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Thermal dissipation force modeling with preliminary results ... - ZARM

ARTICLE IN PRESS2 B. Rievers et al. / Acta Astronautica ( ) –NomenclatureFE finite elementRTG radio isotopic thermal generatorPA pioneer anomalyF abs absorption **force** componentF emis emission **force** componentF ref reflection **force** componenti number of currently active elementj number of currently receiving elementnP emisαβγεσtotal number of surface elementsemitted powerabsorptivityelevation anglereflectivityemissivityazimuth angleStefan–Boltzmann constantdistribution for the processed thermal state of the spacecraft.In a postprocessing the temperatures of the outer elementsurfaces, the nodal positions in the cartesian frame, the materialparameters and the node/element assignment list areexported into text files that can be read in by the **force** computationalgorithm. Using the information stored in the textfiles as database the total **force** resulting from thermal **dissipation**is computed by the algorithm including reflectionand absorption between the different surfaces of the model.In comparison to the existing models [10–12] for the calculationof thermal perturbations two main differences can beidentified:1. The calculation is based on a full thermal finite elementanalysis.2. In the proposed method each surface in the model is consideredas a Lambertian radiation source.The FE analysis enables an improved geometrical accuracyand the inclusion of measurement and sensor data as boundariesfor the FE solution. Compared to other methods (analyticalapproach [12] or nodal models for the computation ofthe equilibrium state [10,11]) this approach enables a greaterlevel of detail and improved solution accuracy. Furthermorethe creation of the model can be realised using standard FEpreprocessors and solvers (e.g. ANSYS) and parameter sensitivityanalysis can be realised easily using macro-based FEeditors (e.g. APDL).The consideration of each model surface as Lambertianradiator leads to an increase in computation time but alsoto an increase in numerical accuracy. Other methods [10,11]consider only the resulting component normal to the emittingsurface for the computation of the thermal recoil **force**.In difference to this the method proposed in this paper uses ahemispherical pattern of a large number of angularly spacedrays to compute absorption and reflection effects thus alsoincluding interaction of surfaces which are not directly facingeach other. The resulting heat fluxes are computed bymeans of view factors for each detected hit. Furthermore,the number of considered reflections can be set freely usingboth specularly and diffuse reflection models.2. Theoretical backgroundFor an emitting grey body that obeys Lamberts law, theintensity distribution in polar direction can be computedfrom the intensity in normal direction [5]I = LA cos β = I n cos β. (1)From Stefan–Boltzmanns law we knowP tot = εAσT 4 . (2)The integration of the intensity over the complete hemisphere(determined by the angles and β **with** 0 2πand 0 β π/2) leads to the total normal component of thepower output∫ π/2 ∫ 2πP ⊥ = I n cos β 2 sin β dβ d = 2 3 P tot. (3)0 0The resulting recoil **force** generated by a radiating body canbe expressed **with** P tot and the speed of light c as [8]F ⊥ = P ⊥c . (4)3. Analytical test modelFor a later comparison between the proposed FE methodand **results** acquired **with** simple node models an analyticaltest case is formulated. Here the radio isotope thermal generators(RTGs) are modeled as pointlike isotropic radiationsources **with** a total power of 1200 W each of which correspondsroughly to the available power at the start of the Pioneer11 mission. The high gain antenna dish is modeled **with**a specified number of nodes distributed evenly according tothe real geometrical shape of the antenna. Fig. 1 shows theconfiguration of the test case.The **force** resulting from an emission in a specific viewingdirection x isF emis =−x P emis. (5)cFor the isotropic radiation pattern assumed for the pointlikeRTGs no resulting **force** **results**. Taking into account the interaction**with** the high gain antenna dish, a fraction of theradiation emitted from the RTGs is absorbed and reflectedat the antenna surface thus leading to a **force** ofF res =−F emis (6)resulting from the asymmetry in the emission pattern of theRTGs.The radiation exchange between antenna nodes and RTGsis computed **with** view factors under the assumption that theRTG power is emitted isotropically. The view factors betweenPlease cite this article as: B. Rievers, et al., **Thermal** **dissipation** **force** **modeling** **with** **preliminary** **results** for Pioneer 10/11, ActaAstronautica (2009), doi:10.1016/j.actaastro.2009.06.009

ARTICLE IN PRESSB. Rievers et al. / Acta Astronautica ( ) – 3Fig. 1. Testcase 1a: configuration **with** main antenna dish and RTG assemblies.antenna nodes and RTGs can then be computed based onthe isotropic luminosity L Iso = P/4πr 2 to i,j = 14πA i∫A i∫A jcos(Θ j )r(i, j) 2 dA j dA i , (7)where i denotes the radiation sources (RTGs) and j denotesthe radiation sinks (antenna nodes). Due to the assumptiongiven above, the cosine for the source term cancels out. Forthe sink (antenna nodes) the orientation and the area associated**with** the node has to be known. The antenna shapecan be characterised **with** the parametrisation√F(x, y, z) = r − g(z) = x 2 + y 2 − αz β = 0 (8)**with**x = r cos , y = r sin , r =√x 2 + y 2 , (9)where the geometrical coefficients α and β are derived fromavailable technical drawings of the Pioneer 11 high gain antennaby a least squares estimate to α=2.047, β=0.512. Thenormal direction of any node x 0 on the antenna surface canbe computed **with** ∇F(x, y, z) atx 0 .(∇F(x, y, z) =x√x 2 + y 2 ;)y√x 2 + y ; 2 αβzβ−1 . (10)The total surface of the high gain antenna can be computedby rotating the function g(z) **with**∫ b √I(g(z)) = 2π g(z) 1 + g ′2 (z) dz (11)ato A HGA = 6.535 m 2 . This area is distributed evenly amongstall nodes assigned to the antenna. For the computation ofthe angle cos(Θ j ) the node connection vectors between RTGand antenna nodes are computed **with**r i,j = x j − x i (12)and the cosine term iscos(Θ j ) =‖∇F(x, y, z)| xj ‖·‖r i,j ‖. (13)Using the equations and values introduced above the net**force** fraction resulting from interaction **with** a specific antennanode can be determined **with**F =‖r i,j ‖P emis k, (14)where k is the effective optical parameter that defines thereflection behaviour of antenna nodes ranging from 1 (totalabsorption) to 2 (ideal reflection). The z-component of the**force** vector is aligned **with** the flight direction and correspondsto the axis of the observed anomalous acceleration.The total resulting **force** can be computed from the sum ofall individual antenna nodal **force**sF res,z =n∑F i,z . (15)iThe analytical test case has been processed using the equationsgiven above by varying the number of antenna nodesin the model. For the mass of the spacecraft the Pioneer drymass of 233 kg [2] is considered, the optical constant k is setto 1.8 which corresponds to a reflection coefficient of thecoated antenna surface of 0.8. Fig. 2 shows the resulting total**force**s for a different number of modeled antenna nodes.As can be seen in the graph, the computation errors arelarge for a low number of antenna nodes and decrease **with**more nodes in the model. This effect is easily understandablebecause a higher number of nodes represent the real antennageometry better and also imply a higher number of surfacedifferent orientations thus reducing the misalignment errorsof each individual nodal surface. The solution converges **with**∼ 400 nodes to a final value of a res =2.263×10 −10 m/s 2 . Lookingat the observed anomalous acceleration of the Pioneerspacecraft of a Pio = 8.74 × 10 −10 m/s 2 [6,7,9,14] this numbercorresponds to about 25.8 percent of the observed anomalyagainst flight direction. This is in good agreement **with** otherestimation methods which predict a resulting RTG disturbanceacceleration against flight direction of ∼ 20 percentPA [15].The **results** points out that a more thorough analysis ofthe thermal recoil **force** **with** respect to the Pioneer anomalyis necessary. In the presented calculation only the RTGs havebeen considered as heat sources, all other surfaces in themodel can only absorb and reflect radiation. In a more realisticmodel all heat sources aboard, including payloads, heaterunits, radiators and louver system will have to be included.As absorbing surfaces only the high gain antenna (which issupposed to receive the major part of the RTG radiation hittingthe craft) has been modeled. The inclusion of equipmentsection and experiment section and the external payloadswill of course change the resulting disturbance acceleration.The **modeling** of the RTGs as pointlike isotropic radiationPlease cite this article as: B. Rievers, et al., **Thermal** **dissipation** **force** **modeling** **with** **preliminary** **results** for Pioneer 10/11, ActaAstronautica (2009), doi:10.1016/j.actaastro.2009.06.009

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