aa r X i v : . [ phy s i c s . a o - ph ] S e p Thunder-cell as Source of Energetic Protons
Aleˇs Berkopec ∗ September 29, 2020
Abstract
In this article we present the following hypothesis:thunder-cell ejects highly energetic protons, eachof which creates a tree-structure of weakly ionizedtrajectories that can develop into a lightning chan-nel. The tree-structure and the channel have thesame geometry so the mean free path of a protoncorresponds to the average length of the channelbetween two successive nodes (branching points).We show this length is around 660 m in lower Earthatmosphere. Effects of Coulomb interaction andvarious outcomes of proton-nucleus reaction aretaken into account. A prediction of CG/CC ra-tio that follows agrees well with the available data,but only measurements of lightning geometry canreveal whether the hypothesis is any closer to thecorrect explanation of the phenomenon.
Keywords : lightning initial phase, CG and CClightning
The reported values of potential gradient beforeand during a lightning strike are at least one or-der of magnitude smaller compared to the ones re-quired to induce sparks under controlled conditionsin laboratories [1, 2, 3]. We tried to explain thisdifference with theoretical prediction [4] involvingfast charged particles that precede stepped leader.This idea was later encouraged by a report aboutweak correlation between lightning frequency andsolar wind intensity [5]. Since every CG (cloud-to-ground) and CC (intra-cloud, inter-cloud or cloud-to-air) lightning channel starts in a thunder-cellthe intense freezing of super-cooled water insidethunder-cell was proposed as a process that mightlead to ejection of charged projectiles [6].In view of the hypothesis, trajectory of the pri-mary projectile and its collision products, all elec-trically charged, determine the geometry of thestepped leader and subsequent lightning channel.Interaction of the projectile with electrons in airslows the projectile down and ionizes the trajec-tory, while collisions with the nuclei may producehigher-order projectiles and result in branching ofthe channel. At the end of the process, weakly ∗ contact: [email protected] ionized tree structure forms a conducting path be-tween cloud and ground (CG lightning, Fig. 1), orcloud and a point in the air (CC lightning). ground thunder-cell l trajectory ofprimary projectile(main channel) trajectory ofsecondaryprojectilecollision withtarget (node) Figure 1:
Thunder-cell ejects a projectile. Whyprotons are best candidates for projectiles is ex-plained in Section 2, for the estimation of lengthbetween two successive collisions l see Sections 3and 5, and for a role of Coulomb interaction seeSection 4. The projectile and the ionized tree-structure itleaves behind provide explanation for at least threeaspects of CG and CC lightnings: a) lower valuesof potential gradient sufficient to initiate lightning,b) tree-structure of a lightning channel with looselydefined direction, and c) dependence of CG/CCratio on height of the cloud-base and consequentlyon latitude.
We reason the projectile is expected to have thefollowing characteristics:a) electrically charged
Only a charged particle leaves behind an ion-ized trajectory. Coulomb interaction has onlya minor effect on the geometry of the channel,as we show later.b) induces secondary projectiles of the same type
Lightning channel is often split, however, thebranches that grow from the split point areindistinguishable.1) elementary particle
Atomic nuclei of elements heavier than hydro-gen or other composite particles do not fit therole. Their collisions with nuclei in the airwould lead to a spallation at such energies [4],so one could observe lightning channels withalso three or more branches continuing froma single node. Such channels have not beenobserved in CG and CC lightnings.The most suitable candidates for the role areprotons: they are electrically charged, they donot decay, pick-up reactions (p + , + ) (proton-projectile hits a nucleus, passes through, and ejectsadditional, secondary proton-projectile) are com-mon, and they are not composite.Other types of particles, like pions and kaons,are here not considered as projectiles. The reasonsfor this are addressed in Discussion. Imagine first a projectile is an electrically neutralrigid ball ejected from an origin in a random di-rection, traveling in a straight line. The volume ofthe cylinder it sweeps after passing length x equals π r x , where r p is the projectile’s radius. Assuminga target particle is a rigid ball of radius r t placedrandomly in the sphere of radius x , the probabilitythat the projectile collides with the target is ratioof volumes V p /V , where V p = π ( r p + r t ) x . Theprobability that the process passes without colli-sion is therefore q = 1 − V p /V (1)If the sphere contains two targets the probabilityequals (1 − V p /V ) , and in case of N targets theprobability is (1 − V p /V ) N . As N becomes largethe probability for survival equals p ( x ) = lim N →∞ (cid:18) − π ( r p + r t ) xN/n (cid:19) N = exp( − x/λ )where 1 /λ = π ( r p + r t ) n and volume density ofthe target particles is n = N/V . For targets ofdifferent types with radii r i and volume densities n i , we find p ( x ) = Y i lim N i →∞ (cid:18) − π ( r p + r i ) xN i /n i (cid:19) N i == Y i exp( − x/λ i ) = exp( − x/λ ) (2)where 1 /λ i = π ( r p + r i ) n i and 1 /λ = P i (1 /λ i ).The mean free path of a proton in lower Earthatmosphere is thus (2) λ p + = 1 P i λ i = 1 π n P i η i ( r + r i ) . = 660 m (3) Classical values for proton radius r = 0 .
875 fmand radii of the target nuclei r i = r A / i with A i = [14 , ,
40] were used in calculation of λ p + .For the volume density of the nuclei we assumed n = 5 . · / m , and for the rates η N = 78 . η O = 21 . η Ar = 0 . Coulomb interaction is responsible for the loss ofprojectile’s kinetic energy and ionization of its tra-jectory. The rate of change in kinetic energy isexpressed by Bethe equation [7]. The range de-pendence for a proton in lower Earth atmosphere isderived in [6] and shown on Fig. 2 for λ p + . = 660 m.Since the height of a thunder-cell is around 1 kmor higher, graph on Fig. 2 suggests the minimuminitial energy for a proton that reaches the groundis about 1 GeV.0 1 2 3 4 5 60246810 W [GeV] L [ k m ] Figure 2:
Range L for protons in lower Earthatmosphere as a function of their initial energy W . W [GeV] h l i [ m ] Figure 3:
Average length h l i between successivenodes (branching points) as a function of proton’skinetic energy W at the first node. The dashedline represents estimation λ p + . = 660 m from (3). W . Since protons with W < W in [1 GeV .. Collision of p + projectile with a nucleus X dis-cussed so far was assumed to be of the pick-up typep + + X → X − + p + + p + , or (p + , + ), plus ar-bitrary number of neutral particles. According tothe hypothesis, such reactions correspond to theobserved binary-tree geometry of lightning chan-nels. Two types of reactions, swap and capture,produce the results of collisions not accounted for:swap can not be distinguished from a part of abranch that has no split, and capture of a protonby a nucleus looks like the end of a branch.Now we extend the survival probability for aprojectile and one target (1) to reactions that arenot necessarily of pick-up type. Let us presumethe pick-up reaction occurs with probability p pu .Then the channel does not split with probability q in case the target is not hit, or with probability(1 − q ) · (1 − p pu ) it the target is hit but the reactionis not of pick-up type, or: q + (1 − q ) · (1 − p pu ) = 1 − p pu V p V Assuming probability p pu is equal for all targets wefind for N targets (1 − p pu V p /V ) N , and (2) changesinto p ( x ) = exp( − p pu x/λ ). From comparison with(3) we see that for p pu < . λ → λp pu (4) It is well documented that the ratio between thenumber of strikes to the ground (CG) and the num-ber of strikes that do not reach the ground (CC)for a given storm depends on its latitude.This dependence can be explained in view of ourhypothesis. We first presume that the average dis-tance R between the channel’s origin in a thunder-cell and its most distant point is independent oflatitude. Second, we take that the projectiles thatreach the ground induce CG lightnings while thosethat do not, induce CC lightnings.For projectiles ejected in a random direction theCG/CC ratio can be estimated from the ratio of hR CCCGground thunder-cell
Figure 4:
Spherical angle below thunder-cell (atthe top of the shaded region) is proportional to theprobability that a lightning strikes the ground. the areas defined by the intersection of the sphere,whose center is a thunder-cell at height h , and theplane, representing the ground (see Fig. 4). Thearea of the sphere below the plane correlates withincidence of CG lightnings (corresponding spheri-cal angle is shaded), the area above the plane toCC lightnings. The ratio of the areas and corre-sponding spherical angles equals η = R − hR + h Taking the ratios η i and η j at two different lat-itudes, where the heights of the thunder-cells are h i and h j , respectively, one finds the estimation forthe ratio of thunder-cell heights reads h i h j = 1 − η i η i · η j − η j (5)Since a thunder-cell is always located close to thebase of its thundercloud we take that in the firstapproximation the ratios of the heights h i /h j and H i /H j are close enough H i /H j ≈ h i /h j . At Equa-tor at Λ = 0 ◦ experimental data gives η = 0 . H ≈ η ( H ) = a − H/H a + H/H (6)where a = (1 + η ) / (1 − η ) ≈ / η (a) (Λ) = 14 .
16 + 2 .
16 cos(3Λ) (7)while Pierce [10] proposes η (b) (Λ) = (cid:20) . .
25 sin Λ − (cid:21) − (8)Values of CG/CC ratios for three latitudes from(7) and (8) along with our prediction (6) are givenin Table 1.3 H [m] η (a) (Λ) η (b) (Λ) η ( H )0 ◦ .
16 0 .
11 0 . ◦
700 0 .
38 0 .
38 0 . ◦
500 0 .
50 0 .
46 0 . Table 1:
Height H of a thunder-cloud base andCG/CC ratios for three latitudes Λ (see [8]). Val-ues of η (a) and η (b) follow from (7) and (8). Ourprediction (6) is in column η ( H ). h l i Let us make a rough Fermi-type estimate aboutaverage length h l i from lightning photos and expe-rience as observers. We aim at higher confidencelevel and are less concerned with error margin.It is fairly safe to assume that less than 20% ofchannels have no nodes, and that no channel hasmore than 20 nodes. Consequently, the remainingchannels with number of nodes between 1 and 19occur with probability between 80% and 100%. If M is the number of the nodes, the minimum andthe maximum expected values for M are E ( M ) min = 0 · . · . ·
20 = 0 . E ( M ) max = 0 · . · . ·
20 = 19Cloud-base heights range from H min = 500 mto H max = 1200 m [8]. It is impossible to findthe height of the cloud for a given lightning fromits photograph, so we take L min ≈ H because thelength of the main channel can not be shorter thanthe minimum distance between the cloud base andthe ground, while the maximum length is taken tobe five times that, or L max ≈ · H (in such case theaverage direction of the channel is 80 ◦ from verti-cal, and the lightning strikes the ground around4.8 km from the cloud).The lower and upper expected values for the av-erage length between successive nodes are then E ( h l i ) min = L min E ( M ) max +1 = = 25 m E ( h l i ) max = L max E ( M ) min +1 = · . ≈ . + , + ) reactions. For longest expectedvalue h l i ≈ p pu accordingto (4) equals p pu ≈ / ≈ Estimation of h l i above has a wide error margin butthe particles that may be involved in the processdo not come in arbitrary sizes. Classical predictionfrom (3) for projectile of zero size gives λ ≈ λ π . = 700 m for pions and λ K . = 810 mfor kaons. Short-lived pions decay in muons, andmuons decay in positrons or electrons, so these canonly contribute a non-branched part to a channel.One specific decay of kaons K ± → π ± may leadto channel branching but it is less likely to occuras kaons themselves are the rarest secondary prod-ucts among the p + , π ± , and K ± , and since thistype of decay for kaons has only 6% rate. [11]Experimental verification of the hypothesisshould involve measurements of the average lengthbetween successive nodes by reconstruction of 3Dchannel geometry. Correlation between intensity offreezing/precipitation and frequency of lightning isexpected, as well as the ejection of elementary par-ticles from super-cooled water during freezing. References [1] R. Gunn, Electric field intensity inside ofnatural clouds.
Journal of Applied Physics ,19(5):481–484, 1948.[2] T.C. Marshall and W.D. Rust, Electricfield soundings through thunderstorms.
Jour-nal of Geophysical Research: Atmospheres ,96(D12):22297–22306, 1991.[3] W.P. Winn et al., Measurements of electricfields in thunderclouds,
Journal of Geophysi-cal Research , 79(12):1761–1767, 1974.[4] A. Berkopec, Fast particles as initiators ofstepped leaders in CG and IC lightnings,
Journal of Electrostatics , 70(5):462–467, 2012.[5] C.J. Scott et al., Evidence for solar wind mod-ulation of lightning,
Environmental ResearchLetters , 9(5), 2014.[6] A. Berkopec, About Geometry and Ini-tial Phase of Cloud-to-Ground Lightning,arXiv:1602.02496, physics.ao-ph, 2016.[7] H. Bethe and W. Heitler, On the stopping offast particles and on the creation of positiveelectrons.
Proceedings of the Royal Society ofLondon. Series A , 146(856):83–112, 1934.[8] V.A. Rakov and M.A. Uman, Lightning:physics and effects, 2003, Cambridge Univer-sity Press.[9] S.A. Prentice, D. Mackerras, The ratio ofcloud to cloud-ground lightning flashes inthunderstorms,
J. Appl. Meteor. , 545–550,1977.[10] E.T. Pierce, Latitudinal variation of lightningparameters,
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