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Model of infrared emission from sprites
\ PERGAMON
Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Model of infrared emission from sprites G[ M[ Milikh \ D[ A[ Usikov\ J[ A[ Valdivia University of Maryland\ Departments of Physics and Astronomy\ College Park\ MD 19631\ U[S[A[ Received 04 December 0886^ accepted 00 February 0887
Abstract A model of the 3[15 mm infrared emission due to red sprites is presented[ The model considers the generation of nitrogen {vibrons| due to the collisions of the nitrogen molecules with the electrons energized by the electric _eld from lightning\ followed by the transition of the nitrogen vibrons to the CO1"990# vibrational level\ with a lifetime much shorter than that of nitrogen[ The infrared photons of wavelength l 3[15 mm radiated by the CO1"990# propagate through the optically thick atmosphere ^ therefore\ this emission could best be observed from space[ The model computes the infrared radiance of sprites\ as well as the energy collected by a state!of!the!art space infrared detector\ and estimates the signal to background ratio[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ Introduction The recently discovered sprite phenomenon manifests itself as optical ~ashes sparkling above the top of giant thunderstorms at heights of from 49Ð84 km "Lyons\ 0883 ^ Sentman et al[\ 0884 ^ Bossipio et al[\ 0884 ^ Winck! ler et al[\ 0885#[ This phenomenon occurs due to the electromagnetic pulse or quasi!static _eld from the light! ning which energizes the ambient electrons[ The electrons then collide with the air molecules exciting their electronic levels followed by the optical emissions[ However\ less than 0) of the total absorbed electromagnetic energy of lightning is released through the optical emissions\ while most of this energy is pumped into the nitrogen vibrational levels having long lifetimes[ This energy then is transferred through resonant molecular collisions into the CO1"990# vibrational level with a much shorter life! time\ t 1[4 ms\ leading to the radiation of infrared photons of wavelength l 3[15 mm[ The radiated IR photons propagate through the optically thick atmo! sphere ^ therefore this emission is almost totally absorbed in the atmosphere beneath the sprite\ but could be detected from space[ Such an observation could provide us with interesting data about the energetics of sprites[ The objective of this paper is to present a model of the generation of the IR emission by red sprites\ and of the propagation of the IR photons through the atmosphere[
Corresponding author[ E!mail ] milikhÝastro!umd[edu
The intensity and angular distribution of the IR emission from red sprites is obtained from this model\ and the amount of energy collected by the state!of!the!art IR space detector along with the signal to background ratio is estimated[ In the next section we describe the model of the gen! eration of IR photons and their propagation through the atmosphere[ In Section 2 we discuss how the IR photons generated by sprites and then escaping from the atmo! sphere could be observed from space[ This is followed by the conclusions[
1[ Model description The model is based on the energization of ionospheric electrons by _elds generated by conventional lightning and applies equally well to energization by elec! tomagnetie "EM# or quasi!static _elds "Milikh et al[\ 0885 ^ Rowland et al[\ 0884 ^ Pasko et al[\ 0884#[ We consider the atmosphere as consisting of a set of hori! zontal slabs each having constant density and compo! sition[ The electromagnetic energy from lightning is released during a short pulse with the duration of a few ms\ inside a localized region in the upper atmosphere "Lyons\ 0883 ^ Sentman et al[\ 0884 ^ Rairden and Mende\ 0884 ^ Winckler et al[\ 0885#[ The model starts with a quasi!static or low frequency electric _eld from lightning at altitudes between 59 and 89 km\ and computes kin! etically the modi_cation of the electron distribution func!
S9906Ð8209:87:,*See front matter Þ 0887 Published by Elsevier Science Ltd[ All rights reserved PII ] S 0 2 5 3 Ð 5 7 1 5 " 8 7 # 9 9 9 2 2 2 Ð 8
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G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
tion by the electric _eld and the excitation of the N1 and CO1 n!th vibrational level by the energized electrons[ Once we have the excitation rates of the N1"n# and CO1"n# we introduce a Monte!Carlo model for the description of the photochemical processes leading to the emission of the CO1 IR photons\ followed by their di}usion through the atmosphere[ 1[0[ Excitation rates of N1"n#\ CO1"n# The basic input to the model is the electric _eld ampli! tude E9 at a particular altitude z characterized by the ambient electron!neutral collision frequency n9[ Here E9 is determined as E9 zSv Ev1 \ where Ev1 is the spectral energy density[ For a given value of E9 and n9 the asymp! totic stationary state of the electron distribution function f "v# is found by solving numerically the local kinetic FokkerÐPlanck equation "Tsang et al[\ 0880# for altitudes such that cos1 a ³ n1:V1 ] 1f "v# e1E91 0 1 v1n 1f −L"v#[ 1 1 1t 2m v 1v n ¦V1 1v
0
1
"0#
Here L"v# is the operator which describes the e}ect of the inelastic collisions\ V is the electron cyclotron frequency\ n is the electron collision frequency in the presence of the electric _eld\ while a is the angle between the electric and geomagnetic _elds[ Equation "0# is general and applies to quasi!static or EM _elds[ There are two ways of exam! ining the results of eqn "0#[ One is to specify an altitude\ and an electric _eld at that altitude\ and solve eqn "0# as a function of E9 in order to obtain the electron distribution function[ The alternative is to _nd the collision frequency n"E9\ z# explicitly from the FokkerÐPlanck code\ and pro! ceed with the self!similar solution for f "v# as a function of the electron quiver energy o½ "Valdivia et al[\ 0886#[ The latter is given by o½
e1E91 1
1
mðV ¦n Ł
$
0¦
V1 1
n
%
cos1 a [
n ex
7 eff s nknex kex
"2#
n 0
where the summation goes over all of the eight vibrational levels of the N1 ground electronic state[ Figure 0 shows the excitation rate coe.cients of the n!th vibrational levels of N1\ and the e}ective excitation rate coe.cient of N1"n 0#\ along with that of the CO1"990# level\ as a function of the electron quiver energy\ found for the height of 79 km[ Here the excitation rate coe.cient is normalized by the factor NCO1:NN1 in order to compare the rate of the direct pumping of the CO1"990# level with that of the excitation through the collisions with N1 "n 0# molecules[ Figure 0 reveals that at the range of the electric _eld which corresponds to red sprites the direct pumping of the CO1"990# level can be neglected[ We next compute the density of vibrationally excited nitrogen molecules N1 "n 0#\ or vibrons\ at di}erent altitudes as a function of the electric _eld and the pulse width tp\ taking into consideration the increase of the vibron abundance due to the electron impact ionization Nvib Ne9NN1
g
tp
e} kex "E9"t## exp "nie}"E9"t##t# dt
"3#
9
The quiver energy is the critical parameter that controls the behavior of the distribution function f "v# under an electric _eld at a given height[ In a case of electron ener! gization by a static electric _eld the parameter o½ reduces to E:n\ which in turn is controlled by the ratio E:N\ where N is the air density[ The next element of the model is the computation of the excitation rate for a particular vibrational level based on the value of f "v# found above[ We _rst compute the excitation rate coe.cient knex of the n!vibrational level of N1 by the electron impact[ knex 3pÐf "v#v2snex"v# dv
tation of the CO1"990# level using the excitation cross section from Register et al[ "0879#[ Note that the nitrogen molecules excited to a high vibrational level "n × 0# are involved in the exchange of vibrational quanta with the nitrogen molecules being in the ground state\ producing the ensemble of N1 "n 0#¦N1 "n 9# particles[ This process occurs much faster than the energy transfer from N1 "n 0# to CO1 due to the relatively low density of CO1 "NCO1: NN1 ¾ 2[2×09−3#[ Therefore the energy pumped into the vibrational N1 level leads to the production of N1 "n 0# molecules which then transmit their energy to CO1[ We introduce the e}ective rate coe.cient for the excitation of N1 "n 0# ]
"1#
where s is the excitation cross section of this level "Shultz\ 0853#[ The same approach is valid for the exci!
where Nvib is the vibrons| density\ Ne9 is the ambient elec! tron density\ NN1 is the density of nitrogen\ and the e}ec! tive ionization rate is nieff ni−na[ Here ni is the ionization rate while na is the rate of dissociative electron attachment to molecular oxygen[ Note that an important role is played by the total amount of vibrons generated in the volume irradiated by the electric _eld from the lightning[ In order to estimate the intensity of the source of vibrons we discuss below two cases[ One is a simple model of the sprite as a source of vibrons\ which occurs inside an uniform column and which is caused by an electromagnetic pulse "EMP# from the lightning[ The other is the spatially integrated source of vibrons due to the electric _eld induced by the fractal lightning\ as described by Valdivia et al[ "0886#[
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
786
Fig[ 0[ Excitation rate coe.cients of the N1"n# vibrational levels "solid lines#\ the e}ective rate constant keff ex de_ned by eqn "2# "points#\ and excitation rate coe.cient of the CO1"990# level "broken line#[ The latter is normalized by the factor NCO1:NN1[
In the _rst case we assume that the sprite occurs inside a cylindrical column centered at 79 km\ having a radius of 09 km and a height of 19 km "Sentman et al[\ 0884#\ and caused by a 4Ð09 ms pulse[ Moreover\ based on spectroscopic observations Mende et al[\ 0884 ^ Hampton et al[\ 0885# and the model by Milikh et al[ "0886#\ we assume that the electric _eld from lightning at 79 km ranges between 24 and 49 V:m\ while atmospheric break! down occurs at E9 × 34 V:m[ In the second case we consider the source of vibrons produced by the low altitude fractal lightning[ For a given fractal lightning we compute _rst the electric _eld in the near and far zones[ Then we consider the _eld propa! gation into the lower ionosphere\ including self!absorp! tion[ Finally\ using the excitation rate coe.cients as func! tion of electron quiver energy given in Fig[ 0\ we compute the time!integrated spatial distribution of the vibron source[ Details of the computations can be found in the paper by Valdivia et al[ "0886#[ Here we present an exam! ple taking a horizontal lightning of fractal dimension D 0[33\ which propagates with the speed of 9[94 times the speed of light at 4 km altitude\ and discharges 099 C[ Figure 1 shows the fractal structure of the lightning along with the electric _eld distribution at the height 59 km normalized over its mean value\ and the vibron density pattern normalized over the density\ averaged over the whole computational box[ The total amount of vibrons in the volume irradiated by the electric _eld from lightning serves as an input to the next element of the model which describes the IR photon emission and propagation through the atmo! sphere[
1[1[ IR Emission and atmospheric propa`ation The process following the excitation of N1 by the elec! tron impact involves a set of photochemical reactions "Kumer and James\ 0863#[ They are shown in Fig[ 2\ where straight lines show the bimolecular reactions while the wavy lines reveal photon emission or absorption[ According to this scheme\ the vibrationally excited N1 molecule either loses the energy in a set of vibrationÐ vibrational transitions\ or vibrationalÐtranslational reac! tions with particles other than CO1\ or transfer the exci! tation to CO1 molecules exciting the 990 state[ In turn\ the CO1"990# molecule either transmits the excitation back to N1 or radiates an IR photon of the 3[15 mm wavelength[ The IR photons propagate through the atmosphere until they are absorbed by CO1 molecules[ The photon absorp! tion and reradiation will go over and over again until they are lost due to thermal heating or escape from the atmosphere[ The escaping photons can reach a space! borne detector[ The above model is treated by using the Monte!Carlo method[ Namely\ some vibrons are gen! erated in the code\ and then their transitions are followed[ The probability density at each transition is chosen by the Monte!Carlo procedure\ which will be discussed below[ First\ we choose between two channels ] energy tran! sition from the excited N1 to CO1 from the one side\ and the loss of the vibron due to a set of collisions with the particles other than CO1\ on the other[ The probability of each of the above channels is proportional to their reaction rates[ In fact\ the rate r of the energy transition of the vibron to CO1 is equal to the rate constant k0 times the local CO1 density[ At the same time\ the loss rate rlos
787
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Fig[ 1[ The discharge patterns of fractal dimension D 0[33 "a#\ the electric _eld distribution "b#\ along with the vibron density pattern "c#[ The charge is Q 099 C and the current propagates with the speed 9[94 c\ c being the speed of light[
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G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Fig[ 2[ Schematic diagram of the relevant set of photochemical reactions[
of the vibron due to collisions with other than CO1 par! ticles is described by the following equation which con! siders reactions revealed by Fig[ 2 ]
0
rlos k1NO¦k2NO1 0−
k?2NN1
1
k?2NN1¦k4NO¦k3NH1O
"4#
where the relevant reaction rate coe.cients are taken from Kumer and James "0863#\ while the vertical dis! tribution of air density and temperature is adopted from the U[S[ Standard Atmosphere "0865#[ Therefore\ the probability of the energy transition from N1 "n 0# to CO1 is r:"r¦rlos#[ We next estimate the time required for this transition[ In order to do this we introduce the probability density P0"t# k0NCO1 exp"−k0NCO1t#\ so t that Ð 9 P0"t# dt 0[ Then we relate the value of Ð9 P0"t?#dt? to the random number {num| between 9 and 0[ Conse! quently\ the time under consideration is determined as 0 log "0−num#[ t − k0NCO1
"5#
Proceeding\ we discuss how the model treats the emis! sion and absorption of the IR photons[ The energy quan! tum transmitted to CO1 from the N1 "n 0# molecules excites the J rotational level of the CO1"990# state[ The J!level is chosen by the Monte!Carlo procedure such as described above using the relative population of
rotational states from Kumer and James "0863#\ and relating the corresponding probability density to the chosen random number[ The probability of the photon radiation is determined by comparing the Einstein coe.cient of the CO1"990# state with the rate of energy transmission from CO1"990# back to N1[ The frequency of the radiated photon is obtained using the selection rules and by taking into account the Doppler frequency shift DnD0 due to the ther! mal motion of the radiating molecule[ We also compute the direction of the photon propagation by using the Monte!Carlo procedure[ The photon could then be absorbed by another CO1 molecule and populate a rotational level J?\ depending on the photon frequency[ The absorption cross section is given by ] 0 s s9 1
"1J?¦0#e−J?"J?¦0#B:T
s
"1J?¦0#e−J?"J?¦0#B:T
J? 9\1\[[[ 1 ge} 1 e}
Dop 0
g ¦"Dn
¦Dn1Dop#1
\ "6#
where s9 1[4×09−03 cm1 "Kumer and James\ 0863#\ B 9[28 cm−0 is the rotational constant\ while Dn0Dop and Dn1Dop are the Doppler shifts of the radiating and absorb! ing molecules[ Finally\ ge} is the e}ective broadening
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G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
which includes both collisional and Doppler broadening[ It can be represented in the following form "Lenoble\ 0882# ] geff"cm−0#
60
6×09−1P"atm#
X 1 0 T 299
¦ nph
1
1
17
vT 0 c zlog 1
\
"7#
where P is the air pressure\ vT and c are the thermal molecular speed and the speed of light\ respectively\ while nph"J?# is the photon frequency[ Note that we consider the main band and the major isotope of CO1[ The role played by the hot bands and minor isotopes was discussed by Nebel et al[ "0883#[ The above model allows us to compute the photon ~ux radiated into space[ It is characterized by the sprite|s IR radiance Lspr\ which represents the amount of energy traversing a unit of surface per unit time in a solid angle inclined at the angel u relative to the source[ In fact\ assuming that the electric _eld does not change in time we obtained using eqn "3# that ½ "u\ t#ÐNvib dV Lspr"u\ t# hnL
"8#
where hn is the energy of the radiated IR photon\ L "u\ t# is the radiance per vibron given in m−1 rad−0 s−0\ which takes into consideration the IR photon losses due to air heating\ and the integral is carried out along the volume irradiated by the electric _eld from the lightning[
2[ Discussion We apply the computational scheme discussed above to _nd the probability for an IR photon to escape from the atmosphere[ First we compute the fraction of photons escaping from di}erent atmospheric heights[ This is shown in Fig[ 3\ which reveals that almost all photons escape into space from the top part of sprite located above 79 km\ while photons radiated by the sprite bottom\ located below 59 km\ are totally absorbed[ We then compute the number of photons escaping during each time interval\ normalized over the total number of vibrons generated in the model\ i[e[\ the probability den! sity of the IR photons released into space by the sprite[ The results of the computations for two di}erent sprite locations\ at 79 and 89 km altitude\ are shown in Fig[ 4[ In this _gure we limit the {observation time| to 0999 s after the sprite occurs[ A short "ms# pulse from lightning produces a broad pulse "½199 s# of IR emission[ This is due to two physical e}ects ] _rst\ the considerable time required for excited N1 molecules to collide with CO1 molecules in order to transmit their energy to the CO1"990# state and\ second\ the long di}usion time of the
IR photons in the optically thick atmosphere[ Notice that the temporal distribution of the probability density of the IR photons radiated into space by the sprite is steeper when the sprite is located at a higher altitude "compare curves 0 and 1 in Fig[ 4#[ Thus\ the time derivative of the IR intensity could provide us with the height of the sprite\ if observed by a detector in space[ Figure 5 shows the angular distribution of the time integrated radiance per photon of a sprite located at 79 km altitude[ This _gure reveals that the photons escape preferentially in the vertical direction along the air density gradient\ which minimizes the number of photon col! lisions with the CO1 molecules leading to IR absorption[ According to our computations\ the angular distribution depends weakly on the sprite|s height[ Proceeding\ we estimate the amount of IR energy from the sprite collected by a space IR detector\ along with the signal to background ratio[ As an example we consider the SPIRIT III IR telescope ~ying on the MSX satellite[ The latter has a nearly Sun!synchronous orbit at an alti! tude of H 777 km "Huebschman and Pardoe\ 0881#[ The telescope|s _eld of view is D8 0> by Dc 2>\ so it views a sprite within the solid angle Vdet 8[02×09−3\ while the dish has an area of Sdet 789 cm1[ Thus\ during the time t that the detector could collect sprite!generated energy\ the amount of IR energy from the sprite given by Ispr SdetVdet
g
t
dt?Lspr"u"t?#\ t?#[
"09#
9
The angle u between the vertical through the sprite and line connecting the sprite and the satellite is related to the angle b\ which describes the satellite location\ as shown by Fig[ 6[ The angle b varies at b b9¦vsatt:"RE¦H#\ where vsat is the satellite velocity\ RE is the Earth radius\ and b9 corresponds to the location of the satellite when the sprite appears[ Correspondingly\ the zenith angle u changes from the initial value uini u"b9# to the _nal zenith angle u_n[ The latter is due to the fact that the IR photons from the sprite cannot propagate over the visible horizon[ Then we compute the sprite|s IR radiance per vibron by using eqn "09#\ modeling the sprite by a vertical column extended from 69Ð89 km\ assuming that the sat! ellite trajectory is located in the same plane as the sprite\ and that the sprite appears at b9 4 and 09>[ This is shown in Fig[ 7"a#\ "b# where the arrows mark the time when the satellite is over~ying the sprite[ Figure 7 reveals that about 79) of the total IR energy from the sprite is collected by the detector ~ying toward the sprite during the _rst 099 or 199 s after the sprite|s appearance[ The amount of IR radiation collected by the detector is reduced signi_cantly when it moves away from the sprite[ This is due to the reduction of the IR ~ux with time and angle "see Figs 4 and 5#\ and due to the increase in the ~ux divergence with the distance from the sprite Rds[ Note that the data in Fig[ 7 are averaged over 19 s[
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
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Fig[ 3[ The transparency of the atmosphere for the 3[15 mm photons radiated from di}erent heights[
Fig[ 4[ Probability density of IR photons radiated into space by a sprite[ Trace 0 corresponds to a sprite located at 79 km\ and trace 1 corresponds to a sprite located at 89 km altitude[
In order to _nd the IR energy which could be collected by the satellite detector observing sprites\ we estimate _rst the intensity of a sprite as a source of vibrons[ Note that the ionosphere electrons\ which excite the vibrons\ are energized by either the QS electric _eld or by the EMP from the lightning[ However\ those processes occur on a di}erent timescale[ While the QS _eld exists in the ionosphere during the local relaxation time trel which ranges between 0 and 9[2 ms at the height 89Ð89 km "Pasko et al[\ 0885#\ the pulse width of the EMP from
lightning is determined by the size and propagation speed of the lightning discharge and weakly depends on the ionospheric ionization[ Furthermore\ the propagation speed during a cloud!to!ground return stroke can reach a speed of about 9[4 of the speed of light "Uman\ 0876#\ while the propagation speed of intracloud discharges is at least an order of magnitude lower[ The current pulse takes a few milliseconds to traverse the fractal\ of l 09 km in size\ with a speed of v:c 9[94 "Valdiva et al[\ 0886#[ Since the duration of the discharge scales at t ½ l:v\
891
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Fig[ 5[ Angular distribution of the time integrated irradiance per photon for a sprite located at 79 km[
Fig[ 6[ Schematic diagram for satellite observations of sprite[
the ionosphere could be heated at the same rate for a longer time if the fractal size is longer than above "a spider discharge can reach 099 km in length "Lyons\ 0885##[ For further estimates we consider the pulse width of about 4 ms[ This is much longer than a pumping time eff of the nitrogen vibrational to levels tpump 0:kex NN1 which ranges between 9[4 and 2 ms at the height 79 to 89 km for the electric _elds of interest "see Fig[ 0#[ Thus the intensity of the IR source is determined by the amplitude of the electric _eld and by the pulse width of the EMP from the lightning[ As mentioned in Section 1[0\ we consider two di}erent sprite models ] a cylindrical column irradiated by a rec! tangular EMP from lightning\ and spatially integrated ionospheric heating by the electric _eld induced by the
fractal lightning[ In fact\ these two models are consistent\ as is evident from the following consideration[ A sprite modeled by a cylindrical column centered at 79 km\ having a radius of 09 km and a height of 19 km\ with the ambient electron density Ne9 099 cm−2\ and if irradiated by a 4 ms EMP of the electric _eld just below the breakdown threshold\ generates the same number of vibrons as does the fractal lightning described in Section 1[0[ The latter lasts for a few ms and induces in the lower ionosphere an electric _eld below the breakdown threshold[ Thus\ from now on\ we use a simple model of a cylindrical sprite[ Using eqns "8# and "09# we _nd that in this case the energy collected by the above detector ranges between 09−4 and 09−3 erg for the electric _eld between 39 and 37 V:m[ Note that in the above cases the total energy pumped into the N1 vibrational levels in the overall irradiated volume ranges between 9[2 and 2 MJ[ The latter value looks like a possible upper limit for the EM energy absorbed in the lower ionosphere[ In fact\ according to Uman "0876# the energy released by a lightning discharge is given by U×Q\ where U is the potential di}erence created over distances of a few km in the vicinity of a thundercloud\ which is of the order of 097 V\ while the charge could reach\ say\ 099 C ^ thus a total of 09 GJ can be released[ Moreover\ about 09−3 of this energy can be converted into electromagnetic radiation[ Thus\ if the EMP radiated due to the lightning stroke is totally absorbed in the ionosphere\ up to a few MJ can be con! verted into electron heating which in turn mostly goes into the excitation of the N1 vibrational levels[ We next evaluate the signal to background ratio for the IR detector[ In order to do this we compute more accurately the density of the N1 vibrons excited by the EMP from lightning at the di}erent altitudes[ Those vib!
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
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Fig[ 7[ Sprite IR radiance per vibron\ time integrated along the satellite trajectory[ The sprite appears at b9 4> "a#\ and b9 09> "b#[
rons are transformed into the IR photons and radiated in space[ From our model "see Fig[ 4#\ on average 79) of the photons are released over a time of 199 s[ Thus\ neglecting the quenching of N1"n# molecules\ the mean intensity of the photon source is Q
0 1 ph
cm2 s
9[7Nvib:199 s[
"00#
Assuming a nighttime pro_le of the middle latitudes ionosphere as Ne9"cm−2# 099 exp ð"z"km#−79#:3[4Ł\ we compute the intensity of the local IR photon source for two di}erent values of the electron quiver energy\ 9[97 and 9[06 eV\ corresponding to under and slightly over the breakdown _elds "for the height 79 km these cor! respond to 39 and 37 V:m\ respectively\ while the break! down threshold is 34 V:m#[ Those intensities are com!
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G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
could help to direct the IR detector toward the sprite and investigate the early stages of the sprite[
Acknowledgments The authors are indebted to A[ V[ Gurevich\ K[ Papa! dopoulos and D[ Book for valuable discussions[ The work was supported by NSF Grant ATM 8311483[
References Fig[ 8[ Signal to background ratio computed for the electron quiver energy 9[97 eV "dashed trace# and 9[06 eV "solid trace#[
pared with the background source of the IR photons found by Kumer et al[ "0867# using the observations made by a rocket!borne radiometer[ The ratio of the IR intensities to the background source is shown in Fig[ 8[ The minimum at 74 km is related to the corresponding peak of the background source[ Notice that the local increase of the IR source due to the lightning discharge ranges between a few percent and a few tens of percent\ depending on the value of the electric _eld and of the pulse width[ The IR signal from a sprite could possibly be detected by a satellite!borne telescope\ using limb observations[
3[ Conclusions We have presented a model for the 3[15 mm infrared emission due to red sprites[ The model discusses the gen! eration of nitrogen vibrons due to the collisions of the nitrogen molecules with the electrons energized by the electric _eld from lightning\ followed by a transition of the nitrogen vibrons to the CO1"990# vibrational level\ which then radiates infrared photons of wavelength l 3[15 mm[ We have evaluated the intensity of the IR source based on the sprite parameters "lightning par! ameters#[ Moreover\ we have considered the propagation of IR photons from sprites through the optically thick atmosphere\ and computed the sprite radiance[ We then computed the amount of the IR energy collected by a satellite detector over~ying the sprite\ and estimated the signal to background ratio[ The model reveals that it is possible that the IR signal from a sprite could be detected by a state!of!the!art space infrared telescope using limb observations[ Such obser! vations could help to estimate the energetics and fre! quency of sprite appearances[ They could be specially helpful if combined with optical observations in the vis! ible range[ In fact a short "ms# optical pulse from a sprite
Bossipio\ D[J[\ Williams\ E[R[\ Heckman\ S[\ Lions\ W[A[\ Baker\ I[T[\ Boldi\ R[\ 0884[ Sprites\ ELF transients\ and positive ground strokes[ Science 158\ 0977Ð0980[ Hampton\ D[L[\ Heavner\ M[J[\ Wescott\ E[M[\ Sentman\ D[D[\ 0885[ Optical spectral characteristics of sprites[ Geophys[ Res[ Lett[ 12\ 78Ð81[ Huebschman\ R[K[\ Pardoe\ C[T[\ 0881[ The midcourse space experiment spacecraft[ Abstracts of 0881 Aerospace Design Conference\ 2Ð5 February 0881[ Irvine\ CA\ American Inst[ Aeronautics and Astronautics\ Inc[ Kumer\ J[B[\ James\ T[C[\ 0863[ CO1"990# and N1 vibrational temperatures in the 49 ³ z ³ 029 km altitude range[ J[ Geophys[ Res[ 68\ 527Ð537[ Kumer\ J[B[\ Star\ A[T[\ Wheeler\ N[\ Baker\ K[D[\ Baker\ D[J[\ 0867[ Evidence for an OH : N1 : CO1"n2# : CO1¦hn"3[2 mm# mechanism for 3[2!mm air~ow[ J[ Geophys[ Res[ 72\ 3632Ð 3636[ Lenoble\ J[\ 0882[ Atmospheric Radiative Transfer[ A[ Deepak Publishing\ Hampton\ Virginia[ Lyons\ W[A[\ 0883[ Characteristics of luminous structures in the stratosphere above thunderstorms as imaged by low!light video[ Geophys[ Res[ Lett[ 10\ 764Ð767[ Lyons\ W[A[\ 0885[ Sprite observations above the U[S[ High Plains in relation to their parent thunderstorm system[ J[ Geophys[ Res[ D[ 090\ 18\530Ð18\541[ Mende\ S[B[\ Rairden\ R[L[\ Swenson\ G[R[\ Lyons\ W[A[\ 0884[ Sprite spectra ^ N1 0 PG band identi_cation[ Geophys[ Res[ Lett[ 11\ 0522Ð1525[ Milikh\ G[M[\ Papadopoulos\ K[\ Chang\ C[L[\ 0884[ On the physics of high altitude lightning[ Geophys[ Res[ Lett[ 11\ 74Ð 77[ Milikh\ G[M[\ Valdivia\ J[A[\ Papadopoulos\ K[\ 0885[ Model of red sprite optical spectra[ Geophys[ Res[ Lett[ 13\ 722Ð725[ Nebel\ H[\ Wintersteiner\ P[P[\ Picard\ R[H[\ Winick\ J[R[\ Sharma\ R[D[\ 0883[ CO1 non!local thermodynamic equi! librium radiative excitation and infrared dayglow at 3[2 mm ] application to spectral infrared rocket experiment data[ J[ Geophys[ Res[ 88\ 09\398Ð09\308[ Pasko\ V[P[\ Inan\ U[S[\ Taranenko\ Y[N[\ Bell\ T[F[\ 0884[ Heating\ ionization\ and upward discharges in the mesosphere due to intense quasi!electric thundercloud _elds[ Geophys[ Res[ Lett 11\ 254Ð257[ Pasko\ V[P[\ Inan\ U[S[\ Bell\ T[F[\ Taranenko\ Y[N[\ 0885[ Sprites as luminous columns of ionization produced by quasi! electrostatic thundercloud _elds[ Geophys[ Res[ Lett[ 12\ 538Ð 541[
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894 Rairden\ R[L[\ Mende\ S[B[\ 0884[ Time resolved sprite imagery[ Geophys[ Res[ Lett[ 11\ 2354Ð2357[ Register\ D[F[\ Nishimura\ H[ and Trajmar\ S[\ 0879[ Elastic scattering and vibrational excitation of CO1 by 3\ 09\ 19 and 49 eV electrons[ J[ Phys[ B ] Atom[ Molec[ Phys[ 02\ 0540Ð 0551[ Rowland\ H[L[\ Fernsler\ R[F[\ Huba\ J[D[\ Bernhardt\ P[A[\ 0884[ Lightning driven EMP in the upper atmosphere[ Geophys[ Res[ Lett[ 11\ 250Ð253[ Sentman\ D[D[\ Wescott\ E[M[\ Osborne\ D[L[\ Hampton\ D[L[\ Heavner\ M[J[\ 0884[ Preliminary results from the Sprites 83 aircraft campaign ] 0[ red sprites[ Geophys[ Res[ Lett[ 11\ 0194Ð0197[
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Shultz\ G[J[\ 0853[ Vibrational excitation of N1\ CO\ and H1 by electron impact[ Phys[ Rev[ 024A\ 877Ð883[ Tsang\ K[\ Papadopoulos\ K[\ Drobot\ A[\ Vitello\ P[\ Wallace\ T[\ Shanny\ R[\ 0880[ RF ionization of the lower ionosphere[ Radio Sci[ 19\ 0234Ð0259[ Uman\ M[A[\ 0876[ The Lightning Discharge[ Academic Press\ Orlando[ Valdivia\ J[A[\ Milikh\ G[M[\ Papadopoulos\ K[\ 0886[ Red sprites ] lightning as a fractal antenna[ Geophys[ Res[ Lett[ 13\ 2058Ð2061[ Winckler\ J[R[\ Lyons\ W[A[\ Nelson\ T[E[\ Nemzek\ R[J[\ 0885[ New high!resolution ground!based studies of sprites[ Geophys[ Res[ Lett[ 090\ 5886Ð6993[
Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Model of infrared emission from sprites G[ M[ Milikh \ D[ A[ Usikov\ J[ A[ Valdivia University of Maryland\ Departments of Physics and Astronomy\ College Park\ MD 19631\ U[S[A[ Received 04 December 0886^ accepted 00 February 0887
Abstract A model of the 3[15 mm infrared emission due to red sprites is presented[ The model considers the generation of nitrogen {vibrons| due to the collisions of the nitrogen molecules with the electrons energized by the electric _eld from lightning\ followed by the transition of the nitrogen vibrons to the CO1"990# vibrational level\ with a lifetime much shorter than that of nitrogen[ The infrared photons of wavelength l 3[15 mm radiated by the CO1"990# propagate through the optically thick atmosphere ^ therefore\ this emission could best be observed from space[ The model computes the infrared radiance of sprites\ as well as the energy collected by a state!of!the!art space infrared detector\ and estimates the signal to background ratio[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ Introduction The recently discovered sprite phenomenon manifests itself as optical ~ashes sparkling above the top of giant thunderstorms at heights of from 49Ð84 km "Lyons\ 0883 ^ Sentman et al[\ 0884 ^ Bossipio et al[\ 0884 ^ Winck! ler et al[\ 0885#[ This phenomenon occurs due to the electromagnetic pulse or quasi!static _eld from the light! ning which energizes the ambient electrons[ The electrons then collide with the air molecules exciting their electronic levels followed by the optical emissions[ However\ less than 0) of the total absorbed electromagnetic energy of lightning is released through the optical emissions\ while most of this energy is pumped into the nitrogen vibrational levels having long lifetimes[ This energy then is transferred through resonant molecular collisions into the CO1"990# vibrational level with a much shorter life! time\ t 1[4 ms\ leading to the radiation of infrared photons of wavelength l 3[15 mm[ The radiated IR photons propagate through the optically thick atmo! sphere ^ therefore this emission is almost totally absorbed in the atmosphere beneath the sprite\ but could be detected from space[ Such an observation could provide us with interesting data about the energetics of sprites[ The objective of this paper is to present a model of the generation of the IR emission by red sprites\ and of the propagation of the IR photons through the atmosphere[
Corresponding author[ E!mail ] milikhÝastro!umd[edu
The intensity and angular distribution of the IR emission from red sprites is obtained from this model\ and the amount of energy collected by the state!of!the!art IR space detector along with the signal to background ratio is estimated[ In the next section we describe the model of the gen! eration of IR photons and their propagation through the atmosphere[ In Section 2 we discuss how the IR photons generated by sprites and then escaping from the atmo! sphere could be observed from space[ This is followed by the conclusions[
1[ Model description The model is based on the energization of ionospheric electrons by _elds generated by conventional lightning and applies equally well to energization by elec! tomagnetie "EM# or quasi!static _elds "Milikh et al[\ 0885 ^ Rowland et al[\ 0884 ^ Pasko et al[\ 0884#[ We consider the atmosphere as consisting of a set of hori! zontal slabs each having constant density and compo! sition[ The electromagnetic energy from lightning is released during a short pulse with the duration of a few ms\ inside a localized region in the upper atmosphere "Lyons\ 0883 ^ Sentman et al[\ 0884 ^ Rairden and Mende\ 0884 ^ Winckler et al[\ 0885#[ The model starts with a quasi!static or low frequency electric _eld from lightning at altitudes between 59 and 89 km\ and computes kin! etically the modi_cation of the electron distribution func!
S9906Ð8209:87:,*See front matter Þ 0887 Published by Elsevier Science Ltd[ All rights reserved PII ] S 0 2 5 3 Ð 5 7 1 5 " 8 7 # 9 9 9 2 2 2 Ð 8
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G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
tion by the electric _eld and the excitation of the N1 and CO1 n!th vibrational level by the energized electrons[ Once we have the excitation rates of the N1"n# and CO1"n# we introduce a Monte!Carlo model for the description of the photochemical processes leading to the emission of the CO1 IR photons\ followed by their di}usion through the atmosphere[ 1[0[ Excitation rates of N1"n#\ CO1"n# The basic input to the model is the electric _eld ampli! tude E9 at a particular altitude z characterized by the ambient electron!neutral collision frequency n9[ Here E9 is determined as E9 zSv Ev1 \ where Ev1 is the spectral energy density[ For a given value of E9 and n9 the asymp! totic stationary state of the electron distribution function f "v# is found by solving numerically the local kinetic FokkerÐPlanck equation "Tsang et al[\ 0880# for altitudes such that cos1 a ³ n1:V1 ] 1f "v# e1E91 0 1 v1n 1f −L"v#[ 1 1 1t 2m v 1v n ¦V1 1v
0
1
"0#
Here L"v# is the operator which describes the e}ect of the inelastic collisions\ V is the electron cyclotron frequency\ n is the electron collision frequency in the presence of the electric _eld\ while a is the angle between the electric and geomagnetic _elds[ Equation "0# is general and applies to quasi!static or EM _elds[ There are two ways of exam! ining the results of eqn "0#[ One is to specify an altitude\ and an electric _eld at that altitude\ and solve eqn "0# as a function of E9 in order to obtain the electron distribution function[ The alternative is to _nd the collision frequency n"E9\ z# explicitly from the FokkerÐPlanck code\ and pro! ceed with the self!similar solution for f "v# as a function of the electron quiver energy o½ "Valdivia et al[\ 0886#[ The latter is given by o½
e1E91 1
1
mðV ¦n Ł
$
0¦
V1 1
n
%
cos1 a [
n ex
7 eff s nknex kex
"2#
n 0
where the summation goes over all of the eight vibrational levels of the N1 ground electronic state[ Figure 0 shows the excitation rate coe.cients of the n!th vibrational levels of N1\ and the e}ective excitation rate coe.cient of N1"n 0#\ along with that of the CO1"990# level\ as a function of the electron quiver energy\ found for the height of 79 km[ Here the excitation rate coe.cient is normalized by the factor NCO1:NN1 in order to compare the rate of the direct pumping of the CO1"990# level with that of the excitation through the collisions with N1 "n 0# molecules[ Figure 0 reveals that at the range of the electric _eld which corresponds to red sprites the direct pumping of the CO1"990# level can be neglected[ We next compute the density of vibrationally excited nitrogen molecules N1 "n 0#\ or vibrons\ at di}erent altitudes as a function of the electric _eld and the pulse width tp\ taking into consideration the increase of the vibron abundance due to the electron impact ionization Nvib Ne9NN1
g
tp
e} kex "E9"t## exp "nie}"E9"t##t# dt
"3#
9
The quiver energy is the critical parameter that controls the behavior of the distribution function f "v# under an electric _eld at a given height[ In a case of electron ener! gization by a static electric _eld the parameter o½ reduces to E:n\ which in turn is controlled by the ratio E:N\ where N is the air density[ The next element of the model is the computation of the excitation rate for a particular vibrational level based on the value of f "v# found above[ We _rst compute the excitation rate coe.cient knex of the n!vibrational level of N1 by the electron impact[ knex 3pÐf "v#v2snex"v# dv
tation of the CO1"990# level using the excitation cross section from Register et al[ "0879#[ Note that the nitrogen molecules excited to a high vibrational level "n × 0# are involved in the exchange of vibrational quanta with the nitrogen molecules being in the ground state\ producing the ensemble of N1 "n 0#¦N1 "n 9# particles[ This process occurs much faster than the energy transfer from N1 "n 0# to CO1 due to the relatively low density of CO1 "NCO1: NN1 ¾ 2[2×09−3#[ Therefore the energy pumped into the vibrational N1 level leads to the production of N1 "n 0# molecules which then transmit their energy to CO1[ We introduce the e}ective rate coe.cient for the excitation of N1 "n 0# ]
"1#
where s is the excitation cross section of this level "Shultz\ 0853#[ The same approach is valid for the exci!
where Nvib is the vibrons| density\ Ne9 is the ambient elec! tron density\ NN1 is the density of nitrogen\ and the e}ec! tive ionization rate is nieff ni−na[ Here ni is the ionization rate while na is the rate of dissociative electron attachment to molecular oxygen[ Note that an important role is played by the total amount of vibrons generated in the volume irradiated by the electric _eld from the lightning[ In order to estimate the intensity of the source of vibrons we discuss below two cases[ One is a simple model of the sprite as a source of vibrons\ which occurs inside an uniform column and which is caused by an electromagnetic pulse "EMP# from the lightning[ The other is the spatially integrated source of vibrons due to the electric _eld induced by the fractal lightning\ as described by Valdivia et al[ "0886#[
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
786
Fig[ 0[ Excitation rate coe.cients of the N1"n# vibrational levels "solid lines#\ the e}ective rate constant keff ex de_ned by eqn "2# "points#\ and excitation rate coe.cient of the CO1"990# level "broken line#[ The latter is normalized by the factor NCO1:NN1[
In the _rst case we assume that the sprite occurs inside a cylindrical column centered at 79 km\ having a radius of 09 km and a height of 19 km "Sentman et al[\ 0884#\ and caused by a 4Ð09 ms pulse[ Moreover\ based on spectroscopic observations Mende et al[\ 0884 ^ Hampton et al[\ 0885# and the model by Milikh et al[ "0886#\ we assume that the electric _eld from lightning at 79 km ranges between 24 and 49 V:m\ while atmospheric break! down occurs at E9 × 34 V:m[ In the second case we consider the source of vibrons produced by the low altitude fractal lightning[ For a given fractal lightning we compute _rst the electric _eld in the near and far zones[ Then we consider the _eld propa! gation into the lower ionosphere\ including self!absorp! tion[ Finally\ using the excitation rate coe.cients as func! tion of electron quiver energy given in Fig[ 0\ we compute the time!integrated spatial distribution of the vibron source[ Details of the computations can be found in the paper by Valdivia et al[ "0886#[ Here we present an exam! ple taking a horizontal lightning of fractal dimension D 0[33\ which propagates with the speed of 9[94 times the speed of light at 4 km altitude\ and discharges 099 C[ Figure 1 shows the fractal structure of the lightning along with the electric _eld distribution at the height 59 km normalized over its mean value\ and the vibron density pattern normalized over the density\ averaged over the whole computational box[ The total amount of vibrons in the volume irradiated by the electric _eld from lightning serves as an input to the next element of the model which describes the IR photon emission and propagation through the atmo! sphere[
1[1[ IR Emission and atmospheric propa`ation The process following the excitation of N1 by the elec! tron impact involves a set of photochemical reactions "Kumer and James\ 0863#[ They are shown in Fig[ 2\ where straight lines show the bimolecular reactions while the wavy lines reveal photon emission or absorption[ According to this scheme\ the vibrationally excited N1 molecule either loses the energy in a set of vibrationÐ vibrational transitions\ or vibrationalÐtranslational reac! tions with particles other than CO1\ or transfer the exci! tation to CO1 molecules exciting the 990 state[ In turn\ the CO1"990# molecule either transmits the excitation back to N1 or radiates an IR photon of the 3[15 mm wavelength[ The IR photons propagate through the atmosphere until they are absorbed by CO1 molecules[ The photon absorp! tion and reradiation will go over and over again until they are lost due to thermal heating or escape from the atmosphere[ The escaping photons can reach a space! borne detector[ The above model is treated by using the Monte!Carlo method[ Namely\ some vibrons are gen! erated in the code\ and then their transitions are followed[ The probability density at each transition is chosen by the Monte!Carlo procedure\ which will be discussed below[ First\ we choose between two channels ] energy tran! sition from the excited N1 to CO1 from the one side\ and the loss of the vibron due to a set of collisions with the particles other than CO1\ on the other[ The probability of each of the above channels is proportional to their reaction rates[ In fact\ the rate r of the energy transition of the vibron to CO1 is equal to the rate constant k0 times the local CO1 density[ At the same time\ the loss rate rlos
787
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Fig[ 1[ The discharge patterns of fractal dimension D 0[33 "a#\ the electric _eld distribution "b#\ along with the vibron density pattern "c#[ The charge is Q 099 C and the current propagates with the speed 9[94 c\ c being the speed of light[
788
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Fig[ 2[ Schematic diagram of the relevant set of photochemical reactions[
of the vibron due to collisions with other than CO1 par! ticles is described by the following equation which con! siders reactions revealed by Fig[ 2 ]
0
rlos k1NO¦k2NO1 0−
k?2NN1
1
k?2NN1¦k4NO¦k3NH1O
"4#
where the relevant reaction rate coe.cients are taken from Kumer and James "0863#\ while the vertical dis! tribution of air density and temperature is adopted from the U[S[ Standard Atmosphere "0865#[ Therefore\ the probability of the energy transition from N1 "n 0# to CO1 is r:"r¦rlos#[ We next estimate the time required for this transition[ In order to do this we introduce the probability density P0"t# k0NCO1 exp"−k0NCO1t#\ so t that Ð 9 P0"t# dt 0[ Then we relate the value of Ð9 P0"t?#dt? to the random number {num| between 9 and 0[ Conse! quently\ the time under consideration is determined as 0 log "0−num#[ t − k0NCO1
"5#
Proceeding\ we discuss how the model treats the emis! sion and absorption of the IR photons[ The energy quan! tum transmitted to CO1 from the N1 "n 0# molecules excites the J rotational level of the CO1"990# state[ The J!level is chosen by the Monte!Carlo procedure such as described above using the relative population of
rotational states from Kumer and James "0863#\ and relating the corresponding probability density to the chosen random number[ The probability of the photon radiation is determined by comparing the Einstein coe.cient of the CO1"990# state with the rate of energy transmission from CO1"990# back to N1[ The frequency of the radiated photon is obtained using the selection rules and by taking into account the Doppler frequency shift DnD0 due to the ther! mal motion of the radiating molecule[ We also compute the direction of the photon propagation by using the Monte!Carlo procedure[ The photon could then be absorbed by another CO1 molecule and populate a rotational level J?\ depending on the photon frequency[ The absorption cross section is given by ] 0 s s9 1
"1J?¦0#e−J?"J?¦0#B:T
s
"1J?¦0#e−J?"J?¦0#B:T
J? 9\1\[[[ 1 ge} 1 e}
Dop 0
g ¦"Dn
¦Dn1Dop#1
\ "6#
where s9 1[4×09−03 cm1 "Kumer and James\ 0863#\ B 9[28 cm−0 is the rotational constant\ while Dn0Dop and Dn1Dop are the Doppler shifts of the radiating and absorb! ing molecules[ Finally\ ge} is the e}ective broadening
899
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
which includes both collisional and Doppler broadening[ It can be represented in the following form "Lenoble\ 0882# ] geff"cm−0#
60
6×09−1P"atm#
X 1 0 T 299
¦ nph
1
1
17
vT 0 c zlog 1
\
"7#
where P is the air pressure\ vT and c are the thermal molecular speed and the speed of light\ respectively\ while nph"J?# is the photon frequency[ Note that we consider the main band and the major isotope of CO1[ The role played by the hot bands and minor isotopes was discussed by Nebel et al[ "0883#[ The above model allows us to compute the photon ~ux radiated into space[ It is characterized by the sprite|s IR radiance Lspr\ which represents the amount of energy traversing a unit of surface per unit time in a solid angle inclined at the angel u relative to the source[ In fact\ assuming that the electric _eld does not change in time we obtained using eqn "3# that ½ "u\ t#ÐNvib dV Lspr"u\ t# hnL
"8#
where hn is the energy of the radiated IR photon\ L "u\ t# is the radiance per vibron given in m−1 rad−0 s−0\ which takes into consideration the IR photon losses due to air heating\ and the integral is carried out along the volume irradiated by the electric _eld from the lightning[
2[ Discussion We apply the computational scheme discussed above to _nd the probability for an IR photon to escape from the atmosphere[ First we compute the fraction of photons escaping from di}erent atmospheric heights[ This is shown in Fig[ 3\ which reveals that almost all photons escape into space from the top part of sprite located above 79 km\ while photons radiated by the sprite bottom\ located below 59 km\ are totally absorbed[ We then compute the number of photons escaping during each time interval\ normalized over the total number of vibrons generated in the model\ i[e[\ the probability den! sity of the IR photons released into space by the sprite[ The results of the computations for two di}erent sprite locations\ at 79 and 89 km altitude\ are shown in Fig[ 4[ In this _gure we limit the {observation time| to 0999 s after the sprite occurs[ A short "ms# pulse from lightning produces a broad pulse "½199 s# of IR emission[ This is due to two physical e}ects ] _rst\ the considerable time required for excited N1 molecules to collide with CO1 molecules in order to transmit their energy to the CO1"990# state and\ second\ the long di}usion time of the
IR photons in the optically thick atmosphere[ Notice that the temporal distribution of the probability density of the IR photons radiated into space by the sprite is steeper when the sprite is located at a higher altitude "compare curves 0 and 1 in Fig[ 4#[ Thus\ the time derivative of the IR intensity could provide us with the height of the sprite\ if observed by a detector in space[ Figure 5 shows the angular distribution of the time integrated radiance per photon of a sprite located at 79 km altitude[ This _gure reveals that the photons escape preferentially in the vertical direction along the air density gradient\ which minimizes the number of photon col! lisions with the CO1 molecules leading to IR absorption[ According to our computations\ the angular distribution depends weakly on the sprite|s height[ Proceeding\ we estimate the amount of IR energy from the sprite collected by a space IR detector\ along with the signal to background ratio[ As an example we consider the SPIRIT III IR telescope ~ying on the MSX satellite[ The latter has a nearly Sun!synchronous orbit at an alti! tude of H 777 km "Huebschman and Pardoe\ 0881#[ The telescope|s _eld of view is D8 0> by Dc 2>\ so it views a sprite within the solid angle Vdet 8[02×09−3\ while the dish has an area of Sdet 789 cm1[ Thus\ during the time t that the detector could collect sprite!generated energy\ the amount of IR energy from the sprite given by Ispr SdetVdet
g
t
dt?Lspr"u"t?#\ t?#[
"09#
9
The angle u between the vertical through the sprite and line connecting the sprite and the satellite is related to the angle b\ which describes the satellite location\ as shown by Fig[ 6[ The angle b varies at b b9¦vsatt:"RE¦H#\ where vsat is the satellite velocity\ RE is the Earth radius\ and b9 corresponds to the location of the satellite when the sprite appears[ Correspondingly\ the zenith angle u changes from the initial value uini u"b9# to the _nal zenith angle u_n[ The latter is due to the fact that the IR photons from the sprite cannot propagate over the visible horizon[ Then we compute the sprite|s IR radiance per vibron by using eqn "09#\ modeling the sprite by a vertical column extended from 69Ð89 km\ assuming that the sat! ellite trajectory is located in the same plane as the sprite\ and that the sprite appears at b9 4 and 09>[ This is shown in Fig[ 7"a#\ "b# where the arrows mark the time when the satellite is over~ying the sprite[ Figure 7 reveals that about 79) of the total IR energy from the sprite is collected by the detector ~ying toward the sprite during the _rst 099 or 199 s after the sprite|s appearance[ The amount of IR radiation collected by the detector is reduced signi_cantly when it moves away from the sprite[ This is due to the reduction of the IR ~ux with time and angle "see Figs 4 and 5#\ and due to the increase in the ~ux divergence with the distance from the sprite Rds[ Note that the data in Fig[ 7 are averaged over 19 s[
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
890
Fig[ 3[ The transparency of the atmosphere for the 3[15 mm photons radiated from di}erent heights[
Fig[ 4[ Probability density of IR photons radiated into space by a sprite[ Trace 0 corresponds to a sprite located at 79 km\ and trace 1 corresponds to a sprite located at 89 km altitude[
In order to _nd the IR energy which could be collected by the satellite detector observing sprites\ we estimate _rst the intensity of a sprite as a source of vibrons[ Note that the ionosphere electrons\ which excite the vibrons\ are energized by either the QS electric _eld or by the EMP from the lightning[ However\ those processes occur on a di}erent timescale[ While the QS _eld exists in the ionosphere during the local relaxation time trel which ranges between 0 and 9[2 ms at the height 89Ð89 km "Pasko et al[\ 0885#\ the pulse width of the EMP from
lightning is determined by the size and propagation speed of the lightning discharge and weakly depends on the ionospheric ionization[ Furthermore\ the propagation speed during a cloud!to!ground return stroke can reach a speed of about 9[4 of the speed of light "Uman\ 0876#\ while the propagation speed of intracloud discharges is at least an order of magnitude lower[ The current pulse takes a few milliseconds to traverse the fractal\ of l 09 km in size\ with a speed of v:c 9[94 "Valdiva et al[\ 0886#[ Since the duration of the discharge scales at t ½ l:v\
891
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
Fig[ 5[ Angular distribution of the time integrated irradiance per photon for a sprite located at 79 km[
Fig[ 6[ Schematic diagram for satellite observations of sprite[
the ionosphere could be heated at the same rate for a longer time if the fractal size is longer than above "a spider discharge can reach 099 km in length "Lyons\ 0885##[ For further estimates we consider the pulse width of about 4 ms[ This is much longer than a pumping time eff of the nitrogen vibrational to levels tpump 0:kex NN1 which ranges between 9[4 and 2 ms at the height 79 to 89 km for the electric _elds of interest "see Fig[ 0#[ Thus the intensity of the IR source is determined by the amplitude of the electric _eld and by the pulse width of the EMP from the lightning[ As mentioned in Section 1[0\ we consider two di}erent sprite models ] a cylindrical column irradiated by a rec! tangular EMP from lightning\ and spatially integrated ionospheric heating by the electric _eld induced by the
fractal lightning[ In fact\ these two models are consistent\ as is evident from the following consideration[ A sprite modeled by a cylindrical column centered at 79 km\ having a radius of 09 km and a height of 19 km\ with the ambient electron density Ne9 099 cm−2\ and if irradiated by a 4 ms EMP of the electric _eld just below the breakdown threshold\ generates the same number of vibrons as does the fractal lightning described in Section 1[0[ The latter lasts for a few ms and induces in the lower ionosphere an electric _eld below the breakdown threshold[ Thus\ from now on\ we use a simple model of a cylindrical sprite[ Using eqns "8# and "09# we _nd that in this case the energy collected by the above detector ranges between 09−4 and 09−3 erg for the electric _eld between 39 and 37 V:m[ Note that in the above cases the total energy pumped into the N1 vibrational levels in the overall irradiated volume ranges between 9[2 and 2 MJ[ The latter value looks like a possible upper limit for the EM energy absorbed in the lower ionosphere[ In fact\ according to Uman "0876# the energy released by a lightning discharge is given by U×Q\ where U is the potential di}erence created over distances of a few km in the vicinity of a thundercloud\ which is of the order of 097 V\ while the charge could reach\ say\ 099 C ^ thus a total of 09 GJ can be released[ Moreover\ about 09−3 of this energy can be converted into electromagnetic radiation[ Thus\ if the EMP radiated due to the lightning stroke is totally absorbed in the ionosphere\ up to a few MJ can be con! verted into electron heating which in turn mostly goes into the excitation of the N1 vibrational levels[ We next evaluate the signal to background ratio for the IR detector[ In order to do this we compute more accurately the density of the N1 vibrons excited by the EMP from lightning at the di}erent altitudes[ Those vib!
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
892
Fig[ 7[ Sprite IR radiance per vibron\ time integrated along the satellite trajectory[ The sprite appears at b9 4> "a#\ and b9 09> "b#[
rons are transformed into the IR photons and radiated in space[ From our model "see Fig[ 4#\ on average 79) of the photons are released over a time of 199 s[ Thus\ neglecting the quenching of N1"n# molecules\ the mean intensity of the photon source is Q
0 1 ph
cm2 s
9[7Nvib:199 s[
"00#
Assuming a nighttime pro_le of the middle latitudes ionosphere as Ne9"cm−2# 099 exp ð"z"km#−79#:3[4Ł\ we compute the intensity of the local IR photon source for two di}erent values of the electron quiver energy\ 9[97 and 9[06 eV\ corresponding to under and slightly over the breakdown _elds "for the height 79 km these cor! respond to 39 and 37 V:m\ respectively\ while the break! down threshold is 34 V:m#[ Those intensities are com!
893
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894
could help to direct the IR detector toward the sprite and investigate the early stages of the sprite[
Acknowledgments The authors are indebted to A[ V[ Gurevich\ K[ Papa! dopoulos and D[ Book for valuable discussions[ The work was supported by NSF Grant ATM 8311483[
References Fig[ 8[ Signal to background ratio computed for the electron quiver energy 9[97 eV "dashed trace# and 9[06 eV "solid trace#[
pared with the background source of the IR photons found by Kumer et al[ "0867# using the observations made by a rocket!borne radiometer[ The ratio of the IR intensities to the background source is shown in Fig[ 8[ The minimum at 74 km is related to the corresponding peak of the background source[ Notice that the local increase of the IR source due to the lightning discharge ranges between a few percent and a few tens of percent\ depending on the value of the electric _eld and of the pulse width[ The IR signal from a sprite could possibly be detected by a satellite!borne telescope\ using limb observations[
3[ Conclusions We have presented a model for the 3[15 mm infrared emission due to red sprites[ The model discusses the gen! eration of nitrogen vibrons due to the collisions of the nitrogen molecules with the electrons energized by the electric _eld from lightning\ followed by a transition of the nitrogen vibrons to the CO1"990# vibrational level\ which then radiates infrared photons of wavelength l 3[15 mm[ We have evaluated the intensity of the IR source based on the sprite parameters "lightning par! ameters#[ Moreover\ we have considered the propagation of IR photons from sprites through the optically thick atmosphere\ and computed the sprite radiance[ We then computed the amount of the IR energy collected by a satellite detector over~ying the sprite\ and estimated the signal to background ratio[ The model reveals that it is possible that the IR signal from a sprite could be detected by a state!of!the!art space infrared telescope using limb observations[ Such obser! vations could help to estimate the energetics and fre! quency of sprite appearances[ They could be specially helpful if combined with optical observations in the vis! ible range[ In fact a short "ms# optical pulse from a sprite
Bossipio\ D[J[\ Williams\ E[R[\ Heckman\ S[\ Lions\ W[A[\ Baker\ I[T[\ Boldi\ R[\ 0884[ Sprites\ ELF transients\ and positive ground strokes[ Science 158\ 0977Ð0980[ Hampton\ D[L[\ Heavner\ M[J[\ Wescott\ E[M[\ Sentman\ D[D[\ 0885[ Optical spectral characteristics of sprites[ Geophys[ Res[ Lett[ 12\ 78Ð81[ Huebschman\ R[K[\ Pardoe\ C[T[\ 0881[ The midcourse space experiment spacecraft[ Abstracts of 0881 Aerospace Design Conference\ 2Ð5 February 0881[ Irvine\ CA\ American Inst[ Aeronautics and Astronautics\ Inc[ Kumer\ J[B[\ James\ T[C[\ 0863[ CO1"990# and N1 vibrational temperatures in the 49 ³ z ³ 029 km altitude range[ J[ Geophys[ Res[ 68\ 527Ð537[ Kumer\ J[B[\ Star\ A[T[\ Wheeler\ N[\ Baker\ K[D[\ Baker\ D[J[\ 0867[ Evidence for an OH : N1 : CO1"n2# : CO1¦hn"3[2 mm# mechanism for 3[2!mm air~ow[ J[ Geophys[ Res[ 72\ 3632Ð 3636[ Lenoble\ J[\ 0882[ Atmospheric Radiative Transfer[ A[ Deepak Publishing\ Hampton\ Virginia[ Lyons\ W[A[\ 0883[ Characteristics of luminous structures in the stratosphere above thunderstorms as imaged by low!light video[ Geophys[ Res[ Lett[ 10\ 764Ð767[ Lyons\ W[A[\ 0885[ Sprite observations above the U[S[ High Plains in relation to their parent thunderstorm system[ J[ Geophys[ Res[ D[ 090\ 18\530Ð18\541[ Mende\ S[B[\ Rairden\ R[L[\ Swenson\ G[R[\ Lyons\ W[A[\ 0884[ Sprite spectra ^ N1 0 PG band identi_cation[ Geophys[ Res[ Lett[ 11\ 0522Ð1525[ Milikh\ G[M[\ Papadopoulos\ K[\ Chang\ C[L[\ 0884[ On the physics of high altitude lightning[ Geophys[ Res[ Lett[ 11\ 74Ð 77[ Milikh\ G[M[\ Valdivia\ J[A[\ Papadopoulos\ K[\ 0885[ Model of red sprite optical spectra[ Geophys[ Res[ Lett[ 13\ 722Ð725[ Nebel\ H[\ Wintersteiner\ P[P[\ Picard\ R[H[\ Winick\ J[R[\ Sharma\ R[D[\ 0883[ CO1 non!local thermodynamic equi! librium radiative excitation and infrared dayglow at 3[2 mm ] application to spectral infrared rocket experiment data[ J[ Geophys[ Res[ 88\ 09\398Ð09\308[ Pasko\ V[P[\ Inan\ U[S[\ Taranenko\ Y[N[\ Bell\ T[F[\ 0884[ Heating\ ionization\ and upward discharges in the mesosphere due to intense quasi!electric thundercloud _elds[ Geophys[ Res[ Lett 11\ 254Ð257[ Pasko\ V[P[\ Inan\ U[S[\ Bell\ T[F[\ Taranenko\ Y[N[\ 0885[ Sprites as luminous columns of ionization produced by quasi! electrostatic thundercloud _elds[ Geophys[ Res[ Lett[ 12\ 538Ð 541[
G[M[ Milikh et al[:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 784Ð894 Rairden\ R[L[\ Mende\ S[B[\ 0884[ Time resolved sprite imagery[ Geophys[ Res[ Lett[ 11\ 2354Ð2357[ Register\ D[F[\ Nishimura\ H[ and Trajmar\ S[\ 0879[ Elastic scattering and vibrational excitation of CO1 by 3\ 09\ 19 and 49 eV electrons[ J[ Phys[ B ] Atom[ Molec[ Phys[ 02\ 0540Ð 0551[ Rowland\ H[L[\ Fernsler\ R[F[\ Huba\ J[D[\ Bernhardt\ P[A[\ 0884[ Lightning driven EMP in the upper atmosphere[ Geophys[ Res[ Lett[ 11\ 250Ð253[ Sentman\ D[D[\ Wescott\ E[M[\ Osborne\ D[L[\ Hampton\ D[L[\ Heavner\ M[J[\ 0884[ Preliminary results from the Sprites 83 aircraft campaign ] 0[ red sprites[ Geophys[ Res[ Lett[ 11\ 0194Ð0197[
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