7.4 The thermal recoil force

It has been long recognized [24Jump To The Next Citation Point27Jump To The Next Citation Point397Jump To The Next Citation Point] that anisotropically emitted thermal radiation can contribute to the acceleration of the Pioneer 10 and 11 spacecraft, and that the available thermal inventory on board (in excess of 2 kW), if directed, is more than sufficient to provide the necessary acceleration (requiring only ∼ 65 W of collimated electromagnetic radiation.)

Nonetheless, in 2002, having only limited thermal data and relevant spacecraft information at hand, the authors of [27Jump To The Next Citation Point] estimated the contribution of the thermal recoil force as −10 2 (− 0.55 ± 0.73) × 10 m ∕s (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 [376Jump To The Next Citation Point] 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 ISSI32 [171Jump To The Next Citation Point172Jump To The Next Citation Point400Jump To The Next Citation Point328Jump To The Next Citation Point329Jump To The Next Citation Point376Jump To The Next Citation Point], 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. [311Jump To The Next Citation Point312Jump To The Next Citation Point] 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., [27Jump To The Next Citation Point]) 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 [397Jump To The Next Citation Point] 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 [170171Jump To The Next Citation Point172Jump To The Next Citation Point]. 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.

7.4.1 General formalism

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 E is p = E ∕c, and thus, the recoil force associated with a collimated beam of electromagnetic radiation with power P is F = P ∕c. 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 [35108181318405427], 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 [380Jump To The Next Citation Point]. This can be accomplished in stages, first by describing heat conduction inside the object using Fourier’s law [165180209]:

q = − k ∇T, (7.1 )
where q is the heat flux (measured in units of power over area), T is the temperature, and k 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
∂T ∇ ⋅ q = b − Chρ---, (7.2 ) ∂t
where b is the volumetric heat release (measured in units of power density), Ch is the material’s specific heat, and ρ is its density. For discrete sources, ∑ b(x,t) = ni=1Bi (t)δ3(x − xi), where Bi is the thermal power of the i-th source, xi is its location, and δ denotes Dirac’s delta function.

At a radiating surface,

q = q ⋅ a, (7.3 )
where a is the unit normal of the radiating surface element, and q 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:
4 q(x, t) = σ 𝜖(x,t,T)T (x,t), (7.4 )
where σ ≃ 5.67 × 10−8 Wm −2 K −4 is the Stefan–Boltzmann constant, while the dimensionless coefficient 0 ≤ 𝜖 ≤ 1 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 (7.4View Equation) 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 Tsky ≃ 2.7 K). Together with internal boundary conditions (i.e., the power and distribution of internal heat sources, represented by b above) the problem becomes fully solvable: values of q inside the object, and q on its boundary surface can be computed [380Jump To The Next Citation Point].

Outside the radiating object, the electromagnetic field is described by the stress-energy-momentum tensor

( ) μν c−2u 𝖕 T = 𝖕 ℙ , (7.5 )
where u is the energy density of the radiation field, 𝖕 is its momentum density, and ℙ is the radiation pressure tensor. The stress-energy-momentum tensor obeys the conservation equation ∇ μTμν = 0, where ∇ μ is the covariant derivative with respect to the coordinate x μ. This allows one to develop an expression for the recoil force F acting on a radiating surface A in the form
∫ F(t) = − ℙ(x,t) ⋅ dA, (7.6 )
which, for an isotropic (Lambertian) emitter yields the well-known law
∫ 2 1 F (t) = − ---- q(x, t) dA. (7.7 ) 3 c
If the geometry of the emitter’s exterior (represented by A) and the radiant intensity q along the exterior surface are known, the recoil force can be computed [380Jump To The Next Citation Point].

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).

7.4.2 Contribution of the heat from the RTGs

View Image

Figure 7.9: Heat generated by RTGs (red, approximately straight line, scale on left) and electrical equipment (green, scale on right) in Pioneer 10 over the lifetime of the spacecraft.

Although each of the spacecraft has four RTGs, their temporal behavior is identical (characterized by the half-life of the 238Pu 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 Brtg(t). The amount of 238Pu fuel on board is well known from pre-launch test data, and the physics of the fuel’s radioactive decay is well understood [378379397Jump To The Next Citation Point]; therefore, the total power of the RTGs, Prtg(t) is known:

Prtg(t) = 2−(t−t0)∕TPrtg(t0), (7.8 )
where t0 is the time when the power Prtg(t0) was measured, and T ≃ 87.74 years is the half-life of the radioactive fuel.

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

Brtg(t) = Prtg(t) − Pelec(t). (7.9 )

The value of Brtg(t) can be computed with good accuracy (Figure 7.9View Image). The initial power Prtg(t0) ≃ 650 W per RTG was reported with a measurement accuracy of 1 W for each RTG; the month, though not the exact date, of t0 is known. Therefore, Prtg(t) 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 σ = 2.1 W rtg; the total power of the four RTGs combined is ∼ 2600 W at the beginning of the Pioneer 10 and 11 missions [380Jump To The Next Citation Point].

7.4.3 Effects of the electrical heat

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:

Belec(t) = known from telemetry. (7.10 )

Therefore, the value of Belec(t) is available from telemetry (Figure 7.9View Image). Uncertainties in the estimate of Belec 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 σelec = 1.8 W in the electrical heat output of the spacecraft body [380Jump To The Next Citation Point].

7.4.4 The thermal recoil force

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 238Pu 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 2.8View Image), 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 [169Jump To The Next Citation Point] and is yielding valuable results. The results of this analysis will be published when available (see also Figures 7.7View Image and 7.8View Image).

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 [380Jump To The Next Citation Point] 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

1- F = c(ξrtgBrtg + ξelecBelec)s, (7.11 )
where ξrtg and ξelec are efficiency factors associated with RTG thermal power Brtg and electrical power Belec, while s is a unit vector in the spin axis direction.

The factors ξrtg and ξelec can be computed, in principle, from the geometry and thermal properties of the spacecraft. However, there also exists another possible approach [380Jump To The Next Citation Point]: after incorporating the force model Equation (7.11View Equation) 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 [380Jump To The Next Citation Point]. 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 2.10View Image). When these louvers were partially open (see Figure 2.11View Image), the effective thermal emissivity of the aft side was significantly higher, and varied as a function of internal temperatures (see Figures 2.12View Image and 2.13View Image). 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 [27Jump To The Next Citation Point] 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 [378379397]. All the tools needed to analyze resulted thermal recoil forces are now built and tested [169171Jump To The Next Citation Point172Jump To The Next Citation Point311Jump To The Next Citation Point312Jump To The Next Citation Point377Jump To The Next Citation Point380395]. 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 7.9View Image, 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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