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authorsotech117 <michael_foiani@brown.edu>2024-02-29 01:52:17 -0500
committersotech117 <michael_foiani@brown.edu>2024-02-29 01:52:17 -0500
commit02756d17bca6f2b3bafa3f7b9fb6e5af438e94a0 (patch)
treeccb65043d06198f8d4246418570e79f74d6d64e2
parentec622e808d84956a75c2b9a9826479cb5737e04a (diff)
finish hw4
-rw-r--r--.DS_Storebin6148 -> 6148 bytes
-rw-r--r--hw4/.DS_Storebin0 -> 8196 bytes
-rw-r--r--hw4/4-10.jl52
-rw-r--r--hw4/4-13.jl2
-rw-r--r--hw4/4-17.jl287
-rw-r--r--hw4/4-19.jl282
-rw-r--r--hw4/SolarSystem3.jl274
-rw-r--r--hw4/comphysHW4-final.pdfbin0 -> 1292987 bytes
-rw-r--r--hw4/images/.DS_Storebin0 -> 6148 bytes
-rw-r--r--hw4/images/4-10-dθvdt-ldl.pngbin0 -> 32065 bytes
-rw-r--r--hw4/images/4-10.pngbin195826 -> 112631 bytes
-rw-r--r--hw4/images/4-17.pngbin0 -> 44694 bytes
-rw-r--r--hw4/images/4-19-.224.pngbin0 -> 159051 bytes
-rw-r--r--hw4/images/4-19-0.pngbin0 -> 120320 bytes
-rw-r--r--hw4/images/4-19-final.pngbin0 -> 46468 bytes
-rw-r--r--hw4/images/4-19-log.pngbin0 -> 41271 bytes
-rw-r--r--hw4/images/4-19.pngbin0 -> 71163 bytes
-rw-r--r--hw4/images/A_alphachange.pngbin0 -> 29380 bytes
-rw-r--r--hw4/images/l_exponent.svg144
19 files changed, 1024 insertions, 17 deletions
diff --git a/.DS_Store b/.DS_Store
index ada1a6c..0994b43 100644
--- a/.DS_Store
+++ b/.DS_Store
Binary files differ
diff --git a/hw4/.DS_Store b/hw4/.DS_Store
new file mode 100644
index 0000000..a5beba9
--- /dev/null
+++ b/hw4/.DS_Store
Binary files differ
diff --git a/hw4/4-10.jl b/hw4/4-10.jl
index fa63926..4335c17 100644
--- a/hw4/4-10.jl
+++ b/hw4/4-10.jl
@@ -1,5 +1,6 @@
#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
# FOR PROBLEM 4.10
+# author: sotech117
# Simulate planet around stationary star
@@ -107,6 +108,11 @@ function linear_regression(x, y)
return (a, b)
end
+function ldlVec(p, r, k)
+ return cross(p, angularMomentum(vcat(r, p))) - k*(normalize(r))
+end
+
+
function do_α_simulations(α_values, a, e, GM_S, t_final, δ)
plots = []
dθvdt = []
@@ -119,6 +125,8 @@ function do_α_simulations(α_values, a, e, GM_S, t_final, δ)
p0 = [0.0, vy1_0, 0.0] # initial velocity in AU / year
rp0 = vcat(r0, p0) # initial condition in phase space
+ ldl_0 = normalize(ldlVec(p0, r0, GM_S))
+
tspan = (0.0, t_final) # span of time to simulate
@@ -168,21 +176,35 @@ function do_α_simulations(α_values, a, e, GM_S, t_final, δ)
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)
+ # # 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 + $b", left_margin=7mm)
+ # # Push to the plots array
+ # push!(plots, dθ)
+
+ # # Store the slope into the return array
+ # push!(dθvdt, a)
+
+ # find the final ldl vector
+ rf = sol[1:3, end]
+ pf = sol[4:6, end]
+ ldl_f = normalize(ldlVec(pf, rf, GM_S))
+ t_final = sol.t[end]
+ println("A_t=0 = ", ldl_0)
+ println("A_t=$(t_final) = ", ldl_f)
+ println("θ between A_t=0 and A_t = ", acos(dot(ldl_0, ldl_f)))
+ dθ = acos(dot(ldl_0, ldl_f)) * rad_to_deg
+ println("Δθ/Δt for α=$(α) = ", (dθ / t_final), "\n")
+
+ push!(dθvdt, (dθ / t_final))
end
return plots, dθvdt
end
+
# Run the simulations
(plots, dθvdt) = do_α_simulations(α_values, a, e, GM_S, t_final, δ)
@@ -194,19 +216,15 @@ 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)")
+p_dθvdt = scatter(α_values, dθvdt, xlabel="α", ylabel="Δθ/Δt (degrees/year)", title="Percession Rate vs. α", lw=2, label="Δθ/Δt (from Laplace-Runge-Lenz vector)")
# 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")
-
+savefig(p_dθvdt, "hw4/4-10-dθvdt-ldl.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")
-
-
-
+println("Percession rate = ", percession_rate, " arcsec/century") \ No newline at end of file
diff --git a/hw4/4-13.jl b/hw4/4-13.jl
index 2a5bd9f..b942ef7 100644
--- a/hw4/4-13.jl
+++ b/hw4/4-13.jl
@@ -1,4 +1,6 @@
#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+# FOR PROBLEM 4.13
+# author: sotech117
# Simulate a solar system
diff --git a/hw4/4-17.jl b/hw4/4-17.jl
new file mode 100644
index 0000000..eb2b3b3
--- /dev/null
+++ b/hw4/4-17.jl
@@ -0,0 +1,287 @@
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+# FOR PROBLEM 4.17
+# author: sotech117
+
+# Simulate a solar system
+
+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+using LinearAlgebra # for dot and cross products
+
+println("Number of threads = ", Threads.nthreads())
+
+G = 4.0*pi^2 # time scale = year and length scale = AU
+
+mutable struct body
+ name::String # name of star or planet
+ m::Float64 # mass
+ r::Vector{Float64} # position vector
+ v::Vector{Float64} # velocity vector
+end
+
+function star(name="Sun", m = 1.0, r = zeros(3), v = zeros(3))
+ return body(name, m, r, v)
+end
+
+function planet(name="Earth", m=3.0e-6, a=1.0, ϵ=0.017, i=0.0)
+ perihelion = (1.0 - ϵ) * a
+ aphelion = (1.0 + ϵ) * a
+ speed = sqrt(G * (1.0 + ϵ)^2 / (a * (1.0 - ϵ^2)))
+ phi = 0.0
+ r = [perihelion * cos(i*pi/180.0) * cos(phi), perihelion * cos(i*pi/180.0) * sin(phi), perihelion * sin(i*pi/180.0)]
+ v = [-speed * sin(phi), speed * cos(phi), 0.0]
+ return body(name, m, r, v)
+end
+
+function momentum(b::body)
+ return b.m * b.v
+end
+
+function angularMomentum(b::body)
+ return b.m * cross(b.r, b.v)
+end
+
+function kineticEnergy(b::body)
+ v = b.v
+ m = b.m
+ return 0.5 * m * dot(v, v)
+end
+
+function potentialEnergy(body1::body, body2::body)
+ r = body1.r - body2.r
+ rSquared = dot(r, r)
+ return -G * body1.m * body2.m / sqrt(rSquared)
+end
+
+function rv(b::body)
+ return [b.r; b.v]
+end
+
+function force(body1::body, body2::body)
+ r = body1.r - body2.r
+ rSquared = dot(r, r)
+ return -G * r * rSquared^(-1.5)
+ #return -G * r / (rSquared * sqrt(rSquared))
+end
+
+mutable struct SolarSystem
+ bodies::Vector{body}
+ numberOfBodies::Int64
+ phaseSpace::Matrix{Float64} # 6N dimensional phase space
+end
+
+function SolarSystem()
+
+ bodies = Vector{body}()
+ push!(bodies, star()) # default = Sun
+ # push!(bodies, planet("Venus", 2.44e-6, 0.72, 0.0068, 3.39))
+ # push!(bodies, planet("Earth", 3.0e-6, 1.0, 0.017, 0.0))
+ push!(bodies, planet("Jupiter", 0.00095, 5.2, 0.049, 1.3))
+ # push!(bodies, planet("Saturn", 0.000285, 9.58, 0.055, 2.48))
+ push!(bodies, body("Astroid 1", 1e-13, [3.0, 0.0, 0.0], [0.0, 3.628, 0.0]))
+ push!(bodies, body("Astroid 2 ", 1e-13, [3.276, 0.0, 0.0], [0.0, 3.471, 0.0]))
+ # push!(bodies, body("Astroid 3 (no gap)", 1e-13,[3.7, 0.0, 0.0], [0.0, 3.267, 0.0]))
+ numberOfBodies = size(bodies)[1]
+
+ phaseSpace = zeros(6, 0)
+ for b in bodies
+ phaseSpace = [phaseSpace rv(b)]
+ end
+
+ return SolarSystem(bodies, numberOfBodies, phaseSpace)
+
+end
+
+function TotalMass(s::SolarSystem)
+ M = 0.0
+ for b in s.bodies
+ M += b.m
+ end
+ return M
+end
+
+function TotalLinearMomentum(s::SolarSystem)
+ P = zeros(3)
+ for b in s.bodies
+ P += momentum(b)
+ end
+ return P
+end
+
+function TotalAngularMomentum(s::SolarSystem)
+ L = zeros(3)
+ for b in s.bodies
+ L += angularMomentum(b)
+ end
+
+ return L
+end
+
+function TotalEnergy(s::SolarSystem)
+ ke = 0.0
+ pe = 0.0
+
+ for body1 in s.bodies
+ ke += kineticEnergy(body1)
+ for body2 in s.bodies
+ if (body1 != body2)
+ pe += 0.5 * potentialEnergy(body1, body2)
+ end
+ end
+ end
+
+ return pe + ke
+end
+
+function CenterOfMassFrame!(s::SolarSystem)
+
+ M = TotalMass(s)
+ P = TotalLinearMomentum(s)
+ V = P / M
+
+ s.phaseSpace = zeros(6, 0)
+ for body in s.bodies
+ body.v -= V #boost to COM frame
+ s.phaseSpace = [s.phaseSpace rv(body)]
+ end
+
+ return nothing
+end
+
+function tendency!(dps, ps, s::SolarSystem, t)
+
+ i = 1 # update phase space of individual bodies
+ for b in s.bodies
+ b.r = ps[1:3, i]
+ b.v = ps[4:6, i]
+ i += 1
+ end
+
+ # find velocities of bodies and forces on them. O(N^2) computational cost
+ N = s.numberOfBodies
+ for i in 1:N
+ b1 = s.bodies[i]
+ dps[1:3, i] = b1.v #dr/dt
+ dps[4:6, i] = zeros(3)
+ for j in 1:i-1
+ b2 = s.bodies[j]
+ f = force(b1, b2) # call only once
+ dps[4:6, i] += b2.m * f
+ dps[4:6, j] -= b1.m * f # Newton's 3rd law
+ end
+ end
+
+ return nothing
+end
+
+function parallel_tendency!(dps, ps, s::SolarSystem, t)
+
+ i = 1 # update phase space of individual bodies
+ for b in s.bodies
+ b.r = ps[1:3, i]
+ b.v = ps[4:6, i]
+ i += 1
+ end
+
+ # find velocities of bodies and forces on them. O(N^2) computational cost
+ N = s.numberOfBodies
+ Threads.@threads for i in 1:N
+ b1 = s.bodies[i]
+ dps[1:3, i] = b1.v #dr/dt
+ dps[4:6, i] = zeros(3)
+ for j in 1:N
+ if (j != i)
+ b2 = s.bodies[j]
+ f = force(b1, b2)
+ dps[4:6, i] += b2.m * f
+ end
+ end
+ end
+
+ return nothing
+end
+
+s = SolarSystem()
+CenterOfMassFrame!(s)
+println(typeof(s))
+println("Initial total energy = ", TotalEnergy(s))
+println("Initial total linear momentum = ", TotalLinearMomentum(s))
+println("Initial total angular momentum = ", TotalAngularMomentum(s))
+println("Number of bodies = ", s.numberOfBodies)
+for b in s.bodies
+ println("body name = ", b.name)
+end
+
+t_final = 200.0 # final time of simulation
+tspan = (0.0, t_final) # span of time to simulate
+prob = ODEProblem(tendency!, s.phaseSpace, tspan, s) # specify ODE
+sol = solve(prob, maxiters=1e8, reltol=1e-10, abstol=1e-10) # solve using Tsit5 algorithm to specified accuracy
+
+println("\n\t Results")
+println("Final time = ", sol.t[end])
+println("Final total energy = ", TotalEnergy(s))
+println("Final total linear momentum = ", TotalLinearMomentum(s))
+println("Final total angular momentum = ", TotalAngularMomentum(s))
+
+function eccentricity(body::Int64, solution)
+ n = size(solution[1, 1, :])[1]
+ samples = 1000
+ interval = floor(Int, n/samples)
+ ϵ = zeros(samples-1)
+ time = zeros(samples-1)
+ for i in 1:samples-1
+ r2 = sol[1, body, i*interval:(i+1)*interval].^2 + sol[2, body, i*interval:(i+1)*interval].^2 + sol[3, body, i*interval:(i+1)*interval].^2
+ aphelion = sqrt(maximum(r2))
+ perihelion = sqrt(minimum(r2))
+ ϵ[i] = (aphelion - perihelion) / (aphelion + perihelion)
+ time[i] = solution.t[i*interval]
+ end
+ return (time, ϵ)
+end
+
+function obliquity(body::Int64, solution)
+ n = size(solution[1, 1, :])[1]
+ samples = 1000
+ interval = floor(Int, n/samples)
+ ob = zeros(samples)
+ time = zeros(samples)
+ for i in 1:samples
+ r = solution[1:3, body, i*interval]
+ v = solution[4:6, body, i*interval]
+ ell = cross(r, v)
+ norm = sqrt(dot(ell, ell))
+ ob[i] = (180.0 / pi) * acos(ell[3]/norm)
+ time[i] = solution.t[i*interval]
+ end
+ return (time, ob)
+end
+
+
+body1 = 2 # Jupiter
+body2 = 3 # Astroid 1
+body3 = 4 # Astroid 2
+# body4 = 5 # Astroid 3
+
+# Plot of orbit
+xy = scatter([(sol[1, body1, :], sol[2, body1, :]), (sol[1, body2, :], sol[2, body2, :]),
+ (sol[1, body3, :], sol[2, body3, :]),
+ # (sol[1, body4, :], sol[2, body4, :])
+ ],
+ xlabel = "x (AU)", ylabel = "y (AU)", title = "Orbit",
+ label = ["Jupiter" "Asteroid not in gap (3.0AU)" "Asteroid 2/1 gap (3.27AU)" "Asteroid not in gap (3.7AU)"],
+ aspect_ratio=:equal, markersize=.5, markerstrokewidth = 0, legend = :bottomright,
+ colors=[:red :blue :green :black]
+ )
+
+# Plot of obliquity
+tilt = obliquity(body2, sol)
+obliquityPlot = plot(tilt, xlabel = "t", ylabel = "tilt", legend = false, title = "obliquity")
+
+# Plot of eccentricity
+eccentric = eccentricity(body2, sol)
+eccentricityPlot = plot(eccentric, xlabel = "t", ylabel = "ϵ", legend = false, title = "eccentricity")
+
+#plot(obliquityPlot, eccentricityPlot)
+plot(xy, aspect_ratio=:equal)
+
+savefig("4-17.png")
diff --git a/hw4/4-19.jl b/hw4/4-19.jl
new file mode 100644
index 0000000..3208b07
--- /dev/null
+++ b/hw4/4-19.jl
@@ -0,0 +1,282 @@
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+# FOR PROBLEM 4.19
+# author: sotech117
+
+# 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]
+ θ = rp[7]
+
+
+ ω = rp[8]
+
+ (G, δ) = param
+
+ r2 = dot(r, r)
+
+ a = -G * r * r2^(-1.5-δ) # acceleration with m = 1
+
+ x_c = r[1]
+ y_c = r[2]
+
+ τ = -3.0 * G * r2^(-2.5 - δ) * (x_c * sin(θ) - y_c * cos(θ)) * (x_c * cos(θ) + y_c * sin(θ))
+
+ drp[1:3] = p[1:3]
+ drp[4:6] = a[1:3]
+ drp[7] = ω
+ drp[8] = τ
+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
+
+# take a list and reduce theta to the interval [-π, π]
+function clean_θ(θ::Vector{Float64})
+ rθ = []
+ for i in 1:length(θ)
+ tmp = θ[i] % (2 * π)
+ if tmp > π
+ tmp = tmp - 2 * π
+ elseif tmp < -π
+ tmp = tmp + 2 * π
+ end
+ push!(rθ, tmp)
+ end
+ return rθ
+end
+
+function simulate(r0x, p0y, θ01)
+ r0 = [r0x, 0.0, 0.0] # initial position in HU
+ p0 = [0.0, p0y, 0.0] # initial velocity in HU / year
+ θ0 = [θ01]
+ pθ0 = [0.0]
+ rp0 = vcat(r0, p0, θ0, pθ0) # 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 θ over time
+ sol[7, :] = clean_θ(sol[7, :])
+
+ return sol
+end
+
+function calculate_vy1_0(a, e, GM_S)
+ return sqrt(GM_S * (1.0 - e)/(a * (1.0 + e)))
+end
+
+function calculate_Δθ(sol1, sol2, ticks=1000)
+ # generate a equal set of times from the smaller of the two simulations
+ time_0 = min(sol1.t[1], sol2.t[1])
+ time_f = min(sol1.t[end], sol2.t[end])
+ # make an even step fucntion over the length of these times
+ sample_times = range(time_0, time_f, length=ticks)
+
+ # interpolate the two simulations to the same time
+ diff = []
+ for i in 1:length(sample_times)-1
+ t = sample_times[i]
+
+ # find the index that is before the time
+ idx1 = -1
+ for j in 1:length(sol1.t)
+ if sol1.t[j] > t
+ idx1 = j - 1
+ break
+ end
+ end
+ idx2 = -1
+ for j in 1:length(sol2.t)
+ if sol2.t[j] > t
+ idx2 = j - 1
+ break
+ end
+ end
+
+ # interpolate the values between the two timestamps
+ θ1 = sol1[7, idx1] + (sol1[7, idx1+1] - sol1[7, idx1]) * (t - sol1.t[idx1])/(sol1.t[idx1+1] - sol1.t[idx1])
+ θ2 = sol2[7, idx2] + (sol2[7, idx2+1] - sol2[7, idx2]) * (t - sol2.t[idx2])/(sol2.t[idx2+1] - sol2.t[idx2])
+
+ push!(diff, sqrt((θ1 - θ2)^2))
+ end
+
+ return (sample_times[1:end-1], diff)
+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 find_l_exponent(eccentricity, GM_S)
+ a = 1.0
+ vy1_0 = calculate_vy1_0(a, eccentricity, GM_S)
+ sol1 = simulate(1.0, vy1_0, 0.0)
+ sol2 = simulate(1.0, vy1_0, 0.01) # small change in θ_0
+ (ts, diff_θ) = calculate_Δθ(sol1, sol2)
+
+ plt = plot(ts, diff_θ, yscale=:ln, xlabel="time (s)", ylabel="Δθ (radians)", title="Δθ vs time", legend=:bottomright, lw=2, label="\$ Δ θ ≡ √{(θ_1 - θ_2)^2} \$")
+
+ # linear regression of the log of the difference
+ (a, b) = linear_regression(ts, log.(diff_θ))
+ a_rounded = round(a, digits=4)
+ b = round(b, digits=4)
+ plt = plot!(ts, exp.(a*ts .+ b), label="exp($a_rounded*t + $b)", lw=1.2, color=:red, linestyle=:dash)
+
+ return a, plt
+end
+
+function process_data!(les, es, plt)
+ # find the first index where the l exponent is greater than .2
+ idx = -1
+ for i in 1:length(les)
+ if les[i] > .2
+ idx = i
+ break
+ end
+ end
+
+ println("e=$(es[idx]) & l=$(les[idx])")
+
+ # add a vertical line at this value of e
+ e_rounded = round(es[idx], digits=2)
+ plot!(plt, [es[idx], es[idx]], [-0, .75], label="e : L > 0 ≈ $(e_rounded)", color=:red, lw=.75, linestyle=:dash)
+
+ # perform a log regression only on values where the l exponent is greater than .2
+ # (a, b) = linear_regression(es[idx:end], log.(les[idx:end]))
+ # a_rounded = round(a, digits=2)
+ # b = round(b, digits=2)
+ # plot!(plt, es[idx:end], exp.(a*es[idx:end] .+ b), label="L ≈ exp($a_rounded*e + $b)", lw=2, color=:orange, linestyle=:dash)
+
+ return
+end
+
+# simualte two elipical
+# eccentricity = .224
+# (a, plt) = find_l_exponent(eccentricity, G)
+
+# generate a span of eccentricities
+
+# eccentricities = range(0.0, .985, length=750) # todo: add more len for final render
+# l_exponents = []
+# for e in eccentricities
+# (a, _) = find_l_exponent(e, G)
+# push!(l_exponents, a)
+# end
+
+# # do a normal plot
+# # p1 = plot(
+# # eccentricities, l_exponents, xlabel="eccentricity (e)", ylabel="Lyapunov Exponents (L)", title="Lyapunov Exponent vs Eccentricity",
+# # lw=.8, label="Lyapunov Exponents (connected)")
+# p1 = scatter(eccentricities, l_exponents,
+# title="Lyapunov Exponent vs Eccentricity", xlabel="Eccentricity (e)", ylabel="Lyapunov Exponent (L)",
+# color=:blue, markerstrokealpha=0.0, markersize=2, msw=0.0,
+# label="Lyapunov Exponents")
+
+# # process the data
+# process_data!(l_exponents, eccentricities, p1)
+
+# savefig(p1, "hw4/4-19-final.png")
+
+
+eccentricity = .224
+vy_elipitical = calculate_vy1_0(1.0, eccentricity, G)
+println("vy1_0 = ", vy_elipitical)
+sol1 = simulate(1.0, vy_elipitical, 0.0)
+sol2 = simulate(1.0, vy_elipitical, 0.01) # small change in θ_0
+(ts, diff_θ) = calculate_Δθ(sol1, sol2)
+# plot the θ over time
+plt_1 = plot(sol1.t, sol1[7, :], xlabel="time (Hyperion years)", ylabel="θ (radians)", title="θ vs time \$(θ_0 = 0, e = .224\$)", legend=false, lw=2, label="θ_0 = 0.0")
+# plot the ω over time
+plt_2 = plot(sol2.t, sol2[7, :], xlabel="time (Hyperion years)", ylabel="θ (radians)", title="θ vs time \$(θ_0 = .01, e = .224)\$", legend=false, lw=2, label="ω_0 = 0.0")
+# plot the difference in θ over time
+plot_3 = plot(ts, diff_θ, yscale=:ln, xlabel="time (Hyperion years) ", ylabel="Δθ (radians)", title="Δθ vs time", legend=:bottomright, lw=2, label="\$ Δ θ ≡ √{(θ_1 - θ_2)^2} \$")
+
+# apply a linear regression to the log of the difference
+(a, b) = linear_regression(ts, log.(diff_θ))
+a_rounded = round(a, digits=2)
+b = round(b, digits=2)
+plot_3 = plot!(ts, exp.(a*ts .+ b), label="exp($a_rounded*t + $b)", lw=1.2, color=:red, linestyle=:dash)
+
+plt_combined = plot(plt_1, plt_2, plot_3, layout=(3, 1), size=(800, 1000))
+savefig(plt_combined, "hw4/4-19-.224.png")
+
+# run another simuation, but only a circle
+e = 0
+vy_circular = calculate_vy1_0(1.0, e, G)
+println("vy1_0 = ", vy_circular)
+sol1 = simulate(1.0, vy_circular, 0.0)
+sol2 = simulate(1.0, vy_circular, 0.01) # small change in θ_0
+(ts, diff_θ) = calculate_Δθ(sol1, sol2)
+# plot the θ over time
+plt_1 = plot(sol1.t, sol1[7, :], xlabel="time (Hyperion years)", ylabel="θ (radians)", title="θ vs time (\$θ_0 = 0, e = 0\$)", lw=2, legend=false)
+# plot the ω over time
+plt_2 = plot(sol2.t, sol2[7, :], xlabel="time (Hyperion years)", ylabel="θ (radians)", title="θ vs time (\$θ_0 = .01, e = 0)\$", lw=2, legend = false)
+# plot the difference in θ over time
+plt = plot(ts, diff_θ, yscale=:ln, xlabel="time (Hyperion years)", ylabel="Δθ (radians)", title="Δθ vs time", legend=:bottomright, lw=2, label="\$ Δ θ ≡ √{(θ_1 - θ_2)^2} \$")
+
+# apply a linear regression to the log of the difference
+(a, b) = linear_regression(ts, log.(diff_θ))
+a_rounded = round(a, digits=2)
+b = round(b, digits=2)
+plt = plot!(ts, exp.(a*ts .+ b), label="exp($a_rounded*t + $b)", lw=1.2, color=:red, linestyle=:dash)
+
+plt_combined = plot(plt_1, plt_2, plt, layout=(3, 1), size=(800, 1000))
+
+savefig(plt_combined, "hw4/4-19-0.png")
+
diff --git a/hw4/SolarSystem3.jl b/hw4/SolarSystem3.jl
new file mode 100644
index 0000000..d423669
--- /dev/null
+++ b/hw4/SolarSystem3.jl
@@ -0,0 +1,274 @@
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+
+# Simulate a solar system
+
+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+using LinearAlgebra # for dot and cross products
+
+println("Number of threads = ", Threads.nthreads())
+
+G = 4.0*pi^2 # time scale = year and length scale = AU
+
+mutable struct body
+ name::String # name of star or planet
+ m::Float64 # mass
+ r::Vector{Float64} # position vector
+ v::Vector{Float64} # velocity vector
+end
+
+function star(name="Sun", m = 1.0, r = zeros(3), v = zeros(3))
+ return body(name, m, r, v)
+end
+
+function planet(name="Earth", m=3.0e-6, a=1.0, ϵ=0.017, i=0.0)
+ perihelion = (1.0 - ϵ) * a
+ aphelion = (1.0 + ϵ) * a
+ speed = sqrt(G * (1.0 + ϵ)^2 / (a * (1.0 - ϵ^2)))
+ phi = 0.0
+ r = [perihelion * cos(i*pi/180.0) * cos(phi), perihelion * cos(i*pi/180.0) * sin(phi), perihelion * sin(i*pi/180.0)]
+ v = [-speed * sin(phi), speed * cos(phi), 0.0]
+ return body(name, m, r, v)
+end
+
+function momentum(b::body)
+ return b.m * b.v
+end
+
+function angularMomentum(b::body)
+ return b.m * cross(b.r, b.v)
+end
+
+function kineticEnergy(b::body)
+ v = b.v
+ m = b.m
+ return 0.5 * m * dot(v, v)
+end
+
+function potentialEnergy(body1::body, body2::body)
+ r = body1.r - body2.r
+ rSquared = dot(r, r)
+ return -G * body1.m * body2.m / sqrt(rSquared)
+end
+
+function rv(b::body)
+ return [b.r; b.v]
+end
+
+function force(body1::body, body2::body)
+ r = body1.r - body2.r
+ rSquared = dot(r, r)
+ return -G * r * rSquared^(-1.5)
+ #return -G * r / (rSquared * sqrt(rSquared))
+end
+
+mutable struct SolarSystem
+ bodies::Vector{body}
+ numberOfBodies::Int64
+ phaseSpace::Matrix{Float64} # 6N dimensional phase space
+end
+
+function SolarSystem()
+
+ bodies = Vector{body}()
+ push!(bodies, star()) # default = Sun
+ push!(bodies, planet("Venus", 2.44e-6, 0.72, 0.0068, 3.39))
+ push!(bodies, planet("Earth", 3.0e-6, 1.0, 0.017, 0.0))
+ push!(bodies, planet("Jupiter", 0.00095, 5.2, 0.049, 1.3))
+ push!(bodies, planet("Saturn", 0.000285, 9.58, 0.055, 2.48))
+ numberOfBodies = size(bodies)[1]
+
+ phaseSpace = zeros(6, 0)
+ for b in bodies
+ phaseSpace = [phaseSpace rv(b)]
+ end
+
+ return SolarSystem(bodies, numberOfBodies, phaseSpace)
+
+end
+
+function TotalMass(s::SolarSystem)
+ M = 0.0
+ for b in s.bodies
+ M += b.m
+ end
+ return M
+end
+
+function TotalLinearMomentum(s::SolarSystem)
+ P = zeros(3)
+ for b in s.bodies
+ P += momentum(b)
+ end
+ return P
+end
+
+function TotalAngularMomentum(s::SolarSystem)
+ L = zeros(3)
+ for b in s.bodies
+ L += angularMomentum(b)
+ end
+
+ return L
+end
+
+function TotalEnergy(s::SolarSystem)
+ ke = 0.0
+ pe = 0.0
+
+ for body1 in s.bodies
+ ke += kineticEnergy(body1)
+ for body2 in s.bodies
+ if (body1 != body2)
+ pe += 0.5 * potentialEnergy(body1, body2)
+ end
+ end
+ end
+
+ return pe + ke
+end
+
+function CenterOfMassFrame!(s::SolarSystem)
+
+ M = TotalMass(s)
+ P = TotalLinearMomentum(s)
+ V = P / M
+
+ s.phaseSpace = zeros(6, 0)
+ for body in s.bodies
+ body.v -= V #boost to COM frame
+ s.phaseSpace = [s.phaseSpace rv(body)]
+ end
+
+ return nothing
+end
+
+function tendency!(dps, ps, s::SolarSystem, t)
+
+ i = 1 # update phase space of individual bodies
+ for b in s.bodies
+ b.r = ps[1:3, i]
+ b.v = ps[4:6, i]
+ i += 1
+ end
+
+ # find velocities of bodies and forces on them. O(N^2) computational cost
+ N = s.numberOfBodies
+ for i in 1:N
+ b1 = s.bodies[i]
+ dps[1:3, i] = b1.v #dr/dt
+ dps[4:6, i] = zeros(3)
+ for j in 1:i-1
+ b2 = s.bodies[j]
+ f = force(b1, b2) # call only once
+ dps[4:6, i] += b2.m * f
+ dps[4:6, j] -= b1.m * f # Newton's 3rd law
+ end
+ end
+
+ return nothing
+end
+
+function parallel_tendency!(dps, ps, s::SolarSystem, t)
+
+ i = 1 # update phase space of individual bodies
+ for b in s.bodies
+ b.r = ps[1:3, i]
+ b.v = ps[4:6, i]
+ i += 1
+ end
+
+ # find velocities of bodies and forces on them. O(N^2) computational cost
+ N = s.numberOfBodies
+ Threads.@threads for i in 1:N
+ b1 = s.bodies[i]
+ dps[1:3, i] = b1.v #dr/dt
+ dps[4:6, i] = zeros(3)
+ for j in 1:N
+ if (j != i)
+ b2 = s.bodies[j]
+ f = force(b1, b2)
+ dps[4:6, i] += b2.m * f
+ end
+ end
+ end
+
+ return nothing
+end
+
+s = SolarSystem()
+CenterOfMassFrame!(s)
+println(typeof(s))
+println("Initial total energy = ", TotalEnergy(s))
+println("Initial total linear momentum = ", TotalLinearMomentum(s))
+println("Initial total angular momentum = ", TotalAngularMomentum(s))
+println("Number of bodies = ", s.numberOfBodies)
+for b in s.bodies
+ println("body name = ", b.name)
+end
+
+t_final = 1000.0 # final time of simulation
+tspan = (0.0, t_final) # span of time to simulate
+prob = ODEProblem(tendency!, s.phaseSpace, tspan, s) # specify ODE
+sol = solve(prob, maxiters=1e8, reltol=1e-10, abstol=1e-10) # solve using Tsit5 algorithm to specified accuracy
+
+println("\n\t Results")
+println("Final time = ", sol.t[end])
+println("Final total energy = ", TotalEnergy(s))
+println("Final total linear momentum = ", TotalLinearMomentum(s))
+println("Final total angular momentum = ", TotalAngularMomentum(s))
+
+function eccentricity(body::Int64, solution)
+ n = size(solution[1, 1, :])[1]
+ samples = 1000
+ interval = floor(Int, n/samples)
+ ϵ = zeros(samples-1)
+ time = zeros(samples-1)
+ for i in 1:samples-1
+ r2 = sol[1, body, i*interval:(i+1)*interval].^2 + sol[2, body, i*interval:(i+1)*interval].^2 + sol[3, body, i*interval:(i+1)*interval].^2
+ aphelion = sqrt(maximum(r2))
+ perihelion = sqrt(minimum(r2))
+ ϵ[i] = (aphelion - perihelion) / (aphelion + perihelion)
+ time[i] = solution.t[i*interval]
+ end
+ return (time, ϵ)
+end
+
+function obliquity(body::Int64, solution)
+ n = size(solution[1, 1, :])[1]
+ samples = 1000
+ interval = floor(Int, n/samples)
+ ob = zeros(samples)
+ time = zeros(samples)
+ for i in 1:samples
+ r = solution[1:3, body, i*interval]
+ v = solution[4:6, body, i*interval]
+ ell = cross(r, v)
+ norm = sqrt(dot(ell, ell))
+ ob[i] = (180.0 / pi) * acos(ell[3]/norm)
+ time[i] = solution.t[i*interval]
+ end
+ return (time, ob)
+end
+
+
+body1 = 2
+body2 = 3 # Earth
+body3 = 4
+body4 = 5
+
+# Plot of orbit
+xy = plot([(sol[1, body1, :], sol[2, body1, :]), (sol[1, body2, :], sol[2, body2, :]),
+ (sol[1, body3, :], sol[2, body3, :]), (sol[1, body4, :], sol[2, body4, :])], colors = ("yellow", "green"), xlabel = "x", ylabel = "y", legend = false, title = "Orbit")
+
+# Plot of obliquity
+tilt = obliquity(body2, sol)
+obliquityPlot = plot(tilt, xlabel = "t", ylabel = "tilt", legend = false, title = "obliquity")
+
+# Plot of eccentricity
+eccentric = eccentricity(body2, sol)
+eccentricityPlot = plot(eccentric, xlabel = "t", ylabel = "ϵ", legend = false, title = "eccentricity")
+
+#plot(obliquityPlot, eccentricityPlot)
+plot(xy, aspect_ratio=:equal)
+
diff --git a/hw4/comphysHW4-final.pdf b/hw4/comphysHW4-final.pdf
new file mode 100644
index 0000000..b37a6ec
--- /dev/null
+++ b/hw4/comphysHW4-final.pdf
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diff --git a/hw4/images/.DS_Store b/hw4/images/.DS_Store
new file mode 100644
index 0000000..5008ddf
--- /dev/null
+++ b/hw4/images/.DS_Store
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diff --git a/hw4/images/4-10-dθvdt-ldl.png b/hw4/images/4-10-dθvdt-ldl.png
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diff --git a/hw4/images/4-10.png b/hw4/images/4-10.png
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diff --git a/hw4/images/4-17.png b/hw4/images/4-17.png
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diff --git a/hw4/images/4-19-0.png b/hw4/images/4-19-0.png
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diff --git a/hw4/images/4-19-final.png b/hw4/images/4-19-final.png
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diff --git a/hw4/images/4-19-log.png b/hw4/images/4-19-log.png
new file mode 100644
index 0000000..b06e7df
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diff --git a/hw4/images/4-19.png b/hw4/images/4-19.png
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diff --git a/hw4/images/A_alphachange.png b/hw4/images/A_alphachange.png
new file mode 100644
index 0000000..10e01a4
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+++ b/hw4/images/A_alphachange.png
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diff --git a/hw4/images/l_exponent.svg b/hw4/images/l_exponent.svg
new file mode 100644
index 0000000..ad76bd5
--- /dev/null
+++ b/hw4/images/l_exponent.svg
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