Transcript 30.4 Gravitational collapse & early protostellar evolution I (HB)

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]
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
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
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
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.)
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.
Relative
dimensions of
outer region
greatly reduced
in sketch.
--> 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
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
Relative
dimensions of
outer region
greatly reduced
in sketch.
--> 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.
Temperatures and dimensions of envelope
Relative
dimensions of
outer region
greatly reduced
in sketch.
- 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
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.
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/m RT
(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.
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.
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.
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.
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 protostellar 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.
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 rT-7/2 --> strong decrease with T
- And one finds: Lcrit M*11/2R*-1/2
--> For growing protostars, Lcrit rises sharply surpassing interior luminosity.
--> Convection then disappears and protostar gets radiative barrier.
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.
Deuterium shell burning II
Deuterium shell burning
injects energy causiing
additional swelling.
Appearance of
radiative barrier
Gravity becomes
dominating force,
R* decreases fast and
T & L rise. At ~107 K
H-burning will begin.
--> stop contraction.
End of
deuterium
burning, beginning
of pre-main-sequence
Evolution.
Onset of full convection
Hydrogen fusion and
second initiation of a
central convection zone.
- 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
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.
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 = 4πR2sBTeff4 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.
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 socalled Hayashi tracks. After the cores become radiative, they start to
increase their temperature (& luminosity) moving left on the radiative tracks.
Hertzsprung Russel (HR) diagram II
Birthline
Palla & Stahler 1990
- 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.
Observable Spectral Energy Distributions (SEDs)
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
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]