1. Test Review 3

2. 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

3. H-R diagram 90% of stars

4. Specific segments of the main sequence are occupied by stars of a specific mass Majority of stars are here

5. The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems. L ~ M 3.5 only for main-sequence stars!

6. star mass (solar masses)time (years)Spectral type 603 millionO3 3011 millionO7 1032 millionB4 3370 millionA5 1.53 billionF5 110 billionG2 (Sun) 0.11000's billionsM7 Lifetime T ~ M/L ~ 1/M p-1 = 1/M 2.5 ; p ~ 3.5 M = 4M  ; Lifetime = Amount of hydrogen fuel Rate of energy loss T ~ 3x10 8 years

7. Estimating the Age of a Cluster The lower on the MS the turn-off point, the older the cluster. Age of a cluster = lifetime of stars on the turnoff point

8. H-R diagram 90% of stars are on the main sequence and obey the mass- luminosity dependence L ~ M 3.5 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

10. Cutoff at masses > 100 M  and < 0.08 M 

11. 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)

12. Luminosity classes Ia bright supergiant Ib Supergiant II bright giant III giant IV subgiant V main-sequence star

13. Example Luminosity Classes Our Sun: G2 star on the Main Sequence: G2V Polaris: G2 star with Supergiant luminosity: G2Ib

14. Luminosity Surface area of the star = 4  R 2 Luminosity, or total radiated power L =  T 4 4  R 2 J/s Intensity, or radiation flux on the Earth: R d

15. It is convenient to compare with the Sun or any other star:

17. Surface temperature and color indices Can be applied to any black-body emitter!

18. 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

19. Estimating Stellar Masses Recall Kepler’s 3rd Law: P y 2 = a AU 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 M A and M B different from 1 solar mass: M A + M B = a AU 3 ____ Py2Py2 (M A and M B in units of solar masses)

20. Visual Binaries The ideal case: Both stars can be seen directly, and their separation and relative motion can be followed directly.

21. Spectroscopic binaries Stars are seen as a single point Spectra of both stars are distinguishable Sometimes spectrum of only one star is seen

22. STELLAR WOBBLE RECEDING: REDDER APPROACHING: BLUER

23. The Doppler Effect The Doppler effect allows us to measure the source’s radial velocity. vrvr  / 0 = v r /c

24. Determining the orbital period

25. 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. M A d A = M B d B V ~ d

26. Over 100 extrasolar planets discovered

27. ...BUT THE PLANET CANNOT BE SEEN MOTIONS OF THE STAR BETRAY ITS PRESENCE !

28. X EARTH X JUPITER 150 000 000 km 30 km/s 450 km 9 cm/s 780 000 000 km 13 km/s 750 000 km 13 m/s

29. 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!

30. 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

32. Coldest spots in the galaxy: T ~ 1-10 K Composition: Mainly molecular hydrogen 1% dust

33. Jeans instability: Thermal pressure cannot support the gas cloud against its self- gravity. The cloud collapses and fragments.

34. Shocks Triggering Star Formation Globules = sites where stars are being born right now! Trifid Nebula

35. Jeans instability: Thermal pressure cannot support the gas cloud against its self- gravity. The cloud collapses and fragments.

36. 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

37. Protostellar Disks and Jets – Herbig Haro Objects Herbig Haro Object HH30

38. The matter stops falling on the star Nuclear fusion starts in the core Planets can be formed from the remaining disk

39. 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.

40. 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.

41. 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

42. Energy Transport Structure Inner radiative, outer convective zone Inner convective, outer radiative zone CNO cycle dominantPP chain dominant

43. 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.

45. Formation of degenerate core

46. 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” 4 He + 4 He  8 Be +  8 Be + 4 He  12 C + 

47. If M > 0.5 M sun, 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!

48. p. 192

49. 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

50. What is left? A stellar remnant: white dwarf, composed mainly of carbon and oxygen

51. Sirius B is very hot: surface temperature 25,000 K Yet, it is 10,000 times fainter than Sirius A It should be very small: R ~ 2 R earth Its mass M ~ 1 M sun It should be extremely dense! M/V ~ 10 6 g/cm 3

52. All atoms are smashed and the object is supported by pressure of degenerate electrons White dwarf should be extremely dense! M/V ~ 10 6 g/cm 3

53. Strange properties of degenerate matter It strongly resists compression: P ~  5/3 Pressure does not depend on temperature Compare with classical gas: P ~  T

54. Evolution of sun-like stars on H-R diagram

55. Chandrasekhar limit: 1.4 M sun 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.

57. Stars > 8 solar masses Reactions proceed faster and faster, until Fe and Ni are synthesized

59. 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

60. Summary of Post Main-Sequence Evolution of Stars M > 8 M sun M < 4 M sun Evolution of 4 - 8 M sun stars is still uncertain. Fusion stops at formation of C,O core. Mass loss in stellar winds may reduce them all to < 4 M sun stars. Red dwarfs: He burning never ignites M < 0.4 M sun Supernova Fusion proceeds; formation of Fe core.

61. 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

62. Type I and II Supernovae Core 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 56 Ni and 56 Co

63. 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

64. Continuing cycle of stellar evolution

65. 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

66. The Remnant of SN 1987A Ring due to SN ejecta catching up with pre-SN stellar wind; also observable in X-rays.

67. 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.

68. Crab nebula: the remnants of supernova 1054

69. Fate of the collapsed core White dwarf if the remnant is below the Chandrasekhar limit 1.4 M sun after mass loss Neutron star if the core mass is less than ~ 3 solar masses Black hole otherwise

70. Neutron Stars The central core will collapse into a compact object of ~ a few M sun. A supernova explosion of a M > 8 M sun star blows away its outer layers.

71. Formation of Neutron Stars Compact objects more massive than the Chandrasekhar Limit (1.4 M sun ) 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

72. Properties of Neutron Stars Typical size: R ~ 10 km Mass: M ~ 1.4 – 3 M sun Density:  ~ 10 14 g/cm 3  Piece of neutron star matter of the size of a sugar cube has a mass of ~ 100 million tons!!!

73. 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

75. The enigma of pulsars Pulse repetition: from a few to 0.03 seconds Pulse duration: ~ 0.001 s Period 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 It must be a neutron star!!

76. General idea of a pulsar emission Exact mechanism of pulsar radiation is still unknown!

78. Schwarzschild radius: event horizon for a spherically symmetric object RsRs Black hole: an object shrinks below its event horizon K. Schwarzschild

80. 2005 is the World Year of PHYSICS 100 th 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

81. General Relativity See 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 GrossmannDavid Hilbert Tullio Levi-Civita

82. 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 B D

83. Urbain Le Verrier 1811-1877 Predicted the presence and position of Neptune from 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 …

84. 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.

85. Mercury: the closest planet to the Sun Sun Mercury Perihelion = position closest to the sun Aphelion = position furthest away from the sun Perihelion: 46 million km; Aphelion: 70 million km

86. 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!

87. 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:

88. 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 m1m1 m2m2 If ball 1 moves, ball 2 instantaneously feels it. Faster than light propagation??

89. Same problem existed in electromagnetic theory, but was solved by Maxwell

90. Gravity is a strange force. It has a unique property: M m R All bodies in the same point in space experience the same acceleration!

91. Acceleration of Gravity Acceleration of gravity is independent of the mass of the falling object! Iron ball Wood ball

92. 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.

93. "You mighta seen a house fly, maybe even a superfly, but you ain't never seen a donkey fly!" Donkey

94. 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

95. 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

96. 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: 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.

97. Light path is bent in the accelerated frame. It should be also bent in the gravitational field!

98. Light path is bent in the accelerated frame. It should be also bent in the gravitational field! t = 0, V = 0 t =  x/c, y 2 -y 1 = at 2 /2 y 2 –y 1 = a(  x) 2 /2 Acceleration a Parabola: Light is emitted from the left wall Light reaches the right wall xx y1y1 y1y1 y2y2 x y

99. 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!

100. M m R1R1 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. R2R2

101. 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.

102. 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

103. The shortest path between two cities is not a straight line Shortest paths are called geodesics; they are not straight lines!

105. 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.

106. First test of General Relativity: precession of perihelion for Mercury (43 arcsec per century)

107. PlanetObserved excessPredicted precession Mercury43.11+-0.4543.03 Venus8.4+-0.488.6 Earth 5.0+-1.2 3.8

108. Bending of light

109. Two British expeditions in 1919 confirmed Einstein’s prediction. The shift was about 2 seconds of arc, as predicted

110. Gravitational lensing

111. Gallery of lenses (Hubble Space Telescope)

113. tt t0t0 As measured by a distant observer, clocks slow down when approaching a black hole Time dilatation  t >  t 0

114. Frequency = 1Period of oscillations Increase in time intervals means decrease in frequency : Gravitational redshift!

115. Gravitational redshift Photons 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.

116. Tidal forces and contraction of space squeeze and stretch the astronaut. Lateral pressure is 100 atm at a distance of 100 R s from the event horizon

117. 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 M sun – this is likely a BH. If it is between 1.4 and 3 M sun – this is likely a neutron star.

118. a – in AU P – in years M 1 +M 2 – 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

119. Accretion from stellar wind Accretion through Roche lobe outflow Two mechanisms of mass transfer in a binary system

120. 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!!

121. 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 ~ 10 6 K X-ray emission

122. 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

123. 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.

124. 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

125. X-ray pulsar: an accreting neutron star Compare with a radio pulsar Main feature: strong magnetic field ~ 10 12 -10 15 G X-ray emission from hot accreting plasma

126. Measurement of binary system parameters gave M ~ 7 M sun

127. Low-Mass X-ray binary: accretion through Roche-lobe overflow

128. Low-mass X-ray binaries are best candidates because the mass of a red dwarf is much less than a black-hole mass

129. Black Hole X-Ray Binaries Strong X-ray sources Rapidly, erratically variable (with flickering on time scales of less than a second) Sometimes: Quasi-periodic oscillations (QPOs) Sometimes: Radio-emitting jets Accretion disks around black holes