Cooling and Magnetic Evolution

A neutron star cools by neutrino emission from its interior and photons from its surface while its magnetic field redirects heat, evolves, and can return magnetic energy as Joule heating. The observable is not the core temperature: an X-ray spectrum is filtered through the envelope, atmosphere, interstellar absorption, gravitational redshift, detector response, and an age estimate. Cooling curves therefore constrain a coupled model of composition, pairing, transport, magnetic geometry, and observation. The Cassiopeia A central compact object illustrates both the promise and the calibration sensitivity of that inference.

Required background. The Einstein Equation supplies redshift and stellar structure for global thermal balance.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies local energy conservation; Relativistic Matter and Compact Objects supplies dense-matter microphysics; Computational Relativity supplies transport and convergence checks.

The redshifted energy budget

After the young star becomes thermally relaxed, its deep interior is close to isothermal in the Tolman sense: $\widetilde T=T(r)e^{\Phi(r)}$ is approximately constant, where $g_{tt}=-e^{2\Phi}$. Integrating the local energy equation gives the schematic balance seen by a distant observer,

$$C(\widetilde T)\frac{d\widetilde T}{dt} =-L_\nu^\infty-L_\gamma^\infty+H^\infty.$$

$C$ is the total heat capacity, $L_\nu^\infty$ the redshifted neutrino luminosity, $L_\gamma^\infty$ the photon luminosity, and $H^\infty$ internal heating. Here $t$ is time at infinity. If $c_V$ and $Q_\nu$ denote local heat capacity per proper volume and local neutrino emissivity, respectively, the relaxed, fixed-background approximation gives

$$C(\widetilde T)=\int c_V(\widetilde T e^{-\Phi})\,dV_{\rm proper}, \qquad L_\nu^\infty=\int Q_\nu e^{2\Phi}\,dV_{\rm proper}.$$

One redshift factor lowers each emitted neutrino’s energy; the second converts proper emission time to time at infinity. In the heat-capacity term, the energy-redshift factor cancels $dT/d\widetilde T=e^{-\Phi}$. These relations assume freely escaping neutrinos, negligible structural evolution, and a nearly uniform redshifted interior temperature.

Modified-Urca reactions and nucleon bremsstrahlung provide relatively slow neutrino cooling. Direct Urca, if momentum conservation and composition allow it, is much faster. Pairing suppresses heat capacity and many reactions at low temperature, but pair formation and breaking opens a transient neutrino channel near the critical temperature.

The heat-blanketing envelope maps a temperature at its base, $T_b$, to a local effective surface temperature $T_s$. Light elements make the envelope more heat-transparent than an iron-like one, and a strong magnetic field makes the map direction dependent. At the surface, $T_s^\infty=e^{\Phi(R)}T_s$ and $R_\infty=R e^{-\Phi(R)}$, so $L_\gamma^\infty=4\pi\sigma R_\infty^2(T_s^\infty)^4$. The transport, neutrino, envelope, and atmosphere layers are distinguished in Pons, Dehman, and Viganò 2026, §§2.2–2.3, especially Eqs. (4)–(5) and §2.2.5.

A transparent cooling estimate

Ignore photon losses and heating temporarily and assume normal degenerate matter, so $C=C_0\widetilde T$. If $L_\nu^\infty=L_0\widetilde T^\alpha$ with $\alpha\gt2$, integration of the energy equation gives

$$\widetilde T(t)= \left[\widetilde T_0^{-(\alpha-2)} +(\alpha-2)\frac{L_0}{C_0}(t-t_0)\right]^{-1/(\alpha-2)}.$$

Away from thresholds and pairing transitions, modified Urca has $\alpha\simeq8$ and direct Urca $\alpha\simeq6$, giving late-time powers $\widetilde T\propto t^{-1/6}$ and $t^{-1/4}$ in this simplified regime. The temperature powers and numerical stiffness are discussed in Pons, Dehman, and Viganò 2026, §§2.2.3 and 2.3. The coefficients are also very different, so powers alone do not compare absolute temperatures. Pairing, photon cooling, an evolving envelope, and heating invalidate this single-power estimate; the measured surface temperature is not $\widetilde T$.

Why magnetic and thermal evolution couple

Electrical conductivity sets Ohmic decay; the Hall term advects magnetic flux through the electron fluid without directly dissipating energy, but it can transfer power to smaller scales where resistivity acts faster. Temperature changes conductivity, and magnetic geometry changes the thermal conductivity tensor. Joule dissipation then heats the crust. Consequently, a model that evolves a scalar field strength independently of temperature can miss both feedback loops and surface-temperature anisotropy.

Initial field topology, crustal impurities, superfluid and superconducting states, core flux transport, and the treatment of the crust–core and surface boundaries remain consequential assumptions. In a weak-field old star, magnetic feedback may be perturbative. In a magnetar-strength field, it can dominate the surface temperature and produce hot regions, making a single blackbody radius a poor proxy for the stellar radius. The thermal and induction equations, their numerical treatment, and the dependence on initial and boundary conditions are developed in Pons, Dehman, and Viganò 2026, §§2.2–2.3, 3.1–3.2, 4.2–4.4, and 6.1–6.3.

A named observational test: Cassiopeia A

Evidence cutoff: 2026-08-18. The named primary dataset is the set of Chandra ACIS-S observations of CXOU J232327.9+584842 in the Chandra Data Archive, separated into FAINT subarray and GRADED observing modes. A published uniform reduction of all then-available spectra used CIAO 4.14 with CALDB 4.9.8, grouped to at least 25 counts per energy bin; the observation list, extraction, spectral model, and joint fits are specified in Shternin et al. 2023, §2, Tables 1–3, and Appendix A. Those event files are the public data product. The fitted temperature decline and pairing constraints are derived posteriors, not additional detector measurements.

The calibration version is scientifically active. ACIS contamination and time-dependent response affect an apparent temperature trend; the official CALDB release history reached version 4.12.5 on 2026-06-08, as recorded by the Chandra X-ray Center 2026, CALDB 4 release table and version 4.12.5 notes. Updating from CALDB 4.9.8 is not a textual correction: it requires reprocessing events, rebuilding responses, and refitting the spectra. The subarray analysis through 2020 found different cooling rates when absorbing column density was fixed or varied and explicitly identified contamination calibration as a systematic in Posselt and Pavlov 2022, §§2–4 and Tables 1–2.

An independent detector comparison is now available. Chandra HRC-S observations spanning 1999–2024, reduced with CIAO 4.16 and CALDB 4.11.5, support a declining absorbed flux. For one background choice, the reported flux decrease is $4.7\%\pm1.5\%$ per decade. The inferred temperature decrease is smaller than the ACIS estimate: about $0.6\%$ per decade with fixed $N_{\rm H}$, or $1.1\%$ with varying $N_{\rm H}$, in Zhao et al. 2025, §§2–3, Tables 1–2, and Figs. 2–3. HRC-S has poor energy resolution, so temperature inference uses a carbon atmosphere and other spectral parameters; the variable-column case borrows $N_{\rm H}$ information from ACIS. The comparison therefore strengthens the evidence for flux decline while leaving the cooling rate and microscopic interpretation sensitive to modeling and calibration.

Selection, likelihood, and systematic alternatives

A reproduction must freeze observation identifiers, detector mode, good-time intervals, event grades, source and background regions, energy band, pileup treatment, CIAO and CALDB versions, response matrices, grouping, and the atmosphere table. It must state whether FAINT and GRADED spectra are combined or fitted separately. The forward model multiplies a redshifted atmosphere spectrum and interstellar absorption by the epoch-specific detector response. One may reproduce the published grouped-spectrum statistic or fit unbinned counts with a Poisson likelihood, but changing the likelihood must be reported as a new analysis.

Priors or fixed choices for mass, radius, distance, age, emitting area, atmosphere composition, and hydrogen column $N_{\rm H}$ propagate into the temperature trend. Important systematic alternatives include fixed versus epoch-dependent $N_{\rm H}$, carbon versus hydrogen atmosphere, detector mode, pileup, background region, contamination model, cross-epoch calibration, and the remnant age. Inferring a pairing gap adds EOS, envelope, neutrino-rate, and superfluid-critical-temperature assumptions. Agreement of two reductions that share the same response model is not an independent calibration check.

This page does not rerun CIAO, generate response matrices, or fit a cooling curve. A reproducible reanalysis would retain the archive query, ObsID list, checksums, CIAO environment, CALDB snapshot, extraction commands, response files, spectral model, priors, likelihood, posterior samples, and a comparison of calibration and atmosphere alternatives. A newer CALDB is a reason to repeat that pipeline, not to relabel an old posterior.

Interpreting a cooling track

A point on a temperature–age diagram mixes measurement and model. The age may come from a supernova remnant, kinematics, or spin-down; the temperature may be a color temperature or atmosphere effective temperature; and nondetections have selection-dependent limits. Population comparisons should retain these distinctions and should not use a heterogeneous compilation as if it had one common response and selection function.

Rapid cooling can indicate an efficient neutrino channel, but several inputs are degenerate: mass controls whether a threshold is reached, pairing can both suppress and enhance emission, envelope composition changes $T_s$ at fixed $T_b$, and magnetic heating can mask core cooling. The robust statement is therefore conditional: a specified thermal model and calibrated data imply a likelihood for its parameters. A unique core composition does not follow from one surface-temperature slope.

Check your understanding

Two otherwise identical spectral fits infer redshifted effective temperatures $T_1^\infty=2.00\times10^6\,\mathrm K$ and $T_2^\infty=1.96\times10^6\,\mathrm K$ ten years apart. What is the fractional temperature change and, with fixed emitting area, the corresponding fractional change in bolometric photon luminosity to first order?

Solution

The fractional temperature change is

$$\frac{\Delta T^\infty}{T_1^\infty} =\frac{1.96-2.00}{2.00}=-0.020,$$

or a 2.0% decline. Because $L_\gamma^\infty\propto(T_s^\infty)^4$ at fixed apparent area, $\Delta L/L\simeq4\Delta T/T=-0.080$, an 8% decline. This conversion is invalid if the inferred emitting area or atmosphere changes between epochs—precisely why response and model choices must accompany the quoted slope.

References

• Chandra X-ray Center. “CALDB 4 Release Notes.” Harvard–Smithsonian Center for Astrophysics, version history through CALDB 4.12.5 (2026). Official release table.
• Pons, José A., Clara Dehman, and Daniele Viganò. “Magnetic, Thermal and Rotational Evolution of Isolated Neutron Stars.” Living Reviews in Computational Astrophysics 12 (2026): 5. DOI.
• Posselt, Bettina, and George G. Pavlov. “The Cooling of the Central Compact Object in Cas A from 2006 to 2020.” The Astrophysical Journal 932 (2022): 83. DOI.
• Shternin, Peter S., Dmitry D. Ofengeim, Craig O. Heinke, and Wynn C. G. Ho. “Constraints on Neutron Star Superfluidity from the Cooling Neutron Star in Cassiopeia A Using All Chandra ACIS-S Observations.” Monthly Notices of the Royal Astronomical Society 518 (2023): 2775–2793. DOI.
• Zhao, Jiaqi, Craig O. Heinke, Peter S. Shternin, Wynn C. G. Ho, Dmitry D. Ofengeim, and Daniel Patnaude. “Verification of Cas A Neutron Star Cooling Rate Using Chandra HRC-S Observations.” The Astrophysical Journal 991 (2025): 136. DOI. Open preprint, v2.