# Additional Solar System Gravitational Anomalies

## Abstract

**:**

## 1. Introduction

^{−11}m

^{3}·kg

^{−1}·s

^{−}

^{2}[5]. Although determinations of G are notoriously complex, its official value has 0.01 percent uncertainty, which is orders of magnitude greater than that for other fundamental constants [6].

_{⊙}, of up to about 0.5 percent and are described as a missing mass problem and/or occurring at low accelerations [7]. An example involves 1I/2017 U1 ‘Oumuamua, which is the first macroscopic object to be observed that came from outside the solar system and had an unexplained non-Newtonian acceleration of 0.1 percent away from the Sun [8], so that—according to a NASA press release (number 18-056 of 28 June 2018)—it was 40,000 km further away when it disappeared from view than if only gravitational force had been acting.

- laboratory determinations of G by 13 teams using target masses up to 5 kg;
- orbital dynamics of the planets and of the moons of Earth and Mars, which assume Keplerian motion;
- non-gravitational acceleration (NGA) of ‘Oumuamua and 70 comets.

## 2. Solar System Gravitational Anomalies

#### 2.1. Experimental Determinations of G

^{3}·kg

^{−1}·s

^{−2})

^{−11}m

^{3}·kg

^{−1}·s

^{−2}when the target mass rose from 63 to 68 g [17].

Panel A: Details of Published Studies | |||||||

Reference | Apparatus | G (×10^{−11} m^{3}⋅kg^{−1}⋅s^{−2}) | Attractor Mass (kg) | Target Mass (kg) | |||

[20] | Torsion balance | 6.67387 ± 0.00027 | 54.00 | 0.5000 | |||

[21] | Torsion pendulum, time of swing | 6.67400 ± 0.0007 | 20.98 | 0.0007 | |||

[22] | Torsion pendulum, time of swing | 6.67239 ± 0.0009 | 12.50 | 0.0032 | |||

[23] | Torsion balance | 6.6729 ± 0.0005 | 8.00 | 0.0044 | |||

[24] | Torsion pendulum, time of swing | 6.674184 ± 0.000078 | 1.56 | 0.0680 | |||

[25] | Angular acceleration feedback | 6.674484 ± 0.0000078 | 34.16 | 0.0400 | |||

[26] | Torsion pendulum, time of swing | 6.67349 ± 0.000026 | 1.56 | 0.0630 | |||

[26] | Torsion pendulum, time of swing | 6.6726 ± 0.0005 | 20.98 | 0.0070 | |||

[27] | Torsion pendulum, time of swing | 6.67433 ± 0.00013 | 117.42 | 0.1060 | |||

[28] | Laser interferometer | 6.67234 ± 0.00014 | 480.00 | 1.5600 | |||

[29] | Torsion balance | 6.67545 ± 0.00018 | 44.00 | 4.8000 | |||

[30] | Torsion balance | 6.67559 ± 0.00027 | 48.00 | 4.8000 | |||

[31] | Torsion pendulum, time of swing | 6.67349 ± 0.00018 | 1.58 | 0.0700 | |||

Panel B: Statistics for OLS Regressions Using Target Mass as the Independent Variable | |||||||

Intercept | Slope | Adj R^{2} | |||||

Value | StandardError | p-Value | Value | StandardError | p-Value | ||

OLS Regression | 6.6743 | 0.00027 | <0.0001 | 0.000204 | 0.000137 | 0.016 | 0.279 |

Bootstrapped standard errors (100,000 repetitions) | 6.6744 | 0.00031 | <0.0001 | 0.000202 | 0.000141 | 0.090 | 0.278 |

#### 2.2. Planets’ Orbital Dynamics

_{⊙}, and planets of mass, m

_{P}, which are in a stable orbit at a distance of R

_{P}, with a period T

_{P}(so orbital velocity, V

_{P}, equals 2.π.R

_{P}/T

_{P}). Assume that planets’ orbits are stable, so that centripetal and centrifugal forces are equal. Under Newton’s Law:

^{30}kg—equal 3.36 × 10

^{18}.

#### 2.3. Moons of Earth and Mars

#### 2.4. NGA of ‘Oumuamua and Comets

^{−6}m·s

^{−2}/R

^{2}, which reflects a reduction in solar gravitation of about 0.1 percent [8]. This is at the highest end of the range of NGAs observed in comets and asteroids [35] and triggered an extensive search for emissions that might explain it. However, none were observed, and ‘Oumuamua was inert [36].

^{3}[38], Figure 4 plots their NGA against estimated mass. Following the same approach as used with ‘Oumuamua:

#### 2.5. Other Possible Gravity Anomalies

## 3. Discussion

- Benchtop experiments $\widehat{\mathrm{G}}$ = {6.674 + 0.000200 × ln(m
_{target})} × 10^{−11} - Planets’ orbits $\widehat{\mathrm{G}}=\left\{6.124+0.01004\times \mathrm{ln}\left({\mathrm{m}}_{\mathrm{planet}}\right)\right\}\times {10}^{-11}$
- Martian moons $\widehat{\mathrm{G}}=\left\{6.517+0.00516\times \mathrm{ln}\left({\mathrm{m}}_{\mathrm{moon}}\right)\right\}\times {10}^{-11}$
- Earth’s Moon $\widehat{\mathrm{G}}=6.739\times {10}^{-11}$
- ‘Oumuamua $\widehat{\mathrm{G}}=6.6742\times {10}^{-11}$
- Comets $\widehat{\mathrm{G}}=\left\{\left(6.6707+0.000104\times \mathrm{ln}({\mathrm{m}}_{\mathrm{comet}}\right)\right\}\times {10}^{-11}$

_{⊙}and over separation distances ranging from a few cm to 30 AU.

^{6}kg (typical of a 20 m diameter asteroid). For solar system bodies above that size, such as Earth, Equation (12) increases gravity by up to about 0.5 percent. For smaller bodies, it will reduce gravity and result in positive (i.e., anti-Solar) NGA. This scale-dependence has numerous corollaries. For example, when Equation (12) is applied to ‘Oumuamua, which had radially outward NGA of about one-thousandth of solar gravitation, it suggests a mass of less than about 10

^{3}kg. While this is many orders of magnitude less than expected by most studies, it approximates the mass of typical manmade spacecraft and would be consistent with a light sail as proposed in reference [43].

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

^{32}, and comet data are from the JPL Small Body Database available at https://ssd.jpl.nasa.gov/sbdb_query.cgi#x (accessed on 1 August 2021).

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Planets’ heliocentric distance and orbital period vs. mass (no uncertainties were provided for data).

Panel A: Planet Characteristics | ||||||||

Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune | |

Mass (kg × 10^{24}) | 0.33 | 4.87 | 5.97 | 0.64 | 1.898 | 568 | 86.8 | 102 |

Distance from Sun (m × 10^{10}) | 5.79 | 10.82 | 14.96 | 22.79 | 77.86 | 143.35 | 287.25 | 449.51 |

Orbital period (sec × 10 ^{7}) | 0.76 | 1.94 | 3.16 | 5.94 | 37.42 | 92.85 | 264.29 | 516.67 |

Panel B: Calculated Parameters | ||||||||

Distance^{3}/Period^{2} × 10^{16} | 335.8 | 336.1 | 336.3 | 336.0 | 337.1 | 341.7 | 339.3 | 340.2 |

Moon | Phobos | Deimos | |
---|---|---|---|

Mass (kg) | 7.35 × 10^{22} | 1.06 × 10^{16} | 2.40 × 10^{15} |

Distance from Planet (m) | 3.84 × 10^{8} | 9.38 × 10^{6} | 2.35 × 10^{7} |

Orbital period (sec) | 2.36 × 10^{6} | 2.76 × 10^{4} | 1.09 × 0^{5} |

Panel B: Calculated Parameters | |||

Distance^{3}/Period^{2} × 10^{12} | 10.193 | 1.086 | 1.085 |

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Coleman, L.
Additional Solar System Gravitational Anomalies. *Symmetry* **2021**, *13*, 1696.
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Coleman L.
Additional Solar System Gravitational Anomalies. *Symmetry*. 2021; 13(9):1696.
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2021. "Additional Solar System Gravitational Anomalies" *Symmetry* 13, no. 9: 1696.
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