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Black HolesASTR 2110
Sarazin
Calculation of Curved Spacetime near Merging Black Holes
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Test #2 Monday, November 13, 11 - 11:50 am Ruffner G006 (classroom) Bring pencils, paper, calculator You may not consult the text, your notes,
or any other materials or any person You may bring a 3x5 card with equations ~2/3 Quantitative Problems (like
homework problems) ~1/3 Qualitative Questions
Multiple Choice, Short Answer, Fill In the Blank questions No essay questions
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Black HolesASTR 2110
Sarazin
Calculation of Curved Spacetime near Merging Black Holes
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Black Holes Gravity crushes star to a point = singularity Surrounded by ``surface’’ from which
nothing can escape = event horizon
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Schwarzschild Metric forBlack Hole
Karl Schwarzschild (1916): Spherical coordinates. Non-rotating black hole
ds( )2= c 1− 2GM
c2rdt
"
#$
%
&'
2
−dr
1− 2GMc2r
"
#
$$$$
%
&
''''
2
− r dθ( )2− r sinθ dφ( )2
ds( )2= c 1− RS
rdt
"
#$
%
&'
2
−dr
1− RSr
"
#
$$$$
%
&
''''
2
− r dθ( )2− r sinθ dφ( )2
RS ≡2GMc2 = 3 km M
M"
#$
%
&'
¤
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Black Hole Physics Nothing Escapes Event Horizon
Radius and Time Reverse Roles!! Radius always goes down, time can go
backwards
ds( )2= c 1− RS
rdt
"
#$
%
&'
2
−dr
1− RSr
"
#
$$$$
%
&
''''
2
− r dθ( )2− r sinθ dφ( )2
r < RS ⇒RSr>1⇒ 1− RS
r imaginary, 1− RS
r
"
#$
%
&'
2
< 0
(dt)2 term negative (normally positive)(dr)2 term positive (normally negative)r↔ t radius and time reverse roles!!
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Infall into Black Hole
infalling person
outside observer
infall slows, v è 0, ∞ redshift, fades away, hovers forever just above horizon
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Black Hole Physics Orbits near Black Hole
In general, complex. Consider circular orbits Newton: any radius possible
GR: No stable circular orbits r ≤ 3 RS
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Black Hole Physics “Black Holes Have No Hair”
No matter what they swallow, only retain: Mass M, Charge Q, Spin Angular Momentum J
Interstellar matter, very high electrical conductivity è “shorts out” charge
Astrophysical Black Holes: Mass M, Spin Angular Momentum J
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Spinning Black Holes Kerr Metric (Boyer-Linquist Coordinates)
ds( )2= 1− RSr
ρ2
"
#$
%
&' c dt( )2
−ρ2
Δdr2 − ρ dθ( )2
− r2 +α 2 +RSrα
2
ρ2 sin2θ"
#$
%
&' sinθ dφ( )2
+2RSrα sin2θ
ρ2 cdt dφ
RS =2GMc2 Schwarzschild radius
α ≡JMc
(units of length)
ρ2 ≡ r2 +α 2 cos2θ (units of length)Δ ≡ r2 − RSr +α
2 (units of length2 )
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Spinning Black Holes Kerr Metric
Exterior Solution Event Horizon Oblate spheroid “ergosphere”
Can escape, but must rotate with black hole
top view
side view
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Spinning Black Holes Kerr Metric
Interior Solution Ring singularity Inner Horizon!? But interior solution is
unstable
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Spinning Black Holes Kerr Metric
Worm holes?? Unstable è closes
as soon as anything enters it!
white hole
black hole
Closed wormhole
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Energy from Black Holes Penrose Mechanism
Rotating BH Throw something out
backwards Eout ≈ Ein + Δmc2 !! May not be important in
nature (too complicated), but …
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Solution to Mankind’s Problems Biggest problems:
Get rid of garbage, pollution
Make energy Build city around
rotating BH
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Energy from Black Holes Natural Method
Drop matter into BH carelessly Bump into, rub against other
matter Friction è KE è heat è light Virial Thm. Heat ≈ (1/2) |PE| ≈ (1/2) GMm/RS ≈ (1/2) GMm / (2GM/c2)
Eout = Heat ≈ mc2/4 !!! Accretion by BHs is biggest
important energy source in Universe
mass m mass m
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Black Hole Physics Black Hole Surface Areas
When BHs merge, mass can increase or decrease Total surface area of BH horizons always
increases in GR 2nd Law of BH Dynamics Entropy always increases
S = c3k4G
A Entropy
T = c3
8πkGM Temperature
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The Death of StarsASTR2110
Sarazin
Cat’s Eye Planetary Nebula
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Low Mass Star (1 M�)
Left as AGB (red supergiant) with C/O core, giant H envelope
H envelope, ~ few AU
C/O core, ~ 109 cm, degenerate & inert
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Asymptotic Giant Branch = Supergiant (I) Red supergiant Inert carbon/oxygen core
Low Mass Star - Death
AC
D
E
F
G
H
MS B
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Low Mass Star (1 M�) - Cont. Inert core, must die
Envelope very extended, cool.
Hydrogen recombines: H+ + e- → H + 13.6 eV
Gravitational binding energy per atom = G M mH / R ≈ 10 ev (M / M�)(R/AU)-1
H shell ejected, (1/2) mHv2 ~ few eV, v ~ 20 km/s
Planetary Nebula
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Low Mass Star - Death
AC
D
E
F
G
H
MS B
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Planetary Nebula & White Dwarf
Degenerate core revealed, becomes a white dwarf
C/O for Sun
other composition for other masses
Core initially very hot, Teff ~ 100,000 K, lots of UV, ionizes planetary nebula
Fast wind from WD sweeps envelope into shell
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Low Mass Star - Death
AC
D
E
F
G
H
MS B
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Planetary Nebula & White Dwarf
Degenerate core revealed, becomes a white dwarf
C/O for Sun
other composition for other masses
Core initially very hot, Teff ~ 100,000 K, lots of UV, ionizes planetary nebula
Fast wind from WD sweeps envelope into shell
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula
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Planetary Nebula & White Dwarf
Degenerate core revealed, becomes a white dwarf
C/O for Sun
other composition for other masses
Core initially very hot, Teff ~ 100,000 K, lots of UV, ionizes planetary nebula
Fast wind from WD sweeps envelope into shell
WD cools, fades away
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Low Mass Star - Death
AC
D
E
F
G
H
MS B
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SupernovaeASTR2110
Sarazin
SN1987A - X-ray and Optical
![Page 38: Black Holes ASTR 2110 Sarazin - University of Virginiapeople.virginia.edu/~cls7i/Classes/astr2110/Lecture28...Test #2 Monday, November 13, 11 - 11:50 am Ruffner G006 (classroom) Bring](https://reader033.vdocuments.us/reader033/viewer/2022043013/5fac16dcccb20b2a0941d4bd/html5/thumbnails/38.jpg)
High Mass Star (>8 M�)
Left as red supergiant with Fe core and many burning shells
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A) Inert Fe core, many burning shells B) Either Red Supergiant or blue WR star
High Mass Star - Death
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Fe Core Fe core, no fuel left, inert core, supported by electron degeneracy pressure
Fe core ~ Fe white dwarf (outer envelope extended, weight not so large)
Mcore ≳ 1.4 M� Chandrasekhar mass → implodes!
Heat, Fe nuclei → p, n
p+ + e- → n + νe , lots of neutrinos
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Neutron Core Core is now mainly neutrons
Core collapses until R ~ 10 km, neutrons become degenerate, degeneracy pressure
Neutron core bounces, hits infalling gas at v ~ c/3
Shock from bounce plus neutrinos blow out envelope of star
Core-Collapse Supernova!!
![Page 42: Black Holes ASTR 2110 Sarazin - University of Virginiapeople.virginia.edu/~cls7i/Classes/astr2110/Lecture28...Test #2 Monday, November 13, 11 - 11:50 am Ruffner G006 (classroom) Bring](https://reader033.vdocuments.us/reader033/viewer/2022043013/5fac16dcccb20b2a0941d4bd/html5/thumbnails/42.jpg)
Core-Collapse Supernova
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Core-Collapse Supernova
![Page 44: Black Holes ASTR 2110 Sarazin - University of Virginiapeople.virginia.edu/~cls7i/Classes/astr2110/Lecture28...Test #2 Monday, November 13, 11 - 11:50 am Ruffner G006 (classroom) Bring](https://reader033.vdocuments.us/reader033/viewer/2022043013/5fac16dcccb20b2a0941d4bd/html5/thumbnails/44.jpg)
Core-Collapse Supernova Core becomes a neutron star (at least initially)
Very, very dense
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Energetics Virial Theorem: ENS = (1/2) PE
During infall, ΔE = ΔPE ≈ PE
So, Ebinding = (1/2) |PE| ≈ (1/2) G M2/R ≈ G(1.4 M� )2 / (10 km) is released
Etot ≈ 3 x 1053 ergs released in supernova!!
Released from dense, NS region, only neutrinos can escape!
Etot ≈ 3 x 1053 ergs in neutrinos, main energy of core-collapse supernova!!
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Energetics (Cont.) Shock and neutrinos blow off envelope of star at ~ 10,000 km/s
Eexplsion ≈ 1051 ergs in kinetic energy
Explosion heats material to 109 K initially. Cools rapidly, but very bright initially
L (optical, max) ≈ 1010 L� !!
Would fade rapidly, but radioactivity keeps bright for months
Elight ≈ 1049 ergs in light!!
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Energetics (Cont.) Amazing result: most of energy comes out in neutrinos
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Heavy Elements If explosion is efficient, most of materials outside of iron core are ejected.
Also, fusion during explosion makes more heavy elements
Supernova make and/or disperse most of the heavy elements in the Universe!!
But, iron core -> neutron star, core-collapse SN don’t disperse much iron
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Final Outcome? Explosion very complex, but . . .
1. If nearly all outer layers are ejected: neutron star, M ≈ 1.4 M�
2. If enough of outer layers remain or fall back so that M ≳ 3 M� è Black Hole
Uncertain theoretical problem, but if initial mass of star is
1. 8 M� ≲ Mstar ≲ 20 M� -> neutron star
2. 20 M� ≲ Mstar -> black hole
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Core-Collapse Supernova
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• Core of massive star (50 M8) collapses, forms black hole
• Rotating material from star falls onto BH
• Black holes shoots out jets of gas at 0.99995 c
• Jets blast through star, blowing it up
• If jets point in our direction, we see a gamma-ray burst
Alternative:Collapsar Model = Supernova Due to Jets
from a Black Hole
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Collapsar Model = Supernova Due to Jets from a Black Hole