TKS · deep-time dynamics
A secular trend read from the JPL DE441 reference states — evaluated and regressed by TKS.
Illustrative constant-Q model (k₂ = 0.299, Q ≈ 11.7). The rate is not constant: ≈ +9 cm/yr one Gyr ago, ≈ +2.5 cm/yr one Gyr hence. Dashed gold = the naive constant-3.8 extrapolation. A real deep-time history needs a coupled model (Q and ocean/continent evolution, resonance).
Illustrative linear trend at the DE441-recovered rate +3.997 cm/yr (blue) and the tidal/LLR rate +3.80 cm/yr (gold dashed): +1.215 km vs +1.155 km over 30,390 yr. Not a full tidal history.
TKS Engine B is Newtonian mutual gravity + J2–J4 + optional 1PN EIH. It contains no tidal bulge lag, no k₂/Q, no Earth-rotation/angular-momentum exchange — so by itself it cannot generate tidal recession. That physics is added separately as Module B below (a secular model, not yet a force term inside Engine B).
| Model | What it computes | Status |
|---|---|---|
| A · pure gravity | Earth + Moon + Sun + planets, no dissipation | ȧ = 0 (exact) · Engine-B control confirms |
| B · tidal | k₂/Q lag, Earth spin, angular-momentum transfer (constant-Q secular model) | +3.80 cm/yr (Q≈11.7) · +3.716 at Q=12 |
| C · full TKS | DE441 reference states, evaluated by TKS | +3.997 cm/yr (this page) |
Constant-Q law da/dt = 3 (k₂/Q)(M☾/M⊕)(R⊕/a)⁵ · n · a. With literature k₂ = 0.299, Q = 12 it predicts +3.716 cm/yr; fitting Q to LLR gives Q ≈ 11.7 → +3.800 cm/yr.
Over ±30,390 yr it gives +1.155 km — vs the DE441-recovered +1.215 km.
Same torque, Earth spin: LOD +2.09 ms/century (observed +1.7 … +2.3) — an independent consistency check, since Q was fixed from the recession.
Reproduce: cargo run --release -p tks-ephemeris --example great_extremes · --example recession_detail · --example tidal_evolution · --example engine_b_moon_gravity · source crates/tks-ephemeris/examples/