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Magnetic Fields and Superbubbles
Brian Daniel prei
A thesis submitted to the Department of Physics
in conformity with the requirements for
the degree of Master of Science
Queen's University
Kingston, Ontario, Canada, K7L 3N6
May, 1997
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ABSTRACT
AU observed superbubbles are reviewed in order to test the success rate of previ-
ously proposed formation models. These models are shown not to be universally
applicable and to be unable to explain the most recently observed bubbles. A new
model involving magnetic fields is proposed to account for these bubbles and is
shown to apply to all other superbubbles as well. A simple Utoy model" is used
to gain insight into the magnetic superbubble problem before a comprehensive
physical model is developed. The feasibility study indicates that this model can
generate correct order of magnitude energy estimates for superbubbles provided
that the global magnetic field is included. The structure of the bubble is produced
by a magnetic field locally perturbed by difFerential rotation.
In light of the results from the feasibility study, a three-dimensional magne-
tohydrodynamic self+hilar time-independent model is developed The simpli-
fying assumptions that were required in the feasibility study are removed in this
model. The analytic formulation of this model is explored and the structure of
the magnetic field is constrained. The Poynting flux and the critical points of the
governing equations are discussed to further characterize the parameter functions.
A preliminary investigation of the parameter space reveals that two of the critical
points will have to be properly addressed before a solution which fits the boundary
conditions will be found.
ACKNOWLEDGEMENTS
The preparation of this thesis would not have been possible without the guidance
of my supervisors, Dick Henriksen and Judith Irwin. My sincerest thanks go to
these people for guiding me in my work, encouraging me to explore and generally
h e h g my ambition to learn.
Special thanks also go to Siow-Wang Lee and Denise King for taking the time
to discuss various aspects of this thesis. Thanks also to Kathy Perrett for teaching
me how to use various sake packages, to JJ Kavelaars and Steven Butterworth
for their help with the computer system and to Andrew Kult for his help with
programming in C.
I can't possibly express my gratitude to Zou Zou K u y k for her support and
encouragement over the last few years. She has shared in all of the triumphs and
defeats inherent in graduate studies. She has stimulated my imagination and in
so doing made great contributions to this work. And yet, all I can say is 'thank
you'.
This thesis is dedicated to two people who are very dear to my heart but did
not live to see the completion of this thesis: my mother and my grandfather. May
they rest in peace.
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Introduction 1
. . . . . . . . . . . 2 . Review of Superbtzbbles: Models and Observations 11
. . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Superbubble Models 14
. . . . . . . . . . . . . . . . . . . . . 2.1.1 Multiple Supernovae 16
. . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Impacting Clouds 21
. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Individual Galaxies 22
. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Galaxy 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 LMC 25
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 NGC 3079 28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 HoII 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 NGC 4631 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.8 NGC 1620 31
2.2.9 .The D w h : NGC 1705 . NGC 5253 . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.10 NGC 3044 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2-11 NGC 3556 33
. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Success of the Models 34
. . . . . . . . . . . . . . . . 3 . The Feasibility of Magnetic Supetbubbles 36
. . . . . . . . . . . . . . . . . . . 3.1 A Superbubble's Magnetic Field 36
. . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Field Constraints
3.1.2 A Simple Field Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Higher Order Solutions
. . . . . . . . . . . . . . . . . . . . . . 3.2 Fitting the Magnetic Field
. . . . . . . . . . . 3.3 Application to the Superbubbles of NGC 3556
. . . . . . . . . . . . . . . . . 3.4 Application to Other Superbubbles
3.5 Results of the Feasibility Study . . . . . . . . . . . . . . . . . . .
4 . framework for a Self-Similar Superbubble . . . . . . . . . . . . . . . . 4.1 Defining the Magnetic Superbubble Problem . . . . . . . . . . . .
4.1.1 The Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Parameter Functions
4.1.3 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Analytic Results
. . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 StreamLines
4.2.2 Poynting Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Critical Points
4.3 Preliminary Numerical Work . . . . . . . . . . . . . . . . . . . . . 4.4 Summary of the Self-similar Model . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Conclusions
References
Appendix 102
A . Critical Point Condition for J=O . . . . . . . . . . . . . . . . . . . . . 103
LIST OF FIGURES
The east superbubble in NGC 3556 . . . . . . . . . . . . . . . . . 3
. . . . . . . . . . . . . Global magnetic field structures of galkes 5
The alpha effect as illustrated by Parker (1992) . . . . . . . . . . 8
. . . . . . . . A self-consistent solution for the pressure variation 43
The Brandt rotation curve fit to NGC 3556 by Giguere (1996) . . 49
. . . . . . . . . . . . . . . . . The shapes of 9 magnetic field lines 53
. . . . . . The coordinate system used for the s e I f ' a r analysis 66
. . . . . . . . . . The family of stream Lines which is investigated 71
. . . . . . . . . . An integrated solution curve for &. CI* and 11' 85
. . . . . . . . . . . . . . An integrated solution curve for x = z, 86
. . . . . . . . . . . . . . . . . . An integrated solution c m for o 87
. . . . . . . . A preliminary integration stazting at a critical point 89
LIST OF TABLES
Properties of known superbubbles . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . Properties of galaxies containing superbubbles 15
. . . . . . . . . . . . . . . . . . . . . . Summary of model success 35
Feasibility study model results for NGC 3556 . . . . . . . . . . . . 50
Model results for the NGC 3044 superbubble . . . . . . . . . . . . 55
Model results for the NGC 1620 superbubble . . . . . . . . . . . . 56
Model results for M33 superbubbles . . . . . . . . . . . . . . . . . 57
Model results for the NGC 4631 superbubbles . . . . . . . . . . . 58
Model results for the Milky Way superbubbles . . . . . . . . . . . 60
. . . . . . . . . . . . . Critical points of the differential equations 76
1. INTRODUCTION
Superbubbles are large expanding shells which have been observed in the inter-
stellar medium (ISM) of many galaxies. In many ways, the shell structure of
superbubbles appears to be the same as extremely large supernova remnants.
However, while a typical remnant may be 50 to 75 pc in diameter (Strom, 1996),
many superbubbles have diameters in excess of one k p c Expanding shells are
therefore observed in a continuum of sizes, with supernova remnants at the small
end and superbubbles at the large end of the spectrum. The energy source of
supernova remnants has been identified and studied in great detail. The energy
source of superbubbles has not been identified and is the focus of this thesis.
Just as water boils, it seems that galactic disks boil. When a pot of water boils,
the bubbles form where the energy is concentrated and transport the energy to
cooler regions. A similar transport of energy and material occurs as a result of
superbubble formation and evolution. A superbubble forms in the galactic disk
and expands into the halo, carrying with it energy and matter from the disk.
This transport of material to the halo is likely to be reciprocated by transport
of material from the halo to the disk; otherwise the gaseous material in galactic
disks would evaporate. The circulation of material between the disk and halo has
been named the galactic fountain (Shapiro and Field, 1976).
Although several other processes, such as nuclear jets and outflows, may con-
tribute to a galactic fountain, superbubbles are particularly important because
they have the potential to disrupt large regions of a galaxy. Figure 1.1 illustrates
the extent of the disruption of the disk by one of the bubbles observed in NGC
I . h troduction 2
3556. This superbubble, one of the largest known, is several kpc across and con-
tains about lo8 MQ of material. Understanding how superbub bles form and evolve
in the ISM is therefore a prerequisite to understanding the galactic fountain and
its role in determining galactic structure. Smaller bubbles and even supernova
remnants may contribute to the galactic fountain, but the greatest contribution
will be born the largest and most energetic bubbles. However, superbubbles re-
main a mystery; a superbubble energy crisis exists because an adequate energy
source has not been found. The largest superbubbles require the equivalent input
energy of 100,000 supernovae (SNe). Current models require either this number
of SNe or some form of interaction such as an impacting cloud. Neither model is
able to account for the energetics of all the observed superbubbles.
There are, however, two other components of galactic structure that may be
able to provide the necessary energy for superbubble expansion. These are differ-
ential rotation and the galactic magnetic field. As a whole, a galaxy contains more
than enough rotational kinetic energy or magnetic energy to produce hundreds of
superbubbles. Can one or both of these sources be responsible for producing the
observed superbubbles, and if so how is the energy tranderred to an expanding
shell? These questions are unanswered because neither Werentid rotation nor
magnetic fields have ever been considered as a superbubble energy source before.
The most likely mechanism by which kinetic energy could be transferred from
differential rotation to a superbubble is via shearing of the magnetic field. If
magnetic fields are to play a role in superbubble formation and evolution, then
differential rotation will also be involved. Differential rotation results in the mag-
netic field being sheared, a process which might transfer energy to superbubble
expansion. Although magnetically formed superbubbles have not been previously
considered, there are two other magnetic processes thought to occur in galax-
ies which might relate to superbubbles: the galactic dynamo, a field generation
mechanism, and the Parker instability, predominantly a magnetic buoyancy ef-
Fig
RIGHT ASCENSION (81 950)
we 1 .l: The East HI superbubble in NGC 3556 is located at RA = llh Ogm los, DEC = 55'59' in both images. The top image (Irwin, private communication) is shown to illustrate the extent of the radio emission (contour liues) relative to the optical disk (grey scale). The superbubble is observed as an HI hole in the grey scale and radio contours of the bottom image ( h i et al., 1997). The radio images are only integrated over approaching (blueshifted) velocity intervals. The velocity i n t d and contow levels are different in both images.
fect. These mechanisms are discussed further in the following pages; however both
processes are dependent on general properties of the galaxy's magnetic field. The
strength and spatial structure of observed galactic magnetic fields are therefore
discussed before returning to the topics of dynamos and the Parker Instability.
The key observational tracers of the magnetic field are synchrotron radiation
intensity and polarization, optical and infrared polarization, Faraday rotation, and
Zeeman splitting (Zweibel and Aeiles, 1997). Whenever possible, two or more of
these tracers are used in conjunction to generate the most complete picture of
a galaxy's field. To date, adequate information to constmct a global picture of
the magnetic field structure exists for only a few nearby galaxies such as M51
(Neininger and Horellou, 1996).
Fkom such studies of magnetic field structure it has become standard to express
the magnetic field as the sum of two components: the uniform field and the random
field,
f i=~ . .+a . (1-1)
For the Galaxy, H d e s (1996) adopts Bu = 2.2 pG and Br = 2.0 pG. For M31,
Berkhuijsen et al. (1993) bind Bu = 4 pG and Br = 3 pG- Estimates for other
galaJdes are consistent with the two field components being approximately equal
in strength and having total field strengths of up to 20 pG (see for example NGC
6946: Beck, 1991).
The global magnetic field structures of gal&= are quite varied. The global
field can be Fourier decomposed with respect to the azimuthal angle $,
Most observed fields are adequately described by either the m = 0 (axisymrnetric,
ASS) or the rn = 1 (bisymmetric, BSS) modes (illustrated in Figure 1.2), but
others can only be described by a combination of several modes. The vertical
structure determines the parity of the field. Even parity structures are symmetric
Figure 1.2: hckymmetric and bisymmetric field3 are shown in (a) and (b) respectively. (c) shows an example of an even pariw field and (d) shows an enample of an odd parity field. This figure was reprinted (with permission) from Zweibel and Heiles (1997).
about the midplane of the galaxy (quadrupolar fields) while odd pariw structures
are antisymmetric (dipolar fields) (Beck et d, 1996). Of course, most galaxies for
which magnetic field observations have been made exhibit such ordered structures
only to a first approximation. In general, magnetic loops and spurs are also seen
extending into galactic halos (see for example Beck et al., 1989). Some galaxies,
however, do show the well defined fields of a single mode. For example, the
magnetic field of M31 is an ASS field (Ruzmaikin et a l , 1990; Beck, 1982) and
the field of M81 is a BSS field (Sokoloff et al., 1992; Krause et al., 1989). Models
of the origin of galactic magnetic fields are therefore tightly constrained by these
geometries.
The two competing theories regarding the origin of galactic magnetic fields
are field generation, via the operation of a dynamo, and primordial fields. The
latter theory contends that any present-day field results from the twisting of a
cosmological field by differential rotation (Beck et al., 1996)- While this theory is
still widely supported, it seems to have little in common with the superbubbles
studied here. The dynamo theory, however, does appear to be relevant to magnetic
superbubbles.
According to dynamo theory, magnetic fields are generated by turbulent mo-
tions of the plasma. Tkbulence is created in a galactic disk by the violent outflows
of jets and SNe, by stellar winds and by local heating (among other events). The
magnetic field is primarily coupled to the di&se ionized gas and anchored by
molecular clouds. In order for the dynamo to generate a magnetic field, the field
must be carried about by the convection produced by turbulent motions of the
gas. Although this motion can not be resolved with current telescopes, general
characteristic flow patterns are expected to be present in a differentially rotating
fluid body.
Within a local convective cell, the magnetic field in a meridional plane (a plane
parallel to the rotation axis) can be written as a h c t i o n of a vector potential
4 (w , z) (Parker, 1979) ,*
where w is the cylindrical radius and t is the distance parallel to the rotation axis.
The collective production of 4 by the smallacale circulation provides the working
basis of the dynamo. In a differentially rotating body, azimuthal magnetic field
lines are deformed by turbulence and twisted by rotation. This effect, illustrated
in Figure 1.3 from Parker (1992), is known as the or-effect; it relates the mean
magnetic field and its derivatives to the electromotive force (Beck et al., 1996),
The current density can then be written as a function of the mean fields 8 , C and
1. Introduction 7
The induction
as foUoWs:
equation, once modified to include the a contribution, is written
The turbulent conductivity is related to the conductivity of the medium according
and the turbulent di&sivity, w, is defined as l/pT. The creffect is the gener-
ation of the magnetic fields due to the a term in the induction equation above.
In its most general form, a is a 2nd rank tensor with nonzero off-diagonal terms
representing anisotropies in the medium due to stratification, rotation, shear, SNe
and stellar winds (Beck et ul., 1996). Determining the various components of the
a tensor is non-trivial and has been the subject of several recent papers (see
for example Hanasz and Lesch, 1996); however to first order a is diagonal for
galactic disks (Zel'dovich et al., 1983). A dynamo operating under these con-
ditions is referred to as an a-dynamo. These dynamos, which are based solely
on turbulent mixing, neglect kinematic effects such as differential rotation. The
d y n a m o mixes the turbulent and kinematic effects and so better describes
the dynamo operating in galactic disks. The u-effect generates a radial field
component, B,, from the azimuthal component, B4, and the w action generates
B4 fiom B, (Hanasz and L e d , 1993). The result is a pmcess which is capable
of generating both axisymrnetric and bisymmetric fields of galactic strengths in
approximately log yt. However, this success is dependent on the assumed tur-
bulent diffusivity, rh. = 0.2wiT, where y. and h are the characteristic velocity
and length scales of the dominant eddies. Models that are able to produce the
observed magnetic field structures require a value of rh. = lo* cm2s-I so that
1. Introduction 8
Figure 1.3: The alpha e f k t as UiUustrated by Parker (1992) (reproduced with permis- sion). The (a) initially unitom field is inflated into (b) R-shaped loops which then get rotated by the cyclonic velocity a (c).
adequate *ion can occur. Parker (1992) has shown that under such conditions
the turbulence is strongly constrained by the tension of the mean magnetic field,
implying that the turbulence is unable to provide the small-scale mixing that it
is assumed to provide. Instead, Parker suggests that the Parker Instability, as-
sisted by cosmic rays, may be able to provide the required mixi~g in place of the
turbulence. The Pazker Instability is a perfect example of our imperfect knowl-
edge of how magnetic fields affect the ISM. Although mathematical evidence for
a magnetic instability in the ISM was provided by Parker in 1966, to date there
have been no conclusive observations of the Parker Instability at work.
This instability is a sort of magnetic Rayleigh-Taylor instability. The ISM is
supported against the gravitational force by t h e d pressure, magnetic pressure
gradients and pressure due to cosmic rays. At equilibrium one assumes a plane
parallel magnetic field embedded in a density distribution dependent only on z
(height above the galactic plane). This configuration is unstable with respect
to horizontal perturbations. A small negative e a t i o n in density at one point
allows the field to rise and thus reinforces the flow of material away fiom the
density depression. As a result, the magnetic field acquires a sort of buoyancy in
the ISM.
The instability criterion developed by Parker (l966), as expressed by Mouschovias
(1996), is
where k is the wave number magnitude of the perturbation, k. = (1 +a +P)4lg, and a and /3 are respectively the ratios of the magnetic and cosmic ray pressures
to the thermal pressure. y is the ratio of specific heats and c, is the sound speed
in the medium.
A connection between the Parker Instability and superbubble formation is
suggested by the resulting field structures. The rising field lines become buoyant
in the ISM and balloon out of the plane to form expanding loops. These Mating
loops could provide additional energy to superbubble expansion. As well, the
regions of increased density may become compact enough to trigger the formation
of cloud complexes (Mouschovias, 1996).
Linear analysis of the Parker Instability in the presence of Metentid rotation
indicates that wavelengths of approximately 2rh, where h is the vertical density
scale height, have the f&est growth rates (Mouschovias, 1996). Similar work by
Foglimo and Tagger (1995) indicates that the differential rotation can transiently
stabilize or arnpliZy the instability depending on the intensity of the shearing. In
this case, the fastest growth rate occurs when the rotation curve decreases more
quickly than a flat rotation curve.
Simulations of the non-linear Parker Instability acting on the ISM by Basu
et al. (1997) indicate that the maximum growth rate occurs for roughly the same
wavelengths (or slightly longer) in the non-linear regime as the linear regime.
1- Introduction 10
Although Basu et al. focus on the density enhancement between inflating lobes,
they do note that shocks are formed as a result of expansion above the galactic
plane. Maximum expansion velocities of these lobes reach 20-30 kms-I. However,
the spatial extent of these expanding Lobes (radii of 300400 pc) is consistent only
with the very smallest superbubbles (radii between 300 pc and 3300 pc). Larger
lobes may result from unmodelled effects of cosmic rays (Parker, 1992).
The Parker Instability alone is incapable of forming superbubbles, as is the
galactic dynamo. However, both processes involve the formation of magnetic
field structures which resemble expanding shells. Whether or not these processes
are involved in superbubble formation, it does appear possible that superbubbles
could form as a result of magnetic processes and differential rotation. This thesis
examines this possibility in order to determine whether or not magnetic fields may
provide a solution to the superbubble energy crisis.
In Chapter 2 the proposed models of superbubble formation are discussed and
the observed superbubbles are reviewed to determine the success of the various
models. The need for another model to explain the formation of the largest su-
perbubbles is illustrated. Chapter 3 proposes a simplistic magnetic field model for
a superbubble's energy source and then sees it applied to several of the bubbles
examined in Chapter 2. This feasibility study is physically incomplete and so
serves only to direct further efforts. A comprehensive self-similar model is devel-
oped in Chapter 4. Several analytic results are discussed both in terms of physical
significance and in terms of significance to numerical modelling. A preliminary
investigation using a numerical integration routine is also carried out to provide
further insight into the nature of this self-sirnilax model. The results of this study
are presented in Chapter 5 and a new view of the superbubble is summarized.
2. R E m W OF SUPERBUBBLES: MODELS AND OBSERVATIONS
As early as 1958 expanding shells were observed in the Galaxy (Menon, 1958).
Westerlund and Mathewson (1966) identiiied a kpc scale ring-like structure of
neutral hydrogen in the LMC. In 1973 a similar structure was observed in MlOl
(Allen et al., 1973). These detections happened by chance however, and not
until Heiles (1976, 1979, 1984) carefully analysed two Galactic HI surveys was the
prevalence of these structures realized.
To date, shell-like structures have been identified in several galaxies. Such
shells are commonly referred to as bubbles, superbubbles, supershells or holes.
They range in size fkom a few hundred parsecs to several kpc and in energy from
los2 to 1055 ergs. In general, the larger structures are named superbubbles or
supershells. In this work, the distinction will be made as follows: only bubbles
with estimated input energies greater than los ergs will be referred to as super-
bubbles. Although this particular energy is somewhat arbitrary, it does represent
an approximate boundary between shelh that can be easily attributed to the col-
lective effects of a few SNe andlor stellar winds of OB stars (see for example
Normandeau et al., 1997) and the largest superbubbles examined in this thesis.
Table 2 lists the known superbubbles by galaxy in order of date of observation.
Each superbubble is designated according to the galaxy in which it is found,
followed by the identihition scheme used by the authors of the original studies
(GS is used to denote Galactic shells). Rgd is the distance between the centre of
the superbubble and the galactic centre, Rsh is the radius of the superbubble, Kr,
2. Review of Superbubbk Models and Observatiom 12
is the expansion velocity, M is the mass content of the supperbubble and Es is the
estimated input energy. Energies for superbubbles denoted with an foUowing
the name have been calculated using equation 2 of Heiles (1979) (equation 2.1,
this thesis) and the data given in the listed references. Unless noted otherwise in
Section 2.2, all of the other input energies were calculated by the original authors
using the same equation.
Table 2.2 summarizes various properties of the galaxies listed in Table 2. As
shown, most of the spiral galaxies that contain one or more superbubbles are
barred galaxies. This fact may suggest that the perturbations produced by the
presence of a bar are important to superbubble formation. However there are too
few examples for calculating any meaningful statistics. As well, the effect of a bar
on a galaxy's magnetic field is not understood. To huther complicate the deter-
mination of any correlation, one model of bar formation is tidal interactions with
a companion galaxy (Lynden-Bell, 1996). The direct importance of companions
to superbubbles will become obvious in Section 2.1.2, the discussion of the im-
pacting cloud model. The distance to these galaxies is important in determining
the role of selection effects in the identification of bubbles. The star formation
rate (SFR) affects the SN rate (SNR) and is relevant to the models of superbubble
formation discussed in the next Sections. The SFR is calculated according to a
direct empirical correlation with the far infrared luminosity discussed by Condon
(1992). The Galactic SFR is used only as a normalization factor because it is not
calculated in the same manner. A detailed review of each galaxy is presented in
Section 2.2 in order to examine the success of the various models.
2. Review of Superbubbles: Models and 0 bservations 13
Thble 2.1: Prope~5es of known superbubbles as reported in the references listed.
NGC 3079-A - NGC 3079-B - NGC 3079-C - NGC 3079-D - NGC 3079-E - HoII - 2 1 2.06 NGC 4631 - 1 - 6.5 NGC 4631 -2 - 6.0 NGC 1620 11
2- Review of Superbubbk Models and Observations 14
Name R log Rsh h log M log & Ref.
NGC 1705 - 2* NGC 1800 - 1' NGC 1800 -2* NGC 3125' NGC 3955 -1* NGC 3955 - 2* NGC 4670' NGC 5253' NGC 3044 NGC 3556 -West NGC 3556 -East
Notes: H, is assumed to be 75 km s-I Mpc-? The asterisk indicates that the energy was calculated using equation 2.1 and data from the rehmce. All of the papers referenced as follows: '~des, 1979; 2~eiles, 1989; 3~eaburn, 1980; 'Brinks and Bajaja, 1986; 5Deul and den Hartog, 1990; %win and Seaquist, 1990; ?~uche et d., 1992; *Rand and van der Hulst, 1993; O~and and Stone, 1996; l0Vader and Chaboyer, 1995; ll~arlowe ef d., 1995; 121,ee and Imin, 1997; 13~iguere, 1996; "King and Irwin, 1997.
2.1 Superbubble Models
Various models have been p r o p d to explain the formation of superbubbles.
Small bubbles are thought to be initiated by a single SN and to be further fuelled
by stellar winds. One such bubble in NGC 2363 is thought to contain - 5 W-R
stars (Drissen et al., 1993). Drissen et al. (1993) estimate that the winds could
provide - 1.5 x lo5* em of the 6 x ergs that Roy et al. (1992) consider
to be the input energy responsible for driving the expansion of this bubble at a
rate of 45 kms-I . However, larger superbubbles with input energy requirements
> 10% ergs, like those listed in Table 2, can not be explained in this manner.
To deal with this energy crisis, two models have been proposed: multiple SNe
2- Review of Superbubbles Models and Observatiom 15
'Ihble 2.2: Properties of &alaxies containing superbubbles.
Name Type= DistP Companions SFRb Ref. W p c )
Galaxy Sb (bar?) - LMC, SMC 1-0 LMC dIrr 0.05 Galaxy, SMC - M31 Sb 0.7 M32, MllO - M33 Scd 0.7 none 0.075 1 NGC 3079 Sm bar 20.4 NGC 3073, 6.4 1
MGC 9-17-1 HOD h IV-V 3.2 M81, dwarfs - 2 NGC 4631 Sd bar 6.9 NGC 4656, 1.3 1
NGC 4627, dwarfs NGC 1620 Sbc bar 45.7 none 1.1 3 NGC 1705 SO pec 6.1 C-d 0.14 4 NGC 1800 SO bar pec 9.2 0.11 4 NGC 3125 dIrr 13.8 1.2 4 NGC 3955 peculiar 21.1 0.55 4 NGC 4670 SO bar pec 14.6 0.62 4 NGC 5253 E/SO pec 3.3 N5128 group 0.31 4 NGC 3044 Sc bar 20.6 f 1.3 1 NGC 3556 Scd bar 11.6 none 1.1 1
Notes: The derencea are denoted as follows: lSo~iz et d., 1989; *~uche et d., 1992; 3Young et d., 1986; 'Marlowe et al., 1995.
a Types and distances are fkom the New Galactic Catalog by W e y (1988) or one of the rdezences listed. AN distances have been scaled for h. = 75 km s-I Mjx-l.
SFR are d e d to that of the Galaxy. The Galactic rate is calculated h m an infrared luminosity of 6 x lo9 Lo given by Mezger (1978). Other rates are calculated fiom infirad flux data (references listed) using equations from Condon (1992).
NGC 1705 is described as "quite isolated on sky survey platesn by Meurer et d. (1992). These galaxies appear to have been previously perturbed by interactions with other
galaxies (Hunter et at., 1993). These galaxies are described by Hunter ef al. (1993) as "not obviously interacting
with another system". f NGC 3044 is described as showing no signs of interaction, but there is at least one g a k y within 10 optical radii and Av = 1000 kms-' (Solomon and Sage, 1988).
2. Review of S u ~ e r b u b b k Models and Observations 16
and impacting clouds. The multiple SNe model suggests that superbubbles are
formed by many SNe. Two variations of this scenario are commonly discussed:
correlated SNe and propagating SNe. The impacting cloud model suggests that
superbubbles are not the result of an internal process, but rather result from an
infalling cloud of material.
2.1 -1 Multiple Supernovae
a) Multiple Correlated SNe
In his paper outlining the Galactic superbubbles, Heiles (1979) suggests that a
possible explanation for superbubbles which could produce the shell structure and
meet the energy requirement is simply many SNe. Assuming that a typical SN
releases los1 ergs of energy, over 1000 spatidy correlated SNe would be required
to produce the largest Galactic superbubbles. Several groups have since studied
the effects of such an energy deposition on the ISM.
Bruhweiler et al. (1980) proposed that a shell was the natural outcome of an
OB association. The combined effects of the stellar winds and sequential SNe
were expected to produce the expanding shell structure. However, the energy
requirement implied that the OB association had to contain far more stars than
the associations which have been observed in the Galaxy. Typical Galactic as-
sociations contain about 40 OB stars, but the largest may contain as many as
400 (Tenorio-Tagle and Bodenheimer, 1988). Tomisaka et al. (1981) applied the
hydrodynamic equations to the expansion of a supernova remnant within a hot
bubble as a further step in the analytical treatment of superbubble formation.
However, it became apparent that numerical analysis would be required.
An early analysis by Tomisaka and Ikeuchi (1986) assumed a SN rate of
5 x lo6 yr-', corresponding to an association containing about 250 OB stars
(Tenorio-Tagle and Bodenheimer, 1988), and ignored the effects of stellar winds.
2- Review of SuperbnbbIes: Models and Observations 17
Simulations were done for superbubbles located at heights above the midplane of
z = 0, 100 and 200 pc and disk densities of 1 and 0.1 It was found that the
superbubble growth in the i direction occurred more quickly than the planar di-
rections, leading to asymmetric shells. Although over 10' yrs 50 SNe contributed
energy (5 x LO^* ergs) the largest simulated shell only reached a radius of - 900
pc and a kinetic energy of expansion of - 3 x losL ergs. McCray and Kafatos
(1987) extended this model by considering the evolution of a superbubble around
an OB association over a longer timescale. After a few x lo7 yrs of nearly con-
stant expansion driven by similar input energy, the shell became gravitationally
unstable and fragmented.
W h e r work approximated the discrete SNe as a continuous energy source
(Mac Low and McCray, 1988). Model superbubbles expanding into an exponential
atmosphere were found to blow out of the disk. This escape occurred after the
bubble had reached a size of a few scale heights (typically a few hundred pc)
and was quickly followed by the onset of gravitational instability (Mac Low and
McCray, 1988). In s following paper Mac Low and McCray increased their SN
rate from 3.4 x yr-' (Mac Low and McCray, 1988) to 5.3 x loa yr-I (Mac
Low and McCray, 1989). Regardless, the sample superbubble generated using a
numerical program for MHD calculations is seen to blow out after - 8 My (total
input energy - 4.6 x lo5* ergs).
Heiles (1990) hrther categorized superbubbles into Ubreakthrough" bubbles
which break through the dense central disk but do not reach the halo and blowout
bubbles which actually inject disk material into the halo. These blowout bubbles
contradict the behaviour of model shells when faced with the shell-destroying
tendencies of gravitational instability (HeiIes, 1990; Igumentshchev et al., 1990).
The problem of forming the largest superbubbles became a problem of preventing
blowout. Several authors looked to the galactic magnetic field as the answer
2. Review of Su~erbubbles Mod& and Observations 18
(Mineshige et at., 1993; Tomisaka, 1992; FemQre et al., 199 1; Tomisaka, 1990).
b) Inclusion of Magnetic Fields
Tomisaka (1990) considered a plane parallel magnetic field of 5 pG and a continu-
ous energy source extended over a period of 10' yrs. The superbubble expansion
was diminished in the i direction, thereby providing the required containment.
The expansion of the shell in such a magnetic field is shown to be quite anisotropic.
The bubble expands more quickly dong the field lines (2) than across them (8). With 5.3 x 1051 ergs injected into the superbubble, the bubble reached a size
of (x , y, z) = (220pc, 1 4 0 ~ , 200pc). Femhe et al. (1991) noted that an external
field would lead to a thicker shell and confirmed the increased expansion along
the field lines. Their model, which incorporated the difference between the inter-
nal and external magnetic pressures on the shell, as w d as cooling, led to a more
rapid deceleration (Femhre et al., 1991) than Tomisaka's model (Tomisaka, 1990).
Tomisaka (1992) extended the analysis of magnetized superbubbles to higher en-
ergy injection rates (1 SN x loJ y-'). At these extreme rates, corresponding to a
few hundred SNe over a superbubble's lifetime, blowout was once again observed.
Thus, although the inclusion of a magnetic field does allow larger bubbles
to be produced without suffering fkagmentation due to gravitational instability,
the largest shells still require too much energy to be contained within the disk.
Tomisaka (1990) also suggests that the Parker Instability may play an important
role during the late stages of superbubble evolution. The timescale for the Parker
Instability, roughly the free fall timescale in a gravitational field (- 2.5 x lo7 yr) , is also the approximate age of most observed superbubbles (estimated by assuming
constant expansion). Recent numerical studies by Kamaya et al. (1996) have
indicated that once the Parker Instability is initiated, the magnetic field can aid
the inflation of the bubble to the point of blowout. No information regarding the
energetics of such a bubble is presented so it is unclear whether the expansion
2. Review of Superbubblec Models and Obsezwations 19
rate would be adequate to explain the observed bubbles.
c) Propagating SNe
Propagating SNe models try to expand the effective size of an OB association by
allowing cyclic star formation. Propagating SNe models were first discussed in the
context of galactic structure by Mueller and Arnett in 1976. Their proposal was
that a chain reaction of SN - SF - SN would dramatically influence the dynamical
evolution of the ISM. The shock fkont of a SN was assumed to compress the
surrounding material, thereby inducing SF. The newly formed stars were assumed
to be massive so that the cycle could repeat. Numerical modelling of this process
in a differentially rot at ing galaxy revealed transient spiral structures without the
incorporation of a spiral density wave (Mueller and Amett, 1976).
Gerola and Seiden (1978) expanded on this model by removing the determinis-
tic SF assumption. Rather than assuming that every SN would give rise to massive
star formation, a finite probability of SF was assigned. I . numerical simulations
this new model revealed more stable spiral structure (Gerola and Seiden, 1978).
Although the appearance of spiral s t ~ c t u r e does not seem directly relevant to
superbubble formation, it does illustrate the potential of such a local process to
elicit a global effkct.
In the context of superbubble formation, propagating SNe are considered to
be an explanation for the apparent energy crisis. The explosion of a SN in a dense
cloud produces an expanding shock front. If this shock compresses gas sufficiently,
several stars could form in a roughly spherical shell around the original SN (most
observed bubbles are not perfectly spherical). After lo7 y t s these stars could
produce SN to drive the expansion Mher . Feitzinger et al. (1981) has done a
simulation of this process acting on the entire LMC and shown that "large scale
holes and shells" do appear; however, no discussion of the energetics of individual
shells is provided. This simulation also demonstrated that propagating SN could
2. Review of Superbubbk Models and Observations 20
preserve the integrity of the superbubble in the presence of shear.
Dopita et al. (1985) have suggested that this process is occurring in the bubble
LMC4. The observed gas dynamics of LMC4 can be explained by a propagating
SN model which began - 15 x 106 yrs ago and expanded at a rate of- 35 kms-I.
However, Vdenari et al. (1993) recently identified associations on the edge of
LMC4 which have twice this age.
LMC4 has also been the target of a detailed gas motion study by Domgiirgen
et al. (1995). Velocities and column densities of Line cf sight gas were found by
analysing spectra of stars within LMC4. These data were then combined with
HZ emission data to locate gas clouds and stars dong a line of sight (Dorng6rgen
et al., 1995). The various superbubble formation models were then applied to
LMC4 in order to determine which best predicted the observations. Because the
propagating SN model best describes the inhomogeneous expansion and the LOS
velocity components observed at the edge of the shell, Domgiirgen et al. (1995)
conclude that this is most likely the process responsible for forming the superbub-
ble. However, both of these observations could also be produced by the turbulence
caused by shell breakup.
Although the propagating SN model appeanr to describe the dynamics of
LMC4 well, the energy of LMC4 is only - 7 x 1p2 ergs, or - 70 SNe (Meabum,
1980). If one applies this model to one of the shells in Table 2, either the required
number of SNe becomes unrealistically large (as much as 10 000) or many cycles
of SF are required, implying a formation time (several x lo7 yrs) approaching
the rotation time of a galaxy (I. lo8 yrs) during which the bubble would be
significantly sheared.
2- Review of Su~erbubbles Models and Obsenations 21
The basic premise of the impacting cloud model is that a nearly spherical shell
could be produced not by an explosion outward from within the disk, but by the
impact of an external cloud of gas. This model is particularly relevant to the
Galaxy because of the observed high velocity clouds (HVC) . Oort (1970) was the
first person to suggest that the high velocity clouds observed in the Galaxy could
be the impact ejeetcz of an infalling cloud. Mirabel (1982) reports observations
of an infalling cloud complex in the anticentre direction which appears to be
interacting with disk matter. Furthermore, Ehlerovh and Palouii (1996) note that
the largest of Heiles' (1979) shells lie at z 2 2 kpc indicating that superbubbles are
surface features rather than deeply embedded objects.
TenorbTagle (1980) first investigated the energy requirements of a superbub-
ble in the context of an HVC impact. His analytical analysis led him to conclude
that a cloud with radius 50 - 100 pc, densit3f 1 and velocity 200 kms-' < V
< 310 kms-I could deposit loa - lo* ergs of energy by impacting a constant
density disk. This work led to the first numerical models which were restricted to
2D simulations and relatively low energies (loa - los1) (Comeron and Torra,
1992; TenorbTagle et id., 1987). Rand and Stone (1996) have done extensive
modelling of the large superbubble in NGC 4631. This superbubble was chosen
because it is one of the most energetic superbubbles observed. The estimated
input energy is the equivalent of 2 lo4 SNe.
The presence of clouds of material surrounding NGC 4631 can not be ques-
tioned due to the 5 companion galaxies and extensive evidence of tidal disruption
(Rand and Stone, 1996). With this evidence in mind, Rand and Stone (1996)
attempt to use parameters inferred from observations to best reproduce the struc-
tural and kinematic properties of the superbubble. The galaxy is modelled ne-
glecting rotation and shear (due to timescales involved) with a scale height of 1
2. Review of Superbubbles Models and Observations 22
kpc inclined a t 8 6 O and a midplane density of p, = 1.6 x lo-** g cm? The best fit
cloud parameters are found to be as follows: mass of 1.2 x 10' Mo (density equal
to the midplane density), diameter of 500 pc and a velocity of 200 kms-' (imply-
ing an input energy of 5 x los4 eqs) . Further analysis indicates that the inclusion
of an azimuthal magnetic field does not greatly Bffect the results, nor does the
basic shape of the impactor (Rand and Stone, 1996).
Rand and Stone (1996) are very successful in explaining the observed structure
and kinematics of the large NGC 4631 superbubble by means of an impacting cloud
model. Observations of Galactic HVCs suggest that they are prominent enough
to have a significant impact on the evolution of the Milky Way's ISM as well
(Mirabel, 1991; Mirabel and Morras, 1984, 1990). However, the recent discovery
of superbubbles in two companionless galaxies, NGC 3044 (Lee and Irwin, 1997)
and NGC 3556 (King and Irwin, 1997), implies either that impacting clouds are
not the only superbubble formation mechanism or that superbubbles result from
impacting clouds formed by means other than galactic interactions.
Schulman et al. (1994) d e d out a survey of nearby galaxies in HI aimed at
detecting HVCs. They find that the high-velocity wings of the HI profiles can, in
many cases, be fit better by a model incorporating HVCs than one using warped
disks. Of the 14 galaxies surveyed, 10 are well described by models including
HVCs. The other 4 galaxies have low infi.ared fluxes corresponding to lower star
formation rates. As a result, Schulman et d. postulate that SNe and superbubbles
provide kinetic energy to HVCs and not the reverse; cause and effect can not be
readily determined.
2.2 Individual Gdaxies
Although each galaxy included in Table 2 contains a t least one superbubble, there
are differences in how the bubbles were detected, in the number of bubbles de-
2. Review of Superbubbles Mod& and Observations 23
tected, in the size and completeness of the sheb, and in the morphology of the
galaxies themselves which must be kept in mind when considering various models.
It is these difFerences that are outlined here. It should also be recognized that
selection effects are responsible for certain trends in Table 2. For example, more
shells are observed in the closest of the spiral galaxies. Limited resolution restricts
the detection of smaller shells in distant galaxies. Inversely, it is potentially easier
to identify larger structures in distant galaxies.
2.2-1 The Galaxy
Carl Heiles (1979) was the first person to present a detailed survey of "HI Shells
and Supershells". In his 1979 paper (with corrections in 1984)' Heiles identifies
46 non-expanding and 17 expanding shells in the Galaxy. The most energetic of
these he refers to as supershells or superbubbles. Heiles' work has been used as a
reference guide for nearly all of the following superbubble surveys. For this reason,
it is necessary to examine closely Heiles' methods and models. Understanding
the way that Heiles determined gdactocentric radii, input energies, densities and
masses is crucial to being able to put the various numbers into context correctly.
The galactacentric radii of the shells reported by Heiles (1979,1984) have been
estimated using the Galactic rotation curve. Heiles adopts 10 kpc as the radius
of the solar circle, &, and 250 kmsoL as the circular velocity at I&,. Within & the rotation curve is well dehed but an object in a particular direction with a
particular velocity could Lie at either of two points (the intersections of a chord
through a circle). Heiles consistently chooses the nearest possible position to the
sun. Outside of & Heiles adopts the mean value obtained from calculating the
position with a flat rotation curve and Schmidt's (1965) extrapolation. Heiles
estimates his own errors in determining the position of these superbubbles to be
35%-
2. Review of S u ~ e r b u b b k Models a d Observations 24
The expansion velocity of a shell is determined by the range of velocities over
which the shell can be seen (see for example fig. 1 of Irwin and Seaquist, 1990).
Ideally this process measures the velocity difference between the approaching and
receding sides of the shell: twice the expansion velocity. These expansion velocities
are then used to calculate the input energy required to form the shell. Heiles
(1979) adopted a model which assumes that the shells are the product of several
SNe. Assuming a SNe origin, the detection of the bubbles in HI indicates that
much energy has been radiated away. Thus, these shells can not be in the Sedov
phase of expansion, The Sedov phase is characterized by very efficient conversion
to kinetic energy with very Little energy being radiated away (Sedov, 1959). The
later stage of expansion, during which much energy is radiated away, dowing
the shell to cool, was studied numerically by Chevalier (1974). Heiles adopts
Chevalier's energy equation given below:
The calculated energy is the necessary input energy (ergs) to produce an ex-
panding (Post-Sedov) shell via a single SN with a radius R (pc) and an expansion
velocity V, (kms-I). The density q (d) is the ambient density at the loca-
tion of the shell b e f m sheil formation. Heiles assumes that Cheder 's numerical
work can be applied to multiple SNe shells without modification.
To determine HI masses, Heiles estimated the column densities at the shell
centres from Weaver and Wllliams (1973). These column densities were then
assumed to be uniform throughout the spherical volume, even when only one
hemisphere of the shell had been observed. Heiles also included a factor of 1.4 in
his shell mass to account for the presence of Helium.
Thus the energy estimate as derived by Heiles (1979) is model dependent.
The energy requirement is the total input energy of the SNe, including radiated
2. Review of Superbubbles= Models and Observations 25
energy but excIuding neutrino energy (Chevalier, 1974), and should therefore only
be used in comparing shells in terms of how many SNe might be required to
produce them. As mentioned in Section 2.1.2, there is some evidence Linking
Galactic superbubbles to particular W C s . The estimated size of the largest OB
associations is not large enough to form Heiles' shells, but the detected HVCs
could provide enough energy.
LMC
Goudis and Meaburn (1978) and Meaburn (1980) identify 9 possible supergiant
shells in the LMC and 1 in the SMC by unsharp masking and high contrast
copying of previously obtained photographic plates. Of these shells, only LMC-2
and LMG3 have estimated input energies > los3 ergs (Meaburn, 1980). The
densities of these two shells were estimated by averaging the shell mass of LMC2
over its volume. The resulting high density, no = 3 an-3, was then used to
calculate the input energy for both superbubbles by applying equation 2.1.
Goudis and Meaburn (1978) describe both of these shells as showing much
filamentary structure surrounded by an HI outer shell. Intense UV emission and
several OB associations are observed within each shell. Meaburn (1980) indicates
that more than 400 OB stars could be contained in each of these superbubbles.
Such a large number of high mass stars is consistent with the multiple SNe model
discussed in Section 2.1.1.
The detection of X-ray emission in LMC2 by Wang and Helfand (1991) indi-
cates that this superbubble formed near the surface of the galactic plane and then
expanded rapidly into the surrounding low density region. The X-ray emission is
concentrated dong what appears to be the shock kont where the superbubble is
interacting with the denser medium of the galactic plane. The only other shells
for which associated X-ray emission has been confidently detected are the LMG4
2. Review of Su~erbubbles Models and Observations 26
bubble (Bomans et al., 1994) and the bubbles in NGC 4631 (Volger and Pietsch,
1996). The location of this superbubble near the surfaee of the galaxy indicates
that it may have been formed by an impacting cloud. However, the presence of
OB stars within the shell and the proximity of the LMC to the Galany favours a
multiple SNe origin for LMC-2.
A Westerbork Synthesis Radio Telescope survey of M31 in neutral hydrogen de-
tected many structures which Brinks and Shane (1984) describe as "HI holes".
M31 has an inclination of 77 (Brinks and Burton, 1984) and so is seen in a less
highly inclined orientation relative to the majority of galaxies with superbubble
detections. As a result, galactocentric radii are more confidently determined, but
position above or below the mid-plane is lost.
In total, 141 holes were identifwd by each of the authors independently (Brinks
and Bajaja, 1986). Velocity channel maps were used as selection criteria by requl-
ing that each hole be visible over at least 3 successive channel maps. Expansion
velocities were derived from the rauge of channel maps over which each shell was
observed. The density was calculated fiom the column density and the input en-
ergy was calculated using equation 2.1. Of 141 holes, 8 are energetic enough to be
deemed superbubbles here. However, only 5 of the 8 were observed as complete
shells while the other 3 consist of a series of arcs which together resemble a shell
(Brinks and Bajaja, 1986).
Most of these holes lie on or near the 10 kpc radius of M31; the same location as
the peak in emission in all wavelengths studied (Brinks and Bajaja, 1986) and the
peak in the distribution of OB associations (van den Bergh, 1964). This suggests
a connection between the HI holes (including the superbubbles) and the locations
of the OB associations reported by van den Bergh (1964). The number of stars in
2. Review of Supet.bubbles= Models and Observations 27
these OB associations is not known, but van den Bergh estimates that the M31
associations are 5 times larger (in spatial extent) than Galactic associations. The
relatively low energy requirements of the bubbles in M31 are consistent with SNe
models provided that the OB associations contain, on average, as many or more
massive stars than those of the Miiky Way.
Deul and den Hartog (1990) followed the approach used by Brinks and Bajaja
(1986) to study the less inclined galaxy M33. Of the 148 holes discovered, 7 are
energetic enough to be included in Table 2. Five of the 7 were observed only as
partial shells. All of the shell parameters (Va, EE, etc.) were estimated in the
same manner as Brinks and Bajaja (1986).
A correlation is found (statistically) between the HI holes and the W regions
and OB associations. The HI1 regions are only marginally correlated with the ob-
served holes, but should, in general, lie within 250 pc of the holes. The correlation
with OB associations is much stronger and indicates that the associations should
lie within most of the shells. These correlations are based on the entire sample of
observed holes, however, and not just the superbubbles. Humphreys (1979) indi-
cates that the linear size of OB associations in M33 is approximately twice that
of recognized OB associations in the Galaxy. The apparent number of OB stm
in the largest associations is on the order of 100 (Humphreys and Sandage, 1980).
If this is the case, more than one association would have to be contributing to the
superbubbles. Since 143 associations were identified by Humphreys and Sandage
(1980) this may be a plausible model.
2. Review of Superbubbk Models and Observations 28
2.2.5 NGC 3079
NGC 3079 is an "active radio lobe spiral galaxy" (Irwin and Seaquist, 1990) which
is presently interacting with a companion, NGC 3073. Irwin et al. (1987) showed
that outflow fiom NGC 3079 is disrupting the morphology of NGC 3073 enough
to produce a ram pressure tail. Nuclear activity has also been detected in the
form of a jet (Irwin and Seaquist, 1988). As shown in Table 2.2, NGC 3079 is a
starburst galaxy and so it is not particularly surprising that 5 superbubbles were
also detected in this galaxy (Irwin and Seaquist, 1990).
The bubbles were observed by Irwin and Seaquist (1990) as M arcs. The centre
velocities of the arcs are not anomalous with respect to the rotation curve, which
suggests an internal origin (Irwin and Seaquist, 1990). However, no correlation
was found between the bubbles and the location of star fonning regions, mainly
because of its edge-on orientation. The energetics of these bubbles would require
OB associations many times larger than Galactic associations, but the existence
of such large OB associations is expected in a starburst galaxy. On this basis the
SNe model appears to be a likely candidate to explain these shells. Alternatively,
it is possible that the nuclear outflow is i n h t 1 y responsibLe for the formation
of these shells. If enough material is ejected by the jet to form clouds which then
fall back to the disk, enough energy could be deposited to form the superbubbles.
This scenario is turther supported by the fact that all 5 shells lie on the same side
of the galaxy as the most prominent jet (see fig. 1 of Irwin and Seaquist, 1990).
Thus, the extreme activity of NGC 3079 presents a confusing picture that does
not allow a conclusion regarding the likely formation mechanism of the observed
superbubbles.
2. Revr'ew of SuperbubbIes: Models and Observations 29
Puche et al. (1992) made VLA observations of the dwarf galaxy HoII in order to
study superbubbles in what they refer to as %ystems which should be simpler."
This simplicity is expected because the rotational velocity and the Likelihood of a
dwarf companion suffering an W C impact fiom its larger group member, in this
case M81, are both reduced. HoII was also chosen because it is close enough to
allow adequate resolution to identify structures to 50 pc. HoII is inclined to the
LOS at 47".
Puche et al. present a rotation curve and a radial density distribution for
HoII. A total of 51 HI holes were independently identified by three of the authors.
These holes range in radii from 50 pc to 850 pc, have expansion velocities between
1 kms-I and 14 kms-I and have input energies estimated between loS0 ergs and
2 x 10% ergs. The largest holes were found at higher galactic radii. This is not
unexpected in any model involving the outward expansion of a shell into a typical
galactic density distribution. As well, Puche et d. (1992) observe that the Ha
emission is restricted to the edges of the large HI holes but fdls the interiors of the
small holes. This observation is consistent with the passage of an ionizing shock
fkont. Puche et d. therefore suggest that the holes observed in HoII are most
likely due to the combined effects of stellar winds and sequential SNe explosions.
The two superbubbles in NGC 4631 have been detected in both Ha (Rand et al.,
1992) and HI (Rand and van der Hulst, 1993). Each shell is seen as a nearly com-
plete ring with some broken structure near the top. Rand and van der Hulst (1993)
estimate the input energies using the same method as Heiles (equation 2.1) and
the kinetic energy by using the column densities to calculate total shell masses.
The Ha images show a largexale disturbance indicative of a nuclear starburst
2. Revfew of Su~erbubbles Models and Observations 30
and several large spurs (20-40 kpc) indicative of tidal interactions (Rand and van
der Hulst, 1993). These structures are echoed in polarization observations of the
magnetic field of NGG 4631. Dumke et al. (1995) observed nearly vertical field ori-
entations near the galactic centre and a more plane-parallel field orientation away
from the centre. The detected magnetic field strengths, as estimated by assuming
equipartition of energy between the cosmic ray electron energy density and mag-
netic fields, are 7 f 1 pG in the disk and 6.5 f 1 pG in the halo. However, Dumke
et al. describe the polarized emission as having "a very patchy distribution."
NGC 4631 is a spiral barred galaxy (Tdey, 1988) with two companions: a
dwarf elliptical, NGC 4627 and an edge-on spiral, NGC 4656 ( b d and van der
Hulst, 1993). Although the SFR fkom Table 2.2 is not much greater than that
of the Galaxy, NGC 4631 has an extensive radio halo. Hummel and Dettmar
(1990) argue that the spectral indices and the change in the textent of the halo
with frequency indicate the presence of a galactic wind. Humrnel and Dettmar
(1990) also suggest that the geometry of the halo is consistent with a previous
gravitational encounter between NGC 4631 and NGC 4627.
ROSAT observations of NGC 4631 detected X-ray emission associated with
both superbubbles (Volger and Pietsch, 1996). Unlike the diffuse X-ray emission
detected in LMG2, the X-ray sources detected in NGC 4631 are point sources.
Volger and Pietsch (1996) identified a heavily absorbed X-ray source inside shell
1 which has a spectrum consistent with either an accretion powered object or a
coincident background active galactic nucleus (AGN). Two X-ray sources were
found within shell 2. The first is the brightest X-ray source in NGC 4631. The
luminosity and time variability of this source suggests a 2 10 Ma blackhole X-ray
binary. The second source may be either a SNR expanding into ionized hydrogen
or an X-ray binary.
The nature of the NGC 4631 system suggests that the ISM has been greatIy
disturbed by both tidal interaction (perhaps even a collision) and a nuclear star-
2. Review of Superbubblest Models and Observations 31
burst. There is no concrete evidence which favours one superbubble formation
mechanism over another in such a complicated system. However, Rand and van
der Hulst (1993) speculate that the most Likely mechanism is impacting clouds
(see Section 2.1.2) based on the extreme energy requirements.
2.2.8 NGC 1620
Vader and Chaboyer (1995) report that a bubble exists in NGC 1620 at a distance
of - 11 kpc &om the galactic centre. This shell is not located within one of the
spiral arms. The appearance of the shell is a fkagmented circle of HI1 regions with
a star cluster at the centre. The acistence of the HII regions outlining the shell
suggests that star formation may have been initiated around the perimeter of the
bubble.
The energy estimates for the formation of this superbubble are based on Mc-
Cray's (1988) model which predicts the onset of gravitatiofial instabiliv. The
resulting collapse phase is thought to initiate star formation. A lower bound to
the energy is dculated by assuming that the existence of the shell implies an
expansion velocity greater than the random velocity of the stars. A value of 20
k m - I then allows Vader and Chaboyer to use McCray's model to determine an
input energy. An upper bound is calculated by estimating the age of the bubble
based on the maximum age of the stars within it. With the age and the radius
one can find an expansion velocity and thereby determine an energy estimate.
To date, the superbubble in NGC 1620 is unique in that it is the only super-
bubble directly correlated to star-forming regions. Propagating SNe seems to be
the most like1y formation mechaDiSm in this case, although it is possible that an
HVC impact formed the superbubble and triggered the star formation. Thus the
presence of the stars is not particularly helpful in differentiating between these
two models. It should also be emphasized that the expansion velocity of the NGC
2. Reviiw of Superbubbk Models and Observations 32
1620 bubble is the result of a model and has not been observed directly.
2.2.9 The D d : NGC 1705 - NGC 5253
Marlowe et al. (1995) chose to study a sample of nearby dwarf galaxies which
show "clear evidence for the presence of a substantial population of young stars."
Although the SFR for these galaxies appear small in Table 2.2, it must be remem-
bered that these rates are scaled to that of the Galaxy which is far larger. All of
the dwarf galaxies studied by Marlowe et al. (1995) are starburst galaxies. In all
these galaxies, superbubbles are observed as "horseshoe shaped structures that
seem to attach to the bright central star forming regions" (Marlowe et al., 1995).
As well, part arcs are observed in several of the galaxies; Marlowe et al. refer to
these as filaments.
The bubble kinematics summarized in Table 5 of Marlowe et aL(1995) and
Table 2.2 of this report show that the observed expansion velocities are very high.
All of the bubbles reported by Marlowe et d. are included in Table 2.2. The
estimated input energies have been calculated using equation 2 of Heiles (1979)
and n,, = 0.3 cm". This density has not been measured but is assumed to be
char-ristic of these galaxies by Marlowe et d. (1995). Evidence for galactic
rotation was observed only in NGC 2915, 3955 and 4670. In all other cases,
Marlowe et d. report that dl gas motions are dominated by noncircular motions.
In NGC 1705, 1800, and 3125, only fragmented arcs are observed and inter-
preted as superbubbles. However, the bubbles in NGC 2915,3955,4670 and 5253
are more confidently identified by Marlowe et al. The bubble in NGC 4670 is
also identified by Hunter et d. (1993). The dwarf nature of these galaxies causes
the impacting clouds model to appear unlikely. The size of the parent OB asso-
ciations and the spatial correlation with star-forming regions makes the multiple
SNe model the most likely formation mechanism for these superbubbles.
2. Review of Su~erbubbla Models and Observations 33
2.2.10 NGC 3044
NGC 3044 is an isolated edge-on galaxy that was recently mapped in HI and CO
by Lee and Irwin (1997). Although NGC 3044 is not currently interacting with
any other galaxy, Lee and Irwin discovered an asymmetry in the HI distribution
of the disk which may be related to a past merger event. The one superbubble
that is observed in this galaxy appears to have formed after the merger. The age
of the superbubble is estimated by assuming a coostant expansion velocity equal
to the present velocity and extrapolating to zero radius. This method gives an
age of - 3 x lo7 yr- If the merger had occurred so long ago, the effects would
still be observable. Thus, Lee and Irwin feel that a merger scenario is not likeIy
to have caused the superbubble directly. It is possible that HVCs were formed
by the merger. However, clouds of d c i e n t mass and velocity (on the scale of
those used by Rand and Stone (1996) to model the shells in NGC 4631) were not
detected by Lee and Irwin (private communication) even though they would have
been well above their detection threshold. The high energy requirement of this
bubble seems to contradict the supernova rate in this galaxy as well. Neither the
impacting cloud model nor the multiple SNe model appears adequate to explain
the existence of the superbubble in NGC 3044.
2.2.11 NGC 3556
King and Irwin (1997) mapped the edge-on galaxy NGC 3556 (M108) in atomic
hydrogen and discovered two exceptionally large and energetic superbubbles. These
two shells are 12 kpc from the galactic centre on the eastern and western ends of
the disk. Unlike superbubbles observed in other galaxies, these shells appear to
have expanded out of the ends of the disk rather than above or below the disk
mid-plane. The eastern shell also appears to have blown out the rear side of the
shell.
2. Review of Superbubbles: Models and Observations 34
The expansion velocities were taken to be half of the velocity range over which
the shell is observed and the ambient density, 4, is determined from a global
model of the galaxy's density and velocity distributions (King and Irwin, 1997).
The input energies were then calculated using equation 2.1. However, since not all
of the eastern shell is observed, the expansion velocity could be an underestimate
of the true velocity. If this were the case, the input energy in Table 2 would be a
lower bound.
The two superbubbles observed in NGC 3556 are perfect for testing a magnetic
superbubble model for two reasons. Their extreme sizes and energy requirements
contradict SNebased models (discussed in Section 2.1.1) and the isolation of NGC
3556 makes the impacting clouds model unlikely (Section 2.1.2).
2.3 Success of the Models
As summarized in the previous sections, most of the observed superbubbles are
well describing by one of the existing models. Table 2.3 illustrates the models
which u p p r to be most successful in explaining the bubbles seen in each of the
galaxies. The most plausible scenario 8ccotding to the authors of the original
studies and from p-nal interpretation is indicated. Other scenarios are not
ruled out conclusively. Thus Table 2.3 does not represent fact, but represents a
"best guess" based on the accumulated data.
The most striking feature of Table 2.3 is the need for some new model to explain
the largest shells in NGC 3556 and NGC 3044. There the multiple SNe model
is ruled out based on energy considerations. The impacting cloud model could
potentially explain these shells, but the obsemtional data do not support the
existence of such clouds. Thus a new model that does not rely on unseen clouds is
required. However, some superbubbles are well explained by the existing models.
In particular, the bubble in NGC 1620 is well explained by SNe models since an
2. Review of Smerbu b bIes= Models azd Observations 35
nble 2.3: The most plausible formation mod& are shown for the superbubbles in each @w-
Most Plausible Model Multi SNe Impacting Clouds Neither Either
Milky Way X LMC M31 M33 NGC 3079 HoII NGC 4631 NGC 1620 dwaxfk NGC 3044 NGC 3556
OB association has been detected within it, and the bubbles in NGC 4631 and
the Galaxy appear to be well explained by infalling clouds because of the obvious
sources for these clouds and apparent spatial correlations. As a result, in order
for one model to explain all of the observed superbubbles as one phenomenon,
it must also explain the correlations observed, at least in some cases, between
superbubbles and both star-forming regions and high velocity clouds. The galactic
magnetic field is the obvious source for such a connection. Studies have already
illustrated that the Parker Instability could initiate star formation (Shibata and
Matsumoto, 1991; Mouschovias, 1974; Mouschovias et d., 1974) while at the same
time producing an inflating field loop. Schulman et al. (1994) have also proposed
that blowout superbubbles expel material into the halo and thereby form W C s .
What remains to be shown is that a galactic-strength magnetic field can self-
consistently provide the necessary energy to power superbubble expansion. An
instructive "toy model" is used to investigate this possibility in the next Chapter.
3. THE FEASIBILITY OF MAGNETIC SUPERBUBBLES
In general, the dynamics of plasmas in a magnetic field are very complicated. At
the star t of a project investigating field effects, much information can be obtained
by first studying a ''toy model". The model must be representative of the prob-
lem under consideration, but is allowed a lot of freedom in terms of simplifying
assumptions. This chapter makes use of such a model to gain insights into the
properties which are important for the development of a comprehensive magnetic
superbubble model. In particular, this simple model illustrates the feasibiliw of
forming superbubbles via magnetic fields instead of SNe or impacting clouds.
3.1 A Superbubble's Magnetic Field
We have fit a magnetic field to the basic geometry of a superbubble in order to
examine the energetics of the field. Previously, magnetic fields have been applied
to superbubble models either to contain the bubble (Tomisaka, 1992; Femdre
et al., 1991; Tomisaka, 1990) or to utilize the Parker InstabiliQ to supplement
energy input from SNe (Kamaya et al., 1996). Here we examine the possibility of
producing superbubbles by harnessing the energy of s sheared (due to Merential
rotation) magnetic field alone. Unlike the previous studies involving magnetic
fields, we give the field a primary role in superbubble formation.
3. 1 . 1 Field Constraints
In order to determine an analytic description of a magnetic field that might be
interpreted as a driving source for superbubble formation, the field must meet
3. The F-bility of Magnetic Superbubbles 37
certain constraints: Maxwell's equations, magnet ohydrodynamic equations and
the geometry of a superbubble. Thge constraints can therefore be used to select
"families" of magnetic fields which may work. Each of these can, in tum, be
compared to observered parameters.
a) Momell's Equations
In the interest of simplicity, certain assumptions are made from the start. First,
the fields are assumed to be stationary. This assumption then limits the model
to an instantaneous look at what the field configuration might be. No time de-
velopment information will be obtained from such a model. Second, equipartition
of energy between the magnetic field and the velocity field is assumed. This as-
sumption has two implications. The velocity field is assumed to be parallel to the
magnetic field everywhere and the kinetic energy is assumed to be equal to the
magnetic energy.
With the above assumptions in mind, the following static form of Maxwell's
equations apply:
Using Ohm's Law for a static conducting medium, f = o 8, and the above
equations, it is possible to show that the r d t i n g constraint equations are:
b) Magnetohydrodynamic Equations
The two equations of particular interest to this study are the equation of continuity
and the equation of motion. These two equations written in a static form are:
3. The Feasibility of Magnetic Superbubbles 38
where v' is the velocity field, fi is the magnetic field, P is the pressure, p is the
density and is the gravitational potential of the galactic disk (self-gravity of the
expanding shell is neglected). An equation of state is not used because we solve
for the pressure variation in the next section.
Applying equipartition of energy to these equations yields
In practice, equation 3.8 can be used to constrain the density function and
equation 3.9 can be used to verify that the solution does not require the pressure
to become negative or divergent at some point within the region of interest.
c) Superbubble leeom*
The h a 1 constraint that the magnetic field must meet is the superbubble geom-
etry. Under the assumption of equipartition of energy, the magnetic field must
resemble an expanding structure: this geometry is that of a velocity field directed
radially outward as if triggered by an explosive event. We choose the centre of
the superbubble as the coordinate orgin to describe such a flow. The field must
be constrained in such a way that it points radially away fkom the origin and that
it is strongest at the centre and diminishes with distance. Since the superbubbles
form in an environment affected by gravity, the velocity field, directed out of the
disk plane near the origin, should reverse so that material is retuned to the disk.
This behaviour is also mimicked by the magnetic field given the assumption of
energy equipartit ion.
3. The Feasibility of Mmetic Su~erbubbles 39
If the differential rotation of the galaxy is to be properly accounted for, the
magnetic field must be constructed in such a manner that the effects of shear can
be included. This implies that the field should vary with azimuthal displacement
from the origin. It is this connection between the field and the Merentid rotation
that must appear in the field equations in order to allow an energy transfer from
the galaxy's rotation to superbubble formation.
3.1 -2 A Simple Field Solution
In the interest of simplicity a first order solution is derived assuming p = amstant
everywhere within the superbubble. This assumption then satisfies equation 3.8.
In order to find sekonsistent solutions to equations 3.4 and 3.5 the magnetic
field is assumed to be separable into components which are functions of each
coordinate. This separation of variables abandons the ability to examine vorticity
in the field. Since we are interested only in the normal mode, this assumption is
adequate. A cartesian strip is adopted as the coordinate system, with the 9 axis
directed radially away from the galactic centre, the @ axis directed azimuthally in
the disk and the i axis directed upwards perpendicular to the disk. The origin for
this local coordinate system, the centre of a bubble, is located at a galactocentric
radius of &d. Since the assumption of a cartesian strip neglects local curvature,
only situations where << 1 can be considered.
The jfi component of the magnetic field is therefore
where Xj Xj(z), Yj Yj(y), and Zj G Zj(z) .
The equations of constraint can then be expressed as the following four equa-
tions (no summation implied):
3. The Feasibility of Mmet ic Superbubbles 40
Any solution which satisfies these four equations is a valid solution to Maxwell's
equations. In order to find a solution which has the potential to describe the field
of a superbubble, we must consider the implications of the superbubble's geometry.
First, the field must be sheared in z and y. The B1 component of the field should
grow with y and the B2 component should grow with z. If Xz, Y1, Y2, and Y3
are taken to be linear functions (or constant), equation 3.11 forces Z2 to be linear
as well. Equation 3.11 reduces to the following form for j' = 1,3:
Again, the superbubble geometry describes how to proceed. The structure
of the bubble indicates that near the edge of the expanding shell the material
must return to the disk. If the Geld lines were to extend out of the galaxy,
material would flow directly into the halo and the model would describe a blown-
out bubble. Under the assumption of energy equipartition, the magnetic field
must also fall towards the disk. Thus, a harmonic structure is required of the
functions of x. To force the B3 component of the field to become negative at the
radius of the shell, the Xs function should be a cosine hc t ion .
I . order to form an explosive structure in the field, exponential functions are
needed in z. To satisfy the above equations the exponential functions must be
written as hyperbolic functions. I . order to make the magnitude of the field
decay with height above the disk a scale height, h, must be introduced into the
hyperbolic Eunctions. The resulting functions are as follows:
XI = s z n ( k x ) ,
X3 = - w s ( k x ) , Zr = w s h ( k ( z - h)),
3. The F-bility of Magnetic Superbubbles 41
Substituting these expressions into equation 3.12 further constrains the possi-
ble expressions for the linear functions.
dY2 kYI C O S ( ~ X ) w s h ( k ( ~ - h)) + X2 - Za - kY3 W S ( ~ X ) ~ s h ( k ( z - h)) = 0 . dy
(3.18)
The simplest solution to this equation is found by dowing Y2 = wnst thereby
forcing Yl = Y3. This solution is not unique however. Other possibilities will
be explored in Section 3.1.3.
The ha1 expression for the magnetic field is:
B1 = A s i n ( k z ) (al y + b l ) w s h ( k ( z - h ) ) ,
Ba = A ( e ~ + b z ) , (3.19)
BJ = - Acos(kx)(aly + b l ) s i n h ( k ( x - h ) ) .
With p constant and a above, equations 343.5 and 3.8 are satisfied. Equation
3.9 still requires that B2
be satisfied. The dynamical equation (above) can be integrated numerically to
describe the pressure variation within the field structure. Figure 3.1 illustrates
that a seIf'o~~~istent solution for the pressure exists without the ptessure becom-
ing negative or infinite. The vertical gravitational force used to generate Figure
3.1 is
F' = -4Goamtan(sinh(z/z,)) , (321)
fkom van der Kmit (1988). a is the surface density of the galaxy and ze is the
HI scale height. The pressure would become negative at some distances greater
than the radius of the shell, but the assumed cartesian strip is also not valid in
this regime. The relative contributions to the pressure equilibrium are consistent
3. Zlze Feasibilit'y of M-etic Superbubbles 42
with those determined for the Galaxy by Bodares and Cox (1990). Since constant
density has been assumed throughout the bubble, the pressure variation shown is
not strictly valid. Figure 3.1 does show, however, that a self-onsistent solution
for the pressure is possible. The magnetic field proposed in equation 3.19 is
therefore a full solution to the non-linear equations of magnetohydrodynamics
given the assumptions that have been made: constant density, equipartition of
energy, infinite conductivity and time independence.
3.1 -3 Higher Order Solutions
The solution developed in Section 3.1.2 represents a first order solution in the
sense that at every opportunity the simplest case was used. In this Section, two
more complicated solutions will be investigated. The first complication will be to
allow Y2 to be a linear function of y rather than merely a constant. The second
complication does not build on the first, but represents a new approach. The
density is allowed to vary spatially, subject to the constraint of equation 3.8.
a) Lcinear Y2
In order to allow Y2 to be a linear firnction of y, eertain expressions for Xi, Yj, Zj
will have to change. However, 4, Xs, ZI, Za are not sffected. In general, the
conclusion from Section 3.1.2 that Yl = Y3 wiU no longer be valid. A start is
then obtained by setting:
Equation 3.12 can be written in the following form which allows one to identify
X2, Y2, and Z2 by matching factom.
Figure 3.1: A seE'nsistent solution for the pressure variation due to the magnetic field given by equation 3.19 is shown to iuustrate that the field satisfies the equations of ideal magnetohydrodynamics. (Reprinted from Ekei et al., 1997)
3. The Feasf'bility of hhgxzetic Superbubbles 44
Equation 3.11 must also be satisfied for j = 2 meaning that Y2 must be
linear in y. Integrating the y factor of equation 3.24 leads to the following:
Thus the new field equations are
B, = A s i n ( k z ) ( a l y + b l ) c o s h ( k ( z - h ) ) , (3 -27)
By = A w s ( ~ x ) ( k ( b 3 - b l ) y + c ) ~ ~ s h ( k ( t - h ) ) (3.28)
Bz = - Ams(kx)(qy + b 3 ) s i n h ( k ( t - h ) ) . (3 -29)
As suggestively simple as these equations appear, the underlying structure is not
geometrically compatible with the observed structure of a superbubble.
b) Variable D e w y
By allowing the density to vary spatially, the model converges on physical reality.
Obviously the density should vary from place to place in a shell. However, a
simple solution which incorporates an equipartition field solution and a variable
density does not exist. If the field is taken to be the same as expressed in equation
3.19, the continuity equation (equation 3.8) can be used to find s characteristic
expression for p. Equation 3.9 can then be integrated in component form to find
an expression for the pressure.
By letting p = X Y Z much the same as in Section 3.1.2, one can determine
that p can be an arbitrary function of sin ( k x ) sinh ( k ( z - h )). The
pressure equations are then integrated to the following form:
3. The Feasibiti& of Mhgnetic Superbubbles 45
where Po is the integration constant.
The second equation above (the y component) clearly indicates that the extra
terms appearing in the x, z equations must be independent of y, otherwise the
integration of P becomes path dependent. However, since p is a function of x and
z and B2 is a function of all three coordinates, the only solution is the trivial case
of p = const. Thus a model incorporating an equipartition field and a Mliable
density does not appear possible. The only clear implication of this result is
that with respect to density variations, the model of Section 3.1.2 is as good
as can be done without adopting a more complete model such as one allowing
non-equipartitioned or timedependent fields.
3.2 Fitting the Magnetic Field
Equation 3.19 provides an expression for a magnetic field of a superbubble that
meets the constraints discussed in Section 3.1.1. In order to fit this magnetic field
to an observed superbubble, several parameters must be identified in terms of
measurable quantities.
The constant k which appears in the sinusoidd and hyperbolic factors deter-
mines the periodicity of the field; varying k wi l l vary the spatid extent of the
turnover of the field or the distance in z between upward flow and downward
flow. k is thus easily identifiable as a function of the bubble radius,
In order to isolate the effects of galactic rotation, the scaling constant A is
taken to be unity. The magnetic field strength is therefore determined only by
the contribution from the equipartitioned velocity field due to galactic rotation.
However, there is the requirement of a scale height in the hyperbolic fundions.
3- The FeasibilW of M a e t i c Su~erbubbies 46
This height h is required to force the field strength to diminish with increasing
I. The inclusion of such a scale height also introduces a scale factor which can
ultimately be incorporated into A in order to achieve a final scale factor of unity, 1
With the scale factor so defined, the effect of selecting a particular value for
the scale height is negligible as long as h > rd. Assigning h = r,, would
force Bz to zero at the uppermost reaches of the bubble rather than having it
merely approach zero. Assigning h < rh has a more serious effect. In this case
Bz passes through zero and begins to increase resulting in a potential blowout
scenario. According to this model, a rapid reacceleration of the gas would result
and continue indefinitely. Clearly this is a non-physical solution so this model
can not be used to describe blowout scenarios in this manner.
The parameters remaining to be identified are all related to the galactic ro-
tation curve. Under the assumption of equipartition of energy, the form of the
magnetic field in the plane of the disk must reflect the form of the galactic rota-
tion. Thus the linear factors in & and B, can be related to parameters obtained
from the rotation curve of the galaxy in question. Since the factors to be mod-
elled are linear over a limited region of space, the local rotation m e is reduced
to a linear function by considering ody a tangent to the rotation curve passing
through the circular speed at R. This tangent can be expressed as
v ( z ) = r ; l ( R + x ) + m . (3.33)
The local velocity field (the velocity due to galactic rotation measwed relative
to z = 0 instead of the galactic centre) can be expressed at the origin to k t
order in y / R as
This leads directly to the definition of the ha1 parameters, where the ,/4?rb factor has been included to satisfy the units:
The magnetic field is now completely identified in terms of measurable quan-
tities and can be easily applied to any observed superbubble. It should be noted,
however, that b1 and 62 are only forced to zero by demanding a pure translation
of energy from rotation to expansion. Nonzero bl and could be related to
(global) external magnetic or velocity fields.
3.3 Application to the Superbubbles of NGC 3556
King and Irwin (1997) mapped the galaxy NGC 3556 in HI and discovered two
shells described in Table 2 (see also Giguere, 1996). As discussed in Section 2.2.11,
the extreme size of the bubbles and the isolated environment of this galaxy are
inconsistent with the models of Section 2.1. These two shells are therefore perfect
test shells for the magnetic field model. Theoretical predictions are made for
the kinetic energies, expansion velocities and average magnetic field strengths by
fitting the magnetic field developed above to each shell. Comparison of these
predictions with the quantities derived from observations (King and Irwin, 1997)
allows an assessment of the feasibility of the magnetic field model.
The parameters which must be identified in order to fit the bld to the bubbles
are the radius of the shell, RHn, the distance of the shell &om the galactic centre,
Rgd, the scale height parameter, h, the density at the shell's position, no, and
the values of a1 and q fkom the galactic rotation curve. The shell radii and
galactocentric distances are taken directly from Giguere (1996). The scale height
3. The F-biE& dMagoetic Superbubbles 48
needs only to be chosen such that h > Rsh- The density (cm-j) can be calculated
from the following density distribution given by Giguere (1996) (r and z measured
in kpc) :
Similarly the values of a1 = -2.9 pG kpc-I and a? = -0.027 pG kpc-' can
be extracted fkom the Brandt rotation curve (shown in Figure 3.2) which Giguere
(1996) fit to NGC 3556.
With the field fit to each of the two superbubbIes, the predicted kinetic energy,
EL, can be calculated from the magnetic energy density. This calculated energy is
strictly the kinetic energy and not total energy required to form the superbubble
using SNe, as calculated using equation 2.1. By integrating the magnetic energy
density over the volume of the bubble, the total magnetic energy is determined.
The bubble voiume is taken to be a cylinder aligned perpendicular to the disk.
Under the assumption of energy equipartition, the magnetic energy is equal to
the kinetic energy. The average magnetic field drength can then be determined
from the average magnetic energy density (total magnetic energy / volume). The
predicted expansion velocity, V& is related to the average field strength using
equipartition of energy again. These calculations are summarized by the following
equations:
These calculations have been used to test the magnetic field model under a
variety of conditions. Table 3.1 summarizes 4 trials for each of the superbubbles in
3. The Fiibility of Magnetic Superbubbles 49
Figure 3.2: The Brandt rotation curve fit to NGC 3556 by Giguere (1996) used to identify the values of a1 and a*.
3. The Feasibility of Magnetic Superbub bles 50
Bble 3.1: The results of s e d variations of the model, fkom Fki et al. (1997), are presented to iuustrate ideas discussed in Section 3.3. was taken to be zero in all cases.
W;est SheU East Shell
h (PG) 0 0 0 3.0 0 0 0 3.0 no (w-~) 0.26 0.10 0.26 0.10 0-26 0.10 0.26 0.10 Ek ( ~ 1 0 ~ ~ ergs) 39 39 39 39 160 160 160 160 Ekt (xloS2 ergs) 5.3 2.6 5.3 43.0 86 43 86 250 V, (kms-l) 41 41 41 41 52 52 52 52 Vd (kms-I) 6.4 6.4 6.4 30 11 11 11 32
NGC 3556. ma1 a) represents the basic model in which aU of the expansion energy
is derived from the galactic rotation. The h scale is chosen to be slightly larger
than the shell radius and bl and are both set to zero. The density is taken to
be the density of the disk mid-plane as given in King and Irwin (1997). In trial b)
the density is halved, corresponding to the density predicted by King and Irwin
at a height of 240 pe. Tkial c) demonstrates that the h scale does not affect the
results as long as h > Rah. nial d) includes a constant azimuthal magnetic field
of 3 pG.
The effect of density on the calculations is obvious fmm comparison of trials a)
and b). The decreased density reduces the total mass contained inside the bubble
and, therefore, the kinetic energy. The expansion velocity does not change with
the density; the magnetic field strength varies to account for the difference. In
this basic model, the density is assumed to be constant over the entire volume of
the shell. Attempts have been made to d o w a variable densiw; however these
are inconsistent with equipartition of energy. The density variations required by
a shell structure would dictate that proportionally more mass would be present
3- The Fea~~*bility of Magnetic Superbubbles 51
in the high velocity regions. This diffaence could imply that the kinetic energies
calculated here are lower bounds-
A comparison between the observed parameters (unprimed) and the modelled
parameters (primed) in Table 3.1 shows that the basic model of extracting energy
from galactic rotation alone is not adequate to produce the observed energies.
There simply is not enough velocity shear present across the diameter of the
bubble. However, the addition of a global azimuthal magnetic field, which is of
the order of those observed in other galaxies, to the local perturbation can provide
more than enough energy. The magnetic field geometry with a 3 pG external field
included is shown in Figure 3.3. Each plot in Figure 3.3 shows 9 magnetic field
lines: 5 begining at Z = 0,3 a t Z = 0.2 kpe and 1 at Z = 1 kpc. The 3 dimensional
structure of the field is shown in (a) and the projections onto the coordinate planes
are shown in (b), (c) and (d). The projections of the field structure which would
be observed for a bubble in the outer region of an edge-on galaxy and in a face-on
galaxy are shown in (b) and (c), respectively. The field structure shown in (b)
represents the outward radial flow of material required to produce an HI hole.
The field structure in this plane is nearly identical to the structure produced by
Kamaya et 1 (1996). (d) shows the field as it would be seen in an edge-on galaxy
if the bubble were in line with the nucleus- A shell structure would be formed
by the convergence of the field lines shown in (d). In this composite magnetic
field, the global field provides about half of the total field strength and local shear
determines the shape of the bubble-
The effects of assuming equipartition of energy on the results of this model are
not entirely clear. Chandrasekhar (1961) shows that equipaxtition solutions are
"stable with respect to small perturbationsn. In a sense, then, the equipartitioned
state may be the ideal end-point of superbubble evolution. Even if during early
stages of superbubble formation the fields are not parallel or equally energetic,
they should evolve to this state (provided sufficiently rapid dissipation). Since
3. The Fessibiliw of M q e t i c Superbubbles 52
Figure 3.3 continued ...
3. The Feasibility of Magnetic Superbubbdes 53
Figure 3.3: The shapes of9 magnetic field lines are shown in 3D and in projection to each of the coordinate planes. The grey scale is Lighter at higher z. (Reprinted &om Frei et aL, 1997)
3. The F-bility of Magnetic Superbubbles 54
our model is time independent, no information is gained regarding this process.
However, the goal of this model is to determine the feasibility of producing super-
bubbles via magnetic fields. To this end, the assumption of energy equipartition
is adequate.
Since the magnetic superbubble model appears to be a feasible energy source
for the largest superbubbles known, it should be adequate for the smaller bubbles
listed in Table 2 as well. The next section explores the model's ability to account
for the kinetic energy of many of these bubbles.
3.4 Application to Other Superbubbles
A full spectrum of expanding bubbles are observed. &om wind-blown bubbles
to supernova remnants to the superbubbles discussed here, the basic geometry is
the same and only the apparent input energy varies. The distinction that has
been made between superbubbles and bubbles is based on an energy "cut-oB,
not on a real acknowledgement of differences. The geometrical similarities of the
superbubbles listed in Table 2 suggests that all of these features are formed by
the same mechanism despite the fact that the SNe-based input energies vary by
over 3 orders of magnitude.
Section 3.3 has shown that the magnetic superbubble model can explain the
most energetic superbubbles known. In this Section, the model is applied to the
bubbles observed in several other galaxies. NGC 3044 is examined since its bubble
is not well explained by any of the other models (see Section 2.2.10). The magnetic
field model is also applied to NGC 1620 and M33 to see how well it predicts the
observed parameters of bubbles which are also explainable using a multiple SNe
model. Similarly, the magnetic field model is applied to the bubbles in NGC 4631
and the Galaxy so that it may be compared to the impacting cloud model. Lastly,
the superbubbles observed in dwarf galaxies are discussed in the context of this
3- The F-ibility of Magnetic Superbubbles 55
Zkble 3.2: The model results using two di.fEere~t magnetic fidd strengths. In general, the model requites a stronger magnetic fidd in order to adequately describe the superbubble in NGC 3044.
Fixed Model Parameters 7- (k?4 1.6 &d ( ~ P c ) 6.6 no (anJ) 0.22 a1 (pG kpc-') -4.8 a2 (pG kp&) 0.43 bl h a 0.0
Model Results h (PG) 3.0 8.0 Ek (x10S2 ergs) 520 520 EL (x1Os2 ergs) 38 207 V, (kms-') 52 52 V& ( k r n ~ - ~ ) 24 55
model.
In Table 3.2, model parameters and results are compared to the observed d u e s
for the bubble in NGC 3044. The value for al was found using the rotation curve
fit to NGC 3044 by Lee and kwin (1997). no was calculated using the density
distribution, also modelled by Lee and Irwin, assuming that the superbubble is
located at z = 0. Results for two Merent global field strengths are shown. As
before, and V& are the model predictions for the kinetic energy and expansion
velocity and El (4mC) and V, are the values obta;ined fiom observations.
The 3 pG global field used previous1y to fit the bubbles in NGC 3556 is inade-
quate to produce the absented energetics of this bubble in NGC 3044. However, an
8 pG global field does provide the correct order of magnitude kinetic energy and
an expansion velocity which is very close to the observed velocity. (A difference
in mass results &om the requirement of constant density.) This result therefore
requires a measurement of the magnetic field strength in NGC 3044 in order to
verify the applicability of the magnetic superbubble model. Unfortunately, the
required observations have not been made. One way to estimate the field strength
is to use equipartition of energy between the cosmic rays and the magnetic field.
Since the SFR affects the cosmic ray energy density, the SFR may be related to
the magnetic field. The SFR of NGC 3044 (Table 2.2) is equivalent to that of
3. The F-bility of Mkgnetic Superbubbles 56
nble 3.3: The model parameters, observed and modeled kinetic energies are shown for the range of masses listed in TabIe 2 for the superbubble in NGC 1620. For these calculations, h = 1.5 kpc and a2 = b1 = 0.
Rgd (kP4 11 I1 11 no (an") 0.3 0.5 1.0 a1 (pG kpc-') -5.0 -6.4 -9.1 bz (PG) 3.0 3.0 3.0 Ek(xlo5*ergs) 5.0 7.9 16.0 EL (x105* ergs) 3.5 3.7 4.1 V, (km-l) 17 14 10
NGC 4631, and so a magnetic field of 7 pG, as observed for NGC 4631, would not
be surprising.
Table 3.3 Lists the results of applying this model to the superbubble seen in
NGC 1620 without changing the strength of the external magnetic field koom that
used for NGC 3556. Three variations of the model are shown, corresponding to
the mais range for this superbubble. The density is calcdated by dividing the
mass (listed in Table 2) by the volume of the bubble (again based on a cylinder out
of the plane). All of the model parameters are shown with the observed kinetic
energy and the modelled kinetic energy. The rotation curve data used to caiculate
the parameters a1 and are from Rubin et d. (1985).
The radius of the shell is assumed to be 0.8 kpc, the logarithmic middle of
the range given by Vader and Chaboyer (1995), and the expansion velocity used
to calculate the observed kinetic energy is 20 kms-'. The range given by Vader
and Chaboyer for the expansion velocity is 20 - 50 kms-', but the lower limit
is most consistent with the results of the magnetic superbubble model. The three
trials in Table 3.3 correspond to the range of shell masses given by Vader and
Chaboyer, the lower limit, mid-range and upper limit, and thus to the range of
3. The Feasibilitv of Mannetic Su~erbubbles 57
Zhble 3.4: The modeled kinetic energies and expansion velocities are compared to the observed values for superbubbles in M33. For these calculations h = 1.5 kpc and bl = 0.
- bubble M33 - 10 19 24 47 74 87 126 7- @PC) 0.6 0.4 0.4 0.5 0.5 0.5 0.4 R@ @PC) 5.7 3.2 1.8 5.3 4.8 4.7 4.5 n o (C7n-3) 0-5 0-8 1.0 1.3 0-7 0.3 0.7 a1 (pG kpcdl) -5.4 -7.6 -15.8 -31.0 -7.7 -5.1 -8.5 a* (pG k p C 1 ) -1.9 -2.5 -2.0 +7.9 -2.4 -1.5 -2.4 b2 ( @ I 3.0 3.0 3.0 3.0 3.0 3.0 3.0 Ek (x1oS2 ergs) 1.7 1.1 2.0 2.5 0.4 0.9 0.8 EL (xUIs2 ergs) 1.7 0.9 0.4 1.2 0.9 0.8 0.4 V, (kms-I) 13 12 20 21 8 19 16 VA (kms-I) 13 11 9 14 12 18 12
energies as well. As shown in Table 3.3, the magnetic superbubble model best
fits the lower limits of the mass / energy ranges. The bubble is best described by
this simple magnetic field model under these conditions. However, if the shell is
more energetic than Vader and Chaboyer's lower limit, this model would require
a stronger magnetic field to accurately describe the bubble. With an average
magnetic field strength measurement for NGC 1620, the magnetic superbubble
model could further constrain the properties of this superbubble.
Similar calculations for the 7 superbubbles observed in M33 are also consistent
with the observed energetics, although these bubbles are in general smaller than
those previously examined. Correct order of magnitude energies are obtained for
every case except the M33-24 bubble. The model results for M33-47 should also
be viewed as suspect due to large uncertainties and variability in this region of
M33's rotation curve. a1 was calculated from the Brandt rotation curve that was
fit to M33 by Zaritsky et d. (1989).
All of the calculations summarized in Table 3.4 were made without varying
the magnetic field strength. More likely, the magnetic field is stronger near the
3. The F-biliw of Magnetic Superbubbles 58
Thble 3.5: Good agreement between the observed parameters and the magnetic super- bubble model predictions is achieved when the global magnetic field strength in NGC 4631 (Dumke et al., 1995) is used. For these calculations, h = 2 kpc
NGC 4631 - 1 2 @PC) 1.5 1.5 0.9 0-9
Rpd @PC) 6.5 6.5 6.0 6.0 n o (mJ) 0.2 0.5 0-6 1.1 a1 (pG kpc-l) 4.7 -7.5 -8.9 -12.0 bz (PG) 7.0 7.0 7.0 7.0 Ek (x1O5* ergs) 200 500 60 100 E& (x105* ergs) 126 133 27 29 V, (kms-I) 45 45 35 35 VA (kms-I) 32 49 29 21
galactic centre. Such a variation in the magnetic field strength would improve the
model results by increasing the calculated kinetic energies for the M33-24 and
M33-47 shells.
As discussed for NGC 3044, the bubbles in NGC 4631 require a stronger mag-
netic field. However, since a field strength of 7 pG has been observed in NGC
4631 (Dumke et d., 1995), the magnetic superbubble model is fully constrained.
Following the summary of rotation curve studies presented by Golla and Htlmmel
(1994), a flat rotation curve with V = 150 kms-' is assumed beyond 2 k p c Table
3.5 clearly indicates that this simple model provides correct order of magnitude
estimates of the kinetic energy of these shells. Particularly good agreement is
made with the lower limits of the shell masses. As well, the success of this model
in describing the shells of NGC 4631 using the observed magnetic field strength
lends credibility to the stronger field required for NGC 3044. Field strengths of
between 5 and 10 pG are considered typical for spiral galaxies (Condon, 1992) so
the fields required by this model to describe the NGC 3044 bubble and the NGC
4631 bubbles are not unreasonable.
3. The F-bit'& of Magnetic Superbubbles 59
The results of applying this model to the Galactic bubbles observed by Heiles
(1979) are mixed. A flat rotation curve is assumed for R 1: 11 kpc with a rotation
velocity of V = 230 kms-' as fit by M d e l d (1992). However, the data points
displayed in Merrifield's figure 9 clearly show that there is substantial variation
about this fit. The shell GS-22 is not modelled due to the uncertainties associated
with the rotation curve within about 2 kpc of the Galactic centre. The density
within each shell is calculated by dividing the shell mass into its volume (the same
cylindrical volume used to find the energy).
As indicated in Table 3.6, three of the nine superbubbles are well described,
three have predicted energies greater than the observed energies and three have
predicted energies less than the observed energies. AU of the bubbles could be
accurately described by the magnetic superbubble model if the global magnetic
field strength were varied. Observations of galactic magnetic fields do show that
apparently random fluctuations in the global field on the order of the field strength
itself do occur (Heiles, 1996; Berkhuijsen et d, 1993), but such variations are
difficult to model. The 3 pG field employed here is in agreement with the average
field strengths measured by Han and Qiao (1994) and references therein.
There does appear to be a correlation between the density and the accuracy
of the model predictions for these shells. The three accurately described shells
have densities of 0.5 an" while the three under-predicted shells have relatively
high densities and the t kee over-predicted shells have comparably low densities.
Since these densities are derived fiom the total shell masses fiom Heiles (1979), any
errors that affected Heiles also affect the results in Table 3.6. In order to calculate
the shell masses, Heiles (1979) assumed that the column density at the centre
was equal to the column density everywhere within the shell. As Heiles indicates,
"this assumption is in fact inconsistent with the typical observed situation ... that
only one hemisphere of a shell is observed." This assumption, coupled with the
intrinsic difEculW of measuring column densities accurately, therefore leads to a
3. The Fasiibility of Magnetic Superbubbles 60
nble 3.6: Application of the magnetic superbubble model to Galaetic superbubbles leads to mixed results. The bubbles are labelled according to the first number of the series in 'hble 2. For these calculations, h = 4 kpc and a2 = b1 = 0.
GS - 57 64 71 88 95 r &PC) 0.6 1.3 2.0 0.6 0.8 Rgd @PI 11.8 16.1 20.7 17.0 17.0 no 0.5 0.1 0.1 1.0 0.5 al(pGkpc-') -6.3 -2.1 -1.6 -6.3 -4.4 b2 (m 3.0 3.0 3.0 3.0 3.0 Ek (x1oS2 ergs) 3.2 6.1 16 11 2.0 EL (x1OS2 ergs) 1.8 13 55 1.8 3.5 V, (kms-l) 18 22 16 24 10 V',, (kms-l) 14 30 30 10 13
GS - 103 123 224 242 9- ( k ~ ) 1.0 1.0 0.6 0.5 &( @PC) 20.4 22.2 16.3 12.1 no (mw3) 0.06 2.6 0.5 3.2
(PC kw') -1.3 -7.7 -4.6 -15.6 h (PG) 3.0 3.0 3.0 3.0 Ek (x1OS2 ergs) 1.9 29 1.9 13 EL (x105* ergs) 13 8.3 1.7 1.1 V, (kms-I) 14 12 14 20 V& (kmsaL) 38 6.4 13 5.8
3. The F-bility of Magnetic Superbubbles 61
large uncertainty in both the shell densities used in our model here and in the
total shell mass used to calculate Ek.
Applying this model to Holmberg II and the other dwarf galaxies is impossible
due to lack of data and to uncertainties in rotation cmes, The rotation curve
for HoII peaks at the galactocentric radius of the observed superbubble so it is
impossible to define an appropriate tangent to determine al and a2. The model
was applied to this bubble assuming the rotation curve to be constant and equal
to the peak value. The r d t i n g energy is on the order of loS2 ergs as expected
for this bubble despite the low rotation rate. Several of the other dwarfs which
contain superbubbles have no detectable rotation, but non-circular gas motions
are observed. The existence of shear flows in such active dwarf galzuies is likely.
In order for this model to describe these bubbles, the properties of these flows
would have to be known. There is not enough data to properly test the magnetic
field model on these superbubbles. It should also be remembered that the energies
of these bubbles have been calculated based on an assumed density. If the density
were lower, as might be expected in the vicinity of active starforming regions, the
kinetic energies of expansion would be lower as well.
3.5 Results of the Feasibility Study
The magnetic superbubble model developed at the beginning of this Chapter has
been applied to the superbubbles observed in several of the galaxies included in
Table 2 in order to test the fegsibility of forming superbubbles via magnetic fields.
As demonstrated in the two preceding Sections, this model accurately predicts the
kinetic energies of observed superbubbles provided that the global magnetic field
of the parent galaxy is included. The magnetic field derived to fit a superbubble
geometry is a superposition of the global field and a l o d y excited field which
is coupled to the differential rotation. The resulting field is therefore a locally
3. The F t b i l i t y of Magnetic Superbubbles 62
shaped global field. Using this field, a seIf-consistent solution for the fully non-
linear equations of ideal magnetohydrodynamics has been shown to exist.
Once fit to a particular superbubble, this simple model is able, in most cases, to
provide correct order of magnitude predictions for the expansion velocities (kinetic
energies) despite the assumptions of energy equipartition and constant density.
The density assumption is the most serious. As Section 3.1.3 illustrates, this
assumption is an essential component of a simple model such as the one employed
here. A more complete model which allows a variable density is explored in the
next Chapter. The assumption of energy equipartition between the magnetic
and velocity fields is an ideal end-point of superbubble evolution. Such a state
would not exist in evolving superbubbles, but the evolution should tend towards
equipartition as time passes. Since only the stationary state has been examined,
no conclusions can be made regarding how the superbubble changes as it evolves,
nor regarding how the magnetic field could come to be in the state predicted by
this model. However, the geometry of the field structure shown in Figure 3.3 is
suggestive of the Parker Instabihy. If the Parker Instability were responsible for
producing the predicted field structure, the superbubble would be expected to
continue to expand due to the bouysncy of the magnetic field lines. This scenario
is consistent with the "toy modeln explored in this Chapter in that the Parker
Instability is a local instability capable of shaping the global field into shell-like
structures (Parker, 1992). Mhermore, the coupling with Merentid rotation
implies a co~ec t ion with the a-mechanism of galactic dynamos. Although the
role of the Parker Instability and dynamos in superbubble formation is speculative
at this point, the scenario is self'onsistent. Should this picture be accurate,
superbubbles would be the result of galaxy-wide processes and not local processes
such as bouts of SNe.
The major success of this model is its ability to accurately describe shells
in all of the galaxies that can be examined. In particular, this "toy model"
3. The F-biliw of Magnetic Superbub bles 63
correctly predicts the energetics of the three bubbles (in NGC 3556 and NGC 3044)
which appear to be inconsistent with the either SNe models or impacting cloud
models. As well, the magnetic superbubble model explains equally well shells
that have previously been explained by either of these previous models. In short,
this model predicts a unified source for ail of the observed superbubbles which is
independent of SNe or impacting clouds. The observed correlations between some
superbubbles and OB associations may be the effect of superbubble formation and
not the cause, as previously assumed. Similarly, observed correlations between
some superbubbles and high velocity clouds might also indicate effect rather than
cause. In these cases, the superbubble may play a role, undetermined in this
feasibility study, in the formation of OB associations and high velocity clouds.
In conclusion, the simple model examined in this feasibility study has proven to
be seIf--consistent given its assumptions. It accurately describes the energetics of a
wide range of superbubbles and also indicates how further studies should proceed.
A more complete physical model is required which includes the global field and
which will d o w an examination of how the Parker Instability and dynamos affect
superbubble formation and how superbubble formation may affect star formation
and high velocity clouds. The andytic formulation of a self-similar model designed
to allow this study is the subject of the next Chapter.
FRAMEWORK FOR A SELF-SIMILAR SUPERBUBBLE
In its most generalized mathematical form, self-similarity is a technique that can
be used to reduce the number of independent Mliables. A simple physical example
of self-simila,r flow is that of a rarefaction wave (Zel'dovich and Raker, 1966). AU
of the flow variables, velocity, density, pressure and sound speed, are functions
of z and t. However, they are not functions of x and t independently; they vary
only with z/t = C. The problem is therefore reduced to only one independent
variable: the similariw variable C. The physical consequences of such behaviour
is that the flow variables change only as a result of changing scale. In the MHD
approximation, self--similar models produce a set of coupled non-linear differential
equations that must be solved.
The purpose of this Chapter is to develop a comprehensive self-similar model
that may be used to further investigate magnetic superbubbies. The formulation
of the problem is guided by the feasibiliity study (previous Chapter) to include
the global magnetic field, to allow non-equipartitioned solutions and to permit
density variations.
4.1 Defining the Magnetic Szqerbubble Problem
The most general form of the magnetohydrodynamic equations includes time de-
pendence. The fully timedependent self4milar problem could, in principle, be
solved; however at this stage only the stationary state is sought. The governing
equations for this model are the same as those used in the feasibility study but
without the initial simplifications resulting from assumptions of constant density
4. h e w o r k for a Selt-Similar Superbubble 65
and energy equipartition. The complete set of equations which are relevant in this
case are the dynamical equations, the equation of continuity, the conservation of
magnetic flux and the induction equation:
In order to find an acceptable solution describing a magnetic superbubbIe, ail 8
of these equations must be simultaneously satisfied and a suitable geometry must
be produced.
4.1 -1 The Coordinate System
The coordinate system chosen for this analysis differs from that of the feasibility
study. Cylindrical coordinates are used because they most logically describe the
arch-like structures observed in superbubbles. A local coordinate system is still
applied, thereby limiting the range over which the azimuthal coordinate is d i d .
Figure 4.1 shows the orientation of this rather unusual coordinate system. As
shown, i is chosen to be the azimuthal direction, in con t r a to the previous
Chapter in which i was the vertical direction. The cylindrical geometry is tipped
on its side in order to allow the arches, extending out of the disk plane, to be
described conventionally by i; and ). Thus Z at 4 = 0 points radially away fkom
the galactic centre and i at t# = 7r/2 points orthogonally to the disk.
With this choice of coordinate system the general form of the parameter func-
tions (velocity, magnetic field, densiw and pressure) can be prescribed. The self-
similarity variable is chosen to be r. Once the dependence on $ is determined,
simply scaling by r will provide the rest of the selfkhilar " f ' y " of solutions.
Figure 4-1: The coordinate system used for the self--similar analysis. GC denotes the galactic centre, BC denotes the bubble centre and %d denotes the galac- tocentric radius of the bubble.
4-1 -2 The Parameter Ehctions
The parameters of the superbubble problem are the functions to be determined,
namely, velocity, magnetic field, density and pressure. These parameters are as-
signed the following self-similar forms:
The constants \k14 are determined fkom the various scales relevant to the
problem such that they carry all of the dimensionality of the parameters. There
are two relevant velocity scales, two length scales and a free scale for the magnetic
field. These scales are the rotational velocity at the coordinate origin, n,R, and
the sound speed, cs, the galactocentric radius R and the shell radius r,h, and the
magnetic field parameter defined as ABR (units of pG). AB has the form of a
magnetic Oort constant.
4- &mework for a S e l f - S e Superbubble 67
Equation 4.1 also contains terms with fi and which are known, but must
also be forced into the seIf-similar form of equations 4.5. As a start, we define
where is the gravitational force within the disk plane and cos(q5) 6 + sin@) i
is the unit vector directed vertically out of the plane of the galactic disk.
The next step is to substitute these definitions into equations 4.1 - 4.4.
4.1.3 The Governing Equations
Relationships among the various exponents can be derived in a straightforward
manner simply by examining the equations which result fiom the application of
the seIf--similar forms to the magnetohydrodynamic equations. The centrifugal
force term is assumed to directly counter I?, thereby removing any effects of these
two forces in the superbubble problem and forcing a dynamic equilibrium along
galactic radii. The rest of the terms are considered to represent dynamicdy
important features. For example, as indicated in Section 3.5, the coriolis term
may be required to excite dynamo-like behaviour. To this end, all of the powers
of (r/rsh) in all of the equations must be set equal. The induction equation, the
equation of continuity and the conservation of magnetic flux trivially reduce to
this condition; however the dynamical equations provide the necessary constraints.
In particular, the radial equation of motion, below, provides a l l of the necessary
relations.
4- h e w o r k k a Seft-Shdar Superbubble 68
The constraints are therefore
The condition that a,, = an + 1 is not surprising since v = rQ. However,
a choice must be made for at least one of the a ' s so we consider ap = 0,
corresponding to a constant gravitational acceleration. The remaining powers
can then be identified as a. = 112, an = -112, a, = a, a b = (a + 1)/2, and a, = a + 1. One unknown variable, a, remains &om the complete set
of six and the equations can be simplified, provided the point r = 0 is excluded.
The induction equations in i, 6 and i are, respectively,
The i and ) components of the induction equation are degenerate and yield
the condition that Ur/U, = Rib6. This requirement implies that the projections
of the velocity and magnetic fields into the +t$ plane are parallel. A simple change
of variables,
br = U, x , b4 = U4 x , b, = U, xZ (4.14)
where x # z,, is therefore useful as it reduces our system from seven equations
with seven unknown functions to six equations with six fnnctions. (a is not
considered a true variable as it is an eigenvalue.) Back substitution of fi and 3
4. Ekamework fir a SefZ4imi.h Superbubble 69
relates z and x, to the r1 4 and z Alfv6nic numbers (the magnetic Mach numbers)
according to
In order to close our system, we require one additional equation, the equation of
state p = p(p, T). We are therefore forced to make an assumption regarding
the relationship between the pressure and the density. An isothermal solution is
adopted along c w e s of constant r by forcing p = g/S2aRZ p.
Finally, the complete set of governing equations is
4- h e w o r k for a S e l f + . Superbubble 70 -
where
are dimensionless constants, d+ denotes differentiation with respect to 4, and 0 = 47tp.
4.2 Analytic Results
This section presents various results that have been derived analytically and dis-
cusses their implications for the selfsimilar model. The stream lines are investi-
gated to constrain the eigendue a, the Poynting Flux is investigated to determine
under what conditions an outward flux of energy is maintained, and critical points
of the solution are examined as possible starting points of numerical integration.
4.2.1 Stream Lines
The stream lines represent the general flow characteristics of the problem. Of
particular interest here, however, are the projections of the stream lines into the
i 6 plane. In this plane, the stream lines are described by
By hypothesizing a particular behaviour of the stream lines at the boundaries,
some conditions on the eigenvalue can be derived.
First, following Fiege and Henriksen (1996), three constants are identified.
The first is a true integral of the system as it applies to every point of parameter
space. The second and third apply only along stream lines.
&om equations 4.19 and 4.20,
from equations 4.19 and 4.23, ..
4. Ramework for a Self-Similar Superbubble n
Figure 4.2: An idealized stream line is shown on a polar plot to illustrate the properties of the fnmily of stream lines investigated in the following Sections. The plot is intended to show only the behaviour at the boundaries 4 = 0 and 4 = 5. The shape of this curve between the boundaries is merely a possibility. The M y represented by this curve is characterbed by the asymptotic approach to both boundaries.
and finally, &om equations 4.20 and 4.23,
Now by forcing the above expressions to remain constant near the boundaries
where the behaviour of the p r o p d stream lines is known, a range can be spec-
ified for the eigendue. The class of stream b e s which is examined is shown in
~ i g & e 4.2. This flow corresponds to an Wow/outflow" scenario (depending on
direction), potentially powered by the coupling between differential rotation and
the magnetic field. An induced Parker Instability would add to the outflow and
would likely link the forming superbubble to a galactic dynamo.
For these stream lines, the velocity component in ) (Ug) tends toward zero at
both 4 = 0 and 4 = f. Since p represents the density, it must remain finite and
non-zero. However, x relates the magnetic field to the velocity field according to
4. Ekamemrk for a Self-Similar Swerbabble 72
bra# = U& z. Thus x -t 0 is acceptable as it describes a very weak magnetic
field. z + m is not acceptable since this would describe a situation involving
.n infinite field.
Equation 4.24 shows that for p relatively constant and Ud + 0, either a -- a --
U+ a+3 + 0 or U# + 00 depending on whether <Oars > 0- - - s k L
Assuming that < 0 , 2 20+3 must approach W t y to maintain q constant;
must therefore be positive. The range of values of a that satisfy both of 2a+3
these conditions is -$ < a < 0. -
-Q+3 Ha > 0 is assumed then z must tend towards zero implying that
< 0; a is therefore required to be an element of the set {(a < -3 or 2a+3
a > 0) and (-3 < cr < 0)), which is the null set. Clearly the only solution
is for x + 0 at the boundaries and for a to be between -: and 0. Equation 3
4.25 partially confirms this result since as U4 + 0, tQ+Z must approach infinity- 3 Since r does not approach zero, a must be greater than -5.
4.2.2 Poyatiag Flux
The Poynting vector = el? /I B' is of importance to this investigation as it
is a measure of energy transport. Since we wish to produce expanding bubbles,
we must be careful to ensure that energy is transported outward.
Using l? = -v'h l?, the components of the Poynting vector are
Several encouraging results are noted after examining the behaviour of 3. First, for a in the range identified in Section 4.2.1, the magnitude of the energy flux
grows with increasing r . When Uo approaches zero, such as along either axis of the
4- Ehmework Tor a Self-Similar Superbubble 73
stream line in Figure 4.2, S4 vanishes. At any point where z = x,, implying that
the magnetic field is parallel to the velocity fidd in a l l 3 dimensions, 9 vanishes.
As a result, we can be sure that any condition which would find x = x, and
d4x = a4xz independent of the other parameters must be excluded from the
most general solution. However, since the field studied in Chapter 3 is necessarily
a field with x = z, everywhere due to the assumption of energy equipartition, it
is reasonable to assume that more energy transfer to bubble expansion should be
possible in a solution which allows non-parallel fields. As well, the full generality
of the 3 dimensional problem is required in order to have a non-zero Poynting flux
In choosing U' = 0 everywhere, not only is the Poynting flux zero, but the i
component of the magnetic field can no longer include the global field component
that was required in the feasibility study. Although setting U, = 0 would simplify
the governing equations substantially, much of the physics deemed important by
the feasibility study would be lost.
The direction of the energy flux can also be used to characterize the behaviour
of various parameters. Since the energy flux should be tangential to the stream
0 and U, c 0 the energy flux should be directed
that S, < 0 forces the relation 2, (x - xz) > 0
lines (see below), when U4 =
towards the origin. Requiring
which in turn guarantees that
Since the six coupled differential equations given by equations 4.16 - 4.21 contain
only first order derivatives, they can be algebraically manipulated to the form
In this form, N,, and D, contain no derivative terms. Despite looking less
4- Ramemrk for a SeU-Similar Saverbubble 74
complicated, this form clearly indicates points that plague numerical integration
with problems. When D,., approaches zero, the derivative becomes infinite. Such
singularities in the system of differential equations are referred to as critical points.
No numerical algorithm will manage to integrate through such a point without
introducing large numerical errors. Physically, such a singulariw corresponds
to a point in phase space where two or more solution c w e s intersect. In a
onedimensional problem, the critical points occur where the flow speed matches
one of the characteristic speeds of the medium: the sound speed or the fast or
slow magnetosonic speeds (Tsinganos et al., 1996). In two-dimensional problems,
the critical points are more complicated. Nonetheless, a proper solution to the
problem must satisfy the boundary conditions and pass smoothly through any
critical points that are encountered. The possibility exists that not all critical
points correspond to physically important solutions and that not all physically
important solutions will pass through a critical point. However, the critical points
do represent loci of solutions and therefore subdivide the possible solutions into
families. The physically important family is that which is able to satisfy the
imp& boundary conditions.
A common strategy for dealing with such points is to choose other parameters
in a manner that forces the numerator to approach zero as wen. The indetermi-
nate derivative can then, in principle, be evaluated using L' Hospital's Rule. This
approach is applied here to equations 4.16 - 4.21. \
The derivative equations are
4. h e w o r k for a Self-Si- Su~erbubble 75
where
By examination, several critical points can be identified for the above set of equa-
tions. These points are summarized in Table 4. l, along with the equations that
are made singular and some comments about the physical meaning of each point.
4- k e w o r k for a S&4hdar Su~erbubble 76
lgble 4.1: The critical points of the system of difkential equations are shown with the corresponding singalar equations and some comments about the interpreta- tion of the point.
Critical Point Singular Equations Comments 0 = 0 4-32 - 4.36 requires zero density ws (4) = 0 all 4 = : is a boundary U, = 0 4.32, 4.34 - 4.37 at boundaries, but elsewhere? & = O 4.36 CT = x2 4.32, 4.34, 4.36 equipartition in f# plane J = 0 all several ways
The first cr i t id point is c = 0 or p = 0 which implies that the density
is zero. Clearly this critical point corresponds to a non-physical solution within
finite radii and so should be avoided. The second critical point is 6 = P which
corresponds to the asymptotic boundary condition. Since the boundary is ex-
pected to display asymptotic behaviour, a numerical solution would be unable
to reach this point. U4 = 0 is also excluded due to the boundary conditions,
but there is the possibility of U4 + 0 at some intermediate angle. The stream
line shown in Figure 4.2 does not allow U', = 0 except at # = 0 and f , but
this stream line is only an idealistic approximation which satisfies the imposed
boundary conditions.
The critical point Uz = 0 is not readily interpreted beyond the recognition
that it either represents a point where v, either reverses sign or is an extremum.
With no boundary conditions to constrain the behaviour of U,, this point must be
explored further by obtaining a finite expression for doz,. In order to find such an
expression for the derivative, the numerator must be set to zero by assuming values
for other parameters. Since the U, = 0 critical point makes only one equation
singular, it is the easiest to analyze. In order to get the 010 form required by
4. h e w o r k for a SeKSidlar Superbubble 77
L'Hospital's Rule, a solution to the following numerator equation must be used:
The two choices are xz = z, corresponding to fully parallel magnetic and velocity
fields, and U4sin(4) = Urws(q5), corresponding to purely vertical Bow.
First using U, = 0 and 2, = x, it is possible to show that
where primes denote that the function has been evaluated at the point of interest.
Thus a h i t e expression is reached that could allow a solution to pass smoothly
through the critical point. However the result that d4x = d4x, at the critical
point where x = zZ is disturbing. Section 4.2.2 has shown that the Poynting
flux Mnishes when the fields are parallel, permitting no energy transport. At this
critical point a zero Poynting flux is acceptable, but the equality of the derivatives
of z and 2, ensures that the Poynting flux will be zero everywhere. In fact,
equations 4.35 and 4.36 show that the only point where it might be possible to
have z = z, at a point but not everywhere else is at a critical point. In the case
of the U, = 0 critical point this possibility fails.
The second possibility is to have Uz = 0 and U,sin(4) = Urcos(#). In this
case it is easy to show that d,#: = 0 and so L'HospitalYs Rule must be applied
at least twice in order to reach a solution. In order to apply this rule for a second
time, the following equation must also be satisfied to ensure the 010 form:
Substitution of d4Ui and d& into the above equation yields a condition that
has a potential zero if o = z2. However, a = 22 is a critical point in itself so
the critical point associated with U' = 0 and Uosin(t,b) = Urcos($) might be
considered a subset of the a = 22 critical point.
4. namework for a SeU-Similar Superbubble 78
o = 9 represents a promising critical point in that it forces Ur,* = br,#/J4k- The poloidal velocities therefore approach the poloidal Alfvh speed. The appar-
ent success of the feasibility study, in which this condition is forced everywhere,
suggests that this point is physically significaui. As well, the clear connection
between the critical point and the Alfir6n speed is consistent with Tsinganos et
ale's (1996) observation that critical points in general are related to characteristic
flow speeds. Since 3 of the derivatives are indeterminate at this critical point,
there are 3 numerator equations which must be satisfied simultaneously. These
are
A' J'COS'/~(~$) - urBx2 = O , (4-44)
where again A', J' and H' have no common factors. Three solutions to these
equations have been found that must be considered:
a) a = 2* & x = z, & solve for U, ,
b) a = X~&U' = Uz = 0 ,
C) o = s2 = I & U4 = &tan(#) & s2, = O .
The first case is complicated by the non-trivial expression obtained for Ur-
However, this critical point is important because it represents a point of complete
energy equipartition between all of the components of the magnetic and velocity
fields. This is a point where the Poynting flux vanishes; thus how the solution
evolves away from this point will determine the nature of the energy transport. If
4. b e w o r k for a Selfairnilkt Su~erbubble 79
d4x = d4xz at this point, as with the U' = 0 critical point, then the Poynting
flux vanishes everywhere. Under this condition, the resulting model would be
closely related to that of the feasibility study in that equipartition holds, but the
density is permitted to vary.
Case b) represents energy equipartition between the 4 components at a point
where the i and i components vanish. Assuming that z, is finite, this critical
point is merely a subset of the critical point represented by a). Case c) also
represents an equipartition scenario although the equipartition is not complete
unless Uz = 0 as well. Otherwise equipartition is d i d only in the i 4 plane.
The requirement that U' = U6tan(#) indicates that the critical point must be
very near to q5 = because this is the only region in which the velocity in the
i 4 plane is expected to be purely vertical (i.e. out of the disk of the galaxy). If
this condition occus at any other value of 4, it would correspond to a blowout
solution. Since this critical point is expected to lie so near to the axis, it should
not affect a numerical integration of the derivative equations.
The only case which must be considered is case a). Before deriving the deriva-
tive equations using L'Hospitalls Rule, the existence of values for x, Ur , U,, W', q5 and a which are within acceptable ranges and which satisfy equations 4.44 and
4.45 must be illustrated. As a first step, these equations can be solved for z in a
simple, although not obvious, way by multiplying the first by Ur and the second
by U, and adding. The resulting condition is that z = 1 at this critical point
for all values of the other parameters. This point is therefore also reduced to a
boundary point and, as such, is excluded &om possible solutions.
The final critical point listed in Table 4.1 is J = 0, a condition which is
achievable in several ways. Since J = 0 makes all of the derivatives indeterminate,
such a point is likely to have some physical significance. In order to identify this
critical point in terms of measwed quantities, the expressions for if, 8, I and K3
are substituted back into equation 4.39. The resulting simplified expression for
4. h e m r k for a S&4im&t Su~erbubble 80
the critical point is
m'A 2 mk4 v; - (4 - -v*) v; + a:, - &u: (G - 1) = 0 (4.47)
4 4
which reduces to the condition derived by Tsinganos (personal communication)
for the critical point in Fiege and Henriksen (1996) where x = z,. The Alfv6n.i~
numbers m~ and m ~ , i correspond to the total velocity and the z* coordinate
velocity. Appendix A explicitly develops this expression and illustrates that when
x = x. alI four of the Alfvhic numbers are equal. The velocities represented by
VA and V A , ~ are the total Alfv6n velocity and the Alfir6.n velocity in the 6 direction
respectively.
Equation 4.47 shows explicitly the complexi* expected for a critical point in
a three dimensional magnetohydrodynamic model. Since J = 0 is expressible
in terms of the various flow speeds, the solution that is sought should acknowl-
edge such points. In particular, solutions which reach the critical point must
pass through it smoothly. Finite expressions for the derivatives at this point are
therefore required. However, the complsdty of the equations renders an analytic
derivation of these derivatives a daunting task. All of the numerators are zero if
the parameters can be chosen such that H = 0. With J = 0 and H = 0, all of the
derivatives have the 010 form required for a critical point.
Applying C'Hospital's Rule to each derivative equation essentially requires
6nding an expression for d+H / 4 J . However, both d4 J and dsH are functions
of all six parameters and their derivatives. Substituting these functions back
into the derivative equations in place of J and H yields a system of six non-
linear equations in six unknowns. (The non-lineariw takes both the y: form
and the yi yj form.) The coefficients of each partial derivative and each product
of partial derivatives are functions of the six parameters. Although this system
of equations is algebraic, the non-linearities make it unsolvable analytically. A
numerical routine for solving N-dimensional sets of non-hear equations, such as
4. b e w o r k for a Self-Shih Superbubble 81
those discussed in Numeriml Recipes (Press et af., 1992), is required to find the
values for each of the derivatives.
En general, values for the parameters will have to be assigned before solving
for the derivatives. So far the only requirements at the critical point are that
J = 0 and K = 0. However, some constraints for the various parameters can be
derived by requiring that they all be real-valued. Solving J = 0 for x requires
that either > K3 or 0 < < min{K3 - q, K3 - @</4). Solving H = 0
for P produces another expression (hear) that must be positive for a suitable
choice of parameters. With these two expressions, it is possible to check whether
a set of initial conditions is valid €or this critical point. Once initial conditions
are selected and the derivatives are evaluated at the critical point, a numerical
integration of equations 4.32 - 4.37 can proceed in both directions fiom the critical
point. If this approach is followed, solution curves which pass smoothly through
the critical point will be obtained. Whether or not any of these curves can be
forced to avoid the other critical points and to match the boundary conditions will
only be known once the work has been camed out. It should be noted, however,
that there remains a four-dimensional parameter space that must be explored b r
valid solutions.
4.3 Preliminary Numerical Work
A small portion of the parameter space has been explored numerically for solutions
which fit the boundary conditions and which produce correct order of magnitude
predictions for various parameters. This Section discusses the computer code used
to integrate equations 4.32 - 4.37, the two approaches used to isolate selected
regions of the parameter space, and the results of the integrations. Some very
encouraging results have been obtained despite the limited volume of parameter
space that was explored. However, in order to fully examine the parameter space,
4. Ekamework for a Self-Simih Superbubble 82
the J = 0 critical point must be included.
The numerical routine employed to integrate equations 4.32 - 4.37 was a
fourth-order Runge-Kutta method with an adaptive stepsize. A modified version
of the appropriate Gcode routines from Numerid Recipes (Press et al., 1992)
was used. In order to reduce computational error, double precision variables were
used throughout. Errors resulting tiom the addition of very small numbers to
very large numbers (rather than summing the small numbers first) could not be
taken into consideration because of the unknown nature of the solution. Essen-
tially, there is no way to know which terms will be small and which will be large.
However, the values obtained for the derivatives at various points in parameter
space were compared to those obtained using MapleV and found to agree to 1
part in 10l0.
Before applying the computer code to the solution of the superbubble prob-
lem, we tested it on several systems of equations for which analytic solutions
were known. This manner of testing verifies that the basic integration routine
is functioning properly. In order to ensure that the routine was able to follow a
rapidly varying solution, one of the test systems was contrived to include a high
frequency sinusoidal function. Even in this case, no discernible difterence was
found between the numerical solution and the curves produced by MapIeV for the
analytic solution.
The system of equations is complex enough that one or all of the variables may
be changing rapidly at any given point. To ensure that the numerical integration
stepped fkom a slowly varying region into a rapidly varying region without incur-
ring large numerical errors, a maximum stepsize was enforced. For most calcula-
tions, this maximum stepsize was requiring at least 1000 steps to complete
the integration. A further accuracy test was made by reducing the maximum
stepsize to and comparing the results. However, using a stepsize which is
too small is not only inefficient but can also allow the global error to become
4. h e w o r k for a S e l f d m Superbubble 83
significant.
The application of the integration routine to the superbubble problem was
carried out using two different approaches. The first approach was simply to
search in a random, but educated, manner for starting conditions which produced
physically realistic solutions. The second approach was the one commonly used
to solve problems of this sort: to start at a critical point and to integrate away
from it in both directions. The results obtained from both approaches provide
valuable insights into which regions of parameter space might result in acceptable
solutions.
The random approach to solving a system of equations as complicated as this
is particularly fnrstrating. All attempted starting conditions resulted in the inte-
gration encountering at least one critical point and more often two. Identifying
which critical point has been reached is necessary, but often difEcult. As discussed
in Section 4.2.3, several of the possible singularities of this system of equations
should occur only at the boundaries. Therefore when one of these points is en-
countered in an integration, the starting conditions must be varied in a manner
that moves the critical point towards a boundary. To make matters worse, the
behaviour of the equations beyond a critical point can not be interpreted &om
the integration even if the numerical routine manages to continue. It is often the
case that the program makes a step large enough to escape the singulari@, but
the resulting solution curve is unrelated to the initial starting conditions unless
the solution passes smoothly through the critical point.
Using this approach, starting conditions were chosen to be consistent with
the expected behaviour of the system. For example, the density is expected to be
greater near the disk and the velocities are expected to be roughly 30 - 40 kms-I . One set of starting conditions was ultimately found that encountered only U' = 0
and J = 0 critical points close to the boundaries. At the II, = 0 points, z was also
equal to x, and so the critical point corresponds to the one studied analytically.
4. h e w o r k for a SeS-Similar Superbubble 84
By incorporating the analytical expressions for d+xz at U' = 0 into the routine,
solutions which pass smoothly through these critical points could be obtained. The
analytical investigation of these critical points revealed that if such a point were
encountered in a solution, x should be equal to z, everywhere. This behaviour
was confirmed in the numerical integration. If the starting conditions did not
include x = z,, this equality was obtained within one step in either direction.
With some additional fine tuning of the starting conditions, the solution curves
shown in Figures 4.3, 4.4 and 4.5 were obtained.
Figure 4.3 shows the resulting behaviour of the velocity functions. If Rn,
is taken to be 150 kms-' then the maximum component velocities are roughly
30 kms-I. The 6 component shows the decline expected near the 4 = f boundary,
but not near the # = 0 boundary. Ideally, U, should also increase more strongly
towards q5 = 0. The asymptotic behaviour of U# near q5 = f is also indicative of
the J = 0 critical point however.
Figure 4.4 illustrates the behaviour of z. The value of AB is unknown and
so can be used to scale the magnetic field strength accordingly. However, AB
also scales the densiw so it is not entirely free. If ABR is chosen to be 1 pG or
0.1 pG, the maximum component magnetic field is 1.7 pG or 0.17 pG respectively.
The maximum and minimum densities that result fiom these two choices for AB
are p = 1.9 cm" or p = 0.19 m-3 and p = 0.7 or p = 0.07 The
asymptotic behaviour of the density in Figure 4.5 near the q5 = f boundary is
non-physical. The J = 0 critical point would need to be addressed before the
behaviour of the density in this region could be believed. However, if addressing
this critical point properly allowed the deIlSif3r to continue to decrease in this
region, it is also Likely that U, would increase more strongly.
An entirely diflerent region of parameter space was explored using a second
approach. Although the critical point produced by o = 22 is expected to occur on
the boundary, it is possible to calculate starting conditions assuming that the crit-
4. h e w o r k for a SdE-Shik Superbubble 85
Figure 4.3: The solution curves obtained for IT,, U4 and Uz by integrating £ram 4 = 0.4, a = -1.22, U, = 0.2, U4 = 0.1 and U, = 0.01. These curves represent the best solutions obtained using the random approach. The docity of the outflow is obtained by multiplying the rotational velocity at R by the vertical axis.
4. Ekamewock for a Self-Similat Superbubble - 86
Figure 4.4: The solution curve obtained for x = z, by integrating fiorn # = 0.4, a = -1.22 and z = z, = 0.9. The magnetic field strength is dependent on the chosen d u e of AB .
4. h e w o r k for a SelfSimilar Superbubble 87
Figure 4.5: The solution curve obtained for a by integrating from 4 = 0.4, a = -1.22 and a = 75. The actuaI density is dependent on the chosen value of AB.
4. h e m k for a Self-Simil=v Smesbubbie 88
ical point is located near the boundary. Figure 4.6 shows one such solution which
was integrated backward fiom 6 = 1.4. Both the U, and IT4 curves exhibit erratic
behaviour starting at q5 = 1.25. The other four parameter cmes (not shown)
also exhibit this behaviour at the same point. This high frequency oscillation
is due to the sign of all the derivatives changing without the derivatives passing
smoothly through zero. The onset of oscillation is an indication that a critical
point has been encountered, in this case a J = 0 point. Since no allowance was
made for the derivative expressions to remain h i t e at this point, the integration
proceeds in a random manner until chance allows it to escape the critical point.
The curves in Figure 4.6 do escape the critical point at q5 = 1.16. From this point,
the integration continues smoothly along a new solution curve which is unrelated
to the initial conditions.
In both approaches, different regions of parameter space have been explored to
the same conclusion: that the J = 0 critical point must be addressed in a solution
that hopes to fit superbubble boundary conditions. The solution obtained born
the random approach displays some of the features expected of a superbubble
model, in particular the outflow velocities of roughly 30 kms-', but fails to meet
all of the required boundary conditions. Varying the starting conditions to include
a stronger magnetic field or a higher disk density only aggravates the problem of
having to include the J = 0 critical point. This behaviour suggests that J = 0
might be a sort of multi-dimensional hyperbolic point. Most solutions appear
only to be able to approach the point closely, but not to actually pass through it.
There should be at least one solution, however, which passes smoothly through this
critical point. The only way to find such solutions wil l be to complete the critical
point analysis by solving for the derivatives. Only by obtaining the conditions
at the critical point itself will the correct solution be filtered out of the six-
dimensional parameter space.
Figure 4.6: The solution curves obtained for Ur and U4 by integrating from the a = 9 = 23 = 1 and Ur = U4tan(t#5) critical point at 4 = 1.4 backward. A second critical point is encountered at 4 = 1.25.
4. h e w o r k for a SeV-Simih Su~erbubble 90
4.4 Summary of the SeIf-Similar Model
A s e I f ' a r model describing a state of superbubble evolution in a realistic galac-
tic environment has been fully developed and partially explored in this Chapter.
AU of the physical effects that were determined to be necessary components of
the feasibility study have been incorporated. This model does not assume en-
ergy equipartition or a constant density, but does include ditferential rotation and
the global magnetic field. This model is limited by the assumptions that the
gravitational acceleration orthogonal to the disk is constant, that the pressure is
isothermal along curves of constant r and that the solutions are time independent.
Several new results regarding magnetic superbubbles are obtained fkom this
study, although the role of the Parker Instability remains speculative. The full
freedom of unrelated velocity and magnetic fields is found not to be necessary as
the induction equation requires that the projections of the two fields into the i d plane be pardel. However, the Poynting flux illustrates the need to have non-
parallel fields in the disk in order to maximize the energy transport. Whether or
not the fields are pardel in three dimensions appears to represent a significant
partitioning of the parameter space of this problem. It has been shown that if
such a state is achieved at some point on a solution curve, then the fields must
be parallel at every point in the solution. Therefore the solution curves for x and
xz may not cross. As well, it has been shown that the Poynting flux is zero when
the fields are fully parallel. The selfsimilar model is then reduced in scope to a
model closely related to that presented in the feasibility study: parallel fields, but
a variable density.
Several critical points, also examined in this Chapter, lead to similar subdivi-
sions of the parameter space. Most of the critical points identified in Section 4.2.3
lie outside, or at the boundary, of the region of interest. However, two critical
points reappear in all of the solution curves examined in this limited numerical
4. h e w o r k for a Se-If-SI'milar Superbubble 91
study. The critical point corresponding to U, = 0 appears to have very little
physical significance beyond making d*c, singular. The J = 0 critical point does
appear to be a physically signiscant point. This condition is satisfied when the
flow speed reaches a particular combination of the sound speed and the M e n
speed. Both points are integral to solutions which fit the boundary conditions
of the magnetic superbubble problem; however it will be very d.ifEcult to deter-
mine darting conditions which will produce solution curves that pass smoothly
through the J = 0 point. Only a preliminaxy numerical study of an isolated
region of parameter space, within which the J = 0 critical point lies at the cal-
culation boundary, has been possible. As shown in Figures 4.3, 4.4 and 4.5, the
solution obtained here is not completely adequate, but is sufficient to warrant the
continued exploration of the six-dimensional parameter space.
5. CONCLUSIONS
Despite nearly 20 years of study, superbubbles remain a mystery. Observations
of superbubbles in several galaxies now indicate that previously proposed models
of superbubble formation and evolution are inadequate. Some superbubbles axe
consistent with multiple SNe models while others are more favourably described
by impacting cloud models and still others can not be explained reasonably by
either model. This thesis presents the first comprehensive study of magnetic field
effects on superbubbles and the first model that is able to describe the energetics
of all observed superbubbles.
The feasibility study, Chapter 3, requires the global magnetic field to be in-
cluded in magnetic superbubble models, in addition to the locally perturbed field.
The local field, shaped by diflerential rotation, provides the geometric structure of
the bubble while the global field is required to provide the energy required for ob-
served expansion velocities. Despite over-simplirying the physics of superbubble
evolution, the "toy model" developed in Chapter 3 is able to predict the correct
order of magnitude kinetic energies for the observed superbubbles if the radius
of the bubble, the global magnetic field strength and the local rotation curve are
provided. The success of this feasibility study justifies the development of a more
sophisticated model.
A self-similar model has been formulated and explored analytically to further
determine the role of magnetic fields in superbubble formation and evolution. Al-
though all of the work in this thesis is timeindependent, stationary states which
match the obsewed states can be studied to determine the impact of magnetic
fields on superbubbles and the magnetic field structures that might result in su-
perbubble formation. The projections of the magnetic and velocity fields within
a superbubble into the plane normal to the azimuthal direction in the disk are
found to be parallel. In general, non-parallel fields in all other planes are required
to maximize the energy transport. However, the success of the feasibility study
suggests that the maximum energy transfer state may not be required to describe
superbu bbles.
A preliminary numerical study of the self-similar model shows that two crit-
ical points are likely to be part of any solutions that fit the proposed boundary
conditions for superbubbles. The first is a singularity in the equations which cor-
responds to a change of sign of the azimuthal component of the velocity. This
critical point has been discussed analytically and incorporated into the numerical
integrations. The second critical point is a more complicated point, at which the
flow speed reaches a particular combination of the sound speed and the Alfv6n
speed. This behaviour has been noted by Tsinganos et al. (1996) in several other
three-dimensional magnetohydrodynamic models. Although a complete expres-
sion for the physical conditions a t which this critical point is encountered has been
developed, further analytical analysis of this point has proven to be cumbersome.
In order to properly include this point in a numerical solution, the integration
will have to begin at the critical point. Determining the initial conditions at the
critical point will require solving a system of six non-lineax algebraic equations
in six unknowns. Although solving such systems is difficult, it has been shown
necessary in order to obtain reasonable solutions to the self-similar model.
The geometry of the field produced by the feasibility study suggests that the
Parker Instability may influence superbubble formation. Kamaya et al. (1996)
have also suggested that this may be the case. Although the work of this thesis is
suggestive, it neither proves nor disproves the Parker Instability hypothesis. The
role of the Parker hstabiliw could be examined if a complete solution for the
self-similar model were found.
The need to include the global field in magnetic superbubble models also sug-
gests that superbubbles may be linked to galactic-scale magnetic processes such
as the galactic dynamo. A link between superbubbles and the oreffect in dynamo
theory has been suggested previously by Ferrike (1992,1996), but in the context
of the superbubble producing turbulence. The link suggested in this thesis is much
more direct. It is possible that superbubbles are the observable manifestation of
the cu-effect induced by a Parker Instability. Once again, though, this hypothesis
can not be tested at present. Synthetic radio observations of a stationary state
solution may be able to address the superbubble - a-effect relation, at least in
part, but a complete time-dependent solution may be required.
The magnetic superbubble model developed in this thesis does not rely on SNe
or on impacting clouds. As a result, any observed correlations between superbub-
bles and OB associations or superbubbles and known high velocity clouds may
be due to the superbubble forming these objects rather than the reverse, as sug-
gested by earlier models. OB associations might be formed by compression of the
medium surrounding a superbubble or by the influences of the Parker Instability
(Mouschovias, 1996). High velocity clouds may form when a superbubble frag-
ments due to gravitational instability (Schulman et al., 1994). The relationship
between superbubbles and OB associations or high velocity clouds, in the context
of this model, is therefore one of effect rather than cause.
In conclusion, this thesis presents the Grst study of a superbubble formation
mechanism based entirely on magnetic fields. The "toy model" developed in
this thesis is the only existing model capable of describing the energetics of alI
observed superbubbles. The analytical formulation of a comprehensive selfsimilar
model further describes the magnetic field structure possible within a superbubble
and lays the foundation for an in-depth numerical study. Preliminary numerical
results indicate that reasonable solutions will be found once the critical points
are properly accounted for. A continuation of this study that finds a solution for
the seIf-similar model that fits the required boundary conditions wil l allow the
roles of the magnetic field, the Parker Instability and the cr-effect in superbubble
formation to be determined. Additional obsemtions of superbubbles in galaxies
are also required to provide more detailed standards by which to judge superbubble
models.
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APPENDIX
A. CRITICAL POINT CONDITION FOR J=O
This appendix develops equations which relate the Alfv6n or Mach numbers for
each coordinate velocity and the total velocity to one another. These relations
are then used to derive the critical point condition (equation 4.47) for the J = 0
critical point discwed in Section 4.2.3.
First, from the definitions of the various components of the magnetic field
The i, 6 and i component Mach numbers are
and
The total Mach number can be expressed as
A, Critcaf Point Condition for J=O 104
which clearly shows that in the case of z = z, all four of the Mach numbers are
equal.
With these relations, the critical point condition for J = 0 can be derived.
Starting with J ,
and substituting the physical variables the following srpression can be found (0 #
Using v: = v2 - v: - v: and mi,# = / v : , ~ so that only the 6 and i
components remain yields
into which a substitution of
gives
Finally, by expanding and collecting terms in vo the critical point condition is
reached,
