1. Our Plan
We want to study rigorously (more than just using the global virial relations) the stability/instability of dense cores. Conditions for instability correspond to finding conditions for the collapse of the core into a star.
Since for cores T <~ U ~ W we can consider only the thermal content U of the cloud to simplify the problem.
Later in the course we will introduce also the contribution of rotational energy and magnetic energy…

2. Isothermal Sphere I “at” = sound speed Spherical case: r=r(r)
Treat cores as SPHERICAL, NON-TURBULENT gas clouds with CONSTANT TEMPERATURE (no bad approximation for dense cores, would be bad for GMCs instead). We want to find the density structure and the potential of such a cloud in equilibrium
“at” = sound speed
Spherical case:
r=r(r)
Boundary condition:
Fg=0 at r=0 and r=
rc at r=0

3. Isothermal Sphere II Boundary conditions: f(0) = 0 f’(0) = 0
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.

4. Isothermal Sphere III Recall r/rc= exp (-f)
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 and the density
at the boundary, r0
Recall r/rc=
exp (-f)

Density and pressure (P=ra2) 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 r=0)
Useful formula to describe density profile (except at very small r).

5. Isothermal Sphere IV With: r = √(at2/(4πGrc))*x r = rc exp(-f)
We now want to isolate the
dependency of m on r/rc,
i.e. the function m(r/rc)

6. 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 x0=0, hence rc/r0=1 and m=0.
For increasing rc/r0 m then increases until rc/r0=14.1, corresponding to the
dimensionless radius x0=6.5.
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 (m=0.798)

7. Jeans Instability
-- Standard (rigorous) derivation considers an infinite, homogeneous gaseous
static self-gravitating medium and studies first order perturbative solutions of the
physical equations valid for a self-gravitating fluid.
-- Heuristic (approximate) derivation; consider a gas sphere and perturb the sphere,
then compare either:
Gravitational (Fg1) and pressure (Fp1) force for the perturbed system and
find condition for Fg1 > Fp1 - instability regime=collapse
(2) Sound crossing time and free fall time in perturbed system
and find condition for which tdyn < tsound - instability regime=collapse
Using either (1) or (2) one finds that the system is unstable (i.e. collapses) on
a scale r given by;
r2 > at2/Gro

8. Infinite homogeneous fluids: can they be static?
(a) Consider a charged fluid (plasma) with no gravitational field; if the fluid is neutral
there is no net electromagnetic force - electrostatic infinite homegeneous plasma
is a sensible physical system
(b) Consider now a charge-free self-gravitating fluid; gravity cannot be neutral since
it is always attractive so a net force will arise  the fluid has to move 
static infinite homogenous self-gravitating fluids cannot exist
However mathematically static homogeneous equilibria are very useful to study
evolution of perturbations.
Need to use Jeans swindle to carry out the analysis = assume that unperturbed
gravitational field is zero, Fo = 0 + Poisson eq. only valid for perturbed quantities
There is no general justification for this but the Jeans swindle is reasonable in
many situations. In fact:
(1) In real systems the unperturbed gravitational field is balanced by other forces
(e.g. pressure, centrifugal rotation) so it is like the net force field is zero (e.g.
pressure/density gradients of Earth’s atmosphere balance Earth’s gravitational field)
and (2) if the scale of these other forces vary is much larger than the scale
of interest (e.g. wavelength of sound waves in Earth’s atmosphere) we can treat
the medium as infinite and homogeneous.

9. Jeans instability Jeans swindle: in homogeneous medium
not generally true, but usually justified
Linearize equations of motion for this medium G G

12. Gravitational stability
Gravity
dominated
Pressure dominated

Low-density contrast cloud: Increasing outer pressure P0 causes a rise
of m and rc/r0. Since P=rat2, 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:
PV=const. --> P * 4/3pr3 = const.

High-density contrast clouds: gravity more important and if rc/r0> 14.1
(x0> 6.5) clouds are gravitationally unstable. The associated critical mass is
the Bonnor-Ebert mass (Ebert 1955, Bonnor 1956)
MBE = (m1at4)/(P01/2G3/2)

13. 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. Similar to Jeans instability but for finite, pressure
bounded system (i.e. with boundary conditions P0 and r0)
Can start from an ansatz for the perturbation, e.g. normal mode (= sinusoidal
mode in which each physical variables oscillate with same frequency in the
indipendent on position 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 < 0 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 < MBE can become unstable as long as it has a strong
enough density contrast (instability to higher order modes)

14. Gravitational stability: The case of B68
Optical Near-Infrared
Starless Bok Globule
x0=6.9 is only
marginally about the
critical value 6.5
gravitational
stable or at the verge
of collapse