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Simulation and uncertainty quantification of electron beams in
active spacecraft charging scenarios∗
Álvaro Romero-Calvo †
Department of Aerospace Engineering Sciences, University of Colorado Boulder, CO, 80303, United States
Gabriel Cano-Gómez‡
Departamento de Física Aplicada III, Universidad de Sevilla, Sevilla, 41092, Spain
Hanspeter Schaub §
Department of Aerospace Engineering Sciences, University of Colorado Boulder, CO, 80303, United States
Novel active sensing methods have been recently proposed to measure the electrostatic po-
tential of non-cooperative objects in geosynchronous equatorial orbit and deep space. Such
approaches make use of electron beams to excite the emission of secondary electrons and X-Rays
and infer properties of the emitting surface. However, the detectability of secondary electrons
is severely complicated in the presence of complex charged bodies, making computationally
efficient simulation frameworks necessary for in-situ potential estimation. The purpose of this
paper is twofold: firstly, to introduce and test a quasi-analytical, uncoupled, and computation-
ally efficient electron beam expansion and deflection model for active charging applications;
and secondly, to characterize the uncertainty in the beam-target intersection properties, which
condition the measurement of secondary electrons. The results show that a combination of sec-
ondary electrons and X-ray methods is highly desirable to yield a robust and accurate measure
of the potential of a target spacecraft.
Nomenclature
𝛼= electrode rotation angle, rad
𝛽= velocity ratio
B= nondimensional external magnetic flux density
𝑩= external magnetic flux density, T
𝒃= internal magnetic flux density, T
∗
An early version of this work was presented at the AIAA Scitech 2021 Virtual Forum with AIAA paper number 2021-1540 on 11–15 & 19–21
January 2021.
†
Graduate Research Assistant, Department of Aerospace Engineering Sciences, University of Colorado Boulder,
alvaro.romerocalvo@
colorado.edu, AIAA Student Member. Corresponding author.
‡Associate Professor, Departamento de Física Aplicada III, Universidad de Sevilla, gabriel@us.es.
§
Professor, Glenn L. Murphy Chair in Engineering, Department of Aerospace Engineering Sciences, University of Colorado Boulder,
hanspeter.schaub@colorado.edu, AIAA Member.
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
𝐶= body capacitance, F
𝑐= speed of light, ms−1
𝛿= initial beam divergence angle, rad
𝜖0= permittivity of free space, Fm−1
E= nondimensional external electric field
𝑬= external electric field, V/m
𝑬𝑗= external electric field of sphere 𝑗, V/m
𝐸𝑏= beam energy, J
𝒆= internal electric field, V/m
F= nondimensional external Lorentz’s force
𝑭= external Lorentz’s force, N
𝒇= internal Lorentz’s force, N
𝛾= Lorentz factor
𝐼𝑏= beam current intensity, A
𝐼0= reference beam current intensity, A
𝑘𝑐= Coulomb constant, Nm2C−2
𝐿𝑐= mean spacecraft separation, m
𝜇0= permeability of free space, Hm−1
𝑚𝑒= electron mass, kg
𝑛= volume density distribution of electrons, m−3
𝒑= position vector, m
𝒒= spheres charge vector, C
𝑞𝑖= sphere 𝑖charge, C
𝑞= electron charge, C
R= dimensionless ratio
𝑅𝑏= beam radius, m
𝑅𝑖= sphere 𝑖radius, m
𝑟= radial beam coordinate, m
𝑟𝑖, 𝑗 = distance between sphere 𝑖and 𝑗, m
[𝑆]= elastance matrix, F−1
𝑠= arc parameter along beam centroid, m
𝜃= beam deflection angle, rad
2
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
𝜏= nondimensional time
𝑡= time, s
v= nondimensional electron velocity
𝑽= spheres potential vector, V
𝑉= potential, V
𝑉𝑖= sphere 𝑖potential, V
𝒗= electron velocity, ms−1
𝑣𝑧= beam propagation velocity, ms−1
x= nondimensional inertial position
𝒙= inertial position, m
{𝒖𝑟,𝒖𝜓,𝒖𝑧}= beam reference system
{ˆ
𝒙,ˆ
𝒚,ˆ
𝒛}= global reference system
Subindices:
𝑓= final
⊥= perpendicular to trajectory
ref = reference value
tar = target
ser = servicer
Operators:
·= scalar product
×= vector product
I. Introduction
The use of secondary electrons (SEs) [
1
] and X-rays [
2
,
3
] has been recently proposed to touchlessly sense the
electrostatic potential of objects in geosynchronous equatorial orbit or deep space. These methods, conceptualized in
Fig. 1, make use of a servicing craft that directs a high-energy electron beam at the target of interest such that low-energy
SEs and X-rays are emitted from the surface. Due to the charge unbalance induced by the electron beam, the SEs are
accelerated toward the servicing craft, arriving with an energy equal to the potential difference between both bodies.
The servicing craft measures the electron and photon energy spectrum and, knowing its own potential with respect to
the ambient space plasma, infers the potential of the target [
4
]. This technology may find application in the electrostatic
3
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
Fig. 1 Conceptual representation of the SEE- and X-Ray active spacecraft potential sensing methods.
detumbling [
5
] and reorbiting [
6
–
8
] of debris, Coulomb formations [
9
], material identification, and the mitigation of
electrostatic perturbations during rendezvous, docking, and proximity operations [
10
,
11
], among others. Potential
levels of the order of 10s of keV and beam currents of up to 1 mA are commonly employed in these scenarios [12].
The validation of SE- and X-ray-based touchless electrostatic potential sensing methods has been thoroughly
addressed in vacuum chamber experiments with flat plates, which simplify experimental procedures and ease data
interpretation [
1
,
3
,
13
]. However, a flat surface is not representative of a standard spacecraft, whose complex geometry
leads to highly inhomogeneous electric fields and well-defined paths where SEs move. The detection of SEs at a
servicing spacecraft is hence conditioned by the target’s geometry, relative position, and source region [
14
]. In fact,
the intersection between the electron beam and the target object defines the area where SEs are generated, and so an
appropriate electron beam propagation model is needed. Past missions have operated electron beams in space, with
some examples being SCATHA [
15
] or the Electron Drift Instruments at GEOS [
16
], Freja [
17
], Cluster [
18
], and MMS
[
19
]. Since beam repulsion effects were negligible or irrelevant in most cases, advanced electron beam models were not
required. However, this may not be true in applications where the electrostatic repulsion plays a more relevant role.
Given the close dependence between beam steering and SE detection processes, the quantification of the beam-target
intersection position uncertainty becomes fundamental for the development of potential sensing technologies based
on SEs and X-rays. The ability to focus the electron beam on a specific spot of the target may also find application
in the identification of surface materials and the characterization of differentially-charged objects. In this regard, it
should be noted that although spacecraft design best practices recommend all exterior surfaces to be connected to a
common ground to prevent electrostatic discharges [
20
], arcing events are far from uncommon, particularly in old
spacecraft [
21
–
24
]. Therefore, the success of these methods largely depends on the quantification and mitigation of the
4
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
uncertainty of the system, the implementation of robust remote sensing strategies, and the development of accurate and
computationally efficient simulation frameworks that support such strategies.
This paper introduces a simplified and computationally efficient electron beam dynamics model in Sec. II, assesses
its validity in active spacecraft charging scenarios in Sec. III, and quantifies the uncertainty in the properties of the
beam-target intersection area in Sec. IV. Monte Carlo simulations are implemented after adopting the perspective of the
servicing spacecraft, unveiling the contribution of each parameter to the uncertainty in the outputs by means of a FAST
sensitivity analysis.
II. Electron beam model
A. Context and strategy
Existing electron beam models may be divided into two families: those that fully implement the space-charge
effects induced by the beam, and those that ignore such interaction [
25
]. In the former, the electric field depends on
the trajectory of the particles and is hence computed by solving Poisson’s equation in the simulation domain, leading
to accurate results but large computational costs. Particle-In-Cell (PIC) simulations are commonly employed for this
purpose, and have been widely used to study the injection and long-term propagation of electron beams in plasma
environments [
26
–
30
]. Charged Particle Optics (CPO
∗
) Boundary Element Method (BEM) [
31
] in combination with
the space-charge cell and tube methods [
32
] has also been applied to all sorts of electrostatic problems [
33
]. In the
models that ignore space-charge effects, on the contrary, the particle trajectories are propagated under the unperturbed
electrostatic potential generated by the electrodes. Some representative approaches are SIMION’s Coulombic and Beam
repulsion models, that approximate the beam expansion dynamics by computing the electrostatic repulsion forces in the
beam cross-section at each time step [
25
]. Simplified analytical results for the beam expansion process can also be
found in the literature [34].
The appropriateness of a certain beam model depends on its scenario of application. In the active spacecraft charging
problem, servicer and target spacecraft are separated a few 10s of meters and employ focused electron beams of 10s
of kV. This implies that the beam will deflect only slightly before reaching the target. In fact, the short propagation
distance makes it remain in the initial expansion phase, where the beam density is much larger than the GEO plasma
density and the expansion dynamics are driven by the radial electric field in the beam cross section [
35
]. With GEO
Debye lengths of 100-1000 m, plasma interactions can be safely ignored, but the beam evolution is determined by the
electric field from nearby charged bodies.
A solution that can be regarded as an intermediate approach between the analytical expansion equations described
by Humphries in Ref.
34
and SIMION’s repulsion models [
25
] is subsequently presented. By taking advantage of
the particular active spacecraft charging environment, a simplified framework of analysis that uncouples electron
∗https://simion.com/cpo/. Consulted on: 06/01/2021
5
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
beam expansion and deflection processes is developed and combined with the Multi-Spheres Method (MSM) for the
estimation of electric fields [
36
]. The result is a computational efficient but accurate particle-tracing-like model that can
be integrated in an onboard flight algorithm. This is highly desirable for the applications here considered, as discussed
in Sec. IV.
B. Physical model
The propagation of electron beams in space is subject to several internal and external electromagnetic interactions.
The quasi-analytical physical model here presented assumes (i) negligible space-charge effects, (ii) small beam deflection
angles
𝜃
, (iii) small radial expansion, (iv) axisymmetric distribution of geometry and loads within the beam cross-section,
and (v) negligible plasma interactions.
The first two assumptions are key for developing a computationally efficient simulation framework, because they
uncouple the beam-electrode system and the expansion and deflection processes. As explained in Sec. IV.A, small
beam deflection angles are produced when the potential difference between servicer and target spacecraft is significantly
smaller than the electron beam energy. This is the case of interest for remote sensing applications; otherwise, the beam
may be deflected enough to completely avoid the target. The third and fourth assumptions reduce the cross-section
electrostatic surface integrals to one dimension by allowing the implementation of an infinite cylindrical beam framework
of analysis. Such approach is appropriate for small beam divergence angles and leads to large computational gains with
respect to existing particle tracing simulations. Finally, and since the separation between servicer and target spacecraft
is of the order of 10s of meters, which represents a fraction of the GEO Debye length of 100-1000 m, the electron beam
dynamics can be reasonably studied without taking into account complex plasma interactions.
C. Mathematical model
In what follows, the deflection of the beam is assumed to be produced by the electromagnetic environment, while
its expansion is a consequence of the distribution of charge in the beam cross-section and the initial beam divergence
angle. The model simultaneously and independently addresses both problems by integrating the trajectories of the beam
centroid (deflection) and a series of electrons distributed along the axisymmetric beam cross-section (expansion). In
both cases, Lorentz’s force defines the electromagnetic force on each particle through
𝑭=𝑞(𝒗×𝑩+𝑬),(1)
with
𝑞
and
𝒗
being the charge and velocity of the electron, and
𝑩
and
𝑬
denoting the magnetic flux density and electric
field, respectively. The relativistic change in momentum of the particle is given by the balance
d(𝛾𝑚𝑒𝒗)
d𝑡
=𝑭,(2)
6
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
where
𝑚𝑒
is the mass of the electron,
𝛾=(
1
−𝛽2)−1/2
is the Lorentz factor,
𝛽=𝑣/𝑐
,
𝑐
is the speed of light, and the
time derivative is inertial. The position 𝒙in the inertial reference frame is computed as
d𝒙
d𝑡
=𝒗.(3)
It should be noted that, in accordance with the special theory of relativity, the inertia of a particle with respect to a
reference frame depends on its speed with respect to such frame. Consequently, the term
𝛾𝑚𝑒
defines the apparent mass
of the particle.
For the sake of clarity, the internal fields, that drive the expansion problem, are subsequently denoted by lowercase
variables, while the external fields, that determine the deflection dynamics, are given by uppercase letters.
1. Expansion of cylindrical electron beams
In the beam expansion problem, the radial trajectories of a set of electrons are integrated at different radii of the
beam cross-section using Eqs. 1-3. The internal electromagnetic fields and forces generated by axisymmetric cylindrical
beams must consequently be computed. This is done under the infinite length approximation, leading to good estimates
when the characteristic longitudinal (propagation) distance is much larger than the characteristic radius of the beam.
The main advantage of this approach is the large reduction in computational cost achieved by expressing a 3D problem
in the axisymmetric domain.
Axisymmetric cylindrical beams generate radial electric and azimuthal magnetic fields. The first is readily derived
from Gauss’s law, resulting in [34]
𝒆(𝑟, 𝑡 )=
𝑞
𝜖0𝑟∫𝑟
0
d𝑟0𝑛(𝑟0, 𝑡)𝑟0𝒖𝑟,(4)
where
𝜖0
is the permittivity of free space,
𝑛(𝑟)
denotes the volume density distribution of electrons, and
{𝒖𝑟,𝒖𝜓,𝒖𝑧}
describes a cylindrical reference system centered in the axis of the beam and whose
𝑧
component is aligned with the
velocity. Similarly, Ampère’s law gives the azimuthal magnetic field [34]
𝒃(𝑟, 𝑡 )=
𝜇0𝑞𝑣 𝑧(𝑡)
𝑟∫𝑟
0
d𝑟0𝑛(𝑟0, 𝑡)𝑟0𝒖𝜓,(5)
with
𝜇0
being the permeability of free space, and
𝑣𝑧
the propagation velocity of the beam (assumed to be uniform in the
cross-section). The modules of the electric and magnetic fields are related through
𝑒=(𝑐/𝛽)𝑏
. By applying Eq.
(1)
to
these fields, the internal electromagnetic force becomes
𝒇(𝑟, 𝑡 )=
𝑞2
𝑟𝜖01−𝛽(𝑡)2∫𝑟
0
d𝑟0𝑛(𝑟0, 𝑡)𝑟0𝒖𝑟,(6)
7
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
where the
𝑧
component of the force, cause by the radial expansion velocity, has been neglected. The magnetic and electric
forces are related through
𝐹mag =−𝛽2𝐹el
. For relativistic electron beams, both terms are approximately compensated
(𝛽→1), allowing long-distance transport at high current levels [34, 37].
The initial beam velocity profile is approximated in two steps. First, the velocity of propagation
𝑣𝑧(
0
)
is computed
from the initial relativistic beam energy,
𝐸𝑏=(𝛾−
1
)𝑚0𝑐2
, by solving for
𝛾
and
𝛽
. Then, the initial divergence angle
𝛿
,
which is not caused by the electromagnetic repulsion between particles but by the optical configuration of the electron
gun itself, is imposed as
𝒗(𝑟, 0) ∼ 𝑟𝛿
𝑅𝑏
𝑣𝑧(0)ˆ
𝒖𝑟+𝑣𝑧(0)ˆ
𝒖𝑧,(7)
where
𝑅𝑏
is the beam radius and a quasi-collimated beam is assumed (
𝛿
1). The initial electron density function,
𝑛(𝑟,
0
)
, is modeled following a pre-defined statistical distribution (e.g. quasi-Gaussian, uniform, etc) that satisfies the
electron beam current intensity 𝐼𝑏and energy 𝐸𝑏. The condition
∫𝑅𝑏
0
d𝑟02𝜋𝑟0𝑛(𝑟0, 𝑡)=
𝐼𝑏
𝑞𝑣 𝑧(𝑡)(8)
is then imposed at each time step to conserve the electron beam current. This expression assumes a uniform
𝑣𝑧
component computed in a plane perpendicular to the axis of the beam, which is consistent with the small radial expansion
assumption of the model. Uniform beams can be discretized with a single external electron in the axisymmetric beam
cross-section, while more complex profiles (e.g. Gaussian) should employ a finer discretization to capture the evolution
of the distribution. A convergence analysis should be carried out in each case; in particular, high-intensity beams require
more points to accurately simulate the electromagnetic repulsion effect.
It should be noted that, although Eqs. 4-8 are given as a function of time (describing the movement of a particle),
they are actually associated with a steady-state solution. Time is related to the arc parameter
𝑠
along the beam centroid
through
𝛿𝑠 =𝑣𝑧𝛿𝑡
. In a straight beam,
𝑠=𝑧
, and each of these expressions can be written in terms of the cylindrical
coordinates
𝑟
and
𝑧
. The ratio
𝛽
also changes depending on the beam propagation velocity, which is computed in the
deflection problem independently of the expansion algorithm.
2. Deflection of cylindrical electron beams
The deflection of the beam is here represented by the trajectory of the centroid of the cross-section, which is
integrated using Eqs. 1-3 for given external electric and magnetic fields. While the first is mainly produced by the
potential difference between both spacecraft, the second is imposed by the magnetic environment.
The charge
𝑞
of a conducting body is related to its capacitance
𝐶
through
𝑞=𝐶𝑉
, where
𝑉
is the potential with
respect to the ambient plasma. The identification of the zero potential with the ambient plasma is a common choice in
the spacecraft charging community [
38
] that has been adopted in this work. If
𝑉
is known, then the capacitance can be
8
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
used to determine the total charge of the conducting body, from which the electric field at distant points can be computed.
However, objects in close proximity exhibit mutual capacitance effects [
39
] which must be accounted for to accurately
determine the total charge, its distribution, and the nearby electric field. Capacitance is a function of the geometry of the
system, but analytical solutions are only available for a limited number of shapes (such as spheres or round plates).
Therefore, a numerical solution scheme must be used to find the capacitance of the system. The Method of Moments is
generally employed for that purpose and, based on its solution, the Multispheres Method (MSM) has been developed as
a computationally efficient alternative to approximate the resulting charge distribution [
36
,
40
]. The MSM performs
such approximation by discretizing the geometry using equivalent charged spheres [
36
,
40
]. Given the potential on each
sphere and its location with respect to the rest, the charge distribution is computed by solving the linear system
©«
𝑉1
𝑉2
.
.
.
𝑉𝑛
ª®®®®®®®®®®®¬
=𝑘𝑐
1/𝑅11/𝑟1,2. . . 1/𝑟1,𝑛
1/𝑟2,11/𝑅2. . . 1/𝑟2,𝑛
.
.
..
.
.....
.
.
1/𝑟𝑛,11/𝑟𝑛,2. . . 1/𝑅𝑛
©«
𝑞1
𝑞2
.
.
.
𝑞𝑛
ª®®®®®®®®®®®¬
,V=[𝑆]q,(9)
where
𝑘𝑐=
1
/(
4
𝜋𝜖0)
is the Coulomb constant,
𝑅𝑖
is the radius of each sphere,
𝑟𝑖, 𝑗
is the distance between spheres
𝑖
and
𝑗
, and
[𝑆]
denotes the elastance matrix [
39
], which is the inverse of the capacitance matrix. If both spacecraft
are assumed to be conducting bodies in electrostatics equilibrium, each of them must have an equipotential surface,
and so all
𝑉𝑖
belonging to the same surface must equal. This assumption is appropriate for a GEO spacecraft since
modern design specifications require all outer surfaces to be electrically connected [
20
], although it can be relaxed
for differential charging studies. The charge vector
q
constitutes a model of the charge distributions on the spacecraft,
which allows calculating the electric field
𝑬
created by these distributions as the superposition of the one produced by
each individual charge 𝑞𝑗, given by
𝑬𝑗(𝑟)=
𝑞𝑗
4𝜋𝜖0𝑝3𝒑, 𝑝 ≥𝑅𝑗,(10)
where
𝒑
denotes the radial position vector, and
𝑅𝑗
is the radius of the sphere. An arbitrary number of spheres can be
placed and their radii adjusted to match the capacitance of the MSM to the true value.
In relation to the magnetostatic interaction, this work assumes an arbitrarily oriented GEO magnetic field of 100
nT. Its large characteristic length of variation (
∼
10
3
km), the small characteristic time of the beam deflection process
(
∼
10
−6
s), and the small influence of the field in the problem under consideration justify its treatment as a fixed
parameter.
9
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
3. Nondimensional formulation
The numerical conditioning of the electron beam expansion and deflection problem can be largely improved by
employing a dimensionless formulation of Eqs. 1-3, which become
F=(v×B+E),(11)
d(𝛾v)
d𝜏
=F,(12)
dx
d𝜏
=v,(13)
where
x=
𝒙
𝑥ref
, 𝜏 =
𝑡
𝑡ref
,v=
𝑡ref
𝑥ref
𝒗,B=
𝑞ref𝑡ref
𝑚ref
𝑩,E=
𝑞ref𝑡2
ref
𝑚ref𝑥ref
𝑬,F=
𝑡2
ref
𝑚ref𝑥ref
𝑭.(14)
The electron mass and charge are taken as a reference (
𝑚ref
,
𝑞ref
), with the characteristic time being
𝑡ref =
10
−6
s. The
characteristic length
𝑥ref
is equal to the initial electron beam radius
𝑅𝑏
and the mean spacecraft separation
𝐿𝑐
for the
expansion and deflection processes, respectively. In other words, two different dimensionless problems are solved
simultaneously.
4. Validity metrics
As noted in Sec. II.B, the analytical model introduced in this section is valid while the beam deflection angle
𝜃=arccos 𝒗(0) · 𝒗(𝑡𝑓)
|𝒗(0)|| 𝒗(𝑡𝑓) | ,(15)
is small, with 𝑡𝑓denoting the final simulation time. The additional dimensionless parameter
R=
𝛾𝑚𝑣2
|𝑞𝐿𝑐(𝒗×𝑩+𝑬)⊥|
=
𝛾v2
| ( v×B+E)⊥|(16)
is defined to describe the ratio between the instantaneous electromagnetic gyroradius and the characteristic spacecraft
separation
𝐿𝑐
, with
⊥
denoting the force component perpendicular to the electron trajectories and the different variables
referring to the deflection problem. The metric
R
reflects the influence of the electromagnetic environment on the
trajectory of the centroid. A small value of
R
implies that its gyroradius is comparable to the characteristic spacecraft
separation, which ultimately leads to the focusing of the beam. The reader may visualize this scenario with a simple
geometrical problem: if two identical circumferences are initially superposed and then separated slightly, two intersection
points will be generated. The same happens with an electron beam when
R≤
1. This effect is not contemplated in the
model, which explains why R(𝜃) must be significantly greater (smaller) than 1.
10
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
5. Numerical integration scheme
The integration of Eqs. 1-3 must conserve the total energy of the system. Common integrators, such as the standard
4th order Runge-Kutta (RK) method, carry a certain truncation error with each time step, resulting in unbounded
divergences in the long term. This has made the Boris algorithm, which is an explicit, time-centered integrator that
conserves the phase space volume and bounds the global energy error, the standard for particle physics simulations [
41
].
However, in short-term applications (like the one discussed in this manuscript) RK integrators still offer an accurate
solution. In the simulations that follow, a variable-step, variable-order Adams-Bashforth-Moulton PECE solver of
orders 1 to 13 is implemented by means of Matlab’s routine
ode113
[
42
], resulting in relative variations of total energy
errors below 0.001%.
III. Performance analysis
A. Verification
Every model should be tested to verify its implementation, a step that is summarized here by independently focusing
on the deflection and expansion processes. As described in Sec. II.C.5, the predicted trajectories pass the energy
conservation test. Besides, they also match the analytical electron gyroradius and gyrofrequency in the presence of a
constant magnetic field. Particle dynamics in combination with the MSM representation of charged bodies have been
thoroughly addressed in previous works [43], leaving the beam expansion dynamics as the last module to be verified.
SIMION’s documentation includes a case of analysis
†
where its Coulombic and Beam repulsion models are validated
with coupled space-charge results from CPO [
25
]. The example consists on an isolated beam of 1eV that originates in a
3 mm circle with an uniform distribution of 1000 electrons and a deflection angle of
𝛿=−16.7°
. The beam current is
set as a multiple of the maximum value
𝐼0=
3
.
47
𝜇
A sustained by the system, leading to the results depicted in Fig. 2.
The same scenario is simulated with the beam model presented in Sec. II, showing an overall excellent agreement with
SIMION. Small differences between both sets of results should be attributed to simplifying assumptions. For instance,
the initial beam velocity profile in Eq. 7, leads to a set of particles with unequal kinetic energies. Although appropriate
for small deflection angles (like the ones used in active spacecraft charging scenarios), this approximation performs
worse with
𝛿
1. However, while the computational cost of each SIMION simulation scales with the square of the
number of particles [
25
], just a few trajectories are required by the proposed framework: the centroid, and a certain
number of points in the axisymmetric cross-section that are employed to recompute the volume distribution of electrons.
Since in this case such distribution is uniform, a single electron is needed to capture the evolution of the beam envelope;
however, 50 particles are simulated for illustrative purposes. This computational advantage, together with the reduction
of a complex problem to a small set of parameters, are the main advantage of the simplified model here introduced.
†The interested reader is referred to the readme.html file in the examples/repulsion folder of SIMION 2020
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
0 5 10 15 20
z [mm]
0
1
2
3
r [mm]
(a) Model, 0.5𝐼0(b) SIMION, 0.5𝐼0
0 5 10 15 20
z [mm]
0
1
2
3
r [mm]
(c) Model, 𝐼0(d) SIMION, 𝐼0
0 5 10 15 20
z [mm]
0
2
4
6
r [mm]
(e) Model, 2𝐼0(f) SIMION, 2𝐼0
Fig. 2 Comparison between simulation framework and SIMION’s beam repulsion model [25] for 𝐸=
1
eV,
𝛿=−16.7°, and 𝐼0=3.47 𝜇A.
B. Validation
The physical mechanisms involved in the electron beam expansion and deflection processes have been very well
understood for decades, and the validation of fundamental particle dynamics has consequently little technical value. On
the contrary, future applications depend on the proper application of the model presented in Sec. II, which relies on a
number of assumptions that limit its validity space. Provided that such assumptions are met, a computationally efficient
and powerful analysis tool is obtained.
With the purpose of exploring the performance of the model in a worst-case scenario, the experimental setup shown
in Fig. 3 is tested in the ECLIPS Space Environment Simulation Facility [
44
]. The assembly exposes an electron beam
from a Kimball Physics EMG-4212D electron gun to the electric field generated by a charged spacecraft-like electrode
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
Fig. 3 Experimental setup inside the ECLIPS chamber
mounted on a rotary stage. The shape and location of the beam spot at approximately 35 cm from the gun orifice are
observed with a 3.81 cm diameter rugged phosphor screen, and the spatial distribution is obtained with a Retarding
Potential Analyzed (RPA) mounted on a linear stage. The beam is configured at 1 keV energy and 10
𝜇𝐴
current, while
the electrode is set at -100 to -500 V employing a Matsusada AU-30R1 high-voltage power supply. The electron flux at
the RPA is measured with a Keithley 2400 multimeter. Finally, the system is automated by means of a LabView VI.
Figure 4 shows the beam spot profiles at the phosphor screen for electrode potentials ranging from -100 to -500
V. Because the gun orifice is slightly below the symmetry plane of the electrode, a voltage decrease leads to a slight
downwards deflection. This is compensated with a fine tuning of the vertical gun deflection settings, which do not
alter the horizontal position or shape of the spot. Figure 4a shows a
∼
13 mm diameter beam cross-section, which is
considerably larger than the initial
∼
3mm diameter beam. Tests with different beam current intensities give the same
spot shape, which demonstrates that the expansion is not induced by the electrostatic repulsion between electrons, but
by the initial beam spread angle
𝛿
. As the voltage decreases, the beam is deflected away from the electrode and its
cross-section is elongated vertically. The spot shape is deformed significantly below -300 V, indicating the existence of
small gyro radii with
R∼
1. These observations are complemented with the electron flux distribution computed with
the RPA in Fig. 5, where the narrowing process reduces the width of the flux peak and its amplitude. Based on the 0 V
case, the spread angle is estimated to be
𝛿∼2.5°
. It should be noted that the apparent beam radius shown in Fig. 4a
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
(a) 0.3 V (b) -100.9 V
(c) -201.2 V (d) -300.9 V
(e) -401.1 V (f) -501.3 V
Fig. 4 Electron beam spot in the phosphor screen under different electrode potentials
V = 0.3 V
100.9 V
201.2 V
300.9 V
401.1 V
501.3 V
Flux [× 1012cm-2s-1]
0
0.5
1
1.5
2
2.5
RPA lateral displacement [cm]
−2 0 2 4 6 8
Fig. 5 Experimental electron flux distribution as a function of the applied electrode potential
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(a) 0°(b) 10°
(c) 20°(d) 30°
(e) 40°(f) 50°
Fig. 6 Electron beam spot in the phosphor screen under different electrode rotation angles at -100 V
Fig. 7 MSM representation of the experimental setup with electron beam propagation at -500 V
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
V = -100 V, θ = 3.06º
-200 V, 6.23º
-300 V, 9.50º
-400 V, 12.88º
-500 V, 16.35º
ℛ
1
10
100
RPA lateral displacement [cm]
0 5 10 15 20 25 30 35
(a) Varying the potential with 𝛼=0
V = -100 V
α = 0º, θ = 3.06º
10º, 3.14º
20º, 3.26º
30º, 3.47º
40º, 3.80º
50º, 4.29º
ℛ
1
10
100
RPA lateral displacement [cm]
0 5 10 15 20 25 30 35
(b) Varying the heading angle with 𝑉=−100 V
Fig. 8 Validation metrics Rand 𝜃as a function of the electrode potential and heading angle
is smaller than the one reported in Fig. 5. This is due to limitations imposed by the power density threshold of the
phosphor screen and the effective aperture of the RPA‡.
The influence of the electrode rotation angle
𝛼
on the beam deflection and spot shape is also explored in Fig. 6 for
𝑉=−
100 V and
𝛼=10°
to
50°
. Although the beam is deflected and the cross section is modified, these effects are
much less pronounced than in Fig. 4, implying that the
R
metric is significantly larger. In other words, the uncoupled
model is far more appropriate for this case.
The framework of analysis introduced in Sec. II is not designed to predict the elongation of the beam cross-section,
but still gives accurate estimations for those cases where the beam deflection angle is small. In order to evaluate the
validity metrics defined in Sec. II.C.4, the experimental setup is reproduced with a 934-spheres MSM representation of
the spacecraft-like electrode. The result is shown in Fig. 7 for an electrode potential of -500 V, that corresponds to the
case in Fig. 4f, and a beam expansion angle 𝛿=2.5°.
The validity metrics
R
and
𝜃
are reported in Fig. 8a as a function of the electrode potential
𝑉
and in Fig. 8b in
terms of the electrode rotation angle
𝛼
. An increase in the electrode potential decreases the minimum
R
value and
increases the deflection angle
𝜃
, reaching
∼3°
and
9.5°
, respectively, for the limit case of -300 V. Larger values lead to
significant beam cross-section deformations, as shown in Figs. 4e-4f. Similarly, the rotation of the electrode creates a
second minimum in the
R
plot (i.e. a second maximum in the electromagnetic force), but since this minimum is larger
than in the -200 V case, its effects on the beam cross-section are less significant. Due to the large beam expansion angle
𝛿
, the magneto-electrostatic repulsion between electrons plays virtually no role in the expansion dynamics of the beam.
The experiment demonstrates the appropriateness of the expansion/deflection decoupling when the validation metrics
R
and
𝜃
adopt sufficiently large values. In such cases, the beam cross-section becomes practically independent of the
‡
The variations in light intensity at the phosphor screen are caused by the Electron-Beam-Induced-Deposition (EBID) of carbon and heavy
molecules over the surface, and not by variations in the distribution of electrons in the beam cross-section.
16
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
external electromagnetic force. Although the assumptions of the model significantly constraint its validity range, it is
precisely in the spacecraft charging scenario where this computationally efficient framework can be better exploited.
IV. Uncertainty in active spacecraft charging scenario
A. Problem statement
Once the validity of the beam model has been contrasted with experimental observations, the base scenario of
analysis is introduced in Fig. 9. The GOES-R
§
and SSL-1300
¶
spacecraft MSM models are shown together with
the
𝑒−
beam centroid evolution in the global reference system
{ˆ
𝒙,ˆ
𝒚,ˆ
𝒛}
. The target spacecraft (-2.5 V) is negatively
charged with respect to the servicer (0 kV) due to the current unbalance induced by the electron beam, generating a
net electrostatic force that tends to deflect and slow down the 5 keV, 10
𝜇
A electrons from 4
.
2
·
10
7
m/s to 3
.
2
·
10
7
m/s. The electron beam energy must be larger than the absolute potential difference to allow the electrons to reach the
target surface. The
R
parameter depends quadratically on the propagation speed and approximately linearly on the
beam energy (see Eq.
(16)
), and hence the physical model here adopted is particularly well suited for high beam energy
applications.
The trade-off between beam energy and spacecraft potential is analyzed in Fig. 10 by comparing the validation
metrics along the beam trajectory in three different scenarios. As expected, an increase in beam energy leads to larger
R
and smaller
𝜃
values, while a decrease in the target spacecraft potential has the opposite effect. In the nominal case
(
𝐸𝑏=
5keV,
𝑉=−
2
.
5kV), a deflection angle
𝜃=5.33°
and a minimum
R=
4are reached, satisfying the validity
§https://www.goes-r.gov/spacesegment/spacecraft.html (Consulted on: 01/06/2021)
¶http://sslmda.com/html/1300_series_platform.php (Consulted on: 01/06/2021)
Fig. 9 Geometry of the 2-SC problem for the basic simulation parameters (see Table 1).
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range of the model. These values are analogous, in terms of
R
and
𝜃
, to the experimental -200 V case pictured in
Fig. 4c and analyzed in Fig. 8.
Figure 11 explores the beam expansion dynamics for different deflection angles. When a stream of collimated
electrons (
𝛿=
0) exits the gun, the magneto-electrostatic repulsion expands the beam radius from 2.5 to 40 mm in the
30 m flight. The trajectory of those electrons is non-linear, but as the initial
𝛿
angle is increased, a linear expansion
is achieved. This qualitatively different behavior reflects the existence of repulsive and inertial expansion regimes.
Although in the second case the expansion dynamics become practically irrelevant, a larger beam-target intersection is
also obtained. This may not be convenient for the characterization of the target.
A discretization of 50 radial points is employed to model the expansion process, deviating less than a 0.01% from
a 200-points model in the worst-case collimated beam regime. An MSM model with 172 spheres is applied to the
deflection problem, resulting in errors below 5 cm in the final beam centroid position with respect to a high-fidelity
1976 spheres MSM simulation. These results are acceptable for the problem here discussed.
B. Uncertainty quantification analysis
The model built in Sec. II is, because of its computational efficiency, particularly well suited to quantify the
uncertainty in the beam-target intersection position in an active spacecraft charging scenario. The analysis is designed
from the perspective of a servicing spacecraft that seeks to steer the beam toward a particular spot of the target. A total
of 702 uncertain variables are considered, with 688 being associated to the MSM spheres that approximate the charge
distribution of the two-spacecraft system. The list of input variables and their distribution is detailed in Table 1. The
outputs of the analysis are (i) the radius of the beam cross section at the end of flight, (ii) the centroid landing position in
the target plane, which is perpendicular to the line of sight between both spacecraft, (iii) the landing energy, and (iv) the
time of flight.
V = -2.5 kV, Eb = 5 keV, θ = 5.33º
V = -2.5 kV, Eb = 10 keV, θ = 2.40º
V = -7.5 kV, Eb = 10 keV, θ = 9.90º
ℛ
0
10
20
30
40
50
60
z [m]
0 5 10 15 20 25 30
Fig. 10 Validation metrics Rand 𝜃as a function of the target spacecraft potential 𝑉and beam energy 𝐸𝑏for
the nominal active spacecraft charging scenario.
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δ = 0º
δ = 0.1º
δ = 0.2º
Beam radius [cm]
0.0
2.5
5.0
7.5
10.0
12.5
z [m]
0 5 10 15 20 25 30
Fig. 11 Beam radius evolution as a function of divergence angle for the nominal active spacecraft charging
scenario.
Table 1 Uncertainty analysis parameters
Variable Distribution Mean STD Unit
Beam current (𝐼𝑏) Normal 10 0.1 𝜇A
Beam energy (𝐸𝑏) Normal 5 0.05 keV
Initial divergence angle (𝛿) Uniform 0.1 Lims: [0, 0.2] deg
Initial particle density STD (𝜎𝑏) Normal 0.83 0.083 mm
Servicer potential (𝑉ser) Normal 0 0.05 kV
Servicer, Euler-313 (𝜓ser,𝜃ser,𝜙ser) Normal [0,90,0] [0.1,0.1,0.1] deg
Target potential (𝑉tar ) Normal -2.5 0.25 kV
Target, Euler-313 (𝜓tar,𝜃tar,𝜙tar) Normal [0,180,0] [5,5,5] deg
Relative Position (𝑟𝑥,𝑟𝑦,𝑟𝑧) Normal [0,10,32] [0.5,0.5,1] m
Capacitances (x172) Normal Dataset 1% C
Spheres pos. (x516) Normal Dataset 1% m
Initial beam radius Fixed 2.5 0 mm
Due to the large number of parameters and reduced computational cost of the simulation, a Monte Carlo analysis is
chosen over other uncertainty quantification methods. The relative influence of each input parameter on the output
metrics is measured by means of sensitivity indices, computed with a Fourier Amplitude Sensitivity Testing (FAST)
suite‖from Ref. 45.
C. Results
The Monte Carlo analysis is carried out with 10
4
random realizations generated from the distributions reported in
Table 1, which are conservative estimations of the different sources of error. Each simulation takes approximately 0.6 s
after parallelizing the code with 7 CPU threads in Matlab 2021 (Intel Core i7-7820HQ CPU at 2.90 GHz, 32 Gb RAM).
‖https://www.mathworks.com/matlabcentral/fileexchange/40759-global- sensitivity-analysis- toolbox
(Consulted on:
01/06/2021)
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
The solution converges in mean and variance for the expansion and deflection problems.
Results in Fig. 12 depict the Probability Density Functions (PDFs) of the model outputs: (a) final beam radius
𝑅𝑏, 𝑓
, (b) final centroid position
𝑝𝑥, 𝑓
and
𝑝𝑦, 𝑓
, (c) time of flight
𝑡𝑓
, and (d) final beam energy
𝐸𝑏, 𝑓
. The first follows a
quasi-uniform distribution, clearly influenced by the uniform sampling of the initial deflection angle
𝛿
, and spans from 4
to 13 cm. These expansion values, computed for
𝛿⊆ [0°,0.2°]
, are small in comparison with the spread of the beam
centroid shown in Fig. 12b, where the target [0.11, -1.26] m is marked as a red cross. The landing positions follow a
multi-Gaussian distribution with mean
[
0
.
07
,−
1
.
20
]
m and covariance
[
0
.
20
,−
0
.
006;
−
0
.
006
,
0
.
28
]
m
2
. This implies
that the beam centroid has a 93.9% probability of intercepting the SSL-1300 solar panel, represented as a rectangle in
the figure, while the chances of hitting a 20 cm diameter circle surrounding the target are just a 0.3%. The time of flight
PDF is represented in Fig. 12c and follows a log-normal distribution with logarithmic mean 14
.
07
𝜇𝑠
and variance
2
.
14
·
10
−4𝜇𝑠2
. This result is relevant for applications employing pulsed beam modulations to filter the returning
secondary electron flux from the target. Modulated electron beams have been employed in previous space instruments,
such as the Electron Drift Instrument of MMS [
19
]. Finally, the landing energy PDF is shown in Fig. 12d and fitted with
a Weibull distribution (scale 3309.98, shape 9.97) with mean 3148
.
55 keV and variance 144294 keV
2
. The landing
energy determines the SE yield, and is hence important for defining the resulting SE flux [
1
]. It also determines the
X-ray spectrum profile and intensity [46].
In order to determine the influence of each input on the outcomes reported in Fig. 12, a Fourier Amplitude Sensitivity
Testing (FAST) Global Sensitivity Analysis (GSA) is conducted. The analysis is limited to the 15 non-MSM inputs
in Table 1 to minimize its computational cost. Although 688 MSM variables are removed, Table 2 shows how the
total variances remain practically identical, denoting that such uncertain inputs have a negligible effect in the final
distributions.
Table 3 reports the sensitivity coefficients for 10
4
realizations. The five outputs of the model (final beam radius
𝐸𝑏, 𝑓
, beam-target intersection coordinates
𝑝𝑥, 𝑓
and
𝑝𝑦, 𝑓
, final energy
𝐸𝑏, 𝑓
, and time of flight
𝑡𝑓
) are listed in the
rows, while the inputs are shown in the columns. Bold fonts are employed to highlight the largest sensitivities, showing
that each output variance can be almost completely explained with less than two inputs. For instance, the final beam
radius is mainly dependent on the initial divergence angle, while the final positions are related to the uncertainties in
Table 2 Comparison of output variances between the full 702 parameters and the reduced 15 parameters MC
analyses
𝑉(𝑅𝑏, 𝑓 )
[m2]
𝑉(𝑝𝑥, 𝑓 )
[m2]
𝑉(𝑝𝑦, 𝑓 )
[m2]
𝑉(𝐸𝑏, 𝑓 )
[keV2]
𝑉(𝑡𝑓)
[s2]
Full 6.830e-4 0.204 0.276 1.387e5 1.301e-16
Reduced 6.790e-4 0.203 0.269 1.456e5 1.319e-16
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
PDF
0
0.02
0.04
0.06
0.08
0.1
Rb,f [cm]
4 6 8 10 12 14
(a) PDF of the final beam radius
beam spot
Max-Min
Solar Panel
y[m]
−6
−4
−2
0
2
x [m]
−4 −2 0 2 4
(b) Final centroid positions
PDF
0
0.02
0.04
0.06
0.08
tf [μs]
0.74 0.76 0.78 0.8 0.82
(c) PDF of the time of flight
PDF
0
0.01
0.02
0.03
0.04
0.05
0.06
Eb,f [keV]
1500 2000 2500 3000 3500 4000 4500
(d) PDF of final beam energy
Fig. 12 Result of the Monte Carlo simulation
their corresponding relative spacecraft position component. The output
𝑝𝑦, 𝑓
is also dependent on the target potential,
which promotes the lateral deflection of the beam, as shown in Fig. 4. Although the results seem to indicate that the
variance in
𝑝𝑥, 𝑓
is also explained by the beam current
𝐼𝑏
, this should be attributed to numerical errors, because the
model uncouples the expansion and deflection problems. The final beam energy
𝐸𝑏, 𝑓
and time of flight
𝑡𝑓
are depend
on the initial beam energy
𝐸𝑏
and target spacecraft potential
𝑉tar
, whose relative influence is strongly influenced by
the uncertainty bands selected in Table 1. The attitude of each spacecraft does not seem to have a large influence in
any output variable; however, this is caused by the small attitude disturbance angles selected in Table 1, which would
increase with less accurate attitude determination sensors.
It should be noted that, among the most influential input parameters, only the target potential and relative positions
are not predefined. An obvious conclusion is that the targeting of specific regions is limited by the accuracy in the
measurement of the relative position between the two spacecraft. Although this problem may be addressed with
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Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
Table 3 Normalized sensitivity indices from FAST sensitivity analysis with 15 inputs and 5 outputs. The values
are scaled by a factor of 10 for convenience.
𝐼𝑏𝐸𝑏𝜎𝑏𝑉tar 𝑉ser 𝑟𝑥𝑟𝑦𝑟𝑧𝜙tar 𝜃tar 𝜓tar 𝜙ser 𝜃ser 𝜓ser 𝛿
𝑅𝑏, 𝑓 0.053 0.112 0.011 0.306 0.044 0.009 0.019 0.121 0.005 0.001 0.015 0.056 0.318 0.365 8.563
𝑝𝑥, 𝑓 1.208 0.204 0.009 0.016 0.004 8.284 0.001 0.003 0.233 0.002 0.005 0.002 0.025 0.002 0.002
𝑝𝑦, 𝑓 0.004 0.030 0.033 1.550 0.354 0.003 7.888 0.109 0.001 0.003 0.001 0.003 0.020 0.001 0.001
𝐸𝑏, 𝑓 0.056 1.541 0.021 7.122 0.554 0.071 0.126 0.226 0.117 0.138 0.006 0.005 0.005 0.013 0.001
𝑡𝑓0.028 3.524 0.005 4.661 0.760 0.006 0.024 0.741 0.010 0.238 0.001 0.000 0.001 0.001 0.001
Fig. 13 Trajectory of 100 secondary electrons generated in the beam-target intersection region described by
the Monte Carlo analysis in Fig. 12b.
better sensing equipment, the strong influence of the target potential raises additional issues. In order to obtain a first
measurement, the electron beam needs to intercept the target, but such interception can only be guaranteed if an estimate
of
𝑉tar
is available. The problem may be solved by temporarily increasing the beam expansion angle
𝛿
to irradiate larger
areas, enhancing the chances of collision, or by employing a more directive beam with higher energy
𝐸𝑏
. An X-ray
sensor oriented toward the irradiated region would then be used to obtain the first target voltage estimation, which would
then be followed by more accurate SE estimations.
However, the availability of target potential measurements using the SE method, which is significantly more accurate
than the X-ray approach [
13
], is strongly dependent on the geometry of the system [
14
]. The spatial distribution reported
in Fig. 12b for the beam-target intersection has a critical influence on the flux of SEs. Figure 13 depicts the trajectories
of 100 SEs uniformly generated in a circle with 1.5 m radius (3𝜎interval) and whose center matches the origin of the
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The final version of this paper can be found in https://doi.org/10.2514/1.A35190
Monte Carlo final beam centroid distribution (
𝑥=
0
.
07 m,
𝑦=−
1
.
20 m,
𝑧=
30 m). Since they are born with energies
of the order of just a few eV [
47
], SEs are assumed to start their trajectory with zero velocity. The SEs are able to reach
the servicer only when the beam hits a very specific area of the target, so it can be readily concluded that a limited
subspace of the Monte Carlo solution domain will be detectable. That is, an RPA mounted in the servicer and aimed at a
suitable target region is not guaranteed to detect SEs with the statistical distributions reported in Table 1, concluding
that the combination of X-ray and SE measurements is necessary to ensure a robust and accurate estimation of the
target spacecraft potential. A feedback control loop may be employed to actively steer the beam and guarantee the
measurement of SEs, following an implementation analogous to the Electron Drift Instrument of MMS [19].
V. Conclusions
This paper introduces a quasi-analytical electron beam model that achieves a great computational efficiency by
decoupling the beam expansion and deflection processes. Although this choice restricts the range of application of the
simulation framework, experimental observations in the ECLIPS Space Environments Simulation Facility [
44
] validate
its use in active spacecraft charging problems, where small deflection angles and radial expansions are expected.
The model is employed to quantify the uncertainty of key metrics in a representative active charging scenario.
The sensitivity of the beam dynamics to a characteristic set of input parameters is studied by means of Monte Carlo
simulations. Although the electron beam centroid is shown to hit the target spacecraft with a 93.85% chance, this
happens within a large
±
3
𝜎
Gaussian interval of 3 m around the target. A FAST sensitivity analysis shows that the
relative spacecraft position and target spacecraft potential account for most of the variance. Furthermore, only a limited
number of solutions ensure that the resulting SEs reach an hypothetical RPA mouted on the servicer, implying that
combined X-ray and SE potential sensing methods are not only desired, but actually required for a robust and accurate
target potential estimation. This setup would benefit from a closed control loop to guarantee the detectability of SEs in
uncertain environments.
Acknowledgments
The authors thank Prof. Alireza Doostan for his assistance in the development of the uncertainty quantification
analysis, and Dr. Miles Bengtson and Dr. Kieran Wilson for fruitful discussions on the setup and operation of the
vacuum chamber experiment. This work was partially supported by U.S. Air Force Office of Scientific Research under
grant FA9550-20-1-0025 and the La Caixa Foundation (ID 100010434), under agreement LCF/BQ/AA18/11680099.
References
[1]
Bengtson, M. T., Wilson, K. T., and Schaub, H., “Experimental Results of Electron Method for Remote Spacecraft Charge
Sensing,” Space Weather, Vol. 18, No. 3, 2020, pp. 1–12. https://doi.org/10.1029/2019SW002341.
23
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
[2]
Wilson, K., and Schaub, H., “X-Ray Spectroscopy for Electrostatic Potential and Material Determination of Space Objects,”
IEEE Transactions on Plasma Science, Vol. 47, No. 8, 2019, pp. 3858–3866. https://doi.org/10.1109/TPS.2019.2910576.
[3]
Wilson, K. T., Bengtson, M. T., and Schaub, H., “X-ray Spectroscopic Determination of Electrostatic Potential and Material
Composition for Spacecraft: Experimental Results,” Space Weather, Vol. 18, No. 4, 2020, pp. 1–10. https://doi.org/10.1029/
2019SW002342.
[4]
Bengtson, M., Hughes, J., and Schaub, H., “Prospects and Challenges for Touchless Sensing of Spacecraft Electrostatic Potential
Using Electrons,” IEEE Transactions on Plasma Science, Vol. 47, No. 8, 2019, pp. 3673–3681. https://doi.org/10.1109/TPS.
2019.2912057.
[5]
Casale, F., Schaub, H., and Douglas Biggs, J., “Lyapunov Optimal Touchless Electrostatic Detumbling of Space Debris
in GEO Using a Surface Multisphere Model,” Journal of Spacecraft and Rockets, Vol. 58, No. 3, 2021, pp. 764–778.
https://doi.org/10.2514/1.A34787.
[6]
Hughes, J., and Schaub, H., “Prospects of Using a Pulsed Electrostatic Tractor With Nominal Geosynchronous Conditions,”
IEEE Transactions on Plasma Science, Vol. 45, No. 8, 2017, pp. 1887–1897. https://doi.org/10.1109/TPS.2017.2684621.
[7]
Bengtson, M., Wilson, K., Hughes, J., and Schaub, H., “Survey of the electrostatic tractor research for reorbiting passive GEO
space objects,” Astrodynamics, Vol. 2, No. 4, 2018, pp. 291–305. https://doi.org/10.1007/s42064-018-0030-0.
[8]
Hughes, J. A., and Schaub, H., “Electrostatic Tractor Analysis Using a Measured Flux Model,” Journal of Spacecraft and
Rockets, Vol. 57, No. 2, 2020, pp. 207–216. https://doi.org/10.2514/1.A34359.
[9]
Schaub, H., Parker, G. G., and King, L. B., “Challenges and Prospect of Coulomb Formations,” Journal of the Astronautical
Sciences, Vol. 52, No. 1–2, 2004, pp. 169–193. https://doi.org/10.1007/BF03546427.
[10]
Wilson, K., and Schaub, H., “Impact of Electrostatic Perturbations on Proximity Operations in High Earth Orbits,” Journal of
Spacecraft and Rockets, Vol. 0, No. 0, 2021, pp. 1–10. https://doi.org/10.2514/1.A35039.
[11]
Wilson, K., Romero-Calvo, A., and Schaub, H., “Constrained Guidance for Spacecraft Proximity Operations Under Electrostatic
Perturbation,” Journal of Spacecraft and Rockets, 2021. under review.
[12]
Schaub, H., and Moorer, D. F., “Geosynchronous Large Debris Reorbiter: Challenges and Prospects,” The Journal of the
Astronautical Sciences, Vol. 59, No. 1, 2012, pp. 161–176. https://doi.org/10.1007/s40295-013-0011-8.
[13]
Wilson, K. T., Bengtson, M., and Schaub, H., “Remote Electrostatic Potential Sensing for Proximity Operations: Comparison
and Fusion of Methods,” Journal of Spacecraft and Rockets, 2021. under review.
[14]
Bengtson, M. T., and Schaub, H., “Electron-Based Touchless Potential Sensing of Shape Primitives and Differentially-Charged
Spacecraft,” Journal of Spacecraft and Rockets, Vol. 0, No. 0, 2021, pp. 1–11. https://doi.org/10.2514/1.A35086.
[15]
Olsen, R., and Cohen, H., “Electron beam experiments at high altitudes,” Advances in Space Research, Vol. 8, No. 1, 1988, pp.
161 – 164. https://doi.org/10.1016/0304-3886(87)90085-4.
24
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
[16]
Melzner, F., Metzner, G., and Antrack, D., “The GEOS electron beam experiment,” Space Science Instrumentation, Vol. 4,
1978, pp. 45–55.
[17]
Paschmann, G., Boehm, M., Höfner, H., Frenzel, R., Parigger, P., Melzner, F., Haerendel, G., Kletzing, C. A., Torbert, R. B.,
and Sartori, G., “The Electron Beam Instrument (F6) on Freja,” Space Science Reviews, Vol. 70, No. 3, 1994, pp. 447–463.
https://doi.org/10.1007/BF00756881.
[18]
Paschmann, G., Melzner, F., Frenzel, R., Vaith, H., Parigger, P., Pagel, U., Bauer, O. H., Haerendel, G., Baumjohann, W.,
Scopke, N., Torbert, R. B., Briggs, B., Chan, J., Lynch, K., Morey, K., Quinn, J. M., Simpson, D., Young, C., Mcilwain, C. E.,
Fillius, W., Kerr, S. S., Mahieu, R., and Whipple, E. C., “The Electron Drift Instrument for Cluster,” Space Science Reviews,
Vol. 79, No. 1, 1997, pp. 233–269. https://doi.org/10.1023/A:1004917512774.
[19]
Torbert, R. B., Vaith, H., Granoff, M., Widholm, M., Gaidos, J. A., Briggs, B. H., Dors, I. G., Chutter, M. W., Macri, J., Argall,
M., Bodet, D., Needell, J., Steller, M. B., Baumjohann, W., Nakamura, R., Plaschke, F., Ottacher, H., Hasiba, J., Hofmann, K.,
Kletzing, C. A., Bounds, S. R., Dvorsky, R. T., Sigsbee, K., and Kooi, V., “The Electron Drift Instrument for MMS,” Space
Science Reviews, Vol. 199, No. 1, 2016, pp. 283–305. https://doi.org/10.1007/s11214-015-0182-7.
[20]
Garrett, H. B., and Whittlesey, A. C., “Spacecraft Design Guidelines,” Guide to Mitigating Spacecraft Charging Effects, John
Wiley & Sons, Ltd, 2012, Chap. 3, pp. 26–61. https://doi.org/10.1002/9781118241400.ch3.
[21]
Olsen, R. C., McIlwain, C. E., and Whipple Jr., E. C., “Observations of differential charging effects on ATS 6,” Journal of
Geophysical Research: Space Physics, Vol. 86, No. A8, 1981, pp. 6809–6819. https://doi.org/10.1029/JA086iA08p06809.
[22]
Grard, R., Knott, K., and Pedersen, “Spacecraft charging effects,” Space Science Reviews, Vol. 34, No. 3, 1983, pp. 289–304.
https://doi.org/10.1007/BF00175284.
[23]
Roeder, J. L., and Fennell, J. F., “Differential Charging of Satellite Surface Materials,” IEEE Transactions on Plasma Science,
Vol. 37, No. 1, 2009, pp. 281–289. https://doi.org/10.1109/TPS.2008.2004765.
[24]
Ferguson, D., White, S., Rast, R., and Holeman, E., “The Case for Global Positioning System Arcing and High Satellite Arc
Rates,” IEEE Transactions on Plasma Science, Vol. 47, No. 8, 2019, pp. 3834–3841. https://doi.org/10.1109/TPS.2019.2922556.
[25]
Manura, D., and Dahl, D., SIMION (R) 8.1 User Manual, Rev-5, Adaptas Solutions, LLC, Palmer, MA 01069, 2008. URL
http://simion.com/manual/.
[26]
Okuda, H., and Berchem, J., “Injection and propagation of a nonrelativistic electron beam and spacecraft charging,” Journal of
Geophysical Research: Space Physics, Vol. 93, No. A1, 1988, pp. 175–195. https://doi.org/10.1029/JA093iA01p00175.
[27]
Okuda, H., and Ashour-Abdalla, M., “Propagation of a nonrelativistic electron beam in three dimensions,” Journal of
Geophysical Research: Space Physics, Vol. 95, No. A3, 1990, pp. 2389–2404. https://doi.org/10.1029/JA095iA03p02389.
[28]
Okuda, H., and Ashour-Abdalla, M., “Injection of an overdense electron beam in space,” Journal of Geophysical Research:
Space Physics, Vol. 95, No. A12, 1990, pp. 21307–21311. https://doi.org/10.1029/JA095iA12p21307.
25
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
[29]
Winglee, R. M., “Simulations of pulsed electron beam injection during active experiments,” Journal of Geophysical Research:
Space Physics, Vol. 96, No. A2, 1991, pp. 1803–1817. https://doi.org/10.1029/90JA02102.
[30]
Koga, J., and Lin, C. S., “A simulation study of radial expansion of an electron beam injected into an ionospheric plasma,”
Journal of Geophysical Research: Space Physics, Vol. 99, No. A3, 1994, pp. 3971–3983. https://doi.org/10.1029/93JA02230.
[31] Harting, E., and F.H., R., Electrostatic Lenses, Elsevier Publishing Company, Amsterdam, 1976.
[32]
Read, F. H., Chalupka, A., and Bowring, N. J., “Charge-tube method for space charge in beams,” Charged Particle
Optics IV, Vol. 3777, edited by E. Munro, International Society for Optics and Photonics, SPIE, 1999, pp. 184 – 191.
https://doi.org/10.1117/12.370129.
[33]
Renau, A., Read, F. H., and Brunt, J. N. H., “The charge-density method of solving electrostatic problems with and
without the inclusion of space-charge,” Journal of Physics E: Scientific Instruments, Vol. 15, No. 3, 1982, pp. 347–354.
https://doi.org/10.1088/0022-3735/15/3/025.
[34] Humphries, S., Charged Particle Beams, Wiley, 1990. Chapter 5: Introduction to Beam-Generated Forces.
[35]
Gendrin, R., “Initial expansion phase of an artificially injected electron beam,” Planetary and Space Science, Vol. 22, No. 4,
1974, pp. 633 – 636. https://doi.org/10.1016/0032-0633(74)90097-X.
[36]
Stevenson, D., and Schaub, H., “Multi-Sphere Method for modeling spacecraft electrostatic forces and torques,” Advances in
Space Research, Vol. 51, No. 1, 2013, pp. 10 – 20. https://doi.org/10.1016/j.asr.2012.08.014.
[37]
Lewellen, J. W., Buechler, C. E., Carlsten, B. E., Dale, G. E., Holloway, M. A., Patrick, D. E., Storms, S. A., and Nguyen,
D. C., “Space-Borne Electron Accelerator Design,” Frontiers in Astronomy and Space Sciences, Vol. 6, 2019, p. 35.
https://doi.org/10.3389/fspas.2019.00035.
[38]
Lai, S. T., Fundamentals of Spacecraft Charging: Spacecraft Interactions with Space Plasmas, Princeton University Press,
2012.
[39] Smythe, W., Static and Dynamic Electricity, 3rd ed., McGraw–Hill, 1968.
[40]
Hughes, J. A., and Schaub, H., “Heterogeneous Surface Multisphere Models Using Method of Moments Foundations,” Journal
of Spacecraft and Rockets, Vol. 56, No. 4, 2019, pp. 1259–1266. https://doi.org/10.2514/1.A34434.
[41]
Qin, H., Zhang, S., Xiao, J., Liu, J., Sun, Y., and Tang, W. M., “Why is Boris algorithm so good?” Physics of Plasmas, Vol. 20,
No. 8, 2013, p. 084503. https://doi.org/10.1063/1.4818428.
[42]
Shampine, L. F., and Reichelt, M. W., “The MATLAB ODE Suite,” SIAM Journal on Scientific Computing, Vol. 18, No. 1,
1997, pp. 1–22. https://doi.org/10.1137/S1064827594276424.
[43]
Bengtson, M. T., “Electron Method for Touchless Electrostatic Potential Sensing of Neighboring Spacecraft,” Ph.D. thesis,
University of Colorado Boulder, 2020. URL http://hanspeterschaub.info/Papers/grads/MilesBengtson.pdf.
26
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
[44]
Wilson, K., Romero-Calvo, A., Bengtson, M., Hammerl, J., Maxwell, J., and Schaub, H., “Development and Characterization
of the ECLIPS Space Environments Simulation Facility,” Acta Astronautica, 2021. under review.
[45]
Cannavó, F., “Sensitivity analysis for volcanic source modeling quality assessment and model selection,” Computers &
Geosciences, Vol. 44, 2012, pp. 52 – 59. https://doi.org/10.1016/j.cageo.2012.03.008.
[46]
Wilson, K. T. H., and Schaub, H., “An X-ray Spectroscopic Approach to Remote Space Object Potential Determination:
Experimental Results,” AIAA SciTech, Orlando, Florida, 2020. https://doi.org/10.2514/6.2020-0049.
[47]
Chung, M. S., and Everhart, T. E., “Simple calculation of energy distribution of low-energy secondary electrons emitted from
metals under electron bombardment,” Journal of Applied Physics, Vol. 45, No. 2, 1974, pp. 707–709. https://doi.org/10.1063/1.
1663306.
27
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.2514/1.A35190
