Reproduction of the flyby anomaly residuals

This repository presents code and Jupyter notebooks that reproduce for the first time a key signature, oscillating Doppler residuals, seen alongside anomalous velocity gains ΔV in NEAR's 1998 flyby of Earth [1, 3], and in Rosetta's in 2005 [2]. The code uses the astropy and poliastro python libraries for orbit propagation, and lmfit for discovering least square fit trajectories that reproduce the pattern of residuals.

More particularly, light-time lags had been anticipated in the two-way Doppler and range data from a deeper mathematical theory revisiting the wave equation, shown implied by the SSN radar range residuals in [5], and subsequently in [6], to closely explain all aspects of the anomalies, including the anomalous velocity gains ΔV to 1%, using range and range rate data directly from JPL Horizons. Copies of the residual graphs from JPL, the Horizons queries and the graph computation and overlay scripts are given and documented in Github: flyby-analysis. See also the brief summary on the anomaly and prior work.

As briefly explained separately, the light-time lags expose a fundamental issue of Doppler under acceleration that was hitherto untreated and renders the very formalism of relativistic space-time fundamentally incomplete for coherent observations, and imply that modulated range codes were delivered at speeds exceeding that of light for days. These aspects, if not the promise of rendering current communication and radar technologies obsolete, make it imperative to verify that the light-time lags are real and the only explanation possible for the anomalies, independently now recognized as already over 100σ [4].

Method and scope

Poliastro defines an Orbit class that encapsulates orbital elements and a method for constructing Orbit objects with reference to a "single attractor" from position and velocity vector coordinates at a given instant in an inertial frame. It also defines an Ephem class that encapsulates an ordered sequence of coordinates and corresponding epochs, and methods to interpolate the coordinates to epochs in-between, and can thus represent a segment of trajectory. A from_horizons constructor allows obtaining a trajectory from JPL Horizons. Version 0.17 added a to_ephem method to the Orbit class to compute the trajectory at arbitrary epochs using any of several proven orbit computation algorithms.

Added here are

• a Station class to encapsulate tracking station coordinates and methods to compute topocentric (station-relative) range, range rate and radial acceleration given the spacecraft position and velocity vectors at a given epoch; and
• an OrbitFitter class that encapsulates lmfit with methods to compute a best fit trajectory (Ephem) in the neighbourhood of an initial Orbit over the orbital elements parameter space, given simulated range or range rate (Doppler) data.

Precession and nutation of the attractor (Earth), gravitational influences of other bodies including the Sun, space-time curvature and cumulative effects of the solar wind, etc. are not modelled. An effort to model these would be not only prohibitive in time and effort, but would at most serve to merely confirm the 1% fit to NEAR's ΔV and zero-noise fit to the SSN residuals already obtained with JPL Horizons data in [6]. Specialist expertise would be required that would also defeat transparency and verifiability of any further improvement or additional result.

This work was therefore meant to explore

• if NASA and ESA orbit determinations (OD) for the flybys could be reproduced more simply without specialist skills;
• whether the Doppler (range rate) residual oscillations also seen could be reproduced and experimented with;
• and if so, whether these residual oscillations also imply the light-time lags or allow other explanations.

In all cases, an initial set of position and velocity vectors is obtained using Ephem.from_horizons from the actual flyby and the official NASA/ESA trajectory, and used to compute a reference set of orbital elements. Range or range rate data are simulated over a trajectory derived from these reference orbital elements for a sample set of pre- or post-encounter epochs. Least square fit is applied to this simulated data to obtain new orbital elements and trajectory, and the final set of residuals computed during the fit are compared to those described in [1, 2].

Note

In NASA's and ESA's investigations, the term Doppler refers to measurements of the carrier frequency, whereas range rate would mean differenced range data, obtained as round trip times of modulated range codes. The computed range rate is simply referred to as the (simulated) Doppler here, with no attempt to actually simulate the corresponding shifts in carrier frequency.

Known limitations

• As stated, the code performs least-square fits of orbital elements rather than solve dynamical equations as in NASA/ESA OD. This approach would be ordinarily meaningful only within a single-attractor sphere of influence (SOI), and thus for satellite rather than deep space missions.

  NEAR's post-encounter residuals extended to 3000 Mm, 3x Earth's SOI. As other forces, including the lunar and solar gravitational influences as well as the solar wind, etc., were already accounted for in the NASA/ESA OD, the residuals represent a fraction of the tracking data that could not also be attributed to such forces in any case.

• Light time corrections are omitted in obtaining the ground station coordinates for computing range, range rate and radial acceleration. Even the 3x SOI distance considered for NEAR is only about 10 s, during which the ground station's relative position and velocity change little.

Results

The main result obtained is that the light-time lags are the necessary and sufficient explanation for the Doppler (range rate) residuals and thereby for the anomalies as a whole, as follows:

• The post-encounter residuals are found very closely reproduced by the least square fit to Doppler (range rate) data simulated with the light-time lags against the Horizons-based reference trajectory for both NEAR, in near_sim_postencounter_doppler.ipynb and for Rosetta in rosetta_sim_postencounter_rangerate.ipynb.

  Similar oscillations are indeed also obtained with range data simulated with the light-time lags, in near_sim_postencounter_range.ipynb. In particular, the fit to Doppler data with lags is shown to produce very similar residuals with range in near_sim_postencounter.ipynb.

  This establishes sufficiency of the light-time lags in the tracking signal.

• The above notebooks also demonstrate that

  • without the light-time lags, the residuals reduce to noise,
  • replacing the light-time lags with a constant lag leads to a different behaviour, and
  • multiplying the light-time lags with a constant scale factor leads to proportionally scaled oscillations.

  This proves necessity of exactly light-time lags as the only possible change in the transponded tracking signal that could lead to the precise form and magnitude of the residuals.

Furthermore, any hypothesis of an external force responsible for the ΔV, including dark matter or relativistic frame-dragging [8] or from the Casimir effect [4], would be incapable of reproducing the residual oscillations, whereas the light-time lags also explain the ΔV to 1% as mentioned.

Investigation history

Following notebooks were developed en route to this conclusion and provide additional details and insight.

Notebook	Result shown
near_deltav.ipynb	Revisits the 1% accounting of NEAR's Δv in the NAECON 2019 paper [6]. Topocentric conversion using poliastro currently achieves a 4% fit.
near_gapcheck.ipynb	Establishes that orbit propagation is too inaccurate to test JPL or ESA OD. Horizons can be used only for initial position and velocity for a reference trajectory, and the tests must be limited to reproducing the published residuals.
near_sim_ssn_residuals.ipynb	Shows that lags added to the LOS-based trajectory accurately predict the SSN residuals. But inserting a lag at LOS does not, making it a mystery how the SSN residuals could be proportional to range against a trajectory extrapolated from LOS without reference to the SSN radars.
near_ssntrack.ipynb	Cesium 3d visualization of the SSN locations and trajectory. This suggested that the SSN residuals could result from a small vertical displacement of the trajectory assuring proportionality to range as a 'similar triangles' property.
near_sim_ssn_fitlosrange_trajectory.ipynb	Validated least square fit from simulated range data with lags in the LOS-based reference trajectory as a means to find a similar trajectory. Very similar values are obtained for the SSN residuals.
near_sim_ssn_fitlosdoppler_trajectory.ipynb	Validated least square fit from simulated Doppler data with lags also as a means to find a similar trajectory. The values predicted for the SSN residuals are much different, however.
near_sim_ssn_revfit_altair.ipynb	Fit to simulated range data lags for Altair. Residuals diverge up to 5 km and 0.5 m/s.
near_sim_ssn_revfit_millstone.ipynb	Fit to simulated range data lags for Millstone. Residuals diverge up to 5 km and 0.5 m/s.
near_sim_ssn_revfit_range.ipynb	Fit to combined range data with lags simulated for the SSNs and Goldstone. The divergence is down to 150 m, finally assuring that (lag induced) error in pre-LOS trajectory sufficed to cause the SSN residuals.
near_sim_ssn_revfit_doppler.ipynb	Fit to combined Doppler data with lags simulated for the SSNs and Goldstone. The range variation is still only 200 m and almost linear. The Doppler variation is down to 50 mm/s. This exceeds the anomaly, but this is without modelling precession, solar wind, etc.
rosetta_sim_approach_rangerate.ipynb	Verifies that fit in reverse to simulated range rate with lags from a post-perigee state reproduces the range rate oscillations before and after perigee.