non-newtonian gravity in the earth’s gravity...

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Non-Newtonian gravity in the Earth’s gravity field

Joel Bergé (ONERA / Paris Saclay University) with P. Brax, M. Pernot-Borràs, J.P. Uzan

Looking for non-Newtonian gravity: Yukawa potential

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Kapner+ 2007, Wagner+ 2012, Masuda+ 2009

Joel Bergé, Rencontres de Moriond, March 27, 2019

Yukawa deviation V(r) = −GM

r(1 + αe−r/λ)

Tests of non-Newtonian gravity in space

!3 Joel Bergé, Rencontres de Moriond, March 27, 2019

From space: e.g. advance of perigee (LAGEOS) or equivalence principle (MICROSCOPE —P. Fayet’s and M. Pernot-Borras’ talks)

Possible lack of consistency: most tests either - assume Earth = point mass (ignore Earth’s shape) - or use/correct lowest shape information (extended sphere,

oblateness) from Earth models obtained under Newtonian gravity assumption


r(1 + αe−r/λ)

Tests of non-Newtonian gravity in space


From space: e.g. advance of perigee (LAGEOS) or equivalence principle (MICROSCOPE —P. Fayet’s and M. Pernot-Borras’ talks)

Possible lack of consistency: most tests either - assume Earth = point mass (ignore Earth’s shape) - or use/correct lowest shape information (extended sphere,

oblateness) from Earth models obtained under Newtonian gravity assumption

Question: how do the shape of the Earth and a Yukawa deviation affect each other?


r(1 + αe−r/λ)

Earth’s shape and gravity field


Two-way problem: invert gravity field to estimate Earth’s mass distribution vs predict gravity field from Earth’s mass distribution

GOCE (ESA)Gravity Field and Steady State Ocean Circulation Explorer, 2009-2013

GRACE (NASA-JPL): Gravity Recovery and Climate Experiment, 2002-2015

× [CNnm cos(mξ) + SN

nm sin(mξ)]

Newtonian gravity!

Spherical harmonics decomposition and Yukawa

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Gravity field of extended body

Taylor expansions

Legendre polynomials Spherical harmonics

JB, P. Brax, M. Pernot-Borras, J.P. Uzan, CQG 35 234001 (2018)


Modified Bessel functions

Spherical harmonics decomposition and Yukawa

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Finally… body’s gravity potential


ylm ∝ Clm − iSlmEasy link to usual Cnm/Snm coefficients

r-dependence!

Yukawa potential brings in a r-dependence

Earth’s shape — Yukawa twist

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Example: homogeneous oblate Earth (only y00 and y20 non-zero)

Form factor

Form factor


Estimating y20 with non-zero Yukawa deviation

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Bias and systematic error if incorrectly assuming Newtonian gravity when inverting gravity field

Yukawa contribution at different altitudes

α α

λ[m] λ[m]Joel Bergé, Rencontres de Moriond, March 27, 2019

Constraining Yukawa from y20(r)

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estimator: compare J2 at two altitudesα

Uncertainty from imperfect knowledge of Earth’s shape and measurement error

uncertainty on oblateness

unce

rtain

ty o

n y 2

0


Current uncertainties

Constraining Yukawa from y20(r): naive GOCE-GRACE comparison

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GOCE vs GRACE’s J2 => significant Yukawa deviation! Excluded for many years => underestimated errors in Earth gravity field models? Time-dependent systematics?


Conclusion

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• Source’s shape and non-Newtonian gravity twisted

• Spherical harmonic decomposition: coefficients ylm are not universal in non-Newtonian gravity, depend on distance to the centre of the source

• Non-Newtonian component implies bias and extra systematic errors when inverting the measured gravity field to reconstruct the source’s shape: still a few orders of magnitude below measurement errors in space, but significant(?) on the ground

• Possibility to test Yukawa deviation by comparing ylm at different altitudes. Ideally, compare measurements taken simultaneously.

• Imperfect knowledge of the source affects constraints on Yukawa parameters: still subdominant compared to measurement errors


non-newtonian gravity in the earth’s gravity...

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