Research Article Modelling of Solar Radiation Pressure ...downloads.hindawi.com/journals/ijae/2015/928206.pdf · Research Article Modelling of Solar Radiation Pressure Effects: Parameter

Research ArticleModelling of Solar Radiation Pressure EffectsParameter Analysis for the MICROSCOPE Mission

Meike List Stefanie Bremer Benny Rievers and Hanns Selig

ZARM University of Bremen Am Fallturm 28359 Bremen Germany

Correspondence should be addressed to Meike List meikelistzarmuni-bremende

Received 30 July 2015 Accepted 19 October 2015

Academic Editor Paolo Tortora

Copyright copy 2015 Meike List et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Modern scientific space missions pose high requirements on the accuracy of the prediction and the analysis of satellite motion Onthe one hand accurate orbit propagationmodels are needed for the design and the preparation of amissionOn the other hand thesemodels are needed for themission data analysis itself thus allowing for the identification of unexpected disturbances couplings andnoises whichmay affect the scientific signalsWe present a numerical approach for Solar Radiation Pressuremodelling which is oneof the main contributors for nongravitational disturbances for Earth orbiting satellites The here introduced modelling approachallows for the inclusion of detailed spacecraft geometries optical surface properties and the variation of these optical surfaceproperties (material degradation) during the mission lifetime By using the geometry definition surface property definitions andmission definition of the FrenchMICROSCOPEmission we highlight the benefit of an accurate Solar Radiation Pressuremodellingversus conventional methods such as the Cannonball model or aWing-Box approach Our analysis shows that the implementationof a detailed satellite geometry and the consideration of changing surface properties allow for the detection of systematics whichare not detectable by conventional models

1 Introduction

Themodelling and propagation of satellite motion are one ofthe central tasks in mission analysis The main driver for theevolution of a satellite orbit is the gravitational field of the cen-tral attracting mass While a spherical symmetric approachfor the gravitational field delivers undisturbed Kepler orbitsmore realistic approaches employ spherical harmonics tomodel the gravitational potential Among others these mod-els implement the effect of Earth oblateness zonal andtesseral variations of themass distribution Consequently theintroduced corrections of the gravitational field can be inter-preted as a gravitational disturbance of an ideal Kepler orbit

However besides these perturbations nongravitationaldisturbance (NGD) effects have a large influence on satellitemotion The largest of these NGDs in low orbit altitudes isthe atmospheric drag resulting from the resistance of residualatmosphere against the satellite body moving at high relativespeed For higher altitudes where the influence of residualatmosphere can be neglected the dominant NGDs resultfrom interaction of the satellite surface with solar photons

causing a drag force known as the Solar Radiation Pressure(SRP) The magnitude of the SRP acting on the satellitedepends on a wide range of parameters The distance to theSun and the position of the satellite with respect to Earthand Sun (regarding possible eclipses) define the intensity ofthe incoming radiation The geometry of the satellite theoptical properties of the external surfaces and the actualorientation with respect to the Sun largely influence theorientation and magnitude of the evolving SRP According tothis any SRP model depends on an accurate implementationof the satellite orbit the attitude and the geometricphysicalproperties of the satellite structure As a consequence a highmodelling effort has to be made in order to obtain preciseresults However if mission planning and analysis for thesatellite mission at hand possess high requirements on orbitmodelling precision a sophisticated SRP model is needed

It has been argued for quite some time that commonlyused SRP models like the Cannonball and the Wing-Boxmodel are not sufficient enough for an accurate SRP analysis[1 2]This is particularly true if the involved geometries differconsiderably from a spherical shape or a standard bus and

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 928206 14 pageshttpdxdoiorg1011552015928206

2 International Journal of Aerospace Engineering

solar panel assemblyThe high gain in modelling accuracy bymeans of a realistic implementation of the satellite geometryhas also been demonstrated with an analysis of NGDs actingduring the cruise phases of the ESA Rosetta spacecraft [3]Here a nonphysical solar constant was measured resultingfrom a parametric fit of the measured contribution of SRPon the total acceleration By means of a sophisticated SRPand thermal radiation pressure (TRP) (TRP results fromphotons emitted by the spacecraft itself) model this offset wasexplained as a nonmodeled TRP correlated with the actingSRP Further examples for a successful implementation ofenhanced SRP models are GNSS satellites where navigationaccuracy directly benefits from an improved SRP modellingapproach [4ndash6]

The modelling effort for an accurate analysis of the SRPeffect on a given satellite is considerably high Consequently atrade-off has to be made between the precision requirementsfor the specific mission the effort that one is willing to takeand the possible gain with respect to an improvement of aprecise implementation of NGDs This paper intends to givean overview on the implications of accurate SRP modellingand the expected improvement of NGD implementationTheparameters for the subsequent SRP analysis are derived fromthe French spacemissionMICROSCOPE [7] which delivers asuitable test case with respect to the specified mission profile

The MICROSCOPE mission requires a very high accu-racy of the spacecraft attitude and attitude stability due tothe specific mission specification In order to realize the highperformance of the differential acceleration measurement ofthe two test masses to test the Weak Equivalence Principle(EP) it is essential to ensure a very low disturbance level(forces and torques acting on the satellite) For this purposeMICROSCOPE will be operated in drag-free mode Anydisturbance will be compensated by forces and torquesgenerated by a cold gas propulsion system in closed loopcontrol The input to the corresponding controller is givenby the common mode acceleration signal of the differentialaccelerometer while the science signal is extracted fromthe differential acceleration signal However in spite ofthe drag-free control the exact modelling of NGDs is stillnecessary due to couplings between the accelerometers andthe satellite structure As a consequence external disturbanceeffects influence the scientific signals since the drag-freecontrol forces and torques needed to compensate the NGDsintroduce a disturbance translated by the coupling

The actual requirements of MICROSCOPE are quitedemanding For the EP measurement sessions the residualacceleration of the spacecraft shall be less than 10

minus12msminus2 Atthe EP test frequency the angular pointing stability shouldbe better than 7 120583rad and the angular velocity stability isrequired to not exceed 10

minus9 radsminus1 respectively [8]The sun-synchronous polar MICROSCOPE orbit leads

to a force vector due to SRP always directed to one side ofthe orbital plane This leads to a linear acceleration normalto the orbital plane which is superimposed by an angularvariation due to seasonal change of the angle between theorbital plane normal and the direction to the sun In caseof simulating a drag-free mission a detailed modelling of

the corresponding SRP forces and torques is important toestimate the actual control forces of the Attitude and OrbitControl System (AOCS) which keep the spacecraft in thefavored state Considering MICROSCOPE NGD effects dueto SRP can easily reach several 120583N Divided by the satellitersquosmass (330 kg) this force induces disturbing accelerations ofsome 10minus8msminus2 which is not negligible in a premission end-to-end simulation and for developing and implementing dataanalysis and data processing strategies

Due to the high demands of the mission the sun-synchronous orbital plane and the LEOcharacter of the orbitMICROSCOPE is an ideal test case scenario for the analysisof the general benefit of accurate SRP modelling for spacemissions After a general introduction of the SRP modellingmethod and the derivation of SRP characteristics for chosenorbit and mission examples we will use MICROSCOPE as atest case scenario for a detailed SRP analysis By looking atdifferent approaches for the implementation of the geometryof the satellite and applying surface degradation models wehighlight the possible benefits and the involved costs of highaccuracy SRP modelling

2 Orbit and Attitude Propagation

Since the evolving SRPmagnitude and orientation depend onthe position and the attitude of the spacecraft a dynamic orbitsimulation including the gravitational acceleration caused bythe Earthrsquos gravitational field is necessary The calculationof the gravitational influence and the integration of theequation of motion are realized within the framework of thegeneric simulation tool High Performance Satellite DynamicsSimulator (HPS) [9]TheHPS is aMATLABSimulink librarywhich is developed at ZARM in cooperation with the DLRInstitute of Space Systems Bremen The main focus of HPSis the propagation of satellite orbits and the computationof the satellitersquos orientation depending on specific initialconditions and the space environment Furthermore thecoupledmotion of up to eight on-board testmasses (arrangedpairwise in up to four accelerometers) can be computed in sixdegrees of freedom

Coupling effects between the satellite and the test massesas well as among the test masses themselves are includedin the implemented differential equation systems of eachconsidered body In the following the satellitersquos equations ofmotion are shown exemplarily The satellite motion is givenby (1)

997888rarr119903

119894

119894119887

= 119892119894

119894119887

( 119903119894

119894119887

) +

119894

control + 119894

dist + 119894

couplsat

119898sat (1)

Here (i) 119898sat is the mass of the satellite (ii) 997888rarr119903

119894

119894119887

is theacceleration of the satellite relative to the ECI frame (iii)119892119894

119894119887

( 119903119894

119894119887

) is the gravitational acceleration (iv) 119894control is thecontrol force (v) 119894dist is the sum of all disturbance forcesacting on the satellite and (vi) 119894couplsat is the force due to thecoupling between the satellite and all considered test massesThe superscript 119894 indicates that all components of (1) are givenin ECI coordinates

International Journal of Aerospace Engineering 3

The satellitersquos rotation and the satellitersquos attitude motionare computed by using (2) and (3)

119887

119894119887

= (I119887119887

)minus1

[119887

control + 119887

dist + 119887

couplsat minus 119887

119894119887

times (I119887119887

119887

119894119887

)]

(2)

q119887119894

=1

2

997888rarr

119887

119894119887

⊙ q119887119894

(3)

Here (i) 120596119887119894119887

is the angular velocity of the satellite relative tothe ECI frame (ii) I119887

119887

is the moment of inertia matrix ofthe satellite (iii)

119887

control is the sum of the control torquesapplied for attitude control (iv)

119887

dist are the disturbancetorques acting on the satellite (v) 119887couplsat are the torques

generated from the satellite-test mass coupling (vi)997888rarr

119887

119894119887

is thequaternion representation of 119887

119894119887

and (vii) q119887119894

represents theEuler symmetric parameters Here the superscript 119887 indicatesa description of the equationsrsquo components in the satellitebody fixed frame

It is obvious that the satellitersquos motion is affected bythe acceleration due to the Earthrsquos gravitational field whichcannot be considered to be spherically symmetric This isdue to the nonuniform mass distribution of the Earth andit results in (i) perturbations of the pure Kepler orbit andin (ii) perturbations of the satellitersquos attitude But apart fromthis for a complete orbit and attitude propagation simulationone has to take into account nongravitational effects actingon the satellite too They force it to go astray from its purelygravitational orbit and induce undesired rotations For manymissions one of the most prominent effects of these NGDs isthe SRP which will be discussed in detail in the next sections

3 SRP Model

The disturbance forces and torques due to SRP originatefrom the interaction of the satellitersquos surface with the photonsemitted by the sun It is assumed that each photon that hits thesatellite is either absorbed or reflected in a specular or diffuseway thus effectively changing the momentum of the satelliteAs a consequence the resulting force acting on an elementalarea 119889119860 can be expressed as the sum of three individualcontributions [10]

119889120572

= minus119875SRP120572 cos (120579) 119890Sun119889119860

119889120574119878= minus2119875SRP120574119878cos

2

(120579) 119890119873

119889119860

119889120574119863

= 119875SRP120574119863 (minus2

3119890119873

minus 119890Sun) cos (120579) 119889119860

(4)

where (i) 119875SRP is the SRP (ii) 119889119860 is the elemental area (iii)119890Sun and 119890

119873

are the unit vector in Sun direction and the unitvector normal on the elemental area 119889119860 respectively and (iv)120579 is the angle between 119890Sun and 119890

119873

Finally (v) 120572 120574119878

and 120574119863

are the coefficients of absorption of specular reflection andof diffuse reflection

With120572+120574119878

+120574119863

= 1 (assuming a nontransparentmaterial)the force due to SRP can be derived as follows

int119889total = minus119875SRP int[(1 minus 120574119878

) 119890Sun

+ 2 (120574119878

cos (120579) + 1

3120574119863

) 119890119873

] cos (120579) 119889119860(5)

Hence the computation of total requires the modelling of

(i) the satellite orbit because the magnitude of 119875SRPdepends on the distance to the Sun

(ii) the satellite attitude in order to derive the correctincident angle 120579 between 119890

119873

and 119890Sun(iii) the satellite geometry for defining appropriate values

of 120574119878

120574119863

119890119873

and 119860

The propagation of the satellite orbit and its orientation is oneof the basic tasks within the simulation software HPS andcan be applied to any Earth orbiting satellite mission Sincemost NGDs such as the SRP are surface-based effects thepropagation needs an input model for the satellite geometryspecific to the actual mission Due to the variations in satellitecomponents general dimensions and external materials it isnot possible to find a suitable standard model that can beused in a flexible way with respect to the variety of spacecraftgeometries This is one of the main setbacks of standardapproaches such as the Cannonball or the Wing-Box model

Instead of using a simplistic approach where total iscalculated with respect to (i) an effective projected satellitesurface area and to (ii) averaged optical surface propertiesthe focus of the HPS SRP interface lies on the capturing of theinfluence of details of the satellite geometry and the involvedmaterial parameters of each component of the satellite Inorder to realize this the HPS SRP approach is divided intotwo main steps Before the actual SRP is calculated thesatellitersquos surface is discretized in small elements 119860

119894

and thedifferent optical properties are assigned to the correspondingelements For a complex geometry this is realized by meansof a finite element (FE) preprocessor where themeshes of theexternal surfaces are exported together with their respectiveoptical property definitions Subsequently theHPS algorithmfor SRP computation evaluates (6) which is the discrete formof (5) for each element that is illuminated by the Sun for thechosen vector 119890Sun

119894

= minus119875SRP [(1 minus 120574119878119894

) 119890Sun

+ 2 (120574119878119894

cos (120579119894

) +1

3120574119863119894

) 119890119873119894

] cos (120579119894

) 119860119894

(6)

The overall force is then derived by computing the sum overall elements

SRP = sum

119894

119894

(7)

By means of geometric criteria (see [11] for details) thealgorithm determines automatically if an element is lit by the


Sun and considers shadowing by other parts of the satellite aswell In combination with an eclipse model the global SRPacting on the satellite is calculated with respect to a realisticillumination scenario

In order to speed up the simulation process resulting SRPmagnitude and directions can be derived in a normalizedform Here a lookup table can be derived where parametersof the stored SRP values are the solar elevation and azimuthconsequently defining the current sun angle When thelookup table is computed in preprocessing the results can beused to determine the dynamical evolution of the SRP duringflight within anHPS simulation For this the normalized SRPvalues are converted to the actual SRP with respect to thecurrent solar distance and orientation of the satellite as wellas the eclipse condition

4 Parameter Analysis

In order to review the systematics of the SRP force modeldiscussed here a parameter analysis is performedThe impli-cations of changes of the relevant input parameters such asorbital elements and geometrical and technical features withrespect to the overall magnitude of the resulting disturbanceforce due to SRP are discussed in the following

41 Solar Radiation Pressure Since the magnitude of theincident solar radiation does depend not only on the orbit ofthe satellite around the Earth but rather on the Earthrsquos orbitaround the Sun too it is sensible to analyse the influence ofthe implications of the central body orbit during the yearTheannual variation of the strength of119875SRP is depicted in Figure 1Exemplarily the resulting SRP in Nmminus2 for the CHAMPmission orbit (see Table 1 for details) is presented

Furthermore the resulting SRP values for the Low EarthOrbit (LEO) missions CHAMP and MICROSCOPE (sun-synchronous orbit (SSO)) the SRP detected for the orbit ofthe geostationary (GEO) mission Meteosat and the disturb-ing radiation pressure for the GALILEO satellites are given inFigure 2 for the timeframe of a single day

The detected strength of the SRP varies on large timescales due to the change of the distance between the Earthand the Sun over the time period of one year (see Figure 1)The variations of the SRP on smaller time scales (see Figure 2)are a consequence of the satellitesrsquo motion around the Earthresulting in additional distance variations with respect tothe Sun Figure 1 shows that in case of CHAMP the SRPchanges about 6 during a half-year period (from winterto summer) The magnitude of the small scale variationsdepends considerably on the type of the satellitersquos orbit thatis LEO GEO and so forth This is demonstrated in Figure 2

Another effect is the variability of the SRPdue to changingsolar activity The main variation of the intensity of solarradiation shows a period of eleven years The correspondingvariation of the amplitude of the solar constant which is thetotal solar irradiance (TSI) at a fixed distance of one AU isonly 01ndash02 percent [12] Due to the relative small variationcompared to the variation induced by the elliptic Earth orbitthis effect is negligible and will not be taken into account inthe following

Mar

201

6

June

201

6

Sept

201

6

Dec

201

6

Mar

201

7

June

201

7

Time in orbit (month)CHAMP

475e minus 06

47e minus 06

465e minus 06

46e minus 06

455e minus 06

45e minus 06

445e minus 06

PSR

P(N

m2)

Figure 1 Annual variation of the SRP for satellites orbiting theEarth Exemplarily the orbit of CHAMP is depicted

June 20 June 21Time (month)

CHAMPMICROSCOPE

MeteosatGALILEO

PSR

P(N

m2)

4421e minus 06

442e minus 06

4419e minus 06

4418e minus 06

4417e minus 06

4416e minus 06

20162016

Figure 2 Resulting SRP for selected satellitemissionsThe variationover one day is shown

The choice of example missions is based on Table 1 whichprovides an overview about the distribution of operatingsatellites on the different mission classes As each category islinked with a typical altitude the above-named conclusionscan be interpreted as a general survey of the evolution of SRPfor a broad range of satellite missions

42 Geometry Models In contrast to the general analysis of119875SRP an investigation of the influence of satellite attitudeand design that is its geometry and the surface materialsrequires higher effort and will be carried out exemplarily forthe MICROSCOPE mission


Table 1 Overview of orbit classes including typical orbit parameters and mission examples

Orbit category Percentage Inclination 119894 [∘] Altitude [km] Semimajor axis [km] Mission name OperationLEO 41

Examples 88940ndash89060 485ndash500 6870ndash6770 GRACE Gravity field recovery87180 454 6823 CHAMP Earth observation

GEO 8

Examples

0028 36000 42162 Meteosat Meteorology0180 35780 42160 GOES Meteorology0020 42161 42165 Arsat-1 Communication0040 35796 42165 Ciel-2 Direct-broadcast

GTO amp HEO 12 7000(Kourou)

250 (perigee)36000 (apogee) 24582 Transfer

SSO 31

Examples

98248 700 7078 MICROSCOPE Science98600 780ndash800 7145 Envisat Earth observation98390 714 7084 SwissCube-1 Science97469 510 6885 Belka 2 Earth observation

MEO 7

Examples56000 23222 29601 GALILEO Navigation64800 19100 25510 GLONASS Navigation55000 20180 26580 GPS Navigation

Molniya 634007378 (perigee)45730 (apogee) 26554 Molniya satellites Communication

MICROSCOPE will be operated on a sun-synchronousorbit at an altitude of 700 km and an inclination of 98248∘In order to provide a stable thermal environment for thepayload and to minimize eclipse phases MICROSCOPE willbe injected in an orbit with 600 hrs or 1800 hrs local solartime at ascending node Figure 3 illustrates the attitude ofMICROSCOPE with respect to its orbital plane

As stated above a usual simplification of the satellitersquosgeometry involves the definition of a reference area withmean values for the optical properties In contrast the HPSconcept utilises FE models which demand a certain effortduring construction Between these approaches a range ofother geometrymodels is of commonusage For example dueto its symmetry a sphere may be used as very simple modelfor the geometry of a satellite This so-called Cannonballmodel [13] results in SRP forces completely independent ofthe attitude if all surfaces share the same optical properties

In reality satellites possess a more or less complex geom-etry The total value of the force can strongly depend on theincident angle even in the case of a homogeneous distributionof optical parameters on the external surfaces In particularflat components like solar panels contribute to this depen-dency For this reason so-called Wing-Box models are usedThey offer the possibility to introduce different optical prop-erties generally for the satellite body and the solar panels [14]

In addition to the complex FE model we generateddifferent geometry models to demonstrate the impact ofgeometric complexity on the resulting SRP effects includingthe most simple approach (disk) a simple box and a Wing-BoxmodelThey are depicted in the upper row of Figure 4 In

order to get the best comparability we set the same projectedsurface area for each model (with 120579 = 90∘ and 120593 = 0∘corresponding to the MICROSCOPE solar panel side) Inaddition a spherical geometry model was chosen to providea global comparative value for this analysis The model isnot shown in Figure 4 for reasons of brevity In the lowerrow of Figure 4 the values of the projected surface areas forthe different geometry models are depicted as a function ofthe incident sunray described in polar coordinates (smallpicture in Figure 3) In each case the comparative value ofthe spherical geometry model appears as constant surfacearea independent of 120593 and 120579 Usually the disk model is onlyapplicable for vertical incident sunlight Therefore it is notexpected to be a good choice for MICROSCOPE as its solarpanels will not be exposed to perpendicular solar irradiationmost of the time This results from the fact that the satellitersquos119909-axis will be aligned with the orbit normal and not with thevector to the Sun During the year the incident angle 120579 variesin the range of about 30∘ which results from the combinationof inclination and obliquity of the eclipticThe simple box theWing-Box and the FEmodels show characteristic results thatrepresent the symmetries of each of the models Obviouslythe simple box model results in large deviations from the FEmodel especially for angles 120579 far from 90∘ because it doesnot take into account the geometry of the solar panels andthe corresponding correct contribution to the total area forthese angles In general the Wing-Box model gives a goodrepresentation of the projected area but the distribution forthe FE model is much smoother Furthermore the definitionof the reference projected area yields an overestimation of


X

Y

Orbit normal

Polar coordinates

X

Y

Z

Z

rsunb

rsunECI

ZECI

XECI

YECI

120579

120593

Figure 3 Illustration of MICROSCOPE orbit with respect to Earth-centred inertial coordinates (ECI) Small picture definition of vector tothe Sun in polar coordinates 120579 and 120593

the projected area for 120579 = 90∘ for both the simple box andthe Wing-Box models In summary the FE model producesthe most accurate results for the projected area

Another reason for using at least simple box modelsis the fact that different optical properties can be assignedto the single satellite surface cells Figure 5 shows the FEmodel of MICROSCOPE in which the different materials arerepresented Each color corresponds to specific values of 120574Sand 120574119863

The influence of the optical properties is demonstrated

in Figure 6 Here the absolute value of SRP is depicted asfunction of 120593 and 120579 for a constant value of 119875SRP In contrastto Figure 4(d) there is a significant difference between thepeak at 120579 = 90

∘ and 120593 = 0∘ corresponding to the solar

panel side and the opposite side (120579 = 90∘ 120593 = 180

∘)although the projected area is nearly the same Looking atthe material distribution the result is not surprising Theback side of the solar panels is covered with White PaintConsequently the corresponding surface cells have higherreflection coefficients and contribute stronger to the absolutevalues of SRP compared to those on the front side Overallthe difference between both sides for perpendicular solarirradiation amounts to approximately 13

When detailed surface models are used the quality of theobtained force considerably depends on the chosenmesh Onthe one hand geometrical features such as spherical bodiescan only be implemented realistically with a considerably

small meshed surface grid The same effect shows in theillumination condition calculations where the shape of theshadow improves with a higher number of elements On theother hand computation time considerably increases witha finer mesh Here the computation of shadowing is thedominant effect Every surface element has to be checkedfor shadowing considering its orientation and position withrespect to each other surface element included in the modelBesides the obvious quadratic increase of the number ofindividual computation steps also the size of the data matri-ces needed to store the shadowing information increasesat the same rate As a consequence a trade-off betweencomputational resources available and accuracy demands hasto be made Keeping in mind that the actual illuminationcondition has to be recalculated for different orientations ofthe satellite to the Sun the surface mesh has to be chosensuch that acceptable computation times can be realizedwhile the quality of the illumination implementation is notcompromised A suitable method to obtain a good trade-off is to calculate the projected illuminated area surfaceat a steep illumination angle (consequently causing longshadows) for different mesh qualities Figure 7 shows theresulting calculated illuminated surface area for a differentnumber of surface elements The mean element edge length119897119890

is used as a mesh criterium and ranges from 50 cm to2 cm resembling the range of the displayed model number 119899(with 119899 = 1 25) where 119897

119890

= 1(119899 sdot 2) Keeping in mind


z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y

minus150minus100 minus50 0 50 100 150120593 (deg)

1500 50 100

120579 (deg)

(a) Disk

z

x

y

minus150 minus100 minus50 0 50 100 150120593 (deg)

150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM

Figure 4 Upper part (a b c and d) different geometry models for the SRP computation with same projected area in 119909-direction Lower part(a b c and d) projected area (119860proj = cos(120579)119860) as function of incident sunray in polar coordinates 120593 and 120579 for the corresponding modelsand the sphere model (constant level)


Radiator surfacesKevlarPolished Aluminum

Solar Panel FrontWhite PaintBlack Paint

Multilayer Insulation

Figure 5 Definition of surface materials in the FE model ofMICROSCOPE

14

13

12

11

1

09

08

07

06

minus150minus100 minus50

times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)

Figure 6 ||SRP as function of incident sunray in polar coordinates120593 and 120579 with a constant value of 119875 for the FE model of MICRO-SCOPE

the bus size of MICROSCOPE (eg +119883 bus side of about11m times 08m) the first model (119899 = 1) translates to fourelements on the +119883 bus face while the last model (119899 = 25)translates to more than two thousand elements on the samesurface A mean value of the obtained area is displayed as agrey dashed line As can be seen the solution converges closeto the mean value for a fine mesh However reducing theelement edge size does not directly lead to a better surfacearea result The actual element size depends (i) on the regionboundary lines (ii) on the meshing sequences and (iii) onthe free parameters specified by the meshing tool (ANSYSclassic preprocessor) The exact values for these parametersmay vary for different element edge lengths When processedfor different illumination angles a mean element edge size119897119890

= 625 cm shows projected areas close to the arithmeticmean Additionally for this chosen value of 119897

119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n

Number of comp stepsBaseline model

Apr

oj(m

2)

Arithmetic mean of Aproj

Aproj

Figure 7 Investigation of convergence for the projected area of theFE model The arithmetic mean of the projected area is representedby the dashed gray line Additionally the appropriate numbers ofcomputational steps are shownThe black asterisk (together with thelight gray dashed line) marks the model of choice

time for a complete assessment of all illumination conditionswith a 5∘ resolution in elevation and azimuth angle is inthe range of 30min for a conventional desktop PC which isstill acceptable Consequently the mesh resulting from 119897

119890

=

625 cm is the baseline for all further calculations in this workHowever since the optimal configuration highly depends onthe actual satellite shape and the positions of its componentsan optimal mesh has to be assessed for each new satellite thathas to be processed

43 Combined Effect of SRP and Geometry Models As seenabove both the geometrical dependency of the SRP force andthe dynamical behaviour of119875SRP determine the resulting totalSRP force acting on the satellite Consequently we investigatethe behaviour of the SRP force acting onMICROSCOPEwithboth effects included in the modelling approach In Figure 8each line in both pictures represents the absolute value ofSRP for a specific 119890Sun that is it is assumed that the satellitersquosorientation is fixed with respect to the Sun over one yearIn the top picture the outcome for normal incident sunlightfor each satellite side is depicted In order to compare morerealistic illumination conditions we considered deviationsfrom the normal vector of the solar panel side of 15∘ and 30

∘respectively which is depicted at the bottom of Figure 8

This resembles the range that is expected for MICRO-SCOPE Naturally all lines show the same characteristics dueto the variation of 119875SRP over the year which yield a maximumdifference in magnitude of about 7 However the influenceof the satellitersquos attitude might result in larger differences forexample 13 for plusmn119883 as seen above For the MICROSCOPEcase differences of roughly 1 are obtained for deviationsfrom the normal axis of +119883 of 15∘ and 9 for deviations of30∘


6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

Mar 2016 June 2016 Sept 2016 Dec 2016Time (month)

+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)

Time (month)Mar 2016 June 2016 Sept 2016 Dec 2016

118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)

Figure 8 Absolute value of disturbance force due to SRP for different illumination conditions depicted over one year (a) Perpendicular solarirradiation of each satellite side (b) Illumination conditions estimated for MICROSCOPE

Finally a simulation of the MICROSCOPE orbit for asimulation time of one year was carried out with all fivegeometry models Figure 9 shows the resulting evolution of||SRP

Only for the sphere model the effect of the changingdistance between Sun and Earth becomes visible For allother models the variation of the incident sunlight is thedominating effect For the FE model the force differs about16 from the maximum in winter to the minimum insummer Besides Figure 9 shows an unexpected result thedisk model performs better for the MICROSCOPE scenariowhich is in contrast to the assumption that aWing-Boxmodelwill resemble the results of a FE model best (according to theprojected area in Figure 4)

Furthermore there are steep changes in the evolutionof the resulting SRP force that only appear for the Wing-Box model Figure 10 reveals the problem that occurs forthis modelling approach Here the calculated illuminationconditions for both the Wing-Box and the FE model aredepicted for two different dates The first one is chosen atthe end of April 2016 right before the steep decrease (upperrow) and the second one only a few days later directly afterthis strong decrease (lower row) The chosen scenarios aremarked with black asterisks in Figure 9 For the FEmodel theshaded area (red elements) changes little due to the modifiedincoming sunlight But for the Wing-Box model the sidepanel changes from fully sunlit to completely shaded andtherefore does not contribute to the force anymore Sucheffects cannot appear for the disk model which yields asmoother evolution of the force This outcome emphasizesthat each scenario has to be investigated individually in orderto obtain the best result

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05

Sept 2016 June 2016 Sept 2016 Dec 2016Time (month)

FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|

Figure 9 Resulting disturbance force due to SRP for differentgeometry models over one year The yellow bar marks the time ofeclipse

44 Degradation Influence In the sections above it wasassumed that the optical properties do not change duringthe mission In a more realistic approach accurate SRPmodels also have to allow for surface degradation effectsthat occur when external surfaces are exposed to the spaceenvironment Material degradation may effectively changethe resulting SRP force magnitude and orientation Differentmaterials show different sensitivity and different degradation


(a) (b)

Figure 10 Illumination conditions of Wing-Box model (a) and FE model (b) for different dates of the simulated MICROSCOPE scenarioUpper row end of April 2016 lower row a few days later in May 2016 Yellow elements are in full sunlight blue elements are not exposed tothe Sun and red elements are shadowed by other parts of the satellite

behaviour with respect to atomic oxygen space debris radi-ation and thermal cycles [15] However for most materialsused in space the mean coefficient of absorptivity (withrespect to the solar spectrum) will increase over time whilethe mean coefficient of emissivity will not show a drasticchange

In order to test the influence of a degradation of theoptical properties of external surfaces to the resulting SRPa variation of solar absorptivity over mission lifetime isconsidered Again MICROSCOPE is used as test case Inorder to define amodel for the degradation rate a logarithmicevolution of the absorptivity is considered Assuming thatsurface degradation leads to a microscopic cratering effec-tively the increase of absorptivity will depend on the increaseof surface area resulting from the roughened surface As aconsequence the rate of change in 120572 will be high during thefirst months of mission and decrease over mission time Asuitable model for this behaviour is a reciprocal dependencyof the time derivative of the mean coefficient of absorptivity120572 on the time 119905

119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)

where 119879 is the total mission lifetime and 119901 is the degradationrate scaling factor The begin-of-life (BOL) and end-of-life

Table 2 Considered BOL and EOL values for 120572 120576 120574119878

and 120574119863

MLIMultilayer Insulation SPF Solar Panel Front WPWhite Paint KVKevlar PA Polished Aluminum RAD Radiator surface and BPBlack Paint

Component MLI SPF WP KV PA RAD BP120572BOL 042 092 024 073 012 008 097120572EOL 05 092 030 073 015 018 097120576 084 085 088 092 004 08 089120574119878BOL 029 00727 038 02455 08 046 0015120574119878EOL 0071 005 0099 0168 053 0116 0004120574119863BOL 029 0007 038 0025 008 046 0015120574119863EOL 0429 0030 0601 0102 03204 0704 00257

(EOL) properties as given by the MICROSCOPE missiondefinition [16ndash18] are listed in Table 2 Note that the specifiedvalues for BOLEOL values of specular and diffuse reflectivityare modeled values since no actual data on their properties isavailable

Since the total coefficient of reflectivity is given by120574 = 120572 (assuming nontransparent surfaces) (8) can also beapplied for an assessment of the evolution of the coefficientof reflectivity However not only the total magnitude ofreflection but also the ratio between specular and diffusereflections may change The individual evolution of 120574

119878

and120574119863

depends on the BOL properties of the respective surfacematerial and the actual conditions experienced in space Dueto the lack of actual data we use a model with a qualitativeapproach Since a roughening of a smooth surface causes adrop in specular reflectivity the ratio between specular and


0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055

0 2 4 6 8 10 12 14 16 18Months in orbit (mdash)


Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

Figure 11 Time evolution of 120574119878

and 120574119863

for different scaling factors MLI values considered for starting reflectivities with a 120583119878119863

of 1

diffuse reflectivities is scaled with an exponential law by thetime of duration in orbit

120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)

where 120582 is a scaling factor for the rate of change fromspecular to diffuse reflectivity Due to lack of actual dataall polished and metal surfaces are assumed to be nearlyperfect specular reflectors at BOL (120583

119878119863

= 10) while MLIis considered to start at a 120583

119878119863

of 1 motivated by the typicalcrinkled surface structure of MLI Painted surfaces (such asthe rear of the solar panel) also start at a 120583

119878119863

of 1 consideringa tarnished coating The EOLBOL values for specular anddiffuse reflectivity as listed in Table 2 are given by

120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)

Thus the coefficients of reflectivity evolve to

120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)

The scaling factor 120582 is now used to model a faster or slowerchanging ratio between specular and diffuse reflections As anexample the evolution of diffuse and specular reflectivity forthe MLI values specified in Table 2 is displayed in Figure 11

For the MICROSCOPE case a moderate change towardsdiffuse reflection is considered Therefore a scaling factor of120582 = 01 has been chosen to obtain the EOL values of diffuseand specular reflectivity as listed in Table 2 Figure 12 showsthe evolution of the coefficient of absorptivity for externalcomponents subjected to degradation following the modeldescribed in this section The assignment of material models

0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)

Figure 12 Considered variation of solar absorptivity over missionlifetime of 18 months for chosen external components of MICRO-SCOPE test case MLI Multilayer Insulation SPF Solar PanelFront WPWhite Paint KV Kevlar PA Polished Aluminum RADRadiator surface and BP Black Paint

to individual components is depicted in Figure 5 Figure 13shows the resulting evolution of diffuse and specular reflectiv-ity for 120582 and 120583

119878119863

values as discussed above As a consequenceof this reflectionmodel a decrease of SRP force over time canbe expected following (5)

However one has to keep in mind that the actual illumi-nation condition also affects the resulting force magnitudeThus a change of satellite attitude and position over timemaylead to a different trend

The resulting SRP force is calculated with a time reso-lution of 1 month Here the position of the MICROSCOPE


0

01

02

03

04

05

06

07

08

0 2 4 6 8 10 12 14 16 18Months in orbit [mdash]

MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)

Figure 13 Considered variation of specular (a) and diffuse (b) reflectivity over mission lifetime of 18 months for chosen external componentsof MICROSCOPE test case MLI Multilayer Insulation SPF Solar Panel Front WPWhite Paint KV Kevlar PA Polished Aluminum RADRadiator surface and BP Black Paint

spacecraft with respect to Sun and Earth as well as thespacecraft attitude is fixed that is 119875 and 119890Sun are constant foreach investigated case

Figure 14 shows the obtained results Again perpendic-ular solar irradiation for each satellite side and illuminationconditions estimated for MICROSCOPE were chosen Thepicture in the center of Figure 14 shows that the forcedecreases and deviates about 5 compared to the BOL valuefor the solar panel side +119883 For all other sides the effect iseven stronger This is due to the fact that the degradationof the solar panels which are the main contributors of side+119883 is small compared to all other materials At the bottomof Figure 14 one can see that the degradation effect is lessstrong for combinations of 120593 and 120579 that form a deviationof 15∘ from the normal axis of +119883 compared to the resultfor normal incident sunlight For deviations about 30∘ theforce even increases during lifetime In all cases differentsides of the satellite contribute to the value of the SRP forcewhereas the dominating effect of the solar panels decreaseswith increasing deviation from the +119883 normal axis

Finally in Figure 15 the degradation effect is applied to theMICROSCOPE scenario for the FE model For comparisonthe evolution of the forcewithout degradation is also depicted(cf Figure 9)The value of the force includingmaterial degra-dation was only evaluated at one point per month because anintegrated degradation algorithm in the simulation processhas not been implemented so far Nevertheless the figureshows that omitting the degradation effect will lead to anover- or underestimation of the actual SRP force

45 Benefits for MICROSCOPE The analysis presented hereshows that a thorough assessment of the influence of SRPis highly relevant with respect to the main scientific goal

of MICROSCOPE The goal of the mission is to detect adifferential acceleration signal at the orbit frequency 119891orbit inthe inertial pointing mode which would imply a violationof the EP At the targeted accuracy of the evaluation of thepossible EP violation (10minus15) the science data has to becleaned from residual accelerations larger then 10

minus12ms2especially at the frequency 119891orbit In order to realize thisMICROSCOPErsquos AOCS is based on a drag-free conceptwhich keeps the spacecraft in the favored state Howeverone has to consider several effects that may lead to exter-nal disturbances influencing the internal inertial sensorsregardless of the drag-free control On the one hand timedelays in controller and actuator responses may cause aninfluence of external disturbances on the science signal whenthe satellite state changes with a high rate (as in tumblingor when the satellite entersleaves eclipse) On the otherhand the inertial sensors are not completely decoupledfromexternal accelerations sincemisalignments anddifferentresponse times of the sensor components cause residualinternal accelerations affecting the measurement in the rangeof the orbit frequency Since themagnitude of the SRP force isin the range of 10minus5N which results in accelerations of about10minus8ms2 residual effects on the science signal cannot be

neglected completely Any SRP residual effect will show upat a frequency of (119891orbit + Δ119891) where Δ119891 is a phase differencecaused by the Earthrsquos orbit around the Sun As a consequenceit might be possible to mitigate this influence by analysinga long time span of science data However this is subject tofurther investigation

Apart from these considerations regarding the later scien-tific data analysis a detailed modelling of the correspondingSRP forces and torques is also important to estimate theactual needed control forces of the AOCS for an evaluation


06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100

Months in orbit (mdash)

+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)

Figure 14 Influence of material degradation on disturbance forcedue to SRP (a) Absolute value for perpendicular solar irradiationof each satellite side (b) Corresponding percentage deviation fromBOL value (c) Percentage deviation from BOL value for illumina-tion conditions that are estimated for MICROSCOPE

100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05


Without degradationWith degradation

SRP

(N)

|F|

Figure 15 Comparison ||SRP for the FE model with and withoutdegradation for the MICROSCOPE scenario

of its performance The analysis procedure described in thispaper reveals explicitly that an incomplete information on theSRP disturbance effect only allows for the identification of itsfrequency But an additional determination of its magnitudefails if no effort is put into detailed surface modellingBoth pieces of information are needed to obtain a goodknowledge of the actual satellite state and thus provide thepossibility of for example taking into account cross-couplingeffects between the sensors due to residual accelerations Asa consequence the resulting disturbances due to the herestudied effects cannot be neglected at the desired level ofMICROSCOPE measurement accuracy

5 Conclusion

The modelling of a realistic disturbance force due to SRPis a complex task which involves a multitude of modellingand simulation steps In our study we used an algorithmfor computing the SRP force which utilises a FE model forestimating the satellitersquos dimensions and surface propertiesinstead of commonly used Cannonball or Wing-Box modelsThis algorithm is embedded in the simulation software HPSthat amongst others propagates the satellite orbit and attitudeMotivated by the high requirements for attitude precision theMICROSCOPEmission served as example for the parameteranalysis of the different contributors to the SRP force Thisstudy reveals that the analysed NGD cannot be neglected atthe desired level of MICROSCOPE measurement accuracyFor this mission case example the magnitude of 119875SRP variesthroughout the year about 7 which is a typical value formany satellite missions For comparison different simplegeometry models of MICROSCOPE were used in additionto the FE model It was shown that the resulting SRP forcevaries due to the yearly changes of the magnitude of the SRPpressure119875SRP whichwerementioned in this paragraph beforeFurthermore it was shown that a second effect appears whichdepends strongly (i) on the geometry model of choice and


(ii) on the satellitersquos orientation This second effect is muchstronger than the impact of the yearly variation of 119875SRP Incase of MICROSCOPE the difference between Solar PanelFront and rear for example amounts to approximately 13for the FE model Although the satellitersquos solar panel willalways point to Sun direction the incident angle will changeduring the year which yields to variations of the SRP force ofat least 9 Combining the results for SRP and the FE modela difference of 16 in the magnitude of the SRP force can beexpected for MICROSCOPE over one year The comparisonwith other geometry models revealed that from the range ofldquosimplerdquo approaches a disk approach resembles the results ofthe FE model the most Another point of the study was theinfluence of the material degradation It was shown that forthe solar panel side the force decreases and deviates about 5from the BOL value at the end of the mission Depending onthe incident angle the degradation effect can even result in anincreasing force over time In summary the study reveals thata simple answer to the question of the main contributor ofSRP force cannot be given easily but depends on the actualmission scenario However the introduced SRP modellingapproach based on FE models enables the highest modellingprecision compared to conventional approaches but alsoimplies a considerably high modelling effort Therefore thebest approach shall be chosen bymeans of a trade-off betweenneeded SRP force accuracy and resources at hand

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors like to thank their colleagues in the MICRO-SCOPE teams from ONERA from Observatoire de la CotedrsquoAzur and from CNES for their support Additionally theauthors like to thank thier colleagues Rene Schwarz andStephan Theil from the DLR Institute of Space Systemsfor the fruitful cooperation on the HPS tool This workis supported by the German Space Agency of DLR withfunds of the BMWi (FKZ 50 OY 1305) and by the DeutscheForschungsgemeinschaft DFG (LA 90512-1)

References

[1] J McMahon ldquoImproving orbit determination with non-cannonball solar radiation pressure modelsrdquo in Proceedingsof the AASAIAA Astrodynamics Specialist Conference HiltonHead Island SC USA August 2013

[2] J WMcMahon and D J Scheeres ldquoImproving space object cat-alog maintenance through advances in solar radiation pressuremodelingrdquo Journal of Guidance Control and Dynamics vol 38no 8 pp 1366ndash1381 2015

[3] B Rievers and C Lammerzahl ldquoHigh precision thermal model-ing of complex systemswith application to the flyby andPioneeranomalyrdquo Annalen der Physik vol 523 no 6 pp 439ndash449 2011

[4] J W McMahon and D J Scheeres ldquoNew solar radiationpressure force model for navigationrdquo Journal of GuidanceControl and Dynamics vol 33 no 5 pp 1418ndash1428 2010

[5] O Montenbruck U Hugentobler R Dach P Steigenbergerand A Hauschild ldquoApparent clock variations of the block IIF-1(SVN62)GPS satelliterdquoGPS Solutions vol 16 no 3 pp 303ndash3132012

[6] O Montenbruck P Steigenberger and U HugentoblerldquoEnhanced solar radiation pressure modeling for Galileosatellitesrdquo Journal of Geodesy vol 89 no 3 pp 283ndash297 2015

[7] J Berge P Touboul and M Rodrigues ldquoStatus of MICRO-SCOPE a mission to test the equivalence principle in spacerdquoJournal of Physics Conference Series vol 610 Article ID 0120092015

[8] P Touboul G Metris V Lebat and A Robert ldquoThe MICRO-SCOPE experiment ready for the in-orbit test of the equiva-lence principlerdquo Classical and Quantum Gravity vol 29 no 18Article ID 184010 2012

[9] M List S Bremer and C Lammerzahl ldquoAdaption of HPSto the MICROSCOPE missionrdquo in Proceedings of the 62ndInternational Astronautical Congress Cape Town South Africa2011

[10] J R Wertz Ed Spacecraft Attitude Determination and ControlKluwer Academic Publishers Dordrecht The Netherlands1978

[11] S Bremer M List H Selig H Rath and H Dittus ldquoModellingand simulation of the space mission MICROSCOPErdquo ActaAstronautica vol 68 no 1-2 pp 28ndash33 2011

[12] Y-M Wang J L Lean and N R Sheeley Jr ldquoModeling thesunrsquosmagnetic field and irradiance since 1713rdquoTheAstrophysicalJournal vol 625 no 1 pp 522ndash538 2005

[13] D P Rubincam ldquoLAGEOS orbit decay due to infrared radiationfrom earthrdquo Journal of Geophysical Research vol 92 no 2 pp1287ndash1294 1987

[14] C J Rodriguez-Solano U Hugentobler and P SteigenbergerldquoAdjustable box-wing model for solar radiation pressureimpacting GPS satellitesrdquo Advances in Space Research vol 49no 7 pp 1113ndash1128 2012

[15] S K R Miller and B Banks ldquoDegradation of spacecraftmaterials in the space environmentrdquo MRS Bulletin vol 35 no1 pp 20ndash24 2010

[16] P Touboul and G Metris ldquoMICROSCOPE mission scenariordquoin Proceedings of the Testing the Equivalence Principle MICRO-SCOPE Colloquium II Palaiseau France January 2013

[17] M Bach ldquoMICROSCOPEmissionamp satellite configurationrdquo inTesting the Equivalence Principle MICROSCOPE ColloquiumII Palaiseau France 2013

[18] C Ingenierie and E Bellouard ldquoMicroscope thermal con-trol definition documentrdquo Tech Rep (MIC-DC-S-1-459-CNS)CNES 2006

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



solar panel assemblyThe high gain in modelling accuracy bymeans of a realistic implementation of the satellite geometryhas also been demonstrated with an analysis of NGDs actingduring the cruise phases of the ESA Rosetta spacecraft [3]Here a nonphysical solar constant was measured resultingfrom a parametric fit of the measured contribution of SRPon the total acceleration By means of a sophisticated SRPand thermal radiation pressure (TRP) (TRP results fromphotons emitted by the spacecraft itself) model this offset wasexplained as a nonmodeled TRP correlated with the actingSRP Further examples for a successful implementation ofenhanced SRP models are GNSS satellites where navigationaccuracy directly benefits from an improved SRP modellingapproach [4ndash6]

The modelling effort for an accurate analysis of the SRPeffect on a given satellite is considerably high Consequently atrade-off has to be made between the precision requirementsfor the specific mission the effort that one is willing to takeand the possible gain with respect to an improvement of aprecise implementation of NGDs This paper intends to givean overview on the implications of accurate SRP modellingand the expected improvement of NGD implementationTheparameters for the subsequent SRP analysis are derived fromthe French spacemissionMICROSCOPE [7] which delivers asuitable test case with respect to the specified mission profile

The MICROSCOPE mission requires a very high accu-racy of the spacecraft attitude and attitude stability due tothe specific mission specification In order to realize the highperformance of the differential acceleration measurement ofthe two test masses to test the Weak Equivalence Principle(EP) it is essential to ensure a very low disturbance level(forces and torques acting on the satellite) For this purposeMICROSCOPE will be operated in drag-free mode Anydisturbance will be compensated by forces and torquesgenerated by a cold gas propulsion system in closed loopcontrol The input to the corresponding controller is givenby the common mode acceleration signal of the differentialaccelerometer while the science signal is extracted fromthe differential acceleration signal However in spite ofthe drag-free control the exact modelling of NGDs is stillnecessary due to couplings between the accelerometers andthe satellite structure As a consequence external disturbanceeffects influence the scientific signals since the drag-freecontrol forces and torques needed to compensate the NGDsintroduce a disturbance translated by the coupling

The actual requirements of MICROSCOPE are quitedemanding For the EP measurement sessions the residualacceleration of the spacecraft shall be less than 10

minus12msminus2 Atthe EP test frequency the angular pointing stability shouldbe better than 7 120583rad and the angular velocity stability isrequired to not exceed 10

minus9 radsminus1 respectively [8]The sun-synchronous polar MICROSCOPE orbit leads

to a force vector due to SRP always directed to one side ofthe orbital plane This leads to a linear acceleration normalto the orbital plane which is superimposed by an angularvariation due to seasonal change of the angle between theorbital plane normal and the direction to the sun In caseof simulating a drag-free mission a detailed modelling of

the corresponding SRP forces and torques is important toestimate the actual control forces of the Attitude and OrbitControl System (AOCS) which keep the spacecraft in thefavored state Considering MICROSCOPE NGD effects dueto SRP can easily reach several 120583N Divided by the satellitersquosmass (330 kg) this force induces disturbing accelerations ofsome 10minus8msminus2 which is not negligible in a premission end-to-end simulation and for developing and implementing dataanalysis and data processing strategies

Due to the high demands of the mission the sun-synchronous orbital plane and the LEOcharacter of the orbitMICROSCOPE is an ideal test case scenario for the analysisof the general benefit of accurate SRP modelling for spacemissions After a general introduction of the SRP modellingmethod and the derivation of SRP characteristics for chosenorbit and mission examples we will use MICROSCOPE as atest case scenario for a detailed SRP analysis By looking atdifferent approaches for the implementation of the geometryof the satellite and applying surface degradation models wehighlight the possible benefits and the involved costs of highaccuracy SRP modelling

2 Orbit and Attitude Propagation

Since the evolving SRPmagnitude and orientation depend onthe position and the attitude of the spacecraft a dynamic orbitsimulation including the gravitational acceleration caused bythe Earthrsquos gravitational field is necessary The calculationof the gravitational influence and the integration of theequation of motion are realized within the framework of thegeneric simulation tool High Performance Satellite DynamicsSimulator (HPS) [9]TheHPS is aMATLABSimulink librarywhich is developed at ZARM in cooperation with the DLRInstitute of Space Systems Bremen The main focus of HPSis the propagation of satellite orbits and the computationof the satellitersquos orientation depending on specific initialconditions and the space environment Furthermore thecoupledmotion of up to eight on-board testmasses (arrangedpairwise in up to four accelerometers) can be computed in sixdegrees of freedom

Coupling effects between the satellite and the test massesas well as among the test masses themselves are includedin the implemented differential equation systems of eachconsidered body In the following the satellitersquos equations ofmotion are shown exemplarily The satellite motion is givenby (1)

997888rarr119903

119894

119894119887

= 119892119894

119894119887

( 119903119894

119894119887

) +

119894

control + 119894

dist + 119894

couplsat

119898sat (1)

Here (i) 119898sat is the mass of the satellite (ii) 997888rarr119903

119894

119894119887

is theacceleration of the satellite relative to the ECI frame (iii)119892119894

119894119887

( 119903119894

119894119887

) is the gravitational acceleration (iv) 119894control is thecontrol force (v) 119894dist is the sum of all disturbance forcesacting on the satellite and (vi) 119894couplsat is the force due to thecoupling between the satellite and all considered test massesThe superscript 119894 indicates that all components of (1) are givenin ECI coordinates



119887

119894119887

= (I119887119887

)minus1

[119887

control + 119887

dist + 119887


119894119887

times (I119887119887

119887

119894119887

)]

(2)

q119887119894

=1

2

997888rarr

119887

119894119887

⊙ q119887119894

(3)

Here (i) 120596119887119894119887


119887


119887


119887



119887

119894119887


119894119887

and (vii) q119887119894



3 SRP Model


119889120572

= minus119875SRP120572 cos (120579) 119890Sun119889119860

119889120574119878= minus2119875SRP120574119878cos

2

(120579) 119890119873

119889119860

119889120574119863

= 119875SRP120574119863 (minus2

3119890119873

minus 119890Sun) cos (120579) 119889119860

(4)


119873


119873

Finally (v) 120572 120574119878

and 120574119863


With120572+120574119878

+120574119863



) 119890Sun

+ 2 (120574119878

cos (120579) + 1

3120574119863

) 119890119873

] cos (120579) 119889119860(5)




119873


of 120574119878

120574119863

119890119873

and 119860



119894


119894

= minus119875SRP [(1 minus 120574119878119894

) 119890Sun

+ 2 (120574119878119894

cos (120579119894

) +1

3120574119863119894

) 119890119873119894

] cos (120579119894

) 119860119894

(6)


SRP = sum

119894

119894

(7)











Mar

201

6

June

201

6

Sept

201

6

Dec

201

6

Mar

201

7

June

201

7


475e minus 06

47e minus 06

465e minus 06

46e minus 06

455e minus 06

45e minus 06

445e minus 06

PSR

P(N

m2)



CHAMPMICROSCOPE

MeteosatGALILEO

PSR

P(N

m2)

4421e minus 06

442e minus 06

4419e minus 06

4418e minus 06

4417e minus 06

4416e minus 06

20162016








GEO 8

Examples




SSO 31

Examples


MEO 7









X

Y

Orbit normal

Polar coordinates

X

Y

Z

Z

rsunb

rsunECI

ZECI

XECI

YECI

120579

120593












119890



z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y


1500 50 100

120579 (deg)

(a) Disk

z

x

y


150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM







14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119887

119894119887

= (I119887119887

)minus1

[119887

control + 119887

dist + 119887


119894119887

times (I119887119887

119887

119894119887

)]

(2)

q119887119894

=1

2

997888rarr

119887

119894119887

⊙ q119887119894

(3)

Here (i) 120596119887119894119887


119887


119887


119887



119887

119894119887


119894119887

and (vii) q119887119894



3 SRP Model


119889120572

= minus119875SRP120572 cos (120579) 119890Sun119889119860

119889120574119878= minus2119875SRP120574119878cos

2

(120579) 119890119873

119889119860

119889120574119863

= 119875SRP120574119863 (minus2

3119890119873

minus 119890Sun) cos (120579) 119889119860

(4)


119873


119873

Finally (v) 120572 120574119878

and 120574119863


With120572+120574119878

+120574119863



) 119890Sun

+ 2 (120574119878

cos (120579) + 1

3120574119863

) 119890119873

] cos (120579) 119889119860(5)




119873


of 120574119878

120574119863

119890119873

and 119860



119894


119894

= minus119875SRP [(1 minus 120574119878119894

) 119890Sun

+ 2 (120574119878119894

cos (120579119894

) +1

3120574119863119894

) 119890119873119894

] cos (120579119894

) 119860119894

(6)


SRP = sum

119894

119894

(7)











Mar

201

6

June

201

6

Sept

201

6

Dec

201

6

Mar

201

7

June

201

7


475e minus 06

47e minus 06

465e minus 06

46e minus 06

455e minus 06

45e minus 06

445e minus 06

PSR

P(N

m2)



CHAMPMICROSCOPE

MeteosatGALILEO

PSR

P(N

m2)

4421e minus 06

442e minus 06

4419e minus 06

4418e minus 06

4417e minus 06

4416e minus 06

20162016








GEO 8

Examples




SSO 31

Examples


MEO 7









X

Y

Orbit normal

Polar coordinates

X

Y

Z

Z

rsunb

rsunECI

ZECI

XECI

YECI

120579

120593












119890



z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y


1500 50 100

120579 (deg)

(a) Disk

z

x

y


150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM







14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















Mar

201

6

June

201

6

Sept

201

6

Dec

201

6

Mar

201

7

June

201

7


475e minus 06

47e minus 06

465e minus 06

46e minus 06

455e minus 06

45e minus 06

445e minus 06

PSR

P(N

m2)



CHAMPMICROSCOPE

MeteosatGALILEO

PSR

P(N

m2)

4421e minus 06

442e minus 06

4419e minus 06

4418e minus 06

4417e minus 06

4416e minus 06

20162016








GEO 8

Examples




SSO 31

Examples


MEO 7









X

Y

Orbit normal

Polar coordinates

X

Y

Z

Z

rsunb

rsunECI

ZECI

XECI

YECI

120579

120593












119890



z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y


1500 50 100

120579 (deg)

(a) Disk

z

x

y


150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM







14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













GEO 8

Examples




SSO 31

Examples


MEO 7









X

Y

Orbit normal

Polar coordinates

X

Y

Z

Z

rsunb

rsunECI

ZECI

XECI

YECI

120579

120593












119890



z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y


1500 50 100

120579 (deg)

(a) Disk

z

x

y


150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM







14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










X

Y

Orbit normal

Polar coordinates

X

Y

Z

Z

rsunb

rsunECI

ZECI

XECI

YECI

120579

120593












119890



z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y


1500 50 100

120579 (deg)

(a) Disk

z

x

y


150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM







14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

2

18

16

14

12

1

08

06

04

02

0

Apr

oj(m

2)

x

y


1500 50 100

120579 (deg)

(a) Disk

z

x

y


150

050

100

120579 (deg)

Apr

oj(m

2)

35

3

25

2

15

1

(b) Simple box

2

18

16

14

12

1

3

28

26

24

22

Apr

oj(m

2)


150

050

100

120579 (deg)

x

y

z

(c) Wing-Box

Apr

oj(m

2)


150

0 50100

120579 (deg)

2

18

16

14

12

1

24

22

x

y

z

(d) FEM







14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














14

13

12

11

1

09

08

07

06


times10minus5

0 50 100 150

150

050

100

120593 (deg) 120579 (deg)

|F| SRP

(N)




119890

the processing

175

180

185

190

195

200

205

210

215

1 5 10 15 20 25

50e + 08

10e + 09

15e + 09

20e + 09

Num

ber o

f com

p st

eps (

mdash)

Model number n


Apr

oj(m

2)


Aproj



119890

=






6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6e minus 06

7e minus 06

8e minus 06

9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05


+X

minusX

+Y

minusY

+Z

minusZ

14e minus 05

|F| S

RP(N

)

(a)

|F| S

RP(N

)


118e minus 05

114e minus 05

116e minus 05

112e minus 05

11e minus 05

108e minus 05

106e minus 05

104e minus 05

102e minus 05

1e minus 05

+X 120579 = 90∘ 120593 =120593 =

0∘

120579 = 75∘ 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

(b)





9e minus 06

1e minus 05

11e minus 05

12e minus 05

13e minus 05

14e minus 05

15e minus 05


FEMWing-BoxBox

DiskSphere

SRP

(N)

|F|




(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)




119889120572

119889119905= 119901 sdot

1

119905 + 1 (8)

leading to

120572 = 120572BOL + 119901 sdot ln (119905 + 1) (9)

with

119901 =120572EOL minus 120572BOLln (119879 + 1)

(10)



and 120574119863





119878

and120574119863



0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

005

01

015

02

025

03

035

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1

025

03

035

04

045

05

055



Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

120582 = 005

120582 = 001120582 = 01120582 = 025

120582 = 1


and 120574119863


of 1


120583119878119863

(119905) =120574119878BOL

120574119863BOL

sdot 119890minus120582119905

(11)


119878119863


119878119863


119878119863


120574119878

(119905) = 120583119878119863

(119905) sdot 120574119863

(119905)

1 minus 120572 (119905) = 120574119878

(119905) + 120574119863

(119905)

(12)


120574119878

(119905) = (1 minus 120572)120583119878119863

(119905)

120583119878119863

+ 1

120574119863

(119905) =1 minus 120572

1 + 120583119878119863

(119905)

(13)



0

01

02

03

04

05

06

07

08

09

1


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f abs

orpt

ivity

120572(mdash

)



119878119863





0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f spe

cula

r refl

ectiv

ity120574 S

(mdash)

(a)

0

01

02

03

04

05

06

07

08


MLISPFWPKEV

PARADBP

Coe

ffici

ent o

f diff

use r

eflec

tivity

120574 D(mdash

)

(b)











06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










06

07

08

09

1

11

12

13

14


+XminusX+Y

minusY+ZminusZ

|F| S

RPlowast10

minus5

(N)

(a)

0 2 4 6 8 10 12 14 16 1892

93

94

95

96

97

98

99

100


+XminusX+Y

minusY+ZminusZ

Dev

iatio

n fr

omSR

PBO

L(

)|F|

(b)

0 2 4 6 8 10 12 14 16 18949596979899

100101102103


Dev

iatio

n fr

omSR

PBO

L(

)

+X 120579 = 90∘ 120593 = 0∘

120579 = 75∘ 120593 = 0∘

120579 = 90∘ 120593 = 15∘

120579 = 105∘ 120593 = 0∘

120579 = 90∘ 120593 = 345∘

120579 = 60∘ 120593 = 0∘

120579 = 120∘ 120593 = 0∘

120579 = 90∘ 120593 = 330∘

120579 = 90∘ 120593 = 30∘

|F|

(c)


100e minus 05

105e minus 05

110e minus 05

115e minus 05

120e minus 05

125e minus 05



SRP

(N)

|F|



5 Conclusion






Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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