Stellar-Mass Black Holes

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1 Stellar-Mass Black Holes General relativity Hawking radiation Gravitational waves 15 February 2018 University of Rochester

2 Escape velocities from stars 15 February 2018 (UR) Astronomy 142 Spring / 36

3 Escape velocities from stars Neglecting relativity, in increasingly bad approximations: E = 1 2 mv2 esc GMm R = 0 2GM v esc = R = 619 km/s = 0.002c Sun = 6970 km/s = 0.023c 1M white dwarf = km/s = 0.514c 1M neutron star And note that v esc = c km/s when R = 2GM c 2 15 February 2018 (UR) Astronomy 142 Spring / 36

4 Beyond the NS maximum mass: Black holes The maximum mass of a neutron star is 2.2M. There is no known physical process that can support a heavier object without internal energy generation. One possibility: quark stars (Ivanenko 1965, Ivanenko 1969)? A non-spinning heavier object will collapse past neutron-star dimensions and soon thereafter becomes a black hole, an object from which even light cannot escape if emitted within a distance R Sch = 2GM c 2 of the object as measured by a distant observer. Schwarzschild radius This spherical surface is called the event horizon or simply the horizon of the black hole. 15 February 2018 (UR) Astronomy 142 Spring / 36

5 Black holes & General Relativity The nonrelativistic result R = 2GM/c 2 is, by accident, the same as R Sch derived with the general theory of relativity (GR). GR is a description of the effect of gravity at any strength, even handling large amounts of mass shrunk to small dimensions. It involves new mathematical concepts beyond the scope of this course. First published by Albert Einstein in 1916 (Einstein 1916), the theory provides a set of field equations to describe gravity: spacetime curvature G µν = 8πG c 2 T µν mass and energy In plain English (Gravitation by Misner 1973): Mass causes space and time to be curved, or warped. The resulting curvature of space determines how masses will move. 15 February 2018 (UR) Astronomy 142 Spring / 36

6 Interesting facts about black holes Time and space are warped substantially near black holes. Time intervals on clocks near black holes appear to distant observers to be slow compared to their own local identical clocks. This effect is known as gravitational time dilation or gravitational redshift: time for a distant observer t = τ proper time near > τ 1 RSch /r the BH Thus, time appears to stop at the event horizon to a distant observer: t as r R Sch This behavior gave the horizon its original name: the Schwarzschild singularity. 15 February 2018 (UR) Astronomy 142 Spring / 36

7 Interesting facts about black holes Near a black hole, a small length L measured simultaneously (as with a measuring tape) between two points on a radial line is greater than the distance r between the points, measured by a distant observer: L = r 1 RSch /r > r Neither r nor the radius r measured by a distant observer of a point near the black hole has meaning as a physical distance to an obesrver near the black hole. r and r are called coordinate distances. However, the circumference C of a circle through that point and centered on the black hole turns out to have the same value in all frames. Think of r only as r = C/2π. 15 February 2018 (UR) Astronomy 142 Spring / 36

8 Circular orbits and their radii in GR Circles in flat spacetime: C = 2πr = πd. That, of course, is the very definition of π. r C d 15 February 2018 (UR) Astronomy 142 Spring / 36

9 Circular orbits in flat space (all in same plane) Concentric circular orbits in flat space. The distance between each orbit is π 4π 8π 10π Local and distant observers report the same distances between concentric orbits. 15 February 2018 (UR) Astronomy 142 Spring / 36

10 Circular orbits in space warped by a black hole Imagine a BH whose horizon has circumference 2π. The distant observer still reports distances of 1 between each orbit π 4π 8π 10π The local observer reports warped distances. Both observes measure the same circumferences! 15 February 2018 (UR) Astronomy 142 Spring / 36

11 Visualization of warped space: Hyperspace To connect these circles with segments of the too long lengths, it is helpful to consider them to be offset from each other along an imaginary dimension x and y but which is not z. If the additional dimension were z, then the circles would not appear to lie on a plane. Such additional dimensions comprise hyperspace. 15 February 2018 (UR) Astronomy 142 Spring / 36

12 Embedding diagrams This is why you often see the equatorial plane of a black hole represented as a funnel-shaped surface, as if made from a stretched rubber sheet. It is important to note that the direction of stretch is in hyperspace. The scene would not look like a funnel to an observer seeing in three spatial dimensions. 15 February 2018 (UR) Astronomy 142 Spring / 36

13 Interesting facts about black holes Orbits outside the BH s horizon, further away than 1.5R Sch (in the coordinate system of a distant observer) still turn out to be ellipses. The resulting coordinate speed in orbit (for the coordinate system of a distant observer) is the same as that obtained in Newtonian gravity: v = r dφ GM dt = r At the horizon, the radial component of the coordinate speed of light is zero: light cannot escape. Thus, no information can reach a distant observer from, or from within, the BH horizon. For non-spinning black holes, orbits with coordinate radius < 3R Sch are unstable to small perturbations. 15 February 2018 (UR) Astronomy 142 Spring / 36

14 Interesting facts about black holes There are no orbits with coordinate radius < 1.5R Sch for a non-spinning black hole. At this radius, the local orbital speed is the speed of light, and smaller orbits would require impossibly higher speeds. You cannot orbit at this close distance because your rest mass is nonzero; if you could, you could train your binoculars straight ahead (in the φ direction) and see the back of your head. To get closer to the horizon, the descent must be radial while balancing gravity with thrust, as in a rocket launch. If the black hole spins, the innermost stable orbit and the photon orbit are smaller than 3R Sch and 1.5R Sch if the particle orbits in the same direction as the spin, and larger if it orbits in the opposite direction. Within r = 1.5R Sch, all geodesics (possible paths for light) terminate at the BH horizon. 15 February 2018 (UR) Astronomy 142 Spring / 36

15 Interesting facts about black holes Thus, from near the horizon, the sky appears to be compressed into a small range of angles directly overhead. The range of angles is smaller the closer one is to the horizon, and vanishes at the horizon. Objects in the sky will appear bluer than their natural colors as well due to the gravitational Doppler shift. Space itself is stuck to the horizon, since one end of all the geodesics are there. If the horizon began to rotate, the ends of the geodesics would rotate with it. (This harmonizes with time stopping there.) Gravitational acceleration turns out to be a = GM 1 r 2 1 R Sch /r 15 February 2018 (UR) Astronomy 142 Spring / 36

16 Interesting facts about black holes This has the familiar Newtonian form at large r but blows up at r = R Sch. Thus, in a vertical descent to a hovering position just above the horizon, very large gravitational accelerations would be encountered. Tidal forces turn out the same near a black hole as in Newtonian gravity, and are finite at the horizon. For an object of length r in the radial direction and x in the crosswise directions, a r = 2GM r 3 r a φ = GM r 3 x For a 2 m person and a 10M BH, the radial tidal acceleration a r at the event horizon is cm/s 2, or g. a r = 1g for a M BH. Thus, if you want to fall freely past the horizon of a BH to see what happens, choose a large one so as not to be torn apart before you get there. 15 February 2018 (UR) Astronomy 142 Spring / 36

17 Hawking radiation 15 February 2018 (UR) Astronomy 142 Spring / 36

18 Hawking radiation: Black holes emit light! Virtual particle-antiparticle pairs created by vacuum fluctuations can be split by the strong gravity near a horizon. Both of the particles can fall into the horizon, but it is also possible for one to fall in while the other escapes. If one particle escapes it looks to a distant observer like the BH is emitting the particle. Question: Doesn t this violate energy conservation? 15 February 2018 (UR) Astronomy 142 Spring / 36

19 Hawking radiation: Black holes emit light! Virtual particle-antiparticle pairs created by vacuum fluctuations can be split by the strong gravity near a horizon. Both of the particles can fall into the horizon, but it is also possible for one to fall in while the other escapes. If one particle escapes it looks to a distant observer like the BH is emitting the particle. Question: Doesn t this violate energy conservation? No. The energy conservation debt created by the un-recombined vacuum fluctuation is paid back by the BH itself; its mass decreases by E/c 2, where E is the energy of the escaping particle. 15 February 2018 (UR) Astronomy 142 Spring / 36

20 Black hole evaporation Hawking radiation is emitted more efficiently if the tides at the horizon are stronger. You will show in recitation that the tides at the horizon are larger for smaller-mass black holes. The emission is the same as a blackbody of temperature T = hc 3 16π 2 kgm ( a r ) Thus an isolated BH will eventually evaporate, as you will show in a future homework. Some calculated evaporation times: BH Mass Evaporation Time 10 9 M yr 2M yr 10 8 g 1 sec 15 February 2018 (UR) Astronomy 142 Spring / 36

21 No-hair theorem After a star has collapsed into a BH, the horizon is smooth. Nothing protrudes from it, and almost everything about the star that gave rise to the BH has lost its identity during the formation of the BH. Metaphor: no hair (information) is left to stick out of the horizon. Any protrusion, prominence, or other departure from spherical smoothness gets turned into gravitational radiation, i.e., it is radiated away during the collapse. Any magnetic field lines emanating from the star close up and get radiated away in the form of light during the collapse. The identity of the matter that made up the star is lost. Nothing about its previous configuration can be reconstructed. Even the matter/antimatter distinction is lost. Two stars of identical mass one of matter and one of antimatter would produce identical black holes. 15 February 2018 (UR) Astronomy 142 Spring / 36

22 No-hair theorem The black hole has only three quantities in common with the start that collapsed to create it: mass, spin, and electric charge. That is, in common with the star as it was immediately before the formation of the horizon. Only very tiny black holes can have much electric charge. Stars are electrically neutral. Spin makes the black hole depart from a spherical shape, but it is still smooth. From an observational standpoint, the bare BH is a lot like a giant elementary particle with only a few measurable properties. 15 February 2018 (UR) Astronomy 142 Spring / 36

23 GRO J : A real stellar mass black hole We know of 40 good candidates for stellar-mass black holes, of which at least 20 are rock-solid cases. GRO J , an X-ray binary in Scorpius, is one example of rock-solid evidence for a black hole. GRO J is a bright, soft X-ray transient (flaring) object discovered by the Compton γ-ray Observatory in It appeared as a nova (Nova Scorpii 1994) in visible light during the X-ray burst. It produced another optical/x-ray outburst in Normal stars, unless very young, and ordinary novae do not emit much light at X-ray wavelengths. To get electric charges to emit X-rays, one has to accelerate them close to c, which if done with gravity would require a neutron star or a black hole. When bursting, GRO J exhibits rapidly variable X-ray emission: the brightness changes by a factor of 2 in t 0.3 ms. Implications: 1. The object is at most 100 km across, which would give a light transit time of 0.3 ms km is far too small to be a star or a white dwarf. 15 February 2018 (UR) Astronomy 142 Spring / 36

24 GRO J : A real stellar mass black hole When it is not bursting, the source looks like a normal star, rather similar to the Sun (V1033 Sco, m 1 = 1.1M ). The star s brightness indicates a distance of 3.2 kpc. The star s spectral lines show it to be a single-line spectroscopic binary system: star and invisible companion in orbit. So the X-ray bright object is the invisible companion. Its period P = 2.62 days and v 1r = 216 km/s. Thus, the mass function (recall HW1) is f (m 1, m 2 ) = Pv3 1r 2πG = 2.7M < m 2 Companion exceeds max NS mass? The star is eclipsed when the system is in outburst but not when it is quiescent, so we view the orbit not exactly edge-on (70 ). Thus we know the mass of the X-ray bright companion precisely (Shahbaz 2003): m 2 = 5.99 ± 0.042M Companion definitely exceeds max NS mass 15 February 2018 (UR) Astronomy 142 Spring / 36

25 GRO J : A real stellar mass black hole GRO J emits relativistic bipolar jets of material, and the speed can be measured from the proper motion of the clumps in the jets (Hjellming 1995). The outflow speed is 0.92c, orientation is 85 from the line of sight, and 15 from the system rotation axis. The shapes of the clumps indicate that the jet is precessing with a period similar to the orbit. Ejection speeds tend to be similar to escape speeds. Nothing but a BH would eject material at 0.92c. 15 February 2018 (UR) Astronomy 142 Spring / 36

26 GRO J : A real stellar mass black hole A 6M non-spinning black hole has a horizon circumference of 111 km and an innermost stable orbit of 333 km. Material in this orbit will circle the black hole at 367 Hz. However, the X-ray brightness of GRO J is often seen at 450 Hz, not 367 Hz, for long stretches of time (Strohmayer 2001). This behavior is called quasiperiodic oscillation. Nothing besides very hot material in a stable orbit can do this so reproducibly at this frequency. Thus there are stable orbits closer to the black hole than they can be if it does not spin. The maximum spin rate of a BH corresponds to a coordinate speed at the horizon of c. GRO J spins at 21% of this rate. 15 February 2018 (UR) Astronomy 142 Spring / 36

27 GRO J : A real stellar mass black hole In blue: the innermost stable orbit for a 5.99 ± 0.42M black hole. In red: the frequency of quasiperiodic oscillations in GRO J (Strohmayer 2001). 15 February 2018 (UR) Astronomy 142 Spring / 36

28 GRO J : A real stellar mass black hole Thus we have many lines of evidence that GRO J is a binary system consisting of a 1.1M main-sequence star and a 6.0M black hole. Until 2016, this was as solid as such a case gets. Let us review the evidence: High-energy radiation (X-rays) Variability size : X-ray object too small to be a star Orbital dynamics: mass of dark companion is precisely determined and far too large to be a neutron star or white dwarf. It is too faint to be an ordinary 6M star at 3.2 kpc. Quasiperiodic X-ray oscillations are easily explained as due to spin in a BH. Relativistic jets are ejected from the system at nearly light speed. 15 February 2018 (UR) Astronomy 142 Spring / 36

29 Gravitational radiation from black hole mergers Cartoon of the spacetime distortion created by two compact objects (neutron stars) in a close binary orbit. Image from LIGO. 15 February 2018 (UR) Astronomy 142 Spring / 36

30 Gravitational radiation from a binary system GW luminosity of a binary system: m 2 L GW = 32 5 G 4 (m 1 m 2 ) 2 (m 1 + m 2 ) c 5 r 5 r 2 r 1 r = r 1 + r 2 Example m 1 = m 2 = 5M and r = 1 AU: m 1 L GW = erg s L But if r = R, then L GW = erg s 1 = 165L 15 February 2018 (UR) Astronomy 142 Spring / 36

31 Properties of gravitational waves (GWs) Polarization modes of GWs (Centrella et al. 2010). 15 February 2018 (UR) Astronomy 142 Spring / 36

32 Properties of gravitational waves (GWs) Effect of GWs on a ring of test particles (Li 2014). 15 February 2018 (UR) Astronomy 142 Spring / 36

33 Detecting gravitational waves Diagram of a LIGO detector (Abbott et al. 2016). 15 February 2018 (UR) Astronomy 142 Spring / 36

34 Black hole merger signal Observation of a BH-BH merger at the two LIGO sites, September 14, 2015 (Abbott et al. 2016). The rise in frequency as a function of time (the chirp ) is characteristic of a binary inspiral. 15 February 2018 (UR) Astronomy 142 Spring / 36

35 Black hole merger signal The estimated strain of the GW signal, the BH-BH separation, and the relative velocity of the merger estimated from numerical GR (Abbott et al. 2016). Note the close separation and highly relativistic velocities of the BHs before the merger and ringdown. Simulation of a BH-BH merger 15 February 2018 (UR) Astronomy 142 Spring / 36

36 Details of the first observed BH-BH merger From numerical simulations of binary BH mergers (Abbott et al. 2016), the September 14, 2015 event corresponded to: m 1 = M m 2 = 29 ± 4M L peak = erg/s M c 2 s 1! Question: How do we know these were not neutron stars? 15 February 2018 (UR) Astronomy 142 Spring / 36

37 Details of the first observed BH-BH merger From numerical simulations of binary BH mergers (Abbott et al. 2016), the September 14, 2015 event corresponded to: m 1 = M m 2 = 29 ± 4M L peak = erg/s M c 2 s 1! Question: How do we know these were not neutron stars? No evidence for corresponding electromagnetic radiation. Best-fit masses from numerical GR of the binary merger far exceed maximum allowed neutron star mass. 15 February 2018 (UR) Astronomy 142 Spring / 36

38 Gravitational wave detections to date Since 2016, we have successfully detected five BH-BH merger events and one NS-NS merger event, thanks to LIGO. Event Date Mergers Final mass GW Sept Two black holes (35.4M and 29.8M ) 62.2M black hole GW Dec Two black holes (14.2M and 7.5M ) 20.8M black hole GW Jan Two black holes (31.2M and 19.4M ) 48.7M black hole GW June 2017 Two black holes (12M and 7M ) 18M black hole GW Aug Two black holes (30.5M and 25.3M ) 53.2M black hole GW Aug Two neutron stars (1.36M and 1.17M ) < 2.74M black hole It is rare in physics for data to be so unambiguous. These provide extremely convincing evidence for the existence of black holes of tens of solar masses! Question: How do we know that the last event involved neutron stars? 15 February 2018 (UR) Astronomy 142 Spring / 36

39 GW170817: a NS-NS merger The first GW detection to be confirmed by non-gravitational means Abbott et al February 2018 (UR) Astronomy 142 Spring / 36