Evidence for BH: Orbits and Stellar SourcesIn the previous lecture we talked about AGN, and the evidence that these are poweredby supermassive black holes. This evidence, though compelling, is indirect. Here we’ll talkabout more direct lines of evidence. Ask class: what is the most direct astronomical wayof measuring mass? Observation of orbits.Star motions in galactic centersOne set of observations, which has had promise for years but has only come into itsown since the 1990s, is the observation of star motions in the centers of galaxies. Oneexpects that the most massive things will settle into the centers of galaxies by gravitationalinteractions, so it is reasonable to look for black holes there. If the hole is not accretingactively, its presence can be sensed by the motion of stars near it. In particular, if manystars are moving rapidly in ways that are consistent with an orbit, then by determinationof their velocity and their radius of orbit one can infer the mass interior to them. This issomething that has plenty of potential hazards. For example, the motion had better be dueto the gravity of the central mass(es), and not something else. A while ago there was apress release announcing the discovery of a 1011M¯black hole in the center of one galaxy,that the authors had to retract when it was discovered that they were actually looking atthe center of two galaxies merging; the high velocities were ballistic, not orbital! However,many examples have by this time been found, particularly with telescopes or techniquesthat allow excellent angular resolution, so one can look at many individual stars as close aspossible to the central object. For our own Galaxy, the mass interior to stars levels off atabout 0.1 pc, at about 3.5 × 106M¯. The leveling off indicates that the mass responsiblefor the orbits is more tightly concentrated yet, which is essentially conclusive evidence thatthis is a black hole. Even more impressively, certain individual stars have been observedto orbit around Sgr A∗(the candidate center of the Galaxy, and a mild radio source) withpericenter distances of as low as 40-60 AU, and an implied mass that is still 3.5 × 106M¯.Can we do this with a cluster of stars? Such a cluster would require a density exceeding1017M¯pc−3. Ask class: why is this a problem? Actual collisions might not be a problem:even at such densities, the average distance between stars is ∼10 AU, and if the stars werestellar remnants such as white dwarfs or neutron stars they wouldn’t collide. It is dynamicalinstabilities that are the problem. (the stars would fling each other out too rapidly). Anotherway out would be to have a (dark!) object that is not a black hole, but has several millionsolar masses of material (several hundred million in other cases). This seems impossible.Movies of the motion of the stars near Sgr A∗, the candidate center of the Galaxy, can befound at http://www.mpe.mpg.de/www ir/GC/prop.html.Black holes in binariesThe above is enough to demonstrate that black holes must exist. However, complemen-tary (and in some ways more convincing) evidence exists from the study of X-ray binaries.In many cases one sees X-rays coming from a region that contains a visible star that isorbiting around something not evident in optical. Ask class: do they know what needs tobe measured to get a constraint on the mass? One can measure the period P of the orbitand the radial velocity v1along the line of sight of the visible companion in its orbit. Herethe “1” indicates that the visible object is labeled 1 (the invisible object is labeled 2). Fromthese observables and Kepler’s third law one can calculate an important quantity called the“mass function”,f(M1, M2, i) =P v312πG=(M2sin i)3(M1+ M2)2. (1)Here M1is the mass of the visible star 1, M2is the mass of the invisible star 2, and i is theinclination angle of the orbital axis relative to the line of sight (e.g., i = 0 for face-on, i = 90for edge-on). You can convince yourself that the minimum mass of the unseen star is just f,which occurs for M1= 0 and i = 90. Therefore, from just these two measured parameters,it is possible to get a lower limit on the mass of the unseen object. Extremely generalconsiderations (the assumption that GR is the correct theory of gravity!) indicate thatneither a neutron star nor any other object with a surface that is supported by degeneracypressure can have a gravitational mass more than 3 M¯(the real limit is probably closer to2 M¯). Therefore, if f > 3 M¯, you’ve got a black hole. This is the case for ∼20 sources inthe Galaxy, and this (in my opinion) is the very best evidence for the existence of black holes.Incidentally, there are some systems where one can say something about the inclination anglei (mainly by modeling the variation in the optical light curve as the distorted star orbitsaround). For those, one can get fairly precise estimates of the black hole mass. The sevenexamples in which f > 3 M¯all have companions with low masses, unlike better-knowncandidates such as Cyg X-1. Ask class: why would it be easier to get mass estimateswhen the hole has a low-mass companion? Because the companion is moved around moreby the gravity of the black hole. For higher-mass stars, the mass function is typically lowbecause M1> M2. Uncertainties in the true mass of the companion then make rigorousidentification of the black hole very difficult if not impossible. There are also significantdifferences between accretion onto a black hole from a low-mass or a high-mass companion.We will now investigate them.Accretion from a low-mass companionIf the companion is a low-mass star, then it has few innate processes by which it losesmass. Mass transfer therefore happens when the star evolves or spirals close enough to theblack hole that the star’s radius exceeds the radius of its Roche lobe (that is, the matterbecomes gravitationally unbound with respect to the companion, and hence flows over to theFig. 1.— Schematic of Roche lobe overflow from a star to a compact object. Fromhttp://www.airynothing.com/high energy tutorial/sources/images/roche lobe.gifblack hole). Ask class: would the mass that makes its way to the black hole therefore havehigh or low angular momentum? High. Ask class: so, what happens to the mass as it spiralsin? An accretion disk forms, as we’ve discussed previously. The long evolutionary times oflow-mass stars means that this kind of accretion can continue for hundreds of