Jeans analysis I

1
Jeans analysis I
Aim study the stabilty of a generic uniform,
isothermal self-gravitating fluid, no
assumptions on geometry or size of cloud A plane
wave travelling in an isothermal gas can be
described as r(x,t) r0 dr
expi(kx - wt) wave number k2p/l Using
this as ansatz to solve the fundamental equations
for an isothermal, self-gravitating fluid
(linearizing in the perturbation) one gets the
so-called dispersion equation
w2 k2at2 - 4pGr0

• For large k high-frequency disturbances
• the wave behaves like sound wave wkat
• isothermal sound speed of background
• However, for low k (kltk0) w2lt0.
• The corresponding Jeans-length is
• lJ 2p/k0 (pat2/Gr0)1/2
• Perturbations with l gt lJ have exponentially
• growing amplitudes --gt unstable
• Using r0 instead of P0, Bonnor-Ebert mass
• is related to the Jeans-Mass MJ (
  4/3p(1/2lj)3)
• MJ 2.47 x m1at3/(r01/2G3/2)
• 2.47 MBE

Solid line dispersion relation Dashed line
dispersion relation for sound waves
2
Jeans analysis II

• This corresponds in physical units to
  Jeans-lengths of
• lJ (pat2/Gr0) 0.19pc (T/(10K))1/2
  (nH2/(104cm-3)-1/2
• and Jeans-mass
• MJ m1at3/(r01/2G3/2) 1.0Msun (T/(10K))3/2
  (nH2/(104cm-3)-1/2
• Clouds larger lJ or more massive than MJ may be
  prone to collapse.
• Conversely, small or low-mass cloudlets could be
  stable if there is
• sufficient external pressure. Otherwise only
  transient objects.
• Example a GMC with T10K and nH2103cm-3
• MJ 3.2 Msun
• Orders of magnitide too low.
• Additional support necessary, e.g., magnetic
  field, turbulence since
• GMCs last at least 20 free-fall times (tstar
  formation in Galaxy is 5 x 107 yr)

3
Protostellar collapse preamble
Aim figure out how to produce a star from the
collapse of a core NOTE huge range of
densities involved, and huge range of
timescales Density of a core 1000 cm -3
Density of star 1024 cm
-3 Radius of core 0.1 pc 3 x 1017 cm
Radius of star 1011 cm
4
Collapse of a core - I
Consider a spherical isothermal core (no
rotation, no magnetic fields) Suppose that the
cloud starts marginally unstable with M MBE
and then it is slightly perturbed. If its density
is increased by a few percent everywhere mass
becomes gt MBE --? the cloud becomes unstable and
begins to collapse. This is the way initial
conditions are setup in typical numerical
experiments -- But in reality collapse initiated
by perturbation due to environment of cloud. --
Good boundary conditions can be used to mimic
how destabilization happens Collapse
studied by solving set of fluid equations Mass
within radius r Mr ?4pr2rdr --gt ?Mr/?r
4pr2r (spherical) Continuity equation ?r/?t
-1/r2 ?(r2ru)/?r u velocity
--gt ?Mr/?t
-4pr2ru (spherical) Momentum eq. ?u/?t u?u/?r
-1/r grad(P) - grad(F) (no equilibrium)
With P rat2 for the ideal isothermal gas and
grad(F) -GMr/r2 momentum eq. becomes ?u/?t
u?u/?r -at2/r ?r/?r - GMr/r2 This set of
equations can be solved numerically for given
boundary conditions.
5
Collapse of a core II

• First rapid transient phase dependent on choice
  of boundary conditions in model (i.e. a
• bit artificial) Example in constant pressure
  boundary case (with say 10 density
• perturbation) 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 (in principle
• a self-gravitating blob) accumulates from the
  mass that is pulled inward by gravity.
• In numerical algorithms the protostellar core is
  treated as a sink particle or sink cell,
• namely a region of the computational volume that
  simply absorbs mass and momentum
• --? too large dynamic range for codes to treat
  protostar directly
• One can introduce the sink when 10 of the mass
  of the cloud has accumulated at
• the center.
• After the core forms collapse continues in an
  inside-out fashion, in the sense that
• infall of matter onto the core is faster the
  closer is such matter to the core.
• This is a natural consequence of how the
  free-fall times scales with (local) density.

6
Collapse of a core III
What is the physical meaning of
this asymptotic accretion rate? Infall
region matter near protostar is in free-fall
because pressure is negligible compared to
gravity. Infall region spreads out with time,
infact Boundary radius of infall region, Rff,
is given by the condition Vff aT which
yields dM/dt (accretion rate)
aT2 dRff/dt /G ----? Rff gt 0 when M gt 0
Consider the phase after protostar formation
(e.g. sink has 10 of mass in cloud) One wants
to determine the increase per unit time of the
sink cell mass (mass accretion rate) limit for
small r of dM/dt Clouds with different structure
(e.g. different density contrast rc/ro) or
different boundary conditions undergo a different
mass accretion history but asymptotic accretion
rate (final leveling off) is universal
--? dM/dt aT3/G

7
Collapse of a core IV
The inner free-falling gas causes an inner
pressure decrease, and a rarefaction produced by
the pressure perturbation moves outward. 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 aT one recovers
the asymptotic accretion rate M aT3/G
.
.
8
Caveat I - non-spherical collapse
If collapse not spherical need to solve fluid
Poisson equations in 3D In this case collapse
has no preferred symmetry and small density
perturbations can arise during the
collapse. However these are erased closed to the
protostar because there the infall becomes
radial again (so spherical approximation still
holds near the protostar!) ---? no localized
collapse (i.e. fragmentation) far from protostar
occurs -?gtgt basic character of collapse
unchanged compared to spherical case
9
Caveat III - turbulence Hydrodynamical turbulent
cloud collapse (no magnetic field) initial
cloud marginally gravitationally unstable (M
MBE) Formation of a star cluster from
the fragmentation of a turbulent molecular clump
(simulation by M.Bate et al 2002)


10
Caveat II - thermodynamics
Central density increases by orders of magnitude
over time as collapse proceeds. Opacity goes up ?
radiative transfer problem

• Collapse NOT
  isothermal in general !
• Heating by cosmic rays negligible inside dense
  core but 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 because
  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

11
The first core II

• Temperature of first core from virial 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.

12
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 GM2/R
• - Hence the released accretion luminosity of the
  protostar can be
• approximated by this energy multiplied by the
  accretion rate
• Lacc G (M/t) 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.)

13
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)

14
Protostellar envelope II
BASICS (1) Radiation
originates from accretion shock at protostar
surface. (2) It is finally absorbed by dust in
the envelope that reradiates the emitted
radiation at far-infrared wavelengths. (3)
Outer envelope largely optically thin so
infrared radiation escapes and reaches observer

• From far too close to the
  protostar
• Dust photosphere (a few AU for typical low-mass
  star) is the effective radiating
• surface observable from outside at that
  evolutionary stage (star not optically visible
  yet).
• Rapid T increase inside 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

15
Protostellar envelope III

• Difference in radiation between
• shocked radiative precursor and
• far-infrared radiation from dust
• photosphere
• In shock region gas parcel (mass m)
• approaches protostar approximately at
• free-fall speed 1/2mvff2 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 highly
  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)

16
Protostellar envelope IV
Optical radiation travels through opacity gap,
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 for the radial drop of temperature
due to absorption by dust -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 ? Lphot Lacc 4pR2photsBT4phot
(conservation of energy) 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.
17
Temperatures and dimensions of envelope

• Full 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