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A
T
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X class copernicus.cls.
Date: 3 September 2014
Simulation study of the plasma brake effect
P. Janhunen
Finnish Meteorological Institute, POB-503, FI-00101, Helsinki, Finland
Correspondence to: Pekka Janhunen
(pekka.janhunen@fmi.fi)
Abstract. The plasma brake is a thin negatively biased tether
which has been proposed as an efficient concept for deor-
biting satellites and debris objects from low Earth orbit. We
simulate the interaction with the ionospheric plasma ram
flow with the plasma brake tether by a high performance
electrostatic particle in cell code to evaluate the thrust. The
tether is assumed to be perpendicular to the flow. We per-
form runs for different tether voltage, magnetic field orienta-
tion and plasma ion mass. We show that a simple analytical
thrust formula reproduces most of the simulation results well.
The interaction with the tether and the plasma flow is lami-
nar (i.e., smooth and not turbulent) when the magnetic field is
perpendicular to the tether and the flow. If the magnetic field
is parallel to the tether, the behaviour is unstable and thrust
is reduced by a modest factor. The case when the magnetic
field is aligned with the flow can also be unstable, but does
not result in notable thrust reduction. We also fix an error in
an earlier reference. According to the simulations, the pre-
dicted thrust of the plasma brake is large enough to make the
method promising for low Earth orbit (LEO) satellite deor-
biting. As a numerical example we estimate that a 5 km long
plasma brake tether weighing 0.055 kg could produce 0.43
mN breaking force which is enough to reduce the orbital al-
titude of a 260 kg object mass by 100 km during one year.
1 Introduction
The plasma brake (Janhunen, 2009, 2010) is an efficient
propellantless concept for deorbiting low Earth orbit (LEO)
satellites. The plasma brake is a very thin negatively charged
tether which, when charged, causes a braking force by cre-
ating enhanced Coulomb drag with ambient ionospheric
plasma ram flow. The plasma brake tether is somewhat simi-
lar to the more well known electrodynamic tether (Sanmartin
et al., 1993), but is much thinner and uses electrostatic rather
than magnetic forces.
Because the plasma brake 4-wire ultrasonically bonded
tether is thin, it is lightweight, 11 grams per kilometre
(Sepp¨
anen et al., 2013) and does not form an appreciable
threat to other satellites even in case of tether breakage. A
broken piece of plasma brake tether deorbits itself quickly
because of electromagnetic forces and neutral drag, and even
if the tether piece would collide with a satellite, the 25-50
µm wires draw only ∼0.1mm deep scratches on its surface.
Said electromagnetic forces are a passive Coulomb drag ef-
fect and to a lesser extent the passive electrodynamic tether
effect. The broken tether piece experiences an orbital mo-
tion induced natural electric field, and whenever this field
has a component along the tether, one end of the tether gets
charged negatively and the other one positively with respect
to the local plasma, causing electrostatic Coulomb drag and
electromagnetic Lorentz drag. Although the passive electro-
magnetic drag effects are weaker than the active ones, the
very low specific mass of the tether (∼0.01 kg/km) makes
passive deorbiting of a broken tether piece fast, typically.
To maintain a tether at positive voltage in the solar wind
requires an electron gun (Janhunen et al., 2010, 2013) that
pumps out negatively charged particles from the system and
thereby cancels the tether’s gathered plasma current. Like-
wise, to maintain a tether at negative voltage can be ac-
complished with a positive ion gun. However, in LEO the
maintenance of the plasma brake tether’s negative voltage
in most cases does not require an ion gun since the satel-
lite’s conducting body can be used as the current balancing
electron gathering surface (Janhunen, 2010). If the satellite’s
grounded and conducting surface area is insufficiently small
for this purpose, a relatively short positively biased tether
made of similar material than the main tether could be used
for gathering the required balancing electron current.
The topic of the paper is to use a realistic electrostatic par-
ticle in cell (PIC) simulation to model the interaction be-
tween the negatively biased plasma brake tether with the
surrounding ionospheric plasma ram flow which moves at a
arXiv:1401.5968v2 [astro-ph.IM] 29 Aug 2014
2 P. Janhunen: Plasma brake PIC simulation
typical satellite orbital speed of 7.5 km/s. We run the code
with different voltages, different magnetic field orientations
and different ion species and determine the thrust per tether
length in each case. We will also present a simple analytical
formula for plasma brake thrust which reproduces the simu-
lation results.
2 Simulation code
We use a two-dimensional explicit electrostatic kinetic PIC
simulation model (Birdsall and Langdon, 1991). The func-
tionality of the code is similar to what we have used earlier
to model the positively biased E-sail tether (Janhunen and
Sandroos, 2007), but the code is vectorised with AgnerVector
library (Fog, 2013) and parallelised with the standard Mes-
sage Passing Interface (MPI) and OpenMP tools. We run the
code with real electron and ion masses.
We simulate a negatively charged tether in plasma flow
which is caused by the satellite’s orbital motion through the
ionosphere. We use a Cartesian coordinate system where
Xis antiparallel with the tether-perpendicular component
of the flow, Zis parallel to the tether and Ycompletes a
right-handed coordinate system. For convenience we take the
plasma flow to be perpendicular to the tether, which implies
no loss in generality. The simulation is two-dimensional in
the tether-perpendicular plane. A constant external magnetic
field is employed in some runs. By default the plasma is cold
oxygen plasma, temperature 0.1 eV, and it flows at 7.5 km/s
which corresponds to 4.7 eV bulk flow ion energy. The tether
bias voltage is typically a few hundred volts negative. The pa-
rameters of all reported runs are given in Table 1. The simu-
lations are initialised from vacuum and the plasma flow starts
entering the box at t= 0.
Above ∼500 km where the plasma brake is relevant
(at lower altitude the neutral drag typically brings satellites
down rapidly enough), ionospheric plasma consists mainly
of O+(16 amu), N+(14 amu), He+(4 amu) and H+(1 amu)
ions. The main species are O+and H+; the minor species N+
can be summed to O+because the masses are approximately
similar. The abundance of O+decreases with altitude. At all
altitudes, solar activity tends to increase the oxygen abun-
dance. For the plasma brake, the most relevant environment
is O+plasma because proton plasma produces order of mag-
nitude smaller thrust and helium is usually not a dominant
species. For this reason, the main emphasis in this paper is
on O+plasma.
Runs reported in this paper used 16 nodes on a Cray XC30.
The XC30 consists of compute nodes, each node has 2 pro-
cessor chips and each chip has 10 2.6 GHz execution cores.
The runs reported here used 16 XC30 nodes and achieved
4.4·109particle propagations per second which corresponds
to ∼1Tflops single precision floating point performance.
3 Results
3.1 Oxygen plasma without magnetic field
We first run the code with oxygen plasma and without mag-
netic field for different values of the tether potential Vw. The
runs are detailed in Table 2. The vacuum potential of the
tether (the voltage difference between the tether and the sim-
ulation box boundary if there is no plasma inside the box)
is an input parameter to the simulation. After the simulation,
the true potential of the tether is calculated by evaluating the
plasma potential at the tether and adding it to the vacuum
potential. For this reason, the tether potential values Vware
not round numbers. We label the runs partly by their vacuum
potential values.
Figure 1 shows the simulated thrust (determined from par-
ticle momentum balance averaged over the last 1/3 of the run)
with open circles. Coloured marks in Fig. 1 are results from
some runs containing a magnetic field and they are described
in later subsections below. The solid line is the following
analytical formula, inspired by our earlier work (Janhunen,
2009, equation 20):
dF
dz = 3.864 ×Pdynso˜
V
eno
exp−Vi/˜
V(1)
where Pdyn =minov2
ois the dynamic pressure, miis the
ion mass (mi= 16 amu for oxygen plasma here), vois the
plasma flow speed relative to spacecraft (assumed to be per-
pendicular to the tether or else vodenotes only the perpen-
dicular component),
˜
V=Vw
ln(λeff
D/r∗
w),(2)
r∗
wis the tether’s effective electric radius (Janhunen and San-
droos, 2007, appendix A), λeff
D=poVw/(eno)is the ef-
fective Debye length and Vi= (1/2)miv2
o/e is the bulk ion
flow energy in voltage units. The effective electric radius is
approximately given by r∗
w=√brwwhere rwis the tether
wire radius, typically 12.5-25 µm, and bis the tether width,
typically 2 cm (a rough value of bis sufficient to know be-
cause r∗
wenters into Eq. (1) only logarithmically). In this
paper we use the value r∗
w= 1 mm. The numerical coeffi-
cient (3.864) in front of the expression has been selected to
give a good fit to the present simulation results. This value is
about 2.25 times larger than the value 1.72 used in our pre-
vious work (Janhunen, 2009). We think that this difference
may arise because in the earlier work we used typical solar
wind parameters so that the ratio Vw/Viwas about 10 while
this ratio is about 100 in the present ionospheric case so that
the works explored different regions of the parameter space.
In the ionospheric case, many ions are deflected backward
by the potential well around the tether which increases the
thrust markedly in comparison to the solar wind case where
P. Janhunen: Plasma brake PIC simulation 3
the much larger bulk flow speed causes the ions to deflect
more modestly.
Figure 2 shows the time history of thrust during the V400
(Baseline) run, computed by two complementary methods:
the direct Coulomb force acting on the tether (by evaluating
the numerical gradient of the plasma potential at the tether
position) and the total momentum Xcomponent lost by par-
ticles between entering and leaving the simulation box. Fig-
ure 3 shows the two-dimensional instantaneous electron and
ion density at the final state of the Baseline run.
Increasing the tether voltage makes the ion sheath larger.
The highest voltage runs (marked with “g” in Table 2) were
performed with extended box size of 768×768 to accom-
modate the larger sheath. As a consistency check, run V500
was performed with both grid sizes. The difference in the
determined thrust was minimal (Table 2, compare V500 and
V500g).
The quasi-monochromatic oscillation seen in Fig. 2 occurs
in runs V300 and higher. In lower voltage runs the oscillation
is absent. When present, the oscillation neither increases nor
decreases with time.
3.2 Effect of magnetic field
We performed three more runs with the same parameters as
V400 (Baseline), but with each of the magnetic field compo-
nents in turn set to 30000 nT, a typical LEO field strength.
The results are shown in Figs. 4-6. It is seen that in runs
Bx and Bz, the oscillation which was already present in the
Baseline run is now increasing in amplitude. In contrast, in
the By run the oscillation is damped and is going to disap-
pear. Thus, By(magnetic field perpendicular to tether and
flow) is stabilising while flow-directed or tether-directed field
is destabilising at the baseline voltage.
The final state of run Bx is shown in Fig. 7. The sheath
surrounding the tether is unstable and radiates plasma waves
in all directions in the XY plane. In the tether-fixed coordi-
nate frame, the waves move approximately at the same phase
speed as the bulk flow (7.5 km/s). The waves appear to move
with the ion flow which is reflected by the tether’s potential
well. The incoming flow has a Mach number of ≈5with
respect to ion acoustic wave speed. Because of energy con-
servation, reflected ions move radially outward with the same
speed as the incoming flow. The tether’s potential well fluctu-
ates, which modulates the flow of outward reflected ions. The
modulations propagate outward approximately at the same
speed as the ions because the ion acoustic speed is less than
the bulk velocity of the outward moving ion population. In
Fig. 7 one can also see that the X-directed magnetic field
tends to restrict electron motion in Y, which creates some
horizontal stripes seen in the figure (reddish stripes on posi-
tive X), visible in both electron and ion density and emanat-
ing from the edges of the electron cavity.
Figure 8 shows the final state of run Bz. Again the state
is unstable and the tether’s potential well radiates plasma
waves. Additionally, the boundaries of the plasma wake be-
hind the tether display some turbulent behaviour. This proba-
bly occurs because now the magnetic field is along the tether
so that electrons respond to Xand Ydirected electric fields
by E×Bdrifting (similar to Kelvin-Helmholtz instability in
plane perpendicular to B). Not surprisingly, the horizontal
stripes which were visible in the Bx run (Fig. 7) are now ab-
sent.
3.3 Effect of ion mass
Figures 9 and 10 show thrust histories for helium and pro-
ton plasma, respectively, and with other parameters in their
Baseline (V400) values. In both light ion cases, an unstable
oscillation is present. The instability grows faster in the pro-
ton run than in the helium run and has time to evolve into
intermittent, nonlinear regime.
Figure 11 shows the final state of the Proton run. Some
ions have become trapped by the potential well which has de-
creased the thrust to some extent (Fig. 10). The sheath emits
plasma waves in all XY directions. The flow Mach num-
ber with respect to ion acoustic speed is low (1.2) which is
probably the reason why the wake behind the tether is rather
short (the wake gets filled quickly from the boundaries be-
cause the ion thermal speed is almost as large as the flow
speed). In Fig. 11 the behaviour of the densities near the in-
flow boundary (right side) shows some Y-directed striping
and does not look completely natural. It is possible that the
finite box size affects the result to some extent in this case. In
the proton case (Fig. 11) it is also noteworthy that the emit-
ted ion waves have short enough wavelength that the electron
density does not follow the ion density, in other words, that
the emitted wavelength is not much larger than the electron
Debye length.
Formula 1 predicts 12.2 nN/m in the helium case and 2.92
nN/m in the proton case. The simulated values at 2.43 ms are
larger (14.6 and 5.05 nN/m, respectively), even though in the
proton run some ion trapping had already occurred which had
reduced the thrust. Probably the smaller dynamic pressure of
light ion bulk flow depresses the sheath less than in the oxy-
gen case, and this effect is not included in Eq. (1) where the
thrust is assumed to be linearly proportional to the ion mass.
In the helium run (not shown) the behaviour is intermediate
between the oxygen and proton runs. In subsection 3.5 below
we will investigate longer time behaviour of proton plasma
where also a magnetic field is included.
3.4 Effect of electron temperature
In all runs thus far presented we have assumed that the elec-
tron temperature is 0.1 eV (the same as ion temperature). In
the 700-900 km altitude range, the electron temperature is
actually often 0.2-0.3 eV. Figure 12 shows thrust behaviour
of a run (named Te) which is otherwise identical to Base-
line (V400), but the electron temperature is set to 0.3 eV. The
4 P. Janhunen: Plasma brake PIC simulation
thrust remains essentially unchanged and the oscillation is
damped. So run Te suggests that a higher electron tempera-
ture improves stability.
3.5 Longer runs with magnetic field
Figure 13 shows longer time behaviour (4·106timesteps,
23.4 ms) of the thrust in the presence of X-directed magnetic
field. The result indicates that after about 7 ms, the thrust
no longer decreases while intermittent unstable wave activity
continues. The asymptotic value of the thrust as determined
from the run is 40.9 nN/m which is only 11 % less than in
V400 (Baseline) when the difference in voltage between the
runs (311 V versus 337 V) is compensated for by a square
root dependence. Hence the simulation suggests that the in-
stability caused by an X-directed magnetic field decreases
the plasma brake thrust only slightly at the baseline voltage.
Figure 14 shows the corresponding result for Z-directed
magnetic field. The behaviour is similar except that the fluc-
tuation spectrum extends to somewhat lower frequencies.
The final state thrust 38.7 nN/m is 17 % lower than voltage-
corrected Baseline.
Figure 15 shows the result with higher voltage (757 V
which is 2.4 times larger than Baseline) and Z-directed field.
The fluctuations are strong and the thrust 53.8 nN/m is 27 %
lower than voltage-corrected Baseline. Thus the relative gap
between the Bz and B= 0 cases increases with voltage. The
increase of the gap is slower than linear, however, since it
increases from 17 % to ∼27 % if the voltage is made 2.4
times stronger. The thrust curve in Fig. 15 might not yet be
completely stabilised at the end of the run so the thrust es-
timate derived from the simulation in this case might some-
what overestimated.
The X-directed magnetic field case with high 856 V volt-
age is shown in Fig. 16. This run provides a positive surprise
since initial transients now die away quickly after which the
state is stable and the thrust is even somewhat larger than
Eq. (1) prediction. Thus, although X-directed field at lower
voltage is unstable and the instability lowers the thrust to a
small extent (run BxLong), at higher voltage the instability
is absent and instead of a small thrust reduction we have a
minor thrust enhancement.
Next we look at the asymptotic state in the strongly un-
stable proton case (see subsection 3.3 above) where in ad-
dition a destabilising Bz magnetic field exists. The result is
shown in Fig. 17. The fluctuations are strong, but the average
magnitude of the thrust gets stabilised during the run. Thus,
even strongly unstable behaviour does not cause a collapse
of the ion sheath which surrounds the tether and whose spa-
tial extent determines the thrust. Interestingly, the determined
thrust of 2.91 nN/m at Vw= 256 V is exactly (within three
decimal places) equal to Eq. (1) prediction. Likely the B= 0
value of the thrust in case of proton plasma would be slightly
above Eq. (1) prediction, while Bztends to slightly lower the
thrust. That the effects cancel out exactly may be fortuitous.
3.6 Miscellaneous runs
We also made a run where the magnetic field components
had equal values: Bx=By=Bz= 30000 nT/√3. The re-
sult is an intermediate case of the Bx, By and Bz runs: a
very slowly growing oscillation appears. It therefore seems
that the behaviour is stable if the magnetic field is predomi-
nantly in the Ydirection and unstable otherwise. The result
suggests that if the magnetic field has general orientation, the
behaviour can be qualitatively interpolated from the X, Y and
Z directed runs.
When the plasma density nois changed, all spatial scales
in the electrostatic PIC simulation model scale naturally by
the electron Debye length i.e. as ∼1/√no. The only thing
that breaks this scale invariance is the fixed value of the
tether’s effective electric radius r∗
w. However, the effect of
r∗
wis minor because it enters in Eq. (1) only logarithmically.
Thus we can say that changing the plasma density nodoes
not affect the stable/unstable nature of the solution and the
thrust scales nearly as proportional to √no. As a consistency
check, we made one run to verify this behaviour.
We also performed a high voltage run with Y-directed
field, V1000By (Table 2) with short 2.43 ms duration be-
cause only a weak stable oscillation is present and no insta-
bility is seen. As in the corresponding lower voltage run By,
the determined thrust 80.7 nN/m is in close agreement with
Eq. 1.
4 Discussion
In Eq. (1) the thrust is nearly linearly proportional to the ion
mass and the simulations are in agreement with this at least
qualitatively.
At ∼320 V, the runs BxLong and BzLong suggest that al-
though the plasma sheath is unstable when an Xor Zdomi-
nant magnetic field is present, the thrust is reduced by the in-
stability only modestly (11 % and 17 %, respectively). When
the voltage is increased to ∼800 V, a Z-directed field is again
unstable (V1000BzLong) and the thrust reduction is larger,
∼27 %. In case of X-directed field, however, the instabil-
ity is absent and there is actually a small thrust enhancement
(V1000Bx). A Y-directed field is always stable and there is
neither thrust reduction nor enhancement.
In other words, Eq. (1) is in most cases able to predict the
thrust well. The only exception is that if the magnetic field
is aligned with the tether (Z-directed case), then there is a
moderate thrust reduction which increases with voltage. The
relative reduction is 17 % at 320 V and ∼27 % at 760 V for
Z-directed magnetic field.
When B= 0, the flow is stable with oxygen plasma and
unstable with proton plasma (runs V400 and Proton). When
an instability sets in, the ion cloud formed by ions passing
near the tether starts to oscillate at ion plasma frequency. In
the oxygen case the ion plasma oscillation is 4 times slower
P. Janhunen: Plasma brake PIC simulation 5
than in the proton case. In the oxygen case the flow has time
to move by more than one ion sheath diameter during one ion
plasma period, while in the proton case it moves only a frac-
tion of the sheath diameter. Thus in the proton plasma case,
if the sheath starts to oscillate, the oscillations have more op-
portunities to disturb the upstream flow and perhaps cause
a positive nonlinear feedback. Maybe this is why lighter ion
mass flow tends to be more unstable. Another way to arrive at
a qualitatively similar conclusion is to note that the ion bulk
flow kinetic energy eVi= (1/2)miv2
ois linearly proportional
to the ion mass. Thus the ratio Vw/Viis 16 times larger in
proton plasma flow than in O+ion flow. Any voltage depen-
dent instability should then occur at 16 times lower voltage
in proton plasma than in O+plasma.
Let us consider a vertical gravity gradient stabilised
plasma brake tether using the untilted dipole approximation
for Earth’s magnetic field. The untilted dipole approximation
can be used in this case because runs where all field com-
ponents had nonzero values seemed to interpolate smoothly
from purely aligned runs (Section 3.6), i.e. exact orientation
of the magnetic field does not seem to matter. In this approx-
imation, in equatorial orbit the magnetic field is Y-directed
all the time and the thrust is consequently predicted well by
Eq. (1). The same is true in polar orbit in low latitudes. Only
for the high latitude portion of a polar orbit the thrust is some-
what smaller than what Eq. (1) predicts.
Let us look at a numeric example. At 1 kV voltage, 3·1010
m−3density and oxygen plasma, the predicted plasma brake
thrust from Eq. 1 is 85 nN/m. A 5 km long tether would pro-
duce 0.43 mN braking force which is equivalent to 13400
Ns impulse per year. At 800 km altitude, reducing the orbital
altitude by 100 km requires 52 m/s of delta-v, thus the exem-
plary 5 km tether (mass 0.055 kg, (Sepp¨
anen et al., 2013))
could lower the orbital altitude of a 260 kg object by 100 km
during one year. The gathered oxygen ion current per tether
length is given by the orbital motion limited (OML) theory
expression as
dI
dz =enor2eVw
mi
dtot
w= 8.4·10−8A/m(3)
where dtot
wis the total width of the four-wire tether (160
µm). Hence a 5 km tether gathers 0.4 mA current and con-
sumes 0.4 W power in this plasma. In reality the current is
expected to be somewhat larger because of production of
some secondary electrons from the tether when hit by oxy-
gen ions. For comparison, the maximal Lorentz force pro-
duced by the tether (Cosmo and Lorenzini, 1997) (using the
length-averaged current of 0.2 mA) is only 0.03 mN which is
14 times smaller than the plasma brake force in this case.
We did not test voltages higher than about 860 V in this
paper. Going to higher voltages tends to increase the needed
computing time because the sheath becomes larger so one
needs more grid cells, particles and timesteps to model its
asymptotic evolution accurately. In a practical device, if one
increases the negative voltage, at some point electron field
emission from the surface of the tether wires starts to become
an issue. Field emission adds to the ionic current gathered
by the tether and hence increases power consumption. We
think that the point where field emission starts to become an
issue is larger than 1 kV but smaller than perhaps 3-5 kV. The
value also depends on the geometry and possible coating of
the tether. Before experimental knowledge about the plasma
brake is obtained, it is not necessarily well motivated to try
simulations with higher voltages than those presented in this
paper.
4.1 Correction of an earlier result
The reference Janhunen (2010) contains an error: the quan-
tities which are in the present paper denoted by ˜
Vand Vw
were confused with each other. Therefore the thrust versus
voltage relationship as given in Janhunen (2010) was too op-
timistic. On the other hand, Janhunen (2010) used an earlier
formula (Janhunen, 2009, equation 20) whose numerical co-
efficient had been found from a test particle calculation using
solar wind parameters. In the ionospheric plasma brake case,
the ratio between the tether voltage Vwand the bulk flow
energy Vi(in voltage units) is much larger than in the so-
lar wind, and consequently ions are reflected backward more
efficiently in the ionosphere than in the solar wind, as is ev-
ident from the simulation results of the present paper. All in
all, the results of Janhunen (2010) are not too far from re-
ality: the error resulting from confusing ˜
Vand Vwand the
inaccuracy resulting from using a formula suitable for solar
wind roughly cancel out each other.
5 Conclusions
According to a series of electrostatic PIC simulations per-
formed at different parameters, Eq. (1) can be used to pre-
dict plasma brake thrust in ionospheric conditions. Only if
the dominant component of the magnetic field is along the
tether, the thrust is reduced, and the relative reduction grows
from 17 % to ∼27 % when the voltage increases from 320 V
to 760 V. The thrust reduction is due to an instability which
has ionic character.
The predicted performance of the plasma brake seems
promising concerning satellite deorbiting applications. For
example in O+plasma with 3·1010 m−3density and using 1
kV voltage, a 5 km long plasma brake tether weighing 0.055
kg could produce 0.43 mN breaking force at altitude ∼800
km, which is enough to reduce the orbital altitude of a 260
kg debris mass by 100 km during one year.
Acknowledgements. The work was partly supported by Academy
of Finland grant 250591.
6 P. Janhunen: Plasma brake PIC simulation
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P. Janhunen: Plasma brake PIC simulation 7
B=0
Bx
By
Bz
0 500 1000 V
0
10
20
30
40
50
60
70
80
nN/m
Fig. 1. Different marks: thrust as function of tether voltage Vwin
oxygen plasma with and without magnetic field, see legend in plot
and Table 2 for run parameters. Solid line: equation (1).
47.9 nN/m
0 0.5 1 1.5 2 ms
0
50
100
150
nN/m
Fig. 2. Time development of thrust in V400 (Baseline) run, tether
Coulomb force computation (black curve) and momentum balance
computation (red curve).
Fig. 3. Electron density (a) and ion density (b) normalised to plasma
stream density in the final state (t= 2.43 ms) of run V400 (Base-
line).
8 P. Janhunen: Plasma brake PIC simulation
51.1 nN/m
0 0.5 1 1.5 2 ms
0
50
100
150
nN/m
Fig. 4. Run Bx: same as Fig. 2, but with Bx= 30000 nT.
48.5 nN/m
0 0.5 1 1.5 2 ms
0
50
100
150
nN/m
Fig. 5. Run By: same as Fig. 2, but with By= 30000 nT.
47.2 nN/m
0 0.5 1 1.5 2 ms
0
50
100
150
nN/m
Fig. 6. Run Bz: same as Fig. 2, but with Bz= 30000 nT.
P. Janhunen: Plasma brake PIC simulation 9
Fig. 7. Electron density (a) and ion density (b) normalised to plasma
stream density in the final state (t= 2.43 ms) of run Bx.
Fig. 8. Electron density (a) and ion density (b) normalised to plasma
stream density in the final state (t= 2.43 ms) of run Bz.
10 P. Janhunen: Plasma brake PIC simulation
14.6 nN/m
0 0.5 1 1.5 2 ms
0
10
20
30
40
50
nN/m
Fig. 9. Run Helium: same as Fig. 2, but in helium plasma.
5.05 nN/m
0 0.5 1 1.5 2 ms
0
10
20
30
40
50
nN/m
Fig. 10. Run Proton: same as Fig. 2, but in proton plasma.
Fig. 11. Electron density (a) and ion density (b) normalised to
plasma stream density in the final state (t= 2.43 ms) of run Pro-
ton.
12 P. Janhunen: Plasma brake PIC simulation
Fig. 13. Run BxLong: Bx= 30000 nT and 23.4 ms duration.
Fig. 14. Run BzLong: Bz= 30000 nT and 23.4 ms duration.
Fig. 15. Run V1000BzLong: Vw= 756.69 V, Bz= 30000 nT and 23.4 ms duration.
P. Janhunen: Plasma brake PIC simulation 13
83.6 nN/m
0 1 2 3 4 5 6 7 8 ms
0
10
20
30
40
50
60
70
80
90
100
nN/m
Fig. 16. Run V1000Bx: Vw= 856.07 V, Bx= 30000 nT and 8 ms duration.
Fig. 17. Run ProtonBzLong: proton plasma, Bz= 30000 nT and 23.4 ms duration.
14 P. Janhunen: Plasma brake PIC simulation
Table 1. Simulation parameters of V400 (Baseline)run.
Parameter Symbol Value
Grid size 512 ×512
Grid spacing ∆x7.3mm
Normalised spacing ∆x/λDe 0.54
X grid domain -2.34 .. 1.4 m
Y grid domain -1.87 .. 1.87 m
Timestep ∆t5.84 ns
Normalised timestep ωp e∆t0.057
Run duration tmax 2.43 ms
Number of timesteps 417000
Electrons per cell N0500 (in plasma stream)
Number of particles ∼52.4M
Plasma density no3·1010 m−3
Ion mass mi16 amu (O+)
Plasma drift vo7.5 km/s
Electron temp. Te0.1 eV
Ion temp. Ti0.1 eV
Magnetic field Bx,By,Bz0 nT
Tether voltage Vw337.436 V
Tether electric radius r∗
w1 mm
P. Janhunen: Plasma brake PIC simulation 15
Table 2. List of performed runs. Only differences to the V400 (Baseline) run are mentioned. Runs ending with “g” have larger 768 ×768
grid. The Long runs have 23.4 ms duration. The ’Rel. thrust’ column tells how much the thrust differs from Eq. (1) prediction.
Run Parameters Thrust/nNm−1Eq. (1)/nNm−1Rel. thrust Nature
V50 Vw= 41.0V 7.06 10.6 -33 % Stable
V100 Vw= 82.4V 19.4 19.8 -2 % Stable
V150 Vw= 125 V 27.3 26.7 +2 % Stable
V200 Vw= 168 V 32.8 32.3 +2 % Stable
V300 Vw= 252 V 40.5 41.2 -2 % Steady oscillation
V400 Table 1, Vw= 337 V 47.9 48.5 -1 % Steady oscillation
V500 Vw= 424 V 54.5 54.9 -1 % Steady oscillation
V500g Vw= 424 V 54.9 54.9 0 % Steady oscillation
V600g Vw= 512 V 60.9 60.6 0 % Steady oscillation
V800g Vw= 689 V 71.4 70.7 +1 % Steady oscillation
V1000g Vw= 869 V 80.1 79.5 +1 % Steady oscillation
Bx Bx= 30µT, Vw= 333 V 51.1 48.2 +6 % Unstable
BxLong Bx= 30µT, Vw= 311 V 40.9 46.4 -12 % Unstable
By By= 30µT, Vw= 338 V 48.5 48.6 0 % Dying oscillation
Bz Bz= 30µT, Vw= 335 V 47.2 48.4 -2 % Unstable
BzLong Bz= 30µT, Vw= 317 V 38.7 46.9 -17 % Unstable
Helium mi= 4 amu, Vw= 350 V 14.6 13.2 +11 % Unstable
Proton mi= 1 amu, Vw= 323 V 5.06 3.24 +56 % Unstable
ProtonBzLong mi= 1 amu, Vw= 256 V 2.91 2.91 0 % Unstable
V1000By By= 30µT, V= 870 V 80.7 79.5 +2 % Steady oscillation
V1000BzLong Bz= 30µT, Vw= 757 V 53.8 74.1 -27 % Unstable
V1000Bx Bx= 30µT, Vw= 856 V 83.6 78.0 +7 % Stable
Te Te= 0.3eV, Vw= 337 V 48.7 48.5 0 % Dying oscillation
