# 2014 Folkner et al: The Planetary and Lunar Ephemerides DE430 and DE431 ## Bibliographic Info - **Year:** 2014 - **Author:** William M. Folkner, James G. Williams, Dale H. Boggs, Ryan S. Park, Petr Kuchynka - **Title:** The Planetary and Lunar Ephemerides DE430 and DE431 - **Publication / Source:** Jet Propulsion Laboratory, IPN Progress Report 42-196 (15 February 2014) - **URL / Archive / DOI:** https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf - **Accessed:** 2026-05-07 ## Source Summary Folkner et al document the construction of JPL's planetary and lunar ephemerides DE430 and DE431. The ephemerides are produced by numerically integrating the equations of motion for the Sun, Moon, eight planet barycenters, Pluto barycenter, and 343 asteroids over a chosen time span, then adjusting the initial conditions and dynamical parameters to minimise residuals against observational data sets including spacecraft tracking ranges, lunar laser ranging (LLR), spacecraft VLBI, and astrometric measurements. The output is stored as 32-day Chebyshev polynomial coefficients giving positions and velocities for each body. DE430 covers years 1550 to 2650; DE431 covers years -13,200 to +17,191 with the lunar core/mantle damping term removed for backward integration stability. The paper documents the coordinate frame (ICRS, tied to ICRF2 by VLBI of Mars-orbiting spacecraft), the time-scale conversions between Barycentric Dynamical Time (TDB) and Terrestrial Time (TT), the parameterised post-Newtonian (PPN) equations of motion used in the integration, the various extended-body and tidal accelerations, the residual format used to compare predictions to observations, and the observational data sets used in the fit. The speed of light c appears explicitly in the SSBC definition (n-body metric), the TDB-TT conversion (at orders c² and c⁴), the PPN point-mass acceleration (at every order from c² to c⁴), the Lense-Thirring acceleration, and the residual format `Δr = (t_meas - t_comp) c / 2` used for both LLR and spacecraft range data. For Jupiter, DE430 used only single 3-D position fixes from Pioneer 10/11, Voyager 1/2, Ulysses, and Cassini flybys (one each), 24 VLBI measurements of Galileo (1996-1997), and ground-based astrometry. There were no spacecraft range data for Jupiter in DE430; Juno had not yet arrived in 2016 when its tracking data became available for DE440. ## Key Claims From the Source ### Claim 1: Ephemerides are integrated dynamical solutions fit to observation **Faithful summary:** DE430 and DE431 are not direct measurements of body positions. They are numerically integrated solutions of the dynamical equations of motion, with the initial conditions and a small set of dynamical parameters adjusted to minimise residuals against observational data. The output is therefore a model. **Direct quote (page 1, abstract):** > "The planetary and lunar ephemerides DE430 and DE431 are generated by fitting numerically integrated orbits of the Moon and planets to observations. The present-day lunar orbit is known to submeter accuracy through fitting lunar laser ranging data with an updated lunar gravity field from the Gravity Recovery and Interior Laboratory (GRAIL) mission. The orbits of the inner planets are known to subkilometer accuracy through fitting radio tracking measurements of spacecraft in orbit about them. ... The orbits of Jupiter and Saturn are determined to accuracies of tens of kilometers as a result of fitting spacecraft tracking data." **Location:** page 1, Abstract **Screenshot:** ![[2014_Folkner_DE430_DE431_p01_abstract.png]] **Contextual note:** The DE430/DE431 output is a least-squares fit to observations within the assumed dynamical model. The dynamical model includes c at the equation-of-motion level. A measurement that disagreed with the assumed c could not appear as a c-anomaly in the output; it would propagate into a position or initial-condition adjustment. ### Claim 2: Solar system barycenter defined via the n-body metric containing c **Faithful summary:** The solar system barycenter (SSBC) is defined using the conserved mass/energy and momentum of the n-body parametrized post-Newtonian metric, where each body's effective gravitational mass parameter μ*A includes a factor `(GM_A) (1 + ½ v_A²/c² - ½ Σ GM_B/(c² r_AB))`. The position of the barycenter is therefore directly dependent on c. **Direct quote (page 5, equation 1-2):** > "The mass/energy of the system M is a conserved quantity where M is defined by [Eq. 1] M = Σ μ*A, where μ*A = GM_A (1 + ½ v_A²/c² - ½ Σ GM_B/(c² r_AB)) [Eq. 2]" **Location:** page 4-5, Section II.B **Screenshot:** ![[2014_Folkner_DE430_DE431_p05_ssbc_n_body_metric.png]] **Contextual note:** The SSBC, used as the origin of the planetary coordinate system in DE430/DE431, is defined in a way that contains c at order v²/c² for every massive body. The c that appears here is held to its CODATA value through the integration. ### Claim 3: TDB-TT conversion is integrated with c² and c⁴ terms **Faithful summary:** Folkner et al present the conversion equation between Barycentric Dynamical Time (TDB) and Terrestrial Time (TT) as a time integral that contains 1/c² and 1/c⁴ terms involving Earth's velocity, the Newtonian potential at Earth from external bodies, the velocity-potential cross terms, and a Δ_E term containing additional 1/c² terms with squared velocity, gravitational potentials, and Earth's acceleration. **Direct quote (page 6, equation 5):** > "The quantity TDB–TT as a function of TDB is given by [Equation 5, with c² and c⁴ terms in the integrand]. ... where ... c is the speed of light, ... v_E is the velocity of the Earth, r_S is the position of the measurement station, and r_E is the position of the Earth." **Location:** page 6, Equation 5 **Screenshot:** ![[2014_Folkner_DE430_DE431_p06_TDB_TT_conversion_eq5.png]] **Contextual note:** The TDB-TT relationship encodes the Lorentz/relativistic transformation between barycentric and terrestrial time scales. Both observational measurements (taken in TT-equivalent UTC) and ephemeris integration (in TDB) must use this conversion. Any change in c would propagate through the time-scale conversion at order v²/c² for every observation. ### Claim 4: PPN n-body point-mass acceleration (Equation 27) contains c at every order **Faithful summary:** The translational equations of motion are derived from the isotropic, parametrized post-Newtonian (PPN) n-body metric. The point-mass acceleration on body A from all other bodies includes the Newtonian term plus relativistic corrections containing 1/c² terms (proportional to βc⁻², γc⁻², v²/c², and v_A · v_B / c²), and 1/c² coupling terms with the accelerations of the other bodies. The Eddington-Robertson-Schiff parameters β and γ are constrained to unity (general relativity). **Direct quote (page 12, Equation 27):** > "The gravitational acceleration of each body due to external point masses is derived from the isotropic, parametrized post-Newtonian (PPN) n-body metric [24-26]. ... [Equation 27 — the full PPN acceleration with terms in 1/c², (v_A/c)², (v_B/c)², v_A · v_B / c², r·a/c² terms, and PPN parameters β and γ] ... where β is the PPN parameter measuring the nonlinearity in superposition of gravity and γ is the PPN parameter measuring space curvature produced by unit rest mass." **Location:** page 12, Equation 27 **Screenshot:** ![[2014_Folkner_DE430_DE431_p12_ppn_eom_eq27.png]] **Contextual note:** This is the dynamical equation that drives the ephemeris integration. c enters at order v²/c² for every body and at every time step. A different c would propagate into different gravitational accelerations and different orbital evolution at every step. The fit is dynamically committed to c = CODATA throughout. ### Claim 5: Range residual format — light-time difference multiplied by c/2 **Faithful summary:** For range observations to spacecraft and lunar laser ranging targets, the residual entering the least-squares fit is `Δr = (t_meas - t_comp) c / 2`. The factor c converts the round-trip light-time into a distance, and the factor 1/2 converts round-trip to one-way. This formulation is identical for LLR (lunar retroreflectors), spacecraft range (MESSENGER, Venus Express, Mars orbiters, Cassini), and any other range data type. **Direct quote (page 19, Section V):** > "LLR data are measurements of the round-trip light-time from an observatory to retroreflectors on the Moon at the Apollo 11, 14, or 15 landing sites or the Lunokhod 1 and 2 rovers. ... Residuals between the measured round-trip light-time t_meas and the value computed from the model t_comp are typically expressed as one-way range residuals Δr = (t_meas - t_comp) c / 2." **Location:** page 19, Section V (LLR description) and same formulation reused for spacecraft range data **Screenshot:** ![[2014_Folkner_DE430_DE431_p19_residual_definition_one_way.png]] **Contextual note:** This formulation hard-codes c as the conversion factor between time and distance during the fit. A measurement that disagreed with c=CODATA could not appear as a c-anomaly in the JPL output; it would propagate into the distance estimate. The same formulation is used for spacecraft range: "the residual light time (measured minus computed) is multiplied by the speed of light, and divided by two to give approximate residual distance in meters" (page 19). The fit is structurally circular in c by design. ### Claim 6: Jupiter orbit constrained by single flyby fixes plus Galileo VLBI plus ground astrometry **Faithful summary:** For DE430/DE431, Jupiter's orbit is constrained by 1 flyby 3-D position fix each from Pioneer 10 (1973), Pioneer 11 (1974), Voyager 1 (1979), Voyager 2 (1979), Ulysses (1992), and Cassini (2000), plus 24 VLBI measurements of Galileo (1996-1997), plus ground-based astrometric data (CCD from Flagstaff and Nikolaev, transit data from La Palma and Washington). There are no spacecraft range data for Jupiter in DE430. **Direct quote (page 21, Table 2):** > Planet — Class — Type — Observatory/Spacecraft — Span — Number > Jupiter — Spacecraft — 3-D — Pioneer 10 — 1973 — 1 > ... Pioneer 11 — 1974 — 1 > ... Voyager 1 — 1979 — 1 > ... Voyager 2 — 1979 — 1 > ... Ulysses — 1992 — 1 > ... Cassini — 2000 — 1 > Jupiter — Spacecraft — VLBI — Galileo — 1996-1997 — 24 > Jupiter — Astrometric — CCD — Flagstaff — 1998-2012 — 342 > ... Nikolaev — 1962-1998 — 2586 > Jupiter — Astrometric — Transit — La Palma — 1986-1997 — 658 > ... Washington — 1914-1994 — 1705 **Location:** page 21, Table 2 **Screenshot:** ![[2014_Folkner_DE430_DE431_p21_jupiter_data_table.png]] **Contextual note:** No DSN range data was available for Jupiter at the time of DE430. The orbit is constrained to "tens of kilometers" accuracy primarily by Galileo's 24 VLBI measurements and the long ground-astrometry baselines. This contrasts sharply with DE440, where 15 Juno range measurements (2016-2020) and 6 Juno VLBA measurements (2016-2019) substantially improved the Jupiter orbit. Any analysis of pre-2016 Jupiter ephemeris should consider that the Jupiter orbit accuracy is fundamentally derived from VLBI angles and astrometric directions, not from ranged distances. ## Equations and Mathematical Material ### Equation 1-2 — Solar system barycenter mass/energy **Source location:** page 5, Equations 1 and 2 **Direct quote or surrounding text:** > "M = Σ μ*A, where μ*A = GM_A (1 + ½ v_A²/c² - ½ Σ_{B≠A} GM_B/(c² r_AB))" **Screenshot:** ![[2014_Folkner_DE430_DE431_p05_ssbc_n_body_metric.png]] **Contextual explanation:** The mass parameter μ*A used in defining the barycenter is the gravitational mass parameter of body A corrected at order 1/c² for the body's kinetic energy and the gravitational potential at A from all other bodies. The conservation of M and the conserved momentum P = Σ μ*A (dr_A/dt) define the centre-of-mass/energy R = (Σ μ*A r_A)/(Σ μ*A) used as the SSBC origin. ### Equation 5 — TDB-TT conversion **Source location:** page 6, Equation 5 **Direct quote or surrounding text:** > "TDB - TT = (L_G - L_B)/(1 - L_B) (TDB - T_0) + (1 - L_G)/(1 - L_B) TDB_0 + (1-L_G)/(1-L_B) ∫(1/c²)(½ v_E² + w_0E + w_LE) dt - (1-L_G)/(1-L_B) ∫(1/c⁴)(-⅛ v_E⁴ - ³⁄₂ v_E² w_0E + 4 v_E · w_iE + ½ w_0E² + Δ_E) dt + (1/c²) v_E · (r_S - r_E) - (1-L_G)/(1-L_B) (1/c⁴)(3 w_0E + ½ v_E²) v_E · (r_S - r_E)" **Screenshot:** ![[2014_Folkner_DE430_DE431_p06_TDB_TT_conversion_eq5.png]] **Contextual explanation:** The barycentric coordinate time TDB and the terrestrial proper time TT differ at order v²/c² and at higher orders containing v⁴/c⁴ and the gravitational potentials w_0E (sum of external GM/r), w_LE (Sun oblateness contribution), and w_iE (sum of GM·v/r). Every observation that has a UTC tag is converted to TDB through this relationship before being compared to the ephemeris. c enters at every order shown. ### Equation 27 — PPN point-mass acceleration **Source location:** page 12, Equation 27 **Direct quote or surrounding text:** > The PPN equation of motion for body A interacting with bodies B contains: Newtonian term (1/r²) + relativistic terms in c⁻² (proportional to β, γ, (v_A · v_B), v², and r·a) + position-velocity cross terms. **Screenshot:** ![[2014_Folkner_DE430_DE431_p12_ppn_eom_eq27.png]] **Contextual explanation:** The acceleration on each body in the integration is the Newtonian acceleration plus order-1/c² and higher relativistic corrections. The PPN parameters β and γ are constrained to unity (the general-relativistic prediction). The integrator advances each body's position and velocity using this acceleration at every time step, with c held fixed at its CODATA value. ### Equation for one-way range residual **Source location:** page 19, Section V **Direct quote or surrounding text:** > "Residuals between the measured round-trip light-time t_meas and the value computed from the model t_comp are typically expressed as one-way range residuals Δr = (t_meas - t_comp) c / 2." **Screenshot:** ![[2014_Folkner_DE430_DE431_p19_residual_definition_one_way.png]] **Contextual explanation:** Range data is fundamentally a measurement of round-trip light-time. The fit converts this to a "residual distance" by multiplying by c and dividing by 2. The conversion uses c as a fixed quantity. A timing residual that, in a world where c was different, would correspond to a different distance, is forced into the c=CODATA distance during the fit. ## Backlinks - [[Romer_Method_Null]] - [[Synodic_Period_Eclipse_Residual]] - [[Dynamic_Heliocentric_Null]] - [[Earth_Anchored_Signals]] - [[2021_Park_DE440_DE441]] - [[2014_OASI_Romer_Revisited]] - [[00_Null_Hypothesis_Index]] ## Tags #source