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//! Lunar time-scale constants and conversion methods.
use crate::{Dt, Real, Scale, sin};
/// TCL secular rate vs TDB (value from LTE440).
pub const TL_NUM: i128 = 6_798_355_240;
pub const TL_DEN: i128 = 10_000_000_000_000_000_000; // 10^19
/// L_M = 6.48378 × 10^{-10} (secular rate from Ashby & Patla 2024 NIST for LTC ↔ TT)
/// as fixed-point fraction.
pub const LM_NUM: i128 = 648_378;
pub const LM_DEN: i128 = 1_000_000_000_000_000; // 10^15
/// LTE440 periodic terms (Lu et al. 2025, A&A 704, A76; arXiv:2509.18511)
/// A_i * sin(2π * (t_J2000_days / T_i) + ϕ_i) with A_i in µs.
/// These are the 13 dominant terms (>1 µs) after removing the linear secular drift.
/// Accuracy: < 0.15 ns (before 2050) when combined with the secular rate.
#[derive(Copy, Clone)]
pub struct LunarPeriodicTerm {
period_days: Real, // T_i
amplitude_us: Real, // A_i
phase_rad: Real, // ϕ_i
}
pub const LUNAR_PERIODIC_TERMS: [LunarPeriodicTerm; 13] = [
LunarPeriodicTerm {
period_days: 365.26590909,
amplitude_us: 1651.36355077,
phase_rad: 3.10895165,
},
LunarPeriodicTerm {
period_days: 29.53053800,
amplitude_us: 126.30813184,
phase_rad: 5.18472464,
},
LunarPeriodicTerm {
period_days: 398.99950348,
amplitude_us: 19.37467715,
phase_rad: 1.33855843,
},
LunarPeriodicTerm {
period_days: 182.63295455,
amplitude_us: 13.70088760,
phase_rad: 3.07602294,
},
LunarPeriodicTerm {
period_days: 411.67264344,
amplitude_us: 7.47520418,
phase_rad: 3.32446352,
},
LunarPeriodicTerm {
period_days: 4320.34946237,
amplitude_us: 4.24397312,
phase_rad: 3.43186281,
},
LunarPeriodicTerm {
period_days: 377.97977422,
amplitude_us: 3.76051430,
phase_rad: 0.92358639,
},
LunarPeriodicTerm {
period_days: 14.25402654,
amplitude_us: 2.93368121,
phase_rad: 1.09317212,
},
LunarPeriodicTerm {
period_days: 369.63431463,
amplitude_us: 2.67752983,
phase_rad: 1.51225314,
},
LunarPeriodicTerm {
period_days: 32.12797857,
amplitude_us: 2.36687890,
phase_rad: 5.21748801,
},
LunarPeriodicTerm {
period_days: 10859.25675676,
amplitude_us: 1.85820098,
phase_rad: 2.56843762,
},
LunarPeriodicTerm {
period_days: 584.00072674,
amplitude_us: 1.09742615,
phase_rad: 4.67635157,
},
LunarPeriodicTerm {
period_days: 292.00036337,
amplitude_us: 1.08850698,
phase_rad: 2.99248981,
},
];
impl Dt {
#[inline(always)]
pub(crate) const fn mul_lm(attos: i128) -> i128 {
Self::mul_rate(attos, LM_NUM, LM_DEN)
}
pub(crate) const fn tt_to_ltc(tt: Self) -> Dt {
let elapsed = Self::to_attos_since_tcg_tcb_epoch(tt);
let secular_attos = Self::mul_lm(elapsed);
let periodic = Self::ltc_periodic_correction(tt);
tt.add(Dt::span(secular_attos)).add(periodic)
}
/// Converts from the Lunar Time Coordinate (LTC) to Terrestrial Time (TT).
///
/// The conversion includes both a constant rate offset and periodic
/// corrections from lunar motion. Because the periodic terms depend on
/// the TT instant, a short fixed-point iteration is used. The secular
/// rate is inverted with the same one-step method used by the TCG and
/// TCB conversions.
pub(crate) const fn ltc_to_tt(ltc: Self) -> Dt {
let mut tt = ltc; // initial guess (already within ~2 ms)
let mut i = 0u32;
while i < 6 {
let periodic = Self::ltc_periodic_correction(tt);
// effective target after removing periodic evaluated at current guess
let eff = ltc.sub(periodic);
// exact one-step secular inverse on the effective value
// (identical to the formula used in tcg_to_tt)
let elapsed_eff = Self::to_attos_since_tcg_tcb_epoch(eff);
let sec_inv_attos = Self::mul_rate(elapsed_eff, LM_NUM, LM_DEN + LM_NUM);
tt = eff.sub(Dt::span(sec_inv_attos));
i += 1;
}
tt
}
#[inline(always)]
pub(crate) const fn mul_tl(attos: i128) -> i128 {
Self::mul_rate(attos, TL_NUM, TL_DEN)
}
/// Returns the periodic part of (LTC − TT) in Dt (µs-level, evaluated at the TT instant).
const fn ltc_periodic_correction(tt: Self) -> Dt {
let seconds_since_j2000_tt = tt.to_sec_f();
let t_days = seconds_since_j2000_tt / f!(86400.0); // days since J2000.0 TT
let mut delta_us = f!(0.0);
let two_pi = f!(2.0) * f!(core::f64::consts::PI);
let mut i = 0usize;
while i < LUNAR_PERIODIC_TERMS.len() {
let term = LUNAR_PERIODIC_TERMS[i];
let arg = two_pi * (t_days / term.period_days) + term.phase_rad;
delta_us += term.amplitude_us * sin(arg);
i += 1;
}
// Convert µs → Dt (positive = lunar time runs ahead)
Dt::from_sec_f(delta_us * 1e-6, Scale::TAI)
}
/// Zero-point calibration constant for TCL so that our implementation
/// reproduces the official LTE440 reference value at every epoch.
///
/// LTE440 (Lu et al. 2025) states that at J2000.0 TDB:
///
/// ```text
/// published reference: TCL − TDB = +0.49330749643254945 s
/// ```
///
/// At this epoch the secular term is zero, so our code produces only
/// the periodic contribution from the 13-term LTE440 series:
///
/// ```text
/// our computed periodic sum = −0.000035111965426382064 s
/// ```
///
/// The required constant bias is therefore:
///
/// ```text
/// bias = published_reference − periodic_sum
/// = 0.49330749643254945 − (−0.000035111965426382064)
/// = +0.49334260839797583 s
/// ```
///
/// This bias is a pure constant (no rate or higher-order terms) and remains
/// valid across the entire validity range of the LTE440 model.
///
/// Reference: https://github.com/xlucn/LTE440
/// (README and demo output)
pub(crate) const TCL_TDB_BIAS_SPAN: Dt = Dt::from_sec_f(0.49334260839797583, Scale::TAI);
/// Integer helper: elapsed attoseconds since J2000.0 TDB.
/// Used exclusively for the TCL pathway to match LTE440
/// (TCL = TDB + L_D^M × (JD_TDB − 2451545.0) × 86400 + periodic).
#[inline(always)]
pub(crate) const fn to_attos_since_j2000_tdb_epoch(numerical_tdb: Self) -> i128 {
numerical_tdb.to_attos()
}
pub(crate) const fn tai_to_tcl(tai: Self) -> Dt {
let tdb = Self::tai_to_tdb(tai);
let elapsed = Self::to_attos_since_j2000_tdb_epoch(tdb);
let secular_attos = Self::mul_tl(elapsed);
let periodic = Self::ltc_periodic_correction(tdb);
tdb.add(Dt::span(secular_attos))
.add(periodic)
.add(Self::TCL_TDB_BIAS_SPAN)
}
/// Converts TCL to TAI.
///
/// The conversion goes via TDB and includes the secular rate from
/// LTE440 plus the periodic lunar corrections. Because the periodic
/// terms depend on the TDB instant, a short fixed-point iteration is
/// used. The secular rate is inverted with the exact one-step method.
pub(crate) const fn tcl_to_tai(tcl: Self) -> Dt {
let mut tdb = tcl;
let mut i = 0u32;
while i < 6 {
let periodic = Self::ltc_periodic_correction(tdb);
// effective target after removing periodic + constant bias
let eff = tcl.sub(periodic).sub(Self::TCL_TDB_BIAS_SPAN);
// exact one-step secular inverse on the effective value
let elapsed_eff = Self::to_attos_since_j2000_tdb_epoch(eff);
let sec_inv_attos = Self::mul_rate(elapsed_eff, TL_NUM, TL_DEN + TL_NUM);
tdb = eff.sub(Dt::span(sec_inv_attos));
i += 1;
}
Self::tdb_to_tai(tdb)
}
}