Launch Window Analysis for Round Trip Mars Missions

The Space Congress® Proceedings 1968 (5th) The Challenge of the 1970's

Apr 1st, 8:00 AM

Larry A. Manning National Aeronautics and Space Administrationm, Office of Advanced Research Technology, Mission Analysis Division

Byron L. Swenson National Aeronautics and Space Administrationm, Office of Advanced Research Technology, Mission Analysis Division

Jerry M. Deerwester National Aeronautics and Space Administrationm, Office of Advanced Research Technology, Mission Analysis Division

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Scholarly Commons Citation Manning, Larry A.; Swenson, Byron L.; and Deerwester, Jerry M., "Launch Window Analysis for Round Trip Mars Missions" (1968). The Space Congress® Proceedings. 1. https://commons.erau.edu/space-congress-proceedings/proceedings-1968-5th/session-6/1

This Event is brought to you for free and open access by the Conferences at Scholarly Commons. It has been accepted for inclusion in The Space Congress® Proceedings by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. LAUNCH WINDOW ANALYSIS FOR ROUND TRIP MARS MISSIONS

Larry A. Manning* Byron L. Swenson* Jerry M. Deerwester*

National Aeronautics and Space Administration Office of Advanced Research Technology Mission Analysis Division Moffett Field, California

SUMMARY In this paper, a comparison of three different tech niques for performing the turning maneuver from cir Round trip missions to Mars have been investigated cular orbit during launch windows at both Earth and to define representative launch windows and associated Mars is made. The techniques consist of optimum one- AV requirements. The 1982 inbound and the 1986 out and two-impulse transfers, and a restricted three- bound Venus swingby missions were selected for analy impulse transfer utilizing an intermediate elliptic orbit sis and serve to demonstrate the influence of the char from which the required plane change is made. In acteristics of the heliocentric trajectories on the launch addition, an assessment of using three similar transfers window velocity requirements. This report presents from elliptical parking orbits at Mars is made and the results indicating the effects on the launch windows of results are compared to the circular orbit results. velocity capability, transfer technique, and of the incli nation, eccentricity, and insertion direction of the orbit. To be most meaningful, the departure analysis The analysis assumed a circular parking orbit at Earth should be mission oriented. Thus, the orbit escape and considers both circular and elliptical parking orbits techniques were applied to two specific Mars stopover at Mars. Use of one-, two-, and three-impulse trans missions. These two missions encompass the four fers were investigated. The three-impulse transfer trajectory classes of primary interest: i.e., direct in employs an intermediate elliptic orbit of 0,9 eccen bound and outbound, and Venus swingby inbound and out tricity. For all cases, insertion at planet arrival was bound transfers. Therefore, while the quantitative re into an orbit coplanar with the arrival asymptote and sults are not directly applicable to other Mars missions, any required plane change was performed during the the conclusions can be generalized to other round trip planet departure phase. trajectory modes by utilizing combinations of these transfer trajectories. The minimum AV requirement to transfer from a circular parking orbit to a hyperbolic asymptote occurs The paper is divided into three major sections. The when the orbits are coplanar and the maneuver is per first discusses the orbit transfer techniques which were formed at periapsis of the hyperbola. The study indi utilized and provides parametric data in support of the cates that, using a three-impulse transfer, the AV assumptions and approximations made in the analysis. penalty for non-coplanar departures, is no more than The second section provides the criteria for the mission 5-10% above the minimum coplanar requirements. analysis and defines the characteristics of the 1982 in Therefore, use in mission analysis of the coplanar AV bound Venus swingby and of the 1936 outbound Venus requirements would not result in large errors if three- swingby which were chosen as representative Mars impulse transfers are acceptable. Use of fewer im missions. The coupling of the 'missions with the trans pulses significantly increases the error. Similar fer'techniques is contained in the third section, which characteristics occur for elliptical parking orbits. How provides contour maps of constant velocity increments ever, due to the low coplanar AV's, they provide a as functions of inclination and staytime in orbit. longer launch window for a given total AV capability. METHOD OF ANALYSIS INTRODUCTION Orbit Geometry Preliminary analysis of interplanetary missions is generally performed with the assumption that the day- In considering the launch window problem, for a to-day launch window penalties can be closely approxi given mission, it is first necessary to establish the mated by the change in the coplanar departure velocity relative positions of the orbit plane and the hyperbolic requirements. Since a parking orbit at Earth and/or escape asymptote with time. For the analysis of plane the objective planet will generally be used, a unique tary launch windows, it was assumed that the parking orbit inclination will exist during the launch period of orbit at planet arrival was coplanar with the arrival interest. The time dependence of the departure hyper asymptote. The resulting orbit elements are shown in • bolic asymptote and the regression of the orbit plane Figure 1. A planet-centered right-hand coordinate due to planetary oblateness allow essentially coplanar system with Z axis at the north pole and X axis at the departures for only short time periods. In general, a * planet vernal equinox was chosen, as indicated. The non-zero angle will exist between the asymptote direc arrival asymptote or hyperbolic excess velocity vector tion and the orbit plane, and the velocity increment is defined conventionally as a planet centered vector necessary to accomplish the required turning maneuver with right ascension, p, and declination, 6*. The orbit can be of such magnitude as to render the coplanar assumption invalid.

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6,1-1 elements of interest, that is, the inclination, i, and magnitude of (p - 12) is greater than 90 v . This departure longitude of the ascending node, ft, are related by is called northern injection. The other plane has the spacecraft departing below or south o:' the excess velocity tan 6 sin (p - £2) = vector. This case has a (p - W less than 9CT and is a tan i southern injection. for all i > 6, while the argument of periapsis. co, is Impulsive Analysis measured from the ascending node 0 The assumption was made for this analysis that the It can be seen that for each inclination there are two velocity increments are applied impulsively. The first orbit planes coplanar with the arrival asymptote. These effect of this assumption is the neglect of gravity losses. two orbits can be distinguished by the relative positions Gravity losses, however, are relatively small for these of the spacecraft approach vector at the time of the missions and can easily be approximated, or even ne final midcourse maneuver and the planet centered glected without significant error. excess velocity vector. If the spacecraft approaches above the excess velocity vector, it moves initially It can be shown that the orientation between the toward the north pole of the planet and the magnitude parking orbit and the escape hyperbola is preserved of (p - 0) will be greater than 90°. This orientation whether the velocity is added impulsively or through a shall be referred, to as a northern insertion. In the finite thrusting time. This is indicated by the example other case, the spacecraft approaches below the excess in Figure 3 where the velocity increment required to velocity vector (i.e., toward the south pole) and the escape to a hyperbolic excess speed of 6.0 km/sec from magnitude of (p - 0) is less than 90°. Figure 1 illus Mars is shown as a function of the turning angle, that is. trates this configuration which is called a southern the angle between the line of apsides and the departure insertion. asymptote. This example is for a departure from an orbit with an eccentricity of 0.9 and 1000 km periapsis The situation at some time after arrival is illus altitude. The initial thrust-to-weight ratio for the trated by Figure 2. The perturbation due to planet finite thrusting maneuver was optimized as a function oblateness causes the orbit plane to regress about the of the true anomaly at the start of thrusting, and varied planet in the manner shown. That is, the inclination of from about 0.03 near apoapsis, to 0.4 near periapsis. the orbit remains unchanged, the longitude of the ascend ing node changes by Aft, and the argument of periapsis The primary difference between the results for changes by Ao> where, to first order in the planet impulsive thrusting and for finite thrusting is in the oblateness, 1 position on the parking orbit at which the velocity is o added. For finite thrust, the start of thrusting occurs prior to (i.e., leads) the true anomaly for impulsive Afl = - 3?r J2 | (cos i) N velocity addition. That is, to obtain the same excess speed and direction as in the impulsive case, the finite = 37rJ2 (~) (2-|sin2 i) N thrusting must be approximately centered about the position of the impulsive velocity addition. Typical lead angles are shown in Figure 4. Here, the velocity incre and ment to escape to an excess speed of 6.0 km/sec from Mars is shown as a function of true anomaly around an orbit with an eccentricity of 0.9 and a periapsis altitude of 1000 km. The true anomaly shown for the finite thrust is that at the start of thrusting. It can be seen that for P = a(l - e2 ) equal velocity increments (and hence equal turning angles) the start of finite thrusting must lead the posi At that time, the orbit plane makes an angle 1^ with the tion for impulsive velocity addition by approximately departure asymptote. 2 5° to 15° in true anomaly for all true anomalies except near apoapsis. Near apoapsis, the associated slow 8111100= cos i [sin 6^ - tan 6r cos 6^ cos (pr - pd - fit)] rotation and finite thrust result in a near zero lead angle requirement. ± [sin2 i- sin2 6r l sec 6r cos 6^ sin (pr - pd - Qt) In summary, the assumption of impulsive velocity It is this angle which must be compensated for during increments is attractive due to the simplification it the departure maneuver. permits and appears justified due to the small resultant errors. The orbit geometry at Earth was analyzed in a similar manner except that the initial orbit (corre Circular Parking Orbit sponding to the orbit at arrival for the planetary case) was chosen in such a manner as to be, through regres A comparison of three methods of transfer from a sion, coplanar with the departure asymptote at the nomi circular parking orbit to a hyperbolic asymptote is nal departure time; that is, zero plane change was provided. These methods are optimal one- and two- chosen for nominal departure. As with the planetary impulse direct transfers and a restricted three-impulse orbits, two orbital planes exist which satisfy this con transfer using an intermediate elliptical orbit. The straint, These planes are distinguished by the relative techniques are briefly delineated in this section, position of the spacecraft departure and the departure excess velocity vector at the nominal departure date. One - Impul s e T r an s f e r s - The single-impulse If the departure is above the excess vector, then the transfer problem is one of solving for the minimum. spacecraft departure is toward the north pole and the velocity increment (AV) to achieve the desired

6.1-2 hyperbolic asymptote from a specific orbit. The exact respectively. These data are for an initial circular solution for minimum AV results in an expression which orbit altitude of 1000 km and an intermediate orbit does nor permit a closed form solution,2,3 However, in with an eccentricity of 0.9, Again, various values of Reference';) an approximate solution is developed which the out-of-plane angle, l^, are shown. The minimum provides very good agreement with the exact solution in departure velocity requirements with hyperbolic for interplanetary velocity ranges, excess speed which occurs for the Earth case at rather low values of excess speed as exhibited by the Two-Impulse Transfers - The two-impulse trans loo = 30° lme is due to an increasing off-periapsis de fer technique considered in this study utilizes one im parture penalty as excess speed is decreased. That is,, pulse at departure from the circular parking orbit and for a given 1^ (non-zero), there is a minimum excess the second impulse at an "infinite" distance from the speed below which a periapsis departure is not possible, planet, i.e., at the sphere-of-influence. The solution used was developed in Reference 3 and employs an opti Elliptical Parking Orbits mal distribution of velocity change and angle change between the two impulses. Reference 2 considered a For the elliptical parking orbit, the sum of the transfer with an optimum distribution of velocity be arrival and the departure AV requirements must be tween the two impulses and with the second impulse sup minimized. This approach is necessary since the posi plying the entire plane change. The parametric results tion of periapsis significantly affects both of the AV are nearly the same as the optimal transfer employed requirements and. the cop!anar arrival maneuver will here. establish that position. In general, both the arrival and the departure 'will be performed off-periapsis to achieve Figure 5 presents the parametric data for one- and the minimum total AV. For all transfer techniques, the two-impulse transfers. The figure indicates the regions orbit insertion point was varied from -f90° to -90° true of velocity and plane change requirements where each is anomaly on the parking orbit, superior in terms of minimum total AV requirement. As with the circular orbits, three transfer tech Three-Impulse Technique - The three-impulse mode niques were considered. 5 They are an optimum, single considered in this study utilizes an in-plane tangential impulse and constrained two impulse and three impulse impulse to insert the spacecraft from a circular parking transfers. The techniques are briefly described, below, orbit into an intermediate highly-elliptical orbit followed by a plane change at apoapsis to rotate the orbit plane so One-Impulse Technique - In this technique, since as to be coplanar with the departure asymptote. The both the departure hyperbolic excess vector and the third impulse is then provided tangentially at the appro characteristics of the parking orbit at departure are priate position on the intermediate ellipse so as to place fixed, the non-coplanar. non-tangential departure the spacecraft on the required escape hyperbola. The maneuver true .anomaly is iterated, to establish the position of the first impulse on the circular parking orbit departure point of minimum AV requirement for each is varied parametric ally and from these results the arrival true anomaly. In general the departure is optimum position is determined by a numerical search. off-periapsis and has a non-minimum plane change In general, the optimum position is such that the plane angle. change maneuver is minimized without an inordinate penalty for an off-periapsis departure from the inter Two-Impulse Technique - The first impulse is mediate ellipse. A complete analysis of this three- applied at apoapsis to perform, the required plane impulse technique has been performed.4 change. The second, impulse is applied tangentially near periapsis and provides the required velocity A brief study was made of off-apoapsis plane-change change to achieve the desired excess velocity. Thus, maneuvers on the intermediate orbit with the motive of this technique is similar to the last two impulses of the rotating the line of apsides in order to reduce off- three-impulse transfer from circular orbit. periapsis departure penalties. It was found that for highly eccentric orbits very little was to be gained by Three-Impulse Technique - The' three-impulse such maneuvers and that the optimum plane-change transfer for elliptical, orbits is. similar to that for cir position was very near apoapsis. This result is pri cular orbits. The first impulse is applied at periapsis marily a reflection of the low off-periapsis departure to increase the orbit eccentricity: again, the 'value of penalties for highly elliptic orbits. The off-apoapsis 0,9 was used for the transfer ellipse, The second im plane-change option was not retained further in this pulse performs the necessary plane change maneuver study. at apoapsis and the third impulse applied near periapsis provides the additional velocity increment necessary to The effect of the eccentricity of the intermediate achieve the desired hyperbolic trajectory. ellipse upon the total departure velocity requirements is shown in Figures 6(a) and 6(b) for Earth and Mars, respectively. These data are for a typical hyperbolic MISSION SELECTION CRITERIA excess speed of 6 km/sec and for an orbit altitude of 1000 km. Curves for various values of the angle be Data to aid in selecting reas<,.,;, «b!.e mission pro tween the orbit plane and the departure asymptote, I^, files is contained in Ref, 6, \vhicr. presents mission are shown. It can be seen, especially for large out-of- characteristics for both direct and, Venus swingby plane angles, that high eccentricities are very attractive. stopover missions to Mars for each Earth-liars oppo sition period from 1980 to the year 2000. Parametric data for the three-impulse departure requirements as a function of hyperbolic excess speed The results of that study indicate that with, the are shown in Figures 7(a) and 7(b) for Earth and Mars, exception of the 1984 and 19 97' oppositions the swingby

6.1-3 mode is more attractive than the direct mode. Propul The technique of maintaining constant velocity is sive velocity requirements are consistently lower, in also applied successfully to the definition of the inbound some cases affording reductions of thirty percent. legs in the face of launch delays at Mars as shown in Earth entry speeds are also consistently lower, afford Figures 9(a) and 9(b). In this case, only the Mars de ing reductions of up to fifty percent. These benefits parture velocity is considered. Constraints could also are realized for mission durations of about 500 days, be imposed on Earth entry speeds, oi course, but as can which represents an increase of about 20 percent over be seen they remain quite low. To maintain either a the durations for direct missions (based on thirty-day constant Mars departure velocity or minimum Mars stopovers). These mission durations were obtained by departure velocity for staytim.es in excess of about minimizing the product of propulsive velocity require 60 days, it is necessary for the inbound leg heliocentric ments and mission duration. These ''nominal" missions transfer angle to exceed 180°. Thus, the inbound leg- from Ref. 6 have mission durations about 25 percent duration must increase accordingly. shorter than minimum energy missions at the expense of less than ten percent increase in total propulsive velocity requirements. As will be demonstrated later, NUMERICAL ANALYSIS such a mission selection procedure contributes to low sensitivity of the mission parameters to launch delays. In order to provide an assessment of the perfor For the above reasons, and to encompass typical in mance of one-, two-, and three-impulse transfers, the bound and outbound swingbys, the launch window analy techniques described for transfer analysis were applied ses were conducted for the "nominal" missions from to the two selected missions. This section presents, Ref. 6 for both the 1982 inbound swingby and the 1986 (1) the results of analysis of the Earth launch window outbound swingby. which employs only circular parking orbits, and (2) the results of the analysis of Mars 1 launch windows em Since the intent of this paper is to assess the in ploying both circular and elliptical parking orbits. fluence of launch delays on mission characteristics, one approach to the selection of the round trip trajec The format selected for presentation of the launch tories could have been the specification of a nominal window data is a contour map, showing lines of constant stopover time. This would have necessitated a further propulsive velocity requirement on an orbit inclination specification on the parking orbit characteristics at versus departure date plot. Thus, for a given orbit, the Mars to ensure that some other criterion be met, e.g., launch window for a given propulsive capability is that the nominal departure from Mars be coplanar, or readily visible. These data are obtained by cross that the plane-change velocity penalty be minimized plotting the complete set of transfer data which is com throughout some arbitrary staytime, etc. The approach puted for fixed inclinations as a function of departure taken here is to vary both stopover time and orbit in date. The data necessary to prepare the contour maps clination parametrically and to divorce the selection of are included in References 4 and 5. the outbound leg from the selection of the inbound leg, Earth Launch Window Representative Earth departure and Mars arrival conditions must nevertheless be specified. For both The launch site considered for these missions was mission years considered, the outbound leg chosen is the Kennedy Space Center (KSC). Therefore, the orbit that of the nominal mission employing a 30-day stop inclinations which can be obtained without a plane over. The Mars departure date was then varied to change are required to be between the site's latitude reflect launch delays. While this may seem restrictive, (« 28°) and the range safety constraint (about 50°). The it is in fact quite reasonable. For outbound swingbys, Earth launch window analysis was, therefore, limited to regardless of mission duration or stopover time, varia posigrade orbits of 30°, 40°, and 50° inclination. tions of no more than about 10 days arise in either the optimum Earth departure or Mars arrival dates. For 1982 Inbound Swingby - The results for the 1982 inbound swingbys, the variation in Mars arrival date launch are shown in Figures 10(a) and 10(b) for a cir can be on the order of 50 days. However, while the use cular orbit of 300 km altitude and using an intermediate of the optimum arrival date would reduce the Mars eccentricity of 0.9 for the three-impulse transfers. coplanar departure velocity requirements, it does not significantly influence the plane-change requirements Since the variation of the Earth departure hyper for the various stopover times, bolic excess velocity for this opportunity is from 4 to 5 km/sec, Figure 6 indicates that the two-impulse trans This paper is not directly concerned with system fer will have lower AV requirements than the one- characteristics, but recognizing that Mars arrival impulse for plane-change angles (1^) over 15°, Figure 10 velocities should not be unconstrained, the outbound shows that the two impulse does indeed lower the AV legs are selected so that the sums of the Earth depar requirement at a given departure date. However, for ture and Mars arrival velocity increments are equal to reasonable velocity penalties associated with the plane the nominal values insofar as possible. When this con change (i.e., 10-20 percent of the minimum coplanar stant value can no longer be maintained for the longer departure AV) the increase in launch window size staytimes, the trajectories are then selected on the (« 1 day) is not of sufficient magnitude to make the basis of minimizing the velocity sum. Note that this two-impulse transfer of practical interest. procedure is possible by virtue of the manner by which the nominal missions were defined. If the nominal The use of three-impulse transfers is seen to sig missions had been selected on the basis of minimum nificantly increase the available launch window. For a energy, then all delays could conceivably cause the AV of 4,5 km/sec, the three-impulse window is over velocity requirements to increase. Figures 3(a) and 24 days for the southern injection (Fig, 10(a)). For the 8(b) illustrate the variation of outbound leg parameters same AV, only a 2-day window is available using one for the 1982 and 1986 missions, respectively. impulse transfers. To achieve a24-day window with the one-impulse transfer would require a AV of 6 to 7 km/sec.

6,1-4 Thus, the increase in launch window flexibility using a 1982 Inbound Swingby - Figures 12(a) through (d) three-impulse transfer is such as to make it worth con present the contour maps of the AV requirements for sideration in mission analyses even with the complexity the 1982 mission. The figures show the data for both of making three separate maneuvers. southern insertions and northern insertions as previ ously defined. The southern injection orbit rotates relative to the departure vectors such that two coplanar departures Figure 12(a) shows that for a one-impulse escape occur during the departure period studied. Departures from a southern insertion orbit,, three regions exist between these points require plane changes and thus having a nearly coplanar departure AV requirement AV penalties. The three-impulse transfer significantly (5 km/sec). These are near-polar orbits for staytimes reduces this plane change AV penalty thus opening the up to 50 days and both low inclination posigrade and region to low AV departures. Comparison of Fig high inclination retrograde orbits at 50 days staytime, ure 10(a) for the southern injection and Figure 10(b) plus or minus 10 days. In order to provide continuous for the northern injection reveals the effect that the departure capability up to about SO days for all avail orbit direction can have upon the launch window. able inclinations, a AV of 9 km/sec is required. The low AV region near 90 degrees inclination occurs since For the northern injection, Figure 10(b), the orbit the arrival and departure right ascensions are nearly rotation relative to the departure vector variation is equal for about 60 days staytime. such as to result in only one coplanar departure in a departure period studied. A second coplanar departure Three-impulse transfers from, a southern insertion occurs at an earlier time. From Figure 8, this earlier orbit. Figure 12(b), make available low inclinations for departure has a higher Earth departure AV requirement. short staytimes (5 days) and a low AV (5 km/sec) capa The plane change penalty for this orbit orientation is bility. When, the AV capability is increased to 6 km/sec also higher than for the southern injection. Therefore, all inclinations obtainable can be achieved for staytimes the use of a three-impulse transfer for the northern up to ?8 f days. This is a significant improvement over injection orbit results in AV requirements about 10% the: one-impulse transfer., A similar increase in avail higher than for the southern injection orbit in 1982 for able staytimes 'does not occur by increasing the AV the same window length. another ,1 km/sec since the copUmar velocity require ment rises rapidly after 60 days staytiine ('Figure 9(a)). 1986 Outbound Swingby - The Earth departure data for the 1986 opportunity is shown in Figure 11 (a) for For tois 1:98121 mission, use of, a northern insertion the southern injection orbit, and ll(b) for the northern at arrival increases the launch, window for a low to injection. Conclusions drawn above as to the relative moderate .AV capability. This is shown in Figures 12(c) value of one, two, and three-impulse transfers for the and (d). The different insertion direction effectively 1982 opportunity apply here as well. The effect of orbit provides ;an additional 20 days of low AV* requirement orientation is different, however. For this opportunity, as can he seen by comparing Figures 12(a) sand (c). it is the northern injection orbit, Figure ll(b), whose This, orbital configuration allows a AV of 6 km/sec rotation, relative to the departure vectors, allows two with, a one-impulse transfer to provide staytimes for coplanar departures over the time period studied. all inclinations of up to 40 days, and for some.'Inclina Again, use of the three-impulse transfer opens the tions of up to ?0! days. The three-impulse transfer entire departure region between these coplanar points provides basically the same launch window 1 capability to low AV requirements. for either orbit insertion since the plane-change penalty is inherently small,, Mars Circular Orbit Launch Window 1986 Outbound Swingby - The. data for the 1986 Mars The minimum inclination which will allow coplanar launch windows are presented in. Figures 13(a) through arrival and departure is defined by the magnitude of the (d). Between, 60 and 70 days, staytime a discontinuity larger of the declinations at arrival and departure. This occurs 'in the' return leg characteristics. This, residts inclination is 28.7 and 20.8 degrees for the 1982 and 1986 from the trajectory selection since the central angle of missions, respectively. The maximum inclination is 'the transfer conic crosses the 180° ridge. Solutions 180° minus the minimum inclination, i.e., a retrograde can,.,, of course, be found in 'this, region by utilizing a orbit. The inclinations were, therefore, varied from finer grid of stay times, than, was done here, or by con 30 to 150 degrees for both missions in 10 degree inter straining the central, angle. vals, assuming a circular parking orbit of 1000 km altitude. Figure 13(a) shows the data for a southern inser tion with a single-impulse departure. The data does Only data for one- and three-impulse transfers are not display the symmetry shown by the 1982 mission shown in the following sections, since the reduction in since the arrival and departure: rigjht ascensions are! AV achieved through use of the two-impulse transfer, at least. 45° apart for the first 60 days of staytime, when a reduction occurred, was small compared to the After 60 days, the symmetry is stronger. For near total AV requirement. The three-impulse transfer coplanar departure AV capability (3.0 km/sec) only technique again employs an intermediate orbit with an »small strips of the inclination-staytime map are avail eccentricity of 0.9 to perform the plane-change able. A AV of 4 km/sec opens up the low inclination maneuver. region for staytimes up to about 80 days. However, to 1 achieve polar 1 orbits for 'this mission, AV's of about Use of the circular parking orbits allows the arrival 5 km/sec would be required 'if' only a one impulse maneuver to be performed tangentially at periapsis of transfer were used. the arrival hyperbola. The insertion AV is, therefore, the same for all staytimes, inclinations and insertion Comparison of these southern insertion data to directions. This AV is 2.7 and 4.3 km/sec for the 1982 Figure 13(b) for a three-impulse transfer 'reveals, the and 1986 Mars arrivals, respectively. effect of reducing the plane-change penalty. 'With,, a

6,1-5 three-impulse transfer, a AV of about 3.1 km/sec will which the arrival and departure conditions are suffi allow use of any obtainable inclination for the staytime ciently near the absolute optimum that essentially no range shown. penalty occurs. A six percent increase (0.4 km/sec) in the total AV results in a staytime of at least 20 days The effect of using a northern insertion at arrival for all inclinations. Using the total AV representative for the 1986 mission is shown in Figures 13(c) and (d). of minimum launch windows for circular orbits (7.7km/ For this mission, the additional low AV region at short sec), a staytime of at least 40 days exists. This is to staytimes, mentioned for 1982, is only about 10 days be compared to the 5-day staytime for the one-impulse and does not significantly change the areas available transfer from circular orbits. for a given AV level. A AV of 5 km/sec is required for either insertion to provide a reasonably large Data for a three-impulse transfer from a 0.3 eccen inclination-staytime spectrum for single-impulse tricity orbit are shown in Figure 14 (b). As in the cir transfers. cular orbit case, an increase in launch window length over the one-impulse transfer occurs. However, the The effect of the different insertion upon the three- increase is relatively less than for the circular orbits. impulse data (Figure 12(d)) is negligible with a AV of This is due to the close positions of the arrival vector about 3.1 km/sec required to open the entire region of and the departure vector and their relative magnitudes. inclinations and staytimes studied. For this year, the arrival and departure pericenters are located so that both maneuvers can be performed Mars Elliptical Orbit Launch Window with one impulse near pericenter since the plane-change requirements are small. A AV of 7.7 would produce a The analysis was performed for parking orbit launch window of about 48 days for any inclination except eccentricities of 0.3, 0.5 and 0.7 for the 1982 and 1986 near equatorial. miss ions. 5 Data for all three transfer techniques was developed, but only one- and three-impulse transfers Figure 15 presents the contour plot for a one- for eccentricities of 0.3 and 0.7 are shown. In the case impulse transfer from a 0.7 eccentricity parking orbit. of the three-impulse transfer, the intermediate orbit For this orbit, the minimum possible AV requirement for the plane-change maneuver has a 0.9 eccentricity. is 5.45 km/sec which is nearly achieved in two regions As with the circular parking orbits, the declinations of of Figure 15. An increase of five percent in AV to the arrival and departure vectors define inclination 5.7 km/sec allows a window of 50 days for near polar limits between which the possibility of coplanar arrival and high inclination posigrade orbits. The particular and departure exist. form of this AV contour in the retrograde region is of interest in that it indicates a departure window of up to In the analysis of elliptical parking orbits two 50 days, but not for staytimes below 20 to 40 days at parameters, in addition to those for the circular parking various inclinations. If this early depart restriction is orbit, are of importance. These are the orbit eccen of concern, then the retrograde AV requirements are tricity and the location of the line of apsides relative to higher than those for posigrade orbits. Use of a AV of the arrival and departure asymptote. For circular or 7.7 km/sec results in a launch window of 75 days. bits, the arrival AV is a function, of the interplanetary trajectory leg and orbit altitude only. However, for The use of a three-impulse transfer from this orbit elliptical orbits, an off-perieenter insertion maneuver does not significantly change the launch window contour at arrival can reduce the off-pericenter requirement at and is, therefore, not shown. departure and thus lower the sum of the AV's at arrival and departure. The value of the optimal off-pericenter 1986 Outbound Swingby - Data for a one-impulse angle at arrival is a function of the inclination of the transfer from a 0.3 eccentricity parking orbit is shown parking orbit and the desired staytime at the planet. in Figure 16(a). A minimum AV requirement of 6.0 km/ In order to assess the launch window requirements, a sec exists for the idealized coplanar, pericenter maneu nominal staytime is selected and the optimum insertion vers; a minimum of 6.1 was actually achieved. A AV of maneuver for that staytime is defined for each inclina 8.3 km/sec must be allowed before a generally available tion. That insertion maneuver is then held fixed for all launch window exists for all staytimes. Even then a staytimes to determine the available departure launch significant benefit over the one-impulse transfer from window. Re selection of the insertion maneuver would circular orbit does not exist. This occurs since the be expected to produce some gains, but based upon the arrival and the departure asymptotes are over 45° dif few cases which were investigated, changes which ferent in right ascension. As a result, the one-impulse resulted in reduced AV's at long staytimes also re case requires a serious compromising of the departure sulted in increased AV T s for short staytimes. In order maneuver. to decrease the volume of data but still present the salient features of elliptical parking orbits, only data Figure 16(b) presents the contour maps for the for the better insertion direction of each year as three-impulse transfer. In this case, as compared to established by the circular orbit data is presented 1982, the plane change and the departure maneuvers (i.e., northern insertion for 1982 and southern inser having been optimized individually allows a significant tion for 1986). increase in the available launch window. The use of a total AV representative of minimum launch windows for 1982 Inbound Swingby - Figure 14(a) presents the circular orbits (7.3 km/sec) results in a launch window total AV contours (arrival plus departure) for a one- of over 60 days for almost all inclinations. impulse transfer for an elliptical orbit of 0.3 eccen tricity as a function of inclination and staytime. If a The contour map for a one-impulse transfer from a coplanar periapsis arrival and departure were possible, 0.7 eccentricity orbit is shown in Figure 17(a). For this a minimum AV of 6.6 km/sec could be achieved. As can orbit, the minimum AV requirement would be 4.85 km/ be seen in Figure 14(a), two small regions exist for sec. A value this low was not achieved anywhere on the

6.1-6 one-impulse transfer plot. A AV of 7.3 km/sec opens 2. Deerwester, J. M., McLaughlin, J. F., and the region below 50° and above 120° for staytimes of Wolfe, J. F., "Earth-Departure Plane Change and 100 days. A AV of 8.0 km/sec allows use of the entire Launch Window Considerations for Interplanetary contour map. Missions," AIAA Journal of Spacecraft and Rockets, February, 1966. The results of using a three-impulse transfer from the 0.7 eccentricity orbit are shown in Figure 17(b). 3. Gunther, P., "Asymptotically Optimum Two-Impulse Here a AV of 6.3 km/sec opens essentially the entire Transfer from Lunar Orbit," AIAA Journal, map to use, while a AV of only 5.3 km/sec would allow February, 1966. any retrograde orbit. 4. Manning, L. A., Swenson, B. L», and Deerwester, J. M,, NASA TM, Analyses of Launch Windows from Cir CONCLUDING REMARKS cular Orbits for Representative Mars Missions, to be published. This paper provides a summary of the variation in launch windows that can be achieved through use of 5. Manning, L. A., and Swenson, B. L., NASA TM, circular and elliptical orbits and different transfer Analysis of Launch Windows from Elliptical Orbits techniques. The data is presented for two opportunities for Representative Mars Missions, to be published. which are representative of future round trip missions. The data can be used to trade-off AV requirements 6. Deerwester, J.M., and Norman, S. M., "Systematic with the complexity of multiple-impulse transfers for Comparison of Venus Swingby Mode with Standard all inclinations and staytimes of interest. Mode of Mars Round Trips," Journal of Spacecraft and Rockets, August, 1967. To provide departure capability on any day for nominal staytimes up to 60 days at Mars and 20 days at Earth can result in AV penalties up to 50 percent of APPENDIX A the nominal coplanar AV (arrival plus departure) if one-impulse transfers are employed from circular NOTATION orbits. However, three-impulse transfers from circular orbits permits the same launch windows a semi-major axis, km (i.e., 60 days at Mars and 20 days at Earth) for AV penalties on the order of 5-10 percent of the total AV AV propulsive velocity increment, km/see requirement. e eccentricity Elliptical parking orbits do not lend themselves to H altitude, km such general conclusions due to the cross coupling of IQO minimum angle between departure vector and the arrival and departure maneuvers. Increasing orbit plane, degs eccentricity of elliptical orbits does, of course, reduce the total AV requirement for a given launch window. i inclination, degs In addition, the use of three impulses to perform the J"2 second harmonic of planetary oblateness transfer generally requires lower AV's than using a single impulse. However, orbit orientation can exist R planet radius, km (e.g., 1982, e = 0.7) for which no gain is shown by the t time, sec three-impulse maneuver. VQQ hyperbolic excess velocity, km/sec As pointed out under the mission selection criteria, x, y, z components of the planet centered coordinate supportable, but somewhat arbitrary, ground rules were system established to define the interplanetary trajectories. Nominal orbit parameters were established based upon d declination, degs these criteria, thus determining the motion of the orbit T unperturbed orbital period, sec as a function of staytime. With an orbit established, res election of the interplanetary trajectory can be p right ascension, degs employed to vary the departure characteristics and fi longitude of ascending node, deg possibly reduce the AV requirements for a particular window. Preliminary data in Reference 4 indicates O " orbital regression rate, degs /day' that reselection of the direct legs does reduce AV a) argument of periapsis, deg requirements for single-impulse transfers. Little effect was apparent for three-impulse transfers due to their already low AV requirements. Reselection of the Subscripts swingby leg was also ineffective since the swingby re quirement severely restricted planet departure c circular characteristics. d departure data i intermediate ellipse for three-impulse REFERENCES transfer p periapsis 1. Roy, Archie E., The Foundations of Astrodynamics, The Macmillan Company, New York, 1965, page 224. r reference data

6.1-7 Equator

Departure V,

Orbit plane at departure Orbit plane at arrival

Figure 1,- Orbit geometry at planet arrival.

Equator Orbit pe naps is

Planet orbit plane

Spacecraft orbit plant

Figure 2,- Orbit geometry at planet departure.

41*8

q q

H-

Hi Hi

Q Q

IA)

ID ID

km/sec increment, increment, Velocity Velocity 6-1*9 Velocity increment, km/sec J_PO ui 4> m CD r- O I I I I

htj

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p. P p, cf (D

d- p.

O P H. OQ p p. d- p. O P

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6.1-10 AV

2- Impulse I Impulse ,— —o—— Cross -over line

Figure 5-- Optimum one-impulse and two-impulse transfers.

6.1-11

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION AMIS RESEARCH CENTER MOFFITT FillD CAU ! FO ! RNlft 81-

7\- deg

,deg

o o

= 6 km/sec Vex) = 6 km/sec = 1000 km H = 1000 km

I I I 0 A ,6 .8 1.0 .4 .6 .8 1.0 Intermediate orbit eccentricity Intermediate orbit eccentricity

Figure6.- Effectof eccentricityon three-impulsetransfers, (a) Earth orbit. (b) Mars orbit. 12

e,=.9 10 H = 1000 km

o 2 o>

o Q. •oO)

_L _L 0 468 10 12 Hyperbolic excess speed, km/.sec I2r-

H = 1000 km 10

^ 0> > 6 0)

o QL •o0)

_L _L I 0 468 10 12 Hyperbolic excess speed, km/sec Figure 7.- Effect of excess speed on three-Impulse transfers, (a) Earth orbit, (to) Mars orbit. 6.1-13

1 1

10 10

i i

0 0

6160 6160 6170

days

(JD-2440000) (JD-2440000)

i i

1 1

date date

-10 -10

delay, delay,

Total

Launch Launch

departure departure

i i

6140 6140 6150

-20 -20

Earth Earth

L L

I I

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characteristics, characteristics,

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1986 1986

days

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(*b) (*b)

Total

ARC/

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i i

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4990 4990

delay, delay,

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i i

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Earth Earth entry entry speed, speed, km/sec

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More More departure departure AV t t km km /sec

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I I I

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30 -

50 r-

40 CD 13 Two impulse v 30 4.5 44 4.5 5 ______I______i

501-

40 Three impulse v 30 4 4 4.5 t J_ I J 2444970 4980 4990 5000 5010 Earth departure date Figure 10.- Earth departure AV contours, (a) 1982 inbound swingby; southern injection. (b) 1982 inbound swingby; northern injection.

50|-

40 One impulse 30 \ 8 AV, km/sec 8

50 r

50 p / , 1 \ 40 - \ , \. Three ' / impulse 30 - \ / 1 4,5 4,5 4, 5 ' 5

till 1 T 2444970 4980 4990 5000 5010 Earth departure date

6,1-16 50r-

40 AV, km/sec One impulse \ 30 8

______I

50 r-

40 Two 7 impulse t 5 4.5

50 r-

40 Three impulse 30 4.5 4.,5 .

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Figure 11.- Earth departure AV contours. (a) 1986 outbound swingby; southern injection. (b) 1986 outbound swingby; northern, injection. 50r-

40 One impulse 30t

50 r

40 Two - 30 impulse t 5 4.5 4,5 5

50 r

40 Three impulse 30

t I I J 2446130 6140 6150 6160 SI 70' Earth departure dote

6.1-17

deg

120

150 150

30 30

60 60

circular circular

(a) (a)

20 20

1982 1982

inbound inbound

parking parking

40 40

orbit; orbit;

swingby; swingby;

Staytime, Staytime,

one one

southern southern

impulse.

days

insertion; insertion;

Declination Declination

Figure Figure

limitation

100

12 12

f f

- -

120

deg

120

150 150

90 60 30

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contours,

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6 6

80 80

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7 7

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(d)1982

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150

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150 150

120

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Declinationation limitationlimitatioi _L -L _L 40 60 80 100 0 20 40 60 80 100 120 Staytime, days Staytime, days

Figure 13.- Mars departure AVcontours.

(a) 1986 outbound swingby; southern insertion; (b) 1986 outbound swingby; southern insertion; •circular parking orbit; one impulse. circular parking orbit; three impulse. 120

3.3

limitation

100

Declination

insertion;

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norhtern

,days

three

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Staytime

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outbound

1986

(d)

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120

150

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120

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100

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days

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1986

(c)

150

120 T3

cu CD

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150 I80

20 60

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Staytime,

(a)

1982

xiay

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limitation

100

1%,~

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150

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6.7 7.7 8.7 9.7 .1~o 90 ~o c

30 Declination limitation

I I 0 20 40 60 80 100 Staytime, day

Figure 15.- Mars AV" contours; 1982 inbound insertion; northern insertio&£ parking orbit eccentricity = 0,7; one impulse,

6,1-23 I80r

insertion;

parking

orbit

Staytime,

eccentricity

(a)

1986

day

outbound

0,3;

Declination

one

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Figure16*-

limitation limitation

180

contours.

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Staytime,

day

Deelination

Declination

limitation

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100

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100

0.7; 0.7;

= =

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1986

limitation

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(b.) (b.)

6.3

eccentricity eccentricity

Declination Declination

5.3

orbit orbit

6.3

5.3

I I

= =

60 60

day

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4.9

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.1 .1

Mars Mars

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17.- 17.-

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southern southern

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swingby; swingby;

Declination Declination

on* on*

7.3

,6.3

60 60

0.7} 0.7}

day

outbound outbound

= =

1986 1986

8.0

(a) (a)

Stayfimg, Stayfimg,

40 40

eccentricity eccentricity

orbit orbit

parking parking

180

150

120

CP CP

"O .1 .1