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Thermal dissipation force modeling with preliminary results for
Pioneer 10/11
Benny Rieversa, ∗ , Stefanie Bremera , Meike Lista , Claus Lämmerzahla , Hansjörg Dittusb
a
b
Center of Applied Space Technology and Microgravity, University of Bremen, Am Fallturm, D-28359 Bremen, Germany
Institute of Space Systems, German Aerospace Center DLR, Robert-Hooke-Strae 7, D-28359 Bremen, Germany
A R T I C L E
I N F O
Article history:
Received 10 February 2009
Received in revised form
27 May 2009
Accepted 18 June 2009
Keywords:
Thermal recoil force
Thermal modeling
Finite elements
Disturbance modeling
Pioneer anomaly
A B S T R A C T
The dissipation of thermal energy can produce disturbance forces on spacecraft surfaces
if the energy is not dissipated in a symmetric pattern. This force can be computed as the
quotient of the radiated power and the speed of light for a plate surface element. Depending on mission and spacecraft design the resulting surface forces have to be included into
the disturbance budget. At ZARM (Center of Applied Space Technology and Microgravity)
a raytracing algorithm was developed that allows the computation of the resulting force
for complex spacecraft geometries. The method is based on the modeling of the spacecraft
geometry in finite elements (FEs). Using an FE-solver the surface temperatures of the
satellite can be derived with geometry and material parameters using heat sources/sinks as
constraints. The outgoing radiation force is computed including reflectivity and absorption
between all elements of the model. As an example for the method a test case model of
the radio isotope thermal generators (RTGs) of Pioneer 11 is processed with this force
computing method. The results show that detailed thermal modeling for the whole craft is
necessary as the simplified test case results in a force that is non-negligible with respect
to the pioneer anomaly.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Perturbation modeling becomes increasingly important
for current and future missions as the requirements for modeling accuracies and perturbation knowledge grow steadily.
For missions like LISA, LISA pathfinder and MICROSCOPE the
models for all perturbing effects have to be as accurate as
possible and new computing methods have to be developed
to reach the necessary performance. At ZARM an elaborated
method for the calculation of the perturbations caused by
∗ Corresponding author. Tel.: +49 421 218 4803; fax: +49 421 218 2521.
E-mail addresses: [email protected] (B. Rievers),
[email protected] (S. Bremer), [email protected]
(M. List), [email protected] (C. Lämmerzahl),
[email protected] (H. Dittus).
anisotropic heat dissipation has been developed which can
be used for any satellite geometry. The method is separated
in two steps:
• Thermal finite element (FE) analysis.
• Force calculation with raytracing.
In the first step a detailed FE model of the craft is generated
to which all material parameters, environmental conditions
according to the mission profile and all heat sources and
sinks are applied. Due to the structure of the force computation algorithm the FE used for the generation of the FE mesh
have to be hexaedric which allows high computation accuracies but also leads to higher modeling effort. For each new
model a trade off between optimal resemblance of the actual
satellite geometry and the work put into the modeling step
has to be made. With the finished FE model a thermal analysis is conducted to acquire the steady state temperature
0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.actaastro.2009.06.009
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Nomenclature
FE
RTG
PA
Fabs
Femis
Fref
i
j
n
Pemis
finite element
radio isotopic thermal generator
pioneer anomaly
absorption force component
emission force component
reflection force component
number of currently active element
number of currently receiving element
distribution for the processed thermal state of the spacecraft.
In a postprocessing the temperatures of the outer element
surfaces, the nodal positions in the cartesian frame, the material parameters and the node/element assignment list are
exported into text files that can be read in by the force computation algorithm. Using the information stored in the text
files as database the total force resulting from thermal dissipation is computed by the algorithm including reflection
and absorption between the different surfaces of the model.
In comparison to the existing models [10–12] for the calculation of thermal perturbations two main differences can be
identified:
1. The calculation is based on a full thermal finite element
analysis.
2. In the proposed method each surface in the model is considered as a Lambertian radiation source.
The FE analysis enables an improved geometrical accuracy
and the inclusion of measurement and sensor data as boundaries for the FE solution. Compared to other methods (analytical approach [12] or nodal models for the computation of
the equilibrium state [10,11]) this approach enables a greater
level of detail and improved solution accuracy. Furthermore
the creation of the model can be realised using standard FE
preprocessors and solvers (e.g. ANSYS) and parameter sensitivity analysis can be realised easily using macro-based FE
editors (e.g. APDL).
The consideration of each model surface as Lambertian
radiator leads to an increase in computation time but also
to an increase in numerical accuracy. Other methods [10,11]
consider only the resulting component normal to the emitting surface for the computation of the thermal recoil force.
In difference to this the method proposed in this paper uses a
hemispherical pattern of a large number of angularly spaced
rays to compute absorption and reflection effects thus also
including interaction of surfaces which are not directly facing each other. The resulting heat fluxes are computed by
means of view factors for each detected hit. Furthermore,
the number of considered reflections can be set freely using
both specularly and diffuse reflection models.
From Stefan–Boltzmanns law we know
Ptot = AT 4 .
For an emitting grey body that obeys Lamberts law, the
intensity distribution in polar direction can be computed
from the intensity in normal direction [5]
(1)
(2)
The integration of the intensity over the complete hemisphere (determined by the angles and with 0 ⱕ ⱕ 2
and 0 ⱕ ⱕ /2) leads to the total normal component of the
power output
P⊥ =
/2 2
0
0
2
In cos sin d d = 23 Ptot .
(3)
The resulting recoil force generated by a radiating body can
be expressed with Ptot and the speed of light c as [8]
F⊥ =
P⊥
.
c
(4)
3. Analytical test model
For a later comparison between the proposed FE method
and results acquired with simple node models an analytical
test case is formulated. Here the radio isotope thermal generators (RTGs) are modeled as pointlike isotropic radiation
sources with a total power of 1200 W each of which corresponds roughly to the available power at the start of the Pioneer 11 mission. The high gain antenna dish is modeled with
a specified number of nodes distributed evenly according to
the real geometrical shape of the antenna. Fig. 1 shows the
configuration of the test case.
The force resulting from an emission in a specific viewing
direction x is
Femis = −x
Pemis
.
c
(5)
For the isotropic radiation pattern assumed for the pointlike
RTGs no resulting force results. Taking into account the interaction with the high gain antenna dish, a fraction of the
radiation emitted from the RTGs is absorbed and reflected
at the antenna surface thus leading to a force of
Fres = −Femis
2. Theoretical background
I = LA cos = In cos .
total number of surface elements
emitted power
absorptivity
elevation angle
reflectivity
emissivity
azimuth angle
Stefan–Boltzmann constant
(6)
resulting from the asymmetry in the emission pattern of the
RTGs.
The radiation exchange between antenna nodes and RTGs
is computed with view factors under the assumption that the
RTG power is emitted isotropically. The view factors between
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Fig. 1. Testcase 1a: configuration with main antenna dish and RTG assemblies.
antenna nodes and RTGs can then be computed based on
the isotropic luminosity LIso = P/4r2 to
cos(j )
1
i,j =
dAj dAi ,
(7)
4Ai Ai Aj r(i, j)2
where i denotes the radiation sources (RTGs) and j denotes
the radiation sinks (antenna nodes). Due to the assumption
given above, the cosine for the source term cancels out. For
the sink (antenna nodes) the orientation and the area associated with the node has to be known. The antenna shape
can be characterised with the parametrisation
(8)
F(x, y, z) = r − g(z) = x2 + y2 − z = 0
with
x = r cos ,
y = r sin ,
r=
x2 + y2 ,
(9)
where the geometrical coefficients and are derived from
available technical drawings of the Pioneer 11 high gain antenna by a least squares estimate to = 2.047, = 0.512. The
normal direction of any node x0 on the antenna surface can
be computed with ∇F(x, y, z) at x0 .
x
y
∇F(x, y, z) = ;
; z−1 .
(10)
x2 + y2
x2 + y2
The total surface of the high gain antenna can be computed
by rotating the function g(z) with
I(g(z)) = 2
b
a
g(z) 1 + g 2 (z) dz
(11)
to AHGA = 6.535 m2 . This area is distributed evenly amongst
all nodes assigned to the antenna. For the computation of
the angle cos(j ) the node connection vectors between RTG
and antenna nodes are computed with
ri,j = xj − xi
(12)
and the cosine term is
cos(j ) = ∇F(x, y, z)|xj · ri,j .
(13)
Using the equations and values introduced above the net
force fraction resulting from interaction with a specific antenna node can be determined with
F = ri,j Pemis k,
(14)
where k is the effective optical parameter that defines the
reflection behaviour of antenna nodes ranging from 1 (total
absorption) to 2 (ideal reflection). The z-component of the
force vector is aligned with the flight direction and corresponds to the axis of the observed anomalous acceleration.
The total resulting force can be computed from the sum of
all individual antenna nodal forces
Fres,z =
n
Fi,z .
(15)
i
The analytical test case has been processed using the equations given above by varying the number of antenna nodes
in the model. For the mass of the spacecraft the Pioneer dry
mass of 233 kg [2] is considered, the optical constant k is set
to 1.8 which corresponds to a reflection coefficient of the
coated antenna surface of 0.8. Fig. 2 shows the resulting total forces for a different number of modeled antenna nodes.
As can be seen in the graph, the computation errors are
large for a low number of antenna nodes and decrease with
more nodes in the model. This effect is easily understandable
because a higher number of nodes represent the real antenna
geometry better and also imply a higher number of surface
different orientations thus reducing the misalignment errors
of each individual nodal surface. The solution converges with
∼ 400 nodes to a final value of ares =2.263×10−10 m/s2 . Looking at the observed anomalous acceleration of the Pioneer
spacecraft of aPio = 8.74 × 10−10 m/s2 [6,7,9,14] this number
corresponds to about 25.8 percent of the observed anomaly
against flight direction. This is in good agreement with other
estimation methods which predict a resulting RTG disturbance acceleration against flight direction of ∼ 20 percent
PA [15].
The results points out that a more thorough analysis of
the thermal recoil force with respect to the Pioneer anomaly
is necessary. In the presented calculation only the RTGs have
been considered as heat sources, all other surfaces in the
model can only absorb and reflect radiation. In a more realistic model all heat sources aboard, including payloads, heater
units, radiators and louver system will have to be included.
As absorbing surfaces only the high gain antenna (which is
supposed to receive the major part of the RTG radiation hitting the craft) has been modeled. The inclusion of equipment
section and experiment section and the external payloads
will of course change the resulting disturbance acceleration.
The modeling of the RTGs as pointlike isotropic radiation
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Fig. 2. Disturbance acceleration with growing number of antenna nodes for analytical testcase 1a.
sources is not an exact representation of the real emission
pattern (dominated by the radiation fins). Looking at these
issues one can formulate three criteria for an improved assessment of the given task.
(1) The resulting graph shows that a better representation of the real geometry of the modeled bodies delivers
more accurate results. Therefore, an exact computation of
the thermal recoil acceleration for the Pioneer spacecraft demands the modeling of the shape of each component in a
very high detail. Furthermore, interaction between the different geometries such as shadowing or multiple reflections
have to be considered.
(2) The assumptions made for the power distributions
of the radiation sources in the analytical test case are too
simplistic. The actual radiation pattern will differ from
the isotropic case due to the geometry and different temperatures on the RTG surfaces. For a computation with
higher accuracy one has to acquire the surface temperature
distribution of the craft based on heat sources, complete
geometrical shape, environmental conditions and material
properties.
(3) The modeling of the heat sources in the analytical test
case is only valid (with the simplifications mentioned above)
for the start of the mission. The main power source are the
Pt 238 fuel rods which decay exponentially with a half-life
of approx. 24 years. Furthermore, not all power generated in
the RTGs is emitted over the radiation fins but a fraction is
used in the electrical compartment and then dissipated via
louver system and shunt radiator. For a precise computation
of the thermal recoil at different mission times all these
issues have to be taken into account especially if one thinks
of the constancy of the anomaly which seems to contradict
a thermal influence (that intuitively should express itself as
an exponential decay in the residuals).
In order to meet the criteria formulated above new modeling and analysis techniques have to be developed. At ZARM
an approach is taken that uses finite element analysis to
acquire high precision surface temperature maps based on
accurate geometry models, material data and heat source
models that include the dynamic behaviour of the radioactive fuel. The results are the exported and processed with
numerical algorithms that use raytracing to compute the resulting surface forces. These algorithms include diffuse, specular and multiple reflections and will be explained in more
detail in the following.
4. Simulation and modeling
In order to improve the modeling accuracy and the level
of geometrical detail that can be processed an innovative
computing method based on FE method has been developed.
The modeling in FE is a time consuming task. The geometry
of the spacecraft for which the thermal perturbations have to
be computed has to be modeled with hexaedral elements in
the detail needed. The determination of necessary and with
respect to thermal effects unnecessary geometrical features
needs strong experience in thermal design. The challenge
is the modeling of different geometrical structures with the
given FE brick shape. It is easy to see that brick elements
cannot be used for round shapes or cut-out without further
processing. The necessary modeling step is the so-called premeshing where all volumes in the model are treated such
that each single volume fulfils the requirements for mapped
hexaedral meshing. Depending on the implemented detail
this can result in extensive cutting and gluing operations
throughout the model because each new node/element constraint has to be continued through adjacent volumes as well.
After the premeshing step the mesh can be generated in the
detail needed. In general a smaller element size will lead to
higher accuracies but also the computation time rises exponentially because radiation exchange is computed for each
model surface. With the meshing the material parameters
are also assigned to the model volumes. On the mesh nodes
and elements constraints such as heat generation, heat sinks
or radiosity can be applied. After this the steady state solution can be acquired with an FE solver (e.g. ANSYS). The
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Fig. 3. Allocation of solid angle elements and force.
premeshing, the meshing and the acquirement of the steady
state temperatures is shown for a test case model of the Pioneer RTGs in the next section.
After the surface temperatures have been computed the
results are read into the force computation algorithm and
the total recoil force generated by thermal dissipation can be
computed. The total force is composed of three major parts:
The resulting force can be computed by the sum of recoil
force without losses and the contributions of absorption and
reflection
(16)
4.1. Emission part
The force contribution of the emission can be computed
for each surface element with the equations introduced in
the previous section. Obeying Lamberts law the resulting
recoil force is normal to the surface of the emitting element.
With the known element node positions the surface normal
vectors en (i) can be computed. Thus the total force resulting
from thermal emission can be summed up over all surface
elements in the model as
Femis =
Femis (i) = −en (i)
i
2 A (i)A(i)T(i)4
.
3
c
d
= sin d d.
(17)
4.2. Absorption part
An exact and detailed spacecraft model also includes
overlapping geometries and surfaces that are shielded by
other surfaces. Thus fractions of the radiation emitted by a
surface element can be absorbed by other surface elements
and vice-versa. Therefore, for each model surface the possible radiation exchange partner surfaces have to be determined. In the force algorithm this is realised by means of
raytracing, sorting procedures and angular criteria.
For this a hemispheric pattern of outgoing ray-vectors is
initialised at each model surface. The ray pattern is modeled by dividing the hemisphere above the radiating surface
(18)
The vector from the radiating element centre coordinate ec (i)
to the centre of a solid angle element ec,d
(, ) is the ray
vector R(i, , ) with
R(i, , ) = ec,d
(,) − ec (i).
• Computation of force due to emission Femis .
• Computation of losses due to absorption Fabs .
• Computation of gains due to reflection Fref .
Ftot = Femis − Fabs + Fref .
into the so-called solid angle elements d
with
(19)
Fig. 3(left) shows the resulting allocation of solid angle elements S, in a tesseral division over the hemispherical surface.
To speed up the computation process, angular criteria
based on the surface normal vectors and the ray vectors
are checked first to reduce the number of surfaces and rays
that actually have to be raytraced. E.g. all surfaces which are
situated behind the element which is currently considered
active cannot receive any radiation because of the hemispheric radiation pattern. All surfaces that pass these angular “visibility” criteria have to be processed with raytracing.
Within this computation all surfaces in the model are considered once as the active element.
Starting with the first active element all other elements in
the model that are “visible” to the active element are sorted
by distance from the active element. This measure is a preparation for the modeling of shadowing. Starting with the first
ray in the pattern it is checked whether the ray intersects
the element which is nearest to the sending element or not.
If a hit is detected the ray is shut down and the next ray is
initialised. If the ray does not intersect the element the next
element in the sorted order will be checked. Thus rays cannot hit elements which are behind other surface elements.
The intersection of the rays and the surface elements is
checked using barycentric coordinates [1]. First the intersection point of the ray and the receiving element plane is
computed solving the following equation:
N1 + r(N2,1 ) + s(N3,1 ) = ec (i) + tR(i, , ),
(20)
where N1 , N2 , N3 are node coordinates of the receiving element. The solution to this system for r, s and t gives the
intersection point P with
P = ec (i) + tR(i, , ).
(21)
The receiving element surface is now divided into two triangles where the triangle nodes are considered as vertex
nodes for the use of barycentric coordinates [1]. The node
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Fig. 4. View factors between two elements.
Fig. 5. Specular and diffuse reflection.
coordinates and the intersection point have to be projected
into the receiving element plane using a quaternion rotation
matrix. With the barycentric coordinate formulation it can
now be checked if the intersection point P lies within the
two triangles or outside. If the intersection point is situated
within the triangles the element has been hit and the ray
is shut down, if not the ray will be checked for the next
element. This procedure is repeated for all radiating surface
elements in the model and delivers the hit-matrix which
stores all element pairs that can exchange radiation.
For all element pairs stored in the hit-matrix the radiation flux between the radiating surfaces has to be computed.
For this the view factors (see Fig. 4) between the radiating
surfaces have to be determined.
1 and 2 are the angles from the connection vector of
the two element middle nodes ec to each element normal
vector, respectively. The view factor from surface Ai to Aj is
defined as [4]
i,j =
1
Ai
cos i cos(j )
Aj
r(i, j)2
dAj dAi
(22)
and the radiation flux from element i to element j can be
computed with [4]
Pi,j = Ptot i,j = i Ai Aj Ti4
cos i cos j
r(i, j)2
.
(23)
Now the force components lost due to absorption can be
computed with
Fabs =
n
n i
ec (i, j) ·
j
Pi,j
.
c
(24)
4.3. Reflection part
For the determination of the influence of reflectivity the
methods presented for the computation of absorption effects
are used as well. The rays hitting an element surface can be
reflected either in a specular or in a diffuse way (see Fig. 5).
For specularly reflected rays the angle of incidence equals
the angle of reflection in the plane. Within the simulation
this is modeled by initialising a new reflection ray with origin
at the centre of the receiving element in the direction ẽc (i, j)
where ẽc (i, j) is −ec (i, j) rotated around the receiving surface
normal vector with the angle .
For an exact modeling of the diffuse reflection the hemispheric ray initialisation described for the absorption part
has to be repeated. This procedure will lead to a significant
increase of computation time because the raytracing is processed for each new reflected ray. For most cases the fraction of diffuse reflection is smaller than the specular fraction. Thus it is acceptable to model the resulting diffuse ray
only (2/3 times the total diffuse energy in surface normal
direction). The power for diffuse and specular reflected rays
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can be computed with
Pref = Pinc ,
(25)
where Pinc is the power received at the reflecting element.
The reflected power is split between specular and diffuse reflection where the ratio is defined by the surface material.
For all reflected rays the raytracing methods described for
the computation of the absorption are applied to detect absorption and further reflection of rays. The number of total
reflections (for a single ray) or a minimum energy threshold can be defined by the user. The resulting reflection force
can be computed by means of an impulse balance. Reflected
rays that are absorbed by another element surface do not
generate a net impulse because the same impulse generated
through emission is subtracted from the system at the absorbing surface. Thus only reflected rays that are emitted out
of the system into space can contribute to a resulting force.
All reflected rays that are not hitting other surface elements
are summed up (with their respective power to determine
the total reflection force)
Fref =
n
n i
Fref,spec (i, j) + Fref,dif (i, j),
(26)
j
where Fref,spec (i, j) and Fref,dif (i, j) are forces assigned to the
specular and diffuse rays send from element i to element j
which leave the system after being reflected. Now the total
resulting thermal dissipation force can be computed with
Eq. (16).
5. Pioneer test case
As a first performance test (testcase 1b) the proposed ray
tracing method is processed with the same geometry and
parameters of testcase 1a (see Fig. 1). This enables the comparison with the results acquired with the analytical method.
All calculations are processed with one single allowed reflection. More reflections will lead to much larger computation times but not change the result significantly due to
the comparably large distance between RTGs and antenna.
The FE model uses a discretisation with a mean element size
of 0.05 m, all other parameters are taken from the analytical test case. The resulting disturbance accelerations in zdirection are shown for a different number of emitted rays
in Fig. 6. The resulting disturbance acceleration converges to
a value of 3.22 × 10−10 m/s2 which corresponds to a value of
36.8 percent PA. This value exceeds the analytically attained
result by about 10 percent PA which can be explained with
the shape of the antenna and the non-sufficient accuracy of
the reflection model in the analytical case. There we took an
effective optical constant k (between 1 and 2) in order to allow for reflection effects. This assumes that the z-component
of the reflected radiation equals the incoming radiation flux
times a reflection coefficient where k = 1 + . Effectively
the reflection contribution in z-direction is 1 + times the
incoming flux. For the ray tracing model the reflection computation is more convenient. There the z-component of the
reflected radiation can exceed the z-component of the incoming flux because the radiation is reflected in different
directions based on the surface normal vector. If the surface
(
)
–
7
is sloped with respect to the global coordinate system then
the reflection z-component can be increased or decreased.
Looking at the shape of the antenna and the global coordinate system we can see that those surfaces nearest to the
RTGs are sloped such that the z-component of the reflection
increases. For those surfaces far away from the respective
RTG the z-component is reduced but the overall contribution of these surfaces is much smaller than that of the near
ones. Thus effectively the total z-force increases, explaining
the higher value acquired with the FE ray tracing method.
Of course the absolute value of the total reflected power is
not affected by this effect. Fig. 7 shows the necessary computation time vs. number of processed rays per element for
the given testcase (1366 elements). The system used was a
conventional Athlon 64 dual core with a computation speed
of 2.6 GHz and 4 Gb RAM. As can be seen the computation
time increases nearly in a linear way with the number of
processed rays. This is quite understandable as the routines
and functions used to compute intersections with the rays
and the surface elements dominate the algorithm. In order to
formulate a precision criterium for the values attained, one
may utilise the symmetry of the model. As the RTGs have
the same thermal boundaries, the same size and possess the
same geometry, the total emission contribution in the model
should ideally be zero as each ray on the RTG surface has
a counterpart ray directed in the opposite direction. Due to
computational errors and the cumulating of these errors the
resulting emission contribution varies from the ideal state
by about 1.4 × 10−14 in the worst case. In general a higher
number of rays will also increase the total computation error which is in particulary true if multiple reflection is processed. Up to convergence state the accuracy gain by using
more rays to scan through the model is bigger than the imprecision in the computations. Ray numbers which lead to
results well beyond the convergence state will decrease the
overall precision again because a higher number of rays will
not process the geometry better but introduce more calculations and thus more inaccuracies that can cumulate.
For a further evaluation of the method and a first assessment of the influence of anisotropic heat dissipation for the
Pioneer 10/11 mission a second test case model (testcase
2) has been defined. This model includes the RTGs of the
Pioneer and an effective cross-sectional area for the implementation of reflection effects at the main spacecraft. Due to
the incomplete representation of the overall geometry and
the many estimates taken for material parameters and heat
sources this model is not applicable for a complete evaluation of the role of thermal effects for the Pioneer spacecraft but it can be used to assess tendencies and order-ofmagnitude hints as will be stated in the conclusion.
Fig. 8 shows the premesh and the mesh for the RTGs included in the test case model. For the meshing preparation
the model volumes had to be cut in several places. The structure contains interior heat source (fuel rods) volumes, heat
shielding, end plugs and radiating fins. The material properties and main geometrical dimensions for the model were
taken from technical documentation of the Pioneer project
[2,3]. As single heat source the volumes representing the fuel
rods are applied with a heat load of 2500 W [13] which corresponds to a mission lifetime of roughly 10 years. The outer
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Fig. 6. FE raytracing method results for testcase 1b.
Fig. 7. Computation time vs. number of rays per element for testcase 1b configuration.
Fig. 8. Hexmesh (left) and premesh (right) of the Pioneer RTG.
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Fig. 9. Temperature distribution for steady state.
Fig. 10. Real Pioneer geometry (left, source: C. Markwardt) and testcase 2 geometry (right).
model surfaces (fins and body) are radiating into space with
an emissivity of 0.9 (white paint). For the calculation of the
steady state temperatures the ANSYS FE solver was used.
The resulting temperature distribution for the equilibrium
is shown in Fig. 9.
The simulated surface temperatures are exported and
read into the force computation algorithm. The antenna
model is identical to the one used in testcase 1b and absorbs/reflects radiation emitted by the RTGs but does not
radiate itself (assumed antenna temperature 0 K). The resulting model geometry and in comparison the real Pioneer
geometry are shown in Fig. 10. In commitment to the more
complex geometry which may lead to multiple reflection
between the RTG fins, the simulation is performed with
three reflections allowed per individual ray.
The result converges for a ray number of about 90 000
rays (spaced after the solid angle formulation as can be seen
in Fig. 3) emitted by each RTG FE surface. For testcase 2 the
resulting force vector is
Fx
Fy
Fz
−2.372 × 10−8 N
−2.102 × 10−12 N
6.823 × 10−8 N
With the same mass assumption as in testcases 1a and 1b
this corresponds to an acceleration of about 33.5 percent PA.
Comparing this result with the spherical RTG model result one notices that the result for testcase 2 is slightly
smaller than the value acquired for testcase 1b. Due to the
preliminary status of the detailed FE model it can not be
stated ultimately if this deviation is due to the difference
between the isotropic source assumption and the more realistic radiation pattern of the complex model. One may however speculate if the reflections between the RTG fins and
the temperature distribution over the RTG surface reduces
Please cite this article as: B. Rievers, et al., Thermal dissipation force modeling with preliminary results for Pioneer 10/11, Acta
Astronautica (2009), doi:10.1016/j.actaastro.2009.06.009
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B. Rievers et al. / Acta Astronautica
the effective flux into the antenna. For a definitive answer
to that more analysis which look in particular at the change
of disturbance acceleration for changing temperature surface maps and more detailed RTG geometries (also in the FE
analysis step) have to be performed. However the result of
test case 2 points out that thermal effects may have a strong
influence on the overall perturbation budget of the Pioneer
11 spacecraft and that a more detailed thermal analysis of
the whole craft including all major geometrical components
and further investigations are necessary.
6. Conclusion
In this paper a new and powerful computation method
for the perturbations caused by anisotropic thermal radiation has been presented. The method has been compared
to simple analytical models and an increased precision
for the assessment of reflection effects has been shown.
Furthermore the method has been used on a test case
model including a more detailed geometrical model of the
Pioneer RTGs and the high gain antenna body in order
to evaluate if thermal aspects can have an influence on
the overall perturbation budget of the Pioneer 11 spacecraft. Due to the exclusion of the main craft (equipment
section, experiment section and outer payloads) in the
analysis and other simplifications (material models, model
used in the FE analysis step, RTGs are the only heated components) the test case model cannot account for an exact
solution of the complete thermal disturbance acceleration
of the Pioneer by any means but can evaluate a tendency
that shows if thermal aspects can have an influence with respect to the anomalous acceleration. The result acquired for
all test case models indeed point out thermal perturbations
cannot be neglected and their influence on the overall perturbation budget needs to be investigated more thoroughly
if a definitive answer to the role of thermal perturbations
with respect to the Pioneer anomaly shall be found. For the
Pioneer mission this will imply the development of a complete detailed FE model which includes material parameters,
internal payload geometries, all main structural elements
and all heat sources aboard (Such as heaters, payloads).
With this model also the influence of material degradation
and other effects on the resulting thermal perturbation
force will be investigated by means of parameter variations. Furthermore the values for the resulting disturbance
accelerations presented in this paper are valid only for the
start of the mission. As optical surface parameters and the
thermal energy available (radioactive fuel with half-life of
approx. 24 years) have changed during mission time the
resulting acceleration may decrease significantly for later
points in time. Using complete FE models future analysis
and calculations will also investigate the influence of the
Pioneer louver system and the effect of surface degradation
with respect to the compatibility of the constancy of the
observed anomaly to a thermal source.
The method presented in this paper is not restricted to
a single mission but can be used for any satellite mission
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where the thermal perturbations have to be assessed.
Especially missions with very high requirements on perturbation knowledge such as LISA, LISA pathfinder or MICROSCOPE will benefit of this new improvement in modeling of
thermal perturbations.
Acknowledgements
Our thanks go to the JPL, in particular Slava Turyshev for
support, valuable discussions about the thermal aspects of
Pioneer and the thermal models and the ongoing cooperation. In addition we also thank Victor Toth, Lou Scheffer,
Craig Markwardt, Orfeo Bertolami and his group for their
contributions to thermal modeling of the Pioneer and their
ongoing investigations which are a great motivation for our
own work. We appreciate the work and contributions of the
members of the Pioneer collaboration and thank many valuable suggestions. Financial support of the German Research
Foundation (DFG) and the German Aerospace Center (DLR)
is gratefully acknowledged.
References
[1] H. Coxeter, Introduction to Geometry, Wiley, USA, 1961.
[2] NASA Ames Research Center, Pioneer Program, NASA Ames Research
Center, Moffet Field, California, 1971, PC-202.
[3] Teledyne Isotopes, SNAP-19 Pioneer F&G Final Report, Teledyne
Isotopes, Energy Systems Division, Timonium, Maryland, 1973, IESD
2873-172.
[4] F. Hell, Grundlagen der Waermeuebertragung, VDI-Verlag,
Duesseldorf, Deutschland, 1982.
[5] S. Kabelac, Thermodynamik der Strahlung, Verlag Vieweg,
Braunschweig/Wiesbaden, Deutschland, 1993.
[6] J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto, S.G. Turyshev,
Indication, from Pioneer 10/11, Galileo, and Ulysses data, of an
apparent anomalous, weak, long-range acceleration, Phys. Rev. Lett.
81 (1998) 2858.
[7] J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto, S.G. Turyshev,
Study of the anomalous acceleration of Pioneer 10 and 11, Phys. Rev.
Lett. D 65 (2002) 082004.
[8] S. Theil, Satellite and test mass dynamics modeling and observation
for drag-free satellite control of the STEP mission, Ph.D. Thesis, ZARM,
University of Bremen, December 2002.
[9] S.G. Turyshev, M.M. Nieto, J.D. Anderson, Study of the Pioneer
Anomaly: A Problem Set, Jet Propulsion Laboratory, California Institute
of Technology, Pasadena, California, 2005, 04.80.-y,95.10.Eg,95.55.Pe.
[10] S. Adhya, Thermal re-radiation modelling for the precise prediction
and determination of spacecraft orbits, Ph.D. Thesis, Department of
Geomatic Engineering, University College London, 2005.
[11] M. Ziebart, S. Adhya, A. Sibthorpe, S. Edwards, P. Cross, Combined
radiation pressure and thermal modelling of complex satellites:
algorithms and on-orbit tests, Adv. Space Res. 36 (2005) 424–430
doi:10.1016/j.asr.2005.01.014.
[12] J. Duha, G.B. Afonso, L.D.D. Ferreira, Thermal re-emission effects on
GPS satellites, J. Geod. 80 (2006) 665–674 doi:10.1007/s00190-0060060-x.
[13] L.K. Scheffer, A conventional explanation for the anomalous
acceleration of Pioneer 10/11, Cadence Design Systems, San Jose,
California, 2006, 04.80.-y,95.10.Eg,95.55.Pe.
[14] V.T. Toth, S.G. Turyshev, Pioneer anomaly: evaluating new data, in:
Recent Development in Gravitation and Cosmology: Third Mexican
Meeting on Mathematical and Experimental Physics, 2008.
[15] O. Bertolami, F. Franciso, P.J.S. Gil, J. Paramos, Thermal analysis of
the Pioneer anomaly: a method to estimate radiative momentum
transfer, Phys. Rev. D. 78 (2008) 103001.
Please cite this article as: B. Rievers, et al., Thermal dissipation force modeling with preliminary results for Pioneer 10/11, Acta
Astronautica (2009), doi:10.1016/j.actaastro.2009.06.009