Searching for Exotrojans in Pulsar Binary Systems

Abstract

Trojan asteroids are found in the equilateral triangle Lagrange points of the Sun-Jupiter system in great number, though they also exist less prolifically in other Sun-planet systems. Despite up to planetary mass Trojans being predicted in extrasolar systems (i.e. exotrojans), they remain largely unconfirmed, though with recent strong candidate evidence emerging. We turn the current search for exotrojans to radio pulsars with low-mass companions ($\sim0.01\,\rm{M}_\odot$) using accurately measured pulse times of arrival. With techniques developed for detecting the reflex motion of a star due to a librating Trojan, we place reasonable mass constraints ($\sim 1\,\rm{M}_\oplus$) on potential exotrojans around binary pulsars observed in the NANOGrav 15-year data set. We find weak evidence consistent with $\sim1\,\rm{M}_{\rm J}$ exotrojans in the PSR~J0023+0923 and PSR~J1705$-$1903 systems, though the signals likely have a different, unknown source. We also place a libration-independent upper mass constraint of $\sim8$\,M$_{\rm J}$ on exotrojans in the PSR~1641+8049 binary system by looking for an inconsistency between the times of superior conjunction as measured by optical light curves and those predicted by radio timing.

Figures (8)

• Figure 1: A schematic of the positions of the Trojan, $m_{\rm T}$, the dominant companion, $m_{\rm c}$, and the effective combined object, $m_{\rm f}$. $\Delta \phi$ is the relative angle between the orbital longitudes of the combined companion and the actual dominant companion. The Trojan and dominant companion are shown orbiting counterclockwise here, meaning the Trojan is in the $L_4$ Lagrange point of the pulsar-dominant companion system. At the time of this figure, the companion is at its closest position to Earth, known as inferior conjunction. The line of sight to the Earth is shown in red. See the text for more details. Inspired by Figure 1 of ford_observational_2006.
• Figure 2: The relevant angles in the triangular three-body system when librations are present. The Earth lies in the $xz$-plane, where $\hat{z} = \hat{x}\times \hat{y}$, with the line of sight making an angle $I$ with the $z$-axis. $m_{\rm f}$ is the fictitious companion mass given by Equation \ref{['eq:mf']}. The Trojan and companion are shown orbiting counterclockwise here, meaning the Trojan is in the $L_4$ Lagrange point. If $m_{\rm T} > m_{\rm c}$, then the companion and Trojan should be interchanged by our convention. $\lambda$ is the angle that the Trojan and dominant companion librate around, which can differ slightly from $\phi$, the angle given by the location of the combined fictitious companion. See the text for more details. Inspired by Figure 1 of leleu_mainpaper_2015.
• Figure 3: HiPERCAM light curve data of the PSR J1641+8049 system for each Sloan band. The data was first presented in mata2023black0023. We show the best-fit model for the $r'$ band as an example, with the remaining model curves shown in kirichenko_black_2024. Every band except $u'$ was used in calculating $T_{\rm o, s}$. The vertical dashed line marks the time of superior conjunction as predicted by radio timing. The vertical cyan band encloses the 68% confidence interval of the time of superior conjunction as calculated from the optical light curve.
• Figure 4: The posterior space of the PSR J0610$-$2100 libration analysis. Each histogram represents the marginalized posterior of the relevant parameter. Each contour plot represents the joint-marginalized posterior of the two relevant parameters. This is a clear non-detection with almost flat posteriors across all parameters. There is a slight preference for $\log{(\mathcal{T}/\rm{s})} \sim 10^{-6}$, which is similar to the noise floor of this pulsar agazie2023nanograv, though $\log{(\mathcal{T}/\rm{s})}$ is agnostic to the other three parameters. Furthermore, the Bayes factor for a non-zero $\log{(\mathcal{T}/\rm{s})}$ is less than 1, so no librations ($\mathcal{T} \equiv 0$) are probabilistically preferred.
• Figure 5: The posterior space of the PSR J0023+0923 libration analysis. Each histogram represents the marginalized posterior of the relevant parameter. Each contour plot represents the joint-marginalized posterior of the two relevant parameters. With a Bayes factor of 11.5(2), we have a tentative detection of the signal in Equation \ref{['eq:libration_residuals']}. However, we find that the origin of this signal is unlikely to be caused by a Trojan due to a nonphysical Gaussian-average value for $\theta$. See the text for more details.