dead stars, failed stars, & stellar-mass black holeskdouglass/classes/ast142/lectures/... ·...

Dead Stars, Failed Stars, &Stellar-Mass Black Holes

Neutron stars, Brown dwarfs, & Giant planetsGeneral relativity

Hawking radiationGravitational waves

February 13, 2020

University of Rochester

Dead stars, Failed stars, &Stellar-mass sized black holes

I Neutron starsI Brown dwarfs and giant planetsI General relativity & its prediction of black

holesI HyperspaceI Hawking radiationI Gravitational radiation

Reading: Kutner Sec. 8.4 & 11.5, Ryden Sec. 18.3

Right: HST image of the Andromeda galaxy with an X-ray imageinset of an ultra-luminous X-ray source (a stellar-mass blackhole). From the MPE.

February 13, 2020 (UR) Astronomy 142 | Spring 2020 2 / 46

https://phys.org/news/2012-02-spectacularly-bright-andromeda-black-hole.html

Theory vs. observation for white dwarfsWhite dwarfs are hard to detect in binariesbecause they are so much fainter thanmain-sequence stars.I Solid circles show white dwarfs in binaries

with precisely known orbits (Provencal1998, Provencal 2002, Parsons 2010, Parsons2011, Parsons 2012a, Parsons 2012b, Pyrzas2012, Hermes 2012).

I The white dwarfs in visual binaries arelabeled with their names.

I These “best” white dwarfs are joined withothers detected in WD-MS systems by theSloan Digital Sky Survey (Rebassa 2012) andshown to be eclipsing (Parsons 2013).


http://adsabs.harvard.edu/abs/1998ApJ...494..759P



http://adsabs.harvard.edu/abs/2010MNRAS.402.2591P

http://adsabs.harvard.edu/abs/2011ApJ...735L..30P

http://adsabs.harvard.edu/abs/2011ApJ...735L..30P



http://adsabs.harvard.edu/abs/2012MNRAS.419..817P


http://adsabs.harvard.edu/abs/2012ApJ...757L..21H

http://adsabs.harvard.edu/abs/2012MNRAS.419..806R


Neutron starsWhat happens in a dead star with M > MSAC?I Such a star cannot be supported by electron degeneracy pressure. Add a little too

much mass and it will collapse under gravity or explode.I During the collapse, the extra energy liberated from gravity, plus the high density,

can help drive some endothermic nuclear reactions, notably

∆E + e− + p→ n + νe

I But neutrons are fermions too, and neutron degeneracy pressure can also balancegravity. A neutron star is formed (Tolman 1939, Oppenheimer 1939).

I The nonrelativistic R-M relation for a NS is

R = 0.0685h2

Gm8/3p

M−1/3

This is the mass of a star and the size of a city.February 13, 2020 (UR) Astronomy 142 | Spring 2020 4 / 46

http://adsabs.harvard.edu/abs/1939PhRv...55..364T

http://adsabs.harvard.edu/abs/1939PhRv...55..374O

Size of a 1M� neutron star


Pulsars

I Most of the neutron stars we observeappear to us as pulars.

I A pulsar is a rapidly rotating NS (orWD) that emits a beam ofelectromagnetic radiation.

I Neutron stars can also be observed asX-ray flaring members of low-massbinary systems (LMXBs).

I PS B1919+21 was the first pulsar to bediscovered in 1967 (Hewish et al. 1968)

I Right: the center of the Crab Nebula(M1), the remnant of SN 1054.

Optical/X-ray overlay from the Hubble and Chandratelescopes.


http://adsabs.harvard.edu/abs/1968Natur.217..709H

Pulsars

X-ray images of the Crab Nebula from the Chandra X-ray Observatory (CXO) taken 0.015 s apart.


http://chandra.harvard.edu/index.html

Neutron star masses

I The maximum mass of a neutron star is not easy to calculate as it requires GeneralRelativity and a strong-interaction equation of state.

I The maximum mass turns out to be about 2.2M� (Lattimer 2012).I The maximum cannot be > 3M� or the equation of state would imply a sound speed

vs > c.I A handful of neutron stars are observed in low-mass X-ray binary (LMXB) systems

and their masses and radii have been measured (see e.g. Steiner et al. 2010).I Neutron stars are so small that eclipses enabling the determination of R from timing

are extremely rare. Instead, we get R from the luminosity and temperaturedetermined from spectra taken in between flares (“quiescent LMXB”).


http://adsabs.harvard.edu/abs/2012ARNPS..62..485L

http://adsabs.harvard.edu/abs/2010ApJ...722...33S

Neutron star R–M relation

I The theoretical calculation of theNS R–M relation is beyond thescope of an undergraduate course.

I Currently the calculation ofLattimer & Prakash (2007) comesclose to matching the data onneutron star binaries (Steiner et al.2010).

I Note that circumference is plottedinstead of radius.


http://adsabs.harvard.edu/abs/2007PhR...442..109L



Degenerate star R–M relations

Gray line: causality cutoffA star smaller than 5.8πGM/c2

turns out to require an EOS with

∂P∂ρ

> c2

which in turn implies a speed ofsound vs > c, violating the order ofcause and effect.


Far below MSAC: Brown dwarfs and giant planets

When stars form, they contract until they are hot enough in the center (about 3× 106 K)to ignite the pp chain fusion reactions. Recall that

Tc =Pcµc

ρck≈ µc

k

(GM2

R4

)(1

150R3

M

)= 15.7× 106

(M

M�

)(R�R

)K

for solar-type stars, if gravity is supported by gas pressure.I For small masses this involves gas pressures that become smaller than the electron

degeneracy pressure, so degeneracy pressure can stop the contraction and preventthe object from reaching fusion temperatures. This imposes a lower mass limit onwhat can become a star.

I The H-burning limit turns out to be 0.08M�.


Far below MSAC: Brown dwarfs and giant planets

Depending upon how they are formed and what their mass is, such objects are calledbrown dwarfs or giant planets.I Because they cannot replace the energy that leaks away in the form of radiation, they

simply remain at the size determined by degeneracy pressure and cool off forever.I Thus, if they are very old, they are very faint. This prevented their detection until

1995.I Today, thousands are known from deep near-IR surveys and from Spitzer Space

Telescope observations.I Once it was thought that these objects could be numerous enough to comprise a

significant (and invisible) component of the mass in the Galaxy.I We will come back to this idea when we discuss Dark Matter.


Escape velocities from starsNeglecting relativity, in increasingly bad approximations:

E =12

mv2esc −

GMmR

= 0

∴ vesc =

√2GM

R= 619 km/s = 0.002c Sun= 6970 km/s = 0.023c 1M� white dwarf= 154000 km/s = 0.514c 1M� neutron star

And note that vesc = c ≈ 3× 105 km/s when

R =2GM

c2


Beyond the NS maximum mass: Black holes

I The maximum mass of a neutron star is ∼ 2.2M�. There is no known physicalprocess that can support a heavier object without internal energy generation.

I A non-spinning heavier object will collapse past neutron-star dimensions and soonthereafter becomes a black hole, an object from which even light cannot escape ifemitted within a distance

RSch =2GM

c2 Schwarzschild radius

of the object as measured by a distant observer.I This spherical surface is called the event horizon or simply the “horizon” of the

black hole.


https://en.wikipedia.org/wiki/Black_hole

Black holes & General Relativity

I The nonrelativistic result R =√

2GM/c2 is, by accident, the same as RSch derivedwith the general theory of relativity (GR).

I GR is a description of the effect of gravity at any strength, even handling largeamounts of mass shrunk to small dimensions. It involves new mathematicalconcepts beyond the scope of this course.

First published by Albert Einstein in 1916 (Einstein 1916), the theory provides a set offield equations to describe gravity:

spacetime curvature→ Gµν =8πG

c2 Tµν ← mass and energy

In plain English (Gravitation by Misner 1973):I Mass causes space and time to be curved, or warped.I The resulting curvature of space determines how masses will move.


https://en.wikipedia.org/wiki/General_relativity

http://adsabs.harvard.edu/abs/1916AnP...354..769E

Interesting facts about black holes

I Time and space are warped substantially near black holes.I Time intervals on clocks near black holes appear to distant observers to be slow

compared to their own local identical clocks.I This effect is known as gravitational time dilation or gravitational redshift:

time measuredby a distantobserver

→ ∆t =∆τ√

1− RSchr

> ∆τ ← proper time nearthe BH

I Thus, time appears to stop at the event horizon to a distant observer:

∆t→ ∞ as r→ RSch

This behavior gave the horizon its original name: the “Schwarzschild singularity.”



I Near a black hole, a small length ∆Lmeasured simultaneously (as with a measuringtape) between two points on a radial line is greater than the distance ∆r between thepoints, measured by a distant observer:

proper lengthnear the BH

→ ∆L =∆r√

1− RSchr

> ∆r← length measured bya distant observer

I Neither ∆r nor the radius r measured by a distant observer of a point near the blackhole has meaning as a physical distance to an observer near the black hole.

I r and ∆r are called coordinate distances.I 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 = C2π .


Circular orbits and their radii in GRCircles in flat spacetime: C = 2πr = πd. That, of course, is the very definition of π.

r

Cd


Circular orbits in flat space (all in same plane)

I Concentric circular orbitsin flat space.

I The distance between eachorbit is 1.

I Local and distantobservers report the samedistances betweenconcentric orbits.

2π1

4π

1

6π

1

8π

1


Circular orbits in space warped by a black hole

I Imagine a BH whosehorizon has circumference2π.

I The distant observer stillreports distances of 1between each orbit.

I The local observer reportswarped distances.

I Both observers measurethe same circumferences!

2π4π

12.296

6π

11.300

8π

11.185


Visualization of warped space: “Hyperspace”

I To connect these circles with segmentsof the “too long” lengths, it is helpful toconsider them to be offset from eachother along an imaginary dimension ⊥x and y but which is not z.

I If the additional dimension were z, thenthe circles would not appear to lie on aplane.

I Such additional dimensions comprisehyperspace.


Embedding diagrams

I This is why you often see the equatorialplane of a black hole represented as afunnel-shaped surface, as if made froma stretched rubber sheet.

I It is important to note that the directionof stretch is in hyperspace.

I The scene would not look like a funnelto an observer of three spatialdimensions.



I Orbits outside the BH’s horizon, further away than 1.5RSch (in the coordinate systemof a distant observer) still turn out to be ellipses.

I The resulting coordinate speed in orbit (for the coordinate system of a distantobserver) is the same as that obtained in Newtonian gravity:

v = rdφ

dt=

√GM

r

I At the horizon, the radial component of the coordinate speed of light is zero: lightcannot escape.

I Thus, no information can reach a distant observer from, or from within, the BHhorizon.

I For non-spinning black holes, orbits with coordinate radius < 3RSch are unstable tosmall perturbations.



I There are no orbits with coordinate radius < 1.5RSch for a non-spinning black hole.At this radius, the local orbital speed is the speed of light, and smaller orbits wouldrequire impossibly higher speeds.

I You cannot orbit at this close distance because your rest mass is nonzero; if youcould, you could train your binoculars straight ahead (in the φ direction) and see theback of your head.

I To get closer to the horizon, the descent must be radial while balancing gravity withthrust, as in a rocket launch.

I If the black hole spins, the innermost stable orbit and the photon orbit are smallerthan 3RSch and 1.5RSch if the particle orbits in the same direction as the spin, andlarger if it orbits in the opposite direction.

I Within r = 1.5RSch, all geodesics (possible paths for light) terminate at the BHhorizon.


Interesting facts about black holesI Thus, from near the horizon, the sky appears to be compressed into a small range of

angles directly overhead.I The range of angles is smaller the closer one is to the horizon, and vanishes at the

horizon.I Objects in the sky will appear bluer than their natural colors as well due to the

gravitational Doppler shift.I Space itself is stuck to the horizon, since one end of all the geodesics are there.I If the horizon began to rotate, the ends of the geodesics would rotate with it. (This

harmonizes with time stopping there.)I Gravitational acceleration turns out to be

a =GMr2

√1

1− RSchr


Interesting facts about black holesI This has the familiar Newtonian form at large r but blows up at r = RSch.I Thus, in a vertical descent to a hovering position just above the horizon, very large

gravitational accelerations would be encountered.I Tidal forces turn out the same near a black hole as in Newtonian gravity, and are

finite at the horizon.I For an object of length ∆r in the radial direction and ∆x in the crosswise directions,

∆ar =2GM

r3 ∆r ∆aφ = −GMr3 ∆x

ExampleI For a 2 m person and a 10M� BH, the radial tidal acceleration ∆ar at the event

horizon is 2× 1010 cm/s2, or 2× 107 g.I ∆ar = 1g for a 4.6× 104M� 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 getthere.


Hawking radiation


Hawking radiation: Black holes emit light!

I Virtual particle-antiparticle pairs created by vacuum fluctuations can be split by thestrong gravity near a horizon.

I Both of the particles can fall into the horizon, but it is also possible for one to fall inwhile the other escapes.

I If one particle escapes it looks to a distant observer like the BH is “emitting” theparticle.

Doesn’t this violate energy conservation?No. The energy conservation debt created by the un-recombined vacuum fluctuation ispaid back by the BH itself; its mass decreases by E/c2, where E is the energy of theescaping particle.


Black hole evaporation

I 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-massblack holes.

I The emission is the same as a blackbody of temperature

T =hc3

16π2kGM

(∝√

∆ar

)I Thus an isolated BH will eventually evaporate, as you will show in a future

homework. Some calculated evaporation times:

BH Mass Evaporation Time109M� 1094 yr

2M� 1067 yr108 g 1 sec


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 duringthe formation of the BH.

Metaphor: no “hair” (information) is left to “stick out” of the horizon.I Any protrusion, prominence, or other departure from spherical smoothness gets

turned into gravitational radiation, i.e., it is radiated away during the collapse.I Any magnetic field lines emanating from the star close up and get radiated away in

the form of light during the collapse.I The identity of the matter that made up the star is lost. Nothing about its previous

configuration can be reconstructed.I Even the matter/antimatter distinction is lost. Two stars of identical mass — one of

matter and one of antimatter — would produce identical black holes.


No-hair theorem

The black hole has only three quantities in common with the start that collapsed to createit: mass, spin, and electric charge.I That is, in common with the star as it was immediately before the formation of the horizon.I Only very tiny black holes can have much electric charge. Stars are electrically

neutral.I 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 particlewith only a few measurable properties.


GRO J1655-40: A real stellar mass black holeWe know of ∼ 70 good candidates for stellar-mass black holes, of which at least 50 arerock-solid cases. GRO J1655-40, an X-ray binary in Scorpius, is one example of rock-solidevidence for a black hole.I GRO J1655-40 is a bright, soft X-ray transient (flaring) object discovered by the

Compton γ-ray Observatory in 1994.I It appeared as a nova (Nova Scorpii 1994) in visible light during the X-ray burst.I It produced another optical/X-ray outburst in 2005.I Normal stars, unless very young, and ordinary novae do not emit much light at

X-ray wavelengths.I 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.I When bursting, GRO J1655-40 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.2. 100 km is far too small to be a star or a white dwarf.February 13, 2020 (UR) Astronomy 142 | Spring 2020 32 / 46

GRO J1655-40: A real stellar mass black holeI When it is not bursting, the source looks like a normal star, rather similar to the Sun

(V1033 Sco, m1 = 1.1M�).I 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 andinvisible companion in orbit.I So the X-ray bright object is the invisible companion.I Its period P = 2.62 days and v1r = 216 km/s.I Thus, the mass function (recall HW1) is

f (m1, m2) =Pv3

1r2πG

= 2.7M� < m2 Companion exceeds max NS mass?

The star is eclipsed when the system is in outburst but not when it is quiescent, so weview the orbit not exactly edge-on (70◦). Thus we know the mass of the X-ray brightcompanion precisely (Shahbaz 2003):

m2 = 5.99± 0.042M� Companion definitely exceeds max NS mass


http://adsabs.harvard.edu/abs/2003MNRAS.339.1031S

GRO J1655-40: A real stellar mass black hole

I GRO J1655-40 emits relativistic bipolar “jets” ofmaterial, and the speed can be measured from theproper motion of the clumps in the jets (Hjellming1995).

I The outflow speed is 0.92c, orientation is 85◦ fromthe line of sight, and 15◦ from the system rotationaxis.

I The shapes of the clumps indicate that the jet isprecessing 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.





A 6M� non-spinning black hole has a horizon circumference of 111 km and an innermoststable orbit of 333 km. Material in this orbit will circle the black hole at 367 Hz.I However, the X-ray brightness of GRO J1655-40 is often seen at 450 Hz, not 367 Hz,

for long stretches of time (Strohmayer 2001).I This behavior is called quasiperiodic oscillation.I Nothing besides very hot material in a stable orbit can do this so reproducibly at this

frequency.I Thus there are stable orbits closer to the black hole than they can be if it does not spin.I The maximum spin rate of a BH corresponds to a coordinate speed at the horizon of

c. GRO J1655-40 spins at 21% of this rate.


http://adsabs.harvard.edu/abs/2001ApJ...552L..49S


I In blue: the innermost stable orbitfor a 5.99± 0.42M� black hole.

I In red: the frequency ofquasiperiodic oscillations inGRO J1655-40 (Strohmayer 2001).


http://adsabs.harvard.edu/abs/2001ApJ...552L..49S


Thus we have many lines of evidence that GRO J1655-40 is a binary system consisting of a1.1M� main-sequence star and a 6.0M� black hole. Until 2016, this was as solid as such acase gets. Let us review the evidence:I High-energy radiation (X-rays)I “Variability size”: X-ray object too small to be a starI Orbital dynamics: mass of dark companion is precisely determined and far too large

to be a neutron star or white dwarf.I It is too faint to be an ordinary 6M� star at 3.2 kpc.I Quasiperiodic X-ray oscillations are easily explained as due to spin in a BH.I Relativistic jets are ejected from the system at nearly light speed.


Gravitational radiation from black hole mergersCartoon of the spacetime distortion created by two compact objects (neutron stars) in a close binary orbit.Image from LIGO.


https://www.ligo.caltech.edu/

Gravitational radiation from a binary system

m2

m1

r2

r1

r = r1 + r2

GW luminosity of a binary system:

LGW =325

G4

c5(m1m2)

2(m1 + m2)

r5

Examplem1 = m2 = 5M� and r = 1 AU:

LGW = 1.4× 1024erg s−1 ≈ 3× 10−10L�

But if r = R�, then

LGW = 6.4× 1035 erg s−1 = 165L�


Properties of gravitational waves (GWs)

Effect of GWs on a ring of test particles (Li 2014).


http://www.2physics.com/2014/10/testing-strong-field-dynamics-of.html

Detecting gravitational waves

Diagram of a LIGOdetector (Abbott et al.2016).


http://adsabs.harvard.edu/abs/2016PhRvL.116f1102A


Black hole merger signal

Observation of a BH-BHmerger at the two LIGO sites,September 14, 2015 (Abbott etal. 2016).

The rise in frequency as afunction of time (the “chirp”) ischaracteristic of a binaryinspiral.




Black hole merger signal

The estimated strain of the GW signal,the BH-BH separation, and the relativevelocity of the merger estimated fromnumerical GR (Abbott et al. 2016).

Note the close separation and highlyrelativistic velocities of the BHs beforethe merger and “ringdown.”

Simulation of a BH-BH merger



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:I m1 = 36+5

−4M�I m2 = 29±4M�I Lpeak = 3.6+0.5

−0.4 × 1056 erg/s ≈ 200+30−20M�c2 s−1!

How do we know these were not neutron stars?I No evidence for corresponding electromagnetic radiation.I Best-fit masses from numerical GR of the binary merger far exceed maximum

allowed neutron star mass.



Gravitational wave detections to date

Since 2015, we havesuccessfully detected 40BH-BH merger events andtwo–four NS-NS mergerevent, thanks to LIGO andVIRGO. It is rare inphysics for data to be sounambiguous. Theseprovide extremelyconvincing evidence forthe existence of blackholes of tens of solarmasses!


GW170817: a NS-NS mergerThe first GW detection to be confirmedby non-gravitational means

Abbott et al. 2017February 13, 2020 (UR) Astronomy 142 | Spring 2020 46 / 46

http://adsabs.harvard.edu/abs/2017ApJ...848L..12A

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