1. Sternentstehung - Star Formation Sommersemester 2006 Henrik Beuther & Thomas Henning 24.4 Today: Introduction & Overview 1.5 Public Holiday: Tag der Arbeit 8.5 --- 15.5 Physical processes, heating & cooling, cloud thermal structure 22.5 Physical processes, heating & cooling, cloud thermal structure (H. Linz) 29.5 Basic gravitational collapse models I 5.6 Public Holiday: Pfingstmontag 12.6 Basic gravitational collapse models II 19.6 Accretion disks 26.6 Molecular outflow and jets 3.7 Protostellar evolution, the stellar birthline 10.7 Cluster formation and the Initial Mass Function 17.7 Massive star formation, summary and outlook 24.7 Extragalactic star formation More Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss06.html beuther@mpia.de

2. Summary outflows HH211, Gueth et al. 1999 Force vs. L

3. The first core I - Contraction of core via ambipolar diffusion initially slow process. - When  /B reaches critical threshold, contraction speeds up --> core becomes opaque --> cooling less efficient --> T & P rise. - Interior still mainly molecular hydrogen --> important for final collapse

4. The first core II - Temperature estimate based on viral theorem: 2T + 2U + W + M = 0 W = -2U (kinetic and magnetic energy approximated as 0) => -GM 2 /R = -3MRT/µ => T = µGM/(3R R) = 850K (M/5e-2M sun ) (R/5AU) -1 --> significantly warmer than original core - With the addition of mass and further shrinking, it soon reaches 2000K, where collisonal dissociation of H 2 starts. --> Temperature increase less steep - However, thermal energy per molecule at 2000K ~ 0.74eV compared to dissociation energy of H 2 of ~ 4.48eV --> Even modest increase of dissociated H 2 absorbs most of the gravitational energy from the collapse --> marginal increase in temperature and pressure - Region of atomic H spreads outward from center - Without significant T & P increase, the first core cannot keep equilibrium, hence the entire core becomes unstable, collapses and forms protostar. --> significant temperature and density increase, sufficient to collisionally ionize most hydrogen --> emerging protostar is now dynamically stable. - A protostar of 0.1M sun has radius of several R star, T~10 5 K and  ~10 -2 g cm -3

5. Accretion shock and Accretion luminosity - The gravitational energy released per unit accreted mass can be approximated by the gravitational potential GM * /R * - Hence the released accretion luminosity of the protostar can be approximated by this energy multiplied by the accretion rate: L acc = G(dM/dt)M * /R * = 61L sun ((dM/dt)/10 -5 M sun /yr) (M * /1M sun ) (R * /5R sun ) -1 - Additional luminosity contributions from contraction and early nuclear fusion are negligable compare to L acc for low- to intermediate-mass stars. - Conventional definition of (low-mass) protostar: “Mass-gaining star deriving most of its luminosity from accretion.” (However, caution for massive stars.).

6. Protostellar envelope I --> Dust photosphere (a few AU for typical low-mass star) is the effective radiating surface observable from outside at that evolutionary stage (not optically visible yet). - Rapid T increase in dust envelope --> dust sublimation at T~1500K. - Inside dust destruction front greatly reduced opacity, and infalling gas almost transparent to protostellar radiation --> opacity gap. - Immediately outside the accretion shock, gas gets collisonally ionized and the opacity increases again --> so-called radiative precursor - Outer envelope largely optically thin. - As infalling gas continues to be compressed (inside-out collapse), the protostellar radiation becomes trapped by high dust grain opacities. - This dust then reradiates the emission at far-infrared wavelengths. Relative dimensions of outer region greatly reduced in sketch.

7. Protostellar envelope II --> Immediate postshock temperature >10 6 K, UV and X-ray regime --> Postshock settling region opaque, quick temperature decrease --> The surface of precursor radiates as approximate blackbody in opacity gap: Stephan-Boltzmann law: L acc ~ 4πR * 2  B T eff 4 Substituting L acc => T eff ~ (GM * (dM/dt)/4πR * 3  B ) 1/4 => T eff ~ 7300K ((dM/dt)/1e-5M sun yr -1 ) (M * /1M sun ) 1/4 (R * /5R sun ) -3/4  Opacity gap is bathed is “optical emission” similar to main-sequence star. Very different to observable dust photosphere. Relative dimensions of outer region greatly reduced in sketch. - Difference in radiation from shocked “radiative precursor” and far-infrared radiation from dust photosphere - In shock region gas approaches protostar approximately at free-fall speed: 1/2mv ff = GM * m/R * => v ff = √2GM * /R * = 280 km/s (M * /1M sun ) 1/2 (R * /5R sun ) -1/2

8. Temperatures and dimensions of envelope - Temperature profile in optically thick dust envelope T(r) ~ r -0.8 - Temperature profile in optically thin outer envelope T(r) ~ r -0.4 - Typical dimensions for a 1M sun protostar: Outer envelope: a few 100 to a few 1000 AU Dust photosphere: ~ 10 AU Dust destruction front: ~ 1 AU Protostar: ~ 5 R * ~ 0.02 AU Relative dimensions of outer region greatly reduced in sketch.

9. Effects of Rotation - Model: M * = 0.5 M sun, dM/dt = 5e-6 M sun yr -1,  = 1.35e-14 s -1,  cen = 0.4 AU outside the dust destruction front. Hoever, most  luminosity stems from accretion shock on protostar or inner disk.  - Temperature structure slightly oblate within  cen because of local density  enhancement there trapping radiation and increasing temperature.  - In contrast, outside  cen T structure more prolate because of this  radiation blockage in the equatorial plane. - Full contours: T decrease outward from 1050K in 50K steps. - Dashed contours: 

10. Protostellar evolution/Stellar Structure equations - The protostellar evolution can be analyzed numerical similarly to stars. --> Stellar Structure equations - The used spatial variable is M r, the mass within shells of radius r M r = 0 ∫ r 4πr 2  dr => ∂r/∂M r = 1/(4πr 2  ) (1) Hydrostatic equilibrium: -1/  grad(P) - grad(  g ) = 0 => ∂P/∂r = -G  M r /r 2 (2) Substituting (1) in (2), one gets => ∂P/∂M r = -GM r /(4πr 4 ) (3) Pressure obeys ideal gas equation: P =  RT (4) (mean molecular weight  depends on state of ionization and is function of T and  ) Thermal structure of opaque interior is described by diffusion equation T 3 ∂T/∂M r = - 3  L int /(256π 2  B r 4 ) (5) (mean opacity  is again funtion of T and  ) And the spatial variation of the internal luminosity L int can be described by the heat equation: ∂L int /∂M r =  ( ,T) - T ∂s/∂t (6) (  ( ,T) rate of nuclear energy release; s( ,T) is the entropy) For a mono-atomic gas, the entropy is: s( ,T) = R/µ ln(T 3/2 /  ) - Using adequate boundary conditions, one can now follow numerically the protostellar evolution.

11. Mass-radius relation - In the absence of nuclear burning, an increasing s(M r ) arises naturally because with rising M *, the velocity of the infalling gas and hence the accretion shock and L acc increases. --> Protostellar radius increases with time. - Suppose initial core very large, then low infall velocity --> low L acc and s(M r ) would dip at beginning of protostellar evolution --> initial decrease of R * - Opposite effect for very small initial state. - Initial size unknown, but that does not matter since it quickly converges. - Adding additional infalling mass shells, the protostar can be described by its entropy profile s(M r ), reflecting the changing conditions at the accretion shock. - Since s represents heat content of each added mass shell, an increase of s(M R ) causes a swelling of the protostar.

12. Convection I - Displace parcel outward. P ext decreases. --> parcel expands and (  int ) 1 < (  int ) 0 - Question: If (  int ) 1 < (  ext ) 1 then parcel gets buoyant --> convection starts and becomes important for heat transfer. If (  int ) 1 > (  ext ) 1 then parcel will sink back down and star remains radiatively stable. - If parcel displacement very quick, heat loss negligable and its specific entropy s stays the same. Conversely, in the absence of nuclear burning, protostar has rising entropy profile s(M r ) --> (s int ) 1 < (s ext ) 1 - Hoever (P int ) 1 = (P ext ) 1 and for ordinary gases  ∂  /∂s) P < 0, i.e., density falls with increasing entropy at constant pressure. --> (  int ) 1 > (  ext ) 1 for a rising entropy profile. --> ∂s/∂M r > 0 implies radiative stability.

13. Convection II - However, the ratio M * /R * rises fast and interior temperatures increase. Nuclear reactions start at center (at ~0.3M sun deuterium burning at ~10 6 K). --> entropy profile overturns ∂s/∂M r < 0, radiative stability criterium not given anymore. - Convection begins because deuterium fusion produces too much energy to be transported radiatively through opaque interior. Since ∂s/∂M r < 0 parcels are underdense ((  int ) 1 < (  ext ) 1 ) and hot. After transfering excess heat to surrounding medium, denser/cooler parcels travel downward again. --> Protostellar interior is well mixed and provides its own deuterium to center for further fusion processes. - Convection is local phenomenon, some regions can be convective whereas others remain radiatively stable.

14. Deuterium burning - 2 H + 1 H --> 3 He + ∆E with ∆E ~ 5.5 MeV, important from 10 6 K - The degree of protostellar size increase depends partly on the accretion rate but the deuterium burning is more important. - The deuterium burning is very temperature sensitive. An increase of T causes more deuterium burning --> more heat --> increase of proto- stellar radius --> lower T again --> Deuterium burning acts as kind of thermostat keeping protostellar core at that evolutionarys stage at about 10 6 K. - Steady supply by new Deuterium from infalling gas via convection necessary to maintain thermostat.

15. Radiative stability again - What happens for protostars gaining more than 1-2M sun of mass? - The critical luminosity L crit is the maximum value to be caried by radiative diffusion. - Continuum opacity from free-free emission (Kramers-law opacity) scales  ff  T -7/2 --> strong decrease with T - And one finds: L crit  M * 11/2 R * -1/2 --> For growing protostars, L crit rises sharply surpassing interior luminosity. --> Convection then disappears and protostar gets radiative barrier.

16. Deuterium shell burning I - For stars more massive than about 2M sun. - Once radiative barrier has formed, no new deuterium to center and residual deuterium is consumed rapidly. - Interior luminosity L int declines below L crit and convection disappears in whole interior volume. - Deuterium accumulates in thick mantle outside radiative barrier. - With no internal fuel R * does not change much anymore --> M * /R * rises more quickly, and temperatures increases rapidly. - Base of deuterium shell reaches 10 6 K, deuterium shell burning starts and convection occurs in this shell structure.

17. Deuterium shell burning II - Deuterium shell burning is accompanied by structural change of protostar. - The shell burning injects heat and rises the entropy s of the outer layers. --> further swelling of the protostellar radius - Adding even more mass, the inevitable rise of L crit drives the radiative barrier and the associated burning layer and convection zone outward. --> The protostar (>~2M sun ) is then almost fully radiatively stable. - The previous scaling relation for L crit should apply to now interior luminosity, and one finds: L int ~ 1L sun (M * /1M sun ) 11/2 (R * /1R sun ) -1/2 Onset of full convection Appearance of radiative barrier Hydrogen fusion and second initiation of a central convection zone. Gravity becomes dominating force, R * decreases fast and Temperature T rises. At ~10 7 K H-burning will begin. --> stop contraction.

18. Protostellar vs. pre-main sequence evolution (mainly for low-mass protostars) - After the deuterium burning has ceased, the protostars start to contract quasi-statically again gaining energy from the gravitational contraction. - For low-mass protostars, the end of deuterium burning roughly coincides with the end of the main accretion phase because no additional deuterium is supplied to the core center. --> From now on, the main luminosity does not stem from the accretion shock anymore but from the gravitational quasi-static contraction. --> One can identify this point with the end of the protostellar and the beginning of the pre-main sequence phase in low-mass stellar evolution.

19. Further contraction until H-burning - After the central deuterium burning in low-mass stars and the deuterium shell-burning in intermediate- to high-mass stars has ceased, the protostars/pre-main sequence stars start to contract quasi-statically gaining energy from the gravitational contraction. - Different evolution for low- and high-mass pre-main sequence stars: - Low-mass: Shrinking releases gravitational energy while surface temperature stays approximately constant. Since L = 4πR 2  B T eff 4  R * 2, the luminosity decreases and falls below L crit. --> Radiative core forms again leaving a shrinking outer convective layer. In this radiative phase, during further slow contraction internal energy, temperature and luminosity rise again until hydrogen burning starts --> ZAMS. Stars below ~0.4M sun reach the ZAMS still fully convective. - Intermediate- to high-mass: Since more massive protostars are not convective anymore, they do not have the phase of decreasing luminosity (although they also shrink). They always gain luminosity and temperature via gravitational energy (and decreasing deuterium shell burning) until Hydrogen ignites --> ZAMS.

20. Hertzsprung Russel (HR) diagram I - The birthline was found first observationally as the locus where stars first appear in the HR diagram emanating from their dusty natal envelope. - Theoretically, one can define the birthline at the time where the main accretion has stopped (no infalling envelope anymore), and the pre-main. sequence star gains the main luminosity from gravitational contraction. - Compared with the theoretical evolution, for low-mass stars, this is about when they begin quasi-static contraction in their still convective phase (after accretion has largely ceased). Hence they move vertically downward the so- called Hayashi tracks. After the cores become radiative, they start to increase their temperature (& luminosity) moving left on the radiative tracks. Open circles: Radiative stability Full circles: Hydrogen burning

21. Hertzsprung Russel (HR) diagram II - Intermediate-mass protostars are already fully radiative when stopping accretion, hence do not have the vertical Hayashi part of the pre-main sequence tracks and move directly on the radiative tracks. - High-mass stars have short Kelvin-Helmholtz contraction time-scale and start nuclear H-burning, hence entering the ZAMS, before ending main accretion phase. Furthermore, they have no (visible) pre-main sequence evolution since the H-burning starts still deeply embedded in their natal cores. Palla & Stahler 1990 Birthline

22. Observable Spectral Energy Distributions (SEDs)

23. Summary - The “first core” contracts until temperatures are able to dissociate H 2 to H. - H-region spreads outward, T and P not high enough to maintain equilibrium, further collapse until H gets collisionally ionized. The dynamically stable protostar has formed. - Accretion luminosity. Definition of low-mass protostar can be “mass-gaining object where the luminosity is dominated by accretion”. - Structure of the protostellar envelope and effects of rotation. - Stellar structure equations: follow numerically the protostellar and then later the pre-main sequence evolution. - Convection and deuterium burning. - End of protostellar/beginning or pre-main sequence evolution --> birthline. - Pre-main sequence evolution in the Hertzsprung-Russel (HR) diagram. - Connection of HR diagram with protostellar and pre-main sequence class scheme.

24. Sternentstehung - Star Formation Sommersemester 2006 Henrik Beuther & Thomas Henning 24.4 Today: Introduction & Overview 1.5 Public Holiday: Tag der Arbeit 8.5 --- 15.5 Physical processes, heating & cooling, cloud thermal structure 22.5 Physical processes, heating & cooling, cloud thermal structure (H. Linz) 29.5 Basic gravitational collapse models I 5.6 Public Holiday: Pfingstmontag 12.6 Basic gravitational collapse models II 19.6 Accretion disks 26.6 Molecular outflow and jets 3.7 Protostellar evolution, the stellar birthline 10.7 Cluster formation and the Initial Mass Function 17.7 Massive star formation, summary and outlook 24.7 Extragalactic star formation More Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss06.html beuther@mpia.de