Nonetheless, in 2002, having only limited thermal data and relevant spacecraft information at hand, the authors of [27] estimated the contribution of the thermal recoil force as (the estimate is based on the values reported for the RTG heat reflected off the craft and nonisotropic radiative cooling of the spacecraft, items 2b) and 2d) in Table 5.2 correspondingly), i.e., only about 6% of the anomalous acceleration.

Since 2002, it has become clear that this figure is in need of a revision. A quantitative estimate was first offered by Scheffer in 2003 [327], who calculated a total of 52 W of directed thermal radiation (not including his estimate on asymmetrical radiation from the RTGs), or about 80% of the thrust required to account fully for the anomalous acceleration. In 2007, Toth [376] presented an argument for expressing the combined recoil force due to electrical and RTG heat as a linear combination of the RTG thermal power and electrical power on board; his coefficients (0.012 for the RTGs, 0.36 for electrical heat) yield a figure of 55 W of directed thermal radiation in the mid-1980s, which, after accounting for the heat from the RHUs (4 W) and the antenna beam (–7 W) as Scheffer did, translates into a result that is similar to Scheffer’s.

Benefiting from the extensive discussions of the topic of thermal
recoil force during the meetings of the Pioneer Explorer Collaboration at
ISSI^{32} [171, 172, 400, 328, 329, 376],
several researchers tried to model effects of this force on a Pioneer-like spacecraft using various computer
tools. In 2008, Bertolami et al. [51] made an attempt to develop a methodology using point-like Lambertian
sources to estimate the thermal recoil force on the Pioneer spacecraft, and obtained an acceleration estimate
that corresponds to 67% of the Pioneer anomaly, or 45 W of directed thermal radiation.
In addition, in 2009, Rievers et al. [311, 312] used finite element modeling and ray tracing
algorithms to compute an acceleration that corresponds to 48 W of directed heat in the
mid-1980s.

These recent results support claims that the contribution of the thermal recoil forces to the Pioneer anomaly was previously (e.g., [27]) underestimated. However, the estimates above used only rough values for on-board thermal and electrical power and are valid only for a particular point in Pioneers’ missions. As such, these estimates can provide only an overall magnitude of the effect and say little on its temporal behavior. On the other hand, the now available telemetry and recovered spacecraft design documentation [397] makes it possible to develop a comprehensive thermal model capable of estimating thermal recoil force throughout the entire mission of both Pioneers. Interim results show that this model is in good agreement with redundant telemetry observations [170, 171, 172]. Therefore, the development of a comprehensive, reliable estimate of the thermal recoil force is now within reach.

Below, we discuss the basic principles of modeling the thermal recoil force, as well as the application of these principles to the case of the Pioneer 10 and 11 spacecraft.

While heat transfer textbooks provide all necessary details about thermal radiation as a mechanism for energy transfer, momentum transfer is rarely covered in any detail. This is perhaps not surprising: the momentum of a photon with energy is , and thus, the recoil force associated with a collimated beam of electromagnetic radiation with power is . For each watt of radiated power, the corresponding recoil force is only 3.33 nN. Such tiny forces rarely, if ever, need to be taken into account in terrestrial applications. This is not so in the case of space applications [35, 108, 181, 318, 405, 427], in particular in the case of Pioneer 10 and 11: the observed anomalous acceleration corresponds to a force of less than 220 nN, which can be produced easily by a modest amount ( 65 W) of electromagnetic radiation.

A formal treatment of the thermal recoil force must establish a relationship between heat sources within the radiating object and the electromagnetic field outside the object [380]. This can be accomplished in stages, first by describing heat conduction inside the object using Fourier’s law [165, 180, 209]:

where is the heat flux (measured in units of power over area), is the temperature, and is the heat conduction coefficient, a tensorial quantity in the general case, but just a number for homogeneous and isotropic materials. Heat flux also obeys the energy conservation equation where is the volumetric heat release (measured in units of power density), is the material’s specific heat, and is its density. For discrete sources, , where is the thermal power of the -th source, is its location, and denotes Dirac’s delta function.At a radiating surface,

where is the unit normal of the radiating surface element, and is the surface element’s radiant intensity. The radiant intensity, or energy flux, of a radiating surface is related to its temperature by the Stefan–Boltzmann law: where is the Stefan–Boltzmann constant, while the dimensionless coefficient is a physical characteristic of the emitting surface. This coefficient can vary not only as a function of location and time, but also as a function of temperature. Equation (7.4) can be used to calculate the radiative coupling between facing surfaces of the object, and between the object and its environment (e.g., the deep sky, which can be modeled as a blackbody with temperature ). Together with internal boundary conditions (i.e., the power and distribution of internal heat sources, represented by above) the problem becomes fully solvable: values of inside the object, and on its boundary surface can be computed [380].Outside the radiating object, the electromagnetic field is described by the stress-energy-momentum tensor

where is the energy density of the radiation field, is its momentum density, and is the radiation pressure tensor. The stress-energy-momentum tensor obeys the conservation equation , where is the covariant derivative with respect to the coordinate . This allows one to develop an expression for the recoil force acting on a radiating surface in the form which, for an isotropic (Lambertian) emitter yields the well-known law If the geometry of the emitter’s exterior (represented by ) and the radiant intensity along the exterior surface are known, the recoil force can be computed [380].The Pioneer 10 and 11 spacecraft have two heat sources that contribute significant amounts to the thermal recoil force - the RTGs and the on-board electrical equipment. The case is further simplified by the fact that the Pioneer 10 and 11 spacecraft are spinning, as it allows a force computation in only one dimension (see Section 4.4).

Although each of the spacecraft has four RTGs, their temporal behavior is identical (characterized by
the half-life of the ^{238}Pu fuel and the decaying efficiency of the thermocouples). The placement of the RTGs
is symmetrical. Consequently, the set of four RTGs can be treated as a single heat source, the power of
which we denote as . The amount of ^{238}Pu fuel on board is well known from pre-launch test data,
and the physics of the fuel’s radioactive decay is well understood [378, 379, 397]; therefore, the total power
of the RTGs, is known:

The amount of power removed from the RTGs in the form of electrical power is measured directly by telemetry and is available for the entire mission durations. Therefore, is given as:

The value of can be computed with good accuracy (Figure 7.9). The initial power per RTG was reported with a measurement accuracy of 1 W for each RTG; the month, though not the exact date, of is known. Therefore, can be calculated with an accuracy of 0.2% or better. Currents and voltages from each RTG are found in the flight telemetry data stream, represented by 6-bit values. The combined error due mainly to the limited resolution of this data set amounts to an uncertainty of 1 W per RTG. When all these independent error sources are combined, the resulting figure is an uncertainty of ; the total power of the four RTGs combined is 2600 W at the beginning of the Pioneer 10 and 11 missions [380].

Electrical power on board the Pioneer spacecraft was 160 W at the beginning of mission [379], slowly decreasing to 60 W at the time when the last transmission was received from Pioneer 10. Some of this power was radiated into space directly by a shunt radiator plate, some power was consumed by instruments mounted external to the spacecraft body, and some power was radiated away in the form of radio waves; however, most electrical power was converted into heat inside the spacecraft body. This heat escaped the spacecraft body through three possible routes: a passive thermal louver system, other leaks and openings, and the spacecraft walls that were covered by multilayer thermal insulation blankets. While the distribution of heat sources inside the spacecraft body was highly inhomogeneous, there is little temporal variation in the distribution of heat inside the spacecraft body, and the exterior temperatures of the multilayer insulation remain linear functions of the total electrical heat. This allows us to treat all electrical heat generated inside the spacecraft body as another single heat source:

Therefore, the value of is available from telemetry (Figure 7.9). Uncertainties in the estimate of are due to several factors. First, telemetry is again limited in resolution to 6-bit data words. Second, the power consumption of specific instruments is not known from telemetry, only their nominal power consumption values are known from documentation. Third, there are uncertainties due to insufficient documentation. When these sources of error are combined, the result is an uncertainty of in the electrical heat output of the spacecraft body [380].

In addition to heat from the RTGs and electrically generated heat, there are other mechanisms producing
heat on board the Pioneer spacecraft. First, there are 11 radioisotope heater units (RHUs) on
board, each of which generated 1 W of heat at launch, using ^{238}Pu fuel with a half-life of
87.74 years. Second, the propulsion system, when used, can also generate substantial amounts of
heat.

Nonetheless, these heat sources can be ignored. The total amount of heat generated by the RHUs is not only small, most of the RHUs themselves are mounted near the edge of the high-gain antenna (see Figure 2.8), and a significant proportion of their heat is expected to be emitted in a direction perpendicular to the spin axis. As to the propulsion system, while it can generate substantial quantities of heat, these events are transient in nature and are completely masked by uncertainties in the maneuvers themselves, which are responsible for this heat generation. These arguments can lead to the conclusion that insofar as the anomalous acceleration is concerned, only the heat from the RTGs and electrical equipment contribute noticeably.

As discussed in Section 7.4.1 above, knowledge of the physical properties (thermal properties and geometry) of the spacecraft and its internal heat sources is sufficient to compute heat, and thus momentum, transfer between the spacecraft and the sky. This can be accomplished using direct calculational methods, such as industry standard finite element thermal-mechanical modeling software. The availability of redundant telemetry (in particular, the simultaneous availability of electrical and thermal measurements) makes it possible to develop a more robust thermal model and also establish reasonable limits on its accuracy. Such a model has recently been developed at JPL [169] and is yielding valuable results. The results of this analysis will be published when available (see also Figures 7.7 and 7.8).

It is also possible to conduct a simplified analysis of the Pioneer spacecraft. First, taking into consideration the spacecraft’s spin means that the thermal recoil force only needs to be calculated in the spin axis direction (see Section 4.4). Second, it has been argued in [380] that for these spacecraft, the total recoil force can be accurately modeled as a quantity that is proportional to some linear combination of the thermal power of the two dominant heat sources, the RTGs and electrical equipment. Thus, one may write

where and are efficiency factors associated with RTG thermal power and electrical power , while is a unit vector in the spin axis direction.The factors and can be computed, in principle, from the geometry and thermal properties of the spacecraft. However, there also exists another possible approach [380]: after incorporating the force model Equation (7.11) into the orbital equations of motion, orbit determination software can solve for these parameters, fitting their values, along with the spacecraft’s initial state vector, maneuvers, and other parameters, to radiometric Doppler observations [380]. While this approach seems promising, its success depends on the extent to which orbit determination code can disentangle the thermal recoil force, solar pressure, and a possible anomalous contribution from one another based on radiometric Doppler data alone.

A further complication arises from the fact that although late in their mission, the physical and thermal configuration of the Pioneer spacecraft were constant in time, this was not always the case. Earlier in their mission, when the spacecraft were closer to the Sun, their internal temperatures were regulated by a thermal louver system located on the aft side (i.e., the side opposite the high-gain antenna; see Figure 2.10). When these louvers were partially open (see Figure 2.11), the effective thermal emissivity of the aft side was significantly higher, and varied as a function of internal temperatures (see Figures 2.12 and 2.13). While this effect is difficult to model analytically, it can be incorporated into a finite element thermal model accurately.

These recent studies and on-going investigations made it clear that the figure published in 2002 [27] is likely an underestimation of the thermal recoil force; far from being insignificant, the thermal recoil force represents a substantial fraction of the force required to generate the anomalous acceleration seen in Pioneer data, and may, in fact, account for all of it. This possibility clearly demands a thorough, in-depth analysis of the thermal environment on-board the Pioneers. Meanwhile, all the needed thermal and power data exist in the form of on-board telemetry [378, 379, 397]. All the tools needed to analyze resulted thermal recoil forces are now built and tested [169, 171, 172, 311, 312, 377, 380, 395]. The analysis now approaches its most exciting part.

If the anomaly, even in part, is of thermal origin, its magnitude must decrease with time as the on-board fuel inventory decreases (see Figure 7.9, for example). Therefore, a thermal model will necessarily predict a decreasing trend in the anomaly. To what extent will this trend contradict the previously reported “constancy” of the effect? Or will it support trends already seen as the jerk terms reported by independent verifications (Section 7.1)? To that extent we emphasize that the primary data set for the new investigation of the Pioneer anomalous acceleration is the much extended set of radiometric Doppler tracking data available for both spacecraft (Section 3.3). It is clear that if the anomaly was found in the navigational data, it must be re-evaluated using data of the same nature. This is why the new set of Doppler data that recently became available, in conjunction with the newly built tools to evaluate thermal recoil forces discussed above, is now being used to evaluate the long-term temporal behavior, direction and other important properties of the Pioneer anomaly (Section 7.2).

Finally, after a period of tedious preparatory work conducted during 2002 – 2009, the study of the Pioneer anomalous acceleration enters its final stages; the results of this work will be reported.

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