© 2021 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society
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Can the Local Bubble explain the radio background?
Martin G. H. Krause? and Martin J. Hardcastle
Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane,
Hatfield, Hertfordshire AL10 9AB, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The ARCADE 2 balloon bolometer along with a number of other instruments have detected what
appears to be a radio synchrotron background at frequencies below about 3 GHz. Neither extra-
galactic radio sources nor diffuse Galactic emission can currently account for this finding. We use
the locally measured Cosmic ray electron population, demodulated for effects of the Solar wind, and
other observational constraints combined with a turbulent magnetic field model to predict the radio
synchrotron emission for the Local Bubble. We find that the spectral index of the modelled radio
emission is roughly consistent with the radio background. Our model can approximately reproduce
the observed antenna temperatures for a mean magnetic field strength B between 3-5 nT. We ar-
gue that this would not violate observational constraints from pulsar measurements. However, the
curvature in the predicted spectrum would mean that other, so far unknown sources would have
to contribute below 100 MHz. Also, the magnetic energy density would then dominate over ther-
mal and cosmic ray electron energy density, likely causing an inverse magnetic cascade with large
variations of the radio emission in different sky directions as well as high polarisation. We argue
that this disagrees with several observations and thus that the magnetic field is probably much
lower, quite possibly limited by equipartition with the energy density in relativistic or thermal par-
ticles (B = 0.2 − 0.6 nT). In the latter case, we predict a contribution of the Local Bubble to the
unexplained radio background at most at the per cent level.
Key words: radio continuum: general – radio continuum: ISM – ISM: bubbles –
Galaxy: local interstellar matter – cosmology: diffuse radiation
1 INTRODUCTION
The balloon-borne precision bolometer ARCADE 2 has re-
ported an excess emission above the Cosmic microwave back-
ground (CMB) of 54 ± 6 mK at 3 GHz (Fixsen et al. 2011).
Together with measurements from the Long Wavelength Ar-
ray at 40-80 MHz and other measurements (Dowell & Taylor
2018), this forms the extragalactic radio background, which
dominates the sky emission below 1 GHz. When the con-
tributions from the CMB and the Milky Way are removed,
an isotropic component with a power law spectrum with in-
dex -2.58 when plotting antenna temperature vs. frequency
remains (α = 0.58 for flux density S ∝ ν−α). The relevant
frequency range includes the 60-80 MHz region, where the
21 cm signal from the epoch of reionisation is expected.
An absorption feature of less than 1 per cent of the radio
background emission has indeed been found by the Experi-
ment to Detect the Global Epoch of Reionization Signature
(EDGES) at these frequencies (Bowman et al. 2018). For
the interpretation of the absorption feature as of cosmolog-
ical origin, it is important to understand whether the radio
? E-mail: M.G.H.Krause@herts.ac.uk
synchrotron background is produced locally or at high red-
shift (e.g., Monsalve et al. 2019; Ewall-Wice et al. 2020).
Since the contribution from the Milky Way has a dis-
tinct geometry and is accounted for already in the aforemen-
tioned results, the most straightforward explanation would
be a large population of known extragalactic radio sources,
namely radio loud active galactic nuclei and star-forming
galaxies. At 3 GHz, measurements with the Karl G. Jan-
sky Very Large Array find a combined antenna tempera-
ture for all such sources of 13 mK, significantly below the
ARCADE 2 result (Condon et al. 2012). A similar measure-
ment has recently been performed with the Low-Frequency
Array (LOFAR) with the similar result that only about
25 per cent of the radio background can be accounted
for by resolved radio sources (Hardcastle et al. 2020). An-
other suggestion that has been put forward is a Galactic
halo of cosmic ray electrons with a scale length of 10 kpc
(Orlando & Strong 2013; Subrahmanyan & Cowsik 2013).
The required particle population would however also pro-
duce X-rays via inverse Compton scattering, which would
violate observational constraints (Singal et al. 2010). Also,
such a prominent radio halo would be atypical for galaxies
like the Milky Way (Singal et al. 2015; Stein et al. 2020),
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even though halos of up to a few kpc at 150 MHz have been
found recently (Stein et al. 2019). These difficulties have in-
spired a number of interesting explanations, including for
example free-free emission related to galaxy formation at
high redshift (Liu et al. 2019) and dark matter annihilation
(Hooper et al. 2012). See Singal et al. (2018) for a recent
review.
We investigate here a comparatively simple explana-
tion: synchrotron emission from the Local Bubble. The
Local Bubble is a low-density cavity in the interstellar
medium around the Solar system (e.g., Cox & Reynolds
1987) The superbubble was likely formed by winds and
explosions of massive stars (Breitschwerdt et al. 2016;
Schulreich et al. 2018). Hot gas in the bubble contributes
significantly to the soft X-ray background (Snowden et al.
1997, 1998; Galeazzi et al. 2014; Snowden 2015). The
boundary is delineated by a dusty shell that has been
mapped with absorption data against stars with known
distances (Lallement et al. 2014; Snowden et al. 2015b;
Pelgrims et al. 2020). Direct observation of the likely
present neutral hydrogen supershell is difficult against the
background of the Milky Way, but the distinct structure
of erosion of the interface towards a neighbouring super-
bubble has been observed (Krause et al. 2018). Similar
features are also known from Nai and Hi absorption
studies (Lallement et al. 2014). Interaction of cosmic
ray particles with the supershell may explain the high-
energy neutrinos observed with IceCube (Andersen et al.
2018; Bouyahiaoui et al. 2020). The superbubble con-
tains high ionisation species (Breitschwerdt & de Avillez
2006), filaments and clouds of partially neutral and
possibly even molecular gas (e.g., Gry & Jenkins 2014,
2017; Redfield & Linsky 2008, 2015; Snowden et al.
2015a; Farhang et al. 2019; Linsky et al. 2019) and is
threaded by magnetic fields (e.g., Andersson & Potter 2006;
McComas et al. 2011; Frisch et al. 2015; Alves et al. 2018;
Piirola et al. 2020). It has already been suggested as the
physical origin of high latitude radio emission by Sun et al.
(2008).
We first make an empirical model based on a compar-
ison to the non-thermal superbubble in the dwarf galaxy
IC 10 (Sect. 2) and then present a detailed model based on
the locally observed population of cosmic ray electrons and
available constraints on the magnetic field in the Local Bub-
ble (Sect. 3). We discuss our findings in the context of the
observational constraints in Sect. 4 and conclude in Sect. 5
that a dominant contribution of the Local Bubble to the
radio background seems unlikely.
2 EMPIRICAL MODEL BY COMPARING TO
THE NON-THERMAL SUPERBUBBLE IN
IC 10
Superbubbles are not usually known to emit a non-thermal
radio synchrotron spectrum. One such object has, however,
been identified in the dwarf galaxy IC 10 (Heesen et al.
2015). The reason why it stands out against thermal and
non-thermal radio emission of the host galaxy might be an
unusually strong explosion, a hypernova, about 1 Myr before
the time of observation. Its size is, similar to the Local Bub-
ble, ∼ 200 pc. The radio spectrum is a power law with the
same spectral index as the radio background, S(ν) ∝ ν−0.6.
The observed non-thermal emission is 40 mJy at 1.5 GHz.
We use these properties of the nonthermal superbubble
in IC 10 to estimate those of the Local Bubble as follows.
First, we scale this by a factor of fs = 0.1 to account for the
fact that likely none of the supernovae that shaped the Local
Bubble was a hypernova. With the given spectral index, this
yields a flux density of 2.7 mJy at 3 GHz. With a distance
of 0.7 Mpc to IC 10, we then get a spectral luminosity of
1.6 × 1017 W Hz−1. Assuming a bubble radius of 100 fr10 pc,
we obtain a volume emissivity of
lν = 1.3 × 10−39
(
fs
0.1
)
f −3r10 WHz
−1m−3 (1)
Placing the Sun at the centre of such a non-thermal bubble
yields a flux contribution from each shell at distance r of
dSν =
4pir2 dr lν
4pir2
= lν dr . (2)
The integral is straightforward and results, for a radius of
the Local Bubble of 100 frLB pc in:
Sν = 4 × 105
(
fs
0.1
)
f −3r10 frLB
( ν
3GHz
)−0.6
Jy . (3)
The antenna temperature follows from this via Tν =
Sνc2/(8pikBν2), and so
Tν = 113
(
fs
0.1
)
f −3r10 frLB
( ν
3GHz
)−2.6
mK . (4)
This overpredicts the radio synchrotron background by
a factor of two and thus demonstrates that the contribution
of the Local Bubble can in principle be very important.
3 DETAILED MODEL OF THE RADIO
SYNCHROTRON EMISSION OF THE
LOCAL BUBBLE
Thanks to a number of measurements unique to the Local
Bubble, it is possible to predict its radio emission with far
better accuracy than we have done in the previous section.
Both elements required to predict synchrotron emission, the
energy distribution of cosmic ray electrons and positrons and
the strength and geometry of the magnetic field are con-
strained by recent experimental data. The Alpha Magnetic
Spectrometer (AMS) onboard the International Space Sta-
tion (ISS) has measured the near-earth energy distribution
for cosmic ray electrons with energies E between 0.5 GeV
and 1.4 TeV(Aguilar et al. 2019). Constraints at lower en-
ergy and outside the volume influenced by the Solar wind
have been provided by Voyager I (Cummings et al. 2016).
The part of this distribution relevant for the radio back-
ground can be calculated once the magnetic field is known,
and constraints are available from pulsar observations. We
review the observational constraints on both, magnetic field
and particle energy spectrum, in the following three subsec-
tions.
3.1 Magnetic field constraints
The magnetic field in the local bubble is constrained by
measurements of the Faraday effect, i.e. the rotation of
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the plane of polarisation of pulses from radio pulsars, com-
bined with the pulse dispersion as a function of frequency.
Such measurements yield magnetic field strength estimates
of B = 0.05 − 0.2 nT (Xu & Han 2019), but the measure-
ments do not contain information whether this field strength
is volume filling or restricted to a small fraction of the path
through the Local Bubble. Field reversals and density inho-
mogeneities affect the estimate. The quantities directly mea-
sured from the pulsar measurements are dispersion measure
DM and rotation measure RM. For eight pulsars at distances
between 90-140 pc, i.e., towards the edge of the Local Bub-
ble, Xu & Han (2019) report a mean dispersion measure of
42 cm−3 pc with a standard deviation of 20 cm−3 pc. This
corresponds to a column of free, thermal electrons of
Ne = (1.3 ± 0.6) × 1024 m−2 . (5)
X-ray measurements of the hot bubble plasma suggest a
thermal electron density of ne,X = (4.68 ± 0.47) × 103 m−3
(Snowden et al. 2014). This value is very typical for super-
bubbles, including X-ray bright ones, as shown in 3D nu-
merical simulations (Krause et al. 2013a, 2014). The contri-
bution to the free electron column in the Local Bubble from
the X-ray emitting plasma, again for a radius of the Local
Bubble of 100 frLB pc is therefore
Ne,X = (1.4 ± 0.1) × 1022 frLBm−2 . (6)
Warm clouds within the Local Bubble have sizes of several
parsecs and electron densities of the order of ne,wc = 105 m−3
(e.g., Gry & Jenkins 2017; Linsky et al. 2019). Assuming a
total warm cloud path length of 10 fwcp pc, we obtain an
estimate for the corresponding free electron column of:
Ne,wc = 3 × 1022 fwcpm−2 . (7)
Hence, neither the hot X-ray plasma nor the warm clouds
and filaments contribute significantly to the pulsar disper-
sion measures. As Xu & Han (2019) note, the dispersion
measure is probably produced predominantly by the bub-
ble wall, an ionised mixing layer between the superbubble
interior and the cold supershell (compare also Krause et al.
2014).
The root mean square rotation measure against the
aforementioned eight pulsars is 33 rad m−2. For a plasma
with electron density ne and line-of-sight magnetic field Blos,
the rotation measure may be expressed as:
RM = 8.1 radm−2
∫ Source
Observer
(
ne
106m−3
) (
Blos
nT
)
dl
pc
, (8)
where dl is the path length element.
For the warm clouds, an estimate for the magnetic
field strength is available from measurements of energetic
neutral atoms that are thought to originate from the so-
lar wind, are scattered by the magnetic field near the he-
liospheric boundary and experience charge exchange reac-
tions (McComas et al. 2011, 2020). For the warm clouds sur-
rounding the heliosphere this leads to an estimate of 0.3 nT
(Schwadron & McComas 2019). Pressure balance with the
volume filling X-ray plasma generally suggest ≈ 0.5 nT for
warm clouds in the Local Bubble (Snowden et al. 2014).
Ignoring field reversals yields an upper limit for the ro-
tation measure for given electron density, magnetic field B
and total path length lpc. For the warm clouds we write this
as:
RM < 4 radm−2
(
ne
ne,wc
) (
Blos
0.5 nT
)
fwcp . (9)
This suggests a perhaps non-negligible, but certainly not
dominant contribution by the warm clouds to the rotation
measure. Scaling to the properties of the X-ray plasma, we
write eq.(9) as:
RM < 38 radm−2
(
ne
ne,X
) (
Blos
10 nT
)
frLB . (10)
Consequently, the X-ray emitting plasma in the Local Bub-
ble may be magnetised up to a level of at least 10 nT without
violating the rotation measure constraint. Since we show be-
low that very small magnetic fields will not lead to an inter-
esting amount of radio emission, we consider in the following
only magnetic field strengths between 0.1 and 10 nT.
3.2 Constraints on the particle energy spectrum
When averaging over the angle between the magnetic field
direction and the isotropically assumed particle directions,
the characteristic frequency for synchrotron emission be-
comes (Longair 2011):
νc = 794MHz
(
E
GeV
)2 ( B
nT
)
. (11)
For magnetic field strengths within the observational limits
(Sect. 3.1), cosmic ray electrons from 50 MeV up to about
6 GeV radiate at frequencies relevant to the radio back-
ground (20 MHz to 3 GHz). Particles at these energies are
strongly affected by the solar modulation, i.e. the energy
spectrum changes during the propagation from interstellar
space through the magnetised Solar wind before reaching
the detector near Earth. The Voyager 1 spacecraft has left
the region influenced by the Solar wind in 2012 and has since
then measured electron energy distributions in the range 2.7-
79 MeV in the local interstellar medium (Cummings et al.
2016). Cosmic ray propagation models constrained by Voy-
ager 1 and AMS data (Aguilar et al. 2019) have been devel-
oped that infer the cosmic ray electron density distribution
in the local interstellar medium, outside the Solar wind bub-
ble for energies between 1 MeV and 1 TeV (Vittino et al.
2019). The resulting distribution can be approximated by
n(E) ∝ E−p, with p = 1.4 (3.1) below (above) 1 GeV.
Orlando (2018) derived a very similar electron energy distri-
bution and showed that the expected inverse Compton emis-
sion is consistent with gamma-ray observations. Positrons,
which are to a large part produced by hadronic interac-
tions (Strong et al. 2011), contribute at a level of several
per cent to the all electron energy spectrum in the relevant
GeV range, and are included in our model.
Turbulent mixing is expected to homogenise the elec-
tron energy spectrum throughout the Local Bubble, even
though tangled magnetic fields may prevent free streaming:
The gyroradius is a function of electron energy E and mag-
netic field B and is given by
rg = 3 × 10−7 pc
(
E
GeV
) (
B
nT
)−1
. (12)
The cosmic ray electrons relevant to the radio background
would hence have gyroradii between 10−9 pc and 10−5 pc.
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The particles are therefore tied to probably tangled mag-
netic field lines locally. Still, mixing is expected to oc-
cur due to gas sloshing caused by off-centre supernovae
(Krause et al. 2014). The characteristic timescale is the
turnover timescale of the bubble, which can be approximated
by the sound crossing time (e.g., Krause et al. 2013b). We
argue in Sect. 3.3 that the Local Bubble has evolved proba-
bly for several crossing times since the last supernova about
1.5-3.2 Myr ago. Therefore, cosmic ray electrons produced by
that supernova or any source that contributed on a similar
timescale are now well mixed throughout the superbubble.
In the following, we use the electron and positron energy
spectra tabulated in Vittino et al. (2019) as representative
for the cosmic ray electron energy spectrum in the Local
Bubble.
3.3 Constraints on the magnetic field geometry
The geometry and intermittency of the magnetic field shapes
the directional dependence of the radio synchrotron emis-
sion. Supernovae in superbubbles drive gas sloshing on the
scale of the superbubble diameter, which leads to decay-
ing turbulence (Krause et al. 2014). Deposits of radioac-
tive 60Fe in deep sea sediments suggest that the last su-
pernova in the Local Bubble occurred 1.5-3.2 Myr ago
(Wallner et al. 2016). The characteristic decay time for tur-
bulence is the sound crossing time. Using a characteris-
tic diameter of 300 pc (Pelgrims et al. 2020) and a sound
speed of 160 km s−1 (for an X-ray temperature of 0.1 keV,
Snowden et al. 2014) gives a sound crossing time of 1.8 Myr.
Superbubbles with sizes comparable to the Local Bubble
may have higher temperatures shortly after the supernova
explosion (Krause et al. 2018). Therefore, turbulence may
have evolved effectively by several decay times since the
last explosion. Additional kinetic energy may currently be
injected by a nearby pulsar wind, which is required to
explain the observed abundance of high energy electrons
and positrons measured by AMS (Lo´pez-Coto et al. 2018;
Bykov et al. 2019).
Observationally, the magnetic field geometry is con-
strained by starlight polarisation. For stars with distances
100-500 pc, a large-scale coherent field is observed towards
galactic coordinates l = 240◦–(360◦)–60◦, whereas a mag-
netic field tangled on small scales is observed for other longi-
tudes (Berdyugin et al. 2014). The directions with coherent
magnetic field structure appear correlated with the direc-
tion towards which the edge of the Local Bubble is nearest
(Pelgrims et al. 2020). It appears therefore plausible that
the coherent structure is a feature of the bubble wall and
that the interior of the Local Bubble has a magnetic field
structure characterised by decaying turbulence, with the
largest magnetic filaments about 40 pc long (Piirola et al.
2020).
3.4 Synchrotron emission model
We therefore model the magnetic field in the Local Bub-
ble as a random field with a vector potential drawn from
a Rayleigh distribution with a Kolmogorov power spec-
trum following, e.g., Tribble (1991) and Murgia et al. (2004).
We use magnetic field cubes with 256 cells on a side.
Most quantities are well converged with this resolution. For
some we obtain meaningful upper limits (compare below).
The approach is well tested for the description of mag-
netic fields in clusters of galaxies with and without ra-
dio lobes (e.g., Guidetti et al. 2010; Huarte-Espinosa et al.
2011; Hardcastle 2013; Hardcastle & Krause 2014). Follow-
ing the experimental data on the fieldaˆA˘Z´s geometry, we set
the 85 per cent largest modes to zero. This is a reasonable
approximation for decaying turbulence in the case of initially
weak magnetic fields that were amplified by a strong driving
event (Brandenburg et al. 2019), e.g., the sloshing following
an off-centre supernova explosion (Krause et al. 2014). The
magnetic field geometry is discussed further in Sect. 4, be-
low. We also show models for the uncut power spectrum and
for a cut at 20 per cent for comparison. We have checked that
varying this cutoff has a negligible effect on the resulting sky
temperature (compare Hardcastle 2013).
We put the observer in the centre of the data cube,
scale the magnetic field to values within the range allowed
by observations and assume a homogeneous distribution of
synchrotron-emitting leptons. We derive the density of non-
thermal electrons and positrons, ne,p, in the local interstellar
medium at a given energy, from the tabulated fluxes Φe,p
from the model of Vittino et al. (2019). The total density of
non-thermal electrons and positrons, n(E), is then obtained
by summing the individual contributions.
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Figure 1. Synthetic radio sky for the detailed Local Bubble model (Sect. 3) with a mean magnetic field of 1.6 nT at 3.3 GHz. The
resolution is 12◦ matching that of the ARCADE 2 radiometer. The top row shows the distribution of the antenna temperature. The
bottom row shows the fractional polarisation for the corresponding image. The left column is for a complete Kolmogorov power spectrum.
The middle (right) one is for a model with the 20 (85) per cent largest modes set to zero.
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6 M. G. H. Krause et al.
In each energy bin, we use the two neighbouring bins
to fit a local power law: n(E) = κE−q . This enables us to use
the synchrotron emissivity for a power law distribution of
electrons (Longair 2011):
J(ν) = A
√
3pie3B
16pi20cme(q − 1)
κ
(
2piνm3ec4
3eB
)− q−12
(13)
with
A =
Γ
(
q
4 +
19
12
)
Γ
(
q
4 − 112
)
Γ
(
q
4 +
5
4
)
Γ
(
q
4 +
7
12
) . (14)
Here, B denotes the magnetic field strength perpendicular
to the line of sight, me and e are, respectively, electron mass
and charge, c is the speed of light and 0 the vacuum per-
mittivity. We divide the sky in a nlon × nlat grid of longitudes
l and latitudes b with spacings ∆l and ∆b. For each cone of
given li and bj , we first select the observing frequency ν. In
each cell, we evaluate the Lorentz factor given the local mag-
netic field and the chosen observing frequency. We then look
up the corresponding non-thermal electron densities and fit
the normalisation and slope of the local power law at the
corresponding energy. After cutting a small region near the
centre of the box (5 per cent of the path length) to avoid res-
olution effects, we find the spectral flux density by summing
the weighted emissivities within a given cone:
Sν(li, bj ) =
∑
cells in cone
jν dV
4pir
, (15)
where each Cartesian cell has the same volume dV and r
is its distance from the centre of the grid, which will be
different for each cell. The intensity is found by dividing
through surface area of the corresponding sky grid cell:
Iν(li, bj ) =
Sν(li, bj )
dl db sin b
. (16)
And, finally, we get the antenna temperature from:
Ti, j =
Iν(li, bj )c2
2kBν2
. (17)
We also calculate polarisation information. The lo-
cal contributions to the Stokes parameters are (compare
Hardcastle & Krause 2014):
©­«
jI
jQ/µ
jU/µ
ª®¬ ∝ (B2φ + B2θ )
q+1
4
©­­«
B2φ + B
2
θ
B2φ − B2θ
2BφBθ
ª®®¬ , (18)
where Bφ and Bθ are the components of the magnetic field
in spherical coordinates that are perpendicular to the line
of sight at a given location. The maximum polarisation µ is
given by
µ =
α + 1
α + 5/3 (19)
with the spectral index of the radio emission α = (q−1)/2. As
q is fitted to for each energy bin, α depends on the observing
frequency. The Stokes parameters are integrated along the
line of sight to obtain I, Q and U for each direction of the
sky grid. The fractional polarisation f is then computed as:
f =
√
Q2 +U2
I
. (20)
3.5 Modelling results
The sky distribution of antenna temperature is shown for
parameters suitable for comparison to the ARCADE 2 ex-
periment in the top row of Fig. 1. The polarisation map for
the corresponding model is shown in the bottom row of the
same figure. The observing frequency is 3.3 GHz and the
spatial resolution is 12◦.
We have chosen three different cuts kmin in the power
spectrum for the magnetic field (compare Sect. 3.4). The left
column is for an uncut Kolmogorov power spectrum. The
middle (right) one for the case where the 20 (85) per cent
largest modes are cut. Large modes in the magnetic power
spectrum lead to differences in antenna temperature of a
factor of a few for different sky directions. Consequently, the
standard deviation of the antenna temperature is almost half
of the mean value. There is little difference between the sky
distributions predicted for kmin = 20 per cent and kmin = 85
per cent. In both cases, the distribution is smooth across the
sky with maximum antenna temperature ratios below two
for any two sky directions and a standard deviation of less
than 10 per cent of the mean.
A noteworthy polarisation signal is only predicted for
the full Kolmogorov power spectrum. The more the large
modes are cut, the lower the polarisation, again with little
difference between kmin = 20 per cent and kmin = 85 per
cent, namely 4 per cent versus 3 per cent. We note that
the polarisation we give for the kmin = 85 per cent case is
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Figure 2. Predicted radio synchrotron emission for the Local Bubble for kmin = 0.85 (Sect. 3.5) and different mean magnetic field
strengths between 0.16 nT (energy equipartition between thermal energy, cosmic ray leptonic internal energy and magnetic energy) and
10 nT (conservative limit from Faraday rotation). Measurements are from Seiffert et al. (2011) and Dowell & Taylor (2018) as indicated
in the legends. Left: antenna temperature against observing frequency. Right: Antenna temperature scaled with (ν/GHz)2.6). A magnetic
field strength between 3 and 5 nT is required in the Local Bubble to fully explain the radio background.
an upper limit as this value was not numerically converged
with our largest grid of 2563 cells.
We plot the mean antenna temperature against observ-
ing frequency in Fig. 2 (left). The Local Bubble has a power
law radio spectrum very similar to that of the radio back-
ground (spectral index α ≈ 0.6). We compare to the measure-
ments discussed above and reported by Seiffert et al. (2011)
and Dowell & Taylor (2018). Seiffert et al. (2011) use AR-
CADE 2 balloon flight and lower frequency radio surveys.
They subtract the Galaxy model from Kogut et al. (2011)
and an estimated contribution from external galaxies from
the data, and then fit a combination of the cosmic microwave
background and the radio synchrotron background to the re-
maining spectrum. Dowell & Taylor (2018) additionally use
data from the Long Wavelength Array and follow similar
methods to obtain the spectrum of the radio background.
Good agreement with the data is found for magnetic
field strengths between 3 and about 5 nT. For more detailed
comparison to the observations, we remove the ν−2.6 scaling
in Fig. 2 (right). There is a slight systematic offset between
the two observational data sets, which Dowell & Taylor
(2018) ascribe to difficulties in the zero-level calibration of
low frequency surveys. There could also be differences due
to the removal of the emission of the Galaxy. This aside, the
Local Bubble model also has difficulties in simultaneously
fitting the data points below and above 100 MHz. For ex-
ample, for the data set by Seiffert et al. (2011), the 45 MHz
data point lies on our 5 nT curve, whereas the 408 MHz data
point is on our 3 nT curve.
For the reference frequency of 400 MHz, our results are
well fit by the power law:
T = 1.44K
(
B
nT
)1.62
(21)
4 DISCUSSION
We used the available data on relativistic particles, mag-
netic fields, and thermal components to model the radio
synchrotron emission of the Local Bubble. We find that the
predicted radio spectra show an approximate scaling of the
antenna temperature with frequency as T ∝ ν−2.6. To pro-
duce the sky temperature of the ARCADE 2 excess, we re-
quire a magnetic field in the Local Bubble of 3-5 nT. This
is consistent with the pulsar rotation measures, as argued in
Sect. 3.1, above.
There are, however, some severe difficulties with this so-
lution. First, the cosmic ray electron spectrum is curved, and
this translates to a clearly visible curvature in our predicted
radio spectra (Fig. 2), but does not show up in the data. The
Local Bubble would of course not be the only contributor
to the radio background. In fact, Condon et al. (2012) and
Hardcastle et al. (2020, submitted) both find a contribution
of about 25 per cent of the emission from discrete extra-
galactic radio sources. Still, if most of the remaining high
frequency emission were explained by the Local Bubble, it
seems that the low frequency data points would require yet
another contributing source. The magnetic field required to
explain 75 per cent of the radio synchrotron background (us-
ing the 408 MHz data point from Seiffert et al. (2011) as a
reference value) would be 2.5 nT.
At this magnetic field strength, radiative losses are still
negligible: For electrons that radiate at a frequency νc, we
can write the loss timescale due to synchrotron radiation as
(Ginzburg & Syrovatskii 1969):
tc,sync = 7Myr
( νc
GHz
)−1/2 ( B
5 nT
)−3/2
. (22)
The dominant radiation field for inverse Compton scatter-
ing is expected to be star light with a wavelength around
1 µm, where the energy density is approximately Urad =
6 × 10−14 J/m3 (Popescu et al. 2017). The inverse Compton
cooling time may then be written as (Fazio 1967):
tc,iC = 0.6Gyr
(
E
GeV
)−1 ( Urad
10−13 Jm−3
)−1
. (23)
These times are long compared to the time since the last
supernova, 1.5-3.2 Myr ago (compare Sect. 3.3), a plausi-
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ble candidate for accelerating the GeV electrons (compare
Sun et al. 2008). Hence, even in scenarios, where the Local
Bubble explains a high fraction of the radio background,
no significant curvature of the radio spectrum would be ex-
pected. Gamma-ray measurements identify a spectral break
at an energy around 1 TeV (Lo´pez-Coto et al. 2018). Iden-
tifying this break with the break expected from synchrotron
cooling fixes the magnetic field to a value of approximately
0.2 nT.
Different magnetic field values mean that different parts
of the particle spectrum are contributing to the observed
emission. Therefore the curvature in the predicted spectra
depends on the magnetic field strength. For magnetic field
strengths around and below 1 nT, the curvature would bet-
ter correspond to the one of the observed radio background.
At this level of magnetic field strength, the Local Bubble
would contribute about 20 per cent of the radio background
between 10 MHz and 10 GHz.
The magnetic field strength for equipartition between
magnetic energy and energy in relativistic leptons in our
Local Bubble model is Beq,rel = 0.16 nT . For equipartition
between magnetic and thermal energy, using the pressure of
1.5 × 10−13 Pa given by Snowden et al. (2014), it is Beq,th =
0.61 nT . A magnetic field strength of 1 nT as discussed in
the previous paragraph would therefore mean an energet-
ically dominant magnetic field. This would create tension
with our assumption of the magnetic power spectrum, be-
cause, if the magnetic energy dominates, one expects an in-
verse cascade for the magnetic power (Christensson et al.
2001; Brandenburg et al. 2015; Reppin & Banerjee 2017;
Sur 2019). The power spectrum would then be expected to
be dominated by such large modes at the current time of
observation. Therefore, the distributions in the left column
in Fig. 1 would approximately apply, i.e., we would predict
large differences of the background emission in different sky
directions and significant polarisation. Given that the radio
background is found as an isotropic component in large sky
surveys, this seems in tension with observations. A magnetic
field ordered on large scales also appears to be in contradic-
tion with the starlight polarisation measurements discussed
in Sect. 3.3, where we argued that the largest coherent scale
for the magnetic field in the Local Bubble was 40 pc. We note
that Singal et al. (2010) have argued against large-scale pat-
terns in polarisation for the radio background from WMAP
data.
For decaying turbulence and an initially weak magnetic
field, we expect magnetic field amplification up to an equilib-
rium with the kinetic energy (Brandenburg et al. 2019). This
growth phase may last several initial crossing (turnover)
times, up to perhaps ten crossing times, depending on the
initial field strength. It is well known that for turbulence
in general, the kinetic energy is converted to thermal en-
ergy, also on a timescale comparable to the crossing time.
The Local Bubble may therefore be in a situation close to
equilibrium between magnetic and thermal energy. For this
situation, we would predict a fairly isotropic contribution of
about 10 per cent to the radio background.
Of course, the magnetic field might still be lower, per-
haps in equipartition with the cosmic ray electrons or even
lower. For a magnetic field strength of 0.16 nT, which inter-
estingly is associated not only with equipartition between
magnetic energy and relativistic leptons, but would also al-
low to interpret the break in the electron energy distribution
at 1 TeV as due to synchrotron cooling, the Local Bubble
contributes to the radio background at a level of about 1 per
cent.
For a magnetic field below equipartition with the ther-
mal energy density, we expect decaying turbulence, which
would lead to a polarisation of at most a few per cent with
no coherent large-scale pattern in polarisation (Fig. 1). This
is very similar to radio polarisation in the Galactic plane in
general (Kogut et al. 2007).
Summarising, a contribution of the Local Bubble to the
radio background at the per cent level appears most likely.
This result is perhaps surprising, given the encourag-
ing scalings from the non-thermal superbubble in IC 10
(Sect. 2). There is clearly a difference in the level of non-
thermal energy and magnetic energy between the two su-
perbubbles, and it would be interesting to understand the
reasons for this better.
5 SUMMARY AND CONCLUSIONS
We have modelled the radio synchrotron emission of the Lo-
cal Bubble, using observational constraints on the energy
distribution of cosmic ray electrons, magnetic fields, X-ray
gas and warm clouds and filaments. We find that in order
to explain the radio synchrotron background remaining af-
ter subtraction of the Galaxy, the cosmic microwave back-
ground and the contribution of known extragalactic point
sources we require a magnetic field of 2.5 nT. This would be
allowed by constraints from Faraday rotation against nearby
pulsars. However, in this case, the magnetic field would dom-
inate energetically, and we would expect an inverse cascade,
leading to large variations of the background emission in dif-
ferent sky direction, significant polarisation with large co-
herence lengths for the magnetic field, and a synchrotron
cooling break in the electron energy spectrum below 1 TeV,
all of which are difficult to reconcile with observations. In
order to avoid an inverse turbulent cascade associated with
large anisotropies of the radio emission and significant polar-
isation, the magnetic energy density should not exceed the
thermal one, and to avoid an unobserved cooling break at
electron energies below 1 TeV, the magnetic field should not
exceed ≈ 0.2 nT. For this case, we predict a smooth emission
with low polarisation and a maximum contribution to the
unexplained background at the per cent level. This leaves
open the possibility that some of the radio background is
produced at very high redshift, which is an important possi-
bility for the interpretation of the EDGES absorption signal
in the context of the epoch of reionisation.
ACKNOWLEDGEMENTS
We thank the anonymous referee for useful comments that
helped to improve the manuscript. MJH acknowledges sup-
port from the UK Science and Technology Facilities Council
(ST/R000905/1).
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DATA AVAILABILITY
The data and code underlying this article are available in
the article and in its online supplementary material.
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