The Effect of General Relativity on Hyperbolic Orbits and Its Application to the Flyby Anomaly

Lorenzo Iorio

doi:10.3814/2009/807695

Lorenzo Iorio

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We investigate the impact of the general relativistic gravitoelectromagnetic forces on hyperbolic orbits around a massive spinning body. The gravitomagnetic field, causing the well-known Lense-Thirring precessions of elliptic orbits, is generated by the spin S of the central body. It deflects and displaces the trajectories differently according to the mutual orientation of S and the orbital angular momentum L of the test particle. The gravitoelectric force, which induces the Einstein precession of the perihelion of the orbit of Mercury, always deflects the trajectories inward irrespective of the L−S orientation. We numerically compute their effect on the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math>, radial and transverse components <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>τ</mml:mi></mml:msub></mml:math> of the velocity, and speed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>v</mml:mi></mml:math> of the NEAR spacecraft at its closest approach with the Earth in January 1998 when it experienced an anomalous increase of its asymptotic outgoing velocity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi>∞</mml:mi><mml:mtext>o</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>13.46</mml:mn><mml:mo>±</mml:mo><mml:mn>0.01</mml:mn></mml:mrow></mml:math> mm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mtext>sec</mml:mtext></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>; while the gravitoelectric force was modeled in the software used to process the NEAR data, this was not done for the gravitomagnetic one. The range rate and the speed are affected by general relativistic gravitoelectromagnetism at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> (gravitoelectric) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> (gravitomagnetic) mm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mtext>sec</mml:mtext></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> levels. The changes in the range are of the order of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> (gravitomagnetic) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math> (gravitoelectric) mm.

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