3OURNAL OF GEOPHYSICALRESEARCH
VOL. 70, NO. ]3
3ULY 1, 1965
Drag and Propulsion of Large Satellites in the Ionosphere'
An Alfv(n PropulsionEngine in Space
S. D. DRELL,1 H. M. FOLEY,
2 AND M. A. RUDERMAN
3
Jason Division, Institute/or De/ense Analyses
Abstract. There is a motionally induced charge separation in a conductor moving across
magnetic field lines. This charge may be conducted away, resulting in adc current flow
through the conductor if it moves through a plasma. The generation of Alfv•n waves is a
mechanism particularly effective for circulating the charge for very large conductorsmoving
in or above the earth's ionosphere.This mechanism is studied in this paper and when applied
to the analysis of the orbit of the Echo satellite is found to give rise to a significant damping
of the motion as mechanical energy is converted to that of Alfv•n radiation. The calculated
drag is comparable to that observed for the orbit of Echo I and attributed in earlier studies
entirely to the mechanical drag of considerablenonionized atmosphericdensity. Perturbations
in electron density associated with this current flow may in appropriate circumstancesbe
detectable even thousands of kilometers away from such a high altitude satellite. The drag
can be changedto a propulsionmechanismwhen a sourceof electrical power is available on
the satellite. Up to fifty per cent of the expended power is available for pushing a space
vehicle acrossan ambient magnetic field.
1. INTRODUCTION
A conductormoving acrossa magneticfield
B0 in a vacuum will have an induced charge
separationsufficientto cancel the electric field
E = (v X B0)/c seenby a co-movingobserver.
When the surroundingmediumis a plasma,there
exist possiblemechanismsfor chargeto be conducted away, resulting in adc current flowing
throughthe conductor.In this paperwe consider
the circulationof chargeby meansof the generation of Alfv•n waves, a mechanismwhich is
particularly effectivefor very large conductors
movingin or abovethe earth'sionosphere.
When
appliedto a study of the Echo satelliteit gives
rise to a significantdamping of the orbit as
mechanicalenergyis convertedto that of Alfv•n
radiation. The calculateddrag is equal to that
observed for the orbit of Echo 1 and attributed
be tappedasa battery. With a sourceof electrical
power the drag force can be converted to a
propulsionmechanism:the satellite pusheson
the earth's magnetic field without any emission
of propellant.
We exploit the qualitative analogy betweena
collisionlessplasma in a magnetic field and a
seriesof transmissionlines parallel to the magnetic field. The moving conductor with its
motionally induced charge separation is, in a
sense,in successivecontact with different transmissionlines as it moves.It inducesan impulse
(Alfv•n wave) traveling alongthe magneticfield
which carriesa chargeseparationand essentially
completes the circuit in which the moving
conductoris adc battery.
Initially we shall idealize the conductor so
that it not only has no internal resistancebut
also no work function to prevent the outward
flow of electronsin responseto an appropriately
in earlierstudiesentirely to the mechanicaldrag
of considerable
nonionizedatmosphericdensity. directed electric field normal to its surface. The
In an appropriatelydesignedsatellite the drag
force can be altered by variationsof an internal proposedmechanismis appropriately modified
for real conductors for which the work function
resistance,or the associateddc current flow can
is significant.
Alfv•n waves,magnetohydrodynamic
disturbx Stanford Linear Accelerator Center, Stanford ances of frequency much less than the ion
University, Stanford,California.
cyclotronfrequency9i = eBo/Mic, propagate
2 Physics Department, Columbia University,
one-dimensionallyalong the direction B0. A
New York, New York.
a Physics Department, New York University, small disturbancepersistswith constant ampliNew York, New York.
tude or until damped by collisionsamong elec3131
3132
DRELL, FOLEY, AND RUDERMAN
Fig. 1. Conductor
movingwithvelocityvcin thex direction
perpendicular
to magnetic
field
linesBoalongthe z directionleadingto a chargeseparationand mottonalelectricfield E in
the y direction.
trons and ions in the plasma. We have not
found the solution offered here in earlier studies
of related problems; these have been twodimensional,or involved only motion parallel
to B0, or posedboundary conditionsthat the
plasma was not disturbed at large distances
from the conductor,none of which are valid for
our solution. Moreover, we find in the rest
frame of the moving conductor a steady dc
motion of an ideal conductorthrough a plasma
with an impressedmagneticfield B0 in order to
establishthe appropriate regime of parameters
and the nature of the Alfv•n disturbance. We
make the followingapproximations:
1. The conductormovingthroughthe plasma
is 'ideal'--i.e., it has zero internal resistance
and work function; it maintains the charge
separationrequiredto producethe field
current rather than a static situation.
r = --(vc X B0)/½
2. IDEAL CONDUCTOR IN A LOSSLESS PLASMA
For the idealized conductor (see section 1)
moving through a collisionlessplasma in a
direction perpendicularto the field lines (Figure
1), the Alfv•n disturbanceextendsout in wings
making an anglea with the field lines suchthat
In the oppositecaseof a conductor
with internal
resistancemovingthrougha mediumof higher
conductivity,the chargeswould be 'bled' from
Top
Bottom
tan
where v• = speed of the conductor, and Va -----Alfv•n speed.The motional electric field
r =
radar
-
t4
z
+ ++++
---m,..
vc
x
in the conductoris cancelledby a chargeseparation as illustrated in Figure 1. In the collisionless
plasma, with infinite conductivity along the B0,
or z, direction,but with zero conductivityperpendicular to B0, current wings as shown in
Figure 2 are createdalongwith a chargeseparation as required to maintain a constant• field
along the wing equal to that at the conductor.
Anomalous
(1)
echoes obtained
from
Yl
iono-
sphericdisturbances
associated
with Echo I may
alsobe related to thesepredictedAlfv•n 'wings,'
althoughfurther studiesare desirableto confirm
their presence.We comment more on this in
section 5.
Let us first analyze the fieldsproducedby the
Fig. 2. Alfv•n wings generatedby an ideal conductor in a collisionlessplasma.
DRAG
AND
PROPULSION
IN THE
the conductorby the mediumat a morerapid
rate than couldbe maintained,and the field E
woulddecrease.
Conditionsfor the applicability
IONOSPHERE
3133
'øøø1
•.\
I •'x
-x•
•'•'l
•'•
•
of the 'ideal' conductor limit are discussed in
I
•,
•
I
Daysunspot
max'mum
----- Night
....•t maximum
•
.....
Daysunset
minimum
E600
'•'%'•
••'• •' Night
sunset
minimu
the following.
2. A linearizedsolutionis obtained;i.e., we
retain terms linear in the Alfv•n fields only.
This requiresin particularthat the magnetic
field h associated with the Alfv•n disturbance
'
.- ....
be smallalongside
the constant
appliedmagnetic
field Bo.
I I I III
I0
Sincewe consideronly Alfv•n disturbances,
we shall consideronly low frequencywaves
/
i l/
I I I I1•1•
102
I I I IIIII
103
104
105
Electron
•llision frequency
Ve, collision/sec
such that
450
4O(
co< 12i----eBo/Mic
35(
•,• 4000 cps(Bo/0.4 gauss)(Mp/Mi)
(2)
%
E
'
Sunspotmaximum
'--'---
Sunspot
minimum
300
ß
where 12i, the ionic gyrofrequency,
is given
numericallyin termsof the ion to protonmass ..• 250
ratioM•/M• andof thefieldstrength
B0divided
20O
by a mean value at the earth's surface. This
meansco• 2000 cpsfor 1000 to 2000-km altitude
150
orbitsin the proionosphere
and co• 200 cps
I00
I JillIll
0,1
for orbits of altitudes as low as 160 km in the
I I IIIIII
1.0
I
I0
-- I I IIIII1'
102
I0 3
104
Ioncollision
frequency
vi, colllsion/sec
F layer. For this reason we are limited to a
3. Electron and ion collisionfrequencies
study of large conductors,since the typical inFig.
the ionosphereversus altitude; taken from
frequencyradiated by a moving source of Johnson [1961].
dimension
L alongits velocityand speedvcis
(3)
and for vc• 7 X 10• cm/secfor a satellitein
earth orbit and • • 1000cps,L • 10 meters.
Electronand ion collisionfrequencies,
r, and
r• in the ionosphere,
are shownas functionsof
altitude in the curvesof Figure 3 taken from
Johnson [1961] and must be included in a
X (,•'1/•2,r, or ,'•,co/•2,)
• 0
(6)
detailedstudy,sincethey may be comparable
•th •. They are, however,negligiblerelative O'o----'n•e•r•/m is the specificelectricalconductivityof the plasma,r, = 1/r,, r• = 1/r•,
to the gyrofrequencies,
i.e.
andn, • n is the electron(ion)numberdensity
rg/• < 10-•
r,/•, < 10-•
(4) in the neutralplasma.•0 is the conductivity
and the ratios (4) are neglectedthroughout. parallel to the magneticfield or in the absence
In this regime the dielectric behavior of the
magnetized
plasmais givenby the tensor
definedby
of a magneticfield and as a function of altitude
variesroughlyas shownin Figure4 takenfrom
Johnson[1961].
In the li•t
Di = eiiEi = E i + 4•j•/i•
(5)
for the •th frequencycomponent'
e•l = ezz•
1 q-•
4•r•o
iw (1 +
1
of a lossless,
i.e., collision-free,
plasma,
e•• 1 -- w•/w•
(7)
wherew•• (4rhea/m)
TMistheplasma
frequency,
and
DRELL,
3134
FOLEY,
AND RUDERMAN
iooo
900
i
800
Day
sunspot
maximum
--.--Night
sunspot
maximum
700
E
','
ß
•
....Day
sunspot
minimum
-----Night
sunspot
minimum
600
/
i •
,!
ß
500
I:,ii
4OO
•.'•'1
•00
.•..•.•;.•'•
2OO
ß
I
I00
10-15
I 0-12
10-II
I 0-10
I 0-9
Specific
electrical
conductivity
%, abmho/cm
I 0-8
Fig. 4. Specific electrical conductivity of the ionosphere (zero field conductivity) versus
altitude; taken from Johnson [1961].
reflectspropagationonly in the directionparallel
4•rnM ;c2
Bo2
to B0.
We want now to solve the Maxwell equations
I + 4•rpic2
-- I + c•'/va
•' (8)
Bo2
V x r = -
•--
V.D
V XB= i)/cd-4•-j,/c (10)
where
(•oV•.,,),• -•
definesthe Alfv•n velocity in terms of the magnetic field strength B0 and the ionic mass
density pi ---- nMi. Numerically the plasma frequency varies from c%• 6 X 107 cps at altitudes of a few hundred km corresponding
= 4•rp.
V.B=0
electrondensitiesof .--5 X 103/cm
s.The Alfv•n
with D and E related by (5) and (6) and with
j. and p, representingthe current and charge
sourceprovided by the moving conductor.The
current in the moving conductor of Figure 1
flows mainly in the y direction to maintain the
charge separation. We work in a linearized
approximation in the field strengths E and
velocities rise from values of va •
h = B-- B0;f• = 1].
to electron densitiesof up to 106/cm
3, to
w• •
4 X 106cps at altitudes of 1600 km and
cm/secat 300-kinaltitudeto v• •
at 1600-km
altitude.
2 X
107
10• cm/sec
The value at 1600 km is
not accuratelyknown, owingto uncertaintiesin
the ionic compositionat the high altitudes. We
return to this point later. The ratio of parallel
to transversecomponentsof e is very large for
the parametersof interest to us,
Fourier transforming (10) we have
k x •(•., •,,,,o)= o•_h
(•_,•,,,,o)(•)
k X h = •oD 4•rij,(k_L,
k,,)•(•0
-- k.vc)
c
c
(lib)
[e,,/•_[> 10•
(9)
eilkllEll
Ji- ezkz'E• = 4•rp.
(11c)
Essentially the electronsare wrapped around
the magneticfield lines,free to move only along Taking the curl of (11a), inserting (lib) and
the Bo direction,as are the ions.This large ratio (11c), and defining Ez = Ez • -t- Ez", where
DRAG AND PROPULSION
k. ßE•
•
ez
k.E. • andkx X E•
k, -- kz2
•
k. X E•'
r,
velocity in the medium.
In this losslessplasma approximation, the
equations for E and h read
4•r 0
(12a)
c2 Ot Vñ'js(x -- Vct,y, z)
--k.k2)E.k
t"
=
4•-i
½
• ',o•e -- •','c) I• X •_•/a
Xv•)
Ex'r)
hi'= c(ki
(k'
(15a)
at
xr•)
0• v•)(V•
4;r 0
(•,)
c20t •_
• ell- k•_-- klleñ]"-0 (12c)
n. = •(• x r)•/(•.•c)
3135
gives a medium at speeds greater than the light
Ez
4srkip.•
IN THE IONOSPHERE
(•)
x •(:• - ,,ct, y, •)
(1,•,)
fi• = -c(V x r)•
(•)
fi, -- --c(Vx X rx)
(15d)
with Ell = 0. Equation 15b describesthe net
radiation of energy into a magnetoionicwave
with velocity va only if vc > va. As we noted,
this radiation is analogous to the Ccrenkov
radiation, or sonicboom. Equation 15a radiates
(12e)
General features of the solution can be identi-
fied in (12). For a plasma with low transverse
conductivity which approachesthe behavior of
a losslessmedium satisfying (9), (12a) reduces
to a one-dimensional wave equation for a
transverse electric field propagating along the
energy out along the direction of B0 for all
values of vc if the idealized conductor moves
acrossthe magneticfield lines and plucks them
like violin strings. This solution satisfies the
equation
B0 directionwith velocityc/eJ/'. •' Vaaccording
Bo.(Vx X r•) = 0
to (8) and with frequency
o• -- k'vc
(13)
and longitudinal wavelength
1 = va
Va
xll--•1-•
• - k.v•
(16)
so that a two-dimensionalsliceperpendicularto
B0 givesthe electricfield from a two-dimensional
chargedistribution(Vx' Ex). Accordingto (15a),
the Alfv•n fields are functions only of the
(14) variables
This is the Alfv•n wave. Accordingto (13), for the
dominanttransversewave numbersk. •' l/L,
Va
where L is the dimension of the current source
and propagateas describedearlier in Figures
along the direction of motion, the frequencyis 1 and 2. The wingsmove out from the conductor
co•, vc/L, as wasclaimedearlierin (3). Thereis, with VaalongB0 and form an anglea suchthat
in addition, according to (12b) an isotropic
tan a =
distu-'rbancewhich is a solution of the ordinary
d'Alcmbcrtian wave equation, also with Alfv•n as illustrated there.
velocity va.For a steady sourcecurrent generated
For the magnitudesof the fields, we know
by a moving conductorwith v• < va,the solution
is a localizeddisturbancefalling off as 1/r'. at
from the first of (10) that the tangential com-
ponent of • is continuousacrossthe conductor
large distancesfrom the source;this is the case surface and is fixed by the motional electric
applyinghere. When, however,v• > va,a radia- field in the conductoraccordingto (1):
tion pattern is created that is the analog of the
Cerenkov radiation for a chargepassingthrough
E = --(v• X Bo)/C
(17)
3136
DRELL, FOLEY, AND RUDERMAN
¾
i
/
-'v',v
-
M
Fig. 5. An/-bar conductoridealized as two fiat rectangularplates with indicated dimensions
chargedto maintain the potential differenceV.
Sincewe are dealingwith longwavelengthmodes different if the moving sourceis the I bar or a
solid plate of the same dimensions.Only the
of field excitationand by (14)
fringingfieldsnear the verticaledgesare different, as is indicatedin Figure6. In particularthe
the detailed three-dimensionalgeometric shape power radiated into the Alfv•n-type wavesis
of the moving conductoris of no relevance.We essentiallythe samein both cases;the effective
radiatingarea neednot be filled with conductor.
can write an exact solution for an /-bar conFor the magneticfield, (15c) and (17) give
ductor described as in Figure 5 by two fiat
1/kll >> 1/k•_ •
L
rectangular plates of dimensions L and N,
separated by a distance M, and charged to
maintain a potential differencein the y direction
of
v =
(
between the plates plus correctionterms near
the edgesfrom the fringingfieldspluscorrections
vanishingas one movesout into the radiation
=
zone far from the body due to local currents.
The solution consistsof growing parallel wings
with a chargesufficientto keep the differencein
potential between the wings always V. The
curlesspart of Ex, which is the solution of (15a),
has the functional
=
A smallratio of the Alfv•n field [hito the earth's
field B0 is a necessarycriterion for the linear
approximationwe have made. For satellitesin
earth orbit, we have
form
_,ct
_,_z,
•
and is the solutionof the two-dimensiOnalproblem correspondingto Figure 5, i.e. two plates
of width L and separationM making an angle
4 X 10-2 at 200-km altitude
10-a at 1600-km altitude
alwaysvery small next to unity.
The sufficiencycondition for the validity of
a = tan-• (Vc/V,,)with B0. A constantpotential
the neglectof higher-ordertermsdependson the
difference V is maintained between the wings, interval over which we need to be sure of an
so that Ell -- 0, and a constant current j8 flows
accurate solution. For computations of the
through the I bar in the y direction. Only out
current flow through the moving satellite, its
along the horizontal arms of the I bar in the x
drag, and the power radiated into the Alfv•n
direction does the current j8 decrease (i.e.,
waves, the solution should be valid out to a
•'x'jx.• • 0) as current flows into the wings.
The
vertical
current
in
the
source
must
be
distanceof about 1 wavclcngth/2•r--• Va/00
away from the conductor.Over this interval hx
exactly sufficient to supply the charge needed
may causea changein directionfor the flow of
for the two growingwingstips, i.e.
charge
awayfromthez axisby anangle•'•h_t./Bo
I., --'•
E•_Lc2v
2•'v•
-
ccL
2,rv•
and a resultant transverse displacement of
Bo
(hzv•)/(Bow).For this not to exceedthe transversedimensionof the Alfv•n wave, we need
We note that the field Ex that moves down
the tube in the B0 direction is not particularly
h•_v,• Bow/kx =Bov,
DRAG AND PROPULSION
IN THE IONOSPHERE
3137
Fig. 6. Fringing fieldsdifferingfor I bar or solid plate conductors.
or
hj_/B0 • •)c/•a
(20)
stant z) of the Alfv•n tube (wing) moves as if
through an incompressiblefluid. In the co-
moving frame the electric field vc X Bo/C
Only for an ideal conductor with no internal
together with that of an infinitesimally thin
resistancemoving in a collisionless
plasma in charge distribution looks like that in Figure 7.
which the inertia of the electronsis negligible The drift velocityv of ionsin the crossedelectric
can the left-hand side of (20) gain even equality and magnetic fields is
with the right-handside.
For an irregularly shaped three-dimensional
cE X Bo
ideal conductorfor which v•/va<< 1, only the
So2
projection onto the x-y plane is relevant: the
wings and their charge distribution form a far from conductor. As we have seen in (15), for
cylindrical tube inside of which there is a uniform
V½•
electric field which, in the co-moving frame,
exactlycancelsv• X Bole. Thus in the moving
frame the net E is zero everywherein the conductor and on its surface,except along the line
wherethe conductortouchesthe extended'paper
thin' cylindrical tube which carries the charge
separation.
As far as the motion of the chargedplasma is
concerned,each two-dimensional slice (at con-
and Etan = 0 on the surfaceof the tube. There-
Va,V_l_ X El = 0 far from the conductor
fore •'.v
= 0 and v• = 0 on the tube surface.
3. VOLTAGE•
POWER•IMPEDANCE•
AND
CURRENT EXPRESSIONS
From the strength of the magneticfield (19)
we readily compute the power radiated in the
Alfv•n
disturbance
to be
¾
E-. Eo= C
Fig. 7. Flow of plasmaaroundAlfv•n tube in co-movingframe.
3138
DRELL, FOLEY, AND RUDERMAN
conductorsmoving through a real plasma such
P= (1/47r)h22(ML)va
(21)as the ionosphere:
- Bø2
vc•
(ML)
Effect of collisionsin the plasma.
2•r
where 2MLv• is the volume filled per secondby
an energydensityh2/4•r;the factor2 takesinto
accountthe existenceof wingsextendingin both
directionsalongB0. Combiningwith (18) for the
potential differencebetween the top and bottom
wing, we computethe current flow:
The effective impedanceof the plasma for this
current flow is then defined as
z-
V
-
Effect of a finite work function limiting the
electroncurrent flow out throughthe surface
of a conductor.
Effect
of finite
internal
resistance in
the
conductor.
Finally we presentsomerealisticdrag calculations, as well as somenovel speculations
on how
to take advantage of these magnetohydrodynamic ionosphericdisturbancesfor orbit control
(acceleration or deceleration and orientation)
and for power generationor storage.
Effect o• collisionsin the plasma. From the
left-hand side of (12a) we extract the dispersion
relation for wave propagationin the medium
2
In terms of thesefamiliar quantities of electrical
circuit theory, the Alfv•n wings can be interpreted as one-dimensionalopen-ended transmission lines of impedance Z across which a
potential V is applied. In this ideal limit of a
losslessmedium, there is an infinite resistance
between the upper and lower line (or Alfv6n
wing) and zeroresistancealongthem. Corrections
due to collisionsleading to a finite transverse
ionic conductivity are computed in the next
= e•-•kl12
co
D- 4•v•
i c2 vc
23c•-4•
c Bo =
esu/cm
•
v•,•__•
--
The l•rge electronic surface charge density
•rises
from
ex •
1 + c:/vd (•th v•lues between10a•nd
the
enormous
dielectric
constants
10•). The totM charge density given by div. E is
smallerby (v•/c):; i.e., •lmost •11the electronic
charge density is c•ncelled by the ions. This
l•rge pol•fiz•bility of the heavy pl•sm• ions
•11 always result in •n ionic displacementmuch
smaller th•n the wing separation•s long as the
verficM dimensionof the body, M, satisfies
v•/M << •
4. R•
P•SM•
•ND COND•CXO•
COmplexioNs
We turn now •o the important practicalfactors
modifying our discussionwhen •pplied to re•l
(25)
In the losslesslimit describedin (7), ($), and
(9), this reducedto
kll = oo/va
(26)
appropriate to a pure Alfv•n wave. Retaining
the dampingtermsin (6), we find the corrections
to (26) are appreciable
if 1/oorior
section.
Finally we compute, using Gauss's law, the
electronicsurface charge density in the wing as
k• •
=
become comparable to unity. The damping of
our Alfv•n waves due to the first factor is always
very smallfor the rangeof parametersof interest;
as is seenfrom Figure3, 1/•y• • 10-• at about
160 km, and •10 -a above altitudes of 30•400
km, for frequenciesin the hundreds of cycles
per secondrange, correspondingto conductors
with dimensions
L • v•/• • 50 meters.Neglecting 1/•
(( 1 and introducing(6) into (25),
we include the effect of the term in (27):
=-•a2
+
-
In (28) the effectsof electroninertia and electron
collisions are both taken into account. The cor-
rection factor arising from electron inertia is
small for large conductorsin 160-500 km orbits,
since• • 5 X 10• cps and
k•c •
(kx• + k••)c•
• •c•
DRAG
AND
PROPULSION
IN
THE
IONOSPHERE
3139
for •0 < 103cpsand for 1/k•,• M > 10 meters.
This means that the wings cannot be considered to have zero thickness,but will spread
to a thickness of perhaps ten meters. The
imaginary (damping) correction from electron
Z=Z1+IZ12+
2ZlZ
2
collisions is also small for these conditions when
1/kj. •> 10 metersaccordingto Figure 3.
For higher altitude orbits above 1000 km, the
dampingterm i/oor•in (28) is < 1 in magnitude
•- 2Z1+ Z2 for Zl/Z 2 << 1
Fig. 8. Transmissionline analogy for the
Alfv•n wings.
for •0 •> 10 eps, correspondingto conductor
dimensionsno larger than L • 1000 meters (for limit of the ionic conductivity perpendicularto
the Echo 1 with L • 30 meters at an altitude of
the magnetic field direction when [2•r• >> 1.
1600 km, the correctionis less than a few per The finite transverseionic conductivity between
cent). The real factor in (28), however, leads to the top and bottom layers of the Alfv•n wings
a major change in the radiated Alfv•n wave in Figures i and 2 allows the damping and
unless1/ki •> 75 meters;the simple Alfv•n termination of the disturbance. Viewing these
solution propagating in one dimension only wings as a simple transmissionline, these terms
appliesonlyto the modeswith ki •< 1/75 meters are, respectively, the resistance cc (ax•)-• and
-• between the
and with frequency•0 •> 10 eps, and we restrict the capacitancecc (k_•o'_•5/O'o)
our attention to these. Thus the wing thickness lines, i.e. the impedanceZ•. in Figure 8 in addi• alongthe line
for these parameters will, if geometrically pos- tion to the inductanceZ• o: Va/C
sible, spread to more than 75 meters in order to through which the current flows.
Even if the moving conductorhas dimensions
carry the required currents,despitethe electron
inertia and the reduced density of electrons suchthat inertial term of (27) is smaller than 1,
the fact that
above 1000 km.
For an object such as Echo 1 with effective
dimensions
of L •
M
•
30 meters
it is nonzero means that
in the
dispersionlaw
for the
+
(28')
current sourceas definedin Figure 5, this means
a reductionin the field strength, sinceonly the the secondterm on the right-hand side plays a
modeswith kx < 1/75 meters• L-•/2.5 can role at large z: the Alfv•n pattern spreadsin the
be described by the Alfv•n solution discussed xy plane as it propagates in the z direction.
here. Since the power spectrum is flat in the For subsequentcalculationsof currentsand drag
transverse wave number kx = (k•, k•.) out to through the conductor, it is sufficient that
Hi/L, the powerradiatedin the Alfv•n disturb- the proposed solution be adequate out to
ancemustbe reducedby a factor• (30/75)•'•
1/6 from the valuecomputedin (21).
z •-• v•/oo•
(v•/v•) L. (In the transmission
line
analogy, the moving conductor remains in
We have presented these numbers in some contact with a given transmission line for a
detail in order to make clear the very approxi- time L/v, duringwhichthe signalhas traveled
mate and qualitative nature of our numerical a distancev• L/v• in the z direction.What hapresults and to show their sensitivity to the pens after that does not affect the forces and
various atmospheric parameters, variable in currents in the conductor.) From (28') the
time and imperfectlyknown, that have beenused. perpendicular spreading of E. is such that for
A useful analogy that may help provide a • = •k•
physic91
pictureof this behavioris suggested
in
the low frequencylimit such that •0r• --* 0 and
wr, --* O. Then (28) can be rewritten as
(M2
= -•----•-ax
t/a
½
-- ki 2
i
0'0
c IZOO•v•jz •< 1
Therefore the Alfv•n wave spreads ultimately
to a lateral
dimension d which increases as zTM.
(29) For ki ,• lfd, v•• 10• cmfsec,v, • 10• cm/sec,
and •%•'--• 10•3sec-•., correspondingto n, •5
wherea0 = ne*r,/rais the freefieldconductivity 103•cc,we have
)<
due to electronsin the plasma,as definedearlier,
and a• = ne•r•/M•(•r•) • is the low frequency
d •> 10•/Sz
•/• cm
(30)
3140
DRELL, FOLEY, AND RUDERMAN
At a distance of z ,-• l0 s km, d •> 200 meters,
which is an increase in size of less than a factor
of 10 for a conductor such as Echo 1 of size
•,30 meters. By conservationof energy, there
is a correspondingreduction by a factor •10
in the field strengthsE x and hx of the Alfv6n
wave at these distances. This result suggests
that, for very high altitude satellites where
collisionsare not dominating,the wingsmay be
detectableout to very great distances.Further,
as the Alfv6n wave penetratesto lower altitudes
where the ion density increasesand Vadecreases,
energy conservationimplies that in the WKB
approximation D•_E•_Va
remains approximately
constant.Thereforethe chargeseparationassociated with the wave increasesas (Va)-•"'. In this
estimate we have neglectedthe possiblegrowth
of the transverse
dimension of the Alfv6n
wave
becauseof the neglect of higher-orderterms in
(•).
Effect o/ finite work/unction limiting thecurrent
flow throughthe sur/aceo/ a conductor. In the
upperwing of the Alfv•n disturbancein Figure 2,
a negative current flowsout from the conductor,
while in the lower wing the negative flow is
consideringthe Alfv•n wave emissiontogether
with the photoelectricemissionduring the daylight hours only. The flux of photonsfrom the
sun deposits0.140 watt/cm•' as the total irradianceabovethe atmosphereat the earth's mean
distancefrom the sun. This correspondsto the
total radiation [Johnson,1961]from a blackbody
at a temperature of 5800øK (although the
spectrum is close to that of a blackbody at
6000øK). The numbers of incident photons per
cm•' with energiesabove 2, 3, and 4 electron
volts are 1.2 X 10•7, 3 X 10•6, and 3.5 X 10•5,
respectively.The corresponding
maximum photoelectric
currents
for these work
functions
are
20•, 5•, and 0.6• milliampere/cm%
respectively,
where • is the efficiencydefinedas the number
of electronsemitted per incident photon.
For the physical parametersof interest here
and with readily achieved efficiencies•, these
photoelectriccurrentsare large enoughto maintain
the Alfv•n
disturbance
and are nowhere
near the space charge limit of Child's Law
[Hamwell, 1949]. If we consider Echo 1, for
example, in a 1600-km orbit where va •'" 109
cm/sec,with effectivedimensions
of L • M • 30
meters, we obtain the following values from
this flow is maintained indefinitely without (19)-(24) for the voltage, power, current,
changein charge density. The plasma electrons impedance,and surfacechargedensity associated
are drawn into the conductor surface, neutral- with the radiation of the Alfv•n wings:
izing the electric field of the ions. In order to
P = 3 watts
maintain the charge separation producing the
V ,-• 3 volts
toward the conductor. In our solution we assume
electricfield E = --(v•/c) B o, the electronsat
the top surfaceof the conductormust be emitted
into the wingsand a current must exist through
the electrically neutral conductorin the steady
state. The problem is that of overcomingthe
work function at the conductor surface so that
the electron current can flow. The alternative
I •
1/2 amp in eachwing
Z •
6 ohmsin each wing
(31)
Y• • 8 X 105electrons/cm
2
An average value of 0.2 gaussat this altitude
for the perpendicularcomponentof the earth's
magnetic field was assumed in writing these
to this is for the current to be carried by the
ions into the surfaceof the conductorat the top,
and the correspondingreduction of the con-
numbers.Since1//•x • 75 metersfor the Alfv•n
ductivity•0 by the massratio m/M; •< 5 X 10-4
be reducedby a factor of (30/75)•' • 1/6 to
P = 1/2 watt. The currentis correspondingly
reducedby a factor of •1/2.5, accordingto
would mean the failure of our simple Alfv•n
solution for the scale of sizes that we have
considered and the dominance of the resistive
solution to be valid according to (28), for
• ,-• 4 X 106 cps, the power of 3 watts must
(22), to a total value of I • 0.2 amp.
The Echo 1 balloon has a good conducting
surface consistingof a few micronsof aluminum
evaporatedon to a mylar base, and, if we take
a very conservative (i.e. high) value of 4 volts
for the work function and an effective emitting
the sun's radiation and calculate the possible surface area for electrons of LSM, where L is
electron current which could be maintained by the width and •M • M the thickness of a
and capacitive corrections discussedin the
precedingsection.
The proposedsolution is valid only if the
current can be maintained by electrons.Since
cold emissionis totally negligible, we turn to
DRAG AND PROPULSION
IN THE IONOSPHERE
3141
¾
X
Fig. 9. Alfv•n wingsof separationM and thickness•M.
wing in the notation of Figure 9, we find that flow before being limited by space charge
the incidentflux of 3.5 X 10•5photons/cm
•-with accordingto Child's law.
energies_>4 ev producesthe required0.2-amp
In (31) we compute a surfacechargedensity
currentif the efficiency• is not lessthan
of •cn - 8 X 105electrons/cm
•' in the wing.
f = 1.2 X 10-3/•M(meters)
For a wing thicknessof bM • 15 meters, the
correspondingvolume density is
For a wingthicknessof •M •< M/2 • 15 meters, 8 >( 10•/1.5 >( 103-- 530electrons/cm
• (32)
the required efficiencyof f • 10-4 is even less
than expected.Sincework functionsand photo- which adds •10% to the ambient density.
electricefficiencies
are relativelysensitive
t(; the There is thus a sizeable perturbation in the
'history' of a surface, we do not probe this ionospherewhich can be observableby radar.
Effect of finite inter•al resistancein the conquestionhere in further detail.
ductor. According to (22), the electric field
Electrons can also be removed from the surface by collisionswith positive ions that are
--(v•/c) B o of the moving conductorcausesa
neutralized
current flow of magnitude
at
the surface.
These ions collide
with the surface by virtue of the motion of the
conductor(seeFigure 7). The maximum current
density soobtained at 1600 kmis • n+evc,•' 10-•
/•amp, much less than the currents maintained
by photoelectronsin daylight. The possible
electron ejection from impacts of the ions
(sputtering) and of the more copious neutral
atoms will also increase the electron emission.
The current of 0.2 amp/4 X 10ø cm•- < 0.1
/•amp/cm2 is well belowthe spacechargelimit.
=
This
current
flow and the electric field are
maintainedif the conductivityof the conductor
itself (so far assumedto be a perfect conductor
with a•. = •o) is high enough.By Ohm'slaw the
flowof currentin the conductorin the y direction
in Figure 5 is
h =
=
To show this, we compute the thicknessof the
The criterion for there to be no reduction of
ion sheath surroundingthe conductingsurface
power or current flow due to internal resistance
of the Echo 1 in order to neutralize the electric
field in the plasma. It is, by Gauss'law,
l --
E
--
2 >( 47rnion•e
vcB
•
87reCnion•
1
0_•cm
wherewe usednio=s• ne• .... • 5 X 10•/cm•
at 1600 km altitude. Thus we have a potential
drop of 3 volts occurring in a distance of 0.1
cm from the surfaceof Echo, and through this
drop a current of hundredsof microamperescan
is obtainedby combining(33) and (34) to get
cvcBo/va'
2w)L
> c2/2rNv•
For Echo 1, with v• •
10• cm/secand N •
30
meters,the requirementis
• > 6 X 10• esu• 6 X 10-• abmho/cm
3142
DRELL, FOLEY, AND RUDERMAN
Since this is well below the specificconductivity
of an aluminum layer a few micronsthick, no
significantcorrectionneed be made. For conductors at lower altitude orbits, the criterion
(35) becomessomewhatmore severeas va drops
4M• (for He+) and --•10-2ø gm/cc if the ions
are H +. Thus there can be, at most, a few per
cent ionization at these altitudes if all the Echo 1
damping comesfrom neutral atoms as in (37).
However, the Alfv•n mode damping which
to •2 X 107 cm/sec'at 160-500 km altitude. requires no neutral atoms may be sufficient to
As is clear from the Maxwell equations, the give the needed damping from the observed
current flow is reducedby a factor
electron(ion) densityn+ • 5 X 10S/cc.
1+
Nac)
(36)
due to internal resistance of the conductor if it
is appreciable.In simpleterms of Ohm's law,
(36) saysmerelythat the effectiveresistance
is
We needa dissipationof • « watt to replace
Patna-drag
in the analysisof the Echo 1 orbit.
In (31) and the subsequentdiscussion,we found
a magnetohydrodynamicbreaking from radia-
tion into the Alfv•n modeof P •
-} watt, very
close to the required damping. Since we must
rely on the sun to provide the energy leading to
increasedfrom the value given in (23) because
photoelectric emission and to maintain the
of adding the internal and plasma resistances
current, we should further reduce P by another
in series.
factor of 2 to a time averagedvalue of
5. DAMPING OF ECHO 10RmT
The detailed orbit of the Echo 1 satellite shows
watt, as Echo 1 is in the daytime sky only onehalf of the time. What we have achieved here,
then, is a new drag mechanism which is of the
right order of magnitude to explain the observed
orbit parametersof Echo 1 without the requirement of a high value of the ionosphericmass
density relative to ion density.
As we follow Echo 1 to lower altitude orbits,
the frictional drag increasesin proportionto the
massdensity and dominatesthe Alfv•n drag by
a complicated
variationof perigee,apogee,and
eccentricityvalueswith time due to a combination of factors.This problemhasbeenstudied
by JastrowandPearse[1957]andby Shapiroand
Jones[1960],whoincludeeffectsof electrostatic
chargingof the bodyin orbit (negligible),
solar
pressure,and atmosphericdrag. This latter
factor plays a significantrole even at tiny the time we are down to altitudes •< 1000 km.
densitiesabove 1000 km, becauseof the very
Direct experimental confirmation of the ideas
abnormally large ratio of surfacearea to mass
presentedhere is highly desirable,particularly in
of Echo 1 (,rR2--• 6 X l06 cm2;weight • 150
view of the very qualitative nature of our results
pounds).In fact the detailedanalysisof Echo's as applied to Echo l, which is somewhat too
orbit has yielded a 'measurement'
of the mass
small to meet the criteria for the Alfv•n regime
densityin the upperionosphere.
To accountfor
of parameters. This confirmation could be
the observeddrag, after allowance for other
achievedby observationof the Alfv•n wingsby
specular reflection of radar from the surface
neededmassdensity [Shapiroand Jones,1960]
chargelayer computedin (32).
is givenasp.... = 1.2>( l0-•sgm/cm
sat 1600 As we saw in (30), an ionosphericdisturbance
km. The corresponding
power dissipation(in
is predicted extending perhaps many hundreds
the approximation
that vc= 7 )• l05cm/secis of kilometers along field lines in both directions
largecompared
withmolecular
andionvelocities; from Echo 1. This may explain why Echo 1
calculable factors such as solar pressure, the
with T ø •
1500øK and effective molecular mass
of M • 8Mp, the averagemolecularvelocityis
v•--" 2 X 105cm/sec)is
Patna-drag
= (71'R2)
P.... •½3
• « watt (37)
Backscatter radar measurementsmade at Lima,
transits
were seen in instances when the radar
crosssectionof the body itself was too small to
be seenabove the instrumental noise [Tiuri and
Kraus, 1963]. In these same measurementsionospheric disturbancesof duration -1-20 minutes
before and after
transit
of Echo
1 above the
Peru,in 1962havefound•5 >( l0selectrons/cm
s radar sighting were often recorded.To explain
at the 1600 km altitude [Bowles,1964; Bowles this in the present context, the Alfv•n wings
etal., 1962].The corresponding
ion massdensity would need to extend as a detectable charge
is •-•3 X 10-20gm/cmsfor a molecularmassof separation along a magnetic field line in one
DRAG
AND
PROPULSION
direction for a distancebetween 5000 and 10,000
kilometers.
Passive detection
because of critical
at sea level is not feasible
reflection
at
the
D
level.
AlfvlSn waves generated by a source whose
geometrical size is not • 1 km will not give a
detectable magnetic anomaly at sea level (Drell,
Foley, and Ruderman, to be published).
IN
THE
IONOSPHERE
3143
there is no difficulty maintaining this current
flow with the sun as the sourceof photoelectrons.
(For 24-hour operation, we could resort to hot
wire filaments or some other active device.)
Once again, this current of •13 /zamp/cm'.is
well
below
the
Child's
law
limit
for emission
through a potential drop of 30 volts occurring
in a fraction
of a centimeter.
The high level of power generated by the
Alfv•n disturbance invites speculationon ways
AND POWER STORAGE USING AN ALrV•N
of making practical use of it. If used passively
PROPULSION ENGINE IN SPACE
as a controllable drag mechanism it can serve
For Echo I the power level generatedin the (1) as a means of converting satellite kinetic
energy to electrical power, (2) as a way of
Alfv•n
disturbance
was measured in watts.
From (21) it is evident that large conductors bringing satellites to lower altitudes without
(L--•
M --• 100 meters) at lower altitudes propellant, and (3) to adjust satellite attitudes
(•160-500 km with va• 2 X 107cm/secand by exploiting torques. If used in conjunction
B0--• 0.4 gauss) can dissipate power at the with an on-boardsourceof electrical power (viz,
level of kilowatts when crossingfield lines; for solar panels or small nuclear reactors), it can
serve (4) to counteractatmosphericdrag effects
example,
on satellites and even propel them to higher
L • M -- 100 meters
altitudes, (5) as a means of storing energy by
converting it temporarily to satellite kinetic
Va= 2 X 107cm/sec
energy, and (6) to maneuver the position and
B0 = 0.4 gauss
attitude of a spacecraft(without propellant) by
pushing on the earth's magnetic field.
lead, by (19), (21), (22), and (23), to
Most simple is the question of utilizing the
PAlfv•n= 8(L/100 m)(M/100 m) kw = 8 kw
power flowing in the current producing the
Alfvdn fields. A low impedance motor with
V = (vc/c)BoM= 30(M/100 m) volts
internal resistance,of --•0.23 ohm could tap
6. SPECULATIONS ON ORBIT MANEUVERABILITY
=
• •X 8 kw - 2 kw of powerfor usein a satellite
30 volts
at the expenseof its kinetic energy. This might
be generated by flying a 'kite,' as illustrated in
Figure 10, consisting of two 100-meter-long
= 130 amp in each wing
(38)
conducting slabs each of •5to 10-meter
diameter, connected by a conducting rod of
Z = 0.23M/Lohm = 0.23ohm
100-meter length oriented perpendicular to B0
According to (28), we are comfortably within through which the current flows. The surface of
the Alfv•n regime with this choiceof parameters. this conductingrod is insulated from the outside
The surfacechargedensity in a wing is, by (24), world, and in serieswith it is connectedthe low
impedance motor delivering the power. Figure
2•a • 2 esu= 4 X 109electrons/cm"
10 showsa schematicdesign.The rigidity must
For a wing thicknessof --•10 meters, this gives be sufficient to withstand some tens of pounds
I = 130(L/100 m) amp
a volumedensityof--•4 X 10• electrons/cc,or
of force
about 10 times ambient
surfaces.A number of photosensitiveslabsmight
at these altitudes.
The
correspondingcurrent flow through the conducting surface of area 10 X 100 m• = 107 cm"
distributed
over
100-meter
rods
and
be desirable to minimize effects of rotation about
the vertical conductor. For short time power
into each wing is 13 /zamp/cma= 8 X 101• needs, we must compare with a very efficient
electrons/cm".Since metallic conductingsur- small gasoline engine which can deliver •1 kw
faceswith work functionsin the range 3-4 volts of power for an hour by burning •1 pound of
and with photoelectricefficienciesof --•1% are fuel. For more sustained use, 500 pounds of
well within the range of comfortabletechnology, solar cells would give an equivalent power.
3144
DRELL,
FOLEY, AND RUDERMAN
estimated that the drag will lower the orbit of a
orbiting laboratory at
I t 6M
~5- 10
meters20,000-pound manned
L~
this altitude by •15 km/day. This represents
100 meters
M + M1•' 100meters
drag force of
dp_ 1
dh/dt
dt-- • mg•
Satellite
/
(39)
whereh is the altitudeanddh/dt = 15km/day •
15 cm/sec,and a powerdissipation
vc
1
Bo
dh
Pdrag
-- 2 mg-• • 7kw
(40)
To compensatefor this drag, we proposeflying
the 'kite,' but with a powersourceaboardto
drive the current backward, thereby gaining the
5-10 kw power required to neutralize the drag
loss and to maintain
Fig. 10. A schematic arrangement for the
Alfv•n propulsion engine.
Short-circuitingthe junction between M and M'
of Figure 10 maximizesthe drag (equivalent to
8 kw of power or a drop of 40 km per day for a
104-pound satellite.) Opening the junction
between M and M ' reduces the drag by an
order of magnitude, since the effective area
for
Alfv•n
wave
radiation
is
reduced
from
(M -5- M•)L to 2(AM)L. Changing the relative
sizes of M and M • gives a torque about the
earth's magnetic field direction and thus rotates
the satellite about B0. Rotations
the altitude of the satellite
with a small increaseof total weight in orbit.
Higher power levels could also be utilized to
make the satellite sail to higher altitudes, the.
gain being•2 kin/day for eachsteadyinput of
•2 kw (available, for example,from about 500
poundsof solar cells).
When the current flow through the conductor
is reversed by an impressedvoltage, the drag
force I X B reversesits direction, leading to an
acceleration. The normal Alfv•n wave gave a
power dissipation(drag)
•
P•u,• Bo
•Vc
(ML) I MBov•
2•' Va
C
__
---.
about the
vertical occur if the center of mass of the satellite
whereI is the currentflow driven by the potential
is slightly displacedfrom the vertical line current
M -- M •. Other geometriesoffer a variety of
similar possibilities.
If a source of electrical power is available on
the satellite, the direction of the drag currents
can be reversed and the drag converted to a
push. The advantage of the Alfv•n propulsion
engine over a small rocket engine lies in those
circumstances
in which the sourceof power does
not involve the consumptionof a heavy fuel (as
in the caseof a gasolineengine) which couldjust
as well be used as a propellant. Rather, the
Alfv•n engine as a propulsionmechanismwould
be most useful for power which originatesfrom
solar panelsor a nuclear generator.
Among the problems in using a manned
orbiting laboratory is that the atmosphericdrag
is a major sourceof power dissipation,setting a
differencev• BM/c. An impressed
voltageV• in
lower limit
of 100 miles on the altitude.
It
is
the opposite direction (say between M and M'
of Figure 9) gives a rate of increaseof satellite
kinetic energy
P•c
= Palfv•n
l'r- (vc/c)BoM
(41)
A sufficiency condition for the validity of our
approximations(in this case, that of linearity
over a distance X from the conductor) restricts
Vz to 2vcBoM/c. It is not yet clearwhetherthe
region of nonlinearity increasesor decreasesthe
linear drag or propulsion.Thus a power source
of 60 volts and 16 kw can give half its power to
generatingAlfv•n waves and half to increasing
the kinetic energy of the system describedin
(38). This is sufficientto lift the mannedorbiting
laboratory about 15 km per day. Clearly this is
alsoa way to storeelectricalor solarpanel energy
DRAG AND PROPULSION
IN THE IONOSPHERE
by usingit to lift a satellite,but the storageand
subsequentreconversionare at most each 50%
PAlœv{n
cc(Bo2/)a)
cc
efficient.
We may speculateon this kind of mechanism
as a propulsionengine for flight to further
reaches of space. For interplanetary travel,
typical parametersare B0 • 10-5 gaussand
densities
•10 -2ag/cma,leadingto Alfv•n speeds
of
v• •
3145
Acknowledqment. We wish to thank Drs. C.
Longmire, M. Rosenbluth, and H. Lewis for their
learned
and useful remarks.
Additional note. M. J. Lighthill has discussed
the theory of Alfv6n radiation from objects of
smaller dimension, for which the unidirectional
mode discussedhere (the vorticity wave, in his
terminology) is not radiated [Liqhthill, 1960a, b].
We wish to thank Professor Lighthill for a very
friendly and illuminating letter discussingthese
10• cm/sec = 10 km/sec
To be in the Alfv•n regimewe require
w< i2• = eB/ M•c
questions.
• 10-1 cps for B•
10-5 gauss
P•EFERENCES
The corresponding
conductordimensionis
Bowles, K. L., Advances in Electronic and Electron Physics, Vol. 19, page 55, Academic Press,
New York, 1964.
For our solutionwe havethe presumed
require- Bowles, K. L., G. R. Ochs, and J. L. Green, On the
absolute intensity of incoherent scatter echoes
ment
from the ionosphere, J. Res. NBS, 66D, 395,
L •, Vc/W> 107cm = 100km
=
Then for vc•
=
1962.
ø om/eo < 1
v•
- (1.6 X 10--11)X (106)(1010)
ß(ML/km2) X 10-7 watt
(42)
• O.02(ML/km
2) watt < } kw
eitherfor the conversion
of electricalenergyto
Hamwell, G. P., Principles of Electricity and
Magnetism, McGraw-Hill Book Company, New
York, 1949.
Jastrow, R., and C. A. Pearse, Atmospheric drag
on the satellite, J. Geophys. Res., 62, 413, 1957.
Johnson,Francis S., Editor, Satellite Environment
Handbook, pp. 40, 41, 42, and 78, Stanford University Press,Stanford, 1961.
Lighthill, M. J., Studiesin magneto-hydrodynamic
waves and other anisotropic wave motions,
Phil. Trans. Roy. Soc. London, A, 252, 397,
1960a.
Lighthill, M. J., Note on waves through gasesat
satellitekineticenergyor vice versa.
pressures small compared with the magnetic
The conversion
of energyfrom gravitational pressure,with applicationsto upper atmosphere,
attraction
nearanother
planetto electrical
energy Fluid Mech., 9, 465, 1960b.
is a reasonablepossibilityif both a reasonable Shapiro, I. I., and H. M. Jones,Perturbations of
the orbit of the echo balloon, Science,132, 1484,
magnetic field and ionosphericplasma are
present.
The phenomenon
discussed
in this papercan
be used to determine ionic mass densities in
regionsof knownfield strength,as both the
Alfv•n fieldstrengthand the powerdissipation
are proportionalto
1960.
Tiuri, M. E., and J. D. Kraus, Ionospheric disturbances
associated
with
Echo
I
as studied
with 19-megacycle-per-second radar, J. Geophys. Res., 68, 5371, 1963.
(Manuscript received December 21, 1964;
revised April 12, 1965.)