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 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
Basic rotational configurations I
Cloud rotating aroud z-axis Further assumption
cloud is axisymmetric (no f-gradients)
reflection symmetry around z0 ---? j depends
only on cylindrical radius w (Poincare-Wavre
theorem)
Centrifugal force pushes fluid elements away from
center ? stabiizes cloud Adding a centrifugal
potential Fcen, the hydrodynamic equation
reads -1/r grad(P) - grad(Fg) - grad(Fcen)
0 For axisymmetric cloud rotating along z-axis
Fcen simply defined as Fcen - ? (j2/w3) dw
j angular momentum
w cylindrical
radius
and jwu with
u the velocity around the rotation axis Rotation
flattens the cores in addition to being a source
of support against collapse --? measuring
flattening gives idea of how important is rotation
4
Basic rotational configurations II
We can generalize the study of stability done for
isothermal spheres to that of isothermal
rotating spheres. Solving the hydrostatic
equilibrium density of the cloud will now be
rrcexp(-Fg Fcen Fg(0))/at2 However need an
additional equation for the shape of j(w) in
order to solve the equation for Fcen Rotation
stability requires j(w) decreases inward. One can
assume cores come from the collapse of an initial
uniform density rigidly rotating clump (W0
constant) angular momentum conservation. In
this case j w and r can be found numerically
using an iterative procedure (guess r, obtain the
Fg (and Fcen from j(w), redetermine r, solve
again for the potentialsuntil the new r and the
previous r differ only by a small error)
5
Basic rotational configurations III

• Compared to the previously discussed Bonnor-Ebert
  models, these rotational
• models now have in addition to the density
  contrast rc/r0 the other parameter
• which quantifies the degree of rotation. b is
  defined as the ratio of rotational
• to gravitational energy
• b W02 R0 3/(3Gm)
  with W0 the angular velocity of the cloud
• 
  and R0
  the initial cloud radius
• Maximum possible b is 1/3 because corresponds to
  breakup speed of the cloud.
• so 0 lt b lt 1/3 (break up speed when Erot Egrav)

6
Basic rotational configurations IV
In realistic clouds, for flattening to
appear, the rotational energy has to be at least
10 of the gravitational energy. Trot/W more
general (recall virial theorem) equals
approximately b (which was defined Specifcally
for the spherical, rigidly rotating
case) Examples
Dense cores aspect ratio 0.6. Estimated Trot/W
10-3 GMCs Velocity gradient of 0.05km/s
representing solid body rotation, 200Msun
and 2pc size imply also Trot/W 10-3
--gt Cloud elongations do not arise from
rotation so centrifugal force
NOT sufficient for cloud stability! Other
stability factors are necessary --gt Magnetic
fields, turbulence (i.e random Kinetic energy)
7
Magnetic fields I
Object Type
Diagnostic B mG

Ursa Major Diffuse cloud HI
10 NGC2024 GMC clump
OH 87
S106 HII region OH
200 W75N Maser
OH
3000 Increasing magnetic field strength with
increasing density indicate field- freezing
between B-field and gas (B-field couples to ions
and electrons, and these via collisions to
neutral gas). Can we understand field
freezing on theoretical grounds?
density
8
MHD equation
It governs the evolution of the magnetic field in
a generic fluid, i.e. a fluid with non-zero
internal motions (so no hydrostatic equilibrium
assumed). Will be derived in next exercise
class It is
dissipative term (resistivity)
which is
a generalization of
Ohms law
(Ohms law is in for a
medium at rest) and depends on
electric conductivity s Ideal MHD neglect
dissipative term. Can do it if conductivity
very large. Indeed Ohmic dissipation timescale
gt sL2/c2 1017 yrs for typical cloud parameters
(conductivity easily estimated once Ionization
level is known, typically ne/n 10-7 in cores)


9
Magnetic fields II
This field freezing is described by ideal MHD
equation
However, ideal MHD must break down at some point.
Example Dense core 1Msun, R00.07pc, B030mG T
Tauri star R15Rsun If
flux-freezing would hold, BR2 should
remain constant
over time --gt B12x107 G, which exceeds
observed values by orders of magnitude! Ambipolar
diffusion neutral and ionized medium decouple,
and neutral gas can sweep through during the
gravitational collapse.
10
Ambipolar diffusion I
In less dense GMCs, the ionization degree is
relatively large and ions and neutrals are
strongly collisionally coupled. Going to denser
molecular cores, the ionization degree
decreases, and neutrals and ions can decouple.
Neutrals stream through the ions as they are
accelerated by gravity. A drag force arises
between ions and neutrals as a result of
collisions ambipolar diffusion happens when
this drag force is small. Lorentz force acts on
ions (and electrons)

• In general there is a drift velocity between ions
  and neutrals is vdrift vi - vn
• because they are not perfectly coupled (drift
  velocity will be large when drag force small)
• And the drag force between ions and neutrals is
  Fdrag nnltsinvdriftgtmnvdrift
• (average number of collision per unit time
  nnltsinvdriftgt times the transferred momentum
  mnvdrift)
• In steady state the drag force has to equal the
  Lorentz force
• niFdrag j x B/c 1/4p (rot B) x B
• (with Amperes law rot
  B 4p/c j)
• vdrift (rot B) x B / (4pninnmn ltsinvdriftgt)

nn number density of neutrals ni number
density of ions sin ion-neutral cross section
mn mass of neutral
11
Ambipolar diffusion II

• For a dense core with a size L, the time-scale
  for ambipolar diffusion is
• tad L/vdrift (4pninnmn ltsinvdriftgt)L /
  ((rot B) x B)
• Approximating (rot B) x B B2/L we get
• tad (4pninnmn ltsinvdriftgt)L2 / B2
• Hence ambipolar diffusion time-scale is
  proportional to ionization degree,
• density and size of the cloud, and inversely
  proportional to magnetic field.
• tad 3x106yr (nH2/104cm-3)3/2 (B/30µG)-2
  (L/0.1pc)2
• Is ambipolar diffusion fast enugh to explain why
  cores collapse into stars?
• Observations star formation in cores takes only
  a few million years.
• Still unclear if ambipolar diffusion time-scale
  sets the rate where star formation
• takes place or whether it is too slow and other
  processes like turbulence are
• Required (note central density of cores can be
  much higher than 10-4 cm-3)