High-Speed Solution of Spacecraft Trajectory Problems Using Taylor Series Integration

It has been known for some time that Taylor series (TS) integration is among the most efficient and accurate numerical methods in solving differential equations. However, the full benefit of the method has yet to be realized in calculating spacecraft trajectories, for two main reasons. First, most applications of Taylor series to trajectory propagation have focused on relatively simple problems of orbital motion or on specific problems and have not provided general applicability. Second, applications that have been more general have required use of a preprocessor, which inevitably imposes constraints on computational efficiency. The latter approach includes the work of Berryman et al., who solved the planetary n-body problem with relativistic effects. Their work specifically noted the computational inefficiencies arising from use of a preprocessor and pointed out the potential benefit of manually coding derivative routines. In this Engineering Note, we report on a systematic effort to directly implement Taylor series integration in an operational trajectory propagation code: the Spacecraft N-Body Analysis Program (SNAP). The present Taylor series implementation is unique in that it applies to spacecraft virtually anywhere in the solar system and can be used interchangeably with another integration method. SNAP is a high-fidelity trajectory propagator that includes force models for central body gravitation with N X N harmonics, other body gravitation with N X N harmonics, solar radiation pressure, atmospheric drag (for Earth orbits), and spacecraft thrusting (including shadowing). The governing equations are solved using an eighth-order Runge-Kutta Fehlberg (RKF) single-step method with variable step size control. In the present effort, TS is implemented by way of highly integrated subroutines that can be used interchangeably with RKF. This makes it possible to turn TS on or off during various phases of a mission. Current TS force models include central body gravitation with the J2 spherical harmonic, other body gravitation, thrust, constant atmospheric drag from Earth's atmosphere, and solar radiation pressure for a sphere under constant illumination. The purpose of this Engineering Note is to demonstrate the performance of TS integration in an operational trajectory analysis code and to compare it with a standard method, eighth-order RKF. Results show that TS is 16.6 times faster on average and is more accurate in 87.5% of the cases presented

Rapid Calculation of Spacecraft Trajectories Using Efficient Taylor Series Integration

A variable-order, variable-step Taylor series integration algorithm was implemented in NASA Glenn's SNAP (Spacecraft N-body Analysis Program) code. SNAP is a high-fidelity trajectory propagation program that can propagate the trajectory of a spacecraft about virtually any body in the solar system. The Taylor series algorithm's very high order accuracy and excellent stability properties lead to large reductions in computer time relative to the code's existing 8th order Runge-Kutta scheme. Head-to-head comparison on near-Earth, lunar, Mars, and Europa missions showed that Taylor series integration is 15.8 times faster than Runge- Kutta on average, and is more accurate. These speedups were obtained for calculations involving central body, other body, thrust, and drag forces. Similar speedups have been obtained for calculations that include J2 spherical harmonic for central body gravitation. The algorithm includes a step size selection method that directly calculates the step size and never requires a repeat step. High-order Taylor series integration algorithms have been shown to provide major reductions in computer time over conventional integration methods in numerous scientific applications. The objective here was to directly implement Taylor series integration in an existing trajectory analysis code and demonstrate that large reductions in computer time (order of magnitude) could be achieved while simultaneously maintaining high accuracy. This software greatly accelerates the calculation of spacecraft trajectories. At each time level, the spacecraft position, velocity, and mass are expanded in a high-order Taylor series whose coefficients are obtained through efficient differentiation arithmetic. This makes it possible to take very large time steps at minimal cost, resulting in large savings in computer time. The Taylor series algorithm is implemented primarily through three subroutines: (1) a driver routine that automatically introduces auxiliary variables and sets up initial conditions and integrates; (2) a routine that calculates system reduced derivatives using recurrence relations for quotients and products; and (3) a routine that determines the step size and sums the series. The order of accuracy used in a trajectory calculation is arbitrary and can be set by the user. The algorithm directly calculates the motion of other planetary bodies and does not require ephemeris files (except to start the calculation). The code also runs with Taylor series and Runge-Kutta used interchangeably for different phases of a mission

A High Power Solar Electric Propulsion - Chemical Mission for Human Exploration of Mars

Recently Solar Electric Propulsion (SEP) as a main propulsion system has been investigated as an option to support manned space missions to near-Earth destinations for the NASA Gateway spacecraft. High efficiency SEP systems are able to reduce the amount of propellant long duration chemical missions require, ultimately reducing the required mass delivered to Low Earth Orbit (LEO) by a launch vehicle. However, for long duration interplanetary Mars missions, using SEP as the sole propulsion source alone may not be feasible due to the long trip times to reach and insert into the destination orbit. By combining an SEP propulsion system with a chemical propulsion system the mission is able to utilize the high-efficiency SEP for sustained vehicle acceleration and deceleration in heliocentric space and the chemical system for orbit insertion maneuvers and trans-earth injection, eliminating the need for long duration spirals. By capturing chemically instead of with low-thrust SEP, Mars stay time increases by nearly 200 days. Additionally, the size the of chemical propulsion system can be significantly reduced from that of a standard Mars mission because the SEP system greatly decreases the Mars arrival and departure hyperbolic excess velocities (V(sub infinity))

Nova Sagittarii 1994 #1 (V4332 Sagittarii): The Discovery and Evolution of an Unusual Luminous Red Variable Star

We report photometry and spectroscopy of the evolution of Nova Sagittarii 1994 #1 (V4332 Sagittarii) during outburst. We compare the photometric and spectral evolution of this outburst to known classes of outbursts -- including classical novae and outbursts occurring on symbiotic stars -- and find this object does NOT conform to any known class of outburst. The closest match to the behavior of this unusual object is M31 RV, an extremely luminous and red variable object discovered in the bulge of M31 in 1988. However, the temporal behavior and maximum luminosity of the two events differ by several orders of magnitude, requiring substantial intrinsic variation if these two events are members the same type of outburst. Our model of the spectroscopic evolution of this outburst shows that the effective temperature cooled from 4400 K to 2300 K over the three month span of our observations. In combination with line diagnostics in our later spectra, including [OI] lambda 5577 and the dramatic increase in the Halpha to Hbeta ratio, we infer the existence of a cool, dense (N_e ~ 10^{8-9} cm^{-3}) envelope that is optically thick in the Hydrogen Balmer recombination lines (case C). We suggest that a nuclear event in a single star, in which a slow shock drove the photosphere outwards, can power the observed luminosity evolution and the emission spectrum.Comment: Accepted for publication in AJ. 24 pages including 8 embedded postscript figures. Also available at http://www.astronomy.ohio-state.edu/~martini/pub

NASA GRC Compass Team Conceptual Point Design and Trades of a Hybrid Solar Electric Propulsion (SEP)/Chemical Propulsion Human Mars Deep Space Transport (DST) Vehicle

NASA has long been conducting studies which apply different in-space propulsion technology assumptions to the mission of sending humans to Mars. Two of the technologies under study that are considered to be the most near-term with respect to technology readiness level (TRL) are traditional chemical propulsion systems and high-power Solar Electric Propulsion (SEP) systems. The benefit of relatively low trip times inherent in using impulsive chemical propulsion systems to perform the full round-trip delta V is hampered by the large propellant mass required to perform these burns for human Mars missions. SEP systems offer the benefit of much lower propellant requirements to perform the same round-trip missions, at the cost of longer trip times. Traditionally, impulsive chemical systems are better suited than SEP when used in a gravity well, and SEP systems are more efficient than traditional impulsive systems when used in interplanetary space. A mission to Mars includes both of these scenarios, and thus several NASA architecture studies performed over the last few years have looked to combine the use of both SEP and chemical propulsion systems where they are the most beneficial to human Mars missions. This combined propulsion system concept has been referred to as a SEP/Chem hybrid Mars Transfer Vehicle and is currently shown as the concept Deep Space Transport (DST) in the March 2017 NASA presentation to the National Aerospace Council (NAC)

Injection Matching Studies using Turn by Turn Beam Profile Measurements in the CERN PS

The very small emittance beam needed for the LHC requires that the emittance blow-up in its injector machines must be kept to a minimum. Mismatch upon the beam transfer from one machine to the next is a potential source of such blow-up. The CERN PS ring is equipped with 3 Secondary Emission Grids (SEM-Grids) which are used for emittance measurement at injection. One of these has been converted to a multi-turn mode, in which several tens of consecutive beam passages can be observed. This allows the study of mismatch between the PS-Booster and the PS. This paper describes the instrument and experimental results obtained during the last year

Neutrino Quasielastic Scattering on Nuclear Targets: Parametrizing Transverse Enhancement (Meson Exchange Currents)

We present a parametrization of the observed enhancement in the transverse electron quasielastic (QE) response function for nucleons bound in carbon as a function of the square of the four momentum transfer ($Q^2$) in terms of a correction to the magnetic form factors of bound nucleons. The parametrization should also be applicable to the transverse cross section in neutrino scattering. If the transverse enhancement originates from meson exchange currents (MEC), then it is theoretically expected that any enhancement in the longitudinal or axial contributions is small. We present the predictions of the "Transverse Enhancement" model (which is based on electron scattering data only) for the $\nu_\mu, \bar{\nu}_\mu$ differential and total QE cross sections for nucleons bound in carbon. The $Q^2$ dependence of the transverse enhancement is observed to resolve much of the long standing discrepancy in the QE total cross sections and differential distributions between low energy and high energy neutrino experiments on nuclear targets.Comment: Revised Version- July 21, 2011: 17 pages, 20 Figures. To be published in Eur. Phys. J.

High energy parton-parton amplitudes from lattice QCD and the stochastic vacuum model

Making use of the gluon gauge-invariant two-point correlation function, recently determined by numerical simulation on the lattice in the quenched approximation and the stochastic vacuum model, we calculate the elementary (parton-parton) amplitudes in both impact-parameter and momentum transfer spaces. The results are compared with those obtained from the Kr\"{a}mer and Dosch ansatz for the correlators. Our main conclusion is that the divergences in the correlations functions suggested by the lattice calculations do not affect substantially the elementary amplitudes. Phenomenological and semiempirical information presently available on elementary amplitudes is also referred to and is critically discussed in connection with some theoretical issues.Comment: Text with 11 pages in LaTeX (twocolumn form), 10 figures in PostScript (psfig.tex used). Replaced with changes, Fig.1 modified, two references added, some points clarified, various typos corrected. Version to appear in Phys. Rev.