10. Stellar Populations and Chemical Evolution
How individual stars evolve with time?
Most stars, during most of their evolution, can be considered as: spherically symmetric with constant mass and in hydrostatic equilibrium.
Under these conditions, the evolution of a star is almost completely determined by its mass and chemical composition through a set of ordinary differential equations that describe the structure of the star. Spherically Symetric?
(i) stars are in hydrostatic state (ii) stars are spherically symmetric.
Structure of a star described by a set of equations in which all quantities depend only on the distance to the center.
Of course stars are not completely spherical; as a gaseous body a star can be flattened by rotation. Consider an element of mass m near the equator of a star.
The centrifugal force on the mass element due to rotation is mω2R, where ω is the angular velocity of the rotation and R is the equatorial radius of the star. This force will not cause significant departure from spherical symmetry if it is much smaller than the gravitational force, GMm/R2, if ω2R3/GM ≪ 1.
For the Sun, ω ≈ 2.5 × 10−6 s−1, ω2R3/GM ≈ 2 × 10−16, and so flattening due to rotation is completely negligible. In general, normal stars can be considered to be spherical to very high accuracy. Hydrostatic Equilibrium?
To check hydrostatic equilibrium, consider the acceleration of a mass element at a radius r in a spherical star:
partial derivative of P is used because the pressure is a function of both radius r and time t. Two time scales based on the ratio between the radius of the star, R, and the acceleration |r ̈|.
−1/2 1) gravitational dynamical time scale, tdyn = (Gρ) , which corresponds to setting the pressure gradient to zero.
2) hydrodynamical time scale, thydro = R/cs (with cs the sound speed of the stellar gas), which corresponds to setting the gravitational term to zero.
−3 10 For the Sun with ρ ∼ 1 g cm , R ∼ 7 × 10 cm and T ∼ 7000K, these two time scales are tdyn ∼ 50min and thydro ∼ 1day, respectively.
These two time scales characterize how fast the gravity and sound waves can adjust a perturbed stellar structure to a new equilibrium configuration - should be compared to any perturbator. Hydrostatic Equilibrium?
Thermal time scale of a star, ratio between
- thermal energy of a star, Eth
- energy loss rate at its surface, L: tth = Eth/L. Gives the rate at which a star changes its structure by radiating its thermal energy.
7 For the Sun, tth ∼ 10 yr. Since this is much longer than both tdyn and thydro, the structure of a normal star can adjust quickly to a new configuration as it radiates.
Main energy source in a normal star is nuclear reactions and, for a star in equilibrium, the energy generation rate is equal to its luminosity. Nuclear energy generation time-scale, ratio between - energy resource in a star
2 −3 - current luminosity, L: tnuc ∼ ηMc /L, η ∼ 10 is the efficiency of nuclear reactions converting rest-mass to radiation For the Sun, this time scale is too long, about 1010 yr.
Thus, for a normal star like the Sun, tnuc ≫ tth ≫ tdyn. Implies that the evolution of a star is determined by nuclear reactions, with thermal and mechanical equilibrium achieved during the evolution. Exceptions are variable stars and stars in their late evolutionary stages, where the luminosity of a star can change significantly over a time scale comparable to the dynamical time scale. 10.1.1 Basic Equations of Stellar Structure
Under the assumption of spherical symmetry and hydrostatic equilibrium, the equation governing the structure of a star is the hydrostatic equation,
Two differential equations, three quantities, P, M and ρ. Another independent relation among these quantities is needed. The equation of state of the stellar material can provide such a relation. In general, the equation of state relates the local pressure P to the local density ρ, temperature T, and mass fractions {Xi} of elements i = 1,2,···,n, and can formally be written as
where the exact form of P(ρ,T,{Xi}) has yet to be specified. Temperature
Two new quantities, T and {Xi}, we need two new equations to complete the description. To get an equation for the temperature, we use the fact that a temperature gradient in a star causes energy transport. Three ways for the energy transport: (i) heat conduction, transfers energy as electrons from hotter regions collide and exchange energy with electrons from cooler regions; (ii) radiation, transfers energy as photons propagate from hotter regions to cooler regions; (iii) convection, transfers energy as material is convected between hot and cold regions.
Consider radiative stars where photon diffusion dominates. The energy flux F, in terms of the temperature gradient and an energy transport coefficient λ is: Temperature
In practice, we work in terms of a quantity called opacity, which is related to λ by
2 −1 where ar is the radiation constant, c is the speed of light, and [κ] = cm g .
In terms of κ, the temperature gradient can be written as
where L ≡ 4πr2F is the luminosity ([L] = ergs−1) at radius r.
To complete this equation, the form of κ has to be specified (then we have the two additional needed). In general κ is a function of ρ, T and {Xi}, and at the moment we denote this function as Luminosity
Now we need an equation for L as well. Suppose that the energy release rate at r is ε ([ε] = ergg−1), then
To complete this equation we need to know ε as a function of ρ, T and {Xi}. At the moment we assume that this function can be obtained and write it as
Basic Equations of Stellar Structure
Eqs. (10.2), (10.3), (10.7), and (10.9), are the four structure equations for the five quantities ρ, M, P, T , L. Together with the equation of state, they provide a complete description of a radiative star with given composition {Xi}, provided that P,
κ and ε are known functions of (ρ,T,{Xi}). Boundary Conditions: Luminosity
The boundary conditions for the structure of a star can be set as follows:
where rs is the radius of the star.
Since by definition the luminosity of a star is related to its radius and effective temperature by
where σSB = arc/4 is the Stefan-Boltzmann constant, we can choose Ts = Teff so that L(r) = L at r = rs. Boundary Conditions: Pressure
The value of Ps can be set by the condition of hydrostatic equilibrium for the stellar atmosphere:
where g is the gravitational acceleration. Since mass in the atmosphere << mass within it, pressure on the stellar surface (bottom of atmosphere) can be written as:
We relate this pressure to the optical depth of the atmosphere,
Boundary Conditions: Pressure
In general, ρ(r) falls rapidly as r increases and the main contribution to P(rs) and τ(rs) comes from layers near rs.
We can therefore write P(rs) and τ(rs) in the following approximate forms: Boundary Conditions: Temperature
Under the ‘gray-atmosphere’ approximation, i.e. the opacity is independent of photon frequency, the temperature is related to the optical depth as
and so τ (rs ) = 2/3 [because T (rs ) = Teff ].
With all these relations, we can finally write the boundary conditions on the surface of a star as
Since T and P on the surface of a star << stellar interior, the outer boundary conditions may be replaced by the so called ‘zero boundary conditions’:
For stars whose outermost layers are in radiative equilibrium, these zero boundary conditions provide a good approximation to the actual boundary conditions. Boundary Conditions: Temperature
Theoretically, the Mass of a star is a constant, while the radius is determined only after the model calculation. It is therefore more convenient to write the structure equations using M(r) (instead of r) as the independent variable, and set the boundary conditions at Ms. Normally the relation between r and M(r) is almost always one-to-one. In terms of M(r), the four stellar structure equations, for a star of mass Ms, can be written as 10.1.2 Stellar Structure Evolution
Stars do evolve with time. As nuclear reactions proceed in the centre and energy is radiated away from its surface, both the structure and chemical composition of the star can change with time. Need dynamical equations in order to describe such evolution? NO
Nuclear reactions drive the evolution of a star. Since in a normal star tdyn ≪ tth ≪ tnuc, any deviation from dynamical and thermal equilibrium caused by nuclear reactions can be adjusted very quickly: At any time the star can be considered to be in an equilibrium state governed by the static structure equations. The evolution comes in only because the star must adjust its structure and thermal state to a new chemical composition determined by the on-going nuclear reactions.
If there is no bulk motion of material in a star (i.e., convection is negligible), the change of chemical composition is localized and is given by the chemical evolution equation,
where the form on the right-hand side takes into account the fact that the rates of nuclear reactions in general depend on ρ, T and chemical composition. It is this change in chemical composition that drives stellar evolution.
Solving the evolution of a star: start with initial assumption for composition as a function of r or M(r) at some time t0, solve the structure equations under the boundary conditions. With the newly obtained ρ, T and the old {Xi} calculate, for each radius in the star, the new composition at a slightly later time t = t0 + δ t :
Solve again 10.1.3 Equation of State, Opacity, and Energy
We must specify the equation of state, P(ρ,T,{Xi}),
the opacity, κ(ρ,T,{Xi}),
and the energy production rate, ε(ρ,T,{Xi}). Equation of State
The pressure in a star consists of two components, gas pressure Pgas and radiation pressure Prad. For a blackbody, the radiation pressure is
Since the separation of gas particles is much larger than the size of atoms, stellar material can be considered as an ideal gas, which allows us to write the gas pressure as:
where n is the number density of particles (i.e. atoms, ions and electrons). In terms of the mean molecular weight,
where mp is the mass of a proton, Eq. (10.29) can be written in the form Equation of State
To express Pgas in the form of P(ρ,T,{Xi}), we need µ in terms of ρ, T and chemical composition. Particle number density in a gas depends on the ionization states of different elements (at given T and rho). For stellar material, T so high that all elements are highly ionized - simple approximations to µ can be made.
Since a fully ionized atom of charge number Qi consists of (Qi + 1) particles, one nucleus and Qi electrons, the number density contributed by such an element is Xi(Qi +1)ρ/(Mimp), where Xi is the mass abundance of element i and Mi is its mass number in units of mp. The total number density can therefore be written as
where X ≡ XH and Y ≡ XHe. The approximation uses the facts that Mi ≈ 2(Qi + 1) for elements heavier than helium and that, by definition, ∑ Xi = 1. Since heavier elements are far less abundant than hydrogen and helium this approximation is very accurate, and allows us to write Opacity
Photons emitted in the stellar interior can be absorbed and scattered before they reach the surface. The opacity of stellar material, κ, in Eq. (10.7) is a measure of the resistance of the material to the passage of photons. Need to know how photons interact with stellar material (ionized gas). The main interactions are:
- Compton or Thomson scattering: Photons can be scattered by ions or free electrons, which changes the directions of photons and slows down the net rate of energy transport.
- Free-free absorption: A photon can be absorbed by a free electron, giving its total energy hPν to the electron.
- Bound-bound absorption: A photon can be absorbed by an atom or an ion, moving an electron from one bound orbit to another bound orbit of higher energy (responsible for the observed spectral absorption lines of stars, but does not contribute much to the opacity in the deep interior of stars where the atoms are highly ionized so that the number of bound states is small).
- Bound-free absorption: A photon can be absorbed by an atom or an ion, removing one electron from its bound orbit (also rare).
Opacity
For given density, temperature and chemical composition, opacity can be different for photons with different frequencies. Opacity should be an average over frequency ν.
2 4 To find such an average, we start by writing Eq. (10.7) as L = (4πr c/3ρκ)(dB/dr), where B = arT is the energy density of radiation. In general, we can write the luminosity within a unit frequency interval near ν as
where Bν is the Planck function. Since L = Lν dν, B(T) = Bν(T)dν and since dBν/dr = (dBν /dT )(dT /dr) (with dT /dr independent of ν), integrating both sides of Eq. (10.36) over ν leads to
is the Rosseland mean opacity, which can be calculated for given T , ρ and chemical composition. Opacity
In practice, stellar opacity calculation must take into account all important atoms and ions. For a given chemical composition, the effective opacity is a function of temperature and density, and it is found that κ is in general low for both very high and very low temperatures: When the temperature is very high, most photons have very high energy and are not absorbed effectively, because the absorption cross sections generally decrease with photon energy. At very low temperatures, many atoms are neutral and only few electrons are available to scatter photons or to take part in the free-free absorption.
The effective opacity can be approximated by power laws in specific ranges of temperature and density:
where κ0 depends on the chemical composition, κ0, α and β depend on the ranges of T and ρ in question. Opacity
At high temperatures, where the main contribution to the opacity is the scattering of photons by free electrons, the opacity is given by:
where σT is the Thomson cross section.
At somewhat lower temperatures, where light elements like hydrogen and helium are fully ionized while heavier elements are partially ionized, the main sources of opacity are free-free (ff) and bound-free (bf) processes. In this case the opacity can be approximated by:
Here the factor (1 + X ) comes from the dependence of the opacity on the electron density given by Eq. (10.35); the factor 2 (X + Y ) comes from the fact that the main contribution to κff ∝ ∑ Q i ni is from H and He; and the factor (1 − X − Y ) comes from the assumption that H and He are fully ionized so that only heavier elements have bound electrons to produce bound-free absorption.
At even lower temperatures (T <~ 104 K), hydrogen becomes partially ionized and the number density of free electrons is a very rapidly increasing function of T in this case. Since both the free-free and bound-free absorption rates increase with ne, the opacity in this temperature range increases very rapidly with temperature: The Effect of Convection
One of the most uncertain aspects in stellar evolution is the treatment of convection. Convection occurs when the specific entropy of the stellar gas increases in the direction of gravity, i.e. when the temperature gradient implied by a constant specific entropy is smaller than that required for photons to carry the heat flux(at given r). material must move for the heat flux to be carried. In this case, blobs of hot gas in the stellar interior move upwards (i.e. away from the center of the star) and are replaced by blobs of cooler gas that move downwards. Very efficient process for transporting energy and material - greatly affects the evolution of a star.
Very difficult to describe rigorously. A standard phenomenology is the ‘mixing-length’ theory, which assumes that a convecting blob typically travels one mixing length lmix before dissolving into the ambient medium.
Because of convection, metals that are produced near the bottom of the convective zone can be effectively transported to the top. In practice a convective zone cannot have a sharp boundary, and so we expect that some metals can be overshot into the region above the top of the convective zone. Such convective overshoot can affect stellar evolution.
Overshooting and mixing length are calibrated so that model calculations best match the observed properties of the Sun and other well-observed stars (unclear how accurate for other stars). Modelling shows that main-sequence stars contain only small convective zones - structures not affected by uncertainties related to convection. Stars in late evolutionary stages can develop large convective zones and model predictions are still very uncertain Stellar Nucleosynthesis and Energy Production
33 −1 Our Sun is currently emitting with a luminosity L⊙≈3.9×10 ergs . Based on radio active dating of rocks on Earth, it can be inferred that the Sun has been shining with such a luminosity for at least about 5 × 109 yr.
The total energy that has been released by the Sun is therefore about 6 × 1050 erg, or about a fraction of 3 × 10−4 of its present total mass. An obvious question is, what is the basic energy source for the Sun?
2 2 7 The gravitational energy, GM⊙ /R⊙, can sustain the Sun at its present luminosity for a period of GM⊙ /R⊙L⊙ ∼ 3 × 10 yr, much too short compared to the age of 5 × 109 yr. (Before dating of rocks on Earth that was believed to be the age of the Sun.) The total thermal energy, which is about half of the gravitational energy, according to the virial theorem, is also far too small to supply the energy. Chemical reactions associated with the burning of material can transform a fraction of ∼10−10 of the rest mass into heat. Much smaller than 3 × 10−4 (also not enough)
The only energy source that can convert rest mass into energy with an efficiency larger than 3 × 10−4 is nuclear reactions. Today we are quite sure that normal stars like the Sun are all powered by nuclear fusion, a process by which light elements synthesize to form heavier ones. Simplest is the fusion of 4 hydrogen nuclei to form a helium nucleus, 4H → He.
−3 Rest mass of He is 3.97mp. Mass converted into energy is therefore 4mp − 3.97mp = 0.03mp, or about 7 × 10 of the original mass, 4mp.
Large enough to power the Sun for 8 × 1010 yr. In fact structure and luminosity of the Sun change drastically once it uses 13% of its total hydrogen. So fusion of H to He can sustain the Sun for about 9 × 109 yr (twice its age) Stellar Nucleosynthesis and Energy Production
Nuclear fusion: energy + main process for synthesizing the heavy elements observed in the Universe. Cosmological primordial nucleosynthesis is effective only in producing light elements such as He and Li. All heavier elements are believed to have been synthesized in the interiors of stars. The binding energy per nucleon in a nucleus with Q protons and N neutrons is defined as
When 4 hydrogen nuclei fuse into a helium nucleus, the binding energy per nucleon increases. From the point of view of thermodynamics, it is therefore more advantageous for hydrogen to fuse into helium than to remain as free protons. Stellar Nucleosynthesis and Energy Production
Binding energy per nucleon becomes larger for heavier elements. Reversed for elements heavier than 56Fe. Balance between the net attraction of the strong nuclear force and the repulsion of the positively charged protons. For small nuclei, the short-range nuclear force wins over the long range electric force, causing the binding energy per nucleon to increase when a nucleon (proton or neutron) is added to the nucleus. However, since the nuclear density is nearly constant, heavier nuclei are also physically bigger. The increase in the binding energy per nucleon with the number of nucleons saturates with iron-56; further addition of nucleons actually reduces the binding energy per nucleon. This is the reason why very heavy nuclei, such as uranium, are unstable. Stellar Nucleosynthesis and Energy Production
If it is more advantageous for nucleons to be synthesized into heavy elements why is most of the cosmic material still in the form of light elements such as H and He?
Nuclear fusion reactions can occur only under certain conditions. Since atomic nuclei are positively charged, the two (or more) nuclei in a fusion reaction tend to repel each other due to the Coulomb force.
In order for the components to come close enough so that the attractive nuclear force becomes dominant, they have to move rapidly towards each other to overcome the Coulomb barrier.
Nucleosynthesis is efficient only where gas density and temperature are both high so that many high-velocity collisions among nuclei can occur. Stellar interiors are such places.
Since the nuclear reactions involved are driven by thermal motions of nuclei, they belong to the category of thermonuclear reactions. Stellar Nucleosynthesis and Energy Production
How thermonuclear reactions depend on temperature?
Consider the collision of two nuclei of charges qi and qj, and masses mi and mj, with a relative speed v at large separation.
2 From classical mechanics, closest separation is r = 2qiqj/mv , where m = mimj/(mi +mj) is the reduced mass.
However, the two particles can penetrate the Coulomb barrier due to quantum tunneling.
2 The probability of this penetration is Pr ∝ exp(−2π r/λ), where λ = hP/mv is the de Broglie wavelength.
For a classical gas with temperature T , the relative velocity v of particles obeys the Maxwell-Boltzmann distribution, Pv ∝ 2 exp(−mv /2kBT). The probability for a fusion to take place is therefore proportional to Pr Pv.
2 1/3 For a given T, this probability is maximized at v = (4π qiqjkBT/mhP) , with a maximum value
3 2 2 where T0 ≡ (3/2) (4π qiqj/hP) (m/kB). Thus, in order for the fusion reaction to proceed at a significant rate, the temperature T must be > T0. T is larger for heavier nuclei, because of the larger values of masses and charges. As a star evolves and its internal temperature rises, light elements will be converted successively into heavier elements, until all the material in the star is converted into elements in the neighborhood of iron in the periodic table. However, this process is slow, because the temperature in a stellar interior does not change much during most of its lifetime. This is the reason why most baryonic material in the Universe is still in the form of H and He. pp chain and CNO cycle
There are two important channels for converting hydrogen into helium. The first is through the pp chain and the second is through the CNO cycle. The pp chain starts with two hydrogen nuclei (i.e. two protons) to form a deuteron (D) which then captures another proton to form a 3 He:
The net effect of the CNO-cycle is to convert four protons into a helium nucleus, with C, N, and O nuclei acting only as catalyst. Note that the completion of the CNO-cycle requires the pre-existence of carbon. pp chain and CNO cycle
The rate with which the pp-chain and the CNO-cycle convert H into He is proportional to the square of the density (because the reactions involved are two-body in nature) and increases rapidly with temperature. The CNO-cycle is more sensitive to temperature because the elements involved are heavier. Rate of energy production (from each reaction) = reaction rate x energy released from each conversion (function of T) The specific energy production rate, ε , is in general a smooth function of T , and can be represented as a power law of T :
Since thermonuclear reactions need high temperature to proceed, there must be systems in which the temperature is never high enough to ignite H-burning. Such stars are called black dwarfs, because they only radiate very little.
The minimum temperature corresponds to a minimum mass, MH ≈ 0.08M⊙.
Might still burn deuterium (which has a lower ignition temperature) - brown dwarf. Helium Burning - triple-alpha
For systems with masses larger than MH, the conversion of H to He first occurs in the central region where the temperature is the highest. 1) There will then be a time when most hydrogen in the central region is exhausted. 2) Star contracts with no thermonuclear reaction to provide pressure support and thermal energy flows out. 3) According to the virial theorem, such contraction leads to increase in the T in the centre of the star. 4) Contraction continues until the next important nuclear reaction takes place to generate sufficient thermal energy to halt the contraction.
As can be inferred from Eq. (10.44), when the temperature reaches ∼108 K, fusion of helium begins to synthesize the next stable nuclei, 12C, through the following reactions:
Called the triple-α process because three α particles (4He) are involved.
Since 8Be is unstable and can decay back to two 4He in a short time(after the end of first reaction), only small amounts of 12C would be produced if the second reaction occurred with a ‘normal’ rate. This is known as the ‘bottleneck’ for the synthesis of heavy elements. A breakthrough of this ‘bottleneck’-problem occurred when it was realized that the second reaction can proceed through a fast (resonant) channel with the production of an excited state 12C∗, which then de-excites to the ground level 12C (Hoyle, 1954). Once 12C is produced, the synthesis of heavy elements can proceed further. Heavier Elements
Two groups of elements are particularly important.
One group is called the α-elements, which includes 16O, 20Ne, 24Mg, 28Si, 32S, 36A, 40Ca (all the elements can be formed by adding α-particles to 12C) The synthesis of these elements is either through the capture of α-particles in reactions such as:
Can be synthesized in a star that starts out with pure hydrogen and helium - abundances are independent of the initial metallicity.
The other group, called the iron-peak elements, includes all elements with atomic numbers in the range 40 < A < 65, i.e. Sc, Ti, V, Cr, Mn, Fe, Co, Ni and Cu. High nuclear charges - form late in a star, when its core becomes extremely hot.
Elements that can be synthesized directly in stars of zero initial metallicity are referred to as primary elements, while elements that can form only in the presence of primary elements are called secondary elements. Main Sequence Lifetimes
4 The Luminosity of a main sequence (hydrogen burning) star scales with mass roughly as Ls ∝ Ms . Massive stars leave less because they are much more luminous. Since a star converts roughly a fixed fraction of its rest mass into radiation before it leaves the main sequence, the main sequence lifetime scales with mass as:
For a star like the Sun, the main sequence lifetime is ∼ 1010 yr, comparable to the age of the Universe.
7 A star with a mass of about 10 M⊙ has a main sequence lifetime of ∼ 10 yr, much shorter than the age of the Universe.
Main sequence lifetime is shorter for stars with lower metallicity, because Ls increases with decreasing Z.
In the mass range between 0.08M⊙ to 100M⊙, and for a metallicity not very different from Z⊙, the main-sequence lifetime can conveniently be approximated by the following formula:
where m is the mass of the star in units of M⊙ Main Sequence Lifetimes
Stars of different mass - different main-sequence lifetimes Massive stars have main sequence lifetimes much shorter than the age of the Universe Very important for galaxy evolution! When we observe a galaxy today (i.e. at redshift z = 0), we are observing the light from the stars that have evolved to the present time.
9 We can see all main sequence stars with Ms ∼ M⊙ formed during the past 9 × 10 yr, but for main sequence stars with Ms ∼ 7 10 M⊙ only those formed during the past 10 yr.
Consequently, the stellar population observed from a galaxy depends strongly on its star formation history. It is possible to learn about the star formation history of a galaxy by studying its stellar population.
Massive stars in their post main sequence evolution can eject material and energy into the interstellar medium either as stellar winds or through supernova explosions.
The fact that massive stars have relatively short lifetimes implies that the chemical composition and thermal state of the gas in a galaxy can be affected almost simultaneously as star formation proceeds. 10.2 Stellar Evolutionary Tracks
The evolution of a star is almost completely determined by its initial mass and chemical composition.
For given initial mass and metallicity, a stellar evolution model should yield all the properties of a star at any time, t, after its birth.
The two most important properties of a star are its luminosity, L, and its effective temperature, Teff, and so the evolution of a star can be represented conveniently by its evolutionary track in the Teff − L plane.
Since Teff and L are related to the color and absolute-magnitude of a star, such evolutionary tracks provide the basis for the understanding of the observed H-R diagrams. Prominent branches heavily populated by stars can be considered as stellar ‘traffic jams’ where stars evolve slowly and many tracks run close to each other. 10.2.1 Pre-Main-Sequence Evolution
A young star is born at the late stage of protostellar collapse, where the star is in a state of quasi-equilibrium and optically visible due to deuterium burning. The star then quickly evolves onto the main sequence. The evolution between the birth and the zero-age main sequence is usually referred to as the pre-main-sequence evolution.
For low-mass stars 0.08 M⊙< Ms < 0.3 M⊙, the pre-main-sequence evolution is characterized by almost vertical tracks. Low surface temperatures - almost entirely convective when they begin to burn deuterium.
More massive stars, become radiative before reaching the main sequence. The late stage of pre-main-sequence is characterized by contraction with nearly constant luminosity.
Stars with masses smaller than ∼ 0.08 M⊙ cannot ignite hydrogen burning in their centers, and so they shine only during the deuterium-burning phase as brown dwarfs. As a brown dwarf runs out of deuterium fuel, it shrinks until it is sup- ported by the degeneracy pressure of electrons, eventually becoming a black dwarf.
Energy released during the pre-main-sequence is from gravitational contraction and deuterium-burning, much smaller than during the main-sequence and post-main-sequence evolution (negligible for galaxy formation). 10.2.1 Pre-Main-Sequence Evolution 10.2.2 Post-Main-Sequence Evolution
Once a star reaches its zero-age main sequence, it will stay on the main-sequence until the hydrogen in its core is burned into helium.
During its main-sequence lifetime the star moves little in the Teff − L plane, because the luminosity of a main-sequence star is limited by the photon-diffusion rate, which depends primarily on the initial mass.
2 The amount of H converted into He at the end of the main-sequence lifetime is LtMS/0.007c , which is about 13% of the total hydrogen mass for a star like the Sun.
The post-main-sequence evolutionary tracks are qualitatively different for low- and high-mass stars. 10.2.2 Post-Main-Sequence Evolution Low Mass Stars
Even after hydrogen has been exhausted in the core of a star, energy transport will continue in its interior. 1) Since the core is still too cold to burn He, there is no thermal energy to counterbalance the energy leak, and so the core contracts. 2) As it contracts, the core is heated up (according to the virial theorem), and so is the gas layer above the He core. 3) Since hydrogen has a lower ignition temperature than helium, the hydrogen in the shell just above the He core will burn first as the temperature increases, while the He core itself still remains dormant. The H shell-burning is quite effective in generating energy. But as long as the star is not convective, the amount of energy that can leak from the star is limited by photon diffusion and is roughly a constant (generates extra heat causing the star to expand). Luminosity almost constant and a slight increase in T. 4) As the temperature in the outer envelope drops (due to expansion), the temperature gradient between the outer and inner regions increases, and the star eventually becomes fully convective. When this happens the effective temperature cannot decrease any further and the star start to ascend almost vertically in the Teff − L plane to the red giant branch (RGB) - Luminosity increases because energy can leak. At this stage, the envelope of the star has been distended so much that substantial mass-loss (stellar wind driven by the radiation pressure) may occur. 5) The He core becomes hotter and hotter as it contracts and as more He ash is added to it from the H-burning shell. When the core temperature reaches ∼ 108 K, helium begins to burn into carbon through the triple-α process; s 6) The He-burning heats the core and causes it to expand, lowering the gravity on the H-burning shell and thereby reducing the strength of shell H-burning. Consequently, the star reaches the tip of the red-giant branch and then settles onto the horizontal branch (HB) where the star is mainly powered by He burning in its core. The He core behaves like a He main-sequence star. Low Mass Stars Low Mass Stars
7) The core He burning will come to a halt once most of the helium nuclei in the core are synthesized into carbon and oxygen. At this stage, the star contains a C/O core, an inner He- burning shell and an outer H-burning shell. 8) As the core temperature is still too low for carbon to burn, the C core contracts. The He-burning shell outside the C core and the H-burning shell outside the He-burning shell also contract in this phase because of the loss of pressure support. Such contraction enhances the double-shell burning, causing the star to ascend the Asymptotic Giant Branch (AGB) in a way similar to what happened during the RGB phase. During the AGB phase, the envelope of the star can be greatly distended, and a large amount of mass loss can occur. When the star reaches the top of the AGB, where it becomes a red supergiant, it can lose all of its blanketing hydrogen layer. 9) The star then makes a sudden move leftwards in the H-R diagram as it peels away its relatively cold mantel. At this stage, the intense ionizing radiation from the star may cause the ejected H envelope to shine as a fluorescent planetary nebula.
10) The He burning will gradually die off, leaving behind a C/O core with a mass in the range 0.55 − 0.6 M⊙ (independent of the initial mass). Because of the small mass, further nuclear burning cannot be ignited in the core. The star will then contract until it is supported by the degeneracy pressure of electrons, becoming a white dwarf.
Valid for all stars with initial masses Ms < 8 M⊙. Massive Stars
For stars with initial masses Ms >8M⊙, core-H exhaustion, He-core contraction, shell-H burning, and core-He ignition proceed in much the same way as for low-mass stars. 1) However, because of the high mass, core-ignition and shell-burning can proceed successively to heavier elements.
During this process, a star in general moves to a higher Teff after a core ignition, and to a lower Teff during core contraction. Since a massive star can maintain radiative equilibrium during the evolution, its luminosity changes only little. As a result, the evolutionary tracks of such stars appear more-or- less horizontal in the H-R diagram. 2) At some stage, all the material in the core may be converted into iron-peak elements that cannot be ignited anymore. The burning in the core will then cease, while Si, O, Ne, C, He, and H continue to burn in shells at successively larger radii. When this happens, the core contracts until it becomes supported by the degeneracy pressure of electrons. 3) Since the shell-burning outside the core keeps dropping ash onto the core, the degenerate core will grow until it reaches
Chandrasekhar’s mass limit, M ≈ 1.4 M⊙, beyond which the iron-core can no longer be supported by the degeneracy pressure of electrons and must collapse further. This collapse can heat the core to such a high temperature (>109 K) that iron nuclei can be photo-disintegrated into α-particles (4He nuclei). 4) Since thermal energy is used in the photo-disintegration, the core will collapse further and be heated to an even higher temperature. The α-particles produced will then also be photo-disintegrated into protons and neutrons, draining more thermal energy from the core. 5) The core then undergoes a phase of catastrophic contraction, and its density can become so high that almost all electrons are squeezed into protons to form neutrons. If the mass is smaller than ∼ 3M⊙, the core will be a neutron star supported by the degeneracy pressure of neutrons; otherwise the core will collapse further to form a black hole. The collapse of the iron-core is so violent that a supernova explosion may take place. Radiation in the Giant Phase
As a star evolves off the MS, it becomes redder, and if its mass is < 2M⊙, it also becomes brighter.
For a star with an initial mass < 1.5M⊙, the total energy radiated in the giant phase is actually larger than that in the MS phase.
Since the MS lifetime of a 1.5M⊙ star is about 2.5 Gyr, the luminosity of a stellar population with an age longer than this is therefore expected to be dominated by giant-branch stars rather than by dwarfs on their MS.
This has important implications. For instance, for a stellar population older than a few Gyr, such as that in an elliptical galaxy, the spectrum is expected to resemble that of giant-branch stars. 10.2.3 Supernova Progenitors and Rates
At the end of their lifetimes some stars will explode as supernovae.
10 These are observed as objects that radiate very intensively (with a luminosity ∼ 10 L⊙) for a period of a few weeks.
The supernovae observed in the Universe are classified into Type Ia, Type Ib and Type II according to their spectral features. Type II supernovae are distinguished from the other two types by the existence of hydrogen lines in their spectra.
The distinction between Type Ia and Type Ib is that the former has significant Si+ absorption in its spectrum while the latter does not. Type II and Ib are rare in early-type galaxies, while Type Ia are observed in all types of galaxies. In spiral galaxies, Type II and Ib supernovae are preferentially found in spiral arms where young (OB) stars form, whereas Type Ia do not reveal such a preference.
These observational results suggest that the progenitors of Type II and Ib supernovae are probably short-lived massive stars, while the progenitors of Type Ia are long-lived low mass stars. Type Ia Supernovae
A star with initial mass Ms < ≈ 8 M⊙ evolves into a C/O white dwarf with a mass ∼ 0.6 M⊙.
If this white dwarf is part of a binary system with a red giant or another white dwarf, it can accrete material from the companion.
When the mass of the white dwarf reaches about 1.38M⊙, carbon is ignited in the center of the degenerate C/O core. This ignition produces a burning wave in the star, converting about half of the C and O into iron-peak elements in about one second, giving rise to a Type Ia supernova.
The light curve and spectrum of the supernova depend on how fast the burning wave propagates through the star. The best agreement with observations is achieved in the carbon-deflagration (C-deflagration) model in which the deflagration wave propagates with a speed slightly smaller than the sound speed.
56 In this model, about 0.6M⊙ of Ni is produced, which powers the light curve of the supernova by the radioactive decay of 56Ni to 56Co and then to 56Fe. Because of the energy release, the star is completely disrupted. Type Ia Supernovae
Consider a binary of mass MB = M1 +M2, with M2 ≤ M1. In order for the primary, the initially more massive one, to result in a Type Ia supernova we have that MB,min ≤ MB ≤ MB,max, with MB,min ∼ 3 M⊙ and MB,max ∼ 16 M⊙.
Upper limit: the mass of each component of the binary cannot exceed 8M⊙, the maximum mass for producing a C/O white dwarf. Lower limit: to ensure that the primary star is massive enough so that the C/O white dwarf can eventually reach Chandrasekhar’s mass by accretion from the companion.
The number of supernova explosions per unit mass per unit time depends on: formation rate of stars in the relevant mass range + binary fraction.
Binary fraction is quite uncertain - use empirical models.
Since the progenitors of Type Ia supernovae (C/O white dwarfs with a narrow mass range) have quite uniform 9 properties, the luminosities of Type Ia supernovae are found to be within a narrow range around LB = 9.6 × 10 L⊙. Because of this uniformity, Type Ia supernovae can be used as standard candles to measure cosmic distances. Type II Supernovae
Type II supernovae are assumed to originate from stars with an initial mass Ms > ≈ 8M⊙.
Such a massive star can develop an iron core during its late evolutionary stage that collapses catastrophically.
Whether the energy released by this implosion results in a Type II supernova depends critically on the exact fraction of this energy (of the order of only ∼ 1%) that is transferred to the stellar envelope.
For a Salpeter IMF, about one Type II supernova is produced for every 100 M⊙ of new stars formed. 10.4 Chemical Evolution of Galaxies
Stars may return substantial fractions of their initial masses to the ISM at the end of their lifetimes through stellar winds and supernova explosions. Since the returned material is enriched in metals, this process can change the metallicity of the interstellar gas, thereby affecting the properties of the ISM.
The mass ejection from a star of initial mass Ms (and an initial chemical composition) can be described by its lifetime (ej) τ(Ms) and by M (Ms), the total mass of element j ejected from the star, where the dependence of these quantities on the initial chemical composition is implicitly implied.
(ej) The values of τ(Ms) and M (Ms) can, in principle, be calculated from stellar evolution models.
However, this involves modeling stars during their late evolutionary stages and some of the results are still fairly uncertain at the present. Metal Production by Low-mass Stars
Stars with initial masses Ms < 8 M⊙ end up as C/O white dwarfs after their AGB phases.
These white-dwarf remnants have masses in the range 0.4 - 1.4M⊙ (with a typical mass ∼ 0.6M⊙), and so the progenitor stars return most of their initial mass to the ISM. A low-mass star can lose mass rapidly once it reaches the red-giant tip of the AGB, where mass loss is driven by stellar winds generated by radiation pressure from the star.
Since metals (C, O and α-elements) are synthesized by He- burning in the stellar core, one might think that these metals are locked up in the white dwarf so that low mass stars can only enrich the ISM with He. However, because of convection, metals produced below the He-burning shell can be brought up to the envelope by a process called dredge-up.
Observationally, the abundances of He, C, N, O in the ejected material from AGB stars can be estimated from the spectra of planetary nebulae in the Milky Way and the Magellanic Clouds.
One has to estimate the difference between the abundances of the ejecta and those of the ISM in the direct vicinity of the planetary nebula (whose abundances are presumably similar to those of the gas out of which the star originally formed). Metal Production by Massive Stars
Massive stars, with masses Ms > 8M⊙, enrich the ISM with metals via both stellar winds and their final explosions as core- collapse (Type II) SNe. Prior to core collapse a massive star consists of an inert iron core surrounded from inside out by shells within which Si, O, Ne, C, He, and H are burning. When a Type II SN explodes, these shells of enriched material may be ejected by shock waves. Exact amount of material that can be blown away is unclear, as it is very hard to determine accurately the small fraction (∼ 1%) of the explosive energy that is transferred to the stellar envelope.
Equally uncertain is the change of chemical composition during and after the core collapse.
Shocks generated by the explosion may induce a burst of nuclear reactions in the shocked layer; in particular, a substantial amount of Si (which is burning in a shell just above the core) may be burnt into iron-peak elements.
In the collapsing core itself, the temperature can become so high (> 109 K) that iron is photo-disintegrated into α particles (He nuclei) and neutrons. These neutrons may collide with the iron-peak elements in the core and convert many of them into r-process elements (ones that are formed via rapid neutron capture). As a result, only a small amount of Fe can be ejected into the ISM by a Type II SN. Metal Production by Type Ia SNe
The progenitor of a Type Ia SN is a C/O white dwarf, which accretes material from a close companion, reaches the Chan- drasekhar limiting mass for a C/O dwarf (1.4M⊙), and explodes. In the carbon-deflagration model, the deflagration wave can convert C and O very quickly into iron-peak elements in the inner region of a star.
56 In particular, about 0.6M⊙ of Ni is produced, which is sufficient to power the light curve of a Type I SN by the radioactive decays of 56Ni (and 56Co) into 56Fe. A substantial amount of α-elements is predicted to be synthesized in the outer layers by the decaying deflagration wave, which is consistent with the observed spectra of Type Ia SNe near the peaks of their light curves.
A Type Ia SN typically ejects about 1.38 M⊙ of material, of which ∼ 0.75 M⊙ is Fe, ∼ 0.05 M⊙ is C, ∼ 0.14 M⊙ is O, ∼
0.005M⊙ is Ne, ∼ 0.01M⊙ is Mg, ∼ 0.15M⊙ is Si, and ∼ 0.09M⊙ is S. Typically, a Type Ia SN produces about 5 to 10 times as much Fe as does a Type II SN.