=0, hence c/0=1 and m=0.
For increasing c/0 m then increases until c/0=14.1, corresponding to the dimensionless radius 0=6.5. Gravitational stability Gravity dominated Pressure dominated ow-density contrast cloud: Increasing outer pressure P 0 causes a rise of m and c/0. With the internal pressure P=at2, it rises more strongly than P0 and the cloud remains stable. nce the physical radius r0 is related to 0 and c like r0 = sqrt(at2/(4Gc)) * 0 and c increases faster than 0, the core actually shrinks with
increasing outer pressure P0. The same as Boyle-Mariotte law for ideal g PV=const. --> P * 4/3r3 = const. l clouds with c/0> 14.1 (0=6.5) are gravitationally unstable, and the critical mass is the Bonnor-Ebert mass (Ebert 1955, Bonnor 1956) Gravitational stability: The case of B6 Optical 0=6.9 is only marginally about the critical value 6.5 gravitational stable or at the verge of collapse
Near-Infrared Jeans analysis I The previous analysis implies that clouds from certain size-scales upwards are prone to collapse --> Jeans analysis early 20th century A travelling wave in an isothermal gas can be described as: (x,t) = 0 + exp[i(kx - t)] wave number k=2/ Using this in all previous equations of the hydrostatic isothermal gas, one gets the so-called dispersion equation 2 = k2at2 - 4G0 For large k high-frequency disturbances the wave behaves like sound wave =kat isothermal sound speed of background 0=(4G0)1/2
k0 = 0/at Solid line: dispersion relation Dashed line: However, for low k (k<=k0) =0. The corresponding Jeans-length is J = 2/k0 = (at2/G0) Perturbations larger J have exponentially growing amplitudes --> instabel Using 0 instead P0, Bonnor-Ebert mass MB is rather known as Jeans-Mass MJ 3 1/2
3/2 Jeans analysis II This corresponds in physical units to Jeans-lengths of J = (at2/G0) = 0.19pc (T/(10K))1/2 (nH2/(104cm-3)-1/2 and Jeans-mass MJ = m1at3/(01/2G3/2) = 1.0Msun (T/(10K))3/2 (nH2/(104cm-3)-1/2 Clouds larger J or more massive than MJ may be prone to collapse. Conversely, small or low-mass cloudlets could be stable if there is sufficient extrenal pressure. Otherwise only transient objects. Example: a GMC with T=10K and nH2=103cm-3 MJ = 3.2 Msun
Orders of magnitide too low. Additional support necessary, e.g., magnetic field, turbulence Virial Analysis What is the force balance within any structure in hydrostatic equilibrium The generalized equation of hydrostatic equlibrium including magnetic fields B acting on a current j and a convective fluid velocity v is: dv/dt = -grad(P) - grad(g) + 1/c j x B Employing the Poisson equation and requiring mass conservation, one gets after repeated integrations the VIRIAL THEOREM
1/2 (2I/t2) = 2T + 2U + W + M I: Moment of inertia, this decreases when a core is collapsing (m*r 2) T: Kinetic energy U: Thermal energy W: Gravitational energy M: Magnetic energy All terms except W are positive. To keep the cloud stable, the other forc have to match W. Application of the Virial Theorem I If all forces are too weak to match the gravitational energy, we get 1/2 (2I/t2) = W ~ -Gm2/r
Approximating further I=mr2, the free-fall time is approximately tff ~ sqrt(r3/Gm) Since the density can be approximated by =m/r3, one can also write tff ~ (G)-1/2 Or more exactly for a pressure-free 3D homogeneous sphere tff = (3/32G)1/2 For a giant molecular cloud, this would correspond to tff ~ 7*106 yr (m/105Msun)-1/2 (R/25pc)3/2 For a dense core with ~105cm-3 the tff is approximately 105 yr. However, no globally collapsing GMCs observed --> add support! Application of the Virial Theorem II f the cloud complexes are in approximate force equilibrium, the momen of inertia actually does not change significantly and hence 1/2 (2I/t2)=0
2T + 2U + W + M = 0 his state is called VIRIAL EQUILIBRIUM. What balances gravitation W bes Thermal Energy: Approximating U by U ~ 3/2nkBT ~ mRT/ U/|W| ~ mRT/ (Gm2/R)-1 = 3*10-3 (m/105Msun)-1 (R/25pc) (T/15K) --> Clouds cannot be supported by thermal pressure alone! agnetic energy: Approximating M by M ~ B2r3/6(cloud approximated as sph M/|W| ~ B2r3/6(Gm2/R)-1 = 0.3 (B/20G)2 (R/25pc)4 (m/105Msun)-2 --> Magnetic force is important for large-scale cloud stability! Application of the Virial Theorem III
The last term to consider in 2T + 2U + W + M = 0 is the kinetic energy T T/|W| ~ 1/mv (Gm2/R)-1 = 0.5 (v/4km/s) (M/105Msun)-1 (R/25pc) nce the shortest form of the virial theorem is 2T = -W, the above numbe imply that a typical cloud with linewidth of a few km/s is in approximate virial equilibrium. he other way round, one can derive an approximate relation between th Observed line-width and the mass of the cloud: 2T = 2* (1/mv) = -W = Gm2/r virial velocity: vvir = (Gm/r)1/2 or virial mass: mvir = v2r/G Basic rotational configurations I
Adding a centrifugal potential cen, the hydrodynamic equation reads -1/grad(P) - grad(g) - grad(cen) = 0 With cen defined as cen = - (j2/3) dj: angular momentum : cylindrical radius and j=u with u the velocity around the rotation axis Rotation flattens the cores and can be additional source of support against Basic rotational configurations II Isothermal sphere, =0
d to the previously discussed Bonnor-Ebert models, these rotational ow have in addition to the density contrast c/0 the other parameter quantifies the degree of rotation. is defined as the ratio of rotational ational energy = 0R0/(3Gm) with 0 the angular velocity and R0 the initial cloud radius corresponds to breakup speed of the cloud. So 0 < < 1/3 Basic rotational configurations III In realistic clouds, for flattening to appear, the rotational energy has to be at least 10% of the gravitational energy. Trot/W
equals approximately (which was defined for the spherical case). Examples: nse cores: aspect ratio ~ 0.6. Estimated Trot/W ~ 10-3 Cs: Velocity gradient of 0.05km/s representing solid body rotation, 200M and 2pc size imply also Trot/W ~ 10-3 --> Cloud elongations do not arise from rotation, and centrifugal forc NOT sufficient for cloud stability! Magnetic fields I Object Type
Diagnostic |B ||| [G] =================================== Ursa Major Diffuse cloud HI 10 NGC2024 GMC clump OH 87 S106 HII region
OH 200 W75N Maser OH 3000 Increasing magnetic field strength with increasing density indicate fieldfreezing between B-field and gas (B-field couples to ions and electrons, and these via collisons to neutral gas). Magnetic fields II This field freezing can be described by ideal MHD:
owever, ideal MHD must break down at some point. Example: ense core: 1Msun, R0=0.07pc, B0=30G Tauri star: R1=5Rsun If flux-freezing would hold, BR2 should remain constant over time --> B1=2x107 G, which exceeds observed values by orders of magnitude mbipolar diffusion: neutral and ionized medium decouple, and neutral ga an sweep through during the gravitational collapse. Magnetic fields III The equation for magneto-hydrodynamic equilibrium now is: -1/grad(P) - grad(g) -1/(c) j x B = 0
Solving the equations again numerically, one gets solutions with 3 fre parameters: the density contrast ratio c/0, the ratio between magnetic to thermal pressure = B02/(8P0) and the dimensionless radius of the initial sphere 0 = (4G0/at2)1/2 * R0 stable unstable Magnetic fields IV Include magnetic field Isothermal sphere
Include rotation he two models on previous slide represent a stable and an unstable case good fit to the numerical results is given by: mcrit = 1.2 + 0.15 1/2 02 Magnetic fields V onverting this to dimensional form (multiply by at4/(P01/2G3/2)), the first ter quals the Bonnor-Ebert Mass (MBE = m1at4/(P01/2G3/2)) Mcrit = MBE + Mmagn with Mmagn = 0.15 1/2 02at4/(P01/2G3/2) = 0.15 2/sqrt(2G) (B0GR02/G1/2) B0
--> the magnetic mass Mmagn is proportional to the B-field! here is a qualitative difference between purely thermal clouds and agnetized clouds discussed here. If one increases the outer pressure P 0 ound a low-mass core of mass M, the Bonnor-Ebert mass will decrease ntil MBE < M, and then the cloud collapses. However, in the magnetic case M < Mmagn the cloud will always remain stable because Mmagn is constant s long a flux-freezing applies. Star and Planet Formation Sommer term 2007 Henrik Beuther & Sebastian Wolf 16.4 Introduction (H.B. & S.W.) 23.4 Physical processes, heating and cooling, radiation transfer (H.B.)
30.4 Gravitational collapse & early protostellar evolution I (H.B.) 07.5 Gravitational collapse & early protostellar evolution II (H.B. 14.5 Outflows and jets (H.B.) 21.5 Pre-main sequence evolution, stellar birthline (H.B.) 28.5 Pfingsten (no lecture) 04.6 11.6 18.6 25.6 02.7 09.7 16.7 23.7
Clusters, the initial mass function (IMF), massive star formation (H.B Protoplanetary disks: Observations + models I (S.W.) Gas in disks, molecules, chemistry, keplerian motions (H.B.) Protoplanetary disks: Observations + models II (S.W.) Accretion, transport processes, local structure and stability (S.W.) Planet formation scenarios (S.W.) Extrasolar planets: Searching for other worlds (S.W.) Summary and open questions (H.B. & S.W.) ore Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss07.html and http://www.mpia.de/homes/swolf/vorlesung/sommer2007.htm Emails: [email protected], [email protected]
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