#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia

# Simulate planet around stationary star 

using Plots # for plotting trajectory
using DifferentialEquations # for solving ODEs
using LinearAlgebra
#using Unitful
#using UnitfulAstro # astrophysical units

G = 4.0*pi^2 # time scale = year and length scale = AU
δ = 0.0 # deviation from inverse-square law

param = (G, δ) # parameters of force law

function tendency!(drp, rp, param, t)
   
   # 6d phase space
   r = rp[1:3] 
   p = rp[4:6] 

   (G, δ) = param

   r2 = dot(r, r)

   a = -G * r * r2^(-1.5-δ) # acceleration with m = 1

   drp[1:3] = p[1:3]
   drp[4:6] = a[1:3]
   
end

function energy(rp, param)

   r = rp[1:3]
   p = rp[4:6]

   (G, δ) = param

   r2 = dot(r, r)

   pe = -G * r2^(-0.5 - δ)/(1.0 + 2.0 * δ)
   ke = 0.5 * dot(p, p)

   return pe + ke

end

function angularMomentum(rp)

   r = rp[1:3]
   p = rp[4:6]
   
   return cross(r, p)

end

r0 = [1.0, 0.0, 0.0] # initial position in AU
p0 = [0.0, 1.0*pi, 0.0] # initial velocity in AU / year
rp0 = vcat(r0, p0) # initial condition in phase space 
t_final = 10.0 # final time of simulation

tspan = (0.0, t_final) # span of time to simulate


prob = ODEProblem(tendency!, rp0, tspan, param) # specify ODE
sol = solve(prob, Tsit5(), reltol=1e-12, abstol=1e-12) # solve using Tsit5 algorithm to specified accuracy

sample_times = sol.t
println("\n\t Results")
println("final time  = ", sample_times[end])
println("Initial energy = ", energy(sol[:,1], param))
println("Final energy = ", energy(sol[:, end], param))
println("Initial angular momentum = ", angularMomentum(sol[:,1]))
println("Final angular momentum  = ", angularMomentum(sol[:, end]))

# Plot of position vs. time
xt = plot(sample_times, sol[1, :], xlabel = "t", ylabel = "x(t)", legend = false, title = "x vs. t")

# Plot of orbit
xy = plot(sol[1, :], sol[2, :], xlabel = "x", ylabel = "y", legend = false, title = "Orbit", aspect_ratio=:equal)

plot(xt, xy)