Inhalt. Einführung JR. Urknall TS. Stellaratmosphere 3.7. TS. He-Verbrennung. r,rp, p Prozesse TS

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1 Inhalt Einführung JR Grundlagen der Kernphysik 8.5. JR Urknall TS Urknall-Nukleosynthese JR Stellaratmosphere 3.7. TS H-Verbrennung JR He-Verbrennung Supernova TS r,rp, p Prozesse TS 1

2 Wiederholung pp cycle 2

3 Wiederholung CNO cycle 3

4 Wiederholung He-Burning This is a fast reaction since a resonance is in the Gamov peak This is a slow reaction. If these resonances were higher it would be faster and there would be no carbon left, if they were lower it would be so slow that there would be no oxygen. He Be dn He Be dn Be dt dt 1 1 dn N dt N He N Be HeBe BeHe BeHe Q 2 He J p = 2 - p=(-1) L Sincethislevelis2 is -, it is not possible for an alpha to be absorbed into this level L p 2 - Otherwise there would be no oxygen left. Only in very heavy stars (very high T) can this proceed (via the 3 - state)

5 Asymptotic giant branch star

6 Degenerated Stars The White Dwarf (which once was the Sun) The sun (today) The Sun as a red giant

7 White Dwarfs in the HR Diagramm

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9 Why are White Dwarfs Stable? If there is no energy generation the star should collapse, but they are stabilized. What have we forgotten? (degeneracy pressure) Our equations for the stable stars is based on the assumption that we can treat the matter inside of a star as an ideal gas Strictly speaking, this is not true since electrons (and protons) are fermions with s=1/2. The proper statistical distribution function is the Fermi distribution (rather than the Boltzmann distribution, which is a good approximation only for sufficiently low density/pressure)

10 Reminder: The Fermi Gas consider an electron gas in the unit volume The density of states N(p) with momenta < p is given by: or, in terms of energy: 4 p 3 N(p) 2 3 h 3 dn(p) 8p 3 dp h 2 N.B. non-relativistic p 2mE p dp 2m E de dn m E de h

11 Integration up to the Fermi Energy p 2mE p dp 2m EdE 16 2 N m E 3h 2 (2 h) E F (3 N) 2m E 2 3 F E F is the Fermi energy. A state of energy E F has a probability of 50% to be occupied. The occupation probability is given by the Fermi Dirac distribution: 1 f(e) (E E F ) kt e 1

12 The Fermi-Dirac Distribution

13 Properties of the Fermi-Dirac Distribution For energies << E F, almost all states t are occupied For energies >> E F, almost all states are vacant The diffuseness of the distribution (the behavior near E F F) is given by the ratio of E F /kt or on the Fermi temperature T F =E F /k at T=0, occupation probability changes abruptly from one to zero Even at T=0 0, the electrons are in non-zero energy states their energy cannot be radiated away, since there are no vacant states at lower energy due to Pauli Blocking compression of the Fermi gas (a reduction of the volume) is only possible at the expense of increased momentum space volume Even at very low temp., there is a non-thermal pressure this is the pressure, which stabilizes a white dwarf! A Fermi gas, where most states below E F are occupied, is called a degenerate Fermi gas A degenerate electron gas is a good thermal conductor: mean free path is large (no vacant levels for electrons after scattering!)

14 Condition for Degeneracy If kt < E F, an average electron will typically not find an unoccupied state to make a transition to lower energy At normal conditions (low pressure, ordinary matter), degeneracy can only be reached at low temperatures For high pressure the situation is different: the situation ti becomes nearly independent d of temperaturet condition for degeneracy: 3 kt EF 2 T 2 2 h N mk

15 Some Estimates first, we express the Fermi energy in terms of the density of the electron gas in a white dwarf A, Z are the number of nucleons and protons, respectively, in the white dwarf s nuclei. Assuming full ionization, we have for the number of electrons N per volume unit : N #electrons # nucleons Z nucleon volume A m (2h) Z E F 3 2m A m H H

16 Onset of Degeneracy T (2h) Z3 23 3mk A m H for Z / A K cm g D Compare the situation in the interior of the Sun and in Sirius B (a white dwarf): Sun : ρ 162 g/cm 3, T c T Kcm /g ρ SiriusB : ρ 10 g/cm In the Sun, electron degeneracy is weak Sirius i B: complete degeneracy! T Kcm /g D 5 3, T c 2 3 ρ D 7 K K

17 Degeneracy Pressure Equation of state (similar to ideal gas) : 2 P 3 We introduce the Fermi momentum : N E E p F E4 pdp pF 3 E p F F 5m p F 2mE With the expression for E F: 0 4 p dp F h P N E F N 5 5m 8 Equation of State of a completely 3 degenerate Fermi gas 5 3

18 Remarks The temperature does not appear in the equation of state! The pressure is not of thermal origin but due to the Pauli principle. The pressure is inversely proportional to the mass the pressure of a proton gas of equal density is smaller by a factor of 2000 as compared to the pressure of the degenerate electron gas! Electron degeneracy pressure is responsible for maintaining hydrostatic equilibrium in a white dwarf A white dwarf can be regarded as a giant atom, bound by the gravitational force (rather than the Coulomb force!)

19 White Dwarfs in the Globular Cluster M4 A view of globular cluster M4 (fourth object in the Messier catalog of star clusters and nebulae). The nearest globular cluster to Earth (7,000 light-years away), and containing more than 100,000 stars A Hubble Space Telescope color image of a small portion of the cluster only 0.63 light-years across reveals eight white dwarf stars (inside blue circles) among the cluster's much brighter population p of yellow sun-like stars and cooler red dwarf stars. Hubble reveals a total of 75 white dwarfs in one small area within M4, out of the total of about 40,000 white dwarfs that the cluster is predicted to contain.

20 White Dwarf Matter Features The weight of 1 cm 3 of white dwarf matter is about 1 ton! size ~ 0.01 R SUN luminosity less than 0.01 L SUN mass ~ 0.5 M SUN initially very hot ~ 100,000 K (surface), rapidly cool to ~ 10,000 K (over millions of years) finally, cool off (and dim) to become a BLACK DWARF

21 Size Comparison

22 Radius Mass Relation for White Dwarfs Hydrostatic equilibrium: (central pressure) P M R ρ ρ M 3 R P M R 2 4 Equation of state: (degeneracy pressure) P 5 3 M 3 R 5 3 Radius Mass Relation: Massive white dwarfs R M 1 3 are aesmaller than light ones!

23 Mass Volume - Relation Follows immediately: R M M V 1 3 V const 1 3 This result is surprising at first glance but clear from the origin of the degeneracy pressure: the larger the mass of a white dwarf, the smaller its volume must be in order to generate sufficiently large momenta for the electrons to keep hydrostatic equilibrium. Does this mean that there are infinitely massive white dwarfs with infinitely it small volume? No! We have ignored some important piece of physics!

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25 What happens if the mass gets large? Let s estimate the electron momentum: N n consider a white dwarf with radius R and a total R number of N electrons. The electron number density is: The volume per electron is proportional 1 to 1/n. Δr According to the uncertainty principle, n 1 3 there are position and momentum uncertainties given by: Δpp Δr n 1 3 If n is sufficiently large, the electrons will eventually get relativistic! 2 P 2 1 3h 5 5m 8 3 N EF 3 3 N 5 3

26 Ultrarelativistic Limit In the limit of complete (ultra)relativistic istic degeneracy we have the relativistic momentum energy relation: E = c p Thus, the kinetic energy arising from the Fermi motion is: E 1 3 c n N 1 3 c N R If the average mass (in u) associated with a fermion is, the gravitational energy per fermion is: GNμ E g R M Nμ

27 Equilibrium Condition For a stable equilibrium, we have to have a minimum of the total energy: E tot E Deg E Grav cn R 1 3 GNμ R 2 1 R cn GNμ Constant Note that both terms scale with 1/R! We can now distinguish 2 cases: 1. E tot > 0. This is the case for small N (mass of white dwarf) 2. E tot < 0. This is the case for large N (mass of white dwarf)

28 Stable Solution for Low Mass White Dwarfs If 1 st case, i.e. E tot > 0 then With increasing radius the Fermi momentum of the electrons drops until they are no longer relativistic. When this occurs, the degenerate electron gas has a different functional relation between energy and density. (E~1/R 2 instead of E~1/R) Since the gravitational energy is still E~1/R, at some radius, it will be larger than the energy in the degenerate non- rel. electron gas. (E<0) As R E0 Thus, there must be a stable minimum! i

29 E E E tot 1 3 g R cn GN R 2 2. E tot < 0. This is the case for large N for R -> 0 : E tot -> - there is no stable minimum! Thus, there must be a limiting mass M max =N. A white dwarf can only be stable, if its mass does not exceed M max We can calculate M max from the condition: max E tot 0 cn R 1 3 max GN R max c Mmax Nmaxμ 15M 1.5M 2 Gμ 3 2 μ 2 sun

30 The Chandrasekhar Limit This limiting mass, above which no white dwarf can be stable, is called the Chandrasekhar limit. A more precise calculation for Z/A = 0.5 yields The reason for the existence of the Chandrasekhar limit is the modification of the equation of state due to the onset of relativistic effects at large stellar masses M Ch = 1.44 M non relativistic : ultra relativistic : P ρ 5 3 P ρ 4 3

31 The Radius of White Dwarfs at T=0

32 Decoupling of Mechanical and Thermal Properties Due to the degeneracy, the equation of state does not include the temperature. One implication is that t if helium burning starts t in the core, it is not accompanied by an increase of pressure, that would normally expand (and thus cool) the core. The resulting rapid rise of temperature leads to runaway production of nuclear energy. This continues until the temperature is so high that the gas is no longer degenerate, then it can expand. (Helium core flash) However, if a star is too small, it will never start to burn helium.

33 The Future of Our Sun

34 What happens if the star s mass is too large to form a white dwarf?

35 Supernova SN98B

36 The Structure of a Pre-Collapse Star

37 Presupernova structure 37

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39 Central Evolution 39

40 Early iron core The core is made of heavy nuclei (iron-mass range A ~ 45 65) The mass of the core Mc is determined by the nucleons There is no nuclear energy source which adds to the pressure. Thus, the pressure is mainly due to the degenerate electrons, with a small correction from the electrostatic interaction between electrons and nuclei. As long as M c < M ch = 1.44(2 Y e ) 2 M sun (plus slight corrections for finite temperature), the core can be stabilized by the degeneracy pressure of the electrons. 40

41 Onset of collapse However, there are two processes which make the situation unstable: 1. Silicon burning is continuing in a shell around the iron core. This adds mass to the iron core, thus Mc grows. 2. Electrons can be captured by nuclei. This reduces the pressure and cools the core, as the neutrinos leave. In other words, Ye and hence the Chandrasekhar mass Mch is reduced. The core finally collapses. 41

42 SN1987A 42

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44 Collapse phase 44

45 The Infall Phase When the mass of the iron core exceeds the maximum for which h the electron degeneracy pressure can support the core, the core begins to collapse gravitationally. This accelerates into a catastrophic collapse on a timescale of less than a second because of pressure loss associated with the dissociation of iron nuclei by highenergy gamma rays production of neutrinos that escape the star in the initial phases of the collapse. The collapse proceeds to subnuclear densities, where the stiff nuclear equation of state causes the core to bounce. Pressure waves propagate outward and steepen into a shock wave at about 50 km from the center. In the prompt shock mechanism this shock wave is then responsible for ejecting the outer layers of the star, producing the light and expansion phenomena associated observationally with supernovae.

46 Acceleration of the Collapse

47 Neutrino trapping 47

48 Neutrino-induced reactions 48

49 Neutrino Interactions

50 Elastic neutrino-nucleus scattering 50

51 Neutrino Mean Free Path

52 Importance of neutrino trapping 52

53 Timescale of the Collapse

54 Collapse history 54

55 Core bounce 55

56 The collapse is stopped 56

57 The Formation of a Neutron Star If M > M Ch, the star will continue to collapse and its electrons will be pushed closer and closer to the nuclei. At some point, a nuclear reaction begins to occur in which electrons and protons combine to form neutrons and neutrinos. A sufficiently dense star is unstable against such an interaction and all electrons and protons are converted to neutrons leaving behind a chargeless and nonluminous star: a neutron star. Just as with electrons, neutrons obey the Pauli Exclusion Principle. If the neutrons are nonrelativistic, the previous calculation for the radius of the white dwarf star will work just the same, with the replacement m e by m n. This change reduces the radius R of the neutron star by a factor of 2000 (the ratio of m n to m e ) : to about 10 km.

58 Energy Release during Neutron Star Formation To form a neutron star, energies of the order of erg (corresponding to the binding energy) should be released in the Supernova explosion. The total light and kinetic energy emitted is only about 1% of the total energy Thus, the rest should have been emitted as an "invisible" i ibl " energy, neutrinos or gravitational waves. Gravitational waves (not yet observed) are supposed to carry at most 1% of this total, this means that neutrinos will carry out the remaining ~98% of erg. The neutron star can be viewes as a giant nucleus, bound by gravitational forces (not by the nuclear force!)

59 Formation of the shock 59

60 Prompt shock scenario 60

61 Shock stagnation 61

62 Neutrinos help the prompt p shock to fail 62

63 Neutrino burst 63

64 Shock revival 64

65 Supernova neutrinos 65

66 Neutrino spheres 66

67 Hierarchy of neutrino spectra 67

68 Gain radius 68

69 Shock revival by neutrino absorption 69

70 The delayed shock model 70

71 Sequence of Events during the Core Collapse

72 Time Sequence of a Supernova Explosion Within about 0.1 second, the core collapses. After about 0.5 second, the collapsing envelope interacts with the outward shock. Neutrinos are emitted. Within 2 hours, the envelope of the star is explosively ejected. When the photons reach the surface of the star, it brightens by a factor of 100 million. Over a period of months, the expanding remnant emits X-rays, visible light and radio waves in a decreasing fashion.

73 Two Types of Supernovae Type I Supernovae: happens, when a white dwarf with a near-by companion star collects matter and eventually exceeds the Chandrasekhar limit probably no remnant Type II Supernovae: happens when a star too massive to form a white dwarf exhausts its fuel leaves a neutron star or a black hole as a remnant

74 Supernova Ia 74

75 Mixing in the explosion 75

76 Successful two-dimensional supernova 76

77 Modeling Supernova Explosions Basic principles p that govern such explosions o s seem to be understood However, detailed computer modeling has not given a completely satisfactory solution to the problem. Generally, one finds that with realistic parameters and assumptions the computer models either don't explode at all or do so with too little energy to be consistent with observations Need for more realistic neutrino transport calculations neutrino physics (masses, oscillations) experimental information on dense nuclear matter experimental data from supernovae light curves

78 Neutrino Driven Supernova Explosions

79 Prior to explosion, infalling gases surround the stalled shock wave

80 Heated by neutrinos released from the boiling core, gases re-energize the shock wave

81 Driven by neutrino heating, the shock wave gains enough energy to explode

82 Light curve 82

83 Supernovae are very, very bright! 83

84 How to distinguish Type I and Type II Supernovae

85 Supernova remnants 85

86 Supernova remnants 86