ENGINEERING RESEARCH INSTITUTE
UNIVERSITY OF MICHIGAN
ANN ARBOR
UPPER AIR RESEARCH PROGRAM
REPORT NO. 3
DYNAMIC PROBE MEASUREMENTS IN THE IONOSPHERE
BY
GUNNAR HOK
N. W. SPENCER
A. REIFMAN
W. G. DCW
Project M824
AIR MATERIEL COMMAND
AIR FORCE CAMBRIDGE RESEARCH LABORATORIES
CAMBRIDGE, MASSACHUSETTS
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the
very important contributions to the success of
this investigation made by the past and present
members of the upper atmosphere research group
of the Department of Electrical Engineering at
the University of Michigan. The construction
of the equipment and the preparation for the
flights, as well as the reduction of the data
from the telemetering records, were possible
only through the wholehearted cooperation of
everybody involved in this work.
Ii1
TABLE OF CONTENTS
Page
ACKNOWTEDGEMENTS ii
LIST OF ILLUSTRATIONS vii
ABSTRACT ix
INTRODUCTION 1
THE ELAYER OF THE IONOSPHERE 3
REVIEW OF THE PROBE THEORY 8
A. Single Probe 8
B. Bipolar Probe 21
C. The Region of Ambipolar Diffusion 28
GENERAL DISCUSSION OF THE IONOSPHERE PROBE 38
PRELIMINARY EXPERIMENTAL RESULTS 42
A. Flight Data and Equipment 42
B. Negative Probe Current and PositiveIon Density 51
C. Positive Probe Current and Electron Energy 58
CONCLUSIONS 68
APPENDIX I PROCEDURE FOR DETERMINING PROBE CURRENT AND
VOLTAGE FROM TELEMETERPING RECORD 70
APPENDIX II CORRECTION OF PROBE VOLTAGE FOR RESISTANCE IN
MEASURING CIRCUITS 75
APPENDIX III THE FIRST TENTATIVE INTERPRETATION OF THE DATA
FROM THE DECEMBER, 1947, FLIGHT 79
BIBLIOGRAPHY 86
LIST OF FIGURES
Fig. Page
1. Cylindrical Probe. Normalized Current vs.?7 13
2. Cylindrical Probe. Asymptotic Relations 15
3. Cylindrical Probe. Parameter p in Equation
in = (2/ n) 7i 17
4. Cylindrical Probe. d [og(_2)]/d [log ~] vs. / 19
5. Schematic Diagram of a Bipolar Probe 22
6. Potential Diagram of a Bipolar Probe 24
7. The SingleProbe VoltAmpere Curves of a Bipolar Probe 2,6
8. The VoltAmpere Curve of a Bipolar Probe 26
9. BipolarProbe System with Guard Electrodes 39
10. Nosepiece with Outer Cone Surfaces Removed 43
11. Probe Voltage vs. Probe Current. Ascent. 86, 90.2,
96.4Km. Altitude 44
12. Probe Voltage vs. Probe Current. Ascent. 99, 100.9,
102Km. Altitude 45
13. Probe Voltage vs. Probe Current. Ascent. 102.8, 103.3Km. Altitude 45
14. Probe Voltage vs. Probe Current. Apex 103.5Km. Altitude.
Descent 103.4Km. Altitude 46
15. Data of the Flight December 8, 1947 47
16. Spherical Probe. d [log(ca)2]/d[log ] vs. v 52
17. PositiveIon Density vs. Altitude. 54
18. PositiveIon Concentration vs. Altitude. Estimate from Data
of December 8, 1947, Flight 57
19. "Electron Temperature" vs. Altitude 59
vii
Fig. Page
20. Probe Current vs. Altitude. Probe Voltage = +10 Volts 60
21. Probe Current vs. Altitude. Probe Voltage = 10 Volts 61
22. Sketch of SaddlePoint Field 61
23. Probe Current vs. Probe Voltage. Feb., 1947, Firing
(a) Time 169.5 Sec. Altitude 107.0 Km.
(b) Time 173.0 Sec. Altitude 107.5 Km. 63
24. Sample of Telemeter Record 70
25. Calibration of Calibration Channel No. 3, Measuring from
Zero Line of Calibration Channel with Probe Grid, Record
No. 1, Dec. 8, 1947, Firing 71
26. Calibration of Probe Channel, Measuring from Probe Base
Line with Probe Grid, Dec. 8, 1947, Firing, Record No. 1 72
27. Distance with Probe Grid Measured from Reference Line vs.
Probe Current, Dec. 8, 1947, Firing 73
28. Distance with Probe Grid Measured from Reference Line vs.
Probe Current, Dec. 8, 1947, Firing 74
29. Distance with Probe Grid Measured from Reference Line vs.
Probe Current, Dec. 8, 1947, Firing 74
30. Probe Circuit (Dec. 8, 1947, Firing) 75
31. Calibration Curve of Cathode Follower of Same Type as Used
on Dec. 8, 1947, V2 Probe Circuit 76
32. Probe Circuit Calibration Curve, Dec. 8, 1947, Firing 77
33. ProbeVoltage Correction (Input Voltage vs. Probe Current) 77
34. ProbeVoltage Correction, Expanded Scale 78
35. The Logarithm of the Electron Current vs. the Potential
Parameter? 82
36. Equilibrium Potential vs. Electron Drift Velocity 83
37. The Logarithm of the Electron Current vs. the Potential
Difference between the Collectors. Dec. 8, 1947, Flight 84
viii
ABSTRACT
A method is discussed in this report whereby data on the characteristics of the Elayer of the ionosphere may be obtained. A bipolar
exploring electrode system is flown on a rocket and its voltampere curves
at various altitudes interpreted in terms of the concentration and energy
distributions of ions and electrons. Such a bipolar probe consists of two
isolated, as far as possible, noninteracting electrodes connected to the
two terminals of a variablevoltage source; it is particularly suitable
for immersion in an ionized medium without any convenient reference electrode, e.g., in an electrodeless discharge, a decaying plasma, or the ionosphere.
In the first part of this paper a brief review is given of our
present knowledge and ideas about the Elayer and it is attempted to postulate a simple "model" Elayer with a minimum number of parameters. In
the seco nd and. third parts Langmuir' s probe theory is reviewed and its
application to the ionosphere and. to a geometry realizable on a rocket
is discussed in detail, as well as numerical and graphical methods for
determining the constants of the model ionosphere from probe data.
The fourth part describes the first rather crude experiments
with bipolar probes on rockets. The reduction of the data has met with
considerable difficulty. The main reasons are held to be tie rather unfavorable geometry and the disturbances caused by the motion of the rocket through the atmosphere. It has not been pos;ible to wo:.rk out a complete and unambiguous interpretation of all the data otLtir.red. during the
flight in terms of the concentrations and energies of the ions and electrons. A detailed discussion of what is considerxe( vie mcst plausible
interpretation is included.
This tentative interpretation requires that r o': concentration
of the negative ions be considerably larger than the conlCentration of the
electrons.
The prospects of obtaining unambiguous reaulls b:y improved probe
design are considered at length and are found to be encourX.. ging. To arrive
at a conclusion in this respect, based on the expeirience gained during the
preliminary experiments, is held to be the main objective cf this report.
ix
DYNAMIC PROBE MEASUREMENTS IN THE IONOSPHERE
INTRODUCTION
Most of our knowledge of the properties of ionized gases has
been derived from experiments with electrical discharges through gasfilled tubes. In the study of the ionized layers of the upper atmosphere
it is natural to try to draw whatever parallels are possible between the
behavior of these layers and the plasma in a gasdischarge tube, with due
regard for the considerable difference in ion densities and processes of
ionization and recombination. Early suggestions in this direction were
made by Tonks.1
When the V2 rocket became available for upper atmosphere research after the end of the second World War the opportunity arose to
apply to the exploration of the lower ionosphere the probe method developed by Langmuir and his associates for study of the properties of
the plasma in a gasdischarge tube.2'3 The singleprobe technique used
in gasdischarge tubes had to be modified to a differential arrangement
involving two collectors with a controlled difference of potential but
with an absolute potential free to assume any equilibrium value required
by the total rate of collection of positive and negative particles. In
connection with a contract between the Air Materiel Command of the U. S.
Air Force and the University of Michigan concerning instrumentation of
2
V2 rockets for measuring pressure, temperature, ion density, and ion energy in the upper atmosphere, one of the present authors, W. G. Dow, suggested this bipolar variablevoltageprobe method for the study of the
characteristics of the lower ionosphere. Such probes of early and not
too adequate design have been in operation on three successful V2 flights,
in November, 1946, in February, 1947, and in December, 1947, respectively.
The first flight proved that the scheme could work, although scale ranges
and other details had not been predicted accurately enough to furnish useful data. The December 1947 flight gave a continuous record of well defined probe curves for altitudes between 80 and 105.5 km. Attempts have
been made to calculate from these data at least rough approximations of
ion concentration and electron energy. The set of probe characteristics
deviates considerably from what would be predicted from probe theory applied to a thin plasma in temperature equilibrium at 250300~K. The main
reasons are believed to be:
1. The shape of the collectors does not approximate
closely enough any of the ideal geometries for
which the current collection is known.
2. The ion sheaths of the two collectors overlap, so
that a complicated interaction takes place between
them.
3. The air flow about the rocket upsets the thermal
equilibrium in the vicinity.
Despite these discrepancies we believe that the experience gained
during these flights and during the analysis of the collected data point
the way to improved design and promise valuable data from future flights.
In this report we shall begin with a summary of the properties
of the lower ionosphere as deduced from observations of various kinds as
5
well as from calculations based on established physical laws. We shall
attempt to specify the state of the ionized air at any point in space in
terms of a minimum number of measurable quantities.
The next task is to relate these quantities to the observed voltampere curves of a bipolar probe and to estimate the errors and disturbing
influences produced by other previously neglected factors and processes.
Optimum design of the collector surfaces and graphical and numerical methods
for the computation of the characteristics of the Elayer will also be
discussed at length.
Finally, the results obtained during the December 1947 flight
will be analyzed and the difficulties referred to above will be investigated. Necessary modifications of the computation procedure are introduced
and the final result presented. Tentative interpretations are given for
unexpected behavior of the probe current, particularly during descent.
TIE ELAYER OF THE IONOSPEERE
At an altitude of 100105 km the composition of the atmosphere
is believed to change rather abruptly: below this altitude the oxygen appears
4
nearly exclusively in molecular form; above, the atomic oxygen predominates.
When the radiation from the sun reaches the molecular oxygen which has a considerably lower ionizing potential than the atmospheric layers above it, an
increased photoionization is to be expected. The resulting ion concentration
reaches a steady state when the loss of ions due to recombination and diffusion
equals the rate of photoelectron ionization. A generally acceptable recombination process to explain the concentrations observed by radiopropagation
measurements has not with certainty been established as yet,' As to the
4
importance of diffusion in the ionosphere, a widely spread opinion seems
to hold diffusion a negligible item in the ionbalance equation of the
Elayer.7'
The ratio of the concentrations of negative ions and electrons
in the Elayer has in general been assumed to be negligible. Theoretical
calculations by Bates and Massey gave an order of magnitude of this ratio
of a few tenths of one per cent. Several independent experimental indications have recently been found of a very much higher ratio. The relatively
long life of each free electron enables it, despite the low gas pressure,
finally to attach itself to a gas molecule, but the abundance of photons
of sufficient energy for photodetachment was considered to make it improbable that it will remain attached an a an ppreciable part of its life.
The probe measurements are not accurate enough to give an independent estimate of the electron concentration. This ratiotherefore, enters the
considerations in this report only in the comparison between the positive
ion concentration obtained from the probe data with the electron density
determined by radiopropagation methods.
The gas temperature at the altitude of the Elayer is estimated
to be 250300~K. It is the predominant opinion among ionosphere physicists
that temperature equilibrium is at least to a rough approximation prevalent in the Elayer. It should be pointed out, however, that a number of
phenomena have been continuously observed which constitute disturbances
of the equilibrium. Such phenomena are steady winds of velocities in the
range 50 to 100 m/sec, random winds of 2 to 3 m/sec, formation and movement of ion clouds, etc. 910 In the interpretation of rocket data it
should also be remembered that the presence and motion of the rocket itself cause a disturbance in its immediate vicinity. It is thus not to be
expected that the ion and electron energies should under all conditions
correspond to the temperature of the air at rest.
Introduction of the concepts of ion and electron temperatures
implies the assumption of a MaxwellBoltzmann distribution of ion and
electron energies. The justification for this assumption may be questioned, particularly as regards the electrons. At ionization the lion's
share of the surplus energy of the photon becomes kinetic energy of the
electron. If this energy is lost in collisions in a small fraction of
the lifetime of the electron, the velocity distribution of the electron
will be very nearly Maxwellian. The electron temperature will be equal
to the gas temperature unless a continuous source of energy, like an impressed electric field, exists. Similarly, a recombination process that
favors a certain energy range till cause only insignificant deviations
from the Maxwellian velocity distribution, if the rate of recombination
is insignificantly small compared with the collision rate.
The characteristics of the Elayer at a certain point should
then be stated in terms of the positiveion density, the negativeion
density, the electron density, the air pressure, and the equilibrium temperature as a function of the altitude. If there are any reasons to
doubt the temperature equilibrium, the gas temperature, ion temperatures,
and electron temperature have to be specified separately.
Experimental observations of the maximum electron concentration
in the Elayer are continuously made by radiopropagation measurements.
These observations give the altitude of this maximum as varying between
100 and 120 km. Pressure measurements have been made from rockets at
this altitude, and the temperature has been calculated by means of the
barometric equation under the assumption of constant composition of the
6
11,12
air. 2 Direct temperature observations by soundvelocity measurements
have been attempted, but no data have so far been obtained up to and above
100km altitude.
Some experiments performed in England to explore the "Luxembourg
effect", i.e., the intermodulation between strong radio signals in the Elayer, are of particular interest because the electron temperature is a
rather important parameter in the theory of this effect.3 The quantitative agreement between observations and calculations based on assumed
temperature equilibrium is reasonably good, giving rather strong support
to the assumed temperature equilibrium, at least in the lower strata of
the Elayer.
The correct approach to an experimental investigation like the
present one is, of course, not to rely on any a priori assumptions of
this kind but to leave it to the probe data to give information as to the
energy distribution of the ions and electrons. However, the problem involves so many variables connected by transcendental relations that we
should take on an unnecessarily severe handicap if we did not make critical use of all previous work in the field, continuously checking its
compatibility with our observations, if possible, as we go along.
The undisturbed Elayer in the ionosphere is assumed to be a
region free of electric fields where in each volume element, on the
average, the concentration of positive charge equals that of negative
charge. In the vicinity of a conducting body immersed in this "plasma",
however, such a state cannot be maintained. If the body has a negative
potential with respect to the plasma, a unipolar, positivespacecharge
sheath is formed at its surface containing very few electrons and negative
7
ions but an appreciable density of positive ions. Outside the sheath
there will be a region of ambipolar diffusion which continuously replaces
the ions and electrons that pass through the sheath and are collected by
the body.
In a quantitative study of the equilibrium state in the vicinity
of the conductor the first step is to derive the relations between sheath
thickness, acrossthesheath potential, and current collected, on the one
hand, and the particle densities and energies at the sheath edge, on the
other. Since these densities and energies are not those of the undisturbed
plasma it is necessary to consider the variations of these quantities in
the region of ambipolar diffusion. At the sheath edge this region is subject to a constant drain of ions and electrons. Since the medium is viscous a concentration gradient is required in the diffusion region. Also,
because of the more rapid diffusion of electrons, a weak electric field
must exist, retarding the negative particles and accelerating the positive
ions. At the sheath edge, therefore, the concentrations of all the charged
particles must be expected to be lower and the positiveion energy higher
than in the undisturbed plasma.
In the next sections the quantitative relations between the current collection and potential of a conductor and the characteristics of
the plasma in which it is immersed will be discussed. Experimental and
computational procedures will be described by means of which these
characteristics can be determined.
REVIEW OF TBE PROBE THEORY
A. Single Probe
When an electrode immersed in an ionized gas is brought to a
potential different from that of the gas, a spacecharge region or "sheath"
forms around it of opposite polarity to the potential of the electrode. If
the gas pressure is so low that the probability is small for an ion or electron entering the sheath to collide with gas molecules before it reaches
the electrode, the current to the electrode can be calculated in terms of
the dimensions of the sheath, the velocity distribution of the ions and electrons arriving at the sheath boundary, and the total potential differences
across the sheath. The current reaching the electrode is independent of
the actual distribution of potential in the sheath, with certain mild reservations. The thickness of the sheath and the potential distribution inside are determined by spacecharge considerations rather than by the initial velocities of the ions and electrons and the potential difference
between the collector and the gas.
Consider a cylinder or wire whose length i is large compared
to its radius r so that axial space variations can be neglected. Assume
that the composition of the ionized gas is uniform about the collector and
that the ion velocities are random. The sheath will then be a circular
cylinder concentric with the collector; let its radius be a meters. Considering the ions of one particular sign only, let N be the number per
unit volume in a small element of volume dt bordering the sheath. In a
plane normal to the axis let u be the radial and v the tangential component of velocity of an ion, u being counted positive when directed toward
9
the center. Then, if
Nf(uv)du dv dV
is the number of ions in dT that can be expected to have velocity components u and v lying in specified ranges u to u + du and v to v + dv,
the total number of ions expected to cross the sheath edge with velocities
within the given limits in unit time will be
2r a N u f(u,v) dudv (1)
On multiplying (1) by the ionic charge e and the length J of
the electrode and integrating between the proper limits, the following
equation is obtained for the expected value of the total current to the
electrode.
i = 2 a N e uf(u,v) dvdu (2)
0',u v1
Here the lower limit of u is to be taken as zero when the
collector potential is accelerating, as ul when it is retarding. The
limits ul and v, are obtained by applying the laws of conservation of
energy and angular momentum to the problem of ion flow to the electrode.
The following values are obtained
2 r2
l2 = 2 r2 (u2 + 2 V) (3)
 (4)
2e mm
10
e = charge of ion
m = mass of ion
V = potential of electrode with respect to the sheath edge,
to be taken positive when the electrode attracts ions.
The expected value of the total current I crossing a unit area at
the edge of the sheath is
I = Ne u f(u,v) dv du (5)
These equations, (2) and (5), give the current to the electrode
and the current per unit area at the sheath edge for any distribution of
the velocity components u and v.
If the velocity components are assumed to have a Maxwellian
distribution the distribution function is
f(uv) =  (m/2kT) (u2 + v2) (6)
f(uv) 2kT' e
where k is Boltzmann's constant and T is the ion temperature in degrees
Kelvin.
When this distribution function is introduced into Eq. (5), the
following expression for the expected ioncurrent density entering the
sheath is obtained:
I = Ne k7/2tm (7)
It is, of course, not strictly correct to assume that this is
the current density at the edge of the sheath. Actually, it represents
the isotropic current density in a space where the velocity distribution
11
is everywhere laxwellian. At the sheath edge, however, the distribution
cannot be Maxwellian, since more ions enter the sheath than leave it; in
extreme cases all ions that enter the sheath will proceed to the collector,
so that no ions leave the sheath. By considering Eq. (7) as the "temperaturelimited emission" from the sheath edge we are treating a transport problem as an equilibrium problem, and the result should be regarded
as an approximation to be justified by experimental observations.
It is convenient at this point to introduce dimensionless variables for current and potential
i i(8)
=; r I = (8)
= eV/kT (9)
where A = 2irA = the area of the collector.
From Eq. (2) we now obtain, for an accelerating potential, that is ~0~
in = yerf (' 4 + et 1  erf f'2 (10)
For a retarding potential, that is 7 0,
in = e= (II)
Here
2 2
erf x =  eY dy (12)
= a/r (13)
For a retarding collector potential the current is independent
of the sheath radius, and according to Eq. (11) its logarithm is a linear
12
function of the collector potential, the slope of the line representing
this function being e/kr.
For an accelerating potential, the current reaching the electrode
depends on the sheath radius, so that another independent relation between
in,, and / is needed to calculate the current. This relation is the
equation of spacechargelimited flow in a cylindrical system.
8=8, _ __v3/2
Vi = — ^ / m D (14)
9 r(p)2
This equation is derived under the assumption that the ions have
no random velocity. An approximate correction for a Maxwellian initial
velocity distribution of temperature T is obtained by replacing V/ by
v5/2 (1 + 0.0247 T. ()2 is a transcendental function of V which
has been tabulated by Langmuir and Blodgett. In terms of the dimensionless
variables introduced in Eqs. (8) and (9) Eq. (14) can be written
s = (P 2 t3/2 (15)
where
8 c kT,i
2P 0 —2 (16)
9r Ne
2
Eqs. (10) and (15) form, with (P) = f(S), a transcendental
system that in general can be solved only by numerical or graphical methods.
A simple graphical approach makes use of a graph of Eq. (10) in loglog coordinates (Fig. 1). For given values of collector radius r and ion concentration N Eq. (15) will be represented in the same coordinates by a
family of straight lines with the slope 3/2. For each Y, the intersection
13
0
0
t_ 1\ I a —___ 00
OD
I —   I I I I —a \ I:I\ — __ __ __
\IT I I I I I \\ I 1_  M 1
X,,r,, C\ _,
"m~, \\    , — L  U
c> < " 11.~ ~>~ \ l lA'I0
r c,  .3 _! tI
crw  _
0>^ — ___   _ _____.
0 0 0: "
0
00 0 0 0 OD PD w LO ra 
Ug
14
between a curve (10) and a line (15) gives a solution to the simultaneous
equations and a curve throu all these itersecions represents the current
vs. the voltage for this collector radius and ion concentration.
Numerical calculations of plasma characteristics can often be
very much simplified by use of one of two asymptotic forms of Eq. (10). One
of these asymptotic relations is represented by the common envelope of the
curves, the other by the horizontal asymptotes of the individual curves for
large values of I.
Along the envelope the current is unaffected by the spacecharge
density in the sheath, and the current is said to be limited by the orbital
motion of the ions. The current is in this case
in = Hi /1 + (17)
These conditions are obtained for low ion density, small collector
radius, or high ion temperature. The quantitative criterion will be given
later.
It is seen from Eq. (17) that iR2 varies linearly with? or the
square of the collector current with the collector potential. If the slope
of the line representing this relation is obtained from experimental data,
the ion density can be calculated from Eqs. (17), (7), (8), and (9).
Denoting this slope by S amps2 per volt, we obtain
N = J V/Sm/2e (18)
Ae
The intercept of the line on the voltage axis Vo gives the
ion temperature
Ti = 11600'Vo (19)
15
This relation is not useful to determine the ion temperature,
however, since Vo is usually too small to be measured accurately.
The second asymptotic form of Eq. (10) is obtained for low ion
temperature, large ion density, or large collector radius. It is
in = / (20)
In this case the current is entirely unaffected by the orbital
motion; all ions that enter the sheath reach the collector. This condition
is described as sheatharealimited current.
When Eq. (20) is combined with Eq. (15) and with the relation between (p)2 and 7, a family of curves is generated with the quantity P as
parameter (Eq. 16). In a graph with logarithmic coordinates these curves
are over a wide range (1.5' i n' 100) nearly identical in shape with the
curve representing Eq. (17) but displaced horizontally by an amount determined by P (Fig. 2). This means that in general the voltampere curve of
400 FIG. 2
CYLINDRICAL PROBE
ASYMPTOTIC RELATIONS
200
1. ini z i+n ORBITAL MOTION
2. i n= 2 P0
100:. — /
80
40 _
17
= 20
//" 2 =.0316
16
the collector for an accelerating potential can be expressed by the equation
in le 2/J Al + d, (21)
where p is a factor smaller than one. When the current is sheatharea
limited, / is approximately 1.5 * p2/3. For conditions intermediate between
sheatharealimited and orbitalmotionlimited current / is given by Fig. 3,
which has been obtained graphically from Eqs. (10) and (15). The inclined
asymptote to the left represents sheatharealimited current, and the second
asymptote i = 1 represents orbitalmotionlimited current. This graph shows
clearly what the criteria are for the validity of these two asymptotic solutions. The current is limited by
sheath area, if T /N r2 104 (22)
orbital motion, if T /N r2 L 102. (23)
It should be noted that according to Eq. (21) a plot of i 2 vs.
7a should approximate a straight line, whether the current is limited by
orbital motion or not. The slope of the line has a different interpretation if f t 1, and the intercept on the axis is larger than the voltage
equivalent of temperature by a factor of 1/.
Eq. (21) is accurate within 10 per cent over most of the range
indicated. The errors increase at the upper end of the range so that the
sheatharealimited current at in = 100 is actually about 40 per cent
larger than the value given by Eq. (21).
The method used for the reduction of data in this report for
sheatharealimited current is based on Eqs. (15) and (20).
N 17
Z D_,O
   l — 0
a. ~
0\ I 101 1  E < 0
Ii: X 1li11 X: 
~II._.______.' l I\il r! 1 I _
4' N \\J c D
I It a NI \1Q I 0
a.o oi O o
 ._ a
_______ _~ _ _~ __ ~_. _ o ___q __o _ q
18
Taking the logarithm of both sides in these equations we get
log i log = log P + logg = l logP(p)2 (24)
By differentiation
d(log id(l(log ) = /2(lo 2 d(log d)  d [log (_)2] (25)
d(log in) d(log v) 1.5
  1.5 (26)
d(log l) d(log ) 1+ d log(P
d log
or
logs( )2 =1.5 (lo =1.5 idV _ (27)
d(log y) d(log in) Vdi
This quantity can thus be computed from experimental data
(coordinates and slope of one point of a currentvs.voltage curve).
A graph of this quantity as a function of Y = a/r (Fig. 4)
yields the ratio of sheath radius to collector radius for the point considered. The graph given here is approximate, in that it has been derived
from Langmuir's (_3)2 by graphical differentiation.
When the resulting value of V is introduced in (21), (8) and (7),
not the ion concentration, but the product of the ion concentration and the
square root of the ion temperature is obtained, so that another independent
relation involving one or both of these quantities is required.
19
w
m
C. >
Ir
J )
10 o 10
__ I __ In'I
i
ii
oi i0 1X~ [[X 01' ]P
C z() O 0l]
20
Since the plasma contains positive ions, negative ions, and
electrons, the collection of all three types of particles determines
the voltampere curve of a cylindrical probe (Fig. 7, collector No. 1).
However, for large negative collector potential the negativeparticle current is negligible, so that the Part AB on the curve represents positiveion current alone. If the radius of the probe is so small
that the ion current is known to be limited by orbital motion along this
part of the curve, the positiveion density can be calculated from the
slope of a line representing the square of the current plotted vs. the
potential of the probe (Eq. 18). An extrapolation of the ion current into
the range BC by means of this line makes it possible to separate positiveion and negativeparticle current in this region. If the positiveion current is known to be sheatharealimited, the procedure involving Eq. (27)
yields the product of the positiveion concentration and the square root
of the positiveion temperature. According to Eq. (21) the positiveion
current can still be extrapolated in the same way into the region BC for
separation of the negative and the positiveparticle currents. The
negativeparticle current varies so much faster with the potential than
the positiveion current that the approximate character of Eq. (21) introduces a negligible error in the negativeparticle current obtained by
the extrapolation procedure. The negativeparticle current is the sum
of the electron current and the negativeion current
i = 2Ir. e /e + In eeV/kTn] (28)
Since the random current density I (Eq.7) is directly proportional to the particle concentration N and inversely proportional to the
21
square root of the particle mass m, the electron component will predominate unless the negativeion density is very much larger than the
electron density. In every case, as long as the negativeion temperature is equal to the electron temperature, this temperature can be determined from experimental data by measuring the slope of the line representing the equation
eV
log (in + ie) = kTV const. (29)
If the ratio between negativeion concentration and electron
concentration were knwnm, it would theoretically be possible to calculate also these concentrations from the same set of data. Because of
the steep exponential variation, however, an extremely accurate knowledge of the potential difference between the gas and the collector is
required. For this reason this method is usually not practically useful
for determination of negativeion and electron concentration.
B. Bipolar Probe
When the singleprobe circuit cannot be connected to an electrode of fixed potential with respect to the ionized gas, such as the
anode or the cathode in a gasdischarge tube under constant operating
conditions, a combination of two probes, each connected to one of the
terminals of a variablevoltage source, can be used. The application
of such bipolar probes to the exploration of the Elayer of the ionosphere is the subject of this report.
The theory of the bipolar probe can be simply derived from
the preceding discussion of the single probe. Consider a combination of
22
two long cylindrical collectors (Fig. 5). Preferably, one of them should
have small enough diameter for the positiveion current to be limited, by
orbital motion, so that the positiveion concentration can be determined
FIG. 5 SCHEMATIC DIAGRAM OF A BIPOLAR PROBE
independently of the positiveion temperature. A considerably larger
second collector has some merits both from practical and theoretical
points of view. The ion current to this electrode (No. 2) shall therefore be considered sheatharealimited. In the following discussion
both ions and electrons are assumed to have a ItxwellBoltzmann energy
distribution at the sheath edge.
From the preceding section and Eqs. (7)(9), (11), (17), and
(21) it is seen that the expected values of the total currents to the
collectors No. 1 and No. 2, respectively, may be given by
iT1 AlIp P lp + 1] 2  AlIe e Aln e (27)
iT2 A2Ip' 2  A2Ie.e 2In e n(28)
where
I = N ~ ekT = random positiveioncurrent density (29)
P'P V2nmp
23
I IT ~ e  = random electroncurrent density (30)
e \ 2(30a)
In "= n e  = random negativeioncurrent density (30a)
Np onn on o o on o
TJn = concentration of positive ions at outer face of sheath
Nn = concentration of negative ions at outer face of sheath
Ne = concentration of electrons at outer face of sheath
Al^ A2 = surface areas of collectors No. 1 and No. 2 respectively
eV1 evl eV
p e n} = k e' +kT n
eV2 eV2 eV2
5pp'l2e' ~n +
1p 2 e} Pp + kT +kTn
Te, Tp, Tn = electron, positiveion and negativeion temperature,
respectively
V1, V2 = acrossthesheath potentials for collectors No. 1 and 2
respectively, also approximately the potentials of these
collectors relative to space potential. Both V1 and V2
are always numerically negative.
24
In regard to the symbols V1, V2 and referring to Fig. 6, it is
evidently convenient to consider collector potentials as being measured
relative to the space potential; therefore, basically, zero potential
(SPACE POTENTIAL)
.lLIii j —! ^ — 0
FIG. 6 POTENTIAL DIAGRAM OF A BIPOLAR PROBE
should be defined as that at the top of the spacepotential plateau.
There is presumably only a small potential drop between plateau
top and sheath edge. Therefore V1 and V2, being defined as numerically
negative, may be thought of as approximating reasonably well the potentials of the respective collectors relative to zero potential at the
top of the plateau. However, rigorously, V1 and V2 describe, negatively,
the potential differences between collectors and sheath edge. Eq.(27)
is a relation only between the potential drop V1 across the sheath, the
positiveion temperature Tp, negativeion temperature Tn, electron temperature Te at the sheath boundary, and the random ion and electron currents, Ip, In, and Ie. Eq. (28) shows, in addition to these variables a
dependence on sheath radius. Each equation consists of a positiveion
current added to an electron and negativeion current that is due to the
electrons and negative ions that succeed in penetrating the retarding
field in the sheath.
One consequence of the difference between the potential of the
plateau"and that of the sheath edge is that the average energy or temperature of the positive ions will be somewhat higher at the sheath edge than
25
in the undisturbed gas. We shall see, however, that if the ionosphere
is initially at temperature equilibrium this increase is only about five
per cent under sheatharealimited conditions and considerably less if the
current is limited by orbital motion.
Since the combination of two collectors in space constitutes an
isolated system, there can be no net flow of current; hence,
iT1 + T2 = 0 (51)
It is reasonable to assume as a very good first approximation
that the potential at the outer edge of the sheath is the same for collector No. 1 as for collector No. 2. This justifies stating the relation
6V = V1  V2 (32)
where SV is the voltage applied between the two collectors.
From Eqs. (27) to (32) it is possible to obtain, in principle at
least, V1 and V2 as functions of 6V and the constants of the ionized gas.
This determines the manner in which the potential drop across the sheath
must vary as the applied voltage SV is changed. Hence, replacing V1 or
V2 by the equivalent expression in terms of 6V provides the desired relationship between iTl and 6V, or between iT2 and 6V. This represents
the experimentally observable voltampere curve for the bipolar probe.
In view of the complicated manner in which the potentials V1
and V2 appear in Eqs. (27) and (28), for cylindrical probes, the exact
dependence of V, or V2 on 6V would be very difficult to find, except by
26
graphical methods. Fig. 7 shows how a graph of the bipolar characteristic
iT = f( V) can be obtained from the two singleprobe plots iT = f(V1)
and iT2 = f(V2). The latter plot is inverted so that the condition
iT1 = iT2 is satisfied by any pair of intersections of the curves with
 . —  
 E   16'iT2"^ ~ COLLECTOR "2
 _ _ _ s^     14
    '_  _  12
8vF.
THE SINGLEPROBE VOLTAMPERE CURVES OF
01,
COPOLAR PROBELECTOR I B
4 8,e
  
,_
 20 6 28 24 4 16 12 8 4 0
VI, V POTENTIAL OF COLLECTOR
WITH RESPECT TO SPACE IN VOLTS
FIG. 7
THE SINGLEPROBE VOLTAMPERE CURVES OF
A BIPOLAR PROBE
a horizontal line. For each value of the current the distance between these
intersections is the corresponding value of $V. The resulting curve is
shown in Fig. 8.
 6 IC
E'14
  
4I
2
C r.A 2
20 16 12 8 4 0 4 8 12 16 20
8V, POTENTIAL OF COLLECTOR #I
WITH RESPECT TO#2 IN VOLTS
FIG. 8
THE VOLTAMPERE CURVE OF A BIPOLAR PROBE
27
The computation of the characteristics of an ionized gas from
such curves for a bipolar probe is facilitated by the fact that some parts
of the bipolarprobe curve have very nearly the same shape as a singleprobe curve. For large negative values of V1, the electron current to
the large collector rises so steeply that V2 can be considered constant.
Except for the different location of the origin, the portion AB of the
bipolarprobe and singleprobe curves are then identical. This will apply
also to the part BC if the ratio of the probe areas A1/A2 is very small.
If this ratio is not too small, V1 may be considered constant for large
positive values of GV, so that the part DE of the bipolar curve (Fig.8)
will be equal to the corresponding part of the curve iT2 = f(V2). The
conditions to be satisfied in the two cases, respectively, are
A1/A2 exp [eV/KTj 4l( (33)
A1/A2 exp [eV/lkTe]?l (34)
It is possible therefore to use the various methods for computation of the characteristics of an ionized gas from experimental data
that were presented in connection with the single probe, if the probe is
so designed that the above criteria are satisfied at the appropriate points
of the bipolar voltampere curves. The part AB of the curve supplies
data for a plot of (ip)2 vs.CV for computation of the ion density from
Eq. (19). The intercept of this line on the horizontal axis represents
V2 + Tp/11,600 and thus roughly calibrates 6V in terms of the gas potential if the current to the probe is known to be limited by orbital motion.
If the criterion for sheatharealimited ion current to collector No. 2, as well as Eq. (34), are satisfied, the method involving Eq. (25)
28
can be used to compute Np JTp. If Np has been determined from the smallprobe characteristic, Tp can thus be obtained. It should be remembered
that these values of N and T represent the conditions at the sheath
p P
edge, not in an undisturbed region of the gas. If different points on
the curve are used for computation, different values of T and N may result, since sheath thickness and acrossthesheath potential difference
will vary with SV.
If the negativeion and electron temperatures are equal, a
plot of Eq. (26) will give this temperature from the slope of the line.
As mentioned above, it is not practical to calculate the electron concentration, or rather the sum of negativeion and electron concentrations
weiglted inversely as the square root of their masses, from the negativeparticle current obtained from the part BC of the probe curve by subtraction of the extrapolated positiveion current. On the other hand, if
these concentrations were known, the space potential could be much more
accurately obtained by this method than by extrapolation of the (ip)2 vs.
 V line.
Johnson and Malter17 have used some convenient procedures to
obtain the electron temperature of a positiveionelectron plasma from
the bipolarprobe curve under circumstances when the condition (33) is not
satisfied. These methods are not reviewed here because they are not easily
adaptable to the probe geometry used so far, nor to the one suggested for
future rocket measurements.
C. The Region of Ambipolar Diffusion
One of the most difficult problems in evaluation of rocket probe
data, even if obtained by a probe of considerably improved design, is the
29
calculation of the difference in chargedpecrticle concentrations and temperatures between the undisturbed ionosphere and the sheath edge. It is
a problem of ion mobility or diffusion, for which no precise theory in
satisfactory agreement with observations exists. The only alternative to
the application of an adequate theory is laboratory experiments so arranged
that they can be extrapolated to the actual probe conditions with strict
adherence to the laws of similitude. The nature of the ionosphere as an
electrodeless discharge with pure volume recombination makes such experiments very difficult. However, a detailed discussion of this question is
beyond the scope of the present report.
We shall here discuss the variations of chargedparticle concentrations and temperature from the point of view of an approximate theory.13,l4 The basic assumptions in this approach are:
1. The flow is completely viscous in the ambipolar diffusion region; this region borders directly on the
sheat, where the viscosity is negligible, certainly
a considerable oversimplification of the actual conditions.
2,. A steady state exists.
3. Np = Nn + Ne, and the ratios between these concentrations are constant.
4. Nn = Nn exp(eV/kTn) (retarding field, V negative) (35)
5. Ne = Neo/xp(eV/kTe) (36)
The positiveion current depends on the positiveion concentration gradient, the positiveion temperature radient and the potential
gradient according to the equation
30
IP = eNp * DpI grad I + 1 (grad Vp + grad V) (37)
P P PLIp V. J
where Dp is the diffusion coefficient of the positive ions and Vp is
the voltage equivalent of the positiveion temperature.
Continuity requires that at every point the divergence of this
current be equal to the difference between the ionization rate and the
recombination rate. The ionization rate is independent of the ion and
electron concentrations while the recombination rate can be considered
to vary with the square of the positiveion concentration.
div Ip =  (Npo2  Np ), (38)
where Npo is the positiveion concentration in the undisturbed plasma.
It is thus evident that the ambipolar diffusion in the ionosphere is quite different from the diffusion in a gasdischarge tube, the
min reasons being the absence of sustaining electrodes and different
predominant processes of ionization and recombination.
To get a first approximate solution it is convenient to assume
temperature equilibrium: T, = Tn = Te = the gas temperature. Then,
according to the assumptions 3 to 5 we can write
grad V
graae(Nn V i'e) grad (Nn + Ne) (59)
e(Nn + Ne)
Ip O eDp[ 1 + (Te/Tp) grad Np 2eDp grad Np (40)
V2 Np P  (po2 2) (41)
pP 2D Po P
31
Since the ambipolardiffusion region is knowm to be very large
compared to the collector dimensions a spherical geometry with
9/$ = a/lQ = 0 will approach the actual conditions over a very
large fraction of the region.
Introducing the new variable
= r(Npo  NP) (42)
we obtain
2a = T + (po + (4) *
Qr2 p+ p
An approximate solution good for very small variations of itp
results if NJp + ITD is replaced by 2. Ito
NpD = Npo  e r (44)
where Ipo and Q are constants to be determined from the boundary conditions at the sheath edge and
 D~ (
= P0 (45)
If numerical values of the order of magnitude encountered in
this experiment are introduced in Eq.(44), it will be found that the variations of INp with r are not always small, so that the linear approximation
to Eq.(41) may be questionable. Since the solution is continuous and monotonic within the range r = a to r = oo, suitable limits for its variation
can be easily established:
2 oar Na (Up~o 26P) z r a (rNp) c 2 czr Npo (Npo  Np,) (46)
32
po  Q1 epr 4 Np Np~ _ Q2 e 2r(47)
r r (47)
where Npo, Q10 and Q2 are constants to be determined from the boundary
conditions and where
P1  a (48) P2 /(49)
Na is the positiveion concentration at the sheath edge (r = a).
The recombination coefficient a has been estimated to 108 cm3/sec
or 1014 m3/sec. The diffusion coefficient is approximately
D = 6 * (50)
9T9n mg
where mg is the mass of a gas molecule and 2 the mean free path of the
ions. For an ion density as high as 10 ions/m3, P is approximately 0.022.
In the part of the diffusion region that is of particular interest
Pr is so small that the difference between the two limits in Eq. (47) is
insignificant.
For very small sheath dimensions a cylindrical solution of the
diffusion problem may be of interest. Linearizing Eq.(41), as before, we
obtain the following solution
N = Npo  Q Ho (jPr) (51)
where H1 is the Hankel function of order zero and P has the same meaning
as in the spherical solution.
In order to determine the relation between the positiveion concentration in the undisturbed ionosphere and that at the sheath edge, we
make use of the continuity or the ioncurrent density at the sheath edge.
33
The spherical solution gives
in 4.r2 I in I Ve
Ve 1 B
 2 — =  e Dp (1 + ) grad p
e Dp (1  ) (Npo INa) a ( + a) (52)
VP
where Ve is the voltage equivalent of the electron temperature.
After introduction of Eqs. (7) and (50) we obtain
NpO a a 1 in (53)
Ila 4i V >p (l+le) (1 + a)'2
Vp
When pa is small, Ve = Vp and the current through the sheath
is limited by sheath area, this reduces to
No  N,
Po Na a
_ — =.265 3 * (54)
When the current is limited by orbital motion in is independent of ~ so that the left member of (53) is inversely proportional to
the sheath radius instead of directly proportional as in (54). A maximum must occur between these two extremes, probably in the neighborhood
of the transition from sheatharealimited current to orbitalmoticnlimited
current.
For very large sheath radius (a'll) Eq. (54) becomes
TNp  Na 0.265
Na Z B  (55)
34
Under ideal cylindrical conditions the corresponding relations
to Eqs. (52) and (53) are
in 2trl I in~ I V
2 ina * = e Dp (1 + V) Q j P Hl(jpa) (56)
Ipo  Ia 3 a Je HOV(joa) n *
I 4 %/J2 + 7,) JPa Hn(jpa7 (*7)
N 'v p (1 at) (jt3(a)?#
Here
H (jpa) 2
ja Hl(jpa) log .71pa for pa c< 1 (58)
A for pa > 1 (59)
For sheatharealimited current, Ve = Vp, and pa <<1 we now
get, in analogy with Eq. (54 ),
N^  N
p Na 0.265 * log 2 ja (60)
Na 1.781a P p
For 0.02  Pa  1.0 the fraction to the left in Eqs. (58) and (59)
lies between the corresponding limits of 4.02 and 0.70. The drop in ion
concentration is consequently appreciably larger in an ambipolar diffusion
region of cylindrical shape than in one of sphericalshape of the same radius.
The two limits obtained by using,B = P1 and P = P32 Eqs. (53) and
(54), may in the cylindrical case differ appreciably even in the neighborhood
35
of the sheath edge and for small values of Pa. It may therefore become
desirable to have a complete solution of the nonlinear equation (41) in
the form of a numerical table or a graphical map.
The assumption of a constant positiveion temperature throuhout
the diffusion region introduces an error whose order of magnitude can easily
be checked after the approximate numerical solution of the diffusion problem has been found according to the procedure outlined above. The potential gradient at the sheath edge is obtained from Eq. (39), and the voltage
equivalent of the positiveion temperature
v = + ( rad V)2 (61)
The product 1 grad V can be calculated from Eqs. (39) and (40)
since the ambipolar diffusion region is very nearly neutral, so that
Np t Ne + Nn. For a spherical sheath,
grad V Ve n * 3 265 Ve (62)
P ye ^ 4 /i 0% 0e
for sheatharealimited current and less for orbitalmotionlimited current. If the electron temperature is equal to the gas temperature, this
means according to Eq. (62) that the ion temperature exceeds the gas
temperature by three per cent or less. Assumption of a cylindrical sheath
leads in this case to the same result. As long as the negativeparticle
temperature is approximately equal to the gas temperature, the increase
in positiveion temperature in the diffusion region is insignificant.
36
The methods for calculation of positiveion density that were
given in the preceding sections assumed an ion density at the sheath edge
independent of the collector potential. In a cylindrical geometry this
condition is not satisfied even if the collector radius is very small (Eq.
60), and the slope of the various plots used will not be so simply related
to the positiveion density at the sheath edge as indicated by Eqs. (18)
and (27). However, if the space potential can be determined from other
data, such as the negativeparticle current, and the current is known to
be limited by orbital motion, the sheathedge positiveion concentration
can be calculated for every point on the probe characteristic. The drop
in the diffusion region can then be obtained from Eq.(57).
When the current is limited by sheath area Eq. (27) in our procedure is invalidated because
d(log i) i) (63)
d(log V) (log) (65)
Instead we have
i = in Na * A (64)
d(log i)
d(log in) d(log V) (65)
d(log ) d(log Na)
1 d(log in)
Since in = 1 we can find
d(log Na) d(log Na) d(log Na) a dNa
d(log in) = d(log ) = d(log a) = Na da (66)
37
from Eq. (57). Since the two unknown variables Na and v are related to
the observed quantities i and V by transcendental equations, numerical or
graphical solutions have to be used. A zeroorder approximation of Na
may be obtained from Eq. (27) and Fig. 4 by considering Na independent of
a, and first and higherorder approximations by using the lowerorder Na
and I in Eqs. (57), (66), (65), and (27).
38
GSIEIEiRL DISCUSSIOiT OF, TM IONOSPHEERE PROBE
The application of the bipolar probe to the determination of
the characteristics of the Elayer of the ionosphere will be discussed
in general terms in this section. After presentation of some preliminary
experimental results a number of sources of error and design precautions
to minimize these will be considered in the next section.
The first consideration is choice of geometry. Analytical solutions of reasonably modest complexity exist only for three typical geometries:
plane, circularly cylindrical, and spherical. The choice of a cylindrical
arrangement has been anticipated in the preceding presentation, for rather
obvious reasons: It is structurally adaptable to mounting on a rocket, and
it has proved its worth during many years of application to gasdischarep
tubes, primarily as single probes but to a lesser extent also as bipolar
probes.15 The structural difficulty connected with the cylindrical probe
is the demand that it be long, not only compared to its diameter, but compared to the sheath diameter. This demand can be somewhat relaxed, if
guard electrodes are used, maintained at the same potential as the collectors and protecting the latter from most of the currents caused by the noncylindrical field at the ends. For any measurements of more than orderofmagnitude accuracy with a structure of practical dimensions, such guard
electrodes are mandatory. When guard electrodes are used iT1 f iT2, and
two currents have to be measured rather than one (Fig. 9). The procedure
for reduction of data will be the same as described above, however, since
singleprobe approximations were used throughout. Dr. Eric Beth has brought
to the authors' attention a suggestion to elaborate the guardelectrode system
39
GUARD RINGS
COLLECTOR #2 GUARD RINGS, sl_ —x y/ COLLECTOR l \
 ;__ X
8v
FIG. 9
BIPOLARPROBE SYSTEM WITH GUARD ELECTRODES
by subdividing the collectors further and measuring the current collection
of each individual section; the approximation to proportionality between
current collected and axial length would show how closely the sheath geometry approached a cylindrical one. Whether or not such an elaborate arrangement can be used on future flights will probably be determined by practical
considerations, such as available telemetering channel space.
The second problem in collector design concerns the choice of
collector diameters. According to the singleprobe theory discussed in a
preceding section a combination of a sheatharealimited collector and an
orbitalmotionlimited collector would have the merit of giving explicit
information of both positiveion density and positiveion temperature. In
order to apply this design criterion we determine from Fig. 3 the transition
point between sheatharealimited current and orbitalmotionlimited current:
T. = 0.7 x 104. (67)
p
40
Thus, if the maximum value of Np is 101 per cubic meter and Tp
= 300~K, the radius of the collector must be smaller than 2 mm in order
to give orbitalmotionlimited current. The design of such a probe for
mounting on a rocket with appropriate guard electrodes meets with structural problems; also, the current collected will be so small that errors
due to leakage, drift, etc., may become appreciable. However, these
difficulties may not be insurmountable.
These design considerations appear in a somewhat different light
when the ambipolar diffusion is taken into account. An accurate knowledge
of the positiveion concentration at the sheath edge is of no particular
value unless the drop in concentration from ambipolar diffusion can be determined with comparable accuracy. This drop is negligible in a spherical
geometry if the sheath radius is small compared to four times the mean
free path of the ions at the transition point between the orbitalmotionlimited and the sheatharealimited current (Eqs.53 and 54). The criterion in a spherical geometry corresponding to the cylindricalgeometry criterion given in Eq.(67) has not yet been worked out by the authors, but
most likely the radius of the spherical probe will be larger but of the
same general order of magnitude as that of a cylindrical probe, say 5 to
10 mm. The practical difficulties connected with such a probe will be
even larger than in the case of the cylindrical probe.
The ambipolar diffusion furnishes an additional reason for designing a cylindrical probe for orbitalmotionlimited current, because
then the iondensity calculation breaks down into two independent operations, one for the sheath and one for the diffusion region, while a sheatharealimited probe gives simultaneous transcendental equations that have
41
to be solved by successive approximations. Since the design of a probe
that has orbitalmotionlimited current at all altitudes appears to meet
practical difficulties, an additional collector whose current is always
limited by sheath area will be helpful to bridge the transition region of
the small probe. At this point it becomes evident that the plan to determine the ion temperature from the data of two collectors should be abandoned,
since the introduction of another variable in the diffusion problem makes
the computation hopelessly complicated. Since it appears that the approximate temperature equilibrium in the ionosphere is or can be established
from other experimental data, this is not a very important sacrifice. Actually the negstiveparticleenergy distribution indicated by the part BC
of the probe curves (Figs. 7 and 8) may prove that such an equilibrium prevails; it is difficult to imagine circumstances under which the positive
ions have substantially higher energies than the electrons.
The diffusion theory is only approximate; consequently experimental verification is required on several points before any ion concentrations calculated from bipolarprobe measurements can be accepted with confidence in their accuracy. The diffusion coefficient (Eq.50) and the mean
free path of the positive ions at 250300~K should be measured in the laboratory under conditions resembling those in the ionosphere as closely as possible. Knowledge of the volume recombination coefficient is also required
in the cylindrical diffusion problem but not in the spherical one; it is
at least roughly known from radiopropagation measurements.
42
PRELIMINABRY EXPERIMENTAL PESULTS
A. Flight Data and Equipment
The previous presentation has been based on a nuimer of idealizations and assumptions that can be reliably justified or improved only by
careful comparison of experimental observations and theoretical predictions
or, possibly, by an interpretation of data in terms of a revised set of
as sumptions.
There have been three V2 rockets on which probes were carried.
The main purpose of the participation of the University of Michigan in
these flights was to measure pressure and tenperatuare in the upper atmosphere, so that the probe was to some extent in the position of a stepchild.
Considerations of space and weight made it necessary to use a rather crude
probe configuration, so that accurate determination of the Elayer characteristics was out of the question. The main purpose of including the
probe was to prove that the scheme was workable and to gain experience
in the various instrumentation problems connected with it. It was hoped
that an attempt to reduce the data would show them to be compatible with
results obtained from radiopropagation measurements and give valuable
hints about secondary phenomena, sources of error, etc. These hopes have
been completely justified.
Of the three V2 firings, November, 1946, February, 1947, and
December, 1947, only the last one provided data that couldbe considered
suitable for analysis. Instrumentation difficulties and rocket spin made
the data from the earlier two flights nearly copleely worthless. The
following will therefore concern the December 1947 flight only. Fig. 10
shows the nose piece of the rocket carrying five ionization gxuages and a
FIG. t0
NOSEPIECE WITH OUTER CONE SURFACES REMOVED
probe. To reduce drag and protect the gauges at low altitudes. the nose
piece was covered bly a smooth cone, which was split and thrown off at a.n
altitude of 45 hm. The probe (collector INo. 1) was a ring in the shape
of a truncated cone nounted on insulal;ors at the top of th;e nose piece.'lhe reqluiremenlts imposed by the pressure measurements leflt room ornly for
this very slorL probe wiich does not by far approach the long cylinder desirable for the purpose of reducing the data by means of formulas derived
for an ideally cylindrical geometry. The approximate dimensions can be
seen from the scaleo at the bottom of the figure. The scanning sawtoothl
voltage generator had a period of 0.6 second and an amplitude of  20 wolts.
44
This voltage was applied between the probe and the nose piece (collector
I'o. 2). A sensitive electronic measuring circuit wras used to record the
current flow between the collectors. The instantaneous value of this current was transmitted to a recording oscillograph on the ground by means
of a radio telemetering system.
The resulting currentversusvoltage oscillogram was transformed
into a set of voltampere curves., representative samples of which are
shown in Figs. 1114. The sawtooth voltage was not continuously measured
but was assumed to vary linearly with time within each cycle and to maintain its amplitude constant. Experiments have confirmed the linearity
within 10 per cent in the range ~ 15 volts. The trajectory of the rocket
FIG. II 50
PROBE VOLTAGE VS. PROBE CURRENT __
ASCENT
96.4 KM.
86, 90.2, 96.4 KM. ALTITUDE C 40   ___
90.2 KM
30
~ ~ —~   ~iC"20.J/
___________ _____ —__
10_
20 16 12 8 4 0 4 8 12 16 20
PROBE VOLTAGE (VOLTS)
45
FIG.12 50
PROBE VOLTAGE VS. PROBE CURRENT _99 KM
ASCENT
99, 100.9, 102 KM ALTITUDE 40 
100.9 KM
20  16  12 8 4 0 4 8 12 16 20
FIIG.13 25i
Wn,.___ __ _ __ _ __ d 0 _ __ _ _ _10 2. KM
20 16 12 8 4 0 4 8 12 16 20
ROE VOLTAGE (VOLTS)
0
10
101___ __
20 16 12 8 4 0 4 8 12 16 20
PROBE VOLTAGE (VOLTS)
FIG. 13 25t
PROBE VOLTAGE VS. PROBE CURRENT
ASCENT
102.8, 103.3 KM ALTITUDE
20
102.8 KM.
420 16 12 8 4 0 4 8 12 16 20
PROBE VOLTAGE (VOLTS)
46
FIG. 14 25
PROBE VOLTAGE VS. PROBE CURRENT
APEX 103.5 KM. ALTITUDE
DESCENT 103.4 KM. ALTITUDE
20
103.5 KM
 15

u1J
20 16 12 8 4 0 4 8 12 16 20
PROBE VOLTAGE (VOLTS)
was recorded simultaneously from a radar station, so that the correct altitude could be associated with each voltampere curve. The radar data
from the December 1947 flight are shown in Fig. 15. This is a plot of
the radar trajectory with the time elapsed since the firing indicated every 10 seconds on the curve. Information concerning rocket aspect, total
velocity, mean free path, and air density is also shown. It can be seen
that the rocket attained a maximum altitude of only 103.5 km, which is appreciably below that anticipated for the probe measurements. The telemetering record shows that the probe current remained too small to be measured until an altitude of about 70 km was reached. From there on, readable
currents were observed for the positive swing of the probe voltage (i.e.,
voltage on collector No. 1) during each scan. This is an indication of
measurable positiveion collection by collector No. 2 (the rocket itself)
above 70 km. It was not until an altitude of about 98 km was reached that
47
110 TOTAL MEAN FREE AIR
VELOCITY PATH DENSITY
MPS CM /CM3X 1013
190 —— 50 rJ' o 10.8 1.6
i80s ok.200 150
7700 X r\ 210
170
100  / I 220  300  6.3 2.8160
230
81 1 1~50 \ 450 3.8 5.2
140 40 10 l l . 0240
90 550 2.0 IQ.O
i55~~/0250
130
260
80 12
270 104 187 199.
180.7
70 \ 177.
ICD~~~~~~~~~ / 280
w  02 i I 172
120 34060 00
60.J_1 ~~~ I I I I I \ I ~~~~~~~~~~163.5
290 100
90
D /
50
80
40
310
70
30
V2 ROCKET`28
WHITE SANDS, N.M.
2:30 PM M.S.T
20 60 _______ DEC. 8,1947 ___
50
10
40
0 10 20 30 40 50 60 70
HORIZONTAL RANGE IN KILOMETERS
FIG. 15
DATA OF THE FLIGHT DEC.8,1947
48
reasonably accurately readable currents were observed during the negative swing of the probe voltage. This indicates that at this altitude
the positiveion collection to the probe or small collector became measurable. Of course, the surface area of No. 2 collector was very much
greater than that of the small No. 1 collector. As the rocket passed the
crest the data show quite unexpected variations, the outstanding effect
being that the positiveswing current dropped to zero. At passage of the
rocket over the crest two of the five pressure gauges mounted adjacent to
the probe (see Fig. 9) indicated a rapid drop in air density, which shows
that the air about the nose piece was violently disturbed by the presence
of the rocket. Under such conditions the collector current is difficult
to predict, and the subsequent data are useless for determination of the
constants of the ionosphere. The range of utilizable data is indicated
by the rectangular inset in Fig. 10; the enlarged inset shows the range
for which experimental voltampere curves were taken as well as the approximate rocket aspect at that time.
It should be stated here that even the data above characterized
as "utilizable" must be expected to deviate considerably from the relations developed in the preceding sections. The most obvious reasons for
this are the noncylindrical geometry and the overlapping of the electric
fields from the two collectors. Other reasons, such as nonMaxwellian
velocity distributions of electrons and ions, may exist, however, and more
than one interpretation of the data is certainly possible. In an advance
publication17 two of the present authors presented a tentative interpretation in terms of electron and ion temperatures considerably higher than
those expected in the undisturbed ionosphere and an electron drift velocity
49
superposed on the Maxwellian distribution. The main supporting evidence
for this interpretation was the difference of potential of about two volts
between the collectors at the sharp bend in the probe curve and the slope
and curvature of the curves of log ie vs.6V. (See Appendix III, Fig 57.)
However, the error in the assumed voltage variation along the saw tooth
plus contact potentials in the collector circuit may very well amount to
two volts. The fact that this difference of potential remains nearly constant while the ion and electron densities vary several orders of magnitude makes an instrumental origin much more probable than an ionospheric
one. The variation of the negativeparticle current with potential difference, on the other hand, is necessarily strongly influenced by the interaction of the sheaths and diffusion regions of the two collectors; deviations from the ideal theory are consequently not surprising. In short,
the evidence supporting an interpretation in considerable disagreement
with the prevalent opinion about the characteristics of the Elayer is
weak. After careful consideration of the errors and uncertainties involved in the experiment the authors have arrived at a different interpretation which is given below. It is by no means complete, and it
leaves many questions unanswered. We still have to revise some of our
assumptions of the characteristics of the Elayer in order to bring this
interpretation in orderofmagnitude agreement with the electron density
data obtained by other methods, but the revisions are less drastic and re*
portedly supported by other independent evidence.
Returning to Fig. 15 it is seen that the molecular mean free
path varies from about 6 cm to 11 cm within the altitude range of interest.
* The authors are indebted to Dr. Eric Beth for information regarding relevent material presented at a seminar at AMC Cambridge Research Laboratories
in the summer of 1950.
50
These figures are based on the pressures simultaneously recorded by the
ionization gauges.
It is clear that the geometry of the probe and the sheaths, being
far from cylindrical, does not permit more than an orderofmagnitude calculation of Elayer constants. Nonetheless, it may be worthwhile to discuss the edge effects and see if some rough corrections can be made. If
we wish to use the cylindrical theory, an "effective length" of the collector, larger than the actual length, has to be used in the computations.
The added length should be a function of the sheath thickness or of the
difference between the thickness of the probesheath and the rocketsheath
thickness.
eff = + 2 + (al  a2) (68)
where' is a factor of the order of unity; in the calculations quoted
below the value 1.0 has been used.
When the correction becomes large there is very little justification for using the cylindrical theory at all. Since there is a tendency for unipolar fields to become spherical far from their source in
terms of the dimensions of this source, the use of the theory of a spherical probe has been considered and some calculations have been made for conmparison. The presence of the large collector, the rocket itself, prevents
the field from the small probe from assuming spherical shape. Because
the latter is located at the extreme tip of the rocket, however, the field
may approach spherical shape over a considerable solid angle. The relations
for a spherical probe that correspond to Eqs. (17) and (27) are
i = 1 + (69)
51
d [log (a)2] d d(log i) 5. dV 2 (70)
3 2 (70)
d(log Y) d(log F) V di
where (cc) is a function of /= a/r analogous to (P), also tabulated
by Langmuir and Blodgett. A plot of the left member of Eq. (70) as a
function of Y is shown in Fig. 16.
Another uncertain element in the reduction of data from the
December 1947 flight is the area A2 of the large collector, that is, the
rocket itself. It appears probable that the paint covering the main body
of the rocket was a sufficiently good insulator to reduce the effective area
to that of the nose piece. Because of the uncertainty this introduces no
use can be made of the data obtained for large positive values of the sawtooth voltage.
The evaluation of the preliminary data obtained from the December,
1947, flight involves two essential operations: calculation of the Elayer
characteristics from the data, and a check whether or not the data as a whole
are in agreement with theory. If not, reasonable explanations of the
deviations should be found. The latter of the two operations can in this
case only be qualitative, but it is nonetheless essential in order to
establish confidence in the bipolarprobe method as a tool for exploration
of the ionosphere.
The two operations are so closely related that they cannot be
treated separately. The presentation of the results will, therefore, be
accompanied by the discussion of the expected or unexpected character of
the data as their relationships suggest.
B. Negative Probe Current and PositiveIon Densit
In order to get an orderofmagnitude estimate of the ion density
in the Elayer from the data collected on the December, 1947, flight, the
52
w I_ I__ __ __I I
S2 f3
I 1 1
[r 2 01.
LLI ) p
w<yo o < O~rt
C')
c:
o~ s a, ic (O V) ~ C) CU
[X oIgoi)p
53
previously reviewed methods of computation have been used under different
sets of idealized assumed conditions. The results will be presented and
discussed in this section. The areas of the data will be selected where
one or the other of the sets of assumptions leads to the best approximation.
The alternative conditions covered by the computations are as
follows:
a) Cylindrical geometry, effective length of probe according
to Eq. (40) ( = 1), current limited by sheath area, ion
temperature 300~K
b) Same as condition a), except for a positiveion temperature
of 3000~K
c) Cylindrical geometry, effective length of probe arbitrarily
assumed to be 13 cm, current limited by orbital motion.
d) Spherical geometry, current limited by sheath area, ion
temperature 300~K
e) Same as condition d), except for an ion temperature of 3000~K
f) Spherical geometry, current limited by orbital motion, ion
temperature 300~K
The data utilized at each altitude are the coordinates and slope
of the probe characteristics at SV = 10 volts. The probe potential,
however, must be referred to space potential rather than rocket potential.
If we assume that the electron temperature is only a few hundred degrees
Kelvin, the space potential will differ only by a few tenths of a volt
from the probe potential for which the sharp upward bend in the curve
occurs. At 6V = 10 volts the actual probe potential is then roughly
12 volts. The results are plotted in Fig. 17.
54
4 FIG. 17
POSITIVEION DENSITY(d)
VS. ALTITUDE /++
ASSUMED CONDITIONS
CYLINDRICAL GEOMETRY
(0) CURRENT LIMITED BY SURFACE
AREA T 300~K x x x x x + (e)
10  b) CURRENT LIMITED BY SURFACE
8 AREA T = 3000~K 0 0 0 0 / oo /
6 (C) CURRENT LIMITED BY ORBITAL x
5 MOTION AAAA __ _/, (a)
4 SPHERICAL GEOMETRY
3 (d) CURRENT LIMITED BY SURFACE 0/ x/
I AREA T 300K + + + + +
2 2 (e) CURRENT LIMITED BY SURFACE 
O AREA T=3000~K a o o a a
m (f) CURRENT LIMITED BY ORBITAL
o 10ll MOTION T300~F * _ /_____
W 8
a. 7
z 5
0 0
4
> ~C(d)
z
w 2
z
0
 + ~(C)
ido _____ 
10' /
98 99 100 101 102 103 104.ALTITUDE IN KILOMETERS
According to Fig. 3 the transition between orbitalmotionlimited
current and sheatharealimited current in an ideal cylindrical geometry
takes place when.71 x 104 (67)
Nprr
55
For T = 300~K and r = 6 cm the critical ion density is 1.2 x 109;
it is ten times larger at 300~K. We should expect that the densityvs.altitude curves calculated under the assumption of orbitalmotionlimited
current and sheath arealimited current, respectively, should intersect
approximately at this critical density, where one solution is just as much
in error as the other. The data do not confirm this very well; the intersections occur at densities 3 to 5 times higher. This could be taken as
an indication of higher ion temperature. As we shall see later this is
not the most likely interpretation.
In order to get from the data presented in Fig. 17 an estimate of
the positiveion concentration in the undisturbed Elayer, we should first
select the curves in this diagram that are likely to give the best appraximation of the sheathedge ion concentration and then make the appropriate
correction for the drop in concentration in the ambipolar diffusion region.
When the sheath is very thin, the curve for a cylindrical probe at 3000K
is undoubtedly the best one. For larger sheaththickness the sphericalprobe curve for 300~K may be a better approximation. At lower altitudes
the sheathedge ion concentration can be expected to approach the values
given by the curve for an orbitalmotionlimited probe. As pointed out
above, this curve appearseto give a somerwhat too high concentration of
positive ions. This may be caused by a variation of the effective solid
angle of the spherical, or rather conical, flow to the probe. If this
angle increases when the probe sheath increases relative to the rocket
sheath, the slope of the probe curve for negative probe voltage would be
increased, thus giving too high positiveion concentration when the ideal
spherical probe relation (Eq 41) is used for computation of ion concentration.
56
Because of the short length of the probe we are justified in
using the spherical solution of the ambipolar diffusion to estimate the
drop in the positiveion density fron free space to the sheath edgre. At
the altitude of 103.5 km the radius of the sheath of the small collector
was about 10 cm, and consequently Npo t 1.25 NIa. This is probably too
low; despite its elongated shape the rocket itself is bound to mod.ify
the flow of ions in the diffusion region. An oppositely extreme value can
be computed by considering the rocket equivalent to a sphere with the same
area. Its radius would be 1.6 m and its sheath radius slightly larger,
say 1.75 m. Then [po AS 4.6 Na. The actual value is probably between
1.5 and 2.
The described process of selecting the most plausible values
from Fig. 17 and applying a rough correction for ambipolar diffusion
leads to the estimated positiveion concentration shown in Fig. 18. At
the highest altitude a correction factor of 1.5 to 2 las been applied
to account for the drop caused by diffusion. This factor has been increased slowly as the altitude is reduced. At the low end the concentration approaches asymptotically a curve somewhat lower than curve f in
Fig. 17.
It is interesting to compare these preliminary results of ionospheric probe measurements with the electron densities and Elayer altitudes determined by radiopropagation tests. The ionosphere recorders
at White Sands, New Mexico, at 2:30 p.m. on December 8, 1947, i.e., at
the time and place of the rocket firing, indicated an Elayer virtual
height of 110 km and a corresponding critical frequency of 3.4 mc, or
a maximum electron density of 1.45 x 1011 per m3. This is roughly 1/10
57
4 — FIG. 18
3 POSITIVEION CONCENTRATION VS. ALTITUDE
ESTIMATE FROM DATA OF DEC.8, 1947 FLIGHT
I52 ___/_,
C 3
I 
03
_o
I
99 98 99 100 101 102 103 104
ALTITUDE IN KILOMETERS
of the maximum positiveion density obtained from the probe data. Crude
as these first probe tests are, careful study of the interpretation makes
it seem unlikely that one should discount completely this evidence that
either the positiveion density appreciably exceeds the electron density
or the ion temperature is considerably higher than the gas temperature.
It has long been assumed that only a negligible number of negative ions,
compared to the number of electrons, exist in the Elayer. This assumption has recently been questioned, however, and other evidence for a considerable concentration of negative ions has been reported. It appears,
therefore, that the most probable explanation of the discrepancy between
58
the anticipated and the observed concentration of positive ions is an
unexpected abundance of negative ions.
In this analysis the motion of the rocket relative to the air
has been disregarded. Actually the rocket had a velocity component perpendicular to its axis of about 150 m/sec. Since the mean thermal velocity of the positive ions is about 390 m/sec, the relative motion is not
negligible, but it is a minor factor in comparison with the uncertainties
introdtuced. by the imperfect geonetry, the limited accuracy of the diffusion
theory, etc.
The positiveionconcentration data from the descent differ rather
radically from those recorded during the ascent. This fact will be discussed later in connection with other anomalies of the probe characteristics
for the descent.
C. Positive Probe Current and Electron Enerry.
The part BC of the voltampere characteristics of an ideal probe
(Fig. 8) can be separated into a positiveioncurrent curve and a negativeparticlecurrent curve, as pointed out in an earlier section (page 20). A
plot of the logarithm of the current vs. the probe potential is in case of
temperature equilibrium a straight line with a slope inversely proportional
to the temperature (Eq. 29).*
_  ~ ~17
 Johnson and. Malter' have criticized the discussion of electron energy
included in a preliminary publication of some material connected with
the present report.19 Their criticism is largely justified as far as
the authors' speculation about a nonltMa: ellian velocity distribution
is concerned. Even with a IIaxwellian distribution the log ievs.V plot
could hardly be expected to be linear over so wide a range as that shown
in a diagram in the preliminary publication, despite the very large ratio
of the collector areas. The main reason, however, is the unfavorable
geometry and consequent overlap of sheaths and diffusion zones. If
Johnson and Malter mean to imply that a method like theirs wouldd give
better results, this is contradicted by the data. The positive part
of the probe curve is mostly of such a shape that it is not possible
to separate the positive and negative collection components to the
large collector with any reasonable degree of certainty.
59
The probe used on the December, 1947, flight was by no means
an ideal one, oand these plots do not very well approximate straight lines.
Use of the maximrun slope close to the foot of the electroncurrent curve
gives an "electron temperature" that varies with altitude as shown in Fig. 19.
7xI03
FIG. 19
"ELECTRON TEMPERATURE"
VS.
e _ _ _. ____ALTITUDE
z / DESCENT + + ASCENT' /    oo DESCENT
I /
7 5 75 80/ 5 0 5 001 —
A. I
( o
03
= 4 / /
z +
+
ASCENT
0 + t/ s
basis on Eiq, (.lj and the p iteio denty c ua in the prearious
section. Teo+ I
+ALTITUDE IN KILOMETERS
spects. The same tqeperature should be observed at aniy specific altitude
whether the rocket is ascending or descending, and its value should not
differ appreciably from the gas temperature; the temperature equilibrilum
lias been fairly well confirmed by quantitative observations of the "Luxembourg effect". The negativeparticle current shows additional peculiarities: at low altitudes it is considerably larger than predicted on the
basis of Eq (11) and the positiveion density calculated in the previous
section. The current observed at positive values of 6V actually decreases
60
as the positiveion density increases (see Figs. 12, 13, 14, 20).
40
FIG. 20
PROBE CURRENT VS. ALTITUDE
PROBE VOLTAGE + 10 VOLTS
32 /
ASCENT 0
24
to 4 78 82 86 90 94 98 102 106
I /! 16 
I 0
0
ALTITUDE IN K RS103.5KM
70 74 78 82 86 90 94 98 102 106
ALTITUDE IN KILOMETERS
The two main causes of these deviations from the ideal probe
theory are believed to be, in addition to the noncylindrical geometry previously discussed: distortion of the fields and sheaths because of the
proximity of the two collectors, and the aerodynamic disturbance caused
by the motion of the rocket. The former factor is helpful to explain the
results obtained during the ascent; the different behavior during the descent is ascribed to the latter cause. (Fig. 22)
When the ion density is small, so that the sheath thickness is
large, the small collector is for a certain range of potentials completely
immersed in the sheath of the large collector. As an illustration, Fig. 22
shows a qualitative map of the equipotentials associated with a small electrode close to the surface of a large electrode in the absence of space
61
ALTITUDE IN KILOMETERS
70 74 78 82 86 90 94 98 102 106
oo^ o FIG.~"' 21AS ENT
4 _______O
5 _. _____ __________________________
I DESCENT'S
~~~ \I~~ I I I I I I ~~~~~ APEX
w 3
PROBE VOLTAGE = 10 VOLTS
4
FIG. 22
SKETCH OF A SADDLEPOINT FIELD
62
charge. If the large electrode has a negative potential with respect to
space, and the potential of the small electrode is positive or closer to
the space potential, the potential has a "saddle point" (P) outside the
small electrode, so that an electron reaching this electrode must have
passed through a potential minimum. Whether or not an electron will reach
the small electrode depends on the depth of this minimum rather than on
the electrode potential. Under such conditions Eq. (11) does not give
the collector current. Because of the accelerating field inside the potential minumum a very large fraction of all electrons that penetrate the
minimum will reach the electrode. The number of electrons exposed to this
accelerating field will be proportional to the "effective area" of the potential minimum. The negative potential of the large electrode, in other
words, has the effect of focusing the electrons on the small electrode.
This is a possible explanation to the observed electron currents that at
the lower altitudes are considerably larger than predicted from Eq. (11).
Since the potential at the minimum varies more slowly than in direct proportion to the potential of the small electrode, the plot of log ie vs. collector potential will give too small a slope and too high an apparent temperature. As the altitude increases, the sheath thickness decreases, and
the errors produced by interactions between the sheaths become negligible.
Even at the highest altitude attained by this rocket the value of the
"electron temperature" calculated from the data is higher than expected,
about 1000~K instead of about 300~K. An error of this magnitude could,
however, very well be introduced in the calibration and in the process cf
reducing the data. The resistance of the currentmeasuring circuit is appreciable, and the correction of the slope for this instrumental error
65
consequently involves taking the difference between approximately equal
numbers with considerable loss in accuracy as a result (see Appendix II).
These first crude measurements by means of a bipolar probe thus neither
confirm nor contradict the electron temperatures calculated from other observations, although the method certainly is potentially capable of a much
more accurate answer than so far obtained.
The rocket had a considerable horizontal velocity component; close
to the apex of the trajectory the motion was consequently nearly horizontal,
and the air surrounding the probe was violently disturbed. Just beyond
the apex the ionization gauge on the axis as well as one of the side gauges
read a pressure lower than that read by the other gauges by a factor of ten.
The correlation between the anomalous probe characteristics (Fig. 14) and
this disturbance suggests a causal colnection. That the correlation is not
a coincidence is indicated by the fact that a very similar correlation was
observed an appreciable time before the apex of an earlier flight, February,
1947, in this case probably caused by a strong wind hitting the rocket at
that moment. That time (Fig. 23) the positive probe current was not completely wiped out as it was in Fig. 14, but the sharp upward bend in the
FIG. 23
PROBE CURRENT VS. PROBE VOLTAGE
(a) TIME 169.5 SEC.
ALTITUDE 107.0 KM.
(b) TIME 173 SEC.
ALTITUDE 107.5 KM.
FEB. 1947 FIRING
PROBE VOLTAGE
Z
o
1a
64
probe curve disappeared and was replaced by a gentle slope. The nose gauge
and two of the side gauges showed a sudden drop in pressure, the former by a
factor of 10, the latter by factors of 5 and 2, respectively. Evidently the
rocket axis was slightly inclined from the vertical, and the wind hit from
the tail end. The instantaneous drop in pressures makes it very unlikely
that the earlier cavitation on one side of the nose piece as compared to the
December 8, 1947 flight was solely due to a different rocket aspect; in contrast the pressure differential between the various gauges about the nose
on the December 8 flight formed quite gradually.
The most remarkable fact about the Fig. 14 and Fig. 23 data is the
high particle energy involved. In Fig. 14 it is seen that a current is forced
through the system against an opposing voltage of up to 20 volts. Temperature
rises caused by adiabatic compression or viscous friction in the boundary layer are obviously inadequate to explain energies of this order of magnitude.
Local electric fields produced by density gradients or by the rapid flow of
plasma through the earth's magnetic field are too small to accelerate the
electrons appreciably during a single transit through the field. It is
necessary to look for a cumulative process whereby the same electrons are
repeatedly exposed to accelerating forces in the same disturbed region.
Bohm and Gross have shown that instability phenomena in a plasma may re20
sult in a considerable acceleration of a group of electrons.2 The density
gradient of atoms and ions indicated by the large difference between the pressures recorded by closely spaced ionization gauges may be a cause of
such instability. Another may be the mixing in the slip stream of the
rocket of plasma streams that have followed different paths about the rocket
and have different velocities and directions. Violent turbulence in the
plasma in conjunction with the earth's magnetic field will produce an
65
equivalent electric field of random direction and intensity. The viscosity
of the air and the magnetic field may keep the same group of electrons long
enough in the disturbed region to raise their temperature substantially, in
spite of the low intensity of the electric field. The fundamental reason
why the electron temperature can rise appreciably above the gas temperature
regardless of the f'requent collisions of the electrons with gas molecules
is of course that the lower concentration of electrons is partly counterbalanced by the smaller mass of the electron and the longer range of the
Coulomb electrontoelectron' interaction as compared to the clectrorntomolecule collision interaction. This enables the aggregate of electrons
to retain some of the increase in kinetic energy and gradually to build up
their temperature. The electrons accelerated by such processes would make
the potential of both collectors more negative with respect to the plasma,
thus increasing the positiveion current above the value observed at the
same altitude during the ascent. The geometry of the collectors and the
asymmetry of the electron and ion flow may very well explain the fact that
the net current to the small collector in Fig. 14 is negative even when
this electrode has a higher potential than the other.
During the ascent a similar phenomenon could be expected to occur
at the tail of the rocket. Since this part of the rocket was covered with
paint it is not too surprising that the probe characteristics do not reveal
its presence then.
Alfven has described another mechanism by which charged particles
in a plasma can be considerably accelerated.21 Charged particles that
because of their thermal energy describe helical paths along the lines of
force of a magnetic field and during part of each loop pass through a stream
66
that has an appreciable velocity relative to the surrounding plasma may
be Gradually accelerated by the electric field caused by the polarisz ation
of the stream. This process does not seem to fit our case, since it requires a mean free path of the electrons several hundred times the length
of one loop of their orbits.
One of the unknown quantities entering the calculation presented
here is the ratio of negativeion to electron concentration. It has been
pointed out above that if we accept the assumption of temperature equilibrium a comparison between the positiveion concentration indicated by the
probe data and the electron concentration determined by radiopropagation
tests suggests that this ratio is considerably larger than unity. Some
qualitative observations nay be made regarding this ratio outside the altitude range of reducible data. When the first trace of a probe current
appears, at a little more than 60ln altitude, the oscillograms are nearly
antisyrmnetrical; no step indicating electron current is observed. This
would mean that the negative charge is carried nearly exclusively by ions.
The electron current begins to appear at about 70km altitude. Similar
observations are made during descent; the ion current, however, is larger
at the same altitude. Also the electron current is observable to a somewhat lower altitude during descent.
It should be pointed out that the interpretation of the first
antisymmetrical probe curves in terns of positive and negative ions is not
the only possible one. For instance, photoelectrons emitted from both
collectors and collected at a different rate, depending on the difference
of potential between the collectors, could possibly give a curve of this
type. This interpretation is less probable; the large ratio between the
photoemissive areas would upset the antisymmetry. Also, it would have to
67
be a pure coincidence that this phenomenon occurs suddenly just before
the ordinary asyrmmetric probe curves begin to appear.
68
CONCLUSIOTIS
The most important question that this report attempts to answer
is whether or not a bipolar probe on a rocket can be so designed that the
data obtained will substantially contribute to our knowledge of the atmospheric strata between 60 and 200 km.
We believe that the material presented in this report justifies
an affirmative answer. The most questionable point in the reduction of
the observed data is the correction for the ambipolar diffusion. In
the transition region between the unipolar sheath and the region of ambipolar diffusion neither of the assumptions made in the analysis of these
two regions holds. This, of course, affects all probe measurements, and
it is not felt that the errors introduced by this circumstance are prohibitive. However, independent experimental determination of diffusion
coefficient, mean free path, and volume recombination coefficient is a
prerequisite for the accurate reduction of data from the probe measurements.
A detailed discussion of all systematic errors affecting probe
measurements in general has been omitted in this report. The subject has
been carefully considered and the data have been examined for evidence of
their presence. In the present set of data they are certainly small compared to the uncertainties introduced by the imperfect geometry and the
interaction between the collectors.
The preliminary exploratory character of the first bipolarprobe
experiments does not permit any firm conclusions from the data obtained
about the characteristics of the ionospheric Elayer. The negativeparticle energy data are not incompatible with an approximate temperature
equilibrium.
69
The positiveion densities estimated from the data are more
likely to be too low than too high; if they are not too high, the positiveion density exceeds the electron density as measured by radiopropagation
methods at least by one order of magnitude. Since it has generally been
assumed until quite recently that the ratio of negativeion concentration
to electron concentration is of the order of 0.1 per cent, it would mean
that future bipolarprobe tests would be very much worth while for the
purpose of settling this question.
The ion density computed from the probe data shows marked random fluctuations with time, but no attempt is at present justified to decide whether this can be altogether attributed to random errors in calibration, telemetering, reading, and reduction of data, or whether it is at
least partly caused by passage of the rocket through ion clouds of varying
density.
70
APPENDIX I
PROCEDURE FOR DETERMINING PROBE CURRENT AND VOLTAGE
FROM TELEMETERING RECORD
Probe Current
To illustrate the procedure used in reducing the telemetering
data, a sample calculation of the probe current will be made. For this
purpose, frequent reference will be made to Fig. 24, which shows Channels
3 and 4 of the Telemetering Record No. 1.
Position 2 of Channel 3 is the variablevoltageprobe current,
and position 3 is a 3.1volt calibration voltage. Position 4 is ground.
Channel 4 is the variablevoltageprobe current.
In order to measure distances on the telemetering record, a
glass plate with a superimposed grid of horizontal and vertical lines
was used. The numerical marking of the grid is arbitrary, but the calibration curves were made on the basis of this same system of units.
GRID
20
16.60
H(wM%"* CHANNEL 3
14.15 2 (CALIBRATION VOLTAGES)
~6 &~6
5 Aw5
11.60 4 GROUND
LINE
192 CYCLE X
RIPPLE
CHANNEL 4
4.~~ 405 ___ ____ ^ (PROBE CURRENT)
CHANNEL 4
0 1.05 9 10 14 REFERENCE LINE
< — D 
D CORRESPONDS TO. APPROXIMATELY 0.6 SEC.
FIG. 24
SAMPLE OF TELEMETER RECORD
71
In Fig. 24, the grid is shown in position on the telemetering
record at 161 seconds. All straight lines in this figure belong to the
grid. The horizontal zero line of the grid is placed on top of the reference line of Channel 4. Vertical grid line No. 14 is placed on the
right end of the probecurrent pulse in Channel 4. The right end was
used as a reference, since it is more sharply defined than the left end,
being relatively free of transients.
All vertical distance readings are taken to the average value
of the 192cycle ripple. (This 192cycle ripple is due to the fact that
each of the 23 channels is sampled at a rate of 192 times per second.)
Readings are made to the nearest half division of the grid, or 0.05 arbitrary units.
The vertical distance to position 3 (3.1volt calibration voltage) is found to be 16.60 units. The distance to the ground or zerovoltage pulse is 11.60 units. Thus, 3.1 volts corresponds to 5.0 units
in the calibration channel. From the "calibration of Channel 3" curve,
Fig. 25, it is seen that 3.1 volts should correspond to 5.3 units.
FIG. 25
CALIBRATION OF CALIBRATION CHANNEL NO. 3
MEASURING FROM ZERO LINE OF CALIBRATION _
CHANNEL WITH PROBE GRID, RECORD NO, I
DEC. 8, 1947 FIRING
0 
I00/
0 1 2'3 4 5 6'7 8 9
DISTANCE
> ~ ~ ~~~~1
72
Therefore, distances in the calibration channel are multiplied by the factor 5.3/5.0 to obtain the corrected reading. This, of course, assumes a
linear error.
At the horizontal distance 9, position 2, vertical distance is
read as 14.15 units. The distance above the ground line is therefore 2.55
units. The corrected distance is then 2.55 (5.3/5.0) = 2.70. From Fig.
25, the corresponding voltage is found to be 1.65 volts.
From Fig. 26, the calibration curve of the probe channel, 1.65
volts is seen to correspond to a distance of 3.95 units. By actual measurement in Channel 4, the distance is found to be 4.05 units. All grid
readings must therefore be multiplied by 3.95/4.05 = 0.975 to obtain the
corrected reading. Again linear error is assumed.
FIG. 26
CALIBRATION OF PROBE CHANNEL
MEASURING FROM PROBE BASE
LINE WITH PROBE GRID
DEC. 8,1947 FIRING, RECORD NO. I
J
1 2 3 4 5 6 7 8 9 10
CORRECTED VERTICAL DISTANCE
This calibration procedure is repeated at a horizontal distance
of 10 units, and for this case the correction factor was found to be 1.00.
The average of the two correction factors is then used to correct all the
73
vertical distances on this cycle. The currents corresponding to these corrected distances are then read from Figs. 27, 28, and 29, which plot vertical distance vs. probe current.
300
FIG. 27
DISTANCE WITH PROBE GRID
200
~ 20C   — MEASURED FROM REFERENCE
LINE VS. PROBE CURRENT
DEC. 8,1947 FIRING
100
80
C')
60 \, 40
L 30 
20
g 20
_z A(Na\ tELECTRON CURRENT
(NEGATIVE)
3 101 23   
0 e 2 3 4 5 6
DISTANCE
Probe Voltage
The total horizontal length of each cycle D corresponds to a
voltage difference of 40.8 volts, from 20.4 to +20.4. The voltagedistance relationship is assumed to be linear. To find the voltage at
any point, x, the horizontal grid scale is read at that point. This
distance is subtracted from 1.4, the width of the probe grid, to obtain
the quantity y. The length D is taken as the distance from the end of
the preceding cycle (1.05 on the grid in this case) to the end of the
cycle being measured (14). Then D = 14  1.0 = 12.95. The ratio y/D
is computed, and the voltage at point x is calculated as $V = (0.5 
y/voltageD) 40.8 volts.
y/D) 40.8 volts.
74
FIG. 28
DISTANCE WITH PROBE GRID
MEASURED FROM REFERENCE
2 LINE VS. PROBE CURRENT
z
w DEC. 8, 1947 FIRING
I  I I  
o
z
z
I 
4 5 6 7 8 9 10
DISTANCE
FIG. 29
DISTANCE WITH PROBE GRID
MEASURED FROM REFERENCE
LINE VS. PROBE CURRENT
6.    DEC. 8,1947 FIRING
wo
a/
0 
0 /___/
w
6 7 8 9 10 11 12
DISTANCE
cr
4~~~~~~I.2
75
APPENDIX II
CORRECTION OF PROBE VOLTAGE
FOR RESISTANCE IN MEASURING CIRCUITS
The difference in potential between the probe and the missile
differs from the potentiometer voltage by the voltage drop across the
input of the cathode follower of the measuring circuit. As may be seen
from the circuit diagram of Fig. 30, this drop will be dependent on the
probe current and the characteristics of the diodes and bias batteries
in the rangeswitching circuit. In the preflight calibration of this
circuit, the output voltage was measured as a function of input current,
but no measurements were made of the input voltage. Since the equipment
was destroyed, no possibility exists to obtain this voltage correction
directly, so two alternatives are available. One is to reconstruct the
input circuit and measure its voltage drop as a function of input current. However, this method will give results which are extremely dependent on the characteristics of the two rangeswitching diodes. It
PROBE
5100 A
+ 105V
TOTAL POT. VOLTS 40.81 6SN7
6AL5a \i TO
800 K  TELEMETERING
SYSTEM
1.57 V
1.56V 580
T6AL5
<< lOOK
22.526K
 MISSILE GROUND
FIG. 30
PROBE CIRCUIT (DEC. 8, 1947 FIRING)
76
is thought that the actual circuit characteristics can be more accurately
reconstructed from the inputcurrentoutputvoltage calibration curve and
measurements on a reproduction of the cathodefollower stage, for which
relatively accurate circuit values were measured at White Sands prior to
the firing. These values, including voltages, are indicated in the diagram of Fig. 30.
The cathodefollower stage was set up in the laboratory using
the constants given, with a 1megohm input resistor, and the output voltage
vs. input voltage was measured (see Fig. 31). Then, using the inputcurrentvs.outputvoltage calibration (Fig. 32), an approximate probecurrentvs.inputvoltagedrop characteristic can be determined (Figs. 33 and 34). This
is the voltage correction that is applied to the potentiometer voltage.
FIG. 31
CALIBRATION CURVE OF CATHODE
FOLLOWER OF SAME TYPE AS
— 5         USED ON DEC.8 1947 V2 PROBE
CIRCUIT.
105 V..g 6SN7
U,,FJ he.^ I IC
~5 4 3 2 1 0 1 2 3 4
E IN
It is observed that the voltage correction has a very noticeable
effect on the calculated electron temperature, reducing the temperature as
calculated without corrections by a factor of approximately two. The correction also reduces the scattering of the data.
77
FIG. 32
PROBE CIRCUIT CALIBRATION CURVE
DEC. 8,1947 FIRING
5
cn
4
180 16 0  120 100 80 60 40 20 0 10
(3
0 L
z
iw
_j
I
X,I
180 160 140  20  00  80  60 40  20 0 10
PROBE CURRENT (MICROAMPS)
FIG. 33,
PROBEVOLTAGE CORRECTION (INPUT VOLTAGE VS. PROBE CURRENT)
bJ
I I /.d
100 90 80 70 60 50 40 30 20 10 0 10
PROBE CURRENT (,ULA)
78
3.0
2.5
FIG. 34
PROBEVOLTAGE CORRECTION
EXPANDED SCALE
2.0
1.5
1'5   / 
1.0 —co
0
0.5
1.5 
6. 5 4 3 2 I 0 2 3
PROBE CURRENT (J1A)
79
APPENDIX III
THE FIRST TENTATIVE INTERPRETATION OF THE
DATA FROM THE DECEMBER, 1947 FLIGHT
In order to arrive at an agreement between the positiveion density calculated from the data and the electron density obtained from radiopropagation tests, it was necessary in the discussion of results presented
in the main body of this report to assume that the ratio of the density of
negative ions to that of electrons was much larger than is generally expected. In an earlier attempt at an interpretation of the data19 this agreement was reached by giving up the idea of an approximate temperature equilibrium in the Elayer and ascribing relatively high energies to the ions
and electrons. It was believed that the data gave evidence not only of
such energies but also of a nonMaxwellian velocity distribution for the
electrons. As pointed out above, it is now the authors' opinion that a
number of errors and uncertainties in the experiment were overlooked or
underestimated in the first evaluation of the data. For the sake of completeness, the theory on which this early interpretation was based is presented in this appendix.
The goal of this theory was to develop a formula with a sufficient number of parameters and of a suitable functional form so that synthetic voltampere curves could be produced identical to those observed.
The probe theory based on Maxwellian velocity distributions failed to do
so in two respects. The electron current to the small collector did not
become observable until the difference of potential between the collectors
was about +2 volts, while the Maxwellian theory predicted a few tenths of
80
a volt. It was also considered that the plots of the logarithm of the
observed electron current to the small collector vs. the collector difference of potential were not linear over as wide a range as predicted
on the same basis. A velocity distribution characterized by a uniform
drift velocity superposed on a Maxwellian velocity distribution was found
to give the desired result.
Proceeding from Eq. (2) in the review of the probe theory above,
we obtain the following expression for the current to a cylindrical collector for such a velocity distribution of the particles:
d. x s co 2 + sin@
AIl; d~ f x exp[(x s cos ) erf (, s sin
0 ().~)l/2
0 ^(1)11
erf 2(L  s sin dx (71)
The notations are the same as those used previously in this report. The random current density I is now referred to a coordinate system
moving with the drift velocity. The dimensionless quantity s is the ratio
of the drift velocity to the most probable velocity of the Maxwellian distribution. The drift velocity is assumed to be perpendicular to the axis
of the cylinder. G is the angle between the radius vector and the direction of the drift velocity. The distortion of the sheath from the shape
of a cylinder coaxial with the collector is assumed to be negligible. As
before, the second integration takes place between the limits 0 to o for
an accelerating field and (_)1/2 to o for a retarding field in the sheath.
In the case of interest here, the potential of the collector is
negative with respect to space, so that a positiveion sheath is formed.
81
The positive ions move in an accelerating field, while the electrons meet
a retarding field on entering the sheath.
If s = 0, the electron current is the same as previously obtained:
ie = A Ie e. (72)
When the drift is not negligible, an asymptotic expression for
Eq. 71 can be obtained by letting / approach infinity. This case was
considered in detail by Langmuir when, in addition, the drift parameter
s is approximately equal to or greater than /r, the result can be expressed in the following form:
ie = 0.384 A Ie / F1(), (73)
where
x = s  VF (74)
1(X) = Fj / 2\ e(Xx)2 dx (75)
0
Here, as in Eq. 72, the current collection in a retarding field
is independent of the, sheath thickness.
When the collector radius is large and the sheath thickness is
moderate or small, a different asymptotic solution is helpful. If / approaches unity, Eq. 71 becomes
_k2
ie = A Ie e + s v/t erf(X) = = AIe F2(Xs), (76)
valid for vJC > s > 1.
82
Eqs. 75 and 76 show that, when the electrons have an appreciable drift velocity, a plot of log ie vs. the collector potential or
e cannot be expected to be a straight line as it is when Eq. 72 holds.
To demonstrate this, the graph in Fig. 35 presents the logarithm of the
righthand side of Eq. 73 vs. 7e for various values of the drift parameter s. For large retarding potentials, the curves are approximately linear, but the slope is considerably smaller than when the drift is zero.
Interpreted on the basis of Eq. 72, they would be held to indicate a much
higher electron temperature than the one actually prevailing.
0 2.0
4
S"5 S'3 9S2 S.l SaO
5 __________
70 60 50 40 30 20 10 0
71
FIG. 35
THE LOGARITHM OF THE ELECTRON CURRENT VS. THE POTENTIAL PARAMETER 77
Another observable quantity of interest is the difference in
isolation potential between the two collectors, which is represented by
the intersection of the bipolarprobe voltampere curve with the voltage
axis. It is also a function of the temperature and drift velocity of
the electrons. Fig. 56 gives the isolation potentials of two cylindrical
83
22 n ~1 Xoo CM3
fla I X 105/C h3
Teo 50000 K
20
1 4
" 18
i r
 14_ _
UwQo COLLECTOR #2
QUILIBR2UM CPC GOLLECTOR 1I
_j (
 1.0 1.5 2.0 25 3
FIG. 36
J —
0 05 1.0 1.5 2.0 2.5 30 3.5 4.0
VELOCITY OF ELECTRON DRIFT IN SQUARE ROOT VOLTS
FIG. 36
EQUILIBRIUM POTENTIAL VS. ELECTRON DRIFT VELOCITY
collectors, one of large radius (V~ ~ 1) and one of small radius (V 0 ),f
as a function of the electron drift velocity.
In this early interpretation of the data the collectors were
assumed to have ideally cylindrical geometry and no interaction between
sheaths. The density of negative ions was assumed negligible. The electron current to the small electrode was obtained by extrapolation of the
positiveion current as previously described. The logarithm of the electron current is plotted in Fig. 37 for a few representative altitudes on
the ascent. It should be remembered that the data at this point had not
yet been corrected for the resistance of the measuring circuit.
84
0.5
0.
< 0.5
0
1.0 —'.2Z~. ~loX ~~~~~~~~~FIG. 37./ / THE LOGARITHM OF THE ELECTRON CURRENT VS..o THE POTENTIAL DIFFERENCE BETWEEN THE
oa/ 15 I/ /COLLECTORS. DEC. 8,1947 FLIGHT
I.5
r~~~/I. — Te r 0sow0 —
0..
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
FV, POTENTIAL OF COLLECTOR #1 WITH RESPECT TO#2 IN VOLTS
Due to the transcendental nature of Eq. 73, the drift velocity
and the electron temperature have to be found from the data by numerical
or graphical methods; for instance, by selecting a pair of values by trial
and error, checking them by calculating the electron current at two different collector potentials, and comparing the result with the data. The electron density was taken equal to the positiveion density determined from the
positiveion current at higher negative collector potential. In this way
an electron temperature of 15000K and a driftvelocity ratio s of approximately 6, or a drift velocity of roughly 1500 km/sec, were arrived at close
to the apex of the rocket trajectory. These figures agree also, at least
as far as the order of magnitude is concerned, with the observed difference
between the isolation potentials of the two collectors.
Incidentally, the sample probe curves shown in Figs. 7 and 8 were
calculated from the theory presented above, using the following "model" data:
density of positive ions and electrons 1011 per m3; electron temperature
85
5000~K; electron drift velocity 2 voltsl/2; area of collector No. 1 0.05
m2; and area of collector No. 2 0.8 m2.
This first interpretation of the data from the December 8, 1947
rocket flight was abandoned because the indications supporting it were considered too weak to contradict evidence obtained elsewhere of an approximate
temperature equilibrium in the lower ionosphere. The imperfect collector
geometry, particularly the overlapping of sheaths and diffusion regions,
and the uncertainty in the collector potential caused by contact potentials
and imperfections in the potentiometer were the main factors considered.
Furthermore, the assumption of a convectioncurrent density of the
order of magnitude consistent with this drift leads to other difficulties.
Diurnal variations of the terrestrial magnetic field caused by currents in
the Elayer are observable, but of a considerably smaller magnitude. According to the most successful theory, they are produced by "dynamo action"
in the earth's magnetic field caused by tidal and thermal motions of the
upper atmosphere. The velocity involved in such motions may be an appreciable fraction of the mean thermal velocity, but the accuracy of the
bipolarprobe measurements is certainly not at present high enough to
justify a correction for such a modest drift motion of ions and electrons..
86
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UNIVERSITY OF MICHIGAN
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