1 ASTR 200 : Lecture 21 Stellar mass Black Holes
High-mass core collapse Just as there is an upper limit to the mass of a white dwarf (the Chandrasekhar limit), there is an upper limit to the mass of a neutron star The maximum mass that it is felt neutron degeneracy pressure could hold up the star is called the Oppenheimer-Volkov limit, estimated to be ~3 M solar 2 This mass is uncertain because the equation of state of extremely dense matter is modified by nuclear forces, and there is several equations of state for degenerate neutron matter exists (uncertainties ~factor 2). BUT, if the mass exceeds this, the relativistic neutron core will collapse, and no known force can stop it.
A Black hole To the current knowledge of physics, such an object will collapse to a 'point' of zero radius, called a `singularity', because the finite mass occupies zero volume and the density becomes infinite. In astronomy and physics, the fundamental property of a `black hole' is that there is a place where the escape speed exceeds the speed of light. Although doing all of this precisely requires general relativity, there are aspects that can be understood with just Newtonian gravity. 3
Escape speed Return to our friend, the vis-viva equation, relating orbital speed v to the mass of the central object M, the current distance r and the orbit's semimajor axis a v 2 =GM ( 2 r 1 a ) If reach some distance r=r inside of which all the mass M is now contained inside that distance R and is spherically symmetric, then pose the question: What speed does one need to travel to get arbitrarily far away? That would correspond to a infinity, and thus 4 v esc= 2GM R which we derived before
The Schwarzschild radius 5 So given the mass M, there is some radius r s that if you pack all the mass interior to that radius, the escape speed reaches the speed of light. Its expression: r s = 2 GM 3 km ( c 2 M M solar ) If a star of mass M falls below this radius, nothing from inside r s could be moving fast enough to escape to great distance, not even a light beam This is called the Schwarzschild radius, after Karl Schwarzschild, who first solved Einstein's field equations for a spherically symmetric mass. Note that this depends only on M...
6 Note that incredible density NOT required Even in the 18 th century, Laplace and Michell realized, after Newton's corpuscular (photons, in modern term) that since light traveled at a finite speed, it might be possible to have v esc > c. Michell (english scientist) suggested there could be 'dark stars' whose gravitational pull would be strong enough that light could not escape Laplace suggested however a simpler idea. Imagine a larger and larger uniform density object; then v esc= 2GM R = 8 π G 3 ρ R 3 R 8ρ G R so even for `normal' densities, a large-enough object has v esc --> c Eg., for water's density, R>3000 AU would have v esc > c
7 The event horizon The surface of radius r s is called the event horizon of the black hole. It is called such because an outside observer cannot `see' (receive photons) from inside this surface; they cannot escape. Even if some unknown quantum mechanical effect were to prevent the central singularity, there is STILL an event horizon, and light and any material object cannot escape. Additionally, any matter which enters the event horizon cannot escape either; it is cut off forever from the rest of the universe. In principle, any object can form a black hole if you force it into a small enough volume, but stellar evolution provides a natural path
8 Stellar Black Hole formation Very high-mass stars may have cores so massive that even neutron degeneracy pressure is unable to hold up their collapsing cores after the supernova explosion. Once the event horizon forms, what happens INSIDE the event horizon becomes un-knowable, and the black hole from the outside actually becomes one of the universe's simplest objects 'Black Holes have no Hair theorem (first version proven by Werner Israel, Univ. of Alberta) In terms of gravity and electromagnetism, a black is completely described by only: It's total mass M, charge Q, and spin.
Observing black holes Well, the problem is...they're black! How does one observe something that emits no radiation? 9
10 Observing black holes Well, the problem is...they're black! How does one observe something that emits no radiation? One needs to detect their effects on other things. Binary motion Like exoplanets...see the motion they induce on something else There may be material in orbit around the black hole (outside the event horizon)
11 Black hole accretion disks The black hole can be surrounded by a disk that feeds matter into it, just like a protostar Local energy dissipation at each radius results in material losing orbital energy and spiraling slowly towards the black hole, getting very hot Because ionized, make a magnetic field which some charged particles escape along `Seeing' this tricky, because black hole bends light and you can see the far side of the disk!
V404 Cygni A binary system of period 6.5 days, with X-ray emission. The visible star is roughly Sun-like (0.6 times the mass), but its periodic Doppler shift (of amplitude ~210 km/sec) shows it orbits an invisible object Minimum mass (because of sin i effect) of invisible companion is 6 solar masses; best estimate is 10-12 solar masses Larger than a neutron star can be Clearest case in our galaxy of a stellar-scale black hole Distance of ~2400 pc X-rays are coming from an `accretion disk' surrounding the black hole. Gas drawn off the companion spirals down and heats up in a disk around black hole, emitting X-rays before it 12 spirals inside the event horizon
13 A brief intro to General Relativity When gravitational fields become very strong, newtonian gravity is only an approximation to how objects accelerate General relativity (GR) is a geometrical description of gravity; it and quantum mechanics are the two pillars of modern physics. There are no known phenomena that are not explained by them. In GR, space and time are considered simultaneously and on an equal footing. The 3 spatial dimensions and one time dimension are combined to form 4d spacetime As time progresses, an object traces out a path through spacetime. If there are no forces, the path is a straight line the shortest distance between two points.
Motion in curved spacetime 14 Objects always move 'locally straight' in spacetime But the presence of a massive objects warps space itself, causing an object's path to appear to `bend' Newton says this bending is because there is a `force' pulling the object In GR, the object is going along a path that is the shortest distance between the two points, along a curve called a geodesic, and a freely moving body is simply following the most logical path
Geodesics on a sphere We can think of motion in curved space Example: motion on a sphere is along geodesics which are great circles - Shortest path between 2 points (why airliners to Europe cross N pole) Here imagine two paths leaving A, on paths b and c. At first they separate, slowing and reaching a maximum distance (marked a here) at B and C; as they continue to the other side of the sphere, they re-converge. If you think of motion 'along the geodesic' as the time dimension, this separation and then re-convergence would be you'd see if two 15 gravitationally attracting objects were initially thrown apart. No forces!
Note that away from the event horizon, motions behave just like before in newtonian physics - attraction to center, orbits, and deflection 16
Clarification :It's just MASS 17 In popular conceptions, black holes are like 'cosmic vacuum cleaners' that are pulling in everything with some of `super gravity' This is not true, once you get `far away' (hundreds of Schwarzchild radii...which is not very far), the black hole's gravity behaves exactly like the gravity of that same object if it was not a black hole If all the Sun's mass was suddenly compressed to within its event horizon, it would have NO gravitational effect on the orbits of the planets. The lack of arriving solar light would, of course, result in things cooling down...