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b = a\sqrt{1 − e^{2}}

tan (\frac{ϕ}{2}) = \sqrt{\frac{1 + e}{1 − e}} tan (\frac{u}{2})

u − e sin u = ωt where: ω = \sqrt{\frac{α}{ma^{3}}}

v_{0} = \sqrt{\frac{α}{ma(1 − e^{2})}}

a = \frac{d}{dt} v(t) = \frac{1}{m} F(r(t)) = − \frac{αr̂}{mr^{2}}

|a| = v_{0}ϕ̇ L = mr^{2}ϕ̇ = mv_{0}(1 + e) a(1 − e) = \sqrt{αma(1 − e^{2})}

L = \sqrt{αma(1 − e^{2})}
|a| = v_{0}ϕ̇
= v_{0} (\frac{v_{0}a(1 − e^{2})}{r^{2}})
= \frac{α}{mr^{2}}

E = − \frac{α}{2a}

Veff(r) = U(r) + \frac{L^{2}}{2mr^{2}}

r = \frac{a(1 − e^{2})}{(1 + e cos ϕ)}

impact parameter: b = a\sqrt{e^{2} − 1}

r = \frac{a(e^{2} − 1)}{1 + e cos ϕ}

tan (\frac{ϕ}{2}) = \sqrt{\frac{e + 1}{e − 1}} tanh (\frac{u}{2})

e sinh u − u = ωt where: ω = \sqrt{\frac{α}{ma^{3}}}

v_{0} = \sqrt{\frac{α}{ma(e^{2} − 1)}} = \frac{v_{∞}}{\sqrt{e^{2} − 1}}

L = mr^{2}ϕ̇ = mv_{∞}b = \sqrt{αma(e^{2} − 1)}

2E = \frac{α}{a} = mv_{∞}^{2}

cos (ϕ_{max}) = − \frac{1}{e} = − sin(θ/2) ϕ_{max} = θ/2 + π/2

tan (\frac{θ}{2}) = \frac{v_{0}}{v_{∞}} = \frac{α/b}{mv_{∞}^{2}} ∼ \frac{PE}{KE}

r = a(e cosh(u) − 1)

r = a(1 − e cos(u))