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

alone would cause outward spiralling as do most cases of tidal drag. c) Moment of Momentum If the additional term is negligible then it can be shown that the moment of momentum is               − = 2 1 2 1 2 1 1 2 exp r r c K h h which depends on separation but is constant when c = infinity. d) Schwarzschild Radius For a constant radius v r = 0 and v = v θ so equation (3) or (3a) becomes r r e e r v c v r K 2 2 2 2 1 θ θ −=       + − so if v θ = c then g r c MG c K r = + = = 2 2 )0 (2 2 which is known as the Schwarzschild Radius. e) Last Stable Orbit Numerical integration of equation (3a) shows that the Last Stable Orbit occurs when the radius of the orbit is 3 times the Schwarzschild Radius, which is the accepted result based on General Relativity. If equation (3) is used then a value of 2.62 r g may be calculated algebraically. However if Q is not unity, as shown in equation (3a), then equation (12), with Q included is, ) ) ( 21( 0 2 0 2 Quur h h g − − ≈ (12a) where the suffix 0 refers to circular motion when u = u o Substituting in equation (8) for h , using equation (12a), equation (13) becomes ( ) 1 2 ) (4 2 2 2 0 0 2 2 0 2 2         −      − +     + − + =+ Q d du u d du h QuuK c K h K u d ud θ θ θ (13a) If ε + = o u u then, for small variations ( ) ( ) ( ) 0 1 2 1 2 = ++ − +′′ Q ur Q ur o g o g ε ε (13b) For circular motion it can be shown that go go ru ru c v − =      2 2 so                 − + = go go ru ru Q 2 1 For a stable near circular orbit then ( ) ( ) ( ) 0 1 2 1 2 > ++ − Q ur Q ur o g o g so when the factor of ε in equation (13b) is zero, algebraic manipulation of (13b) gives r o /r g = 3, which is the accepted value. It also gives a value of 0.5. f) Deflection of Light In equation (3a) terms 2 and 3 are parallel to the velocity so the component normal to the velocity is dt dv c v r K ψ = ⋅              + −= ⋅ n e na r 2 2 1 r R r K s 2 2 −= ⋅ na now 2 2 s R x r + = so ( ) dt dx dx dc dt dc R x KR s s ψ ψ = + −= ⋅ 2/32 2 2 na or, as dx/dt is approximately c ( ) dx R x R c K d s s 2/32 2 2 1 2 + −= ψ integrating gives ( )         + = 2 2 2 2 s s R x x Rc K ψ . When x = 0 ψ = 0 and when x goes to infinity s Rc K 2 2 −= ψ Therefore the total deflection δ = 2 ψ so s Rc K 2 4 = δ (15) With K =M sun G and R s being the radius of the Sun the deflection is 1.75 arcsec. This value agrees with the measured value and with General Relativity. This confirms the assumption that the deflection of light grazing the Sun is small. © 2022 Global Journals Global Journal of Researches in Engineering (A ) Volume XxXII Issue I Version I 6 Year 2022 Gravitomagnetics a Simpler Approach Applied to Dynamics within the Solar System For small variation of the speed of light assume that v = c . Also, for small deflections the scalar product of e r and n can be seen from Figure (2) to be R s / r . Therefore