Dragging of the particle spin and spin–spin coupling effect on its periapsis shift

Abstract The periapsis shift (PS) of spinning test particles in the equatorial plane of an arbitrary stationary and axisymmetric spacetime is studied using the post-Newtonian method. The result is expressed as a half-integer power series of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> , where M is the spacetime mass and p is the semilatus rectum. The coefficients of the series are polynomials of the particle spin, the asymptotic expansion coefficients of the metric functions, and the eccentricity of the orbit. The particle spin is shown to have a similar effect as the Lense-Thirring (LT) effect on the PS, and both of them appear from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> order in the PS. The coupling between the spacetime and particle spins will increase (or decrease) the PS if they are parallel (or antiparallel). For Jupiter and Saturn rotating around the Sun and exceptionally designed satellites around Mercury and Moon, and the space-based gravitational wave observatories LISA and TaiJi around the Sun, the particle spin effect can be comparable to or even larger than the LT one in size. The PS in other spacetime studied is not distinguishable from that in the Kerr spacetime to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> order.