White Star, Black Hole

Frank Hsia-San ShuTaichung First Senior High School

26 February 20051905 = Einstein’s Miracle Year

Surface & Interior of Sun

Pressure at Center of Sun

• Pressure at center of Sun = weight of column of material per unit area above center

• Central pressure 50 times larger: .s m kg 102 -2-116cP

.s m kg 104 ~ pressureMean 211422

Sun

Sun

Sun

Sun

R

M

R

GM

Visualizing Central Pressure of Sun

Solar Plasma

electrons positive ions (nuclei)

• A lot of space between individual particles, unlike the case if unionized atoms were pressed up against each other at mean density of Sun, where most of volume is occupied by the fuzzy electronic shells of the atoms.

• Electrons and nuclei therefore behave as freely moving particles, i.e., as an ideal gas satisfying

P = nkT , where P = pressure, n = number density, k = Boltzmann’s constant, T =

temperature on Kelvin scale.• For the central values, P = and n = we get T = K at the center of the Sun.

16 1 22 10 kg m s , 32 31 10 m , 71.5 10

3-D Random Walk of Photons Out of Sun

Radiative Transfer in the Sun

X-rays free flight optical photons random walk photosphere Slow photon diffusion from interior to surface is

what regulates energy leakage to the rate

watts.104)/()/(

)3/4(~ 26

2

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Sun

SunSun

Source of Energy Output of Sun• Every second, the Sun radiates more energy than lies under the sands of all Arabia.• When will the Sun (and the other stars) run out of energy?• If the Sun shone by burning fossil fuel, and it were entirely made of coal or oil, it

could last only about 10 thousand years. This is far too short – shorter than how long humans have been on Earth (7 million years), much shorter than how long life has existed on Earth (3.5 billion years).

• More powerful energy sources:– Chemical reaction with greatest release of energy per kg: H+H = H H

For Sun, this reaction could last about 25 thousand years, still far too short.– Nuclear reaction: p D + other things p For Sun, this reaction (and others like it) could sustain Sun for 10 billion years. Long

enough.

D.Hpp 2

.H2

2H

Nuclear Energy

• Net reaction: 4H→He (fusion reaction).• Mass deficit: • Energy release: • Implication: If Sun were 100% H, fusion reaction of H into He (“hydrogen

burning”) would provide store of energy sufficient to last 100 billion years at present rate of solar usage:

• Since the Sun is actually only 70% H and since only inner 13% of mass (core) is hot enough to “burn” H during the so-called main-sequence stage, the main-sequence lifetime of the Sun = 9 billion years (see Problem Set 4).

• The Sun, with an age of 4.5 billion years, is half way through its main-sequence life. What it will turn into after another 4.5 billion years is one of the more interesting stories of astronomy.

4 0.03 .H He Hm m m m 2 20.007(4 ).HE mc m c

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Need for High Temperatures

Stability of Solar Thermonuclear Reactor

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Summary of Interior of Sun

From Main Sequence to Subgiant

Horizontal Branch Star

Planetary Nebula to White Dwarf

Some Planetary Nebulae

Motion of Sirius A & B

Theory of White Dwarfs

• R. H. Fowler (1889-1944) proposes white dwarfs, made of element of mean atomic number Z and atomic weight A, are supported against self-gravity by electron degeneracy pressure → mass-radius relationship such that R↓as M↑:

• For M = R is somewhat smaller than the radius of the Earth.• S. Chandrasekhar (1910-1995) includes effect of special relativity, finds

R → 0 as M → finite value (Chandrasekhar’s limit).• Controversy between Eddington and Chandrasekhar. Chandrasekhar

leaves England for United States, settles in University of Chicago. Gave moving tribute in 1982 to commemorate 100th anniversary of Eddington’s birth. Awarded Nobel Prize in Physics in 1984.

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, 1 SunM

Difference Between Chocolate Cakes and White Dwarfs

0.4 kg 0.8 kg

0.4 M 0.8 M

• As M↑, → c, and P increases less quickly when density ↑. Radius → 0 when M → where

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Mass-Radius Relationship of Cold Bodies

Jupiter White Dwarfs

planets

moons

asteroids

310

3010251020101510

410

510

(km)R

210

10

1 (kg)M

ChM

1/ 3R M

1/ 3R M

Jupiter Is Close to BeingLargest Cold Object in Universe

Similarity Between Cooling Ember and White Dwarf

Cooling White

Ember Dwarf

Radiates energy, Radiates energy,

becomes cooler. becomes cooler.

Path to Yet More Compact Objects

• If M were to exceed 1.4 solar masses, would R of a WD really shrink to zero?

• As ρrises, electron degeneracy-energy becomes greater and eventually can make up the mass difference between a proton and a neutron.

• Electrons get “squashed’’ inside protons, and ions of a WD star become converted to neutrons.

• As ρrises even more, neutrons become degenerate and are able to exert pressure at zero T.

• The balance of neutron degeneracy pressure and self-gravity gives rise to a new state of possible equilibrium, a neutron star.

Eta Carina – A Pre-Supernova

Supernova 1987A and its Precursor Star in the LMC

Crab Nebula = SN1054

Crab Pulsar

Pulsars Blink Like Rotating Lighthouses

Neutron Stars

• On average, neutrons stars are smaller than white dwarfs by a factor Actually, by about 103.

• Therefore, mean density is higher by factor of ; i.e., ≈ instead of .• At such densities (nuclear densities), matter is very

incompressible, with the pressure rising rapidly as the density increases.

• Rate of rise of pressure with density = square of speed of sound. Speed of sound cannot exceed speed of light → limit to how much matter, even nuclear matter, can resist gravity →

9 -310 kg m

91018 -310 kg m

3 .nsM M

.1840/ en mm

Cosmic Scale

Atoms Are Mostly Empty Space

Stellar Processing Determines Relative Abundances of the Elements

Summary of Dying Stars

• Red Supergiant:– Dust grains form in cool outer input of atmosphere and expelled by small

ISM radiation pressure. solids

• Planetary Nebula: – Shell of gas and dust illuminated glowing and excited by hot central star, shell of which is exposed core of red ionized supergiant destined to become WD. gas

• Supernova Remnant:– Ejecta of processed matter from neutron high-mass star which impacts with ISM star or and forms supernova remnant that is a black source of cosmic rays. hole

Idea of Black Hole• Consider the escape speed from the surface of a body of mass M and radius R: m M R (See Problem Set 3.)

• For a neutron star of mass and radius , we get

which is 63% the speed of light c!• The above calculation motivates us to ask a question first asked by Pierre Laplace

(1749-1827): For given M, what R would yield an escape speed equal to c? Answer:

• This is the correct answer, but the reasoning is incorrect on two counts.– First, Newtonian mechanics giving the equation for a marginally bound orbit,

does not apply to photons of zero rest mass m and speed c.– Second, Newtonian concepts about gravity do not apply to a situation where the

gravitational field of a body is so strong that even light finds it difficult to escape its clutches.

• Two mistakes combine to give the right looking answer! Correct derivation based on Einstein’s theory of general relativity first given by Karl Schwarzschild (1873-1916).

2 /ev GM R

1.4M M 410 mR

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Event Horizon of Black Hole

Two Views of Gravitation

• Newton:– Gravity is a force which pulls on all things with mass.– Mass acts as the source that generates the force of

gravitation.

• Einstein:– There is no such thing as the force of gravity.

Gravitation arises when spacetime has curvature; indeed gravitation is spacetime curvature.

– Mass-energy and stress (e.g., pressure) act as the sources that generate spacetime curvature.

Basic Postulates ofGeneral Relativity

Bending of Light

Gravitational Bending of Light

Ant Analogy for Bending of Light

Lensing of Background Galaxies by Galaxy Cluster

Observed Lensing of Background Galaxies by Galaxy Cluster

Flight Circles about a Black Hole• = 9 km for a 3 solar-mass BH.• Start with flight circle of circumference = 2π·90 km.• You deduce you’re 90 km radially from BH. Don’t jump to conclusions.• Lower yourself inward radially by 32 km.• Fly around; measure circumference = 2π·60 km. (?)• Lower yourself inward radially by another 33.75 km.• Fly around; measure circumference = 2π·30 km. (??)• Lower yourself inward radially by another 19.8 km.• You compute 32+33.75+19.8 = 85.55. Subtracted from 90, won’t this

bring you inside = 9 km? (!)• Don’t worry; lower yourself by 19.8 km as we requested.• Fly around; measure circumference = 2π·15 km. Whew!• All flight circles are in a single plane. Clearly, presence of a 3 solar-

mass point-mass at center has warped our usual (Euclidean) sense of geometry.

SchR

SchR

Black Holes Are Punctures in Fabric of Spacetime

Behavior Near Event Horizon

Reversal of Space and Time Across Event Horizon of a Black Hole

• Outside event horizon, by exerting enough force on the rope, I can hold your position stationary with respect to center of BH. But there is nothing I can do to stop the forward progression of time for you (or, for that matter, for myself).

• As I lower you toward event horizon, your perception of stars begin to change and blur. Are you getting a sinking feeling?

• When you get close enough to the event horizon, no rope – no matter how strong – can stand the strain. It will snap and break, and you will begin an inexorable fall toward the black hole.

• For you, it takes only a few milliseconds for you to reach and cross the event horizon of the BH. But for me, it seems that you formally take an eternity to reach the event horizon.

• In other words, as you draw near to the event horizon, there is nothing I can do to stop your forward progression through space. But for me, time seems to have stopped moving for you! In some sense, for me on the outside, time and space seem to reverse roles as you approach the event horizon. When you cross it, you will reach a different space and time than the one that we on the outside occupy. In a certain sense, BHs may be portals to other spacetimes and other universes!

Speculation 1-- Wormholes: Shortcuts through Space?

y

x

Speculation 2 -- Wormholes: Machines through Time?

x

t

Detectability of Black Holes

• Since black holes allow nothing to escape from their “surfaces,” not even photons, how can we detect them or verify their existence?

• One technique examines the radiation from matter drawn from a closely orbiting star before this material falls into the black hole in binary X-ray sources. Radiation is associated with an accretion disk surrounding the black hole.

• Famous example -- Cygnus X-1: Inference for compact (unseen) companion in excess of 3 solar masses orbiting a normal star.

• Data accumulated over the past several years show the somewhat surprising result that most stellar-mass black holes found in this manner have a fairly narrow range of masses of around 7 solar masses.

Cygnus X-1 = Interacting Binary Star

Final Stellar States in War between Gravity &Thermodynamics

White Dwarf Truce mediated by quantum behavior of electrons

Neutron Star Truce mediated by quantum behavior of neutrons

Black Hole Final victory for gravity over thermodynamics

Nothing Left Final victory of thermodynamics over gravity (only true equilibrium?)

Speculation 3: Evaporation of BHs?

• Currently popular theoretical view: Proton is a long-lived, but ultimately unstable particle, which will decay into positron plus other particles in some years or so.

• If this speculation is correct, then ultimately all the protons (and neutrons, which will decay into protons if the latter disappear) in WDs and NSs will turn into positrons that will annihilate with the electrons present in these compact objects. The resulting photons (and other massless particles) will then escape at the speed of light and disappear into the universe.

• Even BHs may ultimately evaporate completely away if Stephen Hawking (1943-) is correct. According to Hawking, (nonrotating) BHs of a mass M have a nonzero surface temperature T given by the formula:

3210

.8

3

GM

ckT

True End State of Universe?• Today, BHs preferentially pull in matter and become more massive.

But eventually (and this may take a long, long time), when all the neighboring matter is gone and the night sky has become even darker than it is now, BHs will begin to slowly lose mass and energy back into the universe.

• Since T is inversely proportional to M, this process will accelerate with time as M becomes smaller and smaller, until the BH disappears in a final outburst of light. Need superstring theory to understand this last event.

• The whole story of stellar evolution is to turn normal stars into more and more compact objects, ultimately producing WDs, NSs, or BHs. However, if current theoretical ideas are correct that most forms of matter are ultimately unstable toward decay into lighter particles, with even black holes not immune from the disease, then the end game of the evolution of the universe will see a redispersal of the gravitationally bound “final states” of stellar evolution.

• Is the grand scheme of the cosmos then merely a mechanism to turn matter into energy? This would be an ironic end if current ideas are also correct, that all matter in the universe originally began in the big bang as pure energy.

Thank you, everyone!

Happy New Year!