Transmission Probabilities of Rareﬁed Flows in the
Application of Atmosphere-Breathing Electric Propulsion
T. Binder, P.C. Boldini, F. Romano, G. Herdrich and S. Fasoulas
Institute of Space Systems, University of Stuttgart, Pfaﬀenwaldring 29, 70569 Stuttgart, Germany
Abstract. Atmosphere-Breathing Electric Propulsion systems (ABEP) are currently investigated to utilize the residual atmosphere
as propellant for drag-compensating thrusters on spacecraft in (very) low orbits. The key concept for an eﬃcient intake of such a
system is to feed a large fraction of the incoming ﬂow to the thruster by a high transmission probability Θfor the inﬂow while
Θfor the backﬂow should be as low as possible. This is the case for rareﬁed ﬂows through tube-like structures of arbitrary cross
section when assuming diﬀuse wall reﬂections inside and after these ducts, and entrance velocities ularger than thermal velocities
vth ∝√kBT/m. The theory of transmission for free molecular ﬂow through cylinders is well known for u=0, but less research
results are available for u>0.
In this paper, the desired theoretical characteristics of intakes for ABEP are pointed out, a short review of transmission proba-
bilities is given, and results of Monte Carlo simulations concerning Θare presented. Based on simple algebraic relations, an intake
can be optimized in terms of collection eﬃciency by choosing optimal ducts. It is shown that Θdepends only on non-dimensional
values of the duct geometry combined with vth and u. The simulation results of a complete exemplary ABEP conﬁguration illustrate
the inﬂuence of modeling quality in terms of inﬂow conditions and inter-particle collisions.
Very low Earth orbits (i.e. below ∼250 km) are of great interest for many scientiﬁc, civil, and military purposes. Higher
accuracy for Earth observations and a persistent signal for Earth communications can be achieved, and manufactur-
ing and launching costs can be reduced. Furthermore, equivalent low orbits are also contemplated for future orbiters
around Mars. The drawback of lower altitudes, however, is the higher density of the residual atmosphere. The con-
siderably increased aerodynamic drag dictates the required amount of propellant on-board which is the main limiting
lifetime factor for such a mission. Even with modern Electric Propulsion at most 2 years of drag compensation can be
accomplished for the majority of missions in Earth orbits below 250 km .
Atmosphere-Breathing Electric Propulsion (ABEP) theoretically solves this issue by using the residual atmo-
sphere as propellant. This will decrease, ideally nullify, the on-board propellant requirement and will generate thrust
to partially or fully compensate the drag. A conceptual scheme of a spacecraft with ABEP is shown in Fig. 1a. The
collection process inside the intake is characterized by the highly rareﬁed (mean free path in the order of 0.1 m–1 km
for ABEP-applicable Mars and Earth orbits) and very directed (u>vth) inﬂow. A ﬁrst intuitive conﬁguration based
on continuum ﬂow theories might be a very simple and straightforward funnel-shaped design: a small cylindrical inlet
followed by a cone converging to the thruster. In such a design, though, almost all particles passing through the inlet
section of the intake would be reﬂected back into ﬂight direction, since nearly no inter-molecular collisions occur and
the solid angle towards the outlet as seen from the reﬂection positions along the cone is very small.
Studies from ESA , BUSEK , JAXA , and LIP  are one of the most recent and detailed examples
dealing with the design of ABEP. Their main common feature regarding the intake is the implementation of small
ducts (e.g. in form of a honeycomb structure) inside the very front part of the inlet, as illustrated in Fig. 1b for the
aforementioned funnel-design. The basic principle of inlet-ducts is that of a molecular trap. The transmission through
the ducts is in case of an axially directed inﬂow still high, whereas the backﬂow is signiﬁcantly reduced when assuming
diﬀuse wall reﬂections. That backﬂow corresponds exactly to the theory of transmission as analytically described by
Clausing . For the inﬂow with u>0, however, less data is available as necessary for a detailed design study.
In this paper, ﬁndings regarding those transmission probabilities of rareﬁed ﬂow in the application of an intake
(a) Scheme of spacecraft
(b) Funnel-like intake with inlet-ducts
FIGURE 1. Concept of Atmosphere-Breathing Electric Propulsion
for an Atmosphere-Breathing Electric Propulsion system are presented. First, the desired theoretical characteristics of
such intakes are pointed out with the help of simple relations derived from the balance of mass ﬂows. Afterwards, a
short review of transmission theory is given and, furthermore, the results of Monte Carlo simulations are presented.
Finally, simulation results of a complete exemplary intake are shown which illustrate the inﬂuence of modeling quality
of the ducts on the one hand, and the accuracy of the balance model on the other hand.
In order to understand the dependencies of the ABEP performance on its characteristics such as the transmittances of
the inlet, we previously derived an analytical model for an ABEP intake . The generic design consists of an inlet
section and a chamber section. For the particles inside the chamber it is assumed that (ideally) every single one has
already performed fully diﬀuse wall reﬂections and, therefore, proceeds only with thermal movement. The resultant,
in principle non-directional, particle ﬂows are the backﬂow through the inlet, and the ﬂow through the outlet. The
outlet can represent acceleration grids, an injection device towards the thruster or a further stage of compression. By
balancing all ﬂows, the conditions in the separate sections can be estimated. Figure 2 illustrates the used nomenclature.
FIGURE 2. Balance model scheme
FIGURE 3. Transmission through
cylinder of length Land radius R
The respective cross sections for the inﬂow and outlet are deﬁned by Ain and Aout. The parameters of the inﬂow
are known from the atmospheric model, namely individual species number densities nin, temperature Tin, and the
spacecraft velocity uin. The transmission probability Θrefers to a speciﬁc direction through a single structure. It is the
ratio between the gas particles entering the entrance plane A and the gas particles leaving the exit plane B, see Fig. 3.
A particle can directly reach B from A, or can be scattered along the wall before reaching B. However, a particle
coming from A can also return back to A. For the balance model, three transmittances are set based on which the
respective resultant particle ﬂows can be deﬁned: The particle ﬂow into the chamber section (˙
Ninl.1) passes with Θinl.1
through the inlet section, the backﬂow to the atmosphere after having reached the chamber section ( ˙
Ninl.2) passes with
Θinl.2, and ˙
Nout is the collected net outﬂow with Θout. Main assumptions of the model are:
•Free molecular ﬂow (no inter-particle collisions) in thermodynamic equilibrium;
•Diﬀuse reﬂection at walls;
•Fixed chamber and wall temperature (Tch =Twall);
•Only non-directional, thermal mass ﬂux inside the chamber.
The total particle ﬂow ˙
Nin onto the intake is calculated using free stream conditions and the extruded intake area
Ain. The part reaching the chamber section includes additionally the inﬂow transmittance, i.e.:
NinΘinl.1=nin ¯uinAin Θinl.1.(1)
Based on Tch, the thermal mass ﬂux Γis set, resulting in backﬂow and outﬂow from the chamber. The actively
extracted particle ﬂow ˙
Naccel.depends on the actual thruster. It is expected that a minimum number density nch inside
the chamber is necessary for ignition. Therefore, the focus is on the situation before ignition ( ˙
continuity equation ˙
Nout can be applied which determines Γand, thus, nch. Knowing the state inside
the chamber, the collection eﬃciency ηcand number density ratio in Eqs. 2 can be calculated (mis the particle mass
of the respective species). The eﬃciency ηcis equivalent to the thrust-to-drag-ratio of the spacecraft when considering
only the drag on Ain and constant inﬂow conditions and thruster operation. Hence, ηcis the main ﬁgure of merit;
however, also a minimum nch and ˙
Nout for the thruster has to be ensured. The simplicity of the balance model makes
it a very convenient tool for the study of ABEP, provided that the individual transmission probabilities Θare known.
Ain Θout r2π
The ﬂow of rareﬁed gases through tubes has been a problem of great importance during the whole last century .
After ﬁrst discussed by Knudsen , the transmission probabilities for cylindrical ducts of various aspect ratios was
described by Clausing  by analytical integral equations, providing the fundamental basis for subsequent studies.
As a matter of fact, the balance model through cylinders of ﬁnite length with fast inﬂow and scattered backﬂow
(but without further ”chamber” outﬂow) is basically the same as the one for the free molecular ”Patterson” probe, as
discussed with similar focus on transmission by Hughes . Therefore, Hughes’ work constitutes a good basis for the
analytical description of transmission probabilities. In the following, the theory is subdivided into three points:
•The transmission with u=0 as published by Clausing  (used for Θinl.2in the balance model);
•Hughes’  extension to u>0 (used for Θinl.1);
•The combination of two adjoining Θ’s such as one of an inlet section with ducts and one of a section without.
Clausing case (u=0)
Clausing’s integral equations  are the exact analytical description of the transmission through cylinders with an
inﬂow corresponding to a Maxwell-Boltzmann distribution around u=0 with Tin =Twall and fully diﬀuse wall reﬂec-
tions. Their solution is only achievable with numerical approaches. However, Clausing could derive an approximation
in form of an explicit Θ = f(L/R) dependency based on the assumption that a wall reﬂection leads to a remaining
transmission probability which is linearly dependent on the distance from the inlet and his given function α(L/R).
The ﬁrst values of transmission probabilities with high accuracy were calculated by Cole  in 1977. In the
recent decades the numerical methods were further improved resulting in even more precise results, as summarized
together with their own most recent results by Li et al. . An alternative to methods based on Clausing’s equations
are Monte Carlo simulations applicable to any geometry and also published numerously.
A comparison of Clausing’s approximation with high accuracy results (here by Cole) is shown in Fig. 4a for the
transmission probability of cylindrical ducts in the range of L/R=[0.1,100]. As depicted, Clausing’s values have a
relative error smaller than ∼0.1% until L/R=3, whereas for higher aspect ratios they remain at least below 4%. For
limL/R→∞ and limL/R→0they converge to the exact solution.
0.1 0.20.51 2 510 20 50 100
Aspect ratio L/R, -
rel. deviation, %
Clausing, rel. deviation
(a) Clausing case: Cole  and Clausing 
0.1 0.20.51 2 510 20 50 100
Aspect ratio L/R, -
(b) u>0 for diﬀerent S=u/vth (Hughes’ method )
FIGURE 4. Analytical solutions for transmission probabilities through cylinders of diﬀerent L/R
Non-zero entrance velocity (u>0)
In contrast to Clausing’s assumption of u=0, the inﬂow in the ABEP application corresponds to in-orbit conditions
(u>vth) which motivates to analyze the inﬂuence of a non-zero entrance velocity. Hughes  analytically described
the transmission probabilities for cylindrical ducts with a speed ratio S>0, see Eq. 3, and even arbitrary angle of
attack. He adopted Clausing’s α(L/R) approximation, which ultimately leads in the case of a parallel inﬂow to a single
integration for calculating Θat a speciﬁc L/Rand S.
Figure 4b depicts our results for diﬀerent Sand L/Rcalculated with a numerical integration by the adaptive
Gauss-Kronrod quadrature algorithm quadgk of MATLAB R
. The obtained curves show that with a higher Sthe
Θ(L/R) increases clearly and, therefore, particles are less likely to be scattered at walls or they are scattered at larger
distances from the inlet.
0.05 0.1 0.20.51 2 5
rel. deviation from S=15, %
Θ(X): S=0.5S=15 dev. from S=15: S=0.5S=1S=5S=10
FIGURE 5. Θcylinder (S,L
R) by Hughes’ method  and deviations from S=15 plotted against X
When plotting Θinstead of L/Ragainst a new variable X, combining both L/Rand S(see Eq. 3), the interesting
fact arises that with increasing Sall curves seem to converge to a single one. Figure 5 shows the Θ(X) curves for
S=15 and S=0.5 as black lines. The S=1 curve would lie between, while the ones for higher Swould virtually
lie on top of S=15. For quantifying the actual diﬀerences, the diagram includes also the relative deviations from
the S=15 curve. Due to the logarithmic scale, the dashed parts depict the negative deviation. For small S, e.g. 0.5
and 1, the transmittances are up to 100% greater, while for S=5 and 10 the deviation is until X≈1 below 1%. For
1<X<5, it increases as far as 10%. However, it still might be possible that this increased deviation is only due
to the underlying α(L/R) assumption, since the range with increasing deviation roughly corresponds to L/R>3 (see
previous subsection). All in all, it is shown that for the high Sof interest (for ABEP relevant orbits around Earth and
Mars, Sis in the regime of ∼5–20), the transmission probability approximately becomes from a double dependence a
single one. This makes the regime very convenient for analyses, since the Θ(X) curve has just to be calculated once and
can be evaluated for any Sand L/R. However, the inﬂuence of Clausing’s α(L/R) assumption is unknown for S>0,
by which the presented method of Hughes can so far only indicate the existence of a direct Θ(X,S>5) relation, but
for obtaining precise values, a diﬀerent approach is necessary which motivates, e.g., Monte Carlo simulations.
Combining two adjoining transmittances
As shown in Fig. 2, the inlet section is thought to be ﬁlled with ducts only in its front part. One reason for that
design is a possible optimization in terms of total transmittance. Based on two adjoined tubes with equal radii this
might seem pointless, since independently of the position where an imagined tube is divided into two parts, the
combined geometry, and therefore its transmittance, is always the same. However, one has to bear in mind that a
transmitted ﬂow has gone through a beaming eﬀect. The outﬂow has now a radially, non-equal particle distribution
with preferred small deﬂection angles from the axis; while the inﬂow has an equal distribution and, for the example
of the Clausing case, a velocity distribution according to Lambert’s cosine law. Therefore, the assumption of taking
Nof an outﬂow from a ﬁrst structure I as inﬂow at equilibrium conditions for an adjoined structure II can result
in a total transmittance diﬀerent from the one of a continuous structure. For the derivation of the common theory
of transmittance combination let us assume that a beaming eﬀect can be neglected. Oatley  ﬁrst determined the
combined transmission probability ΘI+II of two adjoining tubes I and II in the Clausing case to be:
1/ΘI+II =1/ΘI+1/ΘII −1.(4)
However, in the case of a non-zero entrance velocity it has to be distinguished between an original Θconsisting of
Θdirect + Θindirect and Θ∗for particles after wall collisions and corresponding to the Clausing case. Thereby, an extended
ΘI+II is derived by separating the transmitted ﬂow into an inﬁnite number of individual parts, as illustrated in Fig. 6.
The sum inside the resultant combined transmittance of Eq. 5 can be formulated as geometric series, as shown in
Eq. 6. Thereby, Eq. 7 follows after some algebraic manipulations, that obviously collapses to Eq. 4 when assuming
Θ = Θ∗. The assumption of inﬂow/outﬂow conditions at equilibrium can be justiﬁed when particularly considering
the transition from a section I ﬁlled with a large number of small ducts to a section II without ducts, because the local,
radial outﬂow distributions of the ducts can be neglected when they are combined to a much larger total inﬂow for II.
ΘI·(1 −ΘII)(1 −Θ∗
ΘI·(1 −ΘII)(1 −Θ∗
FIGURE 6. Derivation of two combined transmittances
ΘI+II = ΘI·ΘII + ΘI·(1 −ΘII)(1 −Θ∗
ΘI+II =ΘIΘII −ΘIΘII(1 −Θ∗
II(1 −ΘII )(1 −Θ∗
The concrete transmittances used in Eq. 7 for calculating the individual values of ΘI+II through the combined
ABEP inlet are listed in Tab. 1. Please note that the direct and indirect parts of Θinl.1are diﬀerently transmitted
through the II section, while Θinl.2already corresponds to the Clausing case.
TABLE 1. Used ΘIand ΘII for the individual values of ΘI+II through the ABEP inlet
Θinl.1,direct for fast, unscattered II-inﬂow: ΘI= ΘI,direct ,ΘII = ΘII,direct + ΘII,ind irect ;
Θinl.1,indirect for scattered II-inﬂow: ΘI= ΘI,indirect ,ΘII = Θ∗
Θinl.2for scattered ”chamber” backﬂow: ΘI= Θ∗
I,ΘII = Θ∗
MONTE CARLO SIMULATIONS
In the previous section it was mentioned that for a set of high speed ratio Sand arbitrary L/R, simple analytical
methods such as Hughes’ might not produce precise results for Θ, and more importantly, they are mostly restricted
to axisymmetric geometries. Therefore, we applied PICLas, a 3-D, highly parallelized, coupled code  including
modules for DSMC and Particle-In-Cell (PIC). The modules for boundary treatment and particle movement can be
used without any inter-particle collision or PIC-related routines, making PICLas for us also a conveniently available
Monte Carlo method for free molecular ﬂow (FMF) simulations. In the following, an exemplary outcome for Θ(X) of
cylindrical ducts is presented, and ﬁnally, the previous results are applied in complete intake simulations.
Determined transmission probabilities for u>0
We conducted simulations on various duct geometries, here however we conﬁne ourselves to cylindrical tubes. The
aforementioned parameter Xwas varied in the range of ∼0.01–2 by diﬀerent uin,Tin /m, and L/R, all in the regime
applicable for ABEP. Additionally, Twall was altered which showed that it does not aﬀect the individual transmittances.
Each simulation included a single duct open at both ends (but particle counting) with an incoming ﬂow parallel to
the cylinder axis and fully diﬀuse wall reﬂections, which additionally mark the reﬂected particles for distinguishing
between Θdirect and Θindirect . Figure 7 shows the results from the simulations.
0.01 0.1 1
L/R(in Mars orbit at 110 km, S=15.89), -
Total transmitted inﬂow,
Θ = f(X)
Total transmitted inﬂow
(analytical, cf. Fig. 5)
inﬂow, Θdirect =f(X)
inﬂow, Θindirect =f(X)
FIGURE 7. Cylinder transmittances Θ(X) determined by Monte Carlo simulations
The black points are the total transmittance Θ(X) as sum of Θdirect (blue) and Θindirect (red). In addition to X
(bottom), also the corresponding L/R-axis is included (top), based on a speed ratio Sof a representative Mars orbit
for ABEP. As comparison, the analytical solutions from Hughes’ method (see previous section) are depicted which
match the simulation results very well until X=1. For larger X, it was shown that the analytical solutions themselves
deviate between each other signiﬁcantly for diﬀerent S, however the inﬂuence of Clausing’s α(L/R) approximation
was unknown. Therefore, the region of Θ(X) near X=2 is depicted magniﬁed in the same diagram. It can be seen that
the points from the simulations lie only 0.2–0.4 % above the analytical approximation. Thus, Hughes’ method seems
to predict the actual transmittances for S>0 even better than for Clausing’s S=0 (the corresponding aspect ratios
are of L/R>3) and the unique Θ(X) relation for S>5 is expected to be used at least up to X=1 for ensuring an
error below 1 %, see Fig. 5.
Application to complete intake simulations and DSMC
One speciﬁc ABEP conﬁguration was analyzed further after an optimization with the balance model for a speciﬁc
small EP thruster being developed at our institute. A Mars orbit at 110 km altitude is chosen  with the inﬂow
conditions shown together with the geometrical parameters in Tab. 2. Internal degrees of freedom are neglected as
well as all other species than CO2(∼90%), resulting in S=15.89. One single case was simulated including inter-
particle collisions with DSMC by VHS cross sections, the remaining cases are FMF. The chosen geometry considers
a square cross section of the inlet converging at the outlet with ∼45◦to the circular section of the thruster. By this
inlet, a ﬁlling with ducts of likewise square cross section can be achieved whose wall thickness was approximated as
zero. The outlet was set to Θout =1 and vacuum condition for focusing on the maximum achievable mass ﬂow. Two
diﬀerent lengths were simulated for the inlet part after the ducts (”back”, see Fig. 2) and four for the ducts themselves.
The complete simulations were compared to ones starting directly after the ducts from which an outﬂow at equi-
librium state is assumed consisting of the original atmospheric condition ( ˙
NinΘinl.1,dir ect) and the scattered particles at
wall temperature ( ˙
NinΘinl.1,indir ect). At these ”equilibrium ducts”, backﬂowing particles were deleted with a probabil-
ity of the respective Clausing-like transmittance, otherwise diﬀusely reﬂected. The individual transmittances for the
diﬀerent L/R(Ris now the half side length) were determined analogously to the cylinders of the previous subsection.
For comparison with the balance model (BM), also the Θ’s of the back part have to be known (”II” in Tab. 1). Here,
the converging part was neglected and a large square duct with respective L/Rwas assumed.
The results are depicted in Fig. 8 in terms of collection eﬃciency plotted against duct aspect ratio. It can be seen
that for both back lengths the curve shapes of all models match surprisingly well. This veriﬁes the approximation of
the combined intake. The relative deviations between BM and simulations are approximately 5–8% which is expected
to be, on the one hand, due to the particle part directly transmitted through the whole intake (in the BM all chamber
particles are assumed to be scattered); on the other hand due to the continuous transition from inlet to the ”chamber”
section which is not considered in the BM. The comparison of Fig. 8a and Fig. 8b shows that, eﬀectively, there is an
optimization possible for the L/Rdistribution between duct and back part of the inlet, but for this case the absolute
gain is only 1–2%. However, this conﬁrms the assumption of setting the outﬂow of the ducts back to equilibrium.
The same interpretation follows when comparing equilibrium ducts with complete simulations - the relative error is
with −1.5% to 5% very small. The deviation between the case without and with inter-particle collisions (L/Rback =4,
L/Rducts =16) is only 0.8% and veriﬁes the FMF assumption for this speciﬁc geometry and inﬂow condition.
TABLE 2. Parameters of intake simulation for Mars at 110 km (Ris half side length of square)
nin (CO2)Tin Twall uin Ain/Aout L/Rduct s Rducts L/Rback Rback =Rin
5.028 ·1017 m−3128 K 300 K 3495 m/s 10 [3,8,16,20] 0.25 cm [4,16] 5.185 cm
ABEP systems are investigated to use residual atmosphere as propellant for drag compensation in low orbits. The
collection eﬃciency ηc(collected part of the ﬂow onto the intake) can be increased by implementing small ducts at
the front part of the intake. The principle is that of a molecular trap, letting most of the fast inﬂow coming through
while reducing the backﬂow. For the optimization of the ABEP performance, an analytical balance model (BM) has
been derived based on the balance of mass ﬂows and transmission probabilities Θthrough the individual sections. The
0510 15 20
L/Rducts , -
Equilibrium ducts (sim.)
Complete geometry (sim.)
Complete, with DSMC
Balance model (analytical)
(a) Inlet part after ducts (back) with L/Rback =4
0510 15 20
L/Rducts , -
Equilibrium ducts (sim.)
Complete geometry (sim.)
Balance model (analytical)
(b) Inlet part after ducts (back) with L/Rback =16
FIGURE 8. Collection eﬃciencies of complete intake simulations compared with balance model
transmission of ﬂows without entrance velocity depends only on the aspect ratio (e.g. L/Rof a tube), while for non-
zero velocities, as present in the inﬂow condition of an ABEP, an additional dependency on the speed ratio Sarises.
However, for the high Sof an ABEP application the double dependence reduces to a single one on X, combining
aspect ratio and S. It was shown that for cylinders the assumption of an unique Θ(X,S>5) relation results in errors
increasing with X, but for X<1 they stay below 1 %. Therefore, just the Θ(X) curve has to be determined and can be
evaluated for any Sand L/R. This might be convenient particularly for ducts with non-circular cross-sections (but still
similar to regular polygons), since they seem to have an analog behavior, but without analytical Θ(X) descriptions.
Together with a derived equation for combining the Θof the duct-including inlet part with the adjoining back part,
determined Θ(X) points for square ducts were applied in a comparison between the BM and 3-D intake simulations.
In terms of ηc, the relative deviations between both approaches are approximately 5–8% which makes the BM well
suited for quick optimizations. Moreover, simulations have been conducted starting directly after the ducts where
corresponding Maxwellian velocities are assumed. The errors compared to the complete simulation are small, showing
that a beaming eﬀect through the ducts is insigniﬁcant, since most of the particles are scattered afterwards anyway.
Last but not least the free molecular ﬂow assumption was veriﬁed for the speciﬁc geometry and inﬂow condition.
For further analyses more detailed gas-surface interactions will be included. However, the fully diﬀuse wall
reﬂections are assumed to be already an useful approximation for ﬁrst evaluations. Since all analytical relations rely
strongly on them, it is not expected that a similarly comprehensive theory could be developed otherwise.
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