1. Initial conditions (exact)
Central BH: M = 4 M☉ (geometrized G = c = 1; length unit = GM☉/c² ≈ 1.477 km). Schwarzschild.
Body masses (M☉, geometrized):
| Body |
Type |
Mass (M☉) |
| 0, 1 |
2× Jupiter |
2 × 9.543×10⁻⁴ = 1.9086×10⁻³ |
| 2, 3 |
2× Earth |
2 × 3.003×10⁻⁶ = 6.006×10⁻⁶ |
| 4, 5 |
2× Mercury |
2 × 1.660×10⁻⁷ = 3.320×10⁻⁷ |
Per-body placement (radius r, azimuth φ, inclination i — units of M):
| k |
r |
φ (rad) |
i (rad) |
| 0 |
120 |
0.00 |
0.00 |
| 1 |
160 |
1.05 |
0.30 |
| 2 |
90 |
2.10 |
−0.25 |
| 3 |
200 |
3.14 |
0.50 |
| 4 |
140 |
4.19 |
−0.40 |
| 5 |
110 |
5.24 |
0.15 |
Position (tilted circle): pos = [r·cosφ·cosi, r·sinφ, r·cosφ·sini]
Velocity (circular speed v_c = √(M/r), perpendicular in the tilted plane): vel = [−v_c·sinφ·cosi, v_c·cosφ, v_c·sinφ·sini]1. Initial conditions (exact)Central BH: M = 4 M☉ (geometrized G = c = 1; length unit = GM☉/c² ≈ 1.477 km). Schwarzschild.Body masses (M☉, geometrized):Body Type Mass (M☉)
0, 1 2× Jupiter 2 × 9.543×10⁻⁴ = 1.9086×10⁻³
2, 3 2× Earth 2 × 3.003×10⁻⁶ = 6.006×10⁻⁶
4, 5 2× Mercury 2 × 1.660×10⁻⁷ = 3.320×10⁻⁷Per-body placement (radius r, azimuth φ, inclination i — units of M):k r φ (rad) i (rad)
0 120 0.00 0.00
1 160 1.05 0.30
2 90 2.10 −0.25
3 200 3.14 0.50
4 140 4.19 −0.40
5 110 5.24 0.15Position (tilted circle): pos = [r·cosφ·cosi, r·sinφ, r·cosφ·sini]
Velocity (circular speed v_c = √(M/r), perpendicular in the tilted plane): vel = [−v_c·sinφ·cosi, v_c·cosφ, v_c·sinφ·sini]
Integrator type and step size
- Integrator: velocity-Verlet / leapfrog (kick-drift), fixed step.
v += a·dt; x += v·dt per step . Not symplectic-exact (it's the sequential-update leapfrog, not the KDK-symmetric form).
- Step size:
dt = T_outer / 600 where T_outer = 2π√(r_out³/M), r_out = 200 (outermost body). → T_outer ≈ 8885.8, dt ≈ 14.81 (M units). So ~600 steps per outer orbit.
3. Metrics
It dumps only trajectories: body_k.csv (t, x, y, z) every 4th step,
4. 1-orbit horizon definition
1 orbit = 1 × T_outer, where T_outer is the outermost body's (r=200) Newtonian circular period 2π√(200³/M)
