Xing’s notes on Carroll & Ostlie, Introduction to Modern Astrophysics

Xing’s notes. Bullet-point summaries of the Big Orange Book, including most of the significant formulas. Unfortunately, I don’t plan on notetaking part III in the foreseeable future.

I. The Tools of Astronomy

	name	center	bandwidth
U	UV	365 nm	68 nm
B	blue	440 nm	98 nm
V	visual	550 nm	89 nm

5. Interaction of Light and Matter

Fraunhofer lines: absorption lines in the Solar spectrum

Kirchoff’s Laws

Hot, dense gas/solid produces continuous, lineless spectrum
Hot, diffuse gas produces emission lines
Cold, diffuse gas in front of a continuous spectrum source produces absorption lines in the spectrum

Spectrographs

Produced via diffraction grating: $d \sin \theta = n\lambda$, $d$ is grating separation; $n$ is spectral order
Resolving power ${\lambda\over\Delta\lambda} = nN$, $\Delta\lambda$ is smallest resolvable wavelength; $N$ is # of gratings illuminated
Low−speed Doppler shift approximation: ${\Delta\lambda\over\lambda} = {u \cos \theta\over c}$

Photoelectric effect

Shining light onto metal ejects electron with energy proportional to frequency, not intensity
Max kinetic energy of an ejected electron $K = h\nu − \phi$
Work function $\phi$ is the metal’s min electron binding energy
Demonstrates quantization of light into packets of energy $E = h\nu$

Compton scattering

Relativistic scattering of electrons by high−energy photons
Photon loses wavelength $\Delta\lambda = {h(1 − \cos \theta)\over m_ec}$, $m_e = 9.109^{-31} \text{ kg}$

Bohr’s semiclassical atom

Assumes quantization of angular momentum $L = \mu v r = n \hbar$, $n$ is principle quantum number
However, still puts electrons in classical circular orbits modeled by Coulombic attraction
Kinetic energy of atom $K = {\mu v^2 \over 2} = {e^2 \over 8 \pi \epsilon_0 r} = {n \hbar^2 \over 2\mu r^2}$
Allowed orbital radii $r = a_0n^2$, Bohr radius $a_0 = {4\pi \epsilon_0 \hbar^2 \over \mu e^2} \approx 5.292^{-11} \text{ m}$
Allowed energy levels: $E = -{E_0 \over n^2}$, $E_0 \approx 13.6 \text{ eV}$

De Broglie waves

Generalizes photon wavelength-frequency-momentum relation to all partilces
$\nu = {E \over h}, \lambda = {h \over p}$

Heisenberg’s Uncertainty Principle

$\Delta x \Delta p \geq {\hbar\over2}$, estimated as $\Delta x \Delta p \approx \hbar$ , equivalent to $\Delta E \Delta t \approx \hbar$
Reduced Planck constant: $\hbar = {h \over 2\pi} = 1.054 \times 10^{-34} \text{ J}\cdot\text{s} = 6.582 \times 10^{-16} \text{ eV}\cdot\text{s}$
Allows for quantum tunnelling when barrier is within a few wavelengths wide
- e.x. total internal reflection can be disrupted by placing a another prism sufficiently close to the boundary

Schrödinger's quantum atom

Schrödinger equation models particles as probability waves
Analytically solvable for H with same result as the Bohr atom
Quantum orbitals are probability density clouds
Adds additional quantum numbers $l \in [0, n-1]$ and $m_l \in [−l, +l]$
Allowed angular momentum magnitudes $L = \hbar \sqrt{ l (l + 1) }$
Allowed angular momentum $z$-components $L_z = \hbar m_l$
Different orbitals with same $n$ are degenerate (i.e. have the same energy) unless under a magnetic field

Normal Zeeman effect

Previously degenerate orbitals split into slightly different energies
3 frequencies: $\nu \in { \nu_0, v_0 \pm {eB \over 4\pi\mu} }$
Allows detection of magnetic field strength around e.g. starspots
Even if field is too weak to cleanly split, different polarizations are still detectable

Anomalous Zeeman effect and Spin

Anomalous Zeeman effect: more complex splitting patterns
Resulted in discovery of the spin quantum number
Allowed spin angular momentum magnitudes $S = \hbar{\sqrt3\over2}$
Allowed spin angular momentum z-component $S_z = \hbar m_s$

Fermions and bosons

Dirac's relativistic Schrödinger's equation divided particles into fermions and bosons
Also predicted antiparticles: opposite electric changes and magnetic moments
Fermions have spins of $\text{odd} \times {\hbar\over2}$; Bosons have spins of $\text{integer} \times \hbar$
Pauli exclusion principle: No two fermions can share the same set of 4 quantum numbers ${n, l, m_l, m_s}$

Complex spectra

Quantum state transitions go from one set of quantum numbers to another
Selection rules significantly lower the probabilities of certain transitions
Restrictions: $\Delta l = \pm1, \Delta m_l \in { 0, \pm1 }, m_l$ of orbitals cannot both be 0
Allowed transitions triggered by collisions or occur spontaneously $\approx 10^{-8} \text{ s}$
Some forbidden transitions occur in very low gas densities
- e.x. diffuse interstellar medium; outer stellar atmospheres

6. Telescopes

Basic optics

Index of refraction $n(\lambda) \equiv {c \over \nu}$
Snell's law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$
Lensmaker's formula: $P = {1\over f} = (n - 1)\left( {1\over R_1} + {1\over R_2} \right) $, $P$ is optical power, $f$ is focal length, $R_n$ are radii of curvature
Focal length is wavelength-independent for mirrors
A lens projects angular separation onto a plate (focal plane)
Linear separation of point sources on focal plane increases with focal length: ${d\theta \over dy} = {1\over f}$

Resolution

Single-slit diffraction of wavefronts through aperture limits resolving power
Airy disk: bright spot around central maximum
Destructive interference creates minimums: $\sin \theta = {m\lambda\over D}, m = 1, 2, 3, …$
Images are “unresolved” when one's central maximum overlaps with another's 1st minimum
Rayleigh criterion defines arbitrary min resolvable distance of a circular aperture $\theta \equiv {1.22\lambda \over D}$, $D$ is aperture diameter
- Possible to differentiate sources within Rayleigh criterion through careful analysis of diffraction patterns
Seeing: ctual ground-based optical resolution worse than ideal due to refraction from atmospheric turbulence, unless corected by adaptive optics
Only lenses suffer from chromatic aberration; somewhat reduceable with correcting lenses
Spherical lenses and mirrors suffer from spherical aberration; mitigated by paraboloids, which are harder to produce
Paraboloid lenses and mirrors suffer from coma: elongation of off-axis images
Astigmatism: different parts of a lens or mirror converge an image at different locations; correction may result in curvature of field issues

Brightness

Illumination $J \propto {1 \over F^2}$, the amount of light energy per second focused onto a unit area of the resolved image
Focal ratio: $F ≡ {f \over D}$
- Larger aperture increases resolution and illumination
- Longer focal length increases image size but decreases illumination
- For fixed focal ratio, larger telescope diameter increases resolution but not illumination

Optical telescopes

Refracting telescope:
- Objective lens focus as much light as possible onto the focal plane
- Sensor is placed on focal plane or eyepiece is placed at its focal length away
- Angular magnification $m = {f_\text{objective} \over f_\text{eye}}$
- Issues:
  - Lens can only be supported near the edge, so gravity deforms heavy ones
- Entire volume must be fabricated precisely
- Slow thermal response and thermal expansion
- Chromatic aberrations
Reflecting telescope:
- Replaces refractor’s objective lens with a mirror
- Minimizes weight, support, and distortion issues, esp. with active support system
- Main issue: Prime focus of the mirror is in the path of the incoming light
  - Newtonian telescope: flat diagonal secondary mirror; suffers from faraway eyepiece that introduces torque
  - Cassegrain telescope: parabolic (or hyperbolic for Ritchey-Chrétien telescope) primary mirror; secondary mirror (usually convex to increase focal length) reflects back through a hole in the primary’s center
  - Coudé telescope: mirror system directs light to an instruments room; very long focal length
  - Schmidt telescope: spheroidal primary mirror minimizing coma; correcting lens to remove spherical aberrations; provides wide field of view (degrees vs arcminutes) with low distortion
Mounts:
- Equatorial: polar axis, easy to adjust, hard to build for massive telecopes
- Altitude-azimuth: easy to build, generally needs computer adjustments
Adaptive optics: small deformable mirror with many piezoelectric actuators counteract atmospheric distortions, constantly calibrated via a guide star
Charge-coupled device (CCD) detects ~100% of incident photons with a linear response across wide wavelength and intensity (dynamic) ranges

Radio telescopes

Strength measured is spectral flux density $S(\nu)$
Typical radio source has $S \approx 1 \text{ jansky (Jy)} = 10^{-26}$ W m$^{-2}$ Hz$^{-1}$; weak sources ~mJy
Integrate $S$ over area and bandwidth for total power received
Long wavelengths require large apertures, but less manufacturing precision
Addition of other telescopes enables interferometry; reduces side lobes and narrows main lobe of antenna sensitivity pattern
Interferometry determines angle of source from phase difference between antennae
Pointing angle $\theta : \sin \theta = {L\over d}$, $L$ is difference in wavefront’s distances to the antennae; $d$ is baseline distance between antennae
Very long baseline interferometry (VLBI) can span multiple continents

IR/UV/X-ray/gamma-ray astronomy

Water vapor is primary contributor to IR absorption; thus low-humidity mountain peaks, balloons and aircraft observatories are used
IR observation also requires very cold telescopes and detectors
Glass is opaque to UV; UV telescopes need very precise reflecting surfaces
X-ray and gamma-ray photons penetrate traditional mirrors; instead imaged by graze-incidence (near 90° incidence) reflections, or by Bragg scattering through crystal lattice inteference

II. The Nature of Stars

7. Binary Systems and Stellar Parameters

In many cases, allows the calculation of stellar masses, highlighting a well-defined mass-luminosity relation for the large majority of stars

Classifications

Optical double: Fake binaries that just look close
(The rest of the classes are not mutually exclusive)
Visual binary: Both stars are resolved
Astrometric binary: Visual oscillatory motion of one star implies companion
Eclipsing binary: Periodic variation in light curves reveal two stars
Spectrum binary: Different spectral classes / Doppler effect reveal superimposed spectra
Spectroscopic binary: Doppler effect in spectra reveal oscillation of radial velocity curves

Mass determination of visual binaries

Mass ratio from ratio of subtended angles: ${m_1 \over m_2} = {r_2 \over r_1} = {\alpha_2 \over \alpha_1}$
If distance to system and inclination are known, masses can be determined: $m_1 + m_2 = {4\pi^2 \over G} {d^3 \over \cos^3 i} {\alpha^3 \over P ^2}$
- Where i is the angle of inclination between the orbit and the sky’s planes

Mass determination of spectroscopic binarie

Mainly detected from binaries with high inclinations
“Double-line” if both spectra are visible; “single-line” otherwise
Radial velocity curves cross at the radial velocity of the center of mass
Radial velocity curve amplitudes scaled uniformly by $\sin i$
Eccentricity skews the curves, but close binaries tend to quickly circularize from tidal forces
Mass ratio: ${m_1 \over m_2} = {v_2 \over v_1}$
- Where v are radial velocities
If inclination is known, masses can be determined: $m_1 + m_2 = {P \over 2 \pi G} \left(v_1 + v_2 \over \sin i\right)^3$,
If only one radial velocity is known (single-line binary), then mass ratio cannot be found, but the mass function still provides rough constraints: ${(m_2 \sin i)^3\over(m_1 + m_2)^2} = {P \over 2\pi G} v_1^3$

Temperature ratio and radii determination of eclipsing binaries

Two dips in the light curve:
- Primary eclipse: hotter star passes behind cooler one
- Secondary eclipse: cooler star passes behind warmer one
- Size doesn’t matter
If smaller star $T_1$ is hotter, temperature ratio from brightness is ${B_0 − B_1 \over B_0 − B_2} = \left( T_1 \over T_2 \right)^4$
Unless extremely close binary, inclination $i \approx 90^\circ$, thus allowing accurate mass and velocity determination
Radius of star $r = {v\Delta t\over2}$
- Where $v$ is relative velocity, and $\Delta t$ is transit time for the smaller star, or eclipse time for the larger

Computer modeling

May incorporate tidal deformations, surface tempreature variations, flux distributions, etc
Generates synthetic light curves to compare against observational data

Extrasolar planets (exoplanets)

Direct observation very difficult due to vast luminosity differences
Indirect methods: spectral radial velocities, astrometric wobbles, and eclipses
First exoplanet around a Sun-like star discovered in 1995
Many rapidly followed due to dramatic advances in detector technology, large-aperture telescopes, and long-term observing campaigns

8. Classification of Stellar Spectra

Stars emit a continuous spectra (the continuum) with absorption lines in certain wavelengths

Harvard spectral types

Pickering and Fleming initially labeled alphabetically by strength of H absorption lines
Different strengths due to different temperatures causing different electron ionization levels and orbitals
- H I (neutral)’s visible spectral (Balmer) lines strongest at A0 ($T_\text{eff} = 9520$ K)
- He I (neutral)’s visible spectral lines strongest at B2 ($T_\text{eff} = 22\,000$ K)
- Ca II (singly ionized)’s visible spectral lines strongest at K0 ($T_\text{eff} = 5250$ K)
Cannon rearranged by temperature (OBAFGKM, O hottest) and added decimal subdivisions: A0 hotter than A9
Additional spectral types for very cool stars and brown dwarfs: OBAFGKMLT

Spectral physics

Statistical mechanics studies macroscopic behavior of stellar atmospheres
Maxwell−Boltzmann velocity distribution $\rho(v) = \left(m \over 2\pi k T \right)^{3/2} e^{−mv^2/2kT} 4\pi v^2$
- Restricted to gasses in thermodynamic equilibrium with $\rho \lesssim 1$ km m$^{-3}$
- Most probable speed $v_\text{mode} = \sqrt{2kT\over m}$
- Root-mean-square speed $v_\text{RMS} = \sqrt{3kT \over m}$ due to right-skewed exponential tail
Electrons’ orbital energies are affected by their atoms’ kinetic energies through collisions: higher orbitals are less likely to be occupied
Probability ratio of electron states: ${P(s_1) \over P(s_2)} = e^{−(E_1−E_2) / kT}$
- As thermal energy $kT \to 0$, $P(\text{ higher energy state }) \to 0$; confined to lowest state
- As $kT \to \infty$, ${P(s_1) \over P(s_2)} \to 1$; all states are equally likely
Statistical weights of some energy levels increased by degenerate orbitals
Boltzmann equation, the probability ratio of electron energies: ${P(E_1) \over P(E_2)} = {g_1 \over g_2} e^{−(E_1−E_2) / kT}$
- Where $g$ is the statistical weight, or number of states with that energy
Balmer lines require exitations from the first excited state $N_2$
Greater fraction of H I are in the excited state $N_2$ at higher temperatures; However, at the same time, significant fractions of H I are ionized to H IIs
Partition function, the number of possible electron states weighted by energy, varies by ionization state of the atom, thus affecting the probability of atoms at different ionization states: $Z = \sum_j g_j e^{−(E_j−E_1) / kT}$
- For H I, the ground state (with two orbitals $s=\pm{1\over2}$) dominates for most temperatures: $Z \approx 2$
- For H II, no electrons means only one possible configuration: $Z = 1$
Saha equation, the probability ratio of ionization states: ${N_+ \over N} = {2Z_+ \over n_e Z} \left( 2\pi m_e k T \over h^2 \right)^{3/2} e^{−\chi / kT}$
- Very sensitive to ionization energy $\chi$, as $kT$ only ranges around 0.5−2 eV
- Also effected by free electron number density $n$, increased by presence of ionized He II&III
Combining the Boltzmann and Saha equations shows varying narrow partial ionization zones for different atoms and ionization states
- For H, there is significant fraction of electrons in the first excited state for temperatures between 8300−11300 K, matching the peaking of Balmer lines $\sim9900$ K

Hertzsprung–Russell diagram

Hertzsprung discovered type G and cooler stars had a range of magnitudes; termed the brighter ones giants; Russell termed the dimmer ones dwarfs
H–R diagram plots absolute magnitude (brighter upward) against spectral type (warmer leftward)
80–90% of stars lie as dwarfs in a main sequence from upper left to lower right
- Width of main sequence due to varying ages and compositions
Initial theory was stars evolved from upper left to lower right; terminology of “earlier” (warmer) and “later” (cooler) stars remain
Simple luminosity–temperature relation reveals fundamental dependence on mass
Iso-radius lines run diagonally roughly parallel to main sequence
- Warmer, more massive stars have a lower average density
Supergiants such as Betelgeuse occupy extreme upper right

Morgan−Keenan luminosity classes

Maury noted subtle line width variations amongst stars with similar effective temperatures and different luminosities; found by Hertzsprung to differentiate main-sequence stars and giants
Morgan and Keenan published atlas appending Roman numeral luminosity classes to Harvard spectral types
- Ia/Ib for supergiants
- V for main requence
- VI for metal-deficient subdwarfs
- Excludes white dwarfs, class D
Luminosity classes roughly correlates with absolute magnitudes: enables placement of star on H−R diagram entirely from its spectrum
Spectroscopic parallax: calculating distance modulus from the spectrally determined absolute magnitude

9. Stellar Atmospheres

Line blanketing: dense series of metallic absorption lines significantly decrease intensity relative to ideal blackbody at certain wavelengths

Radiation field

Radiation energy originates from thermonuclear reactions, gravitational contraction, and core cooling
But observed light comes from gasses in atmosphere, not the opaque interior
Temperature, density, and composition of atmospheric layers determine spectral features
Specific ray intensity: $I(\lambda) \;d\lambda \equiv {\partial I\over\partial\lambda} = {{\partial E \over \partial\lambda} d\lambda \over d\lambda \;dt\;dA \cos \theta \;d\Omega }$, power transmitted at a certain wavelength within a certain solid angle
Mean intensity $\langle I(\lambda) \rangle = I(\lambda)$ for isotropic field; $= B(\lambda)$ for blackbody
Specific energy density $u(\lambda) \;d\lambda = {4\pi\over c} \langle I(\lambda) \rangle \;d\lambda$
- Total energy density for blackbody $u = {4\sigma T^4 \over c}$
- Blackbody radiation pressure $P = {u\over3}$
Specific radiative flux $F(\lambda) \;d\lambda = \int I(\lambda) \;d\lambda \cos \theta \;d\Omega$, net energy at a certain wavelength flowing towards one single direction
For a resolved source, specific ray intensity is measured, thus detector distance does not affect measured intensity, though the angular size decreases
For an unresolved source, specific radiative flux is measured, falling off with ${1\over r^2}$

Temperature

Mean free path $l = {1\over n\sigma}$,
- Average distance traveled by particles and photonsbetween collisions
- Where $n$ is atomic number density and $\sigma \equiv \pi(2a_0)^2$ is collision cross section
Multiple descriptions of temperature:
- Effective temperature: obtained from the Stefan–Boltzmann law
- Excitation temperature: defined by the Boltzmann equation
- Ionization temperature: defined by the Saha equation
- Kinetic temperature: defined by the Maxwell–Boltzmann distribution
- Color temperature: defined by fitting the Planck function on the spectrum
Effective temperature is a global descriptor; rest apply locally, varying by gas location and other conditions
All local temperature definitions result in the same value at thermodynamic equilibrium
Local thermodynamic equilibrium (LTE) occurs when temperature change is very gradual compared to the mean free path length: $H \ll L$
- Temperature scale height $H(T) \equiv {T \over \lvert{ dT \over dr }\rvert}$

Opacity

Both scattering and pure absorption reduce the intensity of directed light: $\;dI = −\kappa\rho I \;ds$
- Where absorption coefficient (opacity) $\kappa$ is dependent on ray wavelength and gas composition, density, and temperature
Characteristic distance $l = {1\over k\rho}$ is equivalent to main free path for photons
Pure absorption decreases intensity exponentially: $I = I_0 e^{−\tau}$
- Where $\tau$ is optical depth with $d\tau = −\kappa\rho \;ds$
Optical depth is equivalent to the number of mean free path lengths along the path of a ray; gas is “optically thick” if $\tau \gg 1$, “optically thin” if $\tau \ll 1$
Balmer jump: opacity of stellar material suddenly increases for wavelengths below the ionization energy of H I in the first excited state
Sources of opacity classified by initial and final quantum states of the interacting electrons:
- Bound–bound transitions (excitation and de-excitation)
  - Small except for wavelengths corresponding to specific excitation energies;
  - If de-excitation returns to the initial state through emission, the photon is essentially scattered
  - If multiple photons are emitted from one absorption, the average photon energy is reduced
  - If de-excitation is instead triggered by collision, then the energy is converted into kinetic/thermal
- Bound-free absorption (photoionization) and free–bound emission (recombination)
  - Wavelength-dependent cross section comparable to that of collisions
  - Adds to continuum opacity, as any photon above an electron’s ionization energy may be absorbed
  - Recombination emits one or more photons, again reducing average photon energy and scattering
- Free–free absorption and bremsstrahlung (“braking radiation”) emission
  - Free electron gains speed from absorption, or loses speed from emission
  - Must take place next to an ion to conserve both energy and momentum
  - Adds to continuum opacity
- Free electron (Thomson) scattering
  - Photon is scattered when the electron is made to oscillate in its EM field
  - Very tiny wavelength-independent cross section
  - Only dominates in stellar interiors and the hottest atmospheres, where near-total ionization eliminates bound-electron processes
- Loosely-bound electron (Compton/Rayleigh) scattering
  - Compton if $\lambda \ll a_0$; very small change in scattered photon wavelength, much like Thomson
  - Rayleigh if $\lambda \gg a_0$; proportional to ${1 \over \lambda^4}$, so only significant in UV for supergiant stars’ extended envelopes, cool main-sequence stars, and planet atmospheres
Electron scattering and photoionization of He are primary sources of continuum opacity for type O
Photoionization of H and free–free absorption primary sources for types B–A
Photoionization of H$^-$ is primary source for type F0 and cooler
- Very low binding energy of 0.754 eV ($\lambda = 1640 \text{ nm}$) for bound–free opacity
- Also contributes to free–free absorption at longer wavelengths
Molecules survive in planetary and cooler stellar atmospheres
- High opacity from large number of discrete absorption lines
- May break apart during absorption through photodissociation
Rosseland mean opacity: weighted harmonic mean of opacity across all wavelengths, accounting for rate of change of the blackbody spectrum with temperature; ${1\over\kappa} \equiv {\int {1\over\kappa_\nu} {\partial B_\nu \over \partial T} \;d\nu \over \int {\partial B_\nu \over \partial T} \;d\nu }$
- Obeys Kramer’s opacity law: $\kappa \propto {\rho\over T^{3.5}}$
- No analytic solution, but approximations exist for bound–free and free–free opacities
- $\kappa_\text{bf} \propto {g_\text{bf} \over t} {Z (1+X) \rho \over T^{3.5}}, k_\text{ff} \propto g_\text{ff} {(1−Z)(1+X) \rho \over T^{3.5}}$
  - $X$, $Y$, and $Z$ are mass fractions of H, He, and metals respectively
  - $g_\text{bf}$ and $g_\text{ff}$ are Gaunt factors correcting for quantum effects; ~1 for visible and UV
  - t is the guillotine factor, typically ranging 1–100

Radiative transfer

No net energy change occurs in any layer of a star that is in steady-state equilibrium
Emission processes complement each of the primary absorption processes, resulting in randomly-directed scattering
Specific intensity: $dI = −\kappa\rho I \;ds + j\rho \;ds$
- Where $\kappa$ is the absorption and $j$ is the emission coefficient; both are $\lambda$-dependent
Equation of radiative transfer: $−{1\over\kappa\rho} {dI\over ds} = {dI\over d\tau} = I − S$
- Where source function: $S \equiv {j\over\kappa}$
- Expresses how light beam composition tends to resemble the local source of photons
In local thermodynamic equilibrium, $S = B$, the blackbody Planck function
- Integrating over all wavelengths, $S = B = {\sigma T^4\over\pi}$
- If $\tau \gg 1, I = B$ as well
Radiation differential “driving” net radiative flux of photons flowing to the surface: ${dP\over d\tau} = −{F\over c}$
Radiative flux throughout star is constant: $F = \sigma T_\text{eff}^4$
In a random walk with $N$ steps of average size $I$, $d = l\sqrt N$
On average, a photon from optical depth $\tau$ needs $\tau^2$ steps to reach the surface
- Photons from $\tau \approx 1$ may escape without scattering
Eddington approximation: $T^4 \approx {3\over4} T_\text{eff}^4 \left(\tau + {2\over3}\right)$
- Star has its effective temperature at optical depth $\tau \approx {2\over3}$
- Average observed photon is emitted from $\tau \approx {2\over3}$, independent of angle
Higher opacity corresponds to shorter pathlength for the same optical depth, thus absorption lines must come from outer, cooler layers
Limb darkening: line of sight reaches $\tau = {2\over3}$ in cooler, dimmer layers when observing closer to the edge of the disk

Spectral line profile

Core of line is formed at higher and cooler layer; formation descends down to continuum region at wings of the line
Line is spectrally thin if radiant flux $F(\lambda)$ is never 0
Equivalent width $W \equiv \int 1 − {F(\lambda) \over F_0} d\lambda$
- Where the integrand is the depth of the line at $\lambda$
Broadening processes:
- Natural broadening: uncertainty principle means momentary occupancy of an excited state for small $\Delta t$ amplifies uncertainty of orbital energy $\Delta E \approx {\hbar \over \Delta t}$, resulting in uncertainty of wavelength $\Delta\lambda \approx {\lambda^2 \over 2\pi c} \left( {1 \over \Delta t_0} + {1 \over \Delta t_1} \right)$, where $\Delta t$ are lifetimes in the two states
- Doppler broadening: random thermodynamic motion results in nonrelativistic Doppler shift $\Delta\lambda \approx {2\lambda \over c} \sqrt{2kT \over m}$
  - in giant and supergiant stars, random large-scale turbulence increases to $\Delta\lambda \approx {2\lambda \over c} \sqrt{2kT\over m + v^2}$, where $v$ is the most probable turbulence speed
  - coherent mass motions such as rotation, pulsation and mass loss also substantial factors
- Pressure broadening: collisions with neutral atoms and pressure from nearby ion electrical fields may perturb the orbitals with $\Delta\lambda = {\lambda^2 \over \pi c \Delta t_0}$, where $\Delta t_0 \approx {1\over n\sigma\sqrt{2kT \over m} }$ is the average collision time
  - Dependence on atomic number density explains narrower lines of sparser giant and supergiant extended atmospheres used for the Morgan–Keenan classification
Damping/Lorentz profile: shape of lines from natural and pressure broadening, characteristic of radiation from electric charge in damped harmonic motion
Voigt profile: total line profile from both Doppler and damping profiles, with Doppler dominating at the core and damping dominating at wings
Schuster–Schwarzchild model: assumes photosphere is a blackbody and atoms above it create absorption lines
Number of atoms involved in absorption per surface area: $fN$
- Where oscillator strength $f$ is the relative likelihood of each transition, and column density $N$ is the area density of absorbing atoms in a column from the surface to the observer
Curve of growth for line width as a function of column density:
- Initially optically thin core: $W \propto N$
- Then, saturated core with optically thin wings: $W \propto \sqrt{\ln N}$
- For high $N$, pressure-broadening profile dominates wings: $W \propto \sqrt N$
Applying Boltzmann and Saha equations to curve of growth finds total number of atoms above continuum layer
- Applying Boltzmann equation also finds excitation temperature
- Applying Saha equation also finds either e$^-$ pressure or ionization temperature from the other

Computer modeling

Construction of an atmospheric model with extensive physics fine-tuned to observations can provide information on line profile, chemical composition, effective temperature, surface gravity, etc

10. The Interiors of Stars

Direct observation only possible with neutrinos and the occasional supernovae
Study requires physically accurate computer models that match observable surface features

Hydrostatic equilibrium

Equilibrium condition requires pressure gradient to counteract gravity: ${dP \over dr} = −\rho g$
- Local gravity $g \equiv {G M_r \rho \over r^2}$, where $M_r$ is mass enclosed within $r$
Mass conservation ${dM_r \over dr} = 4\pi r^2 \rho$
Total pressure $P = {\rho kT \over \mu} + {aT^4\over3}$
- Where first term is ideal gas law and second term is radiation pressure
- With as mean molecular weight

Kelvin–Helmholtz mechanism

Contraction releases gravitational energy $\Delta E \approx {3\over10} {GM^2 \over R}$
Gravitational energy contribution $\epsilon = −{dQ \over dt} = −T {dS \over dt}$
- Where specific entropy: $dS \equiv {dQ \over T}$
- Contraction produces heat and decreases entropy; vise versa for expansion
Kelvin–Helmholtz timescale of star $t = {\Delta E\over L}$, $t_\odot \approx 10^7 \text{ years}$

Nuclear fusion

Releases difference in strong nuclear force binding energy:
- $E = \Delta m_0c^2 = ( Z m_\text{p} + (A−Z) m_\text{n} − m_\text{nucleus} ) c^2$
Timescale $t_\odot \approx 10^{10}$
Must overcome Coulombic repulsion between positive nucleons: $\sim10^6$ eV at $r \approx 10^{-15}$ m
Thermal energy of gas: $E = {\mu v^2 \over 2} = {p^2 \over 2\mu}$
- Where $\mu$ is reduced mass and $v$ is average relative velocity
Requires impossibly high temperatures in classical physics: $E = {3\over2} kT = {q_1q_2 \over 4\pi r \epsilon_0} \to T \approx 10^{10} \text{ K}$ for H–H fusion
With quantum tunneling (de Broglie $\lambda = {h \over p} \approx r$): $E = {(h/\lambda)^2\over2\mu} = {q_1q_2 \over 4\pi\lambda\epsilon_0} \to T \approx 10^7 \text{ K}$, consistent with solar core temperature estimates
Reaction rate depends on velocity (wavelength) distribution, densities, and cross-sectional area of particles 1 and 2: $r = \int n_1 \; n_2 \; \sigma(E) \; v(E) \; { n(E)\over n} \; dE$
- Power-law approximation: $r = r_0 \; X_1 \; X_2 \; \rho^{\alpha'} \; T^\beta$,
  - Where $X$ are mass fractions, $\alpha' \approx 2$, and $\beta$ ranging from ~1 to ≥40
Velocity distribution follows Maxwell–Boltzmann
Cross section $\sigma(E) = {S \over E} e^{−b/\sqrt E}$
- Where de Broglie area is $\pi\lambda^2 \propto {1\over p^2} \propto {1\over E}$, and tunneling across barrier of energy $U$ is $e^{‐2\pi^2 U/E} \propto e^{−b/\sqrt E}$, with $b \propto q_1q_2\sqrt\mu$
- $S$ may be slow-varying function of $E$, or may have sharp peaks from resonance of specific energy levels within the nucleus
Electron screening from sea of free e$^-$ partially hides nuclei, reducing effective charge and Coulomb barrier, sometimes enhancing He production by 10–50%
Likelihood of nuclear reaction: $e^{−b/\sqrt E} e^{−E/kT}$
- Product of velocity distribution’s high-energy tail and the quantum tunneling terms
- Most likely energy ("Gamow peak") $E_0 = \left(bkT\over2\right)^{2\over3}$
Luminosity gradient ${dL_r \over dr} = 4\pi r^2\rho$,
- Where $L_r$ is luminosity enclosed in $r$, and $\epsilon$ is specific power (power released per mass)

Nucleosynthesis

Notation: $^A_Z\text{He}$, where $A$ is atomic mass number and $Z$ is proton number
Simultaneous 4-body collision $4 {\;}^1_1\text{H} + ? \to {\;}^4_2\text{He} + ?$ extremely unlikely
Reaction chain of 2-body interactions more probable
Interactions must obey conservation laws: electric charge, nucleon number, and lepton number
Proton–proton chain (H burning): $4 {\;}^1_1\text{H} \to {\;}^4_2\text{He} + 2\text{e}^+ + 2\nu + 2\gamma$
- Three branches (see Fig. 10.8)
- In Sun, 69% PP I, 31% PP II, 0.3% PP III
- Near $T = 1.5 \times 10^7 \text{ K}$, $\epsilon \approx \epsilon_0 \rho X_\text{H}^2 \left(T \over 10^6 \text{ K}\right)^4$
  - Where $\epsilon_0 = 1.08 \times 10^{-12} \text{ W}\;\text{m}^3\;\text{kg}^{-2}$
CNO cycle (H burning): $\text{C}$, $\text{N}$, and $\text{O}$ are catalysts
- Two main branches, with the second only occuring ~0.04% of the time
- Near $T = 1.5 \times 10^7 \text{ K}$, $\epsilon \approx \epsilon_0 \rho X_\text{H} X_\text{CNO} \left(T \over 10^6 \text{ K}\right)^{20}$
  - Where $\epsilon_0 = 8.24 \times 10^{-31} \text{ W}\;\text{m}^3\;\text{kg}^{-2}$
  - Much more temperature-dependent, only dominating in more massive stars
Triple-alpha process (He burning): $2 {\;}^4_2\text{H} \leftrightarrow {\;}^8_4\text{Be}$, ${\;}^8_4\text{Be} + {\;}^4_2\text{He} \to {\;}^{12}_6\text{C} + \gamma$
- First step produces an extremely unstable $^8_4\text{Be}$, thus combined is essentially a 3-body interaction: $r \propto (\rho Y)^3$
- $\epsilon \approx \epsilon_0 \rho^2 Y^3 \left(T \over 10^8 \text{ K}\right)^{41}$
  - Incredibly strong temperature dependence
Alpha process (C & O burning): $^{12}_6\text{C} + {\;}^4_2\text{He} → {\;}^{16}_8\text{O} + \gamma$, $^{16}_8\text{O} + {\;}^4_2\text{He} \to {\;}^{20}_{10}\text{Ne} + \gamma$
- Alpha capture becomes prohibitive at higher $Z$ due to higher Coulomb barrier
- C–C burning near $6 \times 10^8$ K and O–O burning near $10^9$ K can produce Na, Mg, Si, P, and S; some are endothermic but are normally less likely
Binding energy per nucleon ${E\over A}$
- Relative to atomic number $A$, local maxima are very stable
- Magic nuclei: some elements ($^4_2$He, $^{16}_8$O) have unusually high ${E\over A}$
- Broad peak around $^{56}_{26}$Fe, the most stable nuclei

Energy transport and thermodynamics

Three mechanisms: radiation of photons (affected by opacity), convection, and conduction (generally insignificant)
Radiative temperature gradient ${dT \over dr} = −{3\over4ac} {\kappa\rho \over T^3} {L\over4\pi r^2}$
If temperature gradient becomes too steep, convection takes hold
Convection is much more complicated than radiation
- Strongly coupled to the star’s dynamic behavior
- 3D Navier–Stokes with turbulence is hard compute
- Pressure scale height, convection’s characteristic length scale, is big: $H \equiv −P {dr \over dP} \approx {P\over\rho g} \approx {R_*\over 10}$
First law of thermodynamics: $dU = dQ − dW = dQ − P \;dV$
- $U$ is a state function; $Q$ and $W$ are not — $dQ$ and $dW$ are inexact differentials
Specific heat capacity $C \equiv {\partial Q \over \partial T}$, $C_P = C_V + nR$
- $C_P$ is at constant pressure; $C_V$ is at constant volume
- Heat capacity ratio / adiabatic index $\gamma \equiv {C_P \over C_V}$
- $\gamma = {5\over3}$ for a monoatomic gas; approaches 1 in a partial ionization zone as both specific heats increase
Isochoric process ($dV = 0$): $dU = dQ = C_V \;dT$
Adiabatic process ($dQ = 0$): $dU = −P \;dV$
- Gas law: $P \propto {1 \over V^\gamma} \propto \rho^\gamma \propto T^{\gamma/(\gamma−1)}$
- Speed of sound $v = \sqrt{B\over\rho} = \sqrt{\gamma P \over \rho}$
  - Where bulk modulus $B \equiv −V {\partial P \over \partial V} \;{dQ = 0} $
Adiabatic temperature gradient ${dT \over dr} = −\left(1 − {1\over\gamma}\right) {\mu \over k} {GM \over r^2} = −{g \over C_P}$
If the surrounding temperature gradient is steeper than the bubble’s, even just slightly, the condition becomes superadiabatic, and nearly all luminosity is transferred outwards adiabatically, via convection instead of radiation
- Equivalent criterion: ${d(\ln P) \over d(\ln T)} < {\gamma \over \gamma − 1}$, = 2.5 for ideal monoatomic gas
In general, convection occurs if a region
1. has high opacity (surrounding ${dT \over dr} \propto \kappa$),
2. is ionizing and raising the specific heat capacity (bubble ${dT \over dr} \propto {1 \over C_P}$), or
3. has a highly temperature-dependent fusion process
First two conditions can occur simultaneously; third only occurs deep in the interior with the CNO or triple-alpha processes
Convective flux under mixing-length theory: $F = \rho C_P {k\over\mu} \left({T \over g} \delta\left(dT \over dr\right)\right)^{3/2} \alpha^2 \sqrt\beta$
- Where $0.5 \lesssim \alpha \lesssim 3$ and $0 \lesssim \beta \lesssim 1$ are free parameters
  - $\alpha \equiv {l\over H}$, the ratio of the mixing length (distance traveled by bubble before thermaliing with surrounding) and the pressure scale height
  - $\beta : \beta v^2$ is the average kinetic energy of the bubble as it travels over $l$

Stellar model building

Basic stellar models need constructive relations for $P$, $\bar\kappa$, and $\epsilon$: expressing them in terms of density, temperature, and composition
- $P$ (pressure) generally modelable with ideal gas law and radiation pressure, but is more complex in certain stars’ deep interiors
- $\bar\kappa$ (mean opacity) calculated explicitly from tabular data or fitting functions
- $\epsilon$ (specific power) calculated analytically from reaction networks
Boundary conditions:
- As $r \to 0$, ${M_r, L_r} \to 0$
- And ignoring extended atmospheres and mass loss: as $r \to R_*$, ${T, P, \rho} \to 0$
Vogt–Russell theorem: due to a star’s dependence on nuclear burning, its mass and internal composition uniquely determine its radius, luminosity, internal structure, and subsequent evolution
- Ignores smaller influences such as magnetic fields and rotation
General modeling numeric integrates shell-by-shell with the system of differential equations, often from from a transition point towards both the surface and the center
Polytropes: simplified stellar models in which $P(\rho) \propto \rho^\gamma$
Lane–Emden equation: ${1\over\xi^2} {d \over d\xi} \left(\xi^2 {d(D_n) \over d\xi}\right) = −D_n^n$
- Where the polytropic index $n : \gamma \equiv {n+1 \over n}$
- TODO: too complicated

Main sequence

Vast majority of stars have H mass fraction $X \approx 0.7$ and metal mass fraction $0 \lesssim Z \lesssim 0.03$
Changes in core composition affects observed surface features
Very light stars ($M \lesssim 0.08 M_\odot$) are not hot enough to let fusion stabilize against gravitational contraction
- Highly opaque and fully convective
Lower-initiation-energy PP chain dominates for low-mass stars
- Shallow core thermal gradient leads to radiative core
- High shell opacity leads to convective shell
Highly-temperature-dependent CNO cycle dominates for high-mass stars ($M \gtrsim 1.2 M_\odot$)
- Steep core thermal gradient leads to convective core
- Low shell opacity leads to radiative shell
Very massive stars ($M \gtrsim 90 M_\odot$) have rapid core thermal oscillations affecting fusion rates
Very massive stars may also have radiation pressure exceed gas pressure at outer layers, with maximum stable luminosity given by Eddington limit: $L ≤ {4\pi GcM\over\bar\kappa}$
Main-sequence lifetimes decrease with increasing luminosity

11. The Sun

Loosely treated as two parts: an optically thin atmosphere and an optically thick core
Transition is $\sim600$ km thick

Solar interior

Hydrogen burning below $R \lesssim 0.3 \,R_\odot$; convection above $R \gtrsim 0.7 \,R_\odot$
Heterogeneous composition due to nucleosynthesis, convection, and elmental diffusion
$^4_2$He is more abundant than $^1_1$H below $R \lesssim 0.1 \,R_\odot$
$^3_2$He abundance peaks at the top of the hydrogen-burning region, where cooler temperatures slow the $^3_2$He–$^3_2$He reaction
Convection zone turbulence creates homogeneous composition
Peak energy generation region is shell around $r \approx 0.1\, R_\odot$
- ${dL\over dr}$ affected by shell volume and fuel availability, both smaller at the very center, as well as temperature and pressure
Surface Li abundance somewhat less than expected for the current solar model
Mikheyev–Smirnov–Wolfenstein effect: neutrino oscillation between the $\text{e}^-$, $\mu^-$, and $\tau^-$ flavors explains the solar neutrino problem of only ⅓ as many $\text{e}^-$ neutrinos as expected being detected from the solar core

Photosphere

Where optical photons originate
Starts 100 km below where $\tau_\text{500 nm} = 1$, extending $\sim600$ km
Temperature drops from $\sim9400$ K at base to $\gtrsim4400$ K at top
Continuum opacity partly due to H$^-$ near base of photosphere, as the far more abundant neutral H I cannot contribute much to the continuum
Absorption lines produced in higher, cooler, more opaque regions of the photosphere
Granulation: patches of bright and dark regions ($\sim700$ km) at base of photosphere due to underlying convection zone; Doppler shifts ($\sim0.4$ km/s) cause wiggles in absorption lines
- Characteristic lifetime ($\sim10$ minutes) corresponds to the time a convection eddy takes to rise and fall 1 mixing length
Differential rotation: Doppler shifts at solar limb and solar oscillations show the solar rotation varies by latitude and by radius
- Period of 25 days at equator lengths to 36 days at poles
- Tachocline: base of convection zone ($\sim 0.65 R_\odot$) where differing rotation rates converge, resulting in strong shear theorized to create plasma which generate the solar magnetic field

Chromosphere

Where temperature begins rising again, from 4400 to $\sim10\,000$ K
Starts 525 km above where $\tau_\text{500 nm} = 1$, extending $\sim1600$ km
Density and intensity $\sim10^{-4}$ of the photosphere
Low density and high temperature produce certain absorption and emission lines,
As the blackbody continuum emission peaks $\sim500$ nm, visible-spectrum emission lines are only clearly seen in a flash spectrum of the limb near total eclipse
Strength of $\text{H}\alpha$ emission line allows filters to selectively observe the chromosphere structure
Supergranulation: patches of $\sim30\,000$ km wide from underlying convection
Spicules: vertical gas filaments extending upwards for $\sim10\,000$ km, ejecting mass at $\sim15$ km/s

Transition region

Starts $\sim2100$ km above $\tau_\text{500 nm} = 1$, extending to the corona
Temperature rises rapidly to $\sim10^5$ K in 100 km, then slowly to $\gtrsim10^6$ K
Selectively observed in various UV bands (e.g. Lyman $\alpha$ at $\sim20\,000$ K)

Corona

Faint region ($\lesssim 10^{15} \text{ particles/m}^3$) with vaguely defined outer boundary
High-temperature, high-thermal-conductivity, approximately isothermal plasma
Quiet corona near sunspot minimum (low solar activity)
- More extended at equator than poles, consistent with nearly dipole magnetic field
Active corona near sunspot maximum
- More complex magnetic field shape and structure
Essentially transparent to most EM radiation
Not in local thermodynamic equilibrium thus no strictly definable temperature, but estimated to be $\gtrsim 2 \times 10^6 \text{ K}$
Parker wind model: not in hydrostatic equilibrium as pressure does not vanish at infinity, implying solar wind
Kontinuierlich/continuous K corona:
- From free electron scattering of photospheric light primary between $1 \sim 2.3\, R_\odot$
- Spectral lines essentially blended into continuum from high-velocity Doppler broadening
Fraunhofer F corona:
- From dust scattering of photospheric light beyond $2.3\, R_\odot$
- Slower dust grains have less Doppler broadening and leave detectable Fraunhofer lines
- Merges with zodiacal light from interplanetary dust
Emission E corona:
- From highly ionized coronal atoms, very rich in emission lines in the X-ray spectrum
- Low number density enable forbidden transitions between metastable energy levels, as well as low-energy free-free transitions that produce radio waves
- Radio waves also produced by relativistic electrons’ synchrotron radiation

Solar wind

Stream of escaped ions and electrons
Deflects comets’ ion tails differently than pure radiation pressure on their dust tails
Trapped into Van Allen belts by planetary magnetospheres and create aurorae upon reaching atmosphere
Fast solar wind: $\sim750$ km/s, produced from open magnetic fields
- Associated with coronal holes — darker, cooler coronal regions
Slow solar wind: $\sim300$ km/s, produced by coronal streamers around closed magnetic fields;
- Associated with X-ray bright spots from trapped spiralling charges
Heliopause: outer limit of the Sun’s EM influence; where solar wind meets the interstellar medium and produces a termination shock
Heliosheath: beyond the termination shock, particles slow, magnetic field strengthens, and density increases

Magnetohydrodynamics (MHD)

Longitudinal pressure waves propagate outward from top of convection zone
Drastic drop in medium density turn waves supersonic, creating shock fronts that drastically heat up chromospheric gas as they dissipate
Magnetic energy density and pressure $u = P = {B^2\over2\mu_0}$
Transverse Alfvén waves propagate from oscillations in magnetic field lines
Resistive Joule heating from electrical currents in Alfvén waves also contributes to temperature rise, in particular the steep gradient of the transition region
Open magnetic field lines are dragged by stellar rotation across interplanetary space, slowing stars down significantly over their lifetimes

Sunspots

Zeeman splitting indicates strong magnetic fields, which inhibit convective motion below and create dark spots
Umbra: darkest portion with vertical field lines; diameter $\lesssim 30\,000 $km
Penumbra: border with filament-like structure with field lines becoming horizontal
Generally located in groups, with one dominant sunspot leading several in the direction of rotation
- Lead sunspots in the same hemisphere always have the same polarity during an 11-year cycle
- Opposite polarity in the other hemisphere
Frequency follows 11-year cycle, starting from sunspot minimum
Butterfly diagram: average latitude starts near $\pm40^\circ$, migrating to the equator over cycle
Solar polarity reverses during sunspot minimum, thus technically 22-year cycle
Long-term variations include the Maunder minimum spanning 1645–1715

Plages

Chromospheric regions of higher density and bright H$\alpha$ emission
Located near sunspots, forming before they appear and vanishing after they disappear
Caused by magnetic fields

Flares

Eruptive events releasing $10^{17}\sim10^{25}$ J of energy over timespans from milliseconds to over an hour
May reach 100,000 km in length
Develop in sunspot groups with intense stored magnetic energy
Reconnection of magnetic field lines creates a sheet of current in the plasma, Joule heating the gas up to $10^8$ K
Charged particles are accelerated away from the reconnection point
- $\text{H}\alpha$ line becomes locally in emission from ejected particles recombining by the base of the field lines
- Solar cosmic rays from particles ejected towards outer space
Nonthermal radio waves from synchrotron radiation around field lines
Soft X-rays from high temperatures in loop below reconnection point
Hard X- and gamma-rays from surface nuclear reactions, including spallation of heavy nuclei: $^1_1\text{H} + {\;}^{16}_8\text{O} \to {\;}^{12}_6\text{C}+ {\;}^4_2\text{He} + {\;}^1_1\text{H} + \gamma$

Prominences

Quiescent prominence:
- Curtains of cooler ($\sim8000$ K) ionized gas collected along magnetic field lines
- May be stable for weeks
- Appear as dark filaments against the disk
Eruptive/active prominence:
- Suddenly destabilized magnetic field configuration
- Energy converted into lifting prominence away from the Sun

Coronal mass ejections (CMEs)

Ejects $5 \times 10^{12} \sim 5 \times 10^{13} \text{ kg}$ of material at speeds of $400\sim1000$ km/s
$\sim1$/day, more frequent near sunspot maximum
70% correlated with eruptive prominences; 40% correlated with flares

Magnetic activity in other stars

Flare stars: M-type main-sequence stars whose occasional rapid fluctuation in brightness may be due to large flares on their relatively dimr surface
Starspots may be used to measure stellar rotation
Magnetic field lines measured from Zeeman broadening correlate with luminosity variations

12. The Interstellar Medium and Star Formation

Interstellar medium (ISM)

Gas and dust between the stars
70% H, ~30% He, rest metals (C, Si, etc)
Used in star formation; returned by stellar winds and explosive events

Dust

~1% of molecular clouds by mass, but significant in light absorption and cloud chemistry
Likely facilitate formation of many molecules besides H$_2$, including solid CO, H$_2$O, etc that give the grains icy mantles
Formed in dense envelopes of very cool stars, from supernovae and stellar winds
Unexpected abundance of large grains an area of active research
Responsible for interstellar extinction, a $\lambda$-dependent increase in apparent magnitude from scattering and absorption of starlight: $A = \Delta m$
- Approximately equal to the optical depth: $A \approx 1.086\tau$
- Visual band extinction commonly used as reference: $A_V$
Mie theory applies to IR–visible range
- Shorter wavelengths are scattered more (though spectral lines go unaltered), causing interstellar reddening along direct line of sight, and blue tones in reflection nebulae
- Extinction coefficient: $Q(\lambda) \equiv {\sigma(\lambda)\over\sigma_g}$
- With dust grain radius: $r$ such that grain cross section: $\sigma_g = \pi r^2$
- Where photon cross section approaches 0 for $\lambda \gg r$ and a constant for $\lambda \ll r$: $\sigma(\lambda) \propto {r^3\over\lambda}$ for $\lambda \gtrsim r$, and $\sigma(\lambda) \propto r^2$ for $\lambda \ll r$
Color excesses relative to Mie prediction grow for shorter wavelengths
Bump at 217.5 nm corresponding to resonance of graphite, along with IR emission bands corresponding to C–C and C–H bond vibrations, indicate presence of polycyclic aromatic hydrocarbons (PAHs)
Near-IR absorption bands indicate presence of silicate grains
Slight, $\lambda$-dependent polarization due to anisotropic, somewhat aligned dust grains; alignment most likely due to interactions with a weak magnetic field

Hydrogen

21-cm line: anti-aligned spins ↑↓ has slightly less energy than aligned spins ↑↑
- Very stable, “forbidden” transition that takes millions of years on average to occur per atom
- Only possible (although still rare) for H I in low-density diffuse ISM
- Optically thin, thus optical depth $\propto$ H column density
H$_2$ hard to directly observe except for rovibrational bands; tracer molecules and their isotopomers used instead
Optically thick dust shields H$_2$ from UV photodissociation, as well as enhancing H$_2$ formation
H I generally proportional in number to dust when $A_V \lesssim 1$; displaced by H$_2$ when dust becomes optically thick
Shells of H I clouds surround molecular clouds of H$_2$

Interstellar cloud structure

Most diffuse ISM are hydrogen clouds of ground state H I, only absorbing UV photons and emitting in 21-cm line
Transluscent/diffuse molecular clouds: irregularly shaped; similar in conditions to H I clouds; primarily atomic H with regional concentrations of H$_2$
Giant molecular clouds (GMCs): enormous complexes with clumpy structure
- Contains slightly denser dark cloud complexes, denser clumps and dense cores, and very dense, star-forming hot cores
Bok globules: almost-spherical clouds located outside of larger molecular complexes; possibly dense cores stripped of surrounding gas by stellar winds

thing	$T$ (K)	$M\;(M_\odot)$	$A_V$	$n$ (m$^{-3}$	$D$ (pc)
diffuse molecular cloud	$15\sim50$	$3\sim100$	$1\sim5$	$5 \times 10^8 \sim 5 \times 10^9$	$1\sim10$
giant molecular cloud	$\sim15$	$10^5\sim10^6$	$\gtrsim1$	$1 \times 10^8 \sim 3 \times 10^8$	$\sim50$
· dark cloud complex	$\sim10$	$\sim10^4$	$\sim5$	$\sim5 \times 10^8$	$\sim10$
· clump	$\sim10$	$\sim30$	$\sim10$	$\sim 1 \times 10^9$	$1\sim5$
· dense core	$\sim10$	$\sim10$	$\gtrsim10$	$\sim 1 \times 10^{10}$	$\sim0.1$
· hot core	$100\sim300$	$10\sim1000$	$50\sim1000$	$1 \times 10^{13} \sim 1 \times 10^{15}$	$0.05\sim0.1$
Bok globule	$10\sim$	$1\sim1000$	$\sim10$	$\gtrsim 1 \times 10^{10}$	$\lesssim1$

Interstellar cloud heating and cooling

Primarily heated by cosmic ray protons
- $E$ ranges $10–10^{14}$ MeV, with $10^3–10^8$ MeV common
- Ionizes H and H$_2$, ejecting e$^-$’s that distribute thermal energy throughout ISM via collisions with molecules
Also heated by UV starlight ionizing C and dust grains, X-ray starlight ionizing H, dust grains absorbing starlight, and shocks from supernovae and strong stellar winds
Primarily cooled by IR radiation from post-collision de-excitations, particularly those of C$^+$ and CO after colliding with H and H$_2$, respectively

Protostar formation

Protostars: pre-nuclear-burning objects formed from molecular clouds
Deviation from hydrostatic equilibrium means imbalance in the virial theorem: $2K < |U|$ ⇒ gravitational collapse
Jeans criterion, minimum necessary for spontaneous collapse:
- Jeans mass $M \approx \sqrt{ \left(5kT \over G\mu\right)^3 {3\over4\pi\rho_0} }$
- Jeans length $R \approx \sqrt{ 15kT \over 4\pi G\mu\rho_0}$
- Neglecting rotation, turbulence, magnetic fields, and pressure of surrounding gas
Bonnor–Ebert criterion accounts for external pressure $P_0$
- Bonnor–Ebert mass: $M = { c_BE v^4 \over \sqrt{P_0 G^3} }$
- Where isothermal sound speed $v \equiv \sqrt{kT\over\mu}$, and constant $c_\text{BE} \approx 1.18$
Low-pressure, optically thin first phase of collapse essentially isothermal and in free-fall
- Free-fall timescale $t = \sqrt{3\pi \over 32G\rho_0}$
Homologous collapse: Initially uniform spherical cloud would uniformly contract and condense
Inside-out collapse: Somewhat centrally condensed cloud will have shorter free-fall time and thus condense faster near center
Significant rotation results in disk-like structure
Magnetic pressure keeps clouds balanced; eddy braking slows collapse
- Critical mass with magnetic pressure: $M = {c_B \pi R^2 B \over \sqrt G}$
- Supercritical mass achieved by merger of subcritical clouds, or (more commonly) regional lessening of magnetic field
Ambipolar diffusion: neutral particles drift gravitationally, but are slowed by collisions with magnetically-frozen charged particles
Fragmentation: increasing density with constant $T$ reduces Jeans mass, causing density inhomogeneities to collapse locally, forming large numbers of small protostars in place of a single massive one
Initial mass function (IMF): number of stars formed per mass interval strongly mass-dependent, with abundance of low-mass stars
Increasing density purely adiabatically increases Jeans mass
Fragmentation stops as center becomes optically thick when $\sim10^{-10} \text{ kg}\;\text{m}^{-3}$, trapping radiation and making collapse more adiabatic
Central region reaches near hydrostatic equilibrium and becomes protostar
- Larger dust photosphere has radius and effective temperature where $\tau = {2\over3}$, putting star on H–R diagram
- ~10 AU wide for collapse from $1 M_\odot$ cloud

Protostar evolution

Surrounding material still in free-fall until meeting protostellar core, releasing significant kinetic energy at shock front as they slow from supersonic speeds
- Emits blackbody radiation predominantly in IR
- Luminosity increases with temperature for several thousand years
Dust begins to vaporize and opacity drops ~1000 K
- Substatially reduces photosphere radius to nearly the surface of the hydrostatic core
- Increases effective temperature with little change in luminosity
$\text{H}_2$ dissociates ~2000 K, absorbing thermal energy otherwise used to maintain hydrostatic equilibrium, triggering second collapse
Core radius decreases to $\sim1.3 R_\odot$ for collapse from $1 M_\odot$ cloud, while core mass $\ll 1 M_\odot$, indicating ongoing accretion at a second shock front
Hot core begins burning deuterium ($^2_1\text{H}$), producing ~60% of luminosity, which stays roughly constant
Deuterium burn-out leads to sharp drop in luminosity with small decrease in effective temperature, leading to pre-main-sequence star

Protostar observation

Direct observation shielded by thick dust of molecular cloud, and made rare by relatively short free-fall time
Small IR sources embedded in dense cores or Bok globules indicate collapse
Infalling material around embedded IR objects show twin spectral splitting from Doppler effect

Pre-main-sequence evolution

Increasing temperatures lead to H$^-$ opacity dominating in outer layers, creating envelope to become deeply convective
Hayashi track: convection constrains H–R path to near vertical line
- Luminosity decreases with slight increase in temperature as protostar slows collapse and reaches hydrostatic equilibrium
Formation of $\sim1 M_\odot$ stars:
- Completely convective from high H$^-$ opacity for the first $\sim10^6$ years of collapse, slowed little by scarce deuterium burning
- Rising central temperature decreases opacity and creates expanding radiative core
- Core begins generating energy from first steps in the PP I chain ($2 {\;}^1_1\text{H} \to {\;}^3_2\text{He}$) and CNO cycle ($^{12}_6\text{C} + {\;}^1_1\text{H} \to ^{14}_7\text{N}$),
- Surface luminosity increases with temperature again and moves star off Hayashi track
- Core burning comes to increasingly dominate luminosity generation
- CNO reactions give core steep temperature gradient and some convection
- Core expansion from intense nuclear energy production expends gravitational energy, lowering luminosity and effective temperature to main-sequence values
- After exhaustion of $^{12}_6$C for CNO, core reaches temperature for hydrostatic hydrogen burning via the full PP I chain
Formation of $\lesssim 0.5 M_\odot$ stars:
- Central temperature never reaches efficient $^{12}_6$C-burning temperatures for upward branch between Hayashi track and main sequence
- Fully convection: temperature stays sufficiently low and opacity sufficiently high to never develop radiative core
Formation of brown dwarfs ($0.013 \sim 0.072 M_\odot$):
- Low nuclear burning rate cannot form main-sequence star
- $\text{Li}$ burning $\gtrsim 0.06 M_\odot$
- Deuterium burning $\gtrsim 0.013 M_\odot$ (~13 Jupiter masses)
- Spectral types L and T
Formation of massive stars:
- High central temperatures leads to early, high-luminosity departure from Hayashi track
- Evolve nearly horizontally across H–R diagram
- Full CNO cycle becomes dominant H-burning mechanism, with steep thermal gradient keeping core convective
Zero-age main sequence (ZAMS): when stars begin equilibrium hydrogen burning

Stellar formation effects on medium

OB associations: star groups dominated by O and B main-sequence stars
Massive O and B protostars will first vaporize surrounding dust, then dissociate molecules, and finally ionize immediate surroundings into an $\text{H II}$ region within an $\text{H I}$ region
$\text{H II}$ regions: $\text{H I}$ ionized into $\text{H II}$ by protostar UV radiation fluoresce in visible light during recombination, creating emission nebulae in the visible spectrum during recombination
- Strömgren radius of $\text{H II}$ region: $r \approx \sqrt[3]{3N \over 4\pi\alpha n^2}$
- Where $N$ is rate of ionizing photon production, $\alpha$ is likelihood of recombination, and n is number density of protons and electrons
Radiation pressure from cluster of highly luminous O and B stars drives significant mass loss, disperses surrounding cloud (halting star formation), and weakens gravitational bound of star group (generally unbinding them)
Circumstellar disks
- Formed by spin-up of cloud transferring angular momentum away from collapsing star, possibly via stellar winds coupled to magnetic fields from within the convection zones
- Continuous spectrum from reflection of protostar light
- May be accretion or debris disks, possibly forming protoplanets
- Proplyds: protoplanetary disks $\gtrsim 10^{25} \text{ kg}$
Herbig–Haro objects: narrow beams of supersonic gas jets ejected from poles of young protostars
- Collision with ISM results in excitations, producing bright emission lines
T Tauri stars: low-mass ($0.5\sim2 M_\odot$) pre-main-sequence objects in intermediate phase between IR source and main-sequence star
- Large irregular variations in luminosity with timescales ~ days
- Strong H, Ca II, and Fe emission and Li absorption
- Often forbidden [O I] and [S II] indicative of extremely low gas densities
- Some lines have a P Cygni profile indicative of significant mass loss $\approx 10^{-8} M_\odot$/year: blueshifted absorption trough before an emission peak and a redshifted emission tail
- P Cygni profile sometimes inverts, indicating significant mass accretion in a very unstable environment
FU Orionis stars: T Tauri stars undergoing extreme mass accretion ($\approx 10^{-4} M_\odot$/year) and increase in luminosity ($\gtrsim4$ magnitudes)
- Circumstellar disk instabilities dump $\sim 0.01 M_\odot$ of material onto central star over ~100 years
- Inner disk outshines centural star by $100–1000\times$
- May occur to a T Tauri star several times over its lifetime
Herbig Ae/Be stars: type A and B pre-main-sequence stars with strong emission lines

Modifications to classical model

Classical model neglects rotation, turbulence, magnetic fields, initial inhomogeneities, strong stellar winds, ionizing stellar radiation, pressure-free protostellar collapse, likely smaller initial radii, and upper limits to massive star accretion,
Birth line: evolutionary theories with smaller initial radii place upper limit on observed protostar luminosity
Some observations suggest massive starts $\gtrsim 10 M_\odot$ may instead form from mergers or with accretion disk due to limiting feedback mechanism on non-rotational accretion
Classical model predicts inverse relation of collapse time and mass, implying massive stars would disperse surrounding cloud before any low-mass stars can form

13. Main Sequence and Post-Main-Sequence Stellar Evolution

Generally dominated by nuclear reaction timescale
Kelvin–Helmholtz timescale relevant when transitioning between nuclear sources

Main sequence (MS)

Stars $0.3\sim1.2 M_\odot$:
- Initially, core burns H, predominantly via PP chain
  - Radiative due to low temperature dependence
  - Core burning radius, density, and temperature rise due to fusion’s increase of mean molecular weight causing contraction and release of gravitational energy
  - Surface luminosity, radius, and effective temperature consequently rise over lifetime
- Temperature high enough to burn thick H shell around core immediately after core H depletion
  - Core now isothermal and predominantly He, growing as He ash rains down from shell
  - Shell H burning generates more power than core H burning, but some of it is converted into gravitational potential through slow expansion of envelope
  - Effective temperature decreases slightly
- Schönberg–Chandrasekhar limit, max mass fraction of an isothermal core in hydrostatic equilibrium supported by ideal gas pressure: ${M' \over M} \lesssim 0.37 \left(\mu\over\mu'\right)^2$
  - Where $M'$ and $\mu'$ are mass and mean molecular weights of the core
  - Derived from hydrostatic pressure: ${dP \over dM} = -{Gm \over 4\pi r^4}$
- After reaching SC limit, core contracts on Kelvin–Helmholtz timescale unless mass is low enough to be supported by temperature-independent electron degeneracy pressure: $P \propto \rho^{5\over3}$
Stars $\gtrsim 1.2 M_\odot$:
- Initially, core nearly homogeneous due to convective mixing
- Convection zone decreases in mass during H burning
  - Shrinks faster for more massive stars, disappearing entirely before H exhaustion for stars $\gtrsim 10 M_\odot$
  - Leaves slight composition gradient
- After near-exhaustion of core H ($X \approx 0.05$ for $M \approx 5 M_\odot$), entire star contracts on Kelvin–Helmholtz timescale while increasing luminosity and effective temperature

Subgiant branch (SGB)

Low-mass stars:
- Rapid core contraction releases gravitational energy, expanding envelope and decreasing effective temperature
- Raised H shell temperature and density rapidly increases power generation, again expanding envelope
Massive stars:
- Rising temperatures eventually trigger rapid ignition of thick H shell around core, forcing slight expansion of star envelope, momentarily decreasing luminosity and temperature

Red giant branch (RGB)

Low- and intermediate-mass stars:
- Near-surface convection zone created for due to $\text{H}^-$ ions increasing photospheric opacity
All stars:
- Star rises along Hayashi track as convection zone dominates star interior
- First dredge-up: convection sinks elements from surface ($\text{Li}$, $^{12}_6\text{C}$, etc) and surfaces products of nuclear processes ($^3_2\text{He}$, $^{14}_7\text{N}$, etc)

Red giant tip / start of helium fusion

Stars $\lesssim 1.8 M_\odot$:
- He core becomes strongly electron-degenerate
- Significant neutrino losses create negative temperature gradient near center
- Helium core flash: explosive initiation of triple-alpha process in He core in seconds
  - Core is rapidly lifted out of degeneracy outside-in, with strong temperature dependence of triple-alpha driving extreme thermal runaway
  - Briefly generates $\approx 10^{11} L_\odot$ of power, but most is absorbed in thermalizing core
All stars;
- Core expands from He burning, pushing H-burning shell outward
- Luminosity abruptly decreases due to cooling of H-burning shell
- Effective temperature begins to increase due to envelope contraction

Horizontal branch (HB) / blue loop

Stars $\lesssim 15 M_\odot$:
- Envelope contraction raises effective temperature and compresses H-burning shell, increasing power generation
- Deep convection zone rises towards surface while triple-α makes core convective as well
- Many develop instabilities in outer envelope, leading to periodic pulsations
- Increase in mean molecular mass eventually causes core contraction, expanding envelope and lowering effective temperature
- Depletion of core He rapidly accelerates core contraction
  - Core now mostly C and O, becoming very degenerate and cooling through significant neutrino production
- Contraction and heating of surrounding He shell initiates He shell burning, in turn forcing expansion and cooling of surrounding H shell, temporarily halting H shell burning
Stars $\gtrsim 15 M_\odot$ do not experience the blue loop

Early asymptotic giant branch (E-AGB)

All stars:
- He-burning shell generates most power while H-burning shell is nearly inactive
- Convective envelope absorbs energy and expands, lowering effective temperature
- Second dredge-up: convection deepens again, bringing He and N from interior onto H-rich envelope
- Star back on Hayashi track, approaching previous RGB path asymptotically from the left
- Rapid mass loss due to low surface gravity

Thermally-pulsating AGB (TP-AGB)

All stars:
- As He-burning shell nears exhaustion, H shell reignites
- He ash from H-burning shell builds up and makes base of He shell slightly degenerate, eventually triggering helium shell flash
- Convection zone established between He- and H-burning shells
- He-burning expands and cools H-burning shell, gradually turning it off
- Process repeats with growing amplitude after every pulse
- Cycle evident from abrupt changes in surface luminosity
- Period ranges from $\sim10^3 \text{ years}$ for stars $\sim5 M_\odot$ to $10^5 \text{ years}$ for stars $\sim0.6 M_\odot$
  - Long-Period Variables (LPVs) such as Mira have 100-700-day periods
Stars $\gtrsim 2 M_\odot$
- Third dredge-up: convection envelope merges with inter-shell convection zone and surfaces material from C-synthesizing region
- Inverting number ratio of O and C in stellar atmosphere creates carbon stars, with spectral type C overlapping the traditional K and M
  - Spectral type S inbetween M and C has approximately equal O and C abundances
- Mass loss in cool temperatures (~3000 K) expels into ISM silicate grains from O-rich atmospheres and graphite grains from C-rich atmospheres
- S-process nucleosynthesis: nuclei in deep interior capture neutrons produced by nuclear burning, at a slow enough rate to radioactively decay before their next capture
- S and C-type stars dredge up elements with no stable isotopes (e.g. Tc) into atmosphere, indicating active s-process

Late and post-AGB

Stars $\lesssim 8 M_\odot$:
- He-burning increases mass of CO core, causing contraction until electron degeneracy pressure dominates
- Mass loss accelerates with decreasing mass and increasing radius, developing $\sim10^{-4} M_\odot$/year superwind
  - Precise mechanism unknown
  - Energizes shroud of optically thick dust clouds into OH/IR sources: the OH molecules emit IR as masers
- Mass loss prevents catastrophic core collapse from Chandrasekhar limit, allowing stars between $4\sim8 M_\odot$ to additionally synthesize Ne and Mg in cores
- Surrounding dust cloud eventually becomes optically thin from expansion, revealing an F or G supergiant
- AGB phase ends as envelope is expelled, moving star nearly horizontally blueward
- Luminosity drops rapidly as H and He-burning shells lose pressure and extinguish, revealing hot, degenerate C–O or ONeMg core: a white dwarf
- Planetary nebula: expanding shell of gas around white dwarf
  - Emitting visible spectrum due to UV from central star remnant
  - Complex morphologies due to preferentially equatorial ejection
  - Receding at $10\sim30$ km/s, with character length scales of $\sim0.3$ pc
Stars $\gtrsim 8 M_\odot$: see Ch. 15

Stellar populations

Stars give ISM back material enriched with more heavy elements than given
Population III: stars formed soon after the Big Bang with virtually no metals ($Z = 0$)
- Found in extreme deep field observations to primordial galaxies
Population II: succeeding generations of metal-poor stars ($Z \gtrsim 0$)
- Found outside galactic disk and in globular clusters
Population I: current generations of metal-rich stars ($Z \approx 0.03$)
- Found inside galactic disk and in open clusters

Stellar clusters

Stars formed from the same cloud, with similar compositions and birth times
- Globular clusters: Population II, larger and older
- Galactic/open clusters: Population I, smaller and younger
Graphed onto color-magnitude diagrams: H–R diagrams with B–V indices rather than effective temperatures, and apparent instead of absolute magnitudes if distance is not known
Distance generally calculated with spectroscopic parallax
Isochrone: curve fitting all stars of a cluster at a certain age
Main-sequence turn-off point: color and magnitude where stars are currently leaving main sequence
- Reflects age of cluster as point becomes redder and less luminous over time,
Hertzsprung gap: absence of stars between late main sequence and red giant regions due to rapid Kelvin–Helmholtz-timescale intermediate evolution
Blue stragglers: group of stars found above turn-off point, possibly due to mass exchange with binary companion or collision between stars

14. Stellar Pulsation

Observations

Pulsating stars dim and brighten as radius and temperature change
Long-Period Variables (LPVs; prototype: $\omicron$ Ceti / Mira): thermally-pulsating asymptotic giants with $100\sim700$-day periods and somewhat irregular light curves
- May pulsate in either fundamental or first overtone mode
Classical Cepheids (prototype: $\delta$ Cephei): supergiant Ib stars with 1–50-day periods proportional to their average luminosities
- Period–luminosity relation (Leavitt’s law): $V = –2.81 \log_{10} P – 1.43$, $H = –3.234 \log_{10} P + 16.079$
  - Where $P$ is period in days
  - Measuring magnitude in the IR H-band mitigates some interstellar extinction
- Period–luminosity–color relation: $H = –3.428 \log_{10} P + 1.54(J – K_s) + 15.637$
  - Where J–K_s is an IR color index
  - Adding color term improves data fit
- Used as “standard candles” for measuring intergalactic distances
- Luminosity variation primarily due to $\sim1000$ K variation in temperature, with a phase lag of maximum luminosity occuring behind minimum radius
- Vast majority pulsate in fundamental mode
W Virginis stars: Population II Cepheids, around $4 \times$ less luminous than classical Cepheids of the same period
- Vast majority pulsate in fundamental mode
RR Lyrae stars: horizontal-branch Population II stars with 1.5–24-hour periods, all having nearly the same luminosity
- May pulsate in either fundamental or first overtone mode
$\delta$ Scuti variables: evolved F stars near the main sequence, with both radial and nonradial oscillations on 1–3-hour periods
ZZ Ceti stars: pulsating white dwarfs with 100–1000-second periods
Instability strip: narrow, nearly vertical region on H–R diagram, right of the main sequence, where most pulsating stars are found, including the above types
$\beta$ Cephei stars: luminous (class III–V) blue variables with 3–7-hour periods; found in H–R’s upper left, outside the instability strip

Radial pulsation mechanisms

Oscillations result from standing sound waves in interior
- Sound waves in medium with adiabatic index $\gamma$ travel at $v = \sqrt{\gamma P\over\rho}$
- No displacement at nodes; maximum displacement at antinodes
Period–mean density relation: $\Pi \approx \sqrt{3\pi\over2\gamma G\rho}$
- Denser stars (e.g. white dwarfs) pulsate faster (than e.g. supergiants)
Modes given by conical harmonics
- Fundamental mode: node at star center, antinode at surface
- Each overtone adds a node between center and surface
Surface amplitudes decreases with overtone: ${\delta r\over R} \approx 0.07$ for fundamental, $\lesssim 0.01$ for first, $\approx 0$ for second
Eddington modeled stars as heat engines: layers doing positive net work on surroundings drive oscillations; those doing negative net work dampen them
- Positive work done if layer absorbs heat around max compression; releases heat and reaches max pressure during expansion
Nuclear $\epsilon$-mechanism: compressing center raises temperature and density, increasing power generation
- Amplitude usually too small to drive pulsation
- May contribute to preventing formation of $\gtrsim 90 M_\odot$ stars
Opacity $\kappa$ and $\gamma$ mechanisms: compressing layer increases opacity and traps heat, driving expansion
- For most layers, opacity decreases with increased temperature from compression
- $\kappa$-mechanism: in partial ionization zones however, some compression energy goes into further ionization instead of direct heating, letting opacity increase with the higher density
- $\gamma$-mechanism: heat prefers to flow into partial ionization zone in compression due to its lower relative temperature, reinforcing the $\kappa$-mechanism
Most stars have two main ionization zones
- Hydrogen partial ionization zone: broader and closer to surface; ionizes $\text{H I} \to \text{H II}$ and $\text{He I} \to \text{He II}$ at characteristic temperature of $1–1.5 \times 10^4 \text{ K}$; changing depths during pulsation accounts for phase lag in classical Cepheids and RR Lyrae
- He II partial ionization zone: narrower and deeper; ionizes $\text{He II} \to \text{He III}$ at characterstic temperature of $4 \times 10^4 \text{ K}$; primarily responsible for driving oscillations within instability strip
Temperature-dependent depths of ionization zones determine pulsation properties:
- Blue edge of instability strip (~7500 K): high temperatures puts ionization zone too close to low-density surface, where there is insufficient mass to effectively drive oscillations
- First overtone may be excited ~6500 K
- Fundamental mode takes hold ~5500 K
- Red edge of instability strip (~5000 K): heat transfer by convection bypasses high-opacity ionization zones and quenches pulsation
$\beta$ Cephei stars pulsate due to iron ionization zone
- Effective temperature (20,000~30,000 K) too high for H and He ionization zones
- $\kappa$ and $\gamma$ mechanisms rely on $\text{Fe}$ opacity bump near $10^5 \text{ K}$

Pulsation model

Newton’s second law must be used instead of hydrostatic equilibrium model to account for oscillation of mass shells: $\rho {d^2r \over dt^2} = - {GM\rho \over r^2} - {dP \over dr}$
Nonlinear evaluation can model complex, nonsinusoidal behavior of large-amplitude pulsations, but is very computationally expensive
Linearizable by approximating with small amplitudes, but results in sinusoidal oscillations with no amplitude information
Adiabatic approximation also used to minimize complexity, but nonlinear, nonadiabatic models are necessary for some variable stars
One-zone linear, adiabatic model: $\Pi = {2\pi \over \sqrt{{4\pi \over 3} G\rho(3\gamma–4) }}$
Dynamic stability: star collapses if $\gamma < {4\over3}$

Nonradial pulsation mechanisms

Some regions of surface expand while others contract
Oscillations result from standing sound waves with latitudinal nodal circles and traveling sound waves with longitudinal nodal circles
- No displacement at nodal circles
Angular modes given by real parts of spherical harmonic functions: $Y^m_l(\theta, \phi)$
- There are $l$ nodal circles: $\lvert m\rvert$ longitudes and $l - \lvert m \rvert$ latitudes
- Where $l \in Z^+$ and $m \in [-l, l]$
- Traveling waves take $\vert m\rvert\cdot\Pi$ long to travel around star, with direction dependent on sign of $m$
Pressure waves: p-modes
- Confined to low depths, revealing conditions in stellar surface layers
- Both radial and angular nodes
- Acoustic frequency $S = \sqrt{\gamma P \over \rho} {\sqrt{l(l+1)} \over r}$
- Frequencies split by prograde and retrograde wave motion proportional to star rotation rate: $\Delta S \propto m\Omega$
Internal gravity waves: g-modes
- Reveals movement of stellar material in deep interior
- Only have angular nodes
- Brunt–Väisälä bouyancy frequency: $N = \sqrt{–Ag}$
- Confined to radiative zones, where $A < 0$: $A \equiv {1\over\gamma P} {dP \over dr} – {1\over\rho} {d\rho \over dr}$
Surface gravity waves: f-modes
- Frequency inbetween p- and g-modes

Helioseismology and asteroseismology

All observed solar oscillations are in the p-mode, with 3–8-minute periods and very short (high $l$) horizontal wavelengths
Likely driven by turbulent energy of convection zone
Latitide-dependent rotation rate revealed by m-dependent frequency splitting
Depth-dependent rotation rate revealed by $l$-dependent attenuation below convection zone
Thick convection zone prevents surface observation of of g-modes
$\delta$ Scuti stars tend to pulsate in low-overtone radial modes, low-order p-modes, and possibly g-modes
Rapidly oscillating peculiar A stars (roAp) primarily pulsate in higher-order p-modes, with the pulsation axis aligned with their strong magnetic fields instead of the rotation axis

15. The Fate of Massive Stars

Post-main-sequence evolution

$M\;(M_\odot)$	path to supernova
$\gtrsim85$	O → Of → LBV → WN → WC → SN
$40\sim85$	O → Of → WN → WC → SN
$25\sim40$	O → RSG → WN → WC → SN
$20\sim25$	o → RSG → WN → SN
$10\sim20$	O → RSG → BSG → SN

Luminous blue variables (LBVs; prototype: S Doradus): massive stars whose brightness suddenly erupt from time to time
- Effective temperatures between $15\,000\sim30\,000$ K, masses $\gtrsim 85 M_\odot$, and luminosities $\gtrsim10^6 L_\odot$ approaching the Eddington limit
- Possible mechanisms behind behavior include envelope mass loss from temporarily exceeding Eddington limit, atmospheric pulsation instabilities, and binary companions
Wolf–Rayet stars (WR): very hot, rapidly rotating stars with unusually strong broad emission lines and high mass loss
- Effective temperatures between $25\,000\sim100\,000$ K, masses $\gtrsim 20 M_\odot$, mass loss rates $\gtrsim 10^{-5} M_\odot/\text{year}$, and equatorial rotation speed ~300 km/s
- Atypical spectral composition due to mass loss progressive stripping away outermost layers: WNs emission lines dominate in He and N; WCs in He and C; WOs in O
Blue supergiants (BSG)
Red supergiants (RSG)
- Humphreys–Davidson luminosity limit: the most massive stars never evolve to RSG portion due to max luminosity cutoff
Of stars: O supergiants with pronounced emission lines

Supernova observation and classification

Recorded Milky Way supernovae: SN 1006, SN 1054 with Crab supernova remnant, Tycho’s supernova SN 1572, and Kepler’s supernova SN 1604
SN 1987A in Large Magellanic Clouds (50 kpc from Earth) first nearby supernova in age of modern astronomy
Classified by spectra at maximum light and light curve shape:
- Type I supernova: no H lines
  - H envelope already stripped before SN
  - Dimming $\sim 0.065$/day at 20 days, slowing to a constant rate after $\sim50$ days
  - Type Ia: Si II lines
    - Found in all galaxy types, including low-star-forming ellpticals
    - $M_B \approx -18.4$
    - Dimming $\sim0.015$/day after $\sim50$ days
  - Type Ib/Ic: no Si II lines
    - Only found in star-forming H II regions of spirals
    - $M_B \approx -16$
    - Dimming $\sim0.010$/day after $\sim50$ days
    - Type Ib: He lines
    - Type Ic: no He lines
- Type II supernova: H lines;
  - Type II-P: plateau
    - Temporary plateau in light curve between $30\sim80$ days
    - Occurs $\sim10\times$ as often as Type II-L
  - Type II-L: linear

Core-collapse supernovae

Type Ib, Ic, and II involve collapse of massive, evolved stellar core
He burning eventually adds enough mass and density to CO core to start burning C, creating O–Ne core
O–Ne then starts burning O, creating $^{28}_{14}$Si-dominated core
Si burning at $\sim3\times10^9$ K creates iron core with host of heavy nuclei centered near $^{56}_{26}$Fe
Onion-like interior: Fe core ) Si-burning ) Si-rich ) O-burning ) O-rich ) C-burning ) C-rich ) He-burning ) He-rich ) H-burning ) H–He envelope
Less energy released per mass, thus shorter timescale, with each successive fuel

core-burning fuel	duration for $20 M_\odot$ star
H	$\sim10^7$ years
He	$\sim10^6$ years
C	$\sim300$ years
O	$\sim200$ days
Si	$\sim2$ days

Extreme core temperature ($T_c\approx8\times10^9$ K) leads to photodisintegration: high-energy photons destroy heavy nuclei
- $^{56}_{26}\text{Fe}+\gamma\to13{\;}^4_2\text{He}+4\text{n}$, $^4_2\text{He}+\gamma\to2\text{p}^++2\text{n}$
- Highly endothermic, removing thermal pressure
Extreme core temperature and density ($\rho_c\approx10^{13}$ kg/m$^{3}$) leads to heavy nuclei and photodisintegrated protons capturing free electrons, removing electron degeneracy pressure as well
Loss of hydrostatic pressure causes core to collapse rapidly
Inner core collapses homologously, decoupling from outer core where velocity becomes locally supersonic
Collapse continues until density exceeds $8\times10^{17}$ kg/m$^{3}$ and strong force suddenly becomes repulsive due to neutron degeneracy pressure, sending shock waves to infalling outer core
Supersonic outer core collapses in near free-fall $\sim70\,000$ km/s until heating and photodisintegrating upon encountering shock front
Shock loses energy and stalls, becoming accretionary
Extreme density traps neutrinos below in neutrinosphere
- If neutrino energy reheats base of shock sufficiently quickly, shock resumes traveling outward
- Otherwise, material in shock falls back onto core and no explosion occurs
Shock propels nuclear products and envelope material to kinetic energies $\sim10^{44}$ J
Material becomes optically thin at $\sim10^3$ m $\approx100$ AU, releasing $\sim10^{42}$ J $\approx 10^9 L_\odot \approx L_\text{galaxy}$ of light
$\sim3\times10^{46}$ J of neutrinos released as well
- SN 1987A’s neutrinos traveled relativistically and arrived 3 hours before its photons, placing rest mass limit $m_{\nu_e} \leq 16$ eV
Neutron star forms if remnant of inner core is small enough (neglecting rotation, $m\lesssim2.2M_\odot$ from $M_\text{ZAMS} \lesssim 25M_\odot$) to stablize from neutron degeneracy pressure; collapses into singularity and produces black hole otherwise
Nebulae emit highly polarized synchotron radiation, with electrons supplied by center pulsar
Spectral differencies of types II, Ib, and Ic due to composition and mass of envelope prior to, and radioactive material synthesized during core collapse
- Type II usually from RSGs in H–R’s extreme upper left
  - SN 1987A was subluminous by $\sim3.5$ magnitudes relative to typical Type II supernovae due to being a denser BSG in the blue loop
- Type Ib and Ic likely from WN and WC Wolf–Rayets, respectively
Light curve plateau of Type II-P largely caused by prolonged recombination of H-rich envelope after shock ionization; further supported by radioactive decay of $^{56}_{28}$Ni in shock front
- Light curve slope may be matched to decay of specific radioisotopes

Supernova nucleosynthesis

Solar lithium problem: solar convection zone predicted by stellar models too shallow to adequately burn Li, yet meteorites show primordial Li was $100\times$ the current solar Li abundance
Synthesis of heavy-nuclei by charged particles extremely difficult due to Coulomb barrier, but neutron capture can occur at relatively low temperatures
- Insignificant in energy production, but accounts for abundance ratios of nuclei $A>60$
- Slow s-process yielding stable nuclei if $t_{\beta\text{-decay}} \ll t_\text{n-capture}$; tends to occur in normal phases of stellar evolution
- Rapid r-process yielding neutron-rich nuclei if $t_{\beta\text{-decay}} \gg t_\text{n-capture}$; tends to occur in supernovae

Gamma-ray bursts (GRB)

Brief shower of gamma-ray photons between $1\sim10^6$ keV
Rise time of $\sim10^{-4}$ s followed by exponential decay totaling $10^{-2}\sim10^3$ s
Characteristic length $l = c\cdot t_\text{rise}$ roughly size of neutron star
Emission between $350\sim500$ keV likely gravitationally redshifted pairs of 511-keV photons produced by e$^-$–e$^{+}$ annihilation on neutron star surface
Emission between $20\sim60$ keV likely cyclotron lines from pulsar magnetic field $\sim10^8$ T
Burst distribution isotropic throughout sky, but has radial edge corresponding to extragalactic distances
Total energy released comparable to that of core-collapse supernovae
- Fluence $S$: total energy released per detector surface area
Short–hard GRBs: $\lesssim2$ s, associated with neutron star–neutron star or neutron star–black hole mergers
Long–soft GRBs: $\gtrsim2$ s, associated with supernovae

Long–soft GRBs

Traced to the same locations as some supernovae
Relativistic jets: beams of highly relativistic matter (Lorentz factor $\gamma$ possibly $\gtrsim100$)
- For $\gamma\gg1$, beam cone has half-width angle $\theta\propto1/\gamma$
- Doppler/relativistic beaming: if observer within beam, observed energy and collimation will be greater than what is actually emitted
- Models must avoid slow-down from relativistic speeds after encountering too much (cumulative rest mass $m_0\gtrsim\gamma m_\text{jet}c^2$) baryon-rich material from envelope
Hypernova/collapsar model: black hole formed directly from supernova
- Debris disk and associated magnetic fields collimate relativistic jets emanating from center of supernova
- Gamma rays emitted as jets encounter infalling stellar envelope
Supranova model: rotating neutron star with $M \gtrsim2.2M_\odot$ spins down over weeks to months and collapses into black hole
- Stellar envelope has already been cleared away by supernova
- Debris disk and associated magnetic fields produce both relativistic jets and a GRB

Cosmic rays

Penetrating radiation of charged particles comprising wide range of classes (), masses, and energies ($10^7\sim10^{20}$ eV)
Subatomic particles including e$^\pm$, p$^+$, $\mu^\pm$, and atomic nuclei including C, Ne, Mg, Si, Fe, Ni
Solar cosmic rays relatively low energy, with $\sim750$ km/s protons in fast solar wind having $E\approx 3$ keV, and $\sim0.1c$ protons from CMEs having $E\approx10$ MeV
Supernova-origin cosmic rays travel relativistically $\sim c$ with $E\lesssim10^{16}$ eV
Charged particles orbit magnetic fields at Larmor radius / gyroradius $r = {\gamma m v\over q B}$
- While trapped in field, particle is accelerated to relativistic speeds by successive collisions with advancing shock wave
- With sufficient energy $\sim10^{15}$ eV, $r$ exceeds size scale of supernova remnant $\sim1$ pc and particle is likely ejected
Possible origins for highest-energy cosmic rays;
- Acceleration near neutron stars or black holes
- Collisions involving intergalactic shocks
- Emission from active galactic nuclei around supermassive black hole

16. The Degenerate Remnants of Stars

White dwarfs (WD)

Manufactured in cores of low- and intermediate-mass stars ($M_\text{ZAMS}\lesssim8M_\odot$)
Low luminosity despite high central temperature indicate lack of fuel for thermonuclear reaction
Primarily comprises completedly ionized C and O nuclei, but extreme gravity stratifies interior into CO core ) He layer ) H layer over $\sim100$ years
$0.4\sim0.7M_\odot$, $\sim R_\oplus$, $\rho\approx10^9$ kg/m$^3$, $g\approx 10^6$ m/s$^2$, $T\approx5000\sim80\,000$ K
Interior nearly isothermal due to energy transfer via highly efficient, degenerate electron conduction; steep temperature drop in nondegenerate envelope drive convection mixing in surface layers
Thermal energy primarily resides in kinetic energy of nuclei, as nearly all electrons are in degenerate energy levels
Core cools at $T_\text{core}(t) = T_0\left(1+{5\over2}{t\over\tau_0}\right)^{-2/5}$
- Luminosity $L\propto T_\text{core}^{7/2} \propto T_\text{eff}^4$: interior cools faster than surface
- At $\sim10^{-4}L_\odot$, nuclei begin crystallizing, slowing cooling
Spectral lines significantly broadened due extreme pressure near surface
Spectral type D (dwarf)
- DA: only H absorption lines; $\sim70\%$ of population
- DB: only He absorption lines; $\sim8\%$ of population
- DC: featureless continuum; $\sim14\%$ of population
- DQ: additional C lines
- DZ: additional metal lines
- DO: in between O type and WD
- PNN: planetary nebula nuclei; WD formation phase
- Convective mixing may transform WD between spectral subtypes
Pulsating white dwarfs: on instability strip with periods of $100\sim1000$ s
- Multiple periods with $3\sim125$ different simultaneous frequencies
- Nonradial g-modes resonating in H and He layers cause surface temperature variations without much effect on radius
- DAV stars (prototype: ZZ Ceti): $T_\text{eff}\approx12\;000$ K; resonance driven by H partial ionization zone
- DBV stars: $T_\text{eff}\approx27\;000$ K; resonance driven by He partial ionization zone
- DOV and PNNV stars: $T_\text{eff}\approx10^5$ K

Degenerate matter

Degeneracy of electron fermion gas responsible for maintaining hydrostatic equilibrium in white dwarfs
Fermion gas: obeying the Pauli exclusion principle, only one particle is allowed per state
- At $T = 0$ K, all energy states below the Fermi energy $\varepsilon_F = {\hbar^2\over2m} (3\pi^2n)^{2/3}$ are occupied and the fermion gas is fully degenerate
- For $T > 0$ K, fermion gases are still approximately degenerate if average thermal energy ${3\over2}kT < \varepsilon_F$; for an electron gas, ${T\over\rho^{2/3}}<1261$ K m$^2$ kg$^{-2/3}$
Nonrelativistic degeneracy pressure $P = {(3\pi^2)^{2/3}\over5} {\hbar^2\over m} n^{5/3}$
- Dynamically stable: $P\propto \rho^\gamma$ where adiabatic index $\gamma={5\over3}$
- Yields nonrelativistic mass–volume relation: $V\propto {1\over M}$ and $\rho\propto M^2$
- Largely temperature independent, leading to He flash in degenerate cores
Electrons become relativistic at $\rho\approx10^9$ kg/m$^3$
As $v_e\to c$ at high $M$, adiabatic index $\gamma\to{4\over3}$ and WD becomes dynamically unstable
Chandrasekhar limit: maximum mass supported by electron degeneracy, $M_\text{Ch}\approx0.44M_\odot$

Neutron stars (NS)

$\sim10^{57}$ extremely densely packed neutrons with $\sim M_\text{Ch}$ , $R\approx10\sim15$ km , $\rho\approx10^{17}$ kg/m$^3$, $g\approx10^{12}$ m/s$^2$
Rapid rotation on order of milliseconds to seconds guaranteed due to role in countering collapse
Extremely hot formation temperatures $\sim10^{11}$ K cooled in days to $T_\text{core}\approx 10^9$ K by internal neutron–antineutrino radiation via URCA process: $\text{n}\to\text{p}^++\text{e}^-+\overline{\nu}_\text{e}$, $\text{p}^++\text{e}^-\to\text{n}+\overline{\nu}_\text{e}$
Other neutrino emission processes cool $T_\text{core}$ to $\sim10^9$ K over $\sim10^3$ years, then surface photon radiation (primarily in X-ray) cools $T_\text{core}$ to $\sim10^8$ K and $T_\text{eff}$ to $\sim10^6$ K over $\sim10^4$ years
Outer crust: for low $\rho$ near surface, iron nuclei minimize nucleon energy and electron degeneracy pressure dominates
Neutronization: above $\rho\gtrsim10^{12}$ kg/m$^3$, capture of relativistic electrons $\text{p}^++\text{e}^-\to\text{n}+\nu_\text{e}$ minimizes nucleon energy and produces increasingly neutron-rich nuclei
- Additional neutrons reduce electric repulsion between protons
Inner crust: above $\rho\gtrsim4\times10^{14}$ kg/m$^3$, neutron drip minimizes nucleon energy, creating superfluid of free neutrons outside of the nuclei lattice
- Superfluid: free, degenerate fermions pair up into bosons, which flow without viscosity
Above $\rho\gtrsim4\times10^{15}$ kg/m$^3$, neutron degeneracy pressure dominates
Core: as $\rho\to\rho_\text{nucleus}\approx2\times10^{17}$ kg/m$^3$, nuclei effectively dissolve into superfluid of free neutrons, protons, and electrons; as $\rho$ increases further, $\text{n}:\text{p}^+:\text{e}^-\to8:1:1$
- Superfluid of charged particles are superconducting, flowing without electrical resistance
- Conducting fluids “freeze-in” magnetic field lines, creating magnetic fields $B\approx10^8\sim10^{10}$ T
Physics poorly understood above $\rho>\rho_\text{nucleus}$ at the deep interior
Above $\rho>2\rho_\text{nucleus}$, neutrons spontaneously decay $\text{n}\to\text{p}^++\pi^-$, producing other elementary particles

Pulsars

First discovered by Bell in 1967: two independent interstellar sources of regularly spaced radio pulses
Designated PSR (pulsating source of radio) right ascention + declination
Most periods range $0.25\sim2$ s, spinning down very gradually with ${\dot P}\approx10^{-15}$
- X-ray pulsars have the longest periods and fastest spindown
- High-energy pulsars emitting radio to IR or higher frequencies have typical periods but faster-than-average spindown
- Young millisecond pulsars ($P\lesssim10$ ms) emit radio to gamma rays and display glitches: periods abruptly spinup every several years, possibly from crust shrinking, or separation of core from crust
- Some pulsars, primarily those in binary systems, are speedin up ($\dot P<0$)
Only explainable by rapidly rotating neutron stars
- Oscillation modes of white dwarfs are too long and oscillation modes of neutron stars are too short
- Rapid period rules out all binary systems larger than neutron stars; increasing period rules out inspiraling radiation of NS binaries
Much faster movements through space than normal stars, and rarity of pulsar binary systems, indicate ejection from asymetric core-collapse supernovae
Rotational kinetic energy transformed into accelerating expansion of surrounding nebulae, maintaining strength of magnetic field, and injecting relativistic electrons into the field
Acceleration of relativistic electrons along magnetic field lines (curvature radiation) or circling around them (synchotron radiation) produce light strongly linearly polarized in the plane of acceleration
Individual pulses of a pulsar vary substantially, but integrated pulse profile averaged across 100s of pulses is very stable
- Each pulse comprises several brief subpulses, either distributed randomly in the main pulse window or drifting across the profile
- Sharpness disperses with distance in the interstellar medium, as longer wavelengths slow more below $c_\text{vacuum}$
- Some pulsars abruptly switch between multiple integrated pulse profiles; some pulsars temporarily cease emitting pulses for some periods
Magnetic poles are off-axis and have field strength $B={1\over2\pi R^3\sin\theta}\sqrt{3\mu_0 c^3 I P \dot P\over 2\pi}$
- Where $\theta$ is pole inclination relative to axis of rotation and $I$ is moment of inertia
Rotation rapidly changes B field, inducing changes in E field and generating EM waves radiating away from star as magnetic dipole radiation $\dot E=-{32\pi^5B^2R^6\sin^2\theta\over3\mu_0c^3P^4}$
- Fast-spinning neutron stars lose energy much more quickly
Powerful induced surface E field $\sim6.3\times10^{10}$ V/m rips charged particles from polar regions
- Emit energetic curvature radiation, some of which undergo pair production $\gamma\to\text{e}^-+\text{e}^+$ and generate more charged particles, creating a cascade of subpulses beamed in narrow cone around pole
- Capture by magnetic fields form co-rotating magnetosphere which detaches at light cylinder $R_c=c/\omega$ as a pulsar wind, replenishing charged particles and magnetic field in surrounding nebula
Pulse mechanism may turn off when rotation slows down too much to sustain strong magnetic field, or pulses may simply become too faint

Magnetars

Extremely magnetic neutron stars with $B\approx10^{11}$ T
Stresses in magnetic field cause surface to crack, producing super-Eddington ($10^3\sim10^4L_\text{Eddington}$) releases of energy
Explains soft gamma repeaters (SGRs): objects associated with fairly young ($\sim10^4$ years-old) supernovae emitting bursts of hard X-ray and soft gamma-ray
Relatively slow rotation periods ranging $5\sim8$ s: magnetic field primarily governs behavior rather than rotation

17. General Relativity and Black Holes

General relativity

Geometric description of how intervals in spacetime are measured in the presence of mass and energy
“Mass tell spacetime how to curve, spacetime tells mass how to move”
Spacetime curvature increases distance between points and slow down time
Weak equivalence principle: gravitational charge $m_g$, which determines the force of gravity, is directly proportional to inertial mass $m_i$, which exists even without gravity; thus, every object falls with the same acceleration
Principle of equivalence: All locally inertial (freely falling and nonrotating) frames of reference are physically equivalent, related by the Lorentz transformations between their instantaneous relative velocity
- Only local, as gravity pulls in different magnitudes and directions in different places
Gravitational bending of light: in free-fall reference frame, photons travel in straight lines; observed from the ground, photons curve down with the inertial frame
Gravitational time dilation and gravitational redshift: in free-fall reference frame, upward photons do not lose energy; observed from the ground, the photon took less time and traveled less distance in the inertial frame
- Time dilates and space streches inside gravitational wells
- ${\Delta\nu\over\nu_0}\approx-{gh\over c^2}$ for small $g$
- ${\Delta t_0\over\Delta t_\infty}={\nu_\infty\over\nu_0}=\sqrt{1-{2GM\over r_0c^2}}\approx 1-{GM\over r_0c^2}$

Intervals and geodesics

Einstein field equations of form $\mathcal G = -{8\pi G\over c^4} \mathcal T$ calculates the Einstein tensor $\mathcal G$, describing the curvature of spacetime, from the stress–energy tensor $\mathcal T$, describing the distribution of mass and energy
Event $s$: a point in spacetime with coordinates $(x, y, z, t)$
Worldline $\mathcal W$: path of object moving through spacetime
Spacetime interval $(\Delta s)^2 = c(\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2$
- The “distance” between two events, invariant under Lorentz transformations
- Timelike if $(\Delta s)^2 > 0$
  - Enough time for light to travel between the events
  - Inertial frames exist in which they happen in the same location, separated by proper time $\Delta\tau\equiv\Delta s/c$
- Null/lightlike if $(\Delta s)^2 = 0$
  - Only light can travel between the events
- Spacelike if $(\Delta s)^2 < 0$
  - Too much space for light to travel between the events
  - Inertial frames exist in which they happen simultaneously, separated by proper distance $\Delta\mathcal L = \sqrt{-(\Delta s)^2}$
Light cones: regions of spacetime with causal (timelike/lightlike) connection to an event
- Past light cone: region of past spacetime whose light can reach the event; everything that could influence the event
- Future light cone: region of future spacetime reachable by light from the event; everything the event could influence
- Boundaries are event horizons, dividing the knowable from the unknowable
Metric: formula for measuring differential distance along worldlines, defining $(ds)^2$ such that $\Delta s(\mathcal W) =\int_{\mathcal W}\sqrt{ds)^2}$
- Minkowski metric for flat spacetime: $(ds)^2 = (c\;dt)^2 - (d\ell)^2$, where $(d\ell)^2=(dx)^2+(dy)^2+(dz)^2 = (dr)^2 + (r\; d\theta)^2 + (r \sin\theta \; d\phi)^2$
- Swarzchild metric for single, nonrotating mass at origin: $(ds)^2=\left(c\; dt\sqrt{1-{2GM\over rc^2}}\right)^2-\left(dr\over \sqrt{1-{2GM\over rc^2}}\right)^2 - (r\; d\theta)^2 - (r \sin\theta \; d\phi)^2$
  - Time dilation in $dt$ term and streching of space in $dr$ term
Coordinate speed ${d\ell\over dt}$: rate of change of spatial coordinates
All freely-falling particles follow geodesics: the straightest, shortest possible worldlines
- Massive particles follow timelike geodesics with a locally extremum interval, with either the maximum or minimum $\Delta s$ out of nearby worldlines
  - Ex: orbiting satellites follow maximum interval geodesics in Schwarzchild metric
- Massless particles follow the null geodesic with $\Delta s=0$

Black holes (BH)

Swarzchild radius $R_S = {2GM\over c^2}$: distance from nonrotating mass where escape speed is $c$
At $R_S$, proper time $d\tau$ and coordinate speed of light moving radially ${dr\over dt}$ are $0$: nothing ever happens and light appears to freeze
Black hole: star that collapsed within its $R_S$ , with a physical singularity ($V=0, \rho\to\infty$) at the center enclosed by an event horizon at $R_S$
From distant observer, objects falling into BH emit increasingly time-dilated and redshifted signals
From free-falling reference frames, gravitational differential result in tidal stretching
Below $r<R_S$, nothing can remain at rest and all worldlines converge at singularity
No hair theorem: any black hole can be completely described by its mass, angular momentum, and electric charge
Frame dragging: rotating masses induce rotation in surrounding spacetime; creates ergosphere around rotating black holes within which all particles are forced to move in the direction of rotation
Possible upper limit on angular momentum $L\le{GM^2\over c}$, above which a naked singularity would be exposed
Stellar-mass black holes ($3\sim5\,M_\odot$)
- Formed from direct core-collapse (collapsar), spin-down of neutron star (supranova), or accretion on neutron star from binary companion
Intermediate-mass black holes (IMBH; $10^2\sim10^4\,M_\odot$)
- Formation process unclear; possibly from collapse of supermassive merged star, or merger of stellar-mass BHs
- Observed as ultraluminous X-ray sources (ULXs), correlated with cores of globular clusters and low-mass galaxies
Supermassive black holes (SMBH; $10^5\sim10^9\, M_\odot$)
- Formation process unclear; possibly from galaxy mergers, or growth of IMBHs
- Located at centers of most galaxies
Primordial black holes ($10^{-8}$ kg $\sim10^5\,M_\odot$)
- May have existed in the earliest instants of the universe
Schwarzchild throat / Einstein–Rosen bridge: vacuum solution of field equations with a tunnel at event horizon of nonrotating BH
- Collapses before anything can cross
Wormhole: nonvacuum solution to field equations connecting arbitrary points in spacetime with no event horizon and survival tidal forces
- Requires exotic material (with negative energy density $\rho c^2<0$) to prevent collapse
Hawking radiation $\dot E\propto1/M^2$: pair production near event horizon allows gravitational energy to escape, culminating in high-energy burst of gamma rays and elementary particles at death of black hole

18. Closed Binary Star Systems

Gravitation

Large majority of stars are in multi-star systems, but most of them are detached binaries: sufficiently far apart ($a\gg\max(R_1, R_2)$) to evolve independently
In close systems ($a\lesssim\max(R_1,R_2)$), tidal forces induce pulsation, dissipating orbital and rotational energy until tidal lock is achieved
In corotating center-of-mass reference frame, effective gravitational potential $\Phi=-G\left({M_1\over s_1}+{M_2\over s_2}\right)-{\omega^2r^2\over 2}$
- Where $s_1$ and $s_2$ are distances to the two masses, and angular frequence $\omega={G(M_1+M_2)\over a^3}$
Hydrostatic equilibrium ensures constant pressure and density along equipotential surfaces; expanding stars fill successively larger ones
Gravitational and centrifugal forces balance at Lagragian points, where $\vec F=-m\nabla\Phi=0$
- Local minima are stable; local maxima and saddle-points are unstable
Inner Lagrangian point $L_1$ located between the masses with distances to each $\lbrace \ell_1, \ell_2 \rbrace \approx a\left({1\over2}\pm0.227\log_{10}{M_1\over M_2}\right)$
- Equipotential surfaces meet at $L_1$ as teardrop-shaped Roche lobes
Semidetached binary: secondary star (with $M_2$ not necessarily more massive) fills its Roche lobe and loses mass to the companion primary star
Contact binary: both stars fill (or even expand beyond) their Roche lobes and share a common atmospheric envelope
Mass transfer rate $\dot M\approx\pi R\rho d\sqrt{3kT\over m_\text{H}}$
- Where $R$ is maximum Roche lobe radius and $d$ is overfill distance
- Changes orbital period and orbital separation ${da\over dt}=2a\dot M_1{M_1-M_2\over M_1M_2}$
Infalling material release $\dot E={GM\dot M\over R}$, or $\sim20\%$ of rest mass in energy when falling into neutron stars, compared to $\lesssim1\%$ falling into white dwarfs

Accretion disks

If primary star of semidetached binary has radius $r\lesssim0.05a$, mass stream from secondary will miss its surface and enter orbit at $r_0=a\left(\ell_1\over a\right)^4\left(1+{M_2\over M_1}\right)$
Accretion disk develops; further mass stream impacts at disk hot spot
Viscosity converts orbital kinetic energy into random thermal motion, causing disk mass to inspiral and eventually fall to surface
- Source of strong viscosity poorly understood; possibly convection turbulence or magnetohydrodynamic instabilities
Radius $R_\text{disk}\approx2r_0$ due to conservation of angular momentum: most disk material spiral inward while some spiral outward
If optically thick, temperature $T_\text{disk}(r)=T_\text{0}\sqrt[4]{\left(R\over r\right)^3 \left(1-\sqrt{R-r}\right) }$
- Where characteric temperature $T_0\equiv\sqrt[4]{3GM\dot M\over 8\pi\sigma R^3}\approx 2\max (T_\text{disk})$
Luminosity $L_\text{disk}={GM\dot M\over 2R}={1\over2}\dot E$, with the other half delivered to the primary star
- White dwarf $L_\text{disk}\approx0.22\,L_\odot$, emitting primarily in UV
- Neutron star $L_\text{disk}\approx2400\,L_\odot$, emitting primarily in X-ray
Observational evidence: light curves of eclipsing binaries show steadily rising temperatures toward center, and hot spot on one side of disk

Classification

Algols: two main-sequence or subgiant stars in semidetached binary
- W Serpens stars: active Algols, studied for rapid stages of stellar and binary evolution
- Mass loss may enrich interstellar medium
RS Canum Venaticorum stars & BY Draconis stars: chromospherically active binaries with enhanced magnetic activity; studied for dynamo mechanism in cool (type F and later) stars
W Ursae Majoris contact systems: short-period ($0.2\sim0.8$ days) contact binaries with very high magnetic activity; studied for dynamo mechanism at extreme levels
- Magnetic braking may coalesce them into single stars
Cataclysmic variables (CV) and nova-like binaries: white dwarfs and M-type stars in short-period semidetached binaries; studied for final stages of stellar evolution and accretion phenomena
X-ray binaries: neutron stars or black holes accreting gas from nondegenerate companions and emitting energetic X-rays $L\gt10^{28}$ W; evidence for existence of black holes
$\zeta$ Aurigae systems & VV Cephei systems: G–M-type supergiants and $\sim$B-type companions in long-period interacting binaries; eclipses studied for supergiant atmospheric properties
- Became interacting binaries when more massive star evolved into supergiant
Symbiotic binaries: M-type giants and hot white dwarf/subdwarf/low-mass main-sequence companions in long-period ($200\sim1500$ days) interacting binaries; studied for accretion of cool wind onto hot companion
Barium binaries & S-star binaries: K/M-type giants with cool white dwarf companions that once transferred them nuclear-processed gas; studied for nucleosynthesis and mass loss in evolved stars
Post-common-envelope binaries: cool secondaries and hot white dwarf/subdwarf companions; studied for rapid stages of stellar evolution
AM Herculis stars / polars: semidetached binaries containing white dwarfs with $B\sim2000$ T
- Magnetic torque tidally locks both stars and forms a narrow accretion column with a $\sim10^8$ K shock front emitting hard X-ray at the base
- Photosphere re-emit some hard X-ray as soft X-ray and UV light
- Nonrelativistic electrons in white dwarf magnetosphere also emit visible-light cyclotron radiation
DQ Herculis stars / intermediate polars: far-apart or weakly magnetic polars missing synchronous rotation but having some accretion flow disruption

Cataclysmic variable evolution

Let Star 1 and Star 2 be main-sequence stars in a long-period (months to years) detached binary with $M_1>M_2$
Binary semidetached as Star 1, evolving more rapidly, overflows its Roche lobe after reaching giant/supergiant phase
- Mass transfer where $M_1>M_2$ and $\dot M<0$ enters positive feedback loop with shrinking separation and Roche lobe
First common envelope stage: binary contacts, transferring angular momentum onto common envelope and spiralling inward
- May turn into blue straggler if stellar cores merge
Otherwise, binary detaches as envelope is eventually ejected and Star 1 cools to a white dwarf
Binary again semidetached as Star 2 reaches giant phase
- Mass transfer where $M_1 > M_2$ and $\dot M>0$ faces negative feedback with expanding separation and Roche lobe
- Transfer only maintained if secondary expands faster than Roche lobe or if stellar winds remove sufficient angular momentum
Second common envelope stage: suppose mass transfer continues, binary contacts again and spiral further inward, becoming a cataclysmic variable

Cataclysmic variables

$\sim0.86\, M_\odot$ white dwarf primary and less massive main-sequence (type G or later) secondary in periods mostly ranging 78 minutes $\sim$ 12 hours
- Gap in periods between $1.5\sim3.25$ hours possibly due to abrupt change in angular momentum transfer
Long quiescent intervals punctuated by luminous outbursts, likely due to sudden increases in mass flow rate
Otherwise cool, low-mass, emission-lines-producing accretion disk becomes optically thick with absorption lines during brightening
Dwarf novae: outbursts increase luminosity by $2\sim6$ magnitudes for $5\sim20$ days; separated by quiescent intervals of $30\sim300$ days
- Visible wavelengths precede UV in brightening by $\sim1$ day, indicating outbursts start in cooler, outer disk and spread inward
- Mass transfer rate increases $\sim100\times$ during outburst, possibly due to instability in secondary star or accretion disk
- H partial ionization zone could dam up in secondary and propel material past $L_1$, or dam up in disk and increase viscosity, causing material to fall
Classical novae: outbursts increase luminosity by $7\sim20$ magnitudes
- Luminosity initially rises rapidly, with a brief standstill $\sim2$ magnitudes from maximum
- Fast novae take weeks to dim by 2 visual magnitudes from maximum; slow novae take $\sim100$ days, but are also $\sim3$ visual magnitudes dimmer at maximum
- Bolumetric magnitude remain roughly constant for months: dimming in visual band offset by brightening, first in UV, then in IR
- Accompanied by ejection of $10^{-5}\sim10^{-4}\,M_\odot$ of hot gasses at $\sim10^3$ km/s, with slow novae ejecting them $\sim3\times$ slower
- System returns to pre-eruption luminosity of $M_V\sim4.5$ after decades

Classical nova cycle

White dwarf in semidetached binary accretes H-rich gas onto surface at $10^{-8}\sim10^{-9}\, M_\odot$/year
Convection enriches H shell with surface C, N, and O; pressure heats shell and turns base degenerate
Accumulation of $10^{-4}\sim10^{-5}\, M_\odot$ of H at $\sim10^6$ K triggers runaway CNO cycle at base of shell, reaching $\sim10^8$ K before electrons there are lifted from degeneracy
Hydrodynamic ejection phase, dominant for fast novae: once luminosity exceeds Eddington limit, radiation pressure lifts $\sim10\%$ of H layer into space
Hydrostatic burning phase, dominant or slow novae: CNO cycle stabilizes near Eddington limit and hydrostatic equilibrium is reached; inert layer above H-burning shell becomes fully convective and expands to $\sim10^6$ km
Fireball expansion phase: hydrodynamically ejected gasses form optically thick, glowing “fireball” with $T\approx6000\sim10\,000$ K, resembling a type A/F supergiant; reaches maximum visual brightness in a few days, then becomes optically thin
Optically thin phase: transparent ejecta reveal hydrostatically-burning white dwarf, resembling a blue horizontal-branch giant
Dust formation phase: as $T$ falls below $\lesssim1000$ K, carbon in ejecta condense and form dust shell; turns optically thick $\sim50\%$ of the time and re-radiates energy from white dwarf in IR

Type Ia supernovae

Destruction of white dwarf near Chandrasekhar limit
Maximum brightness very consistently $M_B\approx M_V\approx-19.3\pm0.3$
Peak brightness–rate of decline relation allow accurate distance measurements to faraway host galaxies
- Critical for discovery of dack energy
Absence of H lines indicates evolved nature; P-Cygni profile indicates mass loss; blueshifted absorption lines indicate ejecta speeds $\gtrsim10^4$ km/s $\approx0.1c$
Dobule-degenerate models: two white dwarfs
- Radiation of gravitational waves cause orbits to decay
- Larger and less massive secondary eventually spills over Roche lobe, breaking into thick disk for the primary to accrete
- As primary nears Chandrasekhar limit, nuclear reactions ignite in deep interior and destroy it
- Roughly predict the right number of mergers, but ignitions, if off-center, may result in neutron stars instead of supernovae, and predictions of heavy element abundances are inconsistent with spectra
Single-degenerate models: a C–O white dwarf and an evolving star companion
- Helium-triggered model: He gas from secondary settle on surface with base becoming degenerate; He flash sends shockwave into interior and ignites degenerate C and O
- Direct ignition model: multiple independent ignitions of C and O occur in interior as star nears Chandrasekhar limit
- Require fine-tuning rate of accretion to prevent dwarf novae or classical novae from taking place instead
Unclear if direct ignition single- and double-degenerate models result in subsonic deflagaration or supersonic detonation in shock front

Neutron star binary formation processes

Supernova in existing binary: unbinds system if $\ge{1\over2}$ of total mass is ejected; otherwise, neutron star or black hole now in binary
Tidal capture: neutron star passing within $1\sim3\,R_\text{companion}$ loses significant kinetic energy to tidal dampening and becomes gravitationally bound; most effective in densely populated regions
Disruption of multi-star system: neutron star transfers kinetic energy to another star, ejecting that star while itself is captured
Direct collision with giant: neutron star penetrates star and enters subsurface orbit with its degenerate core, becoming a Thorne–Żytkow object (TŻO); envelope is then quickly expelled, producing neutron star–white dwarf binary

X-ray binaries

Low-mass X-ray binaries (LMXB): short orbital periods ($11$ minutes $\sim35$ days) with low-mass, mostly cool secondary stars
Massive X-ray binaries (MXRB): relatively longer orbital periods ( $0.2\sim580$ days) with O and B giants and supergiants
Alfvén radius $r_A = \sqrt[7]{8\pi^2 B^4 R^{12}\over \mu_0 G M \dot M^2}$
- Where magnetic energy density $u_m = {1\over2\mu_0}B^2\propto1/r^6$ equals kinetic energy density $u_K = {1\over2}\rho v^2$
Below disruption radius $r_d\approx0.5r_A$, ionized gas is channeled onto magnetic poles instead of impacting surface
- If $r_d\approx r_\text{orbit}$, magnetic field prevents formation of disk and confines accretion to narrow streams directed towards poles
- If $r_d\lesssim r_\text{disk}$, magnetic field only distrupts accretion disk near star surface
- Polar-like accretion columns and shock fronts formed above poles
X-ray burster: LMXB with weak magnetic field
- Matter settling on surface of neutron star with weak magnetic field ($\ll10^8$ T) periodically ignites H, He, then possibly C fusion
- He fusion extremely explosive and releases $\sim10^{32}$ J in seconds, predominantly in X-ray
- Accretion disk re-emits some X-ray in visible light seconds after the burst
Companion stars of LMXBs become white dwarfs without disturbing the system
Binary X-ray pulsar: strong magnetic field; $\sim50\%$ of XMRBs
- Matter accreting on poles of neutron star release up to $\sim10^{31}$ W, predominantly in X-ray
- Near-Eddington-limit radiation elevates shock front, resulting in large solid angle for emission
- Period slowly speeding up from transfer of accreting matter’s angular momentum via magnetic torques
  - Pulse arrival times vary with phase of orbit, allowing determination of orbital radius
- Misalignment of magnetic and rotation axes may allow eclipsing of pulses by companion, and thereby complete determination of masses and radii in the system
Some MXRBs ($\gtrsim3\, M_\odot$) may contain a stellar-mass black hole
- Most candidates lack developed accretion disks, making mass determination of compact object difficult
- A0620–00: an X-ray nova with sporatic accretion and a $3.82\pm0.24\, M_\odot$ black hole
- Cygnus X-1: bright MXRB with a $>3.4\, M_\odot$ black hole
- SS 433: three emission lines, with two posessing significant Doppler shifts attributed to relativistic jets of a precessing accretion disk from a neutron star or black hole

X-ray binary remnants

Companion stars of MXRBs may go supernova, either unbinding the system or creating a double neutron star / neutron star–black hole binary
Millisecond radio pulsars: spun up by accretion as LMXBs, but small fraction are missing white dwarf companions
Black widow pulsar: if pulsar beam close to equator, white dwarf may be evaporated
Double neutron star binaries: precise measurements on orbital decay of Hulse–Taylor pulsar (PSR 1913+16) confirmed existence of gravitational waves
Short–hard gamma ray bursts: merger of two neutron stars or neutron star and black hole

Carroll & Ostlie: Introduction to Modern Astrophysics

I. The Tools of Astronomy

1. The Celestial Sphere

Altitude–Azimuth Coordinate System

Equatorial Coordinate System

Precession

Time standards

Motions

Spherical Trigonometry

2. Celestial Mechanics

Ellipses

Kepler’s Laws

Orbital equations

3. The Continuous Spectrum of Light

Parallax and parsecs (pc)

Apparent magnitude

Absolute magnitude

Light

Blackbody radiation

Color index

4. Special Relativity

Einstein’s postulates:

Lorentz transformations:

Relativistic doppler shifts

Relativistic momentum and energy

5. Interaction of Light and Matter

Kirchoff’s Laws

Spectrographs

Photoelectric effect

Compton scattering

Bohr’s semiclassical atom

De Broglie waves

Heisenberg’s Uncertainty Principle

Schrödinger's quantum atom

Normal Zeeman effect

Anomalous Zeeman effect and Spin

Fermions and bosons

Complex spectra

6. Telescopes

Basic optics

Resolution

Brightness

Optical telescopes

Radio telescopes

IR/UV/X-ray/gamma-ray astronomy

II. The Nature of Stars

7. Binary Systems and Stellar Parameters

Classifications

Mass determination of visual binaries

Mass determination of spectroscopic binarie

Temperature ratio and radii determination of eclipsing binaries

Computer modeling

Extrasolar planets (exoplanets)

8. Classification of Stellar Spectra

Harvard spectral types

Spectral physics

Hertzsprung–Russell diagram

Morgan−Keenan luminosity classes

9. Stellar Atmospheres

Radiation field

Temperature

Opacity

Radiative transfer

Spectral line profile

Computer modeling

10. The Interiors of Stars

Hydrostatic equilibrium

Kelvin–Helmholtz mechanism

Nuclear fusion

Nucleosynthesis

Energy transport and thermodynamics

Stellar model building

Main sequence

11. The Sun

Solar interior

Photosphere

Chromosphere

Transition region

Corona

Solar wind