PHYSICS OF PLASMAS
VOLUME 7, NUMBER 11
NOVEMBER 2000
A new basic effect in retarding potential analyzers
Juan R. Sanmartı́na) and O. López-Rebollal
Escuela Técnica Superior de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid,
28040 Madrid, Spain
共Received 25 July 2000; accepted 2 August 2000兲
The Retarding Potential Analyzer 共RPA兲 is the standard instrument for in situ measurement of ion
temperature and other ionospheric parameters. The fraction of incoming ions rejected by a RPA
produces perturbations that reach well ahead of a thin Debye sheath, a feature common to all
collisionless, hypersonic flows past ion-rejecting bodies. This phenomenon is here found to result in
a correction to Whipple’s classical law for the current characteristic of an ideal RPA 共sheath thin;
inverse ram ion Mach number M ⫺1 , and ram angle of RPA aperture ␪, small or moderately small兲.
The current correction increases with the temperature ratio T e /T i , and ranges from a 15%–30%
reduction at M ␪ ⫽0 to a 15%–30% increase at M ␪ ⫽2, for typical values of M , T e /T i and
transparency of aperture grid. Linear analysis of the perturbed plasma beyond the sheath rests on the
fact that a Maxwellian undisturbed ion distribution is Vlasov-stable against quasineutral–
ionacoustic waves. © 2000 American Institute of Physics. 关S1070-664X共00兲04511-0兴
I. INTRODUCTION
lector there is a retarding grid biased at a positive value V P ,
which rejects the less energetic ions. The collected ion current I is registered as a function of V P as this potential is
swept through a broad range of values. For a planar RPA, T i
is determined by fitting the full experimental characteristic
I(V P ) to a formula derived by Whipple.13 Multigrid probes
共Ion Traps兲 may also serve, however, to determine ion density and composition, drifts, and, by inference, electric fields
in the ionosphere.14
Analysis of RPA collection in the Earth’s ionosphere is
simplified by the ordering of characteristic lengths: mean
free path (⬎104 – 105 cm), ion thermal gyroradius (⬃3
⫻102 cm), and satellite size (⬎102 cm) are large compared
to the aperture width (⬃10 cm), which is itself large compared to the distance between grids (⬍1 cm), and the electron Debye length ␭ De (⬃1 cm). Outside a sheath of thickness ␭ De next to the satellite wall and RPA aperture the
plasma is quasineutral. Also, the spacecraft velocity is large
compared with the ion thermal speed. Whipple then took an
one-dimensional approach to derive a formula for the current
reaching the collector. Whipple’s law is actually used with
instruments other than, though derived from, the RPA.9
Early discrepancies between data from planar RPAs and
other measurements led to careful criticism of Whipple’s
ideal law for the characteristic I(V P ). It was found that several effects inside the instrument 共nonuniformity of potential
in grid planes, energy-dependent grid transparencies, space
charge between grids, limited grid width兲 could affect the
characteristic.15 Instrument design was then improved to
avoid problems in using the ideal RPA law for data
interpretation.10,12 The effects of grid-mesh size and relative
alignment on electron motion were analyzed very recently.16
The current characteristic may also be affected by conditions outside the RPA. Although Whipple’s onedimensional approach rests on a condition of planar sheath,
i.e., small ␭ De , 17 RPAs have been used in the Solar Wind,
where the Debye length is large;12 nonplanar sheaths make
Positively biased satellites such as the one used for electron collection by the Tethered Satellite System, flown by the
National Aeronautics and Space Administration in February
1996 共TSS-1R mission兲, produce perturbations that spread
beyond thin Debye sheaths.1 This phenomenon, which had
been noticed both in experiments and in numerical
calculations,2,3 appears to be a fundamental feature of collisionless, hypersonic plasma flows past ion-rejecting bodies:
A low ion-density wake develops behind the body; ions
missing from the wake are those perturbing the plasma far
ahead. We argue here that this same phenomenon may affect
the workings of a Retarding Potential Analyzer in a basic
way.
Accurate values of the temperature T i of ionospheric
ions are needed for establishing a valid energy budget of the
upper atmosphere for the Earth, as well as for other planets,
and may require processing data from a large number of
measurements.4 In a laboratory plasma T i is a parameter hard
to measure. The Retarding Potential Analyzer 共RPA兲 has
been used on board satellites since the beginning of the space
age, having been kept as the standard in situ probe for determining T i in the ionosphere;5 a RPA will soon be flown on
the Chinese ROCSAT-1 spacecraft.6 RPA instruments have
been also used on board rockets at the bottom of the
ionosphere;7 on the Shuttle for measuring the plasma perturbations produced by the Shuttle itself;8,9 in the Venus4,10 and
Mars11,12 ionospheres; and on the TSS-1R electron
collector.1
A RPA is a multigrid electrostatic probe. An entrance
grid in the spacecraft wall is biased negative relative to the
potential in the undisturbed plasma to repel, like the wall
itself, incoming electrons; ions are collected by an electrode
at the back of the instrument. Between the aperture and cola兲
Electronic mail: jrs@faia.upm.es
1070-664X/2000/7(11)/4699/8/$17.00
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© 2000 American Institute of Physics
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Phys. Plasmas, Vol. 7, No. 11, November 2000
J. R. Sanmartı́n and O. López-Rebollal
data analysis difficult, however.17,18 Whipple’s law is also
favored by high values of ion Mach number M of the spacecraft motion through the plasma;19 too low a value of M
results in ‘‘internal shadowing,’’ as in the case of limited
grid width.8,19 Similarly, the law requires not too large an
angle ␪ between normal to RPA sensor and spacecraft
velocity;10 spacecraft spin keeps the angle varying, but measurements with moderately small ␪ are quite usual.10,12,20
The phenomenon studied here, however, affects Whipple’s law in a more basic way in the sense that it holds for an
ideal RPA, too. It was noted that ions rejected by the retarding grid should emerge from the RPA to form a jet with the
same cross section of the entrance grid.21 Unless the transparency ␣ E of that grid is very small, the field disturbance
will not be confined to a sheath and will have a radically
three-dimensional character; this will affect incoming ions
and thus the collected current I. It was independently suggested that ions somehow reflected from the satellite wall
could excite electrostatic Lower Hybrid waves, which might
explain anomalous data in RPA experiments with spacecraft
velocity near parallel to the geomagnetic field.20 Lower Hybrid waves were supposedly excited by the TSS-1R electron
collector, too.22
In Sec. II we recall conditions leading to Whipple’s ideal
law, which appears as the ␣ E ⫽0 limit of a more general
solution. An analysis of the perturbation beyond the RPA
sheath at moderately small ␣ E is presented in Sec. III. A
correction to Whipple’s law is then derived in Sec. IV and
results are discussed in Sec. V. In the Appendix our analysis
is shown to rest on the condition that the undisturbed plasma
be stable in the Vlasov sense.
FIG. 1. Grid schematics and ideal potential profile of a Retarding Potential
Analyzer 共RPA兲; E, P, and G are Entrance, Retarding and Suppressor grids,
C is the collector.
II. THE WHIPPLE MODEL
We use a frame moving with the spacecraft and let the
satellite wall be the 共infinite兲 plane x⫽0, the entrance grid E
being a circle of radius R centered at the origin; the plasma
fills the half-space x⬍0 共Fig. 1兲. Behind the retarding grid P
there is a suppressor G that is biased highly negative to turn
back all electrons able to get past E, and to inhibit photo and
secondary emission from the collector C. For simplicity we
take equal satellite and grid-E bias, V E ⬍0. Potential V(r̄)
and ion distribution function f (r̄, v̄ ) obey Poisson and steady
Vlasov equations, respectively,
冋
冉 冊 冕 册
eV
ⵜ V⫽4 ␲ e N ⬁ exp
⫺
k BT e
2
⳵f
e
⫽0,
v̄ •ⵜ f ⫺ ⵜV•
mi
⳵ v̄
f d v̄ ,
x⬍0,
共1兲
共2兲
x⬍0,
where m i is ion mass, T e electron temperature, and N ⬁ unperturbed ion or electron density. With m e U 2 /k B T e very
small and e 兩 V E 兩 /k B T e moderately large, electrons do follow
Boltzmann’s law except near the wall and grid E, where the
electron density will be exponentially small anyway.
We partition f in the form
f ⬅ f ⫹⫹ f ⫺,
f ⫹ 共 v x ⬍0 兲 ⬅0,
The boundary conditions are
f ⫺ 共 v x ⬎0 兲 ⬅0.
共3a兲
V→0,
⫹
f 共 v x ⬎0 兲 → f ⬁ 共 v̄ 兲
共3b兲
as x→⫺⬁,
共4a兲
V⫽V E ,
f ⫺ 共 v x ⬍0 兲 ⫽H 共 R⫺r⬜ 兲 g 共 r̄⬜ , v̄ 兲
at x⫽0.
共4b兲
Here f ⬁ is the Maxwellian of a single ion species drifting at
velocity Ū,
f ⬁ 共 v̄ 兲 ⫽
N ⬁ m 3/2
i
共 2 ␲ k BT i 兲
冋
3/2 exp
⫺
共 v x ⫺U x 兲 2 ⫹ 共 v̄⬜ ⫺Ū⬜ 兲 2
2k B T i /m i
⬅ f M 共 v x ⫺U x , v̄⬜ ⫺Ū⬜ 兲 ,
册
共5兲
r̄⬜ ( v̄⬜ ) is the position 共velocity兲 vector perpendicular to the
x axis; H is Heaviside’s step function, which takes into account that ions striking the wall at r⬜ ⬎R are neutralized; and
the distribution g depends on ion incidence and RPA model.
We let the drift Ū make an angle ␪ with the normal to the
wall in the x-y plane, U x ⫽U cos ␪ , Ū⬜ ⫽1̄ y U sin ␪ 共Fig. 1兲.
We take the Mach number M ⬅ 冑m i U 2 /k B T i moderately
large. In low Earth orbit, M ranges from 8 to 4 for T i
⬃1600 K and O⫹ and He⫹ ions, respectively; for H⫹, one
has M ⫽3 at T i ⬃750 K. We will also consider moderately
small incidence angles, writing sin ␪⬃␪, cos ␪⬃1; we let M ␪
Phys. Plasmas, Vol. 7, No. 11, November 2000
A new basic effect in retarding potential analyzers
be formally arbitrary, however, the characteristic value for
v⬜ in 共5兲 then being 冑k B T i /m i or U ␪ , whichever is the
largest. The domain in ion phase-space for which collisional
and finite satellite-wall effects are important 共corresponding
to undisturbed distant ions moving away from grid E兲 need
not be considered, those ions making a fraction of order
exp关⫺ 12M 2(1⫺␪2)兴 of the entire population; we shall actually
ignore terms of order M ⫺2 .
For an ideal RPA, as discussed in the Introduction, the
distribution g in boundary condition 共4b兲 takes a simple
form. Since width-to-depth grid ratio and M are moderately
large, and ␪ is small, ions arriving at E with v x such that
1
2
2 m i v x ⬍e(V P ⫺V E ), which are rejected by the retarding grid
P, emerge from E near the point of entry with the same v̄⬜
and opposite v x . This leads to
g⬇ ␣ E2 H
冋冑
册
V P ⫺V E
2e
⫺ 兩 v x 兩 f ⫹ 共 x⫽0,r̄⬜ , v̄⬜ , 兩 v x 兩 兲 ,
mi
v x ⬍0.
共6兲
We consider ␣ E2 formally small, and write expansions
2
⫾
V⫽V 0 ⫹ ␣ E2 V 1 ⫹ . . . , f ⫾ ⫽ f ⫾
0 ⫹ ␣ E f 1 ⫹ . . . , the effect of
the rejected ions proving to be moderately small for actual
values ␣ E2 ⬃1. Since g vanishes with ␣ E2 , and V E is negative,
2
we will have f ⫺
0 ⬅0. The solution of order zero in ␣ E is thus
one-dimensional, Eqs. 共1兲 and 共2兲 reading
冋 冉 冊冕
2
eV 0
d V0
⫺
2 ⫽4 ␲ e N ⬁ exp
dx
k BT e
⳵f⫹
e dV 0 ⳵ f ⫹
0
0
⫺
⫽0,
vx
⳵x
mi dvx ⳵vx
v x ⬎0
册
d v̄ f ⫹
0 ,
共7兲
v x ⬎0.
共8兲
Equation 共8兲 with boundary condition 共3b兲 is trivially solved,
giving
f⫹
0 ⫽fM
v x⬎
冕
冋冑
冑
v 2x ⫹
册
2e
V 共 x 兲 ⫺U x , v̄⬜ ⫺Ū⬜ ,
mi 0
2e
⫺ V 0 共 x 兲 ⬎0,
mi
共9兲
共10兲
Using 共10兲 in 共7兲 to integrate once with boundary condition
共3a兲 yields
冋 冉 冊册 冉 冊
2
␭ De
d eV 0
2 dx kT e
V 0 共0兲⫽V E .
2
Since the particle flux along x is conserved, the lowest
order current to the collector is due to ions with velocity v x
outside the sheath such that 21 m i v 2x ⬎eV P ,
I0
⫽A E e
␣
冕 冕
d v̄⬜
⬁
冑2eV P /m i
⫽A E eN ⬁ U cos ␪
冋
v x d v x f ⬁ 共 v̄ 兲
1⫺erf ⌬
⫹
2
冑
2
册
2 e ⫺⌬
,
␲ 2M cos ␪
冉 冊
eV 0
eV 0
eV 0
Te
⫽exp
⫺1⫺
⫺
kT e
kT e 2M 2 T i kT e
2
,
共7⬘兲
Equation 共7⬘兲 can be readily integrated to determine
V 0 (x)/V E as function of x/␭ De and parameters eV E /k B T e
and M 2 T i /T e ; here, ␭ De is the Debye length,
冑4 ␲ e 2 N ⬁ /k B T e . At large 兩 x 兩 /␭ De , V 0 (x)/V E vanishes as
exp(x/␭De) and f ⫹
0 approaches f ⬁ ( v̄ ).
共11兲
␣ ⫽ ␣ E ⫻ ␣ P ⫻ ␣ G being the overall RPA transparency 共Fig.
1兲 and
⌬⬅
冑
eV P M cos ␪
⫺
.
k BT i
&
共12兲
This is Whipple’s formula.13 As V P is increased, the bracket
in 共11兲 decreases from 1 to 0, the decrease being centered
around ⌬⫽0, or eV P ⫽ 12 m i U 2 cos2 ␪. When limited to this
lowest order perturbation, no small-␪ approximation is used
in the solution, so as to recover Whipple’s result as usually
written.
III. LOW ENTRANCE TRANSPARENCY
PERTURBATIONS
Terms of order ␣ E2 lead to a correction to Whipple’s
formula. Inside the sheath, the equation for f ⫺
1 in 共2兲 is
冉
vx
冊
⳵
⳵
e dV 0 ⳵ f ⫺
1
⫹ v̄⬜ •
f⫺
⫺
⫽0,
⳵x
⳵ r̄⬜ 1 m i dx ⳵ v x
v x ⬍0, 共13兲
which must be solved with boundary condition 共4b兲. Using
f⫹
0 for g to first order in Eq. 共6兲, the second term in the
parenthesis of 共13兲 is of order ␭ De /M R ( ␪ ␭ De /R) relative to
the first, for small 共large兲 M ␪ , and may be dropped. The
solution to 共13兲 is then readily determined; as x/␭ De→⫺⬁
one finds
f⫺
1
冉
冊
x
→⫺⬁,r̄⬜ ⫽H 共 R⫺r⬜ 兲 H
␭ De
冉冑
2eV P
⫺ 兩 v x兩
mi
⫻ f M 共 v x ⫹U x , v̄⬜ ⫺Ū⬜ 兲 .
2
d v̄ f ⫹
0 ⬇N ⬁ 共 1⫹eV 0 /kT i M 兲 .
4701
In the region outside the sheath, the equation for
冉
vx
冊
⳵
⳵
⫹ v̄⬜ •
f ⫺ ⫽0,
⳵x
⳵ r̄⬜ 1
冊
共14兲
f⫺
1
is
共15兲
thermal motion spreading the jet of rejected ions over distances of order MR; if M ␪ is large there is an imbedded
subregion with caracteristic length R/ ␪ along x. Equation
共15兲 is solved by Fourier transforming with respect to r̄⬜ ,
and then integrating with respect to x, to yield
f̃ ⫺
1 共 x,k̄⬜ 兲 ⬅
冕
dr̄⬜ f ⫺
1 共 x,r̄⬜ 兲 exp共 ⫺ik̄⬜ •r̄⬜ 兲
冉
⫽ f̃ ⫺
1 共 0,k̄⬜ 兲 exp ⫺i
冊
k̄⬜ • v̄⬜
x ,
vx
共16a兲
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Phys. Plasmas, Vol. 7, No. 11, November 2000
J. R. Sanmartı́n and O. López-Rebollal
with the r̄⬜ integration extended to the entire y-z plane.
Here, f̃ ⫺
1 (0,k̄⬜ ) must equal the Fourier transform of
(x/␭
f⫺
De→⫺⬁, r̄⬜ ) above,
1
2␲R
J 共 k R 兲H
k⬜ 1 ⬜
f̃ ⫺
1 共 0,k̄⬜ 兲 ⫽
冋冑
⫹U x , v̄⬜ ⫺Ū⬜ 兲 ,
册
v x ⬍0,
共17兲
1⫹erf ⌬
H 共 R⫺ 兩 r̄⬜ ⫹ ␪ x1̄ y 兩 兲 ,
2
共18a兲
N⫺
1 共 兩 x 兩 ⰇM R,z⫽0 兲 ⬇N ⬁
冉 冊
冉 冉 冊冊
1⫹erf ⌬ 1 M R
⫻
2
2 x
⫻exp ⫺
M2
y
␪⫹
2
x
2
2
.
共18b兲
Equation 共18a兲 reduces to 共17兲 unless M ␪ is large. In both
illustrative results 共18a兲, and 共18b兲 we ignored terms of order
1/M .
The density N ⫺
1 gives rise to a first-order potential perturbation, V 1 , which in term produces a first-order perturbation of the incoming ion population, f ⫹
1 . Outside the sheath,
where quasineutrality holds, Eq. 共1兲 to first order reads
N⬁
eV 1
⬇
k BT e
冕
⫺
f⫹
1 d v̄ ⫹N 1 ,
while the equation for
冉
vx
冊
f⫹
1
⫽
冉
共19兲
冊
共20兲
Fourier transforming 共20兲 and using boundary condition 共3b兲
to first order, one finds
f̃ ⫹
1 共 x,k̄⬜ 兲 ⫽
冕
冋
dx ⬘
k̄⬜ • v̄⬜
exp ⫺i
共 x⫺x ⬘ 兲
vx
⫺⬁ v x
⫻
x
再
册
Ñ ⫺
eṼ 1 Ñ ⫹
1
1
␤
⫺
⫽
,
k BT i N ⬁ N ⬁
冎
⳵
f ,
⳵ v̄⬜ ⬁
冕
˜Ṽ 共 k ,k̄ 兲 ⬅
1 x ⬜
⬁
⫺⬁
and then, integrating over v̄ , one arrives at Ñ ⫹
1 in terms of
Ṽ 1 ,
冕
冋
d v̄
v x ⬎0 v x
册
k̄⬜ • v̄⬜ ⳵ f ⬁
.
⳵ v̄
vx
共22兲
共23兲
␤⬅
Ti
,
Te
共24兲
x⬍0.
dx exp共 ⫺ik x x 兲 Ṽ 1 共 x,k̄⬜ 兲 ,
one uses Eq. 共24兲 to solve for ˜Ṽ 1 and then transform back
with respect to k x to find
⫽
冕
⬁
0
冕
˜ ⫺ 共 k̄ 兲
dk x exp共 ik x x 兲 Ñ
1
⬁
⫺⬁
2 ␲ ␤ ⫹L 共 k̄ 兲
N⬁
,
ds exp共 ⫺ik x s 兲 ik̄•L̄ 关 sk̄⬜ 兴 .
共25兲
共26兲
Using 共23兲 in Eq. 共26兲 for L(k̄), the v̄⬜ integration can
be carried out exactly; also, changing variable from s to ␴
⬅ 冑kT i /m i ⫻k⬜ s/ v x & the v x integral can be carried out,
too, when terms exponentially small are neglected. We then
find L(k̄) in terms of a single integral
冕
⬁
0
␨⬅
共21兲
eṼ 1 共 x ⬘ ,k̄⬜ 兲
,
k BT i
Equation 共24兲 is a linear integral equation defined in the
half-space x⬍0; singular; and with a difference kernel as in
the Wiener–Hopf problem.23 Our equation, however, is of
Volterra type. In order to solve it, we consider an extended
problem: Use Eq. 共24兲 with the function Ñ ⫺
1 (x⬍0) on the
right-hand-side continued to the halfspace x⬎0, to find
Ṽ 1 (x,k̄⬜ ) in the entire range ⫺⬁⬍x⬍⬁. We must then
show that our choice of right-hand-side in 共24兲 for x⬎0 has
no effect on the resulting solution for Ṽ 1 in the physical
domain of definition 共x⬍0兲; this is proved in the Appendix.
Here, by taking Fourier transforms with respect to x, e.g.,
L 共 k̄ 兲 ⬅L 关 ␨ 共 k̄ 兲兴 ⬅
e ⳵ Ṽ 1 共 x ⬘ ,k̄⬜ 兲 ⳵
mi
⳵x⬘
⳵vx
⫹Ṽ 1 共 x ⬘ ,k̄⬜ 兲 ik̄⬜ •
k BT i
N ⬁m i
冎
⳵
⳵x⬘
Finally, Fourier transforming Eq. 共19兲 one obtains an equation for Ṽ 1 (x,k̄⬜ )
k BT i
v x ⬎0.
⫺⬁
再
dx ⬘ L x 关共 x⫺x ⬘ 兲 k̄⬜ 兴
⫻exp ⫺i 共 x⫺x ⬘ 兲
L 共 k̄ 兲 ⫽⫺
⳵V1 ⳵
e ⳵V1 ⳵
⫹
•
f ,
m i ⳵ x ⳵ v x ⳵ r̄⬜ ⳵ v̄⬜ ⬁
x
L̄ 关共 x⫺x ⬘ 兲 k̄⬜ 兴 ⬅
eṼ 1 共 x,k̄⬜ 兲
in 共2兲 takes the form
⳵
⳵
⫹ v̄⬜ •
f⫹
⳵x
⳵ r̄⬜ 1
冕
⫹ik̄⬜ •L̄⬜ 关共 x⫺x ⬘ 兲 k̄⬜ 兴
共16b兲
1⫹erf ⌬
H 共 R⫺r⬜ 兲 ,
2
N⫺
1 共 兩 x 兩 ⰆM R,r̄⬜ 兲 ⬇N ⬁
N⬁
⫽
2eV P
⫺ 兩 v x兩 f M 共 v x
mi
where J 1 is the Bessel function of first kind and order. Inte⫺
gration of f̃ ⫺
1 in 共16a兲 over v̄ yields Ñ 1 (x,k̄⬜ ), and Fourier
inversion with respect to k̄⬜ yields N ⫺
1 (r̄) outside the sheath.
In particular, one finds
N⫺
1 共 x⫽0,r̄⬜ 兲 ⫽N ⬁
Ñ ⫹
1 共 x,k̄⬜ 兲
2 ␴ d ␴ exp共 ⫺ ␴ 2 兲 exp共 2i ␴␨ 兲 ,
⫺k̄•Ū
k⬜ 冑2k B T i
,
共27兲
where we now neglected terms of order M ⫺2 ; note that perturbations steady in the satellite frame have frequency
⫺k̄•Ū in the ionospheric frame. Next, using 共16a兲, and 共16b兲
in
Phys. Plasmas, Vol. 7, No. 11, November 2000
冕
˜ ⫺ 共 k̄ 兲 ⫽
Ñ
1
⬁
⫺⬁
dx exp共 ⫺ik x x 兲
冕
v x ⬍0
A new basic effect in retarding potential analyzers
d v̄ f̃ ⫺
1 共 x,k̄⬜ 兲 ,
共28兲
the v̄⬜ and x integrations can be carried out exactly, while
the v x integral can be explicitly evaluated to order M ⫺2
⫻
˜ ⫺ 共 k̄ , ␨ 兲
Ñ
⬜
1
•
N⬁
⬇ 冑2 ␲
3
M RJ 1 共 k⬜ R 兲
k⬜2
冋
⫻ 1⫹erf ⌬⫺
冑
exp关 ⫺ 共 ␨ ⫹&M ␪ sin ␾ 兲 兴
2
2
2 e ⫺⌬
兵 1⫺2
␲ M
册
⫻ 共 ␨ ⫹&M ␪ sin ␾ 兲共 ␨ ⫹ 冑1/2M ␪ sin ␾ 兲 其 ,
共29兲
where we used polar coordinates k⬜ , ␾ for k̄⬜ with sin ␾
⬅ky /k⬜ , and changed from k x to ␨. We thus finally obtain
eṼ 1 共 x,k̄⬜ 兲
⫽
kT i
冕
冋
d␨
k⬜ x
exp ⫺i
共 ␨ &⫹M ␪ sin ␾ 兲
M
⫺⬁ ␤ ⫹L 共 ␨ 兲
⬁
⫻
˜ ⫺ 共 k̄ , ␨ 兲
k⬜ Ñ
⬜
1
M ␲ &N ⬁
册
共30兲
.
Positive v x ions entering the sheath within the cylinder
r⬜ ⬍R with energy 21 m i v 2x ⬎ 12 m i v x min2⬅e(VP⫺␣E2 V*) will
cross the retarding grid P and reach the collector 共Fig. 1兲;
here, V * is V 1 (⫺xⰆM R, R/ ␪ ; r̄⬜ ). Within the sheath, the
ion flux through the lateral surface of that cylinder leads to
corrections of order ␭ De /M R or ␪ ␭ De /R, which we neglect.
To first order in ␣ E2 the current collected is thus given by
I⫽ ␣ e
冕 冕 冕
AE
dr̄⬜
d v̄⬜
⬁
v x min
v x d v x 共 f ⬁ ⫹ ␣ E2 f * 兲
⬇I 0 ⫹ ␣ E2 I 1 ,
共31兲
with f * ⬅ f ⫹
1 (⫺xⰆM R, R/ ␪ , r̄⬜ ). The integral involving f ⬁
yields Whipple’s result, I 0 , plus a small term due to the
perturbed potential V * in v x min . This term is added to the
integral involving f * , where we may set v x min
⬇冑2eV P /m i , to get the full current correction, ␣ E2 I 1
冕 冕
冕
I 1⫽ ␣ e
⫹
AE
dr̄⬜
⬁
冑2eV P /m i
d v̄⬜
冋 冉 冑 冊
册
eV *
f v ⫽
mi ⬁ x
v xd v x f * .
冉
0
⫺⬁
dx exp i
⬁
冑2eV P /m i
dvx
冊
冕
dk̄⬜
exp共 ik̄⬜ •r̄⬜ 兲
4␲2
eṼ 1 共 x ⬘ ,k̄⬜ 兲
k̄⬜ • v̄⬜
x⬘
ik̄⬜
mi
vx
冊
⳵
v̄⬜ ⳵
⫺
f ,
⳵ v̄⬜ v x ⳵ v x ⬁
共32⬘兲
I 1 ⫽⫺ ␣ A E eN ⬁ U
冋
2
册
2
1⫺erf2 ⌬
e ⫺⌬ 1⫹erf ⌬
e ⫺⌬ 1⫺erf ⌬
.
⫻ c1
⫺c 2
⫺c 3
2
M
2
M
2
共33兲
Here c 1 , c 2 , and c 3 are functions of ␤ ⬅T i /T e and M ␪ , and
are given by double integrals involving L( ␨ )
c j 共 ␤ ,M ␪ 兲
冕
2␲
0
d␾
2␲
冕
d ␨ exp关 ⫺ 共 ␨ ⫹&M ␪ sin ␾ 兲 2 兴
hj ,
␤ ⫹L 共 ␨ 兲
⫺⬁ 冑2 ␲
⬁
共34兲
&h 1 ⫽L 共 ␨ 兲 ,
共35a兲
冑␲ h 2 ⫽1⫺ 共 1⫺2 ␨ 2 ⫺ ␨ &M ␪ sin ␾ 兲 L 共 ␨ 兲 ,
共35b兲
冑␲ h 3 ⫽ 关 1⫺2 共 ␨ ⫹&M ␪ sin ␾ 兲共 ␨ ⫹ 冑1/2M ␪ sin ␾ 兲兴 L 共 ␨ 兲 .
共35c兲
Figures 2–4 show c j , j⫽1 – 3; they are real because the real
and imaginary parts of L are even and odd functions of ␨,
respectively.
We note that c 2 approaches some limit function
c 2 ( ␤ , ⬁) 共whereas c 1 , c 3 vanish兲 when formally taking
M ␪ →⬁. That function is easily obtained by writing ␨
⫹ 冑 2M ␪ sin ␸⬅␨⬘⫽O(1), and using the asymptotic approximation for L( ␨ ) at large values of its argument, ⫺L( ␨ )
⬃1/2␨ 2 ⬃1/(4M 2 ␪ 2 sin2 ␸)Ⰶ1 共see the Appendix兲. We then
find
c 2⬇
2eV P
mi
冕
共32兲
⫽
冕
␲
d␾
2␲
0
⫻
We take f * from 共21兲, where we set x⫽0, integrate by
parts the first term within braces, and take the Fourier inverse
with respect to k̄⬜ . The full first order current then becomes
AE
d v̄⬜
dr̄⬜
with Ṽ 1 (x ⬘ ,k̄⬜ ) given by Eq. 共30兲. The resulting multiple
integral for I 1 can be considerably simplified. The r̄⬜ and v̄⬜
integrations are straightforward. Changing variable from x ⬘
to ␴ ⬅⫺ 冑kT i /m i ⫻k⬜ x ⬘ / v x &, the k⬜ integral can be carried out exactly, and the ␴ and v x integrations can be carried
out to order M ⫺2 in terms of the L( ␨ ) function.
The final result is
⫽
IV. MODIFIED CURRENT LAW
冕 冕 冕
冕 ⬘ 冉
I 1⫽ ␣ e
4703
2
⬁
d␨⬘
⫺⬁
冑2 ␲
exp共 ⫺ ␨ ⬘ 2 兲 1⫺ 共 ⫺2 ␨ 2 ⫹ ␨ 2 兲 ⫻ 共 ⫺1/2␨ 2 兲
冑␲
␤
1
2 ␤ 冑2 ␲
.
共36兲
Values at M ␪ ⫽2 in Fig. 3 are already close to the
asymptotic result 共36兲.
4704
Phys. Plasmas, Vol. 7, No. 11, November 2000
J. R. Sanmartı́n and O. López-Rebollal
FIG. 2. Coefficient c 1 ( ␤ ,M ␪ ) in Eq. 共33兲 for the current correction; ␤, M , and ␪ are temperature ratio
T i /T e , ram Mach number for ions, and ram angle for
the RPA, respectively.
One might have surmized full vanishing of the correction I 1 as M ␪ →⬁, the reflected-ion density in Eq. 共18b兲
2 2
being exponentially small 关 N ⫺
1 /N ⬁ ⬃exp(⫺2M ␪ )兴 at the incidence angle y/x⬇ ␪ ; the density maximum lies of course
along the reflected velocity ⫺U x , Ū, i.e., at y/x⫽⫺ ␪ 共Fig.
1兲. Further, with ␤ of order unity and 共␤ ⫹Real part of L兲
⬎0 throughout the integration in 共25兲, V 1 behaves in a way
similar to N ⫺
1 . Ions incident on the RPA should then be
negligibly perturbed at distances 兩 x 兩 /M R⭓O(1). Note,
however, that incident ions, and ions reflected from all gridE points below each particular incidence point in Fig. 1, do
cross over distances 兩 x 兩 ⬃R/ ␪ ⰆM R, where the density N ⫺
1
is given by 共18a兲; this keeps I 1 from vanishing with 1/M ␪ .
Our correction to Whipple’s law is then the replacement
I 0 →I 0 ⫹a E2 I 1 ,
with I 0 and I 1 given by Eqs. 共11兲 and 共33兲, respectively. At
⌬ negative enough few ions are rejected by the RPA and the
correction is negligible ( ␣ E2 I 1 /I 0 →0 as ⌬→⫺⬁). For ⌬
moderately positive the dependence of the ratio ␣ E2 I 1 /I 0 on
all four parameters, ⌬, M , ␤, and M ␪ , is quite complex.
At normal incidence (M ␪ ⫽0) the correction ratio is
negative, its magnitude increasing with increasing M or decreasing ␤. The current reduction goes through a maximum
somewhere between ⌬⫽0 and ⌬⫽2. At ⌬⫽1 in particular,
and taking ␣ E2 ⫽0.9, current is reduced by 12%–20% at ␤
⫽1 and M ⫽4 – 8; and by 20%–33% at ␤ ⫽0.5 and M
⫽4 – 8.
At M ␪ ⫽2, however, the correction ratio is positive;
also, it grows monotonically with ⌬. Its magnitude now increases when either Mach number M or temperature ratio ␤
decreases. At ⌬⫽2, and again taking ␣ E2 ⫽0.9, the corrected
current is larger than Whipple’s current by 11% at ␤ ⫽1,
M ⫽8, and by 33% at ␤ ⫽0.5, M ⫽4.
The greater magnitude of our correction at lower ␤ is
manifest in the first term of 共24兲; at large electron temperature the electrons refuse to cooperate, so to speak, in balancing the reflected-ion density. On the other hand, the change
FIG. 3. Coefficient c 2 ( ␤ ,M ␪ ) in Eq. 共33兲.
Phys. Plasmas, Vol. 7, No. 11, November 2000
A new basic effect in retarding potential analyzers
4705
FIG. 4. Coefficient c 3 ( ␤ ,M ␪ ) in Eq. 共33兲.
of sign in I 1 as M ␪ increases is the result of two competing
effects. Incoming ions are clearly subject to a retarding deceleration, because V 1 , as N ⫺
1 itself, is positive; the second
term in the bracket of Eq. 共32兲 should then be negative,
whereas the first term is positive. One can easily show, in
particular, that, at large M ␪ 关when c 1 , c 3 vanish in Eq. 共33兲,
yielding I 1 ⬎0兴, the integrations in 共32兲 over its first and
second terms yield 2I 1 and ⫺I 1 , respectively.
Our correction to the extremum of the slope dI/V P of
the current characteristic is weaker than the correction for the
current itself; the slope extremum is sometimes used for a
simple estimate of ion temperature. Use of Whipple’s current
共11兲 yields
⫺
dI 0
dV P
⫽
␣ A Ee 2N ⬁
冑2 ␲ m i k B T i
exp共 ⫺⌬ 2 兲 ,
with the extremum at ⌬⫽0 and kT i ⬀1/兩 dI/dV P 兩 2 max . For
M ␪ ⫽2 in our full-current formula the extremum still occurs
at ⌬⬇0 but its value is reduced by a factor of 1— ␣ E2 (c 2
⫺c 3 )/M ; for ␣ E2 ⫽0.9 the corrected temperature is then
smaller than Whipple’s temperature by 4% at ␤ ⫽1, M ⫽8
and by 15% at ␤ ⫽0.5, M ⫽4. For M ␪ ⫽0 the extremum
occurs at ⌬ ext⬇⫺2 ␣ E2 c 1 / 冑 ␲ , with its value increased by a
2
; the actual temperature is now larger than
factor 1⫹⌬ ext
Whipple’s value by 7% at ␤ ⫽1 and by 15% at ␤ ⫽0.5.
V. CONCLUSIONS
We have described a basic correction to Whipple’s classical law 关Eqs. 共11兲 and 共12兲兴 for the current I to a planar
Retarding Potential Analyzer 共RPA兲, which is a standard
multigrid probe serving to determine ion temperature T i and
other ionospheric parameters. Grid-related effects inside the
RPA, which could invalidate Whipple’s law, had been corrected in the past by proper instrument design. The effect
here considered arises from those ions that are rejected by
the RPA and perturb the plasma far ahead of the sheath at its
front. This phenomenon is a fundamental feature of collisionless, hypersonic plasma flows past ion-rejecting bodies,
and was noticed in a broader context both in experiments and
in numerical calculations. Ion rejection is a process essential
to a RPA.
In deriving our correction to the current, we considered
the case of a single ion species, which is easily generalized,
and took the transparency of the aperture grid ␣ E formally
small, the effect of the rejected ions proving to be moderately small for actual values ␣ E ⬃1. We also considered an
ideal RPA as implied in Whipple’s law: We assumed inverse
ion ram Mach number M ⫺1 , ram angle of aperture normal ␪,
and ratio of Debye length to aperture width, small or moderately small, but let M ␪ arbitrary. For a ‘‘nonideal’’ RPA,
rejected ions would still affect the incoming plasma, although our analysis, as Whipple’s law itself, would not hold.
Our correction to Whipple’s law, given by Eqs. 共31兲,
共33兲–共35兲, depends on M ␪ and the temperature ratio ␤
⬅T i /T e , in addition to both M and retarding bias V P . For
typical values ␤ ⫽O(1), perturbations outside the sheath,
which are steady in the spacecraft frame, decay monotonically away from the RPA. The perturbed potential is positive, as the rejected-ion space charge itself, and results in
opposite, competing effects on the current: Incoming ions are
subject to a retarding decceleration, but once in the sheath
they face a reduced potential hill in coming to the retarding
grid. For ␣ E ⬃1 and ␤ ⫽0.5– 1, the correction to the current
amounts to 15%–30%, being negative at M ␪ ⫽0 but positive
at M ␪ ⫽2. From the extremum in the slope of the current
characteristic I(V P ), sometimes used for a quick estimate of
T i , the corrected T i is only 5%–15% larger 共smaller兲 than
Whipple’s value for M ␪ ⫽0 共for M ␪ ⫽2兲.
Our linear analysis can be extended to any ionospheric
ion distribution that is Vlasov-stable against quasineutral
ion–acoustic waves. Actually, the full steady solution, involving the beam of rejected ions, might itself be unstable
against Lower Hybrid perturbations. These need extend,
however, over extremely long distances along the geomagnetic field, which is only possible in the special case of a
spacecraft moving nearly parallel to the field.20
4706
Phys. Plasmas, Vol. 7, No. 11, November 2000
J. R. Sanmartı́n and O. López-Rebollal
ACKNOWLEDGMENT
This work was supported by the Comisión Interministerial de Ciencia y Tecnologı́a of Spain, under Grant No.
PB97-0574-C04-1.
advanced. Finally, note that this equation is the dispersion
relation for electrostatic waves of phase velocity much less
than the electron thermal speed of a Maxwellian plasma at
rest24
2
␭ Di
APPENDIX: UNIQUENESS OF SOLUTION
2
␭ De
We show here that in Eq. 共25兲, written as
eṼ 1 共 x,k̄⬜ 兲
k BT i
⫽
冕
⬁
dk x exp共 ik x x 兲
⫺⬁
2 ␲ ␤ ⫹L 共 k̄ 兲
⫹
冕
⬁
0
dx ⬘ e
⫺ik x x ⬘
册
冋冕
0
⫺⬁
dx ⬘ e ⫺ik x x ⬘
Ñ ⫺
1 共 x ⬘ ,k̄⬜ 兲
N⬁
共A1兲
,
the contribution from the x ⬘ ⬎0 range of integration vanishes
for x⬍0 whatever the function Ñ ⫺
1 , so that our choice of
right-hand-side in 共24兲 for x⬎0 had no effect on the resulting solution in the physical domain of definition. The vanishing of the x ⬘ ⬎0 contribution to 共A1兲 for x⬍0 is a result
of the property
冕
⬁
⫺⬁
dk x exp关 ⫺ik x 共 x ⬘ ⫺x 兲兴
2␲
␤ ⫹L 共 k x ,k̄⬜ 兲
⬅0,
x ⬘ ⫺x⬎0,
共A2兲
which arises from facts 共i兲 L(k x 兲, when continued analytically for complex k x , is analytical in the lower half-plane,
and 共ii兲 the equation ␤ ⫹L(k x )⫽0 has no roots in that halfplane. Then the integral 共A2兲, rewritten as
␦ 共 x ⬘ ⫺x 兲
⫹
␤ ⫹L ⫺ 共 ⬁ 兲
冕
dk x 关 L ⫺ 共 ⬁ 兲 ⫺L 共 k x 兲兴 exp关 ⫺ik x 共 x ⬘ ⫺x 兲兴
,
关 ␤ ⫹L ⫺ 共 ⬁ 兲兴关 ␤ ⫹L 共 k x 兲兴
⫺⬁ 2 ␲
⬁
x ⬘ ⫺x⬎0,
will clearly vanish; here, L ⫺ (⬁)⬅L( 兩 k x 兩 →⬁, Im kx⬍0).
To establish facts 共i兲 and 共ii兲 above, we note the relation
of our function L( ␨ ) in Eq. 共27兲 to the plasma dispersion
function Z( ␨ ), 24
L⫽⫺ 21 dZ/d ␨ ,
冕
冋冕
Z 共 ␨ 兲 ⬅2i
⫽
⬁
0
d ␴ exp共 ⫺ ␴ 2 ⫹2i ␴␨ 兲
dt exp共 ⫺t 2 兲
t⫺ ␨
⫺⬁ 冑␲
⬁
册
for Im ␨ ⬎0 ,
with the large-␨ asymptotic approximation, L( ␨ )⬃⫺ 21 ␨ 2 ,
which was used in Sec. IV. We also note that dZ( ␨ )/d ␨ is an
entire function of ␨, and is equal to a positive number nowhere in the upper half-plane Im ␨⬎0, there existing therefore no Im kx⬍0 solution to the equation ␤ ⫹L(k x )⫽0, as
⫺
1 dZ
2
⫽⫺k 2 ␭ Di
⬇0,
2 d␨
when taking long wave numbers 共kⰆ1/␭ De , 1/␭ Di兲, with ␨ in
共27兲 standing for ␻ /k(2k B T i /m i ) 1/2. Property 共A2兲 relates
then to the fact that a Maxwellian plasma is stable in the
Vlasov sense against quasineutral ion–acoustic waves. This
shows that a solution to the linearized RPA problem can be
determned for any non-Maxwellian ionospheric plasma that
is stable against such waves.
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