Is There a Retrograde Accretion Disk around 4U 1626–67? Tracking Torque Reversals with a State-space Model

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 ≲150 days. Here an unscented Kalman filter is applied to track the ν ( t ) fluctuations observed in 22.7 yr (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 a 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 ≤0.34 dex across the reversal, and one measures ω ( t ) ≈ 0.25 and ω ( t ) ≈ 0.30 during deceleration and acceleration, respectively. The angular acceleration <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> satisfies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>−</mml:mo> <mml:mn>9</mml:mn> <mml:mspace width="0.25em"/> <mml:mo>≲</mml:mo> <mml:mspace width="0.25em"/> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>/</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> <mml:mspace width="0.25em"/> <mml:mi mathvariant="normal">rad</mml:mi> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="0.25em"/> <mml:mo>≲</mml:mo> <mml:mspace width="0.25em"/> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>2</mml:mn> <mml:mspace width="0.25em"/> <mml:mo>≲</mml:mo> <mml:mspace width="0.25em"/> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>/</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> <mml:mspace width="0.25em"/> <mml:mi mathvariant="normal">rad</mml:mi> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="0.25em"/> <mml:mo>≲</mml:mo> <mml:mspace width="0.25em"/> <mml:mn>9</mml:mn> </mml:math> before and after the 2008 reversal, respectively, compared to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> <mml:mo>≈</mml:mo> <mml:mo>−</mml:mo> <mml:mn>3.0</mml:mn> <mml:mspace width="0.25em"/> <mml:mo>×</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> <mml:mspace width="0.25em"/> <mml:mi mathvariant="normal">rad</mml:mi> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> before reversal and <mml:math xmlns