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Thrust and Torque Vector Characteristics of Axially-Symmetric E-Sail
Marco Bassetto, Giovanni Mengali, Alessandro A. Quarta∗
Department of Civil and Industrial Engineering, University of Pisa, I-56122 Pisa, Italy
Abstract
The Electric Solar Wind Sail is an innovative propulsion system concept that gains propulsive acceleration from
the interaction with charged particles released by the Sun. The aim of this paper is to obtain analytical expressions
for the thrust and torque vectors of a spinning sail of given shape. Under the only assumption that each tether
belongs to a plane containing the spacecraft spin axis, a general analytical relation is found for the thrust and
torque vectors as a function of the spacecraft attitude relative to an orbital reference frame. The results are then
applied to the noteworthy situation of a Sun-facing sail, that is, when the spacecraft spin axis is aligned with
the Sun-spacecraft line, which approximatively coincides with the solar wind direction. In that case, the paper
discusses the equilibrium shape of the generic conducting tether as a function of the sail geometry and the spin
rate, using both a numerical and an analytical (approximate) approach. As a result, the structural characteristics
of the conducting tether are related to the spacecraft geometric parameters.
Keywords: Electric solar wind sail, thrust and torque model, Sun-facing sail, sail equilibrium shape
Nomenclature
A,B,C,D= components of the total force, see Eq. (23), [N]
b= dimensionless (shape) coefficient
d= position vector of ds, [m]
E,F,G= components of the total torque, see Eq. (32), [ N m]
f= distance of dsfrom (x, y) plane, [m]
F= total force, with ||F|| ,F, [N]
h= dimensionless abscissa
ˆ
i= unit vector of x-axis
ˆ
ik= unit vector of xk-axis
ˆ
j= unit vector of y-axis
ˆ
k= unit vector of z-axis
K= dimensionless shaping parameter, see Eq. (58)
L= tether length, [m]
mp= proton mass, [kg]
n= solar wind number density, [ m−3]
ˆn= spin velocity unit vector
N= number of tethers
ˆr= Sun-spacecraft unit vector
s= curvilinear abscissa, [m]
∗Corresponding author
Email addresses: marco.bassetto@ing.unipi.it (Marco Bassetto), g.mengali@ing.unipi.it (Giovanni Mengali),
a.quarta@ing.unipi.it (Alessandro A. Quarta)
Published in Acta Astronautica, Vol. 146 May 2018, pp. 134–143. doi: http://dx.doi.org/10.1016/j.actaastro.2018.02.035
S= spacecraft center-of-mass
ˆs= unit vector tangent to the tether
T= total torque, [ N m]
u= solar wind relative velocity vector, with ||u|| ,u, [ m s−1]
V= tether electric potential [V]
Vw= solar wind ions electric potential, [V]
(x, y, z) = axes of the body reference frame
αn= pitch angle, [rad]
β= aperture angle of the right circular cone, [rad]
δn= clock angle, [rad]
0= vacuum permittivity, [ F m−1]
ζk= angle between planes (ˆ
ik,ˆn) and (ˆ
i,ˆn), [rad]
ρ= tether linear mass density, [ kg m−1]
σ= constant, see Eq. (10), [ kg m−1s−1]
τ= tether tension force, [N]
ω= spacecraft spin velocity, with ||ω|| ,ω, [ s−1]
Subscripts
c= conic
k= generic tether
l= logarithmic
max = maximum
p= parabolic
r= root
s= due to solar wind flux
t= tip
ω= centrifugal
Superscripts
0= derivative with respect to x
1. Introduction
The Electric Solar Wind Sail (E-sail) is an innovative propulsion system that exploits the solar wind
particle momentum to generate a propulsive acceleration in the interplanetary space [1]. The incoming ions
interact with an artificial electric field generated on board by means of an electron emitter, which charges a
grid of long tethers at a high voltage level, on the order of some tens of kilovolts [2]. The tethers are deployed
and maintained stretched by spinning the spacecraft and, in a simplified model, they can be assumed to
belong to the same plane orthogonal to the spin axis [3, 4], see Fig. 1. Along with the more classical solar
sail, the E-Sail is one of the most promising propellantless propulsion systems, even though it needs electric
power to produce the required electric field. Unlike a solar sail, whose propulsive force varies as the inverse
square distance from the Sun, a very interesting property of an E-sail is that its maximum thrust modulus
is inversely proportional to the heliocentric distance [5].
A non-negligible portion of the current research is intended for investigating how the geometric features
of such a propulsion system may affect its in-flight performance in terms of thrust and torque vectors [6, 7, 8].
However, the E-sail propulsive characteristics are quite complex to model, as the thrust (or torque) vector and
the sail shape are mutually affected by each other. To get preliminary simulation results, the thrust vector
is often modelled through the simplified assumption of a sail shape resembling that of a rigid disc of given
radius [9, 10, 11, 12, 13]. In some cases such an approximation may be inaccurate, as the actual shape of each
tether depends on the combined effects of the centrifugal force and the solar wind dynamic pressure acting on
it. Moreover, it is known that the actual geometric characteristics of the sail shape may significantly affect
the performance of an E-sail-based spacecraft. Nevertheless, in a preliminary phase of mission design the
mathematical model adopted to describe the sail shape must be simple enough to be successfully implemented
2
Sun
E-sail
solar wind
charged tether
electric field
main body
spin axis
Figure 1: Spinning E-sail conceptual sketch.
within a simulation code, especially when optimal trajectories are investigated [14, 15]. Indeed, in the latter
case, a number of transfer trajectories need to be simulated to minimize a scalar performance index, such
as the flight time [16, 17, 18, 19].
In this context, Toivanen and Janhunen [8] have studied the shape of a rotating E-sail using a numerical
approach, stating that the tether arrangement forms a cone near the spacecraft, while each tether is flattened
near the tip by the centrifugal force. More recently, Huo et al. [20] have obtained a compact and analytical
description of the E-sail thrust vector using a geometric approach and assuming an axisymmetric grid of
tethers belonging to the same plane (the so-called “flat case”). The aim of this paper is to obtain an analytical
expression of both thrust and torque vectors generated by a spinning E-sail of a given (three-dimensional)
shape, under the main assumption that each tether belongs to a plane containing the spacecraft spin axis.
The analytical results are then applied to the noteworthy case of a Sun-facing spinning E-sail [21, 22], thus
obtaining a set of analytical (compact) relations.
The problem of describing the actual E-sail equilibrium shape has indeed a substantial simplification when
the spacecraft spin axis is aligned with the solar wind velocity vector, the latter being nearly parallel to the
Sun-spacecraft direction, see Fig. 2. In that case each tether can be thought of as being aligned with the
force field and belonging to a plane containing the spacecraft spin axis. In particular, this paper shows that
an approximate, analytical, solution to the E-sail equilibrium shape may be found under the assumption of
cylindrical symmetry, that is, when all tethers are the same angle apart from each other. The corresponding
tether equilibrium shape is accurately described by a logarithmic arc whose geometric characteristics are
related (in an analytical form) to the combined effects of centrifugal and solar wind-induced forces. This
result is consistent with the numerical simulations discussed by Toivanen and Janhunen [8]. As such, the new
mathematical relations represent an useful improvement over existing models, as they allow the influence
of tether arrangement on the propulsion system performance to be quantified without the use of numerical
algorithms.
The paper is organized as follows. The resultant force and torque vectors acting on an E-sail of given
shape are firstly analyzed in analytical form, starting from the mathematical model discussed in Ref. [2].
The obtained equations are then applied to the important case of a Sun-facing, axially symmetric, E-sail.
The approximate form of the tether equilibrium shape is then analytically derived, and the resultant root
force is calculated as a function of the tether geometric characteristics and the spacecraft spin rate. Finally,
some concluding remarks are given in the last section.
2. Mathematical description of E-sail thrust and torque
Consider an E-sail-based spacecraft that spins about a body-fixed axis with unit vector ˆnat an angular
velocity ω=ωˆnof constant modulus ω. The E-sail propulsion system consists of N≥2 tethers, each one
being modelled as a planar cable belonging to the plane (ˆ
ik,ˆn), where ˆ
ik(with k∈ {0,1, . . . , N −1}) is a
unit vector orthogonal to ˆn, see Fig. 3.
3
Figure 2: Electric solar wind sail artistic impression. Courtesy of Alexandre Szames, Antigravite (Paris).
0
ˆˆ
=ii
u
tether
ˆ
i
ˆ
j
ˆ
k
ˆ
n
x
y
z
1 tether
st
( +1)-th tetherk
ˆ
k
i
k
x
k
z
sail nominal
plane
ˆˆ
(,)plane
k
ik
S
N-th tether
Figure 3: E-sail geometric arrangement.
The displacement of the generic tether with respect to the spacecraft main body can be evaluated by
introducing a body reference frame T(S;x, y, z) with origin Sat the spacecraft center-of-mass, and unit
vectors {ˆ
i,ˆ
j,ˆ
k}defined as ˆ
k,ˆn,ˆ
i,ˆ
i0,ˆ
j,ˆn׈
i0(1)
Note that the plane (ˆ
i,ˆ
k) contains the first tether, labelled with k= 0, whereas the unit vector ˆ
ikcan be
written as
ˆ
ik= cos ζkˆ
i+ sin ζkˆ
j(2)
where ζkis the angle, measured counterclockwise from the direction of ˆ
i, between the x-axis and the xk-axis
with unit vector ˆ
ik, see Fig. 3. In other words, ζkis the angle between planes (ˆ
i,ˆ
k) and (ˆ
ik,ˆ
k), that is, the
planes that contain the first and the (k+ 1)-th tether, respectively.
2.1. E-sail shape model
Assume that the shape of the generic tether can be described, in the plane (ˆ
ik,ˆ
k), through a continuously
differentiable function fk=fk(xk):[xrk, xtk]→R, where xrk≥0 (or xtk) is the distance of the tether
root (or tip) from the spacecraft spin axis z, see Fig. 4. The position vector dkof a generic infinitesimal
arc-length dskof the conducting tether is given by
dk=xkˆ
ik+fkˆ
k(3)
4
ˆ
k
ˆ
k
i
S
k
r
x
k
x
k
t
x
k
f
dk
s
root
tip
( +1)-th tetherk
spin axis
k
d
Figure 4: Generic tether displacement.
with
dsk=q1+(f0
k)2dxk(4)
where f0
k,dfk/dxk. From Eqs. (3)-(4), the expression of the (local) unit vector ˆsktangent to the generic
tether at point (xk, fk) is
ˆsk,ddk
dsk
=dxkˆ
ik+ dfkˆ
k
p1+(f0
k)2dxk≡ˆ
ik+f0
kˆ
k
p1+(f0
k)2(5)
which can be rewritten, using Eq. (2), as a function of {ˆ
i,ˆ
j,ˆ
k}as
ˆsk=cos ζkˆ
i+ sin ζkˆ
j+f0
kˆ
k
p1+(f0
k)2(6)
2.2. Force acting on tethers
The total force dFkacting on the infinitesimal arc-length dskis the sum of the centrifugal force dFωk,
and that arising from the solar wind dynamic pressure dFsk, viz.
dFk= dFωk+ dFsk(7)
Recalling that xkis the distance of dskfrom the spacecraft spin axis z, the term dFωkcan be written as
dFωk=ρdskxkω2ˆ
ik≡ρdskxkω2cos ζkˆ
i+ sin ζkˆ
j(8)
where ρis the tether (linear) uniform mass density, and ˆ
ikis given by Eq. (2) as a function of {ˆ
i,ˆ
j}. Also,
according to the recent works of Janhunen and Toivanen [3, 6, 8], the thrust dFskgained by dsk, when the
Sun-spacecraft distance is on the order of 1 au, is given by
dFsk=σku⊥kdsk(9)
with
σk,0.18 max(0, Vk−Vw)√0mpn(10)
where Vkis the tether voltage (on the order of 20–40 kV), Vwis the electric potential corresponding to the
kinetic energy of the solar wind ions (with a typical value of about 1 kV), 0is the vacuum permittivity, mp
is the solar wind ion (proton) mass, nis the local solar wind number density, and u⊥kis the component of
the solar wind velocity uperpendicular to the direction of ˆskgiven by Eq. (5).
Assuming a purely radial solar wind stream, that is, u=uˆrwhere ˆris the Sun-spacecraft unit vector
and uis the solar wind velocity modulus, the term u⊥kin Eq. (9) is given by
u⊥k=u(ˆsk׈r)׈sk≡u[ˆr−(ˆr·ˆsk)ˆsk] (11)
5
In particular, according to Fig. 5, the Sun-spacecraft unit vector ˆrcan be written as a function of {ˆ
i,ˆ
j,ˆ
k}
as
ˆr= sin αncos δnˆ
i+ sin αnsin δnˆ
j+ cos αnˆ
k(12)
where δn∈[0,2π] rad is the clock angle, measured counterclockwise from the direction of ˆ
i, between the
x-axis and the projection of ˆron the plane (x, y), while αn∈[0, π] rad is the sail pitch angle, defined as the
angle between ˆrand ˆ
k≡ˆn, viz.
αn,arccos(ˆr·ˆ
k) (13)
ˆ
i
ˆ
j
ˆ
k
x
y
z
1 tether
st
ˆ
k
i
k
x
k
z
sail nominal
plane
S
Sun
ˆ
r
n
d
n
a
Figure 5: Sail pitch angle αn.
Taking into account Eqs. (6) and (12), the dot product ˆr·ˆskin Eq. (11) becomes
ˆr·ˆsk=ˆr·ˆ
ik+f0
kˆr·ˆ
k
p1+(f0
k)2=cos(δn−ζk) sin αn+f0
kcos αn
p1+(f0
k)2(14)
Therefore, with the aid of Eqs. (6), (11) and (14), the thrust dFskgiven by Eq. (9) can be rewritten as
dFsk=σkudskˆr−cos(δn−ζk) sin αn+f0
kcos αn
1+(f0
k)2cos ζkˆ
i+ sin ζkˆ
j+f0
kˆ
k(15)
Substituting Eqs. (8) and (15) into Eq. (7), and bearing in mind Eq. (4), the compact form of the total
force dFkacting on the infinitesimal arc-length dskof the generic tether is
dFk= dAkˆr+ dBkˆ
i+ dCkˆ
j+ dDkˆ
k(16)
where
dAk,σkuq1+(f0
k)2dxk(17)
dBk,ρ xkω2−σkucos(δn−ζk) sin αn+f0
kcos αn
1+(f0
k)2cos ζkq1+(f0
k)2dxk(18)
dCk,ρ xkω2−σkucos(δn−ζk) sin αn+f0
kcos αn
1+(f0
k)2sin ζkq1+(f0
k)2dxk(19)
dDk,−σku f 0
k
cos(δn−ζk) sin αn+f0
kcos αn
p1+(f0
k)2dxk(20)
6
Note that such a decomposition is not unique. Finally, the force Fkacting on the conducting tether is
Fk=Zxtk
xrk
dFk=Akˆr+Bkˆ
i+Ckˆ
j+Dkˆ
k(21)
with
Ak=Zxtk
xrk
dAk,Bk=Zxtk
xrk
dBk,Ck=Zxtk
xrk
dCk,Dk=Zxtk
xrk
dDk(22)
whereas the total force Facting on the E-sail (composed of N≥2 tethers) is given by
F=
N−1
X
k=0
Fk≡ A ˆr+Bˆ
i+Cˆ
j+Dˆ
k(23)
where
A,
N−1
X
k=0 Ak,B,
N−1
X
k=0 Bk,C,
N−1
X
k=0 Ck,D,
N−1
X
k=0 Dk(24)
2.3. Propulsive torque
The torque dTkgiven by an infinitesimal arc-length dskof the generic tether is
dTk=dk×dFk(25)
where the symbol ×denotes the cross product. Taking into account the expressions of dkand dFkgiven
by Eqs. (3) and (16), respectively, and using Eq. (12), dTkmay be written in a compact form, as a function
of {ˆ
i,ˆ
j,ˆ
k}, as
dTk= dEkˆ
i+ dFkˆ
j+ dGkˆ
k(26)
where
dEk,(xksin ζk"σkucos αn−f0
kσku(sin αncos(δn−ζk) + f0
kcos αn)
1+(f0
k)2#+
−fksin ζk"ρ xkω2−σku(sin αncos(δn−ζk) + f0
kcos αn)
1+(f0
k)2#−fkσkusin αnsin δn)q1+(f0
k)2dxk
(27)
dFk,(−xkcos ζk"σkucos αn−f0
kσku(sin αncos(δn−ζk) + f0
kcos αn)
1+(f0
k)2#+
+fkcos ζk"ρ xkω2−σku(sin αncos(δn−ζk) + f0
kcos αn)
1+(f0
k)2#+fkσkusin αncos δn)q1+(f0
k)2dxk
(28)
dGk,σku xksin αnsin(δn−ζk)q1+(f0
k)2dxk(29)
The torque Tkacting on the generic tether is
Tk=Zxtk
xrk
dTk=Ekˆ
i+Fkˆ
j+Gkˆ
k(30)
with
Ek=Zxtk
xrk
dEk,Fk=Zxtk
xrk
dFk,Gk=Zxtk
xrk
dGk(31)
7
whereas the total torque Tacting on the E-sail is
T=
N−1
X
k=0
Tk≡ Eˆ
i+Fˆ
j+Gˆ
k(32)
where
E,
N−1
X
k=0 Ek,F,
N−1
X
k=0 Fk,G,
N−1
X
k=0 Gk(33)
Equations (21) and (32) are the expressions of the total force and torque acting on the E-sail with a given
tether shape, length, and angular separation between tethers. However, some simplifying assumptions need
to be introduced to get a more tractable form of both Fand T, as is thoroughly discussed in the next
section.
3. Case of a Sun-facing E-sail
The previous general results are now specialized to the noteworthy case of a Sun-facing E-sail [21, 22],
which corresponds to when the spacecraft spin axis zcoincides with the Sun-spacecraft line (i.e., ˆ
k≡ˆr). In
this case the pitch angle αnis zero by construction, whereas δncan be set to zero without loss of generality,
because ˆris orthogonal to the plane (x, y), viz.
αn= 0 , δn= 0 (34)
Assuming all tethers to have the same length Land the same voltage Vk(that is, the same value of σk,
see Eq. (10)), the E-sail may reasonably be assumed to have a cylindrical symmetry around the z-axis.
The notation can be therefore simplified by dropping the subscript kin the variables {xk, fk, xrk, xtk, σk}.
Accordingly, all tethers have the same shape (i.e., they are described via the same mathematical function
f=f(x)), and are arranged at the same angle apart from each other, viz.
ζk=2π
Nk(35)
with k= 0,1,...,(N−1).
Taking into account Eqs. (34)–(35), and bearing in mind that ˆ
k≡ˆr, from Eq. (23) the total force F
becomes
F= σ u N Zxt
xr
1
p1+(f0)2dx!ˆr+
+ ρ ω2Zxt
xr
xp1+(f0)2dx−σ u Zxt
xr
f0
p1+(f0)2dx! "ˆ
i
N−1
X
k=0
cos 2π
Nk+ˆ
j
N−1
X
k=0
sin 2π
Nk#(36)
and the total torque (32) is
T=ˆ
i"Zxt
xr
σ u x −ρ ω2x f [1 + (f0)2] + σ u f f 0
p1+(f0)2dx#N−1
X
k=0
sin 2π
Nk+
−ˆ
j"Zxt
xr
σ u x −ρ ω2x f [1 + (f0)2] + σ u f f 0
p1+(f0)2dx#N−1
X
k=0
cos 2π
Nk(37)
whereas the single tether length Lcan be written, as a function of {xr, xt, f0}, as
L=Zxt
xrp1+(f0)2dx(38)
8
According to Ref. [20], when N≥2 the summations in Eqs. (36)-(37) are
N−1
X
k=0
sin 2π
Nk=
N−1
X
k=0
cos 2π
Nk= 0 (39)
and the final form of the total force and torque given by a Sun-facing E-sail reduces to
F= σ u N Zxt
xr
1
p1+(f0)2dx!ˆr,T= 0 (40)
Note that the result T= 0 is consistent with the assumption of an E-sail with cylindrical symmetry with
respect to the spin axis, whereas the actual expression of the total force F(that is, the E-sail propulsive
thrust) depends on the tether shape via the analytical function f0= df /dx. Some noteworthy cases are
now discussed to better investigate the impact of the tether shape f=f(x) on the E-sail total force F.
3.1. Flat shape
When all the tethers are arranged on a flat surface that, in this case, coincides with the E-sail nominal
plane, the condition f0= 0 is to be enforced in the first of Eqs. (40). The total force becomes
F=σ u N L ˆr(41)
where L= (xt−xr), see Eq. (38). In particular, Eq. (41) is consistent with the result discussed in Ref. [20]
for a Sun-facing E-sail (i.e., when αn= 0).
Actually, the case of a purely flat E-sail is only a first approximation of the real sail shape. In fact, all
tethers tend to move away from the E-sail nominal plane (x, y) and to take a three-dimensional arrangement,
while keeping, according to the previous assumptions, a cylindrical symmetry.
3.2. Conic shape
An interesting approximation of the actual E-sail three-dimensional arrangement is given by a conic
shape. In that case, each tether may be analytically described as
f(x) = bcxrx
xr−1with x∈[xr, xt] (42)
where bc>0 is a (constant) dimensionless coefficient, whose value depends on the aperture angle βof the
right circular cone that approximates the E-sail shape through the formula
β=π−2 arctan(bc) (43)
Since f0(x) = bc, from the first of Eqs. (40) the total force results
F=σ u N L
p1 + b2
c
ˆr(44)
where L= (xt−xr)p1 + b2
cis the tether length. Equation (44) is similar to Eq. (41), where a sort
of “effective” tether length (equal to L/p1 + b2
c) is considered in place of the actual length. Note that
L/p1 + b2
cis the tether length when projected on the E-sail nominal plane (x, y).
3.3. Parabolic shape
A simple way to take the tether curvature into account is to consider a parabolic shape. Each tether is
modelled as
f(x) = bpxrx
xr−12
with x∈[xr, xt] (45)
9
where the (constant) dimensionless coefficient bp>0 depends on the tether curvature. In this case
f0(x)=2bpx
xr−1(46)
and, bearing in mind the first of Eqs. (40), the total force becomes
F=σ u N xr
2bp
arcsinh 2bpxt
xr−1 ˆr(47)
where {xr, xt, bp}are related to the tether length Laccording to
L=xr
4bp
arcsinh 2bpxt
xr−1+bpxrxt
xr−1sxt
xr−12
+1
4b2
p
(48)
3.4. Logarithmic shape
An interesting case is obtained when the tether shape f(x) is described through a logarithmic function
of the distance x. Indeed, as will be shown in the next section, the tether equilibrium shape of a Sun-facing
E-sail under the action of the external forces just follows a logarithmic function provided the spin rate ωis
sufficiently large.
Therefore, let the shape function be
f(x) = blxtln x+xt
xr+xtwith x∈[xr, xt] (49)
from which
f0(x) = bl
1 + x
xt
(50)
where the dimensionless coefficient bl>0 is a given parameter. Substituting Eq. (50) into the first of
Eqs. (40), the resultant force vector becomes
F=σ u N xtq4 + b2
l−qb2
l+ (xr/xt+ 1)2ˆr(51)
where {xr, xt, bl}are related to the tether length Lthrough the equation
L=xtq4 + b2
l−blarcsinh bl
2−qb2
l+ (xr/xt+ 1)2+blarcsinh bl
xr/xt+ 1 (52)
The expression (51) is very useful from a practical viewpoint, as is now thoroughly discussed.
4. Tether equilibrium shape of a Sun-facing E-sail
The analytical, approximate, equilibrium shape of a generic tether of a Sun-facing E-sail can be obtained
using the approach discussed in Ref. [8]. Assuming a rotating E-sail, Toivanen and Janhunen [8] describe
the equilibrium tether shape with an integral equation, which is solved numerically. In particular, using an
analytical approximation of the tether shape, Toivanen and Janhunen [8] also obtain closed-form expressions
for both the thrust and torque arising from the solar wind momentum transfer to the E-sail. Their results
essentially state that the tethers form a cone near the spacecraft, while they are (substantially) flattened
around the tip region by the centrifugal force. Note that Toivanen and Janhunen [8] consider a mass at the
tether tip (that is, a mass that models the presence of a remote unit), whereas this work considers the tether
only, without any tip mass.
It will be shown now that the exact tether slope at the tip can be found analytically. An accurate
approximation of the tether equilibrium shape can also be obtained, using the model discussed in the last
10
section. To that end, enforcing the Sun-facing conditions αn= 0 and δn= 0 into Eqs. (16)–(20), the total
force dFkacting on the infinitesimal arc-length dskbecomes
dFk=ρ ω2xk
σku−f0
k
1+(f0
k)2ˆ
ik+1
1+(f0
k)2ˆ
kσkuq1+(f0
k)2dxk(53)
where ˆ
ik, given by Eq. (2), is the unit vector obtained from the projection of dFkon the E-sail nominal
plane (x, y).
Without loss of generality, the notation may be simplified by dropping the subscript kin the variables
{xk, f0
k, σk,ˆ
ik}of Eq. (53). Assume the generic tether to have no bending stiffness, so that only an internal
tension acts tangential to its neutral axis. In this case, according to Toivanen and Janhunen [8], the direction
of the vector tangent to the tether at the generic point Pof abscissa x∈[xr, xt] is parallel to the direction
of the integral of dFfrom xto xt(i.e., the integral of the total force from Pto the tether tip). Therefore,
from Eq. (53), the tether slope f0at point Pis the solution of the following integro-differential equation
f0(x) =
σ u Zxt
x
dy
p1+(f0)2
ρ ω2Zxt
x
yp1+(f0)2dy−σ u Zxt
x
f0dy
p1+(f0)2
(54)
where the numerator (denominator) in the right-hand side is the component along the z-axis (x-axis) of the
resultant force acting on the tether arc between Pand the tip, that is
Fx(x),ρ ω2Zxt
x
yp1+(f0)2dy−σ u Zxt
x
f0dy
p1+(f0)2(55)
Fz(x),σ u Zxt
x
dy
p1+(f0)2(56)
Introduce the dimensionless abscissa h,x/xt, with h∈[hr,1], where hr,xr/xt≥0 is the value at
the root section. Equation (54) can be conveniently rewritten as
f0(h) = Z1
h
dy
p1+(f0)2
KZ1
h
yp1+(f0)2dy−Z1
h
f0dy
p1+(f0)2
(57)
where K > 0 is a dimensionless “shaping parameter” defined as
K,ρ ω2xt
σ u (58)
which relates the tether equilibrium shape of a Sun-facing E-sail to the ratio of electric (σ u) to centrifugal
(ρ ω2xt) effects.
The tether slope at the tip, that is, the exact value of f0(h= 1) ,f0
tcan be obtained from Eq. (57)
using a limit procedure, viz.
f0
t= lim
h→1f0(h) = 1
K[1 + (f0
t)2]−f0
t
(59)
which can be rewritten as f0
t−1
K(f0
t)2+ 1= 0 (60)
11
whose only real solution is
f0
t=1
K≡σ u
ρ ω2xt
(61)
As expected, the tether slope at the tip sharply reduces as the E-sail spin rate increases. The variation of
f0
twith {xt, ω}, when σ= 9.3×10−13 kg/m/s, ρ= 10−5kg/m and u= 400 km/s, is shown in Fig. 6. In
particular, f0
t≤0.1 (or K≥10) when ω≥5 rph and xt≥5 km, which implies a tether slope at the tip less
than 6 deg. Having obtained the exact value of f0
t, it is now possible to calculate the function f0(x) (or f0(h)).
0 5 10 15 20
10-3
10-2
10-1
100
101
102
xt[km]
f
0
t
10
5
4
3
2
1
0.5
[rph]w
Figure 6: Tip slope f0
tas a function of the spin rate ωand the spin axis-tip distance xt, see Eq. (61).
To that end a recursive procedure is necessary, which, starting from the tether tip and backward proceeding
toward the root, numerically solves Eq. (57) for a given value of K. The results of such a procedure are
summarized in Fig. 7 for some values of the shaping parameter K. The figure shows that f0
t= 1/K, in
agreement with Eq. (61). Also note that in the ideal case hr= 0, which amounts neglecting the main body
width and assuming the tether to be attached to the z-axis, the tether slope at root is f0(0) '2f0
t≡2/K
when the shaping parameter is sufficiently large, that is, when K≥5. In that case 1/K ≤f0≤2/K , or
K2+ 1
K2≤1+(f0)2≤K2+ 4
K2(62)
which implies
1+(f0)2'1 (63)
The tether shape may be obtained by means of a numerical integration, and the results are summarized in
Fig. 8 assuming hr= 0. Notably, an accurate analytical approximation may also be obtained, as is discussed
in the next section.
4.1. Tether shape analytical approximation
An accurate analytical approximation of the tether shape can be obtained for a sufficiently large value
of the shaping parameter, for example when K≥5. In that case, substituting Eq. (63) into Eq. (57), the
result is
f0(h)'Z1
h
dy
KZ1
h
ydy−Z1
h
f0dy
=2 (1 −h)/K
1−h2−2[ft−f(h)]
K xt
(64)
12
0 0.2 0.4 0.6 0.8 1
10-2
10-1
100
101
h
f
0
100
50
40
30
20
10
5
4
3
2
1
K
Figure 7: Tether slope f0as a function of the dimensionless abscissa h=x/xtand the shaping parameter K, see Eq. (58).
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
h
fx/
t
100
50
40
30
20
10
5
K
Figure 8: Tether shape as a function of h=x/xtand Kobtained through numerical integration.
Since max{2 [ft−f(h)]/(K xt)} ' 0.11, see Fig. 8, the last relation may be further simplified as
f0(h)'2
K(1 + h)(65)
13
Notably, the approximation of Eq. (65) gives the exact value at tether tip, f0
t= 1/K, and also it captures
the approximate value at tether root, f0(0) = 2/K , in agreement with the estimate obtained in the last
section.
Figure 9 compares the analytic approximation given by Eq. (65) (dash line) with the numerical solution
(solid line) and shows that the two results are nearly coincident when K≥5. Accordingly, an accurate
0 0.2 0.4 0.6 0.8 1
10-2
10-1
100
101
h
f
0
100
50
40
30
20
10
5
4
3
2
1
K
Figure 9: Tether slope f0as a function of hand K: numerical (solid line) vs. analytical approximation (dash line).
analytical solution of the tether shape can be found from Eq. (65). Indeed, using a variable separation and
integrating both sides, it may be verified that
f(h) = 2xt
Kln 1 + h
1 + hrwith h∈[hr,1] (66)
or, using Eq. (58)
f(x) = 2σ u
ρ ω2ln x+xt
xr+xtwith x∈[xr, xt] (67)
The latter coincides with Eq. (49) when
bl=2σ u
ρ ω2xt≡2
K(68)
Equation (67) proofs the importance of a logarithmic shape for describing the equilibrium configuration of
a Sun-facing E-sail. Its actual accuracy is better appreciated with the aid of Fig. 10, which plots Eq. (66)
with hr= 0.01. The obtained results are nearly coincident with those reported in Fig. 8, which correspond
to a numerical integration of the actual tether slope.
14
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
h
fx/
t
100
50
40
30
20
10
5
K
0.3
Figure 10: Tether approximate shape as a function of the dimensionless abscissa h=x/xtand Kwhen hr= 0.01, see Eq. (66).
5. Tether root force
Due to the E-sail rotation, each tether experiences a tension force τwith a maximum value τr, which
occurs at the root section, that is, when x=xr(or h=hr). The value of τris obtained by imposing
the equilibrium condition of all forces acting on the tether at the root, that is τr=pF2
xr+F2
zr, where
Fxr
,Fx(xr) and Fzr
,Fz(xr), see Eqs. (55)-(56). The tension at the root section is therefore
τr=p1+(f0
r)2
f0
r
Fzr(69)
where
Fzr=σ u Zxt
xr
dy
p1+(f0)2(70)
and f0
r≡f0(xr) = Fzr/Fxris the tether slope at the root section. Equation (69) can be rewritten in a
dimensionless form as
τr
σ u xt
=p1+(f0
r)2
f0
rZ1
hr
dy
p1+(f0)2(71)
whose numerical solution is obtained, for a given value of K, using the function f0=f0(h) calculated through
the iterative procedure described in the last section. For example, assuming hr= 0, the dimensionless value
of τris shown in Fig. 11 as a function of K. Note that τr/(σ u xt) has a nearly linear variation with K, with
an angular coefficient equal to 1/2. This same result will now be confirmed by an analytical approximation.
5.1. Analytical approximation of τr
Assuming a shaping parameter K≥5, the tether slope is well approximated by Eq. (65), therefore
Z1
hr
dy
p1+(f0)2=2√K2+ 1 −qK2(1 + hr)2+ 4
K(72)
15
0 20 40 60 80 100
0
5
10
15
20
25
30
35
40
45
50
K
=r/(<ux
t)
Figure 11: Dimensionless tension at tether root as a function of Kwhen hr= 0, see Eq. (66).
Substituting this last relation into Eq. (71) in which f0
r'2/K/(1 + hr), the result is
τr
σ u xt
=qK2(1 + hr)2+ 4
2K2pK2+ 1 −qK2(1 + hr)2+ 4(73)
In the limit as hr→0, the last relation may be further simplified taking into account that K21. The
compact and elegant solution is
τr
σ u xt
=K
2≡ρ ω2xt
2σ u (74)
which is in agreement with the plot shown in Fig. 11. This last relation allows the value of τrto be related
with the tether length L, when its equilibrium shape is a logarithmic function. In fact, assuming xr= 0 and
substituting Eq. (74) into Eq. (68) and then into Eq. (52), it may be verified that
L
xt
=s4 + σ u xt
τr2
−s1 + σ u xt
τr2
+σ u xt
τrarcsinh σ u xt
τr−arcsinh σ u xt
2τr (75)
which is drawn in Fig. 12 when K∈[5,100]. The tension at the root can be expressed as a function of
the pair of design parameters {ω, L}by combining Eqs. (74) and (75). Its maximum value cannot exceed
the tether yield strength, which is about 0.1275 N for a µm-diameter aluminum tether, with a linear mass
density ρapproximately equal to 10 grams per kilometer [23].
For example, assuming V= 20 kV [23], Fig. 13 shows how the tension τrvaries with the tether length
Land spin rate ωwhen xr= 0. Note that each level curve breaks down when the yield strength τmax is
achieved (i.e., when τr=τmax). According to Fig. 13, the tension τrroughly exhibits a parabolic behaviour
with the spacecraft spin rate ωfor a given value of L. In particular, the figure shows that the maximum
allowable spin rate for a baseline tether length of 20km is about ω= 4.57 rph, whereas the value of xtis
19.983 km. In this case, K'34 and the dimensionless root tether is τmax /(σ u xt)'17, in agreement with
the numerical results shown in Fig. 11.
6. Conclusions
The thrust and torque vectors provided by a spinning electric solar wind sail of given shape have been
calculated in a fully analytical form as a function of the spacecraft attitude. This analysis is based on
the hypothesis that each tether is deformed by the external forces such that its shape belongs to a plane
16
1 1.01 1.02 1.03 1.04 1.05
0
5
10
15
20
25
30
35
40
45
50
Lx/t
=
r
/(<ux
t
)
5K<
Figure 12: Dimensionless tether tension at root as a function of the dimensionless tether length when hr= 0.
w[rph]
=
r
[N]
0 2 4 6 8 10
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
max
t
tether failure
5K<
3
5
7
10
15
20
[km]L
Figure 13: Root tension τras a function of Land ωwhen ρ= 10 g/km, xr= 0, and τmax = 0.1275 N. Data adapted from
Ref. [23].
passing through the E-sail spin axis. The general expressions of the thrust and torque vectors have been
then specialized to the case of a Sun-facing sail, with a tether arrangement assumed to be axially symmetric
with respect to the spacecraft spin axis.
The results have been applied to some noteworthy tether shapes, including the flat and the logarithmic
cases. In particular, the equilibrium shape of any tether, when the electric sail axis is parallel to the Sun-
spacecraft direction, is close to a logarithmic arc, in agreement with the numerical results of the recent
literature. The discussed mathematical model allows the geometry of an axially-symmetric Sun-facing sail
to be related to the yield strength of the cable. The problem is that the generation of a high thrust level
requires the tethers to be maintained stretched, but the spin rate must account for the tether structural
load resulting from the centrifugal force.
A natural extension of this work consists in the analysis of the effects of a pitch angle different from zero,
17
that is, when the sail produces an off-axis thrust. The latter assumption breaks the axial-symmetry condition
and, therefore, requires a different approach to analyze the coupling effects between the sail geometry and
the spacecraft attitude. In particular, the actual tether shape can only be checked by simulation through a
finite element analysis, which is left to future analysis.
Acknowledgement
The authors gratefully acknowledge the constructive comments made by the anonymous reviewers that
helped improve the paper.
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