Gravitational stability of a cylindrical plasma with an azimuthal and an axial magnetic field

Abstract

We consider the gravitational stability of a current carrying filamentary
cloud in the presence of both axial and azimuthal magnetic fields using a
simple analytic model. The azimuthal magnetic field is shown to give rise to a
new contribution, dictated by Ampere's law, in the corresponding virial
equation for magnetohydrodynamic equilibrium. From this we obtain a
computationally inexpensive guidance on the gravitational stability of current
carrying filamentary clouds. The approach not only provides a fresh insight
into the essential physical mechanisms involved, but also demonstrates clearly
that, for sufficiently large and yet astronomically realistic currents, the
azimuthal magnetic field can cause filamentary clouds to undergo instability.

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arXiv:1201.0509v1 [astro-ph.SR] 2 Jan 2012
Star formation from dark filamentary clouds
Gravitational stability of a cylindrical plasma
with an azimuthal and axial magnetic field
James A. McLeman1, Charles H.-T. Wang1,2, Robert Bingham2,3
1SUPA Department of Physics, University of Aberdeen, King’s College, Aberdeen AB24
3UE, UK,
2STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK,
3SUPA Department of Physics, University of Strathclyde, Glasgow, G4 ONG, UK.
Abstract
The precise process by which dark filamentary clouds collapse to form
stars is a subject of intense debate. In this paper we consider a cylindrical
distribution of plasma with both axial and azimuthal magnetic field and
examine the resulting gravitational stability. The azimuthal magnetic field
is created from an electric current in the plasma and is found to be dictated
by Ampere’s law. We model this system by using the magnetohydrodynamic
(MHD) equation to derive a new virial theorem. We can reduce it to the
virial theorem due to Chandrasekhar and Fermi (1953) if we remove the
azimuthal magnetic field, as this will represent the case which they have
considered. This new virial theorem gives us a fresh insight into the stability
of the system. We also derive from this new virial theorem the case where
there is only an azimuthal magnetic field. Our generalised stability condition
allows for a possible electric current within realistic astronomical values.
1 Introduction
Understanding the processes involved in the formation of stars presents
many challenges and theoretical explanations. Generally, the process of star
formation begins with a cold, disperse collection of dust, atoms, molecules
and plasma. This cloud of molecular and ionised particles, due to dynamic
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instabilities, collapses down releasing its stored gravitational potential en-
ergy as heat. Eventually we are left with a ball of dense plasma at enormous
temperatures. As the temperature and density continue to increase we begin
to see the necessary conditions required for fusion. However, despite our col-
lective understanding gained, the precise processes involved are still subject
to intense scientific debate. Telescopes like Hubble have provided us with an
invaluable insight into the fundamental workings of celestial bodies but these
cold, disperse molecular clouds are almost invisible to visible light detectors.
They can, however, be seen by looking in the infra-red part of the spectrum
and so telescopes such as Herschel were built to see into otherwise unseen
parts of the universe (Hill et al 2011). This allows us to see the formation
of stars right from their beginnings as Infra-Red Dark Clouds (IRDCs). Due
to these new detectors much has been learned about the formation of stars
ranging from the low mass to high mass nebulae.
One of the fundamental conditions for the onset of the dynamical instabil-
ity causing the disperse, cold gas to collapse was given by the Jeans condition
(Jeans 1902). This gave us a requirement for the instability to occur as an
expression of the velocity of a propagating wave through the gas. When
the velocity of the wave became imaginary we would no longer see oscilla-
tions but an exponential collapse at the boundary instead. This could also
be expressed as a minimum, critical wavelength beyond which we would see
an instability occurring. When considering these molecular clouds they also
arise in cylindrical filamentary mass distributions, such as the “Nessie” Neb-
ula (Jackson & Finn 2010).
To account for this matter distribution the Jeans stability condition was
modified. This modification showed that cylindrical clouds, subject to lon-
gitudinal perturbations, can (under specific conditions) break apart into
smaller pieces (Dibai 1957), the results of which could be used to explain the
creation of star chains. Both of these methods to understand this instability
neglect the magnetic field and its influence on the stability of the system.
It was not until 1953 when Chandrasekhar and Fermi approached this prob-
lem for the first time (Chandrasekhar & Fermi 1953). They realised that for
stationary configurations it is sufficient to consider a less stringent stability
condition called the virial theorem. The method of analysing the stability
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through the virial theorem has the advantage of being much simpler than di-
rectly tackling often cumbersome partial differential equations (PDEs), this
is often the case when considering a method similar to the derivation of the
Jeans condition (Jeans 1928). By considering a plasma, they first derived the
virial theorem in the most general case including a magnetic energy term.
This was done by first considering the Navier-Stokes equation in three di-
mensions with the addition of magnetic force density terms. By integrating
the equation over the volume they obtain an expression involving the en-
ergy terms from the different components, i.e, the magnetic energy and the
molecular energy. This can be used to represent a condition required for the
system to be in equilibrium. Using a similarly fashioned derivation as done
in the most general case, they then did the same for cylindrical distribu-
tions of matter with a prevailing axial magnetic field. From this a stability
condition was reached. It was shown that the average magnetic intensity is
required to be less than a critical value, which is dependent on the average
mass density of the cylinder, in order to maintain stability. However, Dibai
(1958) showed that for longitudinal waves propagating through a cylindrical
medium of plasma, an axial magnetic field does not affect the stability of the
system.
When considering filaments of plasma one would naturally think that this
would set up an azimuthal magnetic field similar to that of a current car-
rying wire, as predicted by Ampere’s law. This additional magnetic field
will influence the stability of the system and to our knowledge a stability
condition involving a azimuthal magnetic field has not yet been rigorously
derived. Therefore, we will be considering cylindrical filaments of plasma
which is not only subject to a prevailing axial magnetic field but also a pre-
scribed azimuthal magnetic field. Starting from the most general fluid flow
equation we derive an equation specific to our problem and then use this as
a basis to reach the new virial theorem. This new form of the virial theorem
will give us information on how the stability is affected by the presence of
this additional magnetic field.
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2 Method
2.1 The Configuration of a Cylindrical Plasma with
Magnetic Fields
We shall idealise a dark filamentary plasma as an infinitely long cylinder
with radius R of fluid having zero resistivity, along whose axis we allow
for a net movement of charge. This sets up an azimuthal magnetic field,
just like that of a current carrying wire, in addition to a possible prevailing
magnetic field in the axial direction. It is natural to adopt the cylindrical
polar coordinates (r,φ,z) where z denotes the axis of the filament. The
cylindrical symmetry requires the all physical qualities involved including
the density ρ, pressure p, velocity v and magnetic field B to be functions of
radial distance from the axis r and time t. In particular both azimuthal and
axial components of magnetic field, Bφand Bzrespectively, depend only on
r and t.
The magnetic fields across the boundary of the cylinder are assumed to match
smoothly (continuous and differentiable) so that there is no surface current
sheet. Exterior to the cylinder the axial magnetic field Bzis assumed to be
zero while the azimuthal magnetic field Bφmay remain non-vanishing.
2.2 Derivation of New Virial Theorem with Axial and
Azimuthal Magnetic Field
We begin this derivation by considering the Navier-Stokes equation for an
inviscid fluid in the MHD approximation subject to the Lorentz force
(∇ × B) × B
µ0
=(B.∇)B
µ0
− ∇(|B|2
2µ0) (1)
as follows
ρdv
dt= −∇(p +|B|2
2µ0) + ρ∇V +(B.∇)B
µ0
(2)
where v is the velocity, t is the time, p is the pressure, V is the gravita-
tional potential, and the density is ρ. Here
derivative of v, which has the following form
dv
dtdenotes the Lagrangian time
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dv
dt= ˆ zdvz
dt+ ˆ rdvr
dt
(3)
due to the cylindrical symmetry.
Let us examine the first term on the RHS of (2). This needs to be converted
to cylindrical coordinates system:
−∇(p +|B|2
2µ0) = −(ˆ r∂
∂r+ˆφ1
r
∂
∂φ+ ˆ z∂
∂z)(p +|B|2
2µ0).
(4)
It follows that we can reduce (4) to:
−(ˆ r∂
∂r+ˆφ1
r
∂
∂φ+ ˆ z∂
∂z)(p +|B|2
2µ0) = −(ˆ r∂
∂r)(p +|B|2
2µ0).
(5)
The second term on the right of (2) is a gravitational term and again it is
to be assumed that it is radially dependent only. This term can be derived
by using a form of Gauss law but for mass distributions instead of charge
distributions, the law states that:
?
g.dA = −4πGMenc.
(6)
Where g is the gravitational acceleration vector, dA is the surface area
unit. Mencis the mass enclosed by the imaginary surface. We must consider
two cases here, the first is the case outside the cylinder and the second is
the case inside the cylinder. Of course the two results must coincide at the
boundary. Let the radius of the cylinder be of length R.
First Case: Outside the Cylinder
Take the imaginary surface to be a cylinder of radius r > R. So due to
the high symmetry of the problem we can say that:
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