# Observational tests in scale invariance III: wide binary stars

André Maeder  Frédéric Courbin

###### Abstract

Recent studies of wide binary stars based on Gaia DR3 suggest that the
relative orbital velocities of objects with separations $s>$ 3’000
astronomical units are statistically larger than the standard Newtonian
predictions. Obviously there is no Dark Matter halo arround binary stars that
could be invoked to explain these high velocities. However, we explore the
properties of two-body systems in the framework of scale invariant vacuum
theory, focusing on the case of objects with extreme separations. In this
regime, the additional acceleration term present in the modified Newton
equation with scale invariance becomes important, and may even dominate the
dynamical evolution at very low gravities. Comparisons with Gaia DR3
observations of wide binaries are performed and suggest that binaries with
separations $s>$ 3’000 astronomical units have experienced such an evolution
for a few Gyr, accounting well for the observed velocity excesses.

##  1 Introduction

The scale invariant vacuum (SIV) theory rests on a fundamental and simple idea
that extends the group of invariances subtending gravitation theory, as
suggested by Dirac [9]. In fact, General Relativity (GR) and Maxwell equations
are indeed invariant in the empty space without charges and currents. The
presence of matter in the Universe, by fixing some scales, tends to kill scale
invariance. Cosmological models have been derived from a general scale
invariant equation confirmed by an action principle [5, 31]. They show that
scale invariant effects fully disappear only above the mean critical density
$\varrho_{\mathrm{c}}=3H^{2}_{0}/8\pi G$ [22]. For typical Universe models
with $\Omega_{\mathrm{m}}=0.2$ or 0.3, the effects are very much reduced,
being negligible in current life. However they do play a significant role in
at least two cases: 1- very low density regions, and 2- dynamical phenomena
occruring at very long timescales.

In paper I of this series [26], the dynamical properties of galaxies in the
scale invariant theory were studied. Among the many results, it was found that
scale invariant dynamics lead to a large reduction of the mass estimates of
galaxy clusters. Account could be given to the relatively flat rotation curves
of galaxies, to the steeper rotation curves observed in redshifted galaxies at
$z=2$ [34], to the omnipresent relation between the apparent amount of dark
and baryonic matters, and also to effects in the ”vertical” velocity
dispersion in the Milky Way. Paper II by Maeder and Courbin [28] was devoted
to strong gravitational lensing. Using the exterior Schwarzschild solution, we
derived the second order geodesic equations and showed that scale invariance
has no significant effects on gravitational lensing. However, we proceeded to
a detailed studied of the system JWST-ER1 [38] and showed that, while the mass
estimates by lensing are totally free of scale invariant effects (the
geodesics are the same as in standard GR), it is not the case for the
spectrophotometric mass determinations, which in standard theory are severely
biased towards too low values. An analysis of the large Sloan Lens ACS (SLACS)
sample of 100 galaxies [1] was further supporting this view.

In the present study, we consider the case of the very wide binaries with
separations $s\sim 10^{4}$ au, noting that such a separation corresponds to an
orbital period of 816’500 yr for a total binary mass of 1.5 M⊙. In such wide-
separation systems, the effective gravity nearly vanishes, becoming as low as
in the outer regions of galaxies so that similar dynamical effects may
possibly be expected. These questions have been boosted by the extraordinarily
accurate results of the Gaia satellite, delivered first in the Gaia eDR3 and
more recently by Gaia DR3, so that since about a decade, a number of most
interesting and detailed investigations have been performed. These are
reviewed in Section 2 below. Section 3 presents the properties of orbital
motions in the scale invariant theory. Section 4 studies the loose and
detaching systems. Section 5 compares observations to the theoretical
predictions. Finally, Section 6 summarizes our conclusions.

##  2 Current status of observational results on very wide binaries

The relative independence of the orbital velocity from the orbital radius
(e.g. the relatively flat rotation curves of galaxies) has generally been
interpreted in term of dark matter, while a minority of voices [32, 24]
suggested possible gravity effects. It makes then sense to envisage that such
effect should be present in systems other than galaxies, and subject to very
low gravitational fiels. We examine here some important recent studies on the
dynamical properties of wide binary stars, where possible deviations from the
Newtonian predictions have been investigated. As is often the case with
studies of new physical effects, at the limit of the experimental or
observational possibilities, this is a domain that is very rich in different
and sometimes controversial views.

Using a sample of 280 wide binaries observed by the Hipparchos satellite
Hernandez [14] showed that relative velocities larger than Newtonian
expectations were measured, independently of their separation. This result was
supported by radial velocity observations [37]. Pittordis and Sutherland [35]
showed the possibility that future observations may solve the question and
pointed out that some MOND generalizations do not seem favourable. Banik and
Kroupa [3] considered the case of Proxima Centauri around A and B Centauri and
showed that a distinction between Newtonian and MOND predictions should become
possible in the future. A sample of 929 carefully selected binary pairs from
the Gaia Early Data Release 3 eDR3 [13] was studied with stringent quality
requirements by Hernandez [16], who found a Newtonian behaviour of the orbital
velocities for gravities above about $a_{0}\geq 1.2\times 10^{-10}$ m s-2,
while below this value the velocities remain constant, independently of the
orbital radius. This value $a_{0}$ is the limit below which the so-called deep
MOND limit, originally derived from galactic rotations, applies with a gravity
given by $g=\sqrt{g_{\mathrm{N}}\,a_{0}}$, i.e. a stronger gravity than the
Newtonian value $g_{\mathrm{N}}$ [32, 33].

The case of undetected tertiaries is always a major concern in such studies.
Clarke [8] considered that a fraction of 50% of tertiaries could account for
the results obtained by [15]. Indeed, many of the significant tertiaries could
be eliminated [16] based on criteria drawn from simulations of multiple
systems by [4]. These authors showed that a RUWE value smaller than 1.4
(Renormalized Unit Weight Error, a value below 1.4 indicates a good
astrometric solution), a high S/N-ratio together with HR diagram selection
criteria maintain the ratio of tertiaries below 5%. A still more extended and
more accurate study of the internal kinematics of wide binaries was published
by Hernandez [17] on the basis of the enhanced DR3 Gaia catalogue with 800’000
binary candidates. In the successive series of severe data quality cuts,
Hernandez used the binary probabilty $B_{p}$ for each star based on spectral,
photometric and astrometric information, the RUWE index, the radial
velocities, the isolation criteria, the Gaia FLAME work package for mass
estimate, etc. Most stringent quality criteria let a set of 450 top-quality
isolated binaries. On this basis, Hernandez [17] confirmed his earlier
results.

Pittordis and Sutherland [36] examined the Newtonian and MOND predictions on
the basis of a large sample of 73’159 wide binaries from the Gaia eDR3 objects
within a distance of 300 pc and with a G magnitude brighter than 17. These
authors critically noticed that, although tight constraints on the sampling
are performed by Hernandez et al., their surviving sample remained very small,
offering only a poor statistical basis. With masses determined by a main-
sequence mass-luminosity relation, the analysis of Pittordis and Sutherland is
based on the histograms of the frequency distributions as a function the
ratios $\upsilon_{p}/\upsilon_{c}(r_{p})$, where $\upsilon_{p}$ is the
pairwise relative projected velocity and where
$\upsilon_{c}(r_{p})=\sqrt{GM_{tot}/r_{p}}$ is the estimated circular orbit
velocity at the current projected separation $r_{p}$. The histograms showed a
peak close to the Newtonian predictions, with in addition a long tail likely
produced by unseen companions. The observed distributions were then fitted by
simulated mixtures of binaries, triple and flyby systems with the Newtonian
and MOND theoretical predictions. They noted, with some reserve, that standard
gravity is preferred.

Chae [6] developed a Monte Carlo method to deproject the observed 2D motions
into the 3D relative space, in order to provide results in the acceleration
plane. In addition, since it is inevitable that some binaries hide close inner
companions, Chae proceeded to a calibration of the fraction
$f_{\mathrm{multi}}$ of multiples within wide binaries. With
$f_{\mathrm{multi}}$ taken as a free parameter, the determinations were
iterated until the deprojected data at high acceleration, i.e. larger than
$>10^{-10}$ m s-2, agreed with Newtonian expectations. A value
$f_{\mathrm{multi}}$ in the range 0.3 to 0.5 was obtained, in agreement with
current literature and then applied to the whole sample. The comparison of the
kinematic acceleration $g=\upsilon^{2}/r$ and the Newtonian gravity
$g_{\mathrm{N}}=GM_{\mathrm{tot}}/r^{2}$ showed a ratio
$g/g_{\mathrm{N}}=1.43$ at $g_{\mathrm{N}}=10^{-10.15}$ m s-2, in agreement
with AQUAL, one of the many generalisations of MOND. Regarding other works,
Chae [6] remarked that Hernandez [17] found deviations from Newtonian
predictions, which are increasing with separation $s$, and would thus be
inconsistent with MOND. Chae [6] also pointed out that it is difficult to test
a gravity theory with a sample displaying a narrow low-acceleration range as
used by Pittordis and Sutherland [36].

Tests of the MOND dynamics on the basis of wide binaries have also been
performed by Banik, Pittordis, Sutherland et al. [2]. They showed that,
locally, the orbital velocities predicted by this dynamics should exceed the
Newtonian prediction by about 20% at large separations (as the external
galactic field limiting the effect). Using 8’611 wide binaries from Gaia DR3
within 250 pc of the Sun, selected according to a set of criteria on the
consistency of the two parallax components, on the angular separations, on the
projected velocity differences and on a number of other refined quality cuts,
they proceeded to a detailed statistical analysis integrating the orbits,
accounting for line-of-sight contamination and undetected close companions.
According to these authors, the uncertainties of the relative velocity between
stars have not been considered in the work by Chae [6]. They also proceeded to
a modeling of the orbits in the Newton and MOND approximations. Their
statistical analysis based on frequency histograms in the line of the above
mentioned work by Pittordis and Sutherland lead them to conclude to a strong
preference for Newtonian gravity, although their best model does not fully
reproduce the observations. They finally suggested that further modifications
to MOND, introducing a new fundamental scale beyond $a_{0}$, could possibly
preserve its galactic successes. We are personnally rather reluctant to accept
further modifications of a theory each time there is an inconsistency with
observations. In this context, we may remark that the scale invariant theory,
which rests on a single and simple clear hypothesis, has no internal
adjustment parameter, no ad’hoc tweaks.

A new complementary study was made by Chae [7], now considering the catalogue
by El-Badry et al. [12] that provides a probability, $\Re$, of chance-
alignment. Chae applied a very strong requirement that $\Re<0.01$
corresponding to ”pure binaries”. He also applied cuts based on RUWE, proper
motions, radial velocities, mass calibrations. Chae pointed out that the
estimate of multiple systems $f_{\mathrm{multi}}$ by Banik et al. [2] was not
calibrated on high acceleration binaries and that these authors found a value
$f_{\mathrm{multi}}=0.70$ that was incompatible with the range currently
obtained between 0.3 and 0.5 by many authors. He also noticed that the
analysis of [2] was compounded by the inclusion of chance-alignment cases.
Testing the dynamical gravity $\upsilon^{2}/r$, Chae [7] obtained a ratio
$g/g_{\mathrm{N}}=1.49$ at $g_{\mathrm{N}}=10^{-10}$ ms-2. As a main result,
the distributions of the sky-projected velocities as a function of the sky-
projected separation $s$ for the so-called pure binaries were also presented.
The deviations from Newton’s starts around $\log s=3.3$ (in au) increasing up
to $\log s=3.8$, keeping then an apparently constant deviation by about 0.1
dex larger. This is shown in Fig. 1, which offers a very interesting
possibility of comparison with theoretical expectations. The effects of
changes in the selection parameters of the sample were examined, from which
the authors concluded that their results were robust.

The above observational situation is well representative of scientific
developments at the frontier of knowledge, with uncertain and often
contradictory conclusions and critical debates. It also reflects the usual
trade-off that needs to be made between precision and accuracy: some
scientists prefer large data samples with high statistical precision but loose
quality cuts while others give preference to smaller samples with stringent
quality requirements leading to high accuracy/reliability but with limited
statistical power. The above discussion is not any exception to the rule, but
our goal is not to add to the debate. We rather focus on presenting a new
interpretation of the data on wide binary stars in the framework of scale
invariant theory, which contains no adjustable parameter and which holds
without any need to invoke a plethora of ad’hoc variations and fixes when
confronted to new observational tests. In fact, all the developments in the
present work can be seen as theoretical predictions independent of data. It
just happens that these predictions match very well the current data on wide
binary stars and given their current precision and accuracy.

##  3 Properties of orbital motions in the scale invariant theory

The basic assumptions of the scale invariant theory and the related
demonstrations and equations have been presented in previous works of this
series, in particular in Maeder [22]. We also recommend the study by Maeder
and Gueorguiev [31] where the fundamental equations for cosmology and weak
fields have been demonstrated from an action principle. While the cosmological
acceleration is important, the current effects in weak fields in the Newtonian
approximation are generally negligible, except for very low densities and long
evolutionary timescales. The modified Newton equation of motion in these weak
fields, that we use here as a starting point, is [27, 25],

$\frac{d^{2}\bf{r}}{d\tau^{2}}\,=\,-\frac{G\,M(\tau)}{r^{2}}\,\frac{\bf{r}}{r}+\frac{\psi(\tau)}{\tau_{0}}\frac{d\bf{r}}{d\tau}\,\,.$
(3.1)

It is expressed in the current time units (seconds, years). In this
expression, $\tau_{0}$ is the age of the Universe, for which we take a value
$\tau_{0}=13.8$ Gyr. There, $\psi(\tau)$ is a numerical factor,

$\psi(\tau)\,=\,\frac{t_{0}-t_{\mathrm{in}}}{[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})]},\quad\quad\mathrm{with}\;\;t_{0}=1,\;t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}},\quad\mathrm{thus}\;\;\psi_{0}=\psi(\tau_{0})\,=1-\Omega^{1/3}_{\mathrm{m}}\,.$
(3.2)

$\Omega_{\mathrm{m}}$ is the usual density parameter and $t$ is a time
parameter appropriate for cosmological models (see [22]), varying between
$t_{\mathrm{in}}$ at the Big-Bang and $t_{0}=1$ at present. The relation
between $t$ and $\tau$ is defined by,

$\frac{\tau-\tau_{\mathrm{in}}}{\tau_{0}-\tau_{\mathrm{in}}}=\frac{t-t_{\mathrm{in}}}{t_{0}-t_{\mathrm{in}}}\,,$
(3.3)

expressing that the age fractions with respect to the present age are
evidently the same in both timescales. This leads to
$t=[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})]$. The
function $\psi(\tau)$ is dimensionless and varies between
$(1-t_{\mathrm{in}})$ at the present time, $\tau_{0}$, and
$(1-t_{\mathrm{in}})/t_{\mathrm{in}}$ at the origin, i.e. when $\tau=0$. We
verify that the time derivative of $\psi(\tau)$ is,

$\dot{\psi}=-\frac{\psi^{2}}{\tau_{0}}.$ (3.4)

In scale invariant theory, the gravitational potential $\Phi=GM/r$ appears as
a more fundamental physical quantity than the mass, as the potential is scale
invariant while the mass is not. As is the case in Special Relativity the mass
may vary. The mass then behaves like,

$M\rightarrow
M_{0}\times\frac{1}{\lambda},\quad\mathrm{with}\;\lambda=\frac{1}{t},\quad\;\mathrm{giving}\;\;M(\tau)=M(\tau_{0})\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})],$
(3.5)

a property already recognized by Dirac [9, 10] and Canuto et al. [5]. However,
while the mass may vary to infinity in Special Relativity, here the variations
of the inertia are very limited, typically of the order of the percent over
the last 400 Myr. Such a behaviour is also required by the action principle in
the scale invariant context [31].

###  3.1 Binet equation, angular mometum conservation and orbital motions

In previous works [27, 25], some properties of motion were examined in a
system with the dimensionless time, $t$, used in cosmological equations and
varying between $t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}}$ at the Big-Bang
and the present time, $t_{0}$. For a cosmology with
$\Omega_{\mathrm{m}}=0.30$, time $t$ would only vary between
$t_{\mathrm{in}}=0.6694$ and $t_{0}=1$. While the cosmological equations are
quite simple in this very constrained system of units, the usual time units
$\tau$, in years, are more adapted for observational applications. In this
case, time varies bewteen $\tau=0$ and $\tau_{0}=13.8$ Gyr. Although the
physics is exactly the same, this brings some changes in the formal writing of
the equations. Starting from the equation of motion in currrent units,
described in Eq. (3.1), we may write the corresponding two equations in plane
polar coordinates $(r,\vartheta)$, which leads to

$\displaystyle\ddot{r}-r\dot{\vartheta}^{2}$ $\displaystyle=$
$\displaystyle-\frac{GM}{r^{2}}+\frac{\psi}{\tau_{0}}\frac{dr}{d\tau},$ (3.6)
$\displaystyle r\ddot{\vartheta}+2\dot{r}\dot{\vartheta}$ $\displaystyle=$
$\displaystyle\frac{\psi}{\tau_{0}}r\dot{\vartheta}.$ (3.7)

When not specified, the quantity $\psi$ is $\psi(\tau)$ and $M$ is $M(\tau)$.
We easily verify that the 2nd of the above equations has for an integral,

$\frac{\psi}{\tau_{0}}\,r^{2}\dot{\vartheta}=\mathcal{L},$ (3.8)

where $\mathcal{L}$ is a constant expressed in $[{\rm cm^{2}\,rad\,s^{-2}}]$.
The above equation is a generalisation of the classical angular momentum
conservation $r^{2}\dot{\vartheta}={\rm const.}$ It shows that the classical
angular momentum, $r^{2}\dot{\vartheta}$, slowly increases with time
following,

$r^{2}\dot{\vartheta}=\frac{\tau_{0}\,\mathcal{L}}{1-t_{\mathrm{in}}}\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})],$
(3.9)

a behaviour quite consistent with the evolution of radius and orbital velocity
(see Sect. 4.1). Our aim now is to express the contents of the three above
equations into one single equation expressing the orbital motion of a test
particle near a massive body of mass $M$. Using Eq.(3.8) the derivative of $r$
with respect to $\vartheta$ can be written,

$r^{\prime}\equiv\frac{dr}{d\vartheta}=\frac{dr}{d\tau}\,\frac{d\tau}{d\vartheta}=\frac{\dot{r}\;r^{2}\psi}{\mathcal{L}\;\tau_{0}}\quad\mathrm{thus}\;\;\dot{r}=\frac{r^{\prime}\mathcal{L}\,\tau_{0}}{r^{2}\,\psi}\,.$
(3.10)

This kind of relation between $r^{\prime}$ and $\dot{r}$ can be extended to
$\ddot{r}$,

$\ddot{r}=\frac{\mathcal{L}^{2}\,r^{\prime\prime}\tau^{2}_{0}}{r^{4}\,\psi^{2}}-\frac{2r^{\prime
2}\mathcal{L}^{2}\tau^{2}_{0}}{r^{5}\psi^{2}}+\frac{r^{\prime}\mathcal{L}}{r^{2}}\,,$
(3.11)

thanks to (3.4). Now, we express the first equation of (3.7) with this last
expression and get,

$\frac{\mathcal{L}^{2}\,\tau^{2}_{0}}{r^{4}\,\psi^{2}}\left(r^{\prime\prime}-2\frac{r^{\prime
2}}{r}\right)-\frac{\mathcal{L}^{2}\,\tau^{2}_{0}}{r^{3}\,\psi^{2}}+\frac{G\,M}{r^{2}}=0\,,$
(3.12)

where remarkably and fortunately the last terms in Eqs. (3.1) and (3.11) have
simplified. Now, writing this last equation with $u=1/r$ is leading to,

$u^{\prime\prime}+u=\frac{GM\,\psi^{2}}{\mathcal{L}^{2}\,\tau^{2}_{0}}\,.$
(3.13)

This is the modified form of the well-known Binet equation, describing
planetary trajectories. The classical form does not contain the term
$\psi^{2}/\tau^{2}_{0}$. If the 2nd member is equal to $0$, the solution is
$u=C\cos\vartheta$, the polar equation of a straight line. Now, we search a
solution of the form $u=p+C\cos\vartheta$ and get

$p=\frac{GM\,\psi^{2}}{\mathcal{L}^{2}\,\tau^{2}_{0}},$ (3.14)

expressed in $[{\rm cm}^{-1}]$. This quantity, $p$, is often called the _latus
rectum_ in the geometrical construction of a conic. $C$ is also a constant in
$[{\rm cm}^{-1}]$. Thus, the solution $r(\vartheta)$ is

$r=\frac{1}{\frac{GM\,\psi^{2}}{\mathcal{L}^{2}\,\tau^{2}_{0}}+C\cos\vartheta}\,\quad\mathrm{or}\;\;r=\frac{r_{0}}{1+e\cos\vartheta},\quad\quad\mathrm{with}\;\;e=\frac{\,C\mathcal{L}^{2}\,\tau^{2}_{0}}{GM\,\psi^{2}}\,,$
(3.15)

where $e$ is the eccentricity essentially determined by the initial
conditions. It is dimensionless and is therefore a scale invariant quantity.
We see that if the eccentricity is zero, then $r_{0}=1/p$,

$r_{0}\,=\,\frac{\mathcal{L}^{2}\,\tau^{2}_{0}}{GM\,\psi^{2}}\,,$ (3.16)

$r_{0}$ being the radius of the circular orbit obtained when $e=0$. We verify
that $r_{0}$ is expressed in $[{\rm cm}]$ and behaves like $1/\lambda$,
implying that it is increasing with time (see next subsection). For elliptical
solutions, with $e<1$, the semi-major and semi-minor axes respectively are
$a=\frac{r_{0}}{(1-e^{2})}$ and $b=\frac{r_{0}}{\sqrt{1-e^{2}}}$, which behave
in time like $r_{0}$. Thus, the solutions are not classical conics with
constant parameters (except for $e$). Instead, as the result of the time
dependence of some quantities like $r_{0}$, they are _expanding conics_ as
originally demonstrated by [27].

###  3.2 Secular properties of the expanding conics and Third Kepler’s Law

Let us examine the time evolution of $r_{0}$, which determines the properties
of the elliptical orbits, noting that similar relations would apply for
parabolic and hyperbolic motions. From (3.16), one has for the relatives
variations of the semi-major axis $a$,

$\frac{\dot{a}}{a}=-\frac{\dot{M}}{M}-2\frac{\dot{\psi}}{\psi}\,.$ (3.17)

From Eq. (3.5), we have

$\dot{M}=\frac{M(\tau_{0})}{\tau_{0}}(1-t_{\mathrm{in}}),\quad\quad\mathrm{and}\;\;\frac{\dot{M}}{M}=\frac{\psi}{\tau_{0}}\,,$
(3.18)

so that with (3.4)

$\frac{\dot{a}}{a}=-\frac{\psi}{\tau_{0}}+2\frac{\psi}{\tau_{0}}=\frac{\psi}{\tau_{0}}\,.$
(3.19)

This implies that the semi-major axis $a(\tau)$ of the elliptical motions
behaves like,

$a(\tau)=a(\tau_{0})\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})].$
(3.20)

The semi-major axis is therefore increasing linearly in time $\tau$, in a way
which is perfectly consistent with the variation of $a(t)=a(t_{0})\times t$
previously obtained in the $t$-scale [27]. Due to the relations recalled above
between the dimensional parameters $a$, $b$ and $r_{0}$, the above dependence
on time $\tau$ also applies to all dimensional quantities.

We now turn to the orbital period $T$, which for simplicity is considered in
the circular case. With the angular velocity $\dot{\vartheta}=2\pi/T$, (for an
elliptical orbit $\dot{\vartheta}$ would be the mean angular velocity) and the
angular momentum conservation
$\dot{\vartheta}=\frac{\mathcal{L}\tau_{0}}{r^{2}\psi}$, one is led to

$\frac{\dot{\vartheta}}{\vartheta}=-2\frac{\dot{r}}{r}-\frac{\dot{\psi}}{\psi}=-2\frac{\psi}{\tau_{0}}+\frac{\psi}{\tau_{0}}=-\frac{\psi}{\tau_{0}},\quad\quad\quad\mathrm{thus}\;\;\frac{\dot{T}}{T}=-\frac{\dot{\vartheta}}{\vartheta}=\frac{\psi}{\tau_{0}}.$
(3.21)

This implies that the period $T$ is varying like,

$T(\tau)=T(\tau_{0})\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]\,,$
(3.22)

also increasing linearly in time $\tau$. Thus, we may note that the mass $M$,
the dimensional parameters $a$, $b$, $r_{0}$ and the orbital period $T$ all
have the same kind of time dependence.

We may wonder what is happening to the Third Kepler’s Law. From the
conservation law (3.8) and for simplicity still considering a test particle
with circular orbits with radius given by (3.16) we have,

$\displaystyle\mathcal{L}=\frac{\psi}{\tau_{0}}\,r^{2}_{0}\frac{2\pi}{T},\quad\mathrm{and}\;\;r_{0}=\frac{\mathcal{L}^{2}\,\tau^{2}_{0}}{GM\,\psi^{2}}$
$\displaystyle=$ $\displaystyle r^{4}_{0}\frac{4\pi^{2}}{T^{2}\,GM},$ (3.23)
$\displaystyle\mathrm{we\;get}\quad\quad\frac{r^{3}_{0}}{T^{2}}$
$\displaystyle=$ $\displaystyle\frac{GM}{4\pi^{2}}\,,$ (3.24)

which corresponds to the Third Kepler’s Law for the case considered. This
applies at any time $\tau$ since all the dependences in the form of the above
$T(\tau)$ cancel each other. Remarkably, the Third Kepler’s Law which
originally has led to the Newton Law is still valid in the scale invariant
theory, even though the Newton Law no longer applies as such. Thus, the Third
Kepler Law is a most fundamental property of gravitation, a Rock in the theory
of gravitation, independent of whether gravitation is scale invariant or not.
This is the law which allows the determination of planetary masses in the
Solar System and it remains unmodified by scale invariance, providing the
correct masses at present time.

##  4 The orbital velocities in bound and loose wide binaries

In relation with the velocity observations in wide binaries reviewed above, we
now specify the behaviour of the orbital velocities in a bound system
consisting of a test particle around a central mass $M$. For masses $M_{1}$
and $M_{2}$, the sum $M_{\mathrm{tot}}$ of the two masses has to be accounted
for. Again we consider the case of circular orbits. We have to distinguish the
case of bound systems where Newtonian gravity $GM/r^{2}$ dominates over the
so-called dynamical gravity $\frac{\psi}{\tau_{0}}\upsilon$ and the opposite
case of loose systems governed by the dynamical gravity.

###  4.1 The orbital velocities in a bound two-body system

Starting from the conservation of the angular momentum
$\frac{\psi}{\tau_{0}}\,r^{2}\dot{\vartheta}=\mathcal{L}\,$ according to
(3.8), we have for the orbital velocity
$\upsilon=r\dot{\vartheta}=\frac{\mathcal{L}\,\tau_{0}}{\psi\,r}$. Again
considering a circular orbit, by introducing the expression of $r_{0}$ given
in Eq. (3.15) one obtains,

$\upsilon=r_{0}\dot{\vartheta}=\frac{\mathcal{L}\,\tau_{0}\,GM\psi^{2}}{\psi\,\mathcal{L}^{2}\,\tau^{2}_{0}}=\frac{\
GM\psi}{\mathcal{L}\tau_{0}}.$ (4.1)

Expressing now the time dependence of both $M$ and $\psi$, we simplify the
above expression,

$\
\upsilon=\frac{G\,M(\tau_{0})}{\mathcal{L}}\left(\frac{1-t_{\mathrm{in}}}{\tau_{0}}\right)=const.$
(4.2)

The time dependences vanish, thus the orbital velocity is a scale invariant
quantity, which is keeping constant in time during secular orbital expansion.
A more usual and convenient form may be obtained from (4.1) and (3.15),

$\upsilon^{2}=\frac{\
G^{2}M^{2}\psi^{2}}{\mathcal{L}^{2}\tau^{2}_{0}}=\frac{GM}{r_{0}}.$ (4.3)

Both $M$ and $r_{0}$ have the same secular variations defined by the evolution
of the scale factor, and both vary with
$[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]$. Thus, the
secular evolution of the scale factor $\lambda$ has no incidence on the
velocity which remains constant, as already pointed out. Moreover, the
resulting velocity in the scale invariant two-body system has the same
expression as the classical one. It is to be emphasized that the fact that the
instantanous expressions of the velocity at time $\tau$ are the same as the
standard ones does not imply that the velocity dispersion
$\overline{\upsilon^{2}}$ of gravitational systems is the same in the
classical and SIV theory.

In standard theory, a body drifting away would see its velocity decrease
according to the standard law of angular momentum conservation
$r_{0}\times\upsilon=const.$ Here in the scale invariant theory, the angular
momentum conservation law is
$\frac{\psi}{\tau_{0}}\,r_{0}\times\upsilon=const.$ according to (3.8). For a
rapid change of the radius (rapid means in a negligible time in front of
$\tau_{0}$), the effect would evidently be the same as in the standard case.
On longer timescales, the secular changes of the radius do leave the velocity
constant, so that the angular momentum $r_{0}\times\upsilon$ just varies like
$r_{0}$, quite consistently with (3.8) and the comments following this
conservation law.

In this section, we have seen (in the current units) the main properties of
the two-body system where both the Newtonian $\frac{GM}{r^{2}}$ and the
dynamical $\frac{\psi(\tau)}{\tau_{0}}\times\upsilon$ gravities are
significant. The extreme case of very wide binaries, where the dynamical
acceleration may largely dominate over gravity, is studied below.

###  4.2 Loose and detached systems

The gravity field of a mass $M$ extends to infinity. For increasing distances
from $M$, the attraction of other masses in the Galaxy makes the local gravity
to vanish. This occurs at the Jacobi radius, also called Roche radius, where
the Galactic field starts dominating; for a typical system a value of about
1.70 pc [19] is estimated for this radius. Long before this happens, the
dynamical acceleration $\frac{\psi}{\tau_{0}}\times\upsilon$ may overcome the
ambiant Newtonian gravity $g_{\mathrm{N}}$. The equality of the two terms of
the equation of motion (3.1) occurs for a gravity $g_{0}$, as given by [25]

$\displaystyle
g_{0}\,=\,\frac{(1-\Omega_{\mathrm{m}})^{2}}{4}\,n\,c\,H_{0}\,\quad\quad\mathrm{or}\quad
g_{0}\,=\,\frac{n\,c\,(1-\Omega_{\mathrm{m}})(1-\Omega^{1/3}_{\mathrm{m}})}{2\,\tau_{0}}\,.$
(4.4)

The product $c\times H_{0}$ is equal to $6.80\times 10^{-8}$ cm s-2. For
$\Omega_{\mathrm{m}}$=0, 0.10, 0.20, 0.30 and 0.50, we get $g_{0}=$ (1.70,
1.36, 1.09, 0.83, 0.43) $\times\,10^{-8}$ cm s-2 respectively. Interestingly
enough, this value of $g_{0}$ well corresponds to the value of $a_{0}$ of
about 1.2 $\times\,10^{-8}$ cm s-2 characterizing the deep MOND limit [32,
33], where the gravity is $g=\sqrt{a_{0}\;g_{\mathrm{N}}}$, where
$g_{\mathrm{N}}$ is the Newtonian gravity. For gravities below $g_{0}$, the
equation of motion (3.1) acccepts, _as an approximation_ , a similar relation
of the form $g=\sqrt{g_{0}\;g_{\mathrm{N}}}$ [25], with $g_{0}$ given by
(4.4). This approximation is valid over the last 400 Myr with an accuracy 1-2%
depending on $\Omega_{\mathrm{m}}$. For a given total mass $M_{\mathrm{tot}}$
of a binary system, the equality of the Newtonian and dynamical accelerations
occurs at a separation,

$s_{\mathrm{trans}}\,=\,\sqrt{\frac{G\,M_{\mathrm{tot}}}{g_{0}}}\,.$ (4.5)

For M${}_{\mathrm{tot}}=1.0$ M⊙, we get a value of $s_{\mathrm{trans}}=7.0$
[kau] or 0.034 [pc] (8.6 [kau] or 0.042 [pc] for a typical value of the total
mass of 1.5 M⊙) [7].

What happens for wide binaries approaching the transition? We may
qualitatively understand the effects of a declining Newtonian gravity by
considering the effects of a decrease of the product $G\times M$ in the basic
parameters $e$ and $r_{0}$. From Eqs. (3.15) and (3.16), we see that both the
eccentricity $e$ and the radius of the circular orbit (or latus-rectum of an
ellipse) $r_{0}\,\rightarrow\,\infty$ for a product $G\times M$ approaching
zero. This means that the orbit first extends in size together with an
increase of the ellipticity up to $e=1$, where it becomes parabolic. At the
parabolic limit, we have,

$C\,=\,\frac{GM\,\psi^{2}}{\mathcal{L}^{2}\,\tau^{2}_{0}}\,\equiv\,\frac{1}{r_{0}}\,.$
(4.6)

Then, for a further decline of Newtonian gravity, the eccentricity becomes
larger than 1 and the orbit is hyperbolic tending to a straight line. It is
interesting to examine the behaviour of the orbital velocity in this limit.
Within SIV theory, it is evidently preferable to use the full equation of
motion and the appropriate relations rather than the MOND-like approximation.
From (3.1), we have,

$\frac{d^{2}\bf{r}}{d\tau^{2}}\,=\frac{\psi(\tau)}{\tau_{0}}\frac{d\bf{r}}{d\tau}\,,\quad\quad\mathrm{or}\;\;\frac{d\upsilon}{d\tau}\,=\,\frac{\psi(\tau)}{\tau_{0}}\;\upsilon\,,$
(4.7)

for the linear motion. This can also be rewritten as,

$\frac{d\upsilon}{\upsilon}\,=\,\frac{\psi(\tau)}{\tau_{0}}\;d\tau\,.$ (4.8)

From Eq. (3.4), one has

$\psi\,=\,-\frac{\dot{\psi}}{\psi}\tau_{0}\,,\quad\mathrm{and\;thus}\;\;\frac{d\upsilon}{\upsilon}\,=\,-\frac{\dot{\psi}}{\psi}\,d\tau\,=\,-\frac{d\psi}{\psi}\,.$
(4.9)

The integration gives according to (3.2),

$\ln\upsilon=-\ln\psi+const.\quad\quad\mathrm{thus}\;\;\upsilon=\frac{k}{\psi}=\frac{k}{(1-t_{\mathrm{in}})}\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]\,,$
(4.10)

where $k$ is a constant with the dimension of a velocity. Setting
$\tau=\tau_{0}$, we have $\upsilon(\tau_{0})=\frac{k}{(1-t_{\mathrm{in}})}$
and the velocity becomes,

$\upsilon(\tau)=\upsilon(\tau_{0})\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})].$
(4.11)

Thus, we see that the velocity is no longer keeping constant with secular
variations as in the bound system, but now the velocity is increasing linearly
in time $\tau$ under the effects of the dynamical acceleration
$\frac{\psi(\tau)}{\tau_{0}}\;\upsilon$. Turning now to the covered distance
$r$ during this type of accelerated motion, we have from (4.11),

$\frac{dr}{d\tau}=\upsilon(\tau_{0})\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})],\quad\mathrm{thus}\;\;r(\tau)=\frac{1}{2}\frac{\tau_{0}}{(1-t_{\mathrm{in}})}\,\upsilon(\tau_{0})\,[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]^{2}.$
(4.12)

This is quite a classical type of accelerated motion, increasing with the
square of the time. Thus, when the dynamical gravity
$\frac{\psi}{\tau_{0}}\upsilon$ significantly dominates over the overall
Newtonian gravity, the motion becomes hyperbolic, with an eccentricity $e>1$,
and a velocity increasing linearly with time. As a result, the covered
distance fastly increases with a quadratic dependence in time,

$\delta(\tau)=[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]^{2}.$
(4.13)

This expression in $t^{2}$ is shared by several equations in scale invariance,
giving the velocity, the orbital radius and the mass, as also seen in Eq.
(3.5). Let us call it the _dynamical regime_. In order to numerically estimate
the possible changes in velocities and in covered distances in this regime,
let us consider a reference time $\tau_{\mathrm{N}}$ schematically marking the
end of the Newtonian regime and the beginning of the dynamical one. We then
get the following ratios,

$\frac{\upsilon(\tau)}{\upsilon(\tau_{\mathrm{N}})}=\frac{[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]}{[t_{\mathrm{in}}+\frac{\tau_{\mathrm{N}}}{\tau_{0}}(1-t_{\mathrm{in}})]},\quad\quad\mathrm{and}\;\;\frac{r(\tau)}{r(\tau_{\mathrm{N}})}=\frac{[t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(1-t_{\mathrm{in}})]^{2}}{[t_{\mathrm{in}}+\frac{\tau_{\mathrm{N}}}{\tau_{0}}(1-t_{\mathrm{in}})]^{2}},$
(4.14)

and thus

$\frac{\upsilon(\tau_{0})}{\upsilon(\tau_{\mathrm{N}})}=\frac{1}{[t_{\mathrm{in}}+\frac{\tau_{\mathrm{N}}}{\tau_{0}}(1-t_{\mathrm{in}})]},\quad\quad\mathrm{and}\;\;\frac{r(\tau_{0})}{r(\tau_{\mathrm{N}})}=\frac{1}{[t_{\mathrm{in}}+\frac{\tau_{\mathrm{N}}}{\tau_{0}}(1-t_{\mathrm{in}})]^{2}}.$
(4.15)

This gives the change in velocity and radius, between the transition time
$\tau_{\mathrm{N}}$ to today in the dynamical regime. Of course, binary
systems are observed locally in the Milky Way at the present time, i.e. at
time $\tau_{0}$.

##  5 Comparisons of observations and theory for very wide binaries

We now proceed to numerical estimates of the above theoretical predictions and
we compare them with the observations by Chae [7] and by Hernandez [17]. Both
works consider a vast range of binary stars samples in Gaia DR3 and show
deviations from Newtonian predictions in the observed relation between
velocities $\upsilon$ and separations $s$.

Figure 1: Projected velocities $\upsilon_{p}$ as a function of separation $s$
for the main sample in Fig. 13 from Chae [7]. The very small red dots are the
observed values and the blue dots are the Newtonian values in the Monte-Carlo
analysis. The larger dots and their connecting lines are the median and
percentiles as indicated. The central black dashed line shows the Keplerian
relation in $1/\sqrt{s}$. The green lines are additional indications of the
mean deviations along the hyperbolic path of the loose systems after 1, 3 and
5 Gyr from the time, $\tau_{N}$, where they depart from the Newtonian relation
evolving under the dynamical acceleration only. The direction of the effects
in velocity and separation is indicated by a green arrow at the bottom on the
left side. The deviation of a given system increases linearly with time as
indicated by Eqs .(4.15). We see that the mean observed relation for the loose
systems correponds to a mean evolution during about 2 to 3 Gyr. In other
words, in only a few Gyrs, loose systems drift away from the Newtonian
relation in a way fully compatible with the dynamical evolution described by
scale invariance (Figure adapted from Chae [7]).

###  5.1 Comparison with the analysis by Chae [7]

This analysis by Chae [7] has been presented in Sect. 2. Fig. 1 shows the
logarithmic relation between $\log\upsilon$ and $\log s$ for a sample of 2’463
binary pairs with masses estimated from relations between the absolute G-band
magnitude, $M_{G}$, and the mass, $M$. The mean of the observations at
different separations, $s$, are given by the continuous red lines. The blue
lines result fom Monte-Carlo (MC) simulations. In practice, from an observed
set of masses $M$ and $s$ values, each MC realization draws the velocity from
the classical Newtonian equations with the appropriate statistical
distributions of the orbital parameters, namely $\vartheta$ the longitude of
the observed star, $\vartheta_{0}$ the longitude of the periastron, the orbit
inclination $i$ and $e$ the eccentricity. A serie of 400 MC simulations are
performed to estimate the statistical error bars, as indicated on the blue
line.

The average total mass of the binary systems is around 1.5 M⊙, meaning that
the Main-Sequence lifetimes are generally several Gyr. Let us consider 3
cases: when the systems start deviating from Newton’s Law 1, 3 and 5 Gyr ago.
This corresponds to 3 values of the past time $\tau_{\mathrm{N}}=$ 12.8, 10.8
and 8.8 Gyr. These ages lead to the following velocity ratios respectively
$\frac{\upsilon(\tau_{0})}{\upsilon(\tau_{\mathrm{N}})}=1.048,1.159$ and
1.296. Thus, according to Eqs. (4.15), the shifts in velocities and distances
since time, $\tau_{\mathrm{N}}$, when the systems entered the dominant free
regime, are the following ones,

$\log\left[\frac{\upsilon(\tau_{0})}{\upsilon(\tau_{\mathrm{N}})}\right]=0.020,0.064,0.113,\quad\quad\mathrm{and}\quad\log\left[\frac{r(\tau_{0})}{r(\tau_{\mathrm{N}})}\right]=0.040,0.128,0.226.$
(5.1)

Figure 2: Comparisons of observations and theory for a sample of 40 very wide
binaries with exceptionally precise radial velocities, with individual
relative errors smaller than 0.005, as selected by Chae [7]. The green broken
lines shows the isochrones corresponding to departures from the Newtonian law
after 1, 3 and 5 Gyr of evolution under the dynamical acceleration in the
scale invariant theory. The departure from Newton’s relation is progressive
and tends towards a value between 3 and 5 Gyr (Data are from Chae [7]).

In Fig. 1 , the above shifts in $\log\upsilon$ and $\log r$ produce a
displacement in the direction indicated by the green arrow in the lower left
corner. The amplitudes of the displacements are determined by the above
numerical values of velocities and distances. We see that the observed mean
shift progressively increases between $s=2$ [kau] to 8 [kau], an interval
which well encompasses the value of $s_{\mathrm{trans}}$ discussed in Sect.
4.2, i.e. $log(s_{\mathrm{trans}})=3.84$ for 1 M⊙ and
$log(s_{\mathrm{trans}})=3.9$ for 1.5 M⊙. Binary systems with ages
corresponding to the green time-lines/isochrones may arise from binaries at
different locations on the reference line $\upsilon\sim 1/\sqrt{s}$, the
amplitudes of the shifts are expected to be significant somewhere above the
limit $s_{\mathrm{trans}}$, as is observed for separations larger than 10’000
au. Note that according to the Third Kepler’s Law, a system with a total mass
of 1.5 M⊙ and a separation 10’000 au, if still bound, would have an orbital
period $T=8.165\times 10^{5}$ yr.

The observed mean shifts corresponds to an evolution in the free regime by
about 2 to 3 Gyr. These shifts are clearly larger than the error bars as
illustrated in the figure. However, we must be careful not to overinterpret
these results because the velocities for separations $s$ larger than $10^{4}$
au are of the order of 170 m/s and the shifts in velocities correspond to
velocity differences of about 27 m/s. Nevertheless, the shifts appear
significant. This is particularly true since all the possible
variations/combinations of the selection criteria by [7] all give similar
shifts between the reference line $\upsilon\sim 1/\sqrt{s}$ and the observed
values near and above $s\sim 10^{3.5}$ au. As a reminder, these selection
criteria include J-band photometry instead of G-band, different magnitude and
mass domains, the use of more stringent astrometric requirements, the change
of eccentricity distributions, and a range of estimated frequency of multiple
systems. We may remark that even the initial general sample of 26’615 binary
pairs shows, according to Chae’s results, some deviation from the Newtonian
behaviour for $s$-values larger than $\log s\geq 3.5$.

Interestingly enough, after considering the statistical properties of large
samples as in Fig. 1, Chae [7] also analysed a smaller sample that includes
only the 40 most highly accurate radial velocity data, with a relative
precision equal or better than 0.005 on the radial velocities, i.e.,
comparable to the astrometric data. Three bins are formed from these
exceptional 40 systems. The statistical uncertainties are evidently larger, as
shown on the blue line, but thanks to the exceptional data quality, the
highest $s$-bin at $\log s=3.7$ now shows a large deviation from Newtonian
relation, which is well compatible with a free motion during nearly 5 Gyr.
This unique highly accurate sample is reinforcing the conclusion that the
existence of significant deviations from Newtonian relation for $\log s\geq
3.5$ and above. The direction and size of the observed effects are compatible
with a dynamical evolution for several Gyr, dominated by the scale invariance
term of Eq. (4.8). At this stage, whether the mean duration of this phase is
actually 3 Gyr, 5 Gyr or more remains uncertain. In any case, it may
correspond to a substantial fraction of the life of our Galaxy.

To summarize, the full sample of 26’615 binary pairs, the sub-sample of 2’463
binaries with excellent data and the very restricted sample of 40 binary pairs
with an extremely high precision of 0.005 on radial velocities all confirm the
occurrence of significant deviations to Newton’s predictions in the velocity
vs. $s$ relations, for separations larger than a few kau. These deviations
correspond to an evolution in the dynamical regime for several Gyr and remain
robust against any change made on statistical parameters when selecting sub-
samples of pairs.

###  5.2 Comparison with the analysis by Hernandez [17]

Let us now turn to the sample of wide binaries assembled by Hernandez [17],
which results from very stringent quality criteria briefly recalled in Sect.
2. In most studies, the mass of the binary components was estimated from
simple magnitude-mass scaling relations. More recently, mass estimates became
available from an internal Gaia FLAME work-package making use of spectroscopic
information in DR3. This provides more accurate masses for 24% of the binary
components. Comparisons of the masses obtained from magnitude-mass scaling
with FLAME masses shows a small offset of about 0.05 M⊙ below 0.7 M⊙, which is
then corrected by the author. The relative velocity $\Delta V$ analysis
restricted to the plane of the sky is primarily determined through Gaia
_proper motions in two perpendicular directions_ , radial velocities playing a
secondary role for separations $s<0.1$ pc. Final cuts are made based on the
signal-to-noise ratios $\Delta V/\sigma_{\Delta V}$, yielding a high-quality
sample of 450 highly-isolated wide binaries, which serve for the astrophysical
study.

Figure 3: Binned root mean square $\langle\Delta V\rangle$ of the relative
velocities in the plane of sky as a function of the 2D projected separation
$s$ for 450 binary pairs, remaining after the application of most stringent
quality criteria by Hernandez [17]. There is a partial overlap of the
intervals of binned binary pairs. The black numbers indicate the number of
binary pairs in the various overlapping means considered. The green lines show
the isochrones corresponding to departures from the Newtonian law after 1, 3
and 5 Gyr of evolution under the dynamical acceleration in the scale invariant
theory (Adapted from Hernandez [17]).

The binned root-mean-square $\langle\Delta V\rangle$ of the relative
velocities between the two components of selected pairs are shown in Fig. 3 as
a function of the 2D projected separations $s$ between the two components of
each binary pair. In the Figure, this blue line shows the Newtonian relation
$\upsilon\sim s^{-1/2}$ resulting of simulations of 50’000 binary pairs
composed of two stars of 1 M⊙ mass [19]. The simulations assume a Newtonian
evolution of 10 Gyr under the galactic field in the solar neighborhood, random
encounters with field stars, random orientation of the lines of sight and
variable eccentricities in agreement with observations. Interestingly enough,
this blue line just follows a Keplerian relation $\langle\Delta V\rangle\sim
s^{-1/2}$ up to the Jacobi radius, also called Roche radius. In the present
case, it corresponds to the galactocentric distance where the Galactic field
would dissolve the binary system. The value of this critical radius is 1.70 pc
in the present case [19], thus well beyond the value of about 0.01 pc where
the observed relation is turning into a plateau in the figure, as analysed in
the next section. Note for comparison with Chae [7] that 0.01 pc corresponds
to 2.063 kau or also $\log(s)=3.31$ (see Fig. 2 for comparison).

If the orbital velocities of the two stars are constant with separation, it is
also the case for their differences, a property that allowed building Fig. 3.
There, the Keplerian relation, consistent with Newton’s Law, is very well
followed for separations up to about $10^{-2}$ pc. However, for larger
separations the mean relative velocities significantly start deviating from
the Keplerian relation and tend to an essentially flat constant value in Fig.
3. We note that this flat appearance is favoured by the graphical
representation of mean binned values of the velocities, as well as possibly by
the objective to perform comparisons with the flat behaviour predicted by
MOND. Indeed, the comparison with the isochrones corresponding to the duration
of the free evolution gives results similar to those by Chae [7]. The result
by [17] also support an evolution in the _dynamical regime_ of the order 5
Gyr, in excellent agreement with the results obtained with the sub-sample of
Chae [7] containing the very best quality observations, while the large
statistical results of Chae are more in favour of 3 Gyr. The key point is that
the different results on the observed velocities of the very wide binary
systems are all supporting significant deviations from the standard Newtonian
predictions, in a way which is consistent with an evolution dominated by Eq.
(4.8) for several Gyr.

A possible difficulty has been raised by Loeb [20]: in wide binaries the
gravitational redshift becomes about similar in amplitude as the Doppler-
Fizeau effect. This might effectively be a problem. However, one has to
consider the differential effects of gravitational redshifts. Following Loeb,
we express the velocity shift $\upsilon_{\mathrm{GR}}$ due to gravitation, and
we apply the mass-radius relation,

$\frac{R}{R_{\odot}}\approx\left(\frac{M}{M_{\odot}}\right)^{0.8},$ (5.2)

for Solar-mass stars close to the Main Sequence. We then obtain:

$\upsilon_{\mathrm{GR}}=-\frac{G\,M}{cR},\quad\quad\mathrm{and\,thus}\quad\upsilon_{\mathrm{GR}}\approx-0.6\left(\frac{M}{M_{\odot}}\right)^{0.2}\;\,\mathrm{km/s}.$
(5.3)

The total binary masses for the studied systems with large $s$ are confined to
a limited interval of 1.2 to 2.0 $M_{\odot}$ [17]. The distribution of the
mass ratios of binaries in the mass range of solar type stars was studied by
[11], who found a mean mass ratio $M_{2}/M_{1}$ of about 0.5. To the power of
0.2, this gives a value of 0.87 for the ratio of the gravitational redshifts
of the two components. For a velocity of 0.60 km/s for the primary, this
corresponds to $\Delta\upsilon_{\mathrm{GR}}$ of about 0.08 km/s, which is
small and cannot be responsible for the observed deviations. Thus, while the
remark by [20] that the gravitational shifts are of the same order as the
Doppler-Fizeau shifts is correct, the differences of gravitational redshifts
between the two components in a binary pair are significantly smaller than the
velocity shifts.

Most importantly, gravitational redshifts do not introduce any systematic
trend, in the sense that sometimes it is the blue shifted approaching star
which has more gravity shift (reducing the $\upsilon$-difference), and
sometimes it is the receding one (increasing the $\upsilon$-difference). Thus,
the velocity differences may be increased or reduced in a non-systematic way.
In the mean of many binary pairs in a given mass interval the differential
effects of the gravitational redshifts of the pairs should statistically
cancel each other. These remarks are important for the sample in [7], while in
the case of [17], the relative motions of the two components are mainly
derived from the proper motions in the plane of the sky, orthogonal to the
direction where gravitational effects is acting, as emphasized by this author.

##  6 Conclusions

In this work, we consider the very accurate data on wide binary stars obtained
from Gaia DR3. We focus in particular on the work by Hernandez [17] and Chae
[7], which both suggest too high velocities with respect to standard Newtonian
predictions at separations larger than about 3000 au, or about 0.015 pc. This
effect is similar to what is observed in the external parts of galaxy rotation
curves, but as stars obsviously do not reside in a dark matter halo, this
standard interpretation does not hold anymore.

Instead, we propose a much more simple explanation in the framework of scale
invariant theory. This theory rests on one single and simple physical
assumption, without any adjustment of any ad’hoc parameter. The direct
consequence of scale invariance is that it naturally adds a dynamical
acceleration that best manifests itself at low gravities in the modified
Newton equation. This is exactly the dynamical regime of wide binary stars.

We find that the dynamical evolution of very wide binary stars indeed appears
to be dominated by the additional dynamical acceleration predicted by the
scale invariant theory. The amplitude of the observed deviations to Newton’s
gravity supports such a dynamical evolution of loose systems, for several
Gyrs. This result is robust against a variety of sub-selections of stars and
quality criteria and is not influenced by gravitational redshift effects that
we find neglible in front of dynamical evolution.

This new result of the present series on observational tests of scale
invariance is the third positive test after rotation curves of galaxies and
galaxy clusters [26] and strong gravitational lensing [28]. It also comes in
strong support of previous work on scale invariance explaining the faster-
than-expected growth of structure in the Universe [29], Lunar recession [30],
and the CMB temperature evolution [23].

## Acknowledgments

A.M. expresses his best thanks to Dr. Vesselin Gueorguiev for constructive
interaction since many years.

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