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Source: Jeans instability

The Jeans instability is a concept in astrophysics that describes an instability that leads to the gravitational collapse of a cloud of gas or dust. It causes the collapse of interstellar gas clouds and subsequent star formation. It occurs when the internal gas pressure is not strong enough to prevent the gravitational collapse of a region filled with matter. It is named after James Jeans.
For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to

d
p

d
r

=
−

G
ρ
(
r
)

M

enc

(
r
)

r

2

,

{\displaystyle {\frac {dp}{dr}}=-{\frac {G\rho (r)M_{\text{enc}}(r)}{r^{2}}},}

where

M

enc

(
r
)

{\textstyle M_{\text{enc}}(r)}

is the enclosed mass,

p

{\textstyle p}

is the pressure,

ρ
(
r
)

{\textstyle \rho (r)}

is the density of the gas (at radius

r

{\textstyle r}

),

G

{\textstyle G}

is the gravitational constant, and

r

{\textstyle r}

is the radius. The equilibrium is stable if small perturbations are damped and unstable if they are amplified. In general, the cloud is unstable if it is either very massive at a given temperature or very cool at a given mass; under these circumstances, the gas pressure gradient cannot overcome gravitational force, and the cloud will collapse. This is called Jeans collapse criterion.
The Jeans instability likely determines when star formation occurs in molecular clouds.

History

Jeans

Jeans mass

Jeans

The resultant Jeans length

λ

J

{\textstyle \lambda _{\text{J}}}

is approximately

λ

J

=

c

s

(
G
ρ

)

1

/

2

≈
0.4

pc

⋅

c

s

0.2

km/s

⋅

(

n

10

3

cm

−
3

)

−
1

/

2

.

{\displaystyle \lambda _{\text{J}}={\frac {c_{\text{s}}}{(G\rho )^{1/2}}}\approx 0.4{\text{ pc}}\cdot {\frac {c_{\text{s}}}{0.2{\text{ km/s}}}}\cdot \left({\frac {n}{10^{3}{\text{ cm}}^{-3}}}\right)^{-1/2}.}

This length scale is known as the Jeans length. All scales larger than the Jeans length are unstable to gravitational collapse, whereas smaller scales are stable. The Jeans mass

M

J

{\textstyle M_{\text{J}}}

is just the mass contained in a sphere of radius

R

J

{\textstyle R_{\text{J}}}

(

R

J

=

1
2

λ

J

{\textstyle R_{\text{J}}={\frac {1}{2}}\lambda _{\text{J}}}

is half the Jeans length):

M

J

=

4
π

3

ρ

R

J

3

=

π
6

⋅

c

s

3

G

3

/

2

ρ

1

/

2

≈
2

M

⊙

⋅

(

c

s

0.2

km/s

)

3

(

n

10

3

cm

−
3

)

−
1

/

2

.

{\displaystyle M_{\text{J}}={\frac {4\pi }{3}}\rho R_{\text{J}}^{3}={\frac {\pi }{6}}\cdot {\frac {c_{\text{s}}^{3}}{G^{3/2}\rho ^{1/2}}}\approx 2{\text{ M}}_{\odot }\cdot \left({\frac {c_{\text{s}}}{0.2{\text{ km/s}}}}\right)^{3}\left({\frac {n}{10^{3}{\text{ cm}}^{-3}}}\right)^{-1/2}.}

= "Jeans swindle"

=
It was later pointed out by other astrophysicists including Binney and Tremaine that the original analysis used by Jeans was flawed: in his formal analysis, although Jeans assumed that the collapsing region of the cloud was surrounded by an infinite, static medium, the surrounding medium should in reality also be collapsing, since all larger scales are also gravitationally unstable by the same analysis. The influence of this medium was completely ignored in Jeans' analysis. This flaw has come to be known as the "Jeans swindle".
Remarkably, when using a more careful analysis taking into account other factors such as the expansion of the Universe fortuitously cancel out the apparent error in Jeans' analysis, and Jeans' equation is correct, even if its derivation might have been dubious.

= Energy-based derivation

=
An alternative, arguably even simpler, derivation can be found using energy considerations. In the interstellar cloud, two opposing forces are at work. The gas pressure, caused by the thermal movement of the atoms or molecules comprising the cloud, tries to make the cloud expand, whereas gravitation tries to make the cloud collapse. The Jeans mass is the critical mass where both forces are in equilibrium with each other. In the following derivation numerical constants (such as

π

{\textstyle \pi }

) and constants of nature (such as the gravitational constant) will be ignored. They will be reintroduced in the result.
Consider a homogeneous spherical gas cloud with radius

R

{\textstyle R}

. In order to compress this sphere to a radius

R
−
d
R

{\textstyle R-dR}

, work must be done against the gas pressure. During the compression, gravitational energy is released. When this energy equals the amount of work to be done on the gas, the critical mass is attained. Let

M

{\textstyle M}

be the mass of the cloud,

T

{\textstyle T}

the (absolute) temperature,

n

{\textstyle n}

the particle density, and

p

{\textstyle p}

the gas pressure. The work to be done equals

p
d
V

{\textstyle pdV}

. Using the ideal gas law, according to which

p
=
n
T

{\textstyle p=nT}

, one arrives at the following expression for the work:

d
W
=
n
T

R

2

d
R
.

{\displaystyle dW=nTR^{2}\,dR.}

The gravitational potential energy of a sphere with mass

M

{\textstyle M}

and radius

R

{\textstyle R}

is, apart from constants, given by the following expression:

U
=

M

2

R

.

{\displaystyle U={\frac {M^{2}}{R}}.}

The amount of energy released when the sphere contracts from radius

R

{\textstyle R}

to radius

R
−
d
R

{\textstyle R-dR}

is obtained by differentiation this expression to

R

{\textstyle R}

, so

d
U
=

M

2

R

2

d
R
.

{\displaystyle dU={\frac {M^{2}}{R^{2}}}\,dR.}

The critical mass is attained as soon as the released gravitational energy is equal to the work done on the gas:

M

2

R

2

=
n
T

R

2

.

{\displaystyle {\frac {M^{2}}{R^{2}}}=nTR^{2}.}

Next, the radius

R

{\textstyle R}

must be expressed in terms of the particle density

n

{\textstyle n}

and the mass

M

{\textstyle M}

. This can be done using the relation

M
=
n

R

3

.

{\displaystyle M=nR^{3}.}

A little algebra leads to the following expression for the critical mass:

M

J

=

(

T

3

n

)

1

/

2

.

{\displaystyle M_{\text{J}}=\left({\frac {T^{3}}{n}}\right)^{1/2}.}

If during the derivation all constants are taken along, the resulting expression is

M

J

=

(

375

k

B

3

4
π

m

4

G

3

)

1

/

2

(

T

3

n

)

1

/

2

,

{\displaystyle M_{\text{J}}=\left({\frac {375k_{\text{B}}^{3}}{4\pi m^{4}G^{3}}}\right)^{1/2}\left({\frac {T^{3}}{n}}\right)^{1/2},}

where

k

B

{\textstyle k_{\text{B}}}

is the Boltzmann constant,

G

{\textstyle G}

the gravitational constant, and

m

{\textstyle m}

the mass of a particle comprising the gas. Assuming the cloud to consist of atomic hydrogen, the prefactor can be calculated. If we take the solar mass as the unit of mass, and use units of

m

−
3

{\textstyle m^{-3}}

for the particle density, the result is

M

J

=
3
×

10

4

(

T

3

n

)

1

/

2

.

{\displaystyle M_{\text{J}}=3\times 10^{4}\left({\frac {T^{3}}{n}}\right)^{1/2}.}

Jeans' length

Jeans' length is the critical radius of a cloud (typically a cloud of interstellar molecular gas and dust) where thermal energy, which causes the cloud to expand, is counteracted by gravity, which causes the cloud to collapse. It is named after the British astronomer Sir James Jeans, who concerned himself with the stability of spherical nebulae in the early 1900s.
The formula for Jeans length is:

λ

J

=

(

15

k

B

T

4
π
G
μ
ρ

)

1

/

2

,

{\displaystyle \lambda _{\text{J}}=\left({\frac {15k_{\text{B}}T}{4\pi G\mu \rho }}\right)^{1/2},}

where

k

B

{\textstyle k_{\text{B}}}

is the Boltzmann constant,

T

{\textstyle T}

is the temperature of the cloud,

μ

{\textstyle \mu }

is the mean molecular weight of the particles,

G

{\textstyle G}

is the gravitational constant, and

ρ

{\textstyle \rho }

is the cloud's mass density (i.e. the cloud's mass divided by the cloud's volume).
Perhaps the easiest way to conceptualize Jeans' length is in terms of a close approximation, in which we discard the factors

15

{\textstyle 15}

and

4
π

{\textstyle 4\pi }

and in which we rephrase

ρ

{\textstyle \rho }

as

M

/

r

3

{\textstyle M/r^{3}}

. The formula for Jeans' length then becomes:

λ

J

≈

(

k

B

T

r

3

G
M
μ

)

1

/

2

.

{\displaystyle \lambda _{\text{J}}\approx \left({\frac {k_{\text{B}}Tr^{3}}{GM\mu }}\right)^{1/2}.}

where

r

{\textstyle r}

is the radius of the cloud.
It follows immediately that

λ

J

=
r

{\textstyle \lambda _{\text{J}}=r}

when

k

B

T
=
G
M
μ

/

r

{\textstyle k_{\text{B}}T=GM\mu /r}

; i.e., the cloud's radius is the Jeans' length when thermal energy per particle equals gravitational work per particle. At this critical length the cloud neither expands nor contracts. It is only when thermal energy is not equal to gravitational work that the cloud either expands and cools or contracts and warms, a process that continues until equilibrium is reached.

Jeans' length as oscillation wavelength

The Jeans' length is the oscillation wavelength (respectively, Jeans' wavenumber,

k

J

{\textstyle k_{\text{J}}}

) below which stable oscillations rather than gravitational collapse will occur.

λ

J

=

2
π

k

J

=

c

s

(

π

G
ρ

)

1

/

2

,

{\displaystyle \lambda _{\text{J}}={\frac {2\pi }{k_{\text{J}}}}=c_{\text{s}}\left({\frac {\pi }{G\rho }}\right)^{1/2},}

where G is the gravitational constant,

c

s

{\textstyle c_{\text{s}}}

is the sound speed, and

ρ

{\textstyle \rho }

is the enclosed mass density.
It is also the distance a sound wave would travel in the collapse time.

Fragmentation

Jeans instability can also give rise to fragmentation in certain conditions. To derive the condition for fragmentation an adiabatic process is assumed in an ideal gas and also a polytropic equation of state is taken. The derivation is shown below through a dimensional analysis:

If the adiabatic index

γ
>

4
3

{\textstyle \gamma >{\frac {4}{3}}}

, the Jeans mass increases with increasing density, while if

γ
<

4
3

{\textstyle \gamma <{\frac {4}{3}}}

the Jeans mass decreases with increasing density. During gravitational collapse density always increases, thus in the second case the Jeans mass will decrease during collapse, allowing smaller overdense regions to collapse, leading to fragmentation of the giant molecular cloud. For an ideal monatomic gas, the adiabatic index is 5/3. However, in astrophysical objects this value is usually close to 1 (for example, in partially ionized gas at temperatures low compared to the ionization energy). More generally, the process is not really adiabatic but involves cooling by radiation that is much faster than the contraction, so that the process can be modeled by an adiabatic index as low as 1 (which corresponds to the polytropic index of an isothermal gas). So the second case is the rule rather than an exception in stars. This is the reason why stars usually form in clusters.