Shapiro Delay Measurements from Fifteen Years of PSR J1231$-$1411 Radio Observations

TL;DR

This work tackles constraining neutron-star masses in the MSP PSR J1231$-$1411 by measuring the relativistic Shapiro delay. It combines ~15 years of radio timing from the GBT and NRT with Fermi-LAT gamma-ray data, applying three analysis frameworks (grid search with fixed $m_c\sin i$, Bayesian inference with astrophysical priors, and fully joint timing-noise modeling) using PINT and ENTERPRISE under general relativity. Bayesian results yield $m_c = 0.23^{+0.09}_{-0.06}\,M_\odot$, $m_p = 1.87^{+1.11}_{-0.67}\,M_\odot$, and $i = 79.80^{+3.47}_{-4.70}$ with a 68.3% CI, and enforcing $m_p \le 3\,M_\odot$ tightens to $m_p = 1.62^{+0.73}_{-0.58}\,M_\odot$. Despite the weak Shapiro signal due to the inclination, these results inform NICER X-ray modeling and illustrate robust strategies for future high-precision mass measurements that constrain the neutron-star interior equation of state.

Abstract

We present 15 years of Nançay and Green Bank radio telescope timing observations for PSR J1231$-$1411. This millisecond pulsar is a primary science target for the Neutron Star Interior Composition Explorer telescope (NICER, which discovered its X-ray pulsations), has accumulated near-continuous $γ$-ray data since the Fermi-Large Area Telescope's launch, and has been studied extensively with the Green Bank and Nançay radio telescopes. We have undertaken a campaign with the Green Bank Telescope targeting specific orbital phases designed to improve our constraint on the pulsar's mass through the detection of a relativistic Shapiro delay. Both frequentist and Bayesian techniques -- the latter incorporating priors from white dwarf binary evolution models -- are applied to fifteen years of radio observations, yielding relatively weak constraints on the companion and pulsar masses of $0.23^{+0.09}_{-0.06}$ M$_{\odot}$ and $1.87^{+1.11}_{-0.67}$ M$_{\odot}$, respectively (68.3% CI from Bayesian fits); however, the orbital inclination is measured to better relative precision ($79.80^{+3.47}_{-4.70}$ degrees). Restricting the maximum allowed pulsar mass to 3 M$_{\odot}$ while simultaneously sampling the noise and timing models improves the constraint and lowers the measured mass to $1.62^{+0.73}_{-0.58}$ M$_{\odot}$. While our radio-derived inclination result has informed recent NICER X-ray studies of PSR J1231$-$1411, the lessons learned from this troublesome pulsar will also bolster future high-precision mass measurement campaigns and resulting constraints on the neutron star interior equation of state.

Figures (1)

• Figure 1: Timing residuals in $$s vs. MJD (top panel) and Orbital Phase (bottom panel) for J1231$-$1411 radio data. All best-fit timing parameters, including those that describe the relativistic Shapiro delay, are included in the model that produces these residuals. Color groups reflect backend and receiver combinations. Dark blue points are from NRT's BON backend and light blue points are from NRT's NUPPI backend; both of these groups use the same L-band receiver. Darker and lighter orange points are GBT 820-MHz TOAs and 350-MHz TOAs, respectively, both of which used the GUPPI backend.