A Proposed Two-Stage Two-Tether
Scientific Mission
at
Jupiter
Mario Charro, Juan
R.
Sanmartin, Claudio Bombardelli, Antonio Sanchez-Torres,
Enrico
C.
Lorenzini, Henry
B.
Garrett,
and
Robin W. Evans
Abstract—A two-stage mission
to
place
a
spacecraft
(SC)
below
the Jovian radiation belts, using
a
spinning bare tether with
plasma contactors
at
both ends
to
provide propulsion
and
power,
is proposed. Capture
by
Lorentz drag
on the
tether,
at the pe-
riapsis
of a
barely hyperbolic equatorial orbit,
is
followed
by a
sequence of orbits
at
near-constant periapsis, drag finally bringing
the
SC
down
to a
circular orbit below
the
halo ring. Although
increasing both tether heating
and
bowing, retrograde motion
can
substantially reduce accumulated dose
as
compared with prograde
motion,
at
equal tether-to-SC mass ratio.
In the
second stage,
the tether
is cut to a
segment
one
order
of
magnitude smaller,
with
a
single plasma contactor, making
the SC to
slowly spiral
inward over several months while generating large onboard power,
which would allow multiple scientific applications, including in situ
study of Jovian grains, auroral sounding of upper atmosphere,
and
space-
and
time-resolved observations
of
surface
and
subsurface.
I.
INTRODUCTION
P
OWER
AND
propulsion needs have been critical issues
in missions
to the
outer planets.
In all
past missions
to Jupiter
in
particular, instruments onboard were powered
by radioisotope thermal
(or
power) generators (RTGs), which
present
a
number
of
problems. Furthermore,
a
recent report
from
its
National Research Council warned that
the
U.S.
was
running
out of
RTG
fuel (Pu-238); production was stopped over
20 years
ago,
although
the
National Aeronautics
and
Space
Administration (NASA) continues
to
acquire about
5 kg
each
year from Russia
[1], The
pending Juno mission
to
Jupiter
chose
to use
solar power, which
is
possible
due to its
limited
mission duration
of one
year,
a
polar Jovian orbit that keeps
its solar panels
in
constant sunlight
and
limits
its
radiation
exposure,
and an
energy-efficient operation plan. Also, the two-
spacecraft
(SC)
Europa Jupiter System Mission, which
was
tentatively planned
by
NASA
and ESA for
2020,
was to
have
used
540 W
from RTGs
in
NASA's Jupiter Europa Orbiter
(JEO)
and 51-m
2
solar panels
in
ESA's Jupiter Ganymede
Orbiter (JGO).
In turn, capture into
and
touring
in the
deep Jovian gravi-
tational well typically require
a
very high wet-mass fraction
if chemical rockets
(in
addition
to
gravity-assist operations)
are used
for
propulsion,
as was
always
the
case
in the
past.
In particular,
the JEO
system mass budget required 2646
kg
of propellant
and 1490 kg of
flight system mass
and
launch
vehicle adapter, with
973 kg
left
for
contingency
and
margin,
for
a
launch mass
of
5040
kg.
Corresponding values
for the
JGO system
are
2562,
1147, 653, and
4362
kg,
which
is
only
1.7
times propellant mass, respectively. Both
JEO and JGO
were
to
have used ballistic trajectories with gravity assists from
Venus and Earth
[2].
Beyond
a
range
of
other applications
[3],
bare electrody-
namic tethers
can
provide both power
and
propulsion, with
just tether hardware accounting
for
tether subsystem mass.
Basically,
a
planetary magnetic field induces a motional electric
field
E
m
in the
orbiting-tether frame
and
exerts
a
Lorentz
force
on the
current that
E
m
drives through
the
tether.
If
used
at
Jupiter, tethers with
a
small tether-hardware/SC mass
ratio
m
t
/M
sc
have been shown able
to
perform Jovian orbit
insertion, followed
by a
moon tour with near-zero
wet
mass,
while providing power throughout
the
entire operation
[4], [5].
A total
of 40
flybys
of
Ganymede, Europa,
and Io (as
against
25 moon flybys planned
for
JEO) could
be
carried
out
before
some limiting radiation dose
did
accumulate.
Radiation dose
is the
main constraint
on
tether
use at
Jupiter
because operation
is
quite requiring ambient conditions,
i.e.,
plasma density
and
magnetic field,
so
orbits must reach near
Jupiter
(the Io
torus density
is
similarly high,
but the
magnetic
field
is
weaker).
For
missions such that orbits skip
the
Jovian
radiation belts, anyway, tethers would
be
greatly useful.
The
Juno case
is a
particular example.
It has
been recently shown
that
a
moderately light tether might provide
the
power that
its
SC requires
[6].
In this paper,
we
consider
a
two-stage two-tether mission.
Following capture
as
described
in
[4],
the
SC tether would have
its apoapsis progressively lowered
to
finally reach
a
circular
or-
bit
at the
periapsis
of
the capture orbit, about
1.3-1.4
times
the
Jovian radius
Rj,
skipping moon flybys
as
considered
in [5] to
reduce dose accumulation.
In the
second stage,
a
short segment
of
the
original tether makes
its SC to
slowly spiral inward,
in
a controlled manner, keeping below
the
belts throughout while
generating power
on
board
for
science applications,
for
which
®
B
\o)
Fig. 1. Relative positions of unit vectors for motional electric field and
spinning tether; A and C are the anodic and cathodic ends, respectively, and
WE
is the unit vector along the motional field E
m
, which is perpendicular to
the relative velocity v' (ag different from
TT/2).
For high tether spin
rate,
angle
f would range over 360° at near-constant orbit position.
the proximity to Jupiter, under the evolving in situ conditions
surrounding the SC, offers a world of opportunities.
Section II deals with evolution through highly elliptical
orbits to an initial circular equatorial orbit below the belts.
Accumulated dose and tether heating and bowing issues are dis-
cussed in Section III, comparing retrograde and prograde orbit
performances. In Section IV, evolution of orbits below the belts,
keeping them near circular throughout, and the power generated
on board are considered. Scientific applications are discussed in
Section V. Conclusions are presented in Section VI.
II. REACHING DOWN TO A LOW JOVIAN ORBIT
Lowering apojove after capture typically requires a large
number of orbits. As the eccentricity decrement per orbit will
be small, calculations are carried out here as if eccentricity e,
although different from unity, is kept constant over each orbit.
Also,
perijove radius r
p
is taken as constant throughout because
drag becomes rapidly small away from perijove. The energy per
unit mass and the eccentricity decrement for each elliptical orbit
can then be written as
6=
•
2r„
(1-
2r
p
2r
p
W
d
Ae =—-Ae= —-——
MJ MJ ™SC
(1)
where
W
dTag
is the work per orbit by the Lorentz drag F
m ag
.
Drag power reads
Wdrag = v
•
F
mag
= vu
t
•
(-I
av
u
x LBk)
= -
LBvI
av
sm(<fi
+ a
E
) (2)
where v is the SC velocity, I
av
is the tether current averaged
over its length L, and B is the Jovian magnetic field. Fig. 1
shows the directions of unit vectors and angles. Note that, since
orbits are equatorial, the no-tilt magnetic field (assumed here)
is perpendicular to both SC and corotating plasma velocities.
Both hollow-cathode contact impedance and radiation im-
pedance for current closure in the Jovian plasma are negligible
[4].
With ohmic effects being typically weak and assuming
relatively low use of power on board, the tether, if bare of
insulation [7], collects electrons over near its entire length.
Since low density N
e
and high temperature T
e
imply a large
Debye length in the Jovian plasma, current I
av
is given by the
orbital-motion-limited law
2 2wL
Ar
2eE
t
L
J-3N =
eN
P
i
5 TT V
Tllp
(3)
where w is the width of the bare tether. The tether is shaped as
a thin tape because this geometry has the lowest cross-sectional
area for a given collecting perimeter. Also, e and m
e
are the
electron charge and mass, respectively, E
t
is the motional-field
vector projection along the tether (Fig. 1)
E
t
= u
•
E
m
= E
m
cos(fi, E
m
= v" x B, (E
m
= v'B)
(4)
and v" = vu
t
± Qjru
9
is the SC-to-plasma relative velocity,
with Q j as the spin rate of Jupiter. Minus/plus signs correspond
to prograde/retrograde orbits, respectively.
The tether would spin in the equatorial orbital plane, perpen-
dicular to the magnetic field, with hollow cathodes at both ends
taking active turns as each end becomes cathodic. For a high-
enough spin rate uj
t
, centrifugal forces will keep it straight, and
the ^-averaged drag power at near-constant orbit position takes
the form
^drag |)
=C*LB^eN
t
2eLB vv' sin
O.E
mV4
(z/2)
C* = (cos
3/2
tp)
« 0.556.
(5)
The tethered system can be spun up by using chemical
thrusters at the tip masses with angular momentum staying
constant once the final speed is attained. After assuming the
parameters in [5] with a final spin period of 12 min, a tether
length of 50 km, and an overall system mass of either 600 or
1000 kg, spinning can be achieved with about 40 or 77 kg of
hydrazine, respectively. As the spin-up time depends on thrust
level, 21 and 39 min will be required for the two options assum-
ing a thrust level of 40 N. For previous studies on alternative
spinning techniques of tethered systems, see [8] and [9].
Using the condition E
m
_L v' and conservation of angular
momentum rv -u
e
= r
p
v
p
, along with eccentricity during each
orbit, yields
v
2
+
Q
2
T
r
2
± 2i\jr
p
v
p
'-J
(6)
vv' smotE =vu
t
•
(vu
t
± iljrug) = v ± iljr
p
v
p
(7)
with v
p
as the SC velocity at perijove and v
2
given by the
vis viva equation
v
2
= ^(^-l + e
(8)
Equations (5)-(7) can be used to show that, as expected,
retrograde orbits produce greater drag than prograde orbits for
any choice of tether and orbital parameters.
To obtain drag work over an entire orbit as required in (1),
we use (6)-(8) in integrating (5) along the drag part of the orbit
(9)
r
\ l^dragl) dr
\W
dlag
\=2j
X
dr/dt
The upper limit is the smaller of the apojove radius r
a
=
r
p
(l + e)/(l - e) and the radius where the tangential rela-
tive velocity vanishes, given by condition v
2
± £ljr
p
v
p
= 0;
clearly, r
u
= r
a
for all retrograde orbits. As previously noticed,
the perijove neighborhood makes a dominant contribution to the
integral. We use a no-tilt no-offset dipole model of the magnetic
field, normalized with its value at the stationary orbit radius
=
(^m)
2^/3
2.24 R;
B/B
s
= B =
a
3
Jr
3
,
B
s
« 0.38 gauss.
(10)
The field at the Jovian surface, i.e., 2.24
3
x 0.38 G, is over one
order of magnitude greater than that at the Earth's surface. For
N
e
, the Divine-Garrett model of the thermal electron density in
the plasmasphere at the equator yields [10]
N
e
/N
s
=N
e
= exp
N
s
« 1.44 x 10
2
cm
r a
s
2
„™-3
r
0
« 7.68i?j.
(11)
Angular momentum conservation and the orbit equation 1
e cos9 = (1 + e)r
p
/r are used to determine dr/dt
dr
~di
Vr,
1
1
1
(12)
Finally, using (5) and (12) in (9), we find (13) and (14),
own a
eB
s
/m
e
is the electron gyro frequency at a
s
shown at the bottom of the page, where r = r/r
p
and Q
III.
TETHER PERFORMANCE
IN
PROGRADE VERSUS
RETROGRADE MOTION
The decrement Ae at fixed e is then given by
16y/2^
m
t
m
e
N
s
fn
es
\
3/2
L
3
/
2
r
p
~
Ae=
E—
C
lvr o~ 37#Wdra
g
(r>,e)
5TT
M
SC
Pt \ttj J haj
(15)
where m
t
, h, and p
t
are the tape mass, thickness, and density,
respectively. Fig. 2 shows the Ae versus e, at r
p
= l.ZRj, for
retrograde and prograde motions, for an aluminum tether. For
0.9
0,7
I
°'
6
x
I
0.4
0.3
O
<n
0,2
0.1
0
\
\
\
\
\
\
\
\
\
\
r
p
/Rj=1.3
RETROGRADE (Al)
PROGRADE (Al)
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7
e
Fig. 2.
(M
S
c/mt)
•
|
Ae| • (50 km/L)
3
/
2
versus
e for r
v
= 1.3 Rj.
Ret-
rograde
and
prograde
cases
are
shown
in
solid
and
broken
lines,
respectively.
equal tether parameters and SC-to-tether mass ratio, retrograde
motion would need fewer orbits to bring the SC down to a circu-
lar orbit. For either motion, performance increases with the ratio
L
3
/
2
/h,
independently of tape width w. Tether dimensions L =
50 km, w = 3 cm, and h = 0.05 mm, for example, yield a mass
m
t
= 202.5 kg, corresponding to Msc = 600 and 1000 kg for
Msc/mt values of about three and Ave, respectively.
The number of perijove passes is a metric for radiation dose.
We use the so-called Galileo interim radiation electron model
in the calculations [11]. Fig. 3 shows the dose increment per
orbit versus eccentricity for four perijove values, considering
an aluminum spherical shield shell of 10-mm thickness for
all 47r steradians. Note that radiation dose first increases as e
is increased and then decreases. Fig. 4 uses calculations for
Figs.
2 and 3 to determine how dose accumulates as apojove
is lowered, for r
p
= 1.3 Rj. Fig. 5 shows the radiation dose
accumulated, when reaching the circular orbit, versus perijove
radius; performance in prograde orbits is worst at the highest
r
p
. Dose can be over 60% higher for prograde motion.
The effect of type of motion on both tether heating and
bowing is opposite, however [4]. In the energy balance, which
is a local one, ohmic effects and solar heating may be ignored.
Furthermore, for a conservative estimate, the tape may be taken
as in quasi-steady equilibrium; radiation loss then balances
heating from the impact of collected electrons. Maximum
5
1 = ^C*m
e
N
s
n
3
J
s
2
x
[^wL
5
/
2
W
dlag
(r
p
,e)
7"
^rNB
3
/
2
dr [{2/7) + e - 1 ± (1 + e)Qjr
p
/jy
p
] Qjr
p
/jy
p
VTT^
~ J
v
/f^I
v
/^F [(2/7) + e - 1 ± 2(1 + e)Sljr
p
/
Vp
+ (1 + e)7
2
(Q
J
r
p
/^
p
)
2
}
\W
(
w,
(13)
(14)
Dose/orbit behind 10 mm Al Spherical Shell Shielding
r
p
/Rj
= 1.2 , 1.3 , 1.4 and 1.5
l.E+06-
W
l.E+05
-
|
1 .
g
l.E+04
*
l.E+03-
-1.2
-1.3
-1.4
-1.5
0.4 0.6
eccentricity
Fig. 3. Dose per orbit versus eccentricity, with an aluminum spherical shell
shielding of 10 mm. Four values of perijove radius are considered.
1x10^
9x10
2
8x10
2
7x10
2
6x10
2
5x10
2
rg.
4x10
2
2
4.
3x10
2
a
<n
Q 2x10
2
1x10
9x10
8x10
7x10
6x10
5x10
4x10
^X
s
\
\
\
\
(Msc/m,) x (h/0.05 mm) x
(SOkm/Lf
2
= 5
RETROGRADE (Al)
PROGRADE (Al)
\
, \
\\
\ \
\
\
\
\
s
\
bs
"
N
5 6 7 8910 20
i-
a
/r
p
(r
p
= 1.3Rj)
30 40 50 607080100
Fig. 4. Radiation dose accumulation from capture to low Jovian orbit for
r
p
/Rj = 1.3 and both retrograde and prograde cases.
temperature occurs around perijove during capture, at each tape
end when anodic, and scales with L
3
/
8
N*
rp^
i
v
s"
£,
e / ^
ly
s
J
-
J
s
J
-
J
\ ~2
,B
S
L\
3/2
2^e
t
na
B
\ m
e
) '
M
x exp
(2.72f^
3
- 3.43^ (f
M
± I)
3
/
2
(16)
where e
t
and o
B
are the tether emissivity and Stefan-
Boltzmann constant, respectively, and v
s
= y
/
2Lij/a
s
is the
parabolic velocity at a
s
, and we introduced r
M
= \/2{a
s
/
r
p
)
3
/
2
[4]. Fig. 6 shows the maximum temperature versus
perijove radius for retrograde and prograde orbits, for e
t
= 0.8.
A simple and conservative estimate of bowing considers the
statics of a rope loaded laterally by the Lorentz force dF/ds =
BI(s) and supported at the two ends [4]
ay
ds
2
BI{s)
T
LBI
av
2600
2400
2200
2000
1800
1600
Jj 1400
» 1200
o
Q
1000
800
600
400
ZOO
0
1,2
/
(Msc/m,) x (h/0.05 mm) x
(SOkm/Lf
2
= 3
RETROGRADE (Al)
PROGRADE (Al)
^^
/
/
/
/
/
/
/
/
/
1,25 1,275 1,3 1,325
Fig. 5. Accumulated radiation dose down to low Jovian orbit.
2000
* 800
IS
700
600
500
400
~^~~- -
^^
--^
RETROGRADE (Al)
PROGRADE (Al)
"~~~
-
~-~-.
~"~"~—"
1,2 1,225 1,25 1,275 1,3 1,325 1,35 1,375 1,4
r
p
/R
J
Fig. 6. Maximum temperature versus perijove radius for both retrograde and
prograde orbits.
where y is the lateral rope deflection assumed small compared
to L, T is the tensile force, and s measures the distance along
the tether. For all parameters equal, motional field and, thus,
Lorentz force and bowing are greater for retrograde cases.
Maximum bowing occurs where Lorentz force is the highest, at
a distance 0.56L from the anodic end [3] for
ip
= 0, at perijove
during capture
%^ = 0.0W^N
s
eB
s
L
Li 1
ev
s
B
s
L\
1/2
^8/
3
L
T
(17)
x exp
(2.72f^
3
- 3.43^ (r
M
± 1)
V2
•
(18)
Fig. 7 shows the maximum bowing versus r
v
, as both pro-
portional to L
5
/
2
/h and inversely proportional to tensile stress
T/(wh). With gravity gradient being negligible,
T/(wh)
scales
!
0.2
1 0,1
5 0,09
jo 0,08
"g.
0,07
£ 0,06
§ 0,05
^ 0,04
E0,03
0,02
k
RETROGRADE (Al)
PROGRADE (Al)
\
•v
\
^s
\
\
\
\
\
^
1,225 1,25
1,275 1,3 1,325
r
pf
R
J
1,35 1,375 1,4
Fig. 7.
Maximum bowing
versus
perijove
radius.
with ptUJ
2
L
2
for fixed Msc/^t- This result is unaffected by
the type of motion, making the maximum bowing scale with
^/L/(p
t
ou
2
h).
A high spin reduces bowing but could result in
too high a tensile stress.
IV. ORBIT EVOLUTION BELOW THE RADIATION BELTS
In this second stage, at the high N
e
and B values prevalent
below the belts, tether lengths on the order of 50 km, as
required for capture and lowering to a low Jovian orbit, would
result in a too rapid deorbiting that is inconvenient for science.
At this point, the tether is cut, leaving a short segment with
just one hollow cathode, to drag the SC down. The cut will
produce a recoil of the tether segment that eventually reaches
an extended configuration because of the centrifugal forces due
to the system spin. The loss of one hollow cathode results in
I
av
being negligible half the time and a reduced ^-average,
(cos
3
/
2
if) « 0.278. For slow inward spiraling, the SC will be
at any time in a near-circular equatorial orbit, just characterized
by its radius a(t).
During each half spin period with the remaining hollow
cathode at the cathodic end, the drag power is
W
dr
V'F^
- LB (a) is (a)
1
av
(a)
cos
cp.
(19)
Since orbit and plasma velocities are now parallel, E
m
is
aligned with the local vertical, making
OLE
=
TT/2
in (2). The
tether would also be generating power, for science applications,
at some useful load of impedance Z\
oa
& characterized by the
point where the tether bias vanishes, i.e., by the length (X(C <
1) of the segment at positive bias (Fig. 8). The bare tether
would collect electrons over that anodic segment, with ( = 1
corresponding to no load. Ion current to the cathodic segment
is negligible. We then have [3], [7]
5-2<^
22wL 2eE
m
(a)Lcoscp
3/2
4v= 7 eJy
e
{a)\ s
f
5 7T V vn
P
3
(20)
hollow
cathode
Fig. 8.
Schematics
of the
short bare
tether
in
generator
mode
with
negligible
ohmic
losses
(straight
tether-potential
line),
during each
half
spin
period
with
the
hollow
cathode
at the
cathodic
end,
throughout
the
second-stage
deorbiting.
%
•^
10-i
5
1
\\
0,5^
0,H
0,05^
0,01-1
«v*
^ . T ,,
tape
width
w =3 cm
£=0.6
* ^ L=3 km
^**s^
* *s>
^>s^ * ***
^^s^^
* ^
^^s„^
* ***
^^ ^
,
*
,
*N*
I
^^ * «v»
^ x^ L=2 kmS^ *"** • ^
L=l km ^ "^^-v*,^
^v ^^
^^
^^^
^^
^^^
*
1 • 1 • 1 • 1
1,1
1,2 1,3
a/Rj
1,4
Fig. 9.
Maximum
load
power
generated
by a
bare
tether
of
given
width
and
lengths
versus
radius
of its
circular
prograde
orbit.
Since eccentricity vanishes throughout, orbit dynamics is just
described by the following equation:
l^drag(a)|
d fnjM
S
c
(21)
The full magnetic power, involving both drag power on the
SC and back power on the corotating and magnetized Jovian
plasma
(—F
mag
)
•
z/
pl
, reads
"mag
-F
mag ' ^
COS (f =
Wdrag
X v'v. (22)
As regards load power, one readily finds [3], [7]
^ioad/|Wmag| = efficiency rj = - _ ^ . (23)
Load power, which is proportional to (1
—
C)C
3
^
2
> is maxi-
mum at ( = 0.6, yielding
r/
max
= 10/19. We then find
is'
d ffijMsc
(24)
which is depicted for prograde motion in Fig. 9, where
(19)—(21) were used. Equation (24) readily yields the energy