4.2 Stellar Evolution After the Giant Branch

Home , Asymptotic giant branch, Hayashi limit

I*M*P*R*S on ASTROPHYSICS at LMU Munich

Astrophysics Introductory Course

Lecture given by:

Ralf Bender and Roberto Saglia

in collaboration with:

Chris Botzler, Andre Crusius-Wätzel, Niv Drory, Georg Feulner, Armin Gabasch, Ulrich Hopp, Claudia Maraston, Michael Matthias, Jan Snigula, Daniel Thomas

Powerpoint version with the help of Hanna Kotarba Fall 2007

IMPRS Astrophysics Introductory Course Fall 2007 1 Chapter 4

Stellar Evolution

IMPRS Astrophysics Introductory Course Fall 2007 2 4.1 Stellar evolution from the main sequence to the giant branch

Immediately after the contraction phase during stellar formation (before nuclear burning sets in) stars have a homogeneous chemical composition. The location of the stars in the HRD where they for the first time start with nuclear fusion in hydrostatic equilibrium is called

Zero Age Main Sequence (ZAMS)

For the period of central hydrogen burning the stars remain on the main sequence near the ZAMS. As the chemical composition changes slowly, the main sequence is not a line but a strip. Massive metal rich stars (M > 20MΘ) in addition loose a significant fraction of their mass due to stellar winds, which moves them a large distance away from the ZAMS even during the hydrogen burning phase. (Lower mass stars also develop winds but only in the later phases of their evolution.)

How does a star move ’inside’ the main sequence? As shown above we have:

IMPRS Astrophysics Introductory Course Fall 2007 3 Evolutionary paths in the HRD up to the point where He burning sets in (from Iben 1967, ARAA 5). The shade of the line segments indicates the time spent in the corresponding phases.

MS (1-3) life-times: 1.0·MΘ: 9.0E9 yrs

2.2·MΘ: 5.0E8 yrs

15·MΘ: 1E7 yrs

GB (5-6) life-times:

1.0·MΘ: 1.0E9 yrs

2.2·MΘ: 3.8E7 yrs

15·MΘ: 1.5E6 yrs (6-10)

IMPRS Astrophysics Introductory Course Fall 2007 4 Hydrogen fusion leads to:

Therefore, following our simple model, the luminosity and the temperatures during hydrogen burning should rise for both pp–chain and CNO–cycle. In reality temperature rises for the pp-chain while it decreases for the CNO-chain (where the core is convective and not radiative as assumed in the model).

Once almost all of the core hydrogen is used up, hydrogen shell burning in a thick shell around the (He-) core starts. The star moves to cooler temperatures to the sub–giant branch with approximately constant luminosity. For M > 1.4MΘ the core is convective and XH so small that contraction has to set in before hydrogen shell burning can start.

The hydrogen burning shell moves outward with time, whereas the helium core slowly contracts and increases its binding energy. The released gravitational energy heats the hydrogen burning shell from below, which increases the fusion (ε ~ T5..17) and forces the star out of equilibrium:

The situation is analogous to the scale height of a planetary atmosphere H = kT/µmpg which is dependent on the temperature. The star expands and its temperature decreases as the energy transport is too inefficient to keep Teff constant.

IMPRS Astrophysics Introductory Course Fall 2007 5 IMPRS Astrophysics Introductory Course Fall 2007 6 IMPRS Astrophysics Introductory Course Fall 2007 7 At the Hayashi–limit the star gets fully convective and the smallest possible temperature is reached. On the way from the main sequence to the Hayashi–limit the luminosity of the 4 2 −1/2 star changes only slightly: L ~ const. ~ Teff · R → Teff ~ R , i.e. R ↑ → Teff ↓. The transition from the main sequence to the Hayashi–limit is determined by the Kelvin– Helmholtz time scale (≈ 107 years — short!). The most massive stars now remain at the Hayashi–limit with constant luminosity. With the onset of helium burning they increase the temperature and move again to the left in the HRD.

Less massive stars keep increasing their radius at approximate constant temperature thus increasing their luminosity ~ R2, they move upwards along the Hayashi line (→ giant branch stars!). The sun for example will expand up to the radius of the earth: Teff ≈ 3500 K, L ≈ 300 LΘ.

IMPRS Astrophysics Introductory Course Fall 2007 8 Evolutionary paths in the HRD up to the point where He burning sets in (from Iben 1967, ARAA 5). The shade of the line segments indicates the time spent in the corresponding phases.

MS (1-3) life-times: 1.0·MΘ: 9.0E9 yrs

2.2·MΘ: 5.0E8 yrs

15·MΘ: 1E7 yrs

GB (5-6) life-times:

1.0·MΘ: 1.0E9 yrs

2.2·MΘ: 3.8E7 yrs

15·MΘ: 1.5E6 yrs (6-10)

IMPRS Astrophysics Introductory Course Fall 2007 9 4.2 Stellar evolution after the giant branch

The post giant branch evolution is dependent on the mass:

M ≤ 0.5MΘ The pressure of the degenerated electron gas stops the contraction of the helium core 8 before Tc ≈ 10 K is reached. → no helium burning. H shell burning slowly stops, the star cools out.

0.5MΘ ≤ M ≤ 2.5MΘ The H shell burning puts more and more He onto the core, which keeps contracting. The core is supported by the pressure of the degenerate electron gas, while the pressure from the He nuclei is negligible. When the temperature has reached ~ 108 K, He–burning 30 suddenly starts (because of ε3α ~ Tc ). Because the pressure from the He nuclei is still negligible relative to the pressure of the degenerate electrons, the nuclear burning can 30 not be moderated via expansion. Again, because of ε3α ~ Tc , the energy production increases dramatically and a large fraction of the He is burned into C and O.

→ thermonuclear runaway!

Only once the temperature of the nuclei is high enough to contribute significantly to the pressure, the core expands and the reaction is slowed down. The star expands on a hydrostatic time scale and reaches its highest luminosity at the tip of the giant branch:

IMPRS Astrophysics Introductory Course Fall 2007 10 Helium flash Note: As helium burning sets in at the same core mass, all mass poor stars have the same luminosity at the tip of the giant branch.

IMPRS Astrophysics Introductory Course Fall 2007 11 After the helium flash, the star is on the horizontal branch. The position on the horizontal branch depends on mass and metallicity. On the horizontal branch we have He → C burning in the core and H → He burning in a shell. In addition we have the following reactions:

12C + 4He → 16O + γ + 7.161 MeV 16O + 4He → 20Ne + γ + 4.730 MeV

The result is a mixture of C and O (≈ 50 : 50) in the stars center. C/O sinks down to the core and replaces He which is driven out of the core. If the He is driven out of the core, helium burning shifts to shell burning. → same procedure as with hydrogen burning. The star expands again to the Hayashi line and then moves to higher luminosities along the asymptotic giant branch (AGB).

IMPRS Astrophysics Introductory Course Fall 2007 12 Evolution of low mass stars with 0.7 and 0.8 solar masses. Times up to the giant branch are in billion years. From Iben 1971, in Boehm-Vitense: Stellar Astrophsics.

IMPRS Astrophysics Introductory Course Fall 2007 13 Massive stars with M > 2.5MΘ Mostly the same evolution as for less massive stars. But, before ignition of helium burning the electron gas is not degenerated, thus we have a continous transition to helium burning. The star performs loops in the HRD at nearly constant luminosity.

Evolution of a 5 solar mass star (Kippenhahn/Weigert).

IMPRS Astrophysics Introductory Course Fall 2007 14 IMPRS Astrophysics Introductory Course Fall 2007 15 An important test for the theory of stellar structure and evolution is the analysis of HRD’s and color–luminosity–diagrams of open star clusters and globular clusters. These represent groups of stars of similar age and chemical composition but different mass and, therefore, the number of stars in the different parts of the HRD can directly be compared with the predictions of the theory. (diagram from Maeder et al., stars from galactic clusters)

IMPRS Astrophysics Introductory Course Fall 2007 16 MS = main sequence TO = turn-off RGB = red giant branch HB = horizontal branch AGB = asymptotic giant b. P-AGB = post-AGB BS = blue stragglers

IMPRS Astrophysics Introductory Course Fall 2007 17 IMPRS Astrophysics Introductory Course Fall 2007 18 4.3 Special effects 4.3.1 The Eddington luminosity

The Eddington luminosity is the maximum luminosity a object of given mass can emit. At higher luminosities, radiation pressure wins over gravitational acceleration. The Eddington luminosity is an upper limit to the luminosity of stars (but also other objects, like accretion disks or active galactic nuclei, see below). Close to the Eddington luminosity, stars will start loosing their outer layers.

For simplicity we work in spherical approximation. We assume we have an object of mass Mc which emits a luminosity L through a spherical surface with Radius R. The flux then is:

resulting in the radiation pressure:

(explanation: a single photon has a momentum of hν/c, summing over all photons and deviding by the area gives the pressure exerted by the photons if absorbed). If the gas is strongly ionized, the interaction between matter and photons can be approximated by the

IMPRS Astrophysics Introductory Course Fall 2007 19 Thomson cross-section of e− and p:

Therefore, the radiation pressure only works on the e− (but the protons get also accelerated because of electromagnetic coupling). A star will start disintegrating, if the accelerating force of the radiation is bigger than the force of gravitation:

This defines the Eddington-Luminosity:

or:

IMPRS Astrophysics Introductory Course Fall 2007 20 4.3.2 Special Effects I: Winds and Thermal Pulses

During the post main sequence phase stars loose a major part of their mass via winds and thermal pulses:

main sequence 1 MΘ → tip of AGB 0.6 MΘ

main sequence 8 MΘ → tip of AGB 0.8 MΘ

The expelled gas is given back to the interstellar medium. At the end of the AGB-phase stars with inital masses less than 8 MΘ lost between 40% and 90% of their mass.

The winds of massive stars are driven by radiation pressure. The higher the metallicity, the more line transitions exist and can absorb photons, the higher is the mass loss in winds. The wind mass loss rates of O,B stars and Wolf-Rayet stars (evolved massive stars which −5 already have suffered significant mass-loss) can be as high as 5 × 10 MΘ/yr. The terminal wind velocities can be up to 4000 km/s. M-type supergiants have mass loss rates up to −6 10 MΘ/yr but slow wind velocities up to 20 km/s.

IMPRS Astrophysics Introductory Course Fall 2007 21 IMPRS Astrophysics Introductory Course Fall 2007 22 IMPRS Astrophysics Introductory Course Fall 2007 23 Spectral signatures of stellar winds are P-Cygni lines (after the prototypical star P Cygni), a combination of absorption and emission line. The extended wind envelope of the star which has τ < 1 emits an emission line. This line is Doppler-broadened due to the wind velocity (regions B and C). From the wind in front of the star we get an absorption line (A). The terminal velocity of the wind can be determined from the blue edge of the absorption line. The mass in the wind is derived from solving the radiative transfer problem. (figure by J. Puls)

IMPRS Astrophysics Introductory Course Fall 2007 24 The driving force behind thermal pulses is different from those of stellar winds. On the AGB the C/O core contracts which increases the efficiency of the He burning in the mass shell above the C/O core. This leads to an expansion of the outer layers which causes the H burning to stop. In the following the He burning moves outward to the region of the hydrogen source and slowly runs out of fuel. The H-enevlope can contract again and H- burning is ignited again. The H fusion produces new fuel for the He burning which after a while sets in again. This process is repeated several times. At each ignition of He burning the envelope expands significantly. During the last phases the star starts to oscillate (a mechanism which is not yet well understood) and pushes off its envelope.

time scales 100. . . 1000 years between the pulses 1. . . 50 years for the puls ≈ 1 year for the oscillation period (Mira stars)

Winds and thermal pulses are important for the transport of the lighter products of nuclear burning to the surface of the stars. From there these elements are subsequently expelled and mixed with the interstellar medium.

In the RGB (red giant branch) phase He and N (and unprocessed H) is lost in winds. In the AGB (asymptotic giant branch) phase mainly C, but also O, He and a little N is lost in thermal pulses (C-stars, Mira-stars, etc).

IMPRS Astrophysics Introductory Course Fall 2007 25 Moreover, thermal pulses are of fundamental importance for the formation of the heaviest elements through n–capture and β–decay. During the thermal pulses convection currents transport H rich material to the core and 12C rich material upwards. This happens several times. During this phase the following nuclear processes provide free neutrons:

And

Furthermore:

The released free neutrons are captured by the most massive nuclei. Between subsequent neutron captures β decay can occur and the next chemical element is built. In this way, starting with iron, a large fraction of all elements up to lead is built up by neutron capture and β–decay.

This process is called s–process because the neutron capture is slow compared to the β–decay. In contrast, in the r–process the neutrons are captured faster than β–decay. This is possible only in explosive nucleosynthesis as it occurs in Supernovae.

IMPRS Astrophysics Introductory Course Fall 2007 26 4.3.3 Special Effects II: stability of stars and the period-luminosity relation of cepheids

In the instability strip of the HRD, stars exhibit radial pulsations. This leads us to the general question of the stability of stars.

If a star is pulsating (or, generally, suffers from instability), the equation of hydrostatic equilibrium has to be extended to account for the inertia of the moving mass:

Multiplication with dV leads to

We assume we know a stationary (but not neccessarily stable) solution to the equations of stellar structure and investigate the effects of a small perturbation:

We have implicitly assumed that the relative change in radius Δ(t) is constant with radius.

IMPRS Astrophysics Introductory Course Fall 2007 27 IMPRS Astrophysics Introductory Course Fall 2007 28 The requirement that the mass within the moving shells stays constant leads to:

and with ∂r = ∂r0·(1+Δ), we have

If we further assume that there is no heat exchange occuring between shells (the process is adiabatic):

For small Δ « 1 we have:

and thus

For ∂P0/∂r0 we know from the stationary solution that:

IMPRS Astrophysics Introductory Course Fall 2007 29 Therefore we obtain for the left-hand side of the hydrostatic equation:

2 neglecting quadratic terms in Δ. For GMr/r ·ρ in the hydrostatic equation we get:

For −ρ·∂2r/∂t2 in the hydrostatic equation we find:

Collecting all terms we finally get:

IMPRS Astrophysics Introductory Course Fall 2007 30 Again, neglecting second order terms in Δ:

iωt which can be solved using the usual ansatz Δ(t) = Δ0e . The frequency of the oscillation ω is: and the period is therefore

2 The instability strip is located at roughly constant Teff, such that L ~ R . The luminosity L varies by a factor of ≈ 1000 along the instability strip, therefore the radii vary by a factor of ≈ 30. In contrast, the mass of stars in the instability strip varies only by a factor of ≈ 10 (L ~ M3). Therefore we have:

IMPRS Astrophysics Introductory Course Fall 2007 31 or in visual magnitudes (ΜV = −2.5 log LV + const)

Cepheids are observed to have

This makes Cepheids one of the most important distance indicators in the nearby universe.

Although our simple assumptions yield a suprisingly good description of the pulsations of stars, we can’t expect the assumption of adiabatic pulsations to hold in reality. During compression, some fraction of the heat will be lost due to radiation, making the pressure smaller than needed for keeping the pulsations going. Therefore, the oscillation will be damped, and this is the reason why most stars do not show pulsations.

We need a mechanism capable of injecting energy in the right moment. Since the pulsations do not penetrate deep enough into the star, nuclear reactions cannot provide such a mechanism in general.

A suitable mechanism called κ-mechanism exists only in the instability strip. It is based on the fact that the mean opacity (Rosseland opacity) κ rises until T ~ 104..5 K and decreases beyond (in the figure κ corresponds to ‘our’ κ devided by ρ). This is due to the ionization/recombination of Helium.

IMPRS Astrophysics Introductory Course Fall 2007 32 IMPRS Astrophysics Introductory Course Fall 2007 33 During compression ρ and T increase implying that the opacity increases significantly. Even less energy can be radiated away than in the equilibrium situation. The retained energy causes the pressure to increase above the level needed to push the star back to equilibrium. The star expands well beyond the equilibrium radius. Now, this implies that ρ, T, and in turn κ decrease again. Radiation can escape easily, causing over-cooling and, con- sequently, over-contraction. The opacity increases again very much and it starts from the beginning...

For stars in the instability strip of the HRD, each small perturbation will immediately trigger pulsations. That’s why all stars in the instability strip pulsate.

RR-Lyrae stars (in old, metal poor populations) are important for the galactic distance scale (MV ≈ 0); Cepheids in young populations (MV > −6) can be used to ~ 20 Mpc (HST) as distance indicators.

The period-luminosity relation was dicovered by Henrietta Leavitt in 1917 and used by Hubble in 1925 to show that Andromeda-Nebula was a galaxy of its own at a distance of several hundred kpc from the Milky Way.

IMPRS Astrophysics Introductory Course Fall 2007 34 HST detection of a -Cepheii star in M 100 in the Virgo cluster (about 17 Mpc distance). by the Hubble-key project: Inlet shows the variable at three different epochs

IMPRS Astrophysics Introductory Course Fall 2007 35 Lightcurve of a δ-Cepheii star in M 31 as ob- Served in the Wendelstein- Calar-Alto Pixel- lensing project (WECAPP).

IMPRS Astrophysics Introductory Course Fall 2007 36 As we have seen, the frequency of the oscillations is

Stars pulsate only if ω > 0, or if 3γ > 4. Instability therefore occurs if

An ideal gas and a non-relativistic Fermi gas have γ = 5/3. Non-adiabatic perturbations usually have γ > 4/3. But a relativistic Fermi gas has γ = 4/3, and radiation pressure dominated stars show γ < 4/3

Therefore, as stars above M > 60MΘ gain significant support from radiation pressure, they become more and more unstable. This is also the reason why extremely massive stars do not exist.

IMPRS Astrophysics Introductory Course Fall 2007 37 4.4 Final evolutionary stages of stars with MZAMS ≤ 8MΘ

As we have seen, the final stages in stellar evolution are almost always accompanied by heavy mass loss via winds and thermal pulses. In addition, explosions are common for massive stars as well. In this way most of the initial mass gets lost. We have:

MZAMS ≤ 8MΘ → white dwarf: Mfinal ≤ 1.4MΘ MZAMS ≥ 8MΘ → neutron star: Mfinal ≤ 3MΘ, or black hole: Mfinal ≥ 3MΘ where ZAMS means zero age main sequence. Depending on the initial mass, the stars reach different final stages.

Stars with MZAMS ≤ 0.5MΘ do not reach the stage of helium burning. After completing hydrogen burning, the star contracts and heats up. The contraction is stopped if the electron gas is completely degenerate; then the star cools out slowly. This could not be observed yet as hydrogen burning lasts for ≥ 40 Gyr for these stars.

Only stars with M > 0.7MΘ have a shorter lifetime than the Hubble time (≈ 15Gyr). Therefore: if white dwarfs are found with M < 0.7MΘ, these originate from stars with MZAMS > 0.7MΘ.

IMPRS Astrophysics Introductory Course Fall 2007 38 If 0.5MΘ < M < 2.5MΘ, the star evolves through giant branch, helium flash, horizontal branch (metal poor) or ‘red clump’ (metal rich) to asymptotic giant branch. Stars which are located in the instability strip during their horizontal branch phase become RR Lyrae stars. Stars with M > 2.5MΘ move almost horizontally in the HRD during their evolution and can cross the Cepheids region.

At the end of the AGB evolution we have a very extended envelope around a compact, contracting C/O core. He-, H-burning instabilities and released gravitational energy cause the loss of the envelope with (vesc ≈ 20 km/s). The core contracts to a white dwarf with 5 Teff ≈ 10 K at nearly constant luminosity.

Mass [MΘ] time τ [years] The evolution time from the tip of the 1 100 AGB to the cooling sequence of white dwarfs is strongly mass depended: 0.8 1000 0.7 10000 0.6 30000 0.55 50000

IMPRS Astrophysics Introductory Course Fall 2007 39 Evolution for low mass stars. Once the white dwarf stage is reached, the stars move downwards along their cooling sequence (labeled WD).

IMPRS Astrophysics Introductory Course Fall 2007 40 IMPRS Astrophysics Introductory Course Fall 2007 41 4.4.1 Planetary Nebulae

If the central star reaches Teff ≥ 25000 K it ionises the envelope expelled in the giant phase. According to Wien’s approximation

already at these temperatures we have enough photons with Eν > 13.6 eV to ionise the hydrogen (in fact kT = 13.6 eV at T ≈ 160000 K). Similarly, O and N can be ionised. The expelled envelope radiates in H-, O-, . . . emission lines as a Planetary Nebula.

Emission of the nebulae is dominated by the forbidden lines of double ionised oxygen OIII at λ = 4959 Å and λ = 5007 Å. These lines were originally attributed to a new element ’nebulium’ and later in 1923 identified as O++. The [OIII] lines emerge on the basis of a + resonance with Lyα of HeII (He ), see Osterbrook: Astrophysics of Gaseous Nebulae.

The life time of the transition is around 10−2 sec because it is a quadrupole and not a dipole transition. Emission of a photon will only occur if the transition is not triggered by a collision with an electron or an other ion.

From the existence of forbidden lines one can determine the density of the emitting gas.

IMPRS Astrophysics Introductory Course Fall 2007 42 Roughly speaking:

For the typical conditions in Planetary Nebulae (T ≈ 10000 K), we determine from the existence of [OIII] λ 5009 Å line:

The expansion velocity can be determined from the Doppler–shifts of the lines:

The radii of Planetary Nebulae reach

From this we get the expansion/life time of

A large fraction of the energy emitted by the central star is re-emitted in the very narrow [OIII] lines; therefore they are relatively easy to detect in narrow-band filters even in external galaxies. They can be used for distance estimates and dynamical investigations of galaxies.

IMPRS Astrophysics Introductory Course Fall 2007 43 4.4.2 White Dwarfs

For white dwarfs we estimate the central pressure and density (see stellar structure) via:

The central pressure is provided by the degenerate electron gas:

Therefore,

and the Mass-radius relation for WDs follows as:

as observed (see chapter 2).

The temperature in the center is T ≈ 108 K and the degenerate electron gas shows high thermal conductivity. A residual atmosphere of H and He gas radiates light, and the WD cools slowly.

IMPRS Astrophysics Introductory Course Fall 2007 44 White dwarfs therefore form a cooling sequence at constant radius and mass. Their cooling time is ~ 10 Gyr (Age of Milky Way from coolest WDs → 10 Gyr for galactic disk). The HRD of white dwarfs looks like this (Scheffler/Elsässer):

IMPRS Astrophysics Introductory Course Fall 2007 45 Relativistic treatment

2/3 The electrons’ Fermi-energy is: εf ~ ne , and therefore, if the density is high enough we 2 2/3 will have εf ≥ mec . In such a case we have to use the relativistic form for P ~ ρ :

We can write the former as:

IMPRS Astrophysics Introductory Course Fall 2007 46 Where Mcrit is the Chandrasekhar mass limit. For M > Mcrit there exists no stable solution and the white dwarf collapses. The result is a Supernova Ia explosion.

4.4.3 SN Ia

Stars with M ≤ 8MΘ end as white dwarfs. Their mass generally falls well below the Chan- drasekhar mass limit. A typical mass is about 0.6MΘ. Nevertheless, these white dwarfs can be brought above the Chandrasekhar mass limit via accretion of mass from a close companion star (for the mechanism see below: cataclysmic variables, accretion disks). Once the Chandrasekhar mass limit is exceeded, the white becomes unstable and collapses. This triggers a run-away fusion reaction where roughly 50% of the white dwarf’s mass is transformed into Fe. The energy released in this process causes the white dwarf to explode as a SN Ia. Each SN Ia produces 0.3 − 1.3MΘ of Fe. This is the most important source of iron in the galaxy!

The light curve of SN Ia is produced by the radioactive decay of Ni via Co to Fe. The energy released in the optical is smaller than in a SN II. We have:

IMPRS Astrophysics Introductory Course Fall 2007 47 An alternative model for SNIa assumes the merger of two white dwarfs.

The typical SNIa light curve. Its shape is determined by radioactive decay.

IMPRS Astrophysics Introductory Course Fall 2007 48 4.5 Final evolutionary stages of stars with MZAMS ≥ 8MΘ

These stars show heavy mass loss on the main sequence due to winds. Stars with M > 40MΘ do not reach the Hayashi line but stay in the area of hot temperatures (Humphries–Davidson limit). For many stars the radiation pressure is larger than the gravitational acceleration, they become instable and start oscillating (LBVs, Luminous Blue Variables). Even small rotation velocities lead to a mass loss along the equator due to the additional centrifugal force. The mass loss causes the star to contract, the gravitation at the surface increases and the star again develops a spherical wind. This is deccelerated in the equatorial plane by the former ejected material so the wind flows along the poles (η Carinae phenomenon). Metal-rich stars with M > 25MΘ loose their H-rich envelope already in the hydrogen burning phase. These stars are called Wolf–Rayet stars and show anomalous abundances of elements in their spectra (e.g. high C abundance).

Whereas stars with M ≤ 8MΘ terminate the stellar evolution with the formation of a C/O white dwarf and the ejection of a Planetary Nebulae (exception: white dwarfs in binary systems → Novae, Supernovae Ia), stars with higher mass go through several cycles of contraction and higher burning phases. A star of ≈ 20MΘ experiences the following evolution:

IMPRS Astrophysics Introductory Course Fall 2007 49 IMPRS Astrophysics Introductory Course Fall 2007 50 fuel T [K] Ρ [g/cm3] cooling duration products core mass after burning

7 7 H 3 · 10 1 γ 10 yr He 10MΘ He 2 · 108 500 γ 106 yr C, O 6MΘ C 6 · 108 105 ν 103 yr Ne, Mg 5MΘ Ne 1.2 · 109 106 ν 10 yr Mg, Si 3MΘ O 2 · 109 107 ν lyr Si, S 2M Si 3 · 109 108 ν hours Fe, Ni Θ 1.5MΘ After the Si–burning to iron, the star has exhausted all its energy sources as iron has the highest binding energy per nucleon. Energy loss from the core due to neutrinos leads to contraction and higher temperatures which in turn causes even more neutrino losses because the temperature rises to 109 K which leads to the following reactions: ↔ ↔ − Moreover, the electrons are pushed into the nuclei via inverse β decay (p + e → n + νe), because this is now energetically favored and entropy can increase during this process. This process reduces the pressure in the star further. The core continues to contract. At ~1011 K, the photons become so energetic that they start to disintegrate the nuclei: ↔ ↔ This is an endothermic reaction, pressure falls dramatically: the core collapses in free fall.

IMPRS Astrophysics Introductory Course Fall 2007 51 Energy production in stellar interiors. A density of 104 g/cm−3 was assumed for hydrogen and helium burning, 105 g/cm−3 for all other cases. The dashed lines are energy loss rates through neutrinos produced in photon- collisions and e+e− annihilation. (Scheffler/Elsässer: Physics of the sun and the stars).

IMPRS Astrophysics Introductory Course Fall 2007 52 Once the density reaches ρ ≈ 1012 g/cm3, a large fraction of the core is already transformed into neutrons and neutrinos which can not escape anymore (!). For ρ ≈ 1014 g/cm3 the density of nuclear matter is reached. Now, either the quantum mechanical degeneracy of the neutrons stops the collapse (Mcore < 2MΘ) and we obtain a neutron star, or, the core collapses into a black hole (Mcore > 2MΘ). If the core forms a neutron star, the compressibility become virtually infinite. The outer stellar material falling onto the neutron star is strongly shocked and heated up causing the explosive expansion of the envelope.

The out-going shockwave loses a lot of energy to photo–desintegration of iron. However, the convective energy transport in the neutron star and the escaping neutrinos presumably supply more energy to sustain the explosion (numerical models are highly difficult; some models require a non–vanishing neutrino mass for explosion). The high observed velocities of neutron stars relative to the galactic plane (up to 1000 km/s) lead to the assumption that the collapse may occure asymetrically and the neutron star may even be ejected from the envelope.

The total collapse of the iron core lasts only milliseconds!

Neutrinos from a Supernova were first detected 1987 for the supernova in the Large Mag- ellanic Cloud (SN 1987a) and confirm this scenario. If the shock wave runs outwards it triggers further fusion processes. This stage is called explosive nucleosynthesis.

IMPRS Astrophysics Introductory Course Fall 2007 53 4.5.1 SN II

Once the out-going shock wave breaks through of outermost layers of the star we have the beginning of a supernova explosion. The luminosity of the star rises by a factor of 108. It can be as bright as a whole galaxy:

Nevertheless, this is only a small fraction of the energy released (≈ 1%):

The neutrino energy ESN,ν is released in about 1 sec. In this second the neutrino luminosity of the supernova is about as high as the optical luminosity of all stars in the visible universe!

Note: The described mechanism of SN–explosion works for massive stars. If a hydrogen rich envelope still exists when the explosion occurs, the supernova is called SN II. Without a hydrogen envelope (progenitors may be Wolf–Rayet stars) the supernova is called SN I b/c. As discussed above, Supernovae of type Ia have a different origin.

IMPRS Astrophysics Introductory Course Fall 2007 54 IMPRS Astrophysics Introductory Course Fall 2007 55 Historic Supernovae in the Galaxy:

185 386 393 1006 1054 Chinese sightings - none in europe - weather? crab nubulae 1181 1572 Tychos supernova (Cassiopeia) 1604 Keplers supernova (Ophichus)

Today: several hundred up to thousand supernovae are discovered, mainly in external galaxies up to distances of some billion light years.

light curves rapid fall-off in luminosity in the first 50 days, thereafter an exponential decay according L ~ e−t/τ with τ ≈ 70...120 days.

spectra depending on type, with hydrogen lines or without. Lines are very broad (high expansion velocity!) and line shapes look partly like those of stellar winds, i.e. have red emission and blue absorption.

IMPRS Astrophysics Introductory Course Fall 2007 56 IMPRS Astrophysics Introductory Course Fall 2007 57 The distance to nearby supernovae (and other objects with expanding envelopes) can be determined using the Baade-Wesselink-method.

We observe an expanding envelope at two different times and determine the arriving fluxes on the earth, Sλ,1 and Sλ,2, the temperatures T1 and T2, as well as the expansion velocity of the shell from the anaysis of the spectrum. We have:

with d the distance to the object. We have 3 equations for the 3 unknown quantities R1, R2 and d, therefore it is possible to determine simultaneously the radii of the envelope and the distance to the object, and from that the luminosity. This method is not very reliable.

IMPRS Astrophysics Introductory Course Fall 2007 58 Bartel et al. 2007 (astroph 0707.881), Distance to M81 = 3.96 (± 0.05 ± 0.29) Mpc from SN 1993J; angular radius of shell ~ 4 mas after 7 years, mean expansion velocity ~ 10000 km/s over 7 years.

IMPRS Astrophysics Introductory Course Fall 2007 59 4.5.2 Gamma Ray Bursts

Gamma-ray bursts (GRBs) are presumably the most powerful explosions that occur today in the Universe. They occur approximately once per day and are brief, but intense, flashes of gamma radiation. They are isotropically distributed over the sky which was early evidence for an origin at cosmological distances (i.e. not from within the galaxy or local supercluster of galaxies). They last from a few milliseconds to a few hundred seconds. For many decades their physical origin was completely enigmatic. Recent observations and in particular follow-up at X-ray and optical-NIR wavelengths have shown that they are indeed located at cosmological distances and associated with star forming galaxies.

IMPRS Astrophysics Introductory Course Fall 2007 60 IMPRS Astrophysics Introductory Course Fall 2007 61 Most GRBs are now considered to be the result of the collapse of very massive stars. In Woosley’s collapsar model of 1993, the central part of a massive star collapses into a black hole on which the surrounding material gets accreted and produces a relativistic jet. This jet breaks through the outermost layers of the star and produces the gamma ray burst. Because of relativistic beaming (see next page), the outburst can be very bright if the jet direction falls on the line of sight.

Presumably not all GRBs are Supernovae. Some may also be produced my merging neutron stars (see below) or by magnetars, i.e.neutron stars with extreme magnetic fields (these may form if the progenitor star has high rotation speed and strong magnetic fields). Magnetars could explain recurring softer burst events. These would be due to star quakes in which the magnetic field accelerates matter to relativistic speed. Some pulsars (see below) may be remnants of magnetars.

Simulation of the collapse of a very massive star that leads to a Gamma Ray Burst (NASA, based on Woosley’s 1993 model)

IMPRS Astrophysics Introductory Course Fall 2007 62 Most GRBs are now considered to be the result of the collapse of very massive stars. In Woosley’s collapsar model of 1993, the central part of a massive star collapses into a black hole on which the surrounding material gets accreted and produces a relativistic jet. This jet breaks through the outermost layers of the star and produces the gamma ray burst. Because of relativistic beaming (see next page), the outburst can be very bright if the jet direction falls on the line of sight.

Simulation of the collapse of a very massive star that leads to a Gamma Ray Burst (NASA, based on Woosley’s 1993 model)

IMPRS Astrophysics Introductory Course Fall 2007 62 Relativistic Beaming:

β=v/c … to observer φ

Consider a source emitting isotropic radiation following a powerspectrum ν−α , and moving with a velocity v and an angle φ to the line-of sight.

Then, when v approaches c, special relativity leads to the so-called beaming effect, i.e. the radiation is emitted pre- dominantly in forward direction (for non-relativistic motions we ‘only’ have the classical Dopplereffect that changes the energy of the photons but leaves them isotropically distributed). Relative to a source at rest, the moving source emits radiation boosted by a factor B:

IMPRS Astrophysics Introductory Course Fall 2007 63 4.5.3 Neutron Stars

A complete description of the structure of neutron stars is significantly more complicated than the one for white dwarfs.

Nuclear interactions are not negligeble anymore, they have to be included in the equation of state 2 General relativistic effects are important (strong grav. field) Egrav ≈ GM /R ≈ 0.1Mc

Densities and radii of NS can be estimated by assuming the typical particle separation to be of the order of the de Broglie wavelength:

As the neutrons form a non-relativistic degenerate gas we have:

IMPRS Astrophysics Introductory Course Fall 2007 64 Note that besides degeneracy, the repulsive component of nuclear forces also helps significantly to stabilize neutron stars. Current best computed results are:

Above a mass of M ≈ 2−3MΘ (Oppenheimer-Volkov limit) NS become relativistically degenerate and collapse to black holes. Practically all observed NS have M ≈ 1.4MΘ (because of stellar evolution).

The following two pages show Figures from Shapiro and Teukolsky: White Dwarfs, Black Holes and Neutron Stars. The first figure illustrates models for the inner structure of neutron stars. The second diagram shows the line for high density (degenerate) equilibrium configurations in the central-density vs. mass plane. Equilibrium configurations only exist on those parts of the curve which show a positive gradient.

IMPRS Astrophysics Introductory Course Fall 2007 65 IMPRS Astrophysics Introductory Course Fall 2007 66 IMPRS Astrophysics Introductory Course Fall 2007 67 4.5.4 Pulsars as neutron stars

The existence of neutron stars had already been suggested by Baade and Zwicky in 1934 and Oppenheimer and Volkoff in 1939. But only in 1967 they were finally discovered by Bell and Hewish through their pulsed radio emission. Because of their very short radio pulsus they were first called pulsars but soon it became clear that the properties of pulsars could only be explained by rapidly spinning neutron stars with strong magnetospheres.

The high rotation velocity and strong magnetosphere are natural features of neutron stars. As the angular momentum of a stellar core is preserved during the final collaps, the resulting neutron stars will be very rapidly spinning objects. Moreover, because of the high conductivity of the stellar material, the magnetic field will be dragged with the collapsing core and highly compressed. In this way magnetic field strengths up to 1012 Gauß can be built up.

The pulses range from milliseconds to seconds, a typical value is half a second, this means that neutron stars rotate with periods below a second. The pulses are very regular. Ne- vertheless, accurate measurements show that the pulse periods are very slowly increasing. This is expected because the magnetosphere accelerates particles which get lost from the system and carry angular momentum with them. The rate of the slow down indicates that pulsars are probably younger than 107 years.

IMPRS Astrophysics Introductory Course Fall 2007 68 IMPRS Astrophysics Introductory Course Fall 2007 69 Spin-up of close binary pulsar. According to General Relativity, a system of masses orbiting each other loses orbital energy due to the emission of gravitational waves. However, the energy loss only becomes significant, if the velocities approach c and the separation is of the order of the Schwarzschild radius. Both conditions are fulfilled for very close binaries of black holes or neutron stars. The loss of orbital energy has been measured accurately for the case of a binary pulsar. It was the first (indirect) evidence that gravitational waves do exist. (from Lon- gair 1993, QJRAS, 34, 157.)

IMPRS Astrophysics Introductory Course Fall 2007 70 4.5.5 Black Holes

Although observational verification is difficult, there is strong evidence for the existence of 6 9 black holes of a few MΘ as remnants of M ≥ 25MΘ stars and of black holes of 10 − 10 MΘ in the centers of galaxies.

The Schwarzschild metric describes spacetime in the presence of a black hole (valid for any point-symmetric non-rotating mass):

with the Schwarzschild radius defined as:

2 (At the Schwarzschild radius the escape velocity becomes c: c /2 = GM/rs). Light escaping from any give radius r is redshifted to

And the relation between the time interval dτ measured at a distance r to the interval dt at

IMPRS Astrophysics Introductory Course Fall 2007 71 r = ∞ is:

The sphere around the black hole defined by r = rs is called the event horizon. The sun has rs = 3km.

There exists relatively strong (indirect) evidence that stellar mass black holes do exist. This is based on the investigation of short period binary stars with strong X-ray emission (X-ray binaries). The X-ray emission is indicative of an accretion disk around a very compact object (see section on accretion disks below). For some of these objects the mass of the dark companion could be determined to be above the maximum mass of neutron stars. So, they ought to be black holes. A few examples (taken from http://astrosun.tn.cornell.edu/courses/astro201/bh candidates.htm):

Cygnus X-1 is an X-ray binary detected already in 1962. The visible object HDE226868 is a 9th magnitude blue supergiant star whose radial velocity curve shows an orbital period of 5.6 days. The fact that the object is a strong X-ray emitter and that the optical and X-ray emission varies on very short time scales (as short as 0.001s) suggest that the companion is extremely compact and might be a black hole. Analysis of the radial velocity variation of the primary under the assumption that it is a normal star suggests that the mass of the companion is about 6 solar masses.

IMPRS Astrophysics Introductory Course Fall 2007 72 LMC X-3 is a powerful source of X-rays located in the Large Magellanic Cloud. The X-ray source is associated with a binary system with an orbital period of 1.7 days. The visible component is a main sequence B3 star whose shape has been severely distorted by the gravitational field of its companion. Although not unambigious, the mass of the compact object is estimated to be at least 3 solar masses and more likely is considerably higher, making this one of the best black hole candidates.

V616 Mon (A0620-00) brightened by more than five orders of magnitude (100,000 times) in the winter of 1975. This X-ray nova event is associated with a main sequence K star known to show optical brightness variations and thus was designated V616 Mon. The K star orbits an unseen companion once every 7.75 hours with a maximum velocity of 457 km/s. It is believed that the light of the K star varies because the star’s shape is distorted by the gravitational pull of its invisible but massive companion. The mass of the compact star must be greater than 3.2 solar masses and may be greater than 7.3 solar masses, making this an excellent candidate for a black hole.

The object SS433 (see artist’s view on next page) is a binary in which the dark companion expells matter and radiation in two jets at velocities close to the speed of light. SS433 serves as a nearby example for the formation of jets by a massive object also believed to power distant quasars.

IMPRS Astrophysics Introductory Course Fall 2007 73 IMPRS Astrophysics Introductory Course Fall 2007 74 4.6 Accretion disks

Accretion disks are a common and important phenomenon in astrophysics. They play a crucial role in the evolution of binary stars, which we will discuss in the next section, and in active galactic nuclei.

When gas is falling towards a compact object, it usually has some angular momentum with respect to the object. This implies that the gas will not just fall directly onto the object but will start orbiting it. Gas ‘particles’ on different orbits will collide and the motion of the gas will be circularized. The gas forms an accretion disk. Through friction, turbulence or magnetic fields in the disk, the gas will continue to drift inward (while angular momentum is transported outward) and will finally fall onto the compact object (see Frank et al.: accretion power in astrophysics, Cambridge university press). We can estimate both the luminosity of an accretion disk and its temperature by a few simple considerations.

We first derive the luminosity that can be generated by accretion. Assume we slowly accrete a small particle of mass dm from a large radius (say ∞) through an accretion disk to a circular orbit (of radius R) around a compact star of mass Mc. From the centrifugal equilibrium we get:

IMPRS Astrophysics Introductory Course Fall 2007 75 The small particle is now strongly bound and has negative total energy. As the energy at ∞ was zero, the excess energy must have been released by friction in the accretion disk:

If we re-write this equation using the Schwarzschild radius of the compact object:

we obtain:

In case of a black hole or neutron star we can accrete the small particle down to the Schwarzschild radius or a few Schwarzschild radii, respectively. This means that a significant fraction of the particle’s rest mass can be transformed into energy. This energy is radiated away by the accretion disk. If we accrete a mass dm per time interval dt, we obtain an accretion luminosity of

IMPRS Astrophysics Introductory Course Fall 2007 76 In case of a black hole (R → RS), the exact general relativisitic calculation yields a smaller factor:

For comparison, the efficiency of hydrogen burning is:

Obviously, accretion is an efficient way to transform mass into energy! For a black hole we get:

The fractional luminosity we receive from a ring with radius R and width dR is:

which yields for the luminosity of the ring (after Taylor expansion):

So, most of the luminosity is emitted close to the compact object. Moreover, if the compact object is not a black hole but a neutron star or white dwarf, we get yet more energy released at the equator of the star, where the accreted material is decelerated to the rotation speed of the object. IMPRS Astrophysics Introductory Course Fall 2007 77 The temperature of the disk can be derived as follows. If we assume that the disk is locally in thermodynamic equilibrium (LTE) and radiates a black body spectrum, the energy 4 emitted per time and surface element is σBT (Stefan-Boltzmann law). This has to be equal to the energy emitted per surface element of the ring 2πRdR. With the equation above we obtain:

or, if we solve for the temperature:

Replacing the mass Mc by its Schwarzschild-radius RS yields:

or:

This shows that the accretion disk around a stellar black hole or neutron star will emit hard X-rays. This makes these objects detectable in X-ray surveys and indeed most stellar black hole candidates were found in this way. White dwarfs will mostly radiate in soft X-rays and the UV.

IMPRS Astrophysics Introductory Course Fall 2007 78 4.7 Stellar evolution in close binaries

about 50% of all stars are in binary systems in well separated binaries, stars evolve like isolated stars in close binaries, the evolution is strongly influenced by tidal forces and transfer of matter

The mass transfer between binaries is determined by the Roche potential:

where ω is the rotation period and s is the distance to the rotation axis.

Within the Roche-surface matter is bound to M1 or M2, outside of it, it is bound to both objects. The inner Lagrange-point L1 is force free. At L1 matter can easily move from one star to the other.

IMPRS Astrophysics Introductory Course Fall 2007 79 The equipotential surfaces between two stars. If one of the stars expands, material can flow through the inner Lagrange point L1 in the direction of the other star. The equipotential surface which encloses both stars and passes through L1 is called the Roche lobe.

IMPRS Astrophysics Introductory Course Fall 2007 80 Depending on the mass and evolutionary stages of the two stars, we can observe the following phenomena:

Cataclysmic Variable: a cool main-sequence star filling its roche lobe and a white dwarf. Mass is transferred on the white dwarf via an accretion disk (see also below). We observe eruptions due to disk instabilities.

Nova: a white dwarf and a low mass star. Mass transfer leads to explosive H-burning on white dwarf → Nova. The so-called thermo-nuclear runaway is due to the degeneracy of the accreted hydrogen rich material on the surface of the white dwarf.

Supernovae Ia: a white dwarf and a low mass companion. Transfer of helium can bring the white dwarf across the Chandrasekhar limit. It explodes as a Supernova Ia.

Low Mass X-Ray Binary (LMXB): neutron star or black hole with low mass companion and roche lobe overflow. We get an accretion stream and a hot X-ray emitting disk.

High Mass X-Ray Binary (HMXB): like LMXB, but with a blue and luminous companion.

IMPRS Astrophysics Introductory Course Fall 2007 81 Evolution of a low mass binary system

IMPRS Astrophysics Introductory Course Fall 2007 82 The previous page shows the evolution of a binary with two low-mass stars: a: the two stars are on the main sequence and orbit each other closely. b: the more massive star evolves faster and fills the Roche lobe during the giant phase c: mass flows from the evolved star onto the companion and increases its mass d: the originally more massive star turns into a white dwarf e: the originally less massive star evolves to a giant and transfers material onto the white dwarf through the inner Lagrange point f: explosive hydrogen burning can lead to a Nova explosion. If the white dwarf is brought above the Chandrasekhar limit, it explodes as a SNIa.

The next page shows an artist’s view of a cataclysmic variable. Matter flows from the companion star through an accretion disk onto a white dwarf. Where the flow hits the disk we observe a so-called hot spot. Instabilities in the flow can cause cataclysmic outbursts. If enough material is collected on the surface of the white dwarf, we can have a thermonuclear runaway resulting in a Nova explosion. If the mass exceeds 1.4 solar masses the white dwarf explodes as a Supernova Ia.

IMPRS Astrophysics Introductory Course Fall 2007 83 IMPRS Astrophysics Introductory Course Fall 2007 84