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

2 2 ) ( r Kv r −= ⋅ av γ (1.4.8) Substituting equation (1.4.8) into equation (1.4.7) gives ( ) 2 1 2 2 2 2 2 r r ev v e a ×× +       − −= cr K c v r K as (2) 2 1 22 2 2 2 v e a cr Kv c v r K r r +       + −= as (3) II. A pplication to T wo M ass P roblem a) Polar Coordinates The following development is based on the conventional treatment of the two body gravitational problem. For the dynamics of bodies in Solar orbits the modified equations are not required. Here, r is the separation and e r is the unit vector in the direction of body 2 as seen from body 1. θ is the orientation of the unit vector with respect to the 'fixed' stars and e θ is the unit vector normal to e r in the plane of the motion. Now θ θ θ θ e e a ) 2 ( ) ( 2     r r r r r + + − = and ( ) θ θ e e v  r r r + = Equation (3a) can now be expressed in component form Q cr Kv c v r K r r r 2 2 2 2 2 2 2 1 +              + −= − θ   (6) ( ) Q cr vKv r dt d r r r r 2 2 2 2 1 2 θ θ θ θ = = +    (7) but for low values of (v/c) the term Q will be taken to be unity. Define ) ( θ  rr h = , the moment of momentum per reduced mass, and u = 1/r. So that h u =  / θ 2 thus θ θ θ d du h dt d d du u r −= −= 2 1  , θ θ d du h d ud hu r   − −= 2 2 2 2 , and hu d du hu r h r rh     2 2 3 2 3 2 2 + = + −= θ θ Equations (6, 7) may now be written 2 2 2 2 2 22 2 2       − + =+ + θ θ θ d du c K c Ku h K u hu h d du d ud  (8) and 2 2 3 2 θ d du hu c K hu −=  (9) Since   h dh d dh d u h = = θ θ θ 2 combining with equation (9) gives, (10) Integrating equation (10) leads to 2 0 /) ( 2 0 c uuK eh h − − = (11) Therefore, for small variations ) /) (41( 2 0 2 0 2 c uuK h h − − ≈ (12) where the suffix 0 refers, in this case, to the position θ = π/2 measured from the periapsis. Substituting in equation (8) for h , using equation (12), we obtain ) (4 2 2 2 0 0 2 2 0 2 2         +     + − + =+ u d du h uuK c K h K u d ud θ θ (13) b) Precession of the Periapsis Equation (4) is very much easier to apply. This equation is equally applicable to the prediction of satellite trajectories. Because in these aces the relative speeds are not close to the speed of light. The equation which was developed in reference [19] for calculating the precession of the perihelion of Mercury per orbit is ( ) ( ) 2 2 2 1 1 6 e ac mmG P − + = π θ (14) where a is the semi-major axis and e is the eccentricity. This generates 42.89 arcsec/century. For the binary pulsar PSR 1913+16, which was discovered by Hulse and Taylor in 1974, (see reference [22]), the accepted data is that the masses of the two stars are 1.441 and 1.387 times the mass of the Sun, the semi-major axis is 1,950,100 km, the eccentricity is 0.617131 and the orbital period is 7.751939106 hr. Using equation (14) we obtain the result 4.22 deg/yr, which is in agreement with the measured value and that predicted by General Relativity. The orbital decay, or inward spiralling, of binary pulsars is said to be simply due to energy loss caused by gravitational wave emission. This may be the case but energy loss alone will not account for the phenomenon. The loss of mass Global Journal of Researches in Engineering (A ) Volume XxXII Issue I Version I 5 Year 2022 © 2022 Global Journals Gravitomagnetics a Simpler Approach Applied to Dynamics within the Solar System 2 2      −= c Kdu h dh