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Radio emission due to simultaneous effect of runaway breakdown and extensive atmospheric showers
26 August 2002
Physics Letters A 301 (2002) 320–326 www.elsevier.com/locate/pla
Radio emission due to simultaneous effect of runaway breakdown and extensive atmospheric showers A.V. Gurevich a,∗ , L.M. Duncan b , Yu.V. Medvedev a , K.P. Zybin a a P.N. Lebedev Institute of Physics, Russian Academy of Sciences, 117924 Moscow, Russia b Thayer School of Engineering, Dartmouth College, NH, USA
Received 9 June 2002; accepted 18 June 2002 Communicated by V.M. Agranovich
Abstract The theory of radio emission generated during thunderstorm by cosmic ray particles having high energy ε is developed. The emission is shown to have a form of bipolar microsecond radio pulse with characteristic frequency 1–10 MHz. Due to combined action of runaway breakdown and extensive atmospheric shower the emission intensity is high. The radio pulse could be observed for ε 1017 eV up to distances 100–300 km. Thus the runaway breakdown in thunderclouds can play a role of a “spark chamber” for radio detection of high energy particles. 2002 Published by Elsevier Science B.V.
1. Introduction A new physical concept of an avalanche type increase of a number of energetic electrons in gas under the action of the electric field was proposed by Gurevich et al. [1]. The avalanche can grow in electric field E Ec . The field Ec is almost an order of magnitude less than the threshold electric field of conventional breakdown Eth . The growth of number of electrons with energies ε > εc ≈ 0.1–1 MeV is determined by the fact that under the action of electric field E > Ec fast electrons could become runaway, what means that they are accelerated by electric field E as suggested by Wilson [2]. Due to collisions with gas molecules they can generate not only large * Corresponding author.
E-mail address: [email protected] (A.V. Gurevich).
number of slow thermal electrons, but the new fast electrons having energies ε > εc as well. Directly this process—acceleration and collisions lead to the avalanche type growth of the number of runaway and thermal electrons, which was called in [1] “runaway breakdown” RB. The detailed kinetic theory of RB was developed in [3–7]. In atmosphere the critical electric field is Nm (z) . Ec ≈ 200 (kV/m) (1) Nm (0) Here Nm (z) is the neutral molecules density at the height z and Nm (0) = 2.7 × 1019 cm−3 —at see level. Ec falls down with z due to exponential diminishing of Nm . At the thundercloud heights z ≈ 4–6 km, the critical field Ec is 100–150 kV/m and exactly these values of electric field are often observed during thunderstorms [8,9]. When the electric field in thunderstorm cloud reaches the critical value E Ec every
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 9 0 0 - 3
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cosmic ray secondary electron (its energy ε > 1 MeV) initiates a microrunaway breakdown (MRB). It serves as a source of intensive ionization of air and manifests itself in a strong amplification of X- and γ -rays emission and effective growth of conductivity in thundercloud [1,10]. These effects were observed and compared with RB theory [11–14]. Extensive atmospheric shower (EAS) is accompanied by a strong local growth of cosmic ray secondaries number [17]. A theory of combined effect of RB–EAS was developed in [15]. It was shown that ionization of atmosphere by the shower in RB conditions is growing strong enough to produce a local highly conductive plasma and can serve for lightning leader initiation. On the other hand the same effect can stimulate the excitation by thundercloud electric field an intensive local pulse of electric current. This short pulse of electric current can generate radio emission. The goal of the present Letter is to develop a theory of radio emission generated in thundercloud due to RB–EAS interaction. The emission would be shown to have a form of microsecond radio pulses with characteristic frequency around 1–10 MHz. As is well-known radio emission is generated during thunderstorm in a wide frequency range. It has a high power and was studied in a multiple observations (see monographs [9,16] and literature cited there). That is why to single out RB–EAS radio pulses in observations is impossible without knowledge of their special features described by the theory. On the other hand the detection and detailed study of this emission has a significant interest either for understanding of lightning generation mechanism or investigation of the fluxes of high energy cosmic ray particles.
2. Electric current Let us calculate the current pulse generated by simultaneous effect of RB and EAS. According to the theory and observations, the seed electrons in EAS are distributed nonuniformly [17]. The typical scale along the direction of motion of relativistic particle z ≈ 5–10 m and in perpendicular direction R⊥ ≈ 100 m [15,17]. It means that current pulse structure in z direction is mostly significant what allows us to develop the theory in one-dimensional
321
approximation. In RB process high energy electrons effectively generate large number of slow thermal electrons [6,7]. One can estimate that namely thermal electrons define the current pulse under the action of electric field. The equations describing this process have a form: Ne ∂Ne ∂(Ne Ve ) + =− + q, ∂t ∂z τatt Ne ∂N− = , ∂t τatt ∂N+ = q, ∂t ∂E = 4πe(N+ − Ne − N− ). ∂z
(2)
We supposed here that electric field is directed along the vertical z. The first equation of system (2) describes the changes of slow electron density Ne induced by drift of electrons in electric field Ve , their birth q due to RB process and death due to attachment. The second and third equations describe the appearance of negative ions N− resulting from electron attachment and generation of positive ions N+ due to RB. The fourth equation define the appearance of polarization electric field. In Eq. (2) Ve is the drift velocity under the action of electric field Ez = E: Ve = eE/mν,
(3)
where ν is electron collision frequency with air molecules. We suppose here that the inclination angle to the vertical of a high energy particle (generating EAS) is small enough, what allows to consider z-component of electric field only. The life-time due to attachment of slow electrons in the air τatt is determined by threebody collisions [18], and the source of secondary electrons q: q = cN0 δ(z − ct)F (z, E/Ec ).
(4)
Here cN0 (ε, r⊥ ) is a flux of newborn slow electrons generated by EAS in the atmosphere in absence of electric field. The number density of secondary electrons N0 in the plane orthogonal to z axis is approximately proportional to the energy of cosmic ray particle ε and falls down with the distance r⊥ from the center of EAS [17]. According to well-known Nishimura–Ksimata–Greizen formulae simplified for
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thundercloud heights [15]: ε/β N0 (ε, r⊥ ) = n0 √ Σ(r⊥ ), ε/β R r⊥ −3.5 Σ(r⊥ ) = , × 1+ r⊥ R β = 72 MeV,
n0 ≈ 1.2 × 10−9 cm−2 .
(5)
Function F (z, E/Ec ) determines slow electron production and their multiplication under the action of electric field E. It depends strongly on the relation E/Ec . If E Ec the factor F is growing exponentially with z due to direct RB process [10]. In usual conditions in the air one runaway electron generates about 50 thermal electrons along 1 cm of its trajectory, i.e., ionization rate η = 30–50 e/cm [20]. An additional logarithmic factor L comes from the reconstruction of distribution function [6,7]. In analytical calculations the function F would be presented in a simple form: z , F (z, E/Ec ) = ηL exp (6) li where li (E/Ec ) is the RB ionization length [10]. Approximation (6) is useful to obtain the main qualitative characteristics, it is correct for E > Ec . In Section 4 the numerical results would be presented, where any value of relation E/Ec will be considered. According to (4), (6) the source q is proportional to growing exponent eλt , where λ = c/ li is the RB increment [5,6], and t = z/c. It means that the solution of the system (2) could be presented in a form: Ne,+,− = ne,+,− eλt ,
E = E eλt .
(7)
After substitution (5)–(7) in Eq. (2) one can find ∂ne ∂(ne Ve ) + λne + ∂t ∂z ne =− + cNδ(z − ct), N = ηLN0 , τatt ∂n− ne + λn− = , ∂t τatt ∂n+ + λn+ = cNδ(z − ct), ∂t ∂E = 4πe(n+ − ne − n− ). (8) ∂z Introducing a new variable χ = ct −z, and denominate parameters la = cτatt , li = c/λ, we find from (8)
ordinary differential equation describing the density changing: Ve 1 1 d 1+ ne = − ne + Nδ(χ), + dχ c la li dn+ n+ dn− n− ne + + = , = Nδ(χ). (9) dχ li la dχ li We note now that the drift velocity of slow electrons is small: Ve /c 1. Using an expansion on this small parameter, one can present the solution of the system (9) in a form: ne = 2Nθ (χ)e−(1/la +1/li )χ , n+ = 2Nθ (χ)e−χ/li , n− = 2Nθ (χ)e−χ/li 1 − e−χ/ la .
(10) E
in (8) is Estimates show that the polarization field not important. Thus the electric current jz associated with electron density (10) takes a form: jz = eVe ne = eVe Nθ (ct − z)e−(ct −z)/ la .
(11)
In conditions (6): N = ηLN0 (ε, r⊥ ), in general case (4) N = ηLN0 f (z), where f (z) = F (z, E/Ec ) × exp(−z/ li ).
3. Radio emission 3.1. Analytic theory To define the radio emission generated by RB–EAS current we have to calculate first the Fourier transform of (11): 1 jω = 2π
+∞ N e−iωz/c j (t)eiωt dt = eVe . 2π iω + c/ la
(12)
−∞
The vector-potential Aω of radio emission determined by the current (12) has a usual form: 1 jω (r ) iωR/c e dr . Aω = c R Here R = (x − x )2 + (y − y )2 + (z − z)2 . After integration over r⊥ we find in wave region (R0 |r |, c/ω R0 ), where R0 = (x 2 + y 2 + z2 )1/2 is the distance to receiver from current region: Aω =
I eiωR0 /c η(ω). cR0 iω + c/ la
(13)
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Here I = eVe Nm r⊥ dr⊥ is the total current across EAS, Nm is maximal value of electron density and η(ω) is the integral +∞ η(ω) = −∞
N(z) −iωz(1−cos θ)/c e dz. Nm
(14)
According to the theory of runaway breakdown the number of electrons is growing exponentially with z up to some point z0 and falls down exponentially at higher z (see [10] Fig. 3). The main input in integral (14) comes from the region around Nm —maximum of N(z). Let us choose it in a form: N(z) = Nm e−(z−z0 )
2 /L2 1
.
The length L1 is of the order li . The integral takes form: η(ω) =
+∞ 2 2 e−iαz−(z−z0 ) /L1 dz,
−∞
α=
ω (1 − cos θ ), c
R L21 ω2 2 I L1 exp iω c − iαz0 − 4c 2 (1 − cos θ ) Aω = . cR0 iω + c/ la (15) The expression (15) define an analytical formula for the spectrum of radio emission. The wave energy dεω emitted in angle dΩ in frequency range dω is given by the formula [19]: c k 2 |Aω |2 R02 sin2 θ dΩ dω. dεω = (16) (2π)2 After substituting (15) in (16) we find
L21 ω2 2 I 2 L21 ω2 exp − 2c 2 (1 − cos θ ) dεω = (2π)2 c3 ω2 + (c/ la )2 × sin2 θ dΩ dω.
323
J (t) = π 3/2 θ (t − R0 /c)e−(ct −R0 )/ la +c /(la ω1 ) c (t − c/R0 )ω1 − × Φ 2 la ω1 c +Φ , la ω1 2c |1 − cos θ |, ω1 = (18) L1 x 2 where Φ(x) = √2π 0 e−t dt is the probability integral. 2
2
3.2. Numerical calculations We have fulfilled the direct numerical calculations which allow to avoid exponential approximation (6) for RB process. The same scheme as in [10] was used. The electric field E(z) was supposed existing in a layer −L0 z L0 in parabolic form: E(z) = Em 1 − a 2 z2 . (19) Here the distance z was measured in ionization length li and parameter a = li /L0 . The distribution function of energetic electrons f (ε, z) was calculated using the kinetic equations (see [10], formula (3)). The density and current of thermal electrons is determined according to (10), (11). The vector-potential is calculated according to the relations (13), (14): I li A(τ ) = cR0
+∞ −∞
∗
eiω τ η(ω∗ ) iω∗ +
li la |1 − cos θ |
dω∗ ,
η(ω∗ )
where = η(ω)/ li and τ = tc/(li |1 − cos θ |). Spectrum Aω and potential A(τ ) are shown at Fig. 1. The vertical component of electric field Ez (τ ) has a form: (17)
It is interesting to note that qualitatively the expressions (15), (17) correspond well to results obtained by Tamm [21] for the emission of “cut” current (the current generated by a particle moving with velocity c on a segment [−L1 , L1 ]). To determine the shape of electromagnetic pulse A(t, R0 ) it is necessary to find inverse Fourier transformation: I L1 A(t, R0 ) = J (t), cR0
Ez (τ ) I sin2 θ cos θ = cz0 1 − cos θ
+∞ −∞
∗
iω∗ eiω τ η(ω∗ ) iω∗ +
li la |1 − cos θ |
dω∗ ,
z0 cos θ = (20) . R0 Here z0 is the height of emission region. The pulse wave form is shown at Fig. 2. It is bipolar, the width of the pulse became wider with distance and the amplitude falls proportional to R0−1 .
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(a) Fig. 2. The wave form of vertical component of electric field as function of normalized time τ = (ct − Ri )/ li . In atmospheric conditions at the heights 5–6 km: li ≈ 100 m.
Characteristic value of RB–EAS radio pulse amplitude according to formulae (20) and Fig. 1b could be approximated as ε 100 km E ≈ 20 1017 eV R0 Em mV . × exp 15 (21) −1 Ec m Here ε is the energy of cosmic ray particle, R0 is the distance from current region to the receiver (15), (20), Em is the maximal value of electric field (19), Ec RB critical field (1), exponential factor is the result of numerical calculations.
4. Discussion (b)
Fig. 1. (a) Frequency spectrum of vector potential Aω for Em /E0 = 1.05; a = li /L0 = 0.2. Frequency ω normalized on c/ li . Black points—real part Aω ; squares—imaginary part of Aω . (b) Vector-potential A(τ ) at Em /Ec = 1.05; a = 0.2, for different angles θ shown at the figures. τ = (ct − R0 )/ li , R0 is the distance to receiver from the point of electric field maximum z = 0 (19). Shift of emission is determined by the shift of maximum of electric current (see Fig. 9 from review [7], the current is proportional to N (z)).
The presented theory demonstrates that in RB conditions EAS generates intensive radio emission. Let us first discuss nowadays state of the observations in thunderstorm atmosphere which confirm the existence of RB. 1. The balloon measurements of electric field E during thunderstorms were compared with the critical field Ec (z) by Marshall et al. [8] (see also [7,9]). The observations clearly show an exponential fall with
A.V. Gurevich et al. / Physics Letters A 301 (2002) 320–326
height z of the maximal value of the electric field Emax in perfect agreement with the formulae (1). It is important also that at Emax (z) ≈ Ec (z) lightning discharges often occur. Electric field in thunderstorm atmosphere never exceed significantly Ec (z). 2. McCarthy and Parks [11] were the first who observed on aircrafts flying in thunderstorm a significant (two–three orders of magnitude) increase in intensity of X-ray quanta with energies 30–100 KeV. The increase lasts for about 10 seconds, what indicates that the region of intensive X-ray emission has several kilometers scale. Analogous result was obtained by Eack et al. (1996) [12] at balloons. Intensive X-ray emission was observed in the same energy range in thundercloud at heights z ≈ 4 km. It lasts approximately 70 s and has sharp changes correlated with lightning. Chubenko et al. (2000) [13] observed X-ray bursts at the same energies in Tien-Shang mountain experiments at heights 3350 m. The bursts lasted about 1 min and were highly correlated at scales 300–500 m. During the last experiments (2001) at Tien-Shang more short (10 s) bursts of X-ray emission highly correlated with the strong electric field amplification were seen. All these observations demonstrate a good agreement with the theory of runaway breakdown [7]. We note also a recent results of correlated with lightning short time amplification of γ -ray emission (ε > 2 MeV) observed by Brunetti et al. (2000) [22], Eack et al. (2001) [23], Alexeenko et al. (2001) [24]. Thus we can conclude that the observations definitely indicate that RB conditions are often fulfilled in thunderstorm atmosphere. From the theory presented here it follows that simultaneous effect of EAS and RB generates an intensive pulse of radio waves in the frequency range 1–10 MHz. RB–EAS radio pulse has quite a definite wave-form, it is strong enough to be observed for high energy cosmic ray particles. The mostly difficult problem is to discriminate RB–EAS and lightning radio emissions. The lightning radio emission has a complicated structure and is generated in a wide frequency range 1 KHz–1 GHz. It is connected with both cloud to ground and intracloud strokes and with their different components: return strokes, preliminary breakdown, step-leader, dart-leader, K, J, M processes an so on. Their basic characteristics could be in some cases close to one, predicted here for RB–EAS emission.
325
For example, according to observations [25,26] the significant part of radio pulses connected with negative step-leader has 1–20 MHz range. The wave-form of the pulses could look also analogous to RB–EAS radio pulses [25]. We emphasize that step-leader emission is tightly connected with return stroke RS—it proceeds RS for 10–50 ms [16]. The RB conditions according to [11– 13] could be fulfilled during much longer time 10– 100 s. So the RB–EAS emission, in principle, can exist in quite a different time interval—far from RS. It can be accompanied also by much smaller changes of electrostatic field, then those connected with lightning leaders. Simultaneously with RB–EAS radio pulse an intensive X-ray and γ -ray emissions could be seen, which give the direct evidence of RB process. A simultaneous measurements of radio pulses, extensive atmospheric showers (EAS), X-ray, γ -ray emission and electrostatic field changes could be performed at high mountain stations. Lebedev Physical Institute group has already began to fulfill complex measurements of these type at Tien-Shang mountain cosmic ray station (LPI MCRS). According to the presented theory the intensity of RB–EAS emission is proportional to the energy of cosmic ray particle. From (21) it follows that for ε 1017–1018 eV the RB–EAS radio emission become in intensity comparable with lightning radio emission. Such a strong radio pulse could be detected at large distances—up to several hundred kilometers. The radio detection method of high energy cosmic ray particles (RADHEP) is now considered as a very perspective one (see [27]). Its goal is to detect a radio emission of a high energy cosmic ray particle generated by electron–positron avalanche due to interaction with the Earth’s magnetic field and atmospheric inhomogeneity. This “transition radiation” type emission was predicted by Askaryan [28]. Of course as RB–EAS radio emission needs thunderstorms, the probability of its generation is much smaller than for the “transition radiation”. On the other hand it is much more intensive. Let us compare intensities. According to [27] for the cosmic ray particles having energy ε = 1017 eV the expected radio pulse amplitude is a few µV/m and it could be seen in a small region at the scale of the 100–200 m. In our case, both the pulse amplitude and the scale of the region is several magnitude higher (see (21)).
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Thus we see, that the RB–EAS radio emission is much stronger. A RB effect in thundercloud works as a “spark chamber” for radio detection of a high energy particles.
The work was supported by Grants EOARD-ISTC N2236p, ISTC-1490, RFBR 00-15-96594, RFBR 0002-16548.
References 5. Conclusion In conclusion we formulate briefly the main results of the theory developed in this Letter and discuss new experimental opportunities which follow from the theory. (1) During thunderstorm in RB conditions EAS generate wide band bipolar pulse of radio emission in frequency range 1–10 MHz. (2) Pulse intensity is approximately proportional to the energy of cosmic ray particle generating EAS. For high energy particles ε 1017 –1018 eV radio pulse is comparable with emission from lightning. It could be detected at large distances—up to several hundred kilometers. (3) The main method for detection calibration, and detailed study of RB–EAS radio pulse is simultaneous observations of radio emission with X-rays, γ -rays and different particles generated by EAS. One can conclude, that the experimental investigation of RB–EAS radio emission is important and can have a significant interest both for study of lightning initiation processes and for possibility of radio detection of a high energy cosmic ray particles.
Acknowledgements The authors are grateful to Prof. V.L. Ginzburg and Prof. B.M. Bolotovskii for useful discussions.
[1] A.V. Gurevich, G.M. Milikh, R. Roussel-Dupre, Phys. Lett. A 165 (1992) 463. [2] C.T.R. Wilson, Proc. Cambridge Philos. Soc. 22 (1924) 34. [3] R. Roussel-Dupre, et al., Phys. Rev. E 49 (1994) 2257. [4] N.G. Lehtinen, et al., Geophys. Res. Lett. 24 (1997) 2639. [5] L.P. Babich, et al., Phys. Lett. A 245 (1998) 460. [6] A.V. Gurevich, et al., Phys. Lett. A 275 (2001) 101. [7] A.V. Gurevich, K.P. Zybin, Phys. Usp. 44 (11) (2001) 1119. [8] T. Marshall, et al., J. Geophys. Res. 100 (1996) 7097. [9] D. MacGorman, W.D. Rust, The Electrical Nature of the Storms, Oxford University Press, New York, 1998. [10] A.V. Gurevich, et al., Phys. Lett. A 282 (2001) 180. [11] M. McCarthy, G. Parks, Geophys. Res. Lett. 12 (1985) 393. [12] K.B. Eack, et al., J. Geophys. Res. 101 (1996) 29637. [13] A.P. Chubenko, et al., Phys. Lett. A 275 (2000) 90. [14] A.V. Gurevich, G.M. Milikh, Phys. Lett. A 262 (1999) 457. [15] A.V. Gurevich, et al., Phys. Lett. A 254 (1999) 79. [16] M.A. Uman, The Lightning Discharges, Academic Press, Orlando, 1987. [17] S.Z. Belenkij, Cosmic Ray Showers (EAS), Atomizdat, Moscow, 1987. [18] A.V. Gurevich, N.D. Borisov, G.M. Milikh, Physics of Microwave Discharges, Gordon and Breach, Amsterdam, 1997. [19] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984. [20] H.A. Bethe, J. Ashkin, in: E. Segre (Ed.), Experimental Nuclear Physics, Vol. 8, Wiley, New York, 1953, pp. 166–375. [21] I.E. Tamm, J. Phys. USSR 1 (5–6) (1939) 439. [22] M. Brunetti, et al., Geophys. Res. Lett. 27 (2000) 1599. [23] K. Eack, et al., Geophys. Res. Lett. 28 (2001). [24] V. Alexeenko, et al., in: Proceedings of ICRC Copernicus Gesselshaft 1, 2001. [25] E. Krider, et al., J. Geophys. Res. 80 (1975) 3801. [26] W. Beasley, et al., J. Geophys. Res. 87 (1982) 4883. [27] J. Rosner, D. Suprun, in: D. Saltzberg, P. Gorham (Eds.), Radio Detection of High Energy Particles, 2001, pp. 81–98. [28] G.A. Askaryan, Sov. Phys. JETP 14 (1962) 441.
Physics Letters A 301 (2002) 320–326 www.elsevier.com/locate/pla
Radio emission due to simultaneous effect of runaway breakdown and extensive atmospheric showers A.V. Gurevich a,∗ , L.M. Duncan b , Yu.V. Medvedev a , K.P. Zybin a a P.N. Lebedev Institute of Physics, Russian Academy of Sciences, 117924 Moscow, Russia b Thayer School of Engineering, Dartmouth College, NH, USA
Received 9 June 2002; accepted 18 June 2002 Communicated by V.M. Agranovich
Abstract The theory of radio emission generated during thunderstorm by cosmic ray particles having high energy ε is developed. The emission is shown to have a form of bipolar microsecond radio pulse with characteristic frequency 1–10 MHz. Due to combined action of runaway breakdown and extensive atmospheric shower the emission intensity is high. The radio pulse could be observed for ε 1017 eV up to distances 100–300 km. Thus the runaway breakdown in thunderclouds can play a role of a “spark chamber” for radio detection of high energy particles. 2002 Published by Elsevier Science B.V.
1. Introduction A new physical concept of an avalanche type increase of a number of energetic electrons in gas under the action of the electric field was proposed by Gurevich et al. [1]. The avalanche can grow in electric field E Ec . The field Ec is almost an order of magnitude less than the threshold electric field of conventional breakdown Eth . The growth of number of electrons with energies ε > εc ≈ 0.1–1 MeV is determined by the fact that under the action of electric field E > Ec fast electrons could become runaway, what means that they are accelerated by electric field E as suggested by Wilson [2]. Due to collisions with gas molecules they can generate not only large * Corresponding author.
E-mail address: [email protected] (A.V. Gurevich).
number of slow thermal electrons, but the new fast electrons having energies ε > εc as well. Directly this process—acceleration and collisions lead to the avalanche type growth of the number of runaway and thermal electrons, which was called in [1] “runaway breakdown” RB. The detailed kinetic theory of RB was developed in [3–7]. In atmosphere the critical electric field is Nm (z) . Ec ≈ 200 (kV/m) (1) Nm (0) Here Nm (z) is the neutral molecules density at the height z and Nm (0) = 2.7 × 1019 cm−3 —at see level. Ec falls down with z due to exponential diminishing of Nm . At the thundercloud heights z ≈ 4–6 km, the critical field Ec is 100–150 kV/m and exactly these values of electric field are often observed during thunderstorms [8,9]. When the electric field in thunderstorm cloud reaches the critical value E Ec every
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 9 0 0 - 3
A.V. Gurevich et al. / Physics Letters A 301 (2002) 320–326
cosmic ray secondary electron (its energy ε > 1 MeV) initiates a microrunaway breakdown (MRB). It serves as a source of intensive ionization of air and manifests itself in a strong amplification of X- and γ -rays emission and effective growth of conductivity in thundercloud [1,10]. These effects were observed and compared with RB theory [11–14]. Extensive atmospheric shower (EAS) is accompanied by a strong local growth of cosmic ray secondaries number [17]. A theory of combined effect of RB–EAS was developed in [15]. It was shown that ionization of atmosphere by the shower in RB conditions is growing strong enough to produce a local highly conductive plasma and can serve for lightning leader initiation. On the other hand the same effect can stimulate the excitation by thundercloud electric field an intensive local pulse of electric current. This short pulse of electric current can generate radio emission. The goal of the present Letter is to develop a theory of radio emission generated in thundercloud due to RB–EAS interaction. The emission would be shown to have a form of microsecond radio pulses with characteristic frequency around 1–10 MHz. As is well-known radio emission is generated during thunderstorm in a wide frequency range. It has a high power and was studied in a multiple observations (see monographs [9,16] and literature cited there). That is why to single out RB–EAS radio pulses in observations is impossible without knowledge of their special features described by the theory. On the other hand the detection and detailed study of this emission has a significant interest either for understanding of lightning generation mechanism or investigation of the fluxes of high energy cosmic ray particles.
2. Electric current Let us calculate the current pulse generated by simultaneous effect of RB and EAS. According to the theory and observations, the seed electrons in EAS are distributed nonuniformly [17]. The typical scale along the direction of motion of relativistic particle z ≈ 5–10 m and in perpendicular direction R⊥ ≈ 100 m [15,17]. It means that current pulse structure in z direction is mostly significant what allows us to develop the theory in one-dimensional
321
approximation. In RB process high energy electrons effectively generate large number of slow thermal electrons [6,7]. One can estimate that namely thermal electrons define the current pulse under the action of electric field. The equations describing this process have a form: Ne ∂Ne ∂(Ne Ve ) + =− + q, ∂t ∂z τatt Ne ∂N− = , ∂t τatt ∂N+ = q, ∂t ∂E = 4πe(N+ − Ne − N− ). ∂z
(2)
We supposed here that electric field is directed along the vertical z. The first equation of system (2) describes the changes of slow electron density Ne induced by drift of electrons in electric field Ve , their birth q due to RB process and death due to attachment. The second and third equations describe the appearance of negative ions N− resulting from electron attachment and generation of positive ions N+ due to RB. The fourth equation define the appearance of polarization electric field. In Eq. (2) Ve is the drift velocity under the action of electric field Ez = E: Ve = eE/mν,
(3)
where ν is electron collision frequency with air molecules. We suppose here that the inclination angle to the vertical of a high energy particle (generating EAS) is small enough, what allows to consider z-component of electric field only. The life-time due to attachment of slow electrons in the air τatt is determined by threebody collisions [18], and the source of secondary electrons q: q = cN0 δ(z − ct)F (z, E/Ec ).
(4)
Here cN0 (ε, r⊥ ) is a flux of newborn slow electrons generated by EAS in the atmosphere in absence of electric field. The number density of secondary electrons N0 in the plane orthogonal to z axis is approximately proportional to the energy of cosmic ray particle ε and falls down with the distance r⊥ from the center of EAS [17]. According to well-known Nishimura–Ksimata–Greizen formulae simplified for
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thundercloud heights [15]: ε/β N0 (ε, r⊥ ) = n0 √ Σ(r⊥ ), ε/β R r⊥ −3.5 Σ(r⊥ ) = , × 1+ r⊥ R β = 72 MeV,
n0 ≈ 1.2 × 10−9 cm−2 .
(5)
Function F (z, E/Ec ) determines slow electron production and their multiplication under the action of electric field E. It depends strongly on the relation E/Ec . If E Ec the factor F is growing exponentially with z due to direct RB process [10]. In usual conditions in the air one runaway electron generates about 50 thermal electrons along 1 cm of its trajectory, i.e., ionization rate η = 30–50 e/cm [20]. An additional logarithmic factor L comes from the reconstruction of distribution function [6,7]. In analytical calculations the function F would be presented in a simple form: z , F (z, E/Ec ) = ηL exp (6) li where li (E/Ec ) is the RB ionization length [10]. Approximation (6) is useful to obtain the main qualitative characteristics, it is correct for E > Ec . In Section 4 the numerical results would be presented, where any value of relation E/Ec will be considered. According to (4), (6) the source q is proportional to growing exponent eλt , where λ = c/ li is the RB increment [5,6], and t = z/c. It means that the solution of the system (2) could be presented in a form: Ne,+,− = ne,+,− eλt ,
E = E eλt .
(7)
After substitution (5)–(7) in Eq. (2) one can find ∂ne ∂(ne Ve ) + λne + ∂t ∂z ne =− + cNδ(z − ct), N = ηLN0 , τatt ∂n− ne + λn− = , ∂t τatt ∂n+ + λn+ = cNδ(z − ct), ∂t ∂E = 4πe(n+ − ne − n− ). (8) ∂z Introducing a new variable χ = ct −z, and denominate parameters la = cτatt , li = c/λ, we find from (8)
ordinary differential equation describing the density changing: Ve 1 1 d 1+ ne = − ne + Nδ(χ), + dχ c la li dn+ n+ dn− n− ne + + = , = Nδ(χ). (9) dχ li la dχ li We note now that the drift velocity of slow electrons is small: Ve /c 1. Using an expansion on this small parameter, one can present the solution of the system (9) in a form: ne = 2Nθ (χ)e−(1/la +1/li )χ , n+ = 2Nθ (χ)e−χ/li , n− = 2Nθ (χ)e−χ/li 1 − e−χ/ la .
(10) E
in (8) is Estimates show that the polarization field not important. Thus the electric current jz associated with electron density (10) takes a form: jz = eVe ne = eVe Nθ (ct − z)e−(ct −z)/ la .
(11)
In conditions (6): N = ηLN0 (ε, r⊥ ), in general case (4) N = ηLN0 f (z), where f (z) = F (z, E/Ec ) × exp(−z/ li ).
3. Radio emission 3.1. Analytic theory To define the radio emission generated by RB–EAS current we have to calculate first the Fourier transform of (11): 1 jω = 2π
+∞ N e−iωz/c j (t)eiωt dt = eVe . 2π iω + c/ la
(12)
−∞
The vector-potential Aω of radio emission determined by the current (12) has a usual form: 1 jω (r ) iωR/c e dr . Aω = c R Here R = (x − x )2 + (y − y )2 + (z − z)2 . After integration over r⊥ we find in wave region (R0 |r |, c/ω R0 ), where R0 = (x 2 + y 2 + z2 )1/2 is the distance to receiver from current region: Aω =
I eiωR0 /c η(ω). cR0 iω + c/ la
(13)
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Here I = eVe Nm r⊥ dr⊥ is the total current across EAS, Nm is maximal value of electron density and η(ω) is the integral +∞ η(ω) = −∞
N(z) −iωz(1−cos θ)/c e dz. Nm
(14)
According to the theory of runaway breakdown the number of electrons is growing exponentially with z up to some point z0 and falls down exponentially at higher z (see [10] Fig. 3). The main input in integral (14) comes from the region around Nm —maximum of N(z). Let us choose it in a form: N(z) = Nm e−(z−z0 )
2 /L2 1
.
The length L1 is of the order li . The integral takes form: η(ω) =
+∞ 2 2 e−iαz−(z−z0 ) /L1 dz,
−∞
α=
ω (1 − cos θ ), c
R L21 ω2 2 I L1 exp iω c − iαz0 − 4c 2 (1 − cos θ ) Aω = . cR0 iω + c/ la (15) The expression (15) define an analytical formula for the spectrum of radio emission. The wave energy dεω emitted in angle dΩ in frequency range dω is given by the formula [19]: c k 2 |Aω |2 R02 sin2 θ dΩ dω. dεω = (16) (2π)2 After substituting (15) in (16) we find
L21 ω2 2 I 2 L21 ω2 exp − 2c 2 (1 − cos θ ) dεω = (2π)2 c3 ω2 + (c/ la )2 × sin2 θ dΩ dω.
323
J (t) = π 3/2 θ (t − R0 /c)e−(ct −R0 )/ la +c /(la ω1 ) c (t − c/R0 )ω1 − × Φ 2 la ω1 c +Φ , la ω1 2c |1 − cos θ |, ω1 = (18) L1 x 2 where Φ(x) = √2π 0 e−t dt is the probability integral. 2
2
3.2. Numerical calculations We have fulfilled the direct numerical calculations which allow to avoid exponential approximation (6) for RB process. The same scheme as in [10] was used. The electric field E(z) was supposed existing in a layer −L0 z L0 in parabolic form: E(z) = Em 1 − a 2 z2 . (19) Here the distance z was measured in ionization length li and parameter a = li /L0 . The distribution function of energetic electrons f (ε, z) was calculated using the kinetic equations (see [10], formula (3)). The density and current of thermal electrons is determined according to (10), (11). The vector-potential is calculated according to the relations (13), (14): I li A(τ ) = cR0
+∞ −∞
∗
eiω τ η(ω∗ ) iω∗ +
li la |1 − cos θ |
dω∗ ,
η(ω∗ )
where = η(ω)/ li and τ = tc/(li |1 − cos θ |). Spectrum Aω and potential A(τ ) are shown at Fig. 1. The vertical component of electric field Ez (τ ) has a form: (17)
It is interesting to note that qualitatively the expressions (15), (17) correspond well to results obtained by Tamm [21] for the emission of “cut” current (the current generated by a particle moving with velocity c on a segment [−L1 , L1 ]). To determine the shape of electromagnetic pulse A(t, R0 ) it is necessary to find inverse Fourier transformation: I L1 A(t, R0 ) = J (t), cR0
Ez (τ ) I sin2 θ cos θ = cz0 1 − cos θ
+∞ −∞
∗
iω∗ eiω τ η(ω∗ ) iω∗ +
li la |1 − cos θ |
dω∗ ,
z0 cos θ = (20) . R0 Here z0 is the height of emission region. The pulse wave form is shown at Fig. 2. It is bipolar, the width of the pulse became wider with distance and the amplitude falls proportional to R0−1 .
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(a) Fig. 2. The wave form of vertical component of electric field as function of normalized time τ = (ct − Ri )/ li . In atmospheric conditions at the heights 5–6 km: li ≈ 100 m.
Characteristic value of RB–EAS radio pulse amplitude according to formulae (20) and Fig. 1b could be approximated as ε 100 km E ≈ 20 1017 eV R0 Em mV . × exp 15 (21) −1 Ec m Here ε is the energy of cosmic ray particle, R0 is the distance from current region to the receiver (15), (20), Em is the maximal value of electric field (19), Ec RB critical field (1), exponential factor is the result of numerical calculations.
4. Discussion (b)
Fig. 1. (a) Frequency spectrum of vector potential Aω for Em /E0 = 1.05; a = li /L0 = 0.2. Frequency ω normalized on c/ li . Black points—real part Aω ; squares—imaginary part of Aω . (b) Vector-potential A(τ ) at Em /Ec = 1.05; a = 0.2, for different angles θ shown at the figures. τ = (ct − R0 )/ li , R0 is the distance to receiver from the point of electric field maximum z = 0 (19). Shift of emission is determined by the shift of maximum of electric current (see Fig. 9 from review [7], the current is proportional to N (z)).
The presented theory demonstrates that in RB conditions EAS generates intensive radio emission. Let us first discuss nowadays state of the observations in thunderstorm atmosphere which confirm the existence of RB. 1. The balloon measurements of electric field E during thunderstorms were compared with the critical field Ec (z) by Marshall et al. [8] (see also [7,9]). The observations clearly show an exponential fall with
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height z of the maximal value of the electric field Emax in perfect agreement with the formulae (1). It is important also that at Emax (z) ≈ Ec (z) lightning discharges often occur. Electric field in thunderstorm atmosphere never exceed significantly Ec (z). 2. McCarthy and Parks [11] were the first who observed on aircrafts flying in thunderstorm a significant (two–three orders of magnitude) increase in intensity of X-ray quanta with energies 30–100 KeV. The increase lasts for about 10 seconds, what indicates that the region of intensive X-ray emission has several kilometers scale. Analogous result was obtained by Eack et al. (1996) [12] at balloons. Intensive X-ray emission was observed in the same energy range in thundercloud at heights z ≈ 4 km. It lasts approximately 70 s and has sharp changes correlated with lightning. Chubenko et al. (2000) [13] observed X-ray bursts at the same energies in Tien-Shang mountain experiments at heights 3350 m. The bursts lasted about 1 min and were highly correlated at scales 300–500 m. During the last experiments (2001) at Tien-Shang more short (10 s) bursts of X-ray emission highly correlated with the strong electric field amplification were seen. All these observations demonstrate a good agreement with the theory of runaway breakdown [7]. We note also a recent results of correlated with lightning short time amplification of γ -ray emission (ε > 2 MeV) observed by Brunetti et al. (2000) [22], Eack et al. (2001) [23], Alexeenko et al. (2001) [24]. Thus we can conclude that the observations definitely indicate that RB conditions are often fulfilled in thunderstorm atmosphere. From the theory presented here it follows that simultaneous effect of EAS and RB generates an intensive pulse of radio waves in the frequency range 1–10 MHz. RB–EAS radio pulse has quite a definite wave-form, it is strong enough to be observed for high energy cosmic ray particles. The mostly difficult problem is to discriminate RB–EAS and lightning radio emissions. The lightning radio emission has a complicated structure and is generated in a wide frequency range 1 KHz–1 GHz. It is connected with both cloud to ground and intracloud strokes and with their different components: return strokes, preliminary breakdown, step-leader, dart-leader, K, J, M processes an so on. Their basic characteristics could be in some cases close to one, predicted here for RB–EAS emission.
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For example, according to observations [25,26] the significant part of radio pulses connected with negative step-leader has 1–20 MHz range. The wave-form of the pulses could look also analogous to RB–EAS radio pulses [25]. We emphasize that step-leader emission is tightly connected with return stroke RS—it proceeds RS for 10–50 ms [16]. The RB conditions according to [11– 13] could be fulfilled during much longer time 10– 100 s. So the RB–EAS emission, in principle, can exist in quite a different time interval—far from RS. It can be accompanied also by much smaller changes of electrostatic field, then those connected with lightning leaders. Simultaneously with RB–EAS radio pulse an intensive X-ray and γ -ray emissions could be seen, which give the direct evidence of RB process. A simultaneous measurements of radio pulses, extensive atmospheric showers (EAS), X-ray, γ -ray emission and electrostatic field changes could be performed at high mountain stations. Lebedev Physical Institute group has already began to fulfill complex measurements of these type at Tien-Shang mountain cosmic ray station (LPI MCRS). According to the presented theory the intensity of RB–EAS emission is proportional to the energy of cosmic ray particle. From (21) it follows that for ε 1017–1018 eV the RB–EAS radio emission become in intensity comparable with lightning radio emission. Such a strong radio pulse could be detected at large distances—up to several hundred kilometers. The radio detection method of high energy cosmic ray particles (RADHEP) is now considered as a very perspective one (see [27]). Its goal is to detect a radio emission of a high energy cosmic ray particle generated by electron–positron avalanche due to interaction with the Earth’s magnetic field and atmospheric inhomogeneity. This “transition radiation” type emission was predicted by Askaryan [28]. Of course as RB–EAS radio emission needs thunderstorms, the probability of its generation is much smaller than for the “transition radiation”. On the other hand it is much more intensive. Let us compare intensities. According to [27] for the cosmic ray particles having energy ε = 1017 eV the expected radio pulse amplitude is a few µV/m and it could be seen in a small region at the scale of the 100–200 m. In our case, both the pulse amplitude and the scale of the region is several magnitude higher (see (21)).
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Thus we see, that the RB–EAS radio emission is much stronger. A RB effect in thundercloud works as a “spark chamber” for radio detection of a high energy particles.
The work was supported by Grants EOARD-ISTC N2236p, ISTC-1490, RFBR 00-15-96594, RFBR 0002-16548.
References 5. Conclusion In conclusion we formulate briefly the main results of the theory developed in this Letter and discuss new experimental opportunities which follow from the theory. (1) During thunderstorm in RB conditions EAS generate wide band bipolar pulse of radio emission in frequency range 1–10 MHz. (2) Pulse intensity is approximately proportional to the energy of cosmic ray particle generating EAS. For high energy particles ε 1017 –1018 eV radio pulse is comparable with emission from lightning. It could be detected at large distances—up to several hundred kilometers. (3) The main method for detection calibration, and detailed study of RB–EAS radio pulse is simultaneous observations of radio emission with X-rays, γ -rays and different particles generated by EAS. One can conclude, that the experimental investigation of RB–EAS radio emission is important and can have a significant interest both for study of lightning initiation processes and for possibility of radio detection of a high energy cosmic ray particles.
Acknowledgements The authors are grateful to Prof. V.L. Ginzburg and Prof. B.M. Bolotovskii for useful discussions.
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