Strong small scale turbulence is observed in the Crab Nebula from the temporal pulse broadening data. It is shown that the strong 30 Hz pulsar wave can parametrically excite instabilities near the lower hybrid frequency in the thermal plasma of the Crab Nebula with a characteristic wavelength of the order of the scale size a of the turbulence observed. These instabilities provided a coupling mechanism between the pulsar wave and the Nebula plasma

Lee, L. C.

English

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SMALL SCALE TURBULENCE 
IN THE CRAB NEBULA: 
X-802-71-89 
PREPRINT 
EVIDENCE OF LOWER HYBRID 
PARAMETRIC INSTABILITIES 
DRIVEN BY THE PULSAR WAVE < 
( N1S1-Tft -I-7129~) S81LL SC1LB 10PBULBNCE IN 871-22030 
THI C&~B lifU11: !Ylt!IC! OF LOWES tiyeBID 
fAB18!lBIC 15S71BILITIBS DBIVEN BY THB 
fOLSJi WAY! (1151) 15 p HC 102/8F 101 Onclas 
CSCL OlB G3/90 25819 
L. C. Lee 
MARCH 1977 
-- GODDARD SPACE FLIGHT CENTER ---
GREENBELT, MARYLAND 
I 
X-602-77-69 
PREPRINT 
SMALL SCALE TURBULENCE IN THE CRAB NEBULA: 
EVIDENCE OF LOWER HYBRID PARAMETRIC INSTABILITIES 
DRIVEN BY THE PULSAR WAVE 
L. C. Lee * 
Theoretical Studies Group 
Goddard Space Flight Center 
Greenbelt, Maryland 
*NAS/NRC Resident Research Associate 
SMALL SCALE TURBULENCE IN THE CRAB NEBULA: 
EVIDENCE OF LOWER HYBRID PARAMETRIC INSTABILITIES 
DRIVEN BY THE PULSAR WAVE 
ABSTRACT 
Strong small scale turbulence with size a '"" 50 km is observed in 
the Crab Nebula from the temporal pulse broadening data. It is 
shown that the strong 30 Hz pulsar wave can parametrically 
excite instabilities near the lower hybrid frequency in the thermal 
plasma of the Crab Nebula with a characteristic wavelength of the 
order ,')f the scale size a of the turbulence observed. The growth 
rate of the excited waves is about 0.2 - 10 SE;C. These instabilities 
provide a coupling mechanism between the pulsar wave and the 
Nebula plasma. 
1 
SMALL SCALE TURBULENCE IN THE CRAB NEBULA: 
EVIDENCE OF LOWER HYBRID PARAME1 RIC INSTABIUTIES 
DRIVEN BY THE PULSAR WAVE 
(I) INTRODUCTION 
Recent observations of angular broadening and temporal broadening of the 
Crab Nebula pulsar's radiation indicate the existence of a scattering region of 
plasma turbulence within the Crab Nebula (Counselman & Rankin 1972. Rankin 
& Counselman 1973, Lyne & Thorne 1975, Vandenberg 1976). UlJmg the results 
of VLBl (very-long-baseline interferometer) data, Vandenberg (1976) was able 
to show that the angular broadening and the temporal broadening of the pulsar 
radiation occur in different locations along the line of sight to the Crab pulsar. 
Vandenberg (1976) then proposed the electron density fluctuation in Filament 
159 of the Crab Nebula (Trimble 1970) as the cause of the temporal pulse 
broadening. However, Vandenberg (1976) made a mistake in estimating the 
scale size and the amount of plasma irregularity in the filament. Lee (1977) 
developed a thin screen scintillation theory for a spherical wave and corrected 
Vandenberg's errors. Lee (1977) pointed out that if the pulse broadening is due 
to the thermal plasma in the Crab Nebula (of thickness -0.5 pc), tl'.en the rms 
electron denSity fluctuation ,1N in the Nebula is 
tlN/cp ~O.OI cm- 3 
o 
(1) 
and the scale size of electron density fluctuation, a, is given by 
• 
. 
l 2!!§!!!!! 
2 
I/~o -15 km (2) 
where ~o is tile rms phase fluctuation caused by the Nebula. From the pulse 
shape observed, one must have ~o;> 1. In the next sectioo, we estimate ~o to be 
-1-5. Thus ~ ~ 0.01-0.05 cm-3 and a ~ 15-75 km. However, if the pulse 
broadening is due to Filament 159 (with thickness -0.01 pc) as proposed by 
Vandenberg, then one has in the filament AN/~o ~ 0.08 cm- 3 and a/tPo ~ 15 km. 
It is the purpose of this paper to propose a mechanism to explain the 00-
served small scale turbulence in the Crab Nebula. We will find in the next 
section that the averaged thermal electron density in the Nebula is N th ~ 0.05 
\!m- 3 • Thus, for the small scale turbulence (a l=t: 50 km), ~N ~ Nth and the 
turbulence is very strong. We believe that some external driving force is 
needed to excite such a strong turbulence at a relatively small scale size. The 
natural candidate for the driving force is the strong 30 Hz pulsar wave. In 
Section II, we briefly describe the phYSical environment in the Crab Nebula. In 
Section III, we show that the 30 Hz pulsar wave can indeed excite several para-
metric instabilities near the lower hybrid frequency in the Crab thermal plasma 
with a characteristic wavelength l=t: a. Thus these inst,r.bilities can explain the 
observed strong turbulence and, in addition, can provide a coupling mechanism 
between the pulsar and the nebula. In Section IV, we discuss and sum up the 
results. 
( 
'J 
(II) CRI'" NEBULA 
(a) THERMA L PLASMA 
The Crab Nebula contains three components: relativistic charged parti-
cles, magnetic field and thermal plasma. The magnetic field in the Crab 
Nebula has been observed to be Bneb - 3 X 10 -4 Gauss (e. g., Fazio et a1. 
1972, Fazio 1973) and the total energy of relativistic electrons is about 10 49 
erg (Trimble & Rees 1970). The thermal electrons in most filaments have a 
density, Nth::: 103 cm- 3 and a temperature, Te ::: 104 oK (Woltjer 1958, 
Trimble 1970~. The thermal plasma density outside filaments is much lower 
than that in filaments. Shklovskii (1966) pointed out that in the Crab Nebula 
the averaged thermal plasma density, Nth" 1 cm- 3 , becs.use of the absence 
of the gross Faraday depolarization of radio emission. From the motion of 
wisps, Scargle (1969) argued that Nth::: 5 X 10-4 cm- 3 near Wisp 1 and 
Nth ::- 0.05 cn:- 3 at the northwest edge. From the fact that the expansion 
speed of the nebula at the outer edge, Uneb "" 1. 5 X 108 cm/sec, much less 
than the speed of light, one would obtain Nth::: 0.05 cm- 3 for Bneb =t:: 10-4 
Gauss near the outer edge of the nebula (Kahn 1971). Assume that Nth > ~N 
3 
(Lee & Jokipii 1976). From Eq. (1), we also find that Nth> 0.01 cm- 3 • Thus 
we take Nth ::- 0.05 cm- 3. If ~N ..;; Nth as assumed, Eq. (1) gives !Po III:< 1-5. 
Therefore, ~N ::: O. 01-0. 05 cm -3 and a ::- 15-75 krn. Assuming energy equi-
patitioll between magnetic field and thermal plasma in the Nebula, we have the 
thermal plasma temperature T ::- 5 X 107 OK. Thus in the following 
e 
i ! 
j~~-~.~~_ .. _~L~ .. ~~~t~~--~--------------------------~r~ 
4 
calculations, we will take Bneb ~ 3 X 1O-4Gauss, Nth ~ 0.05 cm- 3 and 
T e ~ 5 X 107 OK. 
(b) 30 Hz PU LSAR WAVE 
Crab pulsar is generally believed to be a compact oblique magnetic 
rotator and therefore emits strong 30 Hz electromagnetic waves (Ostrfker & 
Gunn, 1969). Just outside the speed-of-light cylinder, the wave electric field 
E is strong and the parameter 
where c is the speed of light. e the electron charge, Me the electro~ mass 
and Wo = 30 Hz is the frequency of the pulsar wave. This 30 Hz wave can 
accelerate particles, leaking from the pulsar magnetosphere, to relativistic 
velocity (Ostriker & Gunn 1969. Ferrari & Trussoni 1975). The lack of cir-
cular polarization in the optical continuum radiation from the nebula strongly 
suggests that the pulsar wave does not propagate freely more than ~20" (or 
6 X 1017 cm) from the pulsar (Landstreet & Angel 1971, Rees 1971). Observa-
tions of a 'valley' in the brightness near the pulsar at both radio and optical 
frequencies suggest there exists a cavity around the pulsar swept clear of 
thermal gas and static magnetic field by the 30 Hz wave (Scargle 1971). 
Thus we have a picture for the Crab Nebula: the 30 Hz pulsar wave 
propagates from the pulsar and interacts, at a distance of several 10 17 cm 
away from the pulsar. with the Crab Nebula, which is composed of thermal 
plasma and magnetic field. At the edges of the central pulsar cavity. the wave 
t 
i 
1 
I I 
• I 
5 
pressure becomes comparable to the thermal and magnetic pressure of thE! 
Nebula and the wave field E -= B
neb • As the wave propagates into the Nebula, 
the pulsar wave will be damped and/or reflected such that E -< B blnside the 
ne 
nebula. 
Several authors have considered the mteraction between the pulsar wave 
and the nebula. Rees & Gunn (1974) suggested that the 30 Hz wave would be 
absorbed due to synchrotron absorption by the downstream shocked high energy 
plasma with a mean free path -3 X 10 12 cm. Ferrari (1974) and Dobrowolny 
& Ferrari (1976) discussed the plasma instabilities excited by the interaction 
between the 30 Hz pulsar wave and the relativistic wind leaking from the pulsar 
magnetosphere. These instabilities occur only near the equatorial plane of the 
pulsar spin axis where wisps are observed. Max (1973) discussed parametric 
instabilities excited by the 30 Hz wave in the nebula thermal plasma, neglecting 
effects of the nebula magnetic field. The existence of a static magnetic field 
allows more possible modes to be excited. In the next section, we will conSider 
the interaction between the 30 Hz wave and the thermal plasma in the presence 
of the nebula magnetic field. 
(III) PARAMETHlC EXCITATION BY THE PULSAR WAVE 
In ~he Crab Nebula where Nih ~ 0.05 cm -3, Bneb ~ 3·X 10-4 Gauss, and 
T e ": 5 X 10 7 OK. we have the electron cyclotron frequency ne::::: 840 Hz, ion 
cyclotron frequency Hi::::: 0.5 Hz, electron plasma frequency wpe::::: 1800 Hz, 
ion plasma frequency w"i:-:- 40 Hz, Debye length >"0": 2 km, and the ion sound 
f 
· , 
6 
speed Vs :: 8 X 10 7 cm/sec. The lower hybrid frequency wLH 1s gi~en by 
(3) 
Thus 'he 30 Hz pulsar wave, when propagating in the thermal plasma of the 
Crab Nebula, acts as a pump wave with frequency Wo in the vicinity of the 
lower hybrid frequency (w
o 
~ wLH ). Parametric decay instabilities with 
pump frequency near wLH has recently been studied (Kindel et a1. 1972, Porkolab 
1974, Hesegawa" Chen 1975, Berger &: Perkins 1976, Ott 1975, Trlpathi et al. 
1977) and observed in the rf heating of tokamak plasma (Hooke &: Bernabei 1972, 
Chu et a1. 1973, Chang" Porkolab 1974, Porkolab et ale 1977). 
Consider now the parametric dec9.y of the pump wave (with w ::0 30 Hz 
o 
and wavenumber .Is 0) into two daughter waves ( wI' ~) and ( w2' ~2). Since the 
wavelength of the incoming pump wave is much greater than that of the daughter 
waves (_lO6 cm), the pump will be treated in the dipole approximation (k "t< 0), 
-0 
(4) 
We then have ~1 + ~2 = !&, "t< O. Under the dipole approximation, Porkolab 
(1974) has investigated possible decay channels for a pump wa"e near lower 
hybrid frequency. According to Kindel et a1. (1972) and Porkolab (1974), the 
pump wave will excite (a) purely growing and (b) decay parametrlc instabilities. 
In both cases, one of the two daughter waves is a lower hybrid wave. Let 
( w 2 • ~2) be the excited lower hybrid wave satisfying the following dispersion 
relation 
where wLH the lower hybrid frequency is given by Eq. (3), Mj Is the proton 
mass and k211 is the component of the wavevector along the magnetic field. 
(a) PURELY GROWING PARAMETRIC INSTArtLlTIES 
7 
(5) 
For the purely g"rowing instabilities which occurs for Wo < ""2 ' the real 
part of WI is zero and the minimum threshold is given by 
(6) 
where U = CE IB b is the E X B drift velocity, V == (kT Im.)V, is the ion 
o ne - - ,. I 
sound speed and 'Y2 is the linear damping rate of the lower hybrid wave. The 
damping rate 'Y 2 for the lower hybrid wave is 
(7) 
where y ;: w/kU Vte and Vte is the electron thermal velocity (Tripathi et al. 
1977). Since y > I, '2 is very small and therefore the condition in Eq. (6) is 
easily satisfied for the pulsar wave propagating in the Nebula. The growth rate 
for the purely growing instabilities well above threshold is given by 
(8) 
where ~ .:: ( Wo - w Lk )/ w lk and x '" kU I w2• Taking x::: 1, (-tJ', = 10-4 for 
example, we havt:! 1-::::: 0.07 wUi ~ 1. 4 sec-I. In general the Lme scale for 
the purely growing instahilities in the nebula is of order of 1 sec. 
I 
Jl 
8 
(b) DECAY PARAMETRIC INSTAmUTIES 
For decay instabilities, the pump wave beaUng with a lower bybrid wave 
form low-frequency 10n quaslmodes with frequencies near or above the Ion 
cyclotron frequency (Porkolab 1974). These quaslmodes may be of two types: 
(0 kinetic modes such that their parallel phase velocities are near the electron 
thermal velocity (w/kll ::= V te)' in which case the driving mechanism is essen-
tially nonlinear electron Landan damping; (li) nonresonant fluid-like modes such 
that w/k II » V Ie' 
For kinetic modes, the threshold is the same as that for the purely grow· 
lng instabilities in Eq. (6). Let WI = wR + 1"". The growth rate wJ well 
above threshold is 
where y :: wR/k ll VIe :::- 1. However, from the numerical calculation of 
Porkolab (1974), WI is peaked around U IV sill: 1. The maximum growth rate 
can then be obtained from Eq. ( ..,) by setting U IV I :; I, "',(max):= 0.02 "'2 
(9) 
::= 0.6 sec -1. In general the time scale for the growth of the kinetic modes is 
about 1-10 sec. 
The flUid-like quasimodes ( wR/k II > V I~ ) can easily be excited even in 
weak pump fields (Porkolab 1974). The growth rate "'I is given by 
(10) 
9 
Again the growth rate wI is peaked at U IV. =:: 1. From Eq. (10), the maximum 
growth rate is wl(max) ~ 0.2 Wo =:: 6 sec-I. Thus the time scale for fluid-
like modes in the nebula is about 0.2-5 sec. 
The wavelength of the excited waveR for both purely growing and decay 
parametric instabilities is given by k AD ~ O. 1-0. 5 (Porkolab 1974, Tripathi 
et a1. 1977). In the Crab Nebula, we have AD =:: 2 lan, and therefore the wave-
length A 0:: 211' /k ~ 20-120 km, which is approximately the scale size a ( -15-75 
km) of the turbulence observed. 
(IV) DISCUSSIONS 
In the above calculation, we assume that the 30 Hz pulsar wave reaches 
the lower hybrid resonant region (wo ~ wLH) directly. If the thermal plasma 
density increases gradually near the inner edge of the Crab Nebula, then before 
reaching the resonant region, the 30 H7 pulsar wave will pass through a layer 
where ""0 ::: ""PI: and may be reflected. However, since the pulsar wave Is 
strong (Q >- 1), it can penetrate through this layer (Max and Perkins 1971) and 
reach regions where ""0 ~ ""lIl' We note that. whereas in the tokamak plasma 
the lower hybrid resonant region is only a thin layer, the lower hybrid resonant 
region in the Nebula is very broad since in the Nebula ""LH is determined 
mostly by the mab'1letic field (wLH ~ (nrO j )~), not by the plasma denaity 
(or ""Pi)' 
Next we consider if the temporal pulse broadening is due to the thermal 
plasma fluctuations in Filament 159 as proposed by Vandenberg (1iJ76). In 
10 
Filament 159 where Tc ~ 10" oK and NIh:"" 100 cm- 3 , the Deby~ length is 
AD :: 70 cm. If waves are parametrically excited in the filament bv the pulsar 
wavt:, then the wavelength A would be only -(1-4) X 104 em which is too small 
to account for the observed turbulence. 
ThJS the 30 Hz pulsar wave can excite parametric instabilities near the 
lower hybrid frequency in the thermal plasma of the Crab Ne')ula w:th a eharac-
teristic wavelength of the order of the scale size a (-15-75 km) of the turbulence 
observed. The time scale for the growth of the excited waves is about 0.2-10 
sec. The excited waves propagate nearly perpendicular to the Nebula field 
(k /I « k) and therefore will propagate radially outward with a phase velocity 
10 8 em Isee superposed on the convective motion of the Nebula plasma. 
Since the dampinll: rate of the lower hybrid waves as given by Eq. (7) is very 
low, these waves will propagate to the outer part of the Nebula. We note that 
as in the rf tOKamak plasma heating, the electron.,. and ions in the nebula will 
be heated b\' this mechanism. If we identify the scale size a of the observed 
turbulence with the wavelength of the excited waves (a - >. 2,,' ~ (0.1-0. !j) x 
2 TT AD)' then >.[} gives the plasma temperature in the Nebula, '.'.: - 107 -108 OK, 
wh ieh is cons islent with I hat detcrm ined by energy equipat l ition in Section II. 
Thus the lower hybrid instabilities also provide a coupling mechanism between 
the :10 Hz wave and the ;urroundin~ Nebula, in which the energv ;lnd momentum 
-~.f 
of thl~ pulsar wave t:'ansferred to the plasma in the t\ebula. 
i 
f 
I 
! 
o 
II 
ACKNOWLEDGMENT 
The author would like to thank Drs. T. J. Birmingham, H. H. Cben, 
J. E. Felten, M. L. Goldstein, R. F. Hubbard, F. C. Jones, T. G. Northrop, 
C. S. Liu, D. P. Stern, and C. S. Wu for belpful discu8sions. 
I 
I 
j aT j It 
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Small scale turbulence in the Crab Nebula: Evidence of lower hybrid parametric instabilities driven by the pulsar wave

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