Secular Evolution of PSR J2021+4026: Long-term <i>γ</i> -Ray Flux and Spin-down Variability beyond State Transitions

Abstract PSR J2021+4026 is a remarkable γ -ray pulsar exhibiting repeated transitions between high γ -ray flux (HGF) and low γ -ray flux (LGF) states. With 17 yr Fermi Large Area Telescope monitoring, we reveal persistent secular evolution and enhanced spin-down rate variability within individual emission states—beneath the quasiperiodic state transitions. After removing discrete jumps, the jump-corrected flux δF γ shows a three-phase evolution: rise ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>+</mml:mo> <mml:mn>2.0</mml:mn> <mml:msubsup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.15</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.17</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>%</mml:mo> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">yr</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> ), decline ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>−</mml:mo> <mml:mn>3.7</mml:mn> <mml:msubsup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.47</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.34</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>%</mml:mo> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">yr</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> ), and rapid rise ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>+</mml:mo> <mml:mn>14</mml:mn> <mml:mo>.</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>9</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4.4</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>6.4</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>%</mml:mo> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">yr</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> ), with all rates quoted relative to the long-term mean flux 〈 F γ 〉 = 7.8 × 10 −10 erg cm −2 s −1 . Moreover, the flux of the LGF state is gradually approaching the stable HGF level at a rate of +0.72% ± 0.11% yr −1 . These results demonstrate that secular flux evolution in PSR J2021+4026 operates largely independently of discrete state transitions, yet jointly with them drives the system toward a stable high-flux equilibrium.