Compact Stars and Relativistic Astrophysics

Derivation. Derives Phenomenon 49.1. What must be checked is that the electrons really are degenerate at this density — that the exclusion principle of Identical Particles, and not thermal agitation, sets the pressure. Take \(\rho=10^{9}\,\mathrm{kg}/\mathrm{m}^{3}\) of fully ionized material with two nucleons per electron, so that

The Fermi energy of a free electron gas at that density, from Quantum Statistics, is

whereas the interior of a white dwarf, at a temperature of order \(10^{7}\,\mathrm{K}\), has \(k_{\mathrm{B}}T\approx1.4\times 10^{-16}\,\mathrm{J}\approx0.9\,\mathrm{keV}\). The ratio \(E_{\mathrm{F}}/\left(k_{\mathrm{B}}T\right)\approx200\) settles the question: essentially every state below the Fermi level is occupied, the gas is degenerate, and the pressure it exerts does not vanish as the star cools. That is why a white dwarf can sit in equilibrium with no energy source whatever, which no thermally supported star can do.

The same arithmetic carries a warning. The Fermi energy already reaches about a third of the electron rest energy \(m_{\mathrm{e}}c^{2}=511\,\mathrm{keV}\), so the non-relativistic formula used above is only an estimate, and a denser — that is, a more massive — white dwarf will be relativistic. That is the regime of Section 49.1.3, and it is where the mass limit comes from.

Derives Proposition 49.2. Free-particle states occupy phase space in cells of volume \((2\pi\hbar)^{3}\), two spin orientations per cell (Identical Particles and Quantum Statistics). At zero temperature the exclusion principle fills the states of lowest energy: a sphere in momentum space of radius \(p_{\mathrm{F}}\) fixed by counting,

which is Equation (49.1). The energy density is the sum of single-particle energies \(\varepsilon(p)\) over the filled sphere,

With \(\varepsilon=p^{2}/2m\) this gives \(u=\left(3/5\right)nE_{\mathrm{F}}\) with \(E_{\mathrm{F}}=p_{\mathrm{F}}^{2}/2m\); with \(\varepsilon=pc\) it gives \(u=\left(3/4\right)nE_{\mathrm{F}}\) with \(E_{\mathrm{F}}=p_{\mathrm{F}}c\).

The pressure follows thermodynamically, with no kinetic argument needed. At zero temperature the total energy of \(N\) particles in volume \(V\) is \(E=Vu(N/V)\), and

For any power law \(u\propto n^{\gamma}\) this is \(P=\left(\gamma-1\right)u\). Non-relativistically \(\gamma=5/3\), so \(P=\tfrac{2}{3}u=\tfrac{2}{5}nE_{\mathrm{F}}\), which is Equation (49.2); ultrarelativistically \(\gamma=4/3\), so \(P=\tfrac{1}{3}u=\tfrac{1}{4}nE_{\mathrm{F}}\), which is Equation (49.3). Substituting Equation (49.1) gives the explicit coefficients. Note what happened between the two limits: relativity does not remove the pressure, it softens it, from \(n^{5/3}\) to \(n^{4/3}\) — one power of \(n^{1/3}\), lost because the velocity saturates at \(c\) and can no longer grow with the Fermi momentum. That lost third of a power is the whole story of the Chandrasekhar limit.

Derives Proposition 49.3. This is Theorem 52.6 in the variables of this chapter; the short derivation is repeated so that the Chandrasekhar argument can lean on the surface-slope mass formula Equation (49.7) without a forward excursion. Hydrostatic equilibrium (Proposition 31.16, specialized to the self-gravitating sphere as in Section 52.1.1) reads \(\dd P/\dd r=-Gm\rho/r^{2}\) with \(\dd m/\dd r=4\pi r^{2}\rho\). Eliminating \(m\) by differentiating the first and substituting the second,

Insert \(P=K\rho^{1+1/n}\) and \(\rho=\rho_{\mathrm{c}}\theta^{n}\): the left side becomes \(\left[\left(n+1\right)K\rho_{\mathrm{c}}^{1/n}/r^{2}\right] \dd\left(r^{2}\,\dd\theta/\dd r\right)/\dd r\), and dividing by \(4\pi G\rho_{\mathrm{c}}\) and rescaling \(r=a\xi\) with Equation (49.5) leaves exactly Equation (49.6). The centre is regular (\(\theta'(0)=0\)) because \(m\propto r^{3}\) there. For the mass, use the equation itself backwards:

since \(\theta'(\xi_{1})<0\). No integration of the density profile is ever needed: the surface slope carries the whole mass integral.

Derives Proposition 49.5. For \(n=3/2\), Equation (49.5) gives \(a\propto\rho_{\mathrm{c}}^{-1/6}\), so Equation (49.7) gives \(M\propto a^{3}\rho_{\mathrm{c}}\propto\rho_{\mathrm{c}}^{1/2}\) and \(R\propto\rho_{\mathrm{c}}^{-1/6}\): eliminating the central density, \(R\propto M^{-1/3}\). Restoring the constants (Equation (49.4) and Remark 49.4),

and eliminating \(\rho_{\mathrm{c}}\) between them gives, for \(\mu_{\mathrm{e}}=2\), the figure quoted (computed here from the constants; e.g. \(M=0.6\,M_{\odot}\) gives \(\rho_{\mathrm{c}}\approx1.5\times 10^{9}\,\mathrm{kg}/\mathrm{m}^{3}\) and \(R\approx1.05\times 10^{7}\,\mathrm{m}\) — precisely the density regime the Sirius B puzzle demanded). The formula overestimates the radius of the more massive white dwarfs, whose electrons are already partly relativistic and whose true radius therefore falls faster than \(M^{-1/3}\); the exact interpolating equation of state joins this curve to the terminating sequence of Section 49.1.3.

Derivation. Derives Phenomenon 49.6. The existence of the limit, and its scale, follow from comparing two energies, with no structure equation solved [Landau:1932]. Let the star hold \(N\) electrons within a radius \(R\), with \(\mu_{\mathrm{e}}\) nucleons per electron, so that its mass is \(M=N\mu_{\mathrm{e}}m_{\mathrm{u}}\) and the electron number density is \(n\sim N/R^{3}\).

Once the electrons are relativistic — the regime Phenomenon 49.1 showed to be near — the Fermi energy is \(E_{\mathrm{F}}\sim\hbar c\,n^{1/3} \sim\hbar c\,N^{1/3}/R\), so the kinetic energy of the gas is

while the gravitational energy is

Both vary as \(1/R\), and that is the decisive fact: contracting the star changes the two terms in exactly the same proportion, so no radius is singled out and equilibrium cannot be reached by adjusting \(R\). The sign of the total energy depends on \(N\) alone. It turns negative — unbounded contraction, with no configuration to stop at — once

The corresponding mass is

which for \(\mu_{\mathrm{e}}=2\) evaluates to \(9.3\times 10^{29}\,\mathrm{kg}\), about \(0.47\,M_{\odot}\). What the argument establishes without any coefficient at all is the substance of the phenomenon: a limit exists, it does not depend on the radius, and it is assembled out of \(\hbar\), \(c\), \(G\) and the nucleon mass — a stellar mass written in fundamental constants. The exact coefficient, close to three, is supplied by the \(n=3\) polytrope in Theorem 49.7 below.

Derives Theorem 49.7. \(P=K\rho^{4/3}\) is the polytrope \(n=3\). Equation (49.5) then gives \(a=\left(K/\pi G\right)^{1/2}\rho_{\mathrm{c}}^{-1/3}\), so in Equation (49.7) the central density cancels identically:

The \(n=3\) sequence is therefore a sequence of one mass at every central density: compressing the star (\(R=a\xi_{1}\propto \rho_{\mathrm{c}}^{-1/3}\to0\)) changes nothing in the balance, which is the structural restatement of the \(1/R\) degeneracy in Landau's argument. A star lighter than \(M\) has a subrelativistic core, a stiffer effective \(n<3\), and finds an equilibrium radius; a star heavier has none. Inserting Equation (49.10) with \(\mu_{\mathrm{e}}=2\) and \(\xi_{1}^{2}\abs{\theta'(\xi_{1})} =2.01824\) (Remark 49.4),

computed here from the CODATA constants; it exceeds the dimensional estimate Equation (49.9) by the factor \(\tfrac{1}{2}\sqrt{3\pi}\, \xi_{1}^{2}\abs{\theta'(\xi_{1})}\approx3.1\). Chandrasekhar's 1931 paper [Chandrasekhar:1931] quotes \(0.91\,M_{\odot}\) because it took \(\mu_{\mathrm{e}}=2.5\), the then-standard guess at stellar composition; the formula is the same, and with the modern \(\mu_{\mathrm{e}}=2\) of helium, carbon and oxygen it gives the value above. General-relativistic corrections and electron capture shave a few per cent off the Newtonian figure, which is why the limit is customarily quoted as \(\approx1.4\,M_{\odot}\) [Chandrasekhar:1939] [Shapiro:1983].

Derivation. Derives Phenomenon 49.8. Two facts organize the problem. First, the degenerate electrons conduct heat so well that the interior is nearly isothermal at a single \(T_{\mathrm{c}}\), and their heat capacity is suppressed by the degeneracy factor \(T/T_{\mathrm{F}}\) (Quantum Statistics), so the heat reservoir is the ions:

with \(A\) the mean nucleon number (\(A\approx14\) for a carbon–oxygen mixture). Second, the leak is a thin non-degenerate envelope in radiative equilibrium: photons diffuse with mean free path \(1/\left(\kappa\rho\right)\), giving the radiative-transfer relation

with \(a\) the radiation constant and \(\kappa=\kappa_{0}\rho T^{-7/2}\) the Kramers opacity of Stellar Structure and Nucleosynthesis. Dividing hydrostatic equilibrium (Equation (31.13), with \(m\approx M\) in the thin envelope) by that relation eliminates \(r\), and the ideal gas law of the envelope, \(\rho=\mu m_{\mathrm{u}}P/\left(k_{\mathrm{B}} T\right)\) with \(\mu\) its mean molecular weight, closes the system:

The envelope ends where the electrons turn degenerate, \(k_{\mathrm{B}}T\approx E_{\mathrm{F}}\propto\rho^{2/3}\), i.e.\ \(\rho\propto T^{3/2}\); and because the degenerate interior is isothermal, the temperature there is \(T_{\mathrm{c}}\) itself. Setting \(\rho^{2}\propto T_{\mathrm{c}}^{3}\) equal to \(\left(M/L\right)T_{\mathrm{c}}^{13/2}\) gives Mestel's law

Now the clock: \(-\dd U/\dd t=L\) reads \(T_{\mathrm{c}}^{-7/2}\,\dd T_{\mathrm{c}}\propto-\dd t\), so \(T_{\mathrm{c}}\propto t^{-2/5}\) and

Cooling decelerates: each decade down in luminosity takes about \(10^{5/7}\approx5\) times longer than the last, which is why the luminosity function rises toward the faint end — stars pile up where they cool slowly — and why its termination is sharp. The absolute age at the cutoff requires the calibrated constant in Equation (49.12) together with the crystallisation and phase-separation corrections, and we quote the result rather than the numerics: \(9(2)\,\mathrm{Gyr}\) for the local disc [Winget:1987]. Crystallisation itself was predicted theoretically in the 1960s [VanHorn:1968]; its latent heat, \(\sim k_{\mathrm{B}}T\) per ion, interrupts Equation (49.13) for of order a gigayear, and the resulting traffic jam in the cooling sequence is what Gaia resolved [Tremblay:2019].

Derives Theorem 49.9. The curvature of the metric Equation (49.15) is computed, Christoffel symbols to Ricci tensor, in Schwarzschild Geometry and Black Holes, where the vacuum case yields the Schwarzschild solution; the two Einstein-tensor components needed here are [Tolman:1939] [Misner:1973]

with the prime denoting \(\dd/\dd r\). The angular components carry no new information: by the contracted Bianchi identity (Section 44.3) they are consequences of these two together with stress–energy conservation, which we impose explicitly below.

The \(tt\) equation. With \(T^{t}{}_{t}=-\rho c^{2}\), Equation (49.14) reads \(\dd\left[r\left(1-\ee^{-2\beta}\right)\right]/\dd r =\left(8\pi G/c^{2}\right)\rho r^{2}\). Substituting the definition \(r\left(1-\ee^{-2\beta}\right)=2Gm/c^{2}\) turns this into Equation (49.17). Regularity at the centre requires \(m(0)=0\), so \(m(r)\) is fixed with no further constant. Note what \(m\) is and is not: \(M=m(R)\) is the mass that governs the exterior orbits — the mass a distant satellite weighs — but it is not the integral of \(\rho\) over proper volume, which carries an extra factor \(\ee^{\beta}>1\). The deficit is the gravitational binding energy, already subtracted; general relativity weighs the star bound.

The \(rr\) equation. With \(T^{r}{}_{r}=P\) and \(1-\ee^{-2\beta}=2Gm/\left(rc^{2}\right)\),

Conservation. The fluid obeys \(\nabla_{\mu}T^{\mu}{}_{\nu}=0\) (Section 44.2.2); take the \(\nu=r\) component. For the static diagonal stress tensor,

because \(\Gamma^{t}{}_{tr}=\alpha'\) while the spatial Christoffel terms multiply \(P-P=0\). Setting this to zero gives Equation (49.19): pressure gradients are what hold the fluid static against the gravitational potential \(\alpha\). Substituting \(\alpha'\) from the \(rr\) equation yields Equation (49.18).

Matching. At \(r=R\) the pressure vanishes and Equation (49.17) makes \(m\) constant; the metric functions become \(\ee^{-2\beta}=1-2GM/\left(rc^{2}\right)\) and, by integrating \(\alpha'\) in vacuum, \(\ee^{2\alpha}=1-2GM/\left(rc^{2}\right)\) — the Schwarzschild exterior of Schwarzschild Geometry and Black Holes, as it must be by Birkhoff's theorem there. Given an equation of state \(P(\rho)\), the system Equations (49.17) and (49.18) integrates outward from a chosen central density \(\rho_{\mathrm{c}}\) until \(P\) reaches zero: one equilibrium star per central density, and the mass–radius curve is the image of that one-parameter family.

Derives Proposition 49.12. For the uniform star the bound is read off Equation (49.21): the central pressure diverges when the denominator vanishes, \(3\sqrt{1-r_{\mathrm{s}}/R}=1\), i.e. at \(r_{\mathrm{s}}/R=8/9\), and no equilibrium exists at or beyond that compactness — infinite pressure is not available, and would only gravitate harder if it were (Remark 49.10). Buchdahl proved that the uniform star is the extreme case: for any equation of state whose density does not increase outward, integrating the field equations from centre to surface bounds the central metric function away from zero only when Equation (49.22) holds [Buchdahl:1959]. We use the theorem in Buchdahl's generality but exhibit the proof only in the uniform case, which saturates it. Two consequences are observable: the gravitational redshift from the surface of any static star is bounded by \(1+z=\left(1-r_{\mathrm{s}}/R\right)^{-1/2}<3\), i.e. \(z<2\); and any object more compact than \(\tfrac{9}{8}r_{\mathrm{s}}\) cannot be a star in equilibrium at all — the operational dividing line between the objects of this chapter and those of Schwarzschild Geometry and Black Holes.

Derivation. Derives Phenomenon 49.14. Three numbers are derived: the energy, its carrier, and its time-scale. Collapse carries a stellar core of mass \(M\) from a radius of order \(10^{6}\,\mathrm{m}\) down to the radius of a neutron star, and the energy released is the difference of gravitational binding energies, which for a uniform sphere is

the initial term being negligible because the initial radius is two orders of magnitude larger. With \(M=1.4\,M_{\odot}=2.8\times 10^{30}\,\mathrm{kg}\) and \(R=12\,\mathrm{km}\),

about a tenth of the rest energy \(Mc^{2}\). Set this beside the two things a supernova is usually described by: the kinetic energy of the ejecta, of order \(10^{44}\,\mathrm{J}\), and the radiated light, of order \(10^{42}\,\mathrm{J}\). The visible supernova is a by-product at the level of a per cent and a hundredth of a per cent respectively; the event itself is a neutrino event, and that is what the SN1987A detections confirmed.

That the energy emerges as neutrinos, over seconds rather than milliseconds, is a statement about opacity. Photons are hopelessly trapped, and even neutrinos are trapped at nuclear density: with the weak-interaction cross section of Flavour Physics and Neutrinos, \(\sigma\sim1.7\times 10^{-48}\,\mathrm{m}^{2} \left(E_{\nu}/m_{\mathrm{e}}c^{2}\right)^{2}\), a \(30\,\mathrm{MeV}\) neutrino in matter of \(n\approx1.8\times 10^{44}\,/\mathrm{m}^{3}\) has a mean free path \(\lambda=1/\left(n\sigma\right)\) of order one metre — against a stellar radius of \(10^{4}\,\mathrm{m}\). The energy therefore leaves by a random walk, in a diffusion time \(t\sim R^{2}/\left(\lambda c\right)\), which with these crude numbers is already of order a second; detailed transport, with the cross sections rising steeply toward the hot centre, lengthens this to the observed ten seconds [Colgate:1966] [Shapiro:1983]. The neutrinos finally escape from a layer where the temperature corresponds to mean energies of order \(10\,\mathrm{MeV}\) — as observed.

The observed event counts check the budget quantitatively. Spreading \(\Delta E\) over six neutrino species at \(\langle E_{\nu}\rangle\approx12\,\mathrm{MeV}\) gives \(\sim10^{58}\) neutrinos; at the LMC distance of \(50\,\mathrm{kpc}=1.5\times 10^{21}\,\mathrm{m}\) the fluence at Earth is \(\sim5\times 10^{14}\,/\mathrm{m}^{2}\) (computed). Kamiokande-II's fiducial \(2.14\times 10^{6}\,\mathrm{kg}\) of water holds about \(1.4\times 10^{32}\) free protons; folding the inverse-beta-decay cross section \(\bar{\nu}_{e}p\to ne^{+}\) at the mean energy, \(\sigma\sim1.5\times 10^{-45}\,\mathrm{m}^{2}\), with the \(\bar{\nu}_{e}\) sixth of the fluence predicts of order ten to twenty detected events — and eleven were seen [Hirata:1987], with eight more in the larger, higher-threshold IMB tank [Bionta:1987]. An order-of-magnitude estimate this crude agreeing this well means the energy, the carrier and the time-scale are all as Equation (49.23) demands. The bounds those nineteen events place on neutrino mass, lifetime and speed belong to Flavour Physics and Neutrinos.

Derivation. Derives Phenomenon 49.16. A self-gravitating body rotating with angular velocity \(\omega\) holds together at its equator only if its own gravity supplies at least the centripetal acceleration required there,

Writing \(M=\tfrac{4}{3}\pi R^{3}\rho\) for the mean density and \(P=2\pi/\omega\) for the period, the factor \(R^{3}\) cancels from both sides and there remains

The radius has dropped out entirely: the observed period bounds the density alone, with no model of the star required. For the Crab pulsar, \(P=33\,\mathrm{ms}\) gives \(\rho\geq1.3\times 10^{14}\,\mathrm{kg}/\mathrm{m}^{3}\), five orders of magnitude above the white-dwarf density of Phenomenon 49.1; for the \(1.396\,\mathrm{ms}\) record holder it gives \(\rho\geq7.3\times 10^{16}\,\mathrm{kg}/\mathrm{m}^{3}\), within a factor of four of nuclear saturation density. No white dwarf can rotate this fast, and no other known state of matter reaches such densities. The bound is crude — it treats the body as uniform and ignores the distortion that rotation itself produces — but it is model-independent, and it is what identified the pulsars with the neutron stars predicted three decades earlier [Baade:1934] [Gold:1968].

Derives Proposition 49.17. Electromagnetic theory gives, for any magnetic moment \(\vect{m}(t)\) varying in time, the radiated power \(\mu_{0}\abs{\ddot{\vect{m}}}^{2}/\left(6\pi c^{3}\right)\) — the magnetic counterpart of the electric-dipole formula, quoted here from its home in Section 65.3.2 [Jackson:1999]. For a moment of fixed magnitude inclined at \(\theta_{m}\) to the rotation axis, the component along the axis is static and the perpendicular component \(m_{\perp}=m\sin\theta_{m}\) rotates rigidly at \(\omega\), so \(\abs{\ddot{\vect{m}}}=\omega^{2}m_{\perp}\) at every instant, giving Equation (49.25). Setting \(\dot{E}=\dd\left(\tfrac{1}{2}I\omega^{2}\right)/\dd t =I\omega\dot{\omega}\) yields \(\dot{\omega}=-K\omega^{3}\) with \(K=\mu_{0}m_{\perp}^{2}/\left(6\pi c^{3}I\right)\); in terms of the period, \(P\dot{P}=4\pi^{2}K\), constant. Integrating \(\dot{\omega}=-K\omega^{3}\) from birth spin \(\omega_{0}\),

which is Equation (49.26). Finally, a dipole of moment \(m\) has polar surface field \(B_{\mathrm{p}}=\mu_{0}m/\left(2\pi R^{3}\right)\); solving \(P\dot{P}=4\pi^{2}K\) for \(m_{\perp}\) gives Equation (49.27).

The Crab pulsar makes every line quantitative. With \(P=33.4\,\mathrm{ms}\), \(\dot{P}=4.2\times 10^{-13}\) and the canonical \(I=10^{38}\,\mathrm{kg}\,\mathrm{m}^{2}\) (a uniform \(1.4\,M_{\odot}\) sphere of \(12\,\mathrm{km}\) gives \(\tfrac{2}{5}MR^{2} \approx1.6\times 10^{38}\,\mathrm{kg}\,\mathrm{m}^{2}\)), Equation (49.25) gives \(\abs{\dot{E}}=4\pi^{2}I\dot{P}/P^{3}\approx4.5\times 10^{31}\,\mathrm{W}\) (computed) — and the Crab nebula is observed to radiate of order \(10^{31}\,\mathrm{W}\) across the whole spectrum. The pulsar's slowing powers the nebula, which settled the identification [Gold:1968] [Staelin:1968]. The characteristic age Equation (49.26) is \(1260\) years, against the \(972\) years actually elapsed since the supernova of AD 1054: the difference measures the birth period — about \(16\,\mathrm{ms}\) for pure dipole braking. And Equation (49.27) gives \(B_{\mathrm{p}}\approx4\times 10^{8}\,\mathrm{T}\) — ten million times the strongest steady laboratory magnet, yet a thousandth of a magnetar (Section 49.3.2).

Derivation. Derives Phenomenon 49.18. The orbital-decay physics is proved in this book once, in The Quadrupole Formula for Gravitational Radiation: energy balance against the quadrupole luminosity Equation (46.6), carried through the Keplerian ellipse harmonics, gives the period derivative Equation (A.83), whose eccentricity enhancement factor \(f(e)=\left(1+\tfrac{73}{24}e^{2}+\tfrac{37}{96}e^{4}\right) \left(1-e^{2}\right)^{-7/2}\) evaluates to \(11.86\) at the measured \(e=0.6171\); with the measured masses the prediction is \(\dot{P}_{b}=-2.40\times 10^{-12}\), evaluated numerically there. We do not repeat the derivation; what remains to state is the correction named in the phenomenon. The observed \(\dot{P}_{b}\) is not purely intrinsic: the pulsar and the solar system accelerate differently in the Galactic potential, and a relative acceleration along the line of sight produces a secular drift of the apparent period (a kinematic Doppler derivative, with a further “Shklovskii” term from proper motion) that adds to \(\dot{P}_{b}\) at the per-cent level for PSR B1913+16. It is computed from the measured distance, proper motion and a Galactic rotation model, and subtracted; the \(0.16\%\) agreement of Section 47.1 is between the prediction and this intrinsic decay, and its error budget is now dominated by the Galactic correction, not by the timing [Weisberg:2016].

The double pulsar sharpens every number. PSR J0737-3039 is a \(2.45\,\mathrm{h}\) orbit holding a \(22.7\,\mathrm{ms}\) recycled pulsar and (until it precessed out of view) a \(2.77\,\mathrm{s}\) young one [Burgay:2003] [Kramer:2006]. Seeing both clocks fixes the mass ratio directly, so each measured post-Keplerian parameter — the periastron advance of about \(16.9^\circ\) per year (a Mercury-century of precession every six hours), the Einstein delay, the two Shapiro-delay parameters (the companion's orbit is seen nearly edge-on), and \(\dot{P}_{b}\) — is a separate test with no freedom left. All are mutually consistent; the Shapiro shape was tested to \(0.05\%\) already in 2006 [Kramer:2006], and with sixteen years of data the observed orbital decay agrees with the general-relativistic prediction to \(1.3\times 10^{-4}\) — the most stringent test of the quadrupole formula in existence [Kramer:2021]. The same fits deliver the component masses to a few parts in \(10^{4}\), \(m_{A}\approx1.34\,M_{\odot}\) and \(m_{B}\approx1.25\,M_{\odot}\) [Kramer:2021], the kind of measurement Section 49.4.1 builds on.

Derivation. Derives Phenomenon 49.19. The lower half of the statement rests on machinery already proved. The masses are Keplerian and post-Keplerian timing measurements — for the Shapiro delay, the excess delay as a function of orbital phase yields the companion mass and inclination separately, and the mass function then fixes the pulsar mass — with no reference to the pulsar's structure; the exclusion of soft equations of state is then immediate from Remark 49.15: the summit of \(M(\rho_{\mathrm{c}})\) computed from Equation (49.18) for the candidate \(P(\rho)\) falls below the measured mass, and no stable configuration of that matter can be the observed star.

The upper half — the Rhoades–Ruffini bound — needs one further argument, and it is worth exhibiting because it uses ignorance optimally. Suppose the equation of state is known up to some matching density \(\rho_{m}\) (nuclear physics vouches for it there) and completely unknown beyond, except for thermodynamic stability \(\dd P/\dd\rho\geq0\) and causality \(\dd P/\dd\rho\leq c^{2}\). Among all admissible continuations, the one supporting the largest mass is the stiffest, \(P=P_{m}+\left(\rho-\rho_{m}\right)c^{2}\): any softer choice lowers the pressure everywhere and with it the maximum mass. The bound is therefore computed by integrating Equations (49.17) and (49.18) with the causal continuation, and its scale is fixed by dimensional analysis alone. Above \(\rho_{m}\) the only quantities in the problem are \(G\), \(c\) and \(\rho_{m}\), and the unique mass they form is

where the dimensionless coefficient (about \(0.09\) of \(c^{3}/\sqrt{G^{3}\rho_{m}}\), computed) is quoted from the numerical integration of Rhoades and Ruffini at their matching density [Rhoades:1974]. The bound moves only as \(\rho_{m}^{-1/2}\) — trusting nuclear physics to twice the density costs merely a factor \(\sqrt{2}\) — which is what makes it robust, and it brackets the measured two-solar-mass stars from above with room to spare. The Buchdahl statement was proved as Proposition 49.12.

Derivation. Derives Phenomenon 49.21. Each method needs one mechanism derived: light bending for the pulse profile, and the entry of \(\Lambda\) into the waveform for GW170817.

Light bending. A photon leaving the surface at angle \(\alpha\) to the outward radial direction carries, by the null-geodesic machinery of Schwarzschild Geometry and Black Holes, an impact parameter \(b=R\sin\alpha/\sqrt{1-r_{\mathrm{s}}/R}\), and arrives at infinity at an angle

from the radius of emission. Grazing emission (\(\alpha=\pi/2\)) fixes the visible cap: everything within colatitude \(\psi_{\max}=\psi(\pi/2)\) of the sub-observer point is seen at once, and \(\psi_{\max}>\pi/2\) whenever \(r_{\mathrm{s}}>0\) — a compact star shows more than half its surface. The integral is done numerically in the fits; the compact approximation \(\cos\alpha\approx u+\left(1-u\right)\cos\psi\) with \(u=r_{\mathrm{s}}/R\), accurate to a per cent for \(R\gtrsim2r_{\mathrm{s}}\) [Beloborodov:2002], exhibits the physics: \(\cos\psi_{\max}=-u/\left(1-u\right)\), so for PSR J0030+0451 (\(u\approx0.33\)) the visible cap extends to \(\psi_{\max}\approx119^{\circ}\) — about \(74\%\) of the surface (computed). A polar hot spot therefore never fully disappears, and the depth of the pulse modulation falls monotonically with \(u\): the profile measures compactness, while the observed flux and temperature scale, blue- and redshifted by \(\left(1+z\right)=\left(1-u\right)^{-1/2}\approx1.2\), tie in the radius itself. Inverting energy-resolved profiles for mass, radius, spot geometry and viewing angle is the numerical likelihood analysis of [Miller:2019] [Riley:2019], run twice with independent codes as the phenomenon records.

Tidal phasing. In a binary at separation \(r\), each star sits in its companion's tidal field, \(\mathcal{E}_{ij}\sim Gm_{2}/r^{3}\), and by Equation (49.29) acquires \(Q_{ij}=-\lambda\mathcal{E}_{ij}\). The induced quadrupole does two things, both scaling with \(\lambda\): it deepens the mutual potential by the induced-moment interaction \(\delta U=-\tfrac{1}{2}\lambda\mathcal{E}_{ij}\mathcal{E}^{ij} \sim-\lambda G^{2}m_{2}^{2}/r^{6}\), and it adds coherently to the quadrupole moment whose time variation radiates (Equation (46.6)). Relative to the point-mass energy \(\sim Gm_{1}m_{2}/r\), the tidal term is smaller by \(\lambda G/r^{5}\sim\Lambda\left(GM/rc^{2}\right)^{5}\): five powers of compactness, i.e. ten powers of \(v/c\) — formally a fifth-post-Newtonian correction, utterly negligible except that \(\Lambda\) is a few hundred, and that the phase of a chirping binary (Equation (46.8)) accumulates over thousands of cycles. Carrying the energy and flux corrections through the phase evolution gives the leading tidal contribution to the frequency-domain phase, proportional to the single mass-weighted combination

with a known numerical coefficient, quoted from the standard computation [Flanagan:2008] [Hinderer:2008]. Matched filtering of GW170817's final orbits therefore measures \(\tilde{\Lambda}\); the discovery analysis bounded \(\Lambda(1.4\,M_{\odot})\leq800\) at \(90\%\) confidence (low-spin prior) [Abbott:2017], and the dedicated reanalysis gives \(\Lambda(1.4\,M_{\odot})=190^{+390}_{-120}\), which through Equation (49.30) with the relativistic \(k_{2}\) of [Hinderer:2008] yields \(R=11.9\pm1.4\,\mathrm{km}\) for both stars [Abbott:2018] — the numbers the phenomenon quotes, in agreement with NICER's, from data that share not a single instrument, wavelength or systematic. The same event's electromagnetic counterpart — the kilonova whose lanthanide spectra realized the r-process site of Stellar Structure and Nucleosynthesis — and its detection history belong to Experiment: Gravitational Waves.