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Three-Dimensional Lunar Mission Studies

Some three-dimensional lunar trajectories have been calculated by integration of the equations of motion of the classical restricted three-body problem of celestial mechanics. The calculations have been used for analysis of several aspects of lunar flight including requirements for achieving lunar impact and for establishment of a close lunar satellite. The allowable errors in initial conditions for lunar missions are strongly dependent on the values of the initial injection velocity and the injection angle. There can be large differences in results obtained from two-dimensional analyses (in which the vehicle trajectory is assumed to remain always in the earth-moon plane) and those obtained from three-dimensional analyses. Some of the accuracy tolerances can be fairly well estimated by use of a two-body analysis which considers the inclination of the plane of the vehicle trajectory to the earth-moon plane. Satisfactory orbits for a relatively close lunar satellite can be obtained with accuracies in the initial conditions approximately equal to those required for lunar impact.

Michael, William H., Jr.

Three-Dimensional Orbits of Earth Satellites, Including Effects of Earth Oblateness and Atmospheric Rotation

The principal purpose of the present paper is to present sets of equations which may be used for calculating complete trajectories of earth satellites from outer space to the ground under the influence of air drag and gravity, including oblateness effects, and to apply these to several examples of entry trajectories starting from a circular orbit. Equations of motion, based on an "instantaneous ellipse" technique, with polar angle as independent variable, were found suitable for automatic computation of orbits in which the trajectory consists of a number of revolutions. This method is suitable as long as the trajectory does not become nearly vertical. In the terminal phase of the trajectories, which are nearly vertical, equations of motion in spherical polar coordinates with time as the independent variable were found to be more suitable. In the first illustrative example the effects of the oblateness component of the earth's gravitational field and of atmospheric rotation were studied for equatorial orbits. The satellites were launched into circular orbits at a height of 120 miles, an altitude sufficiently high that a number of revolutions could be studied. The importance of the oblateness component of the earth's gravitational field is shown by the fact that a satellite launched at circular orbital speed, neglecting oblateness, has a perigee some 67,000 feet lower when oblateness forces are included in the equations of motion than when they are not included. Also, the loss in altitude per revolution is double that of a satellite following an orbit not subject to oblateness. The effect of atmospheric rotation on the loss of altitude per revolution was small. As might be surmised, the regression of the line of nodes as predicted by celestial mechanics is unchanged when drag is included. It is clear that the inclination of the orbital plane to the equator will be relatively unaffected by drag for no atmospheric rotation since the drag lies in the orbital plane in this case. With the inclusion of atmospheric rotation it was found that the inclination of the plane changed about one-millionth of a radian per revolution. Thus the prediction of the position of the orbital plane of an earth satellite is not complicated by the introduction of drag. The line of apsides, which without drag but with oblateness moves slowly in space, tends to move with the satellite when drag is included in the calculations. As a results, the usual linearized solutions based on oblateness alone must be basically altered when drag is included to take into account the rapid movement of the line of apsides. In the second illustrative example the final revolution was calculated to impact for a number of trajectories in an orbital plane inclined at 650 to the equator. Of particular interest is the large effect the oblateness gravitational field and atmospheric rotation can have on the impact point. For a value of CDA/m of unity, and for an initial downward angle at 80 miles altitude of 0.01 radian, such as might be utilized for manned re-entry, oblateness had an influence of about 300 miles in the impact point, and atmospheric rotation had about a 150-mile influence.

Nielsen, Jack N.

Dynamics of the Trans-Neptune Region: Apsidal Waves in the Kuiper Belt

The role of apsidal density waves propagating in a primordial trans-Neptune disk (i.e., Kuiper belt) is investigated. It is shown that Neptune launches apsidal waves at its secular resonance near 40 AU that propagate radially outward, deeper into the particle disk. The wavelength of apsidal waves is considerably longer than waves that might be launched at Lindblad resonances, because the pattern speed, g(sub s), resulting from the apsis precession of Neptune is much slower than its mean motion, Omega(sub s). If the early Kuiper belt had a sufficient surface density, sigma, the disk's wave response to Neptune's secular perturbation would have spread the disturbing torque radially over a collective scale lambda(sub *) approx. = r(2(mu)(sub d)Omega/ absolute value of r dg/dr)(sup 1/2), where mu(sub d)equivalent pi(sigma)r(exp 2)/(1 solar mass) and Omega(r) and g(r) are respectively the mean motion and precession frequency of the disk particles. This results in considerably smaller eccentricities at resonance than had the disk particles been treated as noninteracting test particles. Consequently, particles are less apt to be excited into planet-crossing orbits, implying that the erosion timescales reported by earlier test-particle simulations of the Kuiper belt may be underestimated. It is also shown that the torque the disk exerts upon the planet (due to its gravitational attraction for the disk's spiral wave pattern) damps the planet's eccentricity and further inhibits the planet's ability to erode the disk. Key words: celestial mechanics, stellar dynamics - comets: general minor planets, asteroids

Ward, William R.

Astrometric and Geodetic Properties of Earth and the Solar System

The mass, size and shape of planets and their satellites and are essential information from which one can consider the balance of gravity and tensile strength, chemical makeup and such factors as internal temperature or porosity. Orbits and planetary rotation are also useful clues concerning origin, internal structure and tidal history. The tables compiled here include some of the latest results such as detection of densities of Pluto-Charon from analysis of HST images and the latest results for Venus' shape, gravity field and pole orientation based on Magellan spacecraft data. Data concerning prominent asteroids, comets and Sun are also included. Most of the material here is presented as tables. They are preceded by brief explanations of the relevant geophysical and orbit parameters. More complete explanations can be found in any of several reference texts on geodesy, geophysics and celestial mechanics.

Yoder, Charles F.

Geoid Recovery Using Geophysical Inverse Theory Applied to Satellite to Satellite Tracking Data

This report describes a new method for determination of the geopotential, or the equivalent geoid. It is based on Satellite-to-Satellite Tracking (SST) of two co-orbiting low earth satellites separated by a few hundred kilometers. The analysis is aimed at the GRACE Mission, though it is generally applicable to any SST data. It is proposed that the SST be viewed as a mapping mission. That is, the result will be maps of the geoid or gravity, as contrasted with determination of spherical harmonics or Fourier coefficients. A method has been developed, based on Geophysical Inverse Theory (GIT), that can provide maps at a prescribed (desired) resolution and the corresponding error map from the SST data. This computation can be done area by area avoiding simultaneous recovery of all the geopotential information. The necessary elements of potential theory, celestial mechanics, and Geophysical Inverse Theory are described, a computation architecture is described, and the results of several simulations presented. Centimeter accuracy geoids with 50 to 100 km resolution can be recovered with a 30 to 60 day mission.

Gaposchkin, E. M.

Fixed Future and Uncertain Past: Theorems Explain Why It Is Often More Difficult to Reconstruct the Past Than to Predict the Future

At first glance. it may seem that reconstructing the past is, in general, easier than predicting the future, because the past has already occurred and it has already left its traces, while the future is still yet to come, and so no traces of the future are available. However, in many real life situations, including problems from geophysics and celestial mechanics, reconstructing the past is much more computationally difficult than predicting the future. In this paper, we give an explanation of this difficulty. This explanation is given both on a formal level (as a theorem) and on the informal level (as a more intuitive explanation).

Alefeld, Goetz

Orbit Determination of the Mars Global Surveyor Spacecraft Using Laser Altimetry

Many of the scientific investigations of the Mars Global Surveyor (MGS) mission require high precision orbital information and some are limited entirely by its quality. These include the laser altimeter (MOLA) the Mars gravity field and atmospheric occultation investigations by radio science, and the planetary dynamics and celestial mechanics investigations. The precision of the orbits can usually be assessed by comparing overlapping orbits for a given period; but these results tend to reflect the repeatability rather than the accuracy. The re-constructed orbits from the doppler and range tracking data on MGS are (to date) at the few meter level radially, and a few hundreds of meters horizontally, using the best gravity models, presently available. With the laser altimeter on MGS we have a mechanism to measure the quality and to actually make significant improvements in the orbital accuracy by incorporating the altimetry data as a tracking datatype. By adding the altimeter measurements at orbital cross-over locations we have been able to reduce die radial error to 1 meter of less on average and have reduced the along track and out of plane error by almost 2 orders of magnitude down to a few meters. It is apparent that the altimeter observation provides a geometric strength to the orbit that it is not possible to obtain from the present doppler and the range data alone. We discuss the results obtained for the first year of the MGS mapping orbit. This work is supported by the NASA Mars Program.

Smith, David E.

Geoid Recovery using Geophysical Inverse Theory Applied to Satellite to Satellite Tracking Data

This report describes a new method for determination of the geopotential. The analysis is aimed at the GRACE mission. This Satellite-to-Satellite Tracking (SST) mission is viewed as a mapping mission The result will be maps of the geoid. The elements of potential theory, celestial mechanics, and Geophysical Inverse Theory are integrated into a computation architecture, and the results of several simulations presented Centimeter accuracy geoids with 50 to 100 km resolution can be recovered with a 30 to 60 day mission.

Gaposchkin, E. M.

Long-Term Stability of Planets in Binary Systems

A simple question of celestial mechanics is investigated: in what regions of phase space near a binary system can planets persist for long times? The planets are taken to be test particles moving in the field of an eccentric binary system. A range of values of the binary eccentricity and mass ratio is studied, and both the case of planets orbiting close to one of the stars, and that of planets outside the binary orbiting the systems center of mass, are examined. From the results, empirical expressions are developed for both (1) the largest orbit around each of the stars and (2) the smallest orbit around the binary system as a whole, in which test particles survive the length of the integration (10A4 binary periods). The empirical expressions developed, which are roughly linear in both the mass ratio mu and the binary eccentricity e, are determined for the range 0.0=e=0.7-0.8 and 0.1=mu=0.9 in both regions and can be used to guide searches for planets in binary systems. After considering the case of a single low-mass planet in binary systems, the stability of a mutually interacting system of planets orbiting one star of a binary system is examined, though in less detail.

Holman, Matthew J.

A New Spin

The trouble was that the shuttle was still under development when that schedule was set. As time went on, the Shuttle had problems with its high pressure turbines, thermal protection tiles, engines, and more. The early launch dates had to be scrapped. NASA Headquarters told us, "We re going to delay your launch two years to allow more time for the Shuttle development to take place. You can slow your development accordingly." Right off the bat, we looked into the celestial mechanics and how they would affect us. The difficulty in launching a spacecraft to Jupiter changes on a year-to- year basis, in a cyclical pattern that repeats about every ten or twelve years. In order to achieve the velocity needed to get from low earth orbit to Jupiter, an upper stage is required in the Shuttle. For the 1982 launch the upper stage was adequate, but it could not provide the velocity we would need in 1984. This meant we would have to separate the Galileo probe from the Galileo orbiter before launch and put each of them on separate Shuttles with separate upper stages. When we told the folks at Headquarters this, they told us, "Okay we'll give you two Shuttle launches."

Casani, John

Future Visions for Scientific Human Exploration

Today, humans explore deep-space locations such as Mars, asteroids, and beyond, vicariously here on Earth, with noteworthy success. However, to achieve the revolutionary breakthroughs that have punctuated the history of science since the dawn of the Space Age has always required humans as "the discoverers," as Daniel Boorstin contends in this book of the same name. During Apollo 17, human explorers on the lunar surface discovered the "genesis rock," orange glass, and humans in space revamped the optically crippled Hubble Space Telescope to enable some of the greatest astronomical discoveries of all time. Science-driven human exploration is about developing the opportunities for such events, perhaps associated with challenging problems such as whether we can identify life beyond Earth within the universe. At issue, however, is how to safely insert humans and the spaceflight systems required to allow humans to operate as they do best in the hostile environment of deep space. The first issue is minimizing the problems associated with human adaptation to the most challenging aspects of deep space space radiation and microgravity (or non-Earth gravity). One solution path is to develop technologies that allow for minimization of the exposure time of people to deep space, as was accomplished in Apollo. For a mission to the planet Mars, this might entail new technological solutions for in-space propulsion that would make possible time-minimized transfers to and from Mars. The problem of rapid, reliable in-space transportation is challenged by the celestial mechanics of moving in space and the so-called "rocket equation." To travel to Mars from Earth in less than the time fuel-minimizing trajectories allow (i.e., Hohmann transfers) requires an exponential increase in the amount of fuel. Thus, month-long transits would require a mass of fuel as large as the dry mass of the ISS, assuming the existence of continuous acceleration engines. This raises the largest technological stumbling block to moving humans on site as deep-space explorers, delivering the masses required for human spaceflight systems to LEO or other Earth orbital vantage points using the existing or projected fleet of Earth-to-orbit (ETO) launch vehicles. Without a return to Saturn V-class boosters or an alternate path, one cannot imagine emplacing the masses that would be required for any deep-space voyage without a prohibitive number of Shuttle-class launches. One futurist solution might involve mass launch systems that could be used to move the consumables, including fuel, water, food, and building materials, to LEO in pieces rather than launching integrated systems. This approach would necessitate the development of robotic assembly and fuel-storage systems in Earth orbit, but could provide for a natural separation of low-value cargo (e.g., fuel, water).

Garvin, James

A novel two-step laser ranging technique for a precision test of the theory of gravity

All powered spacecraft experience residual systematic acceleration due to anisotropy of the thermal radiation pressure and fuel leakage. The residual acceleration limits the accuracy of any test of gravity that relies on the precise determination of the spacecraft trajectory. We describe a novel two-step laser ranging technique, which largely eliminates the effects of non-gravity acceleration sources and enables celestial mechanics checks with unprecedented precision.

ranging

Stochastic Representation of Chaos Using Terminal Attractors

A nonlinear version of the Liouville equation based on terminal attractors is part of a mathematical formalism for describing postinstability motions of dynamical systems characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism can be applied to both conservative systems (e.g., multibody systems in celestial mechanics) and dissipative systems (e.g., viscous fluids). The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail

Extending Science from Lunar Laser Ranging

The Lunar Laser Ranging (LLR) experiment has accumulated 50 years of range data of improving accuracy from ground stations to the laser retroreflector arrays (LRAs) on the lunar surface. The upcoming decade offers several opportunities to break new ground in data precision through the deployment of the next generation of single corner-cube lunar retroreflectors and active laser transponders. This is likely to expand the LLR station network. Lunar dynamical models and analysis tools have the potential to improve and fully exploit the long temporal baseline and precision allowed by millimetric LLR data. Some of the model limitations are outlined for future efforts. Differential observation techniques will help mitigate some of the primary limiting factors and reach unprecedented accuracy. Such observations and techniques may enable the detection of several subtle signatures required to understand the dynamics of the Earth- Moon system and the deep lunar interior. LLR model improvements would impact multi- disciplinary fields that include lunar and planetary science, Earth science, fundamental physics, celestial mechanics and ephemerides.

Vishnu Viswanathan

The SIRTF in-orbit checkout and science verification plan

This paper describes a nominal mission plan that progressively establishes SIRTF capabilities during the IOC/SV phases, taking into consideration thermal, cryogrenic, optical, communications, celestial mechanics, and operational designs and constraints.

SIRTF

Disk Tides and Accretion Runaway

It is suggested that tidal interaction of an accreting planetary embryo with the gaseous pre-planetary disk may provide a mechanism to breach the so-called runaway limit during the formation of the giant planet cores.

accretion

Optimizing infrasound observations for sample return capsule re-entry: Insights from OSIRIS-REx and Hayabusa2

The atmospheric entry of meteoroids presents a rare and unpredictable phenomenon, posing challenges for systematic observation and detailed characterization. Such events are nonetheless critical for advancing understanding of acoustic wave propagation, atmospheric structure, and entry dynamics. In contrast, sample return capsules (SRCs) from space missions follow well constrained re-entry trajectories, enabling planned observations of shock wave generation and propagation under controlled conditions. This study compares two SRC atmospheric entries, Hayabusa2 in 2020 and OSIRIS-REx in 2023, to assess how different infrasound array configurations influence shock wave detection and trajectory validation. Hayabusa2’s re-entry was monitored using a distributed network of 28 portable infrasound sensors across seven arrays in Woomera, Australia, permitting three-dimensional reconstruction of the trajectory and analysis of wave characteristics. For OSIRIS-REx, a compact four-sensor array deployed near Eureka Airport provided trajectory confirmation through arrival-time differences and back-azimuth estimates. Spectral and waveform analyses revealed differences in signal properties associated with variations in entry angle and velocity. The results illustrate both the strengths and the limitations of the deployed array configurations. The dense and distributed arrays during Hayabusa2’s re-entry enabled detailed trajectory reconstruction, whereas the compact array at Eureka primarily provided confirmation of signal coherence and back-azimuth consistency. These case studies highlight that even relatively small arrays, if located close to the predicted ground track, can still capture useful information on arrival direction and timing. Such insights provide practical guidance for planning future observational campaigns of SRC returns and other controlled atmospheric entries, and may also inform approaches to opportunistic observations of meteoroid events and other atmospheric acoustic phenomena.

celestial mechanics: orbit determination