Collapse of a core II

1
Collapse of a core II

• First rapid transient phase dependent on choice
  of boundary conditions in model (i.e. a
• bit artificial) in which inward velocities
  develop from the edge to the center until the
• entire cloud accelerates inward and a shock is
  produced (velocities are supersonic).
• Near the center compression is maximum and a
  prostostellar core (a self-
• gravitating blob) accumulates from the mass that
  is pulled in by gravity.
• After the core forms collapse continues in an
  inside-out fashion, in the sense that
• infall of matter onto the core is stronger the
  closer is such matter to the core.
• The infall region (in which matter is in
  free-fall because pressure is negligible
• compared to gravity) spreads out with time ,
• The radius of such region, Rff, is indeed given
  by the condition Vff aT which yields
• M (accretion rate)
  aT2 Rff/G ----? Rff gt 0 when M gt 0

.
.
.
.
2
Collapse of a core III
Isothermal spheres have a universal asymptotic
mass accretion rate (i.e. independent on initial
conditions such as density contrast or boundary
conditions).
r ? 0
r ? 0
.
.
The collapse affects the density structure of the
cloud
Density and velocity profile in the inner
free-fall region r µ r-3/2 ( u µ
r-1/2) Density profile in the outer envelope r µ
r-2 (u/at ltlt 1) (singular isothermal sphere,
as for equilibrium cases)
3
Collapse of a core IV
The inner free-falling gas causes an inner
pressure decrease, and a rarefaction wave moves
outward (resulting from pressure
perturbation) Within the rarefaction wave, the
gas is free-falling because of missing pressure
support. The rarefaction wave is indeed a sound
wave with vrw aT. Indeed setting Rffvrw one
recovers the asymptotic accretion rate M m0
aT3/G
.
.
4
The first core I - thermodynamics
Central density increases over time as collapse
proceeds. slow increase if collapse is magnetized
with ambipolar diffusion

• Collapse NOT
  isothermal !
• Although heating by cosmic rays negligible inside
  dense core gravitational
• collapse is itself also source of heating because
  at any given radius gas
• is compressed by other gas falling in from just
  outside this radius (PdV work)
• collapse increases density to the point where gas
  opaque to cooling (e.g IR
• radiation by dust and CO cooling) --? T goes up,
  equation of state changes!
• Once gas cannot cool collapse stops and central
  region settles in nearly
• hydrostatic equilibrium (accretion damped by
  growing P gradient) ---?
• first
  core forms

5
The first core II

• Temperature of first core from viirial theorem
  2T 2U W M 0
• W -2U (kinetic and magnetic
  energy approximated as 0)
• gt -GM2/R -3MR T/µ gt T µGM/(3R R)
• 850K (M/5x 10-2 Msun) (R/5AU)-1 (m2.4 for
  molecular gas with solar metallicity)
• --gt significantly warmer than
  original core (15-20 K)
• (1)Core begins to shrink again as material
  piles up from outside (still not
• opaque at lower densities) and soon reaches 2000K
  --? collisonal
• dissociation of H2 starts (first core almost
  entirely H2)
• -? Thermal energy per molecule at
  2000K 0.74eV
• compared to dissociation energy of H2 of
  4.48eV
• --gt Even modest increase of
  dissociated H2 absorbs most of the
• heating provided by
  gravitational collapse
• --gt marginal increase in
  temperature and pressure
• (2) Region of atomic H spreads outward from
  center
• Without significant T P increase, the first
  core cannot keep
• equilibrium (similar to an isothermal sphere
  growing until M MBE)
• hence the entire core becomes unstable,
  collapses ? forms protostar.
• --gt significant temperature and density
  increase, sufficient to collisionally
• ionize most hydrogen --gt emerging
  protostar is now dynamically stable.

6
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.

7
Accretion shock and Accretion luminosity
Vff gt aT

• The gravitational energy released per unit
  accreted mass can be
• approximated by the (lost) gravitational
  potential energy 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 negligible compare to Lacc for low- to
  intermediate-mass stars -?
• Lrad sets the radius of protostar (from
  conservation of energy).
• Conventional definition of (low-mass) protostar
  
• Mass-gaining star deriving most of its
  luminosity from accretion.
• (However, caution for massive stars.)

8
Protostellar envelope I

• How does the accretion radiation escape?
• Radiative transfer problem, need opacity as
• a function of radius
• Protostellar envelope is sequence of optically
• thick and optically thin layers, as predicted by
• collapse calculations (that give P, r, T as a
• function of radius)
• -Protostellar radiation re-processed as it goes
• through different layers, ultimately escapes as
• infrared radiation from outer envelope (so
  protostar
• identified as compact infrared source
  observationally)

9
Protostellar envelope II

• - 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 --gt dust
  sublimation at T1500K.
• Inside dust destruction front greatly reduced
  opacity, and infalling gas
• almost transparent to protostellar radiation
  --gt opacity gap.
• Immediately outside the accretion shock, gas
  gets collisonally ionized
• and the opacity increases again --gt so-called
  radiative precursor

10
Protostellar envelope III

• Difference in radiation between
• shocked radiative precursor and
• far-infrared radiation from dust
• photosphere
• In shock region gas approaches
• protostar approximately at free-fall
• speed 1/2mvff GMm/R
• gt vff v2GM/R
• 280 km/s (M/1Msun)1/2 (R/5Rsun)-1/2

• --gt this high kinetic energy implies
  immediate post-shock temperature gt106 K, UV and
  X-ray regime (metal lines, such as Fe IX) from
  Vshock Vff aT
• --gt Post-shock settling region opaque
  because gas collisionally ionized (lots of
• free electrons -? Thomson scattering). Photons
  lose energy, T decreases sharply
• --gt The surface of precursor radiates as
  approximate
• blackbody in opacity gap
  Stephan-Boltzmann law
• Lacc 4pR2sBTeff4 Substituting Lacc
  gt Teff (GM(dM/dt)/4pR3sB)1/4 gt 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
• (hence the name radiative precursor)

11
Protostellar envelope IV
Optical radiation travels through opacity gap and
then enters dust photosphere (optically thick to
optical radiation) past dust destruction
front. - Diffusion approximation for the
radiative transfer equation (valid in media with
t gtgt 1, will discuss in exercise class) can be
used to calculate T profile across photosphere
(eq. 11.9) Assuming (Rosseland mean) opacity law
k Ta, , a 0.8 (given by combination of relevant
absorption processes) one gets T r-g, where g
5/2(4-a) 0.8 -The radiation is thus shifted to
lower frequencies until it becomes optically thin
again (outside dust photosphere). This happens
when mean free path of average photon as large as
distance from star -? 1/rk Rphot-? Rphotrk
1 The photosphere of radius Rphot will emit like
a blackbody with temperature Tphot ? L
Lacc 4pR2photsBT4phot. Solving numerically the
system of two equations replacing with r r-3/2
and k Ta, Lacc GM M/R one gets Tphot
300 K for M10-5 Mo/yr and M 1 Mo -? l
hc/kBTphot 49 mm -? i.e. radiation that finally
escapes the cloud is in far-infrared regime.
12
Temperatures and dimensions of envelope

• Radiative transfer calculations produce global
  temperature profile,
• some aspects quite consistent with approximate
  modeling
• 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

13
Magnetized Collapse I

• Only restricted models no simulation
  simultaneously 3D, self-gravitating,
• non-ideal MHD ambipolar diffusion and MHD
  turbulence
• Magnetized clouds collapse thanks to ambipolar
  diffusion -? neutral matter
• drifts along field lines so gas contracts but
  magnetic field increases much
• less than expected based on flux-freezing (BR2
  constant)
• This is equivalent to damping of magnetic
  pressure (and tension)
• Result dM/dFB (function describing how much
  mass has a given magnetic
• flux) rises towards the center (where FB0) as
  more mass shifts to smaller
• values of FB during the collapse (independent on
  form of FB)
• -? magnetic flux loss there is an increasing
  fraction of mass with small
• magnetic flux see Fig. 10.9 - and overall total
  flux in the collapsing portion
• of the cloud diminishes. Contraction process
  slow, quas-static.
• But what about MHD waves which also support
  clouds against collapse?
• -Ambipolar diffusion leads to damping of Alfven
  waves on scales l lt lmin,
• lmin 0.1 pc for B 10 mg, nH2 103 cm-3 -?
  MHD waves damped in the

Numerical model contraction of
magnetized non-turbulent cloud (w. ambipolar
diffusion)
14
Magnetized collapse II

Neutral matter reaches the center both traveling
along the field lines and across them. Field
lines respond as they are partially dragged in by
matter falling through them (pinching) -Accreti
on along field lines (column A) essentially
like unmagnetized case because magnetic force
small (and no MHD waves in dense, central region)
-? collapse proceeds inside-out with dM/dt
dM/dt of unmagnetized case -Across field lines
(region B) collapse also-inside out but slowed
down by magnetic pressure and tension -gt dM/dt
related to ambipolar diffusion timescale and
protostar gains elogation

15
Magnetic reconnection
What happens to the residual magnetic field that
is dragged in by the collapse? Magnetic
reconnection - effect of the Ohmic dissipation
term in MHD equation

• Field lines of opposite direction are dragged
  together
• --gt antiparallel B field lines annihilate ?
  region with B0 produced (dashed
• rectangle) where magnetic energy is dissipated as
  heat.
• This process was first invoked to explain large
  luminosities observed
• in solar flares (fluid is expelled away from
  reconnecting region)
• In protostellar cores this happens near the
  center (equatorial region B, see
• previous slide)