Magnetism Guide

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Last modified April 15, 2019

Galactic Magnetism PhD Study Guide
Jessica Campbell, Dunlap Institute for Astronomy & Astrophysics (UofT)

Contents
1 Preface

1

2 Terminology

2

3 Variables

28

4 Equations

31

1

Preface

The interstellar medium (ISM) is a complex environment of gas and dust with varying degrees of ionization, densities, and spatial distributions called phases. It accounts for ∼ 10 − 15% of the mass and
1 − 2% of the volume of our Galaxy. From the largest of Galactic scales to the smallest scales of star and
planet formation, this tenuous plasma is threaded with magnetic field lines and exhibits large degrees of
turbulence, both of which are believed to be major driving forces in shaping the structure and dynamics
of our Galaxy. These magnetic fields, along with cosmic rays, exert outward pressures on the ISM which
are counterbalanced by the gravitational attraction provided by the ordinary matter. The ordinary matter, cosmic rays, and magnetic fields are therefore in a state of pressure equilibrium. Turbulence is an
important aspect of the ISM as it not only transports energy from large (∼kpc) to small (∼pc) scales,
but also amplifies magnetic fields and accelerates cosmic rays, explaining the observed µG magnetic field
strengths as well as the ∼GeV cosmic ray energies in our Galaxy. Turbulence itself is a complex nonlinear fluid phenomenon which results in an extreme range of correlated spatial and temporal scales in
the multi-phase ISM, and is driven by a variety of large- and small-scale energy sources. While magnetic
fields and turbulence are generally understood to play fundamental roles in shaping the structure and
dynamics of our Galaxy, the degree to which they do so across different phases and spatial scales of the
ISM remains poorly understood.

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2

Terminology

Advection: The transport of a substance by bulk motion.
Advection operator:
Alfvén Mach number: A commonly used parameter of turbulence used to obtain information on gas
compressibility and magnetization given by
|~v |
,
vA
where |~v | is the local velocity and vA is the Alfvén speed.
Alfvén speed (vA ): Given by
MA ≡

vA =

~
|B|
[m s−1 ],
4πρn

~ is the magnetic field strength and ρ is the density of neutral particles. When the Alfvén speed
where |B|
is greater than the sound speed, the fast and Alfvén wave families are damped at or below the ambipolar
diffusion scale LAD ; when the Alfvén speed is less than the sound speed, the slow and Alfvén wave
families are damped.
Alfvén wave: In 1942, Hannes Alfvén combined the mathematics of fluid mechanics and electromagnetism to predict that plasmas could support wave-like variation in the magnetic field, a wave phenomenon
that now bears his name, Alfvén waves. These are a type of magnetohydrodynamic (MHD) wave in which
ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines.
The waves initially proposed by Alfvén are considered “basic”. They have a characteristic that they are
compressional, which means that magnetic field variation of the Alfvén waves is in the direction of the
wave motion. Charged particles moving through a plasma with these waves have very little alteration
of their trajectory. But Alfvén waves can exhibit more variety. A variant is the “kinetic” Alfvén wave
which is transverse, with strong magnetic field variation perpendicular to the wave motion, so can trade
energy between the different frequencies which might propagate through a plasma. This also means it
can exchange energy with the particles in the plasma, in some cases, trapping particles in the troughs of
the waves and carrying them along.
Ambipolar diffusion: A.k.a. ion-neutral drift or ion-neutral friction. In many situations, ideal MHD is
not a sufficiently good assumption and additional effects need to be accounted for. In many contexts, the
dominant correction is the so-called ambipolar diffusion. Since the neutrals are not charged they are not
subject to the Lorentz force which applies only on the ions. However, through collisions the neutrals and
the ions exchange momentum and therefore the Lorentz force has an influence on the neutrals through
the ions. If the number of ions is large (i.e., if the ionization is high) the number of collisions is expected
to be large and ideal MHD remains a good approximation. However in regions like molecular clouds, the
ionization is usually of the order of 10−7 and therefore the two fluids are not perfectly coupled. The ions
drag the field lines and drift with respect to the neutrals implying that the latter can cross the field lines.
The field is not frozen in the gas anymore. The drift of neutral particles towards the central gravitational
potential through the ionized particles tied to the magnetic field. This is often invoked as a source of
dissipation of the MHD energy cascade. The scale at which ions and neutral particles decouple is called
the ambipolar diffusion scale. The application of ambipolar diffusion extends beyond direct studies of
star formation and to include general studies of magnetic fields. Ambipolar diffusion has been proposed
to damp particular families of MHD waves.
Ambipolar diffusion scale (LAD ): The scale at which ions and neutral particles decouple through
the process of ambipolar diffusion. It can be estimated as the scale at which the Reynolds number, with
diffusivity given by ambipolar diffusivity, is equal to unity. The ambipolar diffusion scale has been thought
to set the dissipation scale of turbulence in molecular clouds and set a fundamental characteristic scale
for gravitational collapse in star formation. When the Alfvén speed is greater than the sound speed, the
fast and Alfvén wave families are damped at or below the ambipolar diffusion scale LAD ; when the Alfvén
speed is less than the sound speed, the slow and Alfvén wave families are damped. On scales larger than
LAD , it was also predicted that two-fluid turbulence (ion-neutral) acts like single-fluid MHD turbulence.
The ambipolar diffusivity is given by
B2
[ms s−1 ],
4πρi ρn α
where ρi and ρn are the density of the ions and neutrals, respectively, B is the magnetic field strength,
and α is the frictional coupling coefficient between the ions and neutrals. The Reynolds number for
ion-neutral drift is defined as
νAD =

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RAD =

LV
[dimensionless],
νAD

B
)
4πρn
= 1. This gives the form of the ambipolar diffusion scale as often found in the

where V is a characteristic velocity (e.g., for trans-Alfvénic turbulence it is the Alfvén speed, vA =
and L = LAD when RAD
literature:

VA
[m].
αρi
It has been shown that the plane-of-sky magnetic field can be estimated using the ambipolar diffusion
length scale.
Autocorrelation: The correlation of a signal with a delayed copy of itself as a function of delay.
Informally, it is the similarity between observations as a function of the time lag between them. The
autocorrelation of an observable A with position r and position increment δr is given by
LAD =

C(δr) = hf (r)f (r + δr)i [dimensionless].
Axi-symmetric spiral (ASS) model:
~ θ ): Often the Galactic magnetic field is expressed in cylindrical coordiAzimuthal magnetic field (B
~ θ ) dominates the Galactic disk where the radial (B
~ r ) and
nates (θ, r, z). The azimuthal component (B
~
vertical (Bz ) components are generally weak.
Balbus-Hawley instability: See magnetorotational instability.
Bandwidth depolarization: A type of external depolarization where the polarization vector is substantially rotated within the observing bandwidth if the Faraday depth is large enough.
Beam depolarization: A type of external depolarization due to fluctuations in the foreground screen
within the observing beam: unresolved density or magnetic field inhomogeneities of the media through
which the radiation propagates induces unresolved spatial variations in the Faraday rotation measure.
Biermann battery mechanism: The process by which a weak seed magnetic field can be produced from
zero magnetic field initial conditions via perpendicular density and temperature gradients. The Biermann
mechanism can explain the generation of a ∼ 10−20 G field after recombination, but it is questionable
whether turbulent dynamo growth alone can explain amplification up to the 10 µG fields seen today
throughout the ISM. A possible solution to this problem may be provided by the potential role played by
kinetic instabilities in the amplification of magnetic fields. One such instability is the Weibel instability.
Temperature gradients form perpendicular to astrophysical shocks (hotter in the center of the shock),
while density gradients form parallel to the shock, once again allowing the Biermann battery to take place.
The Biermann mechanism is also the presumed cause of self-generated magnetic fields (∼ 106 G) found
in laser-solid interaction experiments. The laser generates an expanding bubble of plasma by hitting and
ionizing a solid foil of metal or plastic. This bubble thus has a temperature gradient perpendicular to
the beam (hottest closest to the beam axis), and a density gradient in the direction normal to the foil,
allowing for the Biermann battery mechanism to take place. Assuming a two-fluid description of a plasma
with massless electrons, the magnetic field evolution is given by the generalized induction equation:
2
~
∂B
~ + ηc ∇2 B
~ − 1 ∇ × (~j × B)
~ − c ∇n × ∇Te ,
= ∇ × (~v × B)
∂t
4π
en
ne
~
which shows the evolution of magnetic field B, based on the fluid velocity ~v , the current density ~j =
~
c∇ × B/4π,
the number density n, and the electron temperature Te = Pe /n, where Pe is the electron
plasma pressure. Here, c is the speed of light, η is the resistivity, and e is the charge of an electron. The
terms on the RHS from left to right are the convective term, the resistive term, the Hall term, and the
~ = 0, all terms on the RHS except for the Biermann term can
Biermann battery term. Starting with B
2
~
be ignored, and thus B grows linearly. Based on scaling, given Te = me vth
, we find

B(t) ≈

2
me c vth
t [G],
e LT Ln

where L is more precisely defined by the length of the gradients (Ln ≡ n/∇n, LT ≡ Te /∇Te , which can
be assumed to be approximately comparable).
Birefringence: The optical property of a medium to have a refractive index that is dependent on the
polarization and direction of propagation of light. Birefringence implies that there are two natural wave
modes which may be described by their polarizations, which are necessarily orthogonal to each other, and
by ∆k, the difference in their wavenumbers. The magnetoionic medium is an example of such a medium
as the polarization angle of light becomes rotated as it propagates via Faraday rotation as a function of
frequency.
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Bi-symmetric spiral (BSS) model:
Bonnor-Ebert sphere:
Bremsstrahlung radiation: Also known as “braking radiation”. This radiation is produced by the
deceleration of a charged particle after being deflected by another charged particle, typically an electron
by an atomic nucleus. The particle being deflected loses energy which is lost via radiation of a photon. For
example, when free electrons within an HII region pass near a positive ion (H+ , He+ ) they are accelerated
by the Coulomb field and emit bremsstrahlung radiation. Bremsstrahlung emission is a source of polarized
continuum radiation and is a type of free-free radiation.
β parameter: The squared ratio of the sound speed to the Alfvéns speen:
β≡

c2s
2 [dimensionless].
vA

Cascade rate:
Central molecular zone (CMZ): The inner ∼ 500 pc of the Galaxy which contains . 10% of the
MW’s molecular gas but ∼ 80% of the dense (n & 103 cm−3 ) gas: a reservoir of 2 − 6 × 107 M of
molecular material. The physical properties of the ISM in the CMZ differ substantially from those in
the disc. Gas column and volume densities can be ∼ 2 orders of magnitude greater, velocity dispersions
measured for a given physical size are larger, and although there exists a significant fraction of cold dust,
gas temperatures can range from 50 − 400 K.
Chandrasekhar-Fermi method: Originally proposed by Chandrasekhar & Fermi (1953) to estimate
the field strength in spiral arms, the Chandrasekhar-Fermi method uses starlight polarization to estimate
the average field strength hBi in a region. The method relates the line-of-sight velocity dispersion to
the dispersion of starlight polarization angles about a mean component. Assuming that turbulence
isotropically randomizes the magnetic field in the region, the mean field strength is given by
2
BCF
≡ B̄ 2 = ξ4πρ

σ(vlos )2
,
σ(tan(δ ∗ ))2

where
δ ∗ ≡ θ∗ − θ¯∗ ,
ρ is the gas density, θ∗ is the mean starlight polarization angle, and ξ is a correction factor representing
the ratio of turbulent magnetic to turbulent kinetic energy. The validity of the method depends critically
on the presence of a significant mean field component.
Chaotic system:
Coherent magnetic field: See ordered magnetic field.
Cold neutral medium (CNM): A thermally-stable phase of the atomic ISM with a typical density and
kinetic temperature of n ∼ 7 − 70 cm−3 and TK ∼ 60 − 260 K, respectively. It seems almost certain that
the CNM transitions to the DMM before transitioning to the classical MM. Our current knowledge of
physical conditions and morphology in the CNM depends overwhelmingly on results from the Millennium
survey of Heiles & Troland (2005), who used Arecibo with long integration times suitable for detecting
Zeeman splitting. For the HI line in absorption, they derived column densities, temperatures, turbulent
Mach number, and magnetic fields. HI CNM column densities are usually below 1020 cm−2 with a
median value N (HI) ∼ 0.5 × 1020 cm−2 . The median spin temperature is Ts ∼ 50 K, median turbulent
Mach number is Ms ∼ 3.7, and median magnetic field is ∼ 6 µG. By assuming reasonable values for
the thermal gas pressure and comparing observed column densities, shapes and angular sizes as seen
on the sky, Heiles & Troland (2003) find that interstellar CNM structures cannot be characterized as
isotropic. The major argument is that a reasonable interstellar pressure, combined with the measured
kinetic temperature, determines the volume density; this, combined with the observed column density,
determines the thickness of the cloud along the line of sight. This dimension is almost always much
smaller than the linear sizes inferred from the angular sizes seen on the sky. They characterize the typical
structures as “blobby sheets”, which applies for angular scales of arcseconds to degrees.
It has been shown by numerical studies that CNM can be formed in a shock-compressed layer of the
WNM.
∗
Complex conjugate (f (x) or f¯(x)): The number with an equal real part and an imaginary part equal
in magnitude but opposite in sign.
Compressible turbulence:
Compton scattering: The inelastic scattering of a photon by a free charged particle (usually an
electron) in which energy is lost from the photon (typically a gamma ray or X-ray) which is in part
transferred to recoiling the charged particle.
Cosmic rays: Extremely energetic and electrically charged particles pervading the ISM. The Galactic
origin of the most energetic cosmic rays and their widespread distribution throughout the Milky Way
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was not recognized until the observed Galactic radio emission was correctly identified with synchrotron
radiation emitted by cosmic-ray electrons gyrating about the local Galactic magnetic field. Cosmic rays
impinge on the ISM in three important ways: (1) they contribute to its ionization through direct collisions
with gas particles, (2) they constitute a triple source of heating arising from the excess energy carried
away by the electrons released in cosmic-ray ionization, from Coulomb encounters with charged particles
of gas particles, and from the damping of Alfvén waves excited by cosmic rays streaming along magnetic
field lines, and (3) they are dynamically coupled to the ISM via the magnetic field.
Dark molecular medium (DMM): Molecular gas in which hydrogen is molecular but the usual H2
tracer, CO emission, is absent. Generally, the mass of DMM is found to be comparable to that of the
CO-bright molecular medium (MM). It seems almost certain that the DMM is a transition state between
the CNM and classical MM. Moreover, we expect the details of the DMM transition region to depend
not only on physical conditions but also cloud morphology as it determines whether UV photons can
penetrate to destroy molecules via photodissociation or photoionization. It seems very unlikely that one
can understand the transition between atomic and molecular gas without understanding the effect of UV
photons, and thus cloud morphology. In addition, there are hints that cloud morphology is affected by
the magnetic field; after all, magnetic forces are one of the important forces on the ISM (the others being
turbulent pressure, cosmic ray pressure (coupled to the gas by the magnetic field), thermal pressure, and
gravity).
2
Delta variance (σ∆
(L)): A way to measure power on various scales defined as
2
σ∆
(L)

* 3L/2
Z

+

(A[r + x] − hAi)

=

2

(x) dx ,

0

for a two-step function
 −2 (
1,
L
×
(x) = π
2
−0.125,

if x < (L/2)
if (L/2)x < (3L/2)

.

The delta variance is related to the power spectrum P (k): for an emission distribution with a power
2
spectrum P (k) ∝ k −n for wavenumber k, the delta variance is σ∆
(L) ∝ rn−2 for r = 1/k.
Depolarization: A reduction in the degree of polarization, measured as the ratio of the observed to the
intrinsic polarization, either at a given frequency or when comparing two frequencies. Such depolarization
can be caused by Faraday rotation in two different circumstances: internal depolarization or external
depolarization. In order to distinguish between internal and external depolarization, very high resolution
and sensitive polarization data at multiple frequencies are needed. The key difference is that internal
depolarization should be correlated with the Faraday RM (such that regions with small RM exhibit low
amounts of depolarization) whereas external depolarization should be correlated with the gradient of the
RM.
Depolarization canals: Caused either by resolution-element or line-of-sight effects.
Depth depolarization:
Dispersion measure (DM):
observer
Z

~ [pc cm−3 ]
ne d`

DM = −
source

Dust-to-gas mass ratio (r):
Dust polarization: The polarization angle of dust emission is conventionally taken to be 90◦ from the
orientation of the local Galactic magnetic field.
Dynamo theory: The mechanism by which a magnetic field is produced in which a rotating, convecting,
and electrically conducting fluid can generate and maintain a magnetic field over astronomical timescales.
The dynamo that creates the large-scale magnetic field acts by violation of the reflection symmetry of
the turbulence in rotating galaxies; this violation is associated, in turn, with dominance of, for example,
right-handed helical motions in the turbulent flow.
Eddies:
Eddy interaction rate:
Emission cross section per H nucleon (σe (ν)): Also called the “opacity” of the interstellar material,
the emission cross-section reflects the efficiency of thermal dust emission per unit mass. It is related to
the mass absorption/emission coefficient via
σe (ν)µmH κν [cm2 ].

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Emission measure (EM): The square of the number density of free electrons integrated over the volume
of plasma.
observer
Z

~ [pc cm−6 ]
n2e d`

EM = −
source

Emissivity index (β): A spectral index related to the mass absorption coefficient via:
 β
ν
κν = κ0
[cm2 g−1 ].
ν0
The local ISM SED can be fit well with β = 1.8 for the diffuse medium.
Energy cascade:
Energy spectrum (E(k)): The term energy refers to any squared quantity, not necessarily velocity.
Epicycles: Small oscillations that Galactic disk stars experience about a perfectly circular orbit in the
Galactic plane due to their velocity dispersion of ∼ 10 − 40 km s−1 about their ' 220 km s−1 rotational
velocity.
External depolarization: Depolarization induced by the limitations of the instrumental capabilities.
For example, beamwidth depolarization is due to fluctuations in the foreground screen within the observing
beam: unresolved density or magnetic field inhomogeneities of the media through which the radiation
propagates induces unresolved spatial variations in the Faraday rotation measure. Another form of
external depolarization is bandwidth depolarization which can occur when a signification rotation of the
polarization angle is produced across the observing bandwidth.
Faraday depth (φ): The Faraday depth of a source is defined as
observer
Z

~ [rad m−2 ],
~ · dr
ne B

φ(r) = −0.81
source

~ is an infinitesimal
~ is the magnetic field strength in µG, and dr
where ne is the electron density in cm , B
path length in pc. The negative sign sets the convention that φ is positive for a B direction pointing
towards the observer. Most compact sources like pulsars and extremely compact extragalactic sources
show a single value of φ, called the rotation measure (RM). The Faraday depth (or RM) does not increase
monotonically with distance along the line of sight.
Faraday dispersion: A type of internal depolarization caused by emission at different Faraday depths
along the same line of sight.
Faraday dispersion function (F (φ)): Also referred to as the Faraday spectrum introduced by Burn
(1966) which describes the complex polarization vector as a function of Faraday depth as
−3

Z∞
F (φ) =

2

P (λ2 )e−2iφλ dλ2 [rad m−2 ],

∞

where F (φ) is the complex polarized surface brightness per unit Faraday depth and P (λ2 ) = p(λ2 )I(λ2 )
is the complex polarized surface brightness. To obtain the Faraday dispersion function, rotation measure
(RM) synthesis is used to Fourier transform the observed polarized surface brightness into the Faraday
spectrum. This Faraday dispersion function is not straightforward to interpret; in particular, there is no
direct relationship between Faraday depth and physical depth. Further, the Faraday dispersion function
suffers from sidelobes of the main components caused by limited coverage of the observed wavelength
space. Burn (1966) assumes that F (φ) is independent of frequency. The equation for P (λ2 ) is very similar
to a Fourier transform; a fundamental difference is that P (λ2 ) only has physical meaning for λ ≥ 0. Since
P (λ2 ) cannot be measured for λ < 0, it is only invertible if one makes assumptions about the values of
P for λ2 < 0 based on those for λ2 ≥ 0 (Burn 1966). For example, assuming that P (λ2 ) is Hermitian
corresponds to assuming that F (φ) is strictly real.
Faraday rotation: A frequency-dependent magneto-optical phenomenon (i.e., an interaction between
light and a magnetic field) in which the plane of polarization is rotated by an amount that is linearly
proportional to the strength of the magnetic field in the direction of propagation. The Faraday effect is
very useful in studies of galactic magnetism because it allows one to determine not only the strength but
also the direction of a magnetic field. A linearly polarized electromagnetic wave propagating along the
magnetic field of an ionized medium can be decomposed into two circularly polarized modes: a right~ vector rotates about the magnetic field in the same sense as the free electrons
hand mode, whose E
~ vector rotates in the opposite direction. As a result
gyrate around it, and a left-hand mode, whose E
~
of the interaction between the E vector and the free electrons, the right-hand mode travels faster than

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the left-hand mode; consequently the plane of linear polarization experiences Faraday rotation as the
wave propagates. Faraday rotation results from the fact that right-handed circularly (RHC) and lefthanded circularly (LHC) polarized light experience different phase velocities when traveling through a
magnetoionic medium. Faraday rotation is induced by thermal electrons (not synchrotron radiation)
coincident with a magnetic field which is at least partially oriented along the line of sight between the
source and the observer. Non-thermal electrons have linear (not circular) modes, and produce generalized
Faraday rotation. The Faraday rotation is used to measure the parallel component of the magnetic field
via ionized gas. Given an initial polarization angle χ0 and an observing frequency λ2 , the observed
polarization angle is a function of the Faraday depth φ of the medium at a given physical distance r given
by
χ = χ0 + φ(r)λ2 [rad].
When a straight line is fit to this equation to provide a single value for φ(r), this is traditionally called the
rotation measure (RM). This would only be a valid approximation for the simplest of cases, for example, a
single foreground medium inducing Faraday rotation that is itself not emitting its own polarized emission.
In more complex scenarios, polarized synchrotron emission may originate from volumes that are also
inducing Faraday rotation which leads to polarized synchrotron emission at a range of Faraday depths.
In such situations, rotation measure synthesis (RM-synthesis) must be done to obtain the Faraday depth
as a function of physical distance, φ(r). The Faraday depth φ(r) is a proportionality constant that
encapsulates the physics of the situation:
observer
Z

~ · ~r [rad m−2 ].
ne B

φ(r) = −0.81
source

Additional complications may arise if the Faraday depth is large enough to substantially rotate the
polarization vector within the observing bandwidth, an effect known as bandwidth depolarization.
Faraday spectrum: See Faraday dispersion function.
Faraday screen: A background emitting source of radiation which experiences Faraday rotation by a
foreground source. In this case, and Faraday depth are identical; otherwise Rotation Measure Synthesis
is needed.
Faraday thick: A source is Faraday thick if the wavelength squared times the extent of the object in
units of Faraday depth is much greater than 1: λ2 ∆φ  1. In the case of Faraday thick, objects are
extended in Faraday space and substantially depolarized at wavelength squared. Remember that whether
an object is Faraday thin or Faraday thick is wavelength dependent.
Faraday thin: A source is Faraday thin if the wavelength squared times the extent of the object in
units of Faraday depth is much less than 1: λ2 ∆φ  1. In the case of Faraday thin, objects are well
approximated by a Dirac-delta function in Faraday space. Remember that whether an object is Faraday
thin or Faraday thick is wavelength dependent.
Filling factor: See volume filling factor.
First moment: See mean.
Fluid turbulence:
Flux freezing: The coupling between the cold neutral medium (CNM) and the magnetic field due to its
non-zero ionization fraction.
Forbidden line: Also known as a forbidden mechanism or a forbidden transition. A spectral line
associated with the emission or absorption of light by atomic nuclei, atoms, or molecules which undergo a
transition that is not allowed by a particular selection rule but does occur if the approximation associated
with that rule is not made. Forbidden emission lines have been observed in extremely low-density gas
and plasma in which collisions are infrequent. Under such conditions, once an atom or molecule has been
excited for any reason into a meta-stable state, it is almost certain to decay by emission of a forbiddenline photon. Since meta-stable states are rather common, forbidden transitions account for a significant
percentage of the photons emitted by the ultra-low density gas in space.
Fourier transform (fˆ(x)): Decomposes a function of time (a signal) into the frequencies that make it
up. The Fourier transform (FT) of a function f (x) is given by
fˆ(x) =

Z∞

0

f (x0 )e−2πixx dx0 .

−∞

The reason for the negative sign convention in the definition of fˆ(x) is that the integral produces the
0
amplitude and phase of the function f (x0 )e−2πxx at frequency zero (0), which is identical to the amplitude
and phase of the function f (x0 ) at frequency x0 , which is what fˆ(x) is supposed to represent. The function

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f (x) can be reconstructed from its Fourier transform fˆ(x), which is known as the Fourier inverse theorem.
Fourth moment: See kurtosis.
Fractional polarization (p(α)):
Galactic magnetic field (GMF): Often the GMF is expressed in cylindrical coordinates (θ, r, z). The
~ θ ) dominates the Galactic disk where the radial (B
~ r ) and vertical (B
~ z ) comazimuthal magnetic field (B
~
ponents are generally weak. The toroidal magnetic field refers to the structures (without Bz components)
confined to a plane parallel to the Galactic plane, while the poloidal magnetic field refers to the axisym~ θ components) such as dipole fields. These terms are artificially
metric field structure around z (without B
designed for convenience when studying magnetic fields. Real magnetic fields would be all connected in
space, with all components everywhere.
The RM sky is dominated by the GMF, with negligible contributions from the intergalactic medium and
the RM intrinsic to the radio source (whether it be extragalactic radio sources or pulsars). A prominent
feature of the RM sky is the antisymmetry in the inner Galactic quadrants (|`| < 90◦ ). The positive RMs
in the regions of (0◦ < ` < 90◦ , b > 0◦ ) and (270◦ < ` < 360◦ , b < 0◦ ) indicate that the magnetic fields
point away from us. Such a high symmetry to the Galactic plane as the Galactic meridian through the
Galactic center cannot simply be caused by localized features as previously thought. The antisymmetric
pattern is very consistent with the magnetic field configuration of an A0 dynamo, which provides such
toroidal fields with reversed directions above and below the Galactic plane. The toroidal fields possibly
extend to the inner Galaxy, even towards the central molecular zone (CMZ). This magnetic field model is
also supported by the non-thermal radio filaments observed in the Galactic center region for a long time,
which have been thought to be indications for the poloidal field in dipole form. The antisymmetric RM
sky is also shown by pulsar RMs at high Galactic latitudes (|b| > 8◦ ). This implies that the magnetic
field responsible for the antisymmetry pattern could be nearer than the pulsars.
Generalized Faraday rotation: In a medium whose natural modes are linearly or elliptically polarized,
the counterpart of Faraday rotation, referred to as “generalized Faraday rotation”, can lead to a partial
conversion of linear into circular polarization.
Great circle: A circle on the surface of a sphere that lies in a plane passing through the sphere’s center
which represents the shortest distance between any two points on the surface of a sphere.
Gyration frequency (ω): The angular frequency of circular motion.
Hot ionized medium (HIM): Diffuse interstellar gas with a typical temperature of T ∼ 105 − 106 K.
This hot interstellar gas is believed to have been generated mainly by supernovae and stellar winds from
massive stars, forming as the shock wave sweeps through the interstellar medium.
Hydrogen spectral series: Six named Hydrogen line series describing the emission spectrum of Hydrogen as dictated by the Rydberg equation. Includes the Lyman series (n0 = 1), Balmer series (n0 = 2),
Paschen (or Bohr) series (n0 = 3), Brackett series (n0 = 4), Pfund series (n0 = 5), Humphreys series
(n0 = 6).
HI fibers: Clark et al. (2014,2015) identified slender, linear HI features called “fibers” in the Galactic
Arecibo L-Band Feed Array HI (GALFA-HI) survey data. By developing a new machine vision transformation technique named the Rolling Hough Transform (RHT), they identified the fibers and found that
they are oriented along the interstellar magnetic fields probed by both starlight and dust polarization.
They also showed that angular dispersion of the fibers can be used to measure the magnetic field strength
through the Chandrasekhar-Fermi method. Based on the observed properties such as column density and
line width, the HI fibers are suggested to be the CNM with density ∼ 10 cm−3 , temperature ∼ 200 K,
and width ∼ 0.1 pc and are embedded in a shell of the local bubble. From the theoretical point of view,
neither the origin of the HI fibers nor the mechanism of the alignment of the fibers and magnetic field is
yet understood.
t is known that when gas condensation is triggered by the thermal instability from a static thermally
unstable medium, a CNM is formed that is flattened perpendicular to the magnetic field because magnetic
pressure prevents the gas condensation except in the direction along the local magnetic field. Using threedimensional MHD simulations, Inoue & Inutsuka (2016) showed that shock sweeping of the magnetized
WNM via supernova explosions creates thermally unstable gas in which fragmented HI clouds are formed
as a consequence of the thermal instability. First, the shock compression creates a thermally unstable gas
slab in which magnetic pressure balances upstream ram pressure. Then, the thermal instability develops
to create the CNM in a cooling timescale of ∼ 1 Myr. Because the timescale of the thermal instability
that enhances gas density is governed by the cooling timescale, the thermal pressure is decreased down
to the initial upstream level by the time of the CMN formation. This leads to the formation of a CNM
with density n . 100 cm−3 . The CMN is formed via the thermal instability that drives runaway gas
condensation along the local magnetic field. This is why the CNM clumps basically have a flattened
shape. If the initial magnetic field is in the same direction as the shock propagation, or if the magnetic
field is neglected, the CNM forms at much higher densities.

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HII region: The ionized clouds around massive OB stars which are responsible for ionizing most of the
hydrogen around such stars. The boundaries of HII regions are determined by the volume in which the
rate of UV photoionization equals the rate of recombination of electrons. When free electrons within an
HII region pass near a positive ion (H+ , He+ ) they are accelerated by the Coulomb field and emit radiation
known as “free-free” or bremsstrahlung emission which is a source of polarized continuum emission. The
average electron density of an HII region is ∼ 103 cm−3 .
Ideal magnetohydrodynamics (MHD): Ideal MHD implies that fluid particles are attached to their
field lines, that is to say they can flow along the field lines but cannot go across them. In a turbulent
fluid, given the stochastic nature of the motions, such a situation would lead to a field that would be so
tangled, that motions would quickly become prohibited. This implies that ideal MHD cannot, strictly
speaking, be correct for a turbulent fluid and that some reconnection (i.e., some changes of the field lines
topology) must be occurring. The physical origin of this reconnection is still debated but an appealing
model has been proposed by Lazarian & Vishniac (1999). In this view the reconnection is driven by
turbulence and is a multi-scale process, that is unrelated to the details of the microphysical processes. It
is certainly the case, at least in numerical simulations of MHD turbulence, where the numerical diffusivity
is often controlling the reconnection, that the MHD is far to be ideal. This process in particular induces
an effective diffusion of the magnetic flux, that is therefore not not fully frozen as one would expect if
MHD was truly ideal.
Incompressible turbulence: The Kolmogorov power spectrum for incompressible turbulence in three
dimensions is P (k) ∝ k −11/3 while its energy spectrum is E(k) ∝ k −5/3 . In two dimensions, P (k) ∝ k −8/3
and E(k) ∝ k −5/3 (unchanged). For one dimension, P (k) ∝ k −5/3 and E(k) ∝ k −5/3 (unchanged).
Induction equation: Assuming a two-fluid description of a plasma with massless electrons, the magnetic
field evolution is given by the generalized induction equation
2
~
∂B
~ + ηc ∇2 B
~ − 1 ∇ × (~j × B)
~ − c ∇n × ∇Te ,
= ∇ × (~v × B)
∂t
4π
en
ne
~
which shows the evolution of magnetic field B, based on the fluid velocity ~v , the current density ~j =
~
c∇ × B/4π,
the number density n, and the electron temperature Te = Pe /n, where Pe is the electron
plasma pressure. Here, c is the speed of light, η is the resistivity, and e is the charge of an electron. The
terms on the RHS from left to right are the convective term, the resistive term, the Hall term, and the
Biermann battery term. The induction equation is often simplified by assuming that the system size L
is large compared to all kinetic scales, and only considering the convective term on the RHS.
Infrared polarization: The same large-scale alignment of aspherical, spinning dust grains that causes
starlight polarization also causes polarization of far-infrared emission.
Intensity of thermal dust emission (Iν ): The intensity of the thermal dust emission, when optically
thin, is given by

Iν = τν Bν (T ) [W m−2 Hz−1 sr−1 ]
≡ σe (ν)NH Bν (T ) [W m−2 Hz−1 sr−1 ]
= rµmH NH κν Bν (T ) [W m−2 Hz−1 sr−1 ],
where τν is the dust optical depth at frequency ν, Bν (T ) is the Planck function, σe (ν) is the emission
cross section per H nucleon (or “opacity”), NH is the total hydrogen column density (H in any form), r
is the dust-to-gas mass ratio, µ is the mean molecular weight of the gas, mH is the hydrogen mass, and
κν is the dust mass (or emission) coefficient (or “opacity”).
Intergalactic medium (IGM):
Internal depolarization: Depolarization due to the spatial extent of the source and occurs even if
the intervening media are completely homogenous. Along the line of sight, the emission from individual
electrons within a source arrive from different depths and suffer different Faraday rotation angles due
to different path lengths. For the total radiation emitted by a source, this results in a reduction of the
observed degree of polarization.
Interstellar medium (ISM): A tenuous medium throughout galaxies that consists of three basic constituents: (1) ordinary matter, (2) relativistic charged particles called cosmic rays, and (3) magnetic
fields. These three basic constituents have comparable pressures and are bound together by electromagnetic forces. The ordinary matter itself consists of gas (atoms, molecules, ions, and electrons) and dust
(tiny solid particles) which can exist in a number of phases: molecular, cold atomic, warm atomic, warm
ionized, and hot ionized. Apart from the densest parts of molecular clouds whose degree of ionization
is exceedingly low, virtually all interstellar regions are sufficiently ionized for their neutral component
to remain tightly coupled to the charged component and hence to the local magnetic field. Cosmic rays
and magnetic fields influence both the dynamics of the ordinary matter and its spatial distribution at all
scales, providing, in particular, an efficient support mechanism against gravity. Conversely, the weight
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of the ordinary (i.e., baryonic) matter confines magnetic fields and, hence, cosmic rays to the Galaxy,
while its turbulent motion can be held responsible for the amplification of magnetic fields and for the
acceleration of cosmic rays. Studies of the diffuse (n ∼ 0.1−100 cm−3 ) HI suggests that the magnetic field
strength is relatively independent of its volume density, in contrast to magnetic fields in molecular clouds.
The Galactic origin of the most energetic cosmic rays and their widespread distribution throughout the
Milky Way was not recognized until the observed Galactic radio emission was correctly identified with
synchrotron radiation emitted by cosmic-ray electrons gyrating about the local Galactic magnetic field.
The ISM encloses but a small fraction of the total mass of the Galaxy. Moreover, it does not shine in the
sky as visibly as stars do, yet it plays a vital role in many of the physical and chemical processes taking
place in the Galactic ecosystem. The ISM is not merely a passive substrate within which stars evolve;
it constitutes their direct partner in the Galactic ecosystem, continually exchanging matter and energy
with them and controlling many of their properties. It is the spatial distribution of the ISM together
with its thermal and chemical characteristics that determines where new stars form as well as their mass
and luminosity spectra. These in turn govern the overall structure, optical appearance, and large-scale
dynamics of our Galaxy. Hence understanding the present-day properties of our Galaxy and being able to
predict its long-term evolution requires a good knowledge of the dynamics, energetics, and chemistry of
the ISM. Table 1 outlines the different phases of the interstellar gas, including their typical temperature,
number density, mass density, and total mass.
The MW is not a closed system. The evolution of the MW is significantly impacted by the two-way flow of
gas and energy between the Galactic disk, halo, and IGM. We have long known that the atomic hydrogen
halo extends far beyond the disk of the Galaxy. In recent years we have come to realize that the halo is
also a highly structured and dynamic component of the Galaxy. Although we can now detect hundreds
of clumped clouds in the atomic medium of the halo, we are far from understanding the halo?s origin
and its interaction with the disk of the Galaxy. It seems that a significant fraction of the structure of
gas in the Galactic halo may be attributed to the outflow of structures formed in the disk, but extending
into the halo. An example of such a structure may be an HI supershell. There are several examples
of HI supershells that have grown large enough to effectively outgrow the Galactic HI disk. When this
happens, the rapidly decreasing density of the Galactic halo does not provide sufficient resistance to the
shell’s expansion and it will expand unimpeded into the Galactic halo, creating a chimney from disk to
halo. These chimneys supply hot, metal-enriched gas to the Galactic halo and may act as a mechanism
for spreading metals across the disk. It has been theorized that HI chimneys in the disk of the Galaxy
may be a dominant source of structure for the halo through a Galactic Fountain model. Some Fountain
models predict that cold cloudlets should develop out of the hot gas expelled by chimneys on timescales
of tens of millions of years. Other Fountain theories suggest that the cool caps of an HI supershell will
extend to large heights above the Galactic plane before they break. Once broken, the remains of the
shell caps could be an alternate source of small clouds for the lower halo. Recent observational work
has placed this theory on firmer footing, showing that the clumped clouds that populate the lower halo
are not only more prevalent in regions of the Galaxy showing massive star formation than in less active
regions, but that they extend higher into the halo in these regions.

Table 1: Descriptive parameters of the different components of the interstellar gas. T is the temperature, n is the true (as
opposed to space-averaged) number density of hydrogen nuclei near the Sun, Σ is the azimuthally averaged mass density
per unit area at the solar circle, and M is the mass contained in the entire Milky Way. Both Σ and M include 70.4%
hydrogen, 28.1% helium, and 1.5% heavier elements. All values were rescaled to R = 8.5 kpc Table taken from Ferriére
(2001).
Inverse Compton scattering: The upscattering of radio photons to become optical or X-ray photons
by means of the inelastic scattering of a charged particle (usually an electron).
Inverse Fourier transform (IFT): The reconstruction of a function from its decomposition into the
frequencies that make it up. The inverse Fourier transform (IFT) is given by
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Z∞
f (x) =

0
fˆ(x0 )e2πixx dx0 .

−∞

The fact that a function f (x) can be reconstructed from its Fourier transform fˆ(x) is known as the
Fourier inverse theorem.
Ion-neutral drift: See ambipolar diffusion.
Ion-neutral friction: See ambipolar diffusion.
Kelvin-Helmholtz instability:
Kolmogorov microscale:
Kolmogorov spectrum:
Kurtosis: The fourth order statistical moment which is a measure of whether the data are heavy-tailed
or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy
tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. Kurtosis is
defined as
3
N 
1 X xi − µ
[dimensionless],
Kurt[x] =
N i=1
σ
where µ is the statistical mean and σ is the standard deviation. Like the skewness, it is a dimensionless
quantity.
Large scale magnetic field: How large is the “large” scale? Obviously it should be a scale, relatively
much larger than some kind of standard. For example, the large-scale magnetic field of the Sun refers
to the global scale field or the field with a scale length comparable to the size of the Sun, up to 109 m,
rather than small-scale magnetic fields on the Solar surface. For the magnetic fields of our Galaxy, we
should define the “large” scale as being a scale larger than the separation between spiral arms. That is to
say, large scale means a scale larger than 2 − 3 kpc. Note that in the literature, “large” scale is sometimes
used for large angular scale when discussing the structures or prominent plane-of-the-sky features (e.g.,
the large-scale features in radio continuum surveys). These large angular scale features are often very
localized phenomena, and not very large in linear scale.
Lorentz factor (γ): The factor by which time, length, and relativistic mass change for an object while
that object is moving with speed v:
r
1 − (v/c)2
.
γ=
c2
For non-relativistic motion γ ≈ 1, while for relativistic motion γ > 1. For example, γ(v = 0.9c) ≈ 2 and
γ(v = 0.99c) ≈ 7.
Lorentz force: The combination of electric and magnetic force on a point charge due to electromagnetic
fields. A particle of mass m and charge q moving with a velocity v within a magnetic field B experiences
a force


d
~v
~
~
~
F =
(mγ~v ) = q E + × B [N],
dτ
c
where τ is the retarded time and γ is the Lorentz factor. Of course, the Lorentz force only acts on charged
particles but its effect is then transmitted to neutral particles via ion-neutral collisions.
Luminosity per H atom (L): A key quantity is the luminosity per H atom emitted by dust grains
(equal to the absorbed power) computed by integrating the SED over ν:
Z
L = 4πκ0 (ν/ν0 )β µmH Bν (T ) dν [W H−1 ].
Mach number (M):
Magnetic buoyancy: See Rayleigh-Taylor instability.
Magnetic reconnection:
Magnetohydrodynamic (MHD) equations: The equations of ideal MHD assume that the fluids are
~ and B
~ exert on the fluid
perfect conductors. The Lorentz force, which is the force that the EM fields E
must be taken into account. The EM field evolution is described by Maxwell equations. Written in CGS
units, these equations are

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~ ·B
~ =0
∇
~ ·E
~ = 4πρe
∇
~
~ ×E
~ = − ∂B
c∇
∂t
~ ×B
~ = 4π~j +
c∇

~
∂E
,
∂t

where ρe and ~j are the fluid charge and current densities. The equation for charge conservation links
these two quantities:
∂ρe ~ ~
+ ∇ · j = 0.
∂t
Magnetohydrodynamics (MHD): Magnetohydrodynamics (MHD) denotes the study of the dynamics
of electrically conducting fluids. It establishes a coupling between the Navier-Stokes equations for fluid
dynamics and Maxwell’s equations for electromagnetism. The main concept behind MHD is that magnetic
fields can induce currents in a moving conductive fluid, which in turn create forces on the fluid and
influence the magnetic field itself.
Magnetoionic medium (MIM):
Magnetorotational instability: See Balbus-Hawley instability.
Mass absorption coefficient (κν ): Also called the “opacity” of the interstellar material:
 β
ν
[cm2 g−1 ].
κν = κ0
ν0
It is related to the mass emission cross section per H nucleon via
σe (ν)
[cm2 g−1 ].
µmH
 β
ν
κν = κ0
[cm2 g−1 ]
ν0
κν =

Mean molecular weight (µ):
Meridian: A circle of constant longitude passing through a given place on Earth’s surface and the
terrestrial poles.
Microturbulence:
Milky Way Galaxy (MWG): We see the Milky Way as a narrow band encircling us because the Galaxy
has the shape of a flattened disk within which we are deeply embedded. Our Galaxy comprises a thin disk
with radius ∼ 25 − 30 kpc and effective thickness ∼ 400 − 600 pc, plus a spherical system itself composed
of a bulge with radius ∼ 2 − 3 kpc and a halo extending out to more than 30 kpc from the center. The Sun
resides in the Galactic disk, approximately 15 pc from the midplane and 8.5 kpc away from the center.
The stars belonging to the disk rotate around the Galactic center in nearly circular orbits. Their angular
rotation rate is a decreasing function of their radial distance. At the Sun’s orbital distance, the Galactic
rotation velocity is ' 220 km s−1 , corresponding to a rotational period of ' 240×106 years. Disk stars also
have a velocity dispersion of ∼ 10 − 40 km s−1 which causes them to experience small oscillations about
a perfectly circular orbit both in the Galactic plane (epicycles) and the vertical direction. In contrast,
stars in the bulge and the halo rotate slowly and often have very eccentric orbits. Radio observations
of interstellar neutral hydrogen indicate that the Milky Way possesses a spiral structure similar to those
seen in optical wavelengths of external galaxies. The exact spiral structure of our own Galaxy is difficult
to determine from within; the best radio data to date points to a structure characterized by a bulge of
intermediate size and a moderate winding of the spiral arms. Infrared (IR) images of the Galactic center
clearly display the distinctive signature of a bar. Our position in the spiral pattern can be derived from
local optical measurements which give quite an accurate outline of the three closest arms; they locate
the Sun between the inner Sagittarius arm and the outer Perseus arm, near the inner edge of the local
Orion-Cygnus arm.
Molecular medium (MM): Molecular gas is explored primarily with the 2.6 mm and other radio
emission lines of the second most abundant molecule, CO, and secondarily with radio emission and
absorption lines of less abundant molecules, such as OH, H2 O, and NH3 . Historically, molecular lines
were seen mainly in emission towards the standard dense molecular clouds, and CO was emphasized
to the extent that its presence defined molecular gas. While Spatially, the molecular gas is confined
to discrete clouds, which are roundish, gravitationally bound, and organized hierarchically from large
complexes (size ∼ 20 − 80 pc and mass ∼ 105 − 106 M ) down to small clumps (size . 0.5 pc, mass
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. 103 M ). Along the vertical direction, the molecular gas is the most strongly confined to the Galactic
plane, with a HWHM thickness near the Sun ∼ 70 − 80 pc. Horizontally, all the gas components tend
to concentrate along the spiral arms, this tendency being probably most pronounced for the molecular
gas. Upon averaging along Galactic circles, through spiral arms and interarm regions, it is found that
most of the molecular gas resides in a ring extending radially between 3.5 − 7 kpc from the Galactic
center. It seems almost certain that the DMM is a transition state between the CNM and classical
MM. Moreover, we expect the details of the DMM transition region to depend not only on physical
conditions but also cloud morphology as it determines whether UV photons can penetrate to destroy
molecules via photodissociation or photoionization. It seems very unlikely that one can understand the
transition between atomic and molecular gas without understanding the effect of UV photons, and thus
cloud morphology. In addition, there are hints that cloud morphology is affected by the magnetic field;
after all, magnetic forces are one of the important forces on the ISM (the others being turbulent pressure,
cosmic ray pressure (coupled to the gas by the magnetic field), thermal pressure, and gravity).
nπ ambiguity: The inability to distinguish between polarization angles modulo π radians, rendering
traditional linear RM fits often arbitrary. One way to deal with this ambiguity is to rely on resolving
smooth spatial gradients in the polarization angle at each wavelength. With this assumption, the appropriate value of n can be resolved for each spatial pixel, yielding the correct polarization angle at each
wavelength and thus the true value of RM. This is the basis of the PACERMAN routine developed by
Dolag et al., (2005). However, routines like PACERMAN cannot deal with the second and third problems
listed above since it is ultimately based on fitting a single value of RM along each line of sight.
Open cluster: A rather loose, irregular grouping of 102 − 103 stars confined to the Galactic disk and
therefore also known as Galactic clusters.
Optical depth (τν ):
Ordered magnetic field: A.k.a. “coherent magnetic field”. Whether a magnetic field is ordered or
random depends on the scales concerned. A uniform field at a 1 kpc scale could be part of random fields
on a 10 kpc scale, while it is of a very large scale relative to the pc scale magnetic fields in molecular
clouds. Uniform fields are ordered fields. Regularly ordered fields can coherently change their directions,
so they may not be uniform fields.
Paramagnetism: Paramagnetic materials have unpaired electrons which, when in the presence of an
external magnetic field such as the interstellar magnetic field, align in the same direction causing grain
alignment.
Parker instability:
Partially ionized plasma (PIP): It is assumed that a PIP is composed of multiple kind of particles
such as electrons, ions that can have different ionization states, neutral particles with zero ionization, as
well as dust grains, positively or negatively charged. The concept of a fluid can be applied separately to
each of these components in all the environments of interest. Therefore the behavior of such a plasma
can be described by a set of equations of mass, momentum, and energy conservation for each of the
components.
Passive mixing:
Pitch angle: The angle between a charged particle’s velocity vector and the local magnetic field.
Photodissociation region (PDR): Also known as photon-dominated regions or PDRs. Predominantly
neutral regions of the ISM in which UV photons strongly influence the gas chemistry and act as the
dominant energy source. They occur in any region of interstellar gas that is dense and cold enough to
remain neutral, but that has too low of a column density to prevent penetration of far-UV photons from
massive stars. They are also associated with HII regions. All of the atomic gas and most of the molecular
gas is found in photodissociation regions.
Photon-dominated region (PDR): See photodissociation region.
Pitch angle: The angle between a charged particle’s velocity vector and the local magnetic field.
Planck function (Bν (T ) or Bλ (T )): The spectral radiance of an object at a given temperature as a
function of frequency or wavelength:


1
2hν 3
[W m−2 Hz−1 sr−1 ]
Bν (T ) =
2
hν/k
T −1
B
c
e


2hc2
1
Bλ (T ) =
[W m−2 m−1 sr−1 ].
5
hc/λk
BT − 1
λ
e
Plasma beta (β): The ratio of the gas to magnetic pressure. For either high or low β, it was predicted
that Alfvén waves should should damp at the ambipolar diffusion scale LAD and thus the magnetohydrodynamic (MHD) cascade should damp past the ambipolar diffusion scale.
Poincaré sphere:
Polarized intensity (P ): A measure of the total linear polarization in radio emission given as

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P ≡

p
Q2 + U 2 [Jy],

where Q is the Stokes Q polarization and U the Stokes U polarization. The images of P (as well as Q
and U ) are often filled with complex structures that bear little resemblance to the Stokes I image of total
intensity. The intensity variations seen in P (as well as Q and U ) are the result of small-scale angular
structure in the Faraday rotation induced by ionized gas, and are thus an indirect representation of
turbulent fluctuations in the free-electron density and magnetic field throughout the interstellar medium.
while Q and U exhibit Gaussian noise properties, this is not true for P . The noise in Q and U is squared
when calculating P , having a Ricean distribution, and this causes the observed polarization intensity to
be biased toward larger values
Polarization angle (χ): Given by
 
U
1
[rad],
χ ≡ arctan
2
Q
where Q is the Stokes Q polarization vector and U the Stokes U polarization vector. The arctan2
function is generally used to determine χ over the full ±nπ range. The polarization angle (likewise with
the amplitude of polarized intensity) is not preserved under arbitrary rotationspand translations of the
Q-U plane. In the most general case, then, the observed values of χ (and P ≡ Q2 + U 2 ) do not have
any physical significance; only measurements of quantities that are both rotationally and translationally
invariant in the Q − U plane can provide insight into the physical conditions that produce the observed
polarization distribution.
Polarization fraction (p(λ2 )):
p(λ2 ) =

P (λ2 )
[dimensionless],
I(λ2 )

where P (λ2 ) is the polarized surface brightness and I(λ2 ) is the total surface brightness.
p
~ P~ |): The rate at which the polarized intensity complex vector P~ ≡ Q2 + U 2
Polarization gradient (|∇
traces out a trajectory in the Q-U plane as a function of position on the sky given by
s
2 
2 
2 
2
∂Q
∂U
∂Q
∂U
~ P~ | =
|∇
+
+
+
[Jy beam−1 ],
∂x
∂x
∂y
∂y
or
v
v"
u
u 
u
2 
2 
2 
2 # 2
u
u
∂Q
∂U
∂U
∂U
u
u "
2 
2 
2 
2 #
+
+
+
u
u1
∂x
∂x
∂x
∂y
∂Q
∂U
∂U
∂U
1u
∂ P~
u
=u
+
+
+
+ u


u
∂s
∂x
∂x
∂x
∂y
2t
2
u2
∂Q ∂U
∂Q ∂U
max
t
−4
−
,
∂x ∂y
∂y ∂x
where Q and U are the complex Stokes vectors and x and y are the Cartesian
axes of the image plane.
p
Note that |∇P~ | cannot be constructed from the scalar quantity P ≡ Q2 + U 2 , but is derived from
~ P~ | provides an image of
the vector field P~ ≡ (Q, U ). The amplitude of the polarization gradient |∇
magnetized turbulence in diffuse, ionized gas manifested as a complex filamentary web of discontinuities
in gas density and magnetic field strength. This quantity is rotationally and translationally invariant
in the Q − U plane, and so has the potential to reveal properties of the polarized distribution that
might otherwise be hidden by excess foreground emission or Faraday rotation, or in data sets from which
large-scale structure is missing. The polarization gradient is shown in Figure 1.
Using polarization gradients of isothermal MHD turbulence, the supersonic case (Figure 2c) was found
to show localized groupings of very high-gradient filaments, corresponding to ensembles of intersecting
shocks. By contrast, the subsonic (Figure 2a) and tran-sonic (Figure 2b) cases showed more diffuse
networks of filaments, representing the cusps and discontinuities characteristic of any turbulent velocity
field.
In the ISM, fluctuations in density and magnetic field will occur as a result of MHD turbulence, which will
be visible in polarimetric maps. In the case of taking gradients of a turbulent field, one would expect to
find filamentary structure created by shock fronts, jumps, and discontinuities. Figure 3 shows a schematic
illustrating these three separate cases of a possible profile and its respective derivative. The cases are as
follows:
• A Hölder continuous profile that is not differentiable at a given point (e.g., the absolute value
function at the origin): common for all types of MHD turbulence. It is known that the turbulent

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~P
~ | for an 18-deg2 region of the Southern Galactic Plane Survey which reveals a complex network of tangled
Figure 1: |∇
~ | is high consists of elongated, narrow structures rather than extended
filaments. In particular, all regions in which |∇P
~ | changes
patches. In the inset, the direction of |∇P| is shown for a small subregion of the image, demonstrating that |∇P
most rapidly along directions oriented perpendicular to the filaments. Figure taken from Gaensler (2011).

~P
~ | derived from propagation of linear radio polarization through three different isothermal simulations of
Figure 2: |∇
magnetized turbulence. Figure taken from Gaensler (2011).

velocity field in a Kolmogorov-type inertial range both in hydro and MHD is not differentiable, but
only Hölder continuous. This case can be found in both subsonic- and supersonic turbulence.
• A jump profile: weak shocks, strong fluctuations, or edges (e.g., a cloud in the foreground which
suddenly stops). This case creates a structure in the gradient by a shock jump or a large fluctuation
~ Here again, this type of enhancement in |∇P~ | could be found in supersonicin either ne or B.
and subsonic turbulence, and is due either to large random spatial increases or decreases due to
turbulent fluctuations along the LOS or weak shocks. We expect weak shock turbulence to show a
larger amplitude in |∇P~ | than the subsonic case due to increases in density fluctuations.
• A spike profile (e.g., delta function): strong shock regime. This case is unique to supersonic
~ across a shock front. The
turbulence in that it represents a very sharp spike in ne and/or B
difference between this case and what might be seen in case two is that here we are dealing with
interactions of strong shock fronts, which are known to create delta function-like distributions in
density, creating a “double jump” profile across the shock front.
Of great interest is the question of which quantity is providing the dominant contribution to the structures
~ LOS |, or both equally? Especially in the case of compressible turbulence, the
in |∇P~ |: |∇ne,LOS |, |∇B
magnetic energy is correlated with density: denser regions contain stronger magnetic fields due to the
compressibility of the gas and the potential dynamo amplification of the magnetic field in dense gas. This
causes the magnetic field to follow the flow of plasma if the magnetic tension is negligible. The compressed
regions are dense enough to distort the magnetic field lines, enhance the magnetic field intensity, and
effectively trap the magnetic energy due to the frozen-in condition. Thus, for the supersonic cases, the
intensity of the structures seen in |∇P~ | is more pronounced than in the subsonic case. However, in
the case of subsonic turbulence, there are no compressive motions. In this case, random fluctuations in

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Figure 3: Schematic example of three possible scenarios for enhancements in a generic image “n,” where “n” could be |P~ |,
RM, or ρ/N/EM (density, column density, emission measure). Case one (top row) shows an example of a Hölder continuous
function that is not differentiable at the origin (applicable to all turbulent fields). Case two (middle row) shows an example
of a jump resulting from strong turbulent fluctuations along the LOS or weak shocks. Case three (bottom row) shows a
delta function profile resulting from interactions of strong shocks. In this case, the derivative gives a double jump profile
~|
which produces morphology that is distinctly different from the previous cases. In all cases we show examples from |P
simulations. Figure taken from Burkhart (2012).
density and magnetic field will create structures in |∇P~ |. It’s been shown using MHD simulations that in
~ while in the supersonic case |∇P~ | correlates
the case of subsonic turbulence, |∇P~ | correlates with |∇B|
with density fluctuations. This is because density enhancements are dominant due to shock fronts in the
case of supersonic turbulence, while in subsonic turbulence density is marginally incompressible.
Polarization gradient (radial component): The radial component quantifies how changes in polarization intensity contribute to the directional derivative |∂ P~ /∂s|:
s
∂Q
∂U 2
∂U 2
(Q ∂Q
∂ P~
∂x + U ∂x ) + (Q ∂y + U ∂y )
=
[Jy pc−1 ]
∂s
Q2 + U 2
rad

If changes in polarization intensity are dominant for a feature, then this could imply that the amount
of depolarization due to the addition of polarization vectors along the line of sight varies significantly
between different positions, and it follows that the medium producing the polarized emission may be very
turbulent. This is true for both thermal dust emission and for synchrotron emission.
Polarization gradient (tangential component): The tangential component quantifies how changes
in polarization angle, weighted by polarization intensity, contribute to the directional derivative |∂ P~ /∂s|:
s
∂Q 2
∂Q 2
∂U
(Q ∂U
∂ P~
∂x − U ∂x ) + (Q ∂y − U ∂y )
=
[Jy pc−1 ]
∂s
Q2 + U 2
tan

If changes in polarization angle are dominant, then this could indicate changes in the regular magnetic
field threading the observed region, as this would produce significant changes in the emitted polarization
angle in the case of thermal dust emission or synchrotron emission. Additionally, changes in the regular
magnetic field may also cause the amount of Faraday rotation along different lines of sight to vary
significantly, in the case of synchrotron emission.
~ P~ ): The direction of the polarization gradient at a given spatial
Polarization gradient direction (arg∇
position defined as

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s



2 
+






∂Q
∂Q
∂U
∂U
~ P~ ) ≡ arctan sign
 [rad],
s
arg(∇
+





2
2
∂x ∂y
∂x ∂y


∂Q
∂U
+
∂x
∂x
∂Q
∂y

2



∂U
∂y

where Q and U are the complex Stokes vectors.
Polarization horizon: The furthest distance we can see diffuse polarized emission.
Polarized surface brightness (P (λ2 )):
P (λ2 ) = p(λ2 )I(λ2 ) = Q + iU [Jy],
where p(λ2 ) is the polarization fraction, I(λ) is the total surface brightness, and Q and U are the complex
Stokes vectors. The absolute value of this complex vector is given by
p
||P (λ2 )|| = Q2 + U 2 [Jy],
where Q and U are the complex Stokes vectors.
Poloidal magnetic field: Often the GMF is expressed in cylindrical coordinates (θ, r, z). The poloidal
~ θ components) such as
magnetic field refers to the axisymmetrical field structure around z (without B
dipole fields.
Power spectrum (P (k)): Describes the distribution of power into frequency components composing
that signal defined as
∗
P (k) = fˆ(k)fˆ(k) ,

for wavenumber k where fˆ(k) denotes the Fourier transform (FT) and fˆ(k) its complex conjugate. The
power spectrum is the Fourier transform (FT) of the autocorrelation function.
Prandtl number: The ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity:
µcp
ν
[dimensionless],
Pr = =
α
k
where ν = µ/ρ is the momentum diffusivity (kinematic viscosity) in m2 s−1 , α = k/(cp ρ) is the thermal
diffusivity in m2 s−1 , µ is the dynamic viscosity in Ps s, k is the thermal conductivity in W m−1 K, cp is
the specific heat in J kg−1 K, and ρ is the density in kg m−3 .
Q − U plane: Translations and rotations within the Q − U plane can result from one or more of a smooth
distribution of intervening polarized emission, a uniform screen of foreground Faraday rotation, and the
effects of missing large-scale structure in an interferometric data set.
Random magnetic field: See turbulent magnetic field.
Rayleigh-Taylor instability: Also known as magnetic buoyancy.
Reynolds number (Re): An important dimensionless quantity in fluid mechanics used to help predict
flow patterns in different fluid flow situations which characterizes the relative importance of inertial
(resistant to change or motion) and viscous (heavy and gluey) forces:
Re =

ρvL
[dimensionless],
µ

where ρ is the density in kg m−3 , v is the velocity in m s−1 , L is the characteristic length in m, and µ is the
dynamic viscosity coefficient in Ps s. At low Reynolds numbers, flows tend to be dominated by laminar
(sheet-like) flow, while at high Reynolds numbers, turbulence results from differences in the fluid’s speed
and direction, which may sometimes intersect or even move counter to the overall direction of the flow
(eddy currents).
Rolling Hough Transform (RHT): A machine vision algorithm designed for detecting and parameterizing linear structure in astronomical data, originally applied to HI images. The detection of astronomical
linear structure is approached in various ways depending on the context. Because HI structures are not
objects with distinct boundaries, the problem is fundamentally different from many others. As these diffuse HI fibers were not formed by gravitational forces, there is no reason to require that they must be, or
bridge, local overdensities. Indeed, these fibers are found often to be in groups of parallel structures, very
unlike the cosmic web. Thus, methods developed for gravitationally dominated systems are not optimal
for these purposes. The RHT is, as its name suggests, a modification of the Hough transform. The Hough
transform was first introduced in in a patent for the detection of complex patterns in bubble chamber
photographs. It was soon recognized as a powerful line detection technique, and has found wide applications in image processing and machine vision. The adaption of the Hough transform with the RHT is a
rolling version that is particularly well suited to the detection and quantization of specific linear features

Galactic Magnetism Study Guide

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Figure 4:

Flows at varying Reynolds number Re. In
each panel, a fluid that has
been dyed red is injected from
the top into the clear fluid
on the bottom.
The fluids are glycerin-water mixture,
for which the viscosity can be
changed by altering the glycerin to water ratio. By changing the viscosity and the injection speed, it is possible to
alter the Reynolds number of
the injected flow. The frames
show how the flow develops as
the Reynolds number is varied.
This image is a still from the
National Committee for Fluid
Mechanics Film series (Taylor, 1964), which, once you get
past the distinctly 1960s production values, are a wonderful
resource for everything related
to fluids.

in astronomical data. The RHT does not merely identify fibers; it encodes the probability that any given
image pixel is part of a coherent linear structure. This allows the user to quantify the linearity of regions
of sky without specifying fibers as discrete entities. The RHT operates on two-dimensional data and is
designed to be sensitive to linear structure irrespective of the overall brightness of the region. The first
step is to unsharp mask the image. The image is convolved with a two-dimensional top-hat smoothing
kernel of a user-defined diameter, DK . The smoothed data is then subtracted from the original data and
the resulting map is thresholded at 0 to obtain a bitmask. The subtraction of the smoothed component
can be considered a suppression of large-scale structure, or a high-pass Fourier filter. Each straight line is
parameterized in terms of the angle θ of its normal, and its minimum Euclidean distance from the origin
ρ,
ρ = x cos θ + y sin θ [pixels].
Every possible line in the image space is uniquely specified by a point in the ρ − θ space. The RHT
mapping is performed on a circular domain, diameter DW , centered on each image-space pixel (x0 , y0 )
in turn. Then a Hough transform is performed on this area, limited to ρ = 0. Thus the ρ − θ space is
reduced to a one-dimensional space on θ for each pixel. All intensity over a set intensity threshold Z
is stored as R(θ, x0 , y0 ): RHT intensity as a function of θ for that pixel. Z is a percentage. In every
direction θ, Z × DW pixels must contain signal in order for the transform to record the data in that
direction. We use the canonical binning for the number of theta bins:
#
" √
2
(DW − 1) [dimensionless].
nθ = π
2
By iterating (“rolling”) over the entire image space we produce the RHT output, R(θ, x, y). A visualization of the linear structures identified by the RHT, the backprojection R(x, y), is obtained by
integrating R(θ, x, y) over θ:
Z
R(x, y) = R(θ, x, y)dθ [dimensionless].
One advantage of the RHT is that the input parameters of the transform can be chosen to highlight
specific linear features of interest. One defines, for a given run of the RHT, a smoothing kernel diameter
(DK ), a window diameter (DW ), and an intensity threshold (Z). The rolling nature of the RHT ensures
that linear structure at least as long as DW will be identified. Thus DW , along with the Z, sets a lower
limit for the spatial length of the linear features. Thresholding below 100% (Z < 1) reflects the fact that
structures can be physically coherent even if they are not visibly connected. R(θ, x, y) is intensity as a
function of angle on a domain θ ∈ [0, π), as a 0◦ orientation is equivalent to a 180◦ orientation. R(θ, x, y)
can be sampled in a circular region around each star in the field as
Z Z
R∗ (x, y) =
R(θ, x, y)dxdy [dimensionless].
disk

To estimate the direction of a given region of the backrprojection R∗ (θ, x, y), the expectation value is
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given by
hθi0 =

R

sin(2θ)R∗ (θ)dθ
1
arctan R
[rad]
2
cos(2θ)R∗ (θ)dθ

where the equivalent value is found on the interval θ ∈ [0, π) via
hθi = π − mod(hθi0 + π, π) [rad].
Linear polarization data can be fully described by either a polarization angle χ and
p polarized intensity
P or by the Stokes parameters Q and U , where χ = (1/2) arctan(U/Q) and P = Q2 + U 2 . From the
RHT output, similar ‘Stokes vectors’ can be defined via
Z
QRHT = cos(2θ)R(θ)dθ [Jy]
Z
URHT = sin(2θ)R(θ)dθ [Jy].
This allows for an estimate of the orientation of the magnetic field to be derived solely from HI data via


1
URHT
θRHT = arctan
[rad].
2
QRHT
Rotation measure (RM): Characterizes the amount of Faraday rotation that polarized light experiences while passing through thermal electrons. Most compact polarized sources like pulsars and extremely
compact extragalactic sources show a single value of Faraday rotation which is the RM. This is commonly
defined as the slope of the polarization angle χ versus λ2 plot:
RM =

dχ(λ2 )
[rad m−2 ],
dλ2

where
1
χ = arctan
2



U
Q


[rad].

The RM, then, modifies the polarization angle χ from it’s initial value χ0 as
χ = χ0 + RMλ2 [rad].
The value of RM is the integral of the line-of-sight component of the magnetic field weighted by the
line-of-sight distribution of electron density, given by
observer
Z

~ [rad m−2 ],
~ · dl
ne B

RM = −0.81
source

~ has pc. The RM can be measured towards distant
~ has units of µG, ne has cm , and dl
where B
extragalactic radio sources and pulsars. The RM of an extragalactic radio source consists of an RM
contribution intrinsic to the source, the RM from the intergalactic space from the source to the Galaxy,
and the RM within the Galaxy. The first term (i.e., the extragalactic radio source) should be random
and hence reasonably small on average, because a source can be randomly oriented in space with any
possible field configuration; thus we observe a random quantity. The second term (i.e., the intergalactic
source) is negligible on average. Intergalactic magnetic fields are too weak to be detected with current
technology. Even if there is a weak field in intergalactic space, an integration over the path-length of
the intergalactic magnetic field with random directions with extremely thin gas should give quite a small
contribution. Therefore, the common contribution of RMs of extragalactic radio sources is from our own
Galaxy.
The RM does not increase monotonically with distance along the line of sight. Traditionally, the RM is
obtained by performing a least-squares fit to the data to determine the slope. There are however three
potential problems to this method: (1) the observed polarization angle is only known modulo π radians;
thus with measurements in only a few wavelength bands, the RM is often ambiguous (known as the nπ
ambiguity), (2) polarized emission with different RM values can be present along a single line of sight;
the signal from these regions mix, making a single linear fit inappropriate, and (3) faint sources with high
RM will be undetectable in individual channels due to low signal to noise and will remain undetectable
even after integrating all channels due to bandwidth depolarization; thus, no χ(λ2 ) data will be available
for the traditional linear fit.
Rotation measure (RM) synthesis: A robust method for determining the Faraday dispersion function
−3

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as proposed by Burn (1966). It was seldom used until Brentjens and de Bruyn (2005) who coined the term
RM Synthesis. This technique involves Fourier transforming the observed polarized surface brightness
P (λ2 ) into the Faraday dispersion function F (φ) (also referred to as the Faraday spectrum) which is
the complex polarized surface brightness as a function of Faraday depth. As shown by Burn (1966), the
the observed complex polarization vector can be written as P (λ2 ) = pIe2iχ , where p is the polarization
fraction and I is the Stokes I vector. Substituting the expression χ(λ2 ) = χ0 + φλ2 for the polarization
angle χ as a function of Faraday depth φ, we obtain
P (λ2 ) = pIe2i(χ0 +φλ

2

)

2

= pIe2iχ0 e2iφλ


2
= pIe2iχ0 e2iφλ
2

= F (φ)e2iφλ [Jy],
where F (φ) is the Faraday dispersion function which describes the intrinsic polarized flux as a function
of Faraday depth. Since the observed polarization originates from emission along all Faraday depths,
integrating this over φ gives the final form
Z∞

2

P (λ ) =

2

F (φ)e2iφλ dφ [Jy].

−∞

Thus there is a simple expression that relates the intrinsic quantity F (φ) to the observable P (λ2 ) which
takes the form of a Fourier transform. This can now be inverted to obtain the Faraday dispersion function
Z∞
F (φ) =

2

P (λ2 )e−2iφλ dλ2 [rad m−2 ].

−∞

However, one is confronted with a problem: namely, that we cannot observe at wavelengths λ < 0, nor
do we observe for all wavelengths λ > 0. To resolve this issue, Brentjens & de Bruyn (2005) introduce
a window function W (λ2 ) which is non-zero only at wavelengths sampled by the telescope. They show
that the observed polarized surface brightness can be rewritten as
2

2

2

Z∞

2

P̃ (λ ) ≡ W (λ )P (λ ) = W (λ )

2

F (φ)e2iφ(λ

−λ20 )

dφ [Jy],

−∞

and the reconstructed Faraday dispersion function as
R∞
F̃ (φ) ≡

2

P̃ (λ2 )e−2iφ(λ

−λ20 )

dλ2

−∞

R∞

= F (φ) ∗ R(φ) [rad m−2 ],
W (λ2 )dλ2

−∞

where ∗ denotes convolution. F̃ (φ) is an approximate reconstruction of F (φ). More precisely, it is F (φ)
convolved with R(φ) after Fourier filtering by the weight function W ( lambda2 ). R(φ) is the rotation
measure transfer function (RMTF) which is a crucially important quantity defined by
R∞
R(φ) ≡

W (λ2 )e−2iφ(λ

2

−λ20 )

−∞

R∞

dλ2
[rad m−2 ],

W (λ2 )dλ2

−∞

which is normalized to unity at φ = 0. It is a complex valued function. The real part corresponds
to the response of the transform parallel to the (Q, U ) vector at λ0 = λ while the imaginary part
corresponds to the response orthogonal to it. Taking a look at the equations above for F̃ (φ) and R(φ),
these can be seen as applications of the Fourier shift theorem – since the shift theorem only affects the
argument and not the absolute value of the resulting complex vector function, de-rotating these equations
to λ0 6= 0 does not change their amplitude. The Faraday spectrum is not straightforward to interpret;
in particular, there is no direct relationship between Faraday depth and physical depth. Further, the
Faraday dispersion function suffers from sidelobes of the main components caused by limited coverage of
the observed wavelength space. RM Synthesis is required when multiple emitting and rotating regions
are located along the line of sight, as opposed to a single emitting region (i.e., Faraday screen) behind a
single rotating region. RM synthesis was applied to an entire field of view for the first time by de Bruyn

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(1966) using pulsar observations. When applied to a complete field of view instead of just a single line
of sight, the output of RM synthesis is referred to as an “RM cube”. RM synthesis is characterized by
four parameters: (1) the resolution in Faraday space, which is inversely proportional to the coverage in
wavelength space, (2) the maximum observable of a point-like source in Faraday space, which is inversely
proportional to the width of a single frequency channel, (3) the maximum width of extended structures
in Faraday space (Faraday rotating and synchrotron emitting sources), which is inversely proportional
to the square of the minimum observable wavelength; wide-band observations at long wavelengths yield
high resolution in Faraday space but cannot detect extended structures, and (4) the ratio of maximum to
minimum wavelengths which is crucial to recognize a range of different scales in Faraday space. Table 2
these parameters for a variety of radio telescopes. The highest resolution in Faraday space are for those
with the largest wavelength coverage (LOFAR and SKA) while the largest range of scales in Faraday
space are for those with the greatest ratio of maximum to minimum wavelengths (ATCA, JVLA, SKA).

Table 2: Spectral ranges of various radio telescopes and parameters crucial for RM synthesis. Table from Beck et al.
(2012).

Rotation measure transfer function (RMTF): A crucially important quantity introduced by de
Bruyn (1966) and defined by
R∞
R(φ) ≡

W (λ2 )e−2iφ(λ

2

−λ20 )

dλ2

−∞

R∞

[rad m−2 ],
W (λ2 )dλ2

−∞

which is normalized to unity at φ = 0. For a simple weight function W (λ2 ) that is a top-hat (boxcar)
fuction centered on λ2c with width ∆λ2 = λ22 − λ21 , the corresponding RMTF is a sinc function with a
phase wind:


2
sin(φ∆λ2 )
R(φ) = eiφλc
[rad m−2 ]
φ∆λ2
It is a complex valued function; the real part corresponds to the response of the transform parallel to the
(Q, U ) vector at λ0 = λ while the imaginary part corresponds to the response orthogonal to it. Ideally,
the response in the entire main peak of the RMTF should be parallel to the actual polarization vector at
λ0 . Brentjens & de Bruyn (2005) show that this optimal choice of λ20 is the mean of the sampled λ2 values
weighted by W (λ2 ). However, since the shift theorem of Fourier theory only applies to the argument,
changing the value of λ0 will not change the absolute value of the RMTF. A drawback of having λ0 6= 0
is that the polarization angle that one derives still needs to be transformed to a polarization angle at
λ = 0 if one wants information on the electric or magnetic field direction of the source. In the case of a
high S/N , this is easy:
χ0 = χ(λ20 ) − φλ20 [rad].
However, if the S/N is low, the uncertainty in φ usually prevents accurate de-rotation to λ = 0.
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Second moment: See variance.
Skewness: The degree of asymmetry of a distribution, and can be quantified to define the extent to
which a distribution differs from a normal distribution. Skewness is defined as
4
N 
1 X xi − µ
Skew[x] =
− 3 [dimensionless],
N i=1
σ
where µ is the statistical mean and σ is the standard deviation. It can be negative, positive, zero or
undefined, and like the kurtosis, is a dimensionless quantity.
Sonic Mach number (Ms ): The ratio of the flow speed to that of the sound speed of the interstellar
medium given by
 
|~v |
[dimensionless],
Ms =
cs
where |~v | is the local velocity and cs is the sound speed. This number is a commonly used parameter
of turbulence used to obtain information on gas compressibility and magnetization. Using polarization
gradients of synchrotron radiation, it was shown by Gaensler (2011) that the turbulence of the WIM has
a relatively low sonic Mach number, Ms . 2.
Spectral index (α): A measure of the dependence of radiative flux density on frequency. If flux does not
follow a power-law distribution, the spectral index itself is a function of frequency. In the radio regime,
a spectral index of α = 0 − 2 indicates thermal emission while a steep negative spectrum indicates
synchrotron radiation.
Scintillation:
Spectral correlation function (Sx,y ): The average over all neighboring spectra of the normalized rms
difference between brightness temperatures Tb for pairs of velocity channels (vi , vj ) defined as
n

Sx,y ≡ Svi ,vj =

1X
Tb ([x, y]a , vi )Tb ([x, y]a , vj ),
n a=1

where n is the number of positions in the map. A histogram of Sx,y reveals the autocorrelation properties:
if Sx,y is close to unity, the spectra do not vary much.
Spectral irradiance (Eν (T ) or Eλ (T )): The Planck function multiplied by the total solid angle of a
sphere (4π sr) which describes the power per area per frequency or the power per area per wavelength:


8πhν 3
1
Eν (T ) =
[W m−2 Hz−1 ]
c2
ehν/kB T − 1


8πhc2
1
Eλ (T ) =
[W m−2 m−1 ].
λ5
ehc/λkB T − 1
Starlight polarization: The polarization of initially unpolarized starlight which becomes polarized as it
passes through dust grains aligned by an external magnetic field. Paramagnetic materials have unpaired
electrons which, when in the presence of an external magnetic field such as the interstellar magnetic field,
align in the same direction causing grain alignment. In the case of elongated dust grains, if the shortest
axis is aligned with the direction of the magnetic field, then the grains will absorb light polarized along
the long axis of the grain which is perpendicular to the field. This results in transmitted radiation having
a polarization direction parallel to the magnetic field. Optical polarization of starlight was first imaged
by Hall & Mikesell (1949) and Hiltner (1949). They found that polarized light from nearby stars had
similar orientation, indicating that the polarizing mechanism was a source other than the individual stars.
Shortly thereafter, Davis & Greenstein (1951) published a theory on the origin of polarized starlight. They
proposed that an elongated dust grain spinning about its shortest axis would have a preferred orientation
when located in a magnetic field. Starlight polarization is most useful for detecting magnetic fields in
the Solar neighbourhood out to about 1 − 3 kpc. The same large-scale alignment of spinning, aspherical
dust grains that causes starlight polarization also causes infrared polarization.
Stellar wind bubble (SWB): A cavity light years across filled with hot gas blown into the interstellar
medium by high-velocity (several thousand km s−1 ) stellar wind from a single massive O or B star. The
heliosphere blown by the Solar wind, within which all of the major planets of the Solar System are
embedded, is a small example of a stellar wind bubble.
Stochastic system:
Stokes parameters:
Stokes I (I): A measure of the total intensity of radio emission.
Stokes Q (Q): A measure of the linear polarizationp
of radio emission. The images of Stokes Q (and Stokes
U ) as well as the linear polarized intensity P ≡ (Q2 + U 2 ) are often filled with complex structures

Galactic Magnetism Study Guide

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that bear little resemblance to the Stokes I image of total intensity. The intensity variations seen in Q
(as well as U and P ) are the result of small-scale angular structure in the Faraday rotation induced by
ionized gas, and are thus an indirect representation of turbulent fluctuations in the free-electron density
and magnetic field throughout the interstellar medium.
Stokes U (U ): A measure of the linear polarization
p of radio emission. The images of Stokes U (and Stokes
Q) as well as the linear polarized intensity P ≡ (Q2 + U 2 ) are often filled with complex structures that
bear little resemblance to the Stokes I image of total intensity. The intensity variations seen in U (as well
as Q and P ) are the result of small-scale angular structure in the Faraday rotation induced by ionized
gas, and are thus an indirect representation of turbulent fluctuations in the free-electron density and
magnetic field throughout the interstellar medium.
Stokes V (V ): A measure of circular polarization of radio emission. If V > 0, then the electromagnetic
wave is right-handed circularly polarized (RCP), while if V < 0, then the electromagnetic wave is lefthanded circularly polarized (LCP). Stokes V is related to the line-of-sight magnetic field in the following
manner:
V (ν) ∝ Bk ×

dI
[Jy],
dν

where I is Stokes I and ν is the observed frequency.
Striation:
Strömgren sphere: A sphere of ionized hydrogen (HII) around young OB stars.
Structure function (Sp (δr)): The structure function of order p for an observable A is given by
Sp (δr) = h|A(r) − A(r + δr)|p i,
for position r and position increment δr. The power-law fit to this, Sp (δe) ∝ δrpζ , gives the slope ζp .
Superbubble: Also known as a supershell. A cavity that is hundreds of light years across filled with
1 × 106 K gas blown into the ISM by multiple SNe and stellar winds from clusters and associations
of massive O and B stars. A superbubble behaves qualitatively like an individual supernova remnant,
except that it has a continuous supply of energy. For the first 3 Myrs at least, their energy supply
is exclusively due to stellar winds, whose cumulative power rises rapidly with time. Supernovae start
exploding after & 3 Myrs, and within ∼ 2 Myrs, they overpower the stellar winds. From then on, the
successive supernovae explosions continue to inject energy into the superbubble at a slowly decreasing
rate, depending on the initial mass function (IMF) of the progenitor stars, until ∼ 40 Myrs. Altogether,
stellar winds account for a fraction comprising between ∼ 12% and ∼ 17% of the total energy input. The
Solar System lies near the center of an old superbubble known as the Local Bubble whose boundary can
be traced by a sudden rise in dust extinction.
Supernova: A supernova is a transient astronomical event that occurs during the last stellar evolutionary
stages of a star’s life, either a massive star or a white dwarf, whose destruction is marked by one final,
titanic explosion. Supernovae come in two types. Type-I supernovae arise from old, degenerate low-mass
stars which supposedly are accreting from a companion and undergoes a thermonuclear instability upon
accumulation of a critical mass. Type-II supernovae arise from young stars with initial masses & 8 M
whose core gravitationally collapses once it has exhausted all of its fuel. Like the bright, massive stars,
Type-II supernovae are tightly confined to the spiral arms while Type-I supernovae have a more spreadout distribution similar to the general stellar population. Type-I supernovae are less frequent than their
Type-II counterparts. All of them are uncorrelated in space and have basically the same repercussions
on the ISM as isolated Type-II supernovae. The hot gas created by supernovae explosions (and stellar
winds) in the Galactic disk rises into the halo under the influence of its own buoyancy. In the course
of its upward motion, it cools down (almost adiabatically at first, then by radiative transport), and
eventually condenses into cold neutral clouds. Once formed, these clouds fall ballistically towards the
Galactic plane. This convective cycle of interstellar material between the disk and the halo is known as
the Galactic fountain.
Synchrotron radiation: The radio emission produced by relativistic cosmic rays when accelerated
radially (e.g., by magnetic fields) which is one of the most commonly used tracers of the Galactic magnetic
field. Synchrotron emission has a steep spectral index of α ∼ −2.75, which results in much brighter
emission at lower frequencies (this applies to both linear polarization and total intensity). For a single
electron, the critical frequency of synchrotron radiation can be expressed as
νc =

3 q 2
γ (B sin θ) [Hz],
4π mc

or
νc ∼ γ 2

Galactic Magnetism Study Guide

eB
[Hz],
2πm0
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where γ is the Lorentz factor, e and m0 are the charge and rest mass of the cosmic ray electron, and B is
the magnetic field strength. As the cosmic ray gyrates around the local magnetic field, it generates the
synchrotron radiation beamed within the shape of a cone at an angle of
mc c2
[rad].
E
On the surface of the cone, the radiation is 100% linearly polarized with an electric field E that has
a direction of −v × B. Synchrotron radiation from the Galaxy was the first emission detected in radio
astronomy by Jansky (1933), however, it was not until the 1950’s when detailed radio maps of the Galaxy
were produced that the connection was made between radio emission and the synchrotron mechanism.
Linear polarization of synchrotron radiation can be used to trace ordered magnetic fields in the plane of
the sky (B⊥ ). Unpolarized synchrotron emission can be indicative of a turbulent magnetic field causing
depolarization via Faraday rotation of thermal electrons (synchrotron radiation cannot cause Faraday
rotation). The radiated power from an accelerating charge q with mass m by a magnetic field B at an
observing frequency ν is
θ=±

2 q4
(γν)2 (B sin θ)2 [Jy].
3 m2 c5
Since synchrotron radiation is proportional to the charge-to-mass ratio, electrons dominate synchrotron
radiation over protons and ions by several orders of magnitude. Synchrotron radiation peaks at a characteristic frequency of
P =

3 q 2
γ (B sin θ) [Hz].
4π mc
For radio observations with typical frequencies at or above tens of MHz and with typical magnetic
field strengths of order µG, the observed population of synchrotron electrons have γ  1. Synchrotron
emission has a steep negative spectral index in the radio regime. Assuming that the electrons are moving
isotropically, Le Roux (1961) showed that the maximum intrinsic degree of polarization (i.e., polarization
fraction) of synchrotron radiation from plasma in a uniform magnetic field is a function of the electron’s
spectral index γ given by
νc =

p=

3γ + 3
[dimensionless],
3γ + 7

independent of frequency and observing angle. Using observations of the Crab Nebula, Woltjer (1958)
and Westfold (1959) showed that γ ≈ 5/3 and therefore p ≈ 67%. Le Roux (1961) gives a full derivation
of synchrotron radiation. Figure 5 shows the Galactic synchrotron emission at 408 MHz.
Taylor scale:
Third moment: See skewness.
Toroidal magnetic field: Often the GMF is expressed in cylindrical coordinates (θ, r, z). The toroidal
~ z components) confined to a plane parallel to the Galactic
magnetic field refers to the structures (without B
plane. The toroidal fields possibly extend to the inner Galaxy, even towards the central molecular zone
(CMZ).
Tsallis distribution: A function that can be fit to incremental probability distribution functions (PDFs)
of turbulent density, magnetic field, and velocity. The Tsallis distribution was originally derived by
Tsallis (1988) as a means to extend traditional Boltzmann-Gibbs mechanics to fractal and multifractal
systems. The complex dynamics of multifractal systems apply to many natural environments, such as
ISM turbulence. The Tsallis function of an arbitrary incremental PDF ∆f has the form
−1/(q−1)

∆f (r)2
Rq = a 1 + (q − 1)
.
w2
The fit is described by three dependent variables. The a parameter describes the amplitude while w is
related to the width or dispersion of the distribution. Parameter q, referred to as the “non-extensivity
parameter” or “entropic index” describes the sharpness and tail size of the distribution. The Tsallis fit
parameters are in many ways similar to statistical moments. Moments, more specifically the third- and
fourth-order moments, have been used to describe the density distributions and have shown sensitivities
to simulation compressibility. The first- and second-order moments simply correspond to the mean and
variance of a distribution. Skewness, or third-order moment, describes the asymmetry of a distribution
about its mode. Skewness can have positive or negative values corresponding to right and left shifts of
a distribution, respectively. The fourth-order moment, kurtosis, is a measure of a distributions peaked
or flatness compared to a Gaussian distribution. Like skewness, kurtosis can have positive or negative
values corresponding to increased sharpness or flatness. With regard to the Tsallis fitting parameters, the

Galactic Magnetism Study Guide

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Figure 5: The synchrotron emission at 408 MHz across the entire sky in Galactic coordinates. As expected, the emission
is concentrated along the Galactic plane. However, the feature known as Loop I, is clearly arching up from ` = 55◦ towards
the North Galactic Pole. This Figure is adapted from Haslam et al. (1981). Figure taken from Newton-McGee (2009).

w parameter is similar to the second-order moment variance while q is closely analogous to fourth-order
moment kurtosis. Unlike higher order moments, however, the Tsallis fitting parameters are dependent
least-squares fit coefficients and are more sensitive to subtle changes in the PDF.
Turbulence: Astrophysical turbulence is a complex nonlinear fluid phenomenon that can occur in a
multiphase medium which results in the excitation of an extreme range of correlated spatial and temporal
scales. There are many injection sources on scales ranging from kp down to sub-AU The physical processes
by which kinetic energy is converted into turbulence are not well understood for the ISM. The main sources
for large-scale motions are: (a) stars, whose energy input is in the form of protostellar winds, expanding
HII regions, O star and Wolf-Rayet winds, supernovae, and combinations of these producing superbubbles;
(b) Galactic rotation in the shocks of spiral arms or bars, in the Balbus-Hawley instability (also known
as the magnetorotational instability), and in the gravitational scattering of cloud complexes at different
epicyclic phases; (c) gaseous self-gravity through swing-amplified instabilities and cloud collapse; (d)
Kelvin-Helmholtz and other fluid instabilities; and (e) Galactic gravity during disk-halo circulation, the
Parker instability, and galaxy interactions. Interstellar turbulence has been characterized by structure
functions, correlation functions (e.g., spatial power spectrum), autocorrelations, power spectra, energy
spectra, and delta variance. Sources for small-scale turbulence observed by radio scintillation include sonic
reflections of shock waves hitting clouds, cosmic ray streaming and other instabilities, field star motions
and winds, and energy cascades from larger scales. Turbulence affects the structure and motion of nearly
all temperature and density regimes of the interstellar gas. Magnetohydrodynamic (MHD) turbulence
is a key element in the study of star formation, molecular cloud structure, magnetic reconnection, heat
transport, and cosmic ray propagation. MHD turbulence is known to be different from a collection of
linear Alfvénic waves. Does MHD turbulence, specifically the Alfvén wave modes, damp at the decoupling
scale of the ambipolar diffusion scale LAD ? Cascading rates and the anisotropy of turbulence should be
accounted for carefully before a definitive conclusion about turbulent damping in the partially ionized
media. Despite the importance of turbulence for ISM studies, many mysteries remain including the nature
of turbulence driving and damping. The most common observational techniques for studying turbulence
include scintillation studies (which are limited to fluctuations in ionized plasmas), density fluctuations
via column density maps, and radio spectroscopic observations via centroids of spectral lines. Positionposition-velocity (PPV) spectroscopic data have the advantage over column density maps in that it
contains information on the turbulent velocity field. However, this type of data provides contributions
of both velocity and density fluctuations entangled together, and the process of separating the two has
proven to be a challenging problem. One of the main approaches for characterizing ISM turbulence
is based on using statistical techniques and descriptions (e.g., spatial power spectrum). Although the
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power spectrum is useful for obtaining information about energy transfer over scales, it does not provide
a full picture of turbulence, partly because it only contains information on Fourier amplitudes (i.e., two
substantially different density distributions can have the same power spectrum). Probability distribution
functions (PDFs) of turbulence in PPV and column density data have been studied for decades and have
proven to be very useful to show that turbulence is present in the ISM, providing insight into the effects of
turbulent driving and characterizing the type of turbulence in question. Many studies on turbulence focus
on obtaining parameters such as sonic and Alfvén Mach numbers, injection scale, gas temperature, and
Reynolds number. In particular, the sonic and Alfvén Mach numbers provide much coveted information
on the gas compressibility and magnetization.
Turbulent fragmentation:
Turbulent magnetic field: A.k.a. “random magnetic field”. Deviations from coherent or ordered fields
are taken as turbulent fields. The fluctuations of fields at scales 10 times smaller than a concerned scale
are often also taken as turbulent fields.
Turbulent pressure:
Two-fluid turbulence: The turbulence of a partially ionized plasma where the ions are decoupled from
the neutrals (i.e., at the scale below the ambipolar diffusion scale).
Type-I Supernova: Arises from old, degenerate low-mass stars which supposedly are accreting from a
companion and undergoes a thermonuclear instability upon accumulation of a critical mass. Like Type-II
supernovae, releases an amount of energy of ' 1051 ergs.
Type-II Supernova: Arises from young stars with initial masses & 8 M whose core gravitationally
collapses once it has exhausted all of its fuel. Like Type-I supernovae, releases an amount of energy of
' 1051 ergs.
Unstable neutral medium (UNM): A thermally-unstable phase of the atomic ISM. Figure 6 shows the
thermal pressure function p(n). This curve has several familiar features, described in the classic paper on
this subject by Field, Goldsmith & Habing (1969). These include separate branches for warm intercloud
(red) and cold cloud (green) components, a maximum intercloud thermal pressure (4400 cm−3 K for the
curve shown), a minimum cloud thermal pressure (1700 cm−3 K for this example), and a range of density
(0.8−7 cm−3 ), or more fundamentally temperature (roughly 270−5500 K), between these two (blue) that
is regarded as forbidden in equilibrium because at constant thermal pressure it is thermally unstable. In
the unstable region, gas slightly above the curve is too hot for its density and cools. But at constant
thermal pressure, cooling moves it to the right, to higher density, taking it further from the curve and
making it cool faster, until it reaches the stable cloud branch. Similarly, gas slightly below the curve
has excess heating and at constant thermal pressure moves horizontally to the left until it joins the
stable intercloud component. For the standard curve, the mean midplane interstellar density in the Solar
Neighbourhood (slightly over 1 cm−3 ) is in the unstable regime. This appears to guarantee that both the
cloud and intercloud branches will be populated.
Figure 6:

The Two-Phase
Medium thermal pressure versus density curve is shown in
the red-blue-green curve. The
red segment is the thermally
stable warm component, the
green the corresponding stable
cold component, and the blue
the thermally unstable regime.
The black dashed line is the total midplane pressure, the orange dashed line represents the
magnetic pressure, the dashed
aqua line shows how the p(n)
curve shifts when the heating
rate is raised by a factor of 10,
and the dashed purple rectangle outlines the regime of essentially zero bulk modulus. Figure taken from Cox (2005).

One of the most interesting features of p(n)f is that over a density range of a factor of 350, from 0.2 to
70 cm−3 , the thermal pressure variation is less than a factor of three. In Figure 6, this region is shown
enclosed by a dashed purple rectangle. It is a regime in which the thermal pressure offers essentially
zero bulk modulus. This means that if, for some reason, the ISM had a totally random distribution of
densities within this range, the corresponding distribution of thermal pressures would be observationally
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indistinguishable from one in which thermal pressure had an active role in arranging things, except there
would be no range of forbidden densities. The small variation in thermal pressure between regions of
differing density can be compensated by slight changes in the somewhat stronger magnetic field pressure.
This is true when establishing pressure uniformity across the magnetic field lines, but equilibrium parallel
to the field cannot be established in this way.
Volume filling factor (f ): A correction factor such that a fraction f of the line of sight intersects
clouds of uniform density N . There are three effects of the interstellar magnetic field that conspire to
lower the filling factor of hot cavities: (1) the background magnetic pressure acting on the surrounding
shells directly opposes their expansion, (2) the magnetic tension in the swept-up field lines gives rise to
an inward restoring force, while the associated magnetic pressure prevents the shells from fully collapsing
and, therefore, keeps them relatively thick, and (3) the enhanced external “signal speed” causes the shells
to merge earlier than they would in an unmagnetized medium.
Warm ionized medium (WIM): A diffuse phase of the interstellar medium with a typical temperature
of T ∼ 8, 000 K and average electron density of ∼ 0.2 − 0.5 cm−3 . The ionized gas has mostly been
mapped using the Hα line. One limitation of the Hα line comes from the obscuration caused by dust,
which restricts visibility to within 2 − 3 kpc from the Sun. Using polarization gradients of synchrotron
radiation, it was shown by Gaensler (2011) that the turbulence of the WIM has a relatively low sonic
Mach number, Ms . 2.
Warm neutral medium (WNM): A thermally-stable phase of the atomic ISM with a typical density
and kinetic temperature of n ∼ 0.2 − 0.9 cm−3 and TK ∼ 5000 − 8300 K, respectively.
Warm partially ionized medium (WPIM):
Wave turbulence:
Zeeman effect: The effect of a single spectral line splitting into multiple components in the presence of
a static magnetic field due to the degeneracy between various electron energy levels that become broken
when exposed to an external magnetic field. In an astrophysical context, a magnetic field produces small
frequency shifts in the left- and right-handed circular polarized components of a given spectral line with
respect to the intrinsic central frequency of the atom of molecule. Zeeman splitting is used to measure
the parallel component of the magnetic field via atomic and molecular gas. The first measurement of
an astrophysical magnetic field was using Zeeman splitting. The 21 cm line of atomic hydrogen is the
most widespread, and it provides an opportunity to measure splitting in both emission and absorption.
Zeeman splitting is a powerful tool as the magnetic field can be directly determined from the energy
difference between the electron levels. The separation in energy of the magnetic hyperfine levels from the
unsplit level of the zero magnetic field case is given by
∆E = −µB mF gB [J],
where µB is the Bohr magneton, mF is the quantum number of the Zeeman splitting, B is the magnetic
field strength, and g is the Landé g-factor. However, the effect is difficult to observe since the frequency
shift (∆ν) in the spectral lines is small:
µB mF gB
[Hz].
h
Such a frequency shift is usually smaller than the spectral linewidth, making Zeeman splitting observations
resolution limited. Lower frequencies produce better Zeeman splitting results because, although the
splitting is independent of the line frequency itself, the higher the frequency, the broader the linewidth.
The statistical sample of measured Zeeman splittings is woefully small and desperately needs to be
expanded. Zeeman-splitting observations are time-consuming: typically tens of hours are required for
each source. In addition, Zeeman splitting observations cannot be related to the large-scale GMF.
Zenith: The point in the sky or celestial sphere directly above an observer.
Zeroth moment:
∆ν =

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3

Variables

Alfvén Mach number: MA [dimensionless]
Alfvén speed: vA [m s−1 ]
Ambipolar diffusion scale: LAD [m]
Ambipolar diffusivity: νAD
Angle: θ [rad]
Autocorrelation: C(δr) [dimensionless]
Bandwidth in frequency: ∆ν [Hz]
Bandwidth in wavelength: ∆λ [Hz]
Bohr magneton: µB
β parameter: β
Channel central frequency: νc [Hz]
Channel width in frequency: δν [Hz]
Channel central wavelength: λc [m]
Channel width in wavelength: δλ [m]
Charge density: ρe [C m−3 ]
∗
Complex conjugate: f (x) or f¯(x)
Current density: ~j [A m−3 ]
Delta function: δ(x)
2
Delta variance: σ∆
(L)
Density: ρ [m−3 ]
Density of electrons: ρe [m−3 ]
Density of ions: ρi [m−3 ]
Density of neutrals: ρn [m−3 ]
Diameter of window (RHT): DW [pixels]
Dirac delta function: δ(x)
Dispersion measure: DM [pc cm−3 ]
Dust-to-gas mass ratio: r [dimensionless]
Dynamic viscosity: µ [Ps s]
Electron charge: e [C]
Electron mass: em [kg]
Electron plasma pressure: Pe [Pa]
Electron temperature: Te [K]
Emission cross section per H nucleon: σe (ν) [cm2 ]
Emission measure: EM [pc cm−6 ]
Emissivity index: β [dimensionless]
Energy spectrum: E(k) [Jy m−1 ]
Expectation value of θ on any domain (RHT): hθi0 [rad]
Expectation value of θ on θ ∈ [0, π) (RHT): hθi [rad]
Faraday depth: φ [rad m−2 ]
Faraday dispersion function (no spectral dependence): F (φ) [rad m−2 ]
Faraday dispersion function (general form): F (φ, λ) [rad m−2 ]
Frequency: ν [Hz]
Frictional coupling coefficient between ions and neutrals: α [dimensionless]
FWHM of RMTF main peak: ∆φ [rad m−2 ]
Fourier transform (FT): fˆ(x)
Hydrogen column density: NH [cm−2 ]
Incremental position: δr [m]
Intensity of thermal dust emission: Iν [W m−2 Hz−1 sr−1 ]
Intensity threshold: Z [dimensionless]
Jansky: Jy [unit]
Joule: J [unit]
Landé g-factor: g
Lorentz factor: γ [dimensionless]
Luminosity per H atom: L [W H−1 ]
Mach number: M [dimensionless]
~ [µG]
Magnetic field: B
Mass absorption coefficient: κν [cm2 g−1 ]
Mean: µ(x)
Mean molecular weight: µ [dimensionless]
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Momentum diffusivity: ν [m2 s−1 ]
Number density: n [m−3 ]
Number of θ bins (RHT): n [dimensionless]
Optical depth: τν [dimensionless]
Order: p [dimensionless]
Planck constant: h [J s]
Planck constant (reduced): ~ [J s]
Planck function: Bν (T ) [W m−2 Hz−1 sr−1 ] or Bλ (T ) [W m−2 m−1 sr−1 ]
Polarization angle: χ [rad]
Polarization angle (RHT): χRHT [rad]
Polarization angle at λ = 0: χ0 [rad]
Polarization fraction: p(λ2 ) [dimensionless]
∂ P~
~ P~ | [Jy beam−1 ]
or |∇
Polarization gradient:
∂s
∂ P~
~ P~ |max [Jy beam−1 ]
Polarization gradient (maximum value):
or |∇
∂s
max

Polarization gradient (radial component; maximum value):

∂ P~
∂s

Polarization gradient (tangential component; maximum value):

[Jy beam−1 ]
rad

∂ P~
∂s

[Jy beam−1 ]
tan

~ P~ ) [rad]
Polarization gradient (direction): arg(∇
Polarized intensity: P [Jy]
Polarized surface brightness: P (λ2 ) [Jy]
Polarized surface brightness (observed): P̃ (λ2 ) [Jy]
Position: r [m]
Power spectrum: P (k)
Quantum number of Zeeman splitting: mF
∂P
Radial component of polarization gradient:
[Jy pc−1 ]
∂s rad
Resistivity: η [Ω m]
Rest mass: m0 [kg]
Reynolds number: Re [dimensionless]
Reynolds number for ambipolar diffusion: RAD [dimensionless]
RHT angle: θ [rad]
RHT backprojection: R(x, y) [dimensionless]
RHT backprojection sampling disk around star: R∗ (x, y) [dimensionless]
RHT distance: ρ [pixels]
RHT intensity vs angle: R(θ, x0 , y0 ) [dimensionless]
Rotation measure: RM [rad m−2 ]
Rotation measure transfer function (RMTF): R(φ)[rad m−2 ]
Sonic Mach number: Ms [dimensionless]
Sound speed: cs [m s−1 ]
Specific heat: cp [J kg−1 K]
Spectral correlation function: Sx,y [dimensionless]
Spectral index: α [dimensionless]
Spectral irradiance: Eν (T ) [W m−2 Hz−1 ] or Eλ (T ) [W m−2 m−1 ]
Speed of light: c [m s−1 ]
Standard deviation: σ(x)
Stokes I: I [Jy]
Stokes Q: Q [Jy]
Stokes Q (RHT): QRHT [Jy]
Stokes U: U [Jy]
Stokes U (RHT): URHT [Jy]
Stokes V: V [Jy]
Structure function: Sp
Structure function power-law slope: ζp
∂P
Tangential component of polarization gradient:
[Jy pc−1 ]
∂s tan

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Thermal conductivity: k [W m−1 K]
Thermal diffusivity: α [m2 s−1 ]
Variance: σ 2
Velocity: ~v [m s−1 ]
Viscosity coefficient: µ [Pa s]
Wavelength: λ [m]
Wavelength to which all polarization vectors are de-rotated: λ0 [m]
Wavenumber: k [m−1 ]
Weight function: W (λ2 ) [dimensionless]
Zeeman splitting energy difference: ∆E [J]
Zeeman splitting frequency difference: ∆ν [Hz]

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4

Equations

Alfvén Mach number:


MA =

Alfvén speed:

|~v |
vA


[dimensionless]

~
|B|
vA = √ [m s−1 ]
ρ

Ambipolar diffusion scale:
LAD =
Ambipolar diffusivity:

B2
[ms s−1 ]
4πρi ρn α

νAD =
Autocorrelation:

VA
[m]
αρi

C(δr) = hf (r)f (r + δr)i [dimensionless]

β parameter:
β≡

c2s
2 [dimensionless]
vA

Channel central wavelength:
λ2c

c2
≈ 2
νc

(Brentjens & de Bruyn, 2005)
Charge conservation:



3
1+
4

δν
νc

2 !

[m2 ]

∂ρe ~ ~
+∇·j =0
∂t

Channel width in wavelength:
2c2 δν
δλ ≈
νc3
2

(Brentjens & de Bruyn, 2005)
Current density:

1
1+
2



δν
νc

2 !

[m2 ]

~
~j = c∇ × B [A m−2 ]
4π

(Schoeffler et al., 2016)
Delta variance:
2
σ∆
(L)

* 3L/2
Z

+

(A[r + x] − hAi)

=

2

(x) dx ,

0

Dispersion measure (DM):

observer
Z

~ [pc cm−3 ]
ne d`

DM = −
source

Electron plasma pressure:
Pe = Te η [Pa]
(Schoeffler et al., 2016)
Electron temperature:
Te =

Pe
[K]
η

(Schoeffler et al., 2016)
Emission cross section per H nucleon:
τν
σe (ν) =
[cm2 ]
NH
(Roy et al., 2013)
Emission measure:

observer
Z

~ [pc cm−6 ]
n2e d`

EM = −
source

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Expectation value of θ on any domain (RHT):

R
sin(2θ)R∗ (θ)dθ
1
0
hθi = arctan R
2
cos(2θ)R∗ (θ)dθ
Expectation value of θ on θ ∈ [0, π) (RHT):
hθi = π − mod(hθi0 + π, π) [rad]
Faraday depth:

observer
Z

~ [rad m−2 ]
~ · dr
ne B

φ(~r) = −0.81
source

Faraday depth standard error: 
σφ =

P 2
−1
χi
1
P 2 2 i −1
P 2 2
N − 2 i (λi ) − N ( i λi )
"
#−1

(N − 1)(σχ2 + 
hχi)
1
P
P
=
N − 2 i (λ2i )2 − N −1 ( i λ2i )2
"
#−1
σχ2
N −1
P
P
=
N − 2 i (λ2i )2 − N −1 ( i λ2i )2
"
#−1
2
2
σQ
≈ σU
≈ σ2
≈
[rad m−2 ]
4(N − 2)||P ||2 σλ2 2

(Brentjens & de Bruyn, 2005)
Faraday dispersion function:

Z∞
F (φ) =

2

P (λ2 )e−2iφλ dλ2 [rad m−2 ]

∞

Faraday dispersion function∞(reconstructed):
R
2
2
P̃ (λ2 )e−2iφ(λ −λ0 ) dλ2
−∞
F̃ (φ) ≡
= F (φ) ∗ R(φ) [rad m−2 ]
R∞
W (λ2 )dλ2
−∞

Fourier transform (FT):

Z∞

fˆ(x) =

0

f (x0 )e−2πxx dx0

−∞

Hydrogen column density:
NH = (11.5 ± 0.5) × 1021 E(J − Ks ) [cm−2 ] (NH . 5 × 1021 cm−2 )
(Martin et al., 2012)
Induction equation:
2
~
∂B
~ + ηc ∇2 B
~ − 1 ∇ × (~j × B)
~ − c ∇n × ∇Te
= ∇ × (~v × B)
∂t
4π
en
ne
(Schoeffler et al., 2016)
Intensity of thermal dust emission:
Iν = τν Bν (T ) [W m−2 Hz−1 sr−1 ]
≡ σe (ν)NH Bν (T ) [W m−2 Hz−1 sr−1 ]
= rµmH NH κν Bν (T ) [W m−2 Hz−1 sr−1 ]
Inverse Fourier transform (FT):

Z∞
f (x) =

0
fˆ(x0 )e2πixx dx0

−∞

Kurtosis:
Kurt[x] =
Lorentz factor:

r
γ=

Galactic Magnetism Study Guide

3
N 
1 X xi − µ
[dimensionless]
N i=1
σ
1 − (v/c)2
[dimensionless]
c2

Page 32 / 36

Luminosity per H atom:

Z
L=

4πκ0 (ν/ν0 )β µmH Bν (T ) dν [W H−1 ]

Magnetohydrodynamic (MHD) equations:
~ ·B
~ =0
∇
~ ·E
~ = 4πρe
∇
~
~ ×E
~ = − ∂B
c∇
∂t
~ ×B
~ = 4π~j +
c∇
Mass absorption coefficient:

~
∂E
∂t

β
ν
[cm2 g−1 ]
ν0
σe (ν)
=
[cm2 g−1 ]
µmH


κν = κ0

Number of θ bins (RHT):

" √

2
(DW − 1)
nθ = π
2
Planck function:

#
[dimensionless]


2hν 3
1
Bν (T ) =
[W m−2 Hz−1 sr−1 ]
2
hν/k
T −1
B
c
e


2hc2
1
Bλ (T ) =
[W m−2 m−1 sr−1 ]
5
hc/λk
BT − 1
λ
e


Polarization angle:
χ=

1
arctan
2



U
Q


[rad]

= χ0 + RMλ2 [rad]
Polarization angle (RHT):
χRHT

1
= arctan
2



URHT
QRHT


[rad]

Polarization angle distribution standard
error:
#−1
"
1 X 2
χ − hχi
[rad]
σχ =
N −1 i i
(Brentjens & de Bruyn, 2005)
Polarization angle standard s
error:


2
2
∂χ
∂χ
2
2
σχ =
σQ +
σU
∂Q
∂U
v
!
!2
u
U
u ∂ 1 arctan U 2
∂ 12 arctan Q
2
Q
t
2
2
=
σQ +
σU
∂Q
∂U
s



U2
Q2
1
1
2 +
2
=
σ
σU
Q
4 (Q2 + U 2 )2
4 (Q2 + U 2 )2
=

2
2
U 2 σQ
+ Q2 σU
[rad]
4||P ||4

(Brentjens & de Bruyn, 2005)
Polarization angle at λ = 0 standard
error:
"

#−1
2
2
σQ
≈ σU
≈ σ2 N − 1
λ40
σχ0 =
+ 2
[rad]
4(N − 2)||P ||2
N
σλ2
(Brentjens & de Bruyn, 2005)
Polarization
gradient:
v
u

2 
2 !



2 
2 !
u
~
∂P
∂Q
∂U
∂Q
∂Q
∂U
∂U
∂Q
∂U
2
= tcos2 θ
+
+ 2 cos θ sin θ
+
+ sin θ
+
[Jy beam−1 ]
∂s
∂x
∂x
∂x ∂y
∂x ∂y
∂y
∂y

Galactic Magnetism Study Guide

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(Herron, 2005)
Polarization gradient (direction):


s

2 
+






∂Q ∂Q ∂U ∂U
~
~

 [rad]
s
arg(∇P ) ≡ arctan sign
+




2
2
∂x ∂y
∂x ∂y


∂U
∂Q
+
∂x
∂x

(Herron, 2005)
Polarization
v gradient (maximum value):
u
u
u
u "
2 
2 
2 
2 #
u1
~
∂P
∂U
∂U
∂U
∂Q
u
=u
+
+
+
+
∂s
∂x
∂x
∂x
∂y
u2
max
t

∂Q
∂y

2



∂U
∂y

v"
u 
2 
2 
2 
2 # 2
u
∂Q
∂U
∂U
∂U
u
+
+
+
u
∂x
∂x
∂x
∂y
1u
u


u
2t
2
∂Q ∂U
∂Q ∂U
−
−4
∂x ∂y
∂y ∂x

(Herron, 2005)
Polarization 
gradient (radial
component): 




U
∂ P~
∂Q
∂Q
U
∂U
∂U
= cos arctan
cos θ +
sin θ + sin arctan
cos θ +
sin θ [Jy beam−1 ]
∂s
Q
∂x
∂y
Q
∂x
∂y
rad

(Herron, 2005)
Polarization gradient (radial component;
maximum value):
v



u
u Q ∂Q + U ∂U 2 + Q ∂Q + U ∂U 2
t
∂x
∂x
∂y
∂y
∂ P~
=
[Jy beam−1 ]
∂s
Q2 + U 2
rad,max

(Herron, 2005)
Polarization gradient
(tangential
component):






∂ P~
U
∂Q
U
∂U
∂Q
∂U
= − sin arctan
cos θ +
sin θ + cos arctan
cos θ +
sin θ [Jy beam−1 ]
∂s
Q
∂x
∂y
Q
∂x
∂y
tan

(Herron, 2005)
Polarization gradient (tangential
maximum value):
v  component;



u
u Q ∂U − U ∂Q 2 + Q ∂U − U ∂Q 2
~
t
∂x
∂x
∂y
∂y
∂P
=
[Jy beam−1 ]
∂s
Q2 + U 2
tan,max

(Herron, 2005)
Polarization fraction:
p(λ2 ) =

P (λ2 )
[dimensionless]
I(λ2 )

Polarized intensity:
P ≡

p
Q2 + U 2 [Jy]

Polarized intensity standard error:
s
2

2
∂||P ||
∂||P ||
2 +
2
σQ
σU
σP =
∂Q
∂U
s



Q2
U2
2 +
2
=
σ
σU
Q
Q2 + U 2
Q2 + U 2
s
Q2 2
U2 2
σ
+
σ [Jy]
=
||P ||2 Q ||P ||2 U
(Brentjens & de Bruyn, 2005)
Polarized surface brightness:
P (λ2 ) = ||p(λ2 )||Ie2iχ = p(λ2 )I(λ2 ) = Q + iU [Jy]
Z∞
2
=
F (φ)e2iφλ dφ [Jy]
−∞

Polarized surface brightness (absolute):
p
||P (λ2 )|| = Q2 + U 2 [Jy]

Galactic Magnetism Study Guide

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Polarized surface brightness (observed):
2

2

2

Z∞

2

P̃ (λ ) ≡ W (λ )P (λ ) = W (λ )

2

F (φ)e2iφ(λ

−λ20 )

dφ [Jy]

−∞

Power spectrum:

∗
P (k) = fˆ(k)fˆ(k)

Reynolds number:

ρvL
[dimensionless],
µ
Reynolds number for ambipolar diffusion:
LV
[dimensionless]
RAD =
νAD
Z
RHT backprojection:
R(x, y) = R(θ, x, y)dθ [dimensionless]
Re =

Z around
Z
RHT backprojection sampling disk
star:
R∗ (x, y) =
R(θ, x, y)dxdy [dimensionless]
disk

RHT distance:
ρ = x cos θ + y sin θ [pixels]
Rotation measure:

dχ(λ2 )
[rad m−2 ]
dλ2
observer
Z
~ [rad m−2 ]
~ · dl
RM = −0.81
ne B
RM =

source

Rotation measure transfer function
(RMTF):
R∞
2
2
W (λ2 )e−2iφ(λ −λ0 ) dλ2
−∞
[rad m−2 ]
R(φ) ≡
R∞
2
2
W (λ )dλ
−∞

Rotation measure transfer function (RMTF)
for box-car
weight function W (λ2 ):


2
2
sin(φ∆λ
)
[rad m−2 ]
R(φ) = eiφλc
φ∆λ2
Skewness:

4
N 
1 X xi − µ
Skew[x] =
− 3 [dimensionless]
N i=1
σ

Sonic Mach number:


Ms =

|~v |
cs

Spectral correlation function:

[dimensionless]

n

Sx,y ≡ Svi ,vj =
Spectral irradiance:



1X
T ([x, y]a , vi )T ([x, y]a , vj )
n a=1


8πhν 3
1
Eν (T ) =
[W m−2 Hz−1 ]
2
hν/k
T −1
B
c
e


8πhc2
1
Eλ (T ) =
[W m−2 m−1 ]
5
hc/λk
BT − 1
λ
e


Stokes Q (RHT):

Z
QRHT =

Stokes Q (RHT):

cos(2θ)R(θ)dθ [Jy]
Z

URHT =

sin(2θ)R(θ)dθ [Jy]

Stokes V magnetic field:
V (ν) ∝ Bk ×

Galactic Magnetism Study Guide

dI
[Jy]
dν

Page 35 / 36

Structure function:

Sp (δr) = h|A(r) − A(r + δr)|p i,

Synchrotron radiation angle:

m c c2
[rad]
E
Synchrotron radiation characteristic frequency:
3 q 2
γ (B sin θ) [Hz]
νc =
4π mc
eB
[Hz]
νc ∼ γ 2
2πm0
Synchrotron radiation maximum polarization:
3γ + 3
[dimensionless]
p=
3γ + 7
Synchrotron radiation power:
2 q4
P =
(γν)2 (B sin θ)2 [Jy]
3 m2 c5
Wavelength squared distribution
standard error:
"
!#−1
X
X
1
4
−1
2 2
[m−2 ]
σλ2 =
λi − N (
λi )
N −1
i
i
θ=±

Zeeman splitting energy difference:

∆E = −µB mF gB [J]

Zeeman splitting frequency difference:
µB mF gB
[Hz]
∆ν =
h

Galactic Magnetism Study Guide

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