The Radius of PSR J0437-4715 from NICER Data

TL;DR

This work analyzes NICER observations of PSR J0437-4715 to extract the neutron-star radius by modeling surface hot spots while incorporating a modulated nonthermal emission component suggested by NuSTAR data. Using a Bayesian, two-stage sampling framework and several spot configurations (notably a three-circle model), the authors demonstrate that including the modulated power-law component is essential for a good bolometric fit and for avoiding radius bias. The resulting radius posterior, centered near 13 km with a 68% interval of roughly 11.8–15.1 km at the well-measured mass $M\approx1.418\,M_\odot$, is consistent with other NICER measurements and informs the equation of state of dense matter; when combined with GW data and previous NICER results, it tightens constraints on high-density EOS features. The analysis highlights the importance of robust background modeling and nonthermal emission in NICER pulsar analyses and points to modest but meaningful improvements expected with future data.

Abstract

Neutron star Interior Composition Explorer (NICER) data have been used to estimate the masses and radii of the rotation-powered millisecond pulsars PSR J0030$+$0451, PSR J0740$+$6620, PSR J0437$-$4715, PSR J1231$-$1411, and PSR J0614$-$3329, sometimes in joint analyses with X-ray Multi-Mirror (XMM-Newton) data. These measurements provide invaluable information about the properties of cold, catalyzed matter beyond nuclear saturation density. Here we present the results of our modeling of NICER data on PSR J0437$-$4715 using several different models of hot thermal X-ray emitting spots on the stellar surface. For this pulsar, previous Nuclear Spectroscopic Telescope Array (NuSTAR) observations established that there is also a modulated nonthermal component to the emission, but the previously published analysis of NICER data did not model this component. We find that the Bayesian evidence is significantly higher when the modulated nonthermal component is included, and that omission of this component leads to poor fits to the bolometric NICER data and thus risks bias in the resulting radius estimates. Our models, which we pursue to inferential convergence, therefore have modulated nonthermal emission, and our headline model has in addition three uniform-temperature thermally-emitting circular spots. Using this model, the symmetric 68% credible range in the radius is 11.8 km to 15.1 km, which at the independently-measured mass of $M=1.418\pm 0.044~M_\odot$ is consistent with previous reports of the radius of the $\sim 1.4~M_\odot$ pulsar PSR J0030$+$0451. We discuss the implications of this measurement for the equation of state of dense matter.

Figures (12)

• Figure 1: Radius posteriors for each of our three models with a modulated power law (the vertical axis is linear in the probability density); from top to bottom in the legend the model numbers are 5, 4 and 3. The dotted line shows the prior. There is a large overlap of the posteriors of all three models, which shows that among the models with a modulated power law, the posterior is not especially sensitive to the model chosen.
• Figure 2: Spot locations, sizes, and temperatures for the best-fit of our featured model with three uniform-temperature circular spots plus a modulated power law, i.e., model number 5. The smallest spot has an effective temperature, as measured by a comoving observer on the surface, of 0.17 keV; the middle-sized spot has an effective temperature of 0.031 keV; and the largest spot has an effective temperature of 0.11 keV. The solid black circle indicates the colatitude of the observer, 0.742 radians, which is strongly constrained by radio observations. See the Appendix for the full set of parameter values for this best fit. Bearing in mind that the temperature distribution is surely not actually uniform circles, this could be an indication that the two spots in the hemisphere of the observer represent a single spot with a range of temperatures.
• Figure 3: Phase-channel residuals for the best fit to the data of our model with three uniform-temperature circular spots plus a modulated power law, over the full set of 32 uniformly-spaced rotational phases and NICER PI channels 30--299 inclusive. Here, for a phase-channel bin $i$, if the data have $d_i$ counts and the model predicts $m_i$ counts, we define $\chi\equiv (m_i-d_i)/\sqrt{m_i}$. In addition to the fit having an overall acceptable $\chi^2/{\rm dof}=8335.00/8345$ (for which the probability of this $\chi^2$ or higher for a correct model is 53%), there are no patterns visible and no individual phase-channel bins with an unexpectedly large $|\chi|$. This is a one-way test: our satisfactory fit does not guarantee that the model is correct, but a very low probability would indicate that we would need to look more closely at our model.
• Figure 4: (top panel) Comparison between the bolometric data (histogram) and the best model with three uniform-temperature circular spots plus a modulated power law (red line). (bottom panel) Residuals between the data and the best model. Here we plot two full rotational cycles, with 32 uniformly-distributed rotational phases per cycle. In addition to the overall adequate bolometric $\chi^2/{\rm dof}=33.39/27$ (18% probability if the model is correct), we see no strong outliers and no obvious patterns in the residuals. As with the phase-channel $\chi$, this is a one-way test, which might detect a strong deviation between the model and the data but cannot guarantee the model's correctness if it passes the test.
• Figure 5: Mass and radius posteriors, using our best fit to the data of our model with three uniform-temperature circular spots plus a modulated power law, i.e., our model number 5. The dotted lines show the priors. The approximate upper diagonal boundary in the mass-radius plot corresponds to our $c^2R_{\rm e}/(GM)=8$ prior upper limit, whereas the approximate lower boundary is set by the likelihood rather than by our prior $c^2R_{\rm e}/(GM)=3.2$ lower limit. The mass posterior is single-peaked and is shifted only slightly from the tight prior given by radio observations, whereas the radius posterior is bimodal (with a larger mode at higher radii).