# Artemis II: Spot the Kinematics Artemis II launched 1 April 2026. NASA calls the trajectory a product of "multibody gravitational dynamics", Earth gravity harmonics, lunar gravity fields, solar perturbations, numerical integration to $10^{-12}$ tolerance. The language is dynamical. The mathematics is not. Parent note: [[Cosmological_Dynamics_Null]] ## Null hypothesis **H₁** (assume true): The Artemis II trajectory contains dynamical variables that are independent from kinematics. **H₀** (assume false, try to falsify): The Artemis II trajectory does not contain dynamical variables that are independent from kinematics. If H₁ is true, at least one variable in the trajectory computation (GM, force, mass, field strength) cannot be reduced to positions, velocities, angles, and times. This note traces every published parameter back to its kinematic origin to find that variable. --- ## 1. Mission profile Artemis II is a crewed lunar flyby on a hybrid free return trajectory. The spacecraft does not enter lunar orbit, the Moon's gravity bends the path and returns Orion to Earth without a propulsive manoeuvre after TLI. Total mission duration: ~10 days. | Phase | Event | Time (MET) | |-------|-------|------------| | 1 | Launch, ascent, core separation | T+0 to T+08:18 | | 2 | ICPS coasts to apogee, PRM raises perigee to ~185 km | T+49:00 | | 3 | Apogee raise burn → high Earth orbit (~71,660 × 185 km) | T+01:47:57 | | 4 | ICPS/Orion separation, proximity ops | T+03:24 to T+04:35 | | 5 | Orion perigee raise, then TLI by ESM engine ($\Delta v = 388$ m/s) | Flight Day 2 | | 6 | Outbound transit (3 correction burns) | Days 2–5 | | 7 | Lunar flyby, closest approach ~6,500 km from surface | T+5/01:23 | | 8 | Free return transit (3 correction burns) | Days 7–9 | | 9 | Entry at ~11 km/s, splashdown Pacific | T+9/01:46 | Maximum distance from Earth: ~252,000 miles (405,500 km), exceeding Apollo 13's record. --- ## 2. The gravitational parameters NASA's trajectory design software (Copernicus) uses the following GM values from JPL DE421/DE440 ephemerides: | Body | GM (km³/s²) | Source | |------|-------------|--------| | Earth | 398,600.435507 | JPL DE440 | | Moon | 4,902.800118 | JPL DE440 | | Sun | 1.327 × 10¹¹ | JPL DE440 | These are presented as fundamental physical constants, the product of Newton's gravitational constant $G$ and each body's mass $M$. But where do they come from? ### How GM_Earth is measured $GM_\oplus = 398{,}600.435507$ km³/s² is measured by tracking satellite orbits. The definitive modern value comes from the **GRACE** mission (Gravity Recovery And Climate Experiment, 2002–2017), which flew two satellites in formation and measured their relative positions to micrometre precision. The 8×8 spherical harmonic model (GGM02C) that Artemis uses is derived from GRACE orbital tracking data. What GRACE measured: **the positions and velocities of two satellites as functions of time**. That is kinematics. The "gravity field" is a map of accelerations derived from those kinematics via $\ddot{r} = d^2r/dt^2$. The $GM$ value is the monopole term: $GM = 4\pi^2 a^3/T^2$ averaged over many satellites. ### How GM_Moon is measured $GM_☾ = 4{,}902.800118$ km³/s² comes from **GRAIL** (Gravity Recovery and Interior Laboratory, 2011–2012). Same method: two satellites, relative position tracking, kinematic data converted to an acceleration map. The GRGM660PRIM model that Artemis uses (up to 50×50 harmonics near the Moon) is built from GRAIL satellite orbit tracking. ### The pattern | "Dynamical" input | What was actually measured | Method | |---------------------|---------------------------|--------| | $GM_\oplus$ | Satellite positions vs time | GRACE orbit tracking | | $GM_☾$ | Satellite positions vs time | GRAIL orbit tracking | | $GM_\odot$ | Planet positions vs time | Planetary ephemerides | | Spherical harmonics ($J_2$, etc.) | Satellite orbit perturbations | Same tracking data | Every gravitational parameter used by Copernicus is extracted from orbital kinematics. The "force model" is a kinematic model with dynamical notation. $G$ and $M$ never appear independently, only the product $GM$, which is $4\pi^2 a^3/T^2$ for the reference orbit. --- ## 3. Spot the kinematics: phase by phase ### 3a. Orbital velocity (any altitude) The vis-viva equation gives velocity at radius $r$ in an orbit with semi-major axis $a$: $v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right)$ Substituting $GM = 4\pi^2 a_{ref}^3/T_{ref}^2$ (from any known reference orbit): $v^2 = \frac{4\pi^2 a_{ref}^3}{T_{ref}^2}\left(\frac{2}{r} - \frac{1}{a}\right)$ No $G$. No $M$. One orbit's kinematics $(a_{ref}, T_{ref})$ predicts another orbit's velocity. The vis-viva equation is a kinematic transfer function. For the circular LEO at 185 km altitude: $v_{circ} = \sqrt{\frac{GM}{R_\oplus + 185}} = \sqrt{\frac{398{,}600.44}{6{,}556}} = 7.80 \text{ km/s}$ Or equivalently, using the ISS orbit ($a = 6{,}793$ km, $T = 5{,}560$ s): $v_{circ} = \frac{2\pi \times 6{,}793}{5{,}560} \times \sqrt{\frac{6{,}793^3}{6{,}556 \times 6{,}793^2}} = 7.80 \text{ km/s}$ Same number. The GM route and the kinematic route give the same answer because they are the same equation. ### 3b. Trans-Lunar Injection The TLI burn ($\Delta v = 388$ m/s) boosts Orion from the high Earth orbit perigee velocity to the translunar velocity. The required velocity is set by the vis-viva equation for the transfer orbit: $v_{TLI} = \sqrt{GM\left(\frac{2}{r_{perigee}} - \frac{1}{a_{transfer}}\right)}$ where $a_{transfer} \approx (r_{perigee} + r_{Moon})/2$ for the transfer ellipse reaching the Moon. Every term: $r_{perigee}$ (measured), $r_{Moon}$ (measured), $GM$ (from orbits). The Δv is the difference between two kinematic velocities. NASA reports the post-TLI velocity as ~10.95 km/s (~24,500 mph). This is deliberately slower than the ~11.2 km/s escape velocity to maintain the free return constraint. That constraint is itself kinematic: it requires the specific energy $\xi = v^2/2 - GM/r$ to be slightly negative (bound orbit that reaches the Moon and returns). ### 3c. High Earth orbit After the apogee raise burn, Orion is in a ~71,660 × 185 km orbit: | Parameter | Value | Source | |-----------|-------|--------| | Perigee altitude | 185 km | NASA press kit | | Apogee altitude | 71,660 km | NASA press kit | | $r_p$ | 6,556 km | $R_\oplus + 185$ | | $r_a$ | 78,031 km | $R_\oplus + 71{,}660$ | | $a$ | 42,294 km | $(r_p + r_a)/2$ | | $e$ | 0.845 | $(r_a - r_p)/(r_a + r_p)$ | | $T$ | 23.5 hr | $2\pi\sqrt{a^3/GM}$ | That period calculation uses $GM$. But $GM$ came from GRACE satellite tracking, i.e. from $(a_{GRACE}, T_{GRACE})$. So the period of Orion's orbit is derived from the period of GRACE's orbit. Kinematics borrowing from kinematics. ### 3d. Sphere of influence The lunar sphere of influence determines when trajectory calculations switch from Earth centred to Moon centred. Batcha et al. (AAS 20-649) explicitly describe this transition. $r_{SOI} = a_{Moon} \left(\frac{GM_{Moon}}{GM_{Earth}}\right)^{2/5}$ Both $GM$ values are kinematic ratios. The sphere of influence is: $r_{SOI} = a_{Moon} \left(\frac{4\pi^2 a_{Moon,sat}^3/T_{Moon,sat}^2}{4\pi^2 a_{Earth,sat}^3/T_{Earth,sat}^2}\right)^{2/5}$ The $4\pi^2$ cancels. The sphere of influence is a ratio of orbital kinematics raised to the 2/5 power, times the Earth Moon distance. No $G$. No $M$. $r_{SOI} = a_{Moon} \left(\frac{a_{Moon,sat}^3 \cdot T_{Earth,sat}^2}{a_{Earth,sat}^3 \cdot T_{Moon,sat}^2}\right)^{2/5}$ Artemis II enters the lunar SOI at T+4/06:59 and exits at T+5/19:47. The boundary is kinematic. ### 3e. Lunar flyby Closest approach: ~6,500 km from the lunar surface (~8,237 km from centre). The flyby geometry is governed by the hyperbolic excess velocity $v_\infty$ and the turn angle $\delta$: $\delta = 2 \arcsin\left(\frac{1}{1 + r_p v_\infty^2/GM_{Moon}}\right)$ Substituting $GM_{Moon} = 4\pi^2 a_{GRAIL}^3/T_{GRAIL}^2$: the turn angle is set by the flyby distance and the incoming velocity, relative to a kinematic ratio derived from GRAIL's orbit. The Moon's "gravitational pull" is quantified by the same $(a, T)$ structure as everything else. ### 3f. Free return The free return trajectory is the central safety feature: if the ESM engine fails after TLI, the Moon's gravity redirects Orion back to Earth without propulsion. NASA describes this as a dynamical property, Earth's gravity "still exerts enough influence" at the chosen velocity. But the free return condition is a constraint on the specific orbital energy: $\xi = \frac{v^2}{2} - \frac{GM}{r} < 0 \quad\text{(bound to Earth Moon system)}$ with the additional geometric constraint that the trajectory passes close enough to the Moon for the lunar gravity assist to redirect it. Both terms ($v^2/2$ and $GM/r$) are kinematic. The "dynamical influence of Earth's gravity" is a statement about whether $v < v_{esc} = \sqrt{2GM/r}$, which is whether $v < \sqrt{8\pi^2 a_{ref}^3/(T_{ref}^2 r)}$. Kinematics. ### 3g. Entry velocity Entry Interface at 121.92 km altitude, velocity ~11 km/s (~36,000 ft/s). This is the velocity of a nearly parabolic orbit at radius $R_\oplus + 122$ km: $v_{EI} = \sqrt{GM\left(\frac{2}{R_\oplus + 122} - \frac{1}{a_{return}}\right)}$ Same vis-viva equation. Same kinematic $GM$. The "fastest crewed reentry ever" is a kinematic consequence of the return orbit's semi-major axis. --- ## 4. The gravity models: kinematics all the way down Batcha et al. (AAS 20-649, p.2) describe the force model: > "The Earth departure phases use an 8×8 Gravity Recovery And Climate Experiment (GRACE) GGM02C Earth gravity model. The Moon centred phases use the Gravity Recovery and Interior Laboratory (GRAIL) GRGM660PRIM gravity model. A 50×50 model is used for the close lunar flybys." The spherical harmonic coefficients $J_2, J_3, C_{22}, S_{22}, \ldots$ are corrections to the point mass $GM$ model. They describe how the gravitational acceleration varies with latitude and longitude. But they are measured the same way $GM$ is measured: **by tracking satellite orbits**. - **GRACE**: Two satellites, ~220 km apart, in polar orbit at ~500 km. K-band ranging measured intersatellite distance changes to micrometre precision. The varying distance = varying relative acceleration = varying gravity field. Input: positions and velocities. Output: acceleration map. Kinematics → kinematics. - **GRAIL**: Two satellites (Ebb and Flow) in lunar polar orbit at ~55 km. Same ranging technique. Input: kinematics. Output: lunar "gravity" map. The spherical harmonics do not add dynamics. They add kinematic refinement. The 50×50 GRAIL model means 2,601 coefficients, each fitted from orbital tracking data. More parameters, same kinematic source. ### The numerical integration Copernicus integrates the equations of motion using DDEABM (Adams-Bashforth-Moulton) with $10^{-12}$ error tolerance: $\ddot{\mathbf{r}} = -\frac{GM}{r^3}\mathbf{r} + \text{harmonics} + \text{third bodies} + \text{SRP}$ This looks dynamical: $F = ma$, therefore $\ddot{r} = F/m$. But $m$ cancelled on both sides. What remains is: $\ddot{\mathbf{r}} = -\frac{4\pi^2 a_{ref}^3}{T_{ref}^2 r^3}\mathbf{r} + \ldots$ A differential equation, a rule for updating position from its own rate of change. The acceleration at position $\mathbf{r}$ is determined by $\mathbf{r}$ itself and Kepler's ratio $4\pi^2 a^3/T^2$ from reference orbits. No force, no mass, just a recipe for computing where things go based on where they already are, calibrated by satellite tracking. --- ## 5. The Copernicus pipeline: a kinematic machine From AAS 20-649 and AAS 23-363, here is what the Copernicus trajectory optimizer actually does: 1. **Input**: Launch epoch (a time), vehicle state vector $(x, y, z, \dot{x}, \dot{y}, \dot{z})$ at core separation, **position and velocity** (kinematics) 2. **Force model**: $GM$ values + spherical harmonics + ephemerides, all derived from **orbital tracking** (kinematics) 3. **Integration**: Propagates state vector forward/backward using $\ddot{r} = -(4\pi^2 a^3/T^2)\mathbf{r}/r^3 + \ldots$, a differential equation with $m$ cancelled, $GM$ = Kepler's ratio 4. **Optimization**: Minimises total $\Delta v$ subject to geometry constraints (flyby altitude, entry corridor), **velocity and position constraints** (kinematics) 5. **Output**: Burn plan = thrust direction $(\alpha, \beta)$, burn duration $\Delta t$, and coast times, **angles, times, and velocity changes** (kinematics) 6. **Validation**: Compare to ephemeris (time history of state vectors), **position and velocity vs time** (kinematics) The optimization variables listed in AAS 20-649 (p.9): - Burn angles $\alpha_0, \beta_0$ (geometry) - Burn durations $\Delta t$ (time) - Flight times between events (time) - Flyby parameters: periapsis radius, eccentricity, inclination, ascending node, argument of periapsis, true anomaly, all **orbital elements** (kinematics) - Entry Interface: longitude, latitude, velocity, azimuth, flight path angle, **position and velocity** (kinematics) Not a single optimisation variable is a mass, a force, or a field strength. Every variable is a position, velocity, angle, or time. --- ## 6. The Δv budget | Burn | $\Delta v$ | What determines it | |------|-----------|-------------------| | PRM (perigee raise) | ~50 m/s | Velocity difference between suborbital and circular at 185 km | | Apogee raise | ~2,900 m/s | Velocity at perigee for HEO vs LEO (vis-viva) | | TLI | 388 m/s | Velocity for translunar transfer vs HEO perigee (vis-viva) | | OTC corrections | ~5–115 ft/s each | Velocity adjustments from tracking (position → velocity error) | | Entry | 0 (passive) | Velocity at EI from return orbit (vis-viva) | Every Δv is a difference between two velocities. Velocity is $d\mathbf{r}/dt$. The entire propulsive budget is kinematic. --- ## 7. The accelerometer reads zero The Orion spacecraft carries accelerometers as part of its GN&C (Guidance, Navigation, and Control) system. Woffinden et al. (AAS 23-062, p.12) describe their function: "The accelerometer measures the **nongravitational acceleration**." An accelerometer works by measuring the displacement of a proof mass relative to its housing. In freefall, gravity accelerates the proof mass and the housing identically, zero relative displacement, zero reading. This is the equivalence principle: freefall in a gravitational field is locally indistinguishable from floating in empty space. During the lunar flyby, the most dramatic "gravitational" event of the mission, the accelerometer reads **zero**. The trajectory bends, the velocity changes by kilometres per second in Earth's frame, and the instrument detects nothing. The crew feels nothing. Every atom in their bodies and every atom in the spacecraft are accelerated identically by the Moon's field. There is no differential force to measure or to feel. The "slingshot" velocity gain exists only as a kinematic effect that depends on which reference frame you measure from: Orion enters and exits the Moon's sphere of influence at the same speed relative to the Moon, but the Moon's own orbital velocity donates momentum in the Earth frame. A kinematic frame transformation, not a locally detectable force. What the accelerometer actually detects across the mission: | Phase | Accelerometer | Source | |-------|--------------|--------| | LEO coast | ~zero | Freefall | | Engine burns | Large signal | Thrust on hull | | Cislunar coast | Tiny noise | Venting, solar pressure | | Lunar flyby | **Zero** | Freefall, gravity undetectable | | Entry | Large signal | Atmospheric drag | The gravitational acceleration is added computationally: software inserts $\ddot{r} = -GM\mathbf{r}/r^3$ using $GM = 4\pi^2a^3/T^2$ from GRACE/GRAIL orbit tracking. The total acceleration is: computed gravity + measured nongravitational acceleration. The most important "force" in the entire trajectory, the one that governs every coast arc, bends the path around the Moon, and returns the crew to Earth, is never measured by any instrument on the spacecraft and never felt by any human on board. It is a computation, and the computation uses kinematics. --- ## 8. Where is the dynamics? The Artemis II trajectory design uses: - GM values measured from satellite orbit tracking → kinematics - Spherical harmonics measured from satellite orbit tracking → kinematics - Vis-viva equation with mass cancelled → kinematics - Numerical integration of $\ddot{r} = -GM\mathbf{r}/r^3$ (mass cancelled) → kinematics - Optimisation over positions, velocities, angles, and times → kinematics - Validation against ephemerides (position/velocity vs time) → kinematics The trajectory was computed, optimised, and validated without a single independent dynamical variable entering the calculation. $GM$ was never measured independently of orbital kinematics, it is $4\pi^2 a^3/T^2$ from satellite orbit tracking. The spherical harmonics add refinement to the kinematic map, not dynamical independence. The force equation $F = ma$ collapsed to $\ddot{r} = f(\mathbf{r})$ when $m$ cancelled, leaving a differential equation that updates position from position. And the onboard accelerometer cannot detect gravity at all, it reads zero during the entire lunar flyby ([[2023_Woffinden_Artemis_II_Trajectory_Correction|Woffinden et al. 2023, Eq. 23]]). The most important "force" in the trajectory is never measured by any instrument and never felt by any human on board. NASA sent four people around the Moon using kinematics, orbital elements, velocity differences, and geometric constraints, with dynamical *labels* on every kinematic quantity. --- ## 9. The H₀ test applied **H₁**: Artemis II trajectory predictions require dynamical variables (GM, force, mass) that are independent from kinematics. Therefore removing those variables would change the predicted trajectory. **H₀**: Artemis II trajectory predictions use only kinematic inputs $(a, T, e, \mathbf{r}, \mathbf{v})$ and medium properties $(\varepsilon_0, \mu_0)$. The dynamical labels (GM, force, gravity field) are notational, not computational. | Test | Dynamical claim | Kinematic reality | Independent? | |------|-----------------|-------------------|-------------| | $GM_\oplus$ | Gravitational parameter from $G \times M$ | $4\pi^2 a^3/T^2$ from GRACE orbits | No | | $GM_☾$ | Gravitational parameter from $G \times M$ | $4\pi^2 a^3/T^2$ from GRAIL orbits | No | | Harmonics ($J_2$, etc.) | Gravity field structure | Acceleration map from orbit perturbations | No | | TLI $\Delta v$ | "Escape Earth's gravity" | $v_{transfer} - v_{current}$ (vis-viva, $m$ cancelled) | No | | Free return | "Gravitational influence" | $v < \sqrt{2GM/r}$ i.e. $v < \sqrt{8\pi^2 a^3/(T^2 r)}$ | No | | SOI boundary | "Gravitational dominance" | Ratio of $(a^3/T^2)$ values to 2/5 power | No | | Flyby turn angle | "Lunar gravity assist" | $f(r_p, v_\infty, GM_☾)$ = $f(r_p, v_\infty, a^3/T^2)$ | No | | Entry velocity | "Gravitational acceleration" | Vis-viva at $r_{EI}$ | No | Eight tests. Zero independent dynamical variables. H₀ not refuted. --- ## 10. The rhetorical structure NASA's trajectory papers (Batcha et al. 2020, Eckman et al. 2023) use dynamical language throughout: "force model," "gravity field," "gravitational parameter," "multibody dynamics," "nonlinear force models." The Copernicus software calls its propagation an "environment simulation." Strip the language and look at the mathematics: | They write | They compute | |------------|-------------| | "Gravitational parameter" | $4\pi^2 a^3/T^2$ from satellite tracking | | "Force model" | Acceleration field $\ddot{r} = f(\mathbf{r})$ (mass cancelled) | | "Gravity field" | Map of $\ddot{r}(\mathbf{r})$ from orbital kinematics | | "Multibody dynamics" | Superposition of kinematic fields | | "Numerical integration" | Propagating $(\mathbf{r}, \mathbf{v})$ in time | | "Trajectory optimisation" | Minimising $\sum|\Delta\mathbf{v}|$ over geometric constraints | | "Environment simulation" | Kinematic field evaluation at each timestep | The dynamical vocabulary is a rhetorical layer over a kinematic computation. It serves an institutional function, connecting the calculation to a theoretical framework (Newtonian mechanics, general relativity), but does not contribute information that the kinematics do not already contain. This is not a criticism of the engineering. The engineering works. If it is true that Orion will reach the Moon and return, nobody will know the dynamic cause as to why it happened. The kinematic computation is correct and internally consistent. The question is whether the dynamical interpretation adds predictive content, or whether it is descriptive language applied after the kinematic calculation is complete. Across precession, orbital decay, light deflection, gravitational redshift, and now cislunar trajectory design: the dynamical variables cancel, and the predictions are reproduced by kinematics and medium properties. The pattern holds. --- ## Sources Batcha, A. L. et al. (2020). "Artemis I Trajectory Design and Optimization." AAS 20-649. [[2020_Batcha_Artemis_I_Trajectory_Design]] Eckman, R. A. et al. (2023). "Trajectory Operations of the Artemis I Mission." AAS 23-363. [[2023_Eckman_Artemis_I_Trajectory_Operations]] Woffinden, D., Eckman, R., and Robinson, S. (2023). "Optimized Trajectory Correction Burn Placement for the NASA Artemis II Mission." AAS 23-062. [[2023_Woffinden_Artemis_II_Trajectory_Correction]] NASA (2026). *Artemis II Press Kit*. Kennedy Space Center. JPL Solar System Dynamics. "Astrodynamic Parameters." DE440, Park et al. (2021). https://ssd.jpl.nasa.gov/astro_par.html Tapley, B. D. et al. (2004). "The Gravity Recovery and Climate Experiment." *Space Science Reviews*, 108, 111-145. Zuber, M. T. et al. (2013). "Gravity Field of the Moon from the Gravity Recovery and Interior Laboratory (GRAIL) Mission." *Science*, 339(6120), 668-671. --- ## See also - [[Cosmological_Dynamics_Null]] — Parent note. GM independence test across 10 bodies in 3 systems. - [[Textbook_Orbital_Dynamics]] — The same teardown applied to a generic textbook patched conic derivation. Shows the pattern is not unique to Artemis II but is built into how astrodynamics is taught. - [[2017_Zhang_Theory_Design_Special_Space_Orbits]] — Springer engineering textbook on hovering, cruising, and CR3BP orbits. Active control is also kinematic; mass enters only as a unit conversion at the end. - [[First_Principles_Dynamics]] — Bouw's first principles version: dynamics = kinematics times $m/m$. - [[2013_Bouw_Geocentricity_Forces_and_First_Principles]] — Source note on the mass cancellation argument. - [[00_Null_Hypothesis_Index]] — Master null hypothesis index.