Stellar Masses and the Main Sequence Measurements of main-sequence stars demonstrate that there is a η mass-luminosity relationship, i.e., L µ M . For M > 1 M8 η ~3.88, while at lower masses, the relation flattens out. A good rule-of- thumb is L µ Mη, with η ~ 3.5. Main Sequence Mass-Radius Relation
There is also a mass-radius relation for main-sequence stars. When parameterized via a power law, R µ Mξ, ξ ~ 0.57 for masses M > 1 M8, and ξ ~ 0.8 for M < 1 M8. Stellar Masses for White Dwarfs
The masses of white dwarf stars There is also an inverse mass- are all less than 1.4 M8. Most radius relation for white dwarfs. a are ~ 0.59 M8. The simple theory says M µ R , with a = -1/3. Numbers to Keep in Mind
• τ8 ~ 10 Gyr = Main sequence lifetime of the Sun 33 • 1 M8 ~ 2 × 10 gm = Mass of the Sun 33 • 1 L8 ~ 4 × 10 ergs/sec = Luminosity of the Sun 10 • 1 R8 ~ 7 × 10 cm = Radius of the Sun • Q ~ 6.3 × 1018 ergs/gm = Energy from hydrogen fusion • Δm ~ 27 MeV = mass defect for hydrogen fusion • Δm ~ 0.7% = percent mass defect for hydrogen fusion • X ~ 0.75 = fraction of hydrogen (by mass) in the Sun • Y ~ 0.23 = fraction of helium (by mass) in the Sun • Z ~ 0.02 = fraction of “metals” (by mass) in the Sun Stellar Timescales
Before starting to discuss how stars work, it is important to have an order-of-magnitude feel for stellar timescales. They are the shortcut to everything! • The Nuclear Timescale: How long will a star live? (In other words, how long will it take to use up its nuclear fuel?) Fuel Q M τ = = nuc Rate of Consumption L
For hydrogen fusion, Q = 6.3 × 1018 ergs/gm. (If the star is fusing helium, the coefficient is an order of magnitude smaller.) Notes: • €Main sequence stars will only consume ~ 0.1 of their available fuel before they must adjust their structure. 10 • You can scale to the Sun: τnuc = 10 years. Stellar Timescales
• The Thermal Timescale: How long does it take for a star to adjust its structure? (How long does it take to release its energy?) This is also called the Kelvin-Helmholtz timescale Gravitational Energy GM 2 τ = = KH Rate of Consumption R L
This describes how long a star can shine with no nuclear energy generation. Alternatively, it describes how long it takes energy produced by gravitational contraction to work its way out. For the Sun, this number is ~ 108 years. Stellar Timescales
• The Dynamical Timescale: How long does it take a star’s interior to feel changes at its surface? (For instance, if a star accretes mass, how long would it take the interior to feel the extra weight?) Alternatively, how long does it take an object to “fall” to the star’s core (under constant g)?
1/ 2 1/ 2 Size of the Star # R3 & # R& τdyn = = % ( = % ( Speed of Pressure Wave $ G M ' $ g '
Note: this is equivalent to the free-fall time, and it’s also equivalent rd to Kepler’s 3 law. For the Sun, τdyn ~ 30 minutes. € Stellar Timescales
• The Mass-Loss Timescale: How long does it take for a star to lose all its mass via its wind? Mass of Star M τ = = ML Mass Loss Rate M!
For the Sun today, this is ~ 1013 years.
Note: for most (but not all) stars, the nuclear, thermal, dynamical, and mass-loss timescales are very different. This makes stellar structure and evolution easy to model. The complications arise when two are more timescales become comparable. Stellar Structure The internal structure of most stars can be computed easily(?) by solving 4 simultaneous differential equations with 4 unknowns. dM(r) Mass Conservation: = 4π r2ρ(r) dr dP(r) GM(r) Momentum Conservation: = −gρ = − ρ(r) dr r2 dL(r) Energy Conservation: = 4π r2ρ(r)ε(ρ,T) dr dT(r) dP(r) dT dP T ⎛ d lnT ⎞ dP T Thermal Structure: = = ⎜ ⎟ = ∇ dr dr dP dr P ⎝ d ln P ⎠ dr P where P = P(ρ,µ,T)
ε(ρ,T) = εnuclear −εneutrino +εgravitational = energy generation d lnT 3κ(ρ,T)L P ∇ = = or ∇ d ln P 16π acG M T 4 ad κ(ρ,T) = opacity Stellar Structure The internal structure of most stars can be computed easily(?) by solving 4 simultaneous differential equations with 4 unknowns. dM(r) Mass Conservation: = 4π r2ρ(r) dr dP(r) GM(r) Momentum Conservation: = −gρ = − ρ(r) dr r2 dL(r) Energy Conservation: = 4π r2ρ(r)ε(ρ,T) dr dT(r) dP(r) dT dP T ⎛ d lnT ⎞ dP T Thermal Structure: = = ⎜ ⎟ = ∇ dr dr dP dr P ⎝ d ln P ⎠ dr P
Disclaimer: I wrote these equations for easy of understanding and used radius as the independent variable. Real calculations of stellar structure use mass as the independent variable. Digression: Mass Fraction and Number Fraction In some subfields of astronomy (such as stellar astronomy), people quote the mass fraction of objects (i.e., 75% of the mass of a star is hydrogen) and quote densities in units such as g/cm3. In other subfields (such as ISM studies) the fraction of atoms is used (i.e., 90% of the atoms in the universe is hydrogen), and density is quoted as atoms/cm3. The two values are related via
where ni is the number density of atoms of species i, xi is the species’ mass fraction, Ai is the species’ atomic mass, and ma is atomic mass unit (1.66 ×10-24 gm). Digression: Mean Molecular Weight Mean molecular weight (μ) is defined as mass per unit mole of material, or, alternatively, the mean mass of a particle in Atomic Mass Units. Thus, number density is related to the mass density, ρ, by ρ ρ N n = = avo µ ma µ Often, number density is split into two terms, one for ions $ ' −1 xi ρ Navo xi nI = ∑ni = ρ Navo∑ = where µI = & ∑ ) i € i Ai µI % i Ai ( And one for the (massless) electrons $ ' −1 xi ρ Navo Zi xi f i € ne = ρ Navo∑ f i Zi = where µe = & ∑ ) i Ai µe % i Ai (
where xi is the mass fraction of the species, Zi is its atomic number, Ai, its atomic mass, and fi, the fraction of electrons that are free. € Mean Molecular Weight -- Simplification In stars, the expressions for mean molecular weight can be greatly simplified. For example, the fraction of heavy elements in a star is small (Z ~ 0.02), so ⎛ ⎞−1 −1 −1 xi ⎛ X Y Z ⎞ ⎛ 1− X ⎞ 4 µI = ⎜∑ ⎟ = ⎜ + + ⎟ ≈ ⎜X + ⎟ = ⎝ i Ai ⎠ ⎝ 1 4 ~ 14⎠ ⎝ 4 ⎠ 1+ 3X Also, if we assume that the gas in the interior of a star is (almost)
entirely ionized, fi ~ 1. And, while hydrogen has Zi/Ai = 1, most other elements have Zi/Ai ~ ½. So
# & −1 −1 −1 Zi xi f i # 1 1 & # 1− X & 2 µe = % ∑ ( = % X + Y + Z( = % X + ( = $ i Ai ' $ 2 2 ' $ 2 ' 1+ X The combined mean molecular weight is therefore 1 1 1 € = + µ µI µe
€ Equation of State To solve the equations of stellar structure, one must also known the relationship between pressure, density, and temperature and mean molecular weight. This is called the equation of state. The pressure
comes from 3 sources, i.e., P = Prad + Pion + Pelectron. 1 • Radiation pressure: P = aT 4 rad 3 ρ • Ion pressure (ideal gas): Pion = k T µi ma • Electron pressure€ : In most main sequence stars, electrons act as an ideal gas, but in the cores of giants and very low-mass main- sequence stars, partial€ degeneracy can occur. One must then solve Fermi-Dirac integrals, i.e.,
5 / 2 3 / 2 ( 3 , ∞ 4π (2mekT) η * neh * P = dη where ψ = ln) - e 3h 3 m ∫0 1+ eη −ψ 3 / 2 e +* 2(2π mekT) .*
€ Electron Degeneracy At high densities, the Maxwellian distribution comes up against the 3 Pauli exclusion principle, dV dp = (Δx Δpx)(Δy Δpy)(Δz Δpz) ~ h . This forces electrons to high momentum states, increasing the pressure. Opacity Opacity is the opposite of transmittance. To get the appropriate (Rosseland) mean, you have to integrate over the blackbody function. Opacity is given as digital tables in temperature and density, but there are rough approximations: • Electron (Thomson) scattering: n σ κ = e e = 0.2(1+ X) cm2 g-1 e ρ • Free-free absorption (Kramer’s style opacity): Z 2 ~ 1023 T −7 / 2 cm2 g-1 κff € ρ € µeµI • Bound-free absorption (Kramer’s style opacity):
25 −7 / 2 2 -1 κbf ~ 10 (1+ X)ρT cm g • Bound€ -bound absorption (complex, but roughly Kramers) • H- opacity (H- + h n D H + e-): complicated, but below ~ 104 K € −31 1/2 9 2 -1 κ H − ~ 2.5 ×10 (Z /0.02) ρ T cm g
€ Opacity Massive supercomputer projects have calculated Rosseland mean opacities as a function of density and temperature.
€ Opacity Sources
€ Energy Transport: Radiation In theory, there are 3 ways to transport heat: § Conduction: Only important in white dwarfs and neutron stars § Convection: mixing § Radiation: diffusion of light due to absorption and re-emission. Radiation is nothing more than the random walk of energy, where the mean free path is lph = 1/κρ. The expression for energy transport of this type is fairly simple:
L 1 dU 1 " 3 dT % F = = − clph = − clph $4π T ' 4πr2 3 dr 3 # dr & This is usually re-written using thermodynamical relations to
dT GMρT 3κ L P = ∇ where ∇ = dr R2P rad rad 16π ac G M T 4
€ Energy Transport: Convection In general, convection is more efficient, if it exists. If a blob of material is displaced outward, does it continue to rise (taking its heat with it) or sink back to where it started? In other words, at location r + δr, is the blob denser or less dense than its new environment?
# dρ& # dρ& Therefore, for convection to occur % ( < % ( $ dr ' blob $ dr ' star After a bit of thermodynamic manipulation, this becomes ∂ lnT € = ∇ < ∇ ∂ lnP ad rad
Note: sad is a thermodynamic quantity intrinsic to the gas. For an γ ideal gas (P µ ρ , with γ=5/3), sad = 0.4. For pure radiation 4 pressure (P µ ⅓ aT€) , sad = 0.25. Nuclear Reaction Rates Most light-element nuclear reactions depend on density (squared) and temperature to some power. To compute that power, one has to integrate reaction rate cross-sections over the Maxwellian distribution. The cross sections are energy dependent, since they depend on momentum and tunneling.
2 & 1 ) 1 σ ∝ π λ ∝ ( 2 + ∝ ' p * E
' 2π Z Z e2 * ' KZ Z * σ ∝ exp( − 1 2 + ∝ exp( − 1 2 + E1/ 2 ) ν , ) , € 14.16 2 2 1/ 3 2 σv = 1/ 3 (Z1 Z2 A) − T6 3 €
€ Nuclear Reaction Rates
Most heavy-element nuclear reactions involve nuclear resonances, many of which are poorly known. This introduces uncertainty into some aspects of heavy element nucleosynthesis. Proton-Proton Chain The Sun produces energy by fusing 4 hydrogen atoms into one helium atom, thereby creating 26.73 MeV per helium nucleus. Most of this happens via the proton-proton chain. • PP I: 1 1 2 + • H + H g H + e + νe <εν> = 0.263 MeV • 2H + 1H g 3He + γ • 3He + 3He g 4He + 1H + 1H • PP II: • 3He + 4He g 7Be + g 7 - 7 • Be + e g Li + ne + g <εν> = 0.8 MeV • 7Li + 1H g 4He + 4He • PP III: • 7Be + 1H g 8B + g 8 8 + • B g Be + e + ne <εν> = 7.2 MeV • 8Be g 4He + 4He
CNO Bi-Cycle In higher mass stars, most hydrogen is fused via the CNO by-cycle. This chain fuses hydrogen to helium using CNO as a catalyst. • 12C + 1H g 13N + g 13 13 + • N g C + e + ne <εν> = 0.71 MeV • 13C + 1H g 14N + g • 14N + 1H g 15O + g 15 15 + • O g N + e + ne <εν> = 1.00 MeV • 15N + 1H g 12C + 4He or (once every ~ 2500 times) • 15N + 1H g 16O + g • 16O + 1H g 17F + g 17 17 + • F g O + e + ne <εν> = 0.94 MeV • 17O + 1H g 14N + 4He CNO Bi-Cycle Note: the net effect of the CNO cycle is 4 1H g 4He. In the process, the relative abundances of CNO get changed. In particular, since 14N has the smallest cross section, it tends to get built up, while C is lost. Nuclear Reaction Rates Since the proton-proton chain involves ions with the least amount of charge, it has the smallest temperature dependence of any reaction.
1/3 2 −33.80 /T6 −1 −3 εpp ∝ ρ X e ergs s cm
6 3.5 Thus, at T6 ~ 5, εpp µ T , while at T6 ~ 20, εpp µ T . Because CNO ions have more electrostatic repulsion, the temperature dependence of 23 13 its process€ is stronger, εCNO µ T at T6 ~ 10 to εCNO µ T at T6 ~ 50. The HR Diagram The Central Temperature and Density for Main Sequence Stars Convection on the Main Sequence
PP/CNO transition Importance of CNO Burning versus Stellar Mass Location of the Main Sequence
In general, the opacity of a star is proportional to its metal abundance (due to bound-free transitions and electrons supplied to H-.) The lower the metal abundance, the smaller the opacity, the less energy is trapped in the star doing work, the smaller the star, and therefore the hotter the star. The metal-poor main sequence is bluer than the metal-rich main sequence.