Empirical Test of Local-Density Curvature Response Using GRACE-FO Laser Ranging Data

1. Overview

Scientific rationale and importance

The discrepancy between observed galaxy rotation curves and the gravitational acceleration predicted by visible matter remains one of the central unresolved problems in astrophysics. Stars in the outer regions of galaxies orbit significantly faster than expected under Newtonian gravity when only baryonic matter is considered. The prevailing explanation invokes large quantities of non-baryonic dark matter, although the physical nature of this component remains unknown.

The κ-framework proposes an alternative possibility in which gravitational response depends on the dynamical environment of baryonic matter. In this formulation spacetime curvature exhibits a small environmental response characterized by a scalar coefficient κ(ρ, S), where ρ represents local density and S represents the strain-rate associated with velocity gradients in the surrounding matter distribution.

In the weak-field limit the effective gravitational acceleration takes the form

$g_{\text{eff}}=\frac{GM}{r^{2}}\mathrm{e}^{\kappa r}$

where κ is not a universal constant but a parameter determined by environmental conditions.

Preliminary empirical investigations have already been conducted. Analysis of the SPARC galaxy rotation curve dataset indicates that the curvature response inferred from observations follows a consistent empirical scaling relation between baryonic acceleration and orbital velocity. Independent high-precision N-body simulations of Solar System dynamics demonstrate that the same framework produces only extremely small perturbations to planetary orbits while allowing slow secular perihelion drift.

These results suggest that an environmental curvature response may provide a phenomenological description capable of reproducing both galactic and planetary gravitational behaviour without introducing additional matter components.

The κ-framework also produces a concrete near-Earth prediction. At altitudes of approximately 500 km the model predicts a small radial acceleration residual in satellite motion. This proposal seeks to test that prediction directly using publicly available GRACE-FO satellite laser ranging data.

Why this test is timely

Modern precise orbit determination (POD) pipelines model gravitational and non-gravitational forces with extremely high accuracy. However, several characteristics of standard orbit reconstruction create an opportunity for targeted residual analysis:

small systematic accelerations can be absorbed into empirical parameters
radial components may be partially degenerate with atmospheric drag modelling
standard residual analysis is not typically designed to test specific directional theoretical predictions

This project therefore performs a targeted analysis of GRACE-FO orbital residuals to determine whether a systematic radial acceleration component remains after full force modelling.

Because GRACE-FO laser ranging data is publicly available and modern orbit determination software is mature, this experiment can be performed rapidly and reproducibly.

Funding, facilities, and ethics

Data sources
Public ILRS and GRACE-FO datasets (via CDDIS)
Facilities
Cloud computing resources (AWS / GCP)
Location
Bank House, Cornwall (remote research)
Ethics
Not applicable (no human or animal subjects)

Timeline (6 months)

Month 1-2
Finalize theoretical framework and analysis pipeline
Month 3-4
Extract satellite residuals and perform statistical analysis
Month 5
Validation tests and robustness checks
Month 6
Paper preparation, arXiv submission, and open release of analysis pipeline

Open science commitment

All outputs will be released openly under an MIT license, including:

analysis code and Jupyter notebooks
processed residual datasets
reproducible analysis pipeline
links to raw ILRS ephemerides

2. Introduction

The ΛCDM cosmological model explains large-scale structure formation through the presence of non-baryonic dark matter comprising approximately 85 % of total matter. Although this model successfully reproduces many cosmological observations, the underlying particle nature of dark matter remains unknown.

Modified gravity approaches attempt to reproduce these phenomena without introducing unseen matter. However, many proposed models face challenges including:

tension with Solar System precision tests
reliance on multiple tuneable parameters
absence of falsifiable predictions in near-field environments

The κ-framework explores a different possibility in which gravitational response depends on environmental conditions.

In this formulation

$\kappa(\rho,r) = \kappa_0 + k_v \left( \frac{S}{10^{-12}\,\mathrm{s^{-1}}} \right)^3 \left( \frac{\rho}{\rho_0} \right)^{1/2}$

where:

ρ represents local baryonic density
S represents the strain-rate associated with velocity gradients
s₀ and ρ₀ are reference scales

This structure introduces sensitivity to both density and dynamical shear.

In high-density environments such as planetary interiors the κ-response is strongly suppressed. In extremely diffuse environments such as interplanetary space density is negligible. However, the terrestrial thermosphere represents an intermediate regime in which density remains non-negligible while wave-driven flows and geomagnetic disturbances can generate measurable strain.

Low-Earth orbit therefore provides a natural experimental environment in which the environmental curvature response may become detectable while remaining consistent with existing Solar System constraints.

Key research question

Does a κ-induced radial acceleration signal appear in GRACE-FO satellite orbit residuals after full force modelling?

Hypotheses

H₁ (signal present)
Residuals exhibit a statistically significant radial acceleration bias consistent with κ-framework predictions.
H₀ (null hypothesis)
Residuals remain consistent with zero within measurement uncertainty, establishing an upper bound on the κ-response.

Scientific impact

Positive detection
Would represent the first empirical indication that gravitational response depends on environmental conditions in near-Earth orbit.
Null detection
Would provide the strongest observational constraint to date on environment-dependent curvature response models.

In either case the results will advance understanding of possible extensions to classical gravitational dynamics. The project tests a single quantitative prediction of an environmentally modulated gravitational framework using existing satellite laser ranging data.

The experiment is:

low cost
rapidly executable
fully open and reproducible
decisively falsifiable

By combining theoretical modelling, galaxy rotation curve analysis, Solar System dynamical tests, and satellite orbit residual analysis, the project seeks to determine whether environmental curvature response plays a measurable role in gravitational dynamics.

A positive result would provide the first empirical evidence for environment-dependent curvature response in gravitational physics. A null result would establish strong observational constraints on such models.

Either outcome contributes directly to the broader question of whether gravitational dynamics alone — rather than unseen matter — explain the observed structure of the universe.

Preliminary Results

Galaxy rotation curve analysis
https://github.com/hasjack/OnGravity/tree/feature/rotation-curve-analysis/python/rotation-curves

Analysis of the SPARC dataset (175 disk galaxies) shows that the curvature response inferred from observations follows an empirical relation between baryonic acceleration and orbital velocity.

The inferred scaling

$\frac{\kappa r}{2} = a + b \log_{10}\!\left(g_{\text{bar}}\right)$

produces an effective acceleration relation

$g_{\mathrm{obs}} \propto g_{\mathrm{bar}}^{\gamma}$

with γ ≈ 0.69.

This scaling closely reproduces the observed baryonic acceleration relation found in galaxy rotation curves.

Figure 1. Empirical curvature-response parameter inferred from the SPARC galaxy rotation curve dataset as a function of baryonic gravitational acceleration. The distribution exhibits a clear scaling relation between κ and baryonic acceleration, consistent with the κ-framework prediction that curvature response depends on the local dynamical environment of baryonic matter.

Figure 2. Distribution of the inferred curvature-response parameter as a function of normalized galactic radius for the SPARC galaxy sample. The scaling behaviour indicates that κ evolves across galactic disks in a manner consistent with environmental dependence.

Solar System dynamical tests
https://github.com/hasjack/OnGravity/tree/feature/solar-system-model/python/solar-system

Independent N-body simulations of Solar System dynamics were performed using the REBOUND integrator with the high-precision IAS15 algorithm.

Simulations of Mercury, Earth, and Jupiter demonstrate that:

planetary orbits remain dynamically stable
deviations from Newtonian trajectories are extremely small
slow secular perihelion drift emerges without destabilizing orbital structure

These results indicate that the κ-framework remains compatible with weak-field gravitational behaviour within the Solar System.

Figure 3. Example orbital trajectories from N-body simulations of the Solar System using the REBOUND IAS15 integrator. The κ-framework preserves orbital stability while producing extremely small deviations from Newtonian trajectories.

Figure 4. Perihelion drift diagnostic derived from the Laplace-Runge-Lenz vector in Solar System simulations. The κ-framework introduces a slow secular rotation of planetary perihelia while maintaining stable orbital structure.

3. Methods

Force model reconstruction

The following standard orbit determination components will be implemented:

Earth geopotential: EGM2008 (degree/order ≥120)
Atmospheric drag: NRLMSISE-00 model
Solar radiation pressure
Earth radiation pressure
Third-body perturbations from Sun and Moon

Residual accelerations will be computed after subtracting the full force model.

Data sample

Source: ILRS precise orbit ephemerides for GRACE-FO
Altitude: ~500 km
Inclination: ~89°
Time span: 2023-2025
Selection: ≥3-day continuous orbital arcs

Only quality-filtered ILRS data will be included.

Residual isolation strategy

To mitigate known degeneracies:

analyse radial acceleration component explicitly
compare results across independent orbital arcs
examine dependence on environmental conditions

Sensitivity tests will include:

solar activity variation
geomagnetic conditions
atmospheric density fluctuations

Planned analysis

Outlier rejection: >3σ
Smoothing: 30-second Gaussian kernel
Binning: time-averaged radial residuals

Statistical testing

Primary test: one-sample t-test against predicted acceleration
Significance threshold: p < 0.01
Statistical power: >80 % for a 3-day arc (validated via Monte-Carlo simulation)

Decision criteria

H₁ supported
Consistent positive radial bias detected with p < 0.01 across independent datasets.
H₀ supported
Residuals consistent with zero within uncertainty, establishing upper bounds on κ.

4. Pilot Data

All code, datasets, and analysis pipelines will be released publicly on GitHub immediately upon publication. The project is designed to be fully reproducible using publicly accessible satellite datasets and open-source computational tools.

5. Budget

Research stipend (6 months): $28,000
Compute resources (Monte-Carlo simulations and orbit analysis): $5,000
Open-source tooling and data hosting: $2,000
Contingency: $1,000

Total Budget: $36,000

6. References

Rubin, V. C., Ford, W. K., & Thonnard, N. (1980)
Rotational properties of 21 Sc galaxies with a large range of luminosities and radii
https://ui.adsabs.harvard.edu/abs/1980ApJ…238..471R/abstract

Sofue, Y., & Rubin, V. (2001)
Rotation Curves of Spiral Galaxies
https://arxiv.org/abs/astro-ph/0010594

Lelli, F., McGaugh, S. S., & Schombert, J. M. (2016)
SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves
https://arxiv.org/abs/1606.09251

McGaugh, S. S., Lelli, F., & Schombert, J. M. (2016)
The Radial Acceleration Relation in Rotationally Supported Galaxies
https://arxiv.org/abs/1609.05917

Navarro, J., Frenk, C., & White, S. (1997)
A Universal Density Profile from Hierarchical Clustering
https://ui.adsabs.harvard.edu/abs/1997ApJ…490..493N/abstract

Planck Collaboration (2020)
Planck 2018 results. VI. Cosmological parameters
https://arxiv.org/abs/1807.06209

Milgrom, M. (1983)
A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis
https://ui.adsabs.harvard.edu/abs/1983ApJ…270..365M/abstract

Famaey, B., & McGaugh, S. (2012)
Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions
https://arxiv.org/abs/1112.3960

Jacobson, T. (1995)
Thermodynamics of spacetime: The Einstein equation of state
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.75.1260

Verlinde, E. (2011)
On the Origin of Gravity and the Laws of Newton
https://arxiv.org/abs/1001.0785

Rein, H., & Liu, S.-F. (2012)
REBOUND: an open-source multi-purpose N-body code for collisional dynamics
https://arxiv.org/abs/1110.4876

Rein, H., & Spiegel, D. S. (2015)
IAS15: a fast, adaptive, high-order integrator for gravitational dynamics
https://arxiv.org/abs/1409.4779

Tapley, B. D., Watkins, M., Flechtner, F., et al. (2019)
Contributions of GRACE to understanding climate change
https://www.nature.com/articles/s41586-019-0930-y

Pearlman, M. R., Degnan, J. J., & Bosworth, J. M. (2002)
The International Laser Ranging Service
https://doi.org/10.1029/2000RG000094

Pickett, J. (2026)
On Gravity
https://doi.org/10.55277/researchhub.zwegi9m9.2

Pickett, J. (2026)
An Environmental Curvature Response for Galaxy Rotation Curves: Empirical Tests of the κ-Framework using the SPARC Dataset
https://doi.org/10.55277/researchhub.53yst6oa.1

Pickett, J. (2026)
Environmental Curvature Response in Planetary Dynamics: Solar System Diagnostics of the κ-Framework
https://doi.org/10.55277/researchhub.05wfwlfo.1

Empirical Test of Local-Density Curvature Response Using GRACE-FO Laser Ranging Data

Empirical Test of Local-Density Curvature Response Using GRACE-FO Laser Ranging Data

Funding Progress

Nonprofit Recipient

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