γ
<
θ

=

ℓ
h

⁢

sin
⁢

θ
⁢

❘
"[LeftBracketingBar]"

(

cot
⁢

θ

-

cot
⁢

γ

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❘
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,

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where (θ, γ) ϵ {0, π / 2} .

By combining the calculated probabilities of striking a stage of blades with the emission distribution of particles that collide, a complete description of the particle transmission probability for each stage may be obtained for particles traveling both into and out of the full TASC configuration. In turn, this allows for a calculation of particle collection efficiency and compression ratio in the following formula:

$\begin{matrix} η_{s} = \frac{ψ_{o u t}^{+ s}}{ψ_{0}^{f}}, & (3) \end{matrix}$

$\begin{matrix} r_{s} = \frac{n_{o u t}}{n_{0}} = \frac{A_{i n}}{A_{o u t}} \frac{v_{0}}{v_{t}} \frac{ψ_{o u t}^{+ s}}{ψ_{0}^{f}}, & (4) \end{matrix}$

where

$ψ_{0}^{f}$

is the flux of particles entering the TASC,

$ψ_{out}^{+ s}$

is the flux of particles that are captured by the TASC according to the transmission probabilities of the combined analytical model, A_inis the area of the inlet of the TASC, and A_outis the area of the outlet of the TASC.

In particular embodiments, some of the performance parameters described above may be fixed and some may be allowed to vary. Those performance parameters that are variable may be optimized to maximize the performance of the TASC in terms of η_sand r_s. In particular, embodiments, system size and inlet and outlet size may be constrained by the full space-vehicle requirements. In particular embodiments, the blade angles, aspects ratios, and number of stages may be varied to substantially optimize TASC performance.

In particular embodiments, a TASC's performance may be optimized with particular design decisions for blade angles and aspect ratios for each series of stages in the TASC. In particular embodiments, a TASC performance may be improved by varying other design parameters within other constraints of the system. The following are example design decisions that may, in particular embodiments, facilitate optimization of a TASC's performance:

- To limit air-particle losses and maximize air particle collection, the first stage of blades should be placed as close to the inlet plane as possible.
- The total collection of stages of blades should be oriented so that on the whole, the system of blades maximizes the transparency of particles that enter the scoop and minimizes the losses of particles back out of the scoop. In practice, this may require approximately that the first stages of blades should be designed for maximum transparency to the incoming particles (to allow them to enter the turbomolecular air scoop) and the later stages of blades should be designed to minimize losses of particles that have been captured. In between stages may be oriented in hybrid configurations.
- The angle of the first stage of blades should be oriented to maximize transparency of particles at the orbital speed. This is accomplished by comparing the orbital speed at the altitude of flight, typically near 7800 m/s to the velocity of the blades created by the rotation speed of the turbomolecular air-scoop rotor and the radius length of the blade. The angle formed by these two speeds is the optimal angle of the blades to maximize transparency. This optimal angle, γ, is defined by cot γ=v_b/v_o, where v_b, the blade velocity, is the product of the rotor angular rotation frequency, ω, and the blade radius, v_b=ωr_b, and v_ois the orbital velocity (and thus the average velocity that particles enter the first stage of blades of the turbomolecular scoop.
- The aspect ratio of the first stage of blades should be as wide as possible to maximize the chance of recapture of particles lost by the later stages. This width may be limited by practical concerns in particular embodiments.
- The middle stages of blade should be oriented to simultaneously allow some transparency to fast moving air particles while minimizing the transparency to back streaming particles that have thermalized within the scoop. This requires a shallower blade angle to ensure some retention against lost particles. Since the shallower blade angle is not perfectly transparent to high velocity particles, this requires a blade aspect ratio that is shorter to ensure good transparency.
- In particular embodiments, the middle stages should be oriented with an angle that allows maximum transparency of particles that have thermalized during the first stage. This may be calculated similarly to the calculation for fast moving particles, such that an angle, δ, can be chosen such that by cot δ=v_b/v_t. In particular embodiments, this angle may vary during the optimization process depending on the number of stages used and other parameters.
- The final stages should be oriented to maximize particle retention and compression of the captured air, thus requiring a smaller blade angle than the first and middle stages. These stages do not need to be optimized to be maximally transparent to high-speed incoming particles because most collected particles will thermalize before these stages.
- The entire series of stages of blades for the rotor may be optimized together to substantially maximize both the collection efficiency and the compression ratio of this system.
- In particular embodiments, increasing the rotation speed of the rotor will improve the TASC collection efficiency and compression ratio. When blade angles and aspect ratios have been optimized for a given performance speed, a TASC performance will not increase substantially for rotations speeds above the optimization point. Therefore, it is imperative to optimize the TASC geometry for its designed rotation speed.
- In particular embodiments, TASC performance depends on the size of the outlet hole that transmit captures air to the propellant line and rest of the propellant flow system. A large outlet hole will allow for better capture of collected air at the expense of compression ratio. A small outlet hole will increase the compression of the system at the expense of collection efficiency. FIG. 19 illustrates an example eight-stage TASC with θ_sand α_ssubstantially optimized for r_in=0.25 m, r_out=0.0015 m, and ω_rpm=60,000 rpm, with fixed ω_rpm, θ_s, α_sand plots of η_sand r_sversus varying r_out. The eight-stage system in the example of FIG. 19 is substantially optimized to operate at approximately 60,000 rpm. As output area increases, collection efficiency improves at the expense of compression ratio. For a substantially optimized version of an eight-stage TASC system, a small r_out=0.0015 m may be chosen to reflect the small tubing size that is typically used in the thruster plumbing.
  
  Although particular performance parameters, particular optimizations, particular design decisions, particular design parameters, and particular constraints are described and illustrated herein, this disclosure contemplates any suitable performance parameters, any suitable optimizations, any suitable design decisions, any suitable design parameters, and any suitable constraints.

FIGS. 1A and 1B illustrate an example satellite 100 with an example TASC 102. Although a particular satellite with particular components (including a particular air-scoop) in a particular arrangement is described and illustrated, this disclosure contemplates any suitable satellite with any suitable components (including any suitable air-scoop) in any suitable arrangement. In addition, FIGS. 1A and 1B are not necessarily to scale.

FIG. 1A provides a first view of satellite 100, with the cylinder of TASC 102 illustrated with a cutaway to show the blades of TASC 102. FIG. 1B provides a second, side view of satellite 100, also with the cylinder of TASC 102 illustrated with a cutaway) with components of satellite 102 indicated with reference numbers. The example satellite in FIGS. 1A and 1B includes one or more electric thrusters 104. In particular embodiments, one or more TASCs 102 collect and compress air or other chemical compounds present to be stored for later use or immediately used in electric thruster 104. In particular embodiments, electric thruster 104 produces thrust by converting electric energy into kinetic energy in the propellant and is capable of utilizing a variety of chemical elements and compounds. In particular embodiments, electric thruster 104 operates on air or other chemical elements or compounds either collected or stored on board satellite 100. In particular embodiments, electric thruster 104 includes an electric-thruster cathode that may operate on air and other collected chemical elements and compounds. In particular embodiments, microwave cathodes, radio-frequency (RF) cathodes, electride cathodes, or other oxygen-compatible cathodes may be used to reduce or eliminate oxygen poisoning. In particular embodiments, it may be useful to operate the electric-thruster cathode on xenon or other chemical elements to address this issue.

Satellite 100 includes an example TASC 102 coupled to a TASC controller 106. In particular embodiments, TASC controller 106 includes one or more computers that control operation of TASC 102, such as, for example, controlling the rotation speeds of one or more blades 108 of TASC 102. Satellite 100 also includes a propellant tank 110 coupled to TASC 102 by a propellant inlet line 112 with an inlet valve 114. Propellant tank 110 is also coupled to electric thruster 104 by a propellant outlet line 116. Electric thruster 104 is also coupled to a power processing unit (PPU) 118 by a power-line harness 120. PPU 118 is also coupled to a flow controller 122 on propellant outlet line 116 by a flow-controller cable 124. In particular embodiments, PPU 118 provides power conditioning, discharge supply, and control for electric thruster 104 and flow controller 122. PPU 118 may also provide fault protection for satellite 100. Satellite 100 also includes a payload 126, which may include hardware, software, or other equipment for satellite broadband communication, EO, or other functions. The satellite also includes one or more solar array(s) 128 that provide power to satellite 100.

FIGS. 2A, 2B, and 2C illustrate an example TASC 200. FIG. 2A provides a first view of TASC 200 with its cylinder illustrated with a cutaway to show blades 202 of TASC 200. FIG. 2B provides a second, side view of TASC 200, with the same cutaway. FIG. 2C provides a third view looking into the opening of TASC 200, also with the same cutaway. FIGS. 2A, 2B, and 2C are not necessarily to scale. In the example of FIGS. 2A, 2B, and 2C, TASC 200 includes a cylindrical opening with multiple rotating-blade stages 206 that collect and compress atmospheric gas that streams into the opening at orbital velocity. Particular embodiments may include alternating clockwise (CW) and counter-clockwise (CCW) rotating stages 204. For example, stages 204a, 204c, 204c, and 204g may rotate CW and stage 204b, 204d, 204f, 204h may rotate CCW. TASC 200 ingests particles with orbital velocity (for example approximately 7,800 m/s) that may enter in near parallel trajectories, and blades 202 in TASC 200 have particular orientations and shapes at each stage 204 designed to facilitate collection and compression of high-speed atmospheric gas.

TASC 200 is formed by a cylindrical shell 206 constituting its main body. At the top, the shell is open and this forms an inlet 208 of the device. At the bottom, the shell is closed except for an outlet hole 210 through which collected air is fed into a thruster or propellant tank. An axial shaft 212 driven by one or more motors, which are not shown, constitutes a common connection point for rotating stages 204. One or more of these motors may be brushless electric motors. Attached to axial shaft 212 are rotor stages 204, each designed to substantially maximize collection and compression, as described below. Non-moving stator stages may be placed between rotor stages 204. These stators may be substantially optimized, as described below, to reduce azimuthal velocity of the gas while allowing fast moving orbital-speed particles to pass through efficiently.

To help balance gyroscopic forces on the spacecraft, two TASCs, designed to spin in opposite directions but otherwise substantially identical to each other, may be placed side by side. In particular embodiments, the rotation speed of one or more TASCs may be controlled to use gyroscopic moments for space-vehicle attitude control. Alternatively, in particular embodiments, a single TASC with two counter-rotating sets of blades may be used. The TASC may include two counter-facing motors, with one at the front of the inlet and the other behind the cylindrical chamber. These motors may independently spin blades connected to the shaft and blades connected to the cylinder walls in opposite directions. The rotation speeds may be set so that the net angular momentum of the two rotating systems may be controlled. The counter-rotors may be designed according to the same principles as the rotors, which are described below.

The blades for each stage in the TASC may have a particular angle and aspect ratio. In particular embodiments, the angles and aspect ratios of the rotors cause the rotors to collectively appear substantially transparent to the orbital-speed particles entering the TASC, while simultaneously trapping the back-streaming particles and preventing them from escaping. As described below, this may be accomplished by fixing the angles to be very high for the first stage(s). For the first stage(s), this angle is set to substantially match the angle formed by the expected velocity of the incoming orbital particles and the velocity at which the blades move (set by the rotation speed and radial position of the blade). For example, if the rotors rotate at 60,000 rpm and are 25 cm in length, they may be angled at 79 degrees. Because the tips of the blades move faster than the inner portion, this angle may be varied along the length of the blade. While these specific angles may be applicable for the given example, other appropriate angles may be contemplated for particular embodiments.

The blades in the last stage(s) may be designed with lower angles to trap the particles that impact the back of the TASC and are thus slowed down. The intermediate stage(s) of the TASC may be designed to balance transparency to fast incoming particles with opacity to back-streaming particles trying to escape the TASC. To do this, the blades of the intermediate stage(s) may be set to angles between those of the first and last stages. However, to facilitate greater collection of the fast-moving particles, the intermediate stage(s) may have lower aspect ratios (e.g. the blades may be narrower).

In particular embodiments, the blades of any given stage may have their angle and aspect ratio vary along the radial length of the blades to account for variation in effective linear velocity of the blades as they rotate in the cylindrical geometry. FIGS. 3A and 3B illustrate an example rotor 300 for an example three-stage TASC in which blade angle varies along the radial length of each stage 302 to form a helical geometry. FIGS. 3A and 3B are not necessarily to scale. In the example of FIGS. 3A and 3B, rotor 300 includes three stages 302 that each include multiple blades 304. Stage 302a is positioned toward a front of rotor 300 and, when installed, would be closest among stages 302 to an air inlet of the TASC and farthest among stages 302 from an air outlet of the TASC. Stage 302c is positioned toward a back of rotor 300 and, when installed, would be farthest among stages 302 from the air inlet of the TASC and closest among stages 302 to the air outlet of the TASC. Stage 302b is an intermediate stage 302 positioned between stages 302a and 302c.

In particular embodiments, multiple stages may be connected so that individual blades of a given stage are contiguous with a corresponding blade on an adjacent stage. Such a configuration may be equivalent to setting the axial spacing between two stages to approximately zero and adjusting the rotational offset between the two stages so that the corresponding blades of the two stages are contiguous. Any suitable number of stages may be set to be contiguous in this manner (as shown in the example of 4A and 4B) to form sections of blades with varying blade parameters. In particular embodiments, a section includes either one stage or multiple stages that are aligned and connected. Adjacent sections may, but need not necessarily, be axially spaced from each other. Herein, reference to a stage may encompass a section, and vice versa, where appropriate. In particular embodiments, connected stages may be spaced a finite distance and rotational offset apart and the corresponding blades of adjacent and connected stages may be connected via a small extension of the stage to ensure contiguity of the blades. FIGS. 4A and 4B illustrate an example rotor 400 in which stages 402b, 402c, 402d, and 402e are aligned and connected. FIGS. 4A and 4B are not necessarily to scale. In the example of FIGS. 4A and 4B, rotor 400 includes five stages 402 that each include multiple blades 404. Stage 402a is positioned toward a front of rotor 400 and, when installed, would be closest among stages 402 to the air inlet of the TASC and farthest among stages 402 from the air outlet of the TASC. Stage 402c is positioned toward a back of rotor 400 and, when installed, would be farthest among stages 402 from the air inlet of the TASC and closest among stages 402 to the air outlet of the TASC. Stages 402b, 402c, and 402d are intermediate stages 402 positioned between stages 402a and 402e. Each blade of stage 402b is contiguous with a corresponding blade of stage 402c (e.g. blade 404b is contiguous with blade 404c); each blade of stage 402c is contiguous with a corresponding blade of stage 402b and with a corresponding blade of stage 402d (e.g. blade 404c is contiguous with blade 404b and with blade 404d); each blade of stage 402d is contiguous with a corresponding blade of stage 402c and with a corresponding blade of stage 402e (e.g. blade 404d is contiguous with blade 404c and with blade 404c); and each blade of stage 402e is contiguous with a corresponding blade of stage 402d (e.g. blade 404e is contiguous with blade 404d). Stage 402a may constitute a blade section of rotor 400, and stages 402b, 402c, 402d, and 402e may constitute another blade section of rotor 400. Herein, reference to a blade may encompass multiple connected blades, and vice versa, where appropriate. For example, blades 404b, 404c, 404d, and 404c may collectively referred to as a blade.

In particular embodiments, multiple stages that have been connected into a single contiguous section may be transformed from a series of discrete stages within a section, such as those show in FIGS. 4A and 4B, into a smoothly varying blade geometry where blade angle and aspect ratio may vary along the axial length of the connected section that approximates the discrete-stage configuration.

In the example of FIGS. 1A and 1B, the rotor of TASC 102 includes eight rotatable blade stages and eight rotatable blade sections. Each of those rotatable blade sections includes one rotatable blade stage. In the example of FIGS. 2A, 2B, and 2C, the rotor of TASC 200 includes four rotatable blade stages, four counter-rotatable blade stages, four rotatable blade sections, and four counter-rotatable blade sections. Each of those rotatable blade sections includes one rotatable blade stage, and each of those counter-rotatable blade sections includes one counter-rotatable blade stage. Reference herein to a rotatable blade stage may encompass a counter-rotatable blade stage, and vice versa, where appropriate. Reference herein to a rotatable blade section may encompass a counter-rotatable blade section, and vice versa, where appropriate. In the example of FIGS. 3A and 3B, rotor 300 includes three rotatable blade stages 302 and three rotatable blade sections. Each of those rotatable blade sections includes one rotatable blade stage 302. In the example of FIGS. 4A and 4B, rotor 400 includes five rotatable blade stages 402 and two rotatable blade sections. One of those rotatable blade sections includes rotatable blade stage 402a, and another one of those rotatable blade sections includes rotatable blade stages 402b, 402c, 402d, and 402e. Although particular rotors with particular blades, stages, and sections are described and illustrated herein (e.g. in FIGS. 1A, 1B, 2A, 2B, 2C, 3A, 3B, 4A, and 4B), this disclosure contemplates any suitable rotors including any suitable blades, stages, and sections.

Propulsion requirements for a satellite using air-breathing EP to maintain orbit may be derived as follows. The orbital velocity of a spacecraft circling the earth at an altitude h can be described as

$\begin{matrix} v_{o r b} = \sqrt{\frac{G M_{E}}{R_{E} + h}}, & (5) \end{matrix}$

where G is the gravitational constant and M_Eand R_Eare the earth's mass and mean radius. At LEO, orbital velocity may be approximately 7,800 m/s. The drag force acting on the satellite F_Dis a function of the mean atmospheric density n_atmand atomic mass m, orbital velocity, drag coefficient C_Dand the spacecraft cross-sectional area A_sat.

$\begin{matrix} F_{D} = \frac{1}{2} m n_{a tm} v_{o r b}^{2} (C_{D_{s a t}} A_{sat}), & (6) \end{matrix}$

where m and n_atmare altitude-dependent parameters. In particular embodiments, the air-breathing EP concept involves ingesting the incoming atmospheric flow through the inlet area A_in. The flow rate through the inlet m_inis proportional to the mean atomic mass, atmospheric density, orbital velocity, and the inlet area according to the following relationship

$\begin{matrix} {\dot{m}}_{i n} = m n_{at m} v_{o r b} A_{i n}, & (7) \end{matrix}$

and the total collected propellant is dependent on the air-scoop efficiency η_ssuch that

$\begin{matrix} {\dot{m}}_{c} = η_{s} {\dot{m}}_{i n} . & (8) \end{matrix}$

These relationships allow the expression for the drag force to be rewritten in terms of the collected mass flow:

$\begin{matrix} F_{D} = \frac{{\dot{m}}_{c} v_{o r b}}{2 η_{s}} C_{D}, & (9) \end{matrix}$

where for simplicity, we have adopted

$C_{D} = C_{D_{s a t}} \frac{A_{s a t}}{A_{i n}} .$

In particular embodiments, to maintain orbit, the propulsion system should produce sufficient thrust to overcome the drag force expressed in Equation 9. Particular embodiments may do this by producing thrust continuously throughout the orbit. In that case, the flow rate through the thruster may be as high as the collected flow rate {dot over (m)}_c. Alternatively, in particular embodiments, a satellite may collect gas throughout the orbital period to store in an onboard propellant tank and only thrust during a short portion of the orbit. A duty-cycle parameter, [Image Omitted] , may be defined to represent the ratio of thruster on-off times, such that {dot over (m)}_T=Q{dot over (m)}_c, where for a thruster firing continuously, =1. If a thruster is only fired for 10% of the orbit, =10.

In particular embodiments, thrust necessary to overcome the drag force experienced during a single orbit may be defined as

$\begin{matrix} T > \frac{1}{2} Q {\dot{m}}_{n} v_{o r b} C_{D} . & (10) \end{matrix}$

Using T=g{dot over (m)}_TI_sp, we get:

$\begin{matrix} I_{s p} > \frac{v_{o r b} C_{D}}{2 g η_{s}}, & (11) \end{matrix}$

where g is the earth gravitational acceleration and I_spis the specific impulse. Together, Equations 10 and 11 may be seen as imposing minimum performance requirements on a thruster. This requirement may be generic and applicable to any suitable air-breathing satellite system.

The power consumption of an EP device may be expressed in terms of thrust T, specific impulse I_sp, and thruster efficiency η_t, i.e.,

$\begin{matrix} P_{T} = \frac{Tg I_{s p}}{2 η_{t}} . & (12) \end{matrix}$

Using Equations 10 and 11, the power requirement for an air-breathing EP satellite may be derived as

$\begin{matrix} P_{T} > Q \frac{{\dot{m}}_{i n} v_{o r b}^{2} C_{D}^{2}}{8 η_{t} η_{s}} . & (13) \end{matrix}$

The orbit-averaged power requirement may be written using equations above:

$\begin{matrix} 〈 P_{T} 〉 > \frac{{\dot{m}}_{i n} v_{o r b}^{2} C_{D}^{2}}{8 η_{t} η_{s}} = \frac{m n_{a t m} v_{o r b}^{3}}{8 η_{s} η_{t}} C_{D}^{2} A_{i n} . & (14) \end{matrix}$

The power produced by a spacecraft solar array may be computed as a product of the solar energy flux φ, solar array area A_sa, and solar array efficiency η_saas

$\begin{matrix} P_{s a} = η_{s a} ϕ A_{s a} = ϕ_{s a} A_{sa}, & (15) \end{matrix}$

where A_samay be larger than the front-projected satellite cross-sectional area, A_sat, and φ_sais a maximum power produced by the solar arrays per unit area. The orbit-averaged available power may be written as

$\begin{matrix} 〈 P 〉 = η_{p} ϕ_{s a} A_{sa}, & (16) \end{matrix}$

where η_pis a geometric efficiency factor that takes into account eclipses and non-optimal pointing of the solar arrays. In a perfectly sun-synchronous dusk-dawn orbit η_p=1 and, in the worst-case eclipses and orientations, η_p=1/π.

Combining Equations 14 and 16 may produce

$\begin{matrix} η_{s} η_{t} > \frac{m n_{a t m} v_{o r b}^{3}}{8 η_{p} ϕ_{s a}} C_{D}^{2} \frac{A_{i n}}{A_{s a}} = \frac{{\dot{m}}_{i n} v_{o r b}^{2} C_{D}^{2}}{8 〈 P 〉} . & (17) \end{matrix}$

In particular embodiments,

$C_{D}^{2} \frac{A_{in}}{A_{sa}}$

is constant for a given satellite geometry and orientation, since it only involves areas and the drag coefficient. In particular embodiments, for fixed altitude (and for a given satellite design), the required thruster efficiency may scale inversely with the air-scoop efficiency, as shown in FIG. 6 (which is described below). A less efficient thruster may require a more efficient air-scoop. In particular embodiments, orbital velocity is a weak function of the orbital altitude h, since for LEO orbits R_E>>h. In particular embodiments, atmospheric density n_atmand the total collected propellant flow rate, {dot over (m)}_c, are strong functions of altitude.

Equation 17 may be useful in particular embodiments for selecting appropriate thruster and air-scoop performance for a given mission. The left-hand side may be interpreted as the overall system efficiency, η=η_sη_t, and the right-hand side may be constant for a fixed satellite geometry and mission orbit.

The earth's atmosphere consists of various gas molecules whose concentration changes as a function of altitude, longitude and latitude, solar weather, and other factors. While there are many atmospheric models that predict species concentration, particular embodiments may rely on the NRL MSISE00 model, which relies on the ISO 14222 standard, to derive the overall system requirements. Density distribution as a function of altitude for typical conditions is plotted in FIG. 5, above. FIG. 5 shows model results taken at zero latitude (equator) and zero longitude (prime meridian) at 8:03:20 on 21 Jul. 2009. Above 180 km, atomic oxygen dominates other atmospheric constituents. Nitrogen is the second dominant species and, below 180 km, it dominates atomic oxygen. The total density n_atmand the mean atomic mass m at a given altitude may be used to calculate mass flow rate, where the mean atomic mass is computed

$\begin{matrix} m = \frac{\sum_{s} m_{s} n_{s}}{n_{a t m}}, & (18) \end{matrix}$

where the summation is performed over all species shown in FIG. 5. In particular embodiments, atomic oxygen may degrade satellite components.

In particular embodiments for a given spacecraft geometry, air-scoop and propulsion requirements may be determined by the mission altitude. Since the orbital velocity is only a weak function of altitude at LEO, these requirements are primarily dependent on the density of the atmospheric gas. In particular embodiments, one such requirement is the air-scoop compression ratio r_s. The air-scoop compression ratio r_smay affect the efficiency of the EP system. An EP system may require sufficient gas density to efficiently ignite and sustain a plasma discharge. In particular embodiments, this density is typically around n_ig=3×10¹⁹m⁻³. In particular embodiments, the air-scoop may have a compression ratio determined by the fraction of n_igand the incoming atmospheric gas density, such that

$\begin{matrix} r_{s} > \frac{n_{i g}}{n_{atm} (h)} . & (19) \end{matrix}$

In particular embodiments, four parameters may together substantially determine an air-breathing EP system's performance for a given satellite geometry and also substantially determine whether that performance will be sufficient to overcome drag forces on the satellite: air-scoop efficiency η_s, air-scoop compression ratio r_s, thruster efficiency η_t, and average power available [Image Omitted] P. In particular embodiments, the required values for these parameters may be functions dependent on the orbital altitude h. This dependence may also be expressed through the total mass flow incident to the inlet {dot over (m)}_in, which may reduce the need to explicitly define satellite geometric areas.

FIG. 6 illustrates example air-scoop efficiency η_sas a function of thruster efficiency η_tfor varying lines of constant altitude at fixed power, as expressed by Equation 17. Performance curves of required η_sare illustrated as a function of η_tfor varying altitudes assuming [Image Omitted] P=1 KW, C_D=3.5, and A_in=1 m². These values approximate the actual parameters taken from the GOCE mission (a mission that flew at ULEO altitudes and maintained orbit temporarily with finite stored propellant). The shaded region is the region of viability if η_s<0.8, η_t<0.25, and r_s<15,000. The altitude dependent data is taken from FIG. 5, and the associated compression ratios to reach efficient ionization are listed.

The data in FIG. 6 is replotted in FIG. 7 to illustrate a mission-efficiency metric, η, which may be defined as η=η_s×η_t, as a function of orbit-averaged power for different inlet mass flow rates. For these curves, the orbital velocity is assumed to be 7.8 km/s. Performance curves of required η are illustrated as a function of [Image Omitted] P for varying altitudes, assuming C_D=3.5 and A_in=1 m². The r_sand {dot over (m)}_infor each line of constant altitude are shown. The shaded region is the region of viability if η_s<0.8, η_t<0.25, and r_s<15,000.

The data in example FIG. 6 is also replotted in FIG. 8 in terms of the orbit-averaged power requirement as a function of altitude and mission efficiency, η. Performance curves of required [Image Omitted] P are illustrated as a function of altitude (and required r_s) for varying η, assuming C_D=3.5 and A_in=1 m². The shaded region is the region of viability if η_s<0.8, η_t<0.25, and r_s<15,000.

In particular embodiments, three thruster-performance parameters may particularly influence thruster technology selection for an air-breathing EP application: thrust, specific impulse, and efficiency. Thrust presents an obvious requirement. In particular embodiments, the thruster should produce sufficient force to overcome drag. In particular embodiments, a thruster should produce specific impulse that is at least 780 s (V_orb/g), which means that, on average, particles should leave the system faster than they enter it. In particular embodiments, specific impulse should be significantly higher than that. Equation 11 states that a system should produce specific impulse high enough to offset the momentum loss due to all the incoming particles striking the satellite, i.e., the drag. Thus, in particular embodiments, exhaust velocity on the order of twice the orbital velocity or even more should be achieved by the thruster. For example, assuming C_D=3.5 and air-scoop efficiency η_s=0.8, Equation 11 places a minimum specific-impulse requirement at approximately 1,739 s. Such high specific impulse falls within the domain of EP. For example, in particular embodiments, electrostatic and electromagnetic thrusters are capable of producing specific impulse in that range.

In particular embodiments, thruster designs such as HETs and GITs may be operated on oxygen, nitrogen, and other chemical compounds and elements collected by the air scoop or stored in the propellant tank. In particular embodiments, thruster designs such as HETs and GITs may be capable of producing specific impulses around approximately 2,000 s when operated with xenon. That number may degrade significantly when operated with oxygen or nitrogen at voltages below approximately 300 V. On the other hand, standard HETs, such as the SPT-100, may produce an I_spwith oxygen within approximately 5% of the I_spproduced with xenon at voltages between approximately 300 V and approximately 400 V. Specifically, I_spof approximately 1,800 s may be produced with discharge voltage around approximately 320 V. With oxygen, I_spmay significantly degrade at lower voltages and be only about 40% of that with xenon at approximately 200 V.

Efficiency of particular embodiments of a modern HET may be around 50% when operated with xenon. That number applies for thrusters in the 1-4 kW range and drops to around 30% below roughly 500 W. This may place a limitation on the size of particular embodiments of the air-breathing system. A small system may be less efficient and may be substantially unfeasible below a certain thruster power level. On the other hand, HETs operated at higher power with higher operating voltage (around 400+ V) may be well suited for the air-breathing application. GITs may also provide sufficient performance for an air-breathing system and typically achieve higher I_spthan HETs. In particular embodiments, they may maintain some performance advantages at lower powers.

In particular embodiments, HETs and GITs require cathodes. Hollow BaO or LaB6 cathodes may be prone to oxygen poisoning and might not work well with air-breathing thrusters. Particular embodiments may use an oxygen-compatible cathode, such as an RF or microwave cathode or electride cathode. However, RF and microwave cathodes require input power and reduce overall thruster efficiency.

In particular embodiments, Equation 17 indicates a relationship between the propulsion and air-scoop requirements. Certain propulsion requirements and limitations of practical systems are described above. These limitations, together with Equation 17, drive the air-scoop system requirements in particular embodiments. Combining Equations 17 and 19, it is found that

$\begin{matrix} \begin{matrix} r_{s} η_{s} > \frac{m (h) n_{i g} v_{o r b}^{3} C_{D}^{2} A_{i n}}{8 η_{t} η_{p} ϕ_{s a} A_{s a}} \\ > \frac{m (h) n_{i g} v_{o r b}^{3} C_{D}^{2} A_{i n}}{8 η_{t} 〈 P 〉} . \end{matrix} & (20) \end{matrix}$

This equation combines the air-scoop compression ratio and collection efficiency into one metric, r_sη_s. The compression ratio is the ratio of the density required to ignite the thruster and the ambient density. In particular embodiments, both of those densities are substantially fixed properties once thruster design and the mission altitude are chosen. Thus, in particular embodiments, an air-scoop should be able to compress the incoming gas to at least the compression ratio dictated by these fixed parameters. However, in particular embodiments, compression beyond this value may not improve system performance but may be beneficial for effective propellant storage. For an air-breathing system in ULEO, compression ratios between approximately 3,000 and approximately 40,000 may be needed to fly between approximately 200 km and approximately 300 km. This may be seen in the example of FIG. 8, where compression ratio and altitude are plotted together on the horizontal axis. For example, assuming an approximately 1 m²area with an approximately 500 W orbit-averaged power, an r_sη_sof greater than approximately 4,500 may be required in particular embodiments to make an air-breathing EP system feasible.

In particular embodiments, the stages of a TASC should collectively appear substantially transparent to the high-velocity orbital particles to facilitate high capture rates. In particular embodiments, each individual stage of a TASC may be optimized to ensure this substantial transparency. In particular embodiments, the collective stages substantially minimize the rate at which thermalized particles escape back through the stages toward the inlet. In particular embodiments, each individual stage of a TASC may be optimized to ensure this escape rate is substantially minimized. In particular embodiments, the total number of stages is substantially minimized to reduce size and weight. In particular embodiments, a TASC includes sets of counter-rotating blades to facilitate gyroscopic balance. This may reduce or even eliminate a need for stators and improve overall system performance. In particular embodiments, if counter-rotating blades are impractical, two side-by-side TASCs, with rotor-stator combination, may be used to facilitate gyroscopic balance. In such a geometry, the stators may be substantially optimized using a similar methodology applied to the rotors, to facilitate transparency to the incoming orbital beam and opacity to the back-flowing particles.

Particular embodiments may substantially optimize one or more of the following design parameters:

- 1. Blade angle—This is the attack angle of the blades of a stage of the TASC. At 90 degrees, the blades are vertical. At 0 degrees, the blades are flat.
- 2. Blade aspect ratio—This is the ratio between the width of a single blade and the separation between two blades within a stage.
- 3. Rotation speed—Faster blade speed ensures greater particle capture.
- 4. Number of stages—Each stage may be given different sets of blade angles and aspect ratios.
- 5. Stage offset—Each stage may be offset azimuthally with respect to a previous stage to increase total transparency of the incoming orbital stream to all stages. Blade alignment may be set so that an orbital particle that transmits through the first stage is likely to be transmitted through all stages.
- 6. Stator angle and aspect ratio—These can be similarly optimized, but may, in particular embodiments, sit near 90 degrees to facilitate greater transparency to v_oparticles.
- 7. System size—A larger radius system will have higher blade-tip speeds. This may improve the performance of the TASC, but may consume more power.
- 8. Ratio of inlet size to outlet size—A smaller outlet ensures higher compression at the expense of capture efficiency, and vice versa.
  
  In particular embodiments, with substantial optimization of these design parameters, an η_saround 80% or greater and an r_saround 10,000 or greater may be achieved.

In particular embodiments, a one-dimensional analytical formulation for TASC performance may be derived with the following set of assumptions in the analytical formulation:

- 1. Analysis is 1-D along the TASC axis and based on overall flux conservation.
- 2. Particles enter the TASC at orbital velocity v_osubstantially monoenergetically. The thermal component of the velocity is ignored. It is estimated that particles will have temperature around 1000° C., which corresponds to S=v_o/v_to˜10.
- 3. The TASC will include N stages of blades, and each stage will move with substantially the same velocity, v_b. This velocity will be calculated assuming a stage rotating with rotation speed ω_rpmwith particles interacting with the blades at the mean radius, r=(⅔)r_in, with r_inbeing the inlet radius of the TASC.
- 4. All surface scattering is substantially diffuse with substantially perfect energy accommodation. All scattered particles take on a velocity v_t=√{square root over (8k_bT/πm)}, where k_bis Boltzmann constant, T is the temperature of the TASC walls, and m is atom mass.
- 5. Particle recombination, such as O+O=O₂, is ignored.
- 6. Between each stage, thermalized particles experience a conductance loss based on Clausing factor estimates for the assumed geometry.
  
  In the subsequent analysis, the following fluxes are defined:

$ψ_{0 or 1}^{f} = n_{o} v_{o}$

$ψ_{0 or 1}^{+ s} = n_{+ s} v_{t}$

$ψ_{0 or 1}^{- s} = n_{- s} v_{t},$

where n_ois the density of fast particles entering the TASC at orbital velocity and n_+sand n_−sare densities of thermal (slow) particles streaming right (downwind) and left (upwind) respectively. Subscripts 0 and 1 refer to boundary conditions, which are described below.

In particular embodiments, the TASC includes a number of stages that each include a set of blades rotating on a shaft. The flow of particles through the rotating blades may be computed using transmission probabilities. Representing a stage of blades using a block diagram, as shown in FIG. 9, the following set of flux-conservation equations may be written:

$\begin{matrix} ψ_{1}^{f} = P_{f}^{f} ψ_{0}^{f} & (21) \end{matrix}$

$\begin{matrix} ψ_{1}^{+ s} = P_{+ s}^{f} ψ_{0}^{f} + P_{+ s}^{+ s} ψ_{0}^{+ s} + P_{+ s}^{- s} ψ_{1}^{- s} & (22) \end{matrix}$

$\begin{matrix} ψ_{0}^{- s} = P_{- s}^{f} ψ_{0}^{f} + P_{- s}^{+ s} ψ_{0}^{+ s} + P_{- s}^{- s} ψ_{1}^{- s}, & (23) \end{matrix}$

where the subscripts and superscripts have the following meaning. The subscripts for the fluxes ψ indicate the upstream boundary 0 and the downstream boundary 1. The superscript f represents incoming orbital (fast) particles. The +s represents slower moving thermal particles moving in the positive direction, i.e., rightward further in to the TASC, and −s represents the thermal particles moving backwards in the system. Note, since all particles are assumed to thermalize when colliding with any surface, there are no fast moving particles moving backwards, and so the direction + is implicit.

In the probability terms, P, the superscript represents the incoming flow to the stage and the subscript represents the fraction of that flow that becomes outgoing for the specified flow type. For example,

$P_{- s}^{f}$

is the probability that the incoming fast particles will scatter from the blades and flow to the left as thermal particles. Therefore, by definition, there are three sets of probability terms that must add up to 1.

$\begin{matrix} \begin{matrix} P_{f}^{f} + P_{+ s}^{f} + P_{- s}^{f} = 1 \\ P_{+ s}^{+ s} + P_{- s}^{+ s} = 1 \\ P_{+ s}^{- s} + P_{- s}^{- s} = 1 \end{matrix} & (24) \end{matrix}$

Converting Equations 21-23 into matrix form, results in

$\begin{matrix} P_{s L}^{1} Ψ_{1} = P_{s R}^{1} Ψ_{0}, & (25) \end{matrix}$

where the left and right probability matrices can be written as

$\begin{matrix} P_{sL} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - P_{+ s}^{- s} \\ 0 & 0 & - P_{-}^{- s} \end{matrix}], P_{sR} = [\begin{matrix} P_{f}^{f} & 0 & 0 \\ P_{+ s}^{f} & P_{+ s}^{+ s} & 0 \\ P_{- s}^{f} & P_{- s}^{+ s} & - 1 \end{matrix}], & (26) \end{matrix}$

$and$

$\begin{matrix} Ψ_{i} = [\begin{matrix} ψ_{i}^{f} \\ ψ_{i}^{+ s} \\ ψ_{i}^{- s} \end{matrix}] & (27) \end{matrix}$

Therefore, each stage can be expressed in terms of fluxes and a single probability matrix for that stage by

$\begin{matrix} Ψ_{i + 1} = P_{s}^{i + 1} Ψ_{i} = {(P_{s L}^{i + 1})}^{- 1} P_{s R}^{i + 1} Ψ_{i} & (28) \end{matrix}$

Writing out Equation 28 explicitly for Stage 1 results in the following expression for the downstream fluxes (subscript 1) as a function of the upstream fluxes (subscript 0).

$\begin{matrix} [\begin{matrix} ψ_{1}^{f} \\ ψ_{1}^{+ s} \\ ψ_{i}^{- s} \end{matrix}] = [\begin{matrix} P_{f}^{f} & 0 & 0 \\ P_{+ s}^{f} - \frac{P_{+ s}^{- s}}{P_{- s}^{- s}} P_{- s}^{f} & P_{+ s}^{+ s} - \frac{P_{+ s}^{- s}}{P_{- s}^{- s}} P_{- s}^{+ s} & \frac{P_{+ s}^{- s}}{P_{- s}^{- s}} \\ - \frac{P_{- s}^{f}}{P_{- s}^{- s}} & - \frac{P_{- s}^{+ s}}{P_{- s}^{- s}} & \frac{1}{P_{- s}^{- s}} \end{matrix}] + [\begin{matrix} ψ_{0}^{f} \\ ψ_{0}^{+ s} \\ ψ_{0}^{- s} \end{matrix}] & (29) \end{matrix}$

In Equations 28 and 29, the individual probabilities are calculated below, and the matrix equation is a set of three equations with three inputs and three unknowns. The inputs are the incoming fluxes,

$ψ_{0}^{f}, ψ_{0}^{+ s}, ψ_{1}^{- s},$

from both sides of the stage, and the outputs are the outgoing fluxes,

$ψ_{1}^{f}, ψ_{1}^{+ s}, ψ_{0}^{- s},$

was, also exiting on both sides.

In particular embodiments, the substantially collimated fast particles are transported effectively between each stage and perfect transmission probability may be assumed for them. In particular embodiments, thermalized particles may scatter from the TASC walls. This conductance loss may be computed using Clausing factor probabilities. The flux conservation equation between the stages may be expressed as:

$\begin{matrix} Ψ_{b} = C Ψ_{a}, where C = [\begin{matrix} 1 & 0 & 0 \\ 0 & \frac{2 s - 1}{s} & \frac{1 - s}{s} \\ 0 & \frac{s - 1}{s} & \frac{1}{s} \end{matrix}], & (30) \end{matrix}$

The value of s may be calculated via Clausing factors for the given geometry. In particular embodiments, the space between stages is a cylindrical tube of length l and radius r_in. The Clausing factor is then approximated by the following formulas:

$\begin{matrix} s = \frac{1}{1 + \frac{\begin{matrix} 3 & l' \end{matrix}}{8 r_{i n}}} & (31) \end{matrix}$

$\begin{matrix} l^{'} = l + \frac{l}{3 + \frac{\begin{matrix} 3 & l \end{matrix}}{7 r_{i n}}} . & (32) \end{matrix}$

Regions between blades may not always be a perfectly open cylinder. For example, if stators are required between stages, they may be spaced closely (or far apart) and that spacing may decrease the transmission probability s from its value for an idealized cylinder. In that configuration, the transmission probability for fast moving particles may differ from

$P_{f}^{f} = 1$

and the conductance calculation may become more complicated.

Moreover, in the limit where s=1, the conductance matrix in Equation 30 becomes the identity matrix.

Equation 28 is a linear system of equations. If, between each stage, finite conductance is included according to Equation 30, then a system with an arbitrary number of stages may be created. For example, a three-stage system may be written as

$\begin{matrix} Ψ_{1} = C P_{s}^{1} Ψ_{0} - first stage, \\ Ψ_{2} = C P_{s}^{2} Ψ_{1} - second stage, \\ Ψ_{3} = C P_{s}^{3} Ψ_{2} - third stage, \end{matrix}$

where

$Ψ_{3} = C P_{s}^{3} C P_{s}^{2} C P_{s}^{1} Ψ_{0} .$

More generally, we can write

$\begin{matrix} Ψ_{N} = (\prod_{i = N}^{1} C P_{s}^{N}) Ψ_{0} = M Ψ_{0}, & (33) \end{matrix}$

where M is the matrix product of the various blade stages and the interwoven conductance matrices. Regardless of the number of stages added, the system of equations combines three equations with three input fluxes,

$ψ_{0}^{f}, ψ_{0}^{+ s}, ψ_{N}^{- s},$

and three output fluxes,

$ψ_{N}^{f}, ψ_{N}^{+ s}, ψ_{0}^{- s} .$

This system may be closed by defining the three input fluxes, leaving three equations and three unknowns.

Final closure of the system may be understood by considering the back wall of the compressor. Assume that

$P_{+ s}^{f} = 0,$

so that

$P_{f}^{f} + P_{- s}^{f} = 1,$

i.e., fast moving particles thermalize and scatter backward. The flux coming back into the N^thstage from outside the wall

$(ψ_{1}^{- s})$

is zero. As a result, the wall probabilities

$P_{- s}^{- s}$

and

$P_{+ s}^{- s}$

may be set to an arbitrary value and should not affect the solution. Therefore, without loss of generality,

$P_{- s}^{- s} = 1$

and

$P_{+ s}^{- s} = 0$

may be set. With these simplifications, Equation 29 yields the following:

$\begin{matrix} Ψ_{out} = [\begin{matrix} P_{f}^{f} & 0 & 0 \\ 0 & P_{+ s}^{+ s} & 0 \\ - P_{- s}^{f} & - P_{- s}^{+ s} & 1 \end{matrix}] Ψ_{N} = W Ψ_{N}, & (34) \end{matrix}$

where W is the wall-probability matrix, similar to the stage probability matrices,

$P_{s}^{i} .$

In particular embodiments, the probability for a forward streaming particle to transmit through the outlet into the thruster (or propellant tank) may be defined to be proportional to A_out/A_in. That is,

$\begin{matrix} P_{f}^{f} = P_{+ s}^{+ s} = \frac{A_{o u t}}{A_{i n}}, & (35) \end{matrix}$

where A_outin practice is an effective area that is dependent on the size of the outlet along with the associated conductance losses to the thruster.

Putting together the wall and stage probability matrices may produce the following equation:

$\begin{matrix} Ψ_{out} = W Ψ_{N} = W M Ψ_{0} = T Ψ_{0}, & (36) \end{matrix}$

where T is the total flux probability transmission matrix. This is a system of three equations, so if

$ψ_{0}^{f}, ψ_{0}^{+ s}, ψ_{out}^{- s}$

are known, then

$ψ_{out}^{f}, ψ_{out}^{+ s}, ψ_{0}^{- s}$

may be solved for.

The input flux quantities are not all on the same side of the equation. This may be fixed by explicitly writing out the terms of Equation 36.

$\begin{matrix} [\begin{matrix} ψ_{out}^{f} \\ ψ_{out}^{+ s} \\ ψ_{out}^{- s} \end{matrix}] = [\begin{matrix} T_{11} & T_{12} & T_{13} \\ T_{2 1} & T_{2 2} & T_{2 3} \\ T_{31} & T_{32} & T_{33} \end{matrix}] [\begin{matrix} ψ_{0}^{f} \\ ψ_{0}^{+ s} \\ ψ_{0}^{- s} \end{matrix}] & (37) \end{matrix}$

The above equation may be rearranged so that

$ψ_{0}^{f}, ψ_{0}^{+ s}, ψ_{out}^{- s}$

are all on the right side.

$\begin{matrix} [\begin{matrix} 1 & 0 & - T_{13} \\ 0 & 1 & - T_{2 3} \\ 0 & 0 & - T_{33} \end{matrix}] [\begin{matrix} ψ_{out}^{f} \\ ψ_{out}^{+ s} \\ ψ_{out}^{- s} \end{matrix}] = [\begin{matrix} T_{11} & T_{12} & 0 \\ T_{2 1} & T_{2 2} & 0 \\ T_{31} & T_{32} & - 1 \end{matrix}] [\begin{matrix} ψ_{0}^{f} \\ ψ_{0}^{+ s} \\ ψ_{out}^{- s} \end{matrix}] & (38) \end{matrix}$

This represents a substantially complete solution, but further simplifying assumptions may be made. First, only the fast particles enter the inlet, which means

$ψ_{0}^{+ s} = 0 .$

Second, particles don't return from the outlet back into N^thstage, which means

$ψ_{out}^{- s} = 0 .$

This leaves the following solution:

$\begin{matrix} [\begin{matrix} 1 & 0 & - T_{13} \\ 0 & 1 & - T_{2 3} \\ 0 & 0 & - T_{33} \end{matrix}] [\begin{matrix} ψ_{out}^{f} \\ ψ_{out}^{+ s} \\ ψ_{0}^{- s} \end{matrix}] = [\begin{matrix} T_{11} \\ T_{2 1} \\ T_{31} \end{matrix}] ψ_{0}^{f}, & (39) \end{matrix}$

and inverting, it may be seen that

$\begin{matrix} [\begin{matrix} ψ_{out}^{f} \\ ψ_{out}^{+ s} \\ ψ_{0}^{- s} \end{matrix}] = [\begin{matrix} T_{11} - \frac{T_{13} T_{31}}{T_{33}} \\ T_{21} - \frac{T_{23} T_{31}}{T_{33}} \\ - \frac{T_{31}}{T_{33}} \end{matrix}] ψ_{0}^{f} . & (40) \end{matrix}$

By continuity, it may be expected that T₁₁+T₂₁−T₃₁/T₃₃(T₁₃+T₂₃+1)=1.

Finally, the total density at the outlet of the compressor may be calculated by noting the total output flow and velocity associated with the two outflowing terms from Equation 40:

$\begin{matrix} n_{out} = \frac{A_{in}}{A_{out}} (ψ_{out}^{f} / v_{0} + ψ_{out}^{+ s} / v_{t}) . & (41) \end{matrix}$

Compressor performance may be expressed in terms of its compression ratio and collection efficiency as

$\begin{matrix} r_{s} = \frac{n_{out}}{n_{0}} = \frac{A_{in}}{A_{out}} \frac{ψ_{out}^{f} + (v_{0} / v_{t}) ψ_{out}^{+ s}}{ψ_{0}^{f}}, & (42) \end{matrix}$

$\begin{matrix} η_{s} = \frac{ψ_{out}^{f} + ψ_{out}^{+ s}}{ψ_{0}^{f}} . & (43) \end{matrix}$

These equations may be further simplified by assuming the output flow fully thermalizes before exiting the system, so that

$ψ_{out}^{f} = 0 .$

This assumption is consistent with assuming the back wall fully thermalizing the incoming particle fluxes, so that

$P_{f}^{f} = P_{+ s}^{f} = 0$

and

$P_{- s}^{f} = 1 .$

Therefore,

$\begin{matrix} r_{s} = \frac{n_{out}}{n_{0}} = \frac{A_{i n}}{A_{out}} \frac{v_{0}}{v_{t}} \frac{ψ_{out}^{+ s}}{ψ_{0}^{f}} & (4) \end{matrix}$

$\begin{matrix} η_{s} = \frac{ψ_{out}^{+ s}}{ψ_{0}^{f}}, & (3) \end{matrix}$

where r_sand η_sare each proportional to

$ψ_{out}^{+ s} / ψ_{0}^{f} .$

This assumption simplifies the T matrix, so that T_1i=0, so that the output ratio is

$\begin{matrix} η_{s} = \frac{ψ_{out}^{+ s}}{ψ_{0}^{f}} = T_{2 1} - \frac{T_{2 3} T_{3 1}}{T_{3 3}} = 1 - \frac{T_{3 1}}{T_{3 3}} . & (44) \end{matrix}$

The transmission probabilities of a particle passing through a stage may be derived using a cartesian geometry in which blades of width [Image Omitted] are spaced by distance h at an angle θ, and move in y direction, which is shown as vertical in FIG. 10. In particular embodiments, the probability of a fast particle moving with velocity v_ostriking a stationary (v_b=0) blade is cos θ/h. When the blade is stationary, the forward moving particle will strike the backward facing side of the blade.

In particular embodiments, as blade speed increases, the probability of a strike on the backward facing side of the blades decreases. Once the blade speed is sufficiently fast, particles tend not to strike the backward facing side of the blade and, instead, begin to strike the forward facing side of the blades. At this point, increasing blade speed increases the probability of a strike.

Assume a slowly moving blade, so that particles will always strike the backward-facing side. In this circumstance, the moving blade reduces the probability of a strike by v_bτ/h, as compared to a stationary blade, where the dwell time within the stage τ can be computed from the blade angle and particle speed, v_oτ= [Image Omitted] sin θ. Thus, the probability of a particle striking a slow moving blade is

$\begin{matrix} p_{st}^{f} = \frac{ℓcos θ - \cot γ ℓ \sin θ}{h} & (1) \end{matrix}$

$= \frac{ℓ}{h} \sin θ (\cot θ - \cot γ),$

where angle γ is defined such that cot γ=v_b/v_o. This γ is the angle at which particles enter the stage in the frame of reference of the moving blades, and the assumption that the blades are slowly moving is substantially equivalent to assuming γ>θ. When γ=θ, the particles enter the stage aligned with the blades and the probability of a strike goes substantially to zero.

On the other hand, when γ<θ, the entering particles will strike the forward facing sides of the blades, which corresponds to proportionally faster blade speeds than the previous case. Calculating the probability that entering particles strike the forward-facing sides in this case, it may be seen that the result in this calculation is that the two cotangents simply switch signs:

$\begin{matrix} p_{st}^{f} = \begin{matrix} {\begin{matrix} \frac{ℓ}{h} \sin θ (\cot θ - \cot γ), & ℽ > θ \\ \frac{ℓ}{h} \sin θ (\cot ℽ - \cot θ), & ℽ < θ \end{matrix} \\ = \frac{ℓ}{h} \sin θ ❘ (\cot θ - \cot γ) ❘, \end{matrix} & (2) \end{matrix}$

where (θ,γ)ϵ{0,π/2}.

In both cases,

$p_{st}^{f} > 1$

is possible, but unphysical. In the first case, where γ>θ, this corresponds to blades wide enough to overlap. In the second case, where γ<θ, this corresponds to blades moving sufficiently fast that multiple blades would pass through the particle trajectory. The true probability of the particle striking a blade,

$P_{st}^{f},$

may be confined to physically realizable values and ignore the possibility of multiple blade strikes in a single pass. Therefore,

$\begin{matrix} P_{st}^{f} = {\begin{matrix} p_{st}^{f} & p_{st}^{f} < 1 \\ 1 & p_{st}^{f} > 1 \end{matrix} . & (45) \end{matrix}$

In practice, a particle striking a blade may scatter diffusely according to both a cosine law distribution and some accommodation coefficient. In this derivation, the approach may be simplified by assuming the accommodation coefficient is one, so that particles thermalize and may flow out of the blade region in either direction. The probability of a particle becoming forward (+) or reverse (−) flowing may be estimated by assuming particles scatter with a uniform distribution off of the angled blades. As a result, if a particle strikes the forward-facing surface, its probability of becoming forward-flowing may be (π−θ)/π. Similarly, if it strikes a backward-facing surface, its probability of becoming forward flowing may be θ/π.

Together, there are four possibilities born from the fact that a particle can strike either the forward or reverse facing surfaces and then flow either forward or backwards. Therefore:

$\begin{matrix} P_{+ s}^{f} = {\begin{matrix} \frac{θ}{π} P_{st}^{f}, & ℽ > θ \\ \frac{π - θ}{π} P_{st}^{f}, & ℽ < θ \end{matrix} & (46) \end{matrix}$

$\begin{matrix} P_{- s}^{f} = {\begin{matrix} \frac{π - θ}{π} P_{st}^{f}, & ℽ > θ \\ \frac{θ}{π} P_{st}^{f} & ℽ < θ \end{matrix} . & (47) \end{matrix}$

When γ=θ, the cotangent difference is zero, and particles tend not to strike a blade. When θ=0 (i.e., flat blades), γ<0 is not possible and, so the probabilities are selected from the top rows, ensuring that

$P_{+ s}^{f} = 0$

and

$P_{- s}^{f} = P_{st}^{f} .$

Finally, when γ≈0 and θ≠0 (i.e. fast moving blades compared to the particle velocity), then γ<0 and from Equation 45,

$P_{st}^{f} = 1 .$

In this situation, particles tend to strike the forward-facing blade surface.

The probability of a particle passing the blades without a collision (and therefore remaining a fast particle) is

$P_{f}^{f} = 1 - P_{st}^{f},$

and combining this with Equations 46 and 47 results in

$P_{f}^{f} + P_{+ s}^{f} + P_{- s}^{f} = 1 .$

The transmission probability for slow (thermal) particles moving in the same forward direction as the fast particles can be calculated in the same way. However, these particles have a different velocity, v_t, so the angle δ is defined such that cot δ=v_b/v_t. The non-bounded probability of a strike becomes

$\begin{matrix} p_{st}^{+ s} = \begin{matrix} {\begin{matrix} \frac{ℓ}{h} \sin θ (\cot θ - \cot δ), & δ > θ \\ \frac{ℓ}{h} \sin θ (\cot δ - \cot θ), & δ < θ \end{matrix} \\ = \frac{ℓ}{h} \sin θ ❘ (\cot θ - \cot δ) ❘, \end{matrix} & (48) \end{matrix}$

where δϵ{0,π/2}. And the physically realizable probability of a strike becomes

$\begin{matrix} P_{st}^{+ s} = {\begin{matrix} p_{st}^{+ s} & p_{st}^{+ s} < 1 \\ 1 & p_{st}^{+ s} > 1 \end{matrix} . & (49) \end{matrix}$

A difference with the previous derivation is that the scattering occurs at the same velocity as the incoming flux. Therefore,

$P_{+ s}^{+ s}$

is the sum of the probability of striking a blade and propagating forward and the probability of passing without collision. Finally, the probabilities become:

$\begin{matrix} P_{- s}^{+ s} = {\begin{matrix} \frac{π - θ}{π} P_{st}^{+ s}, & δ > θ \\ \frac{θ}{π} P_{st}^{+ s}, & δ < θ \end{matrix} . & (50) \end{matrix}$

$\begin{matrix} P_{+ s}^{+ s} = 1 - P_{- s}^{+ s} & (51) \end{matrix}$

The final set of probabilities for the backward streaming slow (thermal) particles differs from the forward streaming case. It may be assumed that the particles have the same thermal speed, v_t, but are now moving in the negative direction. The negative velocity flips the sign in front of the cotangent term in Equation 48. Moreover, it may be unnecessary to keep track of which side of the blade the back-streaming particles strike because particles tend to only strike the forward facing side of the blades. Therefore:

$\begin{matrix} p_{st}^{- s} = \frac{ℓ}{h} \sin θ (\cot θ + \cot δ) . & (52) \end{matrix}$

To make the true probability physical:

$\begin{matrix} P_{st}^{- s} = {\begin{matrix} p_{st}^{- s} & p_{st}^{- s} < 1 \\ 1 & p_{st}^{- s} > 1 \end{matrix} . & (53) \end{matrix}$

By analogy to the forward propagating thermal case, the probability of a back-streaming (−s) particle becoming a forward flowing (+s) particle may be calculated.

$\begin{matrix} P_{+ s}^{- s} = \frac{π - θ}{π} P_{st}^{- s} & (54) \end{matrix}$

$\begin{matrix} P_{- s}^{- s} = 1 - P_{+ s}^{- s} & (55) \end{matrix}$

The seven probabilities used in the matrix of Equation 29 may be calculated:

$P_{f}^{f}, P_{+ s}^{f}, P_{- s}^{f}, P_{+ s}^{+ s}, P_{- s}^{+ s}, P_{+ s}^{- s}, and P_{- s}^{- s} .$

The forward streaming probabilities

$P_{*}^{f}$

may be determined from Equations 2 and 45-47. The slow forward probabilities

$P_{*}^{+ s}$

may be determined from Equations 48-51. The slow backward probabilities

$P_{*}^{- s}$

may be determined from Equations 52-55.

Performance of a TASC with a number of stages may be analyzed as a function of blade angles and aspect ratios, rotation speed, and the inlet and outlet areas. In this analysis, assume that the TASC is a cylinder with a radius r_inand that the TASC outlet has a circular cross-section with a radius r_out. With these assumptions, the inlet and outlet areas are

$A_{i n} = π r_{i n}^{2} and A_{out} = π r_{out}^{2}$

respectively.

In the example TASCs described below, blade speed may be taken to be v_b=(⅔)r_inω_s, for a given rotation speed, ω_s=2πω_rpm. The factor of ⅔ comes from the mean-effective blade radius for blades in a cylinder. Conductance matrices may be calculated individually for each stage as described above, and for the example TASCs described below it is assumed that the stages are substantially equidistant from each other with a separation l=0.05 m. Finally, for the example TASCs described below, the orbital velocity and thermal velocity may be assumed to be v_o=7,800 m/s and v_t=630 m/s, where v_tis calculated assuming oxygen at a temperature of 300 K.

In particular embodiments, there may be three globally controllable parameters (r_in, r_out, and ω_rpm) and two additional parameters per stage (blade angle θ_iand aspect ratio α_i= [Image Omitted] /h). The complexity of the problem may increase with the number of stages. For example, a six-stage TASC may have 15 variables to be substantially optimized. This complexity may be reduced by noting that the inlet and outlet radii only appear in the area fractions, and therefore r_inmay be fixed and variations in r_outmay be investigated. The linear blade speed may be proportional to ω_rpmand r_in. Thus, decreasing the diameter of the TASC may be substantially equivalent to decreasing the rotation speed of the stages and may produce self-similar solutions.

In the case of a single-stage TASC, the inlet and outlet fluxes may be connected by the following relationship:

$\begin{matrix} Ψ_{out} = W C P_{s} Ψ_{0} = T Ψ_{0}, & (56) \end{matrix}$

where P_sis the single-rotor transmission probability matrix as calculated above. For the back-wall matrix, W, it may be assumed that all fast particles thermalize before going through the outlet. Thus,

$P_{f}^{f}$

at the wall is zero. To explicitly calculate the transmission matrix, T, it may be assumed that r_in=0.25 m, r_out=0.025 m, and then vary ω_rpm, θ₁, and α₁.

FIG. 11 illustrates example compression ratio r_sversus example collection efficiency η_sfor an example single-stage TASC. According to Equations 42 and 43, compression ratio and collection efficiency are substantially linearly related for a given area ratio if it is assumed that substantially all particles exiting the outlet are thermalized. This assumption is reasonable for an air-breathing EP system in which the outlet of the air-scoop is connected to a thruster through a tube with a relatively small diameter. FIG. 11 demonstrates the r_s-versus-η_sdependence by plotting results of multiple combinations of angle and aspect ratio for a single-stage system. According to Equations 42 and 43, a multi-stage system may continue the same linear trend to higher r_sand η_svalues as long as the area ratio is kept substantially constant. Because of this substantially linear relationship when the area ratio is fixed, only collection efficiency is shown in FIGS. 12A, 12B, 12C, 13, 15-18, and 20-21.

FIGS. 12A, 12B, and 12C summarize example results of the example parametric study above for an example single-stage TASC by plotting collection efficiency contour plots as a function of blade angle and aspect ratio for three different rotation speeds. FIG. 12A is an example contour plot of collection efficiency of an example single stage TASC when ω_rpm=30,000 rpm; FIG. 12B is an example contour plot of collection efficiency of an example single stage TASC when ω_rpm=60,000 rpm; and FIG. 12C is an example contour plot of collection efficiency of an example single stage TASC when ω_rpm=90,000 rpm. The contour color scale of each of these figures is different to highlight the appropriate dependencies within each figure.

In particular embodiments, collection efficiency may increase with increasing rotation speed. In particular embodiments, a substantially optimum blade configuration is a function of speed and the angle and aspect ratio may both decrease with increasing speed. For example, at 60,000 rpm, particular embodiments achieve substantially optimum performance at θ₁≈7.2° and α₁≈0.83 with collection efficiency of η_s=4.3% and compression ration r_s=53. At 30,000 rpm, in particular embodiments, the substantially optimum values are θ₁≈9.9° and α₁≈0.89. However performance degrades to η_s=2.8% and compression ratio to r_s=34. At 90,000 rpm, particular embodiments achieve substantially optimum performance at similar conditions, θ₁≈6.0° and α₁≈0.80, but performance, on the other hand, improves with collection efficiency of η_s=5.7% and compression ratio of r_s=70.

At slower rotation speeds, higher blade angles may be required in order to allow more of the orbital beam through. From the derivations above, it may be noted that with the assumptions made for our 1-D model, η=γ substantially ensures that substantially all fast particles pass through the stage. However, at slow blade speeds, γ is closer to 90°. While η=γ may lead to substantially perfect transparency for the fast particles, there is a countervailing need to reduce the transparency for the back-streaming thermal particles. In particular embodiments, as the number of stages is increased, this conflicting requirement may be mitigated by allowing stage specialization, with large angles at the front stages and low angles at the back stages, with η_s=γ becoming a substantially optimal answer for the front stages.

FIG. 13 illustrates example efficiency of three example one-stage TASC configurations as a function of speed for three separate θ_s/α_scombinations: (1) substantially optimized for 30,000 rpm, θ_s=9.9°, α_s=0.89; (2) substantially optimized for 60,000 rpm, θ_s=7.2°, α_s=0.83; and (3) substantially optimized for 90,000 rpm, θ_s=6.0°, α_s=0.80. In the example of FIG. 13, the substantially optimum performance angles and aspect ratios determined for each speed shown in FIGS. 12A, 12B, and 12C are used and variations in performance as a function of speed are examined. For a fixed blade geometry, increased speed provides consistent improvement in particular embodiments. However, above the speed at which the substantially optimum geometry is found, improvement is limited in particular embodiments. Below the substantially optimum speed, the performance may drop off significantly in particular embodiments. Therefore, in particular embodiments, the blade geometry may be substantially optimized for a slightly lower speed than its intended operation.

While in particular embodiments r_sand η_sotherwise vary proportionally to each other as shown in the example of FIG. 11 and demonstrated by Equations 42 and 43, adjusting the input-output area ratio is a method in which these two parameters distinctly decouple. FIG. 14 illustrates example efficiency and compression ratio of an example one-stage TASC as a function output radius, r_out, for θ_s=7.2°, α_s=0.83, and ω_rpm=60,000 rpm. In the example of FIG. 14, varying r_sand η_sare plotted as a function of r_outfor an otherwise fixed parameter single-stage example TASC. In particular embodiments, the area fraction may be used as a lever to trade off collection efficiency and compression ratio. Moreover, there is a substantially optimum value of r_outthat maximizes the r_sη_s.

In particular embodiments, a single-stage device may be unable to achieve performance necessary for a viable air-breathing system. More stages may be necessary. Below, a configuration with multiple stages is considered, but the blade geometry is fixed to be substantially identical for all stages. By keeping the blade orientations uniform, a single θ_sand α_sare maintained throughout the geometry.

FIG. 15 illustrates example collection efficiency for four different example TASCs with one, two, four, and eight uniform stages with r_in=0.25 m, r_out=0.025 m, and ω_rpm=60,000 rpm.

Keeping r_in=0.25 m, r_out=0.025 m, and ω_rpm=60,000 rpm, a substantially optimal blade configuration may be sought. Example results of that parametric study are shown in FIG. 15. In particular embodiments, adding more stages improves performance of the TASC with uniform stages. Moreover, in particular embodiments, substantially optimum θ_sincreases and substantially optimum as decreases when more stages are added. In particular embodiments, this pattern may hold regardless of the rotation speed or TASC radii selected.

FIG. 16 highlights these trends by plotting example substantially optimal blade parameters and collection efficiency versus number of stages for example TASCs with uniform stages and with ω_rpm=60,000 rpm. In the example TASCs with uniform stages, aspect ratio decreases as the number of stages goes from one to six and then levels off as more stages are added. In the example TASCs with uniform stages, blade angle increases to about 65 degrees with ten stages. In particular embodiments, the substantially optimum angle may converge toward the substantially optimum value θ_s=γ=82.4° as even more stages are added. In the example TASCs with uniform stages, collection efficiency also increases as the number of stages goes from one to six and then relatively slowly reach 80% with ten stages.

In particular embodiments, uniform blade geometry for all stages does not produce a substantially optimal TASC configuration. However, even in this scenario, adding stages may raise performance to levels sufficient for a viable air-breathing system. Further improvements in performance may be achieved by substantially optimizing each stage.

In particular embodiments, instead of a brute-force approach, a standard numerical constrained optimization routine may be used to find a substantially optimal configuration of each stage in a multi-stage device. In the example optimizations performed below, angles may be limited to 5°<θ_s<90° and aspect ratios may be limited to 0.01<α_s<0.99 to simplify the parameter space. Although particular angle and aspect-ratio limits are described, this disclosure contemplates any suitable limits, or even no limits, for angle or aspect-ratio, where appropriate.

Consider an example eight-stage system with the following parameters: r_in=0.25 m, r_out=0.0015 m, l=0.05 m, ω_rpm=60,000 rpm. In particular embodiments, v_o=7,800 m/s and v_t=630 m/s corresponds to 300 K thermalized oxygen atoms. With these velocities, γ=82.4 and δ=31.0 may be calculated. In particular embodiments, an example numerical optimization routine may produce the following angles and aspect-ratio values for each of the eight stages, with the following system performance:

Stage
1
2
3
4
5
6
7
8

θ_i°
82.4
70
31
31
13
6.0
6.0
6.0

a_i
0.99
0.53
0.59
0.59
0.75
0.87
0.87
0.87

r_s= 3.0e5

η_s= 86%

For comparison, with a uniform θ_i, α_iconstraint, performance may drop significantly, as shown below:

Stage
1
2
3
4
5
6
7
8

θ_i°
35
35
35
35
35
35
35
35

a_i
0.57
0.57
0.57
0.57
0.57
0.57
0.57
0.57

r_s= 2.0e5

η_s= 56%

In particular embodiments, when uniform blade geometry is imposed on all stages, the substantially optimal blade angle and aspect ratio—subject to the uniform constraint—may be found through a compromise that substantially maximizes the amount of fast particles allowed through the stages and that substantially minimizes the amount of thermal particles that stream backwards and are lost. However, in the substantial optimization with arbitrary θ, α configurations, further improvements in performance may be obtained by specializing the stages. The first stages may be selected to be substantially transparent to the fast-moving particles and may be oriented to larger θ_s. The substantially optimum angle for the first stage may be θ₁=γ. This corresponds to a transmission probability for the fast particles of

$P_{f}^{f} = 1$

(see Equation 43). In particular embodiments, in other substantial optimizations for slightly different r_out, ω_rpm, and number of stages, the first few stages may substantially optimize to θ=γ.

In particular embodiments, blade angle may decrease in subsequent stages. The primary function of these stages may be to trap thermalized back-streaming particles. In particular embodiments, minimum blade angle may be limited to approximately 5° for convenience. Although a 5° angle limit is used in particular embodiments, this disclosure contemplates other suitable limits or no limits, where appropriate.

In particular embodiments, allowing stage geometry to be specialized improves overall performance of the system. An eight-stage system may provide substantially adequate performance (r_s=30,000 and η_s=86%).

FIG. 17 illustrates a study of an example eight-stage TASC with geometry varied one stage at a time. In FIG. 17, example efficiency is plotted as a function of angle and aspect ratio for each stage. In the example of FIG. 17, variations in θ_iand α_iof each stage are examined, one stage at a time, while keeping the other stages at substantially optimized settings, as described above. In the example of FIG. 17, the substantially optimized configurations are as follows: θ_1-8=82.4°, 70°, 31°, 31°, 13°, 6°, 6°, 6° and α_1-8=0.99, 0.53, 0.59, 0.59, 0.75, 0.87, 0.87, 0.87, with r_in=0.25 m, r_out=0.0015 m, and ω_rpm=60,000 rpm.

In particular embodiments, overall system performance is sensitive to first-stage geometry. Collection efficiency may peak around a blade angle corresponding to 0₁=γ. In particular embodiments, this angle value may be conducive to substantially maximizing collection of fast particles. Deviation of blade angle from this value may, in particular embodiments, degrade performance.

In particular embodiments, system performance of an example eight-stage TASC may be relatively good as the second-stage blade angle reaches θ₂=γ. However, a more optimal value of the blade angle for that stage may be θ₂=71°, which lies along a horizontal “ridge” of high-performance values in the solution space, as shown in FIG. 17. The ridge becomes more prominent in stages 3-5, which may indicate a gradual shift in blade function from collection to compression. In these stages, performance may be substantially optimized near θ_s=δ, where δ is the angle formed by the blade speed and thermal velocity. In essence, these blades substantially maximize the forward probability of thermal particles.

Stages 6-8, which in particular embodiments may be the final three stages, display similar sensitivity to blade geometry variation. Here, FIG. 17 shows a curved ridge of substantially optimized solutions similar to those observed in the one-stage simulations, shown in FIGS. 12A, 12B, and 12C, along which performance increases toward lower angles. For each of these stages, the substantially optimal solution seems to be relatively insensitive to variations in blade geometry.

FIG. 18 illustrates three eight-stage TASC examples with three different blade configurations, where θ_sand α_sare substantially optimized for r_in=0.25 m, r_out=0.0015 m, and ω_rpm=30,000, 60,000, and 90,000 rpm. FIG. 18 shows sensitivity of these three configurations to variations in ω_rpm. The middle curve in FIG. 18 corresponds to the example TASC with blade geometry substantially optimized to operate at approximately 60,000 rpm. That curve shows a substantial performance peak at that speed, with a slight performance drop as speed increases and a larger performance drop as speed decreases from that performance peak. Similar trends may be observed for the example eight-stage systems substantially optimized to operate at approximately 30,000 rpm and approximately 90,000 rpm. While a system optimized for higher speed has better performance at that substantial optimum, it seems that in particular embodiments a robust design should be substantially optimized for a lower rotation speed than its intended operation, which may allow the TASC to operate in a relatively insensitive plateau.

In particular embodiments, outlet radius r_outmay not be well controlled, as effective output area is determined by other factors, such as plumbing geometry and thruster transmission probability. FIG. 19 illustrates an example eight-stage TASC where the r_outis varied. The θ_sand α_swere chosen to substantially optimize the system performance for r_in=0.25 m, r_out=0.0015 m, and ω_rpm=60,000 rpm. The value of r_out=0.0015 m was chosen to reflect the small tubing size that may be typically used in the thruster plumbing. The results follow a similar trend from the single-stage study above. As output area increases, the example collection efficiency improves at the expense of compression ratio.

Another consideration is variation in inlet velocity, v_o, since incoming orbital flux may vary with altitude and thermospheric winds. FIG. 20 illustrates an example eight-stage TASC with θ_sand α_ssubstantially optimized for r_in=0.25 m, r_out=0.0015 m, and ω_rpm=60,000 rpm and v_o=7.8 km/s. Performance η_sis calculated for varying v_ofrom 0-15 km/s, while keeping all other parameters fixed. In particular embodiments system performance is relatively insensitive to changes in inlet velocity. In the example of FIG. 20, when a substantially optimized TASC is subjected to varying v_ofrom 5-10 km/s, the net efficiency drops only a few percent.

In particular embodiments, counter-rotors or stators may be used to reduce spinning of the compressed air in the TASC, which would reduce blade performance. With counter-rotors, each of the stages of the TASC rotates in a direction opposite of the stages adjacent to it, which reduces air rotation. In particular embodiments, the transmission probabilities of particles through a stage may be insensitive to the direction of rotation as long as blade angles are chosen to match the rotation (or counter-rotation) direction.

In particular embodiments, an alternative to counter-rotors is to place two TASCs side-by-side such that each rotates in opposite directions. (Herein, reference to a single TASC may include multiple TASCs, and vice versa, where appropriate.) This design may substantially maintain gyroscopic balance of the satellite while providing a simpler engineering approach. In this configuration, each TASC may include stators to reduce air spin in particular embodiments. These stators may be treated as regular stages with v_b=0 and, for each of these stator stages, γ=δ=90°.

Particular embodiments include interwoven stators with a substantially optimum orientation of θ=90°. This orientation in the first stator(s) may be helpful to improve fast-particle transparency

$P_{f}^{f} .$

Angles of the subsequent stator(s) do not significantly effect system performance in particular embodiments. For example, in an eight-stage, eight-stator design, the eight stage angles and aspect ratios remain similar to those described earlier, as does the system performance r_s=300,000 and η_s=86%. Each of the eight stators may be substantially optimized at θ_s=90° with α_svarying significantly because, at 90°, aspect ratio is substantially irrelevant in the 1D model. In FIG. 21, each α_s=0.5, r_in=0.25 m, r_out=0.0015 m, and ω_rpm=60,000 rpm and v_o=7.8 km/s, and performance, η_scalculated while varying each stator angle individually. In the example of FIG. 21, changing the angle of the first stator away from 90° reduces efficiency significantly. In particular embodiments, for each subsequent stator, progressively smaller sensitivity to the stator angle can be observed.

In particular embodiments, system performance scales with the rotation speed. However, faster speed increases complexity and power requirements. In particular embodiments, maximizing the output area, r_out, increases collection efficiency. However, this benefit may not be realizable as the output area may be limited by other practical considerations, such as the thruster transmission probability and outlet plumbing diameter. In particular embodiments, the first stage (possibly the first few stages) should be oriented so that θ₁=γ, where γ is the angle formed by the orbital velocity and blade velocity. This ensures that

$P_{f}^{f} \sim 1 .$

If this orientation is achievable, wider aspect ratios, a_s, are superior to narrower aspect ratios. In particular embodiments, the last set of stages should be substantially optimized for compression with θ_sbeing small. While better compression can be achieved with more of these stages, this must be balanced with the extra weight and power draw of the system. In particular embodiments, the system should be substantially optimized for a slightly slower speeds than its intended use to avoid large performance fall-offs if the desired speed is not obtained. In particular embodiments, if stators are used, they should be set to substantially vertical orientations, especially for the first sets of stators.

Particular embodiments facilitate the siphoning off of excess propellant into one or more propellant tanks for later use. (Such a storage tank allows for a thruster on-off duty cycle where the electric thruster is only fired during a portion of the orbit. In particular embodiments, this optimal duty cycle to maintain orbit against atmospheric drag forces may be dependent on the specific orbital inclinations.) For example, particular embodiments may use excess propellant to perform orbital inclination changes or excursions to higher altitudes and back. This may facilitate use of an air-breathing satellite, for example, as a reusable space tug or satellite-servicing platform. To function in this capacity, an air breathing electric propulsion system may use some of the collected mass flow to maintain orbit while preserving the rest for later maneuvers.

In particular embodiments, excess fuel can be used for other tasks, in addition or as an alternative to orbit-keeping. For example, particular embodiments may use excess propellant to perform orbital inclination changes or excursions to higher altitudes and back. This may facilitate use of an air-breathing satellite, for example, as a reusable space tug or satellite-servicing platform. To function in this capacity, an air breathing electric propulsion system may use some of the collected mass flow to maintain orbit while preserving the rest for later maneuvers. In particular embodiments, a space-tugging vehicle powered by an air scooping electric thruster system may be used to deliver other space vehicles to their target orbit (so-called ‘last-mile’ deliver); maneuver to defunct space vehicles and other space debris to capture, then de-orbit the target before refueling to collect a new target; maneuver to a space vehicle that has run out of fuel and perform station keeping maneuvers for that vehicle to extend its life; or maneuver to a space vehicle that has run out of propellant and re-fuel that vehicles propellant supply from the propellant store tank that was filled via the atmosphere.

To incorporate space-tugging capabilities into the analyses above, two additional terms may be added to the previous equations. The first term is a propellant divert ratio, D, representing the fraction of the collected mass diverted to the propellant tank for later use. The second term is the total satellite system dry mass, M_sat, which affects how much Δv may be achieved for given propellant mass.

The term M_satincludes the mass of the object to be maneuvered and the dry mass of the space tug. In some situations, this mass may change throughout a mission. For example, when maneuvering to collect a piece of debris to de-orbit, a space tug may first adjust its own orbit to match the target. After collecting the target, the mass considered would be the combined mass of both the system and the target. In particular embodiments, a conservative choice for M_satmay be used generally to understand the dynamics of the equations.

The drag equation, Equation 9, remains unchanged. However, the total propellant available for the thruster is now reduced:

$\begin{matrix} {\dot{m}}_{T} = (1 - D) {\dot{m}}_{c} & (57) \end{matrix}$

where the thruster operational duty cycle, Q, has been ignored since it cancels out later in the calculations. The thrust requirement to maintain orbit, Equation 10, remains unchanged. However, because the mass flow to the thruster has decreased, the I_sprequirement becomes:

$\begin{matrix} I_{s p} > \frac{v_{o r b} C_{D}}{2 g (1 - D) η_{s}} . & (58) \end{matrix}$

With this divert ratio being propagated through the previous analysis, the new mission metric becomes:

$\begin{matrix} 〈 P 〉 η_{t} η_{s} > \frac{{\dot{m}}_{i n} v_{o r b}^{2} C_{D}^{2}}{8 (1 - D)} . & (59) \end{matrix}$

That is, if we wish to divert some fraction of the collected propellant into a propellant tank, we must increase the thruster I_spaccordingly.

In particular embodiments, an orbital maneuver may be characterized by the change in spacecraft velocity required to accomplish that maneuver. That change in velocity may be called Δv. The achievable Δv for an air-breathing system may be calculated as a function of the system performance and divert ratio.

It may be noted that the rate at which mass is stored in the propellant tank is,

$\begin{matrix} {\dot{m}}_{f} = {\dot{m}}_{c} - {\dot{m}}_{T} = D {\dot{m}}_{c} . & (60) \end{matrix}$

The rocket equation that calculates Δv in terms of the total satellite and propellant masses may be

$\begin{matrix} \frac{M_{P}}{M_{s a t}} = \exp^{Δ v / {gI}_{s p}} - 1, & (61) \end{matrix}$

where M_pis the total propellant mass required for a maneuver and M_satis the satellite dry mass that excludes M_p. Solving for Δv produces the following result:

$\begin{matrix} Δ v = {gI}_{s p} \ln (\frac{M_{P}}{M_{s a t}} + 1) & (62) \end{matrix}$

Rewriting M_pin terms of the fill rate and fill time τ_f,

$\begin{matrix} M_{p} = {\dot{m}}_{f} τ_{f} & (63) \end{matrix}$

and using Equation 58 produces

$\begin{matrix} Δ v = \frac{1}{1 - D} \frac{v_{o r b} C_{D}}{2 η_{s}} \ln (τ_{f} {\dot{m}}_{i n} η_{s} D / M_{sat} + 1) . & (64) \end{matrix}$

In Equation 64, it is assumed that I_spis substantially identical to the I_spused during the re-fueling portion of the mission described in Equation 58. In practice, this assumption may not be accurate. During a maneuver, extra thrust may be required beyond what negates drag. If power [Image Omitted] P is fixed and more mass flow is used in the same thruster, I_spmay decrease if η_tremains constant. Alternatively, in order to keep a fixed I_spwith higher mass flow rate, more power may be required.

Equation 64 describes the size of a maneuver (Δv) that may be performed as a function of fill up time, τ_f. It may be solved for τ_f, such that

$\begin{matrix} τ_{f} = \frac{M_{s a t}}{{\dot{m}}_{i n} η_{s} D} (e^{\frac{1 2 η_{s} Δ v}{1 - D C_{D} v_{o r b}}} - 1) & (65) \end{matrix}$

In Equation 65, it may be noted that, as long as Δv<<V_orb, the exponential may be Taylor expanded using e^x≈1+x. The approximation may be more or less reasonable for certain low Δv maneuvers, but may be less so for LEO to GEO maneuvers. This requirement is similar to stating that Δv<<gI_spsince I_spin the derivation above is proportional to and greater than v_orb.

A performance relationship between τ_fand Δv may be derived

$\begin{matrix} Δ v \approx \frac{D}{1 - D} \frac{C_{D}}{2} v_{o r b} \frac{{\dot{m}}_{i n}}{M_{s a t}} τ_{f} . & (66) \end{matrix}$

Equation 66 tells us the maximum Δv achievable in a given fill up time τ_fas a function of the mass flow into the system, the divert ratio, and the total satellite mass.

In particular embodiments, the divert ratio, D, may not be arbitrarily controllable. Because the thruster should produce sufficient I_spto overcome the drag forces, this ratio may be defined by the achievable I_spof the thruster. If

$I_{s p_{\min}} = \frac{v_{o r b} C_{D}}{2 η_{s}}$

is defined as the minimum I_spnecessary to maintain orbit with D=0, the fill up time may be calculated as

$\begin{matrix} Δ v \approx (I_{s p} I / I_{{sp}_{\min}} - 1) \frac{C_{D}}{2} v_{o r b} \frac{{\dot{m}}_{i n}}{M_{s a t}} τ_{f} . & (67) \end{matrix}$

Using Equation 59, a performance requirement may be rewritten as

$\begin{matrix} \dot{Δ} v = \frac{Δ v}{τ_{f}} \approx \frac{4 D}{v_{o r b} C_{D} M_{s a t}} 〈 P 〉 η_{t} η_{s} . & (68) \end{matrix}$

$Alternatively,$

$\begin{matrix} \dot{Δ} v \approx \frac{2 〈 P 〉 η_{f}}{M_{s a t}} (\frac{1}{I_{{sp}_{\min}}} - \frac{1}{I_{s p}}), & (69) \end{matrix}$

where {dot over (Δ)}v is the substantially maximum possible rate at which a propellant tank could be filled in terms of mission Δv requirements.

Consider a satellite similar to GOCE that has a mass M_sat=1,000 kg, an end-of-life orbit-averaged power (accounting for eclipse) of [Image Omitted] P=500 W, and a drag coefficient C_D=3.5. If it is assumed that a mission metric of η=η_tη_s=0.2 and D=0.5 are possible, the substantially maximum {dot over (Δ)}v achievable may be 230 m/s/year. For reference, in particular embodiments, this may be the approximate Av required to get from 300 km up to 500 km and back down to 300 km circular orbits. In particular embodiments, a GOCE-like satellite could fully refuel for a mission from ULEO to LEO and back with just under one year of atmospheric refueling.

Particular embodiments facilitate air-breathing electric propulsion for satellites flying at altitudes below approximately 300 km. Particular embodiments may be used to overcome atmospheric-drag forces associated with such low orbits without running out of propellant. In particular embodiments, system viability may substantially depend on one or more of the following parameters:

- Scoop Efficiency, Θ_s, which may in particular embodiments represent the fraction of incident mass flow collected by the scoop to be fed to the thruster.
- Scoop Compression Ratio, r_s, which may in particular embodiments represent the ratio of the incoming atmospheric density and the gas density fed to the thruster.
- Thruster Efficiency, η_t, which may in particular embodiments represent the fraction of power converted to kinetic energy in the thruster.
- Thruster Specific Impulse, I_sp, which may in particular embodiments represent the average exhaust speed of the thruster which must be greater than the incoming atmospheric orbital flux speed.
- Orbit Average Power, P, which may in particular embodiments represent the power available for orbit-keeping and not reserved for mission payload.

Particular embodiments use a TASC that spins a series of stages of blades at high rotational speeds to capture and compress incoming air, allowing the air-scoop to reach substantially sufficient values for η_sand r_s. In particular embodiments, the TASC may be designed and substantially optimized to reach desirable performance metrics.

Particular embodiments may use one or more the following design principles. Faster rotation speed is usually better, but rotation speed may be limited by technological considerations. In particular embodiments, rotation speeds of 60,000-100,000 rpm are sufficient for an eight-stage TASC. The first stage(s) of a TASC may be important to maintaining high collection efficiency. These stages may be oriented so that they are substantially transparent to the high speed orbital particles. In particular embodiments, this may be accomplished by setting the blade angle, θ, substantially equal to the angle formed by the blade velocity and particle velocity, γ. Later stage(s) may be substantially optimized for compression with smaller θ_s. Middle stage(s) may balance transparency to high-speed particles with high capture of back-streaming thermal particles.

Gyroscopically balancing a TASC on a space craft may be accomplished by mounting one motor to each end of a TASC and spinning sets of stages in opposite directions. Such a set up may obviate use of stators within the TASC. As an alternative, two TASCs may be placed side-by-side and spun in opposite directions. Each of the TASCs may include stators to prevent spin of the compressed air. These stators, especially those near the front of the device, may be oriented vertically to allow fast particle to enter freely.

In particular embodiments, extra propellant may be stored for later use, which may facilitate orbital inclination maneuvers and altitude changes by a satellite. In particular embodiments, the spacecraft may be used as a refuelable space-tug. The satellite may perform a mission and then descend back to ULEO to refuel. In particular embodiments, this may accomplished by diverting at least some incoming mass flow to a fuel storage tank (rather than directly to the thruster). In particular embodiments, refueling times for missions from ULEO to LEO and back may be on the order of approximately year, allowing a single vehicle to perform many such missions over the course of its life.

Particular embodiments of an example satellite mission flying in ULEO at 245 km are presented. In these examples, we assume the satellite cross-section area A_sat=0.2 m²and the drag coefficient C_D=3.5. In this scenario, assuming the atmospheric conditions described above, Equation 6 may be used to determine the drag force on the satellite, F_D=1.5 mN. We further assume that gas collected throughout the orbital period is stored in a propellant tank, and a thruster is fired for 10% of the orbital period, so that Q=10. Finally, let us assume η_t=25% and all collected mass is used when the thruster is fired to exactly balanced the total impulse from drag forces as presented in Equation 10, so that T=15 mN. In this configuration, the required specific impulse derived from Equation 11 is I_sp>1392s/η_s.

An example single-cylinder TASC for the above mission has eight stages where half are counterrotating so that no stators may be required and gyroscopic balance of the satellite is maintained. This TASC is assumed to have r_in=0.25 m, r_out=0.0015 m, and ω_rpm=60,000 rpm. Therefore, its total inlet area is substantially the satellite cross-sectional area assumed above. In this configuration, a substantially optimum solution has already been identified above for the eight stage angles and aspect ratios to produce a TASC with η_s=86% and r_s=300,000. This compression ratio is more than sufficient to efficiently ignite an electric thruster, and the collection efficiency defines the required thruster specific impulse from Equation 11, so that I_sp>1,620 s. Using Equations 13 and 14 above, this embodiment of an air-breathing satellite mission may require 50 W of orbit-averaged power for thruster operation, or 500 W of thruster power during the 10% on-cycle of thruster operation.

Another example TASC for the above mission has two cylinders with eight rotors and eight stators. The axes of each cylinder rotate in opposite directions to ensure gyroscopic balance of the satellite, but the two TASC cylinders of the configuration are otherwise substantially identical. Each cylinder of this TASC is assumed to have r_in=0.17 m, r_out=0.0015 m, and ω_rpm=60,000 rpm. Therefore, the total inlet area for both cylinders together is substantially the satellite cross-sectional area assumed above. In this configuration, a substantially optimal solution is derived using the approach described above and a TASC performance of r_s=130,000 and η_s=81% may be achieved. In this configuration, the substantially optimum blade angles and aspect ratios for each TASC cylinder arc: θ_1-8=84.8°, 73°, 42°, 31°, 14°, 8°, 5°, 5° and ω_1-8=0.99, 0.73, 0.67, 0.68, 0.81, 0.88, 0.91, 0.91, and the eight stators are uniformly set to θ_stator=90° and α_stator=0.5. The substantially optimized compression ratio that may be achieved is more than sufficient to efficiently ignite an electric thruster, and the collection efficiency defines the required thruster specific impulse from Equation 11, so that I_sp>1,720 s. Using Equations 13 and 14 above, this embodiment of an air-breathing satellite mission may require 55 W of orbit-averaged power for thruster operation, or 550 W of thruster power during the 10% on-cycle of thruster operation.

FIG. 22 illustrates an example computer system 2200. In particular embodiments, one or more computer systems 2200 perform one or more steps of one or more methods described or illustrated herein. In particular embodiments, one or more computer systems 2200 provide functionality described or illustrated herein. In particular embodiments, software running on one or more computer systems 2200 performs one or more steps of one or more methods described or illustrated herein or provides functionality described or illustrated herein. Particular embodiments include one or more portions of one or more computer systems