aainstitutetext: Department of Physics, Brown University, Providence, RI,
02912, USAbbinstitutetext: Brown Theoretical Physics Center, Brown University,
Providence, RI, 02912, USA

# Jupiter missions as probes of dark matter

Lingfeng Li a,b JiJi Fan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Jupiter, the fascinating largest planet in the solar system, has been visited
by nine spacecraft, which have collected a significant amount of data about
Jovian properties. In this paper, we show that one type of the in situ
measurements on the relativistic electron fluxes could be used to probe dark
matter (DM) and dark mediator between the dark sector and our visible world.
Jupiter, with its immense weight and cool core, could be an ideal capturer for
DM with masses around the GeV scale. The captured DM particles could
annihilate into long-lived dark mediators such as dark photons, which
subsequently decay into electrons and positrons outside Jupiter. The charged
particles, trapped by the Jovian magnetic field, have been measured in Jupiter
missions such as the Galileo probe and the Juno orbiter. We use the data
available to set upper bounds on the cross section of DM scattering off
nucleons, $\sigma_{\chi n}$, for dark mediators with lifetime of order ${\cal
O}(0.1-1)$s. The results show that data from Jupiter missions already probe
regions in the parameter space un- or under-explored by existing DM searches,
e.g., constrain $\sigma_{\chi n}$ of order $(10^{-41}-10^{-39})$ cm2 for 1 GeV
DM dominantly annihilating into $e^{+}e^{-}$ through dark mediators. This
study serves as an example and an initial step to explore the full physics
potential of the large planetary datasets from Jupiter missions. We also
outline several other potential directions related to secondary products of
electrons, positron signals and solar axions.

##  1 Introduction

Jupiter, fifth in line from the Sun, is the largest planet in our solar system
with a weight more than twice that of all the other planets combined. Its
fascinating properties, including the vivid stripes and swirls on the surface,
the great red spot as a giant storm about twice the size of the earth, and
more than 75 moons, make it a target of continuous investigation and
exploration, which dates all the way back to at least the Babylonian
astronomers in the 7th or 8th century BC. In modern times, Jupiter has been
visited by nine spacecraft, collecting a plethora of information about the
giant. Among them, seven just flew by, such as the Pioneer 1974Sci…183..301H ;
1975Sci…188..447O  and Voyager missions 1979Sci…204..945S ; 1979Sci…206..925S
, while two have orbited Jupiter, the Galileo mission 1992SSRv…60….3J  and the
Juno mission 2017SSRv..213….5B . The flybys provided snapshots of Jupiter. On
the other hand, the Galileo mission entered the Jovian magnetosphere and
released the Galileo probe, which dived into the atmosphere young1998galileo ;
2000JGR…10512093Y , while the Galileo orbiter remained and orbited within the
Jovian radiation belts and provided an extensive survey of the belts. The
latest Juno mission is the second one in NASA’s new frontiers program. The
spacecraft, launched in 2011, was inserted into the orbit around Jupiter in
2016 and will continue its investigation till 2025 (or its end of life). The
trajectories of the Jupiter missions, Galileo probe and Juno orbiter, are
depicted in Fig. 1.

Figure 1: A cartoon of the trajectories of the Galileo probe (blue) and the
Juno orbiter (red).

No doubt that Jupiter is an important object in astronomy and planetary
science. More intriguingly, data collected by the Jupiter missions could
contribute to an apparently unrelated scientific endeavor: the hunt for dark
matter (DM) and the dark sector, which will be our focus in this article. It
has already been pointed out that Jupiter could be a powerful DM capturer
Kawasaki:1991eu ; Adler:2008ky ; Leane:2021ihh ; Leane:2021tjj : DM particles
in the galactic halo could be captured by Jupiter if they scatter with the
Jovian matter and lose enough kinetic energy so that they become
gravitationally bound and accumulate inside Jupiter. Compared to other planets
in the solar system, e.g., our Earth, the gas giant could capture more DM,
enhancing the annihilation signals of captured particles. Compared to the Sun,
Jupiter is much cooler in the core and the captured DM could remain inside,
ideal for searches of (sub-)GeV DM which evaporate away even if captured
initially by the Sun Leane:2021tjj .

So far, the only detection proposal of DM captured in Jupiter is to search for
gamma rays Leane:2021tjj . The class of dark sector models being probed is as
follows: DM particles annihilate into a pair of dark mediators, which are
portals between the dark sector and our visible sector, which subsequently
decay into two gamma-ray photons outside Jupiter, which could be searched for
using 12 years data of Fermi Large Area Telescope (LAT) 2009arXiv0907.0541G ,
which is close to the Earth.111DM capture by celestial objects was proposed
and computed first in Press:1985ug ; Gould:1987ir ; Gould:1987ww ;
1990ApJ…356..302G . In the context of dark sector models with long-lived dark
mediators, studies of signals from DM capture by other celestial objects such
as the Sun and the Earth have been implemented in Liu:2008kz ; Batell:2009zp ;
Schuster:2009au ; Schuster:2009fc ; Bell:2011sn ; FermiLAT:2011ozd ;
GarciaGarcia:2015fol ; Feng:2015hja ; Feng:2016ijc ; Kouvaris:2016ltf ;
Allahverdi:2016fvl ; Brdar:2016ifs ; Smolinsky:2017fvb ; Ardid:2017lry ;
Leane:2017vag ; Arina:2017sng ; Robertson:2017hdw ; HAWC:2018szf ;
Nisa:2019mpb ; Niblaeus:2019gjk ; Cuoco:2019mlb ; Mazziotta:2020foa ;
Bell:2021pyy ; Bose:2021yhz ; Leane:2021ihh ; Zakeri:2021cur ; Bose:2021cou .

Here we propose a new search for a similar dark sector model but with a
different final state: DM annihilating into dark mediators, which decay into a
pair of electron $e^{-}$ and positron $e^{+}$ outside Jupiter, as depicted in
Fig. 2. Such decay channels could be present and important if long-lived dark
mediators couple to standard model fermions. One type of the in situ
measurements, which have been implemented by some Jupiter missions, such as
the Galileo probe and the Juno mission, is to profile and measure fluxes of
relativistic electrons in the Jovian magnetosphere. For the Galileo probe, its
Energetic Particles Investigation (EPI) instrument uses two totally-depleted,
circular silicon surface barrier detectors. It made omnidirectional
measurements of energetic particle (electrons, protons, $\alpha$-particles,
and heavy ions) population in the innermost regions of the Jovian
magnetosphere fischer1996high . For the Juno mission, the Radiation Monitoring
(RM) investigation analyzes the noise signatures from penetrating radiation in
the images of Juno’s cameras and science instruments becker2017juno . Some
Juno instruments, such as the Stellar Reference Unit (SRU), could operate as a
star camera collecting sky images by the silicon charge coupled device (CCD)
focal plane array. The CCD could register impacts by penetrating charged
particles as noise signals, within a cluster of pixels around each hit. These
counts could be used to infer the electron fluxes with energy $\gtrsim 10$ MeV
at different locations in the radiation belts becker2017observations . While
the Jovian electrons as the background of searching for new flux sources are
still to be fully understood, the observed fluxes would allow us to set
conservative upper bounds on the maximum electron flux induced by dark
mediator decays. This could be translated into constraints on un- or under-
explored regions of parameter space in the dark sector models.

Figure 2: A schematic illustration of the process. Captured DM particles
(solid white) annihilate into a pair of dark mediators (dashed blue), which
decay into $e^{+}e^{-}$ (red solid) outside Jupiter. The energetic electrons
or positrons could be measured by the Jupiter missions.

Our study serves as a proof of concept and an initial step to explore the full
potential of the large datasets from the Jupiter missions to search for new
physics beyond the standard model. So far the only other application is to use
the Jovian magnetic profile to constrain photon mass Davis:1975mn  and dark
photon kinetically mixed with the photon Marocco:2021dku , both of which could
induce modifications to the magnetic field. Yet Jupiter missions collect much
richer information beyond the magnetic field. For example, the relativistic
electron flux datasets, the focus of our paper, have not been applied to
search for DM and dark mediators before.

The paper is organized as follows. In Section 2, we review the key formalism
to compute DM captured by Jupiter. In Section 3, we discuss the dark sector
scenario that could be probed by the proposed search. In Section 4, we study
the motions of electrons trapped in the Jovian magnetic fields and show how to
compute the observables related to the electron flux under the approximation
of dipole magnetic field and ignoring electron loss effects. In Section 5, we
take into account of electron losses, i.e., due to irregular magnetic field
lines deviating from the dipole approximation, and classify three possible
electron trapping scenarios. In Section 6, we apply the datasets available to
constrain the parameter space of the DM and dark mediator model in interest.
We conclude and outline several future directions in Section 7.

##  2 Dark Matter Capture in Jupiter

In this section, we will review the formalism to compute the rate of DM
capture in Jupiter. We follow the discussions in Refs. Gould:1987ir ;
Gould:1987ww ; 1990ApJ…356..302G ; Kouvaris:2010vv ; Bramante:2017xlb ;
Dasgupta:2019juq ; Ilie:2020vec ; Leane:2021ihh ; Leane:2021tjj . We only
present the main results and point interested readers to the references above
for more details.

During the capture processes, DM particles, $\chi$’s, from the galactic halo
go through one or multiple scattering with the matter in Jupiter and
decelerate. It is considered to be captured once DM’s relative velocity falls
below the escape velocity of Jupiter. For DM with mass $m_{\chi}$ around the
GeV scale, substantial momentum exchange via DM-nucleon elastic scattering is
possible as the masses of relevant particles are comparable. Jupiter’s
stopping power could be described by its optical depth, $\tau_{J}$:

$\tau_{J}=\frac{3}{2}\frac{\sigma_{\chi n}}{\sigma_{\rm
sat}}~{},~{}\sigma_{\rm sat}\simeq\frac{\pi R_{J}^{2}}{N_{n,J}}\simeq
10^{-34}~{}\text{cm}^{2},$ (1)

where $\sigma_{\chi n}$ is the DM-nucleon scattering cross section, which we
take to be velocity-independent; $\sigma_{\rm sat}$ is the cross section that
saturates the geometric limit; $R_{J}\simeq 7\times 10^{4}$ km is the Jupiter
radius; and $N_{n,J}$ is the total number of Jovian nucleons. After one or
multiple scattering with nucleons, the DM’s velocity becomes comparable or
smaller than Jupiter’s escape velocity
$v_{J}(R_{J})=\sqrt{2GM_{J}/R_{J}}\simeq 59.5$ km/s ($M_{J}=1.9\times 10^{27}$
kg is the Jupiter’s mass). In this work, we are interested in the small
coupling case, namely the the optically thin limit where $\tau_{J}\ll 1$ and
the capture is achieved in one scattering.222The approximate multiple
scattering formula was first proposed in Bramante:2017xlb  and was improved in
Dasgupta:2019juq ; Ilie:2020vec . However, for Jupiter, the $\sigma_{\chi n}$
needed for multiscattering is much larger than the limit our method can probe.
Note that the single scattering limit of the multiscattering formula in
Bramante:2017xlb  does not match the standard single scattering formula first
derived in Gould:1987ir ; Gould:1987ww . In the optically thin limit, we
follow the discussions in Gould:1987ir ; Gould:1987ww . Assuming a Maxwell-
Boltzmann DM velocity distribution and elastic DM scattering all with hydrogen
atoms, the capture rate from single scattering could be simplified as:

$C_{1}=\sqrt{\frac{8\pi}{3}}\frac{n_{\chi}\tau_{J}R_{J}^{2}}{\bar{v}_{\chi}}\int_{0}^{R_{J}}\frac{4\pi
r^{2}n_{n}(r)}{N_{n,J}}v_{J}^{2}(r)\bigg{(}1-\frac{1-e^{-A(r)^{2}}}{A(r)^{2}}\bigg{)}X[A(r)]dr~{},$
(2)

where $n_{\chi}$ is the local DM number density (we take the local DM energy
density to be 0.4 GeV/cm3 Sivertsson:2017rkp ; Buch:2018qdr ;
2020MNRAS.495.4828G ; 2020A&A…643A..75S ; 2021A&A…653A..86W ;
2022MNRAS.511.1977S ), $\bar{v}_{\chi}\simeq$ 267.2 km/s is DM velocity
dispersion Freese:2012xd , $v_{J}(r)$ is the escape velocity at a distance $r$
from Jupiter’s center and is $\lesssim v_{J}(R_{J})$, $m_{n}$ is the nucleon
mass, $n_{n}(r)$ is the nucleon number density at the distance $r$,
$A(r)^{2}\equiv
6v_{J}(r)^{2}m_{n}m_{\chi}/[\bar{v}_{\chi}^{2}(m_{n}-m_{\chi})^{2}]$, and
$X[A(r)]\in[0.37-0.75]$ is the suppression factor due to the relative motion
between Jupiter and the DM halo Gould:1987ww ; Garani:2021feo .

Since Jupiter’s interior density profile is not completely known, we adopt an
approximation by solving the Lane-Emden equation of the polytropic model with
$n=1$ Garani:2021feo . The resulting density is proportional to $R_{J}\sin(\pi
r/R_{J})/\pi r$. The numerical result of Eq. (2) simplifies to:

$C_{1}\gtrsim
0.28\sqrt{\frac{8\pi}{3}}\frac{n_{\chi}\tau_{J}R_{J}^{2}v_{J}^{2}(R_{J})}{\bar{v}_{\chi}}\bigg{(}1-\frac{1-e^{-A(R_{J})^{2}}}{A(R_{J})^{2}}\bigg{)}~{}.$
(3)

To get the right hand side above, we fix $X[A(r)]$ to be the lower end 0.37,
while the integration of Eq. (2) is implemented numerically with the profile
of the $n=1$ polytropic model, which gives $\approx
0.7v_{J}^{2}(R_{J})\bigg{(}1-\frac{1-e^{-A(R_{J})^{2}}}{A(R_{J})^{2}}\bigg{)}$.
We use the lower value of $C_{1}$ for the capture rate throughout the rest of
the paper.

DM particles trapped in the planet lose their kinetic energies and accumulate
around the center of Jupiter. The total number of trapped DM particles,
$N(t)$, evolves as

$\frac{dN(t)}{dt}=C_{1}-\frac{\langle\sigma_{\rm ann}v\rangle}{V_{\rm
ann}}N(t)^{2}~{},$ (4)

where the second term on the r.h.s. is the rate of DM annihilation, which
depletes DM particles; $\langle\sigma_{\rm ann}v\rangle$ and $V_{\rm ann}$ are
the thermally averaged DM annihilation cross section and effective volume
respectively. The system reaches an equilibrium with a maximum number of DM
particles, $N_{\rm max}=\sqrt{C_{1}V_{\rm ann}/\langle\sigma_{\rm
ann}v\rangle}$, after a time scale $t_{\rm eq}=(C_{1}\langle\sigma_{\rm
ann}v\rangle/V_{\rm ann})^{-1/2}$, given $V_{\rm ann}<R_{J}^{3}$. As DM
eventually thermalizes inside Jupiter, it settles within a length scale of
$r_{\rm ann}=\sqrt{9T_{J}/(4\pi G\rho_{J}m_{\chi})}$ Bramante:2017xlb , where
$T_{J}$ and $\rho_{J}$ are the characteristic temperature and energy density
of the planet’s interior region, respectively. Taking the characteristic core
temperature of Jupiter as $1.5\times 10^{4}$ K 2007ess..book…..M  and the
maximum Jupiter core density estimated to be around $2\times 10^{4}$ kg m-3
ni2018empirical , we find that

$r_{\rm ann}\simeq 0.1\,R_{J}\;\sqrt{\frac{T_{J,{\rm core}}}{1.5\times
10^{4}~{}\text{K}}}\sqrt{\frac{1~{}\text{GeV}}{m_{\chi}}}\sqrt{\frac{2\times
10^{4}~{}\text{kg}~{}\text{m}^{-3}}{\rho_{J,{\rm core}}}}\,,$ (5)

which is compatible with the typical size of Jupiter’s core ni2018empirical .
The original discussion of DM evaporation could be found in 1990ApJ…356..302G
. In particular, the DM evaporation rate is sensitive to its exponential tail
originated from either kinematic or thermal distribution even if most DM
particles are trapped in a small region Garani:2021feo . For DM lighter than 1
GeV, it could evaporate away before annihilation happens Garani:2021feo . The
evaporation boundary varies with $\sigma_{\chi n}$ and Jupiter density
profile.

Given the upper limit on the annihilation cross section, $\langle\sigma_{\rm
ann}v\rangle\lesssim 5.1\times 10^{-27}(m_{\chi}/{\rm GeV})$ cm3 s-1 for
$m_{\chi}\geq 1$ GeV from Planck measurements Leane:2018kjk , the time scale
$t_{\rm ann}$ for 1 GeV DM is:

$t_{\rm eq}\simeq 10^{16}~{}\text{s}\,\sqrt{\frac{5\times
10^{-27}~{}\text{cm}^{3}~{}\text{s}^{-1}}{\langle\sigma_{\rm
ann}v\rangle}}\sqrt{\frac{10^{-38}~{}\text{cm}^{2}}{\sigma_{\chi,n}}}\,,$ (6)

which is shorter than the age of Jupiter $t_{J}\simeq 1.5\times 10^{17}$ s.
This also implies that for small $\sigma_{\chi n}$, the equilibrium between DM
capture and annihilation may not be reached, depending on the DM mass. In the
case when $t_{\rm eq}\gg t_{J}$, the accumulated number of DM particles will
be suppressed by a factor of $t_{J}/t_{\rm eq}$. In our following
computations, we assume the equilibrium is reached and we will revisit this
point later at the end of Sec. 6 when discussing constraints from Jupiter
missions. The total annihilation rate is given by, when the equilibrium is
reached,

$\Gamma_{\rm ann}\equiv\frac{\langle\sigma_{\rm ann}v\rangle}{2V_{\rm
ann}}N^{2}~{}\xrightarrow[t\gg t_{\rm eq}]{}~{}\frac{C_{1}}{2}~{}.$ (7)

Since $C_{1}\propto\sigma_{\chi n}$, $\Gamma_{\rm ann}\propto\sigma_{\chi n}$.

For readers’ convenience, we collect the most important notations throughout
the paper and their meanings in Table 1.

$\chi$ | DM particle
---|---
$m_{\chi}$ | DM mass
$\sigma_{\chi n}$ | DM-nucleon scattering cross section
$\Gamma_{\rm ann}$ | DM annihilation rate
$\xi$ | dark mediator particle
$\Gamma_{D}$ | decay width of the mediator
$\gamma$ | mediator boost factor
$\prod$ BR | Branching fraction of $e^{\pm}$ from DM annihilations
$\equiv$ BR($2\chi\to 2\xi$) $\times$ BR($\xi\to e^{+}e^{-}$) |
$\rho_{D}$ | mediator decay rate density:
| decays per unit volume per unit time
$r$ | the radius distance from the Jupiter center
$B$ | magnetic field magnitude
$L$ | McIlwain parameter of the magnetic field lines
$\theta_{p}$ | geomagnetic latitude $\theta_{p}$
$\alpha$ | pitch angle
$\alpha_{\rm eq}$ | equatorial pitch angle
$E$ | energy
$\bar{I}$ | averaged injection rate of $e^{\pm}$ over their trajectories
$f$ | the electron phase space distribution
$\tau_{E}$ | time scale of electron energy loss
$\tau_{y}$ | time scale of pitch angle variation
$\tau_{\rm loss}$ | time scale of electron loss
$J(L,\theta_{p})$ | omnidirectional number flux of relativistic $e^{\pm}$
| integrated over the measured energy range
| in the $L$-shell at $\theta_{p}$
$F$ | geometric factor: effective collecting area
$K(\tilde{K})$ | observed (predicted) count rates:
| number of electron hits recorded per unit time
$J_{\rm inf}(\tilde{J}_{\rm inf})$ | observed (predicted) omnidirectional fluxes
| inferred from count rates
Table 1: Important notations and their meanings.

##  3 Dark Sector with a Long-lived Mediator

In this section, we present and discuss the DM scenario that could be probed
by the Jupiter in situ measurements on the flux of energetic electrons.

In the scenario being considered, captured DM particles annihilate into a pair
of long-lived mediators, $\xi$’s, with decay lengths comparable to or even
longer than Jupiter’s radius $R_{J}\simeq 7\times 10^{4}$ km, which means that
a significant fraction of $\xi$ decay outside the planet. This is possible if
$\xi$ is feebly coupled to and decays to the standard model. Possible
candidates of $\xi$ include either dark photon kinetically mixed with the
standard model photon Kobzarev:1966qya ; Okun:1982xi ; Galison:1983pa ;
Holdom:1985ag ; Holdom:1986eq  or heavy axion-like particles that couple to
leptons (e.g, MeV-10 GeV axion accompanying DM and coupling to leptons could
arise naturally in low-scale supersymmetry breaking model Nomura:2008ru ;
Ibe:2009dx ; Mardon:2009gw ; Fan:2010is ). Decay products with energy
$\sim\mathcal{O}(m_{\chi})$ will be released to the planet’s radiation belt.
In the following, we only focus on the $2\chi\to 2\xi$ annihilation process,
ignoring all other annihilation channels. In principle, the interaction
responsible for capture of DM by nucleons could also lead to DM annihilation
into standard model particles. Yet this may not need to be the dominant
annihilation channel. For example, one simplest effective operator that gives
rise to spin-dependent DM-nucleon scattering is

$\frac{g_{\chi}g_{q}}{\Lambda^{2}}\left(\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\right)\left(\bar{q}\gamma_{\mu}\gamma^{5}q\right)\,,$
(8)

with DM $\chi$ being a Dirac fermion and $q$ the standard model quarks. This
could be generated via integrating out an axial-vector with a mass about the
scale $\Lambda$ and coupling $g_{\chi}$ ($g_{q}$) to DM (quark). The resulting
spin-dependent DM-nucleon scattering cross section is Fan:2010gt ;
Abdallah:2015ter

$\sigma_{\chi n}\approx 3.8\times 10^{-39}\,{\rm
cm}^{2}\,\left(\frac{\mu_{\chi n}}{\rm
GeV}\right)^{2}\,\left(\frac{g_{\chi}g_{q}}{10^{-3}}\right)^{2}\,\left(\frac{10\,{\rm
GeV}}{\Lambda}\right)^{4}\,,$ (9)

where $\mu_{\chi n}$ is the reduced mass of DM and nucleon and we take $g_{q}$
to be the same for $u,d$ and $s$ quarks. The benchmark value here is chosen to
satisfy current direct detection bound for $m_{\chi}$ of order a few GeV and
below, as well as evade collider constraints. The cross section of
annihilation $\bar{\chi}\chi\to\bar{q}q$ through this operator is suppressed
by both the small $(g_{\chi}g_{q})^{2}$ and
$m_{\chi}^{2}m_{q}^{2}/\Lambda^{4}$ and could be significantly below the CMB
bound when we consider GeV-scale DM. On the other hand, we would consider that
the fermionic DM is charged under a dark $U(1)$ gauge symmetry and annihilates
into dark photons $A^{\prime}$ via the $\bar{\chi}\chi\to
A^{\prime}A^{\prime}$ process. The thermally averaged annihilation cross
section is

$\langle\sigma v\rangle(\bar{\chi}\chi\to A^{\prime}A^{\prime})\approx 2\times
10^{-27}{\rm cm}^{3}\,{\rm s}^{-1}\left(\frac{\alpha_{D}}{5\times
10^{-6}}\right)^{2}\,\left(\frac{\rm GeV}{m_{\chi}}\right)^{2}\,,$ (10)

where we assume $m_{A^{\prime}}\ll m_{\chi}$. This annihilation channel could
dominate over the annihilation into the standard model and saturate the CMB
constraint simultaneously. Note that the overall small annihilation cross
section could lead to too much thermal relic abundance but could be diluted in
a non-thermal scenario, e.g., with early matter domination and another
reheating after the inflationary reheating but before BBN. Another possible
mechanism that could achieve a right thermal relic is the interplay of co-
annihilation and co-scattering processes between the DM, its nearly-degenerate
partner, and the dark photon Cheng:2018vaj . For the rest of the discussion,
we will be agnostic of the relic abundance mechanism.

The scattering of $\xi$ with the Jovian matter before its escapes could be
ignored due to its very weak interactions. From the discussion in the last
section, the annihilation region would mostly be in the core of Jupiter. It is
a reasonable approximation that all DM annihilate, and all $\xi$’s are
produced at the center of Jupiter. The number of mediator decays per second
per unit volume, the mediator decay rate density at a distance $r$ from the
Jupiter center takes the simple form:

$\rho_{D}(r)=\frac{2\Gamma_{\rm ann}}{4\pi r^{2}}\frac{\Gamma_{\rm
D}}{\gamma\beta}e^{-\frac{\Gamma_{\rm D}r}{\gamma\beta}}~{},$ (11)

where $\Gamma_{\rm D}$ is the decay width of the mediator; the mediators from
DM annihilations are boosted by a factor of $\gamma\simeq m_{\chi}/m_{\xi}$
with $m_{\xi}$ the mediator mass; and the velocity of the mediator is given by
$\beta=\sqrt{1-\gamma^{-2}}$. The factor of 2 in the numerator is due to the
fact that each annihilation produces two mediators. Fig. 3 depicts $\rho_{D}$
as a function of $\gamma$ and $\Gamma_{\rm D}^{-1}$. From the figure, one
could see that at a given $r$ close to the Jupiter surface, $\rho_{D}$ is
maximized when $\Gamma_{D}^{-1}\times\gamma\sim R_{J}$. For
$\Gamma_{D}^{-1}\times\gamma\ll R_{J}$, $\rho_{D}$ is suppressed by the
exponential factor in Eq. (11) since most decays happen inside Jupiter. For
$\Gamma_{D}^{-1}\times\gamma\gg R_{J}$, $\rho_{D}$ is suppressed by the small
$\Gamma_{D}$ factor in Eq. (11) since not many decays happen in the innermost
radiation belt.

Figure 3: Mediator decay density rate, $\rho_{D}$, at $r=1.2R_{J}$ in unit of
cm-3 s-1, as a function of the mediator lifetime $\Gamma_{D}^{-1}$ in unit of
$R_{J}$, and the boost factor $\gamma$.

One of the most interesting decay channels of a sub-GeV mediator is the decay
to energetic electrons and positrons, $e^{\pm}$. For example, a dark photon,
$A^{\prime}$, decays to $e^{+}e^{-}$ exclusively if its mass $m_{A^{\prime}}$
is below the $2m_{\mu}$ threshold but above the $2m_{e}$ threshold. Above the
$2m_{\mu}$ threshold, the decay products becomes a mixture of $e^{+}e^{-}$,
$\mu^{+}\mu^{-}$ and $\pi^{+}\pi^{-}$, depending on $m_{A^{\prime}}$
Pospelov:2007mp ; Arkani-Hamed:2008kxc ; Reece:2009un ; Bjorken:2009mm ;
Ilten:2018crw . Most non-electron final states will cascade to a pair of
softer electrons and a few neutrinos, leaving almost no other visible
particles. The discussion of decay products above also applies to axion-like
particles coupling to standard model fermions. We will only consider the
channel of dark mediators decaying into $e^{+}e^{-}$ in the following
discussion. The density rate of $e^{+}e^{-}$ final state is then

$\rho_{D}(r)\prod{\rm BR},\quad\prod{\rm BR}\equiv{\rm BR}(2\chi\to
2\xi)\times{\rm BR}(\xi\to e^{+}e^{-})\,,$ (12)

where $\prod{\rm BR}$ is the product of the branching fractions of DM
annihilating into dark mediators and $\xi$ decaying to $e^{\pm}$.

Requiring the mediator’s decay length comparable to $R_{J}$ allows us to
explore the parameter space that is difficult to be probed otherwise. Take the
dark photon as an example. To have a decay length of order $R_{J}$, its decay
width satisfies

$\Gamma_{\rm D}(A^{\prime}\to
e^{+}e^{-})=\frac{\alpha}{3}\epsilon^{2}m_{A^{\prime}}\sqrt{1-\frac{4m_{e}^{2}}{m_{A^{\prime}}^{2}}}\left(1+\frac{2m_{e}^{2}}{m_{A^{\prime}}^{2}}\right)\sim
R_{J}^{-1}~{}\Rightarrow~{}\epsilon^{2}\sim
10^{-20}\times\bigg{(}\frac{0.1~{}\text{GeV}}{m_{A^{\prime}}}\bigg{)}~{},$
(13)

where $\epsilon$ is the kinetic mixing parameter, $m_{A^{\prime}}$ is the dark
photon mass; $m_{e}$ is the electron mass; and $\alpha\simeq 1/137$ is the
standard model fine structure constant. Such a small mixing $\epsilon$ does
not introduce significant interactions between the mediator and the standard
model, leaving $A^{\prime}$ above 0.1 GeV elusive in terrestrial and
cosmological searches Alexander:2016aln ; Fabbrichesi:2020wbt . We also stress
that the cross section of $A^{\prime}$ mediated DM-electron scattering is
suppressed by the tiny $\epsilon^{2}$ and is irrelevant for the current sub-
GeV DM-electron scattering searches. Similarly, the contribution to DM capture
from DM-nucleon scattering processes mediated by $A^{\prime}$ is negligible.
In other words, DM capture (DM-nucleon scattering) and annihilation (DM
annihilates into dark mediators) are two independent processes in our
scenario.

Electrons from $\xi$ decays are injected into the Jovian magnetosphere and can
be trapped in the strong magnetic field for a long time. The phase space
evolution of the high-energy electrons will be detailed in the next two
sections. The long time scale of the trapped electrons compensates for the low
$\rho_{D}$ when the DM-nucleon scattering cross section $\sigma_{\chi n}$ is
small. In addition, the hard electron spectrum originating from the mass scale
of DM is distinctive from the soft astrophysical background sicard2011jose .
These features make several in situ measurements of Jupiter missions sensitive
to this class of models.

##  4 Relativistic $e^{\pm}$ Fluxes in the Innermost Radiation Belt: Dipole
Approximation

In this section, we describe the motion of $e^{\pm}$ from $\xi$ decaying
inside the Jovian magnetosphere and derive the basic formula for the resulting
flux of charged particles. We will first work with the approximation that the
Jovian magnetic field could be described as a dipole. Note that in situ
measurements do not distinguish between electrons and positrons. The electron
in the following discussions includes both $e^{-}$ and $e^{+}$.

As an approximate magnetic dipole, the Jovian magnetic field is significantly
stronger near the polar regions. Charged particles traveling along the
magnetic flux tube will then be reflected due to the magnetic mirror effects
and eventually trapped inside the radiation belts. It is thus convenient to
introduce the McIlwain $L$ parameter mcilwain1961coordinates  to describe the
magnetic field lines. Here, an $L$-shell could be understood as a collection
of magnetic flux tubes within which charged particles drift through. It is
displaced from the Jupiter center by an amount $LR_{J}$ in the magnetic
equatorial plane under the dipole approximation. As the $B$ field only changes
slowly along the field lines, the overall volume of a flux tube in an
$L$-shell $V_{L}=\int_{L}dV$ can be approximated as $\int_{L}dA\cdot dS$, in
which $S$ stands for the length along the field line and the cross sectional
area $A\propto|B|^{-1}$ since the magnetic flux is a constant. Therefore, the
overall rate of electrons injected from the mediator decays into each flux
tube reads $2\int_{L}\rho_{D}~{}dA\cdot dS$, with the factor of 2 taking both
$e^{-}$ and $e^{+}$ into account.

Figure 4: A cartoon of the electron motions in the Jovian magnetic fields
(field lines are denoted by the black dashed curves). The dark mediator will
travel outside Jupiter (blue dashed line) and decay to a pair of $e^{\pm}$
(which we will refer to as electrons). The electrons gyrate around the
magnetic flux tube and travel along the field lines. For the electron with a
large initial pitch angle $\alpha$, as $B$ increases along its trajectory, its
$\alpha$ eventually reaches $\pi/2$ around the mirror point and gets
reflected, as demonstrated by the red trajectory. In contrast, the electron
with a small initial $\alpha$ will hit and be absorbed by the atmosphere
before reaching the mirror point, as shown by the yellow trajectory.

In addition to energy $E$ and $L$, an extra kinematic variable is needed to
describe the phase space of the trapped charged particles. It is taken to be
the equatorial pitch angle $\alpha_{\rm
eq}\equiv\arcsin(|p_{\perp}|/|p|)|_{\rm eq}$, which is the pitch angle between
the momentum of the particle and the magnetic field at the magnetic equator
($p_{\perp}$ is the component of the momentum perpendicular to the magnetic
field line). As the particle’s magnetic moment $M=|p_{\perp}|^{2}/(2mB)$ is
adiabatically invariant along its trajectory, we have the following relation
for each $L$-shell:

$\alpha_{\rm
eq}=\arcsin\bigg(\sqrt{\frac{B(0)}{B(\theta_{p})}}\sin\alpha(\theta_{p})\bigg{missing})~{},$
(14)

where $\alpha(\theta_{p})$ and $B(\theta_{p})$ are the pitch angle and
magnetic field at the geomagnetic latitude angle $\theta_{p}$, while $B(0)$ is
the equatorial magnetic field. As the particle travels to higher
$|\theta_{p}|$, $\alpha$ increases with $B(\theta_{p})$ and eventually the
electron gets reflected around the two mirror points where $\alpha=\pi/2$.
Even with the same energy and $L$-shell value, the trapping time scales and
spatial distributions of electrons vary when their $\alpha_{\rm eq}$’s change.
Moreover, all particles with $\alpha_{\rm eq}<\alpha_{\rm min}$ will be
absorbed by the Jovian atmosphere. Here $\alpha_{\rm min}$ is the minimum
equatorial pitch angle that makes $\sin\alpha(\theta_{p})=\pi/2$ in Eq. (14)
at the $\theta_{p}$ where $r(L,\theta_{p})=R_{J}$. Electrons with even smaller
$\alpha_{\rm eq}$ will have their mirror points inside Jupiter and be absorbed
by the atmosphere. Thus it is necessary to include the pitch angle
distribution when deducing the electron fluxes. We illustrate the two possible
types of electron motions ($\alpha_{\rm eq}>\alpha_{\rm min}$ and vice versa)
in Fig. 4.

On top of the gyration around the magnetic flux tube and the bounce between
mirror points along the magnetic field lines, there is a third motion, i.e.
the drift in the longitudinal direction around the planet schulz2012particle .
The drift stems from the gradient of the magnetic field and has a much longer
timescale compared to the gyration and bounce. Such a motion doesn’t affect
the electron flux as long as the magnetic field is azimuthally symmetric. Its
effect when the azimuthal symmetry is broken will be discussed in Section 5.

The injected electrons from $\xi$ decays serve as an extra source term of
energetic particles in the radiation belt. We denote the phase space
distribution of the $e^{\pm}$ injection rate as $I_{e^{-}}=I_{e^{+}}\equiv I$.
At each local point, it is a function of $r,L,E,\sin\alpha$333Since
$r=L\cos^{2}\theta_{p}$ in a dipole field, one could also think of $I$ as a
function of $\theta_{p},L,E,\sin\alpha$ in this case. and is related to the
mediator decay rate density $\rho_{D}$, which only depends on $r$, as:

$2\int dE\,E^{2}\int d\phi\int
d\alpha\,\sin\alpha\,I(r,L,E,\sin\alpha)=\rho_{D}(r)~{},$ (15)

where $\phi$ is the gyration angle. In practice, the injected electron rate
distribution $I$ is determined by Monte-Carlo simulations for a given DM
model. We will describe the simulations in App. A.

When reaching equilibrium, the diffusion equation of the electron phase space
density $f(L,E,\sin\alpha_{\rm eq})$ is nenon2018rings :

$\displaystyle\frac{df(L,E,\sin\alpha_{\rm eq})}{dt}$
$\displaystyle=\frac{\int Idz}{\int dz}$
$\displaystyle-\frac{1}{G}\frac{\partial}{\partial
E}\bigg{(}\frac{dE}{dt}Gf\bigg{)}-\frac{1}{G}\frac{\partial}{\partial\sin\alpha_{\rm
eq}}\bigg{(}\frac{d\sin\alpha_{\rm eq}}{dt}Gf\bigg{)}$
$\displaystyle+\rm{loss~{}terms}~{}+~{}\rm{diffusion~{}terms}=0\,,$ (16)

where $z$ is the length of electron trajectories, and the first term $\int
Idz/\int dz\equiv\bar{I}$ is the averaged electron injection rate over their
trajectories as the source term. Some benchmark $\bar{I}$ are depicted in Fig.
5 as functions of $E$ and $\sin\alpha_{\rm eq}$. The presented values are in
the unit of $m_{\chi}^{-3}\rho_{D}$ and thus dimensionless. For boosted $\xi$
decays, the energy distribution of injected electrons is a uniform one with
$E$ between $(1\pm\sqrt{1-\gamma^{-2}})m_{\chi}/2$. Consequently, both $I$ and
$\bar{I}$ scales as $\sim E^{-2}$ to keep $E^{2}I(\bar{I})$ approximately a
constant. From Fig. 5, one can also notice that the contribution of large
$\alpha_{\rm eq}$ is bigger. This is because the boost makes injected
electrons closer to the direction perpendicular to the magnetic field lines.
Such an effect becomes more significant when $\gamma$ increases from 3 to 8.

The factor $G$ in Eq. (4) is the Jacobian from the three action variables
defined in schulz2012particle  to ($L,E,\sin\alpha_{\rm eq}$).444Under the
dipole approximation and in the relativistic limit, $G$ can be approximated as
$E^{2}\sin\alpha_{\text{eq}}\left(3.84\sin\alpha_{\text{eq}}+3.84\sqrt{\sin\alpha_{\text{eq}}}-16.56\right)$
times a function of $L$ schulz2012particle . The two terms in the second line
of Eq. (4) stand for the friction effects (e.g., energy loss and pitch angle
change with time), each associated with a characteristic time scale. Notice
that with friction terms only, $L$ will remain constant. The loss term
describes the removal of electrons due to the deviation from the dipole
approximation and hard scattering processes, which will be detailed in Section
5. In a dipole field, the effect of loss terms is insignificant. Diffusion
terms in Eq. (4) include radial, energy, and angular diffusion effects from
various origins like the interchange instability of plasma 1987JGR….92..109S
or interactions with the low-frequency plasma waves 2005JGRA..110.4206G ,
which are orders of magnitude smaller than the friction terms within the
innermost region ($L\lesssim 1.5$) we are interested in nenon2017new . During
the typical trapping time scale of $e^{\pm}$, they will not affect the
solution of $f$ significantly and thus can be safely ignored.

Figure 5: The trajectory-averaged injection rate $\bar{I}$, as a function of
$E$ and $\alpha_{\rm eq}$ for different model parameters. The presented values
are in the unit of $m_{\chi}^{-3}\rho_{D}$ and thus dimensionless. In both
cases, contributions from the large-$\alpha_{\rm eq}$ regime are significant
due to $\gamma$ greater than 1. LEFT: The distribution of $\bar{I}$ at
$L=1.2$, from a DM model with $\gamma$=3. RIGHT: The same as the left, but
with $\gamma=8$.

###  4.1 Friction Time Scales

Figure 6: Time scales of synchrotron radiation with the magnetic dipole
approximation. LEFT: The energy loss rate for 20 MeV electrons with
$\alpha_{\rm eq}=70^{\circ}$ as a function of $L$, the result is compatible
with the result reported in nenon2017new ; nenon2018rings . RIGHT: Contours of
$\tau_{E}^{-1}$ (solid) and $\tau_{y}^{-1}$ (dashed) as functions of
$\alpha_{\rm eq}$. The five colors, from top to bottom for $\tau_{E}^{-1}$ and
from right to left for $\tau_{y}^{-1}$, stand for
$L=1.1,~{}1.2,~{}1.3,~{}1.4$, and $1.5$, respectively.

In this section, we will estimate the friction effects and the associated time
scales in Eq. (4). The energy loss rate $dE/dt$ receives various
contributions, with the leading one being the synchrotron radiation in our
scenario. The energy loss rate from synchrotron radiation is555The energy loss
rates in this section should all be taken as absolute values.

$\displaystyle\frac{1}{E}\frac{dE}{dt}\bigg{|}_{\rm sync}$
$\displaystyle=\sigma_{T}\frac{E}{m_{e}^{2}}B^{2}\sin^{2}\alpha~{}$
$\displaystyle\simeq 6.3\times
10^{-6}\sin^{2}\alpha\bigg{(}\frac{E}{100~{}\text{MeV}}\bigg{)}\bigg{(}\frac{B}{4~{}\text{Gauss}}\bigg{)}^{2}\,\text{s}^{-1}\,~{},$
(17)

where the Thomson cross section $\sigma_{T}=8\pi\alpha^{2}/(3m_{e}^{2})\approx
6.7\times 10^{-29}$ m2. Since a trapped electron bounces inside the $L$-shell
multiple times before its kinetic energy drops significantly, the time scale,
$\tau_{E}$, can be calculated by averaging over the trajectory parametrized by
the trajectory length $z$:

$\tau_{E}^{-1}|_{\rm
sync}\equiv\left\langle\frac{1}{E}\frac{dE}{dt}\bigg{|}_{\rm
sync}\right\rangle=\sigma_{T}\frac{E}{m_{e}^{2}}\frac{\int
B^{2}\sin^{2}\alpha\,dz}{\int dz}~{}.$ (18)

From Eq. (17), one could see that the characteristic time scale of synchrotron
radiation is $\tau_{E}|_{\rm sync}\sim\mathcal{O}(10^{5})$ s for electrons
with energies of ${\cal O}(100)$ MeV. Taking the dipole approximation with the
intensity from the JRM09 magnetic field model connerney2018new , we present
the energy loss rate as a function of $L$ in the left panel of Fig 6. As
$\tau_{E}|_{\rm sync}\propto B^{-2}$ and $B\sim L^{-3}$ in a dipole field,
$\tau_{E}^{-1}|_{\rm sync}$ scales as $L^{-6}$.

The synchrotron radiation also alters the electron’s pitch angle. The time
scale $\tau_{y}$ of the pitch angle variation is proportional to the energy
loss rate santos2001modeling :

$\tau_{y}^{-1}|_{\rm sync}\equiv\left\langle\frac{d\sin\alpha_{\rm
eq}}{dt}\bigg{|}_{\rm sync}\right\rangle=\frac{\cos^{4}\alpha_{\rm
eq}}{\sin^{2}\alpha_{\rm eq}}\left\langle\frac{1}{E}\frac{dE}{dt}\bigg{|}_{\rm
sync}\right\rangle~{}.$ (19)

For $\alpha_{\rm eq}\gtrsim\pi/4$, the timescale of pitch angle change
$\tau_{y}$ is longer than $\tau_{E}$. This could be seen from the right panel
of Fig 6, in which the inverse time scales $\tau_{E}^{-1}|_{\rm sync}$ and
$\tau_{y}^{-1}|_{\rm sync}$ are plotted. This means that for large enough
$\alpha_{\rm eq}>\pi/4$, $\tau_{E}$ is the more relevant time scale compared
to $\tau_{y}$. Another crucial feature shown in Fig. 6 is that as $\alpha_{\rm
eq}$ increases, both the friction rates become smaller, or equivalently,
$\tau_{E}$ and $\tau_{y}$ becomes longer. From Eq. (17), (18) and (19), one
might expect that $\tau_{E}^{-1}|_{\rm sync}$ and $\tau_{y}^{-1}|_{\rm sync}$
increases as $\alpha_{\rm eq}$ increases. However, with larger $\alpha_{\rm
eq}$, electrons tend to stay near the magnetic equator (in the limit that the
initial $\alpha_{\rm eq}=\pi/2$, the electron just stays in the equatorial
plane), where the magnetic field is the weakest. Smaller $\alpha_{\rm eq}$
allows the electrons to travel to regions with denser magnetic field lines
near the poles. It turns out that $\tau_{E}^{-1}|_{\rm sync}$ and
$\tau_{y}^{-1}|_{\rm sync}$ are more sensitive to variation in the $B$ field
and thus decreases as $\alpha_{\rm eq}$ increases.

An electron could also lose energy due to Coulomb scattering with gas and
plasma along the trajectory. Furthermore, frequent hard scatterings
transferring electron energy to gas lead to electron absorption, which will
then be better described as a loss term instead of a friction term in Eq. (4).
This applies to electrons diving deeply into the Jovian atmosphere. For
observations performed in regions where the gas density is sufficiently small
hinson1998jupiter ; bougher2005jupiter , the energy friction term of Coulomb
scattering is significantly smaller than that of the synchrotron radiation and
thus could be safely ignored. The electron loss time scale, on the contrary,
may affect electrons with $r$ up to $\sim 1.3R_{J}$ and will be discussed in
Section 5.

Beyond the synchrotron radiation and Coulomb scattering effects, other sources
may also introduce extra friction terms, like interaction with grains in the
halo ring nenon2018rings . However, for energetic electrons ($E\gtrsim 10$
MeV) within $L\lesssim 1.5$ region, such extra friction terms are all highly
suppressed compared to the synchrotron radiation and can also be safely
ignored for our purpose nenon2018rings .

###  4.2 Electron Flux

The omnidirectional number flux of ultra-relativistic $e^{\pm}$ (the number of
$e^{\pm}$ passing through unit cross section per unit time) with energies
above a threshold $E_{\rm th}$, in the $L$-shell at the geomagnetic latitude
$\theta_{p}$, could be calculated from the phase space distribution $f$:

$J(L,\theta_{p})|_{E>E_{\rm
th}}=2\int_{0}^{2\pi}d\phi\int_{0}^{\frac{\pi}{2}}\sin\alpha_{\rm
eq}\,d\alpha_{\rm eq}\int_{E_{\rm
th}}^{+\infty}E^{2}\,dE\,f(L,E,\sin\alpha_{\rm eq})\frac{dA_{\rm
eq}}{dA}\frac{(dt/dS)}{(dt/dS)_{\rm eq}}~{},$ (20)

where the factor of 2 in front captures the symmetric particle population
moving along both directions of the magnetic flux tube. Here we choose to
integrate over the equatorial pitch angle. Away from the equatorial plane, we
need to include the last two factors in the equation above: $dA_{\rm eq}/dA$
encodes the enhancement of the electron flux as the magnetic flux tube narrows
at higher latitudes; $\frac{(dt/dS)}{(dt/dS)_{\rm eq}}$ takes into account
that the electron’s speed along the magnetic field line varies:
$dS/dt=v_{\parallel}=\cos\alpha$, and the electron spends less time near the
equatorial region. Using Eq. (14) and $dA\propto B^{-1}$, Eq. (20) is further
reduced to:

$\displaystyle J(L,\theta_{p})|_{E>E_{\rm th}}$ $\displaystyle=4\pi\int
E^{2}dE\int\sin\alpha_{\rm eq}d\alpha_{\rm eq}\,f(L,E,\sin\alpha_{\rm
eq})\frac{B(\theta_{p})}{B(0)}\frac{\cos\alpha_{\rm eq}}{\cos\alpha}$
$\displaystyle=4\pi\int E^{2}dE\int f(L,E,\sin\alpha_{\rm
eq})\bigg{(}\sqrt{\frac{B(\theta_{p})}{B(0)}}\sin\alpha_{\rm
eq}\bigg{)}\bigg{(}\sqrt{\frac{B(\theta_{p})}{B(0)}}\frac{\cos\alpha_{\rm
eq}}{\cos\alpha}d\alpha_{\rm eq}\bigg{)}$ $\displaystyle=4\pi\int
E^{2}dE\,\int\sin\alpha d\alpha\,f(L,E,\sin\alpha_{\rm eq})~{},$ (21)

which also applies to non-dipole magnetic fields.

Figure 7: The spatial distribution of the omnidirectional $e^{\pm}$ flux $J$
with an energy threshold $E_{\rm th}=10$ MeV. Presented values are based on
the fully-trapped scenario and in the unit of cm-2s-1. We fix $m_{\chi}=1$
GeV, $\Gamma_{D}^{-1}=R_{J}$, $\gamma=3$, and $\sigma_{\chi n}$ $\prod$ BR =
$10^{-39}$ cm2.

In Fig. 7, we illustrate the overall $e^{\pm}$ flux for a benchmark DM model
in the region of $L\in[1.3,1.5]$. We assume that DM reaches an equilibrium
between capture and annihilation. Then the signal rate is determined by the
product of $\sigma_{\chi n}$, which determines the capture rate, and $\prod$
BR $\equiv$ BR($2\chi\to 2\xi$) $\times$ BR($\xi\to e^{+}e^{-}$), which sets
the fraction of DM annihilation resulting in the $e^{+}e^{-}$ final state. As
shown in the right panel of Fig. 6, as $\alpha_{\rm eq}$ increases,
$\tau_{E}|_{\rm sync}$ becomes larger, which means that the electrons could be
trapped for a longer time period, resulting in a higher flux. Thus the maximum
flux is observed near the equatorial region, where electrons with
high-$\alpha_{\rm eq}$ tend to stay and contribute the most to $J$.

##  5 Scenarios Beyond the Dipole Approximation

In the previous section, the framework of electron flux calculation and major
friction terms are discussed, assuming a dipole magnetic field. The
characteristic time scale associated with the electron loss term is assumed to
be much longer than $\tau_{E}|_{\rm sync}$ or $\tau_{y}|_{\rm sync}$. However,
for measurements close to the Jovian atmosphere, the hard
scattering/absorption by gas, plasma, or small grains could introduce a non-
negligible electron loss. In addition, when $r\lesssim 1.3R_{J}$, the higher
multipole components of the magnetic field become significant, leading to a
breakdown of the dipole approximation and electron loss 2002P&SS…50..277W . If
any of these effects take place, the loss term in Eq. (4) could not be ignored
and is assumed to take the simplified form:

$\text{loss term}\simeq\tau_{\rm loss}^{-1}f~{},$ (22)

where $\tau_{\rm loss}$ is the time scale of electron disappearances and could
be a function of $L$, $E$, and $\sin\alpha_{\rm eq}$. When the electron loss
is faster than the friction processes ($\tau_{\rm loss}\ll\tau_{E}|_{\rm
sync}$), the solution of Eq. (4) converges to $f\simeq\tau_{\rm loss}\bar{I}$
accordingly.

Depending on the level of dipole approximation violation, we consider three
simplified scenarios of electron trapping in the Jovian magnetic field:

  * •

Fully trapped. The scenario corresponds to the situation in Section 4, in
which electrons are fully trapped and bounce back and forth between mirror
points in the Jovian magnetosphere before they lose most of their energies,
with at most minor deviations. This could apply when the observation is
performed in the region of $r\gtrsim 1.3R_{J}$ and near the magnetic equator
2002P&SS…50..277W . Fully trapped electrons do not meet breaking points where
the dipole approximation is badly broken when they drift along the $L$-shell.
In addition, the region also enjoys a low gas/plasma density due to its
distance from the planet. The extended halo ring 2004Icar..172…59T  affects
the region, but the effect is mild nenon2018rings .

Provided no significant electron loss in this case, we expect that the
electron lifetime is mostly determined by $\tau_{E}|_{\rm sync}$ as discussed
in Section 4.1. To be conservative and take into account of various absorption
terms, we still add $\tau_{\rm loss}$ in this case and approximate it as a
constant in the calculation. The minimum value of $\tau_{\rm loss}$ in the
fully trapped scenario is set by the largest $\tau_{\rm
loss}\sim\mathcal{O}(10^{5})$ s found due to the main ring absorption
2018GeoRL..4510838N . Strictly speaking, such a short time scale comparable to
$\tau_{E}|_{\rm sync}$ only applies to the main ring region with $L\in[1.7,2]$
and serves as a conservative lower bound. For $\tau_{\rm loss}$ longer than
$\tau_{E}|_{\rm sync}$, the electron absorption effect is sub-dominant.

  * •

Quasi-trapped. Electrons in this scenario could still be reflected by both
mirror points along the local field lines. However, as the electron drifts
along the azimuthal (longitudinal) direction due to the magnetic field’s
gradient, it lands on irregular field lines that lead to its removal
2002P&SS…50..277W . More concretely, if the electron drifts from one longitude
to another longitude along which the reflecting magnetic mirror points are
inside Jupiter, the electrons will be absorbed by and lost to the atmosphere.
In this case, $\tau_{\rm loss}$ must then be smaller than the electron drift
period. The dipole approximation may still hold except for a few regions where
electrons are absorbed. The azimuthal drift period $\tau_{\rm drift}$, under
the dipole approximation, reads schulz2012particle

$\tau_{\rm drift}\simeq 2.9\times
10^{5}~{}\text{s}\;L^{-1}(0.11\sin^{2}\alpha_{\rm eq}+0.12\sin\alpha_{\rm
eq}+0.77)^{-1}\bigg{(}\frac{100~{}\text{MeV}}{E}\bigg{)}~{}.$ (23)

By definition, a quasi-trapped electron or positron (with opposite drift
directions) will hit at least one region it cannot pass through within its
drift period. The expected time scale $\tau_{\rm loss}$ in this case is capped
by $\tau_{\rm drift}/2$. More realistically, there could be multiple breaking
spots for a certain $L$-shell, which further shortens $\tau_{\rm loss}$.
Therefore, for this scenario, we approximate the loss time scale to be

$\tau_{\rm loss}\sim\frac{\tau_{\rm drift}}{2}\times\text{const}~{},$ (24)

where the constant $<1$ depends on the magnetic field’s structure and position
of the in situ measurements. The flux predicted in this scenario is thus
longitude-dependent. The precise determination of $\tau_{\rm loss}$ is highly
non-trivial. Such modeling may benefit from matching with other in situ data
such as the fluxes of non-relativistic ions 2021JGRA..12628925K  or magnetic
fields 2022JGRE..12707055C . We will not pursue an accurate modeling of
$\tau_{\rm loss}$ in our paper. Instead, we will only use $\tau_{\rm drift}$
multiplied by a few benchmark constants as proxies for $\tau_{\rm loss}$.
Quasi-trapping could happen in regions close to the Jovian atmosphere with
$r<1.3R_{J}$ and away from the magnetic equator. The electron loss time scale,
$\tau_{\rm loss}$, estimated in this case could be shorter than
$\tau_{E}|_{\rm sync}$ and sets the electron trapping time.

  * •

Untrapped. In this scenario, the magnetic field lines electrons travel along
are highly irregular. The field strength may even decrease (instead of
increasing as in the dipole model) when the field line approaches the planet’s
surface. Charged particles may never be mirrored before being absorbed by the
atmosphere. The electron lifetime is extremely short, and the trajectory
length is characterized by the field line length outside the atmosphere, which
is of $\mathcal{O}(R_{J})$.

##  6 Measurements and Constraints on DM Models

Mission | Instrument | Energy Range | Observable | Measured Values | Scenario
---|---|---|---|---|---
Juno | ASC |  $\gtrsim 10$ MeV | $J_{\rm inf}$ |  $\lesssim 1\times 10^{4}$ cm-2s-1 becker2017observations  | Quasi-trapped
SRU |  $\gtrsim 10$ MeV | $J_{\rm inf}$ |  $\sim 4\times 10^{3}$ cm-2s-1 becker2017observations  | Quasi-trapped
Galileo Probe | P1 |  $\gtrsim 30$ MeV | $K$ |  $\sim$(7 - $5\times 10^{4}$) s-1 EPIData  |  Quasi-trapped & Fully trapped†
P2 |  $\gtrsim 100$ MeV | $K$ |  $\sim$(0.2 - $1\times 10^{3}$) s-1 EPIData  |  Quasi-trapped & Fully trapped†

Table 2: List of in situ measurements. The measured values of two Galileo
probe channels are obtained at $L\in[1.1,1.5]$ near the magnetic equatorial
region. For the two Juno RM instruments, the observations span across
$L\in[1.1,1.5]$ but are away from the magnetic equator. $\dagger$ indicates
that the fully trapped scenario only applies to readouts with $L\gtrsim 1.3$.

In this section, we will compare the relativistic electron signals with
observations to derive the in situ constraints on the DM model. There are only
two missions that have probed the Jovian innermost radiation belt ($L\lesssim
1.5$): the Galileo probe young1998galileo ; 2000JGR…10512093Y  and the Juno
mission 2017SSRv..213….5B . The four in situ measurements of electron fluxes
from these two missions are summarized in Table 2, with their corresponding
electron trapping scenarios considered. For measurements performed with
$L\in[1.3,1.5]$ and at small geomagnetic latitude $\theta_{p}$, the most
likely scenario is that the electrons are fully trapped. For regions with
$L\in[1.1,1.3]$ and large $\theta_{p}$, electrons from the decays of dark
mediators have smaller $\alpha_{\rm eq}$’s and a higher chance of entering the
drift loss zone as described in the quasi-trapped case in the previous
section. Thus we assume that for measurements performed in those regions, the
quasi-trapped scenario would apply.666There is also the possibility that most
electrons in those regions are untrapped. In the untrapped case, the electron
lifetime would be too short $\sim 0.2$ s to give a significant flux. The best
limit on $\sigma_{\chi n}\prod$BR is then about $10^{-36}$ cm2 for DM mass
$m_{\chi}$ at 1 GeV, considerably weaker than the bounds in the quasi-trapped
and fully trapped scenarios, which we will discuss in detail, but still
stronger than the other existing bounds for (sub)-GeV DM if the leading DM-
nucleon interaction is spin-dependent.

Detectors carried by the Galileo probe fischer1996high  and Juno orbiter
becker2017observations  do not provide spectra for sub-GeV-scale $e^{\pm}$.
Instead, the readouts are related to their geometric factors, $F(E)$’s,
describing the effective areas that detectors can receive the electron
signals. Harder electrons will have larger $F(E)$’s in general due to their
high shield-penetrating efficiencies.777Since these detectors do not utilize
magnetic fields to distinguish different charges, we take $F(E)$’s for
electrons and positrons to be identical. The $F_{i}(E)$ ($i=$ P1, P2) for the
Galileo EPI channels are reported in nenon2018physical . The observed count
rates $K_{i}$’s (number of electrons recorded per second) could be found in
EPIData  for P1 and P2 channels with different energy ranges, which are listed
in Table 2. In our DM model, the predicted count rates $\tilde{K}_{i}$’s are
given by

$\displaystyle\tilde{K}_{i}(L,\theta_{p})$ $\displaystyle=4\pi\int\sin\alpha
d\alpha\int E^{2}dE\,F_{i}(E)f(L,E,\sin\alpha_{\rm eq})=\int
F_{i}(E)\left(\frac{\partial J}{\partial E}\right)_{\rm dec}dE~{},$
$\displaystyle=J_{\rm dec}\int F_{i}(E)J_{\rm dec}^{-1}\left(\frac{\partial
J}{\partial E}\right)_{\rm dec}dE~{},$ (25)

where we use Eq. (21) in the second equality; $\left(\partial J/\partial
E\right)_{\rm dec}$ is the normalized differential energy spectrum of the
electrons from dark mediator decays, and the total flux $J_{\rm dec}=\int
dE\left(\partial J/\partial E\right)_{\rm dec}$. Requiring $\tilde{K}_{i}\leq
K_{i}$, we could set constraints on DM models, in particular, the combination
$\sigma_{\chi n}\prod{\rm Br}$. Note that this approach is conservative by
being agnostic about the contributions to $K_{i}$ from astrophysical Jovian
electrons and other charged particles, which constitute the backgrounds for
our DM search. Currently, such backgrounds are not well understood. With
further developments in understanding the Jovian radiation belts, we could
hope for stronger constraints on the DM parameters and even hints for new
physics if anomalous features were observed, which could not be explained by
astrophysical sources. Choosing a DM benchmark model with $\gamma=3$, we plot
the ratio between the count rates induced by unit total flux,
$\tilde{K}_{i}/J_{\rm dec}$, at the magnetic equator with a benchmark
$\tau_{\rm loss}=10^{6}$ s in the left panel of Fig. 8. When model parameters
and positions of measurements vary, the spectrum $(\partial J/\partial E)_{\rm
dec}$ also changes and gives rise to similar but different values of
$\tilde{K}_{i}/J_{\rm dec}$. This ratio indicates the averaged effective
geometric factor weighted by the spectrum in the DM model. The figure shows
that $\tilde{K}_{i}/J_{\rm dec}$ increases with $m_{\chi}$ or equivalently
with the electron energy, as expected.

For Juno’s RM data collected by the Advanced Stellar Compass (ASC) and SRU,
only the inferred omnidirectional fluxes with $E>10$ MeV, which we denote as
$J_{\rm inf}$, are available and listed in Table 2. Such inferred fluxes
assume an input spectrum shape of astrophysical electrons around Jupiter,
denoted as $(\partial J/\partial E)_{\rm bkg}$, which is not necessarily
normalized becker2017juno . The spectrum of the Jovian electrons is much
softer than the DM-induced one. The observed count rates $K_{i}$, $i=$ASC,
SRU, which are not published, are related to $J_{\rm inf}$ as

$K_{i}=\frac{J_{\rm inf}}{J_{\rm
bkg}}\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)\left(\frac{\partial J}{\partial
E}\right)_{\rm bkg}dE~{},$ (26)

where $J_{\rm bkg}=\int\left(\frac{\partial J}{\partial E}\right)_{\rm bkg}dE$
and $F_{i}(E)$ encodes the detector efficiency. The ratio $J_{\rm inf}/J_{\rm
bkg}$ fixes the normalization of the spectrum. By matching the count rates for
each probe, the flux $J_{\rm dec}$ induced by the dark mediator decays could
be translated to a predicted inferred flux $\tilde{J}_{\rm inf}$ as:

$\displaystyle\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)\left(\frac{\partial
J}{\partial E}\right)_{\rm dec}dE=\frac{\tilde{J}_{\rm inf}}{J_{\rm
bkg}}\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)\left(\frac{\partial J}{\partial
E}\right)_{\rm bkg}dE$ $\displaystyle\Rightarrow\quad\tilde{J}_{\rm
inf}=J_{\rm
bkg}\frac{\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)\left(\frac{\partial
J}{\partial E}\right)_{\rm
dec}dE}{\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)\left(\frac{\partial
J}{\partial E}\right)_{\rm bkg}dE}=J_{\rm
dec}\frac{\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)J_{\rm
dec}^{-1}\left(\frac{\partial J}{\partial E}\right)_{\rm
dec}dE}{\int_{10~{}\text{MeV}}^{\infty}F_{i}(E)J_{\rm
bkg}^{-1}\left(\frac{\partial J}{\partial E}\right)_{\rm bkg}dE}~{}.$ (27)

We could then set constraints on DM parameters by requiring $\tilde{J}_{\rm
inf}\leq J_{\rm inf}$. The right panel of Fig. 8 plots the ratio between
$\tilde{J}_{\rm inf}$ and $J_{\rm dec}$ for different $m_{\chi}$’s. Although
the injected $e^{\pm}$ spectrum gets softened by the synchrotron radiation,
the average geometric factor of electrons from dark mediator decays exceeds
that of the background, leading to $\tilde{J}_{\rm inf}/J_{\rm dec}>1$ and
higher sensitivities for any $m_{\chi}>100$ MeV.

Figure 8: Relations between the predicted $e^{\pm}$ flux from mediator decays
$J_{\rm dec}$ and corresponding observables as a function of $m_{\chi}$. Both
plots are computed with benchmark values of $\gamma=3$ and $\theta_{p}=0$.
LEFT: the ratio between $\tilde{K}_{i}$ and $J_{\rm dec}$ for the Galileo
probe channels P1 and P2 at $L=1.5$ assuming $\tau_{\rm loss}=10^{6}$ s, which
could be interpreted as the average effective geometric factor. RIGHT: the
ratio between the inferred flux $\tilde{J}_{\rm inf}$ and the DM induced flux
$J_{\rm dec}$, which could be understood as the ratio of the average geometric
factors for $e^{\pm}$ in DM and background models, for the Juno RM instruments
SRU and ASC at $L=1.2$, respectively. We take $\tau_{\rm loss}=0.05\tau_{\rm
drift}$.

As discussed in Section 2 and 3, when the equilibrium between DM capture and
annihilation is reached, the $e^{\pm}$ flux in the DM model is entirely
determined by $\sigma_{\chi n}\prod$ BR. We first consider the constraints on
this combination in the fully trapped scenario, which applies to the Galileo
EPI probes P1 and P2 with readouts at $L\gtrsim 1.3$ near the magnetic
equator. Constraints are calculated by comparing $\tilde{K}_{i}$’s induced by
$\xi$ decays with the observed count rates. The values of $L$ and $\theta_{p}$
for each observation are obtained using the dipole component of the JRM09
magnetic model connerney2018new  for simplicity and concreteness. As discussed
in Section 5, the constraints in the fully trapped case are not very sensitive
to $\tau_{\rm loss}$ since electron loss in this region is a sub-dominant
effect. The results are presented in the left panel of Fig. 9, with the bands
obtained by varying $\tau_{\rm loss}$ from $10^{5}$ to $10^{8}$ s. We fix
$\gamma=3$, and $\Gamma_{D}=R_{J}^{-1}$ so that the proper lifetime of dark
mediator $\xi$ is 0.2 s. As $m_{\chi}$ increases, $\tau_{E}|_{\rm sync}$
becomes more dominating, reducing the differences in bounds introduced by a
varying $\tau_{\rm loss}$ and making the band narrower. For a given $\tau_{\rm
loss}$, the bounds are the strongest, $\sigma_{\chi n}\prod$ Br $\lesssim{\cal
O}(10^{-39})$ cm2, when $m_{\chi}\simeq 1$ GeV, where the DM capture is
efficient and injected electrons are energetic. Direct detection bounds on
both spin-independent (SI) XENON:2018voc ; CDEX:2019hzn ; XENON:2019zpr ;
PandaX-4T:2021bab ; LUX-ZEPLIN:2022qhg  and spin-dependent (SD) DM-nucleon
scattering cross sections Behnke:2016lsk ; SuperCDMS:2017nns ; XENON:2019zpr ;
PICO:2019vsc  are also presented for comparison.888Cosmological bounds on
$\sigma_{\chi n}$ in this mass range, which are not plotted, are weaker
Gluscevic:2017ywp ; Buen-Abad:2021mvc . We see that the Jupiter in situ
electron flux measurements set stronger constraints for $m_{\chi}$ below a few
GeV, in particular, if the interaction between DM and nucleons is spin-
dependent. Note that SD and SI scatterings do not make a difference for DM
capture inside Jupiter since Jupiter is mostly made up of hydrogen atoms.

Figure 9: Observed upper bounds on $\sigma_{\chi n}\times\prod$ Br due to the
relativistic $e^{\pm}$ flux from dark mediator decays in different trapping
scenarios. LEFT: limits from the Galileo probe EPI P1 and P2 observations in
the fully trapped scenario ($L\simeq 1.45$). The band for each channel is
obtained by varying $\tau_{\rm loss}\in[10^{5},10^{8}]$ s (corresponding to
upper and lower ends of the band). RIGHT: limits from the two Galileo EPI
channels as well as Juno ASC and SRU measurements in the quasi-trapped
scenario at $L\lesssim 1.2$. Since the exact time scale of $\tau_{\rm loss}$
in this case is unknown, we show the band from computing $\tau_{\rm loss}$
using Eq. (23) and (24) with the constant $\in[0.01,0.1]$ (leading to upper
and lower band edges). The lighter (darker) grey regions are constraints on
$\sigma_{\chi n}$ from direct detection experiments, assuming SI (SD)
scattering XENON:2018voc ; CDEX:2019hzn ; XENON:2019zpr ; PandaX-4T:2021bab ;
LUX-ZEPLIN:2022qhg ; Behnke:2016lsk ; SuperCDMS:2017nns ; PICO:2019vsc .

The next step is to consider measurements in regions with $L\lesssim 1.3$
where the electrons are quasi-trapped. The method is similar to the fully-
trapped case discussed above but with $\tau_{\rm loss}$ determined by drift
period as in Eq. (23) and (24). In particular, we vary the constant in Eq.
(24) between $0.01$ to $0.1$ as benchmark values. The chosen range of the
constant is arbitrary. It should depend on the measurement position including
the longitude and becomes larger for smaller $L$. Nevertheless, the
simplification allows us to estimate the DM parameter space Jupiter
measurements could probe. We will leave a more precise determination to future
work. The constraints from the Galileo probe data are shown in the right panel
of Fig. 9. Again we fix $\gamma=3$ and $\Gamma_{D}=R_{J}^{-1}$. In the quasi-
trapped case, $\tau_{\rm loss}$ is shorter than that in the fully-trapped
case, and the electron loss becomes important for evaluating the electron
trapping time. However, the Galileo EPI readouts are much smaller in the
innermost region, which compensates for the shorter $\tau_{\rm loss}$,
resulting in stronger limits, $\sigma_{\chi n}\times\prod$ Br $\lesssim{\cal
O}(10^{-41})$ cm2 at $m_{\chi}$ around 1 GeV. Limits from the two Juno RM
investigations are also included in the right panel of Fig. 9, obtained by
comparing $\tilde{J}_{\rm inf}$ with $J_{\rm inf}$ reported in
becker2017observations . Their sensitivities are weaker compared to the
Galileo ones, limited by multiple factors such as starlight backgrounds and
their dynamic ranges becker2017juno .

Figure 10: Similar to the left panel of Fig. 9 but with different model
parameters. The electron loss time scale $\tau_{\rm loss}=10^{6}$ s is applied
in both panels. LEFT:  Comparison between limits with $\gamma=3$ (upper band
edge) and $\gamma=8$ (lower band edge), while the ratio $\Gamma_{D}/\gamma$ is
fixed at $R_{J}^{-1}/3$. RIGHT:  Comparison between limits from different
$\Gamma_{D}$’s. Solid curves are results from $\Gamma_{D}=R_{J}^{-1}$, while
dashed (dot-dashed) curves are results with
$\Gamma_{D}=0.1R_{J}^{-1}(10R_{J}^{-1})$.

Since the constraints in Fig. 9 are obtained with the benchmark $\gamma=3$ and
$\Gamma_{D}=R_{J}^{-1}$, it is necessary to check limits with alternative
model parameters. Here we use the fully-rapped scenario constraints with
$\tau_{\rm loss}=10^{6}$ s as the representative. In the left panel of Fig.
10, we show the comparison between models with $\gamma=3$ and $\gamma=8$. To
keep the proper decay length of $\xi$ the same for a fair comparison, we fix
$\Gamma_{D}/\gamma=R_{J}^{-1}/3$ in both cases. Consequently, the difference
between the two benchmarks is only induced by the injection term $\bar{I}$ and
is minor, as shown in the figure. The effect of varying $\Gamma_{D}$ could be
estimated with Eq. (11) and is presented in the right panel of Fig 10. For
$\Gamma_{D}/\gamma\lesssim R_{J}^{-1}$, the smaller $\Gamma_{D}$ is, the fewer
decays happen, resulting in weaker constraints. When $\Gamma_{D}/\gamma\gg
R_{J}^{-1}$, most decays happen inside Jupiter, and the bounds get weaker as
well.

Finally, we comment on the assumption of the equilibrium between DM capture
and annihilation. We focus on the DM mass range ${\cal O}(0.1-10)$ GeV and
show that Jupiter missions could constrain $\sigma_{\chi n}\prod$ Br of ${\cal
O}(10^{-41}-10^{-39})$ cm2 under different assumptions. From Eq. (6), we see
that the equilibrium time scale is shorter than or comparable to the Jupiter
age for DM about or heavier than 1 GeV. Thus the equilibrium assumption shall
be valid. In the region of $m_{\chi}\lesssim 1$ GeV, the evaporation effect
could become significant, and the limits we plot are only tentative. We leave
a full analysis including both annihilation and evaporation in this regime to
future work.

##  7 Summary and Outlook

In this article, we propose an intriguing connection between studies of Jovian
radiation belts and DM searches. We consider a class of DM models in which DM
captured by Jupiter could annihilate into a pair of long-lived dark mediators,
which subsequently decay into $e^{\pm}$ outside Jupiter. This could happen
when the decay length of the mediator is comparable to the Jupiter radius,
e.g., a sub-GeV scale dark photon with kinetic mixing parameter $\epsilon\sim
10^{-10}$ which is not (fully) covered experimentally. The produced $e^{\pm}$
could be either fully- or quasi-trapped in the innermost radiation belts and
contribute to energetic electron fluxes, which are recorded in the in situ
measurements by various Jupiter missions. We apply the data collected by the
Galileo probe and Juno missions and find powerful constraints on the product
of $\sigma_{\chi n}$, DM-nucleon scattering cross section, and $\prod$ Br, the
branching fraction of DM annihilations ending in $e^{\pm}$ final state, for DM
mass between (0.1 - 10) GeV. In particular, for DM at 1 GeV, the bound on the
product could be as strong as $10^{-41}$ cm2 from the data collected by
Galileo EPI probes at $L\lesssim 1.2$ where the electrons are quasi-trapped.
The quasi-trapping case is subject to uncertainties related to the electron
loss effects, which we adopt a simplified modeling. A weaker but potentially
more reliable constraint could be derived with data collected at larger $L$
($L\approx 1.5$) where the electrons are fully-trapped and the electron loss
is sub-dominant. In this case, the upper limit on $\sigma_{\chi n}\times\prod$
Br is around $10^{-39}$ cm2 for DM at 1 GeV. These bounds could be comparable
to or stronger than current GeV-scale DM direct detection searches, in
particular, for spin-dependent DM-nucleon scattering.

Our study is only an initial effort to apply the Jovian data from the past,
ongoing, and future Jupiter missions to probe new physics beyond the standard
model of particle physics. The analysis could be improved in several aspects:
the investigation in the quasi-trapping scenario could be refined with a more
precise modeling of the electron loss effects; a better understanding of the
astrophysical electron sources could allow us to set stronger constraints on
the DM parameters. We also want to stress that future Jupiter missions may
enable more precise measurements of energetic electron fluxes and the
corresponding spectra at different positions 2021ExA…tmp..136R , which could
greatly strengthen the bound. Beyond the analysis we did, we list and outline
several other (but not all) exciting avenues to explore the power of Jovian
data in new physics searches below.

  * •

X-rays from inverse Compton scattering The electrons from dark mediator decays
could produce X-rays through inverse Compton scattering with solar
photons.999We thank Elias Roussos for pointing out this possibility. The
interaction rate between injected electrons and solar photons is low, leaving
the energy loss due to this effect negligible in calculations of electron’s
flux. However, photons back-scattered by ultra-relativistic MeV-GeV-scale
electrons pick up energies in the keV-MeV range, depending on the electron
spectrum. So far, there are no in situ measurements of Jovian $X$-rays.
Nevertheless, there are data sets on the Jovian X-ray backgrounds from
Chandra, XMM-Newton and Suzaku X-ray telescopes bhardwaj2005solar ;
elsner2005simultaneous ; branduardi2007latest ; branduardi2008spectral ;
dunn2020jupiter , which are all located near the Earth. In particular, in
Ezoe:2010hw , it was pointed out that a diffuse hard X-ray (1-5 keV) emission
around Jupiter could be explained by solar photons scattering with a large
population of energetic electrons trapped within $r\simeq(4$-$8)R_{J}$. A
quick estimate shows that such a large electron source might be provided by
injected electrons from DM annihilations in our model. We will leave a more
detailed study for the future.

There are other secondary conversion products that could be observational
targets, such as ions from electron impact ionization or even higher energy
photons, i.e., $\gamma$ rays from electromagnetic radiation Leane:2022TBA  or
electron-positron annihilations.

  * •

Positron signals The in situ measurements discussed in this paper are
implemented by instruments without magnetic fields and do not distinguish
positrons from electrons. Since Jupiter is not a known active positron source,
a Jovian positron signal could be striking if detected. One possibility is
that the high-energy positrons outside Jupiter predicted in our DM model could
escape the magnetosphere and transfer to the inner heliosphere through the
twisted magnetic field lines, namely the Parker spiral parker1958dynamics .
Then one could use PAMELA PAMELA:2011bbe  or AMS AMS:2014xys  cosmic-ray
detectors to study the Jovian positrons near the earth. The relative motion
between Jupiter and the Earth will also create a $\sim 13$ month period of
positron flux. Such dynamics and detection potential have been studied for the
Jovian electrons transported to the earth orbit potgieter2002effects ;
ferreira2005transport ; DIFELICE20082037 ; strauss2011modeling ;
vogt2018jovian ; vogt2022numerical .

  * •

Solar axion conversion to X-rays This possibility is not related to the DM
model in this paper.101010We thank Ben Monreal for bringing up this
possibility. It is well known that the Sun could produce axions, one of the
most motivated feebly-coupled particles beyond the standard model, with
energies set by the solar core temperature, which is in the keV range
Fukugita:1982ep ; Sikivie:1983ip . These relativistic axions from the Sun,
also referred to as solar axions, have become a standard benchmark scenario
for direct detection experiments Arisaka:2012pb ; XENON100:2014csq ;
LUX:2017glr ; PandaX:2017ock ; XENON:2020rca . Jupiter, with its strong
magnetic field and huge volume, could serve as a giant cavity for oscillations
between solar axions and photons through the axion-photon coupling. The signal
we could look for is that the solar axions propagate through Jupiter, and some
of them convert into X-rays in the keV range at the dark side of Jupiter away
from the Sun. A similar idea was proposed a while back to use the Earth as the
converter Davoudiasl:2005nh ; Davoudiasl:2008fy . The estimated conversion
probability (scaling as $B^{2}R^{2}$ with $B$ the magnetic field strength and
$R$ the distance axion travels) for the Earth could be comparable to the CERN
axion solar telescope (CAST), a leading terrestrial experiment setting strong
bound on axion-photon coupling CAST:2004gzq ; CAST:2007jps ; CAST:2008ixs ;
CAST:2011rjr ; CAST:2013bqn . While the solar axion flux at Jupiter is reduced
by a factor of 25 compared to that at the Earth since Jupiter is further away
from the Sun, the conversion probability is larger since the Jovian $B$ is one
order of magnitude above the Earth one and the size of the magnetic field the
axion could traverse is also larger. The challenges are two-fold: i) there is
no current in situ measurement of X-ray at the dark side of Jupiter, as
mentioned in the first item; ii) the Jovian X-ray background close to the
planet and in regions at low latitudes away from the pole aurorae is not fully
understood.

## Acknowledgements

We thank Heidi Becker, Raghuveer Garani, Peter Kollmann, Rebecca Leane, Matt
Reece, Elias Roussos, and Hai-Bo Yu for useful communication and discussions.
JF and LL are supported by the DOE grant DE-SC-0010010, the NASA grant
80NSSC18K1010 and 80NSSC22K081.

##  Appendix A Monte Carlo Simulations of the Injected Electron Phase Space
Distribution

The injection distribution $I$ depends on $E$ and $\sin\alpha$ in a highly
nontrivial way, due to the complicated $e^{\pm}$ energy spectrum from boosted
mediator decays. In contrast to the isotropic electron injection from slow
neutron decays 1961JGR….66.4027L , the energy spectrum of injected electrons
is strongly correlated with the distribution of the pitch angle $\alpha$. The
analytical form of $I$ also contains divergences near the boundary of
kinematically allowed regions. In order to get the integrated form of $I$, we
adopt the Monte Carlo approach that is relatively stable numerically and
computationally cheap.

To start with, we generate an $e^{\pm}$ sample sets
$\\{D\\}={D_{1}(w_{1}),D_{2}(w_{2}),...,D_{n}(w_{n})}$ of the same $L$-shell.
Each point $D_{i}$ is obtained by sampling the boosted two-body phase space of
a mediator decaying at a distance $r_{i}$ and geomagnetic latitude
$\theta_{p,i}$. The pitch angle $\alpha_{i}$ and $\alpha_{{\rm eq},i}$ for
each $D_{i}$ are calculated using the magnetic dipole field model, while its
energy $E_{i}$ is directly known from the decay kinematics. For each $D_{i}$,
a weight $w_{i}$ is assigned. The Monte Carlo sample $\\{D\\}$ would be a
representative of the injection term when it satisfies:

$\sum_{\mathclap{\begin{subarray}{c}S_{i}\in[S,S+dS],\\\ E_{i}\in[E,E+dE],\\\
\alpha_{i}\in[\alpha,\alpha+d\alpha]\end{subarray}}}w_{i}\simeq 4\pi
I(L,E,\sin\alpha)E^{2}dE\,\sin\alpha d\alpha\,dS~{}.$ (28)

Integrating the equation above over $\alpha$ and $E$ and plugging it into Eq.
(15), we have:

$\sum_{\mathclap{\theta_{p,i}\in[\theta_{p},\theta_{p}+d\theta_{p}]}}w_{i}=\rho_{D}(r)\frac{dS}{d\theta_{p}}d\theta_{p}\simeq\frac{\rho_{D}(L)}{\cos^{4}\theta_{p}}\frac{dS}{d\theta_{p}}d\theta_{p}~{},$
(29)

where we take the conservative approximation of
$\rho_{D}(r)>\frac{\rho_{D}(L)L^{2}}{r^{2}}\frac{\rho_{D}(L)}{\cos^{4}\theta_{p}}$
to make the calculation $\Gamma_{D}$ invariant. Since our $\\{D\\}$ is
obtained by evenly sampling
$\theta_{p}\in[-\arccos\sqrt{L^{-1}},\arccos\sqrt{L^{-1}}]$, we can thus
assign

$w_{i}=\frac{2\arccos\sqrt{L^{-1}}}{N_{\rm
sample}}\frac{\rho_{D}(L)}{\cos^{4}\theta_{p}}\frac{dS}{d\theta_{p}}~{},$ (30)

where $N_{\rm sample}$ is the total number of sample points.

To calculate the $L$-shall average $\bar{I}$, we follow the standard steps in
1961JGR….66.4027L :

$\displaystyle\int_{z}IdzE^{2}dE\sin\alpha_{\rm eq}d\alpha_{\rm eq}$
$\displaystyle=\int_{S}I\frac{dS}{\cos\alpha}E^{2}dE\sin\alpha_{\rm
eq}d\alpha_{\rm eq}=\int_{S}I\frac{dS}{\cos\alpha}\frac{d\cos\alpha_{\rm
eq}}{d\cos\alpha}E^{2}dE\sin\alpha d\alpha$
$\displaystyle=\int_{S}IdS\frac{B(0)}{B(\theta_{p})}\frac{1}{\cos\alpha_{\rm
eq}}E^{2}dE\sin\alpha d\alpha$
$\displaystyle\simeq\frac{1}{4\pi}\sum_{\begin{subarray}{c}E_{i}\in[E,E+dE],\\\
\alpha_{i}\in[\alpha,\alpha+d\alpha]\end{subarray}}\frac{B(0)}{B(\theta_{p})\cos\alpha_{\rm
eq}}w_{i}~{}.$ (31)

Finally, the trajectory length

$\int_{z}dz=\int_{-\arcsin\sqrt{L^{-1}}}^{\arcsin\sqrt{L^{-1}}}\frac{dS}{d\theta_{p}}\frac{1}{\cos\alpha}d\theta_{p}\equiv
2T(\sin\alpha_{\rm eq})~{}.$ (32)

Although $T$ has no analytical form in general, there is a well-known
numerical approximation schulz2012particle

$T(y)\simeq 1.3802-0.3198(y+\sqrt{y})~{}.$ (33)

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