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#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
# FOR PROBLEM 4.10
# Simulate planet around stationary star
using Plots # for plotting trajectory
using Measures # for adding margins to the plots (no cut-off labels)
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
# GM_S = ustrip(uconvert(u"AU^3/yr^2", 1 * UnitfulAstro.GMsun)) # gravitational constant times mass of the Sun
GM_S = 4.0*pi^2 # gravitational constant times mass of the Sun in AU^3/yr^2 (same as above)
a = 0.39 # semi-major axis of Mercury
e = 0.206 # eccentricity of Mercury
t_final = 2 # final time of simulation in years
α_values = [.0004, .0008, .0016, .0032] # values of α to predict Mercury percession with
function tendency!(drp, rp, param, t)
# 6d phase space
r = rp[1:3]
p = rp[4:6]
(G, δ, α) = param
r2 = dot(r, r)
correction = (1 + α / r2) # correction to force law
a = - G * r * r2^(-1.5-δ) * correction # acceleration with M_S = 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, α) = param
r2 = dot(r, r)
# TODO: add correction to potential energy
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
function calculate_vy1_0(a, e, GM_S)
return sqrt(GM_S * (1.0 - e)/(a * (1.0 + e)))
end
function calculate_x_0(a, e)
return a * (1.0 + e)
end
function find_distance_derivative_changes(sol)
i_change = []
xs = sol[1, :]
ys = sol[2, :]
r = sqrt.(xs.^2 + ys.^2)
derivatives = []
for i in 1:length(r)-1
push!(derivatives, r[i+1] - r[i])
end
for i in 1:length(derivatives)-1
if derivatives[i] > 0 && derivatives[i+1] < 0 # if the derviative goes from positive to negative, store it
if abs(derivatives[i]) < abs(derivatives[i+1])
push!(i_change, i)
else
push!(i_change, i+1)
end
end
end
return i_change
end
function linear_regression(x, y)
n = length(x)
x̄ = sum(x) / n
ȳ = sum(y) / n
a = sum((x .- x̄) .* (y .- ȳ)) / sum((x .- x̄).^2)
b = ȳ - a * x̄
return (a, b)
end
function do_α_simulations(α_values, a, e, GM_S, t_final, δ)
plots = []
dθvdt = []
x_0 = calculate_x_0(a, e)
vy1_0 = calculate_vy1_0(a, e, GM_S)
for α in α_values
param = (GM_S, δ, α) # parameters of force law
r0 = [x_0, 0.0, 0.0] # initial position in AU
p0 = [0.0, vy1_0, 0.0] # initial velocity in AU / year
rp0 = vcat(r0, p0) # initial condition in phase space
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
# get the times where the distance derivatives change
i_changes = find_distance_derivative_changes(sol)
# find the angle of percession at these derviate changes
rad_to_deg = 180/π
all_θ = [atan(sol[2, i] / sol[1, i]) * rad_to_deg for i in 1:length(sol.t)]
# Plot of orbit
xy = plot(
sol[1, :], sol[2, :],
xlabel = "x (AU)",
ylabel = "y (AU)",
title = "Percession when α=$(α)",
aspect_ratio=:equal,
label="Orbit",
legend=:bottomright,
lw=.5,
left_margin=5mm
)
# Add the position of the Sun
scatter!(xy, [0], [0], label = "Sun")
# Add the positions where the distance changes
scatter!(
xy,
sol[1, :][i_changes],
sol[2, :][i_changes],
label = "Dist. Derivative Changes",
color=:green
)
# Add to plots array
push!(plots, xy)
# Plot the change in angles over time
dθ = scatter(
sol.t[i_changes],
all_θ[i_changes],
xlabel = "time (yr)",
ylabel = "θ (degrees)",
title = "Orientation vs. Time (α=$(α))",
label="Dist. Derivative Changes",
color=:green,
right_margin=5mm
)
# Add the linear regression of the change in angles
(a, b) = linear_regression(sol.t[i_changes], all_θ[i_changes])
a = round(a, digits=2)
b = round(b, digits=2)
plot!(dθ, sol.t[i_changes], a * sol.t[i_changes] .+ b, label = "θ ≈ $a * t + $b", left_margin=7mm)
# Push to the plots array
push!(plots, dθ)
# Store the slope into the return array
push!(dθvdt, a)
end
return plots, dθvdt
end
# Run the simulations
(plots, dθvdt) = do_α_simulations(α_values, a, e, GM_S, t_final, δ)
println("α_values = ", α_values)
println("dθvdt = ", dθvdt)
# Combine the plots into one plot
p_orbits = plot(plots..., layout=(4, 2), size=(800, 1000))
savefig(p_orbits, "hw4/4-10.png")
# Plot the change in dθ/dt over α
p_dθvdt = scatter(α_values, dθvdt, xlabel="α", ylabel="dθ/dt (degrees/year)", title="Percession Rate vs. α", lw=2, label="dθ/dt (from regression slopes)")
# Perform a linear regression on the data
(a, b) = linear_regression(α_values, dθvdt)
a = round(a, digits=2)
b = round(b, digits=2)
plot!(p_dθvdt, α_values, a * α_values .+ b, label = "dθ/dt ≈ $a * α + $b", left_margin=7mm)
savefig(p_dθvdt, "hw4/4-10-dθvdt.png")
percession_rate = a * 1.1e-8 # degrees per year to arcsec per year
# convert from degrees per year to arcsec per century
percession_rate = percession_rate * 3600 * 100 # arcsec per century
println("Percession rate = ", percession_rate, " arcsec/century")
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