test review 3. test 3 test covers chapters 9,11-14 and section 5.3 part 1: short questions and...
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Test 3
• Test covers Chapters 9,11-14 and section 5.3• Part 1: Short questions and problems• Part 2: bonus problems, extra 30 points• Show your work everywhere• Don’t forget to prepare formula sheet• Bring your calculator• Textbook and lecture notes are not allowed
Less math, but more concepts
Specific segments of the main sequence are occupiedby stars of a specific mass
Majority of stars are here
The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.
L ~ M3.5 only for main-sequence stars!
5.3
sunsun M
M
L
L
star mass (solar masses) time (years) Spectral type
60 3 million O3
30 11 million O7
10 32 million B4
3 370 million A5
1.5 3 billion F5
1 10 billion G2 (Sun)
0.1 1000's billions M7
Lifetime T ~ M/L ~ 1/Mp-1 = 1/M2.5 ; p ~ 3.5
M = 4M; 32
15.2
M
M
T
T sun
sun
Lifetime = Amount of hydrogen fuel
Rate of energy loss
T ~ 3x108 years
Estimating the Age of a Cluster
The lower on the MS
the turn-off point, the older the cluster.
5.2
1~M
T
Age of a cluster = lifetime of stars on the turnoff point
5.2
M
M
T
T sun
sun
H-R diagram
• 90% of stars are on the main sequence and obey the mass-luminosity dependence L ~ M3.5
• Stars on the main sequence generate energy due to nuclear fusion of hydrogen
• In the end of their lives stars move to the upper right corner of the H-R diagram
Spectral Lines of Giants
=> Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars
Pressure and density in the atmospheres of giants are lower than in main sequence stars.
=> From the line widths, we can estimate the size and luminosity of a star.
Distance estimate (spectroscopic parallax)
Luminosity classes
• Ia bright supergiant
• Ib Supergiant
• II bright giant
• III giant
• IV subgiant
• V main-sequence star
Example Luminosity Classes
• Our Sun: G2 star on the Main Sequence:
G2V
• Polaris: G2 star with Supergiant luminosity:
G2Ib
Luminosity
sT
24
m
Jstar theof areaunit fromFlux
Surface area of the star = 4R2
Luminosity, or total radiated power L = T4 4R2 J/s
Intensity, or radiation flux on the Earth:
)mJ/(s4
22 d
LI
R
d
Binary Stars More than 50 % of all stars in our Milky Way
are not single stars, but belong to binaries:
Pairs or multiple systems of stars which
orbit their common center of mass.
If we can measure and understand their orbital
motion, we can
estimate the stellar masses.
Measuring diameters and masses
A
B
B
A
m
m
r
r
Estimating Stellar Masses
Recall Kepler’s 3rd Law:
Py2 = aAU
3
Valid for the Solar system: star with 1 solar mass in the center.
We find almost the same law for binary stars with masses MA and MB different
from 1 solar mass:
MA + MB = aAU
3 ____ Py
2
(MA and MB in units of solar masses)
Visual Binaries
The ideal case:
Both stars can be seen directly, and
their separation and relative motion can be followed directly.
Spectroscopic binaries
Stars are seen as a single point
• Spectra of both stars are distinguishable
• Sometimes spectrum of only one star is seen
The Doppler Effect The Doppler effect allows us to
measure the source’s radial velocity.
vr
/0 = vr/c
1. Below is a radial velocity curve for a spectroscopic binary. Estimate the mass of each star if the mass of the binary system is 6 solar masses.
Time (days)
V (km/sec)r
-15
-20
-25
-10
-5Star A
Star B
MA dA = MB dB
V ~ d
2
1
A
B
A
B
B
A
V
V
d
d
M
M
6 BA MM
2010
2000
2005
1995
1990
2015
2020
0.002”
MOTIONS OF THE SUN VIEWED FROM A STAR 30 LIGHT YEARS AWAY
0.002’’ IS THE ANGULAR SIZE OF A MAN ON THE MOON OR A STANDARD NEWSPAPER FONT 300 KM AWAY Unobservable!
Eclipsing Binaries
From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane.
Example:
Algol in the constellation of Perseus
Jeans instability:
Thermal pressure cannot support the gas cloud against its self-gravity. The cloud collapses and fragments.
Shocks Triggering Star Formation
Globules = sites where stars are being born right now!
Trifid Nebula
Jeans instability:
Thermal pressure cannot support the gas cloud against its self-gravity. The cloud collapses and fragments.
Heating By Contraction
As a protostar contracts, it heats up:
Free-fall contraction→ Heating
Heating does not stop contraction because the core cools down due to radiation
• The matter stops falling on the star• Nuclear fusion starts in the core• Planets can be formed from the remaining disk
The Source of Stellar Energy
In the sun, this happens primarily through the proton-proton (PP) chain
Recall from our discussion of the sun:
Stars produce energy by nuclear fusion of hydrogen into helium.
The CNO Cycle
In stars slightly more massive than the sun, a more powerful
energy generation mechanism than
the PP chain takes over:
The CNO Cycle.
Net result is the same: four hydrogen nuclei fuse to form one helium nucleus; 27 MeV is released.
Why p-p and CNO cycles? Why so complicated?
Because simultaneous collision of 4 protons is too improbable
Energy Transport Structure
Inner radiative, outer convective
zone
Inner convective, outer radiative
zone
CNO cycle dominant PP chain dominant
Evolution off the Main Sequence: Expansion into a Red Giant
Hydrogen in the core completely converted into He:
H burning continues in a shell around the core.
He Core + H-burning shell produce more energy than needed for pressure support
Expansion and cooling of the outer layers of the
star Red Giant
“Hydrogen burning” (i.e. fusion of H into He) ceases in the core.
Red Giant Evolution
4 H → He
He
H-burning shell keeps dumping He
onto the core.
He-core gets denser and hotter until the
next stage of nuclear burning can begin in
the core:
He fusion through the
“Triple-Alpha Process”
4He + 4He 8Be + 8Be + 4He 12C +
If M > 0.5 Msun, the temperature reaches 100 million K. Nuclear fusion of helium starts; carbon and oxygen are produced
Essentially all C and O in the universe are produced in this way!
The Fate of Our Sun and the End of Earth• Sun will expand to a
Red giant in ~ 5 billion years
• Expands to ~ Earth’s orbit radius or more
• Earth will then be incinerated!
• It will be too hot for life in 200 million years
• Sun may form a planetary nebula (but uncertain)
• Sun’s C,O core will become a white dwarf
Sirius B is very hot: surface temperature 25,000 KYet, it is 10,000 times fainter than Sirius A
It should be very small: R ~ 2 Rearth
Its mass M ~ 1 Msun
It should be extremely dense!
M/V ~ 106 g/cm3
V
M
All atoms are smashed and the object is supported by pressure of degenerate electrons
White dwarf should be extremely dense!
M/V ~ 106 g/cm3V
M
Strange properties of degenerate matter
• It strongly resists compression: P ~ 5/3
• Pressure does not depend on temperature
Compare with classical gas: P ~ T
Chandrasekhar limit: 1.4 Msun
This is because gravitational pressure increases with mass. Electron pressure should also increase, and the only way to do it is to compress the star.
The iron core of a giant star cannot sustain the pressure of gravity. It collapses inward in less than a second.
The shock wave blows away outer layers of a star, creating a SUPERNOVA EXPLOSION!
Precise mechanism – still unknown
Summary of Post Main-Sequence Evolution of Stars
M > 8 Msun
M < 4 Msun
Evolution of 4 - 8 Msun stars is still uncertain.
Fusion stops at formation of C,O core.
Mass loss in stellar winds may reduce them all to < 4 Msun stars.
Red dwarfs: He burning never ignites
M < 0.4 Msun
Supernova
Fusion proceeds; formation of Fe core.
• Evolution of sun-like stars: red giant, planetary nebula, white dwarf
• Evolution of massive stars: red giant or supergiant, supernova
• Three types of compact objects – stellar remnants: white dwarfs, neutron stars, black holes. Limits on their masses. Pulsars as rotating neutron stars
• Compact objects in binary systems. Accreting X-ray binaries
Type I and II SupernovaeCore collapse of a massive star:
Type II Supernova
If an accreting White Dwarf exceeds the Chandrasekhar mass limit, it collapses,
triggering a Type Ia Supernova.
Type I: No hydrogen lines in the spectrum
Type II: Hydrogen lines in the spectrum
Energy release due to radioactive decay of 56Ni and 56Co
Stellar nucleosynthesis
• All elements up to Atomic mass ~ 250 u are synthesized!
• S-processes: “slow” synthesis of elements up to iron
• R-processes (r = rapid): rapid neutron capture during SN explosion; all elements heavier than iron
Supernova Remnants
The Cygnus Loop
The Veil Nebula
The Crab Nebula:
Remnant of a supernova
observed in a.d. 1054
Cassiopeia A
Optical
X-rays
The Remnant of SN 1987A
Ring due to SN ejecta catching up with pre-SN stellar wind; also observable in X-rays.
Synchrotron Emission and Cosmic-Ray Acceleration
The shocks of supernova remnants
accelerate protons and electrons to extremely
high, relativistic energies.
“Cosmic Rays”
In magnetic fields, these relativistic
electrons emit
Synchrotron Radiation.
Fate of the collapsed core
• White dwarf if the remnant is below the Chandrasekhar limit 1.4 Msun after mass loss
• Neutron star if the core mass is less than ~ 3 solar masses
• Black hole otherwise
Neutron Stars
The central core will collapse into a compact object of ~ a few Msun.
A supernova explosion of a M > 8 Msun star blows away its outer layers.
Formation of Neutron StarsCompact objects more massive than the
Chandrasekhar Limit (1.4 Msun) collapse further.
Density and T become so high that electrons and protons combine to form stable neutrons throughout the object:
p + e- n + e
Neutron Star
Properties of Neutron Stars
Typical size: R ~ 10 km
Mass: M ~ 1.4 – 3 Msun
Density: ~ 1014 g/cm3
Piece of neutron star matter of the size of a sugar cube has a mass of ~ 100 million tons!!!
• Neutron stars should rotate extremely fast due to conservation of the angular momentum in the collapse
• They should have huge magnetic field due to conservation of the magnetic flux in the collapse
The enigma of pulsarsPulse repetition: from a few to 0.03 secondsPulse duration: ~ 0.001 sPeriod extremely stable: it increases by less than 1 sec in a million years
What could it be???
Only star rotation can be so stable. However: Centrifugal acceleration < gravitational acceleration
km50~3/1
222
GM
RR
GMR
It must be a neutron star!!
2
2
c
GMRs
Schwarzschild radius: event horizon for a spherically symmetric object
Rs
Black hole: an object shrinks below its event horizon
K. Schwarzschild
2005 is the World Year of PHYSICS
100th anniversary of Albert Einstein’s “miraculous year” of 1905
March 1905: the quantum nature of light
May 1905: Brownian motion shows the existence of atoms and molecules
June 1905: Special Relativity as a theory of space, time, and motion
General RelativitySee also Ch. 5 in Seeds
Developed in 1907-1915 in close collaboration with mathematicians: Grossmann, Hilbert, Levi-Civita
... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.
Marcel Grossmann David Hilbert Tullio Levi-Civita
In 1672 Giovanni Cassini together with Jean Richter (1630-1696) made parallel observations of the Mars parallax in Paris (France) and Cayenne (French Guiana, N. coast of South America)
They were also able to determine the solar parallax as ~ 9 arcseconds and find the distance to the Sun (Astronomical Unit) as 140,000,000 km.Current value is 8.8 arcseconds, or 149,597,892 km.
Newton’s theory has been confirmed by increasingly precise observations …
Parallax angle A
A
BD
B
D
Urbain Le Verrier 1811-1877
Predicted the presence and position of Neptunefrom irregularities in Uranus’s orbit
Neptune was found in 1846 exactly at the predicted position
In the eyes of all impartial men, this discovery [Neptune] will remain one of the most magnificent triumphs of theoretical astronomy …
Arago
I do not know whether M. Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most detested.
A contemporary
Newton’s theory has been confirmed by increasingly precise observations …
The advance of the perihelion of Mercury
One little speck on the brilliant face of Newton’s theory:
In 1855 Le Verrier found that the perihelion of Mercury advanced slightly more than the Newtonian theory predicted.
He and others tried to explain it with a new planet Vulcan, new asteroid belt, etc.
Mercury: the closest planet to the Sun
Sun
MercuryPerihelion = position closest to the sun
Aphelion = position furthest away
from the sun
Perihelion: 46 million km; Aphelion: 70 million km
Mercury's perihelion precession: 5600.73 arcseconds per century
Newtonian perturbations from other planets: 5557.62 arcseconds per century
Remains unexplained: 43 arcseconds/century (Le Verrier 1855)
In reality the orbits deviate from elliptical:
This is only 12,000 km per century, or 29 km per one period!
Newton’s theory is a weak-gravity limit of a more general theory: General Relativity
Even in the weak gravity of the Earth and the Sun, there are measurable deviations from Newtonian mechanics and gravitation law!
• Precession of Mercury’s perihelion
• Bending of light by the Sun’s gravity
General Relativity predicts new effects, completely absent in the Newton’s theory: black holes, event horizon, gravitational waves.
Einstein’s idea:
Problem with Action at a Distance
Direct, instantaneous connection between cause and effect!
By the beginning of the XX century, it became clear that Newtonian gravity has other problems
m1m2
0221
21 rr
mGmFF
1F
2F
If ball 1 moves, ball 2 instantaneously feels it.
Faster than light propagation??
Gravity is a strange force. It has a unique property:
M
m
R
2R
mMGF
2R
MG
m
Fa
All bodies in the same point in space experience the same acceleration!
Acceleration of Gravity
Acceleration of gravity is independent of the mass of the falling object!
Iron ball
Wood ball
This means that in the freely-falling elevator cabin you don’t feel any effects of gravity! You and all objects around you experience the same acceleration.
In outer space you can imitate the effect of gravity by acceleration.
In 1907, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life , namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence:-
... we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.
Equivalence Principle
Immediate consequences of the Equivalence Principle:
Time flow and frequency of light are changed in the gravitational field
Bending of light in the gravitational field
Frequency of light is shifted in the accelerated frame.It should be also shifted in the gravitational field!
H
t = 0, V = 0
H
t = H/c, V = aH/c
Acceleration a
Doppler effect:
2c
aH
c
V
Light is emitted from the nose
Light reaches floor
First observed on the Earth by Pound and Rebka 1960: relative frequency shift of 10-15 over the height H = 22 m.
Light path is bent in the accelerated frame.It should be also bent in the gravitational field!
t = 0, V = 0t = x/c, y2-y1 = at2/2
y2 –y1= a(x)2/2
Acceleration a
Parabola:
2/21 axyy
Light is emitted from the left wall
Light reachesthe right wall
x
y1
y1
y2
x
y
If gravity can be eliminated by motion, no special force of gravity is needed!
How to explain that in the absence of any force the trajectories are not straight lines?
Because space and time are curved!
M
m
R1
21
1 R
MGa
All bodies experience the same acceleration, but only in a small region of space. In another region this acceleration is different. Time flows with a different rate, and paths are bent differently in these two regions.
R2
22
2 R
MGa
About 1912 Einstein realized that the geometry of our world should be non-Euclidean.
He consulted his friend Grossmann who was able to tell Einstein of the important developments of Riemann, Ricci and Levi-Civita.
G.F.B. Riemann(1826-1866)
When Planck visited Einstein in 1913 and Einstein told him the present state of his theories Planck said:
As an older friend I must advise you against it for in the first place you
will not succeed, and even if you succeed no one will believe you.
Space-time gets curved by masses. Objects traveling in curved space-time have their paths deflected, as if a force has acted on them.
Main idea:
“Curvature” of time means that the time flows with a different rate in different points in space
"Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move."
John Wheeler
The shortest path between two cities is not a straight line
Shortest paths are called geodesics; they are not straight lines!
Several versions of Einstein’s GR in 1913-1914 were wrong.
Only in November 1915, after correspondence with Levi-Civita and Hilbert, Einstein published a paper with correct equations.
Hilbert also published correct equations, in fact 5 days earlier than Einstein.
On the 18th November Einstein made a discovery about which he wrote For a few days I was beside myself with joyous excitement . He explained the advance of the perihelion of Mercury with his theory.
Planet Observed excess Predictedprecession
Mercury 43.11+-0.45 43.03
Venus 8.4+-0.48 8.6
Earth 5.0+-1.2 3.8
Two British expeditions in 1919 confirmed Einstein’s prediction.
The shift was about 2 seconds of arc, as predicted
t
t0
As measured by a distant observer, clocks slow down when approaching a black hole
Time dilatationt > t0
Frequency = 1
Period of oscillations
Increase in time intervals means decrease in frequency :Gravitational redshift!
R
Rs 10
Gravitational redshiftPhotons always travel at the speed of light, but they lose energy when travelling out of a gravitational field and appear to be redder to an external observer. The stronger the gravitational field, the more energy the photons lose because of this gravitational redshift. The extreme case is a black hole where photons from within a certain radius lose all their energy.
Gravitational redshift is absent in the Newtonian mechanics. It is a general relativity effect.
R
Rs 10
Tidal forces and contraction of space squeeze and stretch the astronaut. Lateral pressure is 100 atm at a distance of 100 Rs from the event horizon
How to observe a stellar remnant if it does not emit radiation?
• Isolated black hole has almost no chance to be seen• Gravitational lensing is possible but very improbable• Isolated neutron star can be detected as a pulsar, or if it is
very close and hot• Isolated white dwarf can be seen if it is close enough and hot• Good news: most stars are in binary systems
– We can detect radiation from matter accreting onto a compact object. Remember, however, this is only an indirect indicator of a black hole
– We can determine the mass of an unseen companion. If it is much larger than 3 Msun – this is likely a BH. If it is between 1.4 and 3 Msun – this is likely a neutron star.
;2
3
21 P
aMM
a – in AUP – in yearsM1+M2 – in solar masses
Binary systems
If we can calculate the total mass and measure the mass of a normal star independently, we can find the mass of an unseen companion
Accretion from stellar windAccretion through Roche lobe outflow
Two mechanisms of mass transfer in a binary system
Initial ring of gas spreads into the disk due to diffusion.
To be able to accrete on the star, matter should lose angular momentum as a result of viscous friction
Friction leads to heating of the disk and intense radiation!!
White Dwarfs in Binary Systems
Binary consisting of WD + MS or Red Giant star => WD accretes matter from the companion
Angular momentum conservation => accreted matter forms a disk, called accretion disk.
Matter in the accretion disk heats up to ~ 1 million K => X-ray emission => “X-ray binary”.
T ~ 106 K
X-ray emission
Nova Explosions
Nova Cygni 1975
Hydrogen accreted through the accretion
disk accumulates on the surface of the WD
Very hot, dense layer of non-fusing hydrogen
on the WD surface
Explosive onset of H fusion
Nova explosion
Compact Objects with Disks and Jets
Black holes and neutron stars can be part of a binary system.
=> Strong X-ray source!
Matter gets pulled off from the companion star, forming an accretion
disk.
Infalling matter heats up to billions K. Accretion is a very efficient process of
energy release.
Zoo of accreting binaries:
• Novae
• X-ray pulsars
• Millisecond pulsars
• High-mass X-ray binaries: Cygnus X-1
• Low-mass X-ray binaries
• X-ray Novae
X-ray pulsar: an accreting neutron star
Compare with a radio pulsar
Main feature: strong magnetic field ~ 1012-1015 G
X-ray emission from hot accreting plasma
Low-mass X-ray binaries are best candidates because the mass of a red dwarf is much less than a black-hole mass
212
3
21 ; MMP
aMM