M =  2 (10^30);  re =  1.5 (10^11); G = 6.67 (10^(-11)); 
    m = 5.98 (10^24);  k = G m M
year = 3.155 (10^7); ve = ((2 π re)/year)

x = re; y = re; vx = -ve; vy = 0;
ro = (Sqrt[(x^2) + (y^2)]); 
    vo = (Sqrt[(vx^2) + (vy^2)]);
Etot = (1/2) m (vo^2) - (k/ro)
L =  m vo re
α   = ((L^2)/(m k))
ϵ  =  (Sqrt[1  +  ((2 Etot (L^2))/(m (k^2)))])
θo = ArcCos[(((α/ro) - 1)/ϵ)]
θodeg = θo (180/π)

x = re; y = re; vx = -ve; vy = 0; t = 0; dt = 50000; orbit = {}; 
    rdata = {}; rmin = re;
While[
      t < 4  year, {ax =  ((-G M x)/((((x^2)  +  (y^2)))^((3/2)))); 
        ay =  ((-G M y)/((((x^2)  +  (y^2)))^((3/2)))); 
        vx = vx + ax dt; vy = vy + ay dt; x = x + vx dt; y = y + vy dt; 
        t = t + dt; AppendTo[orbit, {x, y}]}]
p1 = 
      ListPlot[orbit, AspectRatio -> Automatic, 
        PlotRange -> {{-10 (10^11), 10 (10^11)}, {-10 (10^11), 
              10 (10^11)}}]
p3 = 
      ParametricPlot[ {r Cos[(π/4)  +  θo], 
          r Sin[(π/4) + θo]}, {r, -5 re, 
          5 re}]
Show[p1, p3, 
      PlotRange -> {{-4 (10^11), 7 (10^11)}, {-4 (10^11), 
            7 (10^11)}}]

r[θ_] := (α/(1 + ϵ Cos[θ]))
ParametricPlot[{r[θ] Cos[θ], 
        r[θ] Sin[θ]}, {θ, 0, 2 π}]

ParametricPlot[{r[θ - (θo + (π/4))] Cos[θ], 
        r[θ - (θo + (π/4))] Sin[θ]}, {θ, 
        0, 2 π}]