The Astronomy Module, Stellar Evolution - Lecture Notes | PHYS 2018, Study notes of Physics

Download The Astronomy Module, Stellar Evolution - Lecture Notes | PHYS 2018 and more Study notes Physics in PDF only on Docsity! Physics 2018: Great Ideas in Science: The Astronomy Module Stellar Evolution Lecture Notes Dr. Donald G. Luttermoser East Tennessee State University Edition 1.0 Abstract These class notes are designed for use of the instructor and students of the course Physics 2018: Great Ideas in Science. This edition was last modified for the Fall 2007 semester. c) Classes C, D, E, H, I, J, L, P, and Q were dropped for one reason or another or merged into other classes (see Jaschek and Jaschek 1987, The Classification of Stars, Cambridge Press). d) The R and N stellar classifications corresponded to car- bon stars and now have been merged into one classifica- tion designated C (although not the same as the original C stars, which were spectroscopic binaries). Many peo- ple however still use the R and N classification (including me) to describe carbon stars. R stars are the hotter of the two and correspond to the oxygen-rich K stars in temper- ature. The N-type carbon stars correspond to the coolest oxygen-rich M stars in temperature. Carbon stars dif- fer from oxygen-rich stars in that there visual spectrum is dominated by carbon molecule (i.e., C2, CN, and CH) absorption bands. e) S stars are another special class similar in temperature to late K and M stars. The S star’s spectrum is dominated by LaO, VO, and ZrO molecular bands. 2. Groups were rearranged from the hottest (called early-type stars ) to the coolest (called late-type stars ) and 10 subdivisions for each group introduced. Each “spectral type” is determined by various line strengths and line ratios. A brief overview is displayed in Table II-1. A more complete description can be found in Jaschek and Jaschek (1987) and Kaler (1989, Stars and Their Spectra, Cambridge Press). 3. Later it was found by Saha, that the sequence of spectral types from hottest to coolest stars should follow: O B A F G K M (R N S) II–3 Table II–1: Spectral Classifications Spectral Special Type Classes Temperatures Spectral Characteristics O 5–9 — 50,000–28,000 K He II lines. ” f ” He II (λ4686) & N III (λλ4630–34) in emission. ” e ” He II & N III in absorption, H in emission. ” WC5–WC8 ” Wolf-Rayet (carbon sequence), O-star spectrum, wide emission lines. ” WN6–WN8 ” Wolf-Rayet (nitrogen sequence), O-star spectrum, wide emission lines. B 0–9 — 28,000–9,900 K He I lines, H lines strengthen. ” e ” He I, H emission lines. ” p ” B-star spectrum; Si II, Mn II, Cr II, Eu II, Sr II are strong. A 0–9 — 9,900–7,400 K H lines strong, Ca II H & K strengthen toward later type. ” p ” A-star spectrum; Si II, Mn II, Cr II, Eu II, Sr II are strong. ” m ” A-star spectrum + Fe lines unusually strong. F 0–9 — 7,400–6,000 K Metals, H lines weaken, Ca II strengthen — emission “bumps” appear in H & K line cores at F4. G 0–9 — 6,000–4,900 K Ca II very strong, Na I D line strengthens, metals. ” Ba — (G2-K4) spectrum, Ba II (λ4554) very strong. ” CH — (G5-K5) spectrum, CH band (λ4300) very strong. K 0–7 — 4,900–3,900 K Ca II strong, Ca I (λ4227) strengthens, molecular bands form. M 0–10 — 3,900–2,500 K TiO bands dominate visual spectrum, neutral metals very strong. S 0–9 — 4,200–2,500 K (K5-M) type stars except ZrO, VO, & LaO molecular bands strong. R 0–9 — 4,800–3,200 K C2, CN, CH strong, =C0-C5 classification. N 0–7 — 3,200–2,500 K C2, CN, CH strong, strong violet flux depression, =C6-C9 classification. II–4 a) Classes R, N, and S are special as described above. b) The weakness of the H lines in O stars is due to most of the hydrogen being completely ionized. c) The weakness of the H lines in M stars is due to their cool atmospheres, most of the electrons are in the ground state, with virtually none in the 2nd level where the Balmer lines arise. C. Luminosity Classification. 1. Absorption lines that appear in a star’s spectrum arise in the outer layers of a star’s atmosphere. Spectral lines are broadened (i.e., made thicker = stronger) via a variety of processes. One of these processes is particle collisions. a) Since collisional rates are a function of density and density depends on the surface gravity of a star; bigger, lower gravity stars will tend to have sharper lines than high gravity stars for a given spectral type. b) A broader line “for the same spectral type” generally im- plies higher gravity. 2. As we just saw, luminosity depends both on the radius and tem- perature of a star. 3. Table II-2 displays the luminosity classification system. 4. The Morgan-Keenan (M-K) classification of a star is simply the spectral type along with the luminosity class and defines a star’s location on an H-R diagram (i.e., the Sun is a G2 V star, α Boo a K1 III star, see below). II–5 c) Population III stars: i) Zero metal abundance (Z = 0). ii) They no longer exist in the Galaxy. iii) These were the first stars to form out of the pri- mordial baryons formed during the Big Bang. 3. Besides representing metalicity with the “Z” index (i.e., mass fraction of metals to all particles), metalicity also is defined by a star’s Fe abundance with respect to the Sun following Eq. (II-3): [Fe/H] ≡ log10  n(Fe) n(H)   − log10  n(Fe) n(H)   . (II-5) a) The Sun’s Fe abundance is about log10 ( n(Fe) n(H) ) = 10−5. b) If a star has solar metalicity, then [Fe/H] = 0.0. c) Population I stars range from −0.5 < [Fe/H] < +0.5. d) Population II stars have [Fe/H] < −0.8. e) The few stars that have −0.8 < [Fe/H] < −0.5 are typ- ically called old disk population stars, though they are typically grouped with the Population I stars. f) Population III stars will have [Fe/H] = 0 if they are ever observed in the youngest galaxies at the farthest reaches of the Universe. II–8 Figure II–1: The Hertzsprung-Russell Diagram. E. The Hertzsprung-Russell Diagram 1. In the early 1900s, 2 astronomers found peculiar groupings of stars when they plotted their absolute magnitude (a star’s bright- ness at 10 parsecs distance) to their (B − V ) (blue - visual mag- nitudes) colors (Hertzsprung) or spectral type (Russell) =⇒ the Hertzsprung-Russell (HR) Diagram (see Figure II-1). a) Main sequence stars (∼90% of all stars are of this type). b) Giant stars (most of them red in color). c) Supergiant stars (very luminous, hence large). d) White dwarf stars (very faint, hence small). 2. When plotting the HR diagram for a star cluster, the stars are vir- tually at the same distance, hence differences in observed bright- ness corresponds to actual luminosity differences (and not dis- II–9 tance differences). a) The shape of an HR diagram for a star cluster gives the age of the star cluster. b) The main sequence turn-off point is a very precise gage of the cluster’s age =⇒ the point at which a main sequence star exhausts its supply of hydrogen in its core. 3. HR diagrams come in 2 types: a) Observed HR diagrams, absolute magnitude (MV ), or ap- parent magnitude (V ) for clusters, versus color index (B− V ) or spectral type. i) The color index is the magnitude difference be- tween 2 different filter measurements (see Table I- 1). The color index, or color, is related to the tem- perature of the star. ii) For very hot stars (e.g., O stars), (U − B) (ul- traviolet minus blue magnitudes) gives the most accurate temperature measurements. These stars appear bluish. iii) For very cool stars (e.g., M stars), (V − R) (vi- sual minus red magnitudes) or (R − I) (red minus infrared magnitudes) gives the most accurate tem- peratures. These stars appear reddish. iv) For every other star, (B − V ) gives the most ac- curate temperatures. v) Faint, bluish stars are on the lower left side of an HR diagram, and bright, reddish stars are on the II–10 b) The gravitational potential energy is Eg = − G M2 R = −16π2 G ρ2 R5 . (II-10) c) For a constant density ρ and temperature T in the cloud, Eth increases with radius R like R 3, while Eg ∝ R5, a much stronger dependence on R. 3. Using Eqs. (II-9) and (II-10) in Eq. (II-8) gives RJ = ( kB 12π G µmH )1/2 T 1/2 ρ1/2 , (II-11) which is the Jeans’ Length. If the radius of a gas cloud is R > RJ (keeping ρ and T constant), the cloud becomes unstable and will collapse. 4. Defining MJ ≡ 4π3 ρ R 3 J as the Jeans’ Mass gives MJ = 4π 3 ( k 12π Gµ mH )3/2 T 3/2 ρ1/2 (II-12) and if M > MJ for a gas cloud, the cloud will collapse due to gravitational instabilities. 5. During this stage of star formation, the gas cloud is in free fall. a) Throughout this free-fall stage, the temperature of the gas stays relatively constant =⇒ the collapse is said to be isothermal. b) One can calculate the free-fall time by equating New- ton’s 2nd Law of Motion to his Law of Gravitation: tff = ( 3π 32 1 Gρ◦ )1/2 , (II-13) where ρ◦ is the initial mass density of the cloud. II–13 c) Since this free-fall time depends only on density, all parts of the cloud will collapse at the same as long as the cloud has uniform density =⇒ homologous collapse. 6. A GMC will not necessarily collapse as a single unit, typically just a portion of it will collapse based upon the triggering mechanism (see below). a) The density of the collapsing cloud will increase by orders of magnitude during the free-fall time. b) Though we have developed the collapse criteria under the assumption of constant density, in reality, there will be pockets of inhomogeneities in the cloud. c) As a result, sections of the cloud will independently sat- isfy the Jeans’ mass limit and begin to collapse locally, producing smaller features within the original cloud. d) This cascading collapse could lead to the formation of large numbers of smaller objects. H. Triggers of Star Formation (SF). 1. Observations. a) In the Milky Way Galaxy, SF occurs in GMCs. b) OB stars form in associations at edges of GMCs. c) OB association ionizes the surrounding gas producing an H II region. d) Lower mass (T Tauri-type) stars form throughout the vol- ume of the GMC. e) SF defines the optical spiral arms of the Milky Way. II–14 2. What is the trigger? Any process that can cause a stable (M < MJ) cloudlet to become unstable (M > MJ). a) Agglomeration: Component cloudlets of GMC’s collide and sometime coallace until M > MJ . b) Shock Wave Compression: A shock can be the trigger =⇒ it acts like a snow plow causing ρ to increase, and as a result, MJ drops (see Figure II-3). i) Spiral Density Wave: As Milky Way Galaxy rotates, its two spiral arms can compress a GMC, which then leads to star formation. ii) Ionization Front: O & B stars form very quickly once cloud collapse has started (see below). These produce H II regions from their strong ionizing UV flux, which initially expand outward away from the OB association. This ionization front heats the gas causing a shock to form. The shock can compress the gas such that M > MJ , which once again, leads to star formation. iii) Supernova Shocks: O & B stars evolve very quickly on the main sequence and die explosively as supernovae. The shock sent out by such a su- pernova can excite further star formation. I. The Free-Fall Stage of Stellar Birth. 1. As a portion of a GMC begins to contract, cloud complexes with masses greater than ∼ 50 M become unstable and fragment into smaller cloudlets (see Figure II-4). Each little cloudlet continues to collapse as described above. II–15 axi s Bulge Disk Figure II–6: Formation of protostar and protoplanetary disk. i) In the outer protoplanetary disk, ice crystals con- dense out of the gas along with some dust grains. ii) In the inner protoplanetary disk, it is too hot for ice crystals to form, only dust condenses out. iii) The dust (and ice) begin to conglomerate to- gether in a process known as advection (similar to building a snowman), building bigger and bigger particles. This processed continued until boulder to mountain sized objects existed =⇒ the planetesi- mals. iv) Planetesimal is the name given to big rocks in orbit about a “protostar.” Due to their small size (D < 1000 km), they are not spherical in shape. • The “rocky” planetesimals are now called aster- oids. II–18 Protostar IR Light Protoplanet Figure II–7: Inside the protoplanetary disk of the collapsing cloudlet. • The “icy” planetesimals are now called comets. v) Planetesimals occasionally smash into each other, sometimes destroying each other, sometimes stick- ing together to form even bigger planetesimals =⇒ this is a process known as accretion. c) Numerous such disks have been seen in stellar nurseries with the Hubble Space Telescope. They are especially easy to see at IR wavelengths. d) Unfortunately, we do not have time to elaborate on the formation of planets in this course. 4. During this time, the cloudlet is in free-fall — the so-called free- fall stage. Its luminosity continues to decrease as the cloud collapses, while temperature increases slightly. This stage ends when the protostar becomes opaque to it own radiation (primar- II–19 ily IR photons). This happen when the cloudlet’s central bulge is about the size of Mercury’s orbit. a) The free-fall time depends upon the density that the cloudlet had at initial formation due to fragmentation. i) GMC temperatures are on the order of 50 K. Though the average density of a GMC is on the order of 103 cm−3, when cloudlet fragmentation occurs, the particle density is on the order of 108 cm−3 which corresponds to a mass density of 2×10−16 gm cm−3. ii) For such initial conditions, the free-fall time to an opaque state is about 4700 years. Increasing the initial density by a factor of 10 would reduce the free-fall time to a mere 1500 years — a blink of an eye to the age of the Universe =⇒ cloudlets collapse rapidly to the protostar stage! b) The higher-mass cloudlets have higher densities due to the higher gravitational force associated with them. As such, the more massive the cloudlet, the faster it will collapse. 5. Though there is a slight increase in temperature, this increase is very small, and as such, this free-fall stage is said to correspond to an isothermal collapse of the cloudlet. J. The Hayashi Stage of Stellar Birth — The Protostars. 1. The contraction now continues at a much slower rate once the gas becomes opaque to IR photons. a) This marks the beginning of the Hayashi stage. At this point, the collapsing cloudlet is now called a protostar. II–20 f) As the central temperature continues to rise, increasing levels of ionization of H decrease the total opacity in that region (the cross section of H− opacity is many orders of magnitude bigger than electron scattering and H bf- and H II ff-opacities). i) At this point, a radiative core develops and pro- gressively encompasses more and more of the star’s mass. ii) The radiative core allows energy to escape into the convective envelope more readily, causing the luminosity of the star to increase again. g) The path that a star takes on the H-R Diagram from the initial formation of the surface convection zone when luminosity drops by an order of magnitude to the point where luminosity begins to rise again is referred to as the Hayashi track. h) At about the time that the luminosity begins to increase again, the central temperatures are now high enough for the first two steps of the PP I reaction chain and the steps from 12C to 14N in the CNO cycle to begin in earnest, but not yet at their equilibrium rates. As time progresses, εgrav has a smaller and smaller impact on L as compared to εnuc. i) The surface of the protostar now becomes hot enough to emit a significant amount of its energy flux as visible light photons (still being powered by gravitational contraction). i) The pressure from this light starts to push dust sized particles out of the protoplanetary disk via radiation pressure. II–23 ii) All that remains in the protoplanetary disk are larger planetesimals (rocky ones close in — the ‘as- teroids,’ and icy ones farther out — the ‘comets’) and the protoplanets. iii) This spring-cleaning phase is called the T-Tauri stage of the protostar evolution, named after the prototype of this class, T Tauri (see Figure II-8). iv) This “spring cleaning” phase can continue well into the epoch when thermonuclear reactions start up in earnest, the so-called main sequence phase. v) Since so much material exists in the disk, the mo- mentum exchange between the photons and matter is not efficient. However at the poles, the density is much less which results in a strong bipolar out- flow =⇒ bipolar jets form. vi) These jets can interact with the surrounding ma- terial causing higher density clumps to form. Such clumps seen in a T Tauri star’s jets are call Herbig- Haro objects. j) The rate of nuclear energy production has become so great that the central core is forced to expand somewhat, caus- ing the gravitational energy term in Eq. (II-14) to become negative. This effect is apparent at the surface as the to- tal luminosity decreases towards the main sequence value. This decrease in L forces T to decrease as well. k) At this point, 12C is completely exhausted from the CN reactions of the CNO cycle which reduces the energy pro- II–24 Bipolar Outflow (no disk at poles to restrict wind) T-Tauri Model Herbig-Haro (HH) Object HH Object (bowshock of jet) Planetary Disk Protostar Figure II–8: Model of a T Tauri star. Gas flows easier at the poles than in the disk which results in a bipolar outflow. duction, the core adjusts by contracting which increases the temperature until the point when the 3rd step of the PP I can start occurring in earnest. l) Now the temperatures become high enough to allow the last stages of the CNO cycle to proceed (though not at a very high rate), which replenishes the core with 12C. m) When this occurs εnuc  εgrav, core contraction stops and the PP I chain achieves thermal equilibrium with the gas. At this point, A STAR IS BORN. 3. Protostar Evolution of a 0.4 M (i.e., Low Mass) Star. a) There are a few major differences between the evolution of a low-mass protostar (∼ 0.4 M ) to that of a solar mass protostar: II–25 b) HSE demands that dP dr = −GMρ r2 . c) Combining these two equation and solving for the lumi- nosity, we get LEd = 4πGc k M , (II-15) where this Eddington luminosity is the maximum lumi- nosity a star can have and still satisfy HSE. d) For massive stars, the effective temperatures are 40,000 K or greater which is enough to ionize hydrogen in their pho- tospheres, and as such, electron scattering is the dominant opacity (i.e., k) source. Making use of this and expressing the Eddington luminosity in terms of solar luminosities, we can write LEd L ' 3.8 × 104 M M . (II-16) e) Eddington also showed that a gas sphere in radiative equi- librium will follow the following mass-luminosity relation- ship (see next section): L L ' ( M M )3.5 . (II-17) f) Equating this radiative-equilibrium luminosity with the Eddington limit gives us the maximum mass a star can have without blowing itself apart from radiation pressure: M ' 70 M . (II-18) g) Since gravitational instabilities tend to prevent stars from forming at such a high mass, there should not be too many stars that exceed the Eddington limit. The Wolf-Rayet stars, however, come very close to this Eddington limit. II–28 K. Life on the Main Sequence. 1. Main sequence (MS) stars are also called hydrogen-core burners. Approximately 90% of a star’s nuclear burning lifetime is spent in this stage (hence 90% of all stars fall on the main sequence on the HR diagram). When a star first ignites hydrogen in its core, it resides on the zero-age main sequence (ZAMS). 2. This main sequence lifetime corresponds to the nuclear time scale: tnuc ≡ Energy available from nuclear fuel Rate at which fuel is used up = η f Xinit M c 2 L , (II-19) where η = 0.0071 is the 4H→He reaction efficiency introduced in the Nuclear Physics section of the Physics Module notes, Xinit is the initial mass fraction of H, and f is the fraction of H actually burned on the MS. a) So tMS = tnuc is the amount of time the star remains on the MS. b) f ≈ 0.1 for 1M stars, which gives tMS ≈ 1010 yr M/M L/L . (II-20) c) f increases up to 0.3 for upper main sequence stars. 3. The mass-luminosity relation for stars on the main sequence. a) Eddington showed in 1924, using a radiative diffusion ap- proximation, that a spherical sphere of ideal gas which is in radiative equilibrium should follow a mass-luminosity relationship given by Eq. (II-17) =⇒ more massive stars are much more luminous than lower mass MS stars. II–29 b) Later, based on main sequence stars in binary star systems in the solar neighborhood, a mass-luminosity relationship was empirically found for main sequence stars of L L ≈ ( M M )4 for L > L (II-21) L L ≈ ( M M )2.8 for L ≤ L . (II-22) 4. Using the Eddington M-L relationship, the MS lifetime then can be written as tMS ≈ ( M M )−2.5 × 1010 yr . (II-23) 5. The energy transport mechanism that dominates the interior of a star changes along the MS as was described in the last section on stellar formation. a) High-mass stars (M >∼ 1.3 M , which correspond to stars earlier than F5 V). i) For stars more massive than 1.2–1.3 M , the CNO cycle dominates over the PP chain in the main se- quence star’s energy production. ii) Since the CNO cycle has such a large temperature dependence (anywhere from T 13 to T 20) in compari- son to the PP chain, temperature gradients are very large in the cores of these stars. iii) Due to this large temperature gradient, convec- tive instability is established and energy in the core is transported outward by convection over radiation transport =⇒ hence these stars have convective cores. II–30 • Collapse to a black hole before there is even a chance to ignite H. • Should these objects ever make it to the main sequence, they supernova within a few 100 to a thousand years! b) The low mass limit is about 0.08 M (M8 – M10 V). i) Temperatures never get high enough to ignite H at an equilibrium rate to produce He. ii) Objects with M < 0.08 M are called brown dwarfs. iii) Objects with M < 10 MJupiter(= 0.01 M ) are called planets. L. Fundamental Time Scales on which Stars Evolve. 1. There are 3 basic time scales that are important to stars during their evolution: NAME CONDITIONS SOLAR dynamic time non-HSE ∼1/2 hour thermal (K-H) time HSE, non-TE ∼30 million years nuclear time HSE, TE ∼10 billion years 2. While on the MS, a star’s luminosity will slowly increase and temperature slowly decrease. This gives the main sequence a thickness when viewed on the HR diagram. a) As such, stars do not stay at their ZAMS position for their entire H-core burning life. b) As the hydrogen is fused into helium, µ (the mean molec- ular weight) in the core increases. II–33 i) This will cause the pressure to drop (assuming ideal gas). ii) Following the dynamic time scale, the core com- presses as a result which increases the density and temperature. iii) This in turn increases the thermonuclear reaction rates which increases the surface luminosity. c) For stars of 1.2-1.3 M or less, this increased luminos- ity, increases the surface temperature of the star. These stars get brighter and hotter during their main sequence lifetime. d) For higher mass stars where H is ionized throughout the entire interior, this added flux increases the photon mo- mentum on the free electrons in the outer layers, which expands these layers and results in a cooling in the photo- sphere. These stars get brighter and cooler while on the main sequence. e) Note that these changes are extremely gradual, and as such, stars on the main sequence can be considered to be in hydrostatic and thermal equilibrium. 3. Once the nuclear time scale has been exceeded, hydrogen in the region of thermonuclear reactions (i.e., the core) becomes ex- hausted. This core goes out of TE and HSE and begins to col- lapse. II–34 Table II–3: Stellar Parameters for ZAMS Stars. Spectral tMS Class M/M R/R L/L Teff (K) (years) O5 60 12 7.9 × 105 44,500 7.6 × 105 B0 17.5 7.4 5.2 × 104 30,000 3.4 × 106 A0 2.9 2.4 54 9520 5.4 × 108 F0 1.6 1.5 6.5 7200 2.5 × 109 G0 1.05 1.1 1.5 6030 7.0 × 109 K0 0.79 0.85 0.42 5250 1.9 × 1010 M0 0.51 0.61 7.7 × 10−2 3850 6.6 × 1010 M5 0.21 0.27 1.1 × 10−2 3240 1.9 × 1012 M. Ascent on the Red Giant Branch — Marching Towards Helium Ignition. 1. For this section, we will investigate stellar evolution from main sequence up until the ignition of helium in the core for stars in three mass classifications. We start with the very Very Low- Mass Stars (M <∼ 0.4M ): a) These stars are completely convective while on the main sequence. As such, since H is continuously being replenish into the core from the outer envelop, these stars have f = 1 in Eq. (II-19) and burn H in their cores until the star is completely exhausted of H. b) Once the PP chain runs out of fuel, the He-rich star goes out of HSE and beings to collapse. c) As the star gets smaller, is gets hotter, ionizing much of the interior of the star. d) The star collapses until the free electrons start to “feel” the Pauli Exclusion Principle — two electrons cannot exist in the same quantum state in the same place at the same time. II–35 iii) The star gets larger (and more luminous) and cooler producing an orange subgiant star. The F5 star Procyon is at this stage in its evolution (although Procyon, being a bit more massive than the Sun, is moving from the white region of the main sequence to the yellow region of the subgiant luminosity class). iv) The star is now said to be ascending the red giant branch. g) While the outer envelope is expanding, the core continues to contract and heat. The H-burning shell is producing more helium ash which falls on the core, increasing the core’s mass as a result. i) Helium fusion will not begin until T > 108 K. ii) Prior to these temperatures being obtained in the core, electrons are forced closer and closer together until the gas becomes degenerate. h) These star quickly experience a run-away thermonuclear event in their cores know as the “helium flash,” or to be more specific, the Helium-Core Flash. i) The electron degeneracy pressure counterbalances gravity (as described above), hence resists compres- sion due to gravity. ii) As shown in Eq. (II-24), pressure (P ) in a degen- erate gas is independent of temperature (T ). II–38 iii) The temperature of the helium atoms continues to rise as matter is dumped from the H-burning shell above — but pressure remains constant =⇒ the normal pressure-temperature-gravity ther- mostat is off since the ideal gas law is no longer in effect! iv) T reaches 108 K so helium starts fusing which increases T more (without increasing P ) =⇒ in- crease of T increases reaction rates, which further increases T ! =⇒ Resulting in runaway thermonuclear reactions! =⇒ Hence, the Helium-Core Flash. At this time, the core luminosity increases to 1011L , comparable to that of an entire galaxy in just a few seconds time! Most of this energy never reaches the surface however, instead it is absorbed by the gas in the envelope, possibly causing some mass to be lost as this time from the envelope. v) Soon, T gets high enough to “lift” degeneracy =⇒ HSE is reinstated =⇒ the (now) red giant becomes stable and is called a red giant clump star (if a Population I star) or a horizontal branch star (if a Population II star). vi) The star now has two energy sources, H-shell burning (primarily from the CNO cycle) and He- core burning (via the triple-α process) and will re- main in this state for the final 10% of its thermonu- II–39 clear life. 3. High-Mass Stars (M >∼ 4 M ) follow the same evolutionary sequence as the Low-Mass Stars, except the core never becomes degenerate during collapse. Once temperatures get above 108 K, the triple-α process begins gradually until the rates obtain an equilibrium value — no runaway occurs in these stars, hence no He-core flash. N. Post-Red Giant Evolution. 1. The path a star takes on its road to stellar death depends upon its initial mass. Note that the following masses correspond to the mass of the star while on the main sequence. 2. M < 0.08M : These objects never ignite hydrogen and hence never become main sequence stars. These objects are called brown dwarfs. They are bright at infrared wavelengths from energy liberated from contraction. 3. 0.08M < M < 0.4M : These stars are totally convective, and as such, helium ash gets uniformly mixed throughout the star. No inert He core develops, hence no H-shell burning and no ex- pansion into a red giant. After hydrogen is exhausted, the entire star contracts until the helium becomes degenerate (see the white dwarf section) and the star becomes a helium-rich white dwarf. 4. 0.4M < M < 3M : These stars ignite He in a degenerate core (i.e., the He-flash). During the core He-fusion stage, the star sits in the red giant clump (Population I stars) or the horizontal branch (Population II stars) on the HR diagram. He-fusion creates 12C and 16O ash and once the He-fusion stops, the carbon- oxygen core continues to collapse. II–40 c) The brightness variability is causes by an actual pulsation of the star! 2. Another type a variable star was discovered in 1784, the Cepheid variables, with δ Cephei as the prototype. a) These stars are more luminous and are hotter than the long period variables. b) Their variability period can range anywhere from 1 to 60 days, depending on their overall luminosity. c) These are pulsating stars too. d) Cepheids follow a period-luminosity relationship. The luminosity of the star scales somewhat linearly with the pulsational period =⇒ by measuring a Cepheids period, we can deduce its absolute magnitude MV , then by com- paring MV to its apparent magnitude V , we can deduce its distance d via the distance modulus formula: V − MV = 5 log d − 5 . (II-25) e) Population I star cepheids (called Type I or classical cepheids) have a slightly different period-luminosity rela- tionship than the Population II star cepheids (called Type II cepheids or W Virginis stars). 3. Lower mass versions of Cepheids exist called RR Lyrae type variables, which change in brightness with period shorter than 1 day. These stars are horizontal branch stars, and as such, all have the approximate same luminosity (e.g., 100L ), hence can also be used as distance indicators. II–43 4. All of these different types of variables pulsate due to their inter- nal structure — they all lie on an instability strip on the HR diagram. a) The pulsations are due to the kappa effect, kappa for opacity, which results from an ionization zone that lies just beneath the photospheres of these stars. b) Miras pulsate from a hydrogen ionization zone just be- neath the surface of the star. c) Cepheids and RR Lyr’s pulsate from a helium ionization zone just beneath the surface. P. Stellar Corpses I: White Dwarfs. 1. Stars will end up in one of 3 states: white dwarf, neutron star, or black hole. Stars that are initially ∼4 M or less on the main sequence will wind up as a white dwarf. 2. As the carbon-oxygen core continues to collapse after He-core burning, the outer envelope of the star continues to expand away, helped along by strong stellar winds. a) The outer envelope detaches itself from the collapsing core and becomes a shell of material surrounding the core. b) When the shell be comes thin enough so that the hot core can be seen through the shell, UV photons emitted from the core fluoresce the expanding shell =⇒ a planetary nebula forms. c) This shell will continue to expand, getting thinner and thinner as it gets larger until it begins to intermingle with the surrounding ISM. II–44 d) Since the outer envelope of the star was enhanced with C, N, and O from the nuclear reactions due the shell burning phases, the ISM gets enhanced with these elements. e) Future stars that will be born out of the ISM gas will thus have higher abundances of C, N, and O due to this previous epoch of stellar evolution. 3. The laws of quantum mechanics (QM) dictate the character- istics of electrons. a) Electrons have a variety of quantum states associated with them: i) Principle quantum number (n) =⇒ which elec- tronic shell an electron is in. ii) Orbital angular momentum quantum number (`) =⇒ IDs the subshell. iii) Spin angular momentum quantum number (s) =⇒ s = +12 for counterclockwise spin, s = − 1 2 for clockwise spin. iv) Total angular momentum quantum number (j = ` + s). b) One fundamental law in QM is the Pauli Exclusion Principle (PEP): no two electrons can exist in the same quantum state (i.e., have the same quantum numbers) at the time in the same place. 4. As the stellar core continues to collapse, the free electrons there are forced so close together that they all try to sit in the same II–45 c) The Fe core become degenerate as it collapses but is too massive for the electron degeneracy pressure to hold up the weight of the star. d) Since the Pauli Exclusion Principle dictates that electrons cannot exist in the same state, the only place for them to go is into the Fe nuclei! 3. The electrons interact with protons in an inverse β-decay: e− + p −→ n + ν, forming a stellar core of pure neutrons ! 4. Once the neutrons are formed, they stop the collapse of the core via the strong nuclear force =⇒ neutron degeneracy pressure holds up the weight of the core =⇒ a neutron star (NS) is born! a) This halting of the collapse of the core is rather sudden and causes the core to bounce and rebound a bit. This bounce sets up a shock wave which propagates outward and blows apart the outer envelope of the star in a su- pernova explosion! b) A tremendous amount of energy is released during this explosion over a fraction of a second, increasing the lumi- nosity of the star by a factor of 108! c) However this luminosity increase only corresponds to 1% of the total energy released, most of the energy comes out in the form of neutrinos! d) After the explosion, the outer envelope continues to ex- pand, creating a supernova remnant which can last for a few million years (e.g., the Crab nebula). II–48 5. Supernovae are classified based primarily upon their spectra and their light curves (i.e., how brightness changes with time). The primary classification is based upon the appearance of hydrogen Balmer lines. a) Type I supernovae do not show hydrogen Balmer lines and reach a maximum brightness of MV ≈ −19. They are classified into 3 subclasses: i) Type Ia supernovae display a strong Si II line at 6150 Å. These are white dwarfs in close binary star systems that exceed the Chandrasekhar limit and explode from a runaway thermonuclear reac- tion in the carbon-oxygen core of the white dwarf. These types of supernovae are seen in both elliptical galaxies and throughout spiral galaxies. ii) Type Ib supernovae display strong helium lines. These tyeps of supernovae are only seen in the arms of spiral galaxies near star forming regions. Hence, this implied that short-lived massive stars in binary systems are probably involved. These explosions are similar to that of a Type II supernova, only in a binary star system where the outer H envelope has been transferred to the secondary star in the system before the Fe-core bounce. iii) Type Ic supernovae display weak helium lines and no Si II is seen. Other than this, they are observed in the same location in galaxies as Type Ib. These types of supernovae are likely the same as Type Ib, except the helium-shell has been trans- ferred to the companion in addition to the hydrogen envelope. II–49 b) Type I12 supernovae display weak hydrogen lines. These are isolated stars of main sequence mass of the range 4- 8 M where carbon detonates in a degenerate core and completely disrupts the star. Hydrogen lines are weak due to earlier mass loss on the AGB which results in a lower mass hydrogen envelope in comparisons to the Type II supernovae. c) Type II supernovae display strong hydrogen lines and reach a maximum brightness of MV ≈ −17. These super- novae result from the explosion of an isolated star due to an Fe-core bounce. They are classified into 2 subclasses base upon the shape of their light curve. i) Type II-L supernovae display a linear decline in their light curves after maximum emission is reached. ii) Type II-P supernovae display a plateau in the light curve between 30 and 80 days after maxi- mum light. This plateau results from additional light generated from the radioactive decay of the large amounts of 56Ni that was created in the ex- plosion. These supernova are likely more massive than the Type II-L supernova which accounts for the larger amounts of this radioactive isotope of nickel in these explosions. 6. During the explosion, material is ejected off of the core at very high velocities. Fast moving free neutrons given off by all of the nucleosynthesis occurring in the outward moving shock interact with nuclei in the envelope to make different chemical elements =⇒ such nuclear reactions are called rapid neutron capture or the r-process. II–50 R. Stellar Corpses III: Black Holes. 1. Whereas Newton envisioned gravity as some magical force, Ein- stein describes gravity as a curvature in space and time (called the space-time continuum). a) The general theory is based upon the principle of equiv- alence. You cannot distinguish between acceleration in a gravitational field and accelerating due to a mechanical force (i.e., F = ma) in a gravity-free environment. b) Space and time are interrelated — you cannot have one without the other! c) The Universe can be considered to consist of a fabric of space-time (i.e., the vacuum) with pockets of matter that exist within this fabric, where this matter bends the fabric of space-time. The more mass density, the greater the bending or warping of space-time (see Figure II-9). 2. As a stellar core collapses, it gets denser and denser, and the escape velocity from the surface of the star goes increases. First, let’s look at the conservation of energy from Newtonian mechan- ics for a body of mass m moving in the gravitational field from rest (v = 0) at the surface (h = 0) to a height h above the surface (marked with a ‘◦’ subscript) of a much larger body of mass M and radius R: E = (KE + PE)◦ = (KE + PE)h = constant, (II-26) where E is the total energy, KE = 1 2 mv2 , (II-27) is the kinetic energy, and PE = −GMm R , (II-28) II–53 Sun White Dwarf Neutron Star Curvature of Space- Time Diagrams Figure II–9: The curvature of space-time gets greater and greater as a stellar core gets denser and denser as it collapses. II–54 is the gravitational potential energy. a) The conservation of energy becomes mv2 2 − GMm R = − GMm (R + h) . (II-29) b) We can solve this for h such that h = (v2/2g)   R R − (v2/2g)   , (II-30) where g = GM/R2 is the surface gravity of mass M . c) One can see from this that as v2/2g → R, h → ∞ — the mass m escapes the gravitational field of M . When this occurs, the velocity required to send h → ∞ is called the escape velocity: vesc = √√√√2GM R . (II-31) 3. The escape velocity for various objects: a) At the Earth’s surface: R = R⊕, M = M⊕, so vesc = 11 km/s. b) At the Sun’s photosphere: R = R , M = M , so vesc = 620 km/s. c) At a WD surface: R = R⊕, M = M , so vesc = 6500 km/s = 0.02 c. d) At a NS surface: R = 30 km, M = 2M , so vesc = 230, 000 km/s = 0.77 c. 4. A black hole (BH) is defined when an object reaches a size such that its escape velocity equals the speed of light. The collapse II–55 Our Space-Time Event Horizon SingularityEinstein-Rosen Bridge Distant part of Space-Time Hyperspace Figure II–10: The space-time continuum in the vicinity of a black hole. d) A standard ruler of length dr∞ measured at point r∞ would appear to be of length dr = dr∞ ( 1 − 2GM rc2 )1/2 = dr∞ ( 1 − rs r )1/2 (II-36) at a position r in the vicinity of a BH. e) A result of these two equations is that the position of a particle released at position r◦ in a BH’s gravitational field follows r ' rs + r◦ e−ct∞/2rs (II-37) as seen by a distant observer =⇒ the particle never reaches the event horizon as view by an external observer! 9. Although we won’t describe the details of the curvature of space- time here, we can present a general overview. Instead of our II–58 3-dimensional universe, assume that we live in a 2-dimensional universe. Then according to the general theory, mass bends this 2-dimensional universe into a 3rd dimension. BH’s bend space- time to such an extent that the BH actually rips a hole in the fabric of the Universe and reconnects with a distance part of the Universe or connects to a parallel universe in a different space- time continuum (see Figure II-10). a) A connecting tunnel forms in space-time called an Einstein- Rosen bridge and also called a wormhole. b) Our 3-dimensional Universe bends into a 4th dimension, called hyperspace. c) Tidal forces increase without bound as you approach the singularity =⇒ anything with size, even nuclear particles, get ripped apart from these tidal forces as matter falls into the wormhole. 10. The only way to detect a BH is to observe the effect one has on a companion star in a binary star system (see Figure II-11). As material is stripped away from a normal star by the BH, it spirals down to the BH and heats up to very high temps (due to friction). Before crossing the event horizon, the gas heats to such a high temp that it emits X-rays. Black candidates must satisfy the following observational requirements: a) An X-ray source in a binary star system whose X-rays vary in brightness over the time period of seconds (hence emitting region must be very small in size). b) The unseen companion of the spectroscopic binary has a mass greater than 3 M . II–59 X-Ray Binary Black Hole Candidate X-Ray Photons Normal Star Accretion Disk Black Hole X-Ray Emitting Region Figure II–11: Model for observing a black hole. c) The best candidates are the ones where the mass of the unseen companion is greater than the mass of the visible star. 11. The best of the stellar BH candidates are: a) Cyg X-1 (optical component: 30 M , X-ray component: 7 M ). b) LMC X-3 (optical component: 5 M , X-ray component: 10 M ). c) V404 Cyg (optical component: 1 M , X-ray component: 12 M ) — this is the best candidate. d) SS 433 (BH with jets?). II–60