▁used ▁for ▁hyper bol ic ▁traject ories ▁( e ▁> ▁ 1 ). ▁The ▁radial ▁Ke pler ▁equation ▁is ▁used ▁for ▁linear ▁( rad ial ) ▁traject ories ▁( e ▁= ▁ 1 ). ▁B ark er ' s ▁equation ▁is ▁used ▁for ▁par abol ic ▁traject ories ▁( e ▁= ▁ 1 ). ▁ ▁When ▁e ▁= ▁ 0 , ▁the ▁orbit ▁is ▁circular . ▁In cre asing ▁e ▁causes ▁the ▁circle ▁to ▁become ▁elli pt ical . ▁When ▁e ▁= ▁ 1 , ▁there ▁are ▁three ▁possibilities : ▁ ▁a ▁par abol ic ▁traject ory , ▁ ▁a ▁traject ory ▁going ▁in ▁or ▁out ▁along ▁an ▁infinite ▁ray ▁em an ating ▁from ▁the ▁centre ▁of ▁att raction , ▁ ▁or ▁a ▁traject ory ▁that ▁goes ▁back ▁and ▁forth ▁along ▁a ▁line ▁segment ▁from ▁the ▁centre ▁of ▁att raction ▁to ▁a ▁point ▁at ▁some ▁distance ▁away . ▁ ▁A ▁slight ▁increase ▁in ▁e ▁above ▁ 1 ▁results ▁in ▁a ▁hyper bol ic ▁orbit ▁with ▁a ▁turning ▁angle ▁of ▁just ▁under ▁ 1 8 0 ▁degrees . ▁Further ▁increases ▁reduce ▁the ▁turning ▁angle , ▁and ▁as ▁e ▁goes ▁to ▁infinity , ▁the ▁orbit ▁becomes ▁a ▁straight ▁line ▁of ▁infinite ▁length . ▁ ▁Hyper bol ic ▁Ke pler ▁equation ▁ ▁The ▁Hyper bol ic ▁Ke pler ▁equation ▁is : ▁ ▁where ▁H ▁is ▁the ▁hyper bol ic ▁ecc ent ric ▁anom aly . ▁This ▁equation ▁is ▁derived ▁by ▁re def ining ▁M ▁to ▁be ▁the ▁square ▁root ▁of ▁− 1 ▁times ▁the ▁right -