Impulsive radio events in quiet solar corona and Axion Quark Nugget Dark Matter

Shuailiang Ge,∗ Md Shahriar Rahim Siddiqui,† Ludovic Van Waerbeke,‡ and Ariel Zhitnitsky§

Department of Physics and Astronomy, University of British Columbia, Vancouver, V6T 1Z1, BC, Canada

The Murchison Widefield Array (MWA) recorded [1] impulsive radio events in the quiet solarcorona at frequencies 98, 120, 132, and 160 MHz. We propose that these radio events are the directmanifestation of dark matter annihilation events within the axion quark nugget (AQN) framework.It has been argued [2, 3] that the AQN annihilation events in the quiet solar corona can be identifiedwith the nanoflares conjectured by Parker [4]. We further support this claim by demonstrating thatobserved impulsive radio events [1], including their rate of appearance, their temporal and spatialdistributions and their energetics, are matching the generic consequences of AQN annihilations in thequiet corona. We propose to test this idea by analyzing the correlated clustering of impulsive radioevents in different frequency bands. These correlations are expressed in terms of the time delaysbetween radio events in different frequency bands, measured in seconds. We also make genericpredictions for low (80 and 89 MHz) and high (179, 196, 217 and 240 MHz) frequency bands, thathave been recorded, but not published, by [1]. We finally suggest to test our proposal by studyingpossible cross-correlation between MWA radio signals and Solar Orbiter recording of extreme UVphotons (a.k.a. “campfires”).

I. INTRODUCTION

In this work, we discuss two seemingly unrelated sto-ries. The first one is motivated by the recent observationsof impulsive radio events in the quiet solar corona (at 98,120, 132, and 160 MHz), carried out by the MurchisonWidefield Array (MWA) [1]. The second one is the axionquark nugget (AQN) dark matter model [5] and its pos-sible role in the heating of the solar corona [2, 3]. Thetopic of the present paper is to explain how, and why,these two stories are related.

The observed impulsive radio events [1] appear to haveall the features normally attributed to nanoflares, con-jectured by Parker [4]) as a possible resolution of thecorona heating mystery [6]. On the other hand, [2, 3]have shown that AQNs entering the Sun’s corona leadto impulsive energy injection events, that provide theproper amount of energy needed to heat the corona. Thisled to the identification of AQN annihilation events withnanoflares. Furthermore, most of the AQN-annihilationenergy was shown to be released in the transition region,at an altitude around 2000 km, a region known to bethe most puzzling layer of the solar corona, where thetemperature and the density of the plasma experience adramatic change across a thin layer. We will show thatthe annihilation events proposed in [2, 3] share many fea-tures with the impulsive radio signals observed by [1], interms of their rate of appearance, their temporal andspatial distributions, their energetics, and other relatedobservables.

Let us first start with an overview of the solar coronaheating puzzle. The solar photosphere is in thermal equi-librium at ∼ 5800 K, while the corona has a temperature

∗ slge@phas.ubc.ca† shahriar.naf07@gmail.com‡ waerbeke@phas.ubc.ca§ arz@phas.ubc.ca

of a few 106 K [6]. Physically, this high temperature cor-responds to an energy excess of a few 1027 erg s−1, mostlyobserved in the extreme ultraviolet (EUV) and soft X-ray bands. The conventional view is that the coronaexcess heating is explained by nanoflares, a concept orig-inally invented by Parker [4]. The individual short energybursts associated with these nanoflares are significantlybelow detection limits and have not yet been observed inthe EUV or X-ray regimes. In fact, all coronal heatingmodels advocated so far seem to require the existenceof an unobserved (i.e. unresolved with current instru-mentation) source of energy distributed over the entireSun [7]. Therefore, nanoflares are modelled as invisiblegeneric events, producing an impulsive energy release ata small scale; their cause and their nature are not spec-ified ([8, 9]). [1] adopted this definition of nanoflares toexplain the impulsive radio events they observed in thequiet solar corona (in terms of frequency of appearance,duration, and wait times distribution, at frequencies 98,120, 132, and 160 MHz). They argued that radio obser-vations allow to probe much weaker energy levels, withmuch better temporal and spatial resolutions, in compar-ison to the current generation instrumentation in EUVand X-rays energy bands. In other words, radio obser-vations can potentially ”see” individual nanoflares andtheir ”internal structures”, where more energetic EUVand X-rays instruments cannot.

Second, let us highlight the basic features of the AQNmodel, while deferring a more detailed overview to Sec-tion II. The axion quark nugget (AQN) dark mattermodel [5] was invented long ago with the single objectiveof explaining the proximity of the dark and the visiblematter densities in the Universe, i.e. ΩDM ∼ Ωvisible,without fine tuning. The AQN model construction is,in many respects, similar to the original quark-nuggetmodel suggested by Witten [10] (see [11] for a review).This type of DM is ”cosmologically dark”, not becauseof the weakness of AQN interactions, but because oftheir small cross-section-to-mass ratio, which scales down

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many observables.Two additional elements of the AQN model make it

a viable DM model compared to the original proposal[10, 11]. First, axion domain walls provide a stabilizationfactor to AQNs. They are copiously produced during theQCD phase transition, which helps alleviating a numberof stability problems with the original nugget model. Sec-ondly, axion quark nuggets can be made of matter as wellas antimatter during the QCD transition. Consequently,the DM density ΩDM and the baryonic matter densityΩvisible automatically assume the same order of magni-tude (ΩDM ∼ Ωvisible), without any fine tuning [5]. Oneshould emphasize that AQNs are stable over cosmologi-cal time scales. Antimatter, hidden in the form of thesevery dense anti-nuggets, is unavailable for annihilation,unless an anti-nugget hits a star or a planet. Very rareannihilation events also happen in the center of galaxies,via collisions with single protons, electrons or light nuclei.They may explain some of the observed excess emission inour Galaxy, in different frequency bands (see next Sect.II for references). We will only focus on AQNs madeof antimatter, the ones capable of releasing a significantamount of energy via annihilation when they enter the so-lar corona. As noted by [2], the power required to solvethe corona EUV excess is of the order of 1027 erg s−1.This corresponds to the mc2 energy available from DMfalling on the Sun (by gravity only), assuming a typicalDM mass density of ρDM ' 0.3 GeV cm−3. This corre-spondence motivated the identification of nanoflares withAQN annihilation events. Furthermore, [3] showed thatthe dominant portion of the annihilation energy is de-posited in the corona, before entering the dense regionsof the photosphere, at an altitude of approximately 2000km, known as the Transition Region.

The main goal of the present work is to explore thepossibility that AQN’s annihilations and nanoflares arethe same impulsive energetic events, and the interpreta-tion of the impulsive radio events observed by [1]. Ourpresentation is organized as follows. In section II, weoverview the basic ideas of the AQN model in the con-text of the impulsive radio events. In section III, wehighlight some features related to the solar corona heat-ing, within the AQN framework. In sections IV and V,we present our estimates supporting the main claim ofthis work, i.e. that observations [1] nicely match thecharacteristics of AQN annihilation events, including thefrequency of appearance, the temporal and spatial distri-butions, the energetics, and other related observables inradio frequency bands.

II. THE AQN MODEL: THE BASICS

It is commonly assumed that the Universe began in asymmetric state with zero global baryonic charge, andlater (through some baryon number violating process,non- equilibrium dynamics, and CP violation effects, re-alizing three famous Sakharov’s criteria) evolved into a

state with a net positive baryon number. This is called”baryogenesis”.

The original motivation for the AQN model comes fromthe possibility of an alternative to this scenario, where”baryogenesis” is replaced by a charge separation pro-cess in which the global baryon number of the universeremains zero at all times. In this model, the unobservedanti-baryons come to comprise dark matter in the form ofdense anti-nuggets, made of antiquarks and antigluons,in a colour superconducting (CS) phase. This ”chargeseparation” process results in two populations of AQNs,carrying positive or negative baryon numbers. In otherwords, an AQN can be formed of either matter or an-timatter. However, due to the global CP violating pro-cesses associated with θ0 6= 0, during the early formationstage, the number of nuggets and anti-nuggets was dif-ferent1.) This difference is always an order one effect, ir-respective of the parameters of the theory (i.e. the axionmass ma, or the initial misalignment angle θ0). We re-fer to the original papers [28–31] devoted to axion quarknuggets’ formation, generation of the baryon asymmetry,and survival pattern of nuggets during the evolution inearly Universe.

Antimatter AQNs can interact with regular matter viaannihilations, which leads to electromagnetic radiationswhose spectral characteristics and flux can be calculatedwithin the AQN framework. These emissions are suffi-ciently dim to not violate any known observational con-straints, but are strong enough to offer a possible solutionto some unexplained astrophysical observations. For in-stance, it is known that the galactic spectrum containsseveral excesses of diffuse emission, the best known ex-ample being the strong galactic 511 keV line, the originof which is not well established and remains debated. IfAQNs have a baryon number in the 〈B〉 ∼ 1025 range,they can offer a potential explanation for several of thesediffuse components, in three different spectral domains: radio, X-ray and γ-ray. In all three cases, the pho-ton emission originates from the outer layer of the AQN,known as the electrosphere. All intensities in differentfrequency bands are expressed in terms of a single pa-rameter, 〈B〉, such that all the relative intensities are un-ambiguously fixed, because they are determined by theStandard Model (SM) of particle physics. This consti-tutes a nontrivial consistency check of the AQN model.For further details, see the original works [32–37].

Interestingly, AQNs could also offer a resolution toother seemingly unrelated puzzles, such as the ”Primor-dial Lithium Puzzle” [38] or the annual modulation ob-served by DAMA/LIBRA (see [39]). Furthermore, it mayresolve the puzzling seasonal variation of the X-ray back-ground in the near-Earth environment, in the 2-6 keV

1 . This source of strong CP violation is no longer available at thepresent epoch, as a result of the dynamics of the axion, whichremains the most compelling resolution of the strong CP problem(see [12–18] and recent reviews [19–27].

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energy range [40], as suggested in [41]. AQN annihila-tion events could also explain a mysterious type of ex-plosions in the Earth’s atmosphere, where infrasonic andseismic acoustic waves have been recorded, without anytraces of accompanying meteor-like events ([42]). Finally,AQN annihilation events which occur under a thunder-storm may explain several events observed by the Tele-scope Array collaboration, as discussed in [43]. In thecontext of our study, however, the most important ap-plication of AQNs is a possible explanation of the solarcorona heating ( [2, 3]), which is reviewed in details innext subsection III.

The key parameter, which determines the intensity ofthe effects mentioned above, is the average baryon charge〈B〉 of the AQNs. It is expected that AQNs do not haveall the same B, but rather B is given by a distributionfunction, f(B). There are several constraints on this pa-rameter which are reviewed below. AQNs are macroscop-ically large objects, with a typical size of R ' 10−5cm.They have roughly nuclear density, resulting in masses ofroughly 10 grams. For an AQN with a baryonic chargeB, mass is given by MN ' mp|B|. For the present work,we adopt a typical nuclear density of order 1040 g cm−3,such that a nugget with |B| ' 1025 has a typical radiusR ' 10−5cm. One can view an AQN as a very smallneutron star (NS), with nuclear density. The differenceis that a NS is squeezed by gravity, while an AQN issqueezed by axion domain wall pressure.

Let us now overview the observational constraints onAQNs. The strongest direct detection limit 2 is set bythe Ice Cube Observatory’s observations (see AppendixA in [45]):

〈B〉 > 3 · 1024 [direct (non)detection constraint]. (1)

The authors of [46] use the Apollo data to constrainthe abundance of AQNs, in the region of 10 kg to onetonne. It has been argued that the contribution of suchheavy nuggets must be at least an order of magnitudeless than would saturate the dark matter in the solarneighbourhood [46]. Assuming that AQNs do saturatedark matter, the constraint [46] can be reinterpreted asat least 90% of the AQNs having masses below 10 kg.This constraint can be approximately expressed in termsof the baryon charge:

〈B〉 <∼ 1028 [Apollo constraint ]. (2)

Therefore, indirect observational constraints (1) and (2)suggest that, if the AQNs exist, and saturate the dark

2 There is also an indirect constraint on the flux of dark matternuggets with mass M < 55g (which corresponds approximatelyB ' 1025) based on the non-detection of etching tracks in ancientmica [44]. It slightly touches the lower bound of the allowedrange (1), but does not strongly constraint the entire window(3) because the dominant portion of AQNs lies well above itslower limit (1), assuming the mass distribution 8)

matter density today, the dominant portion of them mustreside in the window:

3 · 1024 <∼ 〈B〉 <∼ 1028 [constraints from observations].(3)

The authors of [47] considered a generic constraintsfor nuggets made of antimatter (ignoring all essentialspecifics of the AQN model, such as quark matter phaseof the nugget’s core). Our constraints (3) are con-sistent with their findings, including the Cosmic Mi-crowave Background (CMB) and Big Bang Nucleosyn-thesis (BBN) and others, with the exception of ”HumanDetectors” 3.

We emphasize that the AQN model, within the abovewindow (3), is consistent with all presently available cos-mological, astrophysical, satellite and ground-based con-straints. Furthermore, it has been shown that thesemacroscopic objects can be formed, and the dominantportion of them will survive highly disruptive events(such as BBN, galaxy and star formation) during thelong evolution of the Universe [28–31]. The AQN modelis very rigid, and predictive, as there is no flexibility, norfreedom to modify any estimates [2, 3, 32–39, 41–43],which have been carried out in drastically different envi-ronments, where densities and temperatures span manyorders in magnitude.

III. THE AQN MODEL: APPLICATION TOTHE SOLAR CORONA HEATING

In this section, we overview the basic characteristics ofnanoflares, from an AQN viewpoint. The correspondingresults will play a vital role in our studies in section IV,where we interpret the radio events analyzed by [1] interms of the AQN annihilation events [2, 3].

3 We think that the corresponding estimates of [47] are oversimpli-fied, and do not have the same status as those derived from CMBor BBN constraints. In particular, the rate of energy depositionwas estimated in [47] assuming that the annihilation processesbetween anti-nuggets and baryons are similar to pp annihilationprocess. It is known that it cannot be the case in general, andit is not the case in particular in the AQN model because theannihilating objects have drastically different structures. It hasbeen also assumed in [47] that a typical X-ray energy is around 1keV, which is much lower than direct computations in the AQNmodel would suggest [42]. Higher energy x-rays have much longermean-free path, which implies that the dominant portion of theenergy will be deposited outside the human body. Finally, [47]assume that an anti-nugget will result in an ”injury similar to agunshot”. It is obviously a wrong picture as the size of a typi-cal nugget is only R ∼ 10−5cm while the most of the energy isdeposited in form of the x-rays on centimeter scales [42] withoutmaking a large hole similar to bullet as assumed in [47]. In thiscase a human’s death may occur as a result of a large dose ofradiation with a long time delay, which would make it hard toidentify the cause of the death.

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A. The nanoflares: observations and modelling

We start with a few historical remarks. The solarcorona is a very peculiar environment. Starting at analtitude of 1000 km above of the photosphere, the highlyionized iron lines show that the plasma temperature ex-ceeds a few 106 K. The total energy radiated away by thecorona is of the order of Lcorona ∼ 1027erg s−1, which isabout 10−6−10−7 of the total energy radiated by the pho-tosphere. Most of this energy is radiated at the extremeultraviolet (EUV) and soft X-ray wavelengths. There isa very sharp transition region, located in the upper chro-mosphere, where the temperature suddenly jumps froma few thousand degrees to 106 K. This transition layer isrelatively thin, 200 km at most. This transition happensuniformly over the Sun, even in the quiet Sun, where themagnetic field is small (∼ 1 G), away from active spotsand coronal holes. The reason for this uniform heatingof the corona is not understood.

A possible solution to the heating problem in the quietSun corona was proposed in 1983 by Parker [48], whopostulated that a continuous and uniform sequence ofminiature flares, which he called “nanoflares”, could hap-pen in the corona. This became the conventional view.The term ”nanoflare” has been used in a series of papersby Benz and coauthors [49–53], and many others, to ad-vocate the idea that these small ”micro-events” might beresponsible for the heating of the quiet solar corona. Wewant to mention a few relatively recent studies [54–61]and reviews [8, 9] which support the basic claim of ear-lier works, i.e. that nanoflares play the dominant role inthe heating of the solar corona.

In what follows, we adapt the definition suggested in[53] and refer to nanoflares as “micro-events” in quiet re-gions of the corona, to be contrasted with “micro flares,”which are significantly larger in scale and observed in ac-tive regions. The term “micro-events” refers to a shortenhancement of coronal emission in the energy range ofabout (1024 − 1028)erg. One should emphasize that thelower limit gives the instrumental threshold for observingquiet regions, while the upper limit refers to the smallestevents observable in active regions. The list below showsthe most important constraints on nanoflares from theobservations of the EUV iron lines with SoHO/EIT:1. The EUV emission is highly isotropic [50, 52], there-fore the nanoflares have to be distributed very “uniformlyin quiet regions”, in contrast with micro-flares and flareswhich are much more energetic and occur exclusively inactive areas [53]. For instance, flares have a highly non-isotropic spatial distribution because they are associatedwith the active regions;2. According to [51], in order to reproduce the measuredEUV excess, the observed range of nanoflares needs to beextrapolated from the observed events interpolating be-tween (3.1 · 1024 − 1.3 · 1026) erg to sub-resolution eventswith much smaller energies, see item 3 below.3. In order to reproduce the measured radiation loss,the observed range of nano flares (having a lower limit

at about 3 · 1024erg due to the instrumental threshold)needs to be extrapolated to energies as low as 1022ergand in some models, even to 1020erg (see table 1 in [51]);4. The nanoflares and microflares appear in a differentrange of temperature and emission measure (see Fig.3 in[53]). While the instrumental limits prohibit observationsat intermediate temperatures, nevertheless the authorsof [53] argue that “the occurrence rates of nanoflares andmicroflares are so different that they cannot originatefrom the same population”. We emphasize this differ-ence to argue that the flares originate at sunspot areas,with locally large magnetic fields B ∼ (102−103) G, whilethe EUV emission (which is observed even in very quietregions where B ∼ 1G) is isotropic and covers the entiresolar surface;5. Time measurements of many nanoflares demonstratea Doppler shift with typical velocities of (250-310) km/s(see Fig.5 in [49]). The observed line width in OV of±140 km/s far exceeds the thermal ion velocity, which isaround 11 km/s [49];6. The temporal evolution of flares and nanoflares alsoappears different. The typical ratio between the max-imum and minimum EUV irradiance during the solarcycle does not exceed a factor of 3 between its maxi-mum in 2000 and its minimum in 2009 (see Fig. 1 from[62]), while the same ratio for flares and sunspots is muchlarger, of the order of 102. If the magnetic reconnectionwas fully responsible for both the flares and nanoflares,then the variation during the solar cycles should be sim-ilar for these two phenomena. It is not what is observed;the modest variation of the EUV with the solar cyclesin comparison to the flare fluctuations suggests that theEUV radiation does not directly follow the magnetic fieldactivity, and that the EUV fluctuation is a secondary, nota primary effect of the magnetic activity.

The nanoflares are usually characterised by the follow-ing distribution:

dN ∝W−αdW 1021erg <∼W <∼ 1026erg (4)

where dN is the number of nanoflare events per unit time,with an energy between W and W +dW . In formula (4),we display the approximate energy window for W as ex-pressed by items 2 and 3, including the sub-resolutionevents extrapolated to very low energies. The distri-bution dN/dW has been modelled via magnetic-hydro-dynamics (MHD) simulations [54, 63] in such a way thatthe Solar observations match the simulations. The pa-rameter α was fixed to fit observations [54, 63], (see thedescription of the different models in next subsection).

B. The nanoflares as AQN annihilation events

It has been conjectured in [2] that the nanoflares can beidentified with AQN annihilation events. This conjecturewas essentially motivated by the fact that the amountof energy available from the dark matter falling on theSun per second, in the form of mass (mc2) , is similar

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to the amount of energy needed to maintain the coronaat its observed temperature (∼ 1027 erg s−1). The darkmatter density in the solar system is estimated to beof the order of ρDM ' 0.3 GeV cm−3, within a factor∼ 2. From this identification, it follows that the baryoncharge distribution (within the AQN framework) and thenanoflare energy distribution (4) must be one and thesame function [2], i.e.

dN ∝ B−αdB ∝W−αdW (5)

where dN is the number of nanoflare events with energybetween W and W + dW , which occur as a result of thecomplete annihilation of the antimatter AQN carrying abaryon charges between B and B + dB.

An immediate self-consistency check of this conjectureis the observation that the allowed window (3) for theAQNs baryonic charge largely overlaps with the approxi-mate energy window for nanoflares, W expressed by (4).This is because the annihilation of a single baryon chargeproduces an energy of about 2mpc

2 ' 2 GeV, which canbe expressed in terms of the conventional units as follows,

1 GeV = 1.6 · 10−10J = 1.6 · 10−3erg, (6)

such that the nanoflare energyW for the anti-nugget withbaryon charge B can be approximated as W ' 2 GeV·B.One should emphasize that this is a highly nontrivial self-consistency check of proposal [2], as the acceptable win-dows (3) and (4) for the AQNs and nanoflares have beenconstrained from drastically different physical systems.

Encouraged by this self-consistency check and thehighly nontrivial energetic consideration, [3] used thepower-law index α entering (4) to describe the baryonnumber distribution dN/dB for the anti-nuggets, whichrepresents the direct consequence4 of the conjecture (5).More specifically, in the Monte Carlo (MC) simulationsperformed in [3] , the baryon number distribution of theAQNs, as given by (8) , is assumed to directly follow thenanoflare distribution dN/dW , with the same index αas the conjecture (5) states.

The nanoflare distribution models proposed in [54, 63]have been adapted by [3]. Three different choices for thepower-law index α have been considered in [54, 63]:

α = 2.5, 2.0, or

1.2 W <∼ 1024erg↔ B <∼ 3× 1026

2.5 W >∼ 1024erg↔ B >∼ 3× 1026.

In addition to the power law index α, different modelsare also characterized by different choices of Bmin: 1023

4 One should note that it has been argued [31] that the algebraicscaling (5) is a generic feature of the AQN formation mechanismbased on percolation theory. The phenomenological parameterα is determined by the properties of the domain wall formationduring the QCD transition in the early Universe, but it cannotbe theoretically computed in strongly coupled QCD. Instead, itwill be constrained based on the observations as discussed in thetext.

and 3×1024. Therefore, a total of 6 different models havebeen discussed in [54, 63] which we expressed in terms ofthe baryon charge B rather than in terms of the nanoflareenergy W . We also fix Bmax = 1028 to be consistent withthe constraint (3).

In this work, we will only use simulations with 〈B〉 >∼1025 in order to be consistent with (3). This means thatwe are excluding two models considered in [54, 63]: theone with Bmin ∼ 1023 and the one with α = 2, 5 andα = 2. We also exclude the model with Bmin ∼ 1023 andthat with α = 1.25 and α = 2.5 to simplify things asit produces results very similar to another model. Theremaining three models are labeled as follows:

Group 1 : Bmin = 3× 1024, α = 2.5 (7)

Group 2 : Bmin = 3× 1024, α = 2.0

Group 3 : Bmin = 3× 1024, α =

1.2, B <∼ 3× 1026

2.5, B >∼ 3× 1026

while Bmax = 1028 for all the models.The average baryon number of the distribution is de-

fined as

〈B〉 =

∫ Bmax

Bmin

dB [B f(B)],dN

dB∝ f(B) ∝ B−α (8)

where f(B) is normalized and the power-law is taken tohold in the range from Bmin to Bmax.

The above estimate reveals an astonishing coincidencebetween the energy/mass windows (3) and (4) for AQNsand nanoflares respectively. This coincidence is a strongsupport of our proposal [2, 3] that the nanoflares and theAQN annihilation events are the same phenomena (seeitems 2 and 3 of Section III A).

We are now in position to present several additionalarguments in favor of our proposal: item 1 (Section III A)is also naturally explained in the AQN framework as DMis expected to be distributed very uniformly over the Sun,making no distinction between quiet and active regions,in contrast with large flares. A similar argument appliesto item 4 , as the strength of the magnetic field andits localization is absolutely irrelevant for the nanoflareevents in form of the AQNs, in contrast with conventionalparadigm where nanoflares are thought to be scaled downconfigurations of their larger cousins, which are muchmore energetic and occur exclusively in active areas andcannot be uniformly distributed.

The existence of a large Doppler shift, with a typicalvelocities (250-310) km/s, mentioned in item 5, can beunderstood within the AQN interpretation as the follow-ing: the typical velocities of an anti-nugget entering thesolar corona is very high, around 700 km/s. The Machnumber M = vAQN/cs is also very large. A shock wavewill be formed and will push the surrounding material tovelocities which are much higher than would normally bepresent at thermal equilibrium.

Finally, as stated in item 6, the temporal modulationof the EUV irradiance over a solar cycle is very small and

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does not exceed a factor ∼ 3, as opposed to the muchdramatic changes in Solar activity, with modulations onthe level of 102 over the same time scale. This suggeststhat the energy injection from the nanoflares is weaklyrelated to solar activity, which is in contradiction withthe picture where magnetic reconnection modulated bythe Sun activity plays an essential role in the formationand dynamics of nanoflares. This is, however, consis-tent with our interpretation of nanoflares being associ-ated with AQN annihilation events, as an external causeof the main source of the EUV irradiance.

IV. THE AQN MODEL CONFRONTS THERADIO OBSERVATIONS

We start in subsection IV A by describing the basicmechanism of the radio emission due to AQN annihila-tion events in the solar corona. We estimate the eventrate in subsection IV B. The role of non-thermal electronsin generation of the radio signal events is discussed insubsection IV C. Finally, in subsection IV D we estimatethe intensity of the radio signal events.

A. Mechanism of the radio emission in solarcorona.

It is generally accepted that the radio emission fromthe corona results from the interaction of plasma oscilla-tions (also known as Langmuir waves) with non-thermalelectrons which must be injected into the plasma [64].An important element for the successful emission of ra-dio waves is that a plasma instability must develop. Itoccurs when the injected electrons have a non-thermalhigh energy component, with a momentum distributionfunction characterized by a positive derivative5 with re-spect to the electron’s velocity. In this case, a plasmainstability develops and radio waves can be emitted.

The frequency of emission ν is mostly determined bythe plasma frequency ωp in a given environment, i.e.

ω2 = ω2p + k2 3T

me, ω2

p =4παneme

, ν =ω

2π, (9)

where ne is the electron number density in the corona,while T is the temperature at the same altitude and kis the wavenumber. For example, the frequency ν =160 MHz considered in [1] will be emitted when ne '3.4 · 108cm−3. One should emphasize that the emissionof radio waves generically occurs at an altitude which isdistinct from the altitude where the AQN annihilationevents occur, and where the energy is injected into the

5 If the derivative has a negative sign it will lead to the so-calledLandau damping.

plasma. This is because the mean-free path λ of the non-thermal electrons being injected into the plasma is verylong λ ∼ 104 km. Therefore, these electrons can travela very long distance before they transfer their energy tothe radio wave, as we discuss in subsection IV C.

We propose that non-thermal electrons are producedby anti-nuggets entering the solar corona, when the an-nihilation processes start. It is known that the numberdensity of the non-thermal (suprathermal in terminology[64]) electrons ns must be sufficiently large ns/ne >∼ 10−7

for the plasma instability to develop, in which case theradio waves will be generated [64]. As the density ns/neapproaches the threshold values at some specific frequen-cies, the intensity increases sharply, which we identifywith the observed impulsive radio events. These thresh-old conditions may be satisfied randomly in space andtime, depending on properties of the injected electrons[64]. All these plasma properties are well beyond thescope of this paper. However, we shall demonstratethat the number density of the non-thermal electrons nsgenerated by the AQNs can easily be in proper rangens/ne >∼ 10−7 for the plasma instability to develop. Tobe more specific, in next subsection IV C we shall arguethat the ratio ns/ne ∼ 10−7 is always sufficiently largefor the plasma instability to develop, which eventuallygenerate the radio waves.

Therefore, our proposal is that the AQN annihilations(identified with nanoflares as explained in Section III B)produce a large number of non-thermal electrons, which,in turn, generate the observed impulsive radio events [1]as a result of plasma instability. In the next subsections,we will support our proposal by estimating a number ofobservables analyzed in [1] . and show that our proposalis consistent with all observed data, including the fre-quency of appearance, the intensity radiation, duration,spatial and wait time distributions, to be discussed innext subsections IV B, IV C, IV D as well as in SectionsV.

B. The event rate

We are now in a position to interpret the radio emissiondata from [1] in terms of AQN annihilation events. Theanti-nuggets start to loose their baryon charge, due to theannihilation, in close vicinity of the transition region, atan altitude of 2150 km ( see Fig. 5 in [3] and also Fig.5 below). However, the radio emission happens at muchhigher altitudes, as we explain in subsection IV C.

In this subsection, we want to compare the maximumradio event rate (33,481 events observed in the 132 MHzfrequency band, during 70 minutes) to the expected rateof AQN annihilation events which are identified withnanoflares, and must be much more numerous (accord-ing to conventional solar physics modelling). Specificnanoflare models [54, 63] (expressed by eq.(7) in terms ofthe baryon charge B) correspond to events rate which isat least few orders of magnitude higher than the observed

7

1025 1026 1027 1028

B

10−1

100

101

102

103

104

Impa

ctR

ate

[s−

1 ]

Group 1Group 2Group 3

1027 1028

B

10−2

10−1

100

101

Group 1Group 2Group 3

FIG. 1. Left: the impact rate of AQNs with the size above B where B varies from Bmin to Bmax for different groups ofAQNs. The horizontal black dashed line is the observed rate of radio events (11). Right: the result from the second-roundsimulation where we focus on large AQNs only. Again, the horizontal black dashed line is (11). The vertical dashed lines arethe corresponding B for different groups. More details about the numerical simulations that lead to these two subfigures canbe found in Appendix A.

1025 1026 1027 1028

B

1025

1026

1027

Lum

inos

ity[e

rg·s−

1 ]

Group 1Group 2Group 3

1027 1028

B

1023

1024

1025

1026

Group 1Group 2Group 3

FIG. 2. Left: the luminosity generated by the annihilation of AQNs with the size above B where B varies from Bmin to Bmax

for different groups of AQNs. Right: the result from the second-round simulation where we focus on large AQNs only. Thevertical dashed lines corresponds to the B determined by (11) in Fig. 1. More details about the numerical simulations thatlead to these two subfigures can be found in Appendix A.

radio event rate, see Fig 8 in [3]. There is no contradic-tion here because it is likely that the dominant portionof the nanoflare events are too small to be resolved. Thispoint has been mentioned in items 2 and 3 in sectionIII A with a comment that all models must include smallbut frequent events which had been extrapolated to sub-resolution region. Therefore, we interpret the low eventrate at radio frequencies as the manifestation that onlythe strongest and the most energetic, but relatively rare,AQN annihilation events can be resolved in radio bands.We define B as the minimum baryonic charge a nuggetmust have in order to generate a resolved radio impulse.

We can compute (in terms of B) the event rate for theenergetic AQNs which are powerful enough to generatethe resolved radio impulses as recorded in [1]. The corre-sponding impact rate can be computed in the same wayas Fig 8 from [3], the only difference being that the lower

bound is determined by B instead of Bmin, i.e.(dN

dt

)B

∝∫ Bmax

B

dB f(B). (10)

Since the maximum number of detected radio eventsin [1] is 33481 at the 132 MHZ band in 70 minutes, theevent rate is

dNobs.

dt∼ 33481

70 minutes× 1/2∼ 16 s−1. (11)

The factor 1/2 accounts for the fact that only half of theSun’s whole surface is visible.

By equalizing (11) and (10) we can estimate the pa-rameter B when sufficiently large radio events originatefrom large nuggets with B >∼ B 6. The results are pre-

6 This estimate does not include the possibility of “clustering”events with very short time scale discussed in Section V B.

8

sented on Fig. 1. It is the intersection of the black dashedline (11) and the simulated line of each group given byeq.(7). The intersections are shown in the right subfig-ure, and the corresponding B are respectively 5.65×1026,2.21 × 1027, and 1.95 × 1027 for the three groups. Weexpect that only AQNs with masses greater than B aresufficiently energetic to generate the observable impulsiveradio events.

The parameter B obviously depends on the size dis-tribution models listed in (7), it corresponds to a detec-tion limit and should not be treated as a fundamentalparameter of the theory. An instrument with differentresolution and/or sensitivity will affect the radio eventsselection criteria and therefore change the value of B,in which case some events from the continuum spectrumwould be considered as impulsive events7.

Our next task is to estimate the total luminosity LB

released as a result of the complete annihilation of thelarge nuggets with B >∼ B . The calculation is similar tothe estimation done for Fig 10 of [3], the only differenceis that the lower bound is determined by B rather thanBmin, i.e.

LB∝∫ Bmax

B

dB B23 f(B). (12)

The results for the models listed in (7) are presentedon Fig. 2. The corresponding L

Bassume the follow-

ing values: 6.17×1025 erg · s−1, 2.05×1026 erg · s−1 and1.70×1026 erg · s−1, which are approximately an order ofmagnitude smaller than the luminosity released by all theAQNs annihilation. This implies that only about 10% ofthe total the AQN- induced luminosity comes from thelarge nuggets with B >∼ B, which are the same AQNsassumed to produce the resolved radio events in [1]. Ourestimates show that while the strong events with B >∼ Bare very rare with an impact rate approximately 3 ordersof magnitude smaller than all AQN annihilation events,their contribution to the luminosity is suppressed onlyby one order of magnitude. This is, of course, due to thefactor B

23 in the expression for the luminosity (12).

The energy flux ΦB

, observed on Earth, coming from

these large nuggets with B >∼ B is estimated as

ΦB'

LB

4π(AU)2' (1.8− 6) · 10−2 erg

cm2 s, (13)

where we used the range of numerical values for LB

es-timated above. In the following we will establish thephysical connection between the energy flux (13) gener-ated by large nuggets with B >∼ B and the flux observed

7 It is known that the continuum contribution in the radio emis-sions is similar in magnitude to the impulses events as we discussin subsection IV D. Some of the events from continuum could betreated in future as impulsive events if a better resolution instru-ment is available. However, this does not drastically modify ourestimate for B.

in radio frequency bands observed in [1]. In order tomake this connection we have to estimate what fractionof the huge amount of energy due to the AQN annihi-lation is transferred to the tiny portion in the form ofradio waves. To compute this efficiency we need to es-timate the relative density of the non-thermal electronswhich will be produced as a result of the AQN annihila-tion events. The estimation of this efficiency is the topicof the next subsection.

C. Non-thermal electrons

The starting point for our analysis is the number ofannihilation events per unit length while the AQN prop-agates through the ionized corona environment:

dN

dl' πR2

effnp, (14)

where np is the baryon number density of the corona(mostly protons) and the effective radius Reff of theAQNs can be interpreted as the effective size of thenuggets due to the ionization characterized by thenugget’s charge Q as explained in [3]. The enhance-ment of the interaction range Reff due to the long rangeCoulomb force is given by (see [3] for the details):(Reff

R

)= ε1ε2, ε1 ≡

√8(meTP )R2

π, ε2 ≡

(TITP

) 32

,(15)

where TI is the internal temperature of the AQN and TPis the plasma temperature in the corona. The estimationof the internal thermal temperature TI is a highly non-trivial and complicated problem which requires an un-derstanding of how the heat, due to the friction and theannihilation events, will be transferred to the surround-ing plasma from a body moving with supersonic speedwith Mach number M ≡ v/cs > 1.

It is known that the supersonic motion will generateshock waves and turbulence. It is also known that ashock wave leads to a discontinuity in velocity, densityand temperature due to the large Mach numbers M 1.It has been argued in [3, 65] that, for a normal shock, thejump in temperature is given by the Rankine–Hugoniotcondition:

TITP'M2 · 2γ(γ − 1)

(γ + 1)2 1, γ ' 5/3, (16)

and, as a result, all the electrons from the plasma whichare on the AQN path within distance Reff will be affected.To be more precise these electrons will experience elasticscattering by receiving the extra kinetic energy ∆E whichlies in the window ∆E ∈ (TP , TI). It is precisely thesenon-thermal electrons which will subsequently interactwith the plasma and be the source of the plasma insta-bility. These non-thermal electrons will transfer theirenergy to the emission of radio waves with frequency νas explained at the end of Section IV A.

9

We are now in position to estimate the parameter ηdefined as the ratio between the energy transferred (perunit length l) to the radio waves and the total energyproduced by a single AQN (per unit length l) as a resultof the annihilation process:

η ≈(∆E) · [πR2

effne]

(2mpc2) · [πR2effnp]

≈ ∆E

2mpc2∼ 10−7, (17)

where the denominator accounts for the total energy dueto the annihilation events with rate (14) and the numer-ator accounts for the kinetic energy received by affectedelectrons. In our estimate of (17), we assume an ap-proximate local neutrality such that ne ≈ np. Further-more, to be on the conservative side, we also assume that∆E ≈ 2 · 102 eV, such that ∆E only slightly exceeds theplasma temperature ≈ TP at high altitudes of order 104

km, where radio emission occurs. Finally, we also assumethat the dominant portion of the ∆E will be eventuallyreleased in the form of radio waves. It is very likely thatthere are few missing numerical factors of order one onthe right hand side in eq. (17) as our assumptions for-mulated above are only approximations. However, webelieve that (17) gives a correct order of magnitude esti-mate for the energy efficiency transfer ratio η. We pro-vide a few numerical estimates in next subsection IV Dsuggesting that (17) is very reasonable and consistentwith observed intensities in radio bands [1].

The next step is the estimation of ns/ne, which mustbe sufficiently large ns/ne >∼ 10−7 for the plasma in-stability to develop [64] (see section IV A). As we shallsee now, the proposed mechanism indeed satisfies thisrequirement. We start with the expression of the totalnumber of electrons ∆Ne to be affected while the AQNtravels over a distance l:

∆Ne ∼ (πR2eff l) · ne(h), l ' vAQN∆t, (18)

where ne(h) is the electron number density at the altitudeh ' 2000 km where annihilation events become efficient[3]. These affected electrons will receive an extra energy∆E and extra momentum mev⊥ with very large veloc-ity component v⊥ perpendicular to the nugget’s path asthe shock front due to M 1 has a form of a cylinderalong the AQN path. A large portion of the AQN’s tra-jectories can be viewed as an almost horizontal path withrelatively small incident angles toward the Sun (skim tra-jectories). These non-thermal electrons will have a com-ponent v⊥ perpendicular to the nugget’s path and travelunperturbed up to a distance of the order of the meanfree path λ ∼ 104 km (to be estimated below).

After a time ∆t, the same non-thermal electrons ∆Newill have spread over a distance r from the AQN’s path,estimated as follows:

∆Ne ∼ (2πr∆rl) · ns(r), (19)

where ∆r is the width of the shock front measured atdistance r. For a non-thermal electron traveling away

from the AQN path with perpendicular velocity v⊥, thedistance r is given by:

r ∼ v⊥∆t, v⊥ '√

2∆E

me' 104

√∆E

2 · 102 eV

km

s.(20)

Equalizing (18) and (19) we arrive to the following esti-mate for the ratio ns/ne:[

ns(r)

ne(h)

]'(R2eff

r∆r

), r <∼ λ. (21)

The expression (21) holds as long as r <∼ λ. For largerdistances r >∼ λ the non-thermal electrons will eventuallythermalize and loose their ability to generate a plasmainstability. One should emphasize that ns(r) entering(21) is taken at the distance r from the AQN path, whilene(h) is taken in the vicinity of the path, i.e. at r ≈ 0.

We are interested in this ratio when both componentsare computed at the same location and we now have tocheck if it is larger than 10−7, the requirement to gener-ate the plasma instability. The relevant configuration forour study corresponds to non-thermal electrons movingupward8. In this case the relation (21) assumes the form[

ns(r)

ne(r)

]' 1

2

[ne(h)

ne(r + h)

]·(R2eff

r∆r

), r <∼ λ, (22)

where the factor 1/2 accounts for upward moving elec-trons and ne(r) ≡ ne(r + h) is the electron density com-puted at distance ∼ r above the AQN’s path (which islocalized at an altitude of h ' 2000 km).

The expression (22) has a conventional form for a cylin-drical geometry with the expected suppression factor r−1

at large distances and constant value for ∆r. However,it is known that the width of the shock ∆r also growthswith time9 as ∆r ∝

√t ∝

√rReff . Therefore, we expect

that a proper scaling at large r assumes the form:[ns(r)

ne(r)

]∼ 1

2

[ne(h)

ne(r + h)

]·(Reffr

) 32

, r <∼ λ, (23)

We will calculate this ratio for large nuggets with B >∼ Bwhich are capable of generating the resolved radio signals.Using our previous parameters estimates for ε1 and ε2from Section IV.C of [3] and using the electron numberdensity in Table 26 of [66] , we arrive at the estimate[

ns(r)

ne(r)

]>∼ 10−7 for r ∼ 104 km. (24)

8 the radio waves emitted at altitudes below h will have muchhigher frequencies than considered in the present work, and shallnot be discussed here.

9 Such scaling is known to occur, for example, when the mete-oroids propagate in the Earth’s atmosphere when the cylindricalsymmetry is also realized. We refer to [42] (with large list ofreferences on the original literature devoted to this topic) wherethis scaling specific for the cylindrical geometry has been usedin the context of the AQN propagation in Earth’s atmosphere.

10

The condition (24) implies that ns/ne is indeed suffi-ciently large for the plasma instability to develop [64] ondistances of order r ∼ 104 km from the nugget’s path.This implies that the non-thermal electrons can propa-gate upward to very large distances before they transfertheir energy to the radio waves at much higher altitudes,of order (h+ r). The scale r ∼ 104 km assumes the sameorder of magnitude value as the mean free path λ, whichat altitude h ' 104 km can be estimated as follows:

λ−1 ' σnp, σ ' α2

(∆E)2, λ ∼ 104 km, (25)

where (∆E) ≈ 2 · 102 eV is the typical kinetic energy ofthe non-thermal electrons at the moment of emission.

One should emphasize that the estimation given aboveassumes a constant density np along the electron’s path.This is clearly not the case for the upward moving non-thermal electrons. One can define an effective mean freepath λ−1

eff (h) as follows10

λ−1eff (h) ≡

∫ h

h0

dh′σnp(h′)

(h− h0), h0 ' 2150km, (26)

which accounts for the density variation with altitude.It reduces to the canonical definition (25) when np is aconstant along the electron’s path. This definition of theeffective mean free path in the context of the presentproposal is very convenient as it explicitly shows at whataltitude most of the energy will be thermalized, and whatportion of the energy can be released in form of the radiowaves.

To be more precise, the portion f(h) of the non-thermal electrons which survives at altitude h can beestimated as follows

f(h) = exp

(−∫ h

h0

dh′

λeff(h′)

), (27)

where mean free path λeff(h) at altitude h is defined byeq. (26). The behaviour for f(h) as a function of thealtitude h is shown on Fig. 3 by blue line for initialkinetic energy of the non-thermal electrons ∆E ≈ 2 · 102

eV. This value for ∆E has been used in all our estimatesthrough the text.

The most important remark here is that the suppres-sion factor f(h) is very modest for altitudes where highfrequency waves are emitted, see Fig.4. We emphasizethat, in this parameters range, the density of the non-thermal electrons remains sufficiently large to satisfy thecrucial condition (24) for the plasma instability to de-velop [64]. Therefore, the dominant portion of the non-thermal electron’s energy will be released in form of theradio waves.

10 We use h0 ' 2150 km precisely because AQNs start to annihi-late at this altitude (shown in Fig. 5) which is the start of thetransition region as the density drastically increases (see Table26 or Fig. 8 of [66]).

5000 10000 15000 20000 25000 30000 35000 40000h [km]

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f(h)

E = 2 102 eVE = 4 102 eVE = 6 102 eV

FIG. 3. Suppression factor f(h) defined by eq. (27). This fac-tor describes the remaining portion of the non-thermal elec-trons at altitude h. The blue line corresponds to the initialkinetic energy ∆E ≈ 2 · 102 eV which has been used in allour estimates through the text. For illustrative purposes wealso presented the same suppression factor f(h) for other val-ues of parameter ∆E. Suppression factor becomes essentialfor h >∼ 4 · 104 km corresponding to low frequency emissionas one can see from Fig.4. In computing (27), we have usednp(h) ≈ ne(h) above h0 where the profile of ne(h) is from [66](the solar profiles needed in the numerical computations inthis work are all from [66]).

1046×103 2×104 3×1044×104

Height [km]

80

100

120

140

160

180

200

Freq

uenc

yν

[MH

z] 160 MHz

132 MHz

120 MHz

98 MHz

FIG. 4. Frequency of the emission ν = ω/2π ≈ ωp/2π (i.e.,eq. (9)) as a function of height. Radio emission occurs atthe altitudes above 104 km while the dominant portion ofthe AQN annihilation events occur at lower altitudes h <2150 km as shown on Fig.5.

At the same time the suppression becomes essential forhigher altitudes where low frequency waves are emitted.At higher altitudes the suppression factor f(h) plays thedominant role and non-thermal electrons loose their en-ergy to thermalization. The density of the non-thermalelectrons is insufficient to satisfy the crucial condition(24) for the plasma instability to develop [64]. At thispoint the radio emission stops completely. One should

11

emphasize that such a sharp cutoff for the radio emissionat lower frequencies is very unique and specific predictionof the proposed mechanism.

One should also mention that the density np drasticallyincreases at slightly lower altitudes h <∼ 2000 km (incomparison with h ' 2150 km), such that the mean-freepath λeff decreases correspondingly, and the condition(24) breaks down. Therefore, the non-thermal electronsemitted at h <∼ 2000 km cannot propagate to very highaltitudes ∼ 104 km where radio emission occurs.

D. Radio flux intensity

In this subsection we estimate the portion of the AQN-induced energy flux which is transferred to the radiowaves Φradio. We express Φradio in terms of the energyflux emitted by the nuggets as radio waves:

Φradio ' ΦB· η(

∆B

B

)(28)

' (0.6− 6) · 10−10 erg

cm2 s, (theoretical prediction)

where the first factor ΦB

, given by (13), reflects the con-

tribution of the large nuggets with B >∼ B to the totalAQN-induced luminosity. The factor η is given by (17)and represents the portion of the energy transferred tothe radio frequency bands through the non-thermal elec-trons leading to the plasma instability. Finally, the factor∆B/B ∼ (0.3 − 1) · 10−1 describes a typical portion ofthe baryon charge annihilated in the altitude range (2000-2150) km. This is precisely the region where the AQNannihilation events effectively start and where the inter-action of the AQNs with surrounding plasma produce thenon-thermal electrons which eventually generate the ra-dio waves. The Monte-Carlo simulations for ∆B/B arepresented on Fig. 5. One can see that the dominantportion of the annihilation events occur at the lower al-titudes h <∼ 2000 km. However, the mean free path λat lower altitudes of the affected electrons is too shortas our estimations (25) suggest. Therefore, the affectedelectrons from altitudes h <∼ 2000 km cannot reach higheraltitudes where the radio waves are generated. This isprecisely the source of the suppression expressed in theratio ∆B/B 1.

We can now compare our estimate (28) to the observedintensities measured in radio frequency bands by [1]:

dΦradio

dω(160 MHz) ' 6 SFU, ∆ω = 2.56 MHz

dΦradio

dω(120 MHz) ' 3 SFU, ∆ω = 2.56 MHz (29)

where

SFU ≡ 104Jy = 10−19 erg

Hz cm2 s. (30)

The observations [1] were done in twelve frequency bandsfrom 80 MHz to 240 MHz with ∆ω = 2.56 MHz band-

width each. It is known [67, 68] that the radio emis-sion occurs in the entire energy band ∼ (0 − 200) MHz,and not specifically in one of the 12 frequency narrowbands. It is also known [67, 68] that the contributionsfrom continuum and impulsive fluxes are approximatelythe same in all frequency bands. Therefore we estimatethe total intensity in radio bands by multiplying (29)with ∼ 200 MHz to account for the entire radio emis-sion associated with short impulsive events as well as thecontinuum:

Φradiototal ' (0.6− 1.2) · 10−10 erg

cm2 s(observation). (31)

Despite the fact that our calculation involves varioussteps and approximations, the total measured flux (31)is consistent with our order of magnitude estimate (28).We consider this as a highly non-trivial consistency checkfor our proposal as it includes a number of very differentelements which were studied previously for a completelydifferent purpose in a different context.

We conclude this section with few important remarks.The occurrence probability shown on Fig 4 in [1] suggeststhat the power-law index α is always large, with α > 2.As explained in the text we cannot predict this indextheoretically, but all the nanoflare models used in ourstudies as expressed by eq. (7) are consistent with theobserved power-law index α because the nuggets generat-ing the resolved radio impulses must be sufficiently largewith B > B, in which case the index α is always large(index α = 1.2 for one of the model from (7) describesthe distribution of small nuggets with B < B which donot produce the resolved radio signals).

The basic picture for the radio emission advocatedhere is that one and the same AQN may generate theemissions in different frequency bands because the non-thermal electrons produced by the AQN and moving inupward direction can emit the radio waves at different al-titudes with different plasma frequencies as long as non-thermal electron density is sufficiently high and satisfiesthe condition (24). As an illustration, we show the fre-quency of emission (9) as a function of height on Fig.4. Inthis example, all the radio emissions must be correlatedwith in time over seconds, which is considerably shorterthan the typical mass loss time scale which is about 10-20 seconds, see Fig 5, 6 in [3].

This generic picture also suggests that the emission athigher frequencies ν must be more intense due to a num-ber of reasons. First, the upward moving non-thermalelectrons are much more numerous at lower altitude (cor-responding to higher ν) in comparison with higher al-titudes (corresponding to lower ν) because ns/ne ratioscales as r−3/2. When this scaling reaches a ratio belowthe required rate (24) the radio wave emission cannotoccur as the density of the non-thermal electrons is notsufficient for the plasma instability to develop [64]. Fur-thermore, the effective mean free path determined by (26)essentially determines the highest altitudes where non-thermal electrons may reach, see Fig.3. After this height

12

0100020003000Height [km]

0.0

0.2

0.4

0.6

0.8

1.0

Mas

sFr

actio

n

2000

km

≈ 0.88

2150

km

Group 1Group 2Group 3

2150 20000.8

0.9

1.0

050010001500200025003000Height [km]

0.0

0.5

1.0

1.5

2.0

dL B/dH

[erg·s−

1·k

m−

1 ]

×1023

2000

km

2150

km

Group 1Group 2Group 3

FIG. 5. Left: Mass fraction 1−∆B/B being annihilated as a function of the altitude. This is plotted by taking the average ofthe mass loss profiles of the AQNs above B (i.e. the AQNs that will generate radio emissions) where B has been determinedby (11). We see that the AQNs start to annihilate at about 2150 km. Right: Luminosity per unit length as a function of thealtitude where the energy is converted from the mass loss according to (6). This is plotted also by taking the average of theAQNs above B, then multiplied by the impact rate of these large AQNs.

the non-thermal electrons will thermalize and cannot bethe source of the radio waves.

Secondly, according to (28) the lower the altitude, thehigher the annihilation rate. This is because the portionof the annihilated baryon charge ∆B/B drastically in-creases when altitude decreases, see Fig. 5. When thefrequency of the radio emission becomes too high, the ra-diation becomes a subject of absorption too strong to bedetectable above the quiet Sun background. Such sup-pression with higher frequency radiation has indeed beenobserved for frequencies ν >∼ 240 MHz, see [68].

The same line of arguments may also explain the ob-served huge difference between the number of observedevents (4748) at smallest frequency band (98 MHz) incomparison to the rate at larger frequency bands wherethe recorded number of events is almost one order of mag-nitude higher [1]. These arguments suggest that count-ing rate at even lower frequencies (such 80 and 89 MHzbands recorded by MWA) should be even lower than 4748events recorded at 98 MHz [1].

V. THE AQN MODEL. WAIT TIMEDISTRIBUTION.

The goal here is to understand the wait time distri-bution reported by [1]. The main observation was thatthe impulsive events are non- Poissonian in nature. Thisnon- Poissonian feature is shown on Fig 7 of [1] wherethe occurrence probability at small wait times (below10 seconds) is linearly growing instead of approachinga constant, which is what is expected for a Poissoniandistribution.

We shall argue below that, in the AQN model, such abehaviour could be explained by the presence of “effec-tive” clustering of events when one and the same AQN inflight may generate a cascade of seemingly independent

events on short time scales. These events however, arenot truly independent, as they result, in fact, from oneand the same AQN when the typical mass loss time ismeasured in 10-20 seconds, see Fig. 6 in [3]. Few shortradio pulses on scales of few seconds could be easily gen-erated during this long flight time. Such “clustering” willviolate the assumption of the Poissonian distribution ofindependent events.

In what follows we develop an approach which can in-corporate such “clustering” at small time scales, whilethe distribution remains Poissonian at larger time scales,i.e. the time scale of distinct AQNs entering theCorona. The corresponding approach is known as a non-stationary Poissonian process which results in Bayesianstatistics, which is the topic of the next subsection.

A. Non- Poissonian processes. Overview.

We start with an overview of the non- Poissonian pro-cesses and outline the conventional technique to describethem, as given in [69–71]. In case of a conventional ran-dom stationary Poissonian process, the waiting time dis-tribution P (∆t) is expressed as an exponential distribu-tion:

P (∆t) = λe−λ∆t,

∫P (∆t)d∆t = 1, (32)

where λ in this section is mean event occurrence rate.For a constant λ, this distribution describes a stationaryPoissonian process. When λ(t) depends on time, one cangeneralize (32) and introduce the probability function ofwaiting times which becomes itself a function of time [69]:

P (t,∆t) = λ(t+ ∆t) exp

[−∫ t+∆t

t

λ(t′)dt′

]. (33)

13

If observations of a non-stationary Poisson process aremade during a time interval [0, T ], then the distributionof waiting times P (∆t) will be, weighted by the numberof events λ(t)dt in each time interval (t, t+ dt), given by:

P (∆t) =1

N

∫ T

0

λ(t)P (t,∆t)dt, N =

∫ T

0

λ(t)dt. (34)

If λ varies adiabatically one can subdivide non-stationaryPoisson processes into piecewise stationary Poisson pro-cesses (Bayesian blocks), take the continuum limit andrepresent the distribution of waiting times as follows [69–71]:

P (∆t) =

∫ T0λ2(t)e−[λ(t)∆t]dt∫ T

0λ(t)dt

. (35)

One can check that the expression (35) reduces to itsoriginal Poissonian expression (32) when λ is time inde-pendent.

It is convenient to introduce f(λ) which describes theadiabatic changes of λ as follows:

f(λ) ≡ 1

T

dt(λ)

dλ, f(λ)dλ =

dt

T,

∫dλf(λ) = 1. (36)

In terms of f(λ) the distribution of waiting times (35)assumes the form

P (∆t) =

∫∞0λ2f(λ)e−[λ∆t]dλ∫∞

0λf(λ)dλ

. (37)

The stationary Poissonian distribution corresponds tof(λ) = δ(λ − λ0) such that the distribution of waitingtimes (37) reduces to the original expression (32) withconstant λ0 as it should.

B. AQN induced clustering events

We are now in position to describe the physics of “ef-fective” clustering events using non-stationary Poissondistribution framework (37) as outlined above. As pre-viously mentioned several, short radio pulses on scalesof few seconds could be easily generated during a singleAQN “relativly” long flight time of the order of 10-20seconds (see Fig. 6 in [3]).

With this picture in mind, we introduce the followingλ(t) dependence to describe non-stationary Poisson pro-cesses. At long time scales t > t0 we keep the constantλ0 corresponding to the stationary Poisson distribution:

λ = λ0 f(λ) ∼ δ(λ− λ0), for t > t0, (38)

while for shorter time scales t < t0 we parameterize f(λ)as follows:

f(λ) = cλβ , λ = λ0

[t

t0

] 1β+1

for t < t0, (39)

where β, λ0 and t0 parameters should be fitted to matchthe observational signal distribution. The parameter-ization for non-stationary Poisson processes (39) is ageneric power law behaviour which satisfies the condi-tion λ(t → 0) → 0 when t → 0. It has been used pre-viously [69–71] for many different systems, including thesolar flares11. In comparison with previous studies weconsider the superposition of two terms (38) and (39)which allows us to quantitatively characterize (by takingan appropriate limit) the level of non-stationary Poissonprocesses and the extend of deviation from the station-ary Poisson distribution. As we shall argue below, thenon-stationary Poisson processes play the dominant rolein our studies, which is the main claim of the presentsection.

We start by explaining the physical meaning of the pa-rameters entering (38) and (39). As we discuss below thet0 will enter the observables in form of the dimensionlessparameter (t0/T ). The physical meaning of this parame-ter (t0/T ) is clear: it determines the time-portion of theclustering events. In case when (t0/T ) 1 the clusteringevents play a very minor role, while for (t0/T ) ∼ 1 theclustering events become essential. In the limit t0/T → 0the physical mean value 〈λ〉 approaches its unperturbedmagnitude λ0 corresponding to the stationary Poissondistribution. However, in case when (t0/T ) ∼ 1 (whichwill be the case as we discuss below) the dimensionlessparameter (〈λ〉/λ0) must be smaller than one as it ac-counts for non-stationary Poisson processes. The param-eter (〈λ〉/λ0) → 1 approaches identity if non-stationaryPoisson processes play the minor role. The deviation ofthis parameter from (〈λ〉/λ0) 6= 1 is a precise quantitativecharacteristic of the non-stationary Poisson processes inthe dynamics of the system.

From the basic features of the AQN model one shouldexpect (t0/T ) to be large, of order one. This is becausea single AQN event could produce a number of radioemission events which should correspond to the cluster-ing events, since they are not independent. Furthermore,we also expect that (〈λ〉/λ0) strongly deviates from theidentity, which represents a quantitative characteristic ofa contribution due to the clustering events as the Poisso-nian distribution is characterized by a single parameterλ0 with 〈λ〉 = λ0.

With this preliminary remarks on physical meaning ofthe parameters we can now proceed with computationswith the main goal to analyze the role of non-stationaryPoisson processes in the radio wave emission as a resultof the AQN annihilation events.

One can combine equations (38) and (39) to represent

11 In particular, in [71] a more general expression for f(λ) =cλβ exp(−γλ) was considered which also includes the exponen-tial tail exp(−γλ). We do not include this exponential factor asit simply shifts the definition for ∆t → (∆t + γ) as one can seefrom eq. (37).

14

f(λ) as follows:

f(λ) =

[(T − t0T

)δ(λ− λ0)

](40)

+

[β + 1

λ0

t0T

(λ

λ0

)βθ(λ0 − λ)

],

where factor (T − t0)/T is inserted in front of delta func-tion δ(λ− λ0) to preserve the normalization (36).

One should emphasize that the λ0 is not the meanevent occurrence rate 〈λ〉 anymore. Instead, the propervalue for 〈λ〉 reads:

〈λ〉 ≡∫λf(λ)dλ = λ0

[1− 1

β + 2

(t0T

)]. (41)

Now we are in position to compute P (∆t) as definedby (37):

P (∆t) =1

〈λ〉

∫ ∞0

λ2f(λ)e−[λ∆t]dλ, (42)

with f(λ) as given by (40). The result can be representedas follows:

P (∆t) =λ2

0

〈λ〉e−[λ0∆t] ·

(T − t0T

)(43)

+(β + 1)λ2

0

〈λ〉 ·(t0T

)[∫ λ0

0

dλ

λ0

(λ

λ0

)β+2

e−[λ∆t]

],

where the first term describes the stationary Poisson dis-tribution while the second term describes the deviationfrom Poisson distribution at small time scales. The sec-ond term in distribution (43) can be expressed in terms ofthe lower incomplete γ(s, x) function defined as follows:

γ(s, x) ≡∫ x

0

us−1e−udu, γ(s, x) = Γ(s)− Γ(s, x),(44)

where Γ(s) is the gamma function and Γ(s, x) is the upperincomplete gamma function. We identify the parametersfrom the integrand entering (43) as follows:

u = λ∆t, x ≡ λ0∆t, s = β + 3 (45)

to arrive to the following expression for P (∆t) in termsof the lower incomplete γ(s, x) function:

P (∆t) =λ2

0

〈λ〉e−[λ0∆t] ·

(T − t0T

)(46)

+λ2

0(β + 1)

〈λ〉 ·(t0T

)·(

1

λ0∆t

)β+3

· γ [β + 3, λ0∆t] .

This expression is correct for any value of t0/T . However,it is very instructive to see explicit dependence on ∆twhen t0/T 1 is small, and the Poisson distribution isrestored.

With this purpose in mind we simplify expression(46)by expanding the incomplete gamma function enter-ing (46). Therefore, the expression (46) can be simplifiedas follows:

P (∆t) ≈ λ20

〈λ〉e−[λ0∆t]

(T − t0T

)(47)

+λ2

0(β + 1)Γ(β + 3)

〈λ〉 ·(t0T

)·(

1

λ0∆t

)β+3

,

where we use the identity (44) and ignored the exponen-tially small contribution coming from incomplete uppergamma function:

Γ(s, x→∞)→ xs−1 exp(−x). (48)

In the limit (t0/T )→ 0 we recover the conventional Pois-son distribution, while (t0/T ) 6= 0 describes the deviationfrom Poisson statistics in this simplified setting.

We are now ready to analyze the non-Poisson distribu-tion given by (47). Important point here is that this dis-tribution is a superposition of two parts: The first termdescribes the Poisson distribution with small correctionin normalization. Most important part for us is the sec-ond term which is parametrically small at (t0/T ) 1.However, it could become the dominant part of the dis-tribution P (∆t) at small ∆t → 0 due to a high power(∆t)−(β+3) in the denominator (47).

It is interesting to note that [1] noticed that their datacan be fitted as a superposition of two terms which haveprecisely the form of two terms entering (47). How-ever, [1] also fitted the observed signal to an expressionwhich represents the product of two terms rather thanin form of sum of two terms entering the eq. (47) withwell-defined physical meaning of the relevant parameterssuch as (t0/T ). In next subsection we fit that data from[1] using exact (43) and simplified (47) expressions forP (∆t). Our main conclusion of this fit is that the clus-tering events play the dominant role in the distributionP (∆t).

C. Wait time distribution. Theory confronts theobservations.

We are now in position to compare the occurrenceprobability presented on Fig. 7 in [1] with our theoreticalformula (43) which deviates from Poisson distribution asit includes the clustering events.

First, we have to comment that the occurrence prob-ability plotted on Fig. 7 in [1] is different from the waittime distribution P (∆t) defined in the previous subsec-tion. It is convenient to explain the difference using thedescription in terms of the discrete bins [∆ti,∆ti+1]. Inthese terms, Fig. 7 of [1] is a histogram, where the bluepoints represent the values ni/N where ni is the numberof events with wait time located in the bin [∆ti,∆ti+1]and N is the total number of events. However, by defi-nition, the wait time distribution P (∆ti) is obtained by

15

1 5 10 50 100 500

10-6

10-5

10-4

0.001

0.010

0.100

1

wait time Δt [s]

occurrenceprobability

FIG. 6. The blue points are extracted from Figure 7 in [1](132 MHz). Dividing the blue points by the correspondingbin width, we get the red points (i.e., the values of P (∆ti) in(49)). The red line is fitted by (50) with A = 0.56s−1, n '1.5, λ ' 0.0049s−1.

dividing ni/N by the bin width [∆ti+1 −∆ti] for propernormalization of P (∆ti). Indeed,

P (∆ti) ≡niN

1

[∆ti+1 −∆ti], (49)∑

i

P (∆ti)[∆ti+1 −∆ti] =∑i

niN

= 1.

As noticed by [1], the data can be nicely fitted usingthe following function

P (∆t) = A(∆t)−n exp(−λ∆t), (50)

where the continuum limit is already assumed. We con-firm that the good match can indeed be achieved, andthe corresponding fit is shown by the red line on Fig. 6.

We are now ready to interpret the results obtainedabove in terms of the two dimensionless parameters(t0/T ) and (〈λ〉/λ0) introduced in (39) as the genericway to parameterize non-stationary Poisson processes.First of all, the acceptable fit shown on Fig 7 alwaysproduces the relatively large value for (t0/T ). Indeed,the first solution corresponds to (t0/T ) ' 0.95, whilethe second solution assumes the value (t0/T ) ' 0.71.We remind that this parameter (t0/T ) describes the por-tion of time when clustering events occur. In case ofstationary Poisson processes, (t0/T ) = 0. Fit in bothcases suggests that non-stationary Poisson processes oc-cur for most of the time, which unambiguously impliesthat non-stationary Poisson processes play the dominantrole in radio wave emission. This is consistent with theAQN proposed mechanism when the non-stationary Pois-son distribution is expected and anticipated.

Another quantitative characteristic which describesthe deviation from conventional Poisson distribution isthe dimensionless parameter (〈λ〉/λ0). The acceptablefit shown on Fig 7 always produces a strong devia-tion of the parameter (〈λ〉/λ0) from identity. Indeed,

1 5 10 50 100 500

10-6

10-4

0.01

1

wait time Δt [s]

P(Δt)[s

-1]

FIG. 7. The red points are the same as those in Fig. 6 (i.e.,the values of P (∆ti) in (49)). The solid line are fitted bythe full expression of P (∆t) given by (43). The solid red linegives β = −0.9, t0 = 4000 s, λ0 = 0.5 s−1. Other choicesaround this group of parameters can also give similar result.For example, the solid black line corresponds to β = −0.6,t0 = 3000 s, λ0 = 0.2 s−1. In comparison, the dashed linesare the simplified P (∆t) given by (47), with the same groupof parameters chosen correspondingly.

(〈λ〉/λ0) ≈ 0.14 for the first solution while (〈λ〉/λ0) ≈ 0.5for the second solution. This represents another strongevidence supporting our claim that non-stationary Pois-son processes play the dominant role in radio wave emis-sion.

One should note that the quantitative estimates ofthe parameters from the first principles within the AQNframework are hard to carry out. We could only antici-pate a large deviation from conventional Poisson distribu-tion as explained above, while any quantitative estimatesof these parameters are not feasible at the moment. Theproblem is that the radio emission by non-thermal elec-trons is a random process, which strongly depends onsurrounding plasma features. Furthermore, the emissionspectrum of the non-thermal electrons also represents achallenging theoretical problem, as emission occurs inthe system which is moving with very large Mach num-ber when the turbulence, shock waves and other non-equilibrium processes dominate the dynamics of the non-thermal electron’s emission.

Our first quantitative prediction is that the parameters(t0/T ) and (〈λ〉/λ0) must be very similar for differentfrequency bands. Our second quantitative prediction, isthat the emission between radio events observed at dif-ferent frequencies must be correlated with time delaysmeasured in seconds. This correlation is very specific tothe AQN mechanism (see item 4 in Conclusion).

It is interesting to note that the data from [1] canbe nicely fitted using the function (50), which exhibitsstructures similar to our formula (47). The importantdifference here is that our formula was derived with welldefined parameters (t0/T ) and (〈λ〉/λ0) , which quantita-tively characterize the non-stationary Poisson processes,

16

while the extraction of A, λ, n from the fitting (50) rep-resented on Fig. 6 does not allow to arrive to any quan-titative conclusion.

As clustering events play a major role, one may wonderif our estimate of B in section IV B may be modified asa result of these events. We think that the correspond-ing variation is numerically mild, and does not modifythe picture advocated in this work12. Therefore, we ig-nore the corresponding modifications in B in the presentstudy.

We have discussed at length that the presence of clus-tering events is a generic feature of the mechanism forimpulsive radio events. We interpret the fit shown onFig. 7 of the data from [1] with our expression (43) as anadditional strong support for our proposal when radioemissions always accompany nanoflare events.

One should emphasize that nanoflares are introducedas generic events, producing an impulsive energy releaseat small scale ( see the review papers [8, 9]). The factthat nanoflares are the consequence of AQN annihilationevents accompanied by the clustering of radio events isa highly nontrivial consistency check of the entire frame-work . Such clustering events supported by data [1] areclearly related to a non-Poissonian character of distribu-tion, and the AQN model provides a natural solution forthis feature.

VI. CONCLUSION AND FUTUREDEVELOPMENT

We proposed that AQN annihilation events can beidentified with nanoflares and showed that they are in-evitably accompanied by radio events. This proposal isconsistent with all observations reported by [1], includingthe frequency of appearance, the temporal and spatialdistributions, their intensity, and other related observ-ables. There are several direct consequences of this pro-posal, which future observations will be able to supportor refute:1. The proposed mechanism suggests that a consider-able portion of radio events, recorded at different fre-quencies, might be emitted by a single AQN continuouslygenerating radio signals, as a result of different plasmafrequencies at different altitudes. This picture suggeststhat there must be a spatial correlation between radioevents in a given local patch (with size ∼ 105km ), in the

12 Indeed, even if each AQN event generates a cluster consisting onaverage, let us say, three radio events, it would change the eventrate (11) by the same factor three. We note, that much largernumber of events within the same cluster would be inconsistentwith total energy estimate (28) which agrees with observations(31). The scaling parameter α ' 2.5 defined by (7) implies thatthe corresponding variation in B does not exceed a factor 2.5

√3 ≈

1.5. These changes are much smaller than the difference in Bbetween distinct acceptable models (7) , as one can see from Fig.1.

different frequency bands, with time delays measured inseconds.

Observations of correlated clustering events, as dis-cussed in subsections V B, V C , are the direct mani-festation of correlations observed in the same frequencyband. We advocate the idea that similar spatial corre-lations, from different frequency bands, must also exist.This prediction can be directly tested by MWA. Theremust also be similar temporal correlations (see item 4below).

2. Lower frequency waves could be emitted from higheraltitudes. The important point here being that the de-pendence of the intensity of the emission on the altitudeis a highly nontrivial function, for several reasons. First,the upward moving non-thermal electrons are much morenumerous at lower altitude (corresponding to higher ν) incomparison to higher altitudes (corresponding to lower ν)because the ns/ne ratio scales as r−3/2. When this scal-ing reaches a ratio below the required rate (24) , the radiowave emission cannot occur, as the density of the non-thermal electrons is not sufficient for the plasma instabil-ity to develop [64]. Furthermore, the effective mean freepath determined by (26) defines the highest altitudes thenon-thermal electrons may reach. Above this altitude,the non-thermal electrons will thermalize and cannot bea source of the radio waves.

As a result of these suppression factors, we expect thatthe low frequency emissions should be, in general, sup-pressed. Of course, the radio emission is related to ran-dom processes, and highly sensitive to some specific localfeatures of the plasma and non-thermal electrons, as dis-cussed in subsection IV A. Therefore, our prediction onsuppression is the subject of possible fluctuations withinsmall frequency bands. This tendency has been indeedobserved for the 98 MHz band, where the recorded num-ber of resolved events is at least one order of magnitudesmaller than for the three other higher frequencies bands.We predict that the emission rate at 80 MHz and 89 MHz,which have been recorded, but are not yet published by[1], should demonstrate even lower rate of resolved events(even in comparison with 98 MHz emission). This is ahighly nontrivial prediction of our proposal, as it is diffi-cult to understand this behavior using alternative mod-els, since the electron density in the corona is a verysmooth function in this region (see Fig. 4). This predic-tion can be directly tested by MWA, as the observationsaccording to [1] were done in 12 frequency bands, includ-ing the low frequencies of the 80 and 89 MHz bands.

3. In contrast with the low frequency bands, the eventrate for higher frequency bands should be higher thanthe rate recorded for the 160 MHz band . This predic-tion can be directly tested in future analysis by studyingemissions with ν >∼ 160 MHz since, according to [1] , someof their observations were done in the 179, 196, 217 and240 MHz bands. One should comment here that at higherfrequencies ( ν >∼ 240 MHz) , radio emission is subject toa strong absorption, and that the observed intensity willexperience suppression [68], limiting our perspectives to

17

study higher frequency emissions.4. The proposed AQN mechanism of radio emission

predicts the presence of correlations between the emis-sions at different frequency bands. These correlationsemerge due to the upward motion of the non-thermalelectrons, with typical velocities v⊥ ' 104km/s accord-ing to (20). The delays in arrival time at different heightsis measured in seconds, when heights vary on the scaleof 104 km, according to Fig. 4. As a result of upwardmotion, the low frequency emissions should be delayedin comparison to the high frequency emissions. Observ-ing these correlated radio emissions in different frequencybands would unambiguously support our proposal, as itis very hard to imagine how such correlations could occurin any alternative scenarios.

5. Solar Orbiter recently observed so-called “camp-fires” in the extreme UV frequency bands. It is temptingto identify such events with the annihilation of large sizedAQNs, as they are capable of generating radio signals suf-ficiently strong to be resolved. We therefore suggest tosearch for a cross correlation between MWA radio sig-nals and recordings of the extreme UV photons by SolarOrbiter.

ACKNOWLEDGEMENTS

This research was supported in part by the NaturalSciences and Engineering Research Council of Canada.

Appendix A: Simulations

The appendix shows the details of the MC simulationimplemented in this work.

1. The simulation setup

The setup of the simulation in the present work followsthat in [3], which can be divided into three steps. Thefirst step is to use the MC method to generate a largenumber of dark matter particles in the solar neighbor-hood and collect the ones that will eventually impact theSun. The second step is to assign AQNs masses to theparticles. We will use different models of the AQN massdistributions (as shown in (7)). The third step is to solvethe multiple differential equations that dominate the an-nihilation process of AQNs in the solar atmosphere.

Step 1. In this step, we simulate the positions andvelocities of dark matter particles in the solar neighbor-hood. The velocity distribution of the dark matter par-ticles, with respect to the solar system frame, follows aMaxwellian distribution:

f~v(~v)d3~v =d3~v

(2πσ2)3/2exp

[−v

2x + v2

y + (vz − v)2

2σ2

](A1)

where the velocity dispersion is σ ' 110 km/s, and thevelocity shift v ' 220 km/s is due to the relative motionbetween the Sun and the dark matter halo.

The positions of particles are generated in such a waythat they initially uniformly populate in a spherical shellaround the Sun. The inner and outer boundaries ofthe spherical shell are respectively Rmin = 1 AU andRmax = 10 AU. Note that our choice of Rmin is differentfrom Ref. [3] where Rmin = R there. Choosing a largerRmin is to reduce the effect of the Sun’s gravity on theinitial velocity distribution (A1). The solar escape ve-locity at 1 AU is ve ≈ 42 km/s, so when a particle movesfrom infinity with the typical velocity v0 = 220 km/sto this distance, the velocity increment due to the Sun’sgravity is ∆v =

√v2

0 + v2e − v0 ≈ 4 km/s which is very

small compared with v0. Similar to Ref. [3], we gener-ated Nsample = 2×1010 such particles. The particles thenmove following Newton’s gravity, attracted by the Sun,using the classical two-body orbit dynamics. The crite-ria to determine whether or not a particle will impactthe Sun are also the same as in [3]. For a given particle,if the perihelion of the hyperbolic trajectory is smallerthan R (and also if the velocity direction is inward),then it will impact the Sun. It turns out that from theinitial sample of 2 × 1010, the number of particles thatwill impact the Sun is Nimp = 30457.13 The trajectoryand impact properties of these impacting particles areshown in Fig. 8.

The expression of the impact parameter b is

b = rp

√1 +

2GMrpv2

0

(A2)

where rp is the perihelion distance. v0 is the parti-cle velocity at infinity that can be extrapolated fromthe initial velocity and position simulated, i.e. v0 =√v2i − 2GM/ri. The impacting requires 0 ≤ rp ≤ R.

If we take rp = R, we get the maximum impact param-eter bmax. The distribution of the impact parameter (inthe form of b/bmax) is shown in the subplot (b) of Fig. 8.

From the distribution of impact time as shown in thesubplot (c) of Fig. 8, we can calculate the impact rate.Following the logic in Ref. [3], we should only use thetime window where the rate is constant. We choose itas timp ∈ [0.5, 1.5] months where the boundaries are de-noted as two vertical lines in the plot. The rate in thetime window is constant because the dominant part ofparticles impacting the Sun are the particles from the

13 In comparison, the number obtained in Ref. [3] is 36123. Thedifference is beyond the statistical fluctuation. This differenceoccurs not only due to our choice of a larger Rmin, the innerboundary of the initially simulated region, but also a technicaldetail that a different method (more appropriate) is chosen indetermining the perihelion. However, all of these changes haveno significant effects on the results as we can see in Fig. 8 bycomparing it with Ref. [3].

18

2 4 6 8 10Distance [AU]

0.000

0.025

0.050

0.075

0.100

0.125

Prob

abili

tyD

ensi

ty

0.0 0.2 0.4 0.6 0.8 1.0Impact Parameter b/bmax

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5Time to Impact [months]

0.0

0.1

0.2

0.3

Prob

abili

tyD

ensi

ty

650 700 750 800 850Impact Velocity [km/s]

0.000

0.005

0.010

0.015

0.020

0.025

FIG. 8. Probability density distributions of the trajectory and impact properties for the Nimp = 30457 impacting particles.Starting from the top-left panel to the bottom-right, the plots represent (a): the initial distance distribution of these impactingparticles. (b): the impact parameter distribution. (c): the velocity distribution when they impact the Sun. (d): the impacttime distribution.

the spherical shell between Rmin and Rmax. Outside thetime window, we see the rate drops because we did notsimulate the particles outside the spherical shell. Theimpact rate is N(∆timp)/∆timp where N(∆timp) is thenumber of particles impacting the Sun in the time win-dow chosen above. Note that this impact rate is not thetrue impact rate because the number of AQNs simulated,Nsample = 2×1010, is not the true number of AQNs insidethe spherical shell.

To convert the impact rate to the true impact rate, weneed to multiply it by the scaling factor fS which is theratio of the true number of AQNs in the spherical shellto Nsample:

fS =43π(R2

max −R2min) · nAQN

Nsample(A3)

where nAQN is the number density of antimatter AQNsin the solar system:

nAQN =

(2

3· 3

5· ρDM

)· 1

mP 〈B〉. (A4)

and ρDM ' 0.3 GeV cm−3 is the dark matter density inthe solar system. 3/5 of the dark matter is in the form ofantimatter AQN; 2/3 mass of an AQN is in the form ofbaryons (the remaining 1/3 is in the form of axions). mP

is the proton mass. 〈B〉 is the average baryon numbercarried by an AQN. It depends on different models of

AQN mass distribution that will be discussed in Step 2.Thus, the true rate of (antimatter) AQNs impacting thesun is

dN

dt=N(∆timp)

∆timp· fS , timp ∈ [0.5, 1.5] months. (A5)

Step 2. We are now assigning masses (baryon num-bers) to all the AQNs collected in Step 1 that will impactthe Sun. For each AQN, its mass is assigned randomlywith the probability following one of the three models ofpower-law distribution, (7). Thus, we have three copiesof all the impacting AQNs with different mass distribu-tions.

Step 3. The evolution of an AQN in the solar at-mosphere is described by a system of differential equa-tions, including the kinetic energy loss due to frictionand the mass loss due to the annihilation events of theantibaryons carried by AQN with the baryons in the at-mosphere. We refer the reader to [3] for a complete list ofthe differential equations needed here, and their deriva-tion. In order to solve these equations numerically, thedensity and temperature profiles of the solar atmosphereabove the photosphere is also needed. In this work, weadopt the profiles presented in Ref. [66] which are moreaccurate than those used in [3].

The mass loss varying with time (or equivalently,height above the solar photosphere) for the Nimp AQNsis then computed numerically.

19

2. Results

The results obtained from the numerical simulationsare presented in the main text. Additional details aregiven here.

Fig. 1 shows the rate of AQNs in the mass range[B, Bmax] impacting the Sun. The rate is calculated as(A5) but with only large AQNs B ≥ B taken into ac-count. By varying the value of the cutoff B from Bmin

to Bmax, we quantify how the impact rate depends onB, as shown in the figure. The rate at B = Bmin is thetotal impact rate. For the three groups, the total impactrates are respectively 4.17× 104 s−1, 1.52× 104 s−1 and3.52 × 103 s−1 which match well Ref. [3] (see Figure 8there).

In addition, the luminosity L can be calculated asL = 2 〈∆m〉 c2 ·dN/dt where ∆m is an AQN’s mass lossalong its trajectory through the solar atmosphere beforeentering the dense region, the photosphere. Similarly,we can compute the luminosity L

Bby counting large

AQNs (B ≥ B) only, and the result is shown in Fig. 2for different groups of mass distribution. The total lumi-nosity is obtained at B = Bmin. For the three groups,the total luminosity are respectively 1.05×1027 erg · s−1,1.07×1027 erg · s−1 and 1.06×1027 erg · s−1 which matchwell Ref. [3] (see Figure 10 there).

One may notice that in the two left subfigure of Fig. 1and Fig. 2, the simulated lines become zigzag at largebaryon numbers. This is because the proportion of largeAQNs is actually very small. Despite the number of allthe impacting AQNs is as large as 30457, the power-law

index α ∼ (2− 2.5) makes the hit rate with large B verytiny when assigning masses to AQNs randomly in Step 2.For example, our statistical result shows that in Group1, the number of AQNs with B ≥ 5×1026 is only 12, andthe the number of AQNs with B ≥ 1027 is only 3. Suchtiny number causes large statistical fluctuation, so we seethe zigzags in the two left subfigures. We have to gen-erate enough large AQNs to remove the large statisticalfluctuation.

We resolve this technical problems as follow. We simu-late another 1010 AQNs by redoing the three steps in thesetup as described above. We call this procedure as thesecond-round simulation. We get 15019 AQNs that willfinally impact the Sun out of the total 1010 AQNs. Themasses (baryon numbers) assigned to these 15019 impact-ing AQNs are constrained in the range B ∈ [BL, Bmax].BL for each group should be chosen well above Bmin toensure that enough large AQNs can be generated, butBL should not exceed the start of the zigzags. Althoughwe did not simulate all AQNs in this second-round sim-ulation, we can extrapolate the “number” of impactingAQNs in the full mass range by looking at the proportionof large AQNs (B ∈ [BL, Bmax]) in the full mass range 14.Furthermore, we can calculate the extrapolated Nsample

and the extrapolated scaling factor fS . Finally, we ob-tain the true impact rate of these large AQNs simulatedin the second-round simulation. Similarly, we obtain theluminosity. The results are shown in the two right sub-figures of Fig. 1 and Fig. 2, where we see that the largestatistical fluctuation disappears.

[1] S. Mondal, D. Oberoi, and A. Mohan, The AstrophysicalJournal 895, L39 (2020).

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