A Code for the Analysis of Missions with Electrodynamic Tethers

The main novelties and capabilities of the second version of the Bare Electro-dynamic Tether Mission Analysis Software (BETsMA v2.0) are presented. Recent advances on Orbital-Motion-Theory have been incorporated to the electric model of Low-Work-function Tethers (LWTs). An electric model that considers a switch to embed a power supply or a resistor between a Bare Electrodynamic Tether (BET) and an electron emitter is also introduced. For both types of tethers (LWT and BET) and modes of operations (active and passive), robust and e(cid:14)cient numerical algorithms to compute the current and voltage pro(cid:12)les were constructed based on a change of variable proposed in a previous work. The capabilities of the code are illustrated by considering two relevant scenarios. For a BET in the passive mode, it was shown that onboard power can enhance tether performance and reduce signi(cid:12)cantly the deorbit time. For a BET in the active mode, a performance map varying the orbit inclination, the length of the insulated tether segment, and the power was constructed. For both cases, the code was used to investigate through simulations the conditions to keep constant the electric current at the electron emitter.


Introduction
The ongoing transformation of the space sector, including new companies, investors, and the launch of mega-constellations, poses new challenges for policymakers due to the proliferation of space debris [1].It is also expected to trigger in the next future new business opportunities related to on-orbit servicing, post-mission disposal, and the use of orbits at very low altitudes.Some of these applications can benefit from the use of Electrodynamic Tethers (EDTs) that, unlike conventional propulsion technologies, exchange momentum with the planet's magnetosphere with no need for propellant (find a review in [2]).In particular, a Bare EDT (BET) equipped with an Active electron Emitter (AEE) operating in the so-called passive mode in Low Earth Orbit (LEO) produces a drag that deorbits the S/C while providing power for onboard use.If onboard power is available, a BET in the active mode produces a thrust that can be used to compensate the air drag or reboost the S/C [3].
Recently, considerable progress has been made to push the development of EDT technology forward.Relevant examples include the transition from round to tape tethers [4,5,6], the appearance of new consumable-less solutions to achieve a good cathodic contact with the ambient plasma such as the Low-Work-Function Tether (LWT) concept [7,8] and new and improved active electron emitters [9], and novel ideas to enhance EDT's performance and simplicity such as the bare-photovoltaic tether [10].Advances in materials science [11] and the miniaturisation of space components [12] are also important.Combined with their versatility, such developments resulted in a renewed interest in EDTs.
TEPCE by the U.S. Naval Research Laboratory, Dragracer by Millennium Space Systems with a tape tether made by Tether Unlimited, DESCENT [13], and MiTEE [12] are some examples of recent missions with tethers.In Europe, the E.T.PACK consortium is preparing an EDT-based deorbit device [14].
Simulation tools play a key role on EDT development at two different levels.
Firstly, they provide support for carrying out the mission design, including the detumbling and the tether deployment preparation phases, and the nominal operation.The results of such analyses help for instance to select appropriate sensors and actuators of the Attitude and Orbit Control System [15].Secondly, accurate and flexible simulation tools are necessary to assess EDT's performance and allow non-experts and potential investors to evaluate the technology, explore the use of EDTs in new scenarios, and make decisions.Accordingly, research groups and companies have developed EDT simulators with different degree of accuracy, complexity and computational cost like TEMPEST [16], FLEX [17], DYNATETHER [18], TetherSim T M [19], and BETsMA [20] among many others [21,22,23,24,25,26,27,28]. A central element of these software is the electric module, i.e. the numerical algorithm in charge of computing the current and voltage profiles throughout the tether.Being necessary at every time step to evaluate the Lorentz force and torque, these algorithms received an important attention in past works [29,30,31,18] and severely affect the computational cost, robustness, and fidelity of the simulators.This work presents a comprehensive description of the second version of the Bare Electrodynamic Tether Mission Analysis Software (BETsMA v2.0).Section 2 explains the EDTs types and modes and sub-modes of operation implemented in the software.A detailed description of the electric module is given in the Appendixes.Among its main novelties, we highlight here the incorporation of new results in probe theory to tether modelling, the extension of a previously proposed parametrization strategy to improve the efficiency of the numerical algorithms, and the incorporation of a power supply to enhance the performances of EDTs in the passive mode.The combination of a power source with a tether was studied in several previous works [3,19,32], but mainly restricted to the so-called active mode for boosting in LEO.This work presents an electric model that combines a power source with a bare tether in passive mode to enhance deorbiting performances.This is relevant for EDT systems like EDDE [32], which are equipped with solar arrays, and in post-mission disposal scenarios if the solar panels of the satellite are still operative.Simulation results in the passive and the active modes are presented in Sec. 3. Finally, Sec.
4 summarizes the conclusions.Complementary to this work, an online interface to submit simulations using BETsMA v2.0 has been published [33].

Basic Information
The first version of the BETsMA software was developed in the framework of the FP7/Space project with acronym BETs (2010-2014).Focused on the simulation of missions by using bare tethers in the passive mode, the software also implemented the Π − algorithm for the optimal sizing of tethers [34].Detailed information about BETsMA v1.0 is available in a manual [35] and a scientific work [20].As part of the activities of the FET project with acronym E.T.PACK (2019-2022) [14], a second version has been prepared.Table 1 summarizes its capabilities and provides appropriate references to the scientific works that explain in detail the physical models and mathematical algorithms.Regarding tether dynamics, tether attitude can be constrained (tether aligned with the local vertical or spinning at a prescribed angular velocity), or be found self-consistently with a dynamic model that substitutes the tether by a set of N inelastic and articulated bars or a set of particles linked by springs and dampers.Since this work is focused on the electric models and the preliminary mission design, all the simulations in this study assumed a straight tether aligned with the local vertical.Other upgrades are the tether thermal model presented in [2], the incorporation of high harmonics of the Earth's gravitational field, and some improvements related to the numerical methods, like adding a numerical integrator with a variable time step and two types of stop conditions in the simulations.
The upgrades of BETsMA v2.0 are highlighted with bold font in table 1.
BETsMA v2.0 computes the cut probability by small debris of round and tape-like tethers using the model in Ref. 4  for the preliminary analysis of missions using EDTs.However, it is desirable that future versions of BETsMA v2.0 will contain updated values of the debris flux for, at least, two years (corresponding to solar minimum and maximum).
Although it does not affect to tether performances in the actual version of the software, BETsMA v2.0 provides an estimation of the total number of ATOX impacting the tether during the mission.Such information is important due to the negative effect of ATOX on certain materials.For a tether of perimeter p t and length L t , the aerodynamic drag is computed by using a cross-sectional area of p t L t /π, which is valid for round and tape-like tethers.All the environmental models depend on time, except the ones related with the debris flux that are found from ORDEM-2000 or MASTER-2009 models.Besides limitations arising from the physical assumptions used to construct the models, which should be known by the user as part of the common know-how on tethers, there are no limitation for any of the physical parameters.The only exception is the maximum duration of the simulations because the mission should finish by the end of 2015 due to limitations related with the actual implementation of the environmental model.However, this is minor issue because an appropriate starting date can be found by taking into account past and actual data of the solar cycle.
BETsMA v2 includes electric models for BETs [3] and LWTs [7,8,40] in both passive and active modes.The operation mode depends on the directions

Gravitational field
Up to 4th order harmonics

Stop conditions Final altitude/
Mission duration of the tether current (I = I(x)u t ) and the motional electric field in the tether frame (E m = v rel × B) that appears in the unperturbed plasma due to the tether-to plasma relative velocity (v rel ) and the ambient magnetic field (B) [3].
Here, x is the coordinate along the straight tether measured from the anodic point A and u t is an unit vector along the tether and pointing from the cathodic point (C) to the anodic point (A) as shown in Figs. 1 and 2. In the passive mode, one has E m • I > 0 and, in LEO, the Lorentz force is a drag (F L • v < 0) that deorbits the S/C.Symbols L a , v, I av denote the length of the tether segment that carries the electric current, the velocity of the S/C, and the average current along the tether.
In the active mode, a power supply is needed to make E m • I < 0 and, in LEO, the Lorentz force is a thrust (F L • v > 0) that can be used to re-boost the S/C or compensate the aerodynamic drag.The average current in Eq. ( 1) is found by integrating the current profile I(x) along the EDT.Therefore, I av depends on the type of EDT, its longitudinal structure, and the operation mode.

Description of the EDT Systems
Figure 1 shows the longitudinal structure of the two EDTs considered by BETsMA v2 in the passive mode.Both of them have an inert (non-conductive) segment between points Z and A of length L inert .This segment does not contribute to the Lorentz force in Eq. ( 1), but increases the aerodynamic drag and is convenient to enhance the gravity gradient restoring torque and mitigate the dynamic instability of EDTs [21].
In the case of the BET (top panel in Fig. 1), the software considers a bare segment of length L b between points A and P, and a conductive and insulated segment of length L i between point P and the S/C (points S i ).The former is in charge of the passive electron collection and the latter is used to avoid an electric  2, the state of the switch is fixed for each sub-mode, except for the current limited and the constant current sub-modes, where the switch state is dynamically changed during the simulation between points S 1 and S 2 and S 2 and S 3 , respectively.The particular operation sub-mode and constants R 0 , W e0 and I * C are part of the configuration of the simulation.Additionally, it is possible to add constraints on the maximum available power and allowed current at the cathode [find more details in Sec. 3 that presents simulations for the sub-modes (i), (iv), and (v)].

Mode
Sub-mode Switch State Characteristics For the LWT (bottom panel in Fig. 1), the full active tether is coated with a Low-W material.The tether itself is responsible for the passive electron collection between point A and the (zero-bias) point B and the passive electron emission between points B and B * in space-charge-limited (SCL) conditions and between B * and C according to the Richardson-Dushman (RD) law.Since coated with a Low-W material, segment A − B * also emits electrons.However, these electrons are captured again by the LWT because it is positively polarized with respect to the plasma in that segment.Similar to the top panel, it would be possible to prepare configurations combining a LWT with a resistor to harvest electric power or with a power supply to improve the performance.Currently, these sub-modes of operation are not implemented in BETsMA v2 for LWTs.
As shown in Fig. 2, the longitudinal structure of both tether types in active mode are very similar.Moving from point Z to S, one finds an inert (nonconductive segment), a bare segment for electron collections, a conductive and insulated segment to avoid electric arcing and improve the efficiency [3], and the positive pole of a power supply that provides an electric power W e .For BETs (LWTs), the negative pole of the power supply is connected to an AEE (a tether segment of length L c and coated with a low-W material).Three sub-modes of operation have been implemented for EDTs in active mode (see Table 2):

Electric Module
As shown in previous works on BETs [3,45,46] and LWTs [7][43][8] [40], computing the current profile I(x) for each of the four configurations in Fig. 1 and 2 requires a specific analysis.For convenience, we moved the mathematical details to the Appendix and enumerate here the main novelties and contributions of this work.Firstly, a recent progress on Orbital-Motion-Theory to determine the SCL/non-SCL transition point [47] was incorporated in the LWT model, thus improving the accuracy of the physical model.Secondly, the parametrization strategy first proposed in Ref. [31] for BETs in the passive mode has been extended to cover BETs in the active mode and LWTs in the passive and the active modes.As shown in Sec. 3, it impacted notably on the efficiency and robustness of the software.We finally highlight the complex longitudinal structures of the tethers shown in Figs. 1 and 2, and the use of a power supply in the passive mode.
For convenience, we now analyze the impact of the input power on the average current I av of a BET in the passive mode.Since this is a novel feature of the electric model, an analysis is needed before presenting simulation results in Sec. 3. As shown in Appendix B, the normalized average current i av of a BET with cross-sectional area A t and conductivity σ t depends on the normalized potential drop of the AEE (φ C ), the dimensionless resistance (r) and input power (w e ), and the normalized lengths of the insulated (ξ i ) and bare (ξ b ) tether segments [find the definitions of these parameters in Eqs.(A. 16) and (A.17)] For a conventional BET (i.e., ξ i = φ C = r = w e = 0), one has i av ≤ 1 for all ξ b .In particular, i av admits the approximate form i av ≈ 0.3ξ of the electric model because they coincide with previous results in the literature [45,31].Panel (c) shows the impact of adding a power supply that works collaboratively with the motional electric field.Clearly, tether performances are improved and, similarly to bare-photovoltaic tethers [10], the average current can exceed the short-circuit current (i av > 1) for a high enough input power.
It is well-known that, as compared with a fully bare EDT, insulating a tether segment close to the power supply of an EDT in active mode improves its efficiency [3,40].This statement holds because the electrons collected close to the power supply contribute little to the force but consume energy from the power supply.In general, the same result applies for an EDT in passive mode and equipped with a power supply.However, the impact of the insulated segment is modest because the bias is the largest near the power supply for a tether in active mode but the lowest for the proposed tether in the passive mode.

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In fact, it is negative (and there is no electron collection) for real AEE (φ C < 0).
These comments are corroborated by panel (d) in Fig. 3, which shows i av versus the ratio L i /L a for r = φ C = 0, L a = L b + L i = L * , and several input powers.
For w e = 0, insulating a tether segment always deteriorates the performance in the passive mode.The larger the insulated fraction, the lower i av .However, for 220 w e = 0, i av exhibits a maximum at a certain fraction of L i /L a .Nonetheless, as advanced, the improvement in the performances is modest.For this reason, we will take L i = 0 in our simulations in the passive mode.

Simulation results
This section presents simulations with BETsMA v2.0 in deorbiting and reboost scenarios.Emphasis is made on the deorbiting maneuver aided by onboard power, and in deorbiting/reboost missions with constant current by varying dynamically the resistance of the resistor and the supplied power.Due to their larger maturity as compared to LWTs, we focus the analysis on BETs.
IRI, IGRF, NRLMSISE-00, MASTER-2009, and up to 4th order harmonics are used for computing the plasma density, the magnetic field, the density of neutrals, the flux of small debris, and the Earth's gravitational field.The tether is always aligned with the local vertical.Table 3 summarizes the parameters used in the simulations.We consider a 500-kg satellite with ballistic coefficient   the EDT deorbits the satellite in less than 2 months without any power and the tether cut probability by small debris is well below 1% [panels (c) and (d)].For higher inclinations, EDT's performances obviously decrease and the benefits of using available onboard power to improve tether performances become evident.
For instance, for the interesting commercial case of Sun-Synchronous orbits (98 • of inclination), the deorbit time for W e = 0, 50, 100, 250 and 500 W are 277, 183, 158, 128 and 108 days and the cut probabilities are 1.25, 0.78, 0.68, 0.54 and 0.45%.As compared with the case with W e = 500 W, increasing the power to 1 kW (not shown) does not improve considerably the performances.These results show that even relatively short EDTs (2 km) can be used for the fast deorbiting (within a few months) of heavy satellites (500 kg) from high-inclined orbits if the available onboard power is used.
For simulations with constant power, the electric current can reach high instantaneous values.For instance, the dashed black line in panel (a) of Fig. 5 shows that the evolution of the current at the cathode for a simulation with i = 60 • and W e = 100 W takes values above several amps.Since the tether conductivity is assumed to be constant, and the AEE can emit any current at the cost of a potential drop that is constant in the actual version of the software, these high currents raise the question about the impact of these hypotheses on the results.To get some insight, we ran a set of simulations varying the inclination and using BETsMA v2.0 in the current limited mode with 100 W of available power.The input power (or the value of the resistor) were adjusted dynamically in the simulation to have a current at the cathode below 1.5 A. It  3.
was found that the impact on the performances is moderate.For instance, in the worst case of i = 90 • , the deorbit time increased less that 10%.
The impact of the solar cycle on the tether performances was investigated by repeating the simulations for W e = 100 W in Fig. 4, but now using 1/1/1995 as the starting date.This date corresponds to a solar minimum and, since the plasma and air densities are lower, the electrodynamic and the aerodynamic drag decrease and the deorbit time increases.For inclinations up to 65 • , we found that the deorbit time is a factor two higher than the one presented in Fig. 4.Such a factor grows linearly up to a maximum of 2.8 for 90 • of inclination and drops to 2.4 at Sun-synchronous orbit.Since the initial altitude is high (800 km), the Lorentz drag is clearly dominant over the aerodynamic drag for most of the simulation.

The constant current submode
Keeping constant the current at the cathode can help to avoid the instabilities that can appear in certain AEE like hollow cathodes for low current levels and also to maximize their lifetime.The scenario is illustrated by considering the parameters of Sec.There are also other phases where I C = I * C with W e = 0.This corresponds to intervals in panel (f) where the resistor is active and it acts as a current limiter.
One concludes that the injection of power, even if small, can be beneficial to reach a constant (or almost constant) current at the AEE.

The Constant Power Submode
The performances of EDTs in reboost scenarios were studied by considering the parameters of Table 3, and initial and final orbits with altitudes 450 km and H F = 600 km, respectively.The total length of the active segment is 325 L a = L i + L b = 2 km.A deep parametric analysis varying the inclination of the initial orbit, the power, and the insulated-to-active tether length ratio f i = L i /L a was carried out (find a summary of the results in Fig. 6).Panels The reboost time is well below one month for low and middle inclinations.
For instance, for 60 • of inclination, f i = 0.65 and W e = 400, 700 and 1000 W, the reboost time is around 22, 15, and 12 days, respectively.The parametric analysis also highlights the importance of making a correct design of the tether to reach high performances.For instance, if we focus on the case with 90 • of inclination and 400 W of power, we observe that the reboost time is 307, 174, and 214 days for f i = 0.5, 0.65, and 0.8.Therefore, the performances are quite sensitive to the length of the insulated segment.This parametric analyses, which requires a low computational cost, can play an interesting role in the design of EDT reboost missions.Since they use precise environmental models, perturbations, etc, they can provide an accuracy that is halfway between the one provided by analytical or semi-analytical studies [3,46,49] and detailed dynamic studies [50,51,52].
To assess the impact of the solar cycle, the set of simulations with f i = 0.65 and W e were repeated but using 1/1/1995 as starting date.As compared with the case of deorbiting shown in Sec.3.1.1,the solar activity has a moderate impact.In fact, the decrease in the aerodynamic drag is beneficial in this case as the force that needs to be overcome for raising the altitude is lower (i.e., a smaller Lorentz force is necessary).For instance, for inclinations up to 65 • , the reboost time is a factor 1.25 larger than the one shown in Fig. 6.Such a factor increases with the inclination and reaches a value around 1.5 for 90 • and Sun-Synchronous orbit.

The Constant Current Sub-mode
The constant current sub-mode for a tether in the active mode is similar to the one presented in Sec.3.1.2,but no resistor is needed as the modulation of the power is enough.The software determines and uses at every time step the  Dashed black lines in panels (a) and (b) of Fig. 7 show the evolution of the current at the AEE in the simulation of Sec.3.2.1,i.e. with a constant power and equal to W e = 400 W .It is evident that operating the EDT with a constant power in the reboost mode produces significant variations of the current at the electron emitter.Besides vanishing for E m • u t > 0, the current varies from around one to two amps, which may not be optimal for the lifetime and the operation of the emitter.Solid red lines in Fig. 7 display the results by using the constant current sub-mode with I * C = 1 A and W max e = 500 W.
A power below 400 W is needed to reach 1 Amp at the emitter during most of the maneuver.Nonetheless, there are phases where a power above 500 W is necessary to reach the target current.These phases are identified as the segments where power W e saturates to 500 W [panel (d)] and the current at   with the local vertical.The computational cost of the N-bar and N-particle dynamic models for the tether that are available in the software (and were not discussed in this work) is much higher.Regarding versatility, the proposed architecture for the EDT system combines a resistor and a power supply.This allows analyses to cover an interesting range of sub-modes like constant power, constant current, current limited, etc.Besides opening the possibility to study a larger variety of missions, having access to these sub-modes in the simulation tool helps to assess the impact on the performances of additional requirements or constraints.For instance, mitigating the dynamic instability of electrodynamic tethers may require to limit the electric current whereas the operation of the electron emitter and its lifetime can pose boundaries to the maximum and minimum current and its rate of variation.Analysis with the sub-modes of the simulation tool provides key information for the design of the system, like the requirements for the resistor and the power supply.
The simulation analysis revealed that adding a power supply between the bare tether and the electron emitter produces important benefits in deorbiting scenarios.Since the tether operates in the passive mode, such a power supply is not essential, but the performances of the system are enhanced.For instance, it was shown that the required time for deorbiting a 500 kg satellite from a circular orbit of 800 km of altitude and 98 • of inclination by a 2 km×2.5 cm×40 µm BET decreases from 277 to 158 days by adding 100 W of power.This conclusion is particularly important in post-mission disposal scenarios where the solar panel of the satellites could still provide a significant amount of power.The use of this power by the tether would require a certain level of integration with the satellite but this would be largely compensated by the potential benefits.Aided by this power, and even at high inclinations, a relatively short tether could deorbit satellites within a few months according to the simulation results.plasma density.Subscripts for i, φ and ξ denotes their values at a specific point in Figs. 1 and 2. For instance, we will write i P = i(ξ b ) and φ A = φ(0).
The dimensionless variables are very convenient because the normalized tether-to-plasma bias is then governed by [3] where the upper and lower signs hold for the passive and active modes, respectively.Equation (A.3) is coupled with the normalized electric current i(ξ), which depends on the particular type of tether segment.If bare, the current profile is governed by [3] where, as usual in bare tether analysis, we assumed Orbital-Motion-Limited (OML) law for high bias (e|V |/kT e >> 1).Current collection beyond the OML regime is incorporated in BETsMA v2 through a re-scaling of the variable L * as explained in Ref. 31.In the cathodic segment, we used that the electron-to-ion mass ratio is very small (m e /m i << 0) to write di/dξ ≈ 0. For a tether segment coated with a low-W material we use where and J RD and J ph the current densities due to the thermionic and the photoelectric effects (find formulas in Ref. to-Debye length ratio (ρ = R t /λ De ), the tether-to-electron temperature ratio (δ t = T t /T e ), the ion-to-electron temperature ration (δ i = T i /T e ) and the emission level with h p the Planck constant, W p the work function of the coating, and we assumed that the probe emits electrons following the Richardson-Dushman law.
For typical LEO and LWT operational conditions, φ * can be approximated by the minimum between zero and the law [47] φ with φ p1 =0.05ρ − 0.67 √ ρ + 0.13, (A.9) Since our work considers tape-like tether, we will use Eq.(A.8) with the equivalent radius R eq = w t /4, which was shown to be appropriate for probe analysis 435 at high bias (e|V |/kT e >> 1) [53].
Regarding the insulated segment, the current is constant (i P = i S ) and the integration of Eq. (A.3) gives The normalized bias difference between points C and S i at the top panel of Fig. 1 is with ξ a = (L b + L i )/L * ≡ ξ b + ξ i and i P = i S = i C .Therefore, the computation of i av does not require the detail of the full current profile but just the two local variables φ A and i P .They are found from the circuit equation and the analysis of the bare segment (see below), which gives the relation φ P (i P ).
As shown in Ref. [10], the integration of Eqs.(A.3) (with the minus sign) and upper (A.4) with the boundary conditions i(0) = 0 and φ(0) = φ A depends on φ A .For φ A = 1, one finds and considering the debris flux of ORDEM-2000 or MASTER-2009.All the results of this work have been found with MASTER-2009 that, as explained below, may be an adequate choice for conservative estimations.For circular orbits with altitude below 800 kilometers and two different inclinations (60 • and 98 • ), we compared the space debris flux for the small debris size (below 1 meter) given by MASTER-2009 and the latest version MASTER 8, which considers 2016 as reference year.Depending on the altitude, the inclination and the debris size, we found that the flux provided by MASTER 8 is a factor between 2 and 8 lower.This is because the minimum and the maximum of the solar cycle 24 happened in 2008 and 2014, respectively.The former are periods where the small space debris accumulates and the latter, i.e. the solar maximum, have a cleansing effect.The Fengyun-1C anti-satellite test in 2007 and Cosmos-Iridium in 2009, that resulted in a high number of small debris, were close to the reference year of MASTER-2009.Many of these fragments were removed during the solar maximum of 2014, which preceded the reference year of Master 8.This data suggests that MASTER-2009 is adequate

Figure 1 :
Figure 1: Longitudinal structures of a BET (top) and a LWT (bottom) in passive mode.
[i] constant power (W e = W e0 ), [ii] constant current (the power is adjusted dynamically to reach I C = I * C ), and [iii] station-keeping (the power is dynamically adjusted to meet the condition (F L + F A ) • v = 0 with F A the total aerodynamic force on the tether and the S/C).

Figure 2 :
Figure 2: Longitudinal structures of a BET+AEE (top) and a LWT (bottom) in active mode.

3 / 2 b
for ξ b << 1 and the exact form i av = 1 − 1/ξ b for ξ b ≥ 4[45,48].A natural question is the effect of w e on i av .The thick solid lines in panels (a)-(c) in Fig.3show i av versus ξ b for ξ i = φ C = r = w e = 0.The other lines correspond to the same parameters, except the ones indicated in the labels.For instance, panels (a) and (b) display the effect of the potential drop of the AEE and the electrical resistance of the resistor, respectively.They provide evidence about the correct implementation
50 kg/m 2 orbiting initially in a circular orbit.The drag coefficients of the satellite and the tether are C D = 2.The tether has a total length of 2 km and a tape-like cross-section of dimensions 2.5 cm × 40 µm.The potential drop of the AEE is V C = −30 V .The starting date of the simulated missions is January 1st, 2000.

3. 1 . EDT in passive mode 3 . 1 . 1 .
The constant power submodeWe first study the deorbiting from an orbit of 800 km of altitude.The mission is considered accomplished when the altitude reaches 300 km.Following the conclusions of Sec.2.3, we set L i = 0 and then L a = L b = 2 km.The input power is constant in the simulations, except when the electric current at the AEE exceeds I max = 5 A. For these particular time steps, the electric module first finds the required power to have I C = I max at the AEE and then compute the current and voltage profiles accordingly.Panels (a) and (b) in Fig.4show the deorbit time obtained in a set of simulations varying the inclination of the initial orbit.For clarity, different limits in the vertical axes were used in these two panels.For inclination below 60 • ,

Figure 4 :
Figure 4: Simulation results of the deorbit time versus initial orbit inclination for the parameters in Table3.
3.1.1and an initial inclination of 60 • .Given a target current I * C and a maximum available power W max e , the software determines at every time step the resistance R of the resistor and the input power W e of the power supply that satisfy the condition I C (t) = I * C under the constraint W e (t) < W max e .It starts by computing I C0 , which is the current at the cathode for R = W e = 0.If I C0 > I * C , then the code finds the value of R to have I C = I * C .If I C0 < I * C , it computes I max C , which is the current at the cathode for the maximum available power W max e .For I max C < I * C , the code sets the power equal to W max e .Otherwise, it determines W e in order to meet the condition I C = I * C .Once the input power and the resistance are known, the code computes the current and voltage profiles needed for the Lorentz force.Solid red lines in panels (a), (c), and (e) of Fig. 5 show the current at the cathode, the power provided by the power supply, and the power dissipated in the resistor for I * C = 1 Amp and W max e = 100 W. The right panels present a detail of the same magnitudes around the tenth day of the mission.For convenience, the results of the simulation in the standard mode (W e = R = 0, dashed black lines) are also included.As shown in panel (a), the electric current has very large variations in the standard mode: from zero (when E m • u t < 0) to several amps.In the constant current sub-mode, it practically varies from zero during short periods with E m • u t < 0 to the target value, where it spends most of the time.A detailed inspection of panels (b), (d), and (f) provides more useful information.For instance, just after the beginning of the tenth day, the current goes to zero because E m • u t < 0 and the code sets W e = RI 2 C = 0. Afterwards, it is observed that the target current is reached with R = 0 and W e varying below the maximum W max e .There are phases in panel (b) where I C is below I * C because W e in panel (d) reaches the maximum available power W max e .

Figure 5 :
Figure 5: Evolution of the current (top), the power (middle), and the power dissipated in the resistor (bottom), in the standard and the constant current modes for inc = 60 • .
(a),(c) and (e) show the required reboost time versus the orbit inclination for different values of the power and f i = 0.5, 0.65 and 0.8, respectively.For clarity, panels (b), (d), and (f) show the same information but with a different upper limit in the vertical axis.

Figure 6 :
Figure 6: Time for reboosting a 500-kg satellite from a circular orbit of 450 km of altitude to 600 km by using an EDT of dimensions 2 km × 2.5 cm × 40 µm.In panels (a)-(c) the fraction of the insulated segment is 0.5, 0.65, and 0.8, respectively.

375 the cathode is below 1
Amp [panel (b)].For this particular set of parameters, moving from the constant power sub-mode to the constant current sub-mode increases the reboost time from 21 to 30 days.In return, the operation of the electron emitter is eased.

Figure 7 :
Figure 7: Current at the cathode (top) and injected power (bottom) for the constant power (dashed black lines) and the constant current modes (solid red lines).

380
The electric module, i.e. the algorithm that computes the current and voltage profiles throughout the tether, is a critical component of EDT soft-ware because it affects its computational cost and versatility.In the case of BETsMA v2.0, its electric module handles BETs and LWTs in active and passive modes and integrates recent results on Orbital-Motion-Theory (OMT) for emissive probes.As shown in the Appendix, a parametrization proposed in a previous work for BETs in passive mode has been used and extended to cover the two tether types and modes.An important conclusion validated by the simulation campaign of this work is that such strategy yields an efficient and robust code.For instance, 1 day of simulation is typically computed in less than 1 second by a regular workstation.All the simulations of this work, which include parametric analyses of deorbiting and reboost scenarios varying orbit inclination, input power, and tether longitudinal design, are performed in less than 2 hour of machine time.These figures only apply to the tether model assumed in this work, which corresponds to a straight and inelastic tether aligned 40).Equation (A.5) considers OML current collection (bottom), and SCL (middle) and non-SCL (top) electron emission.As recently shown by detailed kinetic analysis of emissive probes with cylindrical geometry in the framework of the Orbital Motion Theory[47], the transition bias φ * that separates SCL from non-SCL regimes depends on the tether radius-

Table 1 :
Characteristics and physical and mathematical models of BETsMA v2.SCL and

Table 3 :
Parameters of the simulation.