Appendix A — Math & Technical Guidance

A1. Parametric mass-change rule ΔM≤k⋅M\Delta M \le k \cdot M

where kk is a risk factor set by the Regulatory Authority (default k=10−4k = 10^{-4}, i.e. 0.01%).

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A2. Surface gravity sensitivity
If the body’s radius RR is unchanged, Δgg≈ΔMM.\frac{\Delta g}{g} \approx \frac{\Delta M}{M}.

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A3. Hill (sphere-of-influence) radius
For a body of mass mm orbiting a central mass McM_c at semimajor axis aa, RH≈a(m3Mc)1/3.R_H \approx a\left(\frac{m}{3M_c}\right)^{1/3}.

The fractional change in RHR_H due to Δm\Delta m is approximately ΔRHRH≈13Δmm.\frac{\Delta R_H}{R_H} \approx \frac{1}{3}\frac{\Delta m}{m}.

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A4. Two-body orbital sensitivity
Planetary orbital parameters around a dominant central mass McM_c depend primarily on McM_c. For m≪Mcm \ll M_c, the fractional change in the system mass due to Δm\Delta m is ≈ΔmMc,\approx \frac{\Delta m}{M_c},

which is normally negligible at k≤10−4k \le 10^{-4}.

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A5. Simulation requirement
If ΔM>ksim⋅M\Delta M > k_{\text{sim}}\cdot M

(recommended ksim=10−6k_{\text{sim}} = 10^{-6}), the Operator must submit:

• n-body integrations covering short, mid and long windows (recommended: 10310^{3}, 10510^{5}, 10710^{7} years);
• Monte Carlo / sensitivity runs sampling initial-condition uncertainties;
• worst-case impact probability tables and risk matrices;
• a mitigation plan for cases where trajectories enter unacceptable risk ranges.

All simulations must include code identification, version, input ephemerides and stated assumptions.

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A6. Recommended numeric examples

• Earth: M⊕≈5.972×1024 kgM_{\oplus} \approx 5.972\times10^{24}\ \mathrm{kg}.
  Default ΔM0.01%≈5.97×1020 kg\Delta M_{0.01\%} \approx 5.97\times10^{20}\ \mathrm{kg}.
• Mars: M≈6.417×1023 kgM \approx 6.417\times10^{23}\ \mathrm{kg}.
  Default ΔM0.01%≈6.42×1019 kg\Delta M_{0.01\%} \approx 6.42\times10^{19}\ \mathrm{kg}.
• Example small asteroid: m≈1.3×1013 kgm \approx 1.3\times10^{13}\ \mathrm{kg}.
  Default ΔM0.01%≈1.3×109 kg\Delta M_{0.01\%} \approx 1.3\times10^{9}\ \mathrm{kg}.

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