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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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slide1

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: [email protected], [email protected]

slide2

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
slide3

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
slide4

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
slide5

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.)

.

slide6

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
slide7

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.

slide8

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
slide9

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.
slide10

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.
slide11

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.
slide12

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.
slide13

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.

slide14

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.

slide15

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.
slide16

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.

slide17

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
slide18

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.
slide19

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.
slide20

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.
slide21

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.
slide23

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.
slide24

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: [email protected], [email protected]

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