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Star and Planet Formation

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.)

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Star and Planet Formation

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  1. 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 Protostellar and pre-main sequence evolution (H.B.) 21.5 Outflows and jets (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: http://www.mpia.de/homes/beuther/lecture_ss07.html and http://www.mpia.de/homes/swolf/vorlesung/sommer2007.html Emails: beuther@mpia.de, swolf@mpia.de

  2. Summary last week • Ambipolar diffusion • Turbulence • Cloud collapse of singular isothermal sphere and Bonnor-Ebert sphere • Inside out collapse, rarefaction wave, density profiles, accretion rates • Accretion shock and accretion luminosity • Rotational effects, magnetic locking of “outer” dense core to cloud, • further inside it causes the formation of accretion disks • Observational infall signatures

  3. The first core I • Contraction of core via ambipolar diffusion initially slow process. • When S/B reaches critical threshold, contraction speeds up, high density, • --> 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) • => -GM2/R = -3MRT/µ => T = µGM/(3R R) • = 850K (M/5e-2Msun) (R/5AU)-1 • --> significantly warmer than original core • With the addition of mass and further shrinking, it soon reaches 2000K, • where collisonal dissociation of H2 starts. • --> Temperature increase less steep • However, thermal energy per molecule at 2000K ~ 0.74eV • compared to dissociation energy of H2 of ~ 4.48eV • --> Even modest increase of dissociated H2 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 0.1Msun has radius of several Rsun, T~105K and r~10-2g 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: • Lacc = G(dM/dt)M*/R* • = 61Lsun ((dM/dt)/10-5Msun/yr) (M*/1Msun) (R*/5Rsun)-1 • Additional luminosity contributions from contraction and early nuclear • fusion are negligable compare to Lacc 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. Relative dimensions of outer region greatly reduced in sketch. Protostellar envelope I • - 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. • --> 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

  7. Relative dimensions of outer region greatly reduced in sketch. Protostellar envelope II • 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: 1/2mvff = GM*m/R* • => vff = √2GM*/R* • = 280 km/s (M*/1Msun)1/2 (R*/5Rsun)-1/2 • --> Immediate postshock temperature >106K, 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: • Lacc ~ 4πR*2sBTeff4 Substituting Lacc => Teff ~ (GM*(dM/dt)/4πR*3sB)1/4 • => Teff ~ 7300K ((dM/dt)/1e-5Msunyr-1) (M*/1Msun)1/4 (R*/5Rsun)-3/4 • Opacity gap is bathed in “optical emission” similar to main-sequence star. Very different to observable dust photosphere.

  8. Relative dimensions of outer region greatly reduced in sketch. 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 1Msun 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

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

  10. Protostellar evolution/Stellar Structure equations • The protostellar evolution can be analyzed numerical similarly to stars. • --> Stellar Structure equations • The used spatial variable is Mr, the mass within shells of radius r • Mr = 0∫r 4πr2r dr • => ∂r/∂Mr = 1/(4πr2r) (1) • Hydrostatic equilibrium: -1/r grad(P) - grad(Fg) = 0 • => ∂P/∂r = -GrMr/r2 (2) • Substituting (1) in (2), one gets • => ∂P/∂Mr = -GMr/(4πr4) (3) • Pressure obeys ideal gas equation: P = r/mRT (4) • (mean molecular weight m depends on state of ionization and is function of T and r) • Thermal structure of opaque interior is described by diffusion equation • T3 ∂T/∂Mr = - 3kLint/(256π2sBr4) (5) • (mean opacity k is again function of T and r) • And the spatial variation of the internal luminosity Lint can be described • by the heat equation: ∂Lint/∂Mr = e(r,T) - T ∂s/∂t (6) • (e(r,T) rate of nuclear energy release; s(r,T) is the entropy) • For a mono-atomic gas, the entropy is: s(r,T) = R/µ ln(T3/2/r) • Using adequate boundary conditions, one can now follow numerically the • protostellar evolution.

  11. Mass-radius relation • - 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(Mr), reflecting the • changing conditions at the accretion shock. • Since s represents heat content of each • added mass shell, an increase of • s(MR) causes a swelling of the protostar. • - In the absence of nuclear burning, an increasing s(Mr) arises naturally • because with rising M*, the velocity of the infalling gas and hence the • accretion shock and Lacc increases. • --> Protostellar radius increases with time. • Suppose initial core very large, then low infall velocity --> low Lacc and • s(Mr) would dip at beginning of protostellar evolution --> initial decrease of R* • - Opposite effect for very small initial state.

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

  13. Convection II - However, the ratio M*/R* rises fast and interior temperatures increase. Nuclear reactions start at center (at ~0.3Msun deuterium burning at ~106K). --> entropy profile overturns ∂s/∂Mr < 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/∂Mr < 0 parcels are underdense ((rint)1 < (rext)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 Circles: onset of full convection - 2H + 1H --> 3He + ∆E with ∆E ~ 5.5 MeV, important from 106K - 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 106K. - 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 1-2Msun of mass? • The critical luminosity Lcrit is the maximum value to be caried by radiative • diffusion. • - Continuum opacity from free-free emission (Kramers-law opacity) scales • kff µ rT-7/2 --> strong decrease with T • And one finds: Lcrit µ M*11/2R*-1/2 • --> For growing protostars, Lcrit rises sharply surpassing interior luminosity. • --> Convection then disappears and protostar gets radiative barrier.

  16. Deuterium shell burning I - For stars more massive than about 2Msun. - Once radiative barrier has formed, no new deuterium to center and residual deuterium is consumed rapidly. - Interior luminosity Lint declines below Lcrit 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 106K, deuterium shell burning starts and convection occurs in this shell structure.

  17. Gravity becomes dominating force, R* decreases fast and T & L rise. At ~107 K H-burning will begin. --> stop contraction. Appearance of radiative barrier Hydrogen fusion and second initiation of a central convection zone. Onset of full convection Deuterium shell burning II Deuterium shell burning injects energy causiing additional swelling. End of deuterium burning, beginning of pre-main-sequence Evolution. • 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 Lcrit drives the radiative • barrier and the associated burning layer and convection zone outward. • --> The protostar (>~2Msun) is then almost fully radiatively stable. • - The previous scaling relation for Lcrit should now apply to interior luminosity, • and one finds: Lint ~ 1Lsun (M*/1Msun)11/2 (R*/1Rsun)-1/2

  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 = 4πR2sBTeff4µ R*2, • the luminosity decreases and falls below Lcrit. --> 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 ~0.4Msun 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 Open circles: Radiative stability Full circles: Hydrogen burning • 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.

  21. Birthline Palla & Stahler 1990 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.

  22. Observable Spectral Energy Distributions (SEDs)

  23. Summary • The “first core” contracts until temperatures are able to dissociate H2 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. 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 Protostellar and pre-main sequence evolution (H.B.) 21.5 Outflows and jets (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: http://www.mpia.de/homes/beuther/lecture_ss07.html and http://www.mpia.de/homes/swolf/vorlesung/sommer2007.html Emails: beuther@mpia.de, swolf@mpia.de

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