Known radio pulsars appear to emit short pulses of radio radiation with pulse periods between 1.4 ms and 8.5 seconds. Even though the word pulsar is a combination of "pulse" and "star," pulsars are not pulsating stars. Their radio emission is actually continuous but beamed, so any one observer sees a pulse of radiation each time the beam sweeps across his line-of-sight. Since the pulse periods equal the rotation periods of spinning neutron stars, they are quite stable. Even though the radio emission mechanism is not well understood, radio observations of pulsars have yielded a number of important results because:

(1) Neutron stars are physics laboratories providing extreme conditions (deep gravitational potentials, densities exceeding nuclear densities, magnetic field strengths as high as $B \sim 10^{14}$ or even $10^{15}$ gauss) not available on Earth.

(2) Pulse periods can be measured with accuracies approaching 1 part in $10^{16}$, permitting exquisitely sensitive measurements of small quantities such as the power of gravitational radiation emitted by a binary pulsar system or the gravitational perturbations from planetary-mass objects orbiting a pulsar.

The radical proposal that neutron stars exist was made with trepidation by Baade & Zwicky in 1934: "With all reserve we advance the view that a supernova represents the transition of an ordinary star into a new form of star, the neutron star, which would be the end point of stellar evolution. Such a star may possess a very small radius and an extremely high density." Pulsars provided the first evidence that neutron stars really do exist. They tell us about the strong nuclear force and the nuclear equation of state in new ranges of pressure and density, test general relativity and alternative theories of gravitation in both shallow and relativisitically deep ($GM/(rc^2) \gg 0$) potentials, and led to the discovery of the first extrasolar planets.

Pulsars were discovered
serendipidously in 1967 on chart-recorder records obtained during a
low-frequency ($\nu = 81$ MHz) survey of extragalactic radio sources
that scintillate in the interplanetary plasma, just as stars twinkle in
the Earth's atmosphere. This important discovery remains a warning
against overprocessing data before looking at them, ignoring
unexpected signals, and failing to explore observational "parameter
space" (here the relevant parameter being time). As radio
instrumentation and data-taking computer programs become more
sophisticated, signals are "cleaned up"
before they reach the astronomer and optimal "matched filtering" tends
to suppress the unexpected. Thus clipping circuits are used to remove
the strong impulses that are usually caused by terrestrial
interference, and integrators smooth out fluctuations shorter than the
integration time. Pulsar signals "had been recorded but not recognized"
several years earlier with the 250-foot Jodrell Bank telescope. Most
pulses seen by radio astronomers are just artificial interference from
radar, electric cattle fences, etc., and short pulses from sources at
astronomical distances imply unexpectedly high brightness temperatures
$T_{\rm b} \sim 10^{25}$–$10^{30} {\rm ~K} \gg 10^{12}$ K, the upper
limit for incoherent electron-synchrotron radiation set by
inverse-Compton scattering. However, Cambridge University graduate
student Jocelyn Bell noticed pulsars in her scintillation survey data
because the pulses appeared earlier by about 4 minutes every solar day,
so they appeared exactly once per sidereal day and thus came from
outside the solar system.

Figure
1. "High-speed" chart
recording of the first known pulsar, CP1919. This confirmation
observation showed that the "scruffy" signals
observed previously were periodic.

The sources and emission mechanism
were originally unknown, and even
intelligent transmissions by LGM ("little green men") were seriously
suggested as explanations for pulsars. Astronomers were used to slowly
varying or pulsating emission from stars, but the natural period of a
radially pulsating star depends on its mean density $\rho$ and is
typically days, not seconds. Likewise there is a lower limit to the
rotation period $P$ of a gravitationally bound star, set by the
requirement that the centrifugal acceleration at its equator not exceed
the gravitational acceleration. If a star of mass $M$ and radius $R$
rotates with angular velocity $\Omega = 2 \pi / P$,

$$\Omega^2 R < {G M \over R^2}$$

$${4 \pi^2 R^3 \over P^2} < GM$$

$$P^2 > \biggl( { 4 \pi R^3 \over
3} \biggr) {3 \pi \over G M}$$

In terms of the mean density

$$\rho =
M \biggl( { 4 \pi R^3 \over
3} \biggr)^{-1}~,$$

$$P > \biggl( { 3 \pi \over G \rho}
\biggr)^{1/2}$$

or

$$\bbox[border:3px blue solid,7pt]{\rho
> { 3 \pi \over G P^2}}\rlap{\quad \rm {(6A1)}}$$

This is actually a very conservative
lower limit to $\rho$ because
a rapidly spinning star becomes oblate, increasing the centrifugal
acceleration and decreasing the gravitational acceleration at its
equator.

Example: The first pulsar discovered (CP 1919+21, where the "CP" stands for Cambridge pulsar) has a period $P = 1.3$ s. What is its minimum mean density?

$$\rho > { 3 \pi \over G P^2} = { 3 \pi \over 6.67 \times 10^{-8} {\rm ~dyne~cm}^2 {\rm ~gm}^{-2} (1.3 {\rm ~s})^2 } \approx 10^8 {\rm ~g~cm}^{-3}$$

This density limit is just consistent
with the densities of
white-dwarf stars. But soon the faster ($P = 0.033$ s) pulsar in the
Crab Nebula was discovered, and its period implied a density too high
for any stable white dwarf. The Crab nebula is the remnant of a
known supernova recorded by ancient Chinese astronomers as a "guest
star" in 1054 AD, so the discovery of this pulsar also confirmed the
Baade & Zwicky suggestion that neutron stars are the compact
remnants of supernovae. The fastest known
pulsar has $P = 1.4\times10^{-3}$ s implying $\rho >
10^{14}$ g cm$^{-3}$, the density of nuclear matter. For a star of mass
greater than the Chandrasekhar
mass

$$M_{\rm Ch} \approx \biggl( {h c \over 2 \pi G} \biggr)^{3/2} {1 \over
m_{\rm p}^2} \approx 1.4 M_\odot$$ (compact stars less massive than
this are
stable as white dwarfs), the maximum
radius
is

$$R < \left( { 3 M \over 4 \pi \rho } \right)^{1/3}$$

In the
case of the $P = 1.4 \times 10^{-3}$ pulsar with $\rho >
10^{14}$ g cm$^{-3}$,

$$ R < \left( { 3 \times 1.4 \times 2.0
\times 10^{33} {\rm ~g} \over 4 \pi \times 10^{14} {\rm ~g~cm}^{-3} }
\right)^{1/3} \approx 2 \times 10^6 {\rm ~cm} \approx 20 {\rm ~km}$$

The canonical
neutron star has $M \approx 1.4 M_\odot$ and $R \approx
10$ km, depending on the equation-of-state of extremely dense matter
composed of neutrons,
quarks, etc. The extreme density and pressure turns most of the star
into a neutron superfluid that is a superconductor up to temperatures
$T \sim 10^9$ K. Any star of significantly higher mass ($M \sim 3
M_\odot$ in standard models) must collapse and become a black hole. The
masses of several neutron stars have been measured with varying degrees
of accuracy, and all turn out to be very close to $1.4 M_\odot$.

The Sun and many other stars are known
to possess roughly dipolar
magnetic fields. Stellar interiors are mostly ionized gas and hence
good electrical
conductors. Charged particles are constrained to move along magnetic
field lines and, conversely, field lines are tied to the particle mass
distribution. When the core of a star collapses from a size $\sim
10^{11}$ cm to $\sim 10^6$ cm, its magnetic flux $\Phi \equiv \int
\vec{B} \cdot \vec{n}\, da$ is conserved and the
initial magnetic field strength is multiplied by $\sim 10^{10}$, the
factor by which the cross-sectional area $a$ falls. An initial magnetic
field strength of $B \sim 100$ G becomes $B \sim 10^{12}$ G after
collapse, so young neutron stars should have very strong dipolar
fields. The best models of the core-collapse process show that a
dynamo effect may generate an even larger
magnetic field. Such dynamos are thought to be able to produce
the $10^{14}-10^{15}$G fields in magnetars,
neutron stars having such strong magnetic fields that their radiation
is powered by magnetic field decay.
Conservation of angular
momentum during collapse increases the rotation rate by about the same
factor, $10^{10}$,
yielding initial periods in the millisecond range. Thus young neutron
stars should drive rapidly rotating magnetic dipoles.

Figure 2: A Pulsar. (left or top): A diagram of the traditional magnetic dipole model of a pulsar. (right or bottom) Diagram of a simple dipole magnetic field near the polar caps. The inset figure shows a schematic of the electon-positron cascade which is required by many models of coherent pulsar radio emission (Both figures are from the Handbook of Pulsar Astronomy by Lorimer and Kramer).

If the magnetic
dipole is inclined by
some angle $\alpha > 0$
from the rotation axis, it emits low-frequency electromagnetic
radiation. Recall the Larmor formula for radiation from a rotating
electric dipole:

$$P_{\rm rad} = { 2 q^2 \dot{v}^2
\over 3 c^3} = {2 \over 3} { (q
\ddot{r} \sin \alpha)^2 \over c^3} = {2 \over 3} { ( \ddot{p}_\bot )^2
\over c^3}~,$$

where $p_\bot$ is the perpendicular
component of the electric
dipole moment. By analogy, the power of

the magnetic
dipole radiation from an inclined magnetic
dipole is

$$\bbox[border:3px blue solid,7pt]{P_{\rm rad} = {2 \over 3} { (
\ddot{m}_\bot )^2 \over
c^3}}\rlap{\quad \rm {(6A2)}}$$

where $m_\bot$ is the perpendicular component of the magnetic
dipole moment. For a uniformly magnetized sphere with radius $R$ and
surface magnetic field strength $B$, the magnetic dipole moment is (see
Jackson's * Classical Electrodynamics*)

$$m = B R^3~.$$

If the
inclined magnetic dipole rotates with angular velocity $\Omega$,

$$m =
m_0 \exp ( - i \Omega t)$$

$$ \dot{m} = - i \Omega m_0 \exp ( - i
\Omega t)$$

$$ \ddot{m} = \Omega^2 m_0 \exp ( -i \Omega t) = \Omega^2
m$$

so

$$P_{\rm rad} = {2 \over 3} { m_\bot^2 \Omega^4 \over c^3} = {2
m_\bot^2 \over 3 c^3} \biggl( {2 \pi \over P} \biggr)^4 = {2 \over 3
c^3} ( B R^3 \sin \alpha)^2 \biggl( {2 \pi \over P} \biggr)^4~,$$

where $P$ is the pulsar period. This
electromagnetic radiation will
appear at the * very* low frequency $\nu = P^{-1} < 1$ kHz, so
low that it cannot be observed, or even propagate through the ionized
ISM. The huge power radiated is responsible for pulsar slowdown as it
extracts rotational kinetic energy from the neutron star. The absorbed
radiation can also light up a surrounding nebula, the Crab nebula for
example.

The rotational kinetic energy $E_{\rm rot}$ is related to the moment of inertia $I$ by $$E_{\rm rot} = \frac{1}{2}I \Omega^2 = {2 \pi^2 I \over P^2}~.$$

Example: The moment of inertia of the
"canonical" neutron star (uniform-density sphere with $M \approx
1.4 M_\odot$ and $R \approx 10$ km) is

$$I = \frac{2}{5} M R^2 \approx {2
\cdot 1.4 \cdot 2.0 \times 10^{33} {\rm ~g} \cdot (10^6 {\rm ~cm})^2
\over 5 } \approx 10^{45} {\rm ~gm~cm}^2$$

$$ E_{\rm rot} = { 2 \pi^2 I \over
P^2} \approx {2 \pi^2 \cdot 10^{45} {\rm ~g~cm}^2 \over (0.033 {\rm
~s})^2 } \approx 1.8 \times 10^{49} {\rm ~ergs}$$

Pulsars are observed to slow down
gradually:

$$\dot{P} \equiv {d P
\over d t} > 0$$

Note that $\dot{P}$ is dimensionless (e.g., seconds
per second). From the observed period $P$ and period derivative
$\dot{P}$ we can estimate the rate at which the rotational energy is
decreasing.

$${d E_{\rm rot} \over d t} = {d \over d t} \left( \frac{1}{2} I
\Omega^2 \right) = I \Omega \dot{\Omega}$$

$$\Omega = {2 \pi \over P} \qquad {\rm so} \qquad \dot{\Omega} = 2 \pi
(-P^{-2} \dot{P})$$

and

$${d E_{\rm rot} \over d t} = I \Omega
\dot{\Omega} = I {2 \pi \over P} { 2 \pi ( - \dot{P}) \over P^2 }$$

$$\bbox[border:3px blue solid,7pt]{{d E_{\rm rot} \over d t} =
{-4
\pi^2 I \dot{P} \over P^3}}\rlap{\quad \rm {(6A3)}}$$

Example: The Crab pulsar has $P =
0.033$ s and $\dot{P} =
10^{-12.4}$. Its rotational energy is changing at the rate

$${ d E_{\rm
rot} \over d t} = { - 4 \pi^2 I \dot{P} \over P^3} = { - 4 \pi^2 \cdot
10^{45} {\rm ~g~cm}^2 \cdot 10^{-12.4} {\rm ~s~s}^{-1} \over (0.033
{\rm ~s})^3 } \approx -4 \times 10^{38} {\rm ~erg~s}^{-1}$$

Thus the low-frequency (30 Hz)
magnetic dipole radiation from the
Crab pulsar radiates a huge power $P_{\rm rad} \approx - d E_{\rm rot}
/ d t \approx 10^5 L_\odot$, comparable with the entire radio
output of our Galaxy. It exceeds the Eddington limit, but that is not a
problem because the energy source
is not accretion. It greatly exceeds
the average radio pulse luminosity of the Crab pulsar, $\sim
10^{30}$ erg s$^{-1}$. The long-wavelength magnetic dipole radiation
energy is absorbed by and powers the Crab nebula (a "megawave oven").

Figure 3: Composite image of
the Crab nebula. Blue indicates X-rays (from Chandra), green is
optical (from the HST), and red is radio (from the VLA). Image credit

If we use $- d E_{\rm rot} / d t$ to
estimate $P_{\rm rad}$, we can
invert Larmor's formula for magnetic dipole radiation to find $B_\bot =
B \sin \alpha$ and get a lower limit to the surface magnetic field
strength $B > B \sin \alpha$, since we don't generally know the
inclination angle $\alpha$.

$$P_{\rm rad} = - {d E_{\rm rot} \over
d t}$$

$${2 \over 3 c^3} (B R^3 \sin
\alpha)^2 \biggl( { 4 \pi^2 \over P^2}
\biggr)^2 = {4 \pi^2 I \dot{P} \over P^3}$$

$$B^2 = { 3 c^3 I P \dot{P} \over 2
\cdot 4 \pi^2 R^6 \sin^2\alpha
}$$

$$ B > \biggl( { 3 c^3 I \over 8
\pi^2 R^6} \biggr)^{1/2} (P
\dot{P})^{1/2}$$

Evaluating the constants for the
canonical pulsar in cgs units, we
get

$$ \biggl[ { 3 \cdot (3 \times 10^{10}
{\rm ~cm~s}^{-1})^3 \cdot
10^{45} {\rm ~g~cm}^2 \over 8 \pi^2 (10^6 {\rm ~cm})^6 } \biggr]^{1/2}
\approx 3.2 \times 10^{19}$$

so the minimum
magnetic field strength at the pulsar surface is

$$\bbox[border:3px blue solid,7pt]{\biggl( { B \over {\rm Gauss}}
\biggr) > 3.2 \times 10^{19}
\biggl( { P \dot{P} \over {\rm s} } \biggr)^{1/2}}\rlap{\quad \rm
{(6A4)}}$$

Example: What is the minimum magnetic field strength of the Crab pulsar ($P = 0.033$ s, $\dot{P} = 10^{-12.4}$)?

$$ \biggl( { B \over {\rm Gauss} } \biggr) > 3.2 \times 10^{19} \biggl( { 0.033 {\rm ~s} \cdot 10^{-12.4} \over {\rm s} } \biggr) = 4 \times 10^{12}$$

This is an amazingly strong magnetic
field. Its energy density is

$$U_{\rm B} = { B^2 \over 8 \pi} > 5 \times 10^{23} {\rm
~erg~cm}^{-3}$$

Just one cm$^3$ of this magnetic field contains over $5
\times 10^{16} {\rm ~J} = 5 \times 10^{16} {\rm ~W~s} = 1.6 \times 10^9
{\rm ~W~yr}$ of energy, the annual output of a large nuclear power
plant. A cubic meter contains more energy than has ever been generated
by mankind.

If $(B \sin\alpha)$ doesn't change
significantly with time, we can
estimate a pulsar's age $\tau$ from $P \dot{P}$ by assuming that the
pulsar's initial period $P_0$ was much shorter than the current period.
Starting with

$$B^2 = { 3 c^3 I P \dot{P} \over 8 \pi^2 R^6 \sin^2
\alpha}$$

we find that

$$P \dot{P} = {8 \pi^2 R^6 (B \sin \alpha)^2
\over 3 c^3 I }$$

doesn't change with time. Rewriting the identity $ P
\dot{P} = P \dot{P}$ as $ P dP = P \dot{P} d t$ and integrating over
the pulsar's lifetime $\tau$ gives

$$\int_{P_0}^P P d P = \int_0^\tau
(P \dot{P})\, d t = P \dot{P} \int_0^\tau d t $$

since $P \dot{P}$ is
assumed to be constant over time.

$${P^2 - P_0^2 \over 2} = P \dot{P}
\tau$$

If $P_0^2 \ll P^2$, the
characteristic age of the
pulsar is

$$\bbox[border:3px blue solid,7pt]{\tau \equiv { P \over 2
\dot{P}}}\rlap{\quad \rm {(6A5)}}$$

Note that the characteristic age is
not affected by uncertainties
in the radius $R$, moment of inertia $I$, or $B \sin \alpha$; the only
assumptions in its derivation are that $P_0 \ll P$ and that $P\dot{P}$
(i.e. $B$) is constant.

Example: What is the characteristic age of the Crab pulsar ($P = 0.033$ s, $\dot{P} = 10^{-12.4}$)?

$$ \tau = {P \over 2 \dot{P} } = {0.033 {\rm ~s} \over 2 \cdot 10^{-12.4}} \approx 4.1 \times 10^{10} {\rm ~s} \approx {4.1 \times 10^{10} {\rm ~s} \over 10^{7.5} {\rm ~s~yr}^{-1} } \approx 1300 {\rm ~yr}$$

Its actual age is about 950 years.

Figure 4: P-Pdot
Diagram. The $P \dot{P}$ diagram is
useful for
following the lives of
pulsars, playing a role similar to the Hertzsprung-Russell diagram for
ordinary stars. It encodes a tremendous amount of information
about the pulsar population and its properties, as determined and
estimated from two of the primary observables, $P$ and $\dot P$.
Using those parameters, one can estimate the pulsar age, magnetic field
strength $B$, and spin-down power $\dot E$. (From the Handbook of
Pulsar Astronomy, by
Lorimer and Kramer)

Pulsars are born in supernovae and
appear in the upper left corner
of the $P
\dot{P}$ diagram. If $B$ is conserved and they age as
described above, they gradually move to the right and down, along lines
of constant $B$ and crossing lines of constant characteristic age.
Pulsars with characteristic ages $ < 10^5$ yr are often found in
or near recognizable supernova remnants. Older pulsars are not, either
because their SNRs have faded to invisibility or because the supernova
explosions expelled the pulsars with enough speed that they have since
escaped from their parent SNRs. The bulk of the pulsar population is
older than $10^5$ yr but much younger than the Galaxy ($\sim
10^{10}$ yr). The observed distribution of pulsars
in the $P \dot{P}$ diagram indicates that something changes as pulsars
age. One controversial possibility is that the
magnetic fields of old pulsars must
decay on time scales $\sim 10^7$ yr, causing old pulsars to move almost
straight down in the $P \dot{P}$ diagram until they fall
below into the graveyard below the
death line and cease
radiating radio pulses.

Almost all short-period pulsars below the spin-up line near $\log [\dot{P}/P({\rm sec})] \approx -16$ are in binary systems, as evidenced by periodic (i.e. orbital) variations in their observed pulse periods. These recycled pulsars have been spun up by accreting mass and angular momentum from their companions, to the point that they emit radio pulses despite their relatively low magnetic field strengths $B \sim 10^8$ G (the accretion causes a substantial reduction in the magnetic field strength). The magnetic fields of neutron stars funnel ionized accreting material onto the magnetic polar caps, which become so hot that they emit X-rays. As the neutron stars rotate, the polar caps appear and disappear from view, causing periodic fluctuations in X-ray flux; many are detectable as X-ray pulsars.

Millisecond
pulsars (MSPs) with
low-mass ($M \sim 0.1-1
M_\odot$) white-dwarf companions
typically have orbits with small eccentricities. Pulsars with
extremely
eccentric orbits usually have neutron-star companions, indicating that
these
companions also exploded as supernovae and nearly disrupted the binary
system. Stellar interactions in globular clusters cause a much
higher fraction of recycled pulsars per unit mass than in the Galactic
disk. These interactions can result in very strange systems such
as pulsar–main-sequence-star binaries and MSPs in highly eccentric
orbits. In both cases, the original low-mass companion star that
recycled the pulsar was ejected in an interaction and replaced by
another star. (The eccentricity
$e$ of an elliptical orbit is defined as the ratio of the separation of
the foci to the length of
the major axis. It ranges between $e =0$ for a circular orbit and
$e = 1$ for a parabolic orbit.)

A few millisecond pulsars are
isolated. They were probably recycled via the standard scenario in
binary systems, but the energetic millisecond pulsars eventually
ablated their companions away.

Figure
5: Examples of Doppler variations observed in binary systems containing
pulsars.
(left or top) The Doppler variations of the globular cluster MSP
J1748$-$2446N
in Terzan 5. This pulsar is in a nearly circular orbit
(eccentricity $e =
4.6\times10^{-5}$) with a companion of minimum mass 0.47
M$_\odot$. The difference between the semi-major and semi-minor
axes
for this orbit is only 51$\pm$4 cm! The thick red lines show the
periods as measured during GBT observations. (right or bottom) Similar
Doppler
variations from the highly eccentric binary MSP J0514$-$4002A in the
globular cluster NGC 1851. This pulsar has one of the most
eccentric orbits known ($e = 0.888$) and a massive white dwarf or
neutron-star companion.

The radio pulses originate in the
pulsar magnetosphere. Because the
neutron star is a spinning magnetic dipole, it acts as a
unipolar generator. The total
Lorentz force acting on a charged particle
is

$$\vec{F} = q \biggl(\vec{E} + {\vec{v} \times \vec{B} \over c}
\biggr)~.$$

Charges in the magnetic equatorial region redistribute
themselves by moving along closed field lines until they build up an
electrostatic field large enough to cancel the magnetic force and give
$\vert\vec{F}\vert = 0$. The voltage induced is about $10^{16}$ V in
MKS units. However, the co-rotating field lines emerging from the polar
caps cross the
light cylinder
(the cylinder centered on the pulsar and aligned with the rotation axis
at whose radius the co-rotating
speed equals the speed of light) and these field lines cannot close.
Electrons in the polar cap are magnetically accelerated to very high
energies along the open but curved field lines, where the acceleration
resulting from the curvature causes them to emit
curvature
radiation that is strongly polarized in the plane of curvature.
As
the radio beam sweeps across the line-of-sight, the plane of
polarization is observed to rotate by up to 180 degrees, a purely
geometrical effect. High-energy photons produced by curvature radiation
interact with the magnetic field and lower-energy photons to produce
electron-positron pairs that radiate more high-energy photons. The
final results of this cascade process are bunches of charged particles
that emit at radio wavelengths. The death line in the $P \dot{P}$
diagram corresponds to neutron stars with sufficiently low $B$ and high
$P$ that the curvature radiation near the polar surface is no longer
capable of generating particle cascades. The extremely high brightness
temperatures are explained by
coherent radiation. The electrons do not
radiate as independent charges $e$; instead bunches of $N$ electrons in
volumes whose dimensions are less than a wavelength emit in phase as
charges $Ne$. Since Larmor's formula indicates that the power
radiated by a chage $q$ is proportional to $q^2$, the radiation
intensity can be $N^2$ times brighter than
incoherent radiation from the same total number $N$ of electrons.
Because the coherent volume is smaller at shorter wavelengths, most
pulsars have extremely steep radio spectra. Typical (negative)
pulsar spectral indices are $\alpha \sim$1.7 ($S \propto\nu^{-1.7}$),
although
some can be much steeper ($\alpha >3$) and a handful are almost flat
($\alpha<0.5$).

(Note: the following closely follows
the discussion in the Handbook of
Pulsar
Astronomy by Lorimer and Kramer)

With their sharp and short-duration
pulse profiles and very high brightness temperatures, pulsars are
unique probes of the interstellar medium (ISM). The electrons in
the ISM make up a cold plasma having a refractive
index

$$\mu = \biggl[{1 -
\left(\frac{\nu_{\rm p}}{\nu}\right)^2}\biggr]^{1/2}~,$$

where $\nu$ is the frequency of
the radio waves, $\nu_{\rm p}$ is the
plasma
frequency

$$\bbox[border:3px blue solid,7pt]{\nu_{\rm p} =
\biggl({e^2 n_{\rm e} \over \pi m_{\rm e}}\biggr)^{1/2} \approx 8.97
{\rm ~kHz} \times\biggl({n_{\rm e} \over {\rm
cm}^{-3}}\biggr)^{1/2}}\rlap{\quad \rm {(6A6)}}$$

and $n_{\rm e}$ is
the electron number density. For a typical ISM value $n_{\rm e}
\sim 0.03$ cm$^{-3}$,
$\nu_{\rm p}\sim1.5$
kHz. If $\nu < \nu_{\rm p}$ then $\mu$ is imaginary and
radio waves cannot propagate through the plasma.

For propagating radio waves, $\mu <
1$ and the group
velocity
$v_{\rm g} = \mu c$ of pulses is less than the vacuum speed of
light. For most radio observations $\nu_{\rm p} \ll \nu$ so

$$\bbox[border:3px blue solid,7pt]{v_{\rm g}\approx c\biggl(1 -
\frac{\nu_{\rm p}^2}{2\nu^2}\biggr)}\rlap{\quad \rm {(6A7)}}$$

A broadband pulse moves through a
plasma more slowly at lower frequencies than at higher
frequencies. If the distance to the source is $d$, the dispersion
delay $t$ at frequency $\nu$ is

$$t = \int_0^d v_{\rm g}^{-1} dl - \frac{d}{c} =
\frac{1}{c}\int_0^d \biggl(1 + \frac{\nu_p^2}{2\nu^2}\biggr) dl -
\frac{d}{c}$$

$$ = \frac{e^2}{2\pi m_{\rm e} c} \frac{\int_0^dn_{\rm e}
dl}{\nu^2}.$$

In astronomically convenient units this becomes

$$\bbox[border:3px blue solid,7pt]{\biggl({t \over {\rm sec}}\biggr)
\approx
4.149\times10^3 \biggl({{\rm DM} \over {\rm pc~cm}^{-3}}\biggr)
\biggl({\nu \over {\rm MHz}}\biggr)^{-2}}\rlap{\quad \rm {(6A8)}}$$

where

$$\bbox[border:3px blue solid,7pt]{{\rm DM} \equiv \int_0^dn_{\rm e}
dl}\rlap{\quad \rm {(6A9)}}$$

in units of pc cm$^{-3}$
is called the dispersion
measure.

Figure 6: Pulsar dispersion. Uncorrected dispersive delays for a pulsar observation over a bandwidth of 288 MHz (96 channels of 3 MHz width each), centered at 1380 MHz. The delays wrap since the data are folded (i.e. averaged) modulo the pulse period. (From the Handbook of Pulsar Astronomy, by Lorimer and Kramer)

Measurements of the dispersion measure can provide distance estimates to pulsars. Crude estimates can be made for pulsars near the Galactic plane assuming that $n_{\rm e} \sim 0.03$ cm$^{-3}$. However, several sophisticated models of the Galactic electron-density distribution now exist (e.g. NE2001; Cordes & Lazio 2002, astro-ph/0207156) that can provide much better ($\Delta d / d \sim 30\%$) distance estimates.

Since pulsar observations almost
always cover a wide bandwidth, uncorrected differential delays across
the
band will cause dispersive
smearing
of the pulse profile. For pulsar searches, the DM is unknown and
becomes a search parameter much like the pulsar spin
frequency. This extra search dimension is one of the primary
reasons that pulsar searches are computationally intensive.

Besides directly determining the
integrated electron density along the line of site, observations of
pulsars can be used to probe the ISM via absorption by spectral lines
of HI or molecules
(which can be used to estimate the pulsar distance as well),
scintillation (allowing estimates of the pulsar transverse velocity),
and pulse broadening.

Figure
7: Pulsar HI Absorption Measurement. With a model for the
Galactic rotation, such absorption measurements can provide pulsar
distance estimates or constraints. (From the Handbook of Pulsar
Astronomy, by
Lorimer and Kramer)

Figure
8: Thin Screen Diffraction/Scattering model.
Inhomogeneities in the ISM cause small-angle deviations in the paths of
the radio waves. These deviations result in time (and therefore
phase) delays that can interfere to create a diffraction pattern,
broaden the pulses in time, and make a larger image of the pulsar on
the sky. (From the Handbook of
Pulsar Astronomy, by
Lorimer and Kramer)

Figure
9: Pulse broadening caused by scattering. Scattering of the
pulsed signal by ISM inhomogeneities results in delays that cause a
scattering tail. This scatter-broadening can greatly decrease
both the observational sensitivity and the timing precision for such
pulsars. (From the Handbook of
Pulsar Astronomy, by
Lorimer and Kramer)

Figure 10: Diffractive Scintillation of a Pulsar. The top plots show dynamic spectra of the bright pulsar B0355+54 taken on two consecutive days with the GBT. The bottom plots show the so-called secondary spectra (the Fourier transforms of the dynamic spectra) and the so-called scintillation arcs (and moving arclets). (Figure provided by Dan Stinebring)