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

!1

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

!2

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

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

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

Yukawa deviation V(r) = −GM

r(1 + αe−r/λ)

Earth’s shape and gravity field

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

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

!5

Gravity field of extended body

Taylor expansions

Legendre polynomials Spherical harmonics

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

Joel Bergé, Rencontres de Moriond, March 27, 2019

Modified Bessel functions

Spherical harmonics decomposition and Yukawa

!6

Finally… body’s gravity potential

Joel Bergé, Rencontres de Moriond, March 27, 2019

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

r-dependence!

Yukawa potential brings in a r-dependence

Earth’s shape — Yukawa twist

!7

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

Form factor

Form factor

Joel Bergé, Rencontres de Moriond, March 27, 2019

Estimating y20 with non-zero Yukawa deviation

!8

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)

!9

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

Joel Bergé, Rencontres de Moriond, March 27, 2019

Current uncertainties

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

!10

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

Joel Bergé, Rencontres de Moriond, March 27, 2019

Conclusion

!11

• 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

Joel Bergé, Rencontres de Moriond, March 27, 2019