Page 157 - Intro to Tensor Calculus
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               where p =2q  and the angle φ is known as the true anomaly associated with the orbit. The quantity p is
               called the semi-parameter of the conic section. (Note that when φ =  π ,then ρ = p.) A more general form
                                                                             2
               of the above equation is

                                                p                 1
                                     ρ =                  or u =   = A[1 +   cos(φ − φ 0 )],         (1.5.130)
                                         1+   cos(φ − φ 0 )       ρ
               where φ 0 is an arbitrary starting anomaly. An additional symbol a, known as the semi-major axes of an
               elliptical orbit can be introduced where q, p,  , a are related by

                                             p                                 2
                                                 = q = a(1 −  )  or p = a(1 −   ).                   (1.5.131)
                                            1+

                   To show that the equation (1.5.128) produces a conic section for the motion of mass m with respect to
               mass M we will show that one form of the solution of equation (1.5.128) is given by the equation (1.5.129).
               To verify this we use the following vector identities:
                                                            ~ ρ × b e ρ =0
                                                                         2
                                                       d      d~ρ       d ~ρ
                                                                    ρ
                                                          ~ ρ ×   =~ ×
                                                       dt     dt        dt 2
                                                                                                     (1.5.132)
                                                              d b e ρ
                                                           b e ρ ·  =0
                                                               dt

                                                             d b e ρ   d b e ρ
                                                  b e ρ ×  b e ρ ×  = −   .
                                                              dt       dt
               From the equation (1.5.128) we find that
                                                              2
                                           d      d~ρ        d ~ρ   GM
                                                                                 ~
                                               ~ ρ ×   = ~ρ ×   = −     ~ ρ × b e ρ = 0              (1.5.133)
                                           dt      dt        dt 2    ρ 2
               so that an integration of equation (1.5.133) produces

                                                        d~ρ
                                                             ~
                                                    ~ ρ ×  = h = constant.                           (1.5.134)
                                                        dt
                                                                                                            ~
                                      ~
                            ~
               The quantity H = ~ρ × mV = ~ρ × m d~ ρ  is the angular momentum of the mass m so that the quantity h
                                                 dt
                                                                                          ~
               represents the angular momentum per unit mass. The equation (1.5.134) tells us that h is a constant for our
                                                ~
               two body system. Note that because h is constant we have
                                                 ~
                                   d            dV         GM            d~ρ
                                                     ~
                                          ~
                                       ~
                                      V × h =      × h = −     b e ρ × ~ρ ×
                                   dt           dt          ρ 2          dt
                                                           GM               d b e ρ  dρ
                                                       = −     b e ρ × [~ρb e ρ × (ρ  +  b e ρ )]
                                                            ρ 2              dt   dt
                                                           GM            d b e ρ  2   d b e ρ
                                                       = −   2  b e ρ × ( b e ρ ×  )ρ = GM
                                                            ρ             dt           dt
               and consequently an integration produces
                                                         ~
                                                     ~
                                                                      ~
                                                     V × h = GM b e ρ + C
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