Baryon Number and Binding Energy

“The mass of a star” can mean several inequivalent quantities. The exterior Schwarzschild parameter measures gravitational mass, while a conserved baryon current counts material particles on a curved spatial slice. Their difference contains gravitational binding and internal-energy contributions and is the quantity relevant to assembly, disruption, and transitions at fixed baryon number. Keeping these notions separate prevents a common error: integrating energy density over proper volume and calling the result the mass seen at infinity.

Required background. The Einstein Equation supplies gravitational mass and matter coupling; Killing Fields and Isometry Groups supplies static symmetry and its conserved energy.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies stress–energy and currents; Relativistic Matter and Compact Objects supplies the thermodynamic setting; Computational Relativity explains integral and convergence checks.

Conserved baryon number

Let $n_{\rm B}$ be the baryon number density in the local fluid rest frame. For reactions that conserve baryon number, the current

$$J_{\rm B}^{\mu}=n_{\rm B}u^\mu, \qquad \nabla_\mu J_{\rm B}^{\mu}=0$$

defines a hypersurface-independent charge when there is no baryon flux through the boundary. For a future-directed unit slice normal $s^\mu$, $N_{\rm B}=-\int_\Sigma J_{\rm B}^\mu s_\mu\,dV$; the minus sign makes the count positive in the inherited metric signature. On a static spherical slice with areal radius $r$,

$$N_{\rm B} =4\pi\int_0^R \frac{n_{\rm B}(r)r^2}{\sqrt{1-2Gm(r)/r}}\,dr.$$

The square-root factor is the proper-volume correction. Multiplying by a declared reference baryon mass $m_{\rm b}$ gives the baryonic mass $M_{\rm b}=m_{\rm b}N_{\rm B}$. The choice of neutron mass, atomic mass unit, or another convention changes the numerical label but not the conserved charge; it must therefore be stated when comparing tables.

This construction follows the conserved-current treatment summarized in Gourgoulhon 2011, §4.1, Eqs. (4.1)–(4.6), pp. 59–60, specialized from circular motion to a static fluid and translated from geometrized units to explicit $G$.

Three masses and one binding convention

The gravitational mass is

$$M=m(R)=4\pi\int_0^R \epsilon(r)r^2\,dr.$$

Although this coordinate integral resembles a Euclidean volume integral, it is fixed by the Einstein equation and equals the asymptotic mass parameter. The proper material-energy integral is instead

$$M_{\rm proper} =4\pi\int_0^R \frac{\epsilon(r)r^2}{\sqrt{1-2Gm(r)/r}}\,dr.$$

For a cold equilibrium at fixed baryon number, a useful binding convention is

$$E_{\rm bind}=M_{\rm b}-M.$$

Positive $E_{\rm bind}$ means that separated reference baryons have more energy than the assembled star. This difference mixes gravitational binding with the choice of reference rest mass and nuclear internal energy. The quantity $M_{\rm proper}-M$ isolates a different comparison and should not be substituted without explanation.

Write $\epsilon=m_{\rm b}n_{\rm B}+u$, with $u$ the local internal-energy density in that reference convention. Then

$$E_{\rm bind} =(M_{\rm proper}-M)-\int u\,dV_{\rm proper}.$$

This identity explains why a positive gravitational contribution does not alone fix the total binding. In the Newtonian limit,

$$M_{\rm proper}-M \simeq \int_0^R \frac{Gm(r)}{r}\,dm =-W,$$

where $W=-\int Gm\,dm/r$ is the Newtonian gravitational potential energy. For a uniform weak-field sphere this gives $-W=3GM^2/(5R)$. Thus $E_{\rm bind}\simeq-(U+W)$, with $U=\int u\,dV$: the familiar total energy of assembly has the opposite sign to the positive-binding convention.

Redshifted chemical equilibrium

For a zero-temperature, single-component barotrope, the local first law gives $d\epsilon=\mu_{\rm B}dn_{\rm B}$ and $\epsilon+p=\mu_{\rm B}n_{\rm B}$. Combining this with hydrostatic balance yields $dp=n_{\rm B}d\mu_{\rm B}$ and therefore

$$\frac{d\ln\mu_{\rm B}}{dr}=-\frac{d\Phi}{dr}, \qquad e^{\Phi(r)}\mu_{\rm B}(r)=\mu_\infty.$$

The chemical potential is larger deeper in the gravitational well, but its redshifted value is constant in diffusive equilibrium. Along a neighboring sequence of equilibria for the same EOS,

$$dM=\mu_\infty\,dN_{\rm B},$$

with the reference units understood. This first-law structure is central to turning-point arguments: perturbations may conserve $N_{\rm B}$ even when central density changes. The constrained-sequence logic is made explicit in Friedman, Ipser, and Sorkin 1988, theorem and discussion, pp. 722–724.

For multifluid or finite-temperature matter, each independently conserved current and entropy contribution needs its own conjugate potential. A single $\mu_{\rm B}$ relation then ceases to capture all allowed perturbations.

Checks and limitations

The weak-field limit provides a sign check: $M$ should be smaller than the sum of rest masses for an ordinarily bound cold star. Conservation supplies a second check: evaluate $N_{\rm B}$ on independent slices of a dynamical model; differences beyond flux through the boundary signal numerical or physical baryon nonconservation. Units supply a third: $N_{\rm B}$ is dimensionless, whereas all three masses carry the same mass dimension.

Binding energy is not automatically energy radiated in a particular formation channel. Neutrino losses, ejected baryons, changes of nuclear composition, rotation, and thermal support alter the bookkeeping. Comparisons must hold fixed the conserved quantities appropriate to the process.

Check a binding fraction

A cold model has $M_{\mathrm{b}}=1.60\,M_\odot$ and $M=1.45\,M_\odot$. Compute $E_{\mathrm{bind}}/M_{\mathrm{b}}$ and state what the number does not determine.

Solution

The binding energy in the declared convention is $E_{\rm bind}=0.15\,M_\odot$, so

$$\frac{E_{\mathrm{bind}}}{M_{\mathrm{b}}}=\frac{0.15}{1.60}=0.09375.$$

Thus about $9.4\%$ of the reference baryonic rest mass is absent from the gravitational mass. The fraction does not determine how much energy a real formation event radiated, because the initial composition, ejecta, neutrinos, thermal energy, and baryon conservation history have not been specified.

References

• Friedman, John L., James R. Ipser, and Rafael D. Sorkin. “Turning-Point Method for Axisymmetric Stability of Rotating Relativistic Stars.” Astrophysical Journal 325 (1988): 722–724. DOI.
• Gourgoulhon, Éric. “An Introduction to the Theory of Rotating Relativistic Stars.” COMPSTAR 2010 lecture notes, revised 20 September 2011. arXiv:1003.5015v2.