Star and Planet Formation
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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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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.)

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]


Star and planet formation

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


Star and planet formation

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


Star and planet formation

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


Star and planet formation

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.)

.


Star and planet formation

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


Star and planet formation

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.


Star and planet formation

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


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

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.


Star and planet formation

Observable Spectral Energy Distributions (SEDs)


Star and planet formation

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.


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.)

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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