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Impact of solar wind fluctuations on Electric Sail mission design
Lorenzo Niccolai, Alessandro Anderlini, Giovanni Mengali, Alessandro A. Quarta∗
Department of Civil and Industrial Engineering, University of Pisa, I-56122 Pisa, Italy
Abstract
The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system that generates thrust by exploiting
the interaction between a grid of tethers, kept at a high electric potential, and the charged particles of the
solar wind. Such an advanced propulsion system allows innovative and exotic mission scenarios to be envisaged,
including non-Keplerian orbits, artificial Lagrange point maintenance, and heliostationary condition attainment.
In the preliminary mission analysis of an E-sail-based spacecraft, the local physical properties of the solar wind
are usually specified and kept constant, while the E-sail propulsive acceleration is assumed to vary with the
heliocentric distance, the sail attitude, and the grid electric voltage. However, the solar wind physical properties
are known to be characterized by a marked variability, which implies a non-negligible uncertainty as to whether
or not the solutions obtained with a deterministic approach are representative of the actual E-sail trajectory. The
aim of this paper is to propose an effective method to evaluate the impact of solar wind variability on the E-Sail
trajectory design, by considering the solar wind dynamic pressure as a random variable with a gamma distribution.
In particular, the effects of plasma property fluctuations on E-sail trajectory are calculated with an uncertainty
quantification procedure based on the generalized polynomial chaos method. The paper also proposes a possible
control strategy that uses suitable adjustments of grid electric voltage. Numerical simulations demonstrate the
importance of such a control system for missions that require a precise modulation of the propulsive acceleration
magnitude.
Keywords: Electric Solar Wind Sail, solar wind fluctuations, uncertainty propagation, generalized
polynomial chaos, E-sail trajectory control
Nomenclature
A,B= control strategies
a= propulsive acceleration vector (with a,kak), [ mm/s2]
ac= spacecraft characteristic acceleration, [mm/s2]
f= probability density function
L= tether length, [ km]
L1= collinear Lagrange point
M= dimension of vector ξ
m= spacecraft mass, [ kg]
N= number of tethers
ˆn= unit vector normal to the E-sail nominal plane
P= truncation order of Eq. (8)
p= solar wind dynamic pressure, [ nPa]
R= generic random process, see Eq. (8)
∗Corresponding author
Email addresses: lorenzo.niccolai@ing.unipi.it (Lorenzo Niccolai), alessandro.anderlini@ing.unipi.it (Alessandro
Anderlini), g.mengali@ing.unipi.it (Giovanni Mengali), a.quarta@ing.unipi.it (Alessandro A. Quarta)
Published in Aerospace Science and Technology, Vol. 82–83, November 2018, pp. 38–45. doi: https://doi.org/10.1016/j.ast.2018.08.032
r= position vector (with r,krk), [ au]
T= total flight time, [ days]
t= time, [ days]
V= grid electric voltage, [ kV]
w= weighting function
Vw= solar wind electric potential, [ kV]
v= spacecraft orbital velocity vector, [km/s]
α= shape parameter of the gamma distribution
β= scale parameter of the gamma distribution
Γ = gamma function
= dimensionless tolerance
0= vacuum permittivity, [ F/m]
λ= projection coefficient, see Eq. (10)
µ= Sun’s gravitational parameter, [ km3/s2]
µ⊕= Earth’s gravitational parameter, [ km3/s2]
ξ= random vector of uncertainty parameters
ρ= radial error, [ au]
σ= tether maximum thrust magnitude per unit length, [ N/m]
τ= switching parameter
Ψ = polynomial base
Ω = parameter space
Subscripts
0 = initial conditions
gPC = generalized Polynomial Chaos
H= heliostationary condition
h= generic time step
j= polynomial base index
L= artificial Lagrange point
max = maximum allowable value
req = required value
st = maximum step variation
⊕= value at r= 1 au
Superscripts
−= mean or nominal value
·= time derivative
b= unit vector
1. Introduction
The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system that exploits the interaction
of the charged particles in the solar wind with a spinning grid of tethers, kept at high potential by an
electron gun and stretched by the centrifugal force [1, 2, 3]. The peculiarity of an E-sail allows innovative
and exotic mission scenarios to be feasible, including non-Keplerian orbits and artificial Lagrange points
maintenance [4, 5, 6]. An in-situ test of the E-sail technology has not been performed yet, even though
the lack of experimental data should be overcome by Aalto-1 satellite [7], which is scheduled to validate the
plasma brake technology [8, 9], a derivation of the E-sail concept for spacecraft deorbiting in a planetocentric
mission.
The analysis of E-sail trajectories in a preliminary mission phase is usually addressed within a determin-
istic approach and, in this respect, analytical models exist for estimating the thrust vector [10, 11, 12, 13]
as a function of the E-sail design parameters. The local characteristics of the solar wind, such as plasma
2
number density and particle velocity, are taken as constant parameters in Refs. [10, 11, 12, 13], which corre-
sponds to assuming a fixed value of the local solar wind dynamic pressure. Under this hypothesis, the E-sail
propulsive acceleration vector is a function of the Sun-spacecraft distance, the sail attitude, and the grid
electric voltage, so that the spacecraft trajectory may be obtained by simple numerical integration of the
differential equations of motion. However, the assumption of constant local properties of the plasma is at
odd with data on solar wind characteristics obtained by a number of space missions, such as Voyager 2 [14],
Ulysses [15], and ACE [16]. Indeed, the in-situ measurements of plasma density, bulk speed, and dynamic
pressure exhibit a chaotic and unpredictable behaviour, with time fluctuations comparable to their mean
value [17, 18, 19, 20]. For these reasons, a refined mission analysis requires the solar wind dynamic pressure
for E-sail thrust generation to be described with more realistic models [21, 22, 23]. The aim of this work
is to propose an effective method to evaluate the impact of the solar wind variability on E-Sail interplane-
tary trajectories, and propose a control strategy that counteracts the solar wind-induced perturbations by
adjusting the grid electric voltage. In particular, the solar wind dynamic pressure is modelled as a random
variable with a gamma distribution, and its effects on the E-sail trajectories are simulated with stochastic
methods [24].
Loosely speaking, a stochastic approach refers to a group of different mathematical algorithms capable
of quantifying how the uncertainty on a set of design parameters, modelled as random variables with a
given Probability Density Function (PDF), propagates into a complex model. From a practical standpoint,
a continuous response surface of a given performance index in the parameter space may be approximated at
a computationally inexpensive cost with a reduced number of deterministic evaluations.
In this work, a method based on generalized Polynomial Chaos (gPC) [25] is used to generate the response
surface of the Sun-spacecraft distance as a function of the solar wind dynamic pressure. A gPC is a spectral
projection over a known orthogonal polynomial base, whose result is a sum of polynomials with suitable
coefficients that are evaluated through inner products. Due to its generality, such an approach has been
used in the last years for both aeronautical [26] and aerospace [24] applications.
The paper is organized as follows. The recent E-sail thrust model is first briefly summarized. The plasma
dynamic pressure is modelled as a random variable with a suitable PDF, based on available solar wind real
data. A gPC-based methodology is then used to analyze the effect of the solar wind variability on the
E-Sail trajectory, showing that the assumption of a fixed value of the local solar wind dynamic pressure
may produce inaccurate results even for short flight times. A control strategy is therefore discussed for
overcoming such a problem, where the solar wind-induced perturbations are balanced by suitably adjusting
the grid electric voltage. The effectiveness of the proposed a control law is illustrated by simulating two
advanced mission scenarios. The main outcomes of the work are finally highlighted.
2. Mathematical model
Consider an E-sail-based spacecraft, placed at distance r=r⊕,1 au from the Sun. The spacecraft
equations of motion in a heliocentric-ecliptic (inertial) reference frame are
˙
r=v(1)
˙
v=−µ
r3r+a(2)
where ris the spacecraft position vector (with r,krk), vis the velocity vector, µis the Sun’s gravitational
parameter, and ais the propulsive acceleration vector. To a first order approximation the E-sail shape is
modelled as a disk, and its propulsive acceleration vector is written as [12]
a=τac
2r⊕
r[ˆr+ (ˆr·ˆn)ˆn] (3)
where τ∈ {0,1}is a switching parameter that accounts for the possibility of switching either off (τ= 0)
or on (τ= 1) the E-sail electron gun, ˆr=r/r is the Sun-spacecraft unit vector, ˆnis the unit vector
normal to the sail nominal plane in the direction opposite to the Sun, and acis the spacecraft characteristic
acceleration, defined as the maximum propulsive acceleration at r=r⊕, see Fig. 1. As implied by Eq. (3),
the propulsive acceleration of a flat E-sail depends on its spacecraft attitude, that is, on the orientation of
3
Sun
E-sail
r
ˆ
r
ˆ
n
nominal plane
v
a
Figure 1: E-sail conceptual scheme.
the sail plane with respect to the radial direction. Notably, the same Eq. (3) may also be used to characterize
the acceleration of a three-dimensional sail, provided its shape is modelled as an axially symmetric body,
which spins around the Sun-spacecraft line and has a Sun-facing orientation, that is, ˆr≡ˆn[13].
2.1. E-sail performance parameter
The typical E-sail performance parameter in the preliminary mission phase is the spacecraft characteristic
acceleration ac. Its value is usually assumed to remain constant along the whole spacecraft trajectory [27,
28, 29], and is written as [12]
ac=N L σ⊕
m(4)
where Nis the number of tethers, Lis the (generic) tether length, mis the total spacecraft mass, and σ⊕
is given by [10, 30]
σ⊕,0.18 max (0, V −Vw)√0p⊕(5)
in which Vwis the electric potential of the solar wind ions (with a typical value of about 1 kV), Vis the
grid electric voltage (with a typical value of a few tens of kilovolts), 0is the vacuum permittivity, and p⊕
is the solar wind dynamic pressure at r=r⊕. Note that σ⊕can been thought of as the maximum thrust
per unit length generated by a tether at a distance r=r⊕from the Sun. Assuming VVwin Eq. (5), the
characteristic acceleration (4) may be conveniently approximated as
ac'0.18 N L V
m√0p⊕(6)
In a preliminary mission analysis phase, the solar wind dynamic pressure p⊕is typically assumed to take
the same value at each heliocentric latitude and at any time instant. In fact, a constant value of {p⊕, V }
corresponds to a constant value of acthroughout the flight, see Eq. (6). According to Refs. [15, 31], the
hypothesis of latitude-invariance for p⊕is sufficiently accurate, provided the heliocentric latitude of the
spacecraft does not have substantial variations during the mission. In other terms, as long as the analysis is
confined to trajectories with moderate inclinations relative to the ecliptic plane (or displaced non-Keplerian
orbits of constant latitude), the assumption of latitude-invariance of the solar wind dynamic pressure p⊕is
consistent with the solar wind behaviour. However, at a given heliocentric latitude, the value of p⊕undergoes
significant time fluctuations, comparable to its mean value [15]. An example of hourly variation of the solar
wind dynamic pressure p⊕on the ecliptic plane is shown in Fig. 2, for a time-span ranging from January
1996 to September 2013, according to the data reported by NASA1. Even though it is possible to recognize
a weak periodicity related to the 11-year solar activity cycle, the solar wind time fluctuations are highly
irregular and unpredictable and take place in short time intervals, on the order of a few hours. As a result,
in a refined mission analysis phase, the value of acmay be hardly taken as constant, as implied by Eq. (6).
The local value of p⊕is now therefore modelled as a random variable, to obtain more reliable information
on the actual spacecraft characteristic acceleration and, as such, on the propulsive acceleration vector given
by Eq. (3).
1See https://omniweb.gsfc.nasa.gov/form/dx1.html. Retrieved on May 22, 2018.
4
year
0
1
2
3
4
5
6
7
8
9
10
1996
2000 2005 2010 2013
Figure 2: Hourly in-situ measurements of p⊕from January 1996 to September 2013. Data taken from NASA.
2.2. Statistical evaluation of solar wind dynamic pressure
To describe the solar wind (local) fluctuations, the dynamic pressure p⊕is modelled as a random vari-
able whose instantaneous value is unaffected by the previous ones [18]. In other terms, this corresponds to
neglecting the periodic variations due to the solar activity cycles. Note that such an assumption is conser-
vative, since it overestimates the unpredictability of solar wind properties, and makes the succeeding design
of a control strategy based on the electric voltage variation more difficult. Figure 3 shows the histogram
plot of the PDF of the values of p⊕reported in Fig. 2. Note the existence of a significant asymmetry, with a
mean value of p⊕= 2nPa and a standard deviation of 1.56 nPa, see Fig. 3. These data are relative to a large
time-span (almost 18 years), and no significant periodic behaviour is evident. Accordingly, the hypothesis
that each value of p⊕is unaffected by the previous ones seems to be fairly realistic, since relevant variations
take place even in a few hours, see Fig. 2.
The measurements reported in Fig. 3 may be reasonably approximated with a PDF in the form of a
gamma distribution
f(p⊕) = β−α
Γ(α)pα−1
⊕exp (−p⊕/β) (7)
where f(p⊕) dp⊕is the probability that the dynamic pressure at a distance r⊕from the Sun ranges between
p⊕and p⊕+dp⊕, Γ(x) denotes the gamma function of the variable x, whereas αand βare the two parameters
necessary to define the properties of the PDF. In particular, the mean value and standard deviation of the
data reported in Fig. 3 are suitably fitted with α= 1.6437 and β= 1.2168. The assumption of a PDF
in the form of Eq. (7) implies that the higher-order statistical moments cannot be freely assigned as they
depend on αand β. In practice, the skewness and the kurtosis indexes are both underestimated. However,
the accordance between the gamma distribution and the experimental data, see Figs. 3 and 4, is sufficient
for appreciating the impact of solar wind fluctuations on the E-sail trajectory.
2.3. gPC-based procedure
A gPC-based procedure [25] is used to quantify how the uncertainty on the dynamic pressure propagates
into the spacecraft dynamics and, in particular, in the Sun-spacecraft distance r. The non-intrusive gPC is
5
0246810
p)[nPa]
0
0.5
1
1.5
2
2.5
3
Figure 3: PDF of p⊕from January 1996 to September 2013. Data taken from NASA.
0246810
p)[nPa]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 4: Gamma distribution of p⊕, see Eq. (7).
a projection of a random process Rover a known orthogonal polynomial base, which can be expressed as
R=
∞
X
j=0
λjΨj(ξ) (8)
6
where ξis a random vector of dimension M, belonging to the parameter space Ω, whose components are the
uncertain parameters. In addition, Ψj(ξ) is the gPC polynomial base of index j, while λjis the corresponding
projection coefficient. For practical purposes, the series in Eq. (8) is truncated at order P, viz.
RgPC =
P
X
j=0
λjΨj(ξ) (9)
Due to the orthogonality property of polynomials, the coefficients λjare obtained as
λj=hR, Ψji
hΨj,Ψji(10)
where hg, uidenotes the inner product between two generic functions gand u. In particular, in the scalar
case (M= 1), the inner product is defined as
hg, ui=ZΩ
w(ξ)g(ξ)u(ξ) dξ(11)
Note that gand uare multiplied by a weighting function w, which is the PDF of the random variable
ξ. Since the polynomials are orthogonal to the weighting function, the choice of the polynomial family
depends on the input parameter distribution. In particular, for a gamma distribution, the set of generalized
Gauss-Laguerre polynomials is used. A single uncertain parameter is assumed, that is ξ≡p⊕, and the
summation (9) is truncated at P= 4. The inner products are calculated with a Gaussian Quadrature
Formula using (P+ 1) = 5 quadrature points. Each quadrature point defines a node where the process R
is sampled, so that (P+ 1) deterministic evaluations only are sufficient to propagate the uncertainty in the
dynamic model.
2.4. Test case
The above-described procedure has been used to perform a stochastic simulation of the heliocentric
trajectory of an E-sail-based spacecraft that initially covers a circular orbit of radius r=r⊕. This example
models a spacecraft that escapes from the Earth’s gravitational field using a parabolic orbit with respect
to the planet. The electron gun is assumed to be always switched on (τ= 1), and the E-sail is maintained
at a Sun-facing attitude (ˆn=ˆr) with a constant value of the grid electric voltage. Note that for a Sun-
facing sail, the (local) propulsive acceleration magnitude takes its maximum value, see Eq. (3). The selected
nominal value (denoted by an over-bar symbol) of the spacecraft characteristic acceleration is consistent
with a near-term technology level in E-sail design [32]. Such a characteristic acceleration corresponds to
an E-sail based spacecraft with m= 560 kg, N= 24, V= 25 kV, L'8 km, and p⊕= 2 nPa, see Eq. (6).
In particular, the nominal characteristic acceleration ac= 0.2 mm/s2corresponds to an ideal situation of
solar wind dynamic pressure p⊕constant and equal to its mean value p⊕= 2 nPa. The effect of a solar
wind uncertainty on the E-Sail trajectory, that is, the impact of the variability of p⊕on the Sun-spacecraft
distance, can be evaluated by analyzing the heliocentric polar trajectory dispersion for θ∈[0,180] deg,
where θis the spacecraft polar angle measured counterclockwise from its initial position vector direction.
Figure 5 shows the PDF distribution (normalized with respect to its peak value) of r=r(θ), where dark
lines represent highly probable spacecraft trajectories.
Note that the spacecraft position has a non-negligible uncertainty after just one half revolution around
the Sun (θ= 180deg), even when the spacecraft attitude is fixed with respect to an orbital reference frame
(Sun-facing case). This may constitute a significant issue for missions requiring an accurate estimate of the
thrust vector such as, for example, interplanetary rendez-vous [27] or generation of highly non-Keplerian
orbits [4, 5, 6]. In these cases a suitable control system, capable of adjusting the propulsive acceleration
magnitude as a function of the instantaneous solar wind characteristics is therefore advisable, as is discussed
in the next section.
3. E-sail control strategies
Two possible control algorithms that may be used by an E-sail-based spacecraft to track a nominal
trajectory are now proposed. In so doing, the only control variable to be selected by the guidance system is
7
polar angle [deg]
start
PDF normalized value
r
Å
Sun
Figure 5: Normalized PDF distribution of the Sun-spacecraft distance for a Sun-facing sail with ac= 0.2 mm/s2.
the grid electric voltage [10, 22, 30]. In fact, the sail attitude (that is, the unit vector ˆn) must be adjusted so
as to obtain the desired thrust direction, while the other variables involved in the thrust generation depend
on the Sun-spacecraft distance and on the environmental conditions, see Eq. (6). In this preliminary analysis,
it is assumed that the power consumption of the electron gun can be slightly varied. This hypothesis has
a limited impact on the discussed simulations as it does not significantly affect the propulsive acceleration
generated by the E-sail. Indeed, the power consumption can be estimated by the orbital motion limited
(OML) current collection theory, according to which it is proportional to both the plasma number density
and V3/2[27, 33]. A hypothetical constraint of constant power consumption would imply a voltage reduction
when the plasma number density (and, consequently, the dynamic pressure) increases, with a resultant very
small variation in the generated thrust.
3.1. Dynamic pressure-based control law
The first proposed strategy is to adjust the grid electric voltage Vin response to the measured value of
the (local) solar wind dynamic pressure p(t) = p⊕(t) (r⊕/r)2, in such a way that the spacecraft characteristic
acceleration fits the nominal value given by
ac=0.18 N L V
mp0p⊕(12)
where Vis the nominal grid electric voltage, see Eq. (6).
It is assumed that the spacecraft is equipped with a sensor capable of measuring the instantaneous value
of the solar wind dynamic pressure, such as a particle detector. In case the shielding action due to the charged
grid is very large and makes the sensor ineffective, the latter could be replaced by an accelerometer that
measures the instantaneous generated thrust, from which the dynamic pressure can be calculated through
Eq. (6). The required value of the grid electric voltage Vreq(t) is found from Eq. (4) as
Vreq(t),m ac
0.18 N L p0p⊕(t)≡Vsp⊕
p⊕(t)(13)
If the control system were capable of modifying the grid electric voltage with no delay and without any
voltage limitations, the nominal trajectory could be tracked by simply enforcing V(t) = Vreq(t) for all
t≥t0,0. However, the grid electric voltage Vis constrained by a maximum value Vmax, and may only
be continuously modified by a certain amount ∆V≤Vst with Vst ∈[0, Vmax] [21]. Therefore, assuming
the mission to be divided into legs within which the voltage is kept constant, the E-sail voltage V(th), at a
8
generic time thwhere a leg starts, is a function not only of the solar wind properties through p(t), but also
of {Vmax, Vst }and of the previous value V(th−1), that is
V(th) =
V(th−1)−Vst if Vreq(th)< V (th−1)−Vst
Vreq(th) if Vreq(th)∈[V(th−1)−Vst , V (th−1) + Vst]
V(th−1) + Vst if Vreq(th)> V (th−1) + Vst and V(th−1) + Vst ≤Vmax
Vmax if Vreq(th)> V (th−1) + Vst and V(th−1) + Vst > Vmax
(14)
The staircase control law described by Eq. (14) is initialized by assuming that the E-sail is turned on at t0
with a voltage equal to the nominal value V(t0) = V. The control law of Eq. (14) will be referred to as
control strategy Ain the rest of the work.
3.2. Heliocentric distance-based control law
Another control strategy consists in relating the grid electric voltage with the Sun-spacecraft distance,
provided the spacecraft is able to accurately determining the value of r(t). In this case, at the initial time
of each leg the spacecraft measures the instantaneous Sun-spacecraft distance r(th), which is compared by
the onboard guidance system with the desired, nominal, value r(t).
The control strategy consists in increasing (decreasing) the grid electric voltage when r(th) is smaller
(greater) than the nominal value r(th), that is
V(th) =
Vmax if r(th)< r(th)(1 −) and V(th−1) + Vst > Vmax
V(th−1) + Vst if r(th)< r(th)(1 −) and V(th−1) + Vst ≤Vmax
V(th−1) if |r(th)−r(th)|
r(th)≤
V(th−1)−Vst if r(th)> r(th)(1 + ) and V(th−1)≥Vst
0 if r(th)> r(th)(1 + ) and V(th−1)< Vst
(15)
where is a given small dimensionless parameter (tolerance). The control law reassumed by Eq. (15) is
denoted with the symbol Band is initialized by V(t0) = V. Note that Eq. (15) accounts for a grid electric
voltage that may take a finite number of values only, thus offering a possible simplification of the power
system design.
4. Mission applications
The previous control algorithms Aand Bare now simulated in two advanced mission scenarios, in order
to investigate their performance in maintaining a working orbit that requires an accurate thrust vector
modulation for a long time interval.
4.1. Heliostationary condition generation
An interesting, albeit exotic, mission application for an E-sail-based spacecraft is the generation of a
heliostationary condition [34, 35], characterized by a spacecraft that maintains a fixed position with respect
to an inertial reference frame, as illustrated in Fig. 6. When the spacecraft has reached the heliostationary
point, the propulsive acceleration given by a Sun-facing E-sail is used to balance the Sun’s gravitational
pull. This mission scenario has interesting scientific implications such as, for example, the observation of the
Sun’s polar region by a vantage point, the monitoring of near-Earth objects, or the release of a small solar
probe along a rectilinear trajectory [36, 37]. Note that the assumption of small variation of the heliocentric
latitude is consistent with this mission scenario, since a spacecraft in a heliostationary condition maintains
a time-invariant position with respect to the Sun.
9
Sun
m
2
r
e
()
/
c
ar r
Å
H
r
E-sail
Figure 6: Sketch of the heliostationary condition.
The Sun-spacecraft heliostationary distance rHand the required (nominal) characteristic acceleration ac
are obtained by enforcing the balance between the Sun’s gravitational pull and the Sun-facing E-sail thrust
(see Fig. 6), calculated from Eq. (3) with τ= 1 (electron gun switched on) and ˆn=ˆr, viz.
µ
r2
H
=acr⊕
rH(16)
For a given value of rH, the nominal characteristic acceleration acrequired to maintain the heliostationary
condition is therefore
ac=µ
r⊕rH
(17)
When rH=r⊕= 1 au, the (required) nominal characteristic acceleration given by Eq. (17) is ac=µ/r2
⊕'
5.93 mm/s2, a value corresponding to a high-performance E-sail.
Assuming that the total simulation time of T= 0.25year is divided into legs of about 1 year/(2π×100) '
0.58 days, Tables 1 and 2 report the radial error, defined as
ρH(t),|r(t)−rH|(18)
for the two control strategies as a function of {Vmax, Vst , }. For each triplet, 100 simulations have been
conducted as follows. At the beginning of each leg, the solar wind dynamic pressure is generated as a
random variable with a PDF in the form of Eq. (7), and the grid electric voltage is adjusted according to the
control law A, see Eq. (14), or B, see Eq. (15). The equations of motion are integrated in double precision
using a variable order Adams-Bashforth-Moulton solver scheme [38, 39] with absolute and relative errors of
10−12 until the end of the leg, when a new value of the dynamic pressure is generated and the procedure is
restarted, up to the final time Tis reached.
Table 2 shows that a smallest value of the tolerance gives better performance, but the control law A
significantly outperforms the control law B. However, note that an accurate position maintenance is possible
only for large values of both Vmax and Vst, even for a short mission time of about 3 months only.
4.2. Artificial Lagrange point maintenance
The second mission scenario consists in the generation of a L1-type artificial (collinear) Lagrange point
in the Sun-[Earth+Moon] three-body system [40, 41, 42], at a distance rLfrom the Sun. In particular, rL
is chosen to be less than the actual Sun-L1distance, that is, less than 0.99 au, see Fig. 7. Such a mission is
important for solar observation purposes, since when placed in the artificial point, the spacecraft would be
able to provide an early warning in the event of coronal mass ejections or solar flares [6], thus representing
an improvement with respect to the ACE mission [16], which covers a Halo orbit around the Sun-Earth L1
point, with a warning time capability of about 1 hour.
10
Vmax [ kV] Vst [ kV] ρH[ au] ρH/rH[%]
mean max mean max
40
1 0.0170 0.1442 1.699 14.418
5 0.0123 0.1123 1.227 11.227
10 0.0111 0.0981 1.138 9.807
40 0.0265 0.1427 2.655 14.266
60
1 0.0176 0.2499 1.758 24.986
5 0.0109 0.1338 1.086 13.377
10 0.0114 0.1277 1.1405 12.767
60 0.0076 0.0543 0.763 5.428
80
1 0.0185 0.1492 1.854 14.915
5 0.0132 0.1293 1.320 12.933
10 0.0091 0.0838 0.907 8.377
80 0.0035 0.0291 0.3478 2.906
no control 0.0387 0.2538 3.868 25.381
Table 1: Results of the simulations for the heliostationary condition (control strategy A).
Vmax [ kV] Vst [ kV] ρH[ au] ρH/rH[%]
mean max mean max
40
50 0.0139 0.0748 1.388 7.483
0.01 0.0487 0.1736 4.871 17.361
10 0 0.0082 0.0437 0.823 4.373
0.01 0.0412 0.1581 4.124 15.811
60
50 0.0306 0.1423 3.062 14.234
0.01 0.0683 0.1927 6.826 19.268
10 0 0.0161 0.0851 1.609 8.506
0.01 0.0392 0.1405 3.920 14.054
80
50 0.0450 0.2533 4.495 25.333
0.01 0.1345 0.4135 13.451 41.349
10 0 0.0292 0.1411 2.925 14.107
0.01 0.0808 0.2661 8.077 26.606
no control 0.0387 0.2538 3.868 25.381
Table 2: Results of the simulations for the heliostationary condition (control strategy B)
Sun
0.01 au»
E-sail
L
r
L1Earth
rÅ
Figure 7: Sketch of the artificial (collinear) Lagrangian point maintenance.
In this case, the spacecraft heliocentric trajectory is a nearly circular non-Keplerian orbit, with a radius
of r=rLand a period of 1 year, such that the Sun, the spacecraft, and the Earth are always aligned. The
artificial Lagrange point may be maintained using the continuous propulsive acceleration generated by a
Sun-facing E-sail [4]. The value of rLis found by enforcing an equilibrium condition between the Sun’s and
11
Earth’s gravitational pulls, the propulsive acceleration, and the centrifugal acceleration [41], viz.
−µ
r2
L
+µ⊕
(r⊕−rL)2+acr⊕
rL+µ
r3
⊕
rL= 0 (19)
which can be solved for rLonce the characteristic acceleration acis derived from Eq. (12). Note that Eq. (19)
is in accordance with the results discussed in Ref. [41], if it is assumed that the Sun-[Earth+Moon] system
center of mass coincides with that of the Sun, and provided that the E-sail thrust model is updated with
the recent results given by Ref. [12].
The effectiveness of the proposed control laws has been analyzed assuming an E-sail-based spacecraft
with a nominal characteristic acceleration of ac= 1 mm/s2and a grid electric voltage of V= 25 kV. Using
the mathematical model discussed in Ref. [32], such a characteristic acceleration may be reached with a
total spacecraft mass m= 696kg, a payload mass of 200 kg (about 25% greater than that required by the
ACE mission), and an E-sail having N= 16 tethers with a length of L= 19.4km each. From Eq. (19),
the distance of the L1-type artificial (collinear) Lagrange point from the Sun is rL= 0.9436 au, which is in
accordance with the previous hypothesis of rto be on the order of 1 au. Such a value of rLwould guarantee
a warning time of more than 5.5 hours, that is, five times greater than that given by the ACE mission.
The total mission time of T= 10years is again divided into legs of about 0.58 days, and the simulations
have been conducted with the same procedure described for the heliostationary condition scenario. For a
given value of the pair {Vmax, Vst }, the control strategy effectiveness is quantified by the time variation of
the radial error with respect to the nominal position, that is
ρL(t),|r(t)−rL|(20)
Different values of Vmax and Vst have been considered, and 100 simulations have been conducted for each
combination. Table 3 reports the mean and the maximum value of ρLobtained from simulation with the
control strategy A. The case of Vst ≡Vmax is also included, which corresponds to a power system capable
of instantaneously adjusting the grid electric voltage. A very accurate orbit maintenance is possible only
Vmax [ kV] Vst [ kV] ρL[ au] ρL/rL[%]
mean max mean max
40
1 0.0228 0.1384 2.418 14.660
5 0.0168 0.1120 1.781 11.863
10 0.0144 0.0791 1.526 8.385
40 0.0167 0.0759 1.774 8.044
60
1 0.0234 0.1510 2.483 16.001
5 0.0208 0.1342 2.199 14.218
10 0.0172 0.1084 1.821 11.480
60 0.0107 0.0391 1.131 4.143
80
1 0.0226 0.1539 2.399 16.306
5 0.0188 0.1410 1.994 14.941
10 0.0175 0.1087 1.850 11.519
80 0.0095 0.0294 1.008 3.119
no control 0.0274 0.1193 2.899 8.071
Table 3: Results of the simulations for the artificial Lagrange point maintenance (control strategy A).
for large values of both Vmax and Vst, when the maximum radial error is about 3% of rL. This aspect could
constitute a technological challenge for future E-sail missions, since when the constraints on Vmax and Vst
are relaxed, the capability for the spacecraft to track the nominal trajectory tends to increase.
Using the same values of Vmax and Vst as those of Tab. 3, simulations have been conducted also for the
control strategy B, considering different values of the tolerance ∈ {0,0.01}, but the results are omitted
here for the sake of conciseness, because the performance of control law Bis significantly worse than that of
control law Ain this scenario.
12
5. Conclusions
A statistical model of the solar wind dynamic pressure has been proposed and used to perform a gen-
eralized polynomial chaos-based analysis of the impact of its variability on the E-sail performance. The
simulation results about the spacecraft interplanetary trajectory, demonstrate the necessity of a control
system capable of adjusting the grid electric voltage in response to in-situ measurements of the solar wind
physical characteristics.
Two control laws have been discussed, based on the measurements of the instantaneous dynamic pres-
sure and on instantaneous heliocentric distance, respectively. Both algorithms have been simulated in two
advanced mission scenarios, that is, the generation of a heliostationary condition and an artificial Lagrange
point maintenance. The control strategy based on the plasma pressure measurements shows a significantly
better performance, but the results highlight that both a large maximum grid electric voltage and the ca-
pability of quickly adjusting the voltage are advisable, when an accurate thrust modulation is required. A
future and more in-depth analysis of control laws should also account for a possible constraint on the power
consumption of the E-sail electron gun.
A natural extension of this work consists in analyzing the impact of the solar wind variability in a classical
interplanetary rendezvous in which the sail attitude varies continuously in order to obtain the required thrust
vector. In that case, a change in nominal characteristic acceleration would affect the (optimal) control law,
especially when the actual position of the initial and target planet (or celestial body) is considered during
the preliminary mission design. The proposed mathematical model can also be applied to a more advanced
mission scenario, such as a flight towards a nearby star system (or the Sun’s gravitational lens whose
distance is about 550 au), in which the E-sail propulsion system is used to decelerate the spacecraft through
the interaction with the interstellar wind.
Acknowledgements
The authors wish to thank Dr. Giovanni Vulpetti for his precious advice. Constructive remarks by the
anonymous reviewers are gratefully acknowledged.
Conflict of interest statement
The authors declared that they have no conflicts of interest to this work.
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