Lecture 3
Star formation
Part II
Lecture Universität Heidelberg WS 11/12
Dr. Christoph Mordasini
Based partially on script of Prof. W. Benz
Mentor Prof. T. Henning
Lecture 3 overview
1. Jeans instability
1.1 from force balance
1.2 from intuitive arguments
1.3 from virial theorem
1.4 from perturbation theory
1.5 with rotation
2. Poisson equation and potential energy of a sphere
3. Collapse of a T=0 spherical gas cloud
3.1 Fragmentation
3.2 Opacity limit
1. Jeans instability
Jeans instability
We have seen in the last lectures that the galaxy contains Giant Molecular Clouds, consisting of
cold gas. These clouds are usually stable, with pressure (due to a finite temperature) balancing
self-gravity.
We have also seen when studying perturbations in a self gravitating fluid, that exponential
growth is possible.
Now we want to understand under which conditions small perturbations of a gas cloud grow
exponentially, leading to the collapse of the cloud (and, in the end star formation). The so-called
Jeans instability describes the gravitational instability of a self-gravitating gas cloud. There are
several ways to derive such a criterion, with increasing complexity.
M
T, p
r
The initially stable, static cloud can get initially compressed a bit
(perturbed) by a shock wave due to a nearby supernova, passing
spiral arms of the galaxy, ..
1.1 Jeans instability
from
force balance
Jeans instability from force balance
The simplest, order of magnitude estimation can be obtained by force balance arguments.
FP
Collapse occurs if the inwards directed gravitational force is
bigger than the outwards directed pressure force.
FG
M, V
Consider mean unit forces per volume.
T, p
r
A
Force balance
We note:
so
critical maximal radius
that allows stability.
higher pressure=>more stability
higher density=>less stability
with
dense, cold clouds are unstable
1.2 Jeans instability
from
intuitive arguments
Jeans instability from intuitive arguments
Consider a spherical, isothermal cloud of radius r. Now imagine that a small perturbation
reducing the volume V so that the mean density increases from initially ρ0 by a small
amount (perturbation), say by ρ1=ρ0 α with α <<1. Since for an isothermal gas, p=c2ρ, we
have for the pressure perturbation p1=c2αρ0.
From the Euler equation we know that the general expression for the acceleration due to the
pressure force is given as
The extra acceleration associated with the perturbation can therefore be estimated by
We see that the stronger we compress the cloud, the stronger the pressure pushes back.
But also the gravity increases due to the increased density. The associated extra
gravitational acceleration is
The cloud will be unstable towards collapse if the excess gravitational acceleration exceeds
the excess pressure acceleration. With the assumption that the cloud remains isothermal, if
this condition is once satisfied it will remain satisfied.
Jeans instability from intuitive arguments II
Thus instability occurs if
This any perturbation with a wavelength exceeding the above length scale will be unstable
towards gravitational collapse. This can also be looked at as a ration of timescales since
sound traversal timescale:
dynamical timescale:
This latter timescale is the gravitational free fall timescale in the absence of pressure support:
(we can use mean values like
M/2, r/2, but they cancel out)
This is also the fundamental timescale with which stars oscillate around its equilibrium.
Jeans instability from intuitive arguments III
Therefore, we can write the instability criterion also as
Information transport in the cloud occurs with a velocity equal c. In this picture, the
criterion means that instability occurs if sound waves do not have the time to traverse the
cloud over a dynamical timescale.
This means that the pressure has not the time to react, and cannot increase in time to
counterbalance the gravity, as e.g. the center of the cloud does not yet know what is going
on at the surface.
The comparison of relevant timescales is a very fundamental concept in astrophysics to
understand (at least roughly) physical mechanisms.
1.3 Jeans instability
from the
virial theorem
Jeans instability from the virial theorem
Let us again assume a static, spherical, homogenous cloud of radius r. We now add the fact that
the cloud is in equilibrium with an outside medium at a pressure pext. We further assume that the
cloud does not have any bulk kinetic energy and the the pressure is constant throughout the cloud.
We recall that the general virial theorem is given as
I: moment of inertia
With our assumptions this becomes
where we have assumed an ideal gas with f degrees of freedom. The factor q in the gravitational
potential energy depends on the density distribution. For a homogeneous density, it is 3/5, cf later.
Assuming further f=3 (1 atomic gas) and
We first note that we can use the ideal gas law
, so
This tells us that the pressure in the cloud must be higher in the presence of gravity. Or in
other words, the necessary external pressure to confine the gas is lower. In its absence, the
pressure in the cloud and the one outside are the same.
Jeans instability from the virial theorem II
pext
A
For fixed M and T the external pressure has the
following shape as a function of R. The curve
gives us possible equilibrium states (radii) at a
specified external pressure.
B
R
Rm
We note:
-for too high pext no equilibria exist.
-for sufficiently low pext two equilibra exist. But
only one is stable:
1) At point B, if we decrease R (compression), pext increases (and with it also the internal p).
This is a stable equilibrium.
2) At point A, if we decrease R (compression), pext necessary to confine the cloud decreases
too (and with it also the internal p). This is clearly an unstable equilibrium.
We can find the critical radius Rm which divides the two regimes by setting
which yields
Using
and the ideal gas EOS
is the isothermal sound speed, we find:
where
Jeans instability from the virial theorem III
The Jeans length is from this analysis
The numerical constant is very close to unity
(ca. 0.95), so we find a similar result as before.
The corresponding Jeans mass is
Using typical values
we find
and
The collapse thus begins
with large masses.
This is more than the typical mass of a single star (less than 1 Msun). This indicates that
during the collapse, only part of the gas ends up in stars, and that the cloud fragments
during collapse. Thus many stars form out of one collapsing cloud, which means that
young stars get born in clusters.
1.4 Jeans instability
from the
perturbation theory
Jeans instability from perturbation theory
We have seen above how we can use the linearized hydrodynamic equations to study
the behavior of waves in a self-gravitating fluid.
Here we consider a blob of gas with uniform T and uniform density ρ0 at rest. For such
a configuration, the mass and momentum conservation equations imply that the
gradient of the gravitational potential is vanishing. Poisson’s equation on the other hand
dictates for the Laplacian:
This initial configuration is therefore not a solution to the conservation equations, which is
in principle a requirement for the perturbation analysis. This is often called the Jeans
swindle, but nevertheless quite often used.
The stability of such a configuration can be studied using the dispersion relation for sound
waves in a self-gravitating medium. The dispersion relation gives the relation between
angular frequency ω = 2 π f and the wave number k = 2 π/λ. We have seen above that
Jeans instability from perturbation theory II
For our wave like perturbations,
ω2>0 have an oscillatory nature
ω2<0 describe an exponential growth, i.e. an instability
The critical value is simply found by setting ω2=0, which gives the critical wave length 2 π/k
which is, to a numerical factor of order unity, the same as before.
1.5 Jeans instability
with rotation
Jeans instability with rotation
To derive a more realistic instability criterion it is necessary to take into account rotation.
In a rotation frame of reference it is necessary to add the Coriolis and the centrifugal
accelerations to the hydrodynamic equations.
The initial velocity is then still zero. Let us assume that
and that
perturbations propagate only along the z axis. We then get the perturbation equations
1)
2)
Note: the centrifugal term cancels the
gradient of the potential of the
equilibrium solution: no Jeans swindle
3)
with the equation of state:
Jeans instability with rotation II
We now seek for solutions of the type
and similar perturbations for
all other variables. Inserting these solutions in the above equations yields
This system of equations can be written in a matrix form:
Jeans instability with rotation III
We are looking for non-trivial solutions. This means that the determinant must be zero.
This leads to a bi-quadratic equation in ω for the dispersion relation:
where cos θ is the angle between the direction of propagation of the perturbation
and the rotation axis . Using Vieta’s relation, we can get the following two equations
for the ω solving the equation:
Two cases must be distinguished:
For θ≠90°, from the second line we note that the sign of
depends only on the
instability criterion without rotation, as the other terms are always >0. This means that
if the Jeans instability criterion without rotation is satisfied, namely if
then also in the case with rotation, collapse occurs. For this geometry, rotation cannot
prevent a cloud from collapsing.
Jeans instability with rotation IV
The second case is the important special case where θ=90°, i.e. where the rotation axis
and the perturbation are perpendicular to each other. The dispersion relation takes the
simple form:
ω
(fast rot.)
(slow rot.)
0
k
Clearly, the additional rotation term acts stabilizing. For
sufficiently large rotation, in the direction perpendicular
to the rotation axis, rotation can stabilize the largest
scales from gravitational collapse. This is nothing else
than the fact that angular momentum is an enemy of
star formation, and that instead of collapsing into a
point, the cloud must now collapse into a disk.
unstable
θ≠90°
small wavelength (large k):
stabilized by pressure
large wavelength (small k):
stabilized by rotation
θ
θ=90°
2. Poisson equation
and
potential energy of a sphere
Poisson equation
We have used already several times the Poisson equation which gives the gravitational potential.
From Newton we know the universal law of gravity:
The flux through an arbitrary surface is then:
as
In spherical coordinates
the flux is therefore
=
Any mass is given as
, there fore this last line is also
Poisson equation II
With the Satz von Gauss we can write the equation for the flux also like this:
Therefore we find
This equation must be true for all volumina. Therefore, the integrands themselves must be
identical:
Gravity is a conservative force i.e. the rotation vanishes:
Therefore, a potential must exist which satisfies
So, we finally have the Poisson equation
Potential energy of a sphere
The total gravitational potential (or binding) energy of a sphere
of uniform density can be found by considering the successive
accretion of spherical layers on the matter already there.
The gravitational potential of the planet outside of
the planet is
and a mass shell is
So, in our derivation for the Jeans mass out of the virial theorem, q=3/5.
Potential energy of a sphere
With the above equation, we can estimate the Kelvin Helmholtz timescale of the sun, i.e. the
time on which the sun cools if it would be powered by gravitational contraction. We remember
from the virial theorem that during contraction, half of the potential energy goes into internal
energy, while the other have is to be radiated away.
So we can radiate away the total energy:
At the current luminosity of the sun, we can therefore estimate the time on which the sun
would cool:
We know that the actual energy source of the sun is fusion, not contraction. But before
the discovery that nuclear fusion powers stars (~1920), this short timescale was a very
puzzling result.
3. Collapse of a T=0 K
spherical homogeneous
gas cloud
Collapse of a T=0 K spherical homogeneous
gas cloud
We now consider what is happening after the collapse has started. We assume that
pressure does not play a role (yet), which we (formally) can represent by assuming T=0 K.
Then, the equation of motion of a spherical cloud is simply given as
Integration by separation of the variables with the initial conditions v(0)=0 and r(0)=r0 yields
the velocity as a function of r
Next we look for v(t) and r(t). With the ansatz r=ro cos2(α) we eventually find the non algebraic
equation for α(t)
Collapse of a T=0 K spherical homogeneous
gas cloud II
One defines the free fall time as the time required for complete collapse (r=0). From our ansatz
r=ro cos2(α), we see that this is a case if α=π/2. Solving for t yields
which depends only on the cloud density, but not on the total cloud mass. The higher the
density, the smaller tff. For interstellar densities ρ0~10-22 g/cm3, we get tff~7 Myrs.
This result ignores both angular momentum (rotation) and magnetic fields, which will
counteract collapse, and therefore lengthen the true collapse time. The free-fall collapse
time given above is a lower limit to a more realistic collapse time calculation.
As the total mass of the collapsing cloud does not change, we find for the density
The velocity is
We note the following points:
Collapse of a T=0 K spherical homogeneous
gas cloud III
1) The density remains constant with r
which means that e.g. the center
does not become proportionally
denser than the outer parts.
2) The velocity is linear in r
Collapse of a T=0 K spherical homogeneous
gas cloud IV
3) Most of the action happens in the last moments cf.
Initially, r and ρ change only slowly, then very rapidly.
For example,
only when
4) Density perturbations (e.g. initial enhancements) grow faster, since for
tff,1<tff,0 for δρ>0 i.e. higher density regions collapse quicker.
3.1 Fragmentation
Fragmentation I
During the collapse, ρ increases. As long as the density still remains adequately low for the
cloud to be transparent, the released thermal energy is radiated into the universe and the
temperature remains approximately constant. As
suggests, this leads to a decrease of the Jeans mass. In particular, sub-sections of the
cloud suddenly surpass their own Jeans limit and start collapsing on their own. As also tff is
smaller for higher densities, these sub-collapses proceed faster. This clearly leads to
fragmentation.
Here we calculate the collapse of an density
perturbation in the otherwise homogenous cloud
Such places might be the origin of later individual star
formation, as they decouple.
Fragmentation II
Using the free fall velocity, we can re-write the master equation
The collapse occurs at α=π/2. To compute the difference in
collapse between the background cloud and the perturbation,
we define β as the parameter characterizing the cloud, and θ the
parameter characterizing the perturbation.
Because most of the action happens shortly before the complete collapse, it is convenient to
introduce small angles measuring the difference to the full collapse (where we can use that for
small angles e.g. sin(α)≈α) :
for the background cloud
for the perturbation
Both angles are <<1, and complete collapse occurs when they vanish.
Inserting this into the master equation, we find after some algebra:
Fragmentation III
This means that at moment when the perturbation has fully collapsed (χ=0), we have in the
background cloud:
This corresponds to a density increase in the background cloud equal
For example, an initial density perturbation of 1% (
)will fragment out of the gas
cloud when the mean density of the latter has increased by a factor 7000. Keeping in mind
that the full density increase form a GMC to a star is ~20 orders of magnitude, this is a small
increase only.
In other words, small initial density fluctuations lead to a much faster collapse.
3.2 Opacity limit
Opacity limit
We have neglected up to now the effect of pressure, or equivalently assumed a low,
constant temperature. As the density becomes higher and higher, the gas increasingly
becomes opaque to its own radiation. Therefore, the temperature must start to rise at some
point. This eventually leads to a minimum fragment size into which the cloud can break up.
This is know as the opacity limit.
We have seen that
In the limit that the gas becomes completely radiatively inefficient, i.e. adiabatic at high
opacities, we have for a monoatomic ideal gas
i.e.
With the EOS
this means
so finally,
which says that in the adiabatic case, the Jeans mass increases with density.
The combination of the decrease of MJ with density in the isothermal regime, and the
increase of MJ with it in the adiabatic regime must lead to a critical mass when the two
regimes meet.
Opacity limit II
We can estimate the critical mass by some simple energy considerations.
1)Heating
The gravitational binding energy of a collapsing gas ball is
q=3/5 for cst. density
The collapse happens on a typical timescale tff
The liberated binding energy per second which heats the gas is therefore
2) Cooling
In the same time cools the gas by radiating as a blackbody at the surface of the gas blob.
f: correction factor i.e
radiation efficiency<1
Opacity limit III
The gas will start to heat up (become adiabatic) as soon as the cooling becomes slower
than the heating. In order to prevent this we must have:
i.e.
Solving for M gives the criterion
In the same time, the mass must be bigger than the Jeans mass for collapse to proceed, so
When MJ and Mcrit become equal, we hit the opacity limit.
Opacity limit IV
Using our earlier result for the Jeans mass
and combining the equations above, we finally find for the minimal fragment mass
(=5.24)
Recalling that the isothermal sound speed is
we get using q=3/5
[g]
We see that the result only depends weakly on the temperature. Numerically we get
T= 10 K, f=1 =>
T= 100 K, f=0.01 =>
From this very simple estimate we thus see that cloud fragmentation leads to objects going
from the upper planetary mass domain to low mass stars (M dwarfs, the most frequent type
of stars in the galaxy).
Further reading
R. Kippenhahn & A. Weigert
Stellar structure and evolution,
Springer Verlag, Berlin, 3rd Ed. 1994
S. Stahler, F. Palla
The formation of stars,
Wiley-VCH, Weinheim, 2004
Questions?