Astrophysical Gas Disks - Formation, Characteristics, and
Nonlinearity
Introduction
Giant gas clouds that formed after the Big Bang are the building blocks of the universe. The
only way to understand the current formation of the universe is through the behaviour of gases;
the dynamical nature of gas clouds is what leads to the formation of galaxies and their beautiful,
complex structures.
Gas often behaves nonlinearly whilst forming and shaping galaxies, which will be discussed in
the sections below. Two models of galactic formation will be covered as well as galactic structures
and merging galaxies - all in which gas dynamics play a leading role. To study the structures
of galaxies however, the fundamentals of gas dynamics must first be covered. The dynamics of
the gas will be governed entirely by the conservation laws in physics: conservation of momentum,
conservation of energy, and conservation of mass.
Conservation of Mass
Consider an arbitrary volume V, with bounding surface S, in which the gas flows. V is a constant
volume, so mass M in the volume will only change with time if there is a net flux of mass over S.
If is mass density and v is the local velocity field then,
dM
Z ∂ρ
Z
=
dV = − ρv · ds
dt
V ∂t
S
(1)
From divergence theorem, and since this must be true for any arbitrary volume,
∂ρ
∂t
+ ∆ρv = 0
(2)
which is conservation of mass (Balbus and Potter, 2016).
Conservation of Momentum
The momentum density of the gas is ρv (this is also the maximum flux), and therefore the total
momentum of the gas is the integral of ρv over the volume. In this case we must also take into
consideration the forces that act on the surface of the volume. On the surface this force will be
represented by -Pn. Putting all this together obtains,
1
∂ Z
Z
ρvdV = −
∂t
Z
ρvv · ndS −
Z
−
P ndS
(3)
∂V
Z
∆ · (ρvv + IP )
(ρvv · n + P n)dS = −
(4)
v
where I is the unit tensor, and Pn has become PIn. The equation can now be written as,
∂
(ρv) + ∆ · (ρvv + IP ) = 0
∂t
(5)
∂
(ρv) + ∆ · (ρvv) + ∆P = 0
∂t
(6)
This represents the conservation of momentum within the volume being analysed. The quantity
∆·(ρvv) + ∆P is the stress tensor of the fluid/gas (Dullemond, 2008). Equation (6) shows that the
conservation of momentum in gas has a nonlinear term (ρvv) - this will lead to nonlinear behaviour
in galactic gas disks seen later on.
Conservation of Energy
The properties of gas are strongly linked to thermal energy, and so in analysing the characteristics
of energy in gas dynamics, the thermal and kinetic energy of the system must be considered.
The (internal) specific thermal energy will be specified as e, and kinetic specific energy is ekin =
v2
2 .
Therefore the total energy with be the volume integral the energy density, ρ(e +
v2
2 ).
The
advection - or transfer of heat by the flow of the fluid - is then the surface integral of the ρ(e +
v2
2 )v·n
through the volume V. Finally the first law of thermodynamics, dV = TdS PdV, must be
taken into consideration. This is also the surface integral of Pvn, which is due to the work done by
the exterior on the binding volume. All together the equaiton becomes,
∂ Z
∂t
v2
ρ(e + )dV = −
2
Z
∂V
ρ
ρ(ρe + 2 )v · ndS −
2v
Z
P V · bdS
(7)
∂
Applying the divergence theorem yields,
∂ Z
∂t
v2
ρ(e + )dV = −
2
Z
(ρe +
eV
ρ
+ P )v = 0
2v 2
Again, it can be seen from equation (8) that conservation laws in gas have nonlinear terms.
2
(8)
Equation (8) must be true for all volumes, so the energy conservation equation now becomes
(Dullemond, 2008) :
∂
(ρetotal ) + ∆ · (ρetotal + P )v) = 0
∂t
(9)
Now it can be seen that mass, momentum, and energy are conserved within the gas, and just
how galaxies form and behave according to gas dynamics can be studied.
Gas Dynamics of Galactic Formation
There are two conflicting theories about how galaxies form, both of which rely heavily on gas
dynamics given that only gas was present at the beginning of the universe. What is so interesting
about these two models is that they almost entirely conflict with each other.
Top-Down Theories
In this first model, galaxies are formed from the collapse of a giant gas cloud in the early universe,
hence ”top-down”. This is also called Monolithic Collapse, because it proposes that the oldest stars
and globular clusters must have also began forming here.
First discussed by O. J. Eggen, D. Lynden-Bell, and A. R. Sandage in 1962, the velocities of 221
well-observed dwarf stars were used to compute eccentricities and angular momenta. What they
found was that approximately 1010 years ago, a protogalaxy began to fall together from the galactic
material (the protogalactic gas cloud). As the gas rushed in it eventually came to equilibrium with
the gravitational force of the system. In some regions this equilibrium was violated, and the gas was
able to condense. Here is where the formation of the halo stars - the oldest stars in the galaxy - and
the globular clusters occur. The galactic collapse is halted in the radial direction due to rotation,
but continues on in the Z-direction leading to a thin disk. Suddenly there is a much higher density
of gas and the rate at which stars formed increased. Essentially what the data found is that the
oldest stars were formed from collapsing gas falling towards the galactic centre, then collapsing
onto the plane.
During this process, the gas of the disk became very hot and most of the energy from the
collapse radiated away. At first, the gas orbited in the path of the stars born from it. However,
over time and many collisions with other gas clouds, thg gas disk lost enough energy to settle into
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circular orbits. From this gas, stars with un-eccentric orbits are born (Eggen, Lynden-Bell, and
Sandage, 1962).
This model’s real issues lie in its age; since 1962 quite a few advancements have been made.
At the time of its creation, dark matter and dark energy were not being considered, and better
observational technology have arisen to suggest some contradiction. Nonetheless it remains an
interesting concept in dynamics and galactic formation.
Bottom-Up Theories
A more promising and current model is the broad category of Hierarchical galaxy formation. In
this model, the smallest objects form first, while larger objects are then created from merging (i.e.
galaxies form first then merge to form galaxy clusters). This contradicts the model in Monolithic
collapse theory; here galaxies form from many tiny galaxies, and objects are formed from small
instabilities. What is interesting is the role that gas dynamics plays in the processes of Hierarchical
formation. In traditional N-body simulations of this model, the issue over merging often arises;
dark matter halos that are much too large to be identified with real galaxies emerge. This problem
however, can be overcome by including a gas component. When the gas is added to the same region
with the same density as the dark halos in the simulation, even though the halos still merge, the
gas cores remain distinct as they dissipate. This implies that there are multiple galaxies within the
halo, negating the issue of having to assign each halo to a single galaxy (Katz, 1992).
Galactic Mergers
Despite comprising of a rather small amount of a galaxy over all mass, when it comes to merging
events, gas plays a big role. The actual dynamics of the merger are not significantly affected
by the gas component, but tidal forces during encounters can cause instabilities that lead to bar
formation. The gas in these barred galaxies can then be forced by a gravitational torque to flow
towards the center where it can create starbursts - regions of incredibly high star formations - that
are kiloparsecs across. The consequences of these merging are often chaotic in nature, due to the
nonlinear dynamics of the gas present in the galaxies.
To examine the nonlinear gas dynamics within these mergers, a simulation model done by
Barnes and Hernquist in 1996 will be examined. In this model, they show two galaxies of equal
mass, where one is tilted 71 to the other’s orbital plane. In Figure 2, it can be seen what happends
when the twogalzies collide head on. The time is located in the upper corner of each image, in units
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[H]
Figure 1: This picture shows two galaxies merging and the consequences at different points in time.
of 250 Myr. Before their first encounter both galaxies have developed spiral arms (their nonlinear
characteristics will be discussed later). As the two galaxies approach they exert extreme tidal forces
upon each other. By time t = 1, these forces have begun to distort the shape of the gas.
At t = 1.125 there is a significant amount of gas in the region between the galaxies. This gas
would have been ripped from the disks during the collison.
By t = 1.5 bridge arms have extended from the disks and meet in the middle, dumping gas
and stars onto the inclined plane. The other two arms on the other hand have a high density of
gas, and run along arms of star distrubtions (see fig. 2). At this time we see that the region of
density propogates out along the filaments. As t grows past t = 1.6 the system is dominated by
graviational braking. This causes the orbits to decay and the galaxies to fall into one another. This
process is largely unaffected by the precess of the gas since it makes up such a small amount of the
total mass (∼1.5%). After this point it is impossible to tell the two galaxies apart. What are left
are very dense regions of gas, where the aformentioned starbursts are located.
The mergers simultaneously create and destroy structures in the universe. The aggressive tidal
forces often remove the spiral arms and complete shapes formed in the creation of the disk by the
gas (Barnes and Hernquist, 1996).
Spiral Arms
Most of the galaxies observed in the universe are classified as spiral galaxies. As their name suggests,
the main feature of this type of galaxy is their spiral arms. Initially, it was thought that these arms
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were composed of mostly isothermal gas, the same material that made up the galactic disk itself.
However, this led to a winding problem - after a few orbits, the arms become completely wound up
and the galaxy would quickly lose its spiral structure (Gronwell, 2010).
Galaxy arms are observed to last billions of years, beyond the time it would theoretically take
an arm made of gas to wind itself (Mihos, 2016). This led to Lin and Shu proposing the density
wave theory; spiral arms are not composed of gas nor are they rotating with the stars, but are
instead density waves through which material (stars, gas) moves. Gravitational attraction between
the stars at different radii can maintain the spiral pattern, thus solving the winding problem. The
density wave theory is analogous to vehicles stuck in a traffic jam - the vehicles (comparable to
the gas, stars in a spiral arm) move through the jam, but the jam itself does not move much
(comparable to the density wave and spiral arms itself). This hypothesis explained the large scale
stationary structures in spiral galaxies, but the spirals themselves are actually far from stationary
(Binney and Tremaine, 459). In fact, galactic gas disks are very responsive to small distubances the gravitational field of a point mass travelling on a circular orbit within the disk can induce a
strong spiral-shaped wave in the stars of the disk - which led to the formation of the density wave
theory, central to understanding astrophysical disks (Binney and Tremaine, 460).
Density Waves and Tightly Wound Spiral Arms
Behaviour of density waves in a disk is affected mainly the gravitational potential of the surface
density pattern. This potential affects stellar orbits and alters the surface density in the galaxy,
which can then be matched to the input surface density to obtain a self-consistent density wave
(Binney and Tremaine, 486).
The perturbed surface density of a disk (approximated to have zero thickness) for a tightly
wound spiral can written as
Σ1 (R, φ, t) = H(R, t)ei[mφ+f (R,t)]
(10)
where m is an integer, f(R,t) is a shape function and H(R,t) is a slowly varying function of
radius that gives the amplitude of the spiral pattern (Binney and Tremaine, 486). The parameters
m and f(R,t) define the location of all m arms,
mφ + f (R, t) = constant
where φ is the azimuthal angle.
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(11)
It is also useful to define a radial wavenumber, k, as equal to
∂f (R, t)
k(R, t) =
(12)
∂R
where R is the radius (distance from galactic disk centre).
Equation (10) shows that the surface density variation can be approximated as sinusoidal in
radius (taking the real of of ei[mφ+f (R,t)] ); complicated nonlinear surface density variations can be
decomposed using a Fourier sum (Binney and Tremaine, 487). Now, the gravitational potential to
the the surface density pattern must be determined. In a razor thin disk, as astrophysical gas disks
are often approximated as, the potential of a plane wave is defined as
Φ(R, φ, t) ' Φa eik(R0 ,t)(R−R0 )−|k(R0 ,t)z|
where Φa = -
(13)
2πGΣa
.
|k|
Perturbations in galactic stellar disks can be modelled using fluid disks, as they are dynamically
similar in nature. In tight winding approximation of perturbations in fluid disks, Euler’s equations
can be used to write in cylindrical coordinates,
∂vR
∂t
∂vφ
∂t
+ vR
+ vR
∂vR
∂R
∂vφ
∂R
+
+
vφ ∂vR
R ∂φ
vφ ∂vφ
R ∂φ
+
−
vφ2
vφ
vR
R
1 ∂p
∂Φ
=−
∂R
−
R ∂φ
(14)
1 ∂p
1 ∂Φ
=−
Σd ∂R
−
RΣd ∂φ
(15)
where the perturbation is defined as (v·∆)v, a nonlinear term. The density here is equal to
surface density Σd , since this models a two-dimensional disk. By choosing a simple equation of
state,
p = KΣd γ
(16)
and defining sound speed vs at a density Σ0 as
vs2 (Σ0 ) = (
dp
dΣ
)Σ0 = γKΣγ−1
0
equations (14) and (15) can simplify to
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(17)
∂Φ
−
∂R
1 ∂p
−
∂Φ
Σd ∂R
=−
∂R
∂Σd
− γKΣγ−2
d
∂R
∂
=
∂R
(Φ + h)
(18)
where h is the specific enthalpy equal to
γ
h=
γ−1
KΣγ−1
0 .
(19)
The perturbation term’s angular component, vφ , can be defined as
r
vφ '
R
dΦ
= RΩ(R)
dR
(20)
where Ω(R) is the circular frequency. Substituting everything into equations (14) and (15)
yields
∂vR
∂t
∂vφ
∂t
+Ω
∂vR
∂φ
∂
− 2Ωvφ = −
∂R
(Φ + h)
(21)
d(ΩR)
+
∂vφ
1 ∂
+Ω +Ω
− 2Ωvφ = −
(Φ + h).
dR
∂φ
R ∂R
(22)
All solutions to equations (21) and (22) take the form
vR1 = Re[vRa (R)ei(mφ−ωt) ]
vφ1 = Re[vφa (R)ei(mφ−ωt) ]
Φ1 = Re[Φa (R)ei(mφ−ωt) ]
Σd1 = Re[Σda (R)ei(mφ−ωt) ]
h1 = Re[ha (R)ei(mφ−ωt) ]
(23)
where m ≥ 0 is an integer and ”the perturbation has m-fold rorational symmetry” (Binney and
Tremaine, 490). Solving for vRa and vφa obtains
i
vRa (R) =
d
2mΩ
[(ω − mΩ)
(Φa + ha ) −
(Φa + ha )]
∆
dR
R
8
i
d
vRa (R) = − [2B dR
(Φa + ha ) − m(ω−mΩ)
(Φa + ha )](24)
R
∆
where ∆ = κ2 - (ω − mΩ)2 and κ, Ω, and ∆ are functions of radius (Binney and Tremaine, 489).
The terms proportional to (Φa + ha )/R are much smaller than the terms with d(Φa + ha )/dR, so
they can be neglected when writing the final solution. Equation (24) can then be simplified to
vRa =
(ω − mΩ)
k(Φa + ha )
∆
2iB
vRa = −
∆
k(Φa + ha )
(25)
If the perturbed velocity term, v (has components vR and vφ above), has only linear terms, the
perturbed surface density becomes
∂Σd1
∂t
+ ∆ · (Σd1 v0 ) + ∆ · (Σ0 v1 ) = 0.
(26)
By changing equation (24) into cylindrical coordinates and substituting into the equations (23),
the following can be obtained
1 d
− i(ω − mΩ)Σda +
R dR
(RvRa Σ0 ) +
imΣ0
R
vφa)=0.(27)
Using equation (25) and the equations in (23), a continuity equation can be written in the form
− (ω − mΩ)Σda + kΣ0 vRa = 0.
(28)
In self-consistent density waves with a polarization equal to 1, the dispersion relation for a
tightly wound fluid disk is
(ω − mΩ)2 = κ2 − 2πGΣ|k| + vs2 k 2 .
(29)
The potential of a tightly wound wave can then be written as
Φa (R) = F (R)eif (R) = F (R)ei
9
RR
kdR
(30)
In a cold disk, where vs = 0, equation (29) becomes
ω 2 = κ2 − 2πGΣ|k| + vs2 k 2 .
(31)
The disk’s local stability can then be determined using equation (31), although the dynamics in
the equations above behave nonlinearly. For example, if ω 2 > 0, then ω is real and the disk is stable.
If ω 2 < 0 however, then ω is complex (has roots of ±iω) and there is a nonlinear perturbation with
a exponentially increasing amplitude - the disk is unstable (Binney and Tremaine, 495). Using
the eimφ terms in the many equations above, it can also be seen from DeMoivre’s formula that
increasing m linearly leads to a nonlinear sinusoidal perturbation - eiφ leads to a cos(φ) term, but
changing the perturbation to e2iφ leads to a cos(2φ) term, which is significantly different from a
cos(φ) term (cos(2φ) 6= 2cos(φ)).
Ultra-harmonic Resonances
Galactic shocks in spiral galaxies lead to perturbations of motion in the spiral arms. The change
in frequencies caused by these galactic shocks cause resonances to appear; resonance occurs when
an external force drives a system to oscillate at a specific frequency with increased amplitude. In
gas disks, shock waves causes a specific type of resonance to appear due to nonlinearity, called
ultraharmonic resonance.
The shock perturbations are governed by the density profile, pressure, and dispersion of gases
in astrophysical disks described by the equations in the section above. The perturbation term
usually takes the form eimφ , which is sinusoidal in nature. Even when m is increased linearly, the
perturbation terms behaves nonlinearly, leading to nonlinear dynamical behaviour in the gas disk
and ultraharmonic resonances. An example of an analysis of ultraharmonic resonance in a fluid
disk with a dimensionless spiral perturbing force F is found below.
For spiral galaxies with a pattern speed of Ωp and an angular velocity of Ω, ultraharmonic
resonances of order n was found when mn(Ωp - Ω) = ±κ (Artymowicz and Lubow, 1992). Here,
m is the number of spiral arms in the galaxy and n is an integer - the wavenumber of a free wave
matches n times the wavenumber of the spiral forcing (Artymowicz and Lubow, 1992). It was found
in simulations that when n=2, ultraharmonic resonances were found at 4:1 - this ratio represents
the material frequency relative to the pattern with the epicyclic frequency (Artymowicz and Lubow,
1992).
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Spurs, Feathers, Branches
Spiral arms in galaxies often contain substructures that are called spurs, feathers, or branches. Spurs
can be described as distincally leading spirals, while feathers are produced by local gravitational
instability of a transient kind (Chakrabati, Laughlin, Shu, 2001). Feathers and spurs do not form
the same way and their progression is sensitive to initial conditions - the formations of feathers
and spurs may be chaotic in nature and evolve nonlinearly. Branches in spiral arms form due to
ultraharmonic resonances caused by galactic shock waves.
The existence of these substructures cannot be explained solely due to ultra-harmonic resonances
due to shock waves unless self gravity plays a significant role (Chakrabati, Laughlin, Shu, 2001)
As a result, self gravity in the spiral arms must enhance the intrinsically nonlinear response of gas
dynamics. The effect of gravitational potential of density waves in spiral arms is explained above.
The nonlinear terms in the disk’s self-gravity combined with ultra-harmonic resonances lead to
Conclusion
Astrophysical gas disks behave similarly to fluids, including the many nonlinear dynamical processes
that happen in fluid dynamics. Nonlinearity can be seen most obviously in the behaviour of spiral
arm galaxies, especially in density waves involving tightly wound spiral arms. The ultra-harmonic
resonances caused by the local spiral motion, enhances by the self-gravity of the disk, lead to
formation of spiral substructures such as spurs, feathers and branches. Galactic disk behaviours
observed in the current universe are dominated by nonlinear gas dynamics, which heavily influences
their creation, merging/collision, evolution, and structural formation.
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