CETUP-2023-023 FERMILAB-PUB-24-0284-T

# Does the Sun have a Dark Disk?

Gustavo F. S. Alves  0000-0002-8269-5365 Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Departamento de Física Matemática,
Instituto de Física Universidade de São Paulo, 05315-970 São Paulo, Brasil
Susan Gardner  0000-0002-6166-5546<EMAIL_ADDRESS>Department of Physics
and Astronomy, University of Kentucky, Lexington, KY, 40506-0055, USA  Pedro
Machado  0000-0002-9118-7354 Fermi National Accelerator Laboratory, Batavia,
IL 60510, USA  Mohammadreza Zakeri  0000-0002-6510-5343 Department of Physics
and Astronomy, University of Kentucky, Lexington, KY, 40506-0055, USA

###### Abstract

The Sun is not quite a perfect sphere, and its oblateness, thought to be
induced through its rotation, has been measured using optical observations of
its radius. Its gravitational quadrupole moment can then be deduced using
solar models, or through helioseismology, and it can also be determined from
measurements of its gravitational effects on Mercury’s orbit. The various
assessments do not agree, with the most complete and precise orbital
assessments being in slight excess of other determinations. This may speak to
the existence of a non-luminous disk or ring, where we also note evidence for
a circumsolar dust ring within Mercury’s orbit from the Solar TErrestrial
RElations Observatory (STEREO) mission. Historically, too, a protoplanetary
disk may have been key to reconciling the Sun’s metallicity with its neutrino
yield. The distribution of the non-luminous mass within Mercury’s orbit can
modify the relative size of the optical and orbital quadrupole moments in
different ways. We develop how we can use these findings to limit a dark disk,
ring, or halo in the immediate vicinity of the Sun, and we note how future
orbital measurements of Mercury and near-Sun asteroids can refine these
constraints.

††preprint: …

##  I Introduction

As first noted by Dicke in 1964, an optical measurement of the solar
oblateness of sufficient size could contribute significantly to Mercury’s
perihelion precession and thus test Einstein’s theory of general relativity
(GR) [1] 111 Dicke [1] noted that the existing GR test from the measured
excess of Mercury’s perihelion precession, of claimed $\sim\\!1\%$ precision,
neglected the effects of solar oblateness. Its inclusion could thus open the
possibility of non-GR contributions. . A subsequent measurement of the
oblateness $\Delta_{\odot}$, defined as the difference in the equatorial and
polar radii over the mean solar radius, gave $(5.0\pm 0.7)\times 10^{-5}$,
apparently challenging GR at the sub–10% level [3]. In the intervening
decades, the ability to assess the oblateness has shown steady and significant
progress, and space-based studies have also been made, yielding refined
errors. Moreover, the impact of magnetic-field-correlated brightness
variations on optical measurements of the oblateness have been noted and
quantified [4], removing disagreements and mitigating earlier puzzles [3, 5,
6]. Ultimately, with improved measurements and theory, the gravitational
quadrupole moment $J_{2}$ of the Sun can be determined, with a nonzero value
of about $2\times 10^{-7}$ — and refined determinations of its value from
improved measurements of Mercury’s orbit also support its approximate value,
with the determined errors being some $100-1000$ times smaller than that, as
we shall detail. These new levels of sensitivity open a new frontier, in that
once negligible effects, such as a circumsolar mass, can become appreciable.
Here we compare the visual and gravitational assessments of $J_{2}$ to
constrain the mass and distribution of non-luminous matter in the immediate
vicinity of the Sun.

Different lines of evidence, from different eras in the solar system’s
history, point to the existence of a non-luminous disk. For example, the
measured energy spectrum of the solar neutrino flux determines the strength of
the CNO cycle in the sun [7, 8, 9] with a rate compatible with high
metallicity [10] but not low metallicity [11] solar models. Concomitantly,
there is also a long-standing inconsistency between the element abundances
determined from the spectroscopy of the surface, as in [11], and those
inferred from the interior through helioseismology [12, 13, 11]. This solar
modelling problem can be mitigated if the interior metallicity of the Sun can
differ from that of its surface 222As noted by [15], such would be at odds
with the standard assumption of a homogeneous zero-age Sun., possibly through
the formation of the photosphere with the gas-giant planets in the early solar
system [15]. Later work has shown a metallicity gradient can appear if the sun
formed while within a circumsolar disk [16, 17]. Evidence has also been found
for a circumsolar dust ring at the approximate location of Mercury’s orbit
[18]. The excess mass density is estimated at 5%, and the total mass is not
determined 333R. Howard and G. Stenborg, private communications.. The origins
of such a dust ring are not known, and it could also stem from effects later
in the solar system’s history [20]. A non-luminous circumsolar disk could also
contain a non-Standard Model, or dark matter, component. The capture of such
an exotic component could be considerably enhanced if ordinary matter in a
non-luminous disk exists over a significant period in the solar system’s
history.

We conclude this section with a sketch of the balance of the paper. We first
describe how the observed solar oblateness connects to $J_{2}^{\rm Opt}$, the
visible quadrupole moment, before describing how Mercury’s measured perihelion
precession, along with other data, can be used to determine $J_{2}^{\rm Orb}$,
the quadrupole moment determined through gravitational interactions. Here we
note that, through use of helioseismology, that $J_{2}^{\rm Heli}$ can also be
found, through different methods. We then discuss possible non-luminous
components, including dark matter, and their origin, that could exist within
Mercury’s orbit. We emphasize that the different possibilities influence the
relative sizes of $J_{2}^{\rm Opt}$, $J_{2}^{\rm Heli}$, and $J_{2}^{\rm Orb}$
differently. After considering the world’s data on $J_{2}$, we describe how
the pattern of existing results limit the mass and distribution of non-
luminous matter. Finally we describe future prospects of observational studies
of the inner solar system in making our final summary. In so doing we offer a
perspective both on non-luminous matter constraints, as well as their possible
impact on future, refined GR tests [21]. Although we have emphasized various
determinations of the gravitational quadrupole moment, these studies naturally
yield limits on solar mass loss and on odd gravitational moments as well, and
we note these and their implications as they arise.

##  II Oblateness and the quadrupole moment

A static, isolated, spherical Sun is perturbed by its rotation. If its center
of mass (CM) is at rest and centered at the origin of a parametrized post-
Newtonian (PPN) coordinate system, its gravitational potential, external to
its surface ($r>R_{\odot}$), becomes [22, 23]

$\phi_{\rm
o}(r,\theta)=-\frac{GM_{\odot}}{r}\left[1-\sum_{2n}\left(\frac{R_{\odot}}{r}\right)^{2n}\,J_{2n}\,P_{2n}(\cos\theta)\right]\,,$
(1)

where $\theta$ is the polar angle from the symmetry axis (co-latitude) and the
$P_{2n}$ are Legendre polynomials. The solar mass $M_{\odot}$ is not
absolutely known, nor is $J_{2}$, the quadrupole-moment parameter, in that
they are effective quantities that can be modified by mass in the Sun’s
immediate vicinity. Moreover, if either external forces act on the Sun’s CM in
an appreciable way or if non-luminous matter within Mercury’s orbit shifts the
Sun’s CM from its visible one, then the gravitational potential need no longer
be reflection symmetric with respect to its mid-plane and terms odd in $n$ can
appear, via $2n\to n$ in Eq. (1). (We note [24, 25] finds a non-zero torque on
stars within some $3\,\rm kpc$ of the Sun.) Introducing the effective
potential $\Phi=\phi_{\rm o}-\phi_{\Omega}$ at the surface, where
$\phi_{\Omega}$ is generated by rotation, the oblateness $\Delta_{\odot}$ can
be connected to the gravitational moments via the equipotential condition
$\Phi(R_{e})=\Phi(R_{p})$ to find, working through $n=4$,

$\Delta_{\odot}\equiv\frac{R_{e}-R_{p}}{R_{\odot}}\approx
J_{1}+\frac{3}{2}J_{2}+J_{3}+\frac{5}{8}J_{4}+\frac{\Omega^{2}\,R_{\odot}^{3}}{2GM_{\odot}}\,,$
(2)

where $R_{e}$, $R_{p}$, and $R_{\odot}$ are the equatorial, polar, and mean
solar radius, respectively — and an effective rotation rate $\Omega$ emerges
under the assumption of axially symmetric differential rotation. We refer to
Appendix A for all details. Neglecting all $J_{i}$ save for $J_{2}$ yields the
long-known result of [26], whereas setting $J_{1}=J_{3}=0$ yields that of
[23]. If $J_{2}$ is determined from $\Delta_{\odot}$ with $J_{i\neq 2}=0$, we
term it $J_{2}^{\rm opt}$.

##  III Interpreting the perihelion precession

Some 92% of Mercury’s perihelion precession stems from precisely known
perturbations from the planets, particularly from Venus, Jupiter, and the
Earth/Moon system [27, 28]. Effects from non-Newtonian gravitational effects
and the Sun’s $J_{2}$ are also well-known but are more poorly determined [27,
28] — and their most complete and precise determinations come from analyses
[28, 29] of near-Mercury radio ranging and Doppler tracking data from the NASA
MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER)
spacecraft mission [30]. Here Mercury’s mean orbit frame is defined with
respect to the International Celestial Reference Frame [31], whose variation
in orientation is sufficiently small that its impact on the perihelion
determination lies beyond the sensitivity of the MESSENGER data [28]. With
this choice the perihelion precession rate is ultimately determined from the
slope of the phase angle determined from fitting the data with a function
containing both steadily increasing and periodic time-dependent features. The
quantity $\dot{\varpi}$, the rate of perihelion precession along Mercury’s
orbit plane, appears linearly in time $t$ and contains the gravitoelectric
(GE), Einstein-Lense-Thirring (ELT), and solar quadrupole moment contributions
within it. In the PPN formalism, the GE effect is linear in $\beta$ and
$\gamma$ (and is nonzero in GR [32], in which $\beta=\gamma=1$), with the ELT
effect being numerically nearly negligible if the solar angular momentum
$S_{\odot}$ determined from helioseismology is used [33]. In [28], e.g., the
ELT effect is absorbed within their error analysis. Nevertheless, a
determination of $\dot{\varpi}$ contains $\beta$, $\gamma$, and $J_{2}$, so
that additional information is needed to separate them. Using information on
$\gamma$ from the Cassini mission, $\gamma-1=(2.1\pm 2.3)\times 10^{-5}$ [34]
and the time structure of the precession angle, both $\beta$ and $J_{2}$ can
be determined [28]. The analysis of [28] uses MESSENGER data from its four-
year orbital phase (2011-2015), whereas that of [29] uses all of the MESSENGER
data over its fly-by and orbital phases (2008-2015) and can employ the
Nordtvedt parameter [35, 36] constraint $\eta=4\beta-\gamma-3=0$ as well. The
values of $J_{2}$ from these studies and others are collected in the Tables in
Appendix B; we will also consider the values that emerge if $\beta=\gamma=1$,
as in GR. These studies also offer constraints on time-dependent effects,
particularly through the apparent time-independence of the residuals in the
perihelion precession fit of [28] and of the estimated rate of change in the
solar gravitational parameter $\mu\equiv GM_{\odot}$, namely
$\dot{\mu}/\mu=(-6.13\pm 1.47)\times 10^{-14}\,\rm yr^{-1}$ [29], to be
compared with the planetary ephemerides result of $(-10.2\pm 1.4)\times
10^{-14}\,\rm yr^{-1}$ [37], where we note [38] for a review. We return to the
implications of these results in Sec. VI.

##  IV Possible non-luminous components

Although the dust in the solar system can reflect light [39], it does not emit
visible light, and thus we have classified it as non-luminous matter. A dusty
disk appears to exist throughout much of the solar system [40], and
simulations suggest that its structure differentiates between the inner and
outer solar system [41]. A population of micrometeroids, seeded by collisional
grinding of zodiacal dust, may also be relevant within 1 AU [42]. We have
noted that a ring of dust has been discovered in the path of Mercury’s orbit
[18], from the STEREO mission [43], and a model of its mass distribution
suggests its mass could be about $(1.02-4.05)\times 10^{12\pm(\approx 1)}\,\rm
kg$, roughly equivalent to the mass of a single asteroid [20]. Known asteroids
range from about $10^{10-21}\,\rm kg$ in mass [44].

Yet dust is not the only possibility. Dark matter, e.g., may also contribute.
Studies of stellar tracers in the solar neighborhood suggests that the local
dark matter density is only some $\rho_{\rm dm}=0.4\,\rm GeV/cm^{3}$ [45], and
a spheroid of that density would contribute a mass of $6\times 10^{11}\,\rm
kg$, or $3\times 10^{-19}\rm M_{\odot}$, within Mercury’s orbit. Moreover, it
has long been thought that gravitational focusing mechanisms exist, modifying
the dark-matter velocity distribution and acting to increase the dark matter
density within the solar system [46, 47, 48, 49, 50, 51, 52]. There have also
been studies of the capture of dark matter within celestial bodies, and for
models with sufficient dark-matter–matter interactions, significant
enhancements over the nominal dark-matter density are possible [53, 54, 55,
56, 57, 58], showing that dark matter could contribute non-trivially to the
mass of a celestial body within its observed radius. Measurements of the
Earth’s heat budget do limit, however, the possibility of strong dark-matter-
matter interactions [59, 60], though some of the internal heating of Jovian
planets could speak to planet-bound dark matter [61].

A celestial body can also possess a dark halo, and this can emerge if the
dark-matter particle simply possesses self-interactions. In the “gravi-atom”
mechanism of [62], for example, two-body scattering, particularly of ultra-
light dark matter, in the gravitational field of a celestial body can yield
dark-matter capture and ultimately a dark halo. We will consider this scenario
further in Sec. VII. If the dark matter couples to neutrinos, then neutrino
oscillation data can probe the Earth’s dark halo [63].

The possibility of exotic dark structures, such as exoplanets [64], filaments
[65], or other macroscopic objects [66], have also been suggested. These
macroscopic objects need not have a dark matter component, as, say, in the
case of magnetic black holes [67], but others, such as primordial black holes
(PBHs), can also function as a dark matter candidate and are as yet poorly
constrained in the asteroid mass range [68]. Their transits of the solar
system can be limited through planetary ephemerides [69], as well as via
observational means [70]. Gravitational wave detection can also probe the
possibility of local, macroscopic dark matter [71] and of the solar system’s
mass distribution [72]. The Sun and planets can also emit light dark
particles, as, e.g., in [73, 74]. However, the time rate of change in the
gravitational parameter is smaller than that estimated from the net effect of
dust capture and the solar wind [29], so that no limit on dark-matter emission
is currently possible.

Constraints on all these scenarios emerge from measurements of planetary
motion [75, 76, 77, 78, 79, 80, 81], which show that the maximum density at
the Earth’s orbit is limited to be below $10^{2}-10^{5}$ GeV/cm3, depending
slightly on the profile shape. Somewhat weaker constraints emerge from studies
of the inner planets [81]. We note estimates of dark-matter capture on the
solar system fall rather short of these limits [48, 49]. There are also
constraints on the Earth’s own dark halo [61], and on the dark matter density
by analyzing the propagation of light in the solar system [82]. Dark matter
would cause time delay and frequency shift of the light, though the
constraints are not very stringent [82, 83].

In this paper we constrain the mass and distribution of non-luminous matter
external to the Sun and planets and thus constrain dark-matter models
connected to these possibilities.

##  V Gravitational Quadrupole Moment Determinations

Determinations of $J_{2}$ began decades ago and also range over decades. Three
distinct methods have been used. There have been direct optical measurements
of the solar oblateness $\Delta_{\odot}$, which, when combined with theory, as
in Eq. (2), yield $J_{2}^{\rm Opt}$. Helioseismological data probes the
structure of the Sun; this, when combined with a model for its structure, can
be used to infer $J_{2}^{\rm Heli}$. Finally, as we have noted, precision
measurements of planetary motion can also be used to infer $J_{2}^{\rm Orb}$.
We emphasize that the orbital determinations are sensitive to the existence of
mass within the orbit in question. In this paper we focus on determinations of
$J_{2}^{\rm Orb}$ from measurements of Mercury’s orbit, to address the
possibility of non-luminous matter in the immediate region of the Sun. There
is a long history of solar oblateness measurements [3, 5], with improved
assessments once space-based studies became possible [84]. Surface magnetism
and other effects can impact the observed shape [85, 86], and Fivian _et al._
have provided a corrected assessment that would remove the enhancement from
solar magnetism [86]. However, we note that discussions of the shape
assessment, and the possibility of a time-dependent solar shape, continues
[87, 88, 89].

We summarize the most pertinent determinations here and note Appendix B for a
more complete list. We report the results for $J_{n}$ in units of $10^{-7}$
throughout. From Park _et al._ [28]

$J_{2}^{\rm Orb}=2.25\pm 0.09\,\rm\,;\,J_{2}^{\rm
Orb}|_{\beta=\gamma=1}=2.28\pm 0.06\,,$ (3)

whereas from Genova _et al._ [29]

$J_{2}^{\rm Orb}=2.246\pm 0.022\,\rm\,;\,J_{2}^{\rm
Orb}|_{\beta=1;\eta=0}=2.2709\pm 0.0044\,.$ (4)

Here we quote the quadrupole moment assessment within a PPN framework, as well
as its value upon assuming GR. In the first case, Genova _et al._ also find
the PPN parameter $\eta=(-6.6\pm 7.2)\times 10^{-5}$, noting that nonzero
$\eta$ can be associated with a shift in the solar system barycenter [29] —
and vice versa. Despite the appreciable shift in the central value and smaller
uncertainties associated with the GR limit, we consider both results in our
analysis.

Turning to the optical assessment of $\Delta r\equiv R_{e}-R_{p}=(8.01\pm
0.14)\,\rm milliarcsec\,(mas)$ due to Fivian _et al._ [86] using the space-
based Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) instrument
444We note https://hesperia.gsfc.nasa.gov/rhessi3/ for further information and
Ref. [87] for discussion of this result and their finding of $\Delta r=7.20\pm
0.49\,\rm,mas$., the value of $J_{2}$ can be determined from
$J_{2}=(2/3)(\Delta r-\Delta r_{\rm surf})/R_{\odot}$ [5] with $\Delta r_{\rm
surf}\approx 7.8\,\rm mas$ [26] and $R_{\odot}=9.5963\times 10^{5}\,\rm mas$
[91], its radius at $1\,\rm AU$ 555This value is employed by Fivian et al.
[86], though it has been officially updated to $9.5923\times 10^{5}\,\rm mas$
by the International Astronomical Union in 2015 [163], with additional space-
based measurements from PICARD/SODISM supporting this later result [164]. For
our purposes $J_{2}$ is sufficiently small that this refinement does not
matter.. Noting $\Delta{r_{\rm surf}}/R_{\odot}$ corresponds to the last term
in Eq. (2), they find [86]

$J_{2}^{\rm Opt}=1.46\pm 1.0\,,$ (5)

where the error does not include an error in $\Delta r_{\rm surf}$. If we
were, rather, to employ the $\Delta r$ measurement in [87], $\Delta r=7.20\pm
0.49\,\rm mas$, the associated $J_{2}$ would be negative but consistent with
zero with a much larger error. For clarity we reiterate that in this paper we
define the oblateness as $\Delta_{\odot}\equiv\Delta r/R_{\odot}$, which
evaluates to $(8.35\pm 0.15)\times 10^{-6}$, because, in contrast, Fivian _et
al._ [86] term $\Delta r$ the oblateness.

We now turn to the assessments from helioseismology. The $J_{2}^{\rm Heli}$
comes from observations of the oscillations in the Sun’s surface interpreted
within a solar model. Here we highlight the work of Mecheri and Meftah for
their use of multiple, space-based solar oscillation data sets and multiple
solar evolution models, all within an integral equation approach that
eliminates the need for a solar differential rotation model [93]. This is an
update of Mecheri et al. (2004) that uses data from the Michelson Doppler
Imager on the Solar and Heliospheric Observatory (SoHO/MDI) and models for the
solar differential rotation. Using models (a) and (b), respectively, from [94]
they report [23]

$\displaystyle J_{2}^{\rm Heli}=2.201\,$ ; $\displaystyle\,J_{4}^{\rm
Heli}=-5.601\times 10^{-2}\,.$ (6) $\displaystyle J_{2}^{\rm Heli}=2.198\,$ ;
$\displaystyle\,J_{4}^{\rm Heli}=-4.805\times 10^{-2}\,.$ (7)

In contrast, $J_{2}^{\rm Opt}$ in Eq. (5) follows from setting $J_{4}$ to
zero, so that $J_{4}$’s size indicates the size of the theoretical systematic
error in that procedure. The outcome of [23] is reported as $J_{2}^{\rm
Heli}=2.20\pm 0.03$ by [29] and results from combining all the $J_{2}$’s in
Table II of [23] (including a uniform solar rotation result) 666The error was
assessed by accounting for their differences and multiplying by two. A.
Genova, private communication.. The update of the 2004 analysis by [93]
reports $J_{2n}$ with $n=1\dots 5$ using data from either the Helioseismic and
Magnetic Imager on the Solar Dynamics Observatory (SDO/HMI) or SoHO/MDI using
the online compilation of [96] with the analysis of [97, 98], upon the use of
either the CESAM [99] or ASTEC [100] theoretical (solar evolution) frameworks.
With CESAM, they report

$\displaystyle{\rm SDO/HMI}:J_{2}^{\rm Heli}=2.211\,;\,J_{4}^{\rm
Heli}=-4.252\times 10^{-2}\,;$ (8) $\displaystyle{\rm SoHO/MDI}:J_{2}^{\rm
Heli}=2.204\,;\,J_{4}^{\rm Heli}=-4.064\times 10^{-2}\,.$ (9)

Using ASTEC, they find [93]

$\displaystyle{\rm SDO/HMI}:J_{2}^{\rm Heli}=2.216\,;\,J_{4}^{\rm
Heli}=-4.256\times 10^{-2}\,;$ (10) $\displaystyle{\rm SoHO/MDI}:J_{2}^{\rm
Heli}=2.208\,;\,J_{4}^{\rm Heli}=-4.069\times 10^{-2}\,.$ (11)

The data sets SDO/HMI and SoHO/MDI are independent and correspond to
observations from 2010 Apr – 2020 July and 1996 May – 2008 March,
respectively. The slight differences in the reported outcomes of the two data
sets could stem from a time dependence [101, 93] or from an observational
systematic error that differentiates between the two instruments — or both. We
note that the SDO/HMI and MESSENGER results are roughly contemporaneous.
Computing the average and standard deviation of the results using one data set
or the other, we find

$\displaystyle{\rm SDO/HMI}:\,$ $\displaystyle J_{2}^{\rm Heli}=2.214\pm
0.002\,;\,$ $\displaystyle J_{4}^{\rm Heli}=(-4.254\pm 0.002)\times
10^{-2}\,;$ $\displaystyle{\rm SoHO/MDI}:\,$ $\displaystyle J_{2}^{\rm
Heli}=2.206\pm 0.002\,;\,$ $\displaystyle J_{4}^{\rm Heli}=(-4.067\pm
0.003)\times 10^{-2}\,.$

Here the error stems entirely from the use of different solar models. We note
that the midpoints of the observing periods for the SDO/HMI and SoHO/MDI data
sets are separated by 11 years, where $J_{2}$ changes by $|\Delta J_{2}^{\rm
Heli}|=0.008\pm 0.002$, since the errors are not independent. For reference,
Antia _et al._ [102] use the SoHO/MDI data to find a $J_{2}$ of $2.220\pm
0.009$, where the error denotes the estimated time dependence [102]. Given
this, we suppose that $|\Delta J_{2}^{\rm Heli}|$ could indeed stem from time
dependence, rather than from an observational systematic error. We thus employ
$J_{2}^{\rm Heli}$ in Eq. (V), using SDO/HMI data, as the helioseismological
outcome to be compared with orbital determinations of $J_{2}$ from the
MESSENGER data, because they are approximately contemporaneous. Using the
$J_{2}^{\rm Orb}$ result of [29] that assumes GR, we find a small, albeit
significant, difference in the assessment of $J_{2}$ from orbital and solar
measurements:

$J_{2}^{\rm Orb}|_{\beta=1;\eta=0}-J_{2}^{\rm Heli}=0.057\pm 0.006\,.$ (14)

Of course the error in $J_{2}^{\rm Heli}$ in Eq. (V) that appears in this
difference comes from the use of different solar models, and we can wonder
about how else to estimate it. Alternatively, we estimate the error in
$J_{2}^{\rm Heli}$ by using all four values in Eqs. (V) and (V) demanding that
$\chi^{2}=1$ for a flat-line fit. This gives an average $J_{2}^{\rm Heli}$ of
$2.210\pm 0.004$, so that Eq. (14) becomes $0.061\pm 0.009$ — and we also find
a positive difference at high significance. Separately, as we detail in
Appendix B, if we compute an average and standard deviation of our collected
helioseismological results, we find

$\langle J_{2}^{\rm Heli}\rangle\Big{|}_{\rm all}=2.213\pm 0.002\,,$ (15)

which, interestingly, is also consistent with Eq. (V). The significance of the
result in Eq. (14) does rely, however, on assuming GR values for the PPN
parameters $\beta$ and $\eta$, although this is the usual procedure if the
possibility of dark matter is considered. Nevertheless, if we use the value of
$J_{2}^{\rm Orb}$ in Eq. (4) that does not assume GR, Eq. (14) becomes
$0.032\pm 0.024$, making the difference positive but no longer significantly
so. On the other hand, referring to the Tables in Appendix B, we observe that
each of the modern $J_{2}^{\rm Heli}$ assessments (after 1990) are numerically
smaller than those relying on MESSENGER data [28, 29], even without assuming
GR, though not all are significantly so. The tabulated values of the optical
$J_{2}$ computed from the observed oblateness show that the errors are much
larger, so that a similar comparison with the orbital results is not possible.
We also show alternate orbital $J_{2}$ assessments in Appendix B. Analyses in
which the ELT effect is taken into account and the various PPN parameters are
fitted are arguably better and give a plausibly more robust inference of
$J_{2}$ than studies that do neither. The three different methods we have
considered to assess $J_{2}$ probe the Sun’s shape, but they are also
sensitive to different mass distributions. Their difference can limit the
presence of additional matter within Mercury’s orbit, as we discuss next.

##  VI Limits on mass and distribution of non-luminous matter

Disagreements in the assessment of nominally equal quantities, as we have
noted for $J_{2}$, can signal the presence of non-luminous matter within
Mercury’s orbit. If that matter were a spherical dark matter halo, then we
would expect its presence to reduce $J_{2}^{\rm Orb}$ with respect to visible
assessments, yielding

$J_{2}^{\rm Orb}<J_{2}^{\rm Opt}\,,\,J_{2}^{\rm Heli}\,.$ (16)

On the other hand, if a dark disk co-planar with the planetary orbits were
present, then we would expect, rather,

$J_{2}^{\rm Orb}>J_{2}^{\rm Opt}\,,\,J_{2}^{\rm Heli}\,.$ (17)

We can also use the differences in the $J_{2}$ assessments to limit the excess
mass enclosed, as we determine later in this section. Observational limits can
also be placed on the change in the Sun’s mass with time, namely
$\dot{M}_{\odot}$, but, as we have noted, the current limit is less than the
changes expected from dust deposition or mass loss through the action of the
solar wind [29]. Also we can limit the possibility of unexpected gravitational
perturbations on Mercury’s orbit. Finally, since we have a “visible”
assessment of $J_{2}$ determined in two different ways, we can also use Eq.
(2) to back out a limit on the odd $J_{n}$ moments. We note that $J_{2n+1}$
can be a signed quantity.

We now work through these points in reverse order. To determine a limit on the
odd $J_{n}$ moments we combine $\Delta_{\odot}$ from Fivian _et al._ [86],
which is from 2008, with the roughly contemporaneous assessment using SoHO/MDI
data from Mecheri and Meftah [93], averaged over solar models, as reported in
Eq. (V). With this, using Eq. (2) and $\Delta r_{\rm surf}/R_{\odot}$ for its
last term, as in [86], we thus find

$J_{1}+J_{3}\simeq J_{1}=-1.1\pm 1.5\,.$ (18)

We note that $J_{1}$ can easily be larger than $J_{3}$ because it can be
generated by either external forces on the Sun or by a shift of the solar CM
from its visible one. Although $J_{1}$ is consistent with zero, it is
intriguing that its central value obeys $J_{1}<0$, which is compatible with
the Sun’s location north of the Galactic mid-plane. We recall that the
inverse-square-law nature of gravitational forces ensures that the
acceleration of a star is typically determined by the mass distribution at
large scales, rather than by stars in its immediate vicinity [103].

Now we turn to the question that motivates this paper: how the amount of non-
luminous mass, supposing some distribution, is limited through the different
determinations of $J_{2}$. The constraints we develop in this section concern
limits on the approximate total mass within Mercury’s orbit. For reasons of
simplicity, we consider the possibility of a dark disk or ring and a spherical
dark halo — though some mixture of these, or some more complicated shape,
could also be possible. We have noted that dust studies point to the existence
of a dusty ring coinciding with the average path of Mercury’s orbit [18], and
our studies limit its possible mass, but not just. Since dust, or conventional
matter more generally, could act as a substrate for the ultralight-dark-matter
capture mechanism described in [62], or that other dark-matter enhancement
mechanisms could operate, so that dark matter, i.e., non-standard, non-
luminous matter, could also contribute appreciably to the non-luminous mass
within Mercury’s orbit. We note that the most precisely determined $J_{2}$
values, as in Eq. (14), may differ significantly from zero, suggesting that
Eq. (17) holds and a non-luminous disk is favored. Nevertheless, in what
follows, we suppose either a disk/ring or spherical halo scenario and limit
the associated mass with each.

Limiting the mass of a non-luminous disk or ring: We conceptualize this object
as an axially symmetric circumsolar ring potentially composed of both dust and
dark matter, centered about the CM of the visible Sun. It is characterized by
a mass $M_{r}$, a uniform density $\sigma$, an inner radius $R_{i}$, an outer
radius $R_{o}$, and a height $h$ along the $\hat{\mathbf{z}}$-axis, defined as
perpendicular to the plane of Mercury’s orbit, noting that the Sun’s rotation
axis is tilted about $7.25^{\circ}$ with respect to it [104]. In what follows,
though, we will set that tilt to zero for simplicity. To compute its
gravitational quadrupole moment, we assume the ring is also rigid. In a
Cartesian coordinate system, the moment of inertia tensor is diagonal, and
[105]

$J_{2}^{\rm ring}=\frac{1}{M_{\rm r}{\bar{R}}^{2}}(I_{z}-I_{x})$ (19)

with $\bar{R}\equiv(R_{i}+R_{o})/2$, where the moment of inertia in the
direction $s$ is given by

$I_{s}=\int
d^{3}r\,\rho(\mathbf{r})\left[r^{2}-(\mathbf{r}\cdot\hat{\mathbf{e}}^{(s)})^{2}\right]\,.$
(20)

More concretely,

$\displaystyle I_{z}$ $\displaystyle=$ $\displaystyle\frac{M_{\rm
r}}{2}\left[R_{o}^{2}+R_{i}^{2}\right]\,,$ (21) $\displaystyle I_{x}$
$\displaystyle=$ $\displaystyle M_{\rm
r}\left[\frac{1}{4}\left(R_{o}^{2}+R_{i}^{2}\right)+\frac{h^{2}}{12}\right]\,$
(22)

and thus

$\displaystyle J_{2}^{\rm
ring}=\frac{1}{\bar{R}^{2}}\left[\frac{1}{4}\left(R_{o}^{2}+R_{i}^{2}\right)-\frac{h^{2}}{12}\right].$
(23)

Combining the potential of the dust ring with the Sun’s for $r>R_{o}$, Eq.
(1), yields:

$\phi_{o}^{\rm tot}=\frac{M_{\odot}+M_{\rm r}}{r}-\left(\frac{J_{2}^{\rm
int}M_{\odot}R_{\odot}^{2}+J_{2}^{\rm ring}M_{\rm
r}\bar{R}^{2}}{r^{3}}\right)\left(\frac{3\cos^{2}\theta-1}{2}\right),$ (24)

where $J_{2}^{\rm int}$ is the intrinsic solar quadrupole moment, which may be
measured through a direct optical measurement or via helioseismology. Defining
the fractional mass of the ring, $\epsilon_{\rm r}\equiv M_{\rm r}/M_{\rm
tot}$, with $M_{\rm tot}\equiv M_{\odot}+M_{\rm r}$, the extrinsic solar
quadrupole moment ($J_{2}^{\rm ext}$), which can be measured gravitationally,
becomes

$\displaystyle J_{2}^{\rm ext}=$ $\displaystyle(1-\epsilon_{\rm r})J_{2}^{\rm
int}+\epsilon_{\rm r}\left(\frac{\bar{R}}{R_{\odot}}\right)^{2}J_{2}^{\rm
ring}$ $\displaystyle=$ $\displaystyle(1-\epsilon_{\rm r})J_{2}^{\rm
int}+\epsilon_{\rm
r}\left(\frac{1}{R_{\odot}^{2}}\right)\left[\frac{1}{4}\left(R_{o}^{2}+R_{i}^{2}\right)-\frac{h^{2}}{12}\right]\,,$
(25)

which can be solved for $\epsilon_{\rm r}$:

$\displaystyle\epsilon_{\rm r}=\frac{2\left(J_{2}^{\rm ext}-J_{2}^{\rm
int}\right)}{\left(R/R_{\odot}\right)^{2}-2J_{2}^{\rm int}}\,,$ (26)

where

$\displaystyle R\equiv$
$\displaystyle\sqrt{\frac{R_{o}^{2}+R_{i}^{2}}{2}-\frac{h^{2}}{6}}\,.$ (27)

This reduces to $R=R_{i}=R_{o}$ for a very thin ring of negligible height,
$h/R_{i}\approx 0$. Here we assume $h/R_{i}\ll 1$ and note $R_{i}\geq
R_{\odot}$. Defining $\delta J_{2}\equiv J_{2}^{\rm ext}-J_{2}^{\rm int}$ we
rewrite Eq. (26) as

$\displaystyle\frac{\delta J_{2}}{J_{2}^{\rm int}}=$
$\displaystyle\,\epsilon_{\rm r}\left[\frac{1}{2J_{2}^{\rm
int}}\left(\frac{R}{R_{\odot}}\right)^{2}-1\right]\approx\epsilon_{\rm
r}\left[\frac{1}{2J_{2}^{\rm
int}}\left(\frac{R}{R_{\odot}}\right)^{2}\right]\,.$ (28)

Recalling [18] and thus the ring’s proximity to Mercury’s orbit with
$R_{o}\approx R_{i}\approx 0.38$ AU we estimate:

$\frac{\delta J_{2}}{J_{2}^{\rm int}}\approx\epsilon_{\rm r}\,(1.5\times
10^{10}),$ (29)

implying that a $1\%$ difference in $J_{2}$, that is, $\delta J_{2}\sim
10^{-2}J_{2}^{\rm int}$, could be caused by $M_{\rm r}\sim
10^{-12}\,M_{\odot}$. Such a mass can contain a dark-matter component, though
its value corresponds to that of a relatively large asteroid, significantly
more massive than those more commonly found in the asteroid belt. We note,
however, the dwarf planet Ceres, the most massive known object in the asteroid
belt between Mars and Jupiter, has a mass of about $5\times
10^{-10}M_{\odot}$, whereas Jupiter’s moon Amalthea has a mass of about
$\left(1.04\pm 0.08\right)\times 10^{-12}\,M_{\odot}$ [106]. We now turn to
concrete limits.

Given determinations of $J_{2}^{\rm ext}$ from orbital measurements
$J_{2}^{\rm Orb}$ and of $J_{2}^{\rm int}$ from both optical and
helioseismological measurements $J_{2}^{\rm Opt}$ and $J_{2}^{\rm Heli}$, we
constrain the mass in a circumsolar ring or disk through limits on
$\epsilon_{r}$ in Eq. (26). Since we assign $M_{\odot}$ to mass within
$R_{\odot}$, practically $R/R_{\odot}>1$, and since $J_{2}\sim{\cal
O}(10^{-7})$, we write

$\epsilon_{r}=2\left(\frac{R_{\odot}}{R}\right)^{2}\delta J_{2}\,,$ (30)

so that our ability to determine $\delta J_{2}$ limits the maximum mass in the
ring, regardless of our confidence that $\delta J_{2}>0$. We compute the
maximum value of $\epsilon_{r}$ at 95% CL through different evaluations of the
maximum value of $\delta J_{2}$ at $2\sigma$, and we display the results in
Fig. (1), noting that $R$ can range from within $R_{\odot}$ to the location of
Mercury’s orbit.

To accomplish this, we use $J_{2}^{\rm Orb}$ derived from the latest MESSENGER
analysis [29], given in Eq. (4). This is then combined with either optical
[86], as shown in Eq. (5), or helioseismological measurements [93], noted in
Eqs. (8) and (10). With the most significant difference, given in Eq. (14), we
have

${\rm max}\left(\delta J_{2}\right)\Big{|}_{2\sigma}=0.069\,,$ (31)

whereas combining Eq. (4), using either the $\beta=1\,;\,\eta=0$ constrained
(GR) or unconstrained value, with the optical result of Eq. (5) yields

${\rm max}\left(\delta J_{2}\right)\Big{|}_{2\sigma}=2.82\,(2.83)\,,$ (32)

respectively. We report the limits on $\epsilon_{r}$ with $R$ at 95% CL
($2\sigma$) that come out, using the GR-constrained orbital result in Fig. 1.
Furthermore, we also depict the constraint on $\epsilon_{r}$ that follows from
the limit $J_{2}^{\rm ext}<3\times 10^{-6}$ [107], determined by demanding
that the lunar librations, which are modelled and observed from the analysis
of lunar laser ranging (LLR) measurements of the Earth-Moon distance [108], do
not exceed $3\sigma/2$ of the LLR residuals. We note that this limit is
sufficiently weak that the impact of the use of $J_{2}^{\rm Opt}$ in Eq. (5)
or $J_{2}^{\rm Heli}$ from combining Eqs. (8,10) is indistinguishably small.

Figure 1: Limits at 95% CL ($2\sigma$) on the maximum mass fraction,
$\epsilon_{\rm r}\equiv M_{\rm r}/M_{\rm tot}$, of a dark matter/dust ring or
disk around the Sun, derived from the difference in gravitational and
optical/helioseismological determinations of the gravitational quadrupole
moment $J_{2}$ as in Eq. (30), noting the maximum values of $\delta J_{2}$ at
$2\sigma$ in Eqs. (31, 32), plotted as dashed lines. The horizontal axis, $R$,
is determined by the geometric dimensions of the dark matter/dust ring or
disk, as defined in Eq. (27). The dashed line corresponds to constraints
inferred from the lunar libration limit on $J_{2}^{\rm ext}$ [107]. We refer
to the text for all details.

Thus far we have limited $\epsilon_{r}$ supposing that a dark disk or ring
exists. Now, we assess the likelihood of this scenario using Eq. (17) and the
observed values of $J_{2}^{\rm ext}$ and $J_{2}^{\rm int}$. Adopting a
Gaussian prior for $\delta J_{2}$, the relevant posterior probability is given
by [109]

$P(\delta J_{2}>0\,|\,J_{2}^{\rm ext},J_{2}^{\rm int})=1-F_{\delta J_{2}}(z),$
(33)

where $F_{\delta J_{2}}(z)$ is the cumulative distribution function (CDF) of
the standard normal distribution,

$F_{\delta
J_{2}}(z)=\frac{1}{\sqrt{2\pi}}\int_{z}^{\infty}dt\exp(-t^{2}/2)\,,$ (34)

and $z=-\mu_{\rm post}/\sigma_{\rm post}$. Here, $\mu_{\rm post}$ and
$\sigma_{\rm post}$ are the posterior mean and standard deviation,
respectively. For a non-informative (flat) prior where the standard deviation
of the prior ($\sigma_{\rm prior}$) tends towards infinity, $\mu_{\rm post}$
and $\sigma_{\rm post}$ simplify to yield

$z\approx\frac{\mu_{J_{2}^{\rm int}}-\mu_{J_{2}^{\rm ext}}}{\sigma_{J_{2}^{\rm
ext}}+\sigma_{J_{2}^{\rm int}}}.$ (35)

The deviation of the computed posterior probabilities, as formulated in Eq.
(33), from one are presented for six distinct cases in Table 1. We observe
that even in the unconstrained cases that the probability that a disk exists
is high.

int ext | constrained | unconstrained
---|---|---
Opt | 0.210 | 0.221
Heli | $1.0\times 10^{-21}$ | 0.091
Heli-All | $2.4\times 10^{-17}$ | 0.089
Table 1: The deviation of the disk structure’s presence likelihood from one,
$1-P(\delta J_{2}>0)$, evaluated using Eq. (33). We compared the GR-
constrained (unconstrained) $J_{2}^{\rm ext}$ values from MESSENGER data, as
given by Eq. (4), against the best optical and helioseismological
determinations of $J_{2}^{\rm int}$, given by Eqs. (5, V, 15), respectively.
We refer to the text for all details.

Limiting the mass of a non-luminous spherical halo:

For a spherical halo, setting $J_{2}^{\rm ring}=0$ in Eq. (VI) yields

$\displaystyle\epsilon_{\rm s}=\frac{J_{2}^{\rm int}-J_{2}^{\rm
ext}}{J_{2}^{\rm int}}\,,$ (36)

in which we denote the fractional mass of the spherical halo by $\epsilon_{\rm
s}\equiv M_{\rm s}/M_{\rm tot}$. We apply the Feldman-Cousins [110] method to
establish limits on $\epsilon_{\rm s}$ given pairs of $(J_{2}^{\rm
ext},J_{2}^{\rm int})$ values. The resulting limit on $\epsilon_{\rm s}$,
derived by comparing the constrained (unconstrained) MESSENGER $J_{2}^{\rm
ext}$ values in Eq. (4) with $J_{2}^{\rm Heli}$ from Eq. (V), is given by

$\epsilon_{\rm s}\Big{|}_{2\sigma}=2.5\times 10^{-4}\,(8.3\times 10^{-3}).$
(37)

We now take the existence of Mercury’s circumsolar dust ring into account
[20]. The portion of the dust ring that falls within the orbit of Mercury
would increase $J_{2}^{\rm ext}$, such that Eq. (36) changes to

$\displaystyle\epsilon_{\rm s}=\frac{(1-\epsilon_{r})J_{2}^{\rm
int}-J_{2}^{\rm ext}+(\epsilon_{r}/2)\left(R/R_{\odot}\right)^{2}}{J_{2}^{\rm
int}}\,,$ (38)

which would weaken the bounds on $\epsilon_{s}$ in Eq. (37) as shown in Fig.
2, where we set the dust ring’s radius to $R\approx 0.31$ AU.

Figure 2: Limits at 95% CL ($2\sigma$) on the mass fraction of a spherical
dark matter halo surrounding the Sun, denoted as $\epsilon_{\rm s}\equiv
M_{\rm s}/M_{\rm tot}$, plotted against the mass fraction, $\epsilon_{\rm
r}\equiv M_{\rm r}/M_{\rm tot}$, of a dust ring orbiting within Mercury’s
path. We input the values from Eqs. (4, V) into Eq. (38). The GR-constrained
and unconstrained cases are shown in green and red, respectively. We refer to
the text for all details.

##  VII Implications for dark-matter models

We have found that existing determinations of $J_{2}^{\rm Heli,Opt}$
($J_{2}^{\rm int}$) and $J_{2}^{\rm Orb}$ ($J_{2}^{\rm ext}$) from the
MESSENGER mission favor the existence of a non-luminous disk within Mercury’s
orbit, with Table 1 providing the likelihood that it does not exist. Although
the significance of our claim depends on the results we pick, the limits we
have found on its mass, as shown in Fig. 1, grossly exceed the contributions
we would expect from the cosmological dark matter density and/or the dust ring
that has been discovered near the path of Mercury’s orbit [18]. The observed
dust ring, assessed at an overdensity of $\approx 3-5\%$ [18], is estimated to
be located between heliocentric distances (radii) of $0.356271\,\rm AU$ and
$0.400308\,\rm AU$, with a mass of $(1.02-4.05)\times 10^{12\pm(\approx
1)}\,\rm kg$ [20]. Dust, in grains ranging in size from some 10 (100) $\mu\rm
m$ to 1 cm in diameter with a density of $2000\,\rm kg/m^{3}$, is generally an
expected component [20]. Dust within our Zodiacal Cloud is observed to be
depleted very close to the Sun [111], and we follow [20] in assuming this
effect can be neglected beyond a radius of $0.05\,\rm AU$ to estimate that the
mass in dust within Mercury’s orbit to be no more than about $7\times
10^{15\pm(\approx 1)}\,\rm kg$, or $3\times 10^{-15\pm(\approx 1)}\,\rm
M_{\odot}$. This is still a few orders of magnitude smaller than our most
stringent bound on the maximum mass within the non-luminous disk, though we
can expect other massive objects to appear as well. For example, a spherical
object of the same composition as dust with a radius of 2.5 km would have a
mass of $1.3\times 10^{14}\rm kg$, so that some 100 of them would be needed to
contribute a mass of $10^{16}\rm kg$, crudely comparable to that in dust. We
note the solar system survey from Gaia Data Release 3 reveals 432 near-Earth
objects with a diameter of 5 km or less [112], though the total number and
mass of such objects within Mercury’s orbit is unknown. Much lighter objects,
such as micrometeroids, may also appear in an appreciable way [42]. There
could well be missing matter, of differing origins, within Mercury’s orbit —
and we now turn to how some measure of it could be dark matter.

The cosmic dark-matter density contributes only $10^{-19}\rm M_{\odot}$ of the
mass within Mercury’s orbit, and local effects could modify that total mass.
For example, a macroscopic dark object, such as a PBH, could exist.
Interestingly, in the mass range of $(10^{-15}-10^{-11})\,\rm M_{\odot}$, PBHs
could constitute all of the dark matter [68]; perhaps a PBH of some
$10^{-11}\rm M_{\odot}$ could be in transit through the solar system within
Mercury’s orbit, where we note [69] for solar-system transit rate estimates.

Ultralight dark matter could also play a role. To that end, we consider the
generic halo formation mechanism of [62], arising from dark-matter self-
interactions in the gravitational field of a massive celestial body, such as
the Sun. This gravi-atom picture is promising in that a strong enhancement of
the dark-matter density is possible in particular regions of parameter space;
for example, this model can yield a spherical halo about the Sun with an
overdensity of $\delta\rho_{\rm dm}\equiv\rho_{\rm crit}(0)/\rho_{\rm
dm}\simeq 7\times 10^{3}$ for an axion dark-matter candidate with a mass of
$10^{-14}\,\rm eV$ and an axion decay constant of $5\times 10^{7}\,\rm GeV$,
with the self-interaction strength $\lambda$ given by
$\lambda=-m^{2}/f_{a}^{2}$. The estimated extra mass this halo contributes
within Mercury’s mean orbit is more than fifty times smaller [62] than the
“extra mass” ephemerides constraint of $\rho_{\rm dm}<9.3\times 10^{-18}\,\rm
g/cm^{3}$ [81], which yields an enclosed mass of $<7\times 10^{12}\,\rm kg$ if
$\rho_{\rm dm}$ were uniform. The constraint of [81] includes a solar $J_{2}$
of a suitable size and error [113], so that this mass limit is not at odds
with our own analysis. Nevertheless, the spherical symmetry of this dark halo
is at odds with the evidence we have found for an extended disk-like object.
However, we can adapt the gravi-atom mechanism [62] to our case by noting that
the central mass $M$ need only be much more massive than that of the dark-
matter candidate $m$, with a halo of radius $R_{\ast}$ that exceeds that of
the central mass. Particularly, we evaluate the possibility that halos form
around clumps of conventional matter in the disk, possibly yielding a number
of $\ll 1\rm AU$-sized halo states. The total dark mass in the disk would then
be given by the sum of the masses associated with each of the halo bound
states.

Translating the gravi-atom estimates in $M$ and $m$ to kg and eV scales is
possible because the size of the bound state, as well as the possible
overdensity, are each determined by $Mm^{2}$, where $M_{\odot}(10^{-14}\rm
eV)^{2}\approx 200\,kg\,eV^{2}$. Here, successful halo formation in the gravi-
atom picture requires a suitably sized $R_{*}$, a wave-like dark-matter
candidate so that the de Broglie wavelength $\lambda_{\rm dB}$ exceeds the
dark matter interparticle spacing, and a maximum halo density $\rho_{\rm
crit}$ (at its center) that significantly exceeds $\rho_{\rm dm}$
($\delta\rho_{\rm dm}\gg 1$). There are also dynamical considerations, in that
dark-matter capture should exceed stripping, making $\xi_{\rm
foc}\equiv\lambda_{\rm db}/R_{\ast}\gtrsim 1$, with the density reaching
$\rho_{\rm crit}$ within about 5 Gyr, the lifetime of the Solar system [62].
There are also independent constraints on the axion from direct searches if it
couples to photons or nucleons, and a favored parameter space of
$m\in(0.7-700)\,\mu\rm eV$ with $f_{a}\in(2\times 10^{13}-2\times
10^{10})\,\rm GeV$, if it ought also solve the strong CP problem of QCD [114].
Thus the solar halo suggested in [62] is unlikely to be generated by a QCD
axion, nor can that axion couple to photons. These features will continue to
bear out in the examples to which we now turn.

Different sorts of constituents could potentially act as the nucleus of a
small dark-matter halo. Dust, for example, is thought to exist in grains
ranging in size from some 10 (100) $\mu\rm m$ to 1 cm in diameter with a
density of $2000\,\rm kg/m^{3}$ [20], and larger clumps of matter should also
occur. Here, we explore different scenarios for dark-halo formation,
considering central masses comprising of (i) a small asteroid with
$M=1.3\times 10^{14}\,\rm kg$ (a $2.5\,\rm km$-radius rock of density
$2000\,\rm kg/m^{3}$), of (ii) a much-denser asteroid with $M=10^{16}\,\rm
kg$, or of (iii) Mercury itself with $M=3.3\times 10^{23}\,\rm kg$, with a
radius of $2440\,\rm km$, so that $R_{\ast}$ must be larger than the noted
radius in each case. In these cases $M\ll M_{\odot}$, and we might suppose an
initial encounter of a dark-matter particle with the Sun produces a population
of dark-matter particles that satisfy $v\ll v_{\rm halo}\approx 240\,\rm
km/s$. We do not think it is possible to generate gravi-atoms with dust grain
cores so that $\xi_{\rm foc}\gtrsim 1$ is satisfied.

As detailed in Appendix C, the chosen scenarios for $M$ can satisfy the noted
constraints under the following conditions: (i) $M=1.3\times 10^{14}\,{\rm
kg};m=1.0\times 10^{-2}\,{\rm eV};f_{a}=1.0\times 10^{1}\,{\rm GeV}$ for which
the radius of the ground state is $R_{\ast}=4.0\,{\rm km}$, the gravitational
fine-structure constant $\alpha=4.9\times 10^{-9}$, and the dark-matter self-
coupling is $\lambda=1.0\times 10^{-24}$ — and the estimated maximum
overdensity is $\delta\rho_{\rm dm}=1.2\times 10^{6}$. If this can be
attained, then $M_{\rm enc,max}\approx\delta\rho_{\rm dm}\rho_{\rm dm}4\pi
R_{\ast}^{3}/3=2.4\times 10^{-4}\,\rm kg$. If $v_{\rm halo}$ characterizes the
dark-matter particle speed, then

$\displaystyle\xi_{\rm foc}=3.8\times 10^{-5};\tau_{\rm rel}=9.1\times
10^{8}\,{\rm Gyr}$ (39)

but with $v\approx 8\,\rm m/s$ (just for illustration), $\xi_{\rm
foc}\rightarrow 1.1$ and $\tau_{\rm rel}\rightarrow 1.0\,\rm Gyr$. However, we
do not think there would be enough rocks in situ for this solution, if
attainable, to generate a significant contribution to the disk mass.

We compare this to a heavier object that is six-times denser, finding (ii)
$M=1.0\times 10^{16}\,{\rm kg};m=6.0\times 10^{-5}\,{\rm eV};f_{a}=1.0\times
10^{1}\,{\rm GeV}$ for which the radius of the ground state is
$R_{\ast}=1.5\times 10^{3}\,{\rm km}$, the gravitational fine-structure
constant $\alpha=2.3\times 10^{-9}$, and the dark-matter self-coupling
$\lambda=3.6\times 10^{-29}$. Here $\delta\rho_{\rm dm}=9.6$ and $M_{\rm
enc,max}\approx 8.8\times 10^{-2}\,\rm kg$ with

$\displaystyle\xi_{\rm foc}=1.8\times 10^{-5};\tau_{\rm rel}=2.0\times
10^{2}\,{\rm Gyr}$ (40)

using $v_{\rm halo}$. By making the central mass denser, we see that more mass
can be stored in the halo, but here, too, it would not seem possible to
generate a significant contribution to the disk mass.

Finally we consider the possibility of a halo about Mercury itself, and we
consider (iii) $M=3.3\times 10^{23}\,{\rm kg};m=9\times 10^{-11}\,{\rm
eV};f_{a}=1.0\times 10^{7}\,{\rm GeV}$ so that $R_{\ast}=0.13\,{\rm AU}$,
$\alpha=1.1\times 10^{-7}$, $\lambda=8.1\times 10^{-53}$. Here
$\delta\rho_{\rm dm}=5.3\times 10^{4}$ and $M_{\rm enc,max}\approx 1.2\times
10^{15}\,\rm kg$ with

$\displaystyle\xi_{\rm foc}=8.7\times 10^{-4};\tau_{\rm rel}=6.6\times
10^{8}\,{\rm Gyr}$ (41)

for $v_{\rm halo}$, and if $v\approx 180\,\rm m/s$, $\xi_{\rm foc}\rightarrow
1.2$ and $\tau_{\rm rel}\rightarrow 3.7\times 10^{2}\,\rm Gyr$ — and the
dynamical constraints can be reasonably well satisfied. In this case a
contribution to the mass of the disk could be as large as $6\times
10^{14}\,\rm kg$. We have thus determined that it appears possible to generate
macroscopic contributions to a non-luminous disk from ultra-light dark matter,
even if we have not yet found an example through which it can grossly dominate
the expected non-luminous mass.

##  VIII Summary

In this paper we have considered different determinations of the Sun’s
gravitational quadrupole moment $J_{2}$, carefully comparing optical
assessments of the Sun’s shape with interferences from orbital observations,
particularly using MESSENGER studies of Mercury. The pattern of observations,
considered broadly, very much favor the pattern

$J_{2}^{\rm Orb}>J_{2}^{\rm Opt}\,,\,J_{2}^{\rm Heli}\,$ (42)

given in Eq. (17), which speaks to the existence of a non-luminous disk.
Particularly, if we compare the orbital results assuming GR and using
MESSENGER data [29], Eq. (4), with contemporaneous helioseismological results
from a space-based observatory [93], Eq. (V), we find the difference, given in
Eq. (14),

$J_{2}^{\rm Orb}|_{\beta=1;\eta=0}-J_{2}^{\rm Heli}=0.057\pm 0.006\,$ (43)

is greater than zero with a significance in excess of $5\sigma$. We have also
explored how alternative helioseismological error estimates and outcomes
impact this conclusion, and we find support for a positive result of high
significance. Consequently, we have developed constraints on the total mass
and mass distribution of that discovered, non-luminous disk, and we claim that
the $2\sigma$-limit on its maximum mass is no less than some $10^{-12}\rm
M_{\odot}$ for the best existing $J_{2}$ assessments. We have also carefully
considered its possible components, and we found that the missing (non-
luminous) mass is sufficiently large that conventional sources of mass, from
dust, from asteroids and meteroids of all sizes, would not seem able to
explain it. We also believe the total mass of these various components to be
poorly determined. We nevertheless suppose that dark matter could also
contribute to its mass. Thus our study points to a new “missing mass” problem
within Mercury’s orbit. A macroscopic dark matter candidate, such as a PBH of
a mass typical of an asteroid, is one concrete possibility. We have also
considered how a particular ultra-light dark matter model with self-
interactions [62] can generate a massive dark halo about Mercury itself,
contributing to the appearance of a massive ring in the path of Mercury’s
orbit.

We anticipate that the Bepi-Columbo mission to Mercury, to arrive in 2025,
will refine the results from the earlier MESSENGER mission, that we have
exploited in this paper, with further precision measurements of Mercury’s
orbit and magnetic properties offering the prospect of confirming and refining
our findings — or not. Particularly, precise assessments of short time-scale
perturbations on Mercury’s orbit should constrain massive exotic objects in
transit through the solar system, such as the PBHs candidates discussed in
[69]. Although the dark-matter particles we have particularly considered would
not seem to be axions, we expect that magnetic field studies may well be
discriminating nonetheless, as they can probe the possibility of macroscopic
dark-matter objects with a slight electric charge. Additional studies of dust
are also planned [115]. Ultimately, too, we expect these studies to have
implications for the precision of future relativistic general relativity tests
[21].

###### Acknowledgements.

We thank Antonio Genova, Roni Harnik, Dan Hooper, Russell Howard, Mrunal
Korwar, and Guillermo Stenborg for helpful correspondence. S.G. would like to
thank the Theoretical Division at Fermilab for lively hospitality and to
acknowledge the Universities Research Association for support during a
sabbatical visit in which this project was initiated. G.F.S.A. would like to
thank the hospitality of the Fermilab Theory Group. G.F.S.A., P.M., and M.Z.
are grateful to the Center for Theoretical Underground Physics and Related
Areas (CETUP*), The Institute for Underground Science at Sanford Underground
Research Facility (SURF), and the South Dakota Science and Technology
Authority for their hospitality and financial support. That stimulating
environment was invaluable to us. S.G. and M.Z. also acknowledge support from
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
under contract DE-FG02-96ER40989. P.M. is supported by Fermi Research
Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of
Energy, Office of Science, Office of High Energy Physics. G.F.S.A. is fully
financially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo
(FAPESP) under Contracts No. 2022/10894-8 and No. 2020/08096-0.

##  Appendix A Oblateness and Its Gravitational Effects

Here we show how a relationship between the solar oblateness and its
gravitational moments can be established under the assumption that the solar
surface constitutes an equipotential surface [116, 117], i.e.,
$\Phi(R_{e})=\Phi(R_{p})$. Assuming further that the Sun’s rotation ($\Omega$)
depends solely on the distance from its rotation axis, $l=r\sin\theta$, and
disregarding magnetic stresses, the effective potential can be expressed as
[26]

$\Phi\equiv\phi_{i}-\phi_{\Omega}=\phi_{i}-\int_{0}^{l_{\odot}}l\,\Omega^{2}(l)\,dl,\qquad
l=r\sin\theta,$ (44)

where $\phi_{i}$ and $\phi_{\Omega}$ represent the internal potential of the
Sun and the effective potential due to rotation, respectively. The continuity
of the potential across the Sun’s surface yields $\phi_{i}^{\rm
surf}=\phi_{o}^{\rm surf}$, where $\phi_{o}$ is the gravitational potential
outside the Sun, given by Eq. (1). The equipotential condition
$\Phi(R_{e})=\Phi(R_{p})$ results in

$\frac{\Delta_{\odot}}{\eta_{p}^{2}\left(1+\frac{\Delta_{\odot}}{\eta_{p}}\right)}+\sum_{n}\,\frac{J_{n}}{\eta_{p}^{n+1}}\left(\frac{P_{n}(0)}{\left(1+\frac{\Delta_{\odot}}{\eta_{p}}\right)^{n+1}}-1\right)+\frac{R_{\odot}\,\phi_{\Omega}(R_{e})}{G\,M_{\odot}}=0,$
(45)

where $\eta_{e,p}\equiv R_{e,p}/R_{\odot}$, and $\eta_{e}$ replaced by
$\eta_{e}=\Delta_{\odot}+\eta_{p}$. Noting that
$(\eta_{e}+\eta_{p})/2=1+\mathcal{O}(\Delta_{\odot}^{2})$, $\eta_{p}$ expanded
to linear order in $\Delta_{\odot}$ is given by $\eta_{p}\approx
1-\Delta_{\odot}/2$. Hence, Eq. (45) expanded to linear order in
$\Delta_{\odot}$ and up to the octopole moment ($n=4$) is

$\Delta_{\odot}^{(4)}=\frac{\left(16R_{\odot}\,\phi_{\Omega}(R_{e})/GM_{\odot}\right)-2\left(8J_{1}+12J_{2}+8J_{3}+5J_{4}\right)}{4\left(4J_{1}+3J_{2}+8J_{3}-4\right)+55J_{4}}.$
(46)

Given that all the multipoles are expected to be smaller than the monopole
term, $J_{n}\ll 1$, the oblateness in Eq. (46) can be approximated by keeping
only the linear terms:

$\displaystyle\Delta_{\odot}^{(4)}\approx
J_{1}+\frac{3J_{2}}{2}+J_{3}+\frac{5J_{4}}{8}-\frac{R_{\odot}\,\phi_{\Omega}(R_{e})}{GM_{\odot}},$
(47)

which in the case of a rigidly rotating body reduces to Eq. (2).

##  Appendix B Gravitational Quadrupole Moments

We have performed an extensive review of the previous determinations of the
solar gravitational quadrupole moment. There are three distinct ways by which
the Sun’s shape can be inferred: through the optical measurement of its
oblateness (Table 2), through helioseismology (Table 4), and through
measurements of planetary orbits (Table 4 and Table 5). We base our analysis
on a few values from these Tables, and we detail our motivations in doing so
here. First, among the optical measurements reported in Table 2, only results
#14–#17 take into account corrections for surface magnetism, and [86] (#14)
provides the most precise determination to date, though we note [87] (#17) for
further discussion of that earlier work.

The helioseismological measurements in Table 4 have been categorized based on
the instruments used. The values reported in [93], namely, #10–#11 and #17–#18
in Table 4 figure prominently in our analysis because of their improved
analysis framework and the extensive helioseismological data sets they employ.
We can also consider the results for each distinct instrument and determine
the average value and standard deviation for each case, to find using Table 4
that

$\displaystyle J_{2}^{\rm SoHO/MDI}=$ $\displaystyle 2.135\pm 0.203\,,\quad$
(48) $\displaystyle J_{2}^{\rm GONG}=$ $\displaystyle 2.167\pm 0.019\,,\quad$
(49) $\displaystyle J_{2}^{\rm SDO/HMI}=$ $\displaystyle 2.214\pm
0.002\,.\quad$ (50)

We use these to calculate a combined average value of $J_{2}$ of these
independent measurements, yielding

$\langle J_{2}^{\rm Heli}\rangle\Big{|}_{\rm all}=2.213\pm 0.002\,.$ (51)

Thus we see that we recover a result compatible with Eq. (V) through
completely different means, supporting our earlier analysis, albeit the HMI
results have the smallest dispersion.

Lastly, for orbital determinations we employ only the most recent $J_{2}$
analysis using MESSENGER data [29]. This work does take into account the ELT
correction and makes a simultaneous fit of PPN parameters in their $J_{2}$
determination. These are important additions, and we have to make sure we are
comparing $J_{2}$ assessments that take into account the same effects. Table.
4 serves this purpose as we opt to report only those values that consider
these additions in their $J_{2}$ determination. For completeness, in Table 5
we compile several orbital assessments for comparison indicating whether they
included either the ELT effect, a simultaneous fit of PPN parameters, or both.

Optical $J_{2}$ measurements
---
# | $\Delta_{\odot}(\times 10^{-6})$ | $J_{2}(\times 10^{-7})$ | $\pm J_{2}(\times 10^{-7})$ | Reference
1 | $50\pm 7$ | $279$ | $47$ | Dicke, 66 [3]
2 | $9.6\pm 6.6$ | $9.91$ | $43.8$ | Hill, 73 [118]
3 | $20.0\pm 1.5$ | $79.4$ | $9.73$ | Dicke, 83 [117]
4 | $5.8\pm 1.4$ | $-15.1$ | $9.03$ | Dicke, 84 [5]
5 | $15.2\pm 2.3$ | $47.4$ | $15.3$ | Dicke, 85 [6]
6 | $14.4\pm 1.4$ | $41.9$ | $9.03$ | Beardsley, 83 [119]
7 | $5.6\pm 6.3$ | $-16.7$ | $42.0$ | Maier, 90 [120]
8 | $4.3\pm 2.0$ | $-25.3$ | $13.3$ | Egidi, 92 [121]
9 | $8.5\pm 2.1$ | $2.67$ | $14.0$ | Egidi, 93 [121]
10 | $8.6\pm 1.4$ | $3.33$ | $9.33$ | Egidi, 94 [121]
11 | $10.3\pm 1.9$ | $14.7$ | $12.7$ | Egidi, 95 [121]
12 | $9.0\pm 2.9$ | $6.44$ | $19.5$ | Emilio, 97 [85]
13 | $19.7\pm 1.9$ | $77.3$ | $13.2$ | Emilio, 01 [85]
14 | $8.35\pm 0.15$ | $1.65$ | $0.97$ | Fivian , 02–08 [86]
15 | $8.8\pm 0.3$ | $4.3$ | $2.1$ | Irbah, 11 [122]
16 | $8.19\pm 0.33$ | $0.55$ | $2.22$ | Meftah, 10–11 [123]
17 | $7.5\pm 0.5$ | $-4.0$ | $3.4$ | Kuhn, 10–12 [87]
Table 2:  Summary of optical $J_{2}$ measurements. To translate from visual
oblateness $\Delta_{\odot}$ to $J_{2}$ we use Eq. (2), including only the
$J_{2}$ contribution and assuming $\Delta r_{\rm surf}=7.78\ \rm mas$ [26]. In
the Reference column, we indicate the first author followed by the date(s) of
observation of the corresponding measurement.

Helioseismological $J_{2}$ measurements
---
# | $J_{2}(\times 10^{-7})$ | $\pm J_{2}(\times 10^{-7})$ | Reference | Solar Model
1 | $36$ | – | Gough [124] | [125, 126]
2 | $1.7$ | $0.4$ | Duvall [127] | [125, 126]
3 | $55$ | $13$ | Hill [128] a | –b
1-5[0.1pt/0.5pt] 4 | $2.23$ | $0.09$ | Pijpers [33] | [129]
5 | $2.22$ | $0.02$ | Armstrong [130] | [131]
6 | $1.6$ | $0.04$ | Godier [132, 133]c | [134]
7 | $2.201$ | – | Mecheri [23] |  [94]d
8 | $2.198$ | – | Mecheri [23] | [94]
9 | $2.220$ | $0.009$ | Antia [102]e |
10 | $2.204$ | – | Mecheri [93] | [99]
11 | $2.208$ | – | Mecheri [93] | [100]
1-5[0.1pt/0.5pt] 12 | $2.206$ | – | Roxburgh [135] | [129]
13 | $2.208$ | – | Roxburgh [135] | [136]
1-5[0.1pt/0.5pt] 14 | $2.14$ | $0.09$ | Pijpers [33] | [129]
15 | $2.18$ | – | Antia [137] |
16 | $2.180$ | $0.005$ | Antia [102]e |
1-5[0.1pt/0.5pt] 17 | $2.211$ | – | Mecheri [93] | [99]
18 | $2.216$ | – | Mecheri [93] | [100]

  * a

This is the corrected value after accounting for a missing numerical factor
[4].

  * b

Solar model by Saio, H. (1982), private communication [128].

  * c

The uncertainty is reported in [138].

  * d

Measurements 7 and 8 use Model (a) and (b) for the solar differential rotation
from [94], respectively.

  * e

The central value and error come from the time average of $J_{2}$ measurements
over 10 years.

Table 3: Summary of helioseismological $J_{2}$ measurements. Measurements #1–3
use the rotational splitting data from [139], #4–11 use data from SoHO/MDI
[140, 97], #12–13 use data from SoHO/MDI and the ground-based Global
Oscillation Network Group (GONG) [141], #14–16 use data from GONG, and #17–18
use data from SDO/HMI [98].

Orbital $J_{2}$ measurements
---
# | $J_{2}(\times 10^{-7})$ | $\pm J_{2}(\times 10^{-7})$ | Reference
1 | $2.25$ | $0.09$ | Park ,11–14 [28]
2 | $2.246$ | $0.022$ | Genova, 08–15 [29]
1-4[0.1pt/0.5pt] 3 | $2.165$ | $0.12$ | Fienga, - [142]
4 | $2.206$ | $0.03$ | Fienga, - [142]
Table 4: Summary of orbital $J_{2}$ measurements. Measurements #1 and #2 are
orbital assessments using data almost exclusively from measurements of
Mercury’s orbit. We also include the ephemerides analyses that have taken into
account the ELT effect and fit for $J_{2}$ together with PPN parameters.

Orbital $J_{2}$ measurements
---
# | $J_{2}(\times 10^{-7})$ | $\pm J_{2}(\times 10^{-7})$ | ELT | PPN | Reference
1 | $180$ | $200$ | N | N | Lieske, 49–68 [143]
2 | $13.9$ | $24.7$ | N | Y | Shapiro, 66–71 [144]
3 | $25$ | $16$ | N | Y | Anderson, 11–76 [145]
4 | $12.3$ | $11.5$ | N | N | Anderson, – [113]
5 | $-1.8$ | $4.5$ | N | Y | Eubanks, – [113]
6 | $-11.7$ | $9.5$ | N | Y | Pitjeva, – [113]
7 | $-1.3$ | $4.1$ | N | Y | Pitjeva, 64-89 [146]
8 | $2.4$ | $0.7$ | N | Y | Pitjeva, – [147]
9 | $-5$ | $10$ | N | Y | Williams, 96–00 [148]
10 | $6.6$ | $9.0$ | N | N | Afanaseva, 80–86 [149]
11 | $-6$ | $58$ | N | N | Landgraf, 49–87 [150]
12 | $2.3$ | $5.2$ | N | Y | Anderson, 71–97 [151]
13 | $1.9$ | $0.3$ | N | Y | Pitjeva, 61–03 [152]
14 | $2.22$ | $0.23$ | N | Y | Pitjeva, – [113]
15 | $2.25$ | $0.09$ | Y | Y | Park ,11–14 [28]
16 | $2.246$ | $0.022$ | Y | Y | Genova, 08–15 [29]
17 | $2.46$ | $0.68$ | N | Y | Standish, - [153]
18 | $2.295$ | $0.010$ | N | N | Viswanathan, - [154]
19 | $1.82$ | $0.47$ | N | N | Fienga, - [155]
20 | $1.8$ | ? | N | Y | Konopliv,– [156]
21 | $2.0$ | $0.20$ | N | Y | Pitjeva,– [157]
22 | $2.40$ | $0.25$ | N | Y | Fienga, - [158]
23 | $2.27$ | $0.25$ | N | Y | Fienga, - [159]
24 | $2.22$ | $0.13$ | N | Y | Fienga, - [159]
25 | $2.165$ | $0.12$ | Y | Y | Fienga, - [142]
26 | $2.206$ | $0.03$ | Y | Y | Fienga, - [142]
27 | $2.40$ | $0.20$ | N | Y | Verma, - [160]
28 | $2.010$ | $0.010$ | N | N | Fienga, - [161]
29 | $2.2180$ | $0.01$ | Y | N | Fienga, - [162]
Table 5: Extended summary of the orbital $J_{2}$ assessments including
information as to whether the references included a simultaneous fit of the
PPN parameters or the ELT effect correction in their analysis.

##  Appendix C Dark Halo Formation

We follow [62] to find that a gravi-atom radius is set by

$R_{\ast}=\frac{(\hbar c)^{2}c^{2}}{GM(mc^{2})^{2}}\approx 0.035\,{\rm
AU}\left(\frac{10^{12}\,{\rm kg}}{M}\right)\left(\frac{100\,\mu{\rm
eV}}{m}\right)^{2}\,,$ (52)

noting that $0.035\,{\rm AU}\approx 7.5\,R_{\odot}$. The gravitational
coupling $\alpha$ is determined by $R_{\ast}=(m\alpha)^{-1}$, and the bound-
state escape velocity is $\sqrt{2}\alpha$. The associated de Broglie
wavelength $\lambda_{\rm db}$ of a dark-matter particle is

$\lambda_{\rm db}\approx 1.6\times 10^{1}\,{\rm m}\left(\frac{100\,\mu{\rm
eV}}{m}\right)\left(\frac{240\,{\rm km/s}}{v}\right)\,,$ (53)

where $v$ is the particle speed. Dark matter is wave-like if $\lambda_{\rm
db}$ exceeds the dark inter-particle spacing, possible if $m\lesssim 30\rm
eV$. Moreover,

$\xi_{\rm foc}\approx 2.9\times 10^{-9}\left(\frac{M}{10^{12}\,{\rm
kg}}\right)\left(\frac{m}{100\,\mu{\rm eV}}\right)\left(\frac{240\,{\rm
km/s}}{v}\right)$ (54)

If $\xi_{\rm foc}\gtrsim 1$, the expected maximum overdensity, with
$\delta\rho_{\rm dm}\sim\rho_{\rm dm}$, is

$\delta\rho_{\rm dm}=7\times 10^{13}\left(\frac{f_{a}}{10^{11}{\rm
GeV}}\right)^{2}\left(\frac{M}{10^{12}{\rm
kg}}\right)^{2}\left(\frac{m}{100\mu{\rm eV}}\right)^{4}\,,$ (55)

which can be reached over a time scale of about

$\tau_{\rm rel}\approx 9\times 10^{12}\,{\rm Gyr}\left(\frac{f_{a}}{10^{5}{\rm
GeV}}\right)^{4}\left(\frac{m}{1\mu{\rm eV}}\right)^{3}\left(\frac{240{\rm
km/s}}{v}\right)^{2}\,,$ (56)

where $\tau_{\rm rel}\sim\rho_{\rm dm}^{-2}$. If $\lambda>0$, then the
ultimately formed halo should be stable.

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