(chap:11)= # White dwarfs, neutron stars, and strong-field physics :::{admonition} Chapter opening :class: chapter-opening White dwarfs and neutron stars take quantum physics to astronomical scales. White-dwarf radii are set by electron degeneracy pressure, Coulomb corrections, and envelope structure. Neutron-star light curves project strong gravity, strong magnetic fields, and relativistic magnetospheres into an event table. The measured quantities are still arrival time, energy, polarization, and phase, but they now correspond to magnetic fields from $10^6$ to $10^{15}\,\mathrm{G}$, periods from milliseconds to hours, and emitting regions from $10\,\mathrm{km}$ to Earth-radius scales. ::: ## White dwarfs: how electron degeneracy pressure becomes a measured radius The main scale of a white dwarf can be understood from a cold Fermi gas. A typical DA white dwarf with $M\simeq0.6\,M_\odot$ and $R\simeq0.012\,R_\odot\simeq1.3\,R_\oplus$ already has a mean density of order $10^6\,\mathrm{g\,cm^{-3}}$, and a central density between $10^6$ and $10^9\,\mathrm{g\,cm^{-3}}$. At these densities, thermal electron motion is secondary. The Pauli exclusion principle fills electron momentum space up to the Fermi momentum ```{math} :label: eq:ch11-01 p_F=\hbar (3\pi^2 n_e)^{1/3}, \qquad n_e=\frac{\rho}{\mu_e m_p}. ``` Here $n_e$ is the electron number density, in $\mathrm{cm^{-3}}$; $\rho$ is mass density; $m_p$ is the proton mass; and $\mu_e$ is the number of baryons per electron, usually $\mu_e\simeq2$ for a carbon-oxygen white dwarf. When $p_F\ll m_e c$, the electrons are nonrelativistic and the pressure is ```{math} :label: eq:ch11-02 P_{\rm NR}=\frac{(3\pi^2)^{2/3}}{5}\frac{\hbar^2}{m_e}n_e^{5/3}; ``` when $p_F\gtrsim m_e c$, relativity changes the power to $4/3$, ```{math} :label: eq:ch11-03 P_{\rm ER}=\frac{(3\pi^2)^{1/3}}{4}\hbar c\, n_e^{4/3}. ``` The pressure $P$ is in $\mathrm{dyn\,cm^{-2}}$. In the nonrelativistic formula, higher density rapidly increases the supporting pressure. In the relativistic formula, the support becomes softer; as the mass increases, the radius shrinks quickly and the star approaches the Chandrasekhar limit {cite:p}`1931ApJ....74...81C,1935MNRAS..95..207C,1961ApJ...134..683H`. A useful observer-facing mass-radius approximation is the Nauenberg form, ```{math} :label: eq:ch11-04 R\simeq 0.0112\,R_\odot \left[\left(\frac{M_{\rm Ch}}{M}\right)^{2/3} -\left(\frac{M}{M_{\rm Ch}}\right)^{2/3}\right]^{1/2}, \qquad M_{\rm Ch}\simeq 5.83\,\mu_e^{-2}M_\odot . ``` $M_{\rm Ch}$ is the limiting mass of an ideal cold white dwarf. For $\mu_e=2$, $M_{\rm Ch}\simeq1.46\,M_\odot$. This expression assumes zero temperature, no rotation, and no strong magnetic deformation. It gives the main scale for $0.2$--$1.35\,M_\odot$ white dwarfs. Real radii also depend on core composition, hydrogen and helium envelope thickness, finite temperature, and magnetic fields; precision work needs evolutionary models or binary constraints {cite:p}`1972ApJ...175..417N,2017MNRAS.470.4473P,2020ApJ...899..146C`. ```{figure} ../_static/figures/generated/chapter_11/ch11_white_dwarf_mass_radius.png :name: fig:chapter-11-wd-mr :width: 94.0% White-dwarf radius decreases as mass increases, and the contraction accelerates near the Chandrasekhar limit. The right panel expresses the same mass-radius relation as a spectroscopic observable, the gravitational redshift velocity $v_{\rm gr}=GM/(Rc)$. A typical DA white dwarf has Balmer-line redshifts of tens of km s−1, comparable to or larger than ordinary stellar atmospheric velocity scales. ``` The mass-radius relation enters observations in several ways. In eclipsing binaries, durations and radial velocities give $R/a$, inclination, and the mass function. Parallax plus the spectral energy distribution gives a radius through $F_\lambda=(R/d)^2 H_\lambda(T_{\rm eff},\log g)$. Spectral-line redshift gives the gravitational redshift velocity, ```{math} :label: eq:ch11-05 v_{\rm gr}=\frac{GM}{Rc} \simeq 31.8\,\mathrm{km\,s^{-1}} \left(\frac{M}{0.6\,M_\odot}\right) \left(\frac{0.012\,R_\odot}{R}\right). ``` $v_{\rm gr}$ is the redshift of a line relative to the systemic velocity. Systemic motion, convective line shifts, and Stark shifts contaminate this quantity in individual stars, so large-sample work often averages Gaia radii, SDSS line centers, and a kinematic model. The angular diameter of a white dwarf is tiny, ```{math} :label: eq:ch11-06 \theta_{\rm WD}=\frac{2R}{d} \simeq 11\,\mu{\rm as} \left(\frac{R}{0.012\,R_\odot}\right) \left(\frac{10\,{\rm pc}}{d}\right), ``` so even within $10\,\mathrm{pc}$, ordinary optical imaging cannot resolve it. The more useful path is to keep fast photometry, polarization, spectral-line velocities, and phase information in one event table. ## Accreting and magnetic white dwarfs: phase, polarization, and flickering Magnetic white dwarfs connect white-dwarf structure to photon statistics. Isolated magnetic white dwarfs can have surface fields from $10^6$ to $10^9\,\mathrm{G}$, while polars and intermediate polars in accreting systems often have fields of $1$--$200\,\mathrm{MG}$. The magnetic moment is ```{math} :label: eq:ch11-07 \mu = B_* R_{\rm WD}^3 \simeq 5.1\times10^{33}\,\mathrm{G\,cm^3} \left(\frac{B_*}{10\,{\rm MG}}\right) \left(\frac{R_{\rm WD}}{8\times10^8\,{\rm cm}}\right)^3 . ``` Here $B_*$ is the surface magnetic field. Given $\mu$, mass $M$, and accretion rate $\dot M$, a common scaling for the magnetospheric radius is ```{math} :label: eq:ch11-08 r_m \simeq k \left(\frac{\mu^4}{2GM\dot M^2}\right)^{1/7}, ``` where $k$ is a geometric factor of order $0.5$--1. In cataclysmic variables, $\dot M$ can range from $10^{-11}$ to $10^{-8}\,M_\odot\,{\rm yr^{-1}}$. If $r_m$ is smaller than the corotation radius, ```{math} :label: eq:ch11-09 r_{\rm co}=\left(\frac{GM}{\Omega_s^2}\right)^{1/3}, ``` matter can follow field lines down to the magnetic poles. If $r_m$ approaches or exceeds $r_{\rm co}$, white-dwarf rotation can expel part of the inflow or alter the accretion geometry. Intermediate polars often have $P_{\rm spin}/P_{\rm orb}$ between $0.01$ and 0.6. Polars are nearly synchronous, with $P_{\rm spin}\simeq P_{\rm orb}$ {cite:p}`1970ApJ...160L.147A,2000PASP..112..873W,1991MNRAS.249..460W,2008ApJ...672..524N`. ```{figure} ../_static/figures/generated/chapter_11/ch11_magnetic_cv_flow_map.png :name: fig:chapter-11-mcv-map :width: 78.0% Accretion flow in a magnetic cataclysmic variable is controlled mainly by the white-dwarf magnetic moment and the spin rate relative to the orbit. Systems with low $P_{\rm spin}/P_{\rm orb}$ can form disk-like flows. Intermediate systems often show magnetically controlled streams. Nearly synchronous or high-ratio systems move toward ring-like, stream-fed, or polar geometries. These geometries directly change photometric phase, circular polarization, and correlation functions. ``` The accretion luminosity scale comes from gravitational potential energy, ```{math} :label: eq:ch11-10 L_{\rm acc}\simeq \frac{GM\dot M}{R_{\rm WD}} \simeq 1.0\times10^{33}\,\mathrm{erg\,s^{-1}} \left(\frac{M}{0.8\,M_\odot}\right) \left(\frac{\dot M}{10^{-10}M_\odot\,{\rm yr^{-1}}}\right) \left(\frac{8\times10^8\,{\rm cm}}{R_{\rm WD}}\right) ``` Only part of this luminosity appears in the optical band. The observed light is usually a mixture of disk emission, hot spots, column shocks, irradiated surfaces, and cyclotron radiation. Polarization helps isolate the cyclotron component. If the sign of circular polarization flips with spin phase, the line of sight is usually crossing different magnetic poles or different projected field directions. If linear polarization and total intensity peak at different phases, the brightness maximum and the magnetic-geometry maximum are probably not the same region. Event-table analysis can bin the data by spin or orbital phase. Let $\phi_s$ be the white-dwarf spin phase, and let $I_{\phi_s}(t)$ be the photon count rate inside that phase window. Then ```{math} :label: eq:ch11-11 g^{(2)}_{\phi_s}(\tau)= \frac{\langle I_{\phi_s}(t)I_{\phi_s}(t+\tau)\rangle} {\langle I_{\phi_s}(t)\rangle^2} ``` measures the intensity memory of the same magnetic pole or accretion-stream segment at delay $\tau$. If $\tau$ is milliseconds to seconds, the signal may come from shock cooling, magnetically guided clumps, or detector dead time. If $\tau$ is tens of seconds to minutes, it is more likely ordinary flickering or atmospheric transparency variation. The average in this equation should be taken within the same phase window, energy band, and polarization channel; otherwise orbital modulation can be mistaken for a short-time correlation. ## Neutron stars: light cylinders, phase, and Crab event tables Neutron stars have radii of only $10$--$14\,\mathrm{km}$ and masses commonly between $1.2$ and $2.2\,M_\odot$. Rotation turns the magnetosphere into a phase coordinate. The basic length scale is the light-cylinder radius, ```{math} :label: eq:ch11-12 R_{\rm LC}=\frac{c}{\Omega}=\frac{cP}{2\pi} \simeq 1.6\times10^3\,\mathrm{km} \left(\frac{P}{33\,{\rm ms}}\right). ``` $P$ is the spin period. The Crab pulsar has $P\simeq33\,\mathrm{ms}$; millisecond pulsars have $P\simeq1.5$--$10\,\mathrm{ms}$; magnetars often have $P=2$--12 s. Beyond $R_{\rm LC}$, rigid corotation would require superluminal motion, so open field lines, particle acceleration, and high-energy radiation must be organized inside this scale. The discovery of pulsars and the rotating-neutron-star interpretation established this picture, and the Crab tied rotational energy loss to a supernova remnant {cite:p}`1968Natur.217..709H,1968Natur.218..731G,1968Natur.219..145P,1972ApJ...175..217B`. ```{figure} ../_static/figures/generated/chapter_11/ch11_light_cylinder_phase.png :name: fig:chapter-11-light-cylinder :width: 94.0% The light-cylinder radius grows linearly with spin period. Millisecond pulsars have light cylinders only tens to hundreds of kilometers across. The Crab light cylinder is about 1.6 × 103 km. Magnetars can reach more than 105 km. The right panel shows a Crab-like folded phase profile; the main-pulse and interpulse phase windows decide which photons enter later intensity-correlation or polarization statistics. ``` The magnetic-dipole estimate of the equatorial surface field is ```{math} :label: eq:ch11-13 B_{\rm dip}\simeq 3.2\times10^{19}(P\dot P)^{1/2}\,\mathrm{G}, ``` where $P$ is in seconds and $\dot P$ is dimensionless $\mathrm{s\,s^{-1}}$. Ordinary young pulsars often have $10^{11}$--$10^{13}\,\mathrm{G}$ fields. Magnetars can reach $10^{14}$--$10^{15}\,\mathrm{G}$. The spin-down luminosity is ```{math} :label: eq:ch11-14 \dot E = 4\pi^2 I\frac{\dot P}{P^3}, \qquad I\simeq10^{45}\,\mathrm{g\,cm^2}. ``` For the Crab, $\dot E$ is of order $10^{38}\,\mathrm{erg\,s^{-1}}$, enough to power pulsed and nebular emission from radio to $\gamma$-rays {cite:p}`1984ApJ...283..279H,1992ApJ...397..187B,2001A&A...378..918K,2008ApJ...680.1378H`. When an event table enters pulsar analysis, each photon is first corrected to the Solar-System barycenter and then folded with a timing ephemeris: ```{math} :label: eq:ch11-15 \phi_i = \left[ \nu(t_i-t_0)+\frac{1}{2}\dot\nu(t_i-t_0)^2 +\frac{1}{6}\ddot\nu(t_i-t_0)^3+\cdots \right]\bmod 1 . ``` $t_i$ is the barycentric arrival time of photon $i$, in seconds; $\nu=1/P$ is the spin frequency; and $\dot\nu$ and $\ddot\nu$ describe spin-down and glitch recovery. The Crab main pulse has a width of about 0.04--0.05 in phase, while radio giant pulses can be as narrow as microseconds. If one selects the rotations containing giant radio pulses, the average optical main pulse is enhanced by about $3\%$, while its shape and arrival phase remain almost unchanged. Energy-resolved ARCONS photon counting further showed that early-arriving giant radio pulses can correspond to optical enhancements of several tens of percent {cite:p}`2003Sci...301..493S,2013ApJ...779L..12S`. These results connect coherent radio bursts and incoherent optical radiation to the same magnetospheric plasma. Phase-resolved second-order correlation can be written as ```{math} :label: eq:ch11-16 g^{(2)}_{ab}(\tau|\Phi)= \frac{\langle n_a(t,\Phi)n_b(t+\tau,\Phi)\rangle} {\langle n_a(t,\Phi)\rangle\langle n_b(t,\Phi)\rangle}, ``` where $a,b$ can denote energy bands, polarization channels, or different telescopes, and $\Phi$ is a main-pulse, bridge, or interpulse phase window. For a bright target such as the Crab, $\tau$ can reach microseconds to milliseconds. For fainter optical pulsars, the first feasible products are usually phase-window photometry and polarization stacks, still short of single-pulse statistics {cite:p}`1996ApJ...468..779M,2000ApJ...537..861S,2009MNRAS.397..103S`. ## Strong-field polarization: from Stokes parameters to vacuum birefringence The basic scale for strong-field polarization is the QED critical field, ```{math} :label: eq:ch11-17 B_Q=\frac{m_e^2c^3}{e\hbar} =4.414\times10^{13}\,\mathrm{G}. ``` When $B\ll B_Q$ and photon energies are far below $m_ec^2$, the Heisenberg--Euler effective theory gives the weak-field approximation ```{math} :label: eq:ch11-18 \Delta n \simeq \frac{\alpha_{\rm fs}}{30\pi} \left(\frac{B_\perp}{B_Q}\right)^2 , ``` where $B_\perp$ is the magnetic-field component perpendicular to the ray, and $\alpha_{\rm fs}$ is the fine-structure constant. Magnetar surfaces can have $B\sim10^{14}$--$10^{15}\,\mathrm{G}$, so the weak-field expression is not a precision model there. It does show why polarization is sensitive to strong fields. The phase difference between the two normal modes accumulates along the path, ```{math} :label: eq:ch11-19 \Delta \Phi(E)=\frac{E}{\hbar c}\int \Delta n(s)\,ds . ``` $E$ is photon energy; X-ray polarimeters often operate at 2--8 $\mathrm{keV}$. The integral follows the light path. If propagation is adiabatic, the polarization vector follows the local magnetic-field direction until it freezes out near the polarization-limiting radius. This picture combines strong-field QED, magnetospheric propagation, and neutron-star surface radiation models {cite:p}`1936ZPhy...98..714H,1971AnPhy..67..599A,2000MNRAS.311..555H,2002PhRvD..66b3002H,2003MNRAS.342..134H`. Polarization observations must report intensity, polarization degree, and polarization angle together. A linear-polarization event estimator usually starts from the modulation angle $\eta_i$, ```{math} :label: eq:ch11-20 I=\sum_i w_i,\qquad Q=\sum_i w_i\cos 2\eta_i,\qquad U=\sum_i w_i\sin 2\eta_i, ``` and then forms ```{math} :label: eq:ch11-21 p=\frac{\sqrt{Q^2+U^2}}{I},\qquad \psi=\frac{1}{2}\arctan2(U,Q). ``` $p$ is the linear polarization fraction, between 0 and 1; $\psi$ is the polarization angle; and $w_i$ includes the instrument modulation factor, background weights, and effective-area weights. Optical polarimetry of RX J1856.5-3754 measured $p=16.43\%\pm5.26\%$ for a source with $V\simeq25.5$. Foreground and instrumental polarization had to be removed term by term. Compared with thermal surface models, this type of measurement supports vacuum birefringence because it reduces polarization cancellation between different surface regions {cite:p}`2015MNRAS.454.3254T,2016MNRAS.459.3585G,2017MNRAS.465..492M`. ```{figure} ../_static/figures/generated/chapter_11/ch11_stokes_qu_track.png :name: fig:chapter-11-stokes :width: 92.0% Rotating magnetic geometry projects polarization-angle changes into phase curves of Q/I and U/I. The left panel shows the two Stokes components obtained from the same phase-resolved event table. The right panel shows the corresponding trajectory in the Q–U plane. The trajectory shape often exposes geometric degeneracy and instrumental crosstalk better than the polarization fraction alone. ``` The phase-dependent polarization angle is often described first with the rotating-vector model, ```{math} :label: eq:ch11-22 \tan(\psi-\psi_0)= \frac{\sin\alpha \sin(\phi-\phi_0)} {\sin\zeta\cos\alpha-\cos\zeta\sin\alpha\cos(\phi-\phi_0)} . ``` $\alpha$ is the angle between magnetic and rotation axes, $\zeta$ is the angle between the line of sight and the rotation axis, $\phi$ is spin phase, and $\psi_0,\phi_0$ are zero points. This formula was developed for radio pulsar polarization geometry. In magnetar X-rays it is only a low-dimensional geometric frame, because surface emission, resonant cyclotron scattering, and vacuum birefringence all affect $p(E,\phi)$ and $\psi(E,\phi)$ {cite:p}`1969ApL.....3..225R,2018ApJ...854...98W,2011ApJ...730..131F`. ```{figure} ../_static/figures/generated/chapter_11/ch11_birefringence_energy.png :name: fig:chapter-11-birefringence :width: 92.0% IXPE results for magnetar 4U 0142+61 can be read as a change between two polarization normal modes with energy. The polarization fraction is about 14% at 2–4 keV, rises to about 41% at 5.5–8 keV, and drops close to the minimum detectable polarization near 4–5 keV, where the polarization angle flips by about 90∘. ``` 4U 0142+61 is a concrete example. IXPE measured a total linear polarization of $12\%\pm1\%$ in $2$--$8\,\mathrm{keV}$, with $14\%\pm1\%$ at $2$--$4\,\mathrm{keV}$, $41\%\pm7\%$ at $5.5$--$8\,\mathrm{keV}$, and a dip near $4$--$5\,\mathrm{keV}$ where the polarization angle rotates by about $90^\circ$. The source has $P=8.69\,\mathrm{s}$, $B\sim10^{14}\,\mathrm{G}$, a soft X-ray blackbody-like component with $kT\sim0.5$--1 $\mathrm{keV}$, and a hard tail extending to tens or even hundreds of keV. Interpreting these data can involve a low-energy O-mode, a hard-tail X-mode, a condensed surface or thin atmosphere, and resonant Compton scattering. A single high-polarization measurement is not, by itself, a standalone detection of vacuum birefringence {cite:p}`2022Sci...378..646T,2023A&A...674L..10C,2023MNRAS.519.3681F`. ## Hot spots, light bending, and radius constraints A neutron-star radius cannot be measured as an angular diameter. It has to be inferred from the light-curve shape, spectrum, and relativistic propagation. The compactness ```{math} :label: eq:ch11-23 u=\frac{2GM}{Rc^2} \simeq 0.345 \left(\frac{M}{1.4\,M_\odot}\right) \left(\frac{12\,{\rm km}}{R}\right) ``` sets the strength of light bending. Larger $u$ makes the far-side hot spot easier to see, reducing pulse amplitude and smoothing the profile. A common approximation relates the emission angle $\alpha$ to the angle $\psi$ between the spot center and the observer, ```{math} :label: eq:ch11-24 \cos\alpha \simeq u+(1-u)\cos\psi . ``` It is useful for scale estimates in an external Schwarzschild spacetime and slow or moderate rotation. Rapid millisecond pulsars also require Doppler boosting, aberration, time delays, stellar oblateness, and atmospheric beaming {cite:p}`2002ApJ...566L..85B,2019AIPC.2127b0008W`. ```{figure} ../_static/figures/generated/chapter_11/ch11_hotspot_lightcurve.png :name: fig:chapter-11-hotspot :width: 78.0% Hot-spot light curves are sensitive to compactness. At low compactness, the spot contribution falls rapidly when it rotates to the far side, so the pulse amplitude is large. At high compactness, gravitational light bending keeps more of the surface visible and raises the trough. Radius inference uses this profile change together with mass, inclination, spot location, and atmospheric beaming. ``` For a hot spot with a spectrum, the expected detector count in energy channel $j$ and phase bin $k$ can be written as ```{math} :label: eq:ch11-25 \lambda_{jk}(\Theta)= T\int dE\, R_j(E)\, \left[ F_E(\phi_k;\Theta)+B_E \right]. ``` $T$ is exposure time, $R_j(E)$ is the effective area and energy response, $F_E$ is the model hot-spot flux, $B_E$ is the background, and $\Theta$ contains $M,R$, distance, inclination, spot colatitude and longitude, spot angular radius, temperature, and atmosphere parameters. The observed counts obey Poisson statistics, ```{math} :label: eq:ch11-26 \ln{\cal L}(\Theta)= \sum_{j,k}\left[ N_{jk}\ln\lambda_{jk}(\Theta)-\lambda_{jk}(\Theta)-\ln(N_{jk}!) \right]. ``` NICER radius measurements of PSR J0030+0451 and PSR J0740+6620 use this type of event-table modeling. For J0740, the analysis combines NICER event tables, XMM-Newton imaging-spectroscopy backgrounds, and radio-timing priors on mass and distance. Riley et al. found $R=12.39^{+1.30}_{-0.98}\,\mathrm{km}$ and $M=2.072^{+0.067}_{-0.066}M_\odot$; Miller et al. obtained compatible radius constraints with a different method. The two independent J0030 analyses also show that hot-spot geometry may involve complex multipolar magnetic structure {cite:p}`2019ApJ...887L..21R,2019ApJ...887L..24M,2021ApJ...918L..27R,2021ApJ...918L..28M`. Strong-field compact objects are rarely determined by one projection of the data. White-dwarf mass and radius require spectra, parallax, binary geometry, and gravitational redshift. Magnetic-white-dwarf period peaks need polarization, accretion state, weather, and detector dead time. Optical enhancement in pulsars needs synchronized radio event tables. Magnetar polarization-angle flips have to be modeled with spectrum, phase, and normal modes. Neutron-star radii come from Poisson event likelihoods, response matrices, and background models. Black holes and accretion flows keep the same event-table logic, but the geometry changes from surfaces and magnetospheres to disks, jets, and photon rings.