How Fast Can We Go To Mars Using High Power ...

48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 30 July - 01 August 2012, Atlanta, Georgia

AIAA 2012-3889

How fast can we go to Mars using high power electric propulsion ? Nicolas Bérend1 ONERA – The French Aerospace Lab, Palaiseau, France, F-91761

Downloaded by CNES-Centre National dEtudes Spatiales on December 17, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2012-3889

Elisa Cliquet2, Jean-Marc Ruault3 CNES Directorate of Launchers, Paris, France,F-75615 Richard Epenoy4 CNES Toulouse, France, F-31401

Recent advances in electric propulsion technologies such as magnetoplasma rockets gave new momentum to the study of nuclear electric propulsion concepts for Mars missions. While the interest of nuclear electric propulsion appears clearly with regard to the payload mass ratio (due to a high level of specific impulse), its interest with regard to transfer time is more complex to define, as it depends on many design parameters. In this paper, we perform a general analysis of the mid-term capability of nuclear electric propulsion systems considering both payload mass ratio and transfer time. This study has been performed through a multidisciplinary analysis which combines general performance calculations for power-limited systems, an analysis of nuclear power-source that could be available in the future, and a series of mission analysis including trajectory optimization. The results obtained in this study emphasize and – most importantly - quantify the importance of the specific mass of the power and propulsion system with regards to the objective of a fast transfer. The objective of a very fast Earth-to-Mars transfer in less than six weeks appears unrealistic in a mid-term context, as it depends on a hypothetical breakthrough on the nuclear electric power source, with a specific mass typically lower than 1 kg/kW. The results obtained in this paper draw the contour of performance improvements that could be obtained in a mid-term horizon, using power generation technologies that are challenging but more commonly considered as reasonably optimistic, for example using a high temperature Brayton or Rankine conversion cycle. It appears that the shorter Earth-toMars transfer time that could be expected for missions with sufficient payload is about 120 days, compared to 180 days with chemical propulsion.

Nomenclature g0 Isp k q M Mf Mi MP MPL MW MS

= = = = = = = = = = =

Earth gravity acceleration at sea-level (9.80665 m/s2) specific impulse tank structural mass ratio = MS / MP mass flow-rate (kg/s) mass (kg) total final mass (kg) total initial mass (kg) propellant mass (kg) payload mass (kg) power source mass (kg) structure mass (kg)

1

Senior Expert, ONERA – The French Aerospace Lab; 91761 Palaiseau Cedex, AIAA Member. Propulsion system specialist, CNES, Directorate of Launchers, 52 rue Jacques Hillairet, 75612 Paris Cedex. 3 Project Manager, CNES, Directorate of Launchers, 52 rue Jacques Hillairet, 75612 Paris Cedex.. 4 Trajectory optimization expert, CNES, 18 avenue Edouard Belin 31401 Toulouse Cedex 9, AIAA Member. 1 American Institute of Aeronautics and Astronautics 2

Copyright © 2012 by ONERA and CNES. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


Pe r t T U v

α γT λ μ η

= = = = = = = = = = =

input power delivered by the source (W) position vector time thrust thrust direction vector velocity vector specific mass of the power source and propulsion system (kg/kW) thrust acceleration (m/s2) trajectory characteristic parameter (m2/s3) Earth gravity constant (3.986.1014 km3/s2) global efficiency of the propulsion system

AARC IMLEO LEO MHD PMP VASIMR

: : : : : :

Ad Astra Rocket Compagny Initial Mass in Low-Earth Orbit Low-Earth Orbit Magneto Hydro Dynamic Pontryagin Maximum Principle VAriable Specific Impulse Magnetoplasma Rocket

Glossary

I. Introduction

O

NE of the challenges of human Mars exploration lies in the long duration of the interplanetary transfer, which is about 180 days for Earth-to-Mars using chemical or nuclear thermal propulsion. Although it shall be not considered as strictly mandatory, a reduction of the transfer time would be an improvement, as it would relieve many issues – both technical and psychological - for human Mars missions. High power nuclear electric power systems, whose primary interest is a high level specific impulse, are considered as a possible solution to address this problem. One example of fast Earth-to-Mars transfer has been studied recently by AdAstra Rocket Company, which proposed a mission architecture allowing to send humans to Mars in 39 days using nuclear electrically powered VASIMR (Variable Specific Impulse Magnetoplasma Rocket) engines [1]. This analysis relies on very strong hypothesis concerning the power level (200 MW) and the specific mass of the nuclear electric generator (< 1 kg/kW). The objective of this paper is to investigate the feasibility of an Earth-to-Mars transfer with reduced transfer time using a high power electric propulsion system. This study has been performed through a multidisciplinary analysis which combines general performance calculations for power-limited systems, an analysis of nuclear power-source that could be available in the future, and a series of mission analysis including trajectory optimization. In the first part of the paper (section II), general considerations on mass target for manned Mars mission are given, and the achievable specific mass of the nuclear power generation system is discussed based on bibliography but also on recent CNES-AREVA studies. This allows to suggest some high level hypothesis to design a mission scenario. In the second part of the paper (section III), the performance of power-limited systems for Earth-to-Mars transfers is analyzed and trajectory optimization results are given. Finally, the feasibility of reduced transfer time scenarios (shorter than 180 days) is investigated. This duration is compatible with the assumptions defined previously about realistic specific mass values accessible in a mid-term future.

II. Mass targets for a manned Mars mission using nuclear electric propulsion Classical manned Mars mission scenarios based on chemical or nuclear thermal propulsion typically allow for two types of mission profiles: either a conjunction class mission (180 to 220 days each way for a 500 to 600 days stay) or an opposition class mission (250-300 days to go to Mars, 30 to 60 days stay and 150 to 300 days for the return). Low thrust propulsion could open the way to alternative, more flexible mission profiles, with potentially reduced transfer times and reduced global mission duration. 2 American Institute of Aeronautics and Astronautics

In order to evaluate the feasibility of reduced transfer time scenarios, we start by defining a few important orders of magnitude that will serve as our hypothesis.

A. Total vehicle limit initial mass in low Earth orbit and payload mass


1.

Payload mass order of magnitudes

A martian mission will very likely include an entry descent and landing stage, two habitats (possibly one predeployed on the surface and one for the transit), an Earth return vehicle (and associated stage), life support for the crew, a rover and scientific equipements. For example, the Mars direct original scenario for a crew of 4, which is one the “lightest” scenario [2], planned to bring the equivalent of a little more than 75 tons in Mars orbit in two opportunities, using aerocapture and allowing for pre-deployement of a first habitat, the Earth Return Vehicule and associated stage, and an in-situ propellant production unit for the return. In term of order of magnitude, we can consider that, if the mission is “all-up”, using nuclear electric propulsion for both ways, the mass to be brought to Mars, including orbit insertion and return propellants, as well as an independant ascent stage, would probably amount to a minimum of 100 tons. In case the mission is split to allow a first slow, high payload, cargo mission, the crew transfer vehicle could be lighter and faster. However, the mass to be brought at Mars shall at least include the crew, its supplies, and a transfer habitat. In the calculations that will be presented in section III, the vehicle arrives at Mars with a velocity of 6,8 km/s. Thus, what we call “payload” mass should also include orbit insertion means: extra propellant to brake the whole vehicle into orbit, and potentially another mean (a conventional chemical stage and aeroshell) that could allow a direct entry or faster orbit insertion and landing for the crew. We can assume the order of magnitude of the payload to be then in the order of 30 tons. 2.

Total initial mass in Low Earth Orbit

Initial mass in Low Earth Orbit (LEO) is also an important parameter for the mission feasibility. The International Space Station weights about 390 tons. Initial mass in Low Earth Orbit that can be found in the literature for manned Mars missions can go up to more than 900 tons [3]. Scenarios derived from “Mars direct”, involving the development of technologies like in-situ propellant use and aerocapture offer dramatic mass reductions but would still need at least 2 if not 3 or 4 launches from a yet to be developed Saturn-V like or ARES like heavy lift launch vehicle. In order for a fast scenario to be competitive with a chemical or nuclear thermal based scenario, we believe that its initial mass in low Earth orbit shall be kept strictly below 500 tons. B. Power generation system In the case of very high power electric propulsion, the specific mass (in kg/kW) of the power and propulsion system is a very important parameter in the mission analysis. For example, the “39-days” scenario assumed a specific mass in the order of magnitude of 1 kg/kW, for a power level of 200 MWe. In our approach, we propose to study what is the specific mass we can consider has “reasonably optimistic” in order to design a mission for which the power generation system could be developed in less than two decades. 1.

Power levels and specific masses already achieved or accessible in a relatively short term

Between 1960 and 1992, more than 30 reactors have been used in space. However, their power levels were limited to less than 10 kWe. Those reactors were using Na-K as a coolant and low efficiency thermo-electric or thermionic technologies for conversion. Specific masses made great progress during this period of time but their values remain very high: 570 kg/kW for Romashka in 1967, 400 kg/kW for Buk in 1977 and 180 kg/kW for TOPAZ in 1987 [4]. Today, such specific masses are not competitive compared to solar panels when considering application in Earth orbits. At the 100 kWe power level, as shown in [5] the specific mass can be expected to be in the range 30 to 40 kg/kWe. Such a specific mass for this power could be achievable using thermo-electric conversion but with Lithium coolant instead of Na-K in order to increase efficiency by increasing hot side temperature. Such a system could be developed in a relatively short term. 3 American Institute of Aeronautics and Astronautics

2.

Multimegawatt power level, elements of bibliography


In terms of power level, given today’s state of the art in space reactors and on submarine reactors, power levels that can be considered for mid-term missions (in 20 years from now) should not exceed a few tens of megawatt. Higher power levels might be considered not likely to be available in the same time frame. The following table gives some examples of power systems (reactor, shield, conversion) that have been studied in literature to be used for manned Mars missions (either for the manned spaceship or for a support cargo mission) or other applications. This table does not pretend to be exhaustive and only gives references in which using solid cores are considered.

J.S. Clark B.Cothran R.J. Cassady J.A.Webb

Year of publication 1990 1994 1988 2011

R.J.Litchford G.Woodcock

2011 2002

R.D.Rovang

1992

G.R.Longhurst

2002

First Author

Power level 4 and 8 MWe 40 MWe 8 MWe 15 MWe 100 MWe 1 MWe >MWe 10 MWe

Conversion cycle Brayton Brayton Brayton 1300K

Specific mass

Reference

10 kg/kW 5.4 kg/kW 8 kg/kW 15 kg/kW 9.6 kg/kW (extrapolated) 32 kg/kW 1 to 3 kg/kWe 3.5 to 5kg/kWe

[6] [7] [8] [9]

MHD (1800K) MHD or Brayton (1500K to 2000K) 5.5 MWe Rankine 6.8 kg/kWe (1300K) 15 MWe Rankine 6.88kg/kWe (1500K) 2.58kg/kW Brayton (2100K) Table 1. Some examples of specific masses

[10] [11]

[12] [13]

These results show a large dispersion, not only due to the choice of the conversion cycle (Rankine is lighter than Brayton because of much smaller radiators thanks to the phase change of the coolant). Even though the perimeter included in the specific mass of the system is not homogenous in the publications (some references explicitly take into account power management and distribution, structures and extra margin provisions but some other references do not give such precision), this does not fully explain the dispersion. Indeed, the very small values of specific masses are mainly explained by hypothesis on the temperature considered as acceptable for both solid cores (very high temperature cores proposed in literature often rely on core technologies developed for nuclear thermal rockets, however such systems do not have the same specifications in terms of operating duration) and turbines, but it also relies on evaluation of MHD conversion specific masses that are low compared to turbines ones. There is no certitude today that a MHD converter, that needs superconducting magnets and cooling, would be lighter than a turbine [11]. 3.

Preliminary results on ongoing French studies on nuclear power generation systems for space exploration

CNES, the French space agency, along with AREVA, has been leading studies in the past few years on space nuclear power generation systems for exploration missions. The goal of these studies is to evaluate technologies that could be relevant for various power levels and to assess their specific mass and readiness level in France and in Europe. At megawatt power level, radiator size becomes highly critical because thousands of square meters may be required depending on radiator temperature and conversion system efficiency. Accordingly, trade-off efforts have been focused on the conversion cycle and on radiator technologies more than on the reactor itself. Various conversion cycles, radiator technologies and general architectures have been assessed. 4 American Institute of Aeronautics and Astronautics


Concerning conversion cycles, numerous options have been considered in a first place: thermoelectric, thermïonic, AMTEC®, thermophotovoltaïc for static conversion, along with Stirling cycle, thermo-acoustic, Brayton cycle (including two optional designs: use of supercritical fluid and use of a magnetohydrodynamic generator instead of a turbine), and Rankine cycle (including an optional design using a magnetohydrodynamic generator instead of a turbine) for dynamic conversion. Performances of each cycle in terms of mass, required radiator area, assembly, reliability, operations, complexity of electrical energy distribution for propulsion engines, and uncertainties concerning technical feasibility were evaluated for each conversion technology. Each of them was given a grade, allowing a global ranking. Among them, Brayton option was preferred due to its good performance and large scalability but specifically because, unlike Rankine cycle, it can be tested on the ground in a representative way. Stirling option is also attractive because of its modularity and intrinsic redundancy. It could be considered as an alternative candidate. However, our evaluation, based on extrapolation of US studies for the Stirling, shows that, for megawatt level, its specific mass is higher than the one of the Brayton option. Moreover its layout is complex and requires a system to efficiently distribute heat to the numerous modules. For Brayton, various redundancy schemes have been evaluated in order to try to find a solution that would limit the impact of improved reliability on the specific mass. For instance, for a direct Brayton 5 MWe system, our studies shows that 6 conversion lines of 1MWe could be a good compromize between specific mass and complexity [14]. Both direct option with gas cooled core and indirect option with liquid metal cooled core and a heat exchanger have been preliminary assessed and compared at 5 MWe. Results tend to show no major difference in terms of specific mass for both configurations when using the same high temperature. Our results, considering 1300 K to 1600 K turbine inlet temperature (which is already very challenging), lead to expect a specific mass of about 15 kg/kW at 5 MW. The positive effect of scaling could allow a slight reduction for upper power levels. Rankine cycle could allow further mass reduction (potentially as low as 5 kg/kW for the total system for a few tens of megawatts) but today feasibility of a space propulsion nuclear reactor operating in zero or very low gravity environment with Rankine conversion is still under question. Looking for further reduction of specific mass (< 3 kg/kW) would necessarily need an even higher core temperature which questions the feasibility of a solid core. Vapor core reactors have thus been considered [15] but their feasibility in less than two decades is questionable. 4.

Thrusters performances and specific mass

Several types of electric thrusters could be candidates to propel a manned Mars mission equipped with a power generator of several megawatts. For the sake of comparison with existing studies, we will consider the VASIMR engine (variable specific impulse from 3000 to 30000s [1]) and an alternative solution using thrusters with fixed specific impulse of 3000s and 5000s. Those performances could be achieved, for example with magnetoplasma dynamic thrusters, or clusters of gridded ion engines. Thruster specific mass is also one of the contributors to the total specific mass of the propulsion system. It is difficult today to give a figure of specific mass for thrusters of the MW level because literature on the topic is quite scarce. Current specific mass target for VASIMR flight prototype VF-200 seems to be around 22 kg/kWe [16], however, the final mass target is at least one order of magnitude lower. In other studies [17], specific mass of magnetoplasma dynamic thrusters have been assumed to be lower than 0,5 kg/kWe. They will be considered as already included in the 10 kg/kW rough assessment of a “reasonably optimistic” specific mass for the power generation system.

C. Hypothesis to assess the feasibility of a reduced transfer time trajectory In our studies, we will consider that feasible and potentially interesting scenarios correspond to: A mass brought at Mars of either 30 tons (fast mission of a split scenario) or minimum 100 tons (all-up scenario), An initial mass in low Earth orbit lower than 500 tons, A specific mass for the power generation system and propulsion system of 10 kg/kWe (considered as “reasonably optimistic” for systems using a Brayton conversion) or 5 kg/kWe (longer term option based on Rankine conversion). Three types of thrusters will be considered: variable Isp from 3000 to 30000 s, and fixed Isp of 3000 and 5000 s. 5 American Institute of Aeronautics and Astronautics

III. Feasibility of reduced transfer time scenarios

A. Methodology


1.

Mission definition and objectives

In this chapter, we focus on a Earth to Mars transfer starting from a departure LEO orbit at a 1000 km altitude, and we arrive at Mars with an hyperbolic excess velocity that is defined further. Mass necessary for orbit insertion is taken into account in the “payload” mass (see II.A.1). The mission is considered to be split in two parts: a planetocentric phase in which the spacecraft is brought from the circular LEO to a given departure relative velocity w.r.t. the Earth, denoted by v∞,d, and a heliocentric EarthMars phase that ends when the spacecraft reaches a given arrival relative velocity w.r.t. Mars, denoted by v∞,a. (cf. Figure 1).

(2) Heliocentric phase 9 9 9 909 90 0 0090 9

(1) Geocentric phase (spiraling)

Figure 1.Illustration of a two-phased LEO to Mars transfer strategy The departure and arrival relative velocities considered are given as follows: v∞,d = 5.0 km/s v∞,a = 6.8 km/s

(1) (2)

As explained in section II.C, we consider three different types of propulsion systems: - Variable Isp system in the range 3000-30000 s, - Fixed Isp system with Isp= 5000 s, - Fixed Isp system with Isp= 3000 s. The study of reduced transfer times scenarios will be performed in two steps: - First (section III.B), using simple analytical results from a previous study [20], we will identify the range of achievable transfer times which could be compatible (i.e. feasible) with the hypotheses of specific mass defined in section II.C: o A short/medium term hypothesis: α = 10 kg/kW, o A longer term hypothesis: α = 5 kg/kW. - Secondly (section III.C), we will perform a more detailed performance analysis for the three types of propulsion systems, focusing on the interval of achievable transfer times determined in the first part of the study. Finally, in the last part of the paper (section III.D), reference vehicle designs will be defined for feasible scenarios. 6 American Institute of Aeronautics and Astronautics


2.

Trajectory optimization approach

The whole study relies on analytical results for power-limited propulsion systems (that will be explained in section B and C) as well as numerical results issued from trajectory optimization. For the latter, the optimization problem is solved as follows: - The planetocentric phase is considered as the solution of a continuous-thrust minimum-time optimal control problem. The associated control law, obtained by means of Pontryagin’s Maximum Principle (PMP) [22] and single shooting methods, consists in switching the engine on and in using the maximum available value of the thrust modulus during all the flight. - The heliocentric phase is considered as a minimum-fuel optimal control problem, leading to a “bang-offbang” strategy for the thrust history. This discontinuity in the thrust profile requires the use of a specific numerical solution method [23]. Moreover, in the case of variable Isp engines, the Isp is considered as an additional control variable that has to be optimized at the same time as the thrust modulus and the thrust direction. More precisely, the dynamic model for the spacecraft in an inertial heliocentric frame can be written as follows:

r&(t ) = v(t ) r (t )

v&(t ) = − μ M& (t ) = −

r (t )

3

+

T (t ) U (t ) M (t )

(3)

T (t ) g 0 Isp(t )

where, at each time t, r(t) v(t) M(t) T(t) U(t) Isp(t)

is the spacecraft heliocentric position vector, is the spacecraft heliocentric velocity vector, is the spacecraft mass, is the thrust modulus of the engine, is the thrust direction vector ( ||U(t)||=1 ), is the specific impulse of the engine.

In addition, μ and g0 denote the gravitational constant of the Sun (resp. the acceleration due to gravity at see level, that is, g0 = 9.80665 m/s2) . Therefore, the control variables are T(t), U(t) and, only in the case of variable Isp engines, Isp(t). These variables have to be optimized in the feasible control space F defined as follows:

⎧⎪ F = ⎨(T , U , Isp ) ⎪⎩ Where:

⎫⎪ ⎡ 2ηPe ⎤ 3 T ∈ ⎢0, ⎥, U ∈ ℜ , U = 1, Isp ∈ [Isp1 , Isp 2 ]⎬ ⎪⎭ ⎣ g 0 ⋅ Isp ⎦

(4)

η ∈ ] 0,1] is the global efficiency of the engine, Pe ∈ ℜ+ is the electrical power available, Isp1 and Isp2 are the two bounds of the variable Isp.

The performance index to be maximized can be written as J = m(tf), where tf denotes the arrival date at Mars. The initial and final conditions are given hereafter: v( t 0 ) − v E ( t 0 ) = v∞ ,d , v( t f ) − v M ( t f ) = v∞ ,a

7 American Institute of Aeronautics and Astronautics

(5)

where t0 denotes the departure date from the Earth (15/07/2018 0h00) and where vE and vM are the heliocentric velocities of the Earth and Mars respectively. Notice here that t0 is equal to the departure date in LEO plus the minimum duration obtained when solving the minimum-time planetocentric optimal control problem.


Then, by using the PMP, solving the optimal control problem reduces to finding the zero of the shooting function. This last one is built by considering the costate equations and the transversality conditions associated with Eq. (5) [22]. In all the optimization runs performed, a 1000 km departure LEO orbit is considered so as different values of IMLEO. The departure and arrival relative velocities are constrained to values defined earlier (Eq. ( 1 ) and ( 2 )) and the efficiency coefficient of the engine is assumed to be equal to η =0.6. In the same way, different values of tf and Pe have also been considered in the optimization runs.

B. Identification of achievable transfer times according to specific mass hypothesis All the propulsion technologies considered in this study belong to power-limited propulsion systems. In this kind of propulsion system, the energy used for rocket propulsion (i.e. used to accelerate and eject the mass of the jet) comes from of a power source delivering a power level Pe. The mass flow rate (q) and specific impulse (Isp) are linked to the power level (Pe) through the following equation:

1 2 q ⋅ ( g 0 ⋅ Isp ) = η ⋅ Pe 2

(6)

whereη is the global efficiency of the system and g0 = 9.80665 m/s2 is the Earth gravity acceleration at sea-level. In this paper, we will consider η =0.6 for all the calculations (see Sec. III.A.2), with a variable or fixed Isp systems. Equation ( 6 ) is valid for both variable and fixed Isp propulsion systems. Under the assumption that the power level Pe is constant, the definition of the Isp level determines the mass flow rate level. The Isp level is fixed by the technology of the thruster: it can be either variable in a certain range, or fixed. An important result regarding power-limited systems is the following variant of the rocket propulsion equation [18][24].

∫

tf

0

⎛ 1 1 ⎞⎟ − ⎟ ⎝ M f Mi ⎠

γ T2 ⋅ dt = 2η ⋅ Pe ⋅ ⎜⎜

(7)

Mi and Mf are the total initial and final masses, respectively. γT is the thrust acceleration, which is a function of time (t), and tf is the duration of the transfer. This equation is valid for power-limited systems in general, for both variable and fixed Isp systems. However, it is important to note that Eq. ( 7 ) depends on the assumption that the global efficiency remains constants and do not depends on the Isp level, which is a simplification. The left-hand side term of Eq. ( 7 ) is comparable to the “ideal velocity” (or “propulsive” velocity increment) of the classical rocket equation. It is a function of the thrust acceleration profile, and it characterizes - up to a certain extent - the mission requirements (with regard to propulsion) for a given scenario. We will denote this term λ and we will refer to it as the “trajectory characteristic parameter”: tf

λ = ∫ γ T2 ⋅ dt 0

In this section, as well as in the next one, we will define the vehicle’s mass budget with 4 items, as follows:


(8)

Mass budget element Model or expression Propellant (MP) MP = Mi- Mf Tank structure (MS) MS = k.Mp Power & propulsion. system (MW) MW = α.Pe Payload (MPL) MPL = Mf-MS-MW Total initial mass (Mi) Mi = MPL+Mp+MS+MW Table 2. Mass budget definition


The mass budget relations rely on the two following parameters, which are measures of the technologies used: - The specific mass of the power and propulsion system (α), defined as:

-

α=

MW Pe

(9)

k=

Ms Mp

( 10 )

The tank structural mass ratio (k), defined as:

In this paper, we will consider k=0.1 for all the calculations. With these definitions for the mass budget and hypotheses for α and k, the vehicle is entirely defined through two independent design variables: the power source (Pe) and the propellant mass (MP). Considering the mass budget model defined above, it can be shown [20] that the payload mass fraction (or payload to initial mass ratio) is expressed as follows:

M PL = 1− Mi

k +1 ⎛ 2η ⎞⎛ P 1 + ⎜ ⎟⎜⎜ e ⎝ λ ⎠⎝ M i

⎛ P − α ⋅ ⎜⎜ e ⎞ ⎝ Mi ⎟⎟ ⎠

⎞ ⎟⎟ ⎠

( 11 )

Although, as mentioned above, a single vehicle is defined through two independent design variables (Pe and MP), Eq. ( 11 ) shows that only one design variable is sufficient to define the performance in term of payload mass fraction: the payload to initial mass ratio (Pe/Mi), and that there exists an optimal value (Pe/Mi) which maximizes the payload mass fraction. Also, if we make the assumption that k, η and λ are constant and do not depend on the vehicle design variables (Pe/Mi), the maximum value of the specific mass that makes the mission feasible is expressed by [20]:

2η ⎛⎜ ⎡M ⎤ ⎞ α= ⋅ 1 + k − k + ⎢ PL ⎥ ⎟ λ ⎜⎝ ⎣ Mi ⎦ ⎟⎠

2

( 12 )

Eq. ( 12 ) is simply obtained by extracting α from Eq. (12) then minimizing it with regard to Pe/Mi. This expression yields the specific mass requirement for a mission characterized by a given payload mass fraction (MPL/Mi) and a given trajectory characteristic parameter (λ). Parameter λ is assumed to be representative of the transfer duration. Some examples of order of magnitudes for a LEO-to-Mars transfer mission with a variable Isp propulsion system in the 3000-30000 s range are: - 180 days: λ ~ 15 m2/s3 - 90 days: λ ~ 100 m2/s3 - 40 days: λ ~ 1200 m2/s3 For the “limit” case of a null payload mass, Eq. ( 12 ) becomes:

α max =

2η

λ

(

⋅ 1+ k − k

)

2


( 13 )


Eq. ( 13 ) makes possible to discuss easily the feasibility of a mission with regard to the specific mass. It gives the order of magnitude that is needed for α, provided we have an estimation of the trajectory characteristic (λ) of the mission. It should be reminded, however, that this calculation is obtained with several simplifications: - The mass budget is defined in a very simplified way (cf. Table 2), - As explained previously, the global efficiency (η) is assumed to be constant and does not depend of the current Isp level, - The tank structural mass ratio (k), is assumed to be a constant (independent of the vehicle sizing), so we do not take into account the “scaling” that is typically observed for rocket stages and leads to lower values for bigger stages, - Finally, one important assumption is that we neglect the influence of the vehicle’s design parameter (namely, Pe/Mi) on λ. We consider λ as independent from Pe/Mi, so its numerical value characterizes the propulsion requirement for a given mission (i.e. initial and final conditions, and transfer time) and a given thruster type (i.e. Isp range), regardless of the vehicle’s sizing. Although the latter simplification is important, this approach can be used for preliminary design to obtain orders of magnitude. In this context, the vehicle design and mission analysis can rely on pre-calculated table of λ as a function of the transfer time, for a particular mission (characterized by given initial and final conditions) and a particular thruster technology characterized by a Isp range (cf. for instance [18][19][25]). Moreover, the following fitting law appears to be appropriate to represent λ with an acceptable precision as a function of the transfer time ([20][25]):

λ (t f ) ≈ a ⋅ t f b

( 14 )

The coefficients a and b of the model can be obtained through least-square fitting of a set of trajectory optimization results. Combined with Eq. (15), Eq. ( 12 ) allows to compute easily the specific mass requirement as a function of payload mass-ratio and transfer time. We have performed this calculation for the Earth to Mars mission defined in section D, considering the variable Isp propulsion system (3000-30000 s specific impulse) and the hypothesis detailed earlier for the global efficiency (η = 0.6) and tank structural mass ratio (k = 0.1). In this case, the coefficients of Eq. ( 14) are the following: a = 5.222.107 b = -2.9183 The units of these coefficients are such that tf is in days and λ in m2/s3 in Eq. ( 14). The results for transfer times between 40 and 180 days and mass ratio between 0 and 0.6 are summarized in Figure 2.


Specific mass α (kg/kW)

Global efficiency 0.6 Tank mass ratio 0.1

50.00 Payload mass ratio:

45.00

MPL/Mi= 0

40.00

Unfeasible

35.00

MPL/Mi = 0.05 MPL/Mi = 0.1


30.00

MPL/Mi = 0.15

25.00

MPL/Mi = 0.2

20.00

MPL/Mi = 0.3

15.00

MPL/Mi = 0.4 MPL/Mi = 0.5

10.00

MPL/Mi = 0.6

5.00

Transfer time (days)

0.00 40

60

80

100

120

140

160

180

Short term technology (α ~ 35kg/kW) Advanced Brayton-cycle technology (α ~10 kg/kW) Longer-term Rankine-cycle technology (α~5 kg/kW) Breakthrough technology (α ~ 1 kg/kW)

Figure 2. Maximum specific mass requirements for an Earth-to- Mars Mission using a variable Isp system (3000-30000 s). On this figure, the unfeasible domain is delimited by Eq. ( 13 ), which corresponds to a null payload mass fraction. Figure 2 shows the “big picture” of the trade off between the mission requirements and the technology level of the power and propulsion system. We verify that the specific mass requirement is all the more demanding (i.e. its value is low) that the transfer is short and the payload mass fraction is high. Also, the figure allows to visualize the shortest transfer time that can be accessed with different hypotheses for the specific mass, as defined in section II. Considering a 0.05 payload mass fraction or lower, the shortest transfer time accessible can be estimated as follows: ~ 170 days, - For a short term technology (α ~ 35 kg/kW): ~ 110 days, - For a medium term advanced Brayton-cycle technology (α ~ 10 kg/kW): - For a longer term Rankine cycle technology (α ~ 5 kg/kW): ~ 90 days. We can see that a short term technology with α ~ 35 kg/kW would only bring a minor improvement of the transfer time with regard to chemical propulsion. A significant reduction of the time transfer requires a more advanced technology. Indeed, our medium and longer term assumptions for the specific mass would cut the transfer time by about 40 % or 50 %, respectively. Furthermore, the results of Figure 2 demonstrate that an even shorter transfer time requires a specific mass which exceeds our estimations for medium and long term technologies. For instance, a very short transfer time of about 40 days requires a real breakthrough in power source technologies with a specific mass around 1 kg/kW. This order of magnitude is consistent with AARC’s assumptions for a 39 day-mission using VASIMR [1]. It should be noted that the variable Isp hypothesis – upon which this study relies – was the most interesting case for the identification of accessible reduced transfer times. Indeed, since the Isp level represents an additional “degree of freedom” to control the system, a variable Isp propulsion system necessarily leads to a better (or at least equal) [payload mass fraction, transfer time] envelope than systems with a constant Isp taken in this range. As a consequence of these results, we will focus the next part of the study on mission scenarios between 90 and 180 days, in order to be consistent with our medium and longer term hypotheses for the power source technology, as defined in section C (α = 5 kg/kW and α = 10 kg/kW).


C. Analysis of reduced transfer time scenarios


1.

Methodology

As explained in the conclusion of section B, we now focus on a smaller domain for the transfer time, between 90 and 180 days. Also, we will consider and compare not only the variable Isp propulsion system (which was the focus of section B) but also the two fixed Isp propulsion systems with a 5000 s and 3000 s Isp. For each type of propulsion system, the methodology of the study will be the following. First, we will compute a set of trajectory optimization results for different vehicle designs (defined by their power level Pe and total initial mass Mi) and given transfer times in the considered interval (between 90 and 180 days). These results will allow studying the variation of the payload mass ratio (MPL/Mi) as a function of the design variable (Pe/Mi), whose importance has been emphasized in section B. For each transfer time considered (3 values for each type of propulsion system), we extract from the results the case that leads to the maximum payload mass ratio. This “best” case is described by three data: - The design variable (Pe/Mi), - The trajectory characteristic parameter (λ), - The payload mass fraction (MPL/Mi). Remark: these data do not correspond exactly to the “true” optimum design, since we have performed a limited number of trajectory optimizations (leading to a limited number of Pe/Mi values), but they can be considered as fair estimations. It is important to note that these three data are valid to describe a family of optimum vehicle design, not only the one which has been considered for the trajectory optimization. As long as the (Pe/Mi) is the same, the payload mass fraction will be maximized with the same optimized trajectory, characterized by λ, provided all the other hypothesis are the same (transfer time tf, Isp variation range, global efficiency η and structural mass ratio k). This point can be demonstrated through the expression of the payload mass fraction, as given in section B (Eq. 12):

M PL = 1− Mi

k +1 ⎛ 2η ⎞⎛ P 1 + ⎜ ⎟⎜⎜ e ⎝ λ ⎠⎝ M i

⎛ P − α ⋅ ⎜⎜ e ⎞ ⎝ Mi ⎟⎟ ⎠

⎞ ⎟⎟ ⎠

( 15 )

with: tf

λ = ∫ γ T2 ⋅ dt

( 16 )

0

Indeed, it can be shown that the thrust acceleration γT also depends on the Pe/Mi ratio. The thrust acceleration at time t is:

γ T (t ) =

T (t ) q(t ) ⋅ g0 ⋅ Isp(t ) = M (t ) M i − q(t ) ⋅ t

( 17 )

T is the thrust; M is the total mass and q is the mass flow rate. Using Eq. (7), we can express the mass flow rate as:

q⋅ =

2 ⋅η ⋅ Pe

(g 0 ⋅ Isp )2

By eliminating q from Eq; (18) and Eq. (19), we obtain, after calculation:


( 18 )


⎛ Pe ⎞ g0 ⋅ Isp(t ) ⋅ ⎜ ⎟ ⎝ Mi ⎠ γ T (t ) = (g0 ⋅ Isp(t ))2 − ⎛ Pe ⎞ ⋅ t ⎜ ⎟ 2 ⋅η ⎝ Mi ⎠

( 19 )

Since γT depends on Pe/Mi, instead of Pe and Mi separately, a given Isp(t) profile obtained for a particular vehicle design can be reused to any other vehicle design as long as the Pe/Mi ratio is the same. Provided the Pe/Mi is unchanged, this Isp(t) profile will lead to the same λ (since it is a function of the γT profile) as well as the same payload mass ratio, since it is a function of λ and Pe/Mi (cf. Eq. (8)). More particularly: the optimal Isp(t) profile we obtain by solving the optimization problem described in section A remains optimal for any other vehicle design with the same Pe/Mi. As a consequence, the three optimum design data listed above (Pe/Mi, λ and MPL/Mi) actually define a family of optimal vehicle design, with no particular restriction on the size of the vehicle. The Pe/Mi and MPL/Mi ratios can be used to define a vehicle with either a prescribed payload mass, a prescribed total initial mass or a prescribed power level. Indeed, if one of these three variables is fixed, the two others are derived from these ratios, then the detailed mass budget can be obtained using Eq. (10), (11) and (11) and the definitions of Table 1. 2.

Results for the variable Isp propulsion system (3000-30000 s)

We have performed a series of trajectory optimizations (according to the approach described in section A) for three values of the transfer time: 60, 90 and 120 days. We have considered different values of Pe between 5 and 50 MW, and values of total initial mass between 100 and 500 t. The results, sorted with increasing Pe/Mi, are summarized in table 2. All the results transcribed in this table correspond to feasible cases in terms of trajectory optimization, except one case for each transfer time, which correspond to the highest Pe/Mi (among the values tested) for which no optimal trajectory satisfying the constraints could be found. Indeed, we observe that there exists a minimum value for Pe/Mi under which the mission (with its prescribed transfer time) is not feasible. This can be explained by the fact that Pe/Mi determines the level of acceleration (cf. Eq. ( 19 )). The results of table 2 (as well as those of tables 3 and 4, for fixed Isp propulsion systems) allow to estimate this threshold. For the variable Isp propulsion system considered in this section, we observe that the minimum Pe/Mi that makes the missions feasible is lies in the following intervals: - For a 60 day transfer: [ 100 ; 125 ] W/kg - For a 90 day transfer: [ 66.7 , 100 ] W/kg - For a 120 day transfer: [ 33.3 , 50 ] W/kg We verify that the threshold increases as the time transfer decreases. This is due to the fact that a shorter transfer requires a higher acceleration. For feasible trajectory cases, we compute the payload mass fraction (MPL/Mi) according to the two hypotheses of specific mass (α = 5 kg/kW and α = 10 kg/kW), which may lead to a negative (unfeasible) value. Also, we identify in these results the case with the maximum payload mass fraction (green cells of table 2). We verify from these results (as well as results of table 2 and 3) that trajectory cases with the same Pe/Mi yield the same optimum payload mass fraction and the same trajectory characteristic parameter (λ). As a complement of Table 2, Figure 3 shows the variation of the payload mass fraction (for feasible cases) as a function of Pe/Mi. These results for the variable Isp system show that: - A 60 days transfer appears to be unfeasible with both hypotheses for the specific mass, - A 90 or a 120 days transfer are feasible with α = 5 kg/kW, but not with α = 10 kg/kW. When α = 5 kg/kW, the payload mass fraction drops dramatically as the transfer time decreases: it is about 25 % for a 120 day transfer and less that 2 % for 90 days.


Transfer time (days)

60


90

120

Pe (MWe) 50 50 50 20 50 50 20 10 20 50 50 50 50 50 10 5 10 20 20 10 20 50 50 50 50 50

Mi (t) 500 400 300 100 200 100 300 100 200 500 400 300 200 100 300 100 200 400 300 100 200 500 400 300 200 100

Pe/Mi (W/kg) 100 125.00 166.67 200.00 250.00 500.00 66.7 100.00 100.00 100.00 125.00 166.67 250.00 500.00 33.33 50.00 50.00 50.00 66.67 100.00 100.00 100.00 125.00 166.67 250.00 500.00

Mf (t) unfeasible 76 117 48 107 65 unfeasible 56 112 280 249 201 140 76 unfeasible 49 100.5 200 186 68 136 340 282 217 151 78.5

λ

(m2/s3) 639.47 312.82 260.00 260.75 323.08 94.29 94.29 94.29 90.96 98.51 128.57 189.47 62.45 59.40 60.00 49.03 56.47 56.47 56.47 62.77 76.50 97.35 164.33

MPL/Mi with α=5 kg/kW

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