Trajectories for Human Missions to Mars, Part 2: Low-Thrust Transfers

Transcription

1 JOURNAL OF SPACECRAFT AND ROCKETS Vol. 43, No. 5, Septeber October 26 Trajectories for Huan Missions to Mars, Part 2: Low-Thrust Transfers Daon F. Landau and Jaes M. Longuski Purdue University, West Lafayette, Indiana, DOI: / We copute optial low-thrust transfers (with constant thrust and constant specific ipulse) between Earth and Mars over a range of flight ties (fro 12 to 27 days) and launch years (between 29 and 222). Unlike ipulsive transfers, the ass-optial trajectory depends strongly on the thrust and specific ipulse of the propulsion syste. A low-thrust version of the rocket equation is provided where the initial ass or thrust ay be iniized by varying the initial acceleration and specific ipulse for a given power-syste specific ass and for a trajectory tie of flight. With fixed tie-of-flight transfers there is a iniu thrust and a axiu allowable specific ass; that is, if the available thrust is too low or the specific ass is too large then the desired transfer does not exist. We find the iniu allowable thrust for constrained tie-of-flight issions is on the order of a few Newtons per etric ton of payload for power systes with tens of kg=kw. As expected, the thrust and V requireents of the trajectories decrease with increasing flight ties. By extending the flight tie fro 18 to 27 days the V is reduced by 4% for powered captures and up to 35% for aeroassisted capture trajectories. Noenclature a = initial acceleration due to thrust, =s 2 f t = tankage factor, % g = standard acceleration due to gravity at Earth s surface, 9:8665 =s 2 I sp = specific ipulse, s pay = payload, t = initial ass, t _ = ass flow rate, kg=s P = power, kw T = thrust, N t b = burn tie, s = specific ass, kg=kw = efficiency, % I. Introduction THE concept of traveling to Mars in a low-thrust vehicle has been explored since the 195s [1 8]. Moreover, there have been several decades of low-thrust trajectory optiization to support the design of huan issions to Mars [9 21]. The ost popular use of low thrust in Mars issions is for cargo transfer, where tie insensitive payloads can take years to reach Mars fro Earth. However, these transfers are not suitable for crew transport because the long flight ties expose the crew to excessive radiation and to icrogravity and often entail short stay ties at Mars. [We do not exaine cargo transfers in this paper as they have been sufficiently treated in the literature (see, for exaple, [1,17]).] To fairly copare low-thrust issions with ipulsive ones, the interplanetary tie of flight (TOF) and the Mars stay tie for the crew transfer vehicle should be the sae. For exaple, a low-thrust ission with 1 yr transfers and a 1 yr stay tie puts the crew at ore risk for less return than a high-thrust ission with half-year transfers and 1.5 years on Mars. We copute optial low-thrust trajectories with Received 26 July 25; accepted for publication 21 Deceber 25. Copyright 26 by Daon F. Landau and Jaes M. Longuski. Published by the Aerican Institute of Aeronautics and Astronautics, Inc., with perission. Copies of this paper ay be ade for personal or internal use, on condition that the copier pay the $1. per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 1923; include the code $1. in correspondence with the CCC. Doctoral Candidate, School of Aeronautics and Astronautics, 315 N. Grant St.; landau@ecn.purdue.edu. Student Meber AIAA. Professor, School of Aeronautics and Astronautics, 315 N. Grant St.; longuski@ecn.purdue.edu. Associate Fellow AIAA. 143 TOF fro 12 to 27 days (the sae TOF constraints as the ipulsive transfers in Part 1) so the ission tielines with low thrust are consistent with the ipulsive transfer issions. Results are provided for both powered capture and aeroassisted trajectories. Because Earth Mars trajectory characteristics approxiately repeat every seven synodic periods (14.95 years), we exaine trajectories over a span of seven issions (with launch dates fro ). II. Trajectory Models Low-thrust trajectory legs are odeled as a series of constrained Vs, which iics the effect of continuous low-agnitude acceleration [18]. Prier vector analysis [22] is not required in our low-thrust optiizer as the nuber of (constrained) ipulses is set a priori, and the steering history is a direct optiization vector. We copute direct (Earth Mars and Mars Earth) low-thrust trajectories for both powered and aeroassisted arrivals with the objective of axiizing the final ass of the spacecraft. The initial spacecraft state is the position and velocity of the departure planet. The target planet s position and velocity are atched for powered arrivals (thus V 1 ), but only the position is atched for aeroassisted arrivals (where V 1 ). The planetocentric portion of the trajectory is assued to be fixed (i.e., we assue the spacecraft is launched optially off the surface to reach V 1 ), though better perforance is possible when both the planetocentric and heliocentric portions are optiized as a whole trajectory [19]. These low-thrust trajectories do not necessarily iniize ission cost, but are indicative of the type of trajectory that would optiize a huan ission to Mars. We use a sequential quadratic prograing algorith [23] to copute iniu-v trajectories with bounded TOF. (By bounded we ean the TOF ay be less than or equal to the constrained value. Moreover, the departure and arrival dates can vary freely as long as the TOF between the is within the bound.) The low-thrust optiization proble is ore coplex than the ipulsive optiization proble. The low-thrust transfers are constructed with 5 constrained Vs, which results in 16 design variables (5V plus the tie, V 1, and spacecraft ass at Earth and Mars), 57 nonlinear inequalities (5V agnitudes plus the position, velocity, and ass at a atch point), 3 linear equalities (V 1 and fixed departure ass), and 1 linear inequality (TOF). On the other hand, the optiization proble for direct ipulsive trajectories can require as few as two design variables (Earth and Mars encounter ties) and one linear inequality (bounded TOF). The ore challenging ipulsive cycler optiization proble has 73

2 144 LANDAU AND LONGUSKI variables (25 planetary encounter ties plus the position and tie of 12 deep space aneuvers), 23 nonlinear equalities (flyby V 1 ), 23 nonlinear inequalities (flyby altitudes), and eight linear inequalities (TOF). Ipulsive trajectories are calculated in MATLAB and the low-thrust transfers are calculated using FORTRAN with software that was previously developed at Purdue. (We found that due to the size of the low-thrust proble our MATLAB code rarely converges for low-thrust transfers, while the FORTRAN code alost always converges in a tiely anner.) For each trajectory type, we initially optiized the long TOF (27- day) trajectory for the 29 ouound opportunity. The 27-day TOF trajectory served as the initial guess for the 26-day TOF trajectory, and so on (by 1-day intervals) until the 12-day TOF transfer was optiized. The whole process was then repeated for the subsequent synodic periods up to the 222 ouound opportunity. III. Results Deterining the axiu payload for a low-thrust transfer requires optiization of the propulsion syste along with the trajectory. For exaple, an increase in thrust lowers the V (and the propellant ass fraction) for the vehicle, but the hardware ass ust increase to provide additional power to the thrusters. Consequently, there is a trade between low propellant ass with high power levels and low hardware ass at low power levels. The balance in this trade is usually deterined by the specific ass () and efficiency () of the propulsion syste. The specific ass deterines the ratio between the hardware (power source, thrusters, etc.) and the aount of power it produces hardware =P electric (1) and the efficiency is the ratio of jet power to hardware power P jet =P electric (2) The jet power ay be deterined fro P jet TgI sp =2 (3) Thus, the hardware ass is hardware TgI sp =2 (4) and, for constant and, increases proportionally with the thrust and specific ipulse. The propellant tanks also contribute a significant portion of the spacecraft ass, and the tank ass ay be estiated via the tankage factor f t tank = propellant (5) The final-to-initial ass fraction ay be coputed for low-thrust issions via the rocket equation [24] V gi sp ln adt f T dt _gi sp dt f gi sp d (6) e V=gI sp (9) pay 1 e V=gI sp =2a gi sp f t e V=gI sp 1 Thus, the spacecraft thrust is (the constant value): T a e V=gI sp (1) pay 1 e V=gI sp =2a gi sp f t e V=gI sp 1 Unlike ipulsive trajectories, the V for low-thrust transfers depends on the acceleration and specific ipulse of the spacecraft [i. e., V Va ;I sp ; TOF]. Fro Eqs. (4) and (1) the hardware ass also depends on acceleration and specific ipulse. As a result, the initial ass [Eq. (9)] for low-thrust trajectories is iniized when the V is balanced against the hardware ass both of which depend on the variables a and I sp. The optiization proble is thus to iniize = pay a ;I sp ; Va ;I sp ; TOF; =; f t by varying a and I sp with constant TOF, =, and f t. We note that the trajectory optiization proble (iniize V for constant a, I sp, and TOF subject to a T=) is a subset of the ass optiization proble (in our forulation). Therefore, to iniize the initial ass [Eq. (9)] or to iniize the thrust [Eq. (1)], an accurate eans of deterining the iniu V for a given trajectory TOF and vehicle a and I sp is required. (In our V iniization proble we assue free departure and arrival dates and optiu thrust vector history subject to TOF and acceleration constraints.) ola [11] describes an approxiate analytic ethod to calculate V as a function of a and I sp, provided the V-optial burn tie t b for a trajectory with the sae TOF is known. In contrast to the axiu-acceleration trajectories provided in Part 1, we present the V for a direct transfer fro Earth to Mars (and vice versa) using the iniu possible acceleration during launch years in Fig. 1. In the specific case of Fig. 1 we calculated trajectories that atch the heliocentric orbit of Mars fro the orbit of Earth and set the ass flow rate to zero (corresponding to an infinite I sp ). Because the acceleration is iniized, the burn tie for these trajectories is equal to the TOF (i. e., the thruster is always on). Using this data, we found that ola s heuristic ethod produces values for optiu payload ass fractions to within a few percent (say, 3%). Alternatively, nuerical optiization techniques provide higher fidelity results at the expense of longer coputation tie. Exaple low-thrust trajectories are presented in Fig. 2. For copleteness the stay tie and ission duration for the trajectories in Fig. 1 are provided in Fig. 3. Let us consider a 21-day Earth-to-Mars transfer in 214 with a transfer vehicle ass (payload) of 4 t, an electric propulsion syste with = 3 kg=kw, and a tankage factor of 5%. The lowest possible thrust for this transfer is 11 N with a :712 =s 2, I sp 2; 5 s, and V 14:3 k=s. (We note that the iniu acceleration at I sp 2; 5 s is :639 =s 2 with V 17:3 k=s.) The required thrust is usually on the order of a few Newtons per etric ton of payload for constrained TOF issions, but the exact value depends on the assued hardware The ass ratio ay be written explicitly as e V=gI sp (7) f where we assue that the thrust and specific ipulse are constant and that the thrust ay be coputed by T _gi sp (8) The rocket equation is particularly useful in deterining the ratio of initial ass (payload, hardware, tankage, and propellant) to payload ass Fig. 1 V for iniu-acceleration transfers with powered capture.

3 LANDAU AND LONGUSKI 145 Fig. 4 V for low-thrust transfers with aerocapture or direct entry. Fig. 2 Low-thrust transfers with V 1 at each planetary encounter. V (k/s) V (k/s) TOF (days) TOF (days) a) Earth to ars b) Mars to earth Fig. 5 Arrival V 1 for low-thrust transfers with aerocapture or direct entry. 4 Fig. 3 Stay tie and ission duration for powered capture low-thrust transfers. developent (i.e., =). In addition to a iniu thrust liit, there is also a axiu allowable = for low-thrust issions [when the denoinator of Eq. (9) approaches zero]. For exaple, transfers with TOF around 12 days are enabled only when = < 2 kg=kw (with I sp > 1; s) and regular (i.e., at every launch opportunity) 12-day transits do not exist unless = < 1 kg=kw. We also exained iniu-acceleration transfers for aerocapture issions (for arrival V 1 ) but found that the V 1 at arrival was ipractical (i.e., it is often at least double the ipulsive transfer V 1 ). Instead, we optiized low-thrust trajectories with infinite I sp (i. e., assuing constant ass) and set the acceleration to the levels found in Fig. 1 for a given launch year and TOF cobination. The resulting V and arrival V 1 (where the launch V 1 is zero) are found in Figs. 4 and 5, respectively. Because the sae initial acceleration was used to calculate the powered and aeroassisted capture trajectories, the burn tie is given by t b V A =V P TOF where V P is the powered arrival V (fro Fig. 1) and V A is the aeroarrival V (fro Fig. 4). We note that increasing the acceleration decreases the V and burn tie, and in the liit of infinite thrust and zero burn tie (corresponding to ipulsive V), the V approaches the su of the departure and arrival V 1 for powered capture issions and becoes the departure V 1 for aeroassisted capture. Let us again consider a 21-day transit in 214 with 4 t of payload, 8 t for the heat shield (at arrival), = 3 kg=kw, and f t 5%. Instead of iniizing the thrust (which would result in excessive arrival V 1 ), we set the acceleration to the sae value as the powered-arrival exaple, a :712 =s 2. In this case, an I sp of 2,5 s results in a thrust level of 54 N with a V of 3.2 k/s (again using ola s ethod) and 6.4 k/s arrival V 1 (as in Fig. 5). We find that the required thrust levels for aeroassisted transfers with oderate V 1 are usually around one Newton per etric ton of payload. However, for low-thrust ission optiization we recoend including a constraint to liit the arrival V 1, or as a rule of thub, setting the acceleration to at least the sae level as that of a powered-arrival trajectory. The hardware liits for aeroassisted trajectories are less stringent than those for powered-arrival transfers as the = can be as large as 5 kg/kw for the ost deanding 12- day transfers. While the trajectory characteristics shown in Figs. 1 5 and the vehicle properties given by Eqs. (9) and (1) are produced assuing constant thrust and I sp (as in nuclear electric propulsion), the results also apply (indirectly) to trajectories with variable thrust and I sp (as in solar electric propulsion). Unlike the constant power of nuclear electric propulsion, the power available fro a solar electric propulsion syste drops off with the inverse square of the distance fro the sun. Thus, the power available at Earth is double the power available at Mars. As a result, either the thrust or I sp (or both) ust be lower at Mars than at Earth [fro Eq. (3)]. Even though the thrust and I sp ay vary during an Earth Mars transfer, iniu-v trajectories with bounded TOF have siilar characteristics (e.g., the sae V and V 1 to within a few percent) as long as the average acceleration and I sp are the sae. Thus the average acceleration and specific ipulse (over the burn tie) ay be substituted into constant thrust and constant I sp odels to deterine the V, V 1, thrust, and ass fractions of solar electric systes. For exaple, consider the iniu-acceleration transfers in Fig. 1. The V for these trajectories do not vary significantly with

4 146 LANDAU AND LONGUSKI different thrust profiles. While a solar electric syste produces ore thrust (and ore acceleration) near Earth and less near Mars than a nuclear electric syste (with constant thrust) the total V for a given TOF is very nearly the sae for both systes. Because the V and burn tie (t b TOF) are the sae, the average acceleration a ave V=t b is also the sae. Therefore, for solar electric propulsion trajectories with powered capture, the spacecraft ust provide ore acceleration and I sp near Earth and less acceleration and I sp near Mars than a nuclear electric syste to produce the sae (optial) V. Again, as long as the average acceleration and I sp are the sae, the trajectory V will be nearly equal (even for transfers above the iniu acceleration liit). This property (of coparable V and V 1 when the average acceleration and I sp are the sae) also applies to trajectories with aeroassisted capture. We designed trajectories with a solar electric propulsion syste so that the average acceleration and I sp are the sae as the trajectories in Fig. 4 and found that the V and arrival V 1 agreed to within 3%. For aeroassisted trajectories, the thrusting occurs near Earth for Earth Mars transfers and near Mars for Mars Earth transfers. Thus, the available power (fro the solar panels) does not vary as uch for aeroassisted captures as for powered captures, which require thrusting near both Earth and Mars. Moreover, the actual acceleration ore closely atches the average acceleration for aeroassisted trajectories, which akes it easier to estiate an initial acceleration to size the thrust level of the solar electric syste. For powered arrivals, the initial acceleration is significantly higher (for Earth departures) or lower (for Mars departures) than the average acceleration, which akes it ore difficult to estiate the necessary thrust levels over the trajectory. We note that for any type of electric propulsion syste there is a iniu required thrust to achieve a given TOF and a axiu allowable = to provide argin for payload in the final spacecraft ass. A notable difference between the ipulsive-v curves (in Part 1) and the iniu-acceleration curves (Fig. 1) is that the ipulsive transfers typically flatten out by 27-day TOF (i.e., a local iniu exists below a TOF of 27 days) whereas the iniu-thrust V continue to decrease at 27-day TOF. The aeroassisted low-thrust curves (Fig. 4) do not exhibit the sae behavior because the acceleration was set to a specific value (deterined fro Fig. 1) for each launch year and TOF cobination. Indeed, there is a range of optial V values for a given launch opportunity and axiu TOF as the thrust varies fro a iniu value to infinity (i.e., ipulsive thrusting). The trajectories in Figs. 1 5 are designed to liit the acceleration requireents of an electric propulsion syste to the lowest practical values for the crew transfer vehicles. (Of course, lower acceleration and longer TOF would be appropriate for cargo transfers.) Further, the low-thrust curves do not rise as sharply as the ipulsive-v curves at low TOF. In Fig. 1 the V doubles fro 27- to 15-day TOF (siilar to ost ipulsive transfers), but the 12-day TOF V is only about 2.6 ties the 27-day TOF V across each launch opportunity (the corresponding ipulsive-v ratio is significantly higher). The relative V reduction fro 18- day TOF to 27-day TOF for iniu-acceleration trajectories is about 4%, while the relative reduction fro 18 days to 21 days is closer to 17% across all launch opportunities. The V for aeroassisted capture trajectories increase at a less unifor rate over the different launch opportunities. For exaple, the V ratio fro a TOF of 27 days to 12 days is only about 1.5 when Mars is near perihelion (during ), but the sae ratio is approxiately 2.4 (for Earth Mars transfers) and 3. (for Mars Earth transfers) when Mars is at aphelion (during ). The ipulsive aeroassisted direct trajectories exhibit a relatively steeper rise fro 27- to 12-day TOF, yet retain the sae disparity between trajectories when Mars is at perihelion or aphelion. There is also a broad range in the reduction in V fro 18-day TOF to 27-day TOF aeroassisted trajectories. When Mars is near perihelion the reduction is only about 5% (for the given acceleration levels), but the V is reduced by up to 3% for ouound transfers and 4% for the inbound trajectories when Mars is near aphelion. The ass savings that ay be gained by extending the allowable TOF depends ainly on the vehicle ass and I sp, but a TOF extension of only one or two onths can lead to a considerable reduction in ission ass. Fro Fig. 3 we note that the stay tie decreases linearly and the ission tie increases linearly as a function of TOF for low-thrust issions. (Though not shown, the stay tie and ission duration of aeroassisted trajectories are usually three weeks shorter than the powered capture ties found in Fig. 3.) The ipulsive curves (direct transfers in Part 1) exhibit siilar trends at lower TOF, but begin to flatten out by about 2 days TOF. The difference occurs because ost ipulsive trajectories reach a local iniu for V at TOF below 27 days while low-thrust trajectories continue to lose V as the TOF is extended beyond 27 days. (Unlike ipulsive trajectories, low-thrust transfers are not significantly affected by the coplications of a plane change around 18 transfer angles.) The ission duration for ipulsive and low-thrust transfers are about the sae at low TOF, but the duration for low-thrust issions is about 2 onths longer (out of 2.6 years) at longer flight ties. Copared with ipulsive issions, the stay tie for low-thrust transfers varies over the TOF range, but is usually within 1 onth (out of 1.5 years) of the ipulsive stay tie. For preliinary trade studies, the Mars stay tie and total ission duration are approxiately the sae for lowthrust and ipulsive transfers with constrained TOF. A suary of low-thrust versus ipulsive transfers is provided in Table 1. IV. Conclusions We investigate low-thrust trajectories with short transfers (TOF fro 12 to 27 days) that are suitable for crew transport. Seven issions with launch dates fro are exained to cover an entire 15-year Earth Mars synodic cycle. As with ipulsive transfers, the V is lower when Mars is near perihelion (216 and 218 launch years) and the V is higher when Mars is near aphelion (during the 29 and 222 issions). The V for aeroassisted capture trajectories is significantly lower than the V for powered captures, but the aeroassisted arrival V 1 tend to becoe higher with low-thrust transfers than with ipulsive trajectories. Further reductions in V are available by extending the TOF beyond the NASA-recoended 18 days. By increasing the TOF by 1 onth to 21 days the V decreases by 17%, (by 7.5 k/s). Further if we choose 27-day transfers over 18-day transfers, up to 39%, or 17.5 k/s, of V is eliinated fro the ission. Because the acceleration and specific ipulse directly affect the iniu V for a low-thrust transfer, we recoend optiizing the trajectory and the electric propulsion syste siultaneously. The specific ass-to-efficiency ratio = of the propulsion syste provides an efficient eans to balance the propellant ass against the hardware ass while iniizing the initial ass or required thrust for a given payload and TOF. Fro a trajectory standpoint, the Table 1 Qualitative coparison of low-thrust and ipulsive trajectories a Characteristic Low-thrust Ipulsive Burn tie Weeks Minutes Total V Larger Saller V is function of TOF, a, and I sp TOF V sensitivity to TOF and launch year Less More Mission feasibility drivers Thrust and = Propellant ass fractions a For siilar flight tie and ission duration.

5 LANDAU AND LONGUSKI 147 priary benefit of increasing the TOF or eploying aeroassisted capture is a reduction in V; however, these trajectory choices also play an iportant role in the necessary hardware for a low-thrust vehicle. As the trajectory V decreases, lower thrust levels are required and power systes with larger specific asses ay be used to coplete the transfer. If people are to travel to Mars in low-thust vehicles, then judicious trajectory design will be required to tailor the propulsion systes with the available technology. Acknowledgents Daon F. Landau s work has been sponsored in part by a National Defense Science and Engineering Graduate (NDSEG) Fellowship and a National Science Foundation Graduate Research Fellowship. References [1] Stuhlinger, E., Electrical Propulsion Syste for Space Ships with Nuclear Power Source, Journal of the Astronautical Sciences, Part 1, Vol. 2, winter 1955, pp ; Part 2, Vol. 3, spring 1956, pp ; Part 3, Vol. 3, suer 1956, p. 33. 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