Can the Local Bubble explain the radio background?
MMNRAS , 1–8 (2019) Preprint 14 January 2021 Compiled using MNRAS L A TEX style file v3.0
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 synchrotronbackground at frequencies below about 3 GHz. Neither extragalactic radio sources nor diffuse Galactic emission can currentlyaccount 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 forthe 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 argue that this would not violate observational constraints from pulsar measurements. However, the curvature in the predictedspectrum would mean that other, so far unknown sources would have to contribute below 100 MHz. Also, the magnetic energydensity would then dominate over thermal and cosmic ray electron energy density, likely causing an inverse magnetic cascadewith large variations of the radio emission in different sky directions as well as high polarisation. We argue that this disagreeswith several observations and thus that the magnetic field is probably much lower, quite possibly limited by equipartition withthe energy density in relativistic or thermal particles ( 𝐵 = . − . Key words: radio continuum: general – radio continuum: ISM – ISM: bubbles – Galaxy: local interstellar matter – cosmology:diffuse radiation
The balloon-borne precision bolometer ARCADE 2 has reported anexcess emission above the Cosmic microwave background (CMB) of54 ± 𝛼 = . 𝑆 ∝ 𝜈 − 𝛼 ). The relevant frequency range includes the60-80 MHz region, where the 21 cm signal from the epoch of reioni-sation is expected. An absorption feature of less than 1 per cent of theradio background emission has indeed been found by the Experimentto Detect the Global Epoch of Reionization Signature (EDGES) atthese frequencies (Bowman et al. 2018). For the interpretation ofthe absorption feature as of cosmological origin, it is important tounderstand whether the radio synchrotron background is producedlocally or at high redshift (e.g., Monsalve et al. 2019; Ewall-Wiceet al. 2020).Since the contribution from the Milky Way has a distinct geometryand is accounted for already in the aforementioned results, the moststraightforward explanation would be a large population of known ★ E-mail: [email protected] extragalactic radio sources, namely radio loud active galactic nucleiand star-forming galaxies. At 3 GHz, measurements with the KarlG. Jansky Very Large Array find a combined antenna temperaturefor all such sources of 13 mK, significantly below the ARCADE2 result (Condon et al. 2012). A similar measurement has recentlybeen performed with the Low-Frequency Array (LOFAR) with thesimilar result that only about 25 per cent of the radio backgroundcan be accounted for by resolved radio sources (Hardcastle et al.2020). Another suggestion that has been put forward is a Galactichalo of cosmic ray electrons with a scale length of 10 kpc (Orlando &Strong 2013; Subrahmanyan & Cowsik 2013). The required particlepopulation would however also produce X-rays via inverse Comp-ton scattering, which would violate observational constraints (Singalet al. 2010). Also, such a prominent radio halo would be atypical forgalaxies like the Milky Way (Singal et al. 2015; Stein et al. 2020),even though halos of up to a few kpc at 150 MHz have been foundrecently (Stein et al. 2019). These difficulties have inspired a numberof interesting explanations, including for example free-free emissionrelated to galaxy formation at high redshift (Liu et al. 2019) and darkmatter annihilation (Hooper et al. 2012). See Singal et al. (2018) fora recent review.We investigate here a comparatively simple explanation: syn-chrotron emission from the Local Bubble. The Local Bubble is alow-density cavity in the interstellar medium around the Solar sys-tem (e.g., Cox & Reynolds 1987) The superbubble was likely formedby winds and explosions of massive stars (Breitschwerdt et al. 2016; © a r X i v : . [ a s t r o - ph . H E ] J a n M. G. H. Krause et al.
Schulreich et al. 2018). Hot gas in the bubble contributes signif-icantly to the soft X-ray background (Snowden et al. 1997, 1998;Galeazzi et al. 2014; Snowden 2015). The boundary is delineated bya dusty shell that has been mapped with absorption data against starswith known distances (Lallement et al. 2014; Snowden et al. 2015b;Pelgrims et al. 2020). Direct observation of the likely present neutralhydrogen supershell is difficult against the background of the MilkyWay, but the distinct structure of erosion of the interface towards aneighbouring superbubble has been observed (Krause et al. 2018).Similar features are also known from Nai and Hi absorption stud-ies (Lallement et al. 2014). Interaction of cosmic ray particles withthe supershell may explain the high-energy neutrinos observed withIceCube (Andersen et al. 2018; Bouyahiaoui et al. 2020). The super-bubble contains high ionisation species (Breitschwerdt & de Avillez2006), filaments and clouds of partially neutral and possibly evenmolecular gas (e.g., Gry & Jenkins 2014, 2017; Redfield & Linsky2008, 2015; Snowden et al. 2015a; Farhang et al. 2019; Linsky et al.2019) and is threaded by magnetic fields (e.g., Andersson & Potter2006; McComas et al. 2011; Frisch et al. 2015; Alves et al. 2018;Piirola et al. 2020). It has already been suggested as the physicalorigin of high latitude radio emission by Sun et al. (2008).We first make an empirical model based on a comparison to thenon-thermal superbubble in the dwarf galaxy IC 10 (Sect. 2) and thenpresent a detailed model based on the locally observed population ofcosmic ray electrons and available constraints on the magnetic fieldin the Local Bubble (Sect. 3). We discuss our findings in the contextof the observational constraints in Sect. 4 and conclude in Sect. 5 thata dominant contribution of the Local Bubble to the radio backgroundseems unlikely.
Superbubbles are not usually known to emit a non-thermal radiosynchrotron spectrum. One such object has, however, been identifiedin the dwarf galaxy IC 10 (Heesen et al. 2015). The reason why itstands out against thermal and non-thermal radio emission of the hostgalaxy might be an unusually strong explosion, a hypernova, about1 Myr before the time of observation. Its size is, similar to the LocalBubble, ∼
200 pc. The radio spectrum is a power law with the samespectral index as the radio background, 𝑆 ( 𝜈 ) ∝ 𝜈 − . . The observednon-thermal emission is 40 mJy at 1.5 GHz.We use these properties of the nonthermal superbubble in IC 10to estimate those of the Local Bubble as follows. First, we scale thisby a factor of 𝑓 s = . . × W Hz − . Assuming a bubble radius of 100 𝑓 r10 pc, weobtain a volume emissivity of 𝑙 𝜈 = . × − (cid:18) 𝑓 s . (cid:19) 𝑓 − W Hz − m − (1)Placing the Sun at the centre of such a non-thermal bubble yields aflux contribution from each shell at distance 𝑟 ofd 𝑆 𝜈 = 𝜋𝑟 d 𝑟 𝑙 𝜈 𝜋𝑟 = 𝑙 𝜈 d 𝑟 . (2)The integral is straightforward and results, for a radius of the LocalBubble of 100 𝑓 rLB pc in: 𝑆 𝜈 = × (cid:18) 𝑓 s . (cid:19) 𝑓 − 𝑓 rLB (cid:16) 𝜈 (cid:17) − . Jy . (3) The antenna temperature follows from this via 𝑇 𝜈 = 𝑆 𝜈 𝑐 /( 𝜋𝑘 B 𝜈 ) ,and so 𝑇 𝜈 = (cid:18) 𝑓 s . (cid:19) 𝑓 − 𝑓 rLB (cid:16) 𝜈 (cid:17) − . mK . (4)This overpredicts the radio synchrotron background by a factor oftwo and thus demonstrates that the contribution of the Local Bubblecan in principle be very important. Thanks to a number of measurements unique to the Local Bubble, it ispossible to predict its radio emission with far better accuracy than wehave done in the previous section. Both elements required to predictsynchrotron emission, the energy distribution of cosmic ray electronsand positrons and the strength and geometry of the magnetic field areconstrained by recent experimental data. The Alpha Magnetic Spec-trometer (AMS) onboard the International Space Station (ISS) hasmeasured the near-earth energy distribution for cosmic ray electronswith energies E between 0.5 GeV and 1.4 TeV(Aguilar et al. 2019).Constraints at lower energy and outside the volume influenced bythe Solar wind have been provided by Voyager I (Cummings et al.2016). The part of this distribution relevant for the radio backgroundcan be calculated once the magnetic field is known, and constraintsare available from pulsar observations. We review the observationalconstraints on both, magnetic field and particle energy spectrum, inthe following three subsections.
The magnetic field in the local bubble is constrained by measure-ments of the Faraday effect, i.e. the rotation of the plane of polarisa-tion of pulses from radio pulsars, combined with the pulse dispersionas a function of frequency. Such measurements yield magnetic fieldstrength estimates of 𝐵 = . − . − pc with a standard deviation of 20 cm − pc.This corresponds to a column of free, thermal electrons of 𝑁 𝑒 = ( . ± . ) × m − . (5)X-ray measurements of the hot bubble plasma suggest a thermalelectron density of 𝑛 𝑒, X = ( . ± . ) × m − (Snowden et al.2014). This value is very typical for superbubbles, including X-raybright ones, as shown in 3D numerical simulations (Krause et al.2013a, 2014). The contribution to the free electron column in theLocal Bubble from the X-ray emitting plasma, again for a radius ofthe Local Bubble of 100 𝑓 rLB pc is therefore 𝑁 𝑒, X = ( . ± . ) × 𝑓 rLB m − . (6)Warm clouds within the Local Bubble have sizes of several parsecsand electron densities of the order of 𝑛 𝑒, wc = m − (e.g., Gry &Jenkins 2017; Linsky et al. 2019). Assuming a total warm cloud path MNRAS000
The magnetic field in the local bubble is constrained by measure-ments of the Faraday effect, i.e. the rotation of the plane of polarisa-tion of pulses from radio pulsars, combined with the pulse dispersionas a function of frequency. Such measurements yield magnetic fieldstrength estimates of 𝐵 = . − . − pc with a standard deviation of 20 cm − pc.This corresponds to a column of free, thermal electrons of 𝑁 𝑒 = ( . ± . ) × m − . (5)X-ray measurements of the hot bubble plasma suggest a thermalelectron density of 𝑛 𝑒, X = ( . ± . ) × m − (Snowden et al.2014). This value is very typical for superbubbles, including X-raybright ones, as shown in 3D numerical simulations (Krause et al.2013a, 2014). The contribution to the free electron column in theLocal Bubble from the X-ray emitting plasma, again for a radius ofthe Local Bubble of 100 𝑓 rLB pc is therefore 𝑁 𝑒, X = ( . ± . ) × 𝑓 rLB m − . (6)Warm clouds within the Local Bubble have sizes of several parsecsand electron densities of the order of 𝑛 𝑒, wc = m − (e.g., Gry &Jenkins 2017; Linsky et al. 2019). Assuming a total warm cloud path MNRAS000 , 1–8 (2019) length of 10 𝑓 wcp pc, we obtain an estimate for the corresponding freeelectron column of: 𝑁 𝑒, wc = × 𝑓 wcp m − . (7)Hence, neither the hot X-ray plasma nor the warm clouds and fil-aments contribute significantly to the pulsar dispersion measures.As Xu & Han (2019) note, the dispersion measure is probably pro-duced predominantly by the bubble wall, an ionised mixing layerbetween the superbubble interior and the cold supershell (comparealso Krause et al. 2014).The root mean square rotation measure against the aforementionedeight pulsars is 33 rad m − . For a plasma with electron density 𝑛 𝑒 and line-of-sight magnetic field 𝐵 los , the rotation measure may beexpressed as: 𝑅𝑀 = . − ∫ SourceObserver (cid:18) 𝑛 𝑒 m − (cid:19) (cid:18) 𝐵 los nT (cid:19) d 𝑙 pc , (8)where d 𝑙 is the path length element.For the warm clouds, an estimate for the magnetic field strengthis available from measurements of energetic neutral atoms that arethought to originate from the solar wind, are scattered by the mag-netic field near the heliospheric boundary and experience chargeexchange reactions (McComas et al. 2011, 2020). For the warmclouds surrounding the heliosphere this leads to an estimate of 0.3 nT(Schwadron & McComas 2019). Pressure balance with the volumefilling X-ray plasma generally suggest ≈ . 𝐵 and total path length 𝑙 pc . For the warm clouds we write this as: 𝑅𝑀 < − (cid:18) 𝑛 𝑒 𝑛 𝑒, wc (cid:19) (cid:18) 𝐵 los . (cid:19) 𝑓 wcp . (9)This suggests a perhaps non-negligible, but certainly not dominantcontribution by the warm clouds to the rotation measure. Scaling tothe properties of the X-ray plasma, we write eq.(9) as: 𝑅𝑀 <
38 rad m − (cid:18) 𝑛 𝑒 𝑛 𝑒, X (cid:19) (cid:18) 𝐵 los
10 nT (cid:19) 𝑓 rLB . (10)Consequently, the X-ray emitting plasma in the Local Bubble may bemagnetised up to a level of at least 10 nT without violating the rotationmeasure constraint. Since we show below that very small magneticfields will not lead to an interesting amount of radio emission, weconsider in the following only magnetic field strengths between 0.1and 10 nT. When averaging over the angle between the magnetic field directionand the isotropically assumed particle directions, the characteristicfrequency for synchrotron emission becomes (Longair 2011): 𝜈 c =
794 MHz (cid:18) 𝐸 GeV (cid:19) (cid:18) 𝐵 nT (cid:19) . (11)For magnetic field strengths within the observational limits(Sect. 3.1), cosmic ray electrons from 50 MeV up to about 6 GeVradiate at frequencies relevant to the radio background (20 MHz to3 GHz). Particles at these energies are strongly affected by the solarmodulation, i.e. the energy spectrum changes during the propagationfrom interstellar space through the magnetised Solar wind beforereaching the detector near Earth. The Voyager 1 spacecraft has leftthe region influenced by the Solar wind in 2012 and has since then measured electron energy distributions in the range 2.7-79 MeV in thelocal interstellar medium (Cummings et al. 2016). Cosmic ray propa-gation models constrained by Voyager 1 and AMS data (Aguilar et al.2019) have been developed that infer the cosmic ray electron densitydistribution in the local interstellar medium, outside the Solar windbubble 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 = . 𝐸 and magnetic field 𝐵 and is given by 𝑟 g = × − pc (cid:18) 𝐸 GeV (cid:19) (cid:18) 𝐵 nT (cid:19) − . (12)The cosmic ray electrons relevant to the radio background wouldhence have gyroradii between 10 − pc and 10 − pc. The particles aretherefore tied to probably tangled magnetic field lines locally. Still,mixing is expected to occur due to gas sloshing caused by off-centresupernovae (Krause et al. 2014). The characteristic timescale is theturnover timescale of the bubble, which can be approximated by thesound crossing time (e.g., Krause et al. 2013b). We argue in Sect. 3.3that the Local Bubble has evolved probably for several crossing timessince the last supernova about 1.5-3.2 Myr ago. Therefore, cosmic rayelectrons produced by that supernova or any source that contributedon a similar timescale are now well mixed throughout the superbub-ble. In the following, we use the electron and positron energy spectratabulated in Vittino et al. (2019) as representative for the cosmic rayelectron energy spectrum in the Local Bubble. The geometry and intermittency of the magnetic field shapes the di-rectional dependence of the radio synchrotron emission. Supernovaein superbubbles drive gas sloshing on the scale of the superbubblediameter, which leads to decaying turbulence (Krause et al. 2014).Deposits of radioactive Fe in deep sea sediments suggest that thelast supernova in the Local Bubble occurred 1.5-3.2 Myr ago (Wall-ner et al. 2016). The characteristic decay time for turbulence is thesound crossing time. Using a characteristic diameter of 300 pc (Pel-grims et al. 2020) and a sound speed of 160 km s − (for an X-raytemperature of 0.1 keV, Snowden et al. 2014) gives a sound crossingtime of 1.8 Myr. Superbubbles with sizes comparable to the LocalBubble may have higher temperatures shortly after the supernova ex-plosion (Krause et al. 2018). Therefore, turbulence may have evolvedeffectively by several decay times since the last explosion. Additionalkinetic energy may currently be injected by a nearby pulsar wind,which is required to explain the observed abundance of high energyelectrons and positrons measured by AMS (López-Coto et al. 2018;Bykov et al. 2019).Observationally, the magnetic field geometry is constrained bystarlight polarisation. For stars with distances 100-500 pc, a large-scale coherent field is observed towards galactic coordinates 𝑙 = ◦ – ( ◦ ) –60 ◦ , whereas a magnetic field tangled on small scalesis observed for other longitudes (Berdyugin et al. 2014). The direc-tions with coherent magnetic field structure appear correlated with MNRAS , 1–8 (2019)
M. G. H. Krause et al.
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 ◦ matchingthat of the ARCADE 2 radiometer. The top row shows the distribution of the antenna temperature. The bottom row shows the fractional polarisation for thecorresponding 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 largestmodes set to zero. the direction towards which the edge of the Local Bubble is nearest(Pelgrims et al. 2020). It appears therefore plausible that the coherentstructure is a feature of the bubble wall and that the interior of theLocal Bubble has a magnetic field structure characterised by decay-ing turbulence, with the largest magnetic filaments about 40 pc long(Piirola et al. 2020). We therefore model the magnetic field in the Local Bubble as a ran-dom field with a vector potential drawn from a Rayleigh distributionwith a Kolmogorov power spectrum following, e.g., Tribble (1991)and Murgia et al. (2004). We use magnetic field cubes with 256cells on a side. Most quantities are well converged with this resolu-tion. For some we obtain meaningful upper limits (compare below).The approach is well tested for the description of magnetic fieldsin clusters of galaxies with and without radio lobes (e.g., Guidettiet al. 2010; Huarte-Espinosa et al. 2011; Hardcastle 2013; Hardcas-tle & Krause 2014). Following the experimental data on the field’sgeometry, we set the 85 per cent largest modes to zero. This is a rea-sonable approximation for decaying turbulence in the case of initiallyweak magnetic fields that were amplified by a strong driving event(Brandenburg et al. 2019), e.g., the sloshing following an off-centresupernova explosion (Krause et al. 2014). The magnetic field geom-etry is discussed further in Sect. 4, below. We also show models forthe uncut power spectrum and for a cut at 20 per cent for comparison.We have checked that varying this cutoff has a negligible effect onthe resulting sky temperature (compare Hardcastle 2013). We put the observer in the centre of the data cube, scale themagnetic field to values within the range allowed by observations andassume a homogeneous distribution of synchrotron-emitting leptons.We derive the density of non-thermal electrons and positrons, 𝑛 𝑒, 𝑝 ,in the local interstellar medium at a given energy, from the tabulatedfluxes Φ 𝑒, 𝑝 from the model of Vittino et al. (2019). The total densityof non-thermal electrons and positrons, 𝑛 ( 𝐸 ) , is then obtained bysumming the individual contributions.In each energy bin, we use the two neighbouring bins to fit a localpower law: 𝑛 ( 𝐸 ) = 𝜅𝐸 − 𝑞 . This enables us to use the synchrotronemissivity for a power law distribution of electrons (Longair 2011): 𝐽 ( 𝜈 ) = 𝐴 √ 𝜋𝑒 𝐵 𝜋 𝜖 𝑐𝑚 𝑒 ( 𝑞 − ) 𝜅 (cid:32) 𝜋𝜈𝑚 𝑒 𝑐 𝑒𝐵 (cid:33) − 𝑞 − (13)with 𝐴 = Γ (cid:16) 𝑞 + (cid:17) Γ (cid:16) 𝑞 − (cid:17) Γ (cid:16) 𝑞 + (cid:17) Γ (cid:16) 𝑞 + (cid:17) . (14)Here, 𝐵 denotes the magnetic field strength perpendicular to the lineof sight, 𝑚 𝑒 and 𝑒 are, respectively, electron mass and charge, 𝑐 is thespeed of light and 𝜖 the vacuum permittivity. We divide the sky in a 𝑛 lon × 𝑛 lat grid of longitudes 𝑙 and latitudes 𝑏 with spacings Δ 𝑙 and Δ 𝑏 . For each cone of given 𝑙 𝑖 and 𝑏 𝑗 , we first select the observingfrequency 𝜈 . In each cell, we evaluate the Lorentz factor given thelocal magnetic field and the chosen observing frequency. We thenlook up the corresponding non-thermal electron densities and fit thenormalisation and slope of the local power law at the corresponding MNRAS000
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 ◦ matchingthat of the ARCADE 2 radiometer. The top row shows the distribution of the antenna temperature. The bottom row shows the fractional polarisation for thecorresponding 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 largestmodes set to zero. the direction towards which the edge of the Local Bubble is nearest(Pelgrims et al. 2020). It appears therefore plausible that the coherentstructure is a feature of the bubble wall and that the interior of theLocal Bubble has a magnetic field structure characterised by decay-ing turbulence, with the largest magnetic filaments about 40 pc long(Piirola et al. 2020). We therefore model the magnetic field in the Local Bubble as a ran-dom field with a vector potential drawn from a Rayleigh distributionwith a Kolmogorov power spectrum following, e.g., Tribble (1991)and Murgia et al. (2004). We use magnetic field cubes with 256cells on a side. Most quantities are well converged with this resolu-tion. For some we obtain meaningful upper limits (compare below).The approach is well tested for the description of magnetic fieldsin clusters of galaxies with and without radio lobes (e.g., Guidettiet al. 2010; Huarte-Espinosa et al. 2011; Hardcastle 2013; Hardcas-tle & Krause 2014). Following the experimental data on the field’sgeometry, we set the 85 per cent largest modes to zero. This is a rea-sonable approximation for decaying turbulence in the case of initiallyweak magnetic fields that were amplified by a strong driving event(Brandenburg et al. 2019), e.g., the sloshing following an off-centresupernova explosion (Krause et al. 2014). The magnetic field geom-etry is discussed further in Sect. 4, below. We also show models forthe uncut power spectrum and for a cut at 20 per cent for comparison.We have checked that varying this cutoff has a negligible effect onthe resulting sky temperature (compare Hardcastle 2013). We put the observer in the centre of the data cube, scale themagnetic field to values within the range allowed by observations andassume a homogeneous distribution of synchrotron-emitting leptons.We derive the density of non-thermal electrons and positrons, 𝑛 𝑒, 𝑝 ,in the local interstellar medium at a given energy, from the tabulatedfluxes Φ 𝑒, 𝑝 from the model of Vittino et al. (2019). The total densityof non-thermal electrons and positrons, 𝑛 ( 𝐸 ) , is then obtained bysumming the individual contributions.In each energy bin, we use the two neighbouring bins to fit a localpower law: 𝑛 ( 𝐸 ) = 𝜅𝐸 − 𝑞 . This enables us to use the synchrotronemissivity for a power law distribution of electrons (Longair 2011): 𝐽 ( 𝜈 ) = 𝐴 √ 𝜋𝑒 𝐵 𝜋 𝜖 𝑐𝑚 𝑒 ( 𝑞 − ) 𝜅 (cid:32) 𝜋𝜈𝑚 𝑒 𝑐 𝑒𝐵 (cid:33) − 𝑞 − (13)with 𝐴 = Γ (cid:16) 𝑞 + (cid:17) Γ (cid:16) 𝑞 − (cid:17) Γ (cid:16) 𝑞 + (cid:17) Γ (cid:16) 𝑞 + (cid:17) . (14)Here, 𝐵 denotes the magnetic field strength perpendicular to the lineof sight, 𝑚 𝑒 and 𝑒 are, respectively, electron mass and charge, 𝑐 is thespeed of light and 𝜖 the vacuum permittivity. We divide the sky in a 𝑛 lon × 𝑛 lat grid of longitudes 𝑙 and latitudes 𝑏 with spacings Δ 𝑙 and Δ 𝑏 . For each cone of given 𝑙 𝑖 and 𝑏 𝑗 , we first select the observingfrequency 𝜈 . In each cell, we evaluate the Lorentz factor given thelocal magnetic field and the chosen observing frequency. We thenlook up the corresponding non-thermal electron densities and fit thenormalisation and slope of the local power law at the corresponding MNRAS000 , 1–8 (2019)
Figure 2.
Predicted radio synchrotron emission for the Local Bubble for 𝑘 min = .
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) . ). A magnetic field strength between 3 and 5 nT is required in the Local Bubble to fully explain the radiobackground. energy. After cutting a small region near the centre of the box (5per cent of the path length) to avoid resolution effects, we find thespectral flux density by summing the weighted emissivities within agiven cone: 𝑆 𝜈 ( 𝑙 𝑖 , 𝑏 𝑗 ) = ∑︁ cells in cone 𝑗 𝜈 d 𝑉 𝜋𝑟 , (15)where each Cartesian cell has the same volume d 𝑉 and 𝑟 is itsdistance from the centre of the grid, which will be different for eachcell. The intensity is found by dividing through surface area of thecorresponding sky grid cell: 𝐼 𝜈 ( 𝑙 𝑖 , 𝑏 𝑗 ) = 𝑆 𝜈 ( 𝑙 𝑖 , 𝑏 𝑗 ) d 𝑙 d 𝑏 sin 𝑏 . (16)And, finally, we get the antenna temperature from: 𝑇 𝑖, 𝑗 = 𝐼 𝜈 ( 𝑙 𝑖 , 𝑏 𝑗 ) 𝑐 𝑘 B 𝜈 . (17)We also calculate polarisation information. The local contributionsto the Stokes parameters are (compare Hardcastle & Krause 2014): (cid:169)(cid:173)(cid:171) 𝑗 𝐼 𝑗 𝑄 / 𝜇𝑗 𝑈 / 𝜇 (cid:170)(cid:174)(cid:172) ∝ ( 𝐵 𝜙 + 𝐵 𝜃 ) 𝑞 + (cid:169)(cid:173)(cid:173)(cid:171) 𝐵 𝜙 + 𝐵 𝜃 𝐵 𝜙 − 𝐵 𝜃 𝐵 𝜙 𝐵 𝜃 (cid:170)(cid:174)(cid:174)(cid:172) , (18)where 𝐵 𝜙 and 𝐵 𝜃 are the components of the magnetic field in spher-ical coordinates that are perpendicular to the line of sight at a givenlocation. The maximum polarisation 𝜇 is given by 𝜇 = 𝛼 + 𝛼 + / 𝛼 = ( 𝑞 − )/
2. As 𝑞 isfitted to for each energy bin, 𝛼 depends on the observing frequency.The Stokes parameters are integrated along the line of sight to ob-tain 𝐼 , 𝑄 and 𝑈 for each direction of the sky grid. The fractionalpolarisation 𝑓 is then computed as: 𝑓 = √︁ 𝑄 + 𝑈 𝐼 . (20)
The sky distribution of antenna temperature is shown for parameterssuitable for comparison to the ARCADE 2 experiment in the toprow of Fig. 1. The polarisation map for the corresponding model isshown in the bottom row of the same figure. The observing frequencyis 3.3 GHz and the spatial resolution is 12 ◦ .We have chosen three different cuts 𝑘 min in the power spectrumfor the magnetic field (compare Sect. 3.4). The left column is for anuncut Kolmogorov power spectrum. The middle (right) one for thecase where the 20 (85) per cent largest modes are cut. Large modes inthe magnetic power spectrum lead to differences in antenna temper-ature of a factor of a few for different sky directions. Consequently,the standard deviation of the antenna temperature is almost half ofthe mean value. There is little difference between the sky distribu-tions predicted for 𝑘 min =
20 per cent and 𝑘 min =
85 per cent. Inboth cases, the distribution is smooth across the sky with maximumantenna temperature ratios below two for any two sky directions anda standard deviation of less than 10 per cent of the mean.A noteworthy polarisation signal is only predicted for the fullKolmogorov power spectrum. The more the large modes are cut, thelower the polarisation, again with little difference between 𝑘 min =
20 per cent and 𝑘 min =
85 per cent, namely 4 per cent versus 3 percent. We note that the polarisation we give for the 𝑘 min =
85 per centcase is an upper limit as this value was not numerically convergedwith our largest grid of 256 cells.We plot the mean antenna temperature against observing frequencyin Fig. 2 (left). The Local Bubble has a power law radio spectrumvery similar to that of the radio background (spectral index 𝛼 ≈ . MNRAS , 1–8 (2019)
M. G. H. Krause et al.
Good agreement with the data is found for magnetic field strengthsbetween 3 and about 5 nT. For more detailed comparison to theobservations, we remove the 𝜈 − . scaling in Fig. 2 (right). Thereis a slight systematic offset between the two observational data sets,which Dowell & Taylor (2018) ascribe to difficulties in the zero-levelcalibration of low frequency surveys. There could also be differencesdue to the removal of the emission of the Galaxy. This aside, theLocal Bubble model also has difficulties in simultaneously fitting thedata points below and above 100 MHz. For example, for the dataset by Seiffert et al. (2011), the 45 MHz data point lies on our 5 nTcurve, whereas the 408 MHz data point is on our 3 nT curve.For the reference frequency of 400 MHz, our results are well fitby the power law: 𝑇 = .
44 K (cid:18) 𝐵 nT (cid:19) . (21) We used the available data on relativistic particles, magnetic fields,and thermal components to model the radio synchrotron emission ofthe Local Bubble. We find that the predicted radio spectra show anapproximate scaling of the antenna temperature with frequency as 𝑇 ∝ 𝜈 − . . To produce the sky temperature of the ARCADE 2 excess,we require a magnetic field in the Local Bubble of 3-5 nT. This isconsistent with the pulsar rotation measures, as argued in Sect. 3.1,above.There are, however, some severe difficulties with this solution.First, the cosmic ray electron spectrum is curved, and this translatesto a clearly visible curvature in our predicted radio spectra (Fig. 2),but does not show up in the data. The Local Bubble would of coursenot be the only contributor to the radio background. In fact, Con-don et al. (2012) and Hardcastle et al. (2020, submitted) both finda contribution of about 25 per cent of the emission from discreteextragalactic radio sources. Still, if most of the remaining high fre-quency emission were explained by the Local Bubble, it seems thatthe low frequency data points would require yet another contributingsource. The magnetic field required to explain 75 per cent of theradio synchrotron background (using the 408 MHz data point fromSeiffert 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 𝜈 c , we can write the losstimescale due to synchrotron radiation as (Ginzburg & Syrovatskii1969): 𝑡 c , sync = (cid:16) 𝜈 c GHz (cid:17) − / (cid:18) 𝐵 (cid:19) − / . (22)The dominant radiation field for inverse Compton scattering is ex-pected to be star light with a wavelength around 1 𝜇 m, where theenergy density is approximately 𝑈 rad = × − J/m (Popescuet al. 2017). The inverse Compton cooling time may then be writtenas (Fazio 1967): 𝑡 c , iC = . (cid:18) 𝐸 GeV (cid:19) − (cid:18) 𝑈 rad − J m − (cid:19) − . (23)These times are long compared to the time since the last supernova,1.5-3.2 Myr ago (compare Sect. 3.3), a plausible candidate for accel-erating the GeV electrons (compare Sun et al. 2008). Hence, even inscenarios, where the Local Bubble explains a high fraction of the ra-dio background, no significant curvature of the radio spectrum wouldbe expected. Gamma-ray measurements identify a spectral break at an energy around 1 TeV (López-Coto et al. 2018). Identifying thisbreak with the break expected from synchrotron cooling fixes themagnetic field to a value of approximately 0.2 nT.Different magnetic field values mean that different parts of theparticle spectrum are contributing to the observed emission. There-fore the curvature in the predicted spectra depends on the magneticfield strength. For magnetic field strengths around and below 1 nT,the curvature would better correspond to the one of the observedradio background. At this level of magnetic field strength, the LocalBubble would contribute about 20 per cent of the radio backgroundbetween 10 MHz and 10 GHz.The magnetic field strength for equipartition between magneticenergy and energy in relativistic leptons in our Local Bubble modelis 𝐵 eq , rel = . 𝑛𝑇 . For equipartition between magnetic and thermalenergy, using the pressure of 1 . × − Pa given by Snowden et al.(2014), it is 𝐵 eq , th = . 𝑛𝑇 . A magnetic field strength of 1 nT asdiscussed in the previous paragraph would therefore mean an ener-getically dominant magnetic field. This would create tension with ourassumption of the magnetic power spectrum, because, if the magneticenergy dominates, one expects an inverse cascade for the magneticpower (Christensson et al. 2001; Brandenburg et al. 2015; Reppin& Banerjee 2017; Sur 2019). The power spectrum would then beexpected to be dominated by such large modes at the current time ofobservation. Therefore, the distributions in the left column in Fig. 1would approximately apply, i.e., we would predict large differencesof the background emission in different sky directions and significantpolarisation. Given that the radio background is found as an isotropiccomponent in large sky surveys, this seems in tension with observa-tions. A magnetic field ordered on large scales also appears to be incontradiction with the starlight polarisation measurements discussedin Sect. 3.3, where we argued that the largest coherent scale for themagnetic field in the Local Bubble was 40 pc. We note that Singalet al. (2010) have argued against large-scale patterns in polarisationfor the radio background from WMAP data.For decaying turbulence and an initially weak magnetic field, weexpect magnetic field amplification up to an equilibrium with thekinetic energy (Brandenburg et al. 2019). This growth phase may lastseveral initial crossing (turnover) times, up to perhaps ten crossingtimes, depending on the initial field strength. It is well known thatfor turbulence in general, the kinetic energy is converted to thermalenergy, also on a timescale comparable to the crossing time. TheLocal Bubble may therefore be in a situation close to equilibriumbetween magnetic and thermal energy. For this situation, we wouldpredict a fairly isotropic contribution of about 10 per cent to the radiobackground.Of course, the magnetic field might still be lower, perhaps inequipartition with the cosmic ray electrons or even lower. For amagnetic field strength of 0.16 nT, which interestingly is associatednot only with equipartition between magnetic energy and relativisticleptons, but would also allow to interpret the break in the electronenergy distribution at 1 TeV as due to synchrotron cooling, the LocalBubble contributes to the radio background at a level of about 1 percent.For a magnetic field below equipartition with the thermal energydensity, we expect decaying turbulence, which would lead to a polar-isation of at most a few per cent with no coherent large-scale patternin polarisation (Fig. 1). This is very similar to radio polarisation inthe Galactic plane in general (Kogut et al. 2007).Summarising, a contribution of the Local Bubble to the radiobackground at the per cent level appears most likely.This result is perhaps surprising, given the encouraging scalingsfrom the non-thermal superbubble in IC 10 (Sect. 2). There is clearly MNRAS000
44 K (cid:18) 𝐵 nT (cid:19) . (21) We used the available data on relativistic particles, magnetic fields,and thermal components to model the radio synchrotron emission ofthe Local Bubble. We find that the predicted radio spectra show anapproximate scaling of the antenna temperature with frequency as 𝑇 ∝ 𝜈 − . . To produce the sky temperature of the ARCADE 2 excess,we require a magnetic field in the Local Bubble of 3-5 nT. This isconsistent with the pulsar rotation measures, as argued in Sect. 3.1,above.There are, however, some severe difficulties with this solution.First, the cosmic ray electron spectrum is curved, and this translatesto a clearly visible curvature in our predicted radio spectra (Fig. 2),but does not show up in the data. The Local Bubble would of coursenot be the only contributor to the radio background. In fact, Con-don et al. (2012) and Hardcastle et al. (2020, submitted) both finda contribution of about 25 per cent of the emission from discreteextragalactic radio sources. Still, if most of the remaining high fre-quency emission were explained by the Local Bubble, it seems thatthe low frequency data points would require yet another contributingsource. The magnetic field required to explain 75 per cent of theradio synchrotron background (using the 408 MHz data point fromSeiffert 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 𝜈 c , we can write the losstimescale due to synchrotron radiation as (Ginzburg & Syrovatskii1969): 𝑡 c , sync = (cid:16) 𝜈 c GHz (cid:17) − / (cid:18) 𝐵 (cid:19) − / . (22)The dominant radiation field for inverse Compton scattering is ex-pected to be star light with a wavelength around 1 𝜇 m, where theenergy density is approximately 𝑈 rad = × − J/m (Popescuet al. 2017). The inverse Compton cooling time may then be writtenas (Fazio 1967): 𝑡 c , iC = . (cid:18) 𝐸 GeV (cid:19) − (cid:18) 𝑈 rad − J m − (cid:19) − . (23)These times are long compared to the time since the last supernova,1.5-3.2 Myr ago (compare Sect. 3.3), a plausible candidate for accel-erating the GeV electrons (compare Sun et al. 2008). Hence, even inscenarios, where the Local Bubble explains a high fraction of the ra-dio background, no significant curvature of the radio spectrum wouldbe expected. Gamma-ray measurements identify a spectral break at an energy around 1 TeV (López-Coto et al. 2018). Identifying thisbreak with the break expected from synchrotron cooling fixes themagnetic field to a value of approximately 0.2 nT.Different magnetic field values mean that different parts of theparticle spectrum are contributing to the observed emission. There-fore the curvature in the predicted spectra depends on the magneticfield strength. For magnetic field strengths around and below 1 nT,the curvature would better correspond to the one of the observedradio background. At this level of magnetic field strength, the LocalBubble would contribute about 20 per cent of the radio backgroundbetween 10 MHz and 10 GHz.The magnetic field strength for equipartition between magneticenergy and energy in relativistic leptons in our Local Bubble modelis 𝐵 eq , rel = . 𝑛𝑇 . For equipartition between magnetic and thermalenergy, using the pressure of 1 . × − Pa given by Snowden et al.(2014), it is 𝐵 eq , th = . 𝑛𝑇 . A magnetic field strength of 1 nT asdiscussed in the previous paragraph would therefore mean an ener-getically dominant magnetic field. This would create tension with ourassumption of the magnetic power spectrum, because, if the magneticenergy dominates, one expects an inverse cascade for the magneticpower (Christensson et al. 2001; Brandenburg et al. 2015; Reppin& Banerjee 2017; Sur 2019). The power spectrum would then beexpected to be dominated by such large modes at the current time ofobservation. Therefore, the distributions in the left column in Fig. 1would approximately apply, i.e., we would predict large differencesof the background emission in different sky directions and significantpolarisation. Given that the radio background is found as an isotropiccomponent in large sky surveys, this seems in tension with observa-tions. A magnetic field ordered on large scales also appears to be incontradiction with the starlight polarisation measurements discussedin Sect. 3.3, where we argued that the largest coherent scale for themagnetic field in the Local Bubble was 40 pc. We note that Singalet al. (2010) have argued against large-scale patterns in polarisationfor the radio background from WMAP data.For decaying turbulence and an initially weak magnetic field, weexpect magnetic field amplification up to an equilibrium with thekinetic energy (Brandenburg et al. 2019). This growth phase may lastseveral initial crossing (turnover) times, up to perhaps ten crossingtimes, depending on the initial field strength. It is well known thatfor turbulence in general, the kinetic energy is converted to thermalenergy, also on a timescale comparable to the crossing time. TheLocal Bubble may therefore be in a situation close to equilibriumbetween magnetic and thermal energy. For this situation, we wouldpredict a fairly isotropic contribution of about 10 per cent to the radiobackground.Of course, the magnetic field might still be lower, perhaps inequipartition with the cosmic ray electrons or even lower. For amagnetic field strength of 0.16 nT, which interestingly is associatednot only with equipartition between magnetic energy and relativisticleptons, but would also allow to interpret the break in the electronenergy distribution at 1 TeV as due to synchrotron cooling, the LocalBubble contributes to the radio background at a level of about 1 percent.For a magnetic field below equipartition with the thermal energydensity, we expect decaying turbulence, which would lead to a polar-isation of at most a few per cent with no coherent large-scale patternin polarisation (Fig. 1). This is very similar to radio polarisation inthe Galactic plane in general (Kogut et al. 2007).Summarising, a contribution of the Local Bubble to the radiobackground at the per cent level appears most likely.This result is perhaps surprising, given the encouraging scalingsfrom the non-thermal superbubble in IC 10 (Sect. 2). There is clearly MNRAS000 , 1–8 (2019) a difference in the level of non-thermal energy and magnetic en-ergy between the two superbubbles, and it would be interesting tounderstand the reasons for this better.
We have modelled the radio synchrotron emission of the Local Bub-ble, using observational constraints on the energy distribution ofcosmic ray electrons, magnetic fields, X-ray gas and warm cloudsand filaments. We find that in order to explain the radio synchrotronbackground remaining after subtraction of the Galaxy, the cosmicmicrowave background and the contribution of known extragalacticpoint sources we require a magnetic field of 2.5 nT. This would beallowed by constraints from Faraday rotation against nearby pulsars.However, in this case, the magnetic field would dominate energet-ically, and we would expect an inverse cascade, leading to largevariations of the background emission in different sky direction, sig-nificant polarisation with large coherence lengths for the magneticfield, and a synchrotron cooling break in the electron energy spectrumbelow 1 TeV, all of which are difficult to reconcile with observations.In order to avoid an inverse turbulent cascade associated with largeanisotropies of the radio emission and significant polarisation, themagnetic energy density should not exceed the thermal one, and toavoid an unobserved cooling break at electron energies below 1 TeV,the magnetic field should not exceed ≈ . ACKNOWLEDGEMENTS
We thank the anonymous referee for useful comments that helped toimprove the manuscript. MJH acknowledges support from the UKScience and Technology Facilities Council (ST/R000905/1).
DATA AVAILABILITY
The data and code underlying this article are available in the articleand in its online supplementary material.
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