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 Clusters, the initial mass function (IMF), massive star formation (H.B.) 11.6 Protoplanetary disks: Observations + models I (S.W.) 18.6 Gas in disks, molecules, chemistry, keplerian motions (H.B.) 25.6 Protoplanetary disks: Observations + models II (S.W.) 02.7 Accretion, transport processes, local structure and stability (S.W.) 09.7 Planet formation scenarios (S.W.) 16.7 Extrasolar planets: Searching for other worlds (S.W.) 23.7 Summary and open questions (H.B. & S.W.) More Information and the current lecture files: and s:

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 NH 3 T kin = T rot (  11  22 T 11 T 22 )

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 Boundary conditions:  (0) = 0  ’(0) = 0 Gravitational potential and force are 0 at the center. --> Numerical integration

Isothermal Sphere III Density and pressure (P=  a 2 ) 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) = a 2 /(2  Gr 2 ) 

Isothermal Sphere IV With: r = √(a t 2 /(4πG  c ))*   =  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 Low-density contrast cloud: Increasing outer pressure P 0 causes a rise of m and  c /  0. With the internal pressure P=  a t 2, it rises more strongly than P 0 and the cloud remains stable. Since the physical radius r 0 is related to  0 and  c like r 0 = sqrt(a t 2 /(4  G  c )) *  0 and  c increases faster than  0, the core actually shrinks with increasing outer pressure P 0. The same as Boyle-Mariotte law for ideal gas: PV=const. --> P * 4/3  r 3 = const. All 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) M BE = (m 1 a t 4 )/(P 0 1/2 G 3/2 ) Pressure dominated Gravity dominated

Gravitational stability: The case of B68 Optical Near-Infrared  0 =6.9 is only marginally about the critical value 6.5  gravitational  stable or at the verge  of collapse

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 = k 2 a t  G  0 For large k high-frequency disturbances the wave behaves like sound wave  =ka t  isothermal sound speed of background However, for low k (k<=k 0 )    =0. The corresponding Jeans-length is  J = 2  /k 0 = (  a t 2 /G  0 ) Perturbations larger J have exponentially growing amplitudes --> instabel Using  0 instead P 0, Bonnor-Ebert mass M BE is rather known as Jeans-Mass M J M J = m 1 a t 3 /(  0 1/2 G 3/2 )  0  =  (4  G  0 ) 1/2 k 0 =  0 /a t Solid line: dispersion relation Dashed line:

Jeans analysis II This corresponds in physical units to Jeans-lengths of J = (  a t 2 /G  0 ) = 0.19pc (T/(10K)) 1/2 (n H2 /(10 4 cm -3 ) -1/2 and Jeans-mass M J = m 1 a t 3 /(  0 1/2 G 3/2 ) = 1.0M sun (T/(10K)) 3/2 (n H2 /(10 4 cm -3 ) -1/2 Clouds larger J or more massive than M J 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 n H2 =10 3 cm -3  M J = 3.2 M sun 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 (  2 I/  t 2 ) = 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 forces  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 (  2 I/  t 2 ) = W ~ -Gm 2 /r Approximating further I=mr 2, the free-fall time is approximately t ff ~ sqrt(r 3 /Gm) Since the density can be approximated by  =m/r 3, one can also write t ff ~ (G  ) -1/2 Or more exactly for a pressure-free 3D homogeneous sphere t ff = (3  /32G  ) 1/2 For a giant molecular cloud, this would correspond to t ff ~ 7*10 6 yr (m/10 5 M sun ) -1/2 (R/25pc) 3/2 For a dense core with  ~10 5 cm -3 the t ff is approximately 10 5 yr. However, no globally collapsing GMCs observed --> add support!

Application of the Virial Theorem II If the cloud complexes are in approximate force equilibrium, the moment of inertia actually does not change significantly and hence 1/2 (  2 I/  t 2 )=0 2T + 2U + W + M = 0 This state is called VIRIAL EQUILIBRIUM. What balances gravitation W best? Thermal Energy: Approximating U by U ~ 3/2nk B T ~ mRT/  U/|W| ~ mRT/  (Gm 2 /R) -1 = 3*10 -3 (m/10 5 M sun ) -1 (R/25pc) (T/15K) --> Clouds cannot be supported by thermal pressure alone! Magnetic energy: Approximating M by M ~ B 2 r 3 /6  (cloud approximated as sphere) M/|W| ~ B 2 r 3 /6  (Gm 2 /R) -1 = 0.3 (B/20  G) 2 (R/25pc) 4 (m/10 5 M sun ) -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/  m  v  (Gm 2 /R) -1 = 0.5 (  v/4km/s) (M/10 5 M sun ) -1 (R/25pc) Since the shortest form of the virial theorem is 2T = -W, the above numbers imply that a typical cloud with linewidth of a few km/s is in approximate virial equilibrium. The other way round, one can derive an approximate relation between the Observed line-width and the mass of the cloud: 2T = 2* (1/  m  v  ) = -W = Gm 2 /r  virial velocity: v vir = (Gm/r) 1/2  or virial mass: m vir = v 2 r/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 = - ∫ (j 2 /  3 ) d   j: 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 collapse.

Basic rotational configurations II Compared to the previously discussed Bonnor-Ebert models, these rotational models now have in addition to the density contrast  c /  0 the other parameter  which quantifies the degree of rotation.  is defined as the ratio of rotational to gravitational energy  =  0 R 0 /(3Gm) with  0 the angular velocity of the cloud  and R 0 the initial cloud radius  > 1/3 corresponds to breakup speed of the cloud. So 0 <  < 1/3 Isothermal sphere,  =0

Basic rotational configurations III Dense cores: aspect ratio ~ 0.6. Estimated T rot /W ~ GMCs: Velocity gradient of 0.05km/s representing solid body rotation, 200M sun and 2pc size imply also T rot /W ~ > Cloud elongations do not arise from rotation, and centrifugal force NOT sufficient for cloud stability! Other stability factors are necessary --> Magnetic fields In realistic clouds, for flattening to appear, the rotational energy has to be at least 10% of the gravitational energy. T rot /W equals approximately  (which was defined for the spherical case). Examples:

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 “field- freezing” 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: However, ideal MHD must break down at some point. Example: Dense core: 1M sun, R 0 =0.07pc, B 0 =30  G T Tauri star: R 1 =5R sun If flux-freezing would hold, BR 2 should remain constant over time --> B 1 =2x10 7 G, which exceeds observed values by orders of magnitude Ambipolar diffusion: neutral and ionized medium decouple, and neutral gas can 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 free parameters: the density contrast ratio  c /  0, the ratio  between magnetic to thermal pressure  = B 0 2 /(8  P 0 ) and the dimensionless radius of the initial sphere  0 = (4  G  0 /a t 2 ) 1/2 * R 0 stable unstable

Magnetic fields IV The two models on previous slide represent a stable and an unstable case. A good fit to the numerical results is given by: m crit =  1/2  0 2 Isothermal sphere Include rotation Include magnetic field

Magnetic fields V Converting this to dimensional form (multiply by a t 4 /(P 0 1/2 G 3/2 )), the first term equals the Bonnor-Ebert Mass (M BE = m 1 a t 4 /(P 0 1/2 G 3/2 )) M crit = M BE + M magn with M magn = 0.15  1/2  0 2 a t 4 /(P 0 1/2 G 3/2 ) = /sqrt(2π) (B 0 πR 0 2 /G 1/2 )  B 0 --> the magnetic mass M magn is proportional to the B-field! There is a qualitative difference between purely thermal clouds and magnetized clouds discussed here. If one increases the outer pressure P 0 around a low-mass core of mass M, the Bonnor-Ebert mass will decrease until M BE < M, and then the cloud collapses. However, in the magnetic case, if M < M magn the cloud will always remain stable because M magn is constant as 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 Clusters, the initial mass function (IMF), massive star formation (H.B.) 11.6 Protoplanetary disks: Observations + models I (S.W.) 18.6 Gas in disks, molecules, chemistry, keplerian motions (H.B.) 25.6 Protoplanetary disks: Observations + models II (S.W.) 02.7 Accretion, transport processes, local structure and stability (S.W.) 09.7 Planet formation scenarios (S.W.) 16.7 Extrasolar planets: Searching for other worlds (S.W.) 23.7 Summary and open questions (H.B. & S.W.) More Information and the current lecture files: and s: