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Home , 1620 Geographos, Apsis, Heliocentric orbit, Highly elliptical orbit

I I A NEW PROPAGATION I PROGRAM FOR DSPSE MISSION SUPPORT

I Jay W. Middour* I Naval Center for Space Technology Naval Research Laboratory I Washington, D.C.

I Abstract ronment. DSPSE has two secondary science missions in which it will fly in lunar for A trajectory propagation program two and then depart the for a I called N-Body is being developed to support four cruise to rendezvous with the near the upcoming Deep Space Program Science Ex­ . In accom­ periment (DSPSE) mission which is also known plishing its secondary missions, DSPSE will I as . The program will support have made the first complete, multi-spectral the activities of the mission operations center mapping of the lunar surface, and executed the which is run by The Naval Research Labora­ first close encounter with a minor (ap­ I tory. The computer program is designed to proximately 100 km). predict a orbit through the various Mission operations for DSPSE will be con­ I phases in which the DSPSE vehicle will fly. ducted by the Naval Research Laboratory's These orbital phases include low Earth, trans­ Naval Center for Space Technology at the lunar, lunar, and deep space. The design of DSPSE Mission Operations Center in Alexan­ I N.Body, its functionality, and the mathemati­ dria, Virginia. During the various mission cal methods it employs are presented. phases, the DSPSE space vehicle will fly in , with apogee I at lunar distances, , and deep space 1 Introduction . The DSPSE Mission Op­ erations Center, or DMOC is tasked to coor­ I The Deep Space Science Experiment Pro. dinate the science activities with the business gram (DSPSE), also known as Clementine, of navigation, orbi t determination, command is a joint Naval Research Laboratory, NASA, and telemetry, and satellite housekeeping func­ I and Lawrence Livermore National Laboratory tions. The operations center must therefore project. DSPSE is funded by the Ballistic Mis­ possess a trajectory integration program which sile Defense Office. The goal of the can accurately integrate the satellite orbit in I DSPSE mission is to demonstrate the opera­ each of these unique space environments. tion of lightweight sensors in the space envi- It was decided that a program developed by I * Aerospace Engineer, Member AIAA members of the operations team itself could This paper is declared a work of the U.S. Government I and is not subject to copyright protection in the United States. 1 I I be tailored expressly to meet the DSPSE re­ space, the force model must contain the ef­ quirements in areas such as engineering and fects of the Earth and Moon gravitational po­ I science support, contingency planning, maneu­ tential fields, Earth atmospheric drag, solar ver design and verification and tracking sta­ radiation pressure and planetary gravitation. tion scheduling. Furthermore, by using a prod­ Several gravitational potential models for the I uct developed in-house, software modifications Earth and Moon are incorporated into the pro­ can be effected more rapidly and less expen­ gram. Alternatively, a user defined potential sively than if an off-the-shelf product were may be specified. Atmospheric drag requires I used. The trajectory propagation program, a model of the density variations of the Earth called N-Body, is being developed by the au­ atmosphere and a physical model of the satel­ thor to satisfy this requirement. lite. A model of the vehicle shape and its I Strictly speaking, the program does not ad­ specular characteristics are needed to compute dress the classical n-body problem in which the the acceleration due to solar radiation pres­ I motion of n objects under mutual gravitation sure impinging on the spacecraft. A plane­ is solved for. Rather, the program attempts to tary is used to obtain the third body solve a restricted problem in which the trajec­ gravitational forces. The planetary ephemeris I tories of n-l objects are known and the motion is also used to predict eclipsing events which af­ of the massless nth object is solved for. In this fect the solar radiation pressure computation. case the n - 1 objects are the , Moon and Finally, the force model allows for impulsive I and the nth object is the space vehicle. spacecraft maneuvers. N-Body is written in FORTRAN and is be­ N-Body employs a Cowell method in which I the total accelerations of all external forces act­ ing developed on VAX workstations running ing on the vehicle are directly integrated. Cow­ under the VMS operating system. The com­ ell's method was selected due to its ease of im­ puter code is largely complete and consists of about 10,000 lines. The program is presently I plementation. For instance, Encke's method undergoing verification testing by the DSPSE in which only the perturbations about the two body problem are integrated can be more effi­ mission operations team. It is anticipated that I cient since the integration step size can usually the program will be fully operational in time to be increased over that of the Cowell method support DSPSE flight operations starting with without any associated loss of accuracy. How­ the planned launch in January, 1994. I ever the Encke method requires that the per­ turbed orbit be rectified with a new reference 2 Design and Operation orbit whenever the difference between the per­ I turbed orbit and the reference orbit grows too N-Body is comprised of a program configura­ large. The rectification process logic is depen­ tion procedure, an executive procedure, and I dent on the particular an extensive mathematical subroutine library. procedure being used. For instance a multi­ A series of key words and variables is used to step integration method would require restart­ control the execution flow! define the input and I ing. By using Cowell's method, the executive output coordinate frames and times, configure algorithms can be independent of the integra­ the force model, select and configure the in­ tion procedures. The result is a major simpli­ tegrator, set the duration of the propagation I fication in software design and maintenance. and select the destination and frequency of the Since the must be propagated in output data. The executive procedure trans­ low Earth orbit, trans-lunar, lunar and deep forms the input coordinates and time to I 2 I I I

a standard internal fonnat the coordinates of nitions, force model parameters, integrator se­ I which are mean equator and of J2000.0 lection and print options. Each output record and TAl (International Atomic Time) seconds. contains the current time in Julian and calen­ Using the mathematical subroutines! the exec­ dar dates, Cartesian position and velocity, Ke­ I utive procedure carries out the orbit propaga­ plerian and Earth referenced tion according to the key word configuration. position of the space vehicle trajectory being I While the propagation is in progress the execu­ integrated. tive procedure monitors its status to detennine An ASCII ephemeris file may be created con­ when output records should be produced. Out­ sisting of a header followed by time, position I put records can be requested at regular time in­ and velocity data records. The header and tervals! after each integration step! at the first data in the ephemeris file conform to the Stan­ and last times in the run or they can be trig­ dard Ephemeris Record Fonnat (SERF) which I gered by orbital events such as apoapsis and will be used by the DSPSE mission operations peri passages. The executive procedure and science teams. The SERF header contains also can be configured to detect the passage of a description of the parameters, constants and I the space vehicle into the umbra and penumbra models used to create the ephemeris. The in­ of a planet, and the of the vehicle put vector coordinate system, origin of coordi­ by a planet as viewed from the Earth. nates, coordinate type, and position, velocity, I The program configuration procedure reads and time units are listed followed by the actual an input file containing key words and vari­ input vector. Generation information is also I ables, as well as the initial condition for the provided which includes the time the run was trajectory that is to be propagated. Each key made, the propagation start and stop times, word and variable is assigned a default value so the interval between ephemeris records, and I that the only item that is required to be in the the number of points in the ephemeris. The input file is the initial condition. The initial force model used for the run is summarized by condition may be specified in tenns of Carte­ listing on I off indicators for Earth harmon­ I sian position and velocity coordinates or clas­ ics, atmospheric drag, solar radiation pressure, sical Keplerian orbital elements. In the case third body perturbations, lunar harmonics and of Keplerian elements, the in-plane position of maneuvering. The output vector coordinates, I the satellite may be specified with either true time, etc. are described in the same manner as anomaly, or time since periap­ the input vector. Finally, The radius and grav­ sis passage. Any of the physical constants used itational parameter of the Earth Moon and Sun I by the program, including the Earth and Moon are listed along with the first four zonals for the models, may be changed by placing the Earth and Moon. I appropriate key words and parameters into the If it has been requested, the executive pro­ input file. In this way the user may re-configure gram will monitor the instantaneous state of the orbit propagation model without changing the space vehicle in order to detect the oc­ I any computer code. currences of shadows and occul tat ions. Shad­ The program output data may be directed ows occur when a planet passes between the to the screen or to a disk file. The output data satellite and the Sun. are defined I begins with a listing of the input file used for to occur when a planet passes between the the current run. Next a summary of the pro­ satellite and an observer located at the center gram configuration is listed with respect to the of the Earth. Accurate knowledge of shadow I input and output coordinate frames, time defi- times is required for operating vehicle power I 3 I I and thermal subsystems. Knowledge of occul­ of the figure of the central body, ~ is the at­ tation times is important for communications mospheric drag acceleration, as is the acceler­ I scheduling. Exact start and stop times of these ation from solar radiation pressure and at are events are computed by iterative methods that the third body gravitational accelerations. No are described below. vehicle thrust acceleration appears in Equation I (3) because maneuvers are accounted for in N­ Body as instantaneous changes in velocity and I 3 Implementation of Cow­ not as vehicle accelerations. ell's Method The user may select one of three different nu­ merical integration procedures: RK4, a fourth I Cowell's method is a straightforward solution order Runge Kutta with constant step size, technique used in the special perturbations so­ RK45 a fourth order Runge Kutta with vari­ lution technique of orbital motion. The formu­ able step size, or SP, the modified Adams Pece I lation was first set down by Gauss but was used formulation of Shampine and Gordon 11. RK4 by Cowell at the beginning of this century for is the simplest method and requires only the integrating planetary ephemerides in rectangu­ step size to be specified. RK45 dynamically I lar coordinates. The term Cowell's method is controls its step size by monitoring the differ­ used now to describe any step by step numer­ ence between the current solution from a fourth ical integration in which the equations of mo­ order formulation with that of a fifth order for­ I tion are integrated in rectangular coordinates. mulation. RK45 requires an initial step size The technique has become more popular and and an error tolerance parameter. SP is the I useful with the advent of modern digital com­ sophisticated numerical integration pro­ puters. The method consists of writing the per­ cedure provided with N-body. It uses a multi­ turbed equations of motion as step Adams method with automatic step size I control. SP is very efficient and has the advan­ (1) tage that very large step sizes may be taken without an associated loss in accuracy. SP re­ I where r is the space vehicle position vector and quires both absolute and relative error toler­ a is the sum of all the accelerations acting on ance parameters. Because of the large integra­ the vehicle. Equation (1) may be rewritten as a tion steps which are taken, the periapsis and I system of two first order differential equations) apoapsis print modes are not permitted when r=v, v=a (2) using the SP integrator. I When further reduced to its rectangular com­ ponents) Equation (2) represents six first or­ 4 Force Model der differential equations which may be numer­ I To compute the acceleration of the space ve­ ically integrated by a variety of techniques. hicle and integrate the equations of motion, For the force model contained in N-Body, the all of the external forces acting on the vehi­ I acceleration vector may be written explicitly in cle must be modeled. The major forces acting terms of its various constituents as on a space vehicle in orbit are gravitational at­ traction from the Sun and planets, atmospheric I drag, and solar radiation pressure due to ab­ Where 8c is the central body acceleration, ah sorbtion and re-emission of the Sun's photons is the acceleration from the harmonic model by the spacecraft surfaces. Each of these forces I

4 I I I may be selected individually through the pro­ to those used in DE200. However, if a poten­ I gram configuration procedure. The Cartesian tial model is selected, the gravitational param­ components of each force are computed and eter associated with that model replaces the transformed to the mean equinox and equator default value. Any of the gravitational param­ I of J2000.0 coordinate frame in which the equa­ eters may be changed from its default value or tions of motion are cast. The accelerations are from the value in the potential model via the computed from the forces and the vehicle mass input file. I and summed to yield the total effective accel­ Atmospheric drag may be included in the eration. force model when propagating a space vehicle I The origin of the coordinates of integration in Earth orbit. The model used to compute may be selected to be the Sun or any of the atmospheric drag is: planets. When the central body is the Earth 1 p v 3 ~ = --CdA-- (4) I or the Moon, the force model allows the use of a 2 m v harmonic expansion of the gravitational poten­ In this model Cd is the coefficient of drag, A is tial in order to account for perturbations due the effective cross-sectional area of the vehicle, I to the or Moon. Two Earth p is the atmospheric density, m is the mass of potential models are provided: The 1984 World the vehicle, and v is the velocity of the vehicle I Geodetic Survey (WGS84)1 truncated to order in a coordinate frame which is rotating with the and degree 12, and the Goddard Earth Model Earth. The use of the Earth relative velocity (GEM9)2 truncated to order and degree 7. For of the space vehicle is equivalent to assuming I about the Moon, The Bills and Ferrari the atmosphere is rotating with the Earth. lunar potential modeP of order and degree 16 The atmospheric density may be computed is included. To provide greater flexibility the using either of two models selectable through I user may specify the coefficients of any nor­ the input file. The first is a simple exponential malized Earth or Moon potential model of up function of altitude and may be useful for ap­ to order and degree 21. The coefficients are plications which do not require high accuracy. I defined through use of an input file in which The second density model is the J acchia 1971 5 the non-zero terms in the potential model are in which atmospheric density is computed as a placed. function of space vehicle position, day of , I Gravitational forces exerted on the vehicle solar flux and the geomagnetic index. by bodies other than the central body are re­ Solar radiation pressure is accounted for by ferred to as third-body forces. Third-body assuming a constant value for solar flux, and I forces may be selected independently for the scaling it by the space vehicle area to mass ra­ Sun and planets. The third-body gravitational tio. The vector force acts in the direction from I forces are computed assuming point mass at­ the Sun to the vehicle. Solar radiation is not traction. Thus the third body forces are func­ computed when the vehicle is determined to be tions of the relative positions of the central in the shadow of the central body. This simple I body, the perturbing body and the space ve­ model will probably not be of sufficient accu­ hicle as well as the gravitational parameters of racy for propagating the trajectory of a deep the central and third bodies. The relative plan­ space, heliocentric orbit. An upgrade of this I etary positions are computed using the JPL module is planned wherein the solar flux will be Planetary Ephemeris System DE2004. By de­ modeled as a function of heliocentric distance fault, the gravitational parameters for each of and an index of reflectivity will be applied to I the bodies in the are set equal the solar radiation pressure scale factor. I 5 I I 5 Coordinates of date, rotations of precession and nutation are applied in order. The precession rotation I Coordinate frame defini tions are of primary takes place about the axis. The angle importance in any trajectory propagation sys­ being defined in reference 6 page SIS. The tem. The fundamental frame chosen for N­ nutation rotation matrix is computed accord­ I Body is the mean equator and equinox of ing to the 1980 IAU Theory of Nutation and J2000.0. J2000.0 refers to the standard epoch Reduction1. I at 0 hours, on January 1at, in the year 2000. An additional rotation about the pole The X, y plane is defined by the Earth's mean through the Greenwich hour angle yields the equatorial plane of J2000.0. The X direction in geographic (Earth fixed) inertial coordinate I the x, y plane is defined by the intersection of frame. When the velocity coordinates are cor­ the plane with the mean Earth equa­ rected for the rotation of Earth the geographic tor of J2000.0. The terminology mean equa­ rotating frame results. I tor of J2000.0 refers to the actual orientation Selenographic (Moon fixed) inertial coordi­ of the Earth equatorial plane at J2000.0 cor­ nates are computed via a transformation di­ rected for nutation. The origin of the reference rectly from J2000.0. Two such transformations I frame may be chosen to be any planet in the are included. In the first transformation, the solar system or the Moon. optical of the Moon is included8• In The mean equator and equinox of J2000.0 the second, the physical as well as optical libra­ I coordinate system is a convenient one in which tions are included. The analytic theory for the to carry out the trajectory integration. How­ libration of the Moon is that of Eckhardt (1981, I ever, depending on the application, the state 1982) and the computer code was provided by of the vehicle may have to be referred to an­ the U.S. Naval Observatory. The reason for other reference frame. For instance, the po­ maintaining both models is that most lunar I sition and velocity of the space vehicle rela­ potential models are reduced without includ­ tive to the true of date stellar background may ing the physical libration. Thus when evalu­ be of interest. In addition, true of date ge­ ating lunar gravitation one should likewise not I ographic coordinates are required to evaluate include the physical libration. However when the gravitational force from a harmonic poten­ it is desired to obtain the true position of the tial model. N-body allows space vehicle co­ space vehicle over the lunar surface to high I ordinates to be input and output in 1) mean accuracy, the physical libration should be in­ equator and equinox of J2000.0, 2) true equa­ cluded. Selenographic rotating coordinates are tor and equinox of date, 3) true ecliptic and computed by assuming a mean rotational rate I equinox of date, 4) mean equator and equinox for the Moon and applying the appropriate cor­ of B1950.0, 5) geographic (Earth fixed) and 6) rection to the velocity coordinates. selenographic (Moon fixed) coordinate frames. For the true ecliptic of date reference frame, I The Earth and Moon referenced velocity co­ precession and nutation are applied to the ordinates may be defined either as inertial or J2000.0 x axis to yield the true equinox of I with respect .to. the .rotatingbody... Theseleno­ date. Then a rotation about the true equinox graphic coordinates may be computed with or through the true obliquity of date is made to without including the physical libration of the establish the fundamental plane as the ecliptic I Moon. plane. In transforming from mean equator and Transformation from J2000.0 to B1950.0 co­ equinox of J2000.0 to true equator and equinox ordinates is computed according to the rota- I 6 I I I

tion matrix provided in reference 7. The rota­ timescales. I tion accounts for a change in the value of nodal Universal Time (UT) is the timescale which precession as well as a correction to the origin is most commonly used in every day life. It is of the B1950.0 coordinate frame. closely related to the motion of the Sun, which I involves the rotational and orbital motion of the Earth, and may be derived directly from 6 Time observations of the transit times of and I radio sources. UT is used internally by the pro­ The epoch times of the initial condition and gram in order to compute Sidereal time. Side­ propagated state vectors may be specified in real time is the measure of the hour angle of the I several different timescales. Input and out­ intersection of the true equator of date with the put state vector times may be independently ecliptic of date. Sidereal time must be known chosen from four different timescales: Inter­ in order to relate positions in an Earth-fixed I national Atomic Time, Coordinated Univer­ reference frame to the J2000.0 reference frame. sal Time, Terrestrial Time, or Barycentric Dy­ When the observationally derived UT is cor­ namical Time. Universal time and Sidereal rected for longitudinal errors caused by polar I time are also used internally by the program motion, it is designated UTI. UTI is related to to compute Earth referenced coordinate trans­ sidereal time by means of a numerical formula. formations. The program must be capable of I UTI and Sidereal Time are not available as translating from one timescale to another be­ input and output timescales, rather they are cause the various theories of motion incorpo­ used internally for the purpose of transform­ rated by the program to compute coordinate I ing from geographic coordinate frames to the frame orientations, planetary ephemerides, etc. J2000.0 frame. use different reference timescales for their in­ I dependent variables. For instance, most mis­ Coordinated Universal Time (UTC) is the sion planning is done with respect to the UTC timescale which is distributed by all broadcast timescale. The independent variable in the time services world wide. UTC differs from I JPL planetary ephemeris is TDB. In the the­ TAl by an integer number of seconds, and the ory of Earth nutation it is TT. In addition, fundamental unit of UTC is the Sl second. for some applications such as comparing propa­ UTC is kept to within 0.90 seconds of UTI I gated trajectories to data from other programs by adding (or subtracting) leap seconds. As or astronautical tables, it is convenient to se­ of July 1993, UTC is 28 seconds behind TAL lect different timescale for the input and output N-Body keeps track of the difference between I state vector times. UTC and TAl through use of a data base which The independent variable over which the must be updated each time an additional leap I equations of motion are integrated is, of course, second occurs. time. The internal timescale used in the in­ Terrestrial Time (TT) is a theoretical, ideal­ tegration is International Atomic Time (TAl). ized timescale which is the independent param­ I The fundamental unit of TAl is the Systeme eter in the apparent geocentric ephemerides. International (Sl) second. The Sl second is In the past this timescale has been called defined to be a fixed number transitions of Ephemeris Time (ET) and more recently Ter­ I a particular energy state of the cesium-133 restrial Dynamical Time (TDT). The funda­ atom. The TAl timescale provides a stable, mental unit of TT is the Sl second. The value very nearly uniform measure of time. It is of TT is TAl + 32.184 seconds. I used as the basis for prediction of the other Barycentric Dynamical Time (TDB) is an- I 7 I I other theoretical timescale which is the time be well defined based on the osculating states. argument in the equations of motion of the so­ Therefore the program may report multiple ap­ I lar system with respect to its . sis passages per revolution for orbits with very TDB is determined from 'IT depending on the low eccentricity. Also since the Shampine inte­ motion bodies in the solar system and the par­ gration technique can take very large step sizes, I ticular theory being used. The variation of 'IT the periapsis and apoapsis print modes may and TDB is periodic and involves no secular only be selected when using the Runge Kutta terms. An approximate formula is given in ref­ integration techniques. I erence 7: The times of passage of the space vehicle into TDB = 'IT + 0.0016588 sin 9 + 0.0000145 sin 29 and out of the shadows of the Moon and plan­ I (5) ets may also be determined. Subsequent inte­ where grated states are checked to detect the passages from full sun light, penumbra, and umbra. The I 9 = 357.53° + 0.9856003(JD - 2451545.0) computation is carried out using the values of and JD is the Julian date. the planetary radii defined in the JPL Develop­ mental Ephemeris System, DE200. The plane­ I tary radii values, as with most other constants 7 Searching for Orbital used in N-Body, may be changed via the in­ put file. When a shadow transition occurs, the I Events space vehicle trajectory is integrated in a sec­ During the course of an integration, the pro­ ondary procedure from the previous state to I gram executive can be instructed to watch for the current state with a small constant step various orbital events which when detected are size in order to better approximate the exact included part of the program output. Tech­ shadow entrance or exit time. The accuracy I niques are employed to search for periapsis and of the approximation is thus equal to the step apoapsis passages, umbra and penumbra pas­ size of the secondary integration. The step size sages, and occultations of the satellite by the may be selected using the appropriate input I planets with respect to the center of the Earth. file parameters. It is possible, depending on When the program is operated in the periap­ the step size of the primary trajectory integra­ sis or apoapsis print modes, it will attempt to tion process, that a shadow event of sufficiently I detect the radius of closest or farthest approach small duration could be missed by this detec­ to the central body. When it has determined tion method. Thus if all such shadow events I such an event, the exact time of passage is ap­ are to be found it may be advisable to con­ proximated by an iterative bisection method figure the program executive to use a Runge which converges very rapidly to an accuracy Kutta integration method with an appropri­ I which may be specified through the input file. ately small, constant step size. The passages are detected by comparing from The beginning and ending times of occulta­ one integration step to the next, the dot prod­ tions of the space vehicle by the Moon, Sun I uct of the position and velocity vectors. When and planets may be determined using the same the dot product changes sign, an apsis has been bisection process used in determining the peri­ passed. Whether it is periapsis or apoapsis is apsis and apoapsis passages. The occultation I determined by the direction of the sign change. condition is computed for an observer located It must be noted that for a near circular orbi t at the center of the Earth. Therefore the par­ the periapsis and apoapsis locations may not allax effect for an observer located somewhere I 8 I I I 4~------on the surface of the Earth is neglected. In I this computation, the radii of the planets de­ 3+------~~~~- fined in DE200 are used but may be overridden through the appropriate input file key words 2+------~------I and parameters. Note that in the periapsis and apoapsis print modes the vehicle state is written to the I ephemeris and program output files. When oc­ 0~--+_--~--~--_4--~~~ cultation or shadow detection is enabled only 0.00 4.00 8.00 12.00 16.00 20.00 24.00 the times of occurrence are reported to the pro­ Integration Time (hours) I gram output file. The ephemeris file is not up­ dated when these events are found to occur. Figure 1: One Day RSS Position Comparison I To TRIP Propagation 8 Code Verification LEO injection state is propagated for one day I Verification of the computer code is being ac­ by N-Body and TRIP and the resulting mean complished in part by comparing its trajectory equator and equinox of J2000.0 positions are predictions to that of other propagation pro­ compared by computing the norm of the po­ I grams. In this process the orbital elements sition vector differences. The orbit parame­ of a space object are propagated in time by ters at injection are that of a 140 by 160 nau­ I each program and the predicted positions and tical mile orbit inclined at approximately 66 velocities are directly compared. To facilitate degrees. The force model in both programs this process care must be taken that all phys­ includes Moon and Sun third body gravita­ I ical constants such as Earth and Moon gravi­ tion and uses the Goddard Earth Model G EM9 tational potential models, planetary constants truncated to order and degree 4. and the like are identical in each program. The results of the comparison of the inte­ I All such constants in N-Body may be changed grated space vehicle position between N-Body through the input file without any code revi­ and TRIP are plotted in Figure L It is seen sion. Thus input files may be tailored to cause that after one day of integration the two pro­ I the force model of N- Body to emulate that of grams predicted the space vehicle position to most other trajectory propagation programs. under 4 meters of each other. This small dif­ Obviously when doing such comparisons, care ference can likely be attributed to differences I must be taken to ensure that the comparison in evaluation of the geopotential model (TRIP states from each program are referred to simi­ uses an unnormalized form, while N- Body uses lar coordinate reference frames and timescales. a normalized form of the geopotential), to dif­ I Extensive verification tests of N-Body have ferences in the implementation of the numeri­ been conducted by comparing its output to cal integration procedures, and to the addition I the TRIP trajectory propagation program de­ of leap seconds which have occurred in since veloped by Kaufman at the Naval Research the latest update of TRIP. This small differ­ Laboratory9. One such comparison exercise ence is viewed to be acceptable and provides a I is presented here in which the trajectory of a validation of the N-Body computer code. space vehicle in low Earth orbit (LEO) is com­ In another verification test a comparison was puted. The reference orbit is that planned for made using the JPL computed ephemeris of I the LEO phase of the DSPSE mission. The the asteroid GeographoslO. The initial state of I 9 I I the ephemeris is propagated for six months at The force model for solar radiation pressure which time it is compared again to the JPL will be upgraded to better model the varia­ I ephemeris. In this instance the force model in­ tion of solar flux with heliocentric distance. cluded the third body perturbations of the ma­ The solar radiation pressure model will also jor planets and solar radiation pressure. The be modified to include the spacecraft reflectiv­ I computed RSS position difference was found to ity coefficient and to check for solar occulta­ be approximately 24 kilometers after the end tion from planetary bodies other than the cen­ of the six month integration. This discrep­ tral body. The Earth atmospheric drag model I ancy can likely be attributed to variations in will be modified to accept as input current or the solar radiation pressure modeL Also, the predicted values of solar flux and the geomag­ I JPL ephemeris is computed using a relativistic netic index. The JPL developed Earth gravi­ gravitational theory whereas N-body uses clas­ tational potential model JGM2 and the God­ sical Newtonian gravitation and the relativistic dard GEMT3 model will be added to the avail­ I masses from DE200. able selections. For lunar orbit propagation the Konopliv Lunar gravity model will be added. The capability to automatically switch the I 9 Execution Performance central body of integration will also be added. The body switching logic will use the plane­ Program development and testing is being car­ tary gravitational parameters to compute the I ried out on a VaxStation 4000 model 90 work­ spheres of influence of each of the bodies in the station. The low Earth orbit verification test solar system. Body switching will automati­ case described above was carried out on this cally take place when the space vehicle trajec­ I machine and the execution run time was cal­ tory leaves the sphere of influence of one body culated to serve as an indication of program and enters that of another. I execution performance. The force model used in the integration consisted of the GEM9 Earth model truncated to order and degree 4, as 11 Summary I well as Moon and Sun third body forces. The fourth order Runge K utta numerical integra­ Development and verification of the orbit prop­ tion method was selected. The force model was agation program N-Body is underway and is planned for completion in time to support the I evaluated four times per integration step. The total elapsed time for the one day propagation Deep Space Program Science Experiment mis­ was 22.8 seconds and a total of 19.1 CPU sec­ sion planned to begin in January 1994. The I onds were used. program contains a high fidelity orbital force model that will support orbit propagation in all phases of the mission: Low Earth orbit, cis­ I lunar orbit, lunar orbit and deep space. 10 Planned The program is designed for flexibility by al­ Enhancements lowing the selection of several different force I model options as wen as the use of user defined The N-Body program is still undergoing de­ physical constants. Optional specifications for velopment and verification. Several enhance­ time and coordinate reference systems enable I ments are planned for the short term in order the program to be useful for a wide variety of support the beginning of the DSPSE mission space mission planning and support applica­ in January 1994. tions. I 10 I I I Code verification is progressing through Seidelmann, P.K., Sinclair, A.T., Tj ufiin , I comparison of computed ephemerides with Y.S., "Report of the IAU/IAG/COSPAR those generated by other programs. Initial Working Group on Cartographic Coor­ testing indicates very good agreement with dinates and Rotational Elements of the I other accepted models. Planets and : 1991," Celes­ tial Mechanics and Dynamical Astronomy, Vol 53, No.4, 1992 I References [9] Kaufman, Bernard, "TRIP (Trajectory p.] "Departmemt of Defense World Geodetic Integration Program) Program Descrip­ I System, 1984/' Defense Mapping Agency tion and Use," NRL Report 7436, July 18, Technical Report 8350.2, 30 September 1972 1987 I [10] Yeomans, D.K., Keesey, M.S. Personal [2] Lerch, Francis J., Klosko, Steven M., Communication, November, 1992 Laubscher, Roy E., Wagner, Carl A., I "Gravity Model Improvement Using Geos [ll] Shampine, L. F., Gordon, M. K., "Com­ 3 (GEM 9 and 10)," Journal of Geophys­ puter Solution of Ordinary Differential ical Research, Vol 84, No. B8, July 30, Equations The Initial Value Problem," I 1979 W.H. Freeman and Company, San Fran­ cisco, California, 1975 [3] Bills, B.G. and Ferrari, A.J., "A Har­ I monic Analysis of Lunar Gravity," Jour­ nal of Geophysical Research, Vol 85, No. I B2, February 10, 1980 [4] Standish, E. Myles Jr., Newhall, X. X. , "The JPL Export Planetary Ephemeris," I June 29, 1990 (Revision 1) [5} Jacchia, L.G., "Revised Static Models of I the Thermosphere and Exosphere with Empirical Temperature Profiles," Smith­ sonian Astrophysical Observatory Spe­ I cial Report No. 332, Cambridge, Mas­ sachusetts, May 1971 I [6] "The Astronomical Almanac for the Year 1984," U.S. Government Printing Office I [7] Seidelmann, P. K., Ed., "Explanatory Supplement to the Astronomical Al­ manac," University Science Books, Mill I Valley, California, 1992 [81 Davies, M.E., Abalakin, V.K., Brahic, A., I Bursa, M., Chovitz, B.B., Lieske, J.H., I 11 I