Chapter 6 Pulsars

Pulsars were discovered [50] serendipitously in 1967 on chart recordings (Figure 6.1) obtained during a low-frequency ($\nu =81$ MHz) search for compact extragalactic radio sources that scintillate in the interplanetary plasma, just as stars twinkle in the Earth’s atmosphere.

This history of this important discovery is a warning against overprocessing data before looking at them, ignoring unexpected signals, and failing to explore the observational “parameter space” covered by the data—here the relevant parameter being time. The strong pulsar PSR B0329+54 was clearly visible in 1954 survey data from the Jodrell Bank 250-foot radio telescope [70], but nobody paid any attention to it at the time. The X-ray pulsar in the Crab Nebula (Figure 8.10) was present in data taken several months before the discovery of radio pulsars, but only after the radio pulsar in the Crab Nebula was announced were the X-ray pulses extracted [39]. As radio instrumentation and data-processing software become more sophisticated, more data are “cleaned up” automatically before they reach the astronomer. Matched filtering that brings out the expected signal usually suppresses the unexpected. Thus, clipping circuits or software remove the strong impulses that are usually caused by terrestrial interference, and integrators smooth out fluctuations shorter than the integration time. Most pulses seen by radio astronomers are just artificial interference from radar, electric cattle fences, etc., and short pulses from sources at interstellar distances imply unexpectedly high brightness temperatures $T_{\mathrm{b}}>{10}^{25}\mathrm{K}$, which is much higher than the $\sim {10}^{12}$ K upper limit for incoherent electron-synchrotron radiation imposed by synchrotron self-Compton cooling (Section 5.5.3).

Cambridge University graduate student Jocelyn Bell recognized that pulsars are astronomical sources where others had failed because she noticed that some “scruffy” pulses in her chart-recorder data (Figure 6.1) didn’t look like other forms of interference and they reappeared exactly once per sidereal day, indicating an origin outside the Solar System. Fortunately, she and her supervisor, Antony Hewish, “decided initially not to computerize the output because until we were familiar with the behavior of our telescope and receivers we thought it better to inspect the data visually, and because a human can recognize signals of different character whereas it is difficult to program a computer to do so.” Read http://www.bigear.org/vol1no1/burnell.htm for the full discovery story in her own words.

The sources of the pulses were originally unknown, and even intelligent transmissions by “LGM” (Little Green Men) were seriously suggested as explanations for pulsars. Their short periods imply very compact sources such as white dwarf stars, black holes, and neutron stars; their stable periods rule out black holes. Astronomers were familiar with 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. There is a comparable 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 nearly spherical star of mass $M$ and radius $R$ rotates with angular velocity $\mathrm{\Omega }\equiv 2\pi /P$,

(6.1)

implies

$$P^2>\left(\frac{4\pi R^3}{3}\right)\frac{3\pi }{GM}.$$

(6.2)

In terms of the mean density

	$$\rho \equiv M{\left(\frac{4\pi R^3}{3}\right)}^{-1},$$		(6.3)
	$$P>{\left(\frac{3\pi }{G\rho }\right)}^{1/2}$$		(6.4)

and

$$\boxed{\rho >\frac{3\pi }{G{P}^2}.}$$

(6.5)

Equation 6.5 gives a conservative lower limit to the mean density because a rapidly spinning star is oblate, which increases the centrifugal acceleration and decreases the gravitational acceleration at its equator.

The first pulsar discovered has a period $P=1.3$ s, so its mean density is at least

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

This limit is just consistent with the known densities of white dwarf stars. But soon the faster ($P=0.033$ s) pulsar in the Crab Nebula was discovered, and its period implies a density much higher than any stable white-dwarf star supported by electron-degeneracy pressure. Also, the Crab Nebula (Figure 8.10) is the remnant of a supernova recorded by Chinese astronomers as a “guest star” in 1054 AD, so the discovery of this pulsar confirmed the suggestion by Baade and Zwicky [5] that neutron stars are the compact remnants of supernovae. The fastest known pulsar has $P=1.4\times {10}^{-3}$ s implying $\rho >{10}^{14}$ g cm${}^{-3}$, the density of atomic nuclei.

A star whose mass is greater than the Chandrasekhar mass

$$M_{\mathrm{Ch}}\sim {\left(\frac{\mathrm{\hslash }c}{G}\right)}^{3/2}\frac{1}{m_{\mathrm{p}}^2}\approx 1.4M_{\odot }$$

(6.6)

cannot be supported by electron degeneracy pressure and will collapse to become a neutron star. Equation 6.2 implies the maximum radius

(6.7)

of a $P=1.4\times {10}^{-3}$ s pulsar with mass $M\approx 1.4M_{\odot }$ is

	$R$
		$\approx 2\times {10}^6\mathrm{cm}=20\mathrm{km}.$		(6.8)

These numbers motivate the definition of the canonical neutron star as being a uniform density sphere with mass $M\approx 1.4M_{\odot }$, radius $R\approx 10$ km, and moment of inertia $I=2M{R}^2/5\approx {10}^{45}\mathrm{g}{\mathrm{cm}}^2$. “Canonical” is perhaps too strong a term for what is really no more than a convention for the typical properties of neutron stars; individual neutron stars can have different masses and radii. The extreme density and pressure turn most of the star into a neutron superfluid that is a superconductor at temperatures up to $T\sim {10}^9$ K. The masses of individual pulsars have been measured with varying degrees of accuracy, and many are close to the canonical $1.4M_{\odot }$. The highest accurately measured pulsar masses are $M\approx 2.0M_{\odot }$, and they rule out all currently proposed hyperon or boson condensate equations of state and “free” quarks [34]. Any neutron star of significantly higher mass ($M\sim 3M_{\odot }$ in standard models) must collapse and become a black hole.

The Sun and many other stars are known to possess approximately dipolar magnetic fields. Stellar interiors are fully ionized and hence good electrical conductors. Charged particles are constrained to move along magnetic field lines, and magnetic field lines are tied to the charged particles. When a star collapses from a radius $\sim {10}^6$ km to $\sim 10$ km, its cross-sectional area $a$ is divided by $\sim {10}^{10}$, its magnetic flux $\mathrm{\Phi }\equiv \int \overrightarrow{B}\cdot \widehat{n}𝑑a$ (where $\widehat{n}$ is the unit vector normal to each infinitesimal surface area $da$) is conserved, and the magnetic field strength is multiplied by $\sim {10}^{10}$. An initial magnetic field strength $B\sim 100$ 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 can generate even stronger magnetic fields. Such dynamos may be able to produce the ${10}^{14}$–${10}^{15}$ G fields observed in magnetars, which are 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 rotation periods $P_0$ in the millisecond range. Thus young neutron stars should contain rapidly rotating magnetic dipoles (Figure 6.2).

If a rotating magnetic dipole is inclined by some inclination angle $\alpha >0$ from the rotation axis, it emits electromagnetic radiation at the rotation frequency. Rewriting the Larmor formula (Equation 2.143) in terms of power radiated by a rotating electric dipole gives

$$P_{\mathrm{rad}}=\frac{2q^2{\dot{v}}^2}{3c^3}=\frac{2}{3}\frac{{(q\ddot{r}\sin \alpha )}^2}{c^3}=\frac{2}{3}\frac{{({\ddot{p}}_{\perp })}^2}{c^3},$$

(6.9)

where $p=qr$ is the electric dipole moment and $p_{\perp }=p\sin \alpha$ is its component perpendicular to the rotation axis. J. J. Thomson’s derivation of the Larmor formula in terms of electric field lines (Section 2.7) is also valid for magnetic field lines. The Gaussian CGS units for magnetic and electric field are the same, so the power of magnetic dipole radiation is

$$\boxed{P_{\mathrm{rad}}=\frac{2}{3}\frac{{({\ddot{m}}_{\perp })}^2}{c^3},}$$

(6.10)

where $m_{\perp }=m\sin \alpha$ is the perpendicular component of the magnetic dipole moment. For a uniformly magnetized sphere with radius $R$ and surface magnetic field strength $B$, the magnitude of the magnetic dipole moment is [55]

$$m=B{R}^3.$$

(6.11)

If the inclined magnetic dipole rotates with angular velocity $\mathrm{\Omega }=2\pi /P$, then

$$P_{\mathrm{rad}}=\frac{2}{3}\frac{m_{\perp }^2{\mathrm{\Omega }}^4}{c^3}=\frac{2}{3c^3}{(B{R}^3\sin \alpha )}^2{\left(\frac{2\pi }{P}\right)}^4,$$

(6.12)

where $P$ is the pulsar period. This electromagnetic radiation will appear at the very low radio frequency kHz, so low that it cannot propagate through the surrounding ionized nebula or ISM. Magnetic dipole radiation extracts rotational kinetic energy from the neutron star and causes the pulsar period to increase with time. The absorbed radiation deposits energy in the surrounding nebula, the Crab Nebula (Figure 8.10) being a prime example.

The rotational kinetic energy $E$ of a spinning object is related to its moment of inertia $I$ by

$$E=\frac{I{\mathrm{\Omega }}^2}{2}=\frac{2{\pi }^{2}I}{P^2}.$$

(6.13)

The moment of inertia of a small mass element about any rotation axis is its mass multiplied by the square of its radial distance $r$ from the rotation axis. The moment of inertia of a sphere with radius $R$, mass $M$, and uniform density $\rho =3M/(4\pi R^3)$ spinning about its $z$-axis is obtained by summing over its mass elements:

$$I={\int }_{-R}^R\left({\int }_0^{{(R^2-z^2)}^{1/2}}\rho r^2\mathrm{\hspace{0.17em}2}\pi r𝑑r\right)𝑑z,$$

(6.14)

where $r$ is the distance from the rotation axis and $z$ is the height in cylindrical coordinates centered on the sphere. Then

$$I=\pi \rho {\int }_0^R{(R^2-z^2)}^2𝑑z=\frac{8\pi \rho R^5}{15}=\frac{2M{R}^2}{5}.$$

(6.15)

The moment of inertia of a canonical neutron star is

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

(6.16)

The rotational kinetic energy of a canonical neutron star with the rotation period $P=0.033$ s of the Crab pulsar is

$$E=\frac{2{\pi }^{2}I}{P^2}\approx \frac{2{\pi }^2\cdot {10}^{45}\mathrm{g}{\mathrm{cm}}^2}{{(0.033\mathrm{s})}^2}\approx 1.8\times {10}^{49}\mathrm{ergs}.$$

As magnetic dipole radiation extracts rotational energy, it slowly increases the period of a pulsar:

$$\dot{P}\equiv \frac{dP}{dt}>0.$$

(6.17)

Note that the period derivative $\dot{P}$ is a dimensionless (seconds per second) pure number. Combining the observed period $P$ and period derivative $\dot{P}$ yields an estimate of the rate $\dot{E}$ at which the rotational energy is changing. The quantity

$$-\dot{E}\equiv -\frac{d{E}_{\mathrm{rot}}}{dt}=-\frac{d}{dt}\left(\frac{1}{2}I{\mathrm{\Omega }}^2\right)=-I\mathrm{\Omega }\dot{\mathrm{\Omega }}$$

(6.18)

is called the spin-down luminosity. It is not a measured luminosity; it is the measured loss rate of rotational energy, which is presumed to equal the luminosity of magnetic dipole radiation. The spin-down luminosity is usually expressed in terms of the pulse period $P$:

$$\mathrm{\Omega }=\frac{2\pi }{P}\mathit{\hspace{1em}\hspace{1em}}\mathrm{so}\mathit{\hspace{1em}\hspace{1em}}\dot{\mathrm{\Omega }}=2\pi (-P^{-2}\dot{P}),$$

(6.19)

and Equation 6.18 becomes

$$\boxed{-\dot{E}=\frac{4{\pi }^{2}I\dot{P}}{P^3}.}$$

(6.20)

The Crab pulsar has $P=0.033$ s and $\dot{P}={10}^{-12.4}$. If $I={10}^{45}\mathrm{g}{\mathrm{cm}}^2$ (Equation 6.16), its spin-down luminosity is

$$-\dot{E}=\frac{4{\pi }^{2}I\dot{P}}{P^3}=\frac{4{\pi }^2\cdot {10}^{45}\mathrm{g}{\mathrm{cm}}^2\cdot {10}^{-12.4}\mathrm{s}{\mathrm{s}}^{-1}}{{(0.033\mathrm{s})}^3}\approx 4\times {10}^{38}\mathrm{erg}{\mathrm{s}}^{-1}\approx {10}^5{L}_{\odot }.$$

If $P_{\mathrm{rad}}\approx -\dot{E}\approx {10}^5{L}_{\odot }$, the luminosity of the low-frequency ($\nu =P^{-1}\approx 30\mathrm{Hz}$) magnetic dipole radiation from the Crab pulsar is comparable with the entire radio output of our Galaxy! It even exceeds the Eddington luminosity limit (Section 5.4.2) of the neutron star, which is possible because the energy source is not accretion. It greatly exceeds the average radio pulse luminosity of the Crab pulsar, $\sim {10}^{30}\mathrm{erg}{\mathrm{s}}^{-1}$. The long wavelength ($\lambda =c/\nu \sim {10}^7$ m) magnetic-dipole radiation is absorbed by and heats up the surrounding Crab Nebula (Figure 8.10), making it the “megawave oven” counterpart of a kitchen microwave oven. The observed bolometric luminosity of the Crab Nebula is comparable with the spin-down luminosity, supporting the model that rotational kinetic energy is converted to magnetic dipole radiation that is absorbed by the surrounding ionized nebula and later reradiated at radio through X-ray wavelengths.

If $-\dot{E}\approx P_{\mathrm{rad}}$, Equations 6.12 and 6.20 can be combined to yield a lower limit to the magnetic field strength $B>B\sin \alpha$ at the neutron star surface, where $\alpha$ is the unknown inclination angle between the rotation and magnetic axes:

	$$P_{\mathrm{rad}}=-\dot{E},$$		(6.21)
	$$\frac{2}{3c^3}{(B{R}^3\sin \alpha )}^2{\left(\frac{4{\pi }^2}{P^2}\right)}^2=\frac{4{\pi }^{2}I\dot{P}}{P^3},$$		(6.22)
	$$B^2=\frac{3c^{3}IP\dot{P}}{2\cdot 4{\pi }^2{R}^6{\sin }^2\alpha },$$		(6.23)
	$$B>{\left(\frac{3c^{3}I}{8{\pi }^2{R}^6}\right)}^{1/2}{(P\dot{P})}^{1/2}.$$		(6.24)

Inserting the constants for the canonical pulsar in CGS units yields for the first factor

$${\left[\frac{3\cdot {(3\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1})}^3\cdot {10}^{45}\mathrm{g}{\mathrm{cm}}^2}{8{\pi }^2{({10}^6\mathrm{cm})}^6}\right]}^{1/2}\approx 3.2\times {10}^{19},$$

(6.25)

so the minimum magnetic field strength at the surface of a canonical pulsar is

$$\boxed{\left(\frac{B}{\mathrm{gauss}}\right)>3.2\times {10}^{19}{\left(\frac{P\dot{P}}{\mathrm{s}}\right)}^{1/2}.}$$

(6.26)

It is sometimes called the characteristic magnetic field of a pulsar.

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

$$\left(\frac{B}{\mathrm{gauss}}\right)>3.2\times {10}^{19}{\left(\frac{0.033\mathrm{s}\cdot {10}^{-12.4}}{\mathrm{s}}\right)}^{1/2}=4\times {10}^{12}.$$

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

$$U_B=\frac{B^2}{8\pi }>6\times {10}^{23}\mathrm{erg}{\mathrm{cm}}^{-3}.$$

(6.27)

Just $1{\mathrm{cm}}^3$ of this magnetic field contains over $6\times {10}^{16}\mathrm{J}=6\times {10}^{16}\mathrm{W}\mathrm{s}\approx 2\mathrm{GW}\mathrm{yr}$ of energy, the annual output of a large nuclear power station.

If the spin-down luminosity equals the magnetic dipole radiation luminosity and $(B\sin \alpha )$ doesn’t change significantly with time, a pulsar’s age $\tau$ can be estimated from $P\dot{P}$ on the further assumption that the pulsar’s initial period $P_0$ was much shorter than its current period. Solving Equation 6.23 for $P\dot{P}$ shows that

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

(6.28)

also doesn’t change with time. Rewriting the identity $P\dot{P}=P\dot{P}$ as $PdP=P\dot{P}dt$ and integrating over the pulsar’s age $\tau$ gives

$${\int }_{P_0}^{P}P𝑑P={\int }_0^{\tau }(P\dot{P})𝑑t=P\dot{P}{\int }_0^{\tau }𝑑t$$

(6.29)

because $P\dot{P}$ is constant. Integrating Equation 6.29 gives

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

(6.30)

In the limit $P_0^2\ll P^2$, the characteristic age of a pulsar defined by

$$\boxed{\tau \equiv \frac{P}{2\dot{P}}}$$

(6.31)

should be close to the actual age of the pulsar. The characteristic age depends only on the observables $P$ and $\dot{P}$; it does not depend on the unknown radius $R$, moment of inertia $I$, or perpendicular magnetic field $B\sin \alpha$. However, a newly formed neutron star with small $P_0$ may be quite oblate and will initially be slowed down by emitting quadrupole gravitational radiation. This can cause the characteristic age to be somewhat larger than the true age for young pulsars.

For example, the characteristic age of the Crab pulsar ($P=0.033$ s, $\dot{P}={10}^{-12.4}$) is

$$\tau =\frac{P}{2\dot{P}}=\frac{0.033\mathrm{s}}{2\cdot {10}^{-12.4}}\approx 4.1\times {10}^{10}\mathrm{s}\approx \frac{4.1\times {10}^{10}\mathrm{s}}{{10}^{7.5}\mathrm{s}{\mathrm{yr}}^{-1}}\approx 1300\mathrm{yr}.$$

It is slightly larger than the actual age of the Crab pulsar, which is known to be just under $1000\mathrm{yr}$ because the Crab supernova was observed in 1054 AD.

If magnetic dipole radiation is solely responsible for pulsar spin down, then $P_{\mathrm{rad}}=-\dot{E}$ and Equations 6.12 and 6.18 together imply

$$\dot{\mathrm{\Omega }}\propto {\mathrm{\Omega }}^3.$$

(6.32)

The pulsar braking index $n$ is defined by

$$\dot{\mathrm{\Omega }}\propto {\mathrm{\Omega }}^n=C{\mathrm{\Omega }}^n,$$

(6.33)

where $C$ is the constant of proportionality. The braking index is completely determined by the observables $P$, $\dot{P}$, and $\ddot{P}$, so comparing its observed value with the expected $n=3$ provides a useful check on pulsar spin-down models. The time derivative of Equation 6.33 is

$$\ddot{\mathrm{\Omega }}=Cn{\mathrm{\Omega }}^{n-1}\dot{\mathrm{\Omega }}=n\left(\frac{C{\mathrm{\Omega }}^n}{\mathrm{\Omega }}\right)\dot{\mathrm{\Omega }}=n\left(\frac{\dot{\mathrm{\Omega }}}{\mathrm{\Omega }}\right)\dot{\mathrm{\Omega }}=\frac{n{\dot{\mathrm{\Omega }}}^2}{\mathrm{\Omega }},$$

(6.34)

so

$$n=\mathrm{\Omega }\ddot{\mathrm{\Omega }}/{\dot{\mathrm{\Omega }}}^2.$$

(6.35)

Converting from angular velocities to periods with

	$$\mathrm{\Omega }=2\pi P^{-1},\dot{\mathrm{\Omega }}=-2\pi P^{-2}\dot{P},\mathrm{and}$$
	$$\ddot{\mathrm{\Omega }}=-2\pi P^{-2}\ddot{P}+\dot{P}4\pi P^{-3}\dot{P}=2\pi \left(\frac{\ddot{P}}{P^2}+\frac{2{\dot{P}}^2}{P^3}\right)$$
yields
	$$n=\left(\frac{2\pi }{P}\right)2\pi \left(\frac{\ddot{P}}{P^2}+\frac{2{\dot{P}}^2}{P^3}\right){(-2\pi P^{-2}\dot{P})}^{-2},$$		(6.36)
	$$n=2-\frac{P\ddot{P}}{{\dot{P}}^2},$$		(6.37)

the braking index in terms of the observables $P$, $\dot{P}$, and $\ddot{P}$.

Braking indices in the range have been observed and used to investigate alternative spin-down mechanisms and to make more sophisticated estimates of pulsar ages and initial spin periods [59].

Pulsars are born in supernovae and appear in the upper left corner of the pulsar $P\dot{P}$ diagram (Figure 6.3). 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 yr are often found in or near recognizable supernova remnants (SNRs). 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 timescales $\sim {10}^7$ yr, causing old pulsars to move almost straight down in the $P\dot{P}$ diagram until either their magnetic field is too weak or their spin rate is too slow to produce radio emission via the normal, and as yet still highly uncertain, emission mechanism. Rotating Radio Transients (RRATs) are pulsars that emit so sporadically that they are more easily detected in searches for single pulses rather than for periodic pulse trains. RRATs with measurable periods usually have $P>1\mathrm{s}$ (Figure 6.3). Some RRATs may be old pulsars on the verge of becoming radio quiet.

Almost all short period ( s) pulsars having fairly weak magnetic field strengths ( G) are in binary systems, as evidenced by periodic orbital variations in their observed pulse periods (Figure 6.4). These recycled pulsars have been spun up by accreting mass and angular momentum from their stellar companions, to the point that they emit radio pulses despite their relatively low magnetic field strengths $B\sim {10}^8$ G. (Apparently, currents in the accreting plasma “bury” the magnetic field of the neutron star itself.) The magnetic fields of neutron stars funnel the ionized accreting material onto the magnetic polar caps, which become so hot that they, as well as the hot accretion disk, emit X-rays. We observe these systems as the Low Mass X-ray Binaries (LMXBs). As the neutron stars rotate, their inclined magnetic polar caps can appear and disappear from view, causing periodic fluctuations in X-ray flux, and making some of these systems detectable as X-ray pulsars.

Millisecond pulsars (MSPs) with low mass ($M\sim 0.1$–$1M_{\odot }$) white-dwarf companions typically have orbits with small eccentricities owing to strong tidal dissipation during the accretion phase which spins up the pulsars. 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. Pulsars with extremely eccentric orbits usually have neutron-star companions, indicating that these companions exploded as asymmetric 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.

About 15% of millisecond pulsars are isolated. They were probably recycled via the standard scenario in binary systems, but the energetic MSPs eventually ablated their companions away. Recently, many new “black widow” and “redback” systems have been detected which seem to be doing exactly this. They exhibit eclipses of their radio MSP emission, likely owing to free–free absorption by the ionized gas in the systems being blown off the companion stars.

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

$$\overrightarrow{F}=q\left(\overrightarrow{E}+\frac{\overrightarrow{v}\times \overrightarrow{B}}{c}\right).$$

(6.38)

Charges in the magnetic equatorial region redistribute themselves by moving along closed magnetic field lines until they build up an electrostatic field large enough to cancel the magnetic force and give $|\overrightarrow{F}|=0$. The induced voltage is about ${10}^{16}\mathrm{V}$ in MKS units. However, the corotating field lines emerging from the polar caps cross the light cylinder (Figure 6.2), an imaginary cylinder centered on the pulsar and aligned with the rotation axis at whose radius the corotating speed equals the speed of light, so 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. Normal “slow” pulsars usually have quite large amounts of typically linear polarization. There are several relativistic effects in the rapidly rotating magnetospheres of millisecond pulsars that complicate this emission-geometry-based polarization, yet millisecond pulsars are typically highly polarized as well. High-energy photons produced by curvature radiation interact with the magnetic field and lower-energy photons to produce electron–positron pairs that radiate additional high-energy photons. The final results of this cascade process are bunches of charged particles that emit at radio wavelengths. Pulsars “die” in the lower right corner of the $P\dot{P}$ diagram (Figure 6.3) when they have 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 of pulsars 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 $Ne$. Larmor’s formula indicates that the power radiated by a charge $q$ is proportional to $q^2$, so the radiation intensity can be $N$ 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 pulsar spectral indices (defined using the positive convention, see Equation 4.55) are $\alpha \sim -1.7$ (i.e., $S\propto {\nu }^{-1.7}$), although some can be much steeper (), and a handful are almost flat ($\alpha >-0.5$).