Invitation to a Contemporary Physics (2004)

are ruled out because they are large enough to have their maximum vibrational, rotational, or orbital periods more than 1 s (and we have many millisecond-pulsars). Black holes do not have a solid surface to which to attach a light beacon, and so cannot produce rotational or vibrational signals. Black holes or neutron stars in a binary could produce the required range of orbital periods, but they would move closer to their companions very quickly, causing their own orbital periods to decrease, not increase. Finally, vibrations of 10 km objects produce periods of milliseconds, not seconds (and we have many second-pulsars). So, the only possibility left is rotating neutron stars as compact objects capable of producing the kind of clockwork mechanism associated with pulsars.

How fast can a neutron star rotate? Clearly there must be a limit to the angular velocity, just as there is a limit to the linear velocity. Even before this limit is reached, however, if the star rotates more rapidly than some critical value, it induces an unstable non-axisymmetric mode of oscillation. This oscillation emits gravitational radiation which takes angular momentum away from the star, thereby pulling it back below the critical value. So there should be a lower bound on the rotation period; calculations show that the minimum possible period of neutron stars ranges from 0.4 to 1.5 ms (or 2 500 to 660 rotations per second). We may have already detected one of the fastest possible spins in the pulsar PSR 1937+214, whose period is 1.55 ms. (The numbers in the pulsar’s identification label indicate its position in the sky.)

As we have already mentioned, when a supergiant’s core collapses, the resulting neutron star inherits a sharply increased spin and magnetization. This combination of a rapid rotation and strong magnetic fields induce enormous electric fields near the surface of the neutron star. The induced electric fields force the electrons and other charged particles to flow o the surface, especially at the two magnetic poles of the star (where the magnetic field density is at its greatest), into space along trajectories parallel to the magnetic field lines. As the electrons are accelerated to extremely high speeds along curved trajectories, they radiate high-energy photons in their directions of motion. These photons interact with the strong magnetic fields and produce jets of electron–positron and other high-energy particle pairs, which proceed to produce more photons, which generate in turn more particles, and so on. The intense light thus emitted covers the whole electromagnetic spectrum, but most of the radiated energy is in the form of gamma and X-rays (with the highest energies exceeding 100 MeV), and only a small fraction (about 10−5) goes into radio emission.

If the star’s magnetic fields are aligned with the spin axis, light is emitted in two thin beams parallel to this axis through the north and south magnetic poles. But usually they are not (for a similar reason that the true north and the magnetic north are at di erent locations on earth), so radiation streams out in two hollow cones centered about the rotation axis, as if two narrow rotating beams of light exiting at the magnetic poles made circles through their sky. As the neutron star rotates and one of its beams points in our direction, the beam will swing away from

Magnetic axis

Spin axis

Earth signal “off”

Magnetic axis

x-rays

Figure 8.14: Light-house e ect in pulsar radiation.

us and not return until the star has completed a rotation. So the light appears to flash on and o at a regular interval as the beam comes in and out of our line of sight. We will not see a neutron star ‘pulsing’ unless one of its beams happens to sweep across us (see Fig. 8.14). Pulsars are most easily observed using radio telescopes, because their emission is brightest in radio waves — no-one knows why.

The large mass of a neutron star means that it takes a lot of energy to speed up or slow down the star’s rotation. So, the timing of its flashes must have exceptional stability, which makes a radio pulsar an extremely accurate time-keeping device in space, as accurate as an atomic clock is on earth. Space travelers could use distant pulsars for navigation.

Any systematic changes in this precise timing are caused by unforeseen external forces, and their observations have led to significant astronomical discoveries. For instance, in 1974 Russell Hulse and Joseph Taylor observed a systematic variation in the arrival times of the pulses from a distant radio source; they reasoned that these smooth periodic variations must be caused by the changing strength of the gravitational field of two compact objects of nearly equal masses in tight orbits around each other. In other words, the radio source must be a pulsar associated with a compact companion, which is either a white dwarf or another neutron star, forming a binary system, now labeled PSR 1913+16. Hulse and Taylor analyzed the star’s motion by examining the arrival times of individual pulses. Their careful analysis not only yielded precise values of the parameters of the double-star system, but also revealed the presence of significant e ects of gravitational time dilatation. Because a binary system loses energy with time, as orbital energy is converted to gravitational radiation, the orbital period of PSR 1913+16 is decreasing, by 2.4 × 10−12 s/s, in perfect accord with theoretical calculations. This agreement provides us with good

circumstancial evidence for the existence of gravitational radiation. A more direct proof could be provided, in the near future, by results gathered by instruments like the Laser Interferometer Gravitational-Wave Observatory (LIGO) and other ground-based observatories under construction worldwide, or sensitive detectors aboard space missions like the Laser Interferometer Space Array (LISA).

Another example based on observations of this kind is the astonishing discovery by Alexander Wolszczan and Dale Frail in 1992 that the radio pulsar PSR 1257+12 has two earth-sized planets as well as a moon-sized satellite. Their existence has been confirmed by the detection that their mutual interaction perturbs their Keplerian orbital motion, resulting in subtle changes in the pulsar timing. At present, this is the only confirmed existence of planets around a neutron star.18

Neutron stars may be rotating very fast at birth, but they spin down afterwards. This is why the youngest pulsars, such as the Crab pulsar (33 ms) and the Vela pulsar (80 ms), have unusually short periods, whereas older neutron stars have longer periods. The neutron star spin-down is due to magnetic braking, a mechanism by which the star’s magnetic field exerts a strong torque on its surroundings, which are then forced to corotate with the star. This causes the star to decelerate and transfer its rotational energy to the radiation of electromagnetic energy into space at a rate that we can calculate.19 Take the Crab pulsar as an example of such rotationpowered neutron stars: the rate at which it loses rotational energy is 6.4 × 1031 W, which is similar to the energy requirements of the surrounding supernova remnant in non-thermal radiation and bulk kinetic energy of expansion, 5 × 1031 W.

The spin-down rate is a fundamental observational parameter and is known with great precision (typically 10−15 second per second in most cases, but 10−13 s/s for a young pulsar like the Crab). From such measurements, researchers have estimated the magnetic fields at the surfaces of radio pulsars to be 106–108 T. If this sounds unbelievably high,20 wait until you hear about the magnetars!

8.5.3The Missing Pulsars

Of the billion or so neutron stars thought to exist in our Galaxy, only a thousand have actually been observed. Where have all the others gone?

Until recently, the main way to find a newly formed neutron star is to detect its radio pulsations. But with sophisticated instruments now at their disposal, astronomers are realizing that neutron stars need not only be radio pulsars. For example, some of the missing ones might be quiet, isolated, cooling compact stars that are

18However, we know of at least 100 planets around normal stars outside our solar system. Most have masses in the 1–10 Jupiter mass range, most have highly eccentric orbits and are extremely

close to their parent star. For the latest developments, go to http://exoplanets.org/science.html 19A star of radius R rotating at angular frequency Ω = 2π/P , with its magnetic moment misaligned from the spin axis by an angle θ, radiates energy at the rate of dE/dt = B2R6Ω4 sin2 θ/6c3,

where c is the speed of light and B is the magnetic field strength at a magnetic pole.

20The highest known field in ordinary stars is 100 T. On earth, we can produce sustained fields of 50 T in a small volume, and fields of 1000 T for milliseconds.

308 Bright Stars and Black Holes

not seen pulsing, because, for unknown reasons, they never ‘turned on.’ However, their hot surfaces glow quietly in X-rays, and they can be seen only in this part of the spectrum.

Other missing neutron stars might be the unpredictable soft gamma-ray repeaters (SGRs), the recurrent soft X-ray transients discovered in the 1980s. Although there are only a few examples known at present,21 SGRs have attracted a lot of attention lately because of their unique characteristics. They emit frequent, but randomly spaced in time, outbursts of low-energy gamma rays of very short duration, usually tenths of seconds. Each outburst sends out 1033–1035 W in luminosity at its peak. In between the bursts, they produce persistent softer radiation, with a clear period of 5–8 s consistent with rotational modulation; and the period is lengthening with each successive eruption at a rapid rate of about 10−12 s/s.

Analyses of the data collected by several satellite-borne instruments (such as ASCA, BATSE and RXTE) show unambiguously that SGRs possess the characteristics of neutron stars. In particular, the locations of SGR 0525–66 and SGR 1806–20 are shown to coincide with the faint X-ray sources found in the young supernova remnants N49 in the Large Magellanic Cloud and G 10.0–0.3 in our Galaxy, respectively. Thus, if SGRs are associated with supernova remnants, then they must be neutron stars some tens of thousands of years old — although neutron stars that young with 8 s periods are rather puzzling. The question is, what causes the gamma-ray bursts?

There is an even more intriguing class of objects called anomalous X-ray pulsars (AXPs) — ‘anomalous’ because we do not understand why they shine. We have to date five confirmed objects of this class,22 all in the galactic plane; they pulse with periods of 6–12 s, which increase with time at the rates of about 10−12 s/s; and, finally, they radiate X-rays at modest luminosities. Most oddly of all, at least two are associated with supernova remnants. They exhibit many similarities with SGRs, and might even erupt in sudden emissions, although they show clearly less activity than their cousins. Again, the question is, where does their energy come from?

The two most obvious possibilities — rotation and accretion of material from a nearby star — can be ruled out, both being inadequate to explain the huge power needed for the brief outbursts observed. In the 1990s Robert Duncan and Christopher Thompson advocated another energy source, that of very strong magnetic fields, greater than 1010 T. Such powerful fields in neutron stars, dubbed magnetars, could provide the energy source for SGR bursts and also provide a mechanism that would quickly spin the stars down to their characteristically long rotational periods. At the quantum level, these strong fields would also modify the atomic energy structure so drastically that the stellar matter becomes essentially transparent to electrons, thereby allowing the huge luminosities observed in SGR outbursts.

21SGR 0525–66, SGR 1806–20, SGR 1900+14 and SGR 1627–41 are known in 2003.

22As of 2003, the AXP catalog consists of 4U 0142+615, 1E 1048–5937, RXS 1708–4009, 1E 1841–045 and 1E 2259+586.

When we apply these ideas to SGR 1806–20 — a specially prolific SGR that exhibits 7.5 s pulsations with a long-term increase of 8 × 10−11 s/s — we find that the neutron star has a spin-down age of 8 000 years and a surface magnetic field strength of 2 × 1010 T. Estimates of the number and ages of SGRs in the Galaxy suggest that magnetars may form one tenth of the entire galactic neutronstar population. Similarly, the spin-down rates of AXPs imply magnetizations in the range of (1–7)×1010 T, which would place them among the strongest fields inferred for neutron stars.

So, are SGRs and AXPs magnetars? The evidence is convincing for SGRs, but less strong for AXPs. The situation will certainly be clarified in the near future with the new data coming back from space missions, such as the Rossi X-Ray Timing Explorer (RXTE) and the Chandra X-Ray Observatory.

8.5.4Neutron Stars in Binaries

Not all neutron stars lead a solitary life; many are found as the optically invisible partners of normal stars in X-ray binary sources. Some of these neutron stars are born in binaries that survive the supernova events; others may have captured (or have been captured by) ordinary stars in dense stellar regions such as globular clusters. But there is no reason to suppose that they are di erent in nature from the isolated variety most often observed as radio pulsars, and, therefore, any distinctive phenomena they might exhibit must derive simply from their close association with another star.

Binary X-ray sources, in which the neutron star components are always optically invisible, come in two broad classes. The first is associated with very luminous and massive (late O or early B type) stars, which are among the youngest stars and have rather short main-sequence lifetimes. These binaries are called HMXBs, or high-mass X-ray binaries. The second class is formed by the LMXBs, or low-mass X-ray binaries, which contain cooler low-mass main-sequence stars, having masses, luminosities, and temperatures similar to those of the sun. In either case, the companion star provides the prime additional source of energy through accretion. This mechanism of mass transfer to the compact star proceeds in di erent ways depending on the mass of the donor star — hence the two-class distinction.

Why do we believe that accretion of matter from the normal (primary) star onto the neutron star (or black hole) is responsible for the X-ray emission from these binary X-ray sources?

First, because when matter falls from the primary star on the secondary star, the kinetic energy it gains is equivalent to the gravitational binding energy of matter on the compact star. Most of this energy is converted into thermal energy when the infalling matter hits the surface, releasing up to 15% of the rest mass of the infalling material as radiation. This represents an excellent source of energy, 20 times more e cient than nuclear energy sources. If we assume that the neutron star gains additional mass in this way at a rate of 10−10 M per year, it can radiate

energy with a power of 1030 W, which is consistent with the observed luminosity of a strong X-ray binary. If this energy is emitted as black-body radiation, then the star’s e ective temperature should be no less than 107 K, and the star should emit in the X-ray waveband.

Another argument in favor of accretion as the mechanism of powering X-ray binaries concerns the existence of an upper limit to their luminosities: what prevents them from accreting matter at an inordinate rate? The answer is, when the luminosity of the accreting object is too great, the radiation pressure acting on the infalling matter (e.g., scattering of the emergent high-energy photons by infalling electrons) increases and can reach a level such that it can e ectively prevent any further matter falling onto the surface of the compact object. Thus, there exists a critical luminosity above which accretion shuts o , known as the Eddington limit, given by

LEdd = 1.3 × 1031 (M/M ) W .

This result agrees quite well with the observed cut-o in the X-ray luminosity distributions of the X-ray binary sources in the Galaxy and the nearby Magellanic Clouds.

The neutron stars in HMXBs are intense X-ray pulsars rotating with periods of 1–1000 s. The accretion mechanism operating here must also explain why their emission is pulsed. All stars emit stellar winds, quiescent mass loss in sun-like stars and more powerful winds in the cases of the more massive stars. In HMXBs associated with luminous, massive O and B stars, the orbiting magnetized compact star is embedded in the strong outflow of the primary (Fig. 8.15). As the infalling matter approaches the surface of the neutron star, the two masses spinning at di erent speeds will try to come into equilibrium with each other, such that the star’s rotation period nearly matches the orbital period of the circling matter just outside the magnetosphere, the region where the magnetic field dominates the accretion flow. From the physics of this spin equilibrium situation, one can infer that these neutron stars must have strong magnetic fields, in excess of 108 T (similar to values found in typical isolated pulsars). Accreting matter is forced to flow along the strong magnetic field lines and can only funnel down onto the star’s surface through the magnetic poles, where the field density is the greatest. So the X-ray emission is concentrated in these two ‘hot spots’ where the accreted particles hit the surface. If the magnetic field axis and the star’s spin axis are misaligned, as is usually the case, then the radiation in our direction sweeps past us once every rotation and we see X-ray pulsations.

From recent observations by orbiting X-ray telescopes, astronomers have detected another remarkable property of these systems: bucking the overall tendency to spin down, some of them occasionally speed up while others display abrupt accelerations and decelerations. These features can be understood as being associated with variations in the mass flow about the magnetosphere of the neutron star

Figure 8.15: Two possible modes of mass transfer in a binary X-ray source. In (a) the normal primary lies inside the Roche lobe (a surface beyond which the primary will begin to shed gas onto its compact companion). The mass transfer is e ected via a stellar wind. In (b) the primary begins to expand to and overflow its Roche lobe. (Based on Shapiro and Teukolsky, p. 399.)

and the subsequent transfer of angular momentum from matter to the star. If there is no accretion, there is no acceleration and the star slows down. But, at times, a disk of swirling matter may form briefly around the star and, because the stellar wind is unstable, it may dissipate to reform later, going the other way.

The neutron stars in LMXBs have companions with masses less than 1 M . They are older than those in HMXBs — 109 years compared to 107 years — and have accreted more matter. They show no apparent evidence of dipolar accretion or pulsations in their radiative fluxes, and so must have relatively weak magnetic fields of about 105 T or less, being reduced naturally to these values by eons of ohmic decay.

The neutron star in an LMXB usually has a gravitational field stronger than the field of its less massive companion. If the outer envelope of the latter is close enough, the compact star pulls the closer parts of the envelope towards itself, creating a narrow stream of gas rushing out of the envelope. As the compact star is very small compared to its companion, the flowing gas has too much angular momentum to fall directly onto the star, and so must orbit around it, forming a thin accretion disk of hot matter. Within the disk, the gas will spiral in swiftly towards the center until it reaches the magnetosphere, where the magnetic field at the neutron star’s surface

takes control and forces it to flow along the field lines to the magnetic poles. As it swirls in towards the neutron star and collides against itself, the gas is heated to millions of degrees, dissipating its gravitational binding energy into radiation which we observe as a continuous X-ray emission.

Since a typical neutron star in this class is old, its magnetic field is expected to be relatively weak and its magnetosphere small. This means that the swirling gas can orbit very close to the star and acquire high orbital frequencies before being picked up by the magnetic field. Also, as the star has been accreting long enough to gather a great amount of additional material (up to 0.1 M ), the angular momentum it acquires from this much high-spin accretion can spin it up to very rapid rotation when the stellar magnetic field is weaker than about 105 T. However, this spin is notoriously di cult to detect, and it was only in 1998 that the first accretionpowered millisecond pulsar was discovered with RXTE, when the neutron star SAX J1808.4–3658 was revealed to have 2.5 ms pulsations in its persistent X-ray flux and an inferred surface field in the range of (2–8)×104 T, comparable to the magnetic strengths observed in millisecond radio pulsars.

Neutron stars in LMXBs show another distinctive phenomenon: they display repeated X-ray bursts, which are brief, intense emissions of X-rays caused by unstable thermonuclear reactions in the accreted layer of hydrogen and helium. Due to thermal instability on their surfaces, the stars are unable to burn the accreted matter as fast as it is gathered. Instead, they accumulate hydrogen and helium for hours or days until instability sets in, then burn the fuel in a few seconds when the local temperature exceeds 109 K. The runaway thermonuclear process starts at a localized site, probably near a magnetic pole, then spreads around the compact star, eventually burning all the nuclear fuel. When astronomers examine closely the power spectra of these outbursts, they discover pulsations in a range (1.7–3 ms) very similar to the spin periods of millisecond radio stars. This provides strong evidence for the existence of rotating neutron stars in LMXBs, supporting the lone direct observation mentioned above.

In summary, the accreting neutron stars in many LMXBs are rotating with millisecond periods; some of them may be magnetized enough to become millisecond radio pulsars once accretion shuts o ; but most others, for unknown reasons, do not show persistent pulsations at all.

8.5.5Summary

Neutron stars are end states of massive stars. They owe their stability to the degenerate-neutron pressure of a very dense neutron-rich matter that makes up its interior. They appear to have the maximum possible mass of 3 M , and are characterized by a rapid rotation and a very strong magnetic field, two properties that determine to a large extent their interactions with the surrounding medium. Whether neutron stars are isolated or associated with a companion star also gives

rise to distinctive phenomena. Thus, we find isolated neutron stars in the form of radio pulsars, soft-gamma-ray repeaters or anomalous X-ray pulsars; and neutron stars in high-mass X-ray binaries or in low-mass X-ray binaries, which are strong X-ray sources.

8.6 Black Holes

When a massive star runs out of nuclear fuel, it contracts into a compact inert core. If the core has a mass greater than 3 M , no known forces can prevent it from collapsing relentlessly under the inward pull of gravity until it falls into a region of space from which no light, matter or signal of any kind can escape. This warp in spacetime is called a black hole. Although black holes were first studied in 1939, by Robert Oppenheimer and Hartland Snyder, evidence of their occurrences did not exist until recently, thanks especially to advances in X-ray astronomy. Nowadays, it is generally believed that black holes occur commonly; they may be found not only in the stellar-mass range, as end states of normal stellar evolution, but also as supermassive objects produced by the coalescence of many dense stars or even of an entire conglomerate of stars and holes. In fact, they may have any mass.

Gravitation becomes so overwhelming in black holes that it changes the nature of spacetime itself, giving it curvature and causing it to produce disturbances that can propagate in waves. These phenomena, unknown to Newtonian physics, can be studied in the framework of Einstein’s general theory of relativity, the most satisfactory theory of gravitation available to us at present.

8.6.1Gravitational Collapse

Let us consider first the collapse of a massive spherical star. When the star, having exhausted its nuclear fuel and having contracted slowly inward, begins to squeeze its electrons onto the atomic nuclei, weakening thereby its source of pressure, it becomes unstable. As the instability develops quickly into a full-scale implosion, the stellar core falls inward on itself. If the degenerate-neutron pressure, which must appear at appropriately high densities in the interior, fails to stop the fall, the collapse keeps pressing on, pulling the star’s surface through its gravitational radius, where the star’s gravitational binding energy becomes comparable to its total mass energy. Once this critical surface is passed, the implosion will not stop until the compressed star has reached a ‘point’ of zero volume and infinite density

— a singularity. Within a short lapse of time, about 10−2 (M/M ) s, the structure of total mass M has matured, and the dynamical behavior of spacetime has settled down into a stationary situation described by an exact solution to Einstein’s field equations called the Schwarzschild spacetime (Figs. 8.16 and 8.17).

This solution represents the geometry of curved spacetime exterior to a spherical static source of total energy Mc2 and which is also asymptotically flat. It depends

314 Bright Stars and Black Holes

Figure 8.16: Representations of the light fronts at successive times: (a) Spatial representation;

(b) Spacetime representation.

only on this one parameter, but is otherwise universal, independent of the nature of the source that has produced it. And it even covers situations more general than stated, for it can be applied to any spherical mass distribution, whether static, collapsing, expanding or pulsating. The gravitational field surrounding the sun, a neutron star, and that of a black hole right up to its singularity are identical, provided their masses are the same. The Schwarzschild solution forms the basis of our understanding of the physics of spherical black holes, which we now summarize.

A spherically symmetric black hole of mass M possesses a characteristic radius known as the gravitational (or Schwarzschild) radius rg = 2GM/c2, which represents the e ective radius of the black hole.23 The non-material spherical surface of radius rg is called the event horizon; it has the defining property that signals emitted inside cannot escape, whereas signals emitted outside can escape to infinity. So it acts as an e ective physical boundary of the black hole. Putting in the values of the constants G and c, we find rg = 3 (M/M ) km, so that solar-mass black holes have rg = 3 km.

The gravitational field inside the event horizon becomes so powerful that even light is pulled ineluctably inward toward the center regardless the direction in which it is emitted. The paths of all motions terminate at the central singularity. Outside the horizon, light may escape, but as the emitter approaches the surface from a distance, light rays find it harder and harder to move away. The rays that can

23In the following, we will use units in which G = c = 1, so that M designates not only mass,

but also energy, length and time. The corresponding expressions in conventional units are Mc2 = 2 × 1047(M/M ) J, GM/c2 = 1.5 (M/M ) km and GM/c3 = 5 (M/M ) µs.