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# Solar System limits on gravitational dipoles

Indranil Banik1 and Pavel Kroupa1,2
1Helmholtz-Institut für Strahlen und Kernphysik (HISKP), University of Bonn,
Nussallee 14$-$16, D-53115 Bonn, Germany
2Charles University, Faculty of Mathematics and Physics, Astronomical
Institute, V Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic Email:
<EMAIL_ADDRESS>(Indranil Banik)
<EMAIL_ADDRESS>(Pavel Kroupa)

###### Abstract

The gravitational dipole theory of Hadjukovic (2010) is based on the
hypothesis that antimatter has a negative gravitational mass and thus falls
upwards on Earth. Astrophysically, the model is similar to but more
fundamental than Modified Newtonian Dynamics (MOND), with the Newtonian
gravity $g_{{}_{N}}$ towards an isolated point mass boosted by the factor
$\nu=1+\left(\alpha/x\right)\tanh\left(\sqrt{x}/\alpha\right)$, where $x\equiv
g_{{}_{N}}/a_{{}_{0}}$ and $a_{{}_{0}}=1.2\times 10^{-10}$ m/s2 is the MOND
acceleration constant. We show that $\alpha$ must lie in the range ${0.4-1}$
to acceptably fit galaxy rotation curves. In the Solar System, this
interpolating function implies an extra Sunwards acceleration of ${\alpha
a_{{}_{0}}}$. This would cause Saturn to deviate from Newtonian expectations
by ${7000\left(\alpha/0.4\right)}$ km over 15 years, starting from known
initial position and velocity on a near-circular orbit. We demonstrate that
this prediction should not be significantly altered by the postulated dipole
haloes of other planets due to the rather small region in which each planet’s
gravity dominates over that of the Sun. The orbit of Saturn should similarly
be little affected by a possible ninth planet in the outer Solar System and by
the Galactic gravity causing a non-spherical distribution of gravitational
dipoles several kAU from the Sun. Radio tracking of the Cassini spacecraft
orbiting Saturn yields a ${5\sigma}$ upper limit of 160 metres on deviations
from its conventionally calculated trajectory. These measurements imply a much
more stringent upper limit on $\alpha$ than the minimum required for
consistency with rotation curve data. Therefore, no value of $\alpha$ can
simultaneously match all available constraints, falsifying the gravitational
dipole theory in its current form at extremely high significance.

###### keywords:

gravitation – dark matter – ephemerides – celestial mechanics – space vehicles
– solar neighbourhood

††pubyear: 2020††pagerange: Solar System limits on gravitational
dipoles–References

##  1 Introduction

One of the great mysteries in contemporary astronomy is the true cause of the
very large dynamical discrepancies between the observed rotation curves of
galaxies and the predictions of Newtonian gravity applied to their luminous
matter distributions (e.g. Babcock, 1939; Rubin & Ford, 1970; Rogstad &
Shostak, 1972). These acceleration discrepancies are usually attributed to
dark matter haloes surrounding each galaxy (Ostriker & Peebles, 1973). This
interpretation is challenged by continued null detection of the constituent
dark matter particles in sensitive searches, for instance in 11 years of Fermi
data on dwarf spheroidal satellites of the Milky Way (Hoof et al., 2020).
Moreover, the galactic acceleration discrepancies follow some remarkable
regularities (Famaey & McGaugh, 2012) that can be summarised as a unique
relation between the acceleration inferred from the rotation curve and that
expected from the baryonic distribution (McGaugh et al., 2016).

Such a radial acceleration relation (RAR) was predicted several decades
earlier using Modified Newtonian Dynamics (MOND, Milgrom, 1983). In MOND, the
supposed dynamical effect of dark matter is instead provided by an
acceleration dependence of the gravity law. The gravitational field strength
$g$ at distance $r$ from an isolated point mass $M$ transitions from the
Newtonian $GM/r^{2}$ law at short range to

$\displaystyle g~{}=~{}\frac{\sqrt{GMa_{{}_{0}}}}{r}~{}~{}~{}\text{for
}~{}r\gg r_{{}_{M}}\equiv\sqrt{\frac{GM}{a_{{}_{0}}}}\,.$ (1)

MOND introduces $a_{{}_{0}}$ as a fundamental acceleration scale of nature
below which the deviation from Newtonian dynamics becomes significant. For a
point mass, this corresponds to distances beyond its MOND radius $r_{{}_{M}}$.

Empirically, $a_{{}_{0}}=1.2\times{10}^{-10}$ m/s2 to match galaxy rotation
curves (Begeman et al., 1991; McGaugh, 2011). Remarkably, this is the same
order of magnitude as the acceleration at which the classical energy density
of a gravitational field (equation 9 of Peters, 1981) becomes comparable to
the dark energy density $u_{{}_{\Lambda}}\equiv\rho_{{}_{\Lambda}}c^{2}$ that
conventionally explains the accelerating expansion of the Universe (Efstathiou
et al., 1990; Ostriker & Steinhardt, 1995; Riess et al., 1998).

$\displaystyle\frac{g^{2}}{8\mathrm{\pi}G}~{}<~{}u_{{}_{\Lambda}}~{}~{}\Leftrightarrow~{}~{}g~{}\la~{}2\mathrm{\pi}a_{{}_{0}}\,.$
(2)

MOND could thus be a result of poorly understood quantum gravity effects that
also underlie dark energy (e.g. Milgrom, 1999; Pazy, 2013; Verlinde, 2016;
Smolin, 2017).

Regardless of the microphysical explanation, MOND can accurately match the
rotation curves of a wide variety of spiral and elliptical galaxies across a
vast range in mass, surface brightness and gas fraction using only their
luminous matter (fig. 5 of Lelli et al., 2017). This only became apparent long
after the MOND field equation was first published (Bekenstein & Milgrom,
1984), making these achievements successful a priori predictions. It is
difficult to explain the success of these predictions in a conventional
gravity context, even with the observational facts in hand (Desmond, 2017a, b;
Ghari et al., 2019). In particular, the latter work showed that it is still
very difficult to keep a tight RAR despite a diversity of rotation curve
shapes at fixed peak velocity (Oman et al., 2015).

Despite its successes, the empirical MOND approach lacks a firm theoretical
foundation. The gravitational dipole (GD) model seeks to explain the MOND
phenomenology using more fundamental concepts (Hajdukovic, 2010). In this
model, antimatter particles have a negative gravitational mass with the same
magnitude as the corresponding particle. The gravitational properties of
antimatter are under investigation at CERN thanks to the experiments known as
GBAR (Pérez et al., 2015), AEgIS (Brusa et al., 2017) and ALPHA-g (Bertsche,
2018). Experimental uncertainties are currently $\approx 100\times$ too large
to test the hypothesis that antimatter falls upwards in the terrestrial
gravitational field (Alpha Collaboration & Charman, 2013). Decisive results
should be available by the end of 2021.

If antimatter anti-gravitates in the sense described, matter-antimatter pairs
resulting from quantum vacuum (QV) fluctuations would be virtual GDs. An
important consequence is that the QV contains no net gravitational mass,
leading to a predicted cosmological constant of zero. In the GD model, the
dynamical discrepancies in galaxies are an effect of gravitational
polarization of the GDs. The divergence of this polarized field of GDs leads
to extra effective mass, similarly to polarization in electrostatics (Equation
12).

In this contribution, we focus on what the GD model implies for the Solar
System. The accurate data within it place tight constraints on modified
gravity theories (Hees et al., 2016). In the context of MOND as modified
gravity, this is mainly due to the Galactic external field effect (Milgrom,
1986) causing the Solar ‘phantom’ dark matter halo to be non-spherical at
large distances. Thus, even without any phantom dark matter within the inner
Solar System, deviations from Keplerian motion are generically expected due to
a non-zero tidal stress (Blanchet & Novak, 2011).

The GD model originally predicted a fixed extra Sunwards acceleration of
$\approx 7a_{{}_{0}}$ in the Solar System (equation 12 of Hajdukovic, 2010).
This was in line with the ‘Pioneer anomaly’, a then-unexplained Sunwards
acceleration of the Pioneer spacecraft (Turyshev & Toth, 2010). However, this
anomaly is likely caused by thermal forces (Turyshev et al., 2012). The
expected anomalous acceleration was later revised down to $a_{{}_{0}}$
(Hajdukovic, 2012) and then to the range ${\left(0.1-1\right)}a_{{}_{0}}$
(equation 6 of Hajdukovic, 2013). A theoretically motivated interpolating
function was then provided between the ‘saturated’ (high-acceleration) region
near a massive body and the low-acceleration region further away, where the
extra gravity of the GDs is expected to dominate (section 2 of Hajdukovic,
2014b). We make use of equation 2 in that work, which was also restated in
equation 24 of Hajdukovic (2014a). These equations are analogous to our
Equation 3. Importantly for our work, the latest publication advocating the GD
model predicts that Solar System planets should experience an extra Sunwards
acceleration of $\left(0.4-0.5\right)a_{{}_{0}}$, with the most likely value
being $5a_{{}_{0}}/12$ (section 6.1 of Hajdukovic, 2020).

The effects of radiation pressure can be minimized by accurately tracking the
motion of a gas giant planet, as done by the Cassini mission (Matson, 1992).
Using radio tracking data from orbit around Saturn, Hees et al. (2014)
constrained the tidal stress on the Solar System to be ${\left(3\pm
3\right)}\times 10^{-27}/s^{2}$ in the direction towards the Galactic Centre
(see their equation 6). At the 9.58 AU distance of Saturn, this corresponds to
an acceleration of ${\left(4.3\pm 4.3\right)}\times 10^{-15}$ m/s2. While this
constraint is not directly applicable to an additional Sunwards force, it does
suggest that any such anomalous acceleration should be $\la
10^{-4}a_{{}_{0}}$. This casts serious doubt on the GD model (Iorio, 2019). We
investigate the model in more detail to see if there is any way around this
apparent falsification.

After introducing the GD model and its context (Section 1), we explore what a
GD force law consistent with galaxy rotation curves (Section 2) implies for
the motion of a planet in the Solar System (Section 3). We find that the GD
model predicts very large deviations (Sections 3.1-3.2) that greatly exceed
observational upper limits (Section 3.3), as previously found by Iorio (2019).
We then explore whether this conclusion can be weakened once we take into
account the gravity of other planets and their dipole haloes (Section 3.4),
the possible presence of a ninth planet in the outer Solar System (Section
3.5), and the Galactic gravity (Section 3.6). Our conclusions are given in
Section 4.

##  2 Galaxy rotation curve constraint

In this section, we consider galaxy rotation curve constraints on the isolated
point mass force law in the GD model, which can be expressed as (equation 24
of Hajdukovic, 2014a):

$\displaystyle g~{}=~{}g_{{}_{N}}+\,\alpha
a_{{}_{0}}\tanh\left(\frac{r_{{}_{M}}}{\alpha r}\right)\,.$ (3)

This reduces to Newtonian dynamics when the Newtonian gravity
$g_{{}_{N}}\equiv GM/r^{2}\gg a_{{}_{0}}$. It also reproduces the correct
asymptotic behaviour for $g_{{}_{N}}\ll a_{{}_{0}}$ (Equation 1).

Equation 3 can be written in the standard MOND terminology with interpolating
function

$\displaystyle\nu_{{}_{GD}}~{}=~{}1\,+\,\frac{\alpha}{x}\tanh\left(\frac{\sqrt{x}}{\alpha}\right),$
(4)

where $x\equiv g_{{}_{N}}/a_{{}_{0}}$ and the actual gravity $g~{}\equiv~{}\nu
g_{{}_{N}}$, with $\nu$ being the factor by which a model enhances the
strength of gravity towards an isolated point mass. The subscript GD indicates
that Equation 4 is motivated by the GD model.

Figure 1: Dipole dark matter-based interpolating functions (Equation 4) for
different values of $\alpha$ (curves with higher $\alpha$ have higher $\nu$).
We also show _top_ : the binned RAR data from Lelli et al. (2017) with error
bars showing (conservatively) the dispersion in $\log_{10}g$ for each bin, and
_bottom_ : the empirical fit to their data (Equation 5) with 0.05 dex error
bars, based on fits to galaxy rotation curves (Li et al., 2018). It is clear
that $\alpha$ should lie in the range $\left(0.4-1\right)$, with $\alpha<0.1$
strongly excluded.

The $\nu$ function is constrained by the baryonic mass distributions and
rotation curves of disc galaxies, with the former determining $g_{{}_{N}}$
while the latter measures the true radial gravity $g$. The empirical relation
between $g$ and $g_{{}_{N}}$ (the RAR) can be fit very well by (equation 11 of
Lelli et al., 2017):

$\displaystyle\nu_{{}_{RAR}}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\mathrm{e}^{-\sqrt{x}}}\,.$ (5)

This RAR-inspired interpolating function is rather similar to the ‘simple’
form $\nu_{s}=\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{1}{x}}$ (Famaey & Binney,
2005) for $x\la 10$ (e.g. fig. 1 of Chae et al., 2019). The main difference is
that the exponential term causes much faster convergence to Newtonian dynamics
at large $x$, minimizing the impact on the Solar System.

The RAR has an intrinsic scatter $<0.05$ dex once observational uncertainties
are taken into account (Li et al., 2018). We take this as the uncertainty in
the mean relation i.e. in the value of $\nu$. Figure 1 shows the binned RAR
data (Lelli et al., 2017) and their fit (Equation 5) with 0.05 dex error bars
for $g_{{}_{N}}=\left(0.01-100\right)a_{{}_{0}}$, the approximate range of the
dataset. Any viable modification to gravity should have an interpolating
function which passes through this region. We also show $\nu_{{}_{GD}}$ for
different values of the parameter $\alpha$ (Equation 4). To match constraints
from galaxy rotation curves, ${\alpha=0.4-1}$ since
$a_{{}_{0}}=1.2\times{10}^{-10}$ m/s2 is fixed by the relation between galaxy
baryonic masses and flatline rotation curve levels (Begeman et al., 1991).

##  3 Solar System constraint

Equation 1 indicates that the MOND radius of the Sun is $r_{{}_{M}}=7000$
astronomical units (7 kAU), implying that the whole Solar System lies well
inside $r_{{}_{M}}$ and is not much affected by MOND. However, the vastly
superior data quality within the Solar System makes it important to explore
what the GD model predicts for $g_{{}_{N}}\gg a_{{}_{0}}$. We focus on
spacecraft tracking data at heliocentric distances $\la 10$ AU. Planets in
this region should feel an extra Sunwards acceleration of $\alpha a_{{}_{0}}$
(Equation 3). This aspect of the dipole model was also discussed in section
4.1 of Hajdukovic (2020) and in its abstract, where it was claimed that such a
small effect (estimated at $6\times{10}^{-11}$ m/s2) cannot be detected in the
Solar System. We will see that this is incorrect.

###  3.1 Order of magnitude estimate

After some observing duration $t$, the anomalous acceleration of $\alpha
a_{{}_{0}}$ implies a position drift of $\approx\alpha a_{{}_{0}}t^{2}/2$
starting from the same initial position and velocity. For $\alpha=0.4$, the
expected deviation from Keplerian motion is 600 km after 5 years for a planet
like Saturn with a much longer orbital period.

Iorio (2019) first pointed out that this casts serious doubt on the GD model
since radio tracking data for the Cassini spacecraft has an accuracy of 32 m,
but we have not detected such large deviations from its conventionally
calculated trajectory (table 11 of Viswanathan et al., 2017).

###  3.2 Epicyclic approximation

We now consider how a planet’s trajectory would be affected by a small non-
Newtonian force. Our derivation will be similar to that in Banik & Kroupa
(2019).

Consider a planet on a near-circular orbit around the Sun at heliocentric
radius $r_{p}$. Its Keplerian orbital frequency
$\Omega=\sqrt{GM_{\odot}/{r_{p}}^{3}}$. Small radial perturbations $r$ can be
understood using the epicyclic approximation
$\ddot{r}=-\Omega^{2}r$.111Epicyclic and orbital frequencies are the same only
for an inverse square central force. Since this is linear in $r$, we may
subtract whatever epicyclic oscillations the planet is undergoing and
concentrate on the evolution of any remaining anomaly $\delta r$. This anomaly
is sensitive to any additional non-Keplerian forces acting on the planet.
Here, we focus on the extra Sunwards acceleration of $\alpha a_{{}_{0}}$ in
the GD model.

$\displaystyle\ddot{\delta r}~{}=~{}-\Omega^{2}\delta r-\alpha a_{{}_{0}}\,.$
(6)

This should be solved subject to the condition $\delta r=\dot{\delta r}=0$ at
some initial time $t=0$ when precise observations of the planet start. There
is no way to know if its position and velocity have been altered by non-
Keplerian forces at earlier times. Such forces are relevant only in so far as
they cause the planet to deviate from the Keplerian trajectory corresponding
to its osculating orbital elements at $t=0$. Thus, the relevant solution to
Equation 6 is

$\displaystyle\delta r$ $\displaystyle=$ $\displaystyle
x_{0}\left(\cos\phi-1\right),\,\text{where}~{}\phi~{}\equiv~{}\Omega
t~{}\text{and}$ (7) $\displaystyle x_{0}$ $\displaystyle=$
$\displaystyle\frac{\alpha a_{{}_{0}}}{\Omega^{2}}~{}=~{}\frac{\alpha
a_{{}_{0}}{r_{p}}^{3}}{GM_{\odot}}\,.$ (8)

For Saturn orbiting at $r_{{}_{Sat}}=9.58$ AU, $x_{0}=1065$ km.

The GD model yields an extra Sunwards force, implying angular momentum is
still conserved. Thus, the angular velocity $\propto\left(r_{p}+\delta
r\right)^{-2}$, leading to a perturbation $\delta p$ parallel to the circular
orbital velocity. This is governed by

$\displaystyle\dot{\delta p}~{}=~{}-2\Omega\delta r\,.$ (9)

For reasons just explained, we also require $\delta p=\dot{\delta p}=0$ at
$t=0$, though the latter is guaranteed if $\delta r=0$ then. Thus, the
relevant solution to Equation 9 is

$\displaystyle\delta p~{}=~{}2x_{0}\left(\phi-\sin\phi\right)\,.$ (10)

The orbital period of Saturn is ${\approx 30}$ years and the Cassini
spacecraft orbited it for almost half this time. The GD-induced displacement
of Saturn when $\phi=\mathrm{\pi}$ is

$\displaystyle\sqrt{\delta r^{2}+\delta
p^{2}}~{}=~{}2x_{0}\sqrt{1+\mathrm{\pi}^{2}}~{}=~{}7025\,\text{km}.$ (11)

The displacement at earlier times is illustrated in Figure 2.

Figure 2: The difference between Newtonian and GD predictions (former
subtracted) for the position of Saturn if $\alpha=0.4$, the minimum needed for
consistency with galaxy rotation curves (Figure 1). The displacements are
$\propto\alpha$. Azimuthal displacements are positive in the direction Saturn
rotates around the Sun. The much smaller orbit of Earth is much less affected
by GDs (Equation 8). Thus, the difference between the models in the Earth-
Saturn range is approximately that in Saturn’s heliocentric distance.

###  3.3 Comparison with observations

The latest planetary ephemerides for Saturn are accurate to 32 m (table 11 of
Viswanathan et al., 2017) thanks to radio tracking data from the Cassini
spacecraft, which orbited Saturn for 13 years (Matson, 1992). The null
detection of deviations from its conventionally calculated trajectory implies
$\alpha\la 10^{-5}$. Since $\alpha\la 0.4$ violates galaxy rotation curve data
(Figure 1), the GD model must be discarded in its present form. This
conclusion was previously reached by Iorio (2019) based on the rate of
perihelion precession, but frequent radio tracking of Cassini provides much
more information than just this time-averaged quantity.

Our results completely rule out MOND interpolating functions of the form in
Equation 4. This form is fundamental to the whole GD approach (section 2 of
Hajdukovic, 2014b). In particular, the $\tanh$ function is derived directly
from the partition function of the dipoles (section 4.2 of Hajdukovic, 2020).
In the remainder of this contribution, we explore the GD model in more detail
to check if it contains features that might weaken this falsification.

###  3.4 Effects of other planets and their dipole haloes

In the Solar System, the GDs would be fully saturated as $g_{{}_{N}}\gg
a_{{}_{0}}$. This approximation $-$ also made in section 5.1 of Hajdukovic
(2020) $-$ implies that the polarization density $\bm{P}$ has its maximum
strength of $P_{max}$ and is aligned with the gravitational field $\bm{g}$.
Thus, the QV effective density

$\displaystyle\rho_{qv}~{}=~{}-\nabla\cdot\bm{P},~{}\text{where}~{}\bm{P}~{}=~{}P_{max}\,\widehat{\bm{g}}~{}\left(g\gg
a_{{}_{0}}\right).$ (12)

We use the convention that $v\equiv\left|\bm{v}\right|$ for any vector
$\bm{v}$ and $\widehat{\bm{v}}$ is the unit vector in the same direction. In
the Solar System, GDs have only a small effect on $\bm{g}$, so
$\widehat{\bm{g}}\approx\widehat{\bm{g}}_{{}_{N}}$. To get an extra Sunwards
acceleration of $\alpha a_{{}_{0}}$, we must have that

$\displaystyle P_{max}~{}=~{}\frac{\alpha a_{{}_{0}}}{4\mathrm{\pi}G}\,.$ (13)

We now find the total mass of the GD halo associated with a planet of mass
$M_{p}$ embedded in an external gravitational field $\bm{g}_{{}_{ext}}$ from
the Sun. The Solar gravity is assumed to be uniform over the whole planetary
halo, an assumption we will justify later.

We use spherical polar co-ordinates $\left(r,\theta\right)$ centred on the
planet with axis parallel to $\widehat{\bm{g}}_{{}_{ext}}$ i.e. towards the
Sun. Applying the divergence theorem to a spherical region of radius $r$
centred on the planet, we get that the volume integral

$\displaystyle\int\rho_{qv}\,dV~{}=~{}-P_{max}\int_{0}^{\mathrm{\pi}}\left(\widehat{\bm{g}}\cdot\widehat{\bm{r}}\right)2\mathrm{\pi}r^{2}\sin\theta\,d\theta\,.$
(14)

We can simplify this by taking $r$ to be sufficiently large such that the
Solar gravity $\bm{g}_{{}_{ext}}$ dominates over the planetary gravity
$\bm{g}_{{}_{int}}=-GM_{p}\widehat{\bm{r}}/r^{2}$. Without the planetary
gravity, $\widehat{\bm{g}}=\widehat{\bm{g}}_{{}_{ext}}$ and so
$\widehat{\bm{g}}\cdot\widehat{\bm{r}}=\cos\theta$, implying
$\int\rho_{qv}\,dV=0$. Thus, we need the first-order perturbation to
$\widehat{\bm{g}}$ caused by $\bm{g}_{{}_{int}}$. The situation is illustrated
in Figure 3.

Figure 3: The total gravitational field $\bm{g}$ at planet-centric position
$\bm{r}$ is the vector sum of the external Solar gravity $\bm{g}_{{}_{ext}}$
and the planetary gravity $\bm{g}_{{}_{int}}$, which deflects the direction of
$\bm{g}$ by a small angle $\phi$ in the region where the Solar gravity
dominates ($g_{{}_{int}}\ll g_{{}_{ext}}$). $\theta$ is the angle between
$\bm{g}_{{}_{ext}}$ and $\bm{r}$.

As $g_{int}\ll g_{{}_{ext}}$, the planetary gravity rotates $\widehat{\bm{g}}$
towards the planet by a small angle

$\displaystyle\phi~{}\approx~{}\frac{g_{int}\sin\theta}{g_{{}_{ext}}}~{}=~{}\frac{GM_{p}\sin\theta}{r^{2}g_{{}_{ext}}}\,.$
(15)

Since $\phi\ll 1$, we get that

$\displaystyle\widehat{\bm{g}}\cdot\widehat{\bm{r}}~{}\equiv~{}\cos\left(\theta+\phi\right)~{}\approx~{}\cos\theta-\frac{GM_{p}\sin^{2}\theta}{r^{2}g_{{}_{ext}}}\,.$
(16)

Substituting this into Equation 14 and using
$\int_{0}^{\mathrm{\pi}}\sin^{3}\theta\,d\theta=4/3$, we obtain a planetary QV
halo mass of

$\displaystyle
M_{halo,p}~{}=~{}\frac{8\mathrm{\pi}GM_{p}P_{max}}{3g_{{}_{ext}}}\,.$ (17)

This result is independent of $r$, implying that the GD halo has a finite
mass. This is similar to how the MOND phantom dark matter halo is truncated in
the presence of an external field (e.g. Wu et al., 2008). The $r$-independence
of our result shows that the effective mass does not lie in the region where
$g_{int}\ll g_{ext}$, so it must lie closer to the planet in the region where
its gravity dominates. This means that the planetary GD halo has an
approximate size of $r_{h}$, defined by the distance where $g_{int}=g_{ext}$.
A planet at heliocentric distance $r_{p}$ has

$\displaystyle r_{h}~{}=~{}r_{p}\sqrt{\frac{M_{p}}{M_{\odot}}}\,.$ (18)

For Jupiter, $r_{h}=0.16$ AU. As other Solar System planets have even less
mass, their GD haloes are concentrated within a very small fraction of their
orbital radius. This justifies our assumption that the Solar gravity can be
treated as uniform over the planetary halo.

Substituting in Equation 13 and the fact that
$g_{ext}=GM_{\odot}/{r_{p}}^{2}$, Equation 17 can be written as

$\displaystyle M_{halo,p}~{}=~{}\frac{2\alpha
a_{{}_{0}}M_{p}{r_{p}}^{2}}{3GM_{\odot}}~{}=~{}\frac{2}{3}\alpha
M_{p}\left(\frac{r_{p}}{r_{{}_{M}}}\right)^{2}.$ (19)

The Solar GD halo interior to the planetary orbit has an effective mass of

$\displaystyle M_{halo,\odot}\left(<r_{p}\right)~{}=~{}\frac{\alpha
a_{{}_{0}}{r_{p}}^{2}}{G}~{}=~{}\alpha
M_{\odot}\left(\frac{r_{p}}{r_{{}_{M}}}\right)^{2}.$ (20)

As the Solar dipole halo is spherically symmetric, a planet is not affected by
$\rho_{qv}$ outside its orbit (though see Section 3.6). We can combine
Equations 19 and 20 to find the ratio between the Jovian effective halo mass
and that of the Solar halo interior to the orbit of Saturn.

$\displaystyle\frac{M_{halo,Jup}}{M_{halo,\odot}\left(<r_{{}_{Sat}}\right)}~{}=~{}\frac{2M_{Jup}}{3M_{\odot}}\left(\frac{r_{{}_{Jup}}}{r_{{}_{Sat}}}\right)^{2}=~{}1.9\times
10^{-4}\,.$ (21)

Clearly, the Solar dipole halo is several thousand times more important to
Saturn than the dipole haloes of other planets.

This result can be understood by approximating that the planetary gravity is
completely dominant inside the region $r<r_{h}$ and completely negligible
outside, such that $\rho_{qv}\neq 0$ only if $r<r_{h}$. In this case, we would
get that

$\displaystyle M_{halo,p}\approx 4\mathrm{\pi}{r_{h}}^{2}P_{max}=\frac{\alpha
a_{{}_{0}}M_{p}{r_{p}}^{2}}{GM_{\odot}}=\alpha
M_{p}\left(\frac{r_{p}}{r_{{}_{M}}}\right)^{2}.$ (22)

This quick estimate agrees rather well with our result in Equation 19. The
halo mass $\propto{r_{h}}^{2}$ because the total amount of GDs scales with
volume but it is their divergence which is relevant, leading to an additional
$1/r_{h}$ geometric factor. This is why the GD theory predicts a fixed extra
acceleration towards a point mass in the saturated regime (Equation 3). As a
result, Jupiter’s dipole halo has an effective mass
$\approx\left(0.16/9.6\right)^{2}\times$ smaller than that in the Sun’s dipole
halo interior to the 9.6 AU radius of Saturn’s orbit.

Therefore, the anomalous Sunwards acceleration of Saturn is almost entirely
determined by the Solar GD halo. Neglecting the haloes of other planets leads
to a fractional uncertainty of order $10^{-4}$, while neglecting its own halo
is justified by the fact that a planet should not be accelerated by its own
halo. Moreover, increasing the effective mass of Jupiter (or any other planet)
yields an extra force towards the planet rather than towards the Sun. Hence,
an extra Sunwards acceleration of $\alpha a_{{}_{0}}$ cannot be cancelled out
by arbitrarily adjusting the effective masses of other planets.

The only possible exception is if we adjust the masses of the terrestrial
planets, as one could argue that these are in a similar direction to the Sun
as perceived from Saturn. Equation 20 implies that we would need to reduce the
mass of e.g. Venus by $\Delta M=0.25M_{\bigoplus}$, almost $5\times$ the mass
of Mercury. We would need a different $\Delta M$ in order to correctly ‘tune’
the orbit of a different gas giant since
$M_{halo,\odot}\left(<r_{p}\right)\propto{r_{p}}^{2}$. In any case, such large
values of $\Delta M$ greatly exceed observational uncertainties because all
known Solar System planets have rather well-known masses thanks to data from
close spacecraft flybys. This also applies to the Sun $-$ the Solar
gravitational parameter $GM_{\odot}$ has been measured to an accuracy of
$2.5\times 10^{-4}GM_{\bigoplus}$ (Pitjeva, 2015).

Hajdukovic (2019) claim that including the gravity of dipoles which “do not
belong to any individual halo” might alleviate the above issues. Since
unpolarized (randomly aligned) GDs have no divergence and thus contribute
nothing to $\rho_{qv}$ (Equation 12), this might refer to regions where the
gravity from a planet is comparable to that from the Sun. It is precisely
these regions which we have just analysed in detail. There is another
situation where it is difficult to identify the dominant object $-$ at
heliocentric distances of several kAU, the Galactic gravity becomes important.
In Section 3.6, we consider the possible impact of GDs in this region on the
orbits of Solar System planets.

###  3.5 Additional planets in the outer Solar System

An undiscovered ninth planet (P9) several hundred AU from the Sun (Batygin &
Brown, 2016) would raise tides on the known Solar System, affecting planetary
orbits. The tidal effect of P9 on the Sun-Saturn relative acceleration is

$\displaystyle
g_{tide}~{}\approx~{}\frac{GM_{{}_{9}}r_{{}_{Sat}}}{{r_{{}_{9}}}^{3}}\,,$ (23)

where $r_{9}$ is the present heliocentric distance of P9. For $g_{tide}$ to be
strong enough to correctly tune the orbit of Saturn, it must be comparable to
$\alpha a_{{}_{0}}$, implying that

$\displaystyle M_{{}_{9}}~{}\approx~{}\frac{\alpha
a_{{}_{0}}{r_{{}_{9}}}^{3}}{Gr_{{}_{Sat}}}~{}=~{}281M_{\bigoplus}\left(\frac{r_{{}_{9}}}{100\,\text{AU}}\right)^{3}\,.$
(24)

However, the proposed P9 has a mass of only ${M_{9}\approx 10\,M_{\bigoplus}}$
and is more distant (${r_{9}\ga 200}$ AU, Batygin & Brown, 2016). In fact, the
latest analysis indicates that P9 must be ${\ga 370}$ AU away if
${M_{9}=10\,M_{\bigoplus}}$ (section 2.6 of Batygin et al., 2019). Such a
distant planet should create only a very small tidal effect on the known Solar
System.

In the GD model, part of the effective mass of P9 would come from its dipole
halo. However, Equation 22 shows that

$\displaystyle\frac{M_{halo,p}}{M_{p}}\,=\,\frac{\alpha
a_{{}_{0}}{r_{p}}^{2}}{GM_{\odot}}\,=\,5.4\times
10^{-5}\left(\frac{r_{p}}{100\,\text{AU}}\right)^{2}\left(\frac{\alpha}{0.4}\right).$
(25)

Moreover, the $M_{9}$ estimate in Batygin & Brown (2016) should already
include its dipole halo since it is based on how P9 perturbs the orbits of
distant Kuiper Belt Objects.

Even if tides from P9 could affect the Sun-Saturn relative acceleration by
$\alpha a_{{}_{0}}$, P9 would not always pull Saturn directly away from the
Sun over a 15 year period. Cassini tracking data would easily reveal any
lateral force of order $\alpha a_{{}_{0}}$ maintained for a few years (Section
3.1). Moreover, extreme fine-tuning would be required for P9 to have the right
tidal effect on the orbit of Saturn (Equation 24).

The effect would be different on other planets as it would scale with the size
of their orbit. However, the GD model predicts that they too should feel an
extra Sunwards acceleration of $\alpha a_{{}_{0}}$ due to the Solar dipole
halo interior to their orbit (Equation 3). This means that different P9 masses
are required to counter this effect for different planets. Consequently, the
possible existence of P9 cannot reconcile the GD model with observations.

###  3.6 Effect of the Galactic gravity

The Solar dipole halo exterior to the orbit of Saturn does not affect its
dynamics because the halo is spherically symmetric. This symmetry eventually
breaks down to mere axisymmetry at distances $\ga 5$ kAU due to the Galactic
gravitational field. As a result, $\rho_{qv}$ in this region can raise tides
on the Solar System. Since the Galactic gravity on the Sun is of order
$a_{{}_{0}}$ (e.g. section 3.6 of Banik & Zhao, 2018), we expect order unity
deviations from Newtonian dynamics at the Sun’s MOND radius of 7 kAU (Equation
1). In other words, we expect tides from the non-spherical $\rho_{qv}$
distribution to have a strength of order $a_{{}_{0}}$ at this distance. Given
that Saturn is $<10$ AU from the Sun, the Sun-Saturn relative acceleration
should only be affected by ${\la 10^{-3}a_{{}_{0}}}$.

Even if these tides were much stronger, tides raised by such distant
$\rho_{qv}$ cannot always cause a Sun-Saturn repulsion while Saturn completes
half an orbit. This is because the tidal field due to $\rho_{qv}$ outside the
Solar System must be divergence free within the Solar System. Repulsive tides
can be maintained around half a Saturnian orbit only if the $\rho_{qv}$
distribution is axisymmetric and the symmetry axis aligns with the orbital
pole of Saturn. However, there is a $95.5^{\circ}$ angle between the North
Ecliptic Pole and the direction towards the Galactic Centre. Therefore, tides
raised by the non-spherical $\rho_{qv}$ distribution several kAU from the Sun
cannot cancel a constant extra Sunwards acceleration of order $a_{{}_{0}}$.
Moreover, tidal accelerations scale with the size of the planetary orbit while
the extra Sunwards acceleration in the GD model is the same for all planets,
making it infeasible for tides to consistently cancel out this effect.

##  4 Conclusions

The GD theory (Hajdukovic, 2010) is similar to MOND with an interpolating
function given by Equation 4. It seeks to explain the MOND phenomenology by
postulating that antimatter has a negative gravitational mass and so falls
upwards in the terrestrial gravitational field. This should be directly
testable in the next few years.

The model also predicts that Solar System planets experience an extra Sunwards
acceleration of ${\alpha a_{{}_{0}}}$, where the MOND parameter
$a_{{}_{0}}=1.2\times{10}^{-10}$ m/s2 in order to match rotation curve
constraints. These require $\alpha\ga 0.4$ (Figure 1), consistent with the
estimate that $\alpha=5/12$ and should certainly lie in the range $0.4-0.5$
(Hajdukovic, 2020). Starting from the same initial position and velocity, this
extra acceleration would cause the position of Saturn to deviate from
conventional expectations by $7000\left(\alpha/0.4\right)$ km over 15 years
(Section 3.2). The Cassini mission at Saturn (Matson, 1992) has been tracked
over roughly this timespan to a precision of 32 m (Viswanathan et al., 2017).
No significant deviations were found from its conventionally calculated
trajectory, casting serious doubt on the GD model (Iorio, 2019).

One possible complication is that in addition to the Solar dipole halo
responsible for an extra Sunwards acceleration, each planet would have its own
dipole halo (Hajdukovic, 2019). However, we showed that these haloes are
truncated when the Solar gravity dominates over that of the planet (Section
3.4). Consequently, the Solar dipole halo interior to the orbit of Saturn
contains vastly more effective mass than the dipole halo of Jupiter, or indeed
any other planet. Even if the effective masses of other planets were
arbitrarily adjusted or new ones invented, their time-varying direction as
viewed from Saturn makes it impossible for them to precisely cancel out an
extra Sunwards acceleration. While this may be possible for a time-averaged
quantity like the rate of Saturn’s perihelion advance, there would still be
km-scale deviations from its conventional trajectory during the course of an
individual orbit. Since Cassini tracking data provides several thousand range
measurements over half its orbital period, such deviations should have been
detected. Moreover, the Solar dipole halo interior to Saturn has an effective
mass of $0.25M_{\bigoplus}$, vastly exceeding observational uncertainties on
the masses of Solar System bodies and the effective masses of their dipole
haloes.

The Galactic gravity causes the Solar dipole halo to depart from spherical
symmetry at distances ${\ga 5}$ kAU. However, $\rho_{qv}$ in this region
should have only a small tidal effect on the Solar System (Section 3.6).
Additionally, the force would not always be aligned with the Sun-Saturn line
while Saturn completes half an orbit.

An extra Sunwards acceleration of order $a_{{}_{0}}$ is thus a strong
prediction of the GD model for any planet in the Solar System. This prediction
is robust to the possible presence of a ninth planet in the outer Solar System
(Section 3.5). Since the anomalous acceleration should be the same for all
planets, it cannot be consistently cancelled by tides raised by sources
outside the Solar System.

Our work confirms the finding of Iorio (2019) that the GD model is falsified
in its present form because it predicts unobserved effects within the Solar
System that exceed observational upper limits by $\approx 5$ orders of
magnitude. This falsification applies to any MOND-like force law with
interpolating function given by Equation 4 $-$ galactic rotation curve fits
require $\alpha>0.4$ while Solar System constraints imply $\alpha<10^{-4}$ at
very high significance. More generally, Solar System ephemerides rule out any
theory that predicts an enhancement to gravity by $\ga 10^{-5}a_{{}_{0}}$
arbitrarily deep into the Newtonian regime. Even though MOND need not imply
such an effect ($\nu$ can be 1 for high $g_{{}_{N}}$), Hees et al. (2016)
constrained its interpolating function based on the fact that the Galactic
external field causes the Solar phantom dark matter halo to become non-
spherical at large distances, imposing a tidal stress (Blanchet & Novak,
2011).

We conclude that the gravitational dipole model proposed by Hajdukovic (2020)
is unable to satisfy both galaxy rotation curve and Solar System constraints.
The impressive successes of Milgromian dynamics must then be explained in some
other way and other solutions must be found to the cosmological constant
problem.

## Acknowledgements

IB is supported by an Alexander von Humboldt postdoctoral research fellowship.
The authors are grateful to D. S. Hajdukovic for visiting them to discuss his
GD model. They also thank the referee for comments which helped to improve
this publication. The graphs were produced using
matlab${}^{\text{\textregistered}}$.

## References

  * Alpha Collaboration & Charman (2013) Alpha Collaboration, Charman A. E., 2013, Nature Communications, 4, 1785
  * Babcock (1939) Babcock H. W., 1939, Lick Observatory Bulletin, 19, 41
  * Banik & Kroupa (2019) Banik I., Kroupa P., 2019, MNRAS, 487, 4565
  * Banik & Zhao (2018) Banik I., Zhao H., 2018, MNRAS, 480, 2660
  * Batygin & Brown (2016) Batygin K., Brown M. E., 2016, AJ, 151, 22
  * Batygin et al. (2019) Batygin K., Adams F. C., Brown M. E., Becker J. C., 2019, Physics Reports, 805, 1
  * Begeman et al. (1991) Begeman K. G., Broeils A. H., Sanders R. H., 1991, MNRAS, 249, 523
  * Bekenstein & Milgrom (1984) Bekenstein J., Milgrom M., 1984, ApJ, 286, 7
  * Bertsche (2018) Bertsche W. A., 2018, Philosophical Transactions of the Royal Society of London Series A, 376, 20170265
  * Blanchet & Novak (2011) Blanchet L., Novak J., 2011, MNRAS, 412, 2530
  * Brusa et al. (2017) Brusa R. S., et al., 2017, in Journal of Physics Conference Series. p. 012014, doi:10.1088/1742-6596/791/1/012014
  * Chae et al. (2019) Chae K.-H., Bernardi M., Sheth R. K., Gong I.-T., 2019, ApJ, 877, 18
  * Desmond (2017a) Desmond H., 2017a, MNRAS, 464, 4160
  * Desmond (2017b) Desmond H., 2017b, MNRAS, 472, L35
  * Efstathiou et al. (1990) Efstathiou G., Sutherland W. J., Maddox S. J., 1990, Nature, 348, 705
  * Famaey & Binney (2005) Famaey B., Binney J., 2005, MNRAS, 363, 603
  * Famaey & McGaugh (2012) Famaey B., McGaugh S. S., 2012, Living Reviews in Relativity, 15, 10
  * Ghari et al. (2019) Ghari A., Famaey B., Laporte C., Haghi H., 2019, A&A, 623, A123
  * Hajdukovic (2010) Hajdukovic D. S., 2010, Ap&SS, 330, 207
  * Hajdukovic (2012) Hajdukovic D. S., 2012, Ap&SS, 337, 9
  * Hajdukovic (2013) Hajdukovic D. S., 2013, Ap&SS, 343, 505
  * Hajdukovic (2014a) Hajdukovic D. S., 2014a, Physics of the Dark Universe, 3, 34
  * Hajdukovic (2014b) Hajdukovic D. S., 2014b, Ap&SS, 349, 1
  * Hajdukovic (2019) Hajdukovic D. S., 2019, Ap&SS, 364, 174
  * Hajdukovic (2020) Hajdukovic D. S., 2020, MNRAS, 491, 4816
  * Hees et al. (2014) Hees A., Folkner W. M., Jacobson R. A., Park R. S., 2014, Physical Review D, 89, 102002
  * Hees et al. (2016) Hees A., Famaey B., Angus G. W., Gentile G., 2016, MNRAS, 455, 449
  * Hoof et al. (2020) Hoof S., Geringer-Sameth A., Trotta R., 2020, JCAP, 2, 012
  * Iorio (2019) Iorio L., 2019, Ap&SS, 364, 126
  * Lelli et al. (2017) Lelli F., McGaugh S. S., Schombert J. M., Pawlowski M. S., 2017, ApJ, 836, 152
  * Li et al. (2018) Li P., Lelli F., McGaugh S., Schombert J., 2018, A&A, 615, A3
  * Matson (1992) Matson D. L., 1992, in Kaldeich B., ed., ESA Special Publication Vol. 338, Symposium on Titan. pp 281–286
  * McGaugh (2011) McGaugh S. S., 2011, Physical Review Letters, 106, 121303
  * McGaugh et al. (2016) McGaugh S., Lelli F., Schombert J., 2016, Phys. Rev. Lett., 117, 201101
  * Milgrom (1983) Milgrom M., 1983, ApJ, 270, 365
  * Milgrom (1986) Milgrom M., 1986, ApJ, 302, 617
  * Milgrom (1999) Milgrom M., 1999, Phys. Lett. A, 253, 273
  * Oman et al. (2015) Oman K. A., et al., 2015, MNRAS, 452, 3650
  * Ostriker & Peebles (1973) Ostriker J. P., Peebles P. J. E., 1973, ApJ, 186, 467
  * Ostriker & Steinhardt (1995) Ostriker J. P., Steinhardt P. J., 1995, Nature, 377, 600
  * Pazy (2013) Pazy E., 2013, Phys. Rev. D, 87, 084063
  * Pérez et al. (2015) Pérez P., et al., 2015, Hyperfine Interactions, 233, 21
  * Peters (1981) Peters P. C., 1981, American Journal of Physics, 49, 564
  * Pitjeva (2015) Pitjeva E. V., 2015, Journal of Physical and Chemical Reference Data, 44, 031210
  * Riess et al. (1998) Riess A. G., et al., 1998, AJ, 116, 1009
  * Rogstad & Shostak (1972) Rogstad D. H., Shostak G. S., 1972, ApJ, 176, 315
  * Rubin & Ford (1970) Rubin V. C., Ford Jr. W. K., 1970, ApJ, 159, 379
  * Smolin (2017) Smolin L., 2017, Physical Review D, 96, 083523
  * Turyshev & Toth (2010) Turyshev S. G., Toth V. T., 2010, Living Reviews in Relativity, 13, 4
  * Turyshev et al. (2012) Turyshev S. G., Toth V. T., Kinsella G., Lee S.-C., Lok S. M., Ellis J., 2012, Physical Review Letters, 108, 241101
  * Verlinde (2016) Verlinde E. P., 2016, SciPost Physics, 2, 16
  * Viswanathan et al. (2017) Viswanathan V., Fienga A., Gastineau M., Laskar J., 2017, NSTIM, 108, 1
  * Wu et al. (2008) Wu X., Famaey B., Gentile G., Perets H., Zhao H., 2008, MNRAS, 386, 2199