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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

11

Accepted Manuscript

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

12

Accepted Manuscript

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

13

Accepted Manuscript

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

14

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The final version of this paper can be found in https://doi.org/10.2514/1.A35190

(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

15

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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

Accepted Manuscript

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).

17

Accepted Manuscript

The final version of this paper can be found in https://doi.org/10.2514/1.A35190

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.

18

Accepted Manuscript

The final version of this paper can be found in https://doi.org/10.2514/1.A35190

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)

19

Accepted Manuscript

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

20

Accepted Manuscript

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

21

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

22

Accepted Manuscript

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.

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