to Mars in 39 days using nuclear electrically powered VASIMR® (Variable ... with different hypothesis concerning IMLEO (100 T, 200 T, 300 T, 400 T and 500 T),.
4TH EUROPEAN CONFERENCE FOR AEROSPACE SCIENCES (EUCASS)
Trajectory Optimisation for Fast Earth-Mars Transfers with High Power Nuclear Electric Propulsion Richard Epenoy*, Julien Laurent-Varin **, Sandrine Avril***, Elisa Cliquet**, Pascal Bultel** *CNES Toulouse 18 avenue Edouard Belin 31401 Toulouse Cedex 9, France **CNES Evry rond point de l’espace - Courcouronnes 91023 Evry Cedex, France ***ATOS ORIGIN 6 impasse Alice Guy BP 43045 31024 Toulouse Cedex 3, France
Abstract Ad Astra Rocket Company (AARC) recently 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 strong hypothesis concerning the power level (200 MW) and the specific mass of the nuclear electric generator (< 0.6kg/kW). On-going French studies [2] tend to show that, in a mid-term context, specific masses of nuclear reactors for exploration missions, even at very high power levels, are expected to be clearly above this high value assumed by AARC in [1]. Thus, CNES decided to evaluate the feasibility of fast Mars transfers with heavy payloads (several tens of tons) depending on reactor power level and associated specific mass hypothesis taking into account feedback from national studies. Firstly, a methodology for designing fuel-optimal heliocentric trajectories is presented. The Pontryagin’s Maximum Principle is used considering the specific impulse (Isp) as an additional control variable. This approach allows us to cross-check AARC results and then to widen the spectrum of possible architectures. In a second time, various power levels up to 50 MW, combined with either variable or fixed Isp are considered. Slightly longer transfer durations (including minimum-time Earth’s escape) are also explored (60, 90 and 120 days). Moreover, sensitivity calculations are performed on some parameters including the fixed specific impulse (of the escape phase) or the departure date. Finally, a synthesis of these results is presented and can be used to design a vehicle based on technologies that could be available in a mid or long-term frame [3].
Copyright © 2011 by First Author and Second Author. Published by the EUCASS association with permission.
4TH EUROPEAN CONFERENCE FOR AEROSPACE SCIENCES (EUCASS)
1. Mission strategy and trajectory computation 1.1 Reference mission and consideration on specific masses Before going into computation of various trajectories for fast manned Mars mission using high power electric propulsion, we started by cross-checking AARC results for the “39 days – 200 MW scenario” given in [1]. Main hypothesis concern a departure from a circular Low Earth Orbit (LEO) of 1000 km altitude, a relative velocity w.r.t. the Earth (V-Infinity) equal to 2.5 km/s at the Earth’s departure and a heliocentric phase with variable specific impulse (this last one being considered as a control variable). AARC describes a two ships mission in [1]: - a first spaceship is used for cargo and - a second spaceship is used for human transfer to Mars. In the present paper, we only focus on the second spaceship which has to be the fastest of both. Moreover, we do not consider the return trajectory from Mars to the Earth. The departure date from LEO is assumed to be July 7th , 2018 and the initial ship mass is equal to 600 T. However, AARC calculations (confirmed by ours) show that with this hypothesis and with the arbitrary value of 20 T payload, the total specific mass available for the propulsion system (including power plant, engines, dry mass of tanks) would be only 0.6 kg/kW. As proposed further in [1], situation can be slightly improved by departing with 600 T from L1 instead of a 1000 km LEO. However, specific mass still needs to be around 1 kg/kW. AARC admits this value is very optimistic. Thanks to CNES studies carried out on nuclear electric technologies [2], we were able to use values of specific masses and power levels that we believe to be more realistic in a mid-term context: for power levels in the order of magnitude of tenth of megawatts, the choice of a Brayton conversion cycle for the power plant could lead to a value of roughly 10 kg/kW. With more complex technologies, like Rankine conversion cycle (challenging to master with two phase flows in low or zero-G environment, in particular for transient phases) a value of roughly 5 kg/kW could be assumed. Both cases are, however, far from the state of the art (qualified space reactors that have successfully flown were using static conversion with low efficiency and delivered less than 10 kWe). Power levels that we have considered are also limited to 50 MW. More details about the justification for those values can be found in [2] and [3]. Thus; in order to be consistent with specific mass values of the nuclear electric generator that we believe to be easier to achieve than 1 kg/kW, we have computed optimal trajectories with various hypothesis concerning the transfer duration. After a first trajectory computation dedicated to the cross-check of AARC results, we have carried out some other calculations associated with different hypothesis concerning IMLEO (100 T, 200 T, 300 T, 400 T and 500 T), different values of the transfer duration (60, 90, 120 and 180 days) plus additional cases with constant Isp as in Magneto-Plasma Dynamic (MPD) technology. By combining most interesting results with our specific masses hypothesis, we will then be able to identify scenarios of fast Mars transfers that could be achievable in a shorter term that a 39 days mission.
1.2 Trajectory segmentation and model An Earth-Mars trajectory is commonly split into different phases : two planetocentric phases (one for the Earth’s departure, the other one for Mars arrival) and one heliocentric phase. For the present work we neglect the second planetocentric phase. For the (first) planetocentric phase, we consider a constant maximum thrust transfer, associated with a fixed Isp equal to its lower value, Isp1 = 3000 s. For the heliocentric phase, the specific impulse is optimised (except for MPD cases) between Isp1 = 3000 s and Isp2 = 30000 s. This optimal tuning of the specific impulse is not classical, thus we present hereafter the necessary optimality conditions induced by this additional degree of freedom.
Copyright © 2011 by First Author and Second Author. Published by the EUCASS association with permission.
SESSION 01.10 NUCLEAR PROPULSION FOR SPACE EXPLORATION
2. Optimal control method for optimising the specific impulse during the heliocentric phase Usually, space electric thrusters have a fixed vacuum specific impulse in steady-state for a given power level. Thus, when optimizing trajectory, thrust direction is controlled but neither are flow rate nor specific impulse. In the frame of AARC technology VASIMR®, thrust and specific impulse can be both optimised. We present hereunder, the application of Pontryagin’s maximum principle to this dynamic model.
2.1 Dynamic model
° r(t ) = v(t ) ° ° r (t ) T (t ) + U (t ) ® v(t ) = − μ 3 m r (t ) ° ° T (t ) − °m (t ) = g 0 Isp (t ) ¯
(1)
Where
r (t ) v(t ) m(t ) T (t )
is the position vector at time t is the velocity vector at time t is the spaceship mass at time t is the thrust value at time t
U (t ) Isp (t )
μ
is the thrust direction vector at time t (
U (t ) = 1 )
is the specific impulse at time t
g 0 are the gravitational constant of the Sun (resp. the acceleration due to gravity at see level
and
g 0 = 9.80665 m / s 2 ) . The control variables are then T(t), U(t) and Isp(t). These variables have to be optimised in the feasible control space F defined as follows:
F = ®(T ,U , Isp) ¯
η ∈]0,1] P ∈ℜ
+
T ∈ [0,
½ 2ηP ],U ∈ ℜ3 , U = 1, Isp ∈ [ Isp1 , Isp2 ]¾ g 0 Isp ¿
(2)
is the efficiency coefficient of the engine, is the electrical power.
2.2 Pontryagin’s maximum principle According to the maximum principle, the optimal control is given by the minimising control of the Hamiltonian:
H = λr v − μ T
where λr ,
2
λv
λvT r r
3
ª λ TU λ º +T« v − m » g 0 Isp ¼ ¬ m
and λm are respectively the co-states associated with position r , velocity
(3)
v and mass m .
R. Epenoy, J. Laurent-Varin, S. Avril, E. Cliquet, P. Bultel. TRAJECTORY OPTIMISATION FOR FAST EARTH-MARS TRANSFERS WITH HIGH POWER NUCLEAR ELECTRIC PROPULSION
We do not present the well-known adjoint equations of the co-states. Remark: In the minimisation process, we avoid undefined situations corresponding to singular arcs, by assigning arbitrary values to undefined control variables.. The table given hereunder gathers the optimal values of control variables (T,U,Isp) as functions of co-states values :
λv
λm
=0
>0 =0 0 =0
−
≠0
