Density waves in a composite system of stellar and gaseous singular isothermal disks

1
Density waves in a composite system of stellar
and gaseous singular isothermal disks
Yue Shen The Tsinghua Center for Astrophysics
2
OUTLINES

• General Introduction
• Objectives of our previous work and achievements
• Prospects and undergoing tasks

3
General Introduction

• Spiral structures of galaxies in our universe
• Material arms hypothesis and the winding problem
• Density waves and the WKBJ regime

4
1. Spiral structures of galaxies in our universe
X-ray
M100
Milky Way
M51
5
2. Material arms hypothesis and the winding
problem
The initial idea about the spiral arms was
intuitionistic, which considered the arms were
composed of material originally making up the
arms. Unfortunately, this concept will inevitably
suffer the so-called winding problem in disks
with differential rotations. Spiral rotation
curves are flat Vc constant. Therefore the
angular rotation rate goes like Vc/r, so that the
outer edge takes longer to complete an orbit than
the inner regions. So no solid arm could last
very long -- it would quickly get "wound up"
Clearly this winding problem calls for more
sophisticated solutions to the structure of
spiral arms. Any material spiral arms would last
a few galactic years (complete revolutions of the
galaxy at some radius) at most.
6
3. Density waves and the WKBJ regime
The first major attack was made by the Swedish
astronomer Bertil Lindblad, who correctly
recognized the spiral structure arises through
the interaction between the orbits and the
gravitational forces of the star of the disk, and
thus should be investigated using stallar
dynamics. But his methods were not well suited to
quantitative analysis. The crucial step was made
by C. C. Lin and Frank Shu (Lin Shu 1964, 1966,
1968), who clearly declared that the spiral
structure is actually a wave pattern, which is
named density wave. The density-wave theory is
now widely accepted and developed by other
theorists.
7
The WKBJ approximation The basic fluid equations
in a differentially rotating disk of gas are as
follows, in the cylindrical coordinates
and the Poissons equation
8
Suppose a small perturbation on all the physical
variables and linearize the equations
and the Poissons equation
where we use subscript 1 to denote the
perturbations. And we have used the polytropic
assumption that , where a is the
sound speed in the disk.
9
(No Transcript)
10
Objectives of our previous work and achievements
Shu et al. (2000) published a paper, where they
investigated stationary configurations in a
singular isothermal disk (SID) system. They
claimed that these zero-frequency configurations
are bifurcations from stable regions to instable
regions, for both aligned and spiral
disturbances. In their model, the SID is assumed
to be infinitesimally thin and with a background
equilibrium of power-law (?1/r) surface density
and a flat rotation curve. Different form the
usual WKBJ approximation, they accepted a more
accurate form of the surface density perturbation
which leads to an accurate solution to the
Poissons integral. Based on the fluid formalism,
they derived stationary solutions (that is, with
zero frequency) for m0,1,2,3cases and made
physical interpretation for the results.
11
However, their focus in on a single SID system
which is made up only by a stellar disk. In real
disk galaxies, the stellar disk is always
accompanied with the interstellar gas. For
primordial disk galaxies, the content of gas may
far exceed that of the stellar mass. So it is
more practical to treat the disk galaxy as a
composite system of one stellar disk and one
gaseous disk. To include the effects of the
gaseous disk on the whole system may yield
different results from the single SID system. And
this motivated our previous work. There have
been numerous studies on a gravitational coupled
composite system, most of which are in the WKBJ
regime. In our model, we search for the accurate
solutions to this problem and make comparison
with the single SID case.
12

• For convenience the composite system is
  explored using a two-fluid formalism. Both disks
  are infinitesimally thin and are in background
  equilibria of power-law (?1/r) surface densities
  and flat rotation curves. We make a different
  assumption here that the flat rotation curves of
  stellar and gaseous SID respectively, are not
  necessary to be the same. We will show later that
  the rotations of the two SIDs are coupled and
  have a relationship. The two SIDs are coupled
  only through gravitational terms.
• The main equations are similar with those
  shown in an earlier time, except that
• There will be two sets of equations for two SIDs.
  The same physical variables are denoted by
  superscripts or subscripts s and g.
• One should replace the potential F appears in the
  previous equations by the total potential.
• We use velocity dispersion in the stellar disk to
  mimic the sound speed that describes the pressure
  term.

13
From the basic equations we can derive the
equilibrium properties
where d is the ratio of surface densities of the
two SIDs. Note the last equation reveals the
relation between the rotations of the two SIDs.
Only when the sound speeds are the same can the
two rotation speeds be the same. We also use
to denote the ratio of the two sound
speeds.
14
(No Transcript)
15
The aligned cases By definitions, aligned
disturbances mean that the maximum and minimum
perturbations happen at different radii line up
in the azimuth. In contrast, when the maximum and
minimum perturbations suffer a systematic shift
in the azimuthal angle from one radial location
to the next, such disturbances are unaligned or
spiral patterns (Kalnajs 1973). For aligned cases
we choose the perturbations of the forms as
where ss and sg are constant coefficients.
Substitution them into the final equations yields
a quadratic equations about . Then we can
solve this equation under different m. The m0
case is trivial and it is just a rescaling of the
axisymmetric background equilibrium. The m1
case can be satisfied with any positive value of
. The mgt1 cases are different, described as
follows
16
As in the single SID case, there only exists one
solution. While in the two-SID system, two
solutions are found with initial parameters d and
ß known as constants. One solution has been
recognized as the counterpart of the single SID
case but is smaller, and may become physically
unmeaningful when ß exceeds a critical value.
However, as a novel solution to this two-SID
system, the second solution is always valid and
greater than the other solution. Moreover, it is
interesting to find that of the two solutions,
the smaller one (i.e. the counterpart of the
single SID case) makes the surface density
perturbations of the stellar and the gaseous
disks in-phase while the larger one (i.e. the
novel one) makes the two surface density
perturbations out-of-phase. We note that the
out-of-phase solution is valid, at least
mathematically. Whether or not such perturbations
can exist in real disk galaxy systems is remained
to be an open question. (Say, if we can
discriminate the stellar arms and the gaseous
arms, respectively.)
17
The spiral or unaligned cases Now we turn to the
unaligned cases. This time we choose the the
perturbations of the forms as
where Nm(a) is the Kalnajs function (Kalnajs
1971), ss and sg are constant coefficients, and a
is the radial wavenumber. Substitution them into
the final equations yields a quadratic equation
about . Then we can solve this equation
under different m. For m0, which means the
axisymmetric disturbances, we have computed the
marginal stability curves of versus a,
invoking the same physical interpretations of Shu
et al (2000) for regimes of collapse, stable
oscillations, and ring fragmentation. For
different parameters d and ß, we find these
marginal stability curves may vary significantly
in reference to those obtained by Shu et al
(2000), as shown in the following figures.
18
The marginal stability curve when m0, d1, ß10.
The increase of d shrinks the collapse regime,
while the increase of both d and ß lowers the
marginal stability curve of ring fragmentation.
The marginal stability curve when m0, d0.2,
ß1.5. As d is small and ß is slightly greater
than 1, the curve does not differ significantly
from the single SID case shown in Fig. 2 of Shu
et al (2000).
19
The marginal stability curve when m0, d10, ß5.
For a sufficiently large d, the collapse regime
disappear while the danger to ring fragmentation
increases.
The marginal stability curve when m0, d1, ß30.
The ring fragmentation curve crosses the a axis
such that the ring fragmentation occurs for all
when a is sufficiently large. Such system is
the most instable against axisymmetric
disturbances.
20
To illustrate the physical meaning of these
marginal stability curves more clearly, we have
carried out a WKBJ analysis for the two-SID
system about the axisymmetric instability, but is
not included in the previous paper. We now just
give the results here. We know that for a
time-dependent perturbation with frequency ?, if
? is real, then the disturbance is stable
oscillating if ? is imaginary, the disturbance
will grow with time, hence the system is
instable. So from the WKBJ dispersion relation
for the two-SID system, we have plotted the
contour of ? square as a function of radial
wavenumber K and Ds square, and filled the ?
square lt0 region (i.e. the instable region) in
grey and the ? square gt0 region (i.e. the stable
region) in white. The basic configurations of
the contours are very similar with the marginal
stability curves we obtained in the previous
work. Hence we confirm the physical meaning of
those stationary marginal curves we achieved in
our previous paper.
21
The two figures are very similar with the
marginal stability curves we have obtained
previously. The main difference rises from the
collapse region, where the wavenumber K is small
so that the requirement for the WKBJ
approximation is poorly satisfied.
22
Now lets move to the mgt0 cases. The m1 case is
again similar to the single SID problem, with
only one solution physically meaningful (i.e. the
counterpart of the single SID system). For the
mgt1 cases we have obtained analytical results
almost in the same forms of aligned cases. This
is easy to be understood since the aligned case
is just a special one complementary to spiral
cases, all of which involve density waves in a
composite system of two coupled SIDs. For each
sets of d and ß we can obtain two curves of Ds
square versus wavenumber a. While the counterpart
one may be negative when ß exceeds a critical
value for fixed values of a. The novel one is
always positive and also larger than the other
solution. Meanwhile, just as the aligned cases,
both branches of solutions give the surface
density perturbations either in-phase or
out-of-phase.
23
Two solutions of stationary spiral configurations
for Ds square versus radial wavenumber a. The
lower branch is recognized as the counterpart of
the single SID case and has the possibility to
turn into physically unmeaningful the upper
branch is novel due to a composite system.
24
Actually, the similarities between the aligned
and spiral cases are due to the same nature of
density waves. We denote here that the aligned
density waves propagate purely azimuthally while
the spiral density waves also involve radial
propagations. (Lou 2002) We also investigated a
composite partial SIDs system. By partial disks
we mean that there is a massive dark halo which
contributes gravitational potential to the system
but does not take part in the perturbations. Such
a model is even practical. With this assumption
we have made modifications to all the above
analysis. In particular, the aligned m1 case is
no longer a solution for arbitrary positive Ds
square, that is, such stationary eccentric
displacements are forbidden in a composite system
of two partial SIDs.
25
Prospects and undergoing tasks

• Our next studies include the following parts
• The density wave solutions in the composite SIDs
  system with isopedic magnetic field (by Yue Wu)
• The same problem but with azimuthal magnetic
  field (by Yue Zou)
• The wave propagation and the angular momentum
  flux (by Yue Shen)

26
References

• Bertin G., Lin C.C., 1996, Spiral Structure in
  Galaxies (Cambridge MIT Press)
• Binney J., Tremaine S., 1987, Galactic Dynamics.
  Princeton University Press, Princeton, New Jersey
• Jog C.J., Solomon P.M., 1984a, ApJ, 276, 114
• Jog C.J., Solomon P.M., 1984b, ApJ, 276, 127
• Kalnajs A.J., 1971, ApJ, 166, 275
• Lin C.C., Shu F.H., 1964, ApJ, 140, 646
• Lin C.C., Shu F.H., 1966, Proc. Nac. Acad. Sci.
  USA, 73, 3785
• Lin C.C., Shu F.H., 1968, in Chretian M., Deser
  S., Goldstein J., eds, Summer Institute in
  Theoretical Physics, Brandeis Univ., Astrophysics
  and General Relativity. Gordon and Breach, New
  York, p. 239
• Lou Y.Q., 2002, MNRAS, 337, 225
• Shu F.H., Laughlin G., Lizano S., Galli D., 2000,
  ApJ, 535, 190
• Toomre A., 1964, ApJ, 139, 1217

Thank You