Is there a retrograde accretion disk around 4U 1626$-$67? Tracking torque reversals with a state-space model

TL;DR

The study investigates torque reversals in the accretion-powered pulsar 4U 1626$-$67 by applying an unscented Kalman filter (UKF) to 3340 pulse-frequency samples from CGRO/BATSE and Fermi/GBM, formulating magnetocentrifugal accretion in a state-space framework with hidden variables $Q(t)$ and $S(t)$ and a sign-changing torque via $\epsilon(t)$. By comparing prograde-prograde and retrograde-prograde disk configurations through Bayesian model selection, the authors find moderate evidence favoring a retrograde disk during the deceleration phase and a prograde disk during acceleration ($\mathcal{M}_{\rm RP}$). Time-resolved reconstructions show a smooth transition of $Q(t)$ and a slow evolution of the fastness $\omega(t)$ around the 2008 reversal, with $\omega(t)$ remaining near $\sim0.25$–$0.30$ under $\mathcal{M}_{\rm RP}$, consistent with a weak propeller regime. The inferred angular accelerations and the behavior of $Q(t)$ and $\omega(t)$ provide indirect, time-resolved support for a retrograde accretion disk and demonstrate the Kalman-filter approach as a powerful tool for probing disk–magnetosphere dynamics in accreting pulsars.

Abstract

X-ray timing studies of the persistent, Galactic, accretion-powered pulsar 4U 1626$-$67 reveal torque reversals, during which the pulse frequency $ν(t)$ alternates between multiyear episodes of secular acceleration and deceleration, separated by transitions lasting $\lesssim 150 \, \rm{days}$. Here an unscented Kalman filter is applied to track the $ν(t)$ fluctuations observed in 22.7 years (3340 samples) of publicly available Compton Gamma-Ray Observatory and Fermi Gamma-Ray Space Telescope data to test the canonical picture of magnetocentrifugal accretion for consistency with prograde-prograde and retrograde-prograde accretion disk configurations on either side of the 2008 torque reversal. It is found that the retrograde-prograde model is preferred, with a log Bayes factor equal to 0.44 and maximum a posteriori log likelihood ratio equal to 2.5. The mass accretion rate $Q(t)$ and magnetocentrifugal fastness $ω(t)$ transition smoothly between episodes of deceleration and acceleration; $Q(t)$ shifts by $\leq 0.34 \, {\rm dex}$ across the reversal, and one measures $ω(t) \approx 0.25$ and $ω(t) \approx 0.30$ during deceleration and acceleration, respectively. The angular acceleration $\dotΩ(t)$ satisfies $-9 \lesssim \dotΩ(t)/(10^{-12} \, \rm{rad \, s^{-2}}) \lesssim -5$ and $2 \lesssim \dotΩ(t)/(10^{-12} \, \rm{rad \, s^{-2}}) \lesssim 9$ before and after the 2008 reversal, respectively, compared to $\dotΩ \approx -3.0 \times 10^{-12} \, \rm{rad \, s^{-2}}$ before reversal and $\dotΩ \approx 2.5 \times 10^{-12} \, \rm{rad \, s^{-2}}$ after reversal, as inferred from previous long-term X-ray timing and spectral analysis of 4U 1626$-$67.

Figures (7)

• Figure 1: Pulse frequency $\nu(t)$ observations (units: mHz) of the low-mass X-ray binary 4U 1626--67 versus time $t$ (units: MJD). The data span $\approx 47$ yr of publicly available, post-processed timing observations. Points plotted in magenta and cyan correspond to the $N_{\rm{B}} = 706$ BATSE and $N_{\rm{F}} = 2634$ GBM $\nu(t)$ samples analyzed in Section \ref{['Sec:Torque']}. The magenta and cyan points bracket the 2008 torque reversal near MJD 54500 Camero_2009
• Figure 2: One-dimensional marginalized posterior distribution of the mass accretion rate $\bar{Q}$ (units: $\rm{g \, s^{-1}})$ of the persistent X-ray pulsar 4U 1626--67 for $\mathcal{M}_{\rm PP}$ (gray histograms) and $\mathcal{M}_{\rm RP}$ (red histograms). The left and right panels summarize the $\bar{Q}$ posterior distributions during the BATSE deceleration and the FGRST acceleration episodes, respectively. The vertical and horizontal axes are on linear and $\log_{10}$ scales, respectively. The acronym PDF stands for probability density function.
• Figure 3: Mass accretion rate $Q(t)$ (units: $\rm{g \, s^{-1}}$) and aperiodic X-ray flux $F_X(t)$ (units: $\rm{erg \, cm^{-2} \, s^{-1}}$) fluctuation histories of 4U 1626$-$67. (Top and middle panels.) Time-resolved mass accretion rate $Q(t_{n'})/\bar{Q}$ (units: dimensionless; magenta points) versus $t_{n'}$ (units: MJD) assuming prograde [$\mathcal{M}_{\rm PP}$; $\epsilon(t_{n'}) = 1$; top panel] and retrograde [$\mathcal{M}_{\rm RP}$; $\epsilon(t_{n'}) = -1$; middle panel] disks during deceleration with $1 \leq n' \leq N_{\rm B}$. The time-resolved $Q(t_{n'})$ histories are normalized by the mode of the one-dimensional marginalized posteriors $\bar{Q}$, visible as gray ($\mathcal{M}_{\rm PP}$) or red ($\mathcal{M}_{\rm RP}$) histograms in the left panel of Figure \ref{['fig2']}. (Bottom panel.) Measured aperiodic X-ray flux $F_X(t_{m'})/F_{\rm{HEAO}}$ (units: dimensionless; energy range: 0.7--60 keV) versus $t_{m'}$ (units: MJD) for the seven MJD values in Table \ref{['tab1']}. Measurements plotted as black stars, squares, and diamonds, are reported in Table 2 of Chakrabarty_19974U, Section 3.1 of Orlandini_1998, and Table 2 of Krauss_2007, respectively, as well as the fourth and fifth columns of Table \ref{['tab1']}. The 1$\sigma$ error bars are reported for the Ginga and ASCA measurements (black stars) only Chakrabarty_19974U. The $F_X(t_{m'})$ measurements are normalized by $F_{\rm HEAO} = 26 \times 10^{-10} \, \rm{erg \, cm^{-2} \, s^{-1}}$, the aperiodic X-ray flux measured by HEAO 1 in the 0.7--60 keV energy range Chakrabarty_19974U, consistent with previous analyses of flux measurements associated with 4U 1626$-$67 Chakrabarty_19974UCamero_2009Camero_2012. The horizontal and vertical axes are plotted on linear scales.
• Figure 4: Time-resolved angular acceleration histories $\dot{\Omega}(t_{n'})$ (units: ${\rm rad \, s^{-2}}$; BATSE observations; magenta points, left column) versus $t_{n'}$ (units: MJD) and $\dot{\Omega}(t_{n"})$ (units: ${\rm rad \, s^{-2}}$; FGRST observations; cyan points, right column) versus $t_{n"}$ (units: MJD) assuming $\mathcal{M}_{\rm PP}$ [top row; $\epsilon(t_{n'}) = \epsilon(t_{n"}) = 1$ during deceleration (left column) and acceleration (right column)] and $\mathcal{M}_{\rm RP}$ [bottom row; $\epsilon(t_{n'}) = -1$ during deceleration (left column) and $\epsilon(t_{n"}) = 1$ during acceleration (right column)] respectively, with $1 \leq n' \leq N_{\rm B}$ and $N_{\rm B} + 1 \leq n" \leq N_{\rm B} + N_{\rm F}$. The top right and bottom right panels are identical, because ${\cal M}_{\rm PP}$ and ${\cal M}_{\rm RP}$ both assume, that the disk is prograde during the acceleration episode (right column). The horizontal and vertical axes are plotted on linear scales.
• Figure 5: Time-resolved fastness histories $\omega(t_{n'})$ (BATSE observations; magenta points, left column) versus $t_{n'}$ (units: MJD) and $\omega(t_{n"})$ (FGRST observations; cyan points, right column) versus $t_{n"}$ (units: MJD) assuming $\mathcal{M}_{\rm PP}$ (top row) and $\mathcal{M}_{\rm RP}$ (bottom row) respectively, with $1 \leq n' \leq N_{\rm B}$ and $N_{\rm B} + 1 \leq n" \leq N_{\rm B} + N_{\rm F}$. We divide each panel into the ordered unstable [$\omega(t) \lesssim 0.45$], chaotic unstable [$0.45 \lesssim \omega(t) \lesssim 0.6$], and stable [$0.6 \lesssim \omega(t) \lesssim 1.0$] accretion regimes identified by numerical simulations Blinova_2016 using three, black, dashed horizontal lines. We extend the vertical axes in all panels to accommodate departures into the weak propeller regime $0 \leq \omega(t) \leq 1.25$ (all panels except the top left) and the strong propeller regime $1.25 \leq \omega(t) \leq 4.5$ (top left panel). The top right and bottom right panels are identical, because ${\cal M}_{\rm PP}$ and ${\cal M}_{\rm RP}$ both assume, that the disk is prograde during the acceleration episode (right column).