Written by a former member of NASA’s Jet Propulsion Laboratory navigation team, this upper undergraduate/graduate textbook covers the theories and procedures of planetary spacecraft navigation, along with the mathematics behind digital navigation programs.

The equations of motion may be numerically integrated to obtain the trajectory of a spacecraft or the path of a photon or radio wave. These equations may also be solved for certain parameters that can be verified by experiment or used to verify the results of numerical integration. The parameters of most interest are the precession of Mercury’s orbit, the bending of light by the sun’s gravity field, the delay in a signal transmitted from a spacecraft to Earth, clock time keeping, solar pressure, and planetary and stellar aberration. Formulas for these parameters are compared with the results of numerical integration.

The Near Earth Asteroid Rendezvous spacecraft passed behind the sun in 1997 on the way to rendezvous, orbiting and landing on the asteroid 453 Eros. As the spacecraft passed behind the sun, the path of the radio link with the Earth came close to the surface of the sun and was eventually occulted by the sun. This provided an opportunity to observe the contribution of general relativity to the time delay in the signal or the path length from the spacecraft to Earth tracking stations. An experiment was performed with the primary objective of verifying operational navigation software and with a secondary objective of verifying Einstein’s General Theory of Relativity. The two-way Doppler and range data provided a measure of the path length that was accurate to a few centimeters. This enabled a determination of the general relativity delay once the signal path length was calibrated to remove other contributors to the delay. These other contributors included determination of position of the spacecraft and Earth tracking stations relative to the sun, troposphere and ionosphere delay, solar plasma delay, tracking station locations, and cabling delays. The experiment enabled verifying Einstein’s theory to an accuracy of about 0.1%.

General relativity field equations are a set of differential equations that can be solved for the metric tensor which defines curved space. They are derived from two physical assumptions that the speed of light is constant in any frame of reference that may be moving relative to one another and space is curved by the mass of physical bodies. Some other assumptions related to mathematics such as symmetry and continuity are also needed. First, the metric tensor is defined as a transformation from Euclidean space to curved space. The equation of geodesics describes the equations of motion of a particle within a gravitational field. The Riemann tensor and Ricci tensor are obtained by differentiation of the metric tensor with respect to an assumed coordinate system. The Einstein tensor scales the Riemann tensor proportional to the mass present. The mathematics use Einstein summation notation and are well defined in many sources. Since navigators use matrix notation, some of the key derivations are repeated in matrix notation. Also, a computer solution for the metric tensor and equations of motion is obtained by processing simulated data in an orbit determination filter thus bypassing the analytic solution obtained from the Einstein tensor.

Once the field equations are defined, they are solved for the metric tensor. For spherical symmetry, the solution may be obtained directly from the Ricci tensor, which is a contraction of the Riemann tensor. In general the solution is obtained from the Einstein tensor. A solution for the metric tensor obtained by Schwarzschild for the case of spherical symmetry is described. This metric tensor is inserted into the equation of geodesics to obtain the equations of motion in free space outside the body. The equations of motion are derived for both a point mass and a photon. The solution for the gravity field inside a body is also described. In curved space a volume element is not cubical. A transformation is defined that makes the volume element cubical. The result is called isotropic Schwarzschild coordinates that make the resulting geometry look more Euclidean and are the coordinates used for spacecraft navigation.

The round trip propagation time of a radio signal passing close to the Sun is affected by the charged particle environment of the solar plasma and the increased path length associated with the curvature of space predicted by general relativity. It has been difficult to separate estimates of the individual contributions to the signal delay. In January 1997, the Near Earth Asteroid Rendezvous (NEAR) spacecraft passed through a deep solar conjunction. For a month before and after conjunction, the spacecraft was pointed at the Sun and attitude operations were suspended. Since the NEAR spacecraft is equipped with two way X-band Doppler and range and the spacecraft attitude was favorable, a high precision spacecraft ephemeris was obtained. Data from the solar conjunction period of the NEAR mission is used to solve explicitly for the total electron content of the solar plasma along the signal path by using the signal delay in the range data to calibrate the signal advance in the Doppler data. The result is a precise estimate of the general relativity delay, and of the total electron content along the signal path, which can be used to improve models of the plasma emanating from the Sun. The general relativity parameter y is determined to less than 0.5 percent, which is competitive with the most accurate current verifications of general relativity.

The classical gravity model used for planetary navigation is a harmonic expansion of Legendre polynomials and associated functions. The harmonic expansion diverges when the orbital radius is less than that of the Brillouin sphere. An alternative method for computing gravitational acceleration is described, and shape models of a brick and the asteroid Eros are used as examples. Computation is reduced to a surface integral by first analytically integrating with respect to r, assuming constant density. The result is evaluated far more rapidly. Results are compared to those generated from a polyhedron model and to those generated from a classical harmonic expansion.

The flight control teams of two Low Earth Orbit missions at EUMETSAT present an overview of the automated tools and methodologies being used to analyse and report on spacecraft health; including trend analysis, data mining and outlier detection. A qualitative analysis of the techniques is provided based on in-flight experience, and proposals for future development of such toolsets are presented. This paper focuses on the experiences of the Copernicus Sentinel-3 and EPS MetOp flight control teams in using the EUMETSAT CHART framework, which allows engineers to define automated reports and perform ad-hoc analysis on large datasets with multiple input sources. Arguments are also presented regarding whether or not it may be appropriate for future missions to consider applying some of these techniques directly onboard as an extension of the currently in-place FDIR mechanisms.

As Phobos orbits Mars, it is subjected to a torque that gives rise to an oscillation about the line connecting the center of Mars with the center of Phobos. This oscillation is referred to as libration and has an amplitude of about one degree. The source of the torque is the variation or gradient of Mars gravity across Phobos. The part of Phobos closer to Mars is accelerated toward Mars more than the part further away. This gradient of the gravity field is what causes tides on Earth and forces the dark side of the Moon to always face away from the Earth whether it is illuminated by the sun or not. The Moon also librates. It was shown on previous missions that a small departure from principal axis rotation or what is called free precession can make determination of the gravity field difficult. An unknown oscillation of a few degrees can cause the orbit determination filter to fail to converge. This is a problem that could complicate navigation in the vicinity of binary asteroids or comets.

A number of gravity models are used for trajectory design, scientific investigations and navigation of spacecraft. Some gravity models are approximate and others are exact. Approximate models are adequate for trajectory design and some scientific investigations, but high precision models are required for navigation, particularly for orbit determination. The most demanding gravity model requirements are for orbit determination around large irregularly-shaped bodies. A harmonic expansion of Legendre polynomials and associated functions, while satisfactory for spherically shaped planets, does not work well for large irregularly-shaped bodies such as comets or asteroids. Inside the sphere of maximum radius, the harmonic expansion diverges. Even outside the sphere of maximum radius, the harmonic expansion requires a high degree and order to obtain accurate orbit determination solutions. In this paper, several gravity models are analyzed and compared to determine their suitability for navigation. Accuracy, number of parameters that need to be determined by the orbit determination filter and speed of computation that must include acceleration and variational partial derivatives are factors that must be considered. Another important factor is the ability to change the computation speed as a function of the required accuracy. For a harmonic expansion, this may be accomplished by changing the degree and order of the expansion. Changing the degree and order introduces an abrupt jerk to the trajectory, which probably would not affect trajectory design, but could seriously confuse the orbit determination filter. Gravity models that use point masses or mascons also have singularities that are a problem for orbit determination. Gravity models that involve numerical integration can be designed to make a seamless transition using error control.

Navigation of spacecraft requires science and mathematics equations to be programmed onto digital computers. For deep space navigation, the science content is about ten percent and the mathematics content is about 90 percent. Science is here defined as any mathematical expression that is observed and cannot be proved. We start with these science mathematical expressions and other mathematical expressions that are accepted as true by inspection and may be regarded as axiomatic. The mathematics presented here involve manipulation of the given mathematical expressions until we obtain a result that is useful. The equations of motion are a simple example. The resultant derivation is regarded as a proof if the given mathematics are generally accepted as true. In this paper, a number of derivations are described that are representative of the mathematics that have been used for navigation of spacecraft in deep space, beyond the orbit of the Moon. The selected derivations are by no means complete but emphasize those incorporated in computer algorithms and differ from the conventional mathematics used for obtaining analytic solutions.

The kinetic theory of gasses is used in deep space navigation for modeling small forces acting on a spacecraft. Acceleration of a spacecraft occurs as a result of the energy and momentum transfer from molecules leaving the spacecraft through evaporation or gas leaks or from molecules impacting the spacecraft from an external source such as a cometary atmosphere. In performing analysis of these events, a model of molecules in a closed container was developed. This model revealed a probability distribution of molecular velocity magnitudes that differs from the Maxwell-Boltzmann distribution. At first it was assumed that the model is not correct. However, further analysis has not revealed an error source. The difference is small but can be detected by direct measurement. A definitive measurement performed in 1955 confirms a small error which was attributed to the apparatus. The error is consistent with the computer model developed for navigation and seems to confirm that the Maxwell-Boltzmann theory is not correct. Over the years, considerable evidence has been developed that supports the computer model. However, there has been no independent confirmation. Furthermore, it seems to upset settled science and questions the application of entropy to molecules in a container. The absence of a mathematical proof hinders acceptance of the computer model, but has not hindered its application to deep space navigation. Finally, a mathematical proof has been developed and is the subject of this paper. The complete probability distribution is not described as a closed form mathematical equation. The proof only applies to a single point where the error is greatest. However, the approach developed here may lead to a new mathematical function to replace the Maxwell-Boltzmann distribution.