Threedimensional velocity dispersion of molecular clouds decreases with

1
Virial relation for varying cloud sizes
Three-dimensional velocity dispersion of
molecular clouds decreases with decreasing
molecular cloud size (observational fact, no
theory) Larsons law DV
DV0 (L/L0)n n0.5 DV0 1
km/s Lo 1 pc
Smallest clouds (below 1 pc) have three
dimensional velocity dispersions comparable to
gas thermal speed (0.3-0.5 km/s). This means
that for these clouds the support from internal
kinetic energy is not as important as for large
clouds. These small clouds are called cores and
are thought to be the precursors for
stars. Thermal energy alone is enough to support
the cloud since
T U W (T W for all cloud
masses) The transition happens at a scale
Ltherm 3RTLo/mDV02 0.1 pc(T/10 K) (obtained
by setting DV vs in Larsons law )
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Observational aspects of cores
Cores have M of a few solar masses and L 0.1
pc, so nH 104 cm-3 densest component of
molecular gas phase Cores are found inside
larger molecular cloud complexes but also in
isolation (Bok Globules) Internal structure of
cores -- some indication that inside cores the
level of turbulence diminishes with increasing
local density (local extension of Larsons
law?). -- there are starless cores but also
cores with embedded point-like infrared
sources associated with an embedded star. This is
(1) proof that cores are the ultimate site of
star formation and (2) shows that the star
formation process is long enough that we can
observe cores in different evolutionary stages
(theoretical models will have to reproduce that)
and (3) cores do not evaporate immediately after
the star is born and heats the surrounding gas
with ultraviolet and X-ray radiation.
3
Bok globules (isolated cores)
200 known Bok globules within 500 pc ? proximity
allows detailed study so these dense cores are
ideal tool for studying the star formation
process High-resolution mapping of cores. Many
have embedded stars (e.g. B335 has a 1 Lo star
that is driving a powerful molecular outflow
traced by CO, see Fig. 3.20) Molecular line
emission from interior of core allows to
determine the temperature profile T(r).T(r)
increases outward despite presence of star at the
center and T nearly constant where most of the
mass (highest density) (Fig. 3.21). T not maximum
near center because (1) stellar heating confined
to radii too small for the resolution of current
observations (from models) and (2) outer region
more efficiently heated by cosmic rays (lowest
density, no outer layers to absorb cosmic
rays). Puzzle where are the sites of massive
star formation? Known globules all contain
low-mass stars (like the sun), no massive stars
( 3 Mo). Hypothesis (to be verified) Bok
globules might be the only surviving part of a
much larger molecular cloud complex that was
destroyed by heating, radiation pressure
and winds by one or more massive stars (cores in
Orion, that has many massive stars, are different
(mass, T, size) than cores in regions with few
massive stars, e.g Taurus)
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Our
Plan We want to study in more detail (more than
given by just the global virial relation) the
stability and collapse (instability) of dense
cores Since for cores U T W we can
consider only the thermal content of the cloud to
simplify the problem. Later we will introduce
also the contribution of magnetic fields to the
stability and collapse of clouds.
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Isothermal Sphere I
Treat cores as spherical gas clouds with constant
temperature (no bad approximation for dense
cores, would be bad for GMCs instead)
at sound speed
Spherical case rr(r) Boundary condition Fg0
at r0 and r rc at r0
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Isothermal Sphere II
Note different notation. compared to book
for adimensional potential
Boundary conditions f(0) 0 f(0)
0 Gravitational potential and force are 0 at the
center.
-- L-E equation must be integrated numerically
with the above boundary cs.
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Isothermal Sphere III
Figure describes an infinite sequence
models parametrized by r/rc (density
contrast) i.e. for any r/rc I can build infinite
models with arbitrary values of rc
Recall r/rc exp (-f)

• Density and pressure (Pra2) increase
  monotonically towards the center.
• -- important to offset inward pull from
  gravity for grav. collapse.
• After numerical integration of the Lane-Emden
  equation,
• one finds that the density r/rc approaches
  asymptotically 2/x2 (x 1)
• Hence the dimensional density profile of the
  singular isothermal sphere is
• r(r) a2/(2pGr2) (called singular because
  density diverges for r0)
• Useful formula to describe density profile
  (except at very small r).

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Isothermal Sphere IV
With r v(at2/(4pGrc))x r rc exp(-f)
We now want to isolate the dependency of m on
r/rc, i.e. the function m(r/rc)
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Isothermal Sphere V

• Read the above function in the following way
  mass function of density
• which in turn is function of radius (all
  expressed via the adim. variables)
• Left boundary value of function is for a
  radius x00, hence rc/r01 and m0.
• For increasing rc/r0 m then increases until
  rc/r014.1, corresponding to the
• dimensionless radius x06.5.
• The function has an oscillatory behaviour to
  the right with a sequence of
• maxima and minima and tends asymptotically to the
  (nondimensional) mass
• of a singular isothermal sphere (m0.798)

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Gravitational stability
Gravity dominated
Pressure dominated
Low-density contrast cloud Increasing outer
pressure P0 causes a rise of m and rc/r0.
Since Prat2, the internal pressure P rises more
strongly than P0 and the cloud remains
stable (no collapse) However the physical
radius r0 (radius of the cloud) is related to x0
and rc like r0
sqrt(at2/(4pGrc)) x0 Since rc increases
faster than x0 (e.g. fig. 9.1), the core actually
shrinks with increasing outer pressure P0.
The same as Boyle-Mariotte law for ideal gas
PVconst. -- P 4/3pr3 const.
High-density contrast clouds gravity more
important and if rc/r0 14.1 (x0clouds are gravitationally unstable. The
associated critical mass is the Bonnor-Ebert
mass (Ebert 1955, Bonnor 1956)
MBE (m1at4)/(P01/2G3/2)
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To see that MBE is the maximum mass for a
gravitationally stable cloud one has to perturb
the Euler (no magnetic field), Poisson and
continuity equations for a spherical cloud and
solve the linearized perturbed equations to
determine the time evolution of perturbations
(i.e. in which condition they grow
exponentially instability and in which
conditions they keep oscillatingstability). Can
start from an ansatz for the perturbation, e.g.
normal mode ( sinusoidal mode in which each
physical variable oscillates with same frequency
in the cloud but different variables (e.g. r, P)
can have different phases of the
oscillation) r(r,T) req(r) dr(r) exp(iwt)
req(r) unperturbed function
Solving the linearized equations one determines
dr(r) (eigenfunction) and w2 (eigenvalue).
Different modes are identified by different
values of w2 (w2 0 stability, w2 instability). Unstable modes have nodes, i.e
radii where the amplitude of the perturbation is
zero. The first order or fundamental mode
(smallest value of w2) has zero nodes which
means entire cloud breathes (like a hearthbeat)
in and out. Cores with M MBE are unstable to
this breathing mode so the entire cloud
contracts as a response to the global
oscillation. Higher order modes (i.e. modes with
increasing number of nodes) are activated as a
new peak of the function m is encountered going
right on the rc/ro axis. Therefore a cloud with
M strong enough density contrast (instability to
higher order modes)
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Gravitational stability The case of B68

• Starless Bok Globule
• x06.9 is only
• marginally about the
• critical value 6.5
• gravitational
• stable or at the verge
• of collapse
• This makes sense,
• otherwise star would have
• formed already inside!