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14.13 BLACK HOLES

14.13 BLACK HOLES.

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14.13 BLACK HOLES

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  1. 14.13 BLACK HOLES Neutron stars are the closest objects to black holes which have been found to date on the stellar scale. The upper mass limit of ~ 2M0 is rather small and we know that the collapsing core in more massive stars can readily exceed this value. No known physics is capable of stopping the collapse (if there were a quantum theory of gravity then a singularity should not occur) and it is probable that a black hole is created. This singularity will be causually disconnected with the outside world. The Schwartzschild radius RS = 2GM/c2 determines the point at which not even photons can escape since the observed frequency n0 of light becomes where ne is the emitted frequency A useful numerical value for astronomers is When the motion of test particles are considered then the angular momentum has to be taken into account. One important result is the existence of a last stable orbit about the point mass which has radius R = 3RS. Circular orbits with radii less than this cannot exist, the particles spiral rapidly to r = 0. Observable Properties of Black Holes Given the likely existence of black holes, how and by what means can we recognise them. Let us try and identify their usable properties. The stationary solutions to the Einstein field equations depend on three parameters : the mass M, the angular momentum J, and the charge Q. All other information about the initial state is radiated away in the form of electromagnetic and gravitational waves during the collapse.. Mass The Schwartzschild radius confines the mass of the object to a given volume of space. Whereas photons cannot communicate to the outside world gravitational forces can. Thus we will be able to measure the mass in e.g. a binary system. Rotation - Kerr Black Holes Since all astronomical objects rotate, so that we can expect black holes formed by gravitational collapse from a rotating parent to be rotating, probably quite rapidly. Observationally we have already seen this for the case of Pulsars. PHYS3010 - STELLAR EVOLUTION

  2. disk disk Rotating black holes are often referred to as Kerr black holes, from the case of the Kerr metric(Q = 0) ( As opposed to the Schwarzschild metric, Q = 0, J = 0). They are of particular interest in the context of binary X-ray sources. Spin axis A key parameter in the Kerr metric is the angular momentum of the BH per unit mass, a. Event Horizon Some key factors for Kerr black holes are : • No BH can be formed for J > GM2/c • For a maximally rotating BH the horizon R+ = GM/c2 • i.e. half the Static value of RS • The rotation causes the ‘dragging’ of inertial frames i.e. the rotation drags all objects near it into orbital motion in the same direction as the hole rotates. A surface exists within which no observer can remain at rest, and must rotate in the same direction as the hole. • The region of space-time between this surface and R+ is called the ergosphere. See figure. It is possible that the Penrose process takes place, in which a particle enters the ergosphere and splits into two sub-particles. If one of them falls down the hole then the other escapes to infinity with greater energy than the original particle had when it fell in. The source of energy comes from the rotation of the BH. • The last stable orbit depends on the direction : it is R = GM/c2 for co-rotation and is much larger, R = 9GM/c2 for counter-rotation and outside the ergosphere. • The maximum amount of energy loss (i.e. the binding energy) in order that the material reaches the innermost bound orbit is ~ 40% of the mass energy. This is the process which provides the energy source which makes BH astronomical objects visible. • It is also possible to tap the rotational energy from Kerr BHs. About 30% of the rest mass energy can be made available to power astrophysical phenomena. Ergosphere The Electrodynamics of Black Holes It is unlikely that charged black holes will be important astrophysically since it is expected that a charged astrophysical object is rapidly neutralised by the surrounding. plasma. A magnetic field can be associated with the BH provided it is tied to the surrounding medium and not the BH itself. The field lines can for example be linked to the accretion disk and the surrounding medium and create a geometrical field structure which is determined by the presence of the black hole. PHYS3010 - STELLAR EVOLUTION

  3. 15. CLOSE BINARY SYSTEMS Since ~ 50% of all stars are originally found in binary systems, and since compact objects (WD, NS, BH) are derived from main sequence stars, we may expect to find these objects in binary systems. We should be able to estimate their masses. Close Binary Systems are defined as systems in which pairs of stars are so close that mass transfer takes place between the two objects. Clearly this will effect their evolution which will be different to normal (i.e. free) stars of the same masses. 15.1 THE GRAVITATIONAL POTENTIAL - ROCHE LOBES W The gravitational potential is modified for the case of two stars rotating about each other in close proximity. v M1 M2 The equipotentials for the Newtonian gravitation + centrifugal potential in the orbit plane of a binary star system with a circular orbit is shown below : PHYS3010 - STELLAR EVOLUTION

  4. In the figure the ratio of the masses of the ‘normal’ star to the compact object is 10 : 1, and the normal star is on the left. The values of the equipotentials are labelled in units of G(M1 + M2)/a, where a is the separation of the centres of mass of the two stars. • The equipotentials close to the stars are approximately spherical and dominated by the 1/r dependence, as if the other star is not there • At very large distances the equipotentials again are spherical for a mass of M1 + M2 • The potentials have local stationary points (DF = 0), called Lagrangian points. These are at the locations marked L. The inner Lagrangian L1, for example, represents the easiest point for material from the normal star to escape to the other object. • The two volumes enclosed by the equipotentials passing through L1 are called the Roche lobes, they can be used to define the enclosed material as belonging to one star or the other. 15.2 MASS TRANSFER - ACCRETION ONTO COMPACT OBJECTS At particular stages in the lifetime of stars we have seen that they can suffer considerable mass loss. The two most notable cases in the immediate context are when they suffer extensive stellar wind losses (e.g. OB supergiants) and when (e.g. Red Giants) they expand outside the limits of their Roche lobes Stellar Wind Losses High Mass X-Ray Binaries (HMXRB) The adjacent figure illustrates the situation in which stellar wind losses dominate. The primary star lies inside the Roche lobe but loses mass via a stellar wind. The orbiting compact star is an obstacle in the wind and a bow-shaped shock front is formed around it by the action of its gravitational field. Some of the shocked material is captured by the compact object. PHYS3010 - STELLAR EVOLUTION

  5. W L2 r2 r1 v1 v2 Roche Lobe Overflow : Low Mass X-Ray Binaries (LMXRB) If the primary star expands to overfill its Roche lobe material the material will flow through the L1 saddle point between the two stars and much of it is quickly captured by the compact object. This gas has a considerable amount of angular momentum and will spiral down to the secondary object in the form of a disk structure - the so-called Accretion Disk. A fraction will find an easy escape route via the L2 point. The material in the accretion disk moves in approximately Keplerian orbits as it spirals down the gravitational potential well. The velocities will be As the material swirls in there will be large shear forces between adjacent parts of the disk which are dissipated in the form of viscous heating as the gas flow has a differential velocity and collisions/turbulence is created. Note that the total energy available from the gravitational field is very high for compact objects such as neutron stars, and can amount to 0.1 mp c2 ~ 100 MeV for each proton. PHYS3010 - STELLAR EVOLUTION

  6. Luminosity from Accretion Consider a proton of mass mp falling onto a star with a mass M and a radius R. The energy gain from the gravitational field is : The emergent luminosity is If we write the efficiency of the radiative emission as e = RS/R Then where RS = GM/c2 is the Schwarzschild radius 15.3 THE EDDINGTON LUMINOSITY LIMIT Now the luminosity cannot be made arbitrarily high since the outflowing photons will drive the infalling matter back and hence cut off the prime source of the radiation. We may estimate the maximum luminosity under these conditions : The downward force on the protons due to gravity is Thompson cross section The upward force due to Compton scattering by the outgoing photons is Equating we obtain Leading to a critical luminosity of PHYS3010 - STELLAR EVOLUTION

  7. Rough Estimate of the Luminosity and Energy of the Emitted Photons The material at the surface of a normal star emits photons at hn ~ 1 eV, so that we may expect the temperature of the inner reaches of an accretion disk to achieve higher temperatures and higher energy photons to emerge : where R is the inner radius of the accretion disk Now so that with a reasonable accretion rate for close binary systems of dM/dt ~ 10-9 M0 y-1 we can make approximate estimates as follows : NOTE • The spectral range of emission from the various objects • Neutron stars and black holes are X/g-ray sources which should emit at the Eddington luminosity limit PHYS3010 - STELLAR EVOLUTION

  8. 15.4 ACCRETION ONTO WHITE DWARFS A wide range of available geometries exist for accretion to take place onto WD, as a result a number of different types of phenomena occur. NOTE:all Cataclysmic Variables(CVs) are strong uv sources. Classical Novae are the most dramatic, exhibiting typically 10 magnitudes of brightening over a few days and remaining luminous for many days. The luminosity is typically Eddington for a WD with a total energy release of 1038 J. In the 1960s Novae were finally observed to be related to WD in binary systems. They are thought to be related to the accretion of a critical mass of H-rich material onto the WD, followed by degenerate thermonuclear runaway which eventually stabilises itself. The light curve could well be sustained by the 22Na g-ray emitting radioactive isotope. About 30 Novae occur in the Galaxy per year, although only a small fraction are observed due to optical extinction. Recurrent Novae are less luminous, but as the name suggests the outbursts take place on the timescales of decades to months and with a great deal of variety between the various objects. These dwarf novae have been found to contain accretion disks. The binary periods of cataclysmic variables are typically of a few hours and all less than 1/2 day, a fact which dictates that the companion is a low mass star which is undergoing Roche lobe overflow. Accretion stream White dwarf Polars : Here the magnetic field is very strong ( ~ 103 T) and the rotation of the WD star is phase locked to the orbit of the binary. AM Herculis is the archetypal example. The accreting matter flows directly along the magnetic field lines from the primary star onto the poles of the WD, no accretion disk is formed. Companion star Intermediate Polars : These objects have weaker magnetic fields and since the rotation can not be synchronised with the orbit the rotation period of the white dwarf is shorter than the orbital period. The magnetic field is not strong enough at large distances to force the accreting material directly down the field lines and an accretion disk is formed. Close to the star the magnetic field will disrupt the accretion disk and matter will accrete onto the star only at the poles. Both of these classes of objects exhibit high and low states of activity which are related to the different states of mass accretion. PHYS3010 - STELLAR EVOLUTION

  9. N Star B Hole yes Spin yes ~108T M-Field Not coupled no Ergosphere yes < 2M0 Mass >6M0 16 km Radius <10km 103Wmax104? W W 15.5 ACCRETION ONTO NEUTRON STARS AND BLACK HOLES Since both NS and BH will emit in the X-ray region of the spectrum, what differences may we expect to find to distinguish them from one another? The mass is clearly the main factor to look for, however there may be more subtle observational differences based indirectly on such aspects as the ergosphere of magnetic field configuration. Regularly Pulsating X-ray Binary Sources. The discovery of regularly pulsating X-ray binary sources by the UHURU satellite was a major milestone in the understanding of neutron star systems. The two archetypal objects are Her X-1 and Cen X-3. Counts per 0.096 s BIN BINS The presence of 4.8 second X-ray pulsations in Cen X-3 as revealed by the UHURU satellite, thus confirming that it includes a rapidly spinning neutron star. PHYS3010 - STELLAR EVOLUTION

  10. Period P Delay Dt Doppler t value IX Detailed Study of the Pulsations • Whereas the X-ray pulsations may, at a first glance, look to be extremely regular, in fact they are not completely stable as for the case of a free neutron star in the form of a pulsar. However their variability is not random and is linked to the fact that the neutron star, whilst pulsating regularly, is in fact rotating around the primary star. Thus we have an accurate clock on one of the members of the binary system, and this may be used to gain more information about the overall system. The pulses experience three kinds of modulation (for an eclipsing binary system) which are related to the phase of the orbit as illustrated in the above figure for Cen X-3: • The modulation of the X-ray intensity IX as the compact object passes behind the primary (Cen X-3 has a period of 2.087 d) PHYS3010 - STELLAR EVOLUTION

  11. The arrival time delay varies sinusoidally with the period P due to the extra distance the radiation has to travel when the secondary is furthest away from the Earth. This enables us to evaluate the diameter D of the secondary orbit i.e. for CenX-3 Dt = 39.7 s so that D = 2 x 39.7c = 2.38 1010 m which means the NS orbit is very close to the surface of the primary star. • The x-ray pulse period fluctuates sinusoidally throughout the orbit due to a periodic Doppler effect The period decreases as the neutron star approaches the Earth and increases as it recedes. For Cen X-3 t0 = 4.84 s which is consitent with a projected velocity of i Mx MP The Estimated Masses of Neutron Stars There are a number of X-ray binaries for which enough observational information is available, including an estimate of the angle i, to permit the mass of the neutron star to be evaluated with some accuracy. PHYS3010 - STELLAR EVOLUTION

  12. Neutron Star Primary Star Neutron Star Equation of State Note that the masses lie within the range discussed earlier. However if we are to obtain an equation of state for neutron stars then we have to evaluate their sizes. Measurement of the red shift by fine spectroscopy of any nuclear emission lines (including the electron-positron annihilation line at 511 keV) emitted from the surface may enable this to be done. Orbital Geometry Many of the neutron stars are very close to the primary star. the diagramme on the left gives some idea of how close they are, and makes it easy to understand why such a large fraction of them are eclipsing. Below is a sketch of an accreting binary system. PHYS3010 - STELLAR EVOLUTION

  13. Period P Time Pulsation Mechanism The masses of the pulsating X-ray binaries indicate that the compact object is a neutron star. The pulsations are thought to be due to the strong dipole magnetic field. Accretion Column Accretion Disk Emergent X-ray beam The accreting material is constrained to flow along the magnetic field lines towards the polar caps. As the material hits the neutron star surface a hot shock is formed in which X-rays are produced. The column above the emitting region means that the emission will not be uniform but shadowed into a fan beam. If the polar caps are displaced from the rotation axis then modulation of the emission will be synchronised with the rotation period of the neutron star. Again, as for pulsars, a kind of lighthouse effect causes the pulsations. Pulse Period Changes The periods of pulsating X-ray binaries are found to decrease as a function of time. This spin-up must be related to the transfer of angular momentum from the material of the accretion disk to the neutron star during the infall. PHYS3010 - STELLAR EVOLUTION

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