Global Journal of Researches in Engineering, A: Mechanical & Mechanics, Volume 22 Issue 1

When the new theory is applied to Gravity Probe B the following equations are derived algebraically using equation (5). The modified equation is not required because the relative velocities are not close to the speed of light. ( ) East) (West to Thirring. - Lense or dragging frame cos 32 2 South) (North to Sitter. de or Geodesic 2 3 2 2 α φ α φ − Ω = = c E z c E x Rc GI Rc Gm    where R E = Radius of the Earth, m E = Mass of the Earth, I E = Moment of inertia of the Earth, R c = Radius of orbit, G = Gravitational constant and c = Speed of light. Also, α is the location of the satellite in its orbit and Ω is the angular velocity of the Earth. These equations yield, ∅̇ 4.4 ± 0.0 arcsec yr ( 6.6 ± 0.0 based on GP − B update ) ∅̇ = 0.02 ± 0.06 arcsec yṙ (0.037 ± 0.007 based on GP-B update) Gravity Probe B Status Update 2011 Ref [27}. Note that one arc second equals 1/3600 degrees of angle. Also, one year is a long time. All four of the gyros give results for de Sitter value which are very close to each other. However two of the gyros give results for the Lense -Thirring value which are very close to the value quoted in this paper. c) Precession of the Periapsis of a small body orbiting a large rotating mass This problem is similar to the discussion of the precession of the perihelion of Mercury except that now the rotation of the Earth is taken into account. For Mercury the rotation of the Sun has negligible effect. The LAGEOS satellites yield results for the so called frame dragging, or Lense Thirring effect, which results from the rotation of the Earth. The Earth is regarded as a uniform spherical body which can be regarded as a set of uniform spherical shells. The sphere, of mass M and moment of inertia I, rotates about the Z axis at a constant angular speed Ω. Consider a test body in orbit around the Earth performing an elliptical orbit where e is the eccentricity and a is the semi-major axis and a period of T . The plane has an incli ation (inc) relative to the equatorial XY plane of the Earth. Again, based on equation (5).the rate of precession of the periapsis, as seen from the plane of the orbit, in radians per orbit, is ( ) ( ) T inc e ac GI e ac GM P ) cos( 1 2 1 6 2/32 3 2 2 2 − Ω − − = ∆ π φ (19) The first term, the de Sitter precession, has been derived algebraically from equation (2). It agrees exactly with the generally accepted form and agrees with the measured results for the precession of the perihelion of Mercury and for the binary pulsar PSR 1913+16. However, the second term, the Lense-Thirring term, justified by numerical integration, is only half of the generally accepted value. d) Anti-Gravity Weight loss of a rotating ring. Consider a ring rotating about a horizontal axis above a large body, such as the Earth. Using equation (3a) evaluate the component of the acceleration in the radial direction. 2 2 1 3 2 22 2 2 2 t v e a       + +       + −= c v c v r K cr Kv c v r K r r r (3a) Noting that v / v t = means that (3a) may be written as v e a r              + +       + −= 2 2 2 2 2 2 1 2 1 c v cr Kv c v r K r or / 2 = − �1 + 2 2 � + �1 + 2 2 � 2 Also as K is G times the total mass therefore as the mass of the ring is very small compared to the large body its mass is negligible. So, K = G (mass of the large body). Now = also ∙ = , where θ is the angle between the velocity and the radius from the large mass. ∙ / 2 = �1 + 2 2 � �−1 + 2 � � 2 � This equation is the radial acceleration divided by the magnitude of the Newtonian gravity acceleration. If v/c is negligible then the right hand side is -1, as Global Journal of Researches in Engineering (A ) Volume XxXII Issue I Version I 9 Year 2022 © 2022 Global Journals Gravitomagnetics a Simpler Approach Applied to Dynamics within the Solar System