hand ▁side ▁of ▁the ▁elli pt ical ▁equation : ▁ ▁( in ▁which ▁E ▁is ▁now ▁imag inary ) ▁and ▁then ▁replacing ▁ ▁by ▁. ▁ ▁Rad ial ▁Ke pler ▁equation ▁The ▁Rad ial ▁Ke pler ▁equation ▁is : ▁ ▁where ▁t ▁is ▁proportional ▁to ▁time ▁and ▁x ▁is ▁proportional ▁to ▁the ▁distance ▁from ▁the ▁centre ▁of ▁att raction ▁along ▁the ▁ray . ▁This ▁equation ▁is ▁derived ▁by ▁multip lying ▁Ke pler ' s ▁equation ▁by ▁ 1 / 2 ▁and ▁setting ▁e ▁to ▁ 1 : ▁ ▁and ▁then ▁making ▁the ▁substitution ▁ ▁In verse ▁problem ▁▁ ▁Calcul ating ▁M ▁for ▁a ▁given ▁value ▁of ▁E ▁is ▁straightforward . ▁However , ▁solving ▁for ▁E ▁when ▁M ▁is ▁given ▁can ▁be ▁consider ably ▁more ▁challeng ing . ▁There ▁is ▁no ▁closed - form ▁solution . ▁ ▁One ▁can ▁write ▁an ▁infinite ▁series ▁expression ▁for ▁the ▁solution ▁to ▁Ke pler ' s ▁equation ▁using ▁Lag range ▁in version , ▁but ▁the ▁series ▁does ▁not ▁conver ge ▁for ▁all ▁combinations ▁of ▁e ▁and ▁M ▁( see ▁below ). ▁ ▁Conf usion ▁over ▁the ▁sol v ability ▁of ▁Ke pler ' s ▁equation ▁has ▁pers isted ▁in ▁the ▁literature ▁for ▁four ▁centuries . ▁Ke pler ▁himself ▁expressed ▁doubt ▁at ▁the ▁possibility ▁of ▁finding ▁a ▁general ▁solution : ▁ ▁In verse ▁Ke pler ▁equation ▁ ▁The ▁inverse ▁Ke pler ▁equation ▁is ▁the ▁solution ▁of ▁Ke pler ' s ▁equation ▁for ▁all ▁real ▁values ▁of ▁: ▁▁▁▁ ▁E valu ating ▁this ▁yields : ▁▁▁▁ ▁These ▁series ▁can ▁be