In the presentation below of the standard modeling of small nongravitational forces, we generally follow the discussion in [27], starting with nongravitational forces that originate from sources external to the spacecraft, and followed by a review of forces of on-board origin. We also discuss effects acting on the radio signal sent to, or received from, the spacecraft.

Most notable among the sources for the forces external to the Pioneer spacecraft is the solar pressure. This force is a result of the exchange of momentum between solar photons and the spacecraft, as solar photons are absorbed or reflected by the spacecraft. This force can be significant in magnitude in the vicinity of the Earth, at 1 AU from the Sun, especially when considering spacecraft with a large surface area, such as those with large solar panels or antennas. For this reason, solar pressure models are usually developed before a spacecraft is launched. These models take into account the effective surface areas of the portions of the spacecraft exposed to sunlight, and their thermal and optical properties. These models offer a computation of the acceleration of the spacecraft due to solar pressure as a function of solar distance and spacecraft orientation.

The simplest way of modeling solar pressure is by using a “flat plate” model. In this case, the spacecraft is treated as a flat surface, oriented at same angle with respect to incoming solar rays. The surface absorbs some solar heat, while it reflects the rest; reflection can be specular or diffuse. A flat plate model is fully characterized by three numbers: the area of the plate, its specular and its diffuse reflectivities. This model is particularly applicable to Pioneer 10 and 11 throughout most of their mission, as the spacecraft were oriented such that their large parabolic dish antennas were aimed only a few degrees away from the Sun, and most of the spacecraft body was behind the antenna, not exposed to sunlight.

In the case of a flat plate model, the force produced by the solar pressure can be described using a combination of several force vectors. One vector, the direction of which coincides with the direction of incoming solar radiation, represents the force due to photons from solar radiation intercepted by the spacecraft. The magnitude of this vector is proportional to the solar constant at the spacecraft’s distance from the Sun, multiplied by the projected area of the flat plate surface:

where is the Sun-spacecraft vector, is the unit vector in this direction with , is the effective area of the spacecraft (i.e., flat plate), is a unit normal vector to the flat plate, is the solar radiation constant at 1 AU from the Sun and is the speed of light. The standard value of the solar radiation constant is when is measured in units of m and , in units of AU. According to Equation (4.8), approximately 65 W of intercepted sunlight can produce a force comparable in magnitude to that of the Pioneer anomaly; in contrast, in the vicinity of the Earth, the Pioneer 10 and 11 spacecraft intercepted 7 kW of sunlight, indicating that solar pressure is truly significant, even as far away from the Sun as Saturn, for precision orbit determination.Equation (4.8) reflects the amount of momentum carried by solar photons that are intercepted by the spacecraft body. However, one must also account for the amount of momentum carried away by photons that are reflected or re-emitted by the spacecraft body. These momenta depend on the material properties of the spacecraft exterior surfaces. The absorptance coefficient determines the amount of sunlight absorbed (i.e., not reflected) by spacecraft materials. The emittance coefficient determines the efficience with which the spacecraft radiates (absorbed or internally generated) heat relative to an idealized black body. Finally, the specularity coefficient determines the direction in which sunlight is reflected: a fully specular surface reflects sunlight like a mirror, whereas a diffuse (Lambertian) surface reflects light in the direction of its normal. Together, these coefficients can be used in conjunction with basic vector algebra to calculate the force acting on the spacecraft due to specular reflection:

and the force due to diffuse reflection:Lastly, the force due to solar heating (i.e., re-emission of absorbed solar heat) can be computed in conjunction with the recoil force due to internally generated heat, which is discussed later in this section.

The solar wind is a stream of charged particles, primarily protons and electrons with energies of 1 keV, ejected from the upper atmosphere of the Sun. Solar wind particles intercepted by a spacecraft transfer their momentum to the spacecraft. The acceleration caused by the solar wind has the same form as Equation (4.8), with replaced by , where is the proton density at 1 AU and is the speed of the wind (electrons in the solar wind travel faster, but due to their smaller mass, their momenta are much smaller than the momenta of the protons). Thus,

Because the density can change by as much as 100%, the exact acceleration is unpredictable. Nonetheless, as confirmed by actual measurement

When a spacecraft is in the vicinity of a planetary body, it interacts with that body in a variety of ways. In addition to the planet’s gravity, the spacecraft may be subjected to radiation pressure from the planet, be slowed by drag in the planet’s extended atmosphere, and it may interact with the planet’s magnetosphere.

For instance, for Earth orbiting satellites, the Earth’s optical albedo of [235]:

yields typical albedo accelerations of 10 – 35% of the acceleration due to solar radiation pressure. On the other hand, when the spacecraft is in the planetary shadow, it does not receive direct sunlight.Atmospheric drag can be modeled as follows [235]:

where is the spacecraft’s position, its velocity, is its cross-sectional area, its mass, is the atmospheric density, and the coefficient has typical values between 1.5 and 3.The Lorentz force acting on a charged object with charge traveling through a magnetic field with field strength at a velocity is given by

Considering the velocity of the Pioneer spacecraft relative to a planetary magnetic field during a planetary encounter (up to 60 km/s during Pioneer 11’s encounter with Jupiter) and a strong planetary magnetic field (up to 1 mT for Jupiter near the poles), if the spacecraft carries a net electric charge, the resulting force can be significant: up to 60 N per Coulomb of charge. In actuality, the maximum measured magnetic field by the two Pioneers at Jupiter was 113.5 T [269]. An upper bound of 0.1 C exists for any positive charge carried by the spacecraft [269], but a possible negative charge cannot be excluded [269] and a negative charge as high as 10The long-term accelerations of Pioneer 10 and 11, however, remain unaffected by planetary effects, due to the fact that except for brief encounters with Jupiter and Saturn, the two spacecraft traveled in deep space, far from any planetary bodies.

The interplanetary magnetic field strength is less than 1 nT [27]. Considering a spacecraft velocity of
10^{4} m/s and a charge of 10^{–4} C, Equation (4.14) gives a force of 10^{–9} N or less, with a corresponding
acceleration (assuming a spacecraft mass of 250 kg) of or less. This value is two
orders of magnitude smaller than the anomalous Pioneer acceleration of
(see Section 5.6).

While there have been attempts to explain the anomalous acceleration as a result of a drag force induced by exotic forms of matter (see Section 6.2), no known form of matter (e.g., gas, dust particles) in interplanetary space produces a drag force of significance.

The drag force on a sail was estimated as [261]:

where is the spacecraft’s velocity relative to the interplanetary medium, its cross sectional area, is the density of the interplanetary medium, and is a dimensionless coefficient that characterizes the absorptance, emittance, and transmittance of the spacecraft with respect to the interplanetary medium. Using the Pioneer spacecraft’s 2.74 m high-gain antenna as a sail and an approximate velocity of
10^{–4} m/s relative to the interplanetary medium, and assuming to be of order unity, we can estimate a
drag force of

The density of dust of interstellar origin has been measured by the Ulysses probe [189, 262] at . The average interplanetary dust density, which also contains orbiting dust, is believed to be almost two orders of magnitude higher according to the consensus view [262]. However, higher dust densities are conceivable.

On the other hand, if one presumes a model density, the constancy of the observed anomalous acceleration of the Pioneer spacecraft puts upper limits on the dust density. For instance, an isothermal density model yields the limit .

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