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Application of a modified generalized sail model on a solar sail with wrinkles
Available online at www.sciencedirect.com
ScienceDirect Advances in Space Research xxx (2020) xxx–xxx www.elsevier.com/locate/asr
Application of a modified generalized sail model on a solar sail with wrinkles Nikolay Nerovny a,⇑, Irina Lapina b a
Department of Spacecraft and Launch Vehicles, Bauman Moscow State Technical University, Moscow 105005, Russia b Department of Mathematical Modelling, Bauman Moscow State Technical University, Moscow 105005, Russia Received 18 October 2019; received in revised form 12 January 2020; accepted 16 January 2020
Abstract In this paper, we present an analysis of effect of wrinkles on the solar sail performance. We describe different analytical, semianalytical and numerical approaches to the calculation of general large-scale curvature of a solar sail as well as parameters of socalled wrinkled domains, and introduce the impact of such wrinkles on the thrust and torque of the solar sail. Finally, we present a model of an optically-orthotropic surface for such non-ideal sail, providing a connection with the Generalized Sail Model, and other solar sail thrust models. Ó 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Solar sail; Radiation pressure; Wrinkles; Generalized sail model
1. Introduction
E F ¼ C R A ^s; c
There are various models of light pressure (solar radiation pressure, SRP) on objects in deep space. From the engineering point of view, the models of light pressure can be divided into three categories. The first category includes a priori physical models of the interaction of radiation with a surface. Taking into account the shape and optical properties of the surface, it is possible to calculate the principal vector and torque of light pressure, or even consider a problem of helioelasticity. These models include a ‘‘Cannonball” model, where the object is presented as a flat mirror plate, always turned towards the sun by one side, and the light falls on it perpendicularly to the surface (Lucchesi, 2001; McInnes, 2004). Generally speaking, this model may be formulated as follows:
Where C R – some experimental constant describing the reflection properties and shape of the body; A – cross section of the body; E – solar flux; c – light speed in vacuum; ^s – unit vector of direction from light source to the body. For the ideal flat solar sail, C R ¼ 1 þ q, where q – total reflectivity. Within the framework of this model, it is impossible to solve the problem of determining the resultant torque of SRP if the center of gravity of the spacecraft coincides with its geometric center. There is also a model of a flat solar sail whose surfaces have specular-diffuse properties. This model can be analytically formulated using the following equation: 2 F ¼ A Ec a1 ð^n ^sÞ^s þ a2 ð^n ^sÞ^n 2a3 ð^n ^sÞ ^n ; a1 ¼ 1 qs;
⇑ Corresponding author.
E-mail address: [email protected] (N. Nerovny).
a2 ¼ Bf qð1 sÞ þ ð1 qÞ ef Bf eb Bb = ef þ eb ; a3 ¼ qs;
https://doi.org/10.1016/j.asr.2020.01.020 0273-1177/Ó 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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Nomenclature Xw wrinkled domain of a surface ^e0x ; ^e0y ; ^e0z orientation unit vectors for a global frame ^ex ; ^ey ; ^ez orientation unit vectors for a local frame r position vector of dA in a global frame ^ m unit vector of the orientation of optical axis ^ n unit vector of the local normal to dA ^s unit vector from the light source to dA Bm ; Bq Lambertian coefficients s specularity q total reflectivity e emissivity k model parameter for back reflection
where ^ n – unit vector of local normal to the surface; s – specularity coefficient; ef ; eb – emissivity for the front and back surfaces; Bf ; Bb – Lambertian coefficient for front and back surfaces. There is a problem with this model – eventually, the configuration of surfaces may change: the front surface may be shadowed, and the back one may be lightened. This model is often used to determine the SRP on bodies that consist of several planar segments, the so-called macromodels (Marshall and Luthcke, 1994). For these models, one can determine the resultant vector of SRP as follows: F¼
N X Fi H i ð^sÞ; i¼1
where Fi – light pressure on facet i, the total number of facets is N; H i – visibility function that shows if the given facet is lightened or not. For bodies of complex shape, it is also possible to use a specular-diffuse model by integrating expressions for the light pressure over the entire surface. A generalized model of the solar sail (Generalized Sail Model, GSM) was obtained using this approach (Rios-Reyes and Scheeres, 2005; Rios-Reyes, 2006; Jing et al., 2012, 2014): F ¼ A I 2 ^s þ ^s I 3 ^s ; where I 2 ; I 3 – some tensors of rank 2 and 3 respectively that describe the combination of shape and reflection properties of a body. This model works only when the lightened surface of a body does not change in time and only for the bodies of a convex shape. This GSM may be extended in the form of tensor series (Nerovny et al., 2017) to account the non-convex bodies with arbitrary orientation towards the light source. It is also possible to calculate the SRP on a spacecraft of arbitrary type by performing ray-tracing in a way similar to that used in the calculation of radiative heat transfer by Monte Carlo methods (Ziebart, 2004; Howell and Menguc, 2015).
Superscripts S light pressure from thermal self emission A light pressure from absorption R light pressure from reflection Subscripts iso isotropic part of light pressure ortho orthotropic part of light pressure ^ 1 optical parameter among m ^ 2 optical parameter perpendicular to m
The second category includes empirical models of light pressure (Springer et al., 1999; Bar-Sever and Kuang, 2004). These models do not depend on a priori information on the nature of the interaction of radiation with the surface of the spacecraft and are necessary for accurate prediction of orbits of space vehicles from accumulated observational data. Usually these models have an analytical form of some trigonometric series. The third category includes combined models, which are based on a priori physical models of light pressure and also contain features that allow for refinement of the model parameters based on actual observations (McMahon and Scheeres, 2010, 2015). The surface of a thin film space structure may be in a planar stress state, in which there is a possibility of formation of wrinkles (Wong and Pellegrino, 2006a). These wrinkles can form a so-called wrinkled domain (Fig. 1), which can cover a significant part of a whole surface: it was shown that solar sail membrane has general curvature, both regular and semi-random (smoothness), and also small wrinkles (Alhorn et al., 2011; Tsuda et al., 2011; Kawaguchi, 2014; Ridenoure et al., 2015). The mechanical properties of thin films in planar stress state has been studied rigorously, Wong and Pellegrino (2006a,b,c). Regarding solar sails, there is a good experimental study that determines the averaged optical parameters of a model
Fig. 1. Typical exterior of a solar sail with the frame during spaceflight.
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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solar sail with wrinkles and folds (Heaton and ArtusioGlimpse, 2015). In this paper, we present a new physical model of light radiation pressure in the context of GSM, an a priori model. This paper is organized as follows. First, we present the model of SRP upon an optically-orthotropic surface, that represents wrinkles on a solar sail, which is a method we use in this paper. Second, we show that this model may be represented using the similar tensor decomposition as for a GSM. In the simulation section, we illustrate the applicability of this work by simulating of the SRP on some model surface with wrinkles. In the last section we discuss the simulation results and possible applications of described method. 2. Model of optical parameters for a surface with wrinkles This section contains a summary of a developed model of a surface with optically orthotropic parameters (Nerovny et al., 2017). The surface A is represented as a two-manifold, A M2 . Xw A is a surface domain that has wrinkles. The global frame is O0 x0 y 0 z0 with unit vectors ^e0x ; ^e0y ; ^e0z (Fig. 2). The local coordinate frame Oxyz is associated with an infinitesimal element dA Xw , the unit vectors are ^ex ; ^ey ; ^ez . The position of dA in the global frame is defined by a vector r. ^ n is a unit vector of the local normal to the surface element dA; ^n ¼ ^ez . For arbitrary vector ^r we used direction angles ðb; hÞ in the local frame as follows: b 2 ½0; p=2 — angle between the vector ^r and þz; h 2 ½0; 2p — angle between the axis Ox and a projection of ^r to the plane Oxy, counterclockwise around Oz. In this study, the incoming light flux is considered to be parallel. Some questions regarding SRP from finite Sun disc were considered previously by different authors, e.g., McInnes and Brown (1990b,a), Koblik et al. (2011) and others, but we are not aware of any works regarding
3
GSM for systems with non-point light source. The ^s is a unit vector from the light source to dA; ^sðb0 ; h0 Þ ¼ sin b0 cos h0^ex sin b0 sin h0^ey cos b0^ez . We neglect the spectral dependency of the surface’s optical parameters. Kezerashvili (2014) and Ancona and Kezerashvili (2016) studied the implications of spectral dependency of optical parameters on the efficiency of solar sails. From the practical point of view, it is possible to consider such spectral dependencies using optical coefficients integrated over real Sun spectrum, however there is still a problem with difference between direct and reflected spectrum. On the surface there are two perpendicular directions ^ is a unit vector of the with different optical parameters. m ^ ¼ orientation of optical axis in the local frame, m cos hm^ex þ sin hm^ey . In the following formulas, E is a light flux, c the light speed in a vacuum, r Stefan-Boltzmann constant, T the temperature of dA. For a solar sail case, the temperature of an element dA can be obtained by the equilibrium equation of thermal flux considering that this temperature does not change among the thickness of the film and that the thermal conductivity is substantial (Rios-Reyes, 2006). The equation for the force from the emitted light flux from the orthotropic surface is the following formula (Nerovny et al., 2017): e1 Bm rT 4 ^ndA; c e1 þ e2 Bm ¼ ; 3e1
dFS ¼
ð1Þ ð2Þ
where Bm is a modified Lambertian coefficient, e1 is an ^ and e2 is an emissivity in peremissivity in the direction m, pendicular direction in the plane of dA. The equation for the force from the absorbed light: E dFA ¼ ð^n ^sÞ^sdA c
ð3Þ
Fig. 2. Coordinate frames. Gray area is a wrinkled domain of a surface.
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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For reflection we modify Maxwell’s model of speculardiffuse reflection (Howell and Menguc, 2015). We introduce the following parameters: q1 and s1 are total reflectiv^ on the plane of dA; q2 ity and specularity in the direction m and s2 are total reflectivity and specularity in perpendicular direction. The equation for the force from diffusely reflected light for the orthotropic surface can be represented in the following vector form: dFR1 ¼
E q ð1 s1 ÞBq ð^ n ^sÞ^ ndA c 1
ð4Þ
where P is from Eq. (20), and: a 1 ¼ 1 q2 s 2 ; a2 ¼ Bq q1 ð1 s1 Þ; a3 ¼ q2 s2 ;
Radiation force upon dA from specularly reflected light flux: 2 dFR2m ¼ Ec ð^ n ^sÞ^s 2ð^ n ^sÞ ^ n ð5Þ q1 cos2 ðhm ho Þ þ q2 sin2 ðhm ho Þ s1 cos2 ðhm ho Þ þ s2 sin2 ðhm ho Þ dA We also introduce the back reflection term with an additional empirical parameter k (Nerovny et al., 2017). Radiation force from back reflection is: ð6Þ
3. Generalization 3.1. Generalized sail model In the GSM, one can analytically separate the description of surface geometry and its optical parameters from its orientation towards the light source. During this process, one can precompute some constants which describe the geometry and optical properties of surface using tensor representation (GSM, Rios-Reyes and Scheeres (2005)). After this, it is possible to calculate the light pressure on a body using scalar products of tensors by Sun vector ^s.
b1 ¼ q1 s1 þ q2 s2 q1 s2 q2 s1 ; b2 ¼ q1 s2 þ q2 s1 2q2 s2 : See Appendix A for expansion of orthotropic term into power series. Eq. (8) is a standard equation for radiation pressure upon a flat surface with specular-diffuse properties (McInnes, 2004), however the optical parameters are different. 3.3. Tensor series for an orthotropic model According to Nerovny et al. (2017), one can write the following tensor series for resultant force F and torque M from isotropic component of radiation pressure upon an optically orthotropic convex surface: ! N max X E Fiso ¼ I 1iso þ S I niso ; ^s ; r ð10Þ c n¼2 ! N max X n E 1 Kiso þ S Kiso ; ^s ; ð11Þ Miso ¼ c n¼2 where I iso and Kiso – some tensors that are only depended on geometry and optical properties of the surface and are invariant to the change of the relative position of the light source, and S is a vector operator with the following index representation: S J n ; ^b ¼ J ni1 ...in1 k bi1 . . . bip1 ; k
3.2. Generalization of an orthotropic model Combining of terms for light pressure from self emission (1), absorption (3), diffuse reflection (4), specularly reflected light pressure (5) and light pressure from back reflection (6), it is possible to write an equation for the infinitesimal radiation force as a combination of a isotropic and orthotropic parts:
where:
ð9Þ
4
ð4 3s1 s2 Þq1 þ ð4 s1 3s2 Þq2 Bq ¼ 12ð1 s1 Þq1
dF ¼ dFiso þ dFortho ;
ð8Þ
a0 ¼ e1 BmErT ;
where
dFR2b ¼ Ec k^sðq1 cos2 ðhm ho Þ þ q2 cos2 ðhm ho ÞÞ ðs1 cos2 ðhm ho Þ þ s2 cos2 ðhm ho ÞÞdA
dFiso ¼ Ec ða0 ^n a1 ð^n ^sÞ^s þ a2 ð^n ^sÞ^nþ 2 þ2a3 ð^n ^sÞ ^n dA; dFortho ¼ Ec ð^n ^sÞ^s 2ð^n ^sÞ2 ^n þ k^s ^ ^sÞ4 þ b1 1 þ 2P 2 þ P 4 ðm ^ ^sÞ2 dA; þb2 1 þ P 2 ðm
ð7Þ
where ^b is an arbitrary unit vector, i1 . . . in1 ; k ¼ x; y; z, and we used the Einstein summation convention. For an orthotropic component we used the visibility function to obtain a equation of radiation force upon dA in convex case: V ¼ H ð^n ^sÞ;
ð12Þ
where H is a Heaviside step function. The derivation of power series for (12) is presented in the Appendix B. These are the equations for orthotropic components of resultant force and torque upon a surface with wrinkles:
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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Fortho ¼
max E NX S I northo ; ^s ; c n¼4
Mortho ¼
ð13Þ
max E NX S Knortho ; ^s ; c n¼4
ð14Þ
where I ortho and Kortho are tensors that described in the Appendix C. For the non-convex structures, it is possible to approximate the components of these tensors, using leastsquares approximations following the procedure described in the article (Nerovny et al., 2017). 4. Simulation Different authors suggested that wrinkles on the thin solid film can be described using sine or cosine laws (Epstein, 2003; Wong and Pellegrino, 2006b). It was shown both numerically (Wong and Pellegrino, 2006c; Wang et al., 2009; Xiao et al., 2011), and experimentally (Wong and Pellegrino, 2006a). Regarding solar sails, wrinkles were described by Murphey et al. (2002), Zeiders (2005, 2007) and Campbell and Thomas (2014), and others. In this work, we describe the magnitude of wrinkles using the following equation: 2px02 ; k where d ¼ d x01 ; x02 – smooth field of amplitude of wrinkles over the wrinkled domain Xw ; k ¼ k x01 ; x02 – smooth field of wavelength of wrinkles over the wrinkled domain Xw . We used the followingparameters ofwrinkled surface: x01 ; x02 2 ½0:5; 0:5; x03 ¼ d cos 8px02 0:5 . There are 4 waves, wavelength k ¼ const ¼ 0:25m, amplitude d ¼ const ¼ 0:2m. Optical parameters of this surface are uniform and have the following values: reflectivity q0 ¼ 1; specularity s0 ¼ 0; Lambertian coefficient B ¼ 2=3. Each surface element is entirely diffuse. Software from Nerovny and Grigorjev (2017) performed the ray-tracing, and results are available in the Mendeley Data (Nerovny, 2017). The range for b is f9 ; 18 ; . . . ; 72 ; 81 g, and the range for h is
x03 ¼ d sin
5
f0 ; 18 ; . . . ; 342 ; 360 g. The angle definitions are the same as on Fig. 2. The number of rays was 100000 in each simulation, 189 simulations total. In our analysis, we used the R code combined with the JAGS toolkit for Markov Chain Monte Carlo (MCMC) approximation of posterior probability (Plummer, 2013). The structure of this code is similar to examples in Kruschke (2015). For the Bayesian inference, we used the following likelihood probability distribution function: 8 jFF j2 iso > 2 > > m ¼ 1; e 2rF ; > > < jFFortho j2 1 P ðdatajm; parametersÞ / exp 2r2 F ; m ¼ 2; >e rF > > > jFFGSM j2 > : 2r2 F ; m ¼ 3; e where rF – standard deviation; F – data obtained from Monte Carlo simulation; Fiso ; Fortho ; FGSM – data predicted by the corresponding model. 5. Results and discussion The summary of simulation results is presented in Table 1. On that table, Isotropic is a summary of MCMC approximation considering the isotropic model of optical parameters; values are median of posterior probability distributions, limits are 95% highest probability density intervals (HDI). Orthotropic is a summary of MCMC approximation considering an orthotropic model of optical parameters, values are medians of distributions, limits are ^ ¼ ð1; 0; 0ÞT . LS GSM is a summary of least 95% HDI, m squares approximation of tensors of extended GSM. The last row is a mean-square deviation between Monte Carlo simulated data and simulation results of a corresponding model. Isotropic and orthotropic data are from Nerovny et al. (2017). On Fig. 3 there is a graphical comparison between different models. On the figure, vertical axis is a light pressure, m2 . Horizontal axis is h. Black lines and dots – F 01 ; m2 (projection on O0 x01 of global frame). Dark gray lines and dots – F 02 ; m2 (projection on O0 x02 of global frame). Light gray
Table 1 Model parameters for isotropic and orthotropic models of SRP, and for a modified GSM. Surface with wrinkles, q0 ¼ 1; s0 ¼ 0; B0 ¼ 2=3. All values are rounded up to three significant digits after the point. The values for LS GSM case are omitted intentionally because that components does not have any good determined geometrical of physical meaning. The generative software for all of the data in analysis may be given to third-party via the private communication with the authors. Isotropic
Deviation
Orthotropic
LS GSM
Median
Upper limit
Lower limit
Median
Upper limit
Lower limit
Value
q ¼ 0:611 s ¼ 0:005 B ¼ 0:707 rF ¼ 0:190
þ0:327 þ0:018 þ0:293 þ0:021
0:196 0:005 0:240 0:018
q1 ¼ 0:104 q2 ¼ 0:369 s1 ¼ 0:677 s2 ¼ 0:331 k ¼ 2:440 rF ¼ 0:097
þ0:034 þ0:083 þ0:308 þ0:197 þ0:790 þ0:009
0:023 0:085 0:240 0:136 0:510 0:009
N max ¼ 4 J 2; J 3; J 4 (117 parameters)
0:101
0:039
0:032
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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Fig. 3. Comparison between ray-tracing results, isotropic model, orthotropic model and GSM approximation for surface with wrinkles, q ¼ 1; s ¼ 0; B ¼ 2=3.
lines and dots – F 03 ; m2 (projection on O0 x03 of global frame). Crosses – ray-tracing results. Solid lines – least squares approximation for GSM. Dashed lines – orthotropic model based on Bayesian median values of parameters (q1 ¼ 0:12; q2 ¼ 0:387; s1 ¼ 0:394; s2 ¼ 0:251; k ¼ 2:84). Dotted lines – isotropic model based on Bayesian median values of parameters (q ¼ 0:611; s ¼ 0:005; B ¼ 0:707). We showed that least squares approximation of modified GSM has the minimal deviation compared with the ray-tracing results, the classical model of flat solar sail
and a previously proposed model of optically orthotropic solar sail. However, it has much more free parameters (117 in the example) that should be obtained by the least squares approximation. For the modified GSM we approximated the components of tensors J 2 ; J 3 , and J 4 . We neglected the J 1 because it is only used for the calculation of a thermal self-emission, which was not performed in this example. As it was shown by McMahon and Scheeres (2010), one can transit from the tensor representation of SRP in a body
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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frame to the Fourier expansion of SRP over solar longitude and latitude to calculate both orbital dynamics and dynamics around the center of gravity. The simulation results showed that it is possible to take into account structural inhomogeneities on the surface of the solar sail by refining the model of reflection of radiation from the surface. This model can be used to simulate the orbital dynamics of a solar sail with wrinkles on the surface, especially in the case of a weak pretension of the film of a solar sail. Another possible application of the model is the analysis of the dynamics of thin-film structures around the center of mass in the presence of wrinkles. The advantage of the model in question is that it is possible to use the GSM and extended GSM relations without having to consider the coefficients of the corresponding tensors directly – it is possible to approximate the specified components, as was shown above. Since the combined models of light pressure, e.g., McMahon and Scheeres (2010), are weakly dependent on the a priori physical models, it is possible to develop a similar combined model within the framework of the proposed physical model. Another potential field of application of this model is the analysis of the flow of spacecraft in the upper atmosphere of the Earth. The analytical expression of the interaction force of a surface with a free-molecular flow (Kn > 1) partially coincides with the expression of the light pressure (Van der Ha, 1986; Storch, 2002). It is also of interest to use the proposed physical model of radiation reflection for analyzing the thermal state of thin-film spacecraft. The deviation for the empirically determined optical parameters range from 0:005 to 0:05 (Heaton and Artusio-Glimpse, 2015) which is far less than the calculated HDI for all of presented models. This empirical deviation may be included into the Bayesian inference process to refine the model, specifically to update the rF parameter. And, the general rule is still applies here – the more data we have, the less deviation we get. Considering Eq. (23), one can assume that wrinkles have smaller influence that general curvature of a solar sail: the effect of orthotropical term starts from the 4th order of approximation. However, due to the nature of the power series, this additional term may influence on the resultant force and torque of radiation pressure in some special cases that has to be studied additionally. 6. Conclusion
7
and emissivity e1 among the general axis, reflectivity q2 , specularity s2 , and emissivity e2 in the perpendicular direction, for the thermal state, absorptivities a1 and a2 (Nerovny et al., 2017), and coefficient k of back reflection. Using these parameters, one can adequately represent the wrinkled domain on the surface of a solar sail by a surface with general large-scale curvature without small wrinkles. Acknowledgements The reported study was funded by Russian Foundation of Basic Research according to the research project No. 1838-00001. Appendix A. Power series for the orthotropic term of force from reflection Let us simplify the additional term in (5): q1 cos2 ðhm ho Þ þ q2 sin2 ðhm ho Þ s1 cos2 ðhm ho Þ þ s2 sin2 ðhm ho Þ ¼ b1 cos4 ðhm ho Þ þ b2 cos2 ðhm ho Þ þ a3 ;
ð15Þ
where b1 ¼ q1 s1 þ q2 s2 q2 s1 q1 s2 ; b2 ¼ q2 s1 þ q1 s2 2q2 s2 ; a3 ¼ q2 s2 . In Eq. (15) the term cos2 ðhm ho Þ can be represented in the following way: cos2 ðhm ho Þ ¼ cos2 hm cos2 h0 þ2 cos hm cos h0 sin hm sin h0 þ sin2 hm sin2 h0 ¼ ¼ cos2 hm cos2 h0 cos2 b0 þ sin2 b0 þ2 cos hm cos h0 sin hm sin h0 cos2 b0 þ sin2 b0 þ sin2 hm sin2 h0 cos2 b0 þ sin2 b0 ¼ 2
2
2
2
¼ ðcos hm Þ ð cos h0 cos b0 Þ þ ðcos hm Þ ð cos h0 sin b0 Þ þ þ2ðcos hm Þðsin hm Þð cos h0 cos b0 Þð sin h0 cos b0 Þþ þ2ðcos hm Þðsin hm Þð cos h0 sin b0 Þð sin h0 sin b0 Þþ 2
2
2
2
þðsin hm Þ ð sin h0 cos b0 Þ þ ðsin hm Þ ð sin h0 sin b0 Þ ¼ ¼ m2x ð cos h0 cos b0 Þ2 þ m2x s2x þ2mx my ð cos h0 cos b0 Þð sin h0 cos b0 Þþ þ2mx my sx sy þ m2y ð sin h0 cos b0 Þ2 þ m2y s2y : ð16Þ
We showed the conformity of a model of an optically orthotropic surface with the Generalized Sail Model. This modified model assumes the anisotropy of reflection of the incoming flux on the macroscale, which can be caused by the wrinkles on a solar sail membrane. In the resulting equations, the wrinkled domain on the surface is considered to have the following optical properties: unit vector of direction of general axis of optical ^ reflectivity q1 , specularity s1 parameters on the surface m,
The cos b0 can be expanded into the Fourier sine series (for b0 2 ð0; pÞ): cos b0 ¼ p8
1 X
n 4n2 1
sin 2nb0 ¼
n¼1
lim 8 N !1 p
N X
n 4n2 1
ð17Þ sin 2nb0 :
n¼1
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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In the following equations, we will limit the maximum number of terms by N ¼ N max . The Eq. (17) may be rewritten using Chebyshev polynomials of the second kind: U 2n1 ðcos b0 Þ ¼
N max X
N max X
n U 2n1 ðcos b0 Þ 4n2 1
I 5ortho ¼ p2 b2 kJ 122 Q0 b2 J 221 þ 12 b2 J 122 ;
¼
I 6ortho ¼ p4 b2 J 321 Q0 p2 b2 J 222 Q0 þ 12 b2 kJ 222 P 21 þ 12 b1 kJ 042 ; I 7ortho ¼ p2 b2 kJ 322 Q1 p2 b2 kJ 322 P 21 Q0 p2 b1 kJ 142 Q0
n 4n2 1
n¼1
b2 J 421 P 21 þ 12 b2 J 322 P 21 b1 J 241 þ 12 b1 J 142 ;
ð18Þ
n1 X k 2n2k1 Þ! 22n2k1 ðcos b0 Þ ¼ ð1Þ k!ðð2nk1 2n2k1Þ!
I 8ortho ¼ p4 b2 J 521 Q1 p2 b2 J 422 Q1 þ p4 b2 J 521 P 21 Q0 p2 b2 J 422 P 21 Q0 þ p4 b1 J 341 Q0 p2 b1 J 242 Q0 þ b1 kJ 242 P 21 ;
k¼0
¼ sin b0
N max X
ð23Þ
I 4ortho ¼ 12 b2 kJ 022 ;
n¼1
¼ sin b0 p8
14 EX S I northo ; ^s : c n¼4
where
sin 2nb0 ; sin b0
cos b0 sin b0 p8
Fortho ¼
P m ðcos b0 Þ2m1 ;
I 9ortho ¼ p2 b2 kJ 522 P 21 Q1 p2 b1 kJ 342 Q1 p4 b1 kJ 342 P 21 Q0 2b1 J 441 P 21 þ b1 J 342 P 21 ;
m¼1
2 2 4 2 4 I 10 ortho ¼ p b2 J 721 P 1 Q1 p b2 J 622 P 1 Q1 þ p b1 J 541 Q1
where max 22mþ2 NX nð1Þnm ðn þ m 1Þ! Pm ¼ : pð2m 1Þ! n¼m ðn mÞ!ð4n2 1Þ
p2 b1 J 442 Q1 þ p8 b1 J 541 P 21 Q0 p4 b1 J 442 P 21 Q0 þ 12 b1 kJ 442 P 41 ;
ð19Þ
Assuming cos b0 ¼ ^ n ^s, one can derive from Eq. (16) the following representation: 0 !2 1 N max X 2m1 Að m ^ ^sÞ2 ¼ cos2 ðhm ho Þ ¼ @1 þ n ^sÞ P m ð^ ð20Þ m¼1 2 ^ ^sÞ : 1 þ P 2 ðm
2 4 4 4 2 I 11 ortho ¼ p b1 kJ 542 P 1 Q1 p b1 kJ 542 P 1 Q0 b1 J 641 P 1 þ 12 b1 J 542 P 41 ; 2 2 4 8 4 4 I 12 ortho ¼ p b1 J 741 P 1 Q1 p b1 J 642 P 1 Q1 þ p b1 J 741 P 1 Q0
I 13 ortho
p2 b1 J 642 P 41 Q0 ; ¼ p2 b1 kJ 742 P 41 Q1 ;
4 4 4 2 I 14 ortho ¼ p b1 J 941 P 1 Q1 p b1 J 842 P 1 Q1 ; ^n . . . ^n m ^ ... m ^ q ^n; J pq1 ¼ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} p
Appendix B. Power series for visibility function H ð^ n ^sÞ ¼ 12 p2
1 X sinðð2kþ1Þ^ n^sÞ 2kþ1
^ ... m ^ q E2: J pq2 ¼ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n^ . . . n^ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} m p
¼
k¼0
¼ 12 p2
1 X 1 X ð1Þn ð2kþ1Þ2nþ1 ð2kþ1Þð2nþ1Þ!
ð^ n ^sÞ
2nþ1
¼
k¼0 n¼0
¼
N1 X
1 p2 lim 2 N !1 1
k¼0
lim
N 2 !1
N2 X ð1Þn ð2kþ1Þ2nþ1 ð2kþ1Þð2nþ1Þ!
! ð^ n ^sÞ
2nþ1
ð21Þ
n¼0
ðassuming N 1 ¼ N 2 ¼ N max Þ N max X 2nþ1 n ^sÞ 12 p2 Qn ð^ ;
References
n¼0
where max ð1Þ NX ð2k þ 1Þ ð2n þ 1Þ! k¼0 2k þ 1
n
Qn ¼
The similar formulas may be obtained for higher order approximations. However, it is preferable to utilize the approximation method described in Nerovny et al. (2017) since the equation for wrinkled sail has the similar form as for a flat sail.
2nþ1
:
Appendix C. Tensor representation of orthotropic term Assuming N max ¼ 1 for Eqs. (19)–(22), one can obtain the following tensor series:
Alhorn, D., Casas, J., Agasid, E., Adams, C., Laue, G., Kitts, C., O’Brien, S., 2011. NanoSail-D: The Small Satellite That Could! (SSC11-VI-1). In: Presentation at AIAA/USU Conference on Small Satellites, Utah State University, Logan, Utah, USA, 2011. http://digitalcommons. usu.edu/smallsat/2011/all2011/37. Ancona, E., Kezerashvili, R.Y., 2016. Orbital dynamics of a solar sail accelerated by thermal desorption of coatings. arXiv preprint arXiv:1609.03131, 2016. http://arxiv.org/abs/1609.03131. Bar-Sever, Y., Kuang, D., 2004. New empirically derived solar radiation pressure model for global positioning. In: System Satellites, IPN Progress Report, pp. 42–159. Campbell, B.A., Thomas, S.J., 2014. Realistic solar sail thrust. In M. Macdonald, editor, Advances in Solar Sailing, pages 407–435. Springer, Berlin Heidelberg, Berlin, Heidelberg, 2014. ISBN 978-3642-34906-5 978-3-642-34907-2. doi: 10.1007/978-3-642-34907-2_27..
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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Nerovny, N., 2017. Data for: Light radiation pressure upon an optically orthotropic surface, 2017. URL. Nerovny, N., Grigorjev, A., 2017. locutus3009/srp2: version of software used in paper Light radiation pressure upon an optically orthotropic surface. Zenodo. https://doi.org/10.5281/zenodo.571409, URL: https://github.com/locutus3009/srp2/tree/JQSRT-2017. Nerovny, N., Lapina, I., Grigorjev, A., 2017. Light radiation pressure upon an optically orthotropic surface. J. Quant. Spectrosc. Radiat. Transf. 202, 64–73. https://doi.org/10.1016/j.jqsrt.2017.07.016. Nerovny, N., Zimin, V., Fedorchuk, S., Golubev, E., 2017. Representation of light pressure resultant force and moment as a tensor series. Celestial Mech. Dynam. Astron. 128 (4), 483–513. https://doi.org/ 10.1007/s10569-017-9758-8. Plummer, M., 2013. JAGS Version 3.4.0 user manual. 2013. URL . Ridenoure, R., Munakata, R., Diaz, A., Wong, S., Plante, B., Stetson, D., Spencer, D., Foley, J., 2015. LightSail Program Status: One Down, One to Go (SSC15-V-3). Presentation at AIAA/USU Conference on Small Satellites, Utah State University, Logan, Utah, USA, 2015. http://digitalcommons.usu.edu/smallsat/2015/all2015/32. Rios-Reyes, L., 2006. Solar Sails: Modeling, Estimation, and Trajectory Control. PhD thesis, University of Michigan, 2006. URL . Rios-Reyes, L., Scheeres, D.J., 2005. Generalized model for solar sails. J. Spacecr. Rock. 42 (1), 182–185. https://doi.org/10.2514/1.9054. Springer, T.A., Beutler, G., Rothacher, M., 1999. A new solar radiation pressure model for GPS satellites. GPS Solut. 2 (3), 50–62. https://doi. org/10.1007/PL00012757. Storch, J.A., 2002. Aerodynamic disturbances on spacecraft in freemolecular flow. Technical Report TR-2003(3397)-1, 2002. URL . Tsuda, Y., Mori, O., Funase, R., Sawada, H., Yamamoto, T., Saiki, T., Endo, T., Kawaguchi, J., 2011. Flight status of IKAROS deep space solar sail demonstrator. Acta Astronaut. 69 (9–10), 833–840. https:// doi.org/10.1016/j.actaastro.2011.06.005. Van der Ha, J.C., 1986. Attitude perturbations induced by free-molecular flow interactions in perigee region. Acta Astronaut. 13 (6–7), 301–309. https://doi.org/10.1016/0094-5765(86)90085-8. Wang, C.G., Du, X.W., Tan, H.F., He, X.D., 2009. A new computational method for wrinkling analysis of gossamer space structures. Int. J. Solids Struct. 46 (6), 1516–1526. https://doi.org/10.1016/j.ijsolstr.2008.11.018. Wong, W., Pellegrino, S., 2006a. Wrinkled membranes I: Experiments. J. Mech. Mater. Struct. 1 (1), 3–25. Wong, W., Pellegrino, S., 2006b. Wrinkled membranes II: analytical models. J. Mech. Mater. Struct. 1 (1), 27–61 http://msp.org/jomms/ 2006/1-1/p03.xhtml. Wong, W., Pellegrino, S., 2006c. Wrinkled membranes III: numerical simulations. J. Mech. Mater. Struct. 1 (1), 63–95 http://msp.org/ jomms/2006/1-1/p04.xhtml. Xiao, W.-W., Chen, W.-J., Fu, G.-Y., 2011. Wrinkle analysis of the space inflatable paraboloid antenna. J. Shanghai Jiaotong Univ. (Sci.) 16 (1), 24–33. https://doi.org/10.1007/s12204-011-1090-y. Zeiders, G., 2007. Forces and moments on a square solar sail (AIAA 2007–1828). In: Presentation at 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii, USA, 2007. https://doi.org/10.2514/6.2007-1828. Zeiders, G.W., 2005. Design rules and scaling for solar sails. Technical report, 2005. URL . Ziebart, M., 2004. Generalized analytical solar radiation pressure modeling algorithm for spacecraft of complex shape. J. Spacecraft Rock. 41 (5), 840–848. https://doi.org/10.2514/1.13097.
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ScienceDirect Advances in Space Research xxx (2020) xxx–xxx www.elsevier.com/locate/asr
Application of a modified generalized sail model on a solar sail with wrinkles Nikolay Nerovny a,⇑, Irina Lapina b a
Department of Spacecraft and Launch Vehicles, Bauman Moscow State Technical University, Moscow 105005, Russia b Department of Mathematical Modelling, Bauman Moscow State Technical University, Moscow 105005, Russia Received 18 October 2019; received in revised form 12 January 2020; accepted 16 January 2020
Abstract In this paper, we present an analysis of effect of wrinkles on the solar sail performance. We describe different analytical, semianalytical and numerical approaches to the calculation of general large-scale curvature of a solar sail as well as parameters of socalled wrinkled domains, and introduce the impact of such wrinkles on the thrust and torque of the solar sail. Finally, we present a model of an optically-orthotropic surface for such non-ideal sail, providing a connection with the Generalized Sail Model, and other solar sail thrust models. Ó 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Solar sail; Radiation pressure; Wrinkles; Generalized sail model
1. Introduction
E F ¼ C R A ^s; c
There are various models of light pressure (solar radiation pressure, SRP) on objects in deep space. From the engineering point of view, the models of light pressure can be divided into three categories. The first category includes a priori physical models of the interaction of radiation with a surface. Taking into account the shape and optical properties of the surface, it is possible to calculate the principal vector and torque of light pressure, or even consider a problem of helioelasticity. These models include a ‘‘Cannonball” model, where the object is presented as a flat mirror plate, always turned towards the sun by one side, and the light falls on it perpendicularly to the surface (Lucchesi, 2001; McInnes, 2004). Generally speaking, this model may be formulated as follows:
Where C R – some experimental constant describing the reflection properties and shape of the body; A – cross section of the body; E – solar flux; c – light speed in vacuum; ^s – unit vector of direction from light source to the body. For the ideal flat solar sail, C R ¼ 1 þ q, where q – total reflectivity. Within the framework of this model, it is impossible to solve the problem of determining the resultant torque of SRP if the center of gravity of the spacecraft coincides with its geometric center. There is also a model of a flat solar sail whose surfaces have specular-diffuse properties. This model can be analytically formulated using the following equation: 2 F ¼ A Ec a1 ð^n ^sÞ^s þ a2 ð^n ^sÞ^n 2a3 ð^n ^sÞ ^n ; a1 ¼ 1 qs;
⇑ Corresponding author.
E-mail address: [email protected] (N. Nerovny).
a2 ¼ Bf qð1 sÞ þ ð1 qÞ ef Bf eb Bb = ef þ eb ; a3 ¼ qs;
https://doi.org/10.1016/j.asr.2020.01.020 0273-1177/Ó 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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N. Nerovny, I. Lapina / Advances in Space Research xxx (2020) xxx–xxx
Nomenclature Xw wrinkled domain of a surface ^e0x ; ^e0y ; ^e0z orientation unit vectors for a global frame ^ex ; ^ey ; ^ez orientation unit vectors for a local frame r position vector of dA in a global frame ^ m unit vector of the orientation of optical axis ^ n unit vector of the local normal to dA ^s unit vector from the light source to dA Bm ; Bq Lambertian coefficients s specularity q total reflectivity e emissivity k model parameter for back reflection
where ^ n – unit vector of local normal to the surface; s – specularity coefficient; ef ; eb – emissivity for the front and back surfaces; Bf ; Bb – Lambertian coefficient for front and back surfaces. There is a problem with this model – eventually, the configuration of surfaces may change: the front surface may be shadowed, and the back one may be lightened. This model is often used to determine the SRP on bodies that consist of several planar segments, the so-called macromodels (Marshall and Luthcke, 1994). For these models, one can determine the resultant vector of SRP as follows: F¼
N X Fi H i ð^sÞ; i¼1
where Fi – light pressure on facet i, the total number of facets is N; H i – visibility function that shows if the given facet is lightened or not. For bodies of complex shape, it is also possible to use a specular-diffuse model by integrating expressions for the light pressure over the entire surface. A generalized model of the solar sail (Generalized Sail Model, GSM) was obtained using this approach (Rios-Reyes and Scheeres, 2005; Rios-Reyes, 2006; Jing et al., 2012, 2014): F ¼ A I 2 ^s þ ^s I 3 ^s ; where I 2 ; I 3 – some tensors of rank 2 and 3 respectively that describe the combination of shape and reflection properties of a body. This model works only when the lightened surface of a body does not change in time and only for the bodies of a convex shape. This GSM may be extended in the form of tensor series (Nerovny et al., 2017) to account the non-convex bodies with arbitrary orientation towards the light source. It is also possible to calculate the SRP on a spacecraft of arbitrary type by performing ray-tracing in a way similar to that used in the calculation of radiative heat transfer by Monte Carlo methods (Ziebart, 2004; Howell and Menguc, 2015).
Superscripts S light pressure from thermal self emission A light pressure from absorption R light pressure from reflection Subscripts iso isotropic part of light pressure ortho orthotropic part of light pressure ^ 1 optical parameter among m ^ 2 optical parameter perpendicular to m
The second category includes empirical models of light pressure (Springer et al., 1999; Bar-Sever and Kuang, 2004). These models do not depend on a priori information on the nature of the interaction of radiation with the surface of the spacecraft and are necessary for accurate prediction of orbits of space vehicles from accumulated observational data. Usually these models have an analytical form of some trigonometric series. The third category includes combined models, which are based on a priori physical models of light pressure and also contain features that allow for refinement of the model parameters based on actual observations (McMahon and Scheeres, 2010, 2015). The surface of a thin film space structure may be in a planar stress state, in which there is a possibility of formation of wrinkles (Wong and Pellegrino, 2006a). These wrinkles can form a so-called wrinkled domain (Fig. 1), which can cover a significant part of a whole surface: it was shown that solar sail membrane has general curvature, both regular and semi-random (smoothness), and also small wrinkles (Alhorn et al., 2011; Tsuda et al., 2011; Kawaguchi, 2014; Ridenoure et al., 2015). The mechanical properties of thin films in planar stress state has been studied rigorously, Wong and Pellegrino (2006a,b,c). Regarding solar sails, there is a good experimental study that determines the averaged optical parameters of a model
Fig. 1. Typical exterior of a solar sail with the frame during spaceflight.
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solar sail with wrinkles and folds (Heaton and ArtusioGlimpse, 2015). In this paper, we present a new physical model of light radiation pressure in the context of GSM, an a priori model. This paper is organized as follows. First, we present the model of SRP upon an optically-orthotropic surface, that represents wrinkles on a solar sail, which is a method we use in this paper. Second, we show that this model may be represented using the similar tensor decomposition as for a GSM. In the simulation section, we illustrate the applicability of this work by simulating of the SRP on some model surface with wrinkles. In the last section we discuss the simulation results and possible applications of described method. 2. Model of optical parameters for a surface with wrinkles This section contains a summary of a developed model of a surface with optically orthotropic parameters (Nerovny et al., 2017). The surface A is represented as a two-manifold, A M2 . Xw A is a surface domain that has wrinkles. The global frame is O0 x0 y 0 z0 with unit vectors ^e0x ; ^e0y ; ^e0z (Fig. 2). The local coordinate frame Oxyz is associated with an infinitesimal element dA Xw , the unit vectors are ^ex ; ^ey ; ^ez . The position of dA in the global frame is defined by a vector r. ^ n is a unit vector of the local normal to the surface element dA; ^n ¼ ^ez . For arbitrary vector ^r we used direction angles ðb; hÞ in the local frame as follows: b 2 ½0; p=2 — angle between the vector ^r and þz; h 2 ½0; 2p — angle between the axis Ox and a projection of ^r to the plane Oxy, counterclockwise around Oz. In this study, the incoming light flux is considered to be parallel. Some questions regarding SRP from finite Sun disc were considered previously by different authors, e.g., McInnes and Brown (1990b,a), Koblik et al. (2011) and others, but we are not aware of any works regarding
3
GSM for systems with non-point light source. The ^s is a unit vector from the light source to dA; ^sðb0 ; h0 Þ ¼ sin b0 cos h0^ex sin b0 sin h0^ey cos b0^ez . We neglect the spectral dependency of the surface’s optical parameters. Kezerashvili (2014) and Ancona and Kezerashvili (2016) studied the implications of spectral dependency of optical parameters on the efficiency of solar sails. From the practical point of view, it is possible to consider such spectral dependencies using optical coefficients integrated over real Sun spectrum, however there is still a problem with difference between direct and reflected spectrum. On the surface there are two perpendicular directions ^ is a unit vector of the with different optical parameters. m ^ ¼ orientation of optical axis in the local frame, m cos hm^ex þ sin hm^ey . In the following formulas, E is a light flux, c the light speed in a vacuum, r Stefan-Boltzmann constant, T the temperature of dA. For a solar sail case, the temperature of an element dA can be obtained by the equilibrium equation of thermal flux considering that this temperature does not change among the thickness of the film and that the thermal conductivity is substantial (Rios-Reyes, 2006). The equation for the force from the emitted light flux from the orthotropic surface is the following formula (Nerovny et al., 2017): e1 Bm rT 4 ^ndA; c e1 þ e2 Bm ¼ ; 3e1
dFS ¼
ð1Þ ð2Þ
where Bm is a modified Lambertian coefficient, e1 is an ^ and e2 is an emissivity in peremissivity in the direction m, pendicular direction in the plane of dA. The equation for the force from the absorbed light: E dFA ¼ ð^n ^sÞ^sdA c
ð3Þ
Fig. 2. Coordinate frames. Gray area is a wrinkled domain of a surface.
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For reflection we modify Maxwell’s model of speculardiffuse reflection (Howell and Menguc, 2015). We introduce the following parameters: q1 and s1 are total reflectiv^ on the plane of dA; q2 ity and specularity in the direction m and s2 are total reflectivity and specularity in perpendicular direction. The equation for the force from diffusely reflected light for the orthotropic surface can be represented in the following vector form: dFR1 ¼
E q ð1 s1 ÞBq ð^ n ^sÞ^ ndA c 1
ð4Þ
where P is from Eq. (20), and: a 1 ¼ 1 q2 s 2 ; a2 ¼ Bq q1 ð1 s1 Þ; a3 ¼ q2 s2 ;
Radiation force upon dA from specularly reflected light flux: 2 dFR2m ¼ Ec ð^ n ^sÞ^s 2ð^ n ^sÞ ^ n ð5Þ q1 cos2 ðhm ho Þ þ q2 sin2 ðhm ho Þ s1 cos2 ðhm ho Þ þ s2 sin2 ðhm ho Þ dA We also introduce the back reflection term with an additional empirical parameter k (Nerovny et al., 2017). Radiation force from back reflection is: ð6Þ
3. Generalization 3.1. Generalized sail model In the GSM, one can analytically separate the description of surface geometry and its optical parameters from its orientation towards the light source. During this process, one can precompute some constants which describe the geometry and optical properties of surface using tensor representation (GSM, Rios-Reyes and Scheeres (2005)). After this, it is possible to calculate the light pressure on a body using scalar products of tensors by Sun vector ^s.
b1 ¼ q1 s1 þ q2 s2 q1 s2 q2 s1 ; b2 ¼ q1 s2 þ q2 s1 2q2 s2 : See Appendix A for expansion of orthotropic term into power series. Eq. (8) is a standard equation for radiation pressure upon a flat surface with specular-diffuse properties (McInnes, 2004), however the optical parameters are different. 3.3. Tensor series for an orthotropic model According to Nerovny et al. (2017), one can write the following tensor series for resultant force F and torque M from isotropic component of radiation pressure upon an optically orthotropic convex surface: ! N max X E Fiso ¼ I 1iso þ S I niso ; ^s ; r ð10Þ c n¼2 ! N max X n E 1 Kiso þ S Kiso ; ^s ; ð11Þ Miso ¼ c n¼2 where I iso and Kiso – some tensors that are only depended on geometry and optical properties of the surface and are invariant to the change of the relative position of the light source, and S is a vector operator with the following index representation: S J n ; ^b ¼ J ni1 ...in1 k bi1 . . . bip1 ; k
3.2. Generalization of an orthotropic model Combining of terms for light pressure from self emission (1), absorption (3), diffuse reflection (4), specularly reflected light pressure (5) and light pressure from back reflection (6), it is possible to write an equation for the infinitesimal radiation force as a combination of a isotropic and orthotropic parts:
where:
ð9Þ
4
ð4 3s1 s2 Þq1 þ ð4 s1 3s2 Þq2 Bq ¼ 12ð1 s1 Þq1
dF ¼ dFiso þ dFortho ;
ð8Þ
a0 ¼ e1 BmErT ;
where
dFR2b ¼ Ec k^sðq1 cos2 ðhm ho Þ þ q2 cos2 ðhm ho ÞÞ ðs1 cos2 ðhm ho Þ þ s2 cos2 ðhm ho ÞÞdA
dFiso ¼ Ec ða0 ^n a1 ð^n ^sÞ^s þ a2 ð^n ^sÞ^nþ 2 þ2a3 ð^n ^sÞ ^n dA; dFortho ¼ Ec ð^n ^sÞ^s 2ð^n ^sÞ2 ^n þ k^s ^ ^sÞ4 þ b1 1 þ 2P 2 þ P 4 ðm ^ ^sÞ2 dA; þb2 1 þ P 2 ðm
ð7Þ
where ^b is an arbitrary unit vector, i1 . . . in1 ; k ¼ x; y; z, and we used the Einstein summation convention. For an orthotropic component we used the visibility function to obtain a equation of radiation force upon dA in convex case: V ¼ H ð^n ^sÞ;
ð12Þ
where H is a Heaviside step function. The derivation of power series for (12) is presented in the Appendix B. These are the equations for orthotropic components of resultant force and torque upon a surface with wrinkles:
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
N. Nerovny, I. Lapina / Advances in Space Research xxx (2020) xxx–xxx
Fortho ¼
max E NX S I northo ; ^s ; c n¼4
Mortho ¼
ð13Þ
max E NX S Knortho ; ^s ; c n¼4
ð14Þ
where I ortho and Kortho are tensors that described in the Appendix C. For the non-convex structures, it is possible to approximate the components of these tensors, using leastsquares approximations following the procedure described in the article (Nerovny et al., 2017). 4. Simulation Different authors suggested that wrinkles on the thin solid film can be described using sine or cosine laws (Epstein, 2003; Wong and Pellegrino, 2006b). It was shown both numerically (Wong and Pellegrino, 2006c; Wang et al., 2009; Xiao et al., 2011), and experimentally (Wong and Pellegrino, 2006a). Regarding solar sails, wrinkles were described by Murphey et al. (2002), Zeiders (2005, 2007) and Campbell and Thomas (2014), and others. In this work, we describe the magnitude of wrinkles using the following equation: 2px02 ; k where d ¼ d x01 ; x02 – smooth field of amplitude of wrinkles over the wrinkled domain Xw ; k ¼ k x01 ; x02 – smooth field of wavelength of wrinkles over the wrinkled domain Xw . We used the followingparameters ofwrinkled surface: x01 ; x02 2 ½0:5; 0:5; x03 ¼ d cos 8px02 0:5 . There are 4 waves, wavelength k ¼ const ¼ 0:25m, amplitude d ¼ const ¼ 0:2m. Optical parameters of this surface are uniform and have the following values: reflectivity q0 ¼ 1; specularity s0 ¼ 0; Lambertian coefficient B ¼ 2=3. Each surface element is entirely diffuse. Software from Nerovny and Grigorjev (2017) performed the ray-tracing, and results are available in the Mendeley Data (Nerovny, 2017). The range for b is f9 ; 18 ; . . . ; 72 ; 81 g, and the range for h is
x03 ¼ d sin
5
f0 ; 18 ; . . . ; 342 ; 360 g. The angle definitions are the same as on Fig. 2. The number of rays was 100000 in each simulation, 189 simulations total. In our analysis, we used the R code combined with the JAGS toolkit for Markov Chain Monte Carlo (MCMC) approximation of posterior probability (Plummer, 2013). The structure of this code is similar to examples in Kruschke (2015). For the Bayesian inference, we used the following likelihood probability distribution function: 8 jFF j2 iso > 2 > > m ¼ 1; e 2rF ; > > < jFFortho j2 1 P ðdatajm; parametersÞ / exp 2r2 F ; m ¼ 2; >e rF > > > jFFGSM j2 > : 2r2 F ; m ¼ 3; e where rF – standard deviation; F – data obtained from Monte Carlo simulation; Fiso ; Fortho ; FGSM – data predicted by the corresponding model. 5. Results and discussion The summary of simulation results is presented in Table 1. On that table, Isotropic is a summary of MCMC approximation considering the isotropic model of optical parameters; values are median of posterior probability distributions, limits are 95% highest probability density intervals (HDI). Orthotropic is a summary of MCMC approximation considering an orthotropic model of optical parameters, values are medians of distributions, limits are ^ ¼ ð1; 0; 0ÞT . LS GSM is a summary of least 95% HDI, m squares approximation of tensors of extended GSM. The last row is a mean-square deviation between Monte Carlo simulated data and simulation results of a corresponding model. Isotropic and orthotropic data are from Nerovny et al. (2017). On Fig. 3 there is a graphical comparison between different models. On the figure, vertical axis is a light pressure, m2 . Horizontal axis is h. Black lines and dots – F 01 ; m2 (projection on O0 x01 of global frame). Dark gray lines and dots – F 02 ; m2 (projection on O0 x02 of global frame). Light gray
Table 1 Model parameters for isotropic and orthotropic models of SRP, and for a modified GSM. Surface with wrinkles, q0 ¼ 1; s0 ¼ 0; B0 ¼ 2=3. All values are rounded up to three significant digits after the point. The values for LS GSM case are omitted intentionally because that components does not have any good determined geometrical of physical meaning. The generative software for all of the data in analysis may be given to third-party via the private communication with the authors. Isotropic
Deviation
Orthotropic
LS GSM
Median
Upper limit
Lower limit
Median
Upper limit
Lower limit
Value
q ¼ 0:611 s ¼ 0:005 B ¼ 0:707 rF ¼ 0:190
þ0:327 þ0:018 þ0:293 þ0:021
0:196 0:005 0:240 0:018
q1 ¼ 0:104 q2 ¼ 0:369 s1 ¼ 0:677 s2 ¼ 0:331 k ¼ 2:440 rF ¼ 0:097
þ0:034 þ0:083 þ0:308 þ0:197 þ0:790 þ0:009
0:023 0:085 0:240 0:136 0:510 0:009
N max ¼ 4 J 2; J 3; J 4 (117 parameters)
0:101
0:039
0:032
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Fig. 3. Comparison between ray-tracing results, isotropic model, orthotropic model and GSM approximation for surface with wrinkles, q ¼ 1; s ¼ 0; B ¼ 2=3.
lines and dots – F 03 ; m2 (projection on O0 x03 of global frame). Crosses – ray-tracing results. Solid lines – least squares approximation for GSM. Dashed lines – orthotropic model based on Bayesian median values of parameters (q1 ¼ 0:12; q2 ¼ 0:387; s1 ¼ 0:394; s2 ¼ 0:251; k ¼ 2:84). Dotted lines – isotropic model based on Bayesian median values of parameters (q ¼ 0:611; s ¼ 0:005; B ¼ 0:707). We showed that least squares approximation of modified GSM has the minimal deviation compared with the ray-tracing results, the classical model of flat solar sail
and a previously proposed model of optically orthotropic solar sail. However, it has much more free parameters (117 in the example) that should be obtained by the least squares approximation. For the modified GSM we approximated the components of tensors J 2 ; J 3 , and J 4 . We neglected the J 1 because it is only used for the calculation of a thermal self-emission, which was not performed in this example. As it was shown by McMahon and Scheeres (2010), one can transit from the tensor representation of SRP in a body
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frame to the Fourier expansion of SRP over solar longitude and latitude to calculate both orbital dynamics and dynamics around the center of gravity. The simulation results showed that it is possible to take into account structural inhomogeneities on the surface of the solar sail by refining the model of reflection of radiation from the surface. This model can be used to simulate the orbital dynamics of a solar sail with wrinkles on the surface, especially in the case of a weak pretension of the film of a solar sail. Another possible application of the model is the analysis of the dynamics of thin-film structures around the center of mass in the presence of wrinkles. The advantage of the model in question is that it is possible to use the GSM and extended GSM relations without having to consider the coefficients of the corresponding tensors directly – it is possible to approximate the specified components, as was shown above. Since the combined models of light pressure, e.g., McMahon and Scheeres (2010), are weakly dependent on the a priori physical models, it is possible to develop a similar combined model within the framework of the proposed physical model. Another potential field of application of this model is the analysis of the flow of spacecraft in the upper atmosphere of the Earth. The analytical expression of the interaction force of a surface with a free-molecular flow (Kn > 1) partially coincides with the expression of the light pressure (Van der Ha, 1986; Storch, 2002). It is also of interest to use the proposed physical model of radiation reflection for analyzing the thermal state of thin-film spacecraft. The deviation for the empirically determined optical parameters range from 0:005 to 0:05 (Heaton and Artusio-Glimpse, 2015) which is far less than the calculated HDI for all of presented models. This empirical deviation may be included into the Bayesian inference process to refine the model, specifically to update the rF parameter. And, the general rule is still applies here – the more data we have, the less deviation we get. Considering Eq. (23), one can assume that wrinkles have smaller influence that general curvature of a solar sail: the effect of orthotropical term starts from the 4th order of approximation. However, due to the nature of the power series, this additional term may influence on the resultant force and torque of radiation pressure in some special cases that has to be studied additionally. 6. Conclusion
7
and emissivity e1 among the general axis, reflectivity q2 , specularity s2 , and emissivity e2 in the perpendicular direction, for the thermal state, absorptivities a1 and a2 (Nerovny et al., 2017), and coefficient k of back reflection. Using these parameters, one can adequately represent the wrinkled domain on the surface of a solar sail by a surface with general large-scale curvature without small wrinkles. Acknowledgements The reported study was funded by Russian Foundation of Basic Research according to the research project No. 1838-00001. Appendix A. Power series for the orthotropic term of force from reflection Let us simplify the additional term in (5): q1 cos2 ðhm ho Þ þ q2 sin2 ðhm ho Þ s1 cos2 ðhm ho Þ þ s2 sin2 ðhm ho Þ ¼ b1 cos4 ðhm ho Þ þ b2 cos2 ðhm ho Þ þ a3 ;
ð15Þ
where b1 ¼ q1 s1 þ q2 s2 q2 s1 q1 s2 ; b2 ¼ q2 s1 þ q1 s2 2q2 s2 ; a3 ¼ q2 s2 . In Eq. (15) the term cos2 ðhm ho Þ can be represented in the following way: cos2 ðhm ho Þ ¼ cos2 hm cos2 h0 þ2 cos hm cos h0 sin hm sin h0 þ sin2 hm sin2 h0 ¼ ¼ cos2 hm cos2 h0 cos2 b0 þ sin2 b0 þ2 cos hm cos h0 sin hm sin h0 cos2 b0 þ sin2 b0 þ sin2 hm sin2 h0 cos2 b0 þ sin2 b0 ¼ 2
2
2
2
¼ ðcos hm Þ ð cos h0 cos b0 Þ þ ðcos hm Þ ð cos h0 sin b0 Þ þ þ2ðcos hm Þðsin hm Þð cos h0 cos b0 Þð sin h0 cos b0 Þþ þ2ðcos hm Þðsin hm Þð cos h0 sin b0 Þð sin h0 sin b0 Þþ 2
2
2
2
þðsin hm Þ ð sin h0 cos b0 Þ þ ðsin hm Þ ð sin h0 sin b0 Þ ¼ ¼ m2x ð cos h0 cos b0 Þ2 þ m2x s2x þ2mx my ð cos h0 cos b0 Þð sin h0 cos b0 Þþ þ2mx my sx sy þ m2y ð sin h0 cos b0 Þ2 þ m2y s2y : ð16Þ
We showed the conformity of a model of an optically orthotropic surface with the Generalized Sail Model. This modified model assumes the anisotropy of reflection of the incoming flux on the macroscale, which can be caused by the wrinkles on a solar sail membrane. In the resulting equations, the wrinkled domain on the surface is considered to have the following optical properties: unit vector of direction of general axis of optical ^ reflectivity q1 , specularity s1 parameters on the surface m,
The cos b0 can be expanded into the Fourier sine series (for b0 2 ð0; pÞ): cos b0 ¼ p8
1 X
n 4n2 1
sin 2nb0 ¼
n¼1
lim 8 N !1 p
N X
n 4n2 1
ð17Þ sin 2nb0 :
n¼1
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
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N. Nerovny, I. Lapina / Advances in Space Research xxx (2020) xxx–xxx
In the following equations, we will limit the maximum number of terms by N ¼ N max . The Eq. (17) may be rewritten using Chebyshev polynomials of the second kind: U 2n1 ðcos b0 Þ ¼
N max X
N max X
n U 2n1 ðcos b0 Þ 4n2 1
I 5ortho ¼ p2 b2 kJ 122 Q0 b2 J 221 þ 12 b2 J 122 ;
¼
I 6ortho ¼ p4 b2 J 321 Q0 p2 b2 J 222 Q0 þ 12 b2 kJ 222 P 21 þ 12 b1 kJ 042 ; I 7ortho ¼ p2 b2 kJ 322 Q1 p2 b2 kJ 322 P 21 Q0 p2 b1 kJ 142 Q0
n 4n2 1
n¼1
b2 J 421 P 21 þ 12 b2 J 322 P 21 b1 J 241 þ 12 b1 J 142 ;
ð18Þ
n1 X k 2n2k1 Þ! 22n2k1 ðcos b0 Þ ¼ ð1Þ k!ðð2nk1 2n2k1Þ!
I 8ortho ¼ p4 b2 J 521 Q1 p2 b2 J 422 Q1 þ p4 b2 J 521 P 21 Q0 p2 b2 J 422 P 21 Q0 þ p4 b1 J 341 Q0 p2 b1 J 242 Q0 þ b1 kJ 242 P 21 ;
k¼0
¼ sin b0
N max X
ð23Þ
I 4ortho ¼ 12 b2 kJ 022 ;
n¼1
¼ sin b0 p8
14 EX S I northo ; ^s : c n¼4
where
sin 2nb0 ; sin b0
cos b0 sin b0 p8
Fortho ¼
P m ðcos b0 Þ2m1 ;
I 9ortho ¼ p2 b2 kJ 522 P 21 Q1 p2 b1 kJ 342 Q1 p4 b1 kJ 342 P 21 Q0 2b1 J 441 P 21 þ b1 J 342 P 21 ;
m¼1
2 2 4 2 4 I 10 ortho ¼ p b2 J 721 P 1 Q1 p b2 J 622 P 1 Q1 þ p b1 J 541 Q1
where max 22mþ2 NX nð1Þnm ðn þ m 1Þ! Pm ¼ : pð2m 1Þ! n¼m ðn mÞ!ð4n2 1Þ
p2 b1 J 442 Q1 þ p8 b1 J 541 P 21 Q0 p4 b1 J 442 P 21 Q0 þ 12 b1 kJ 442 P 41 ;
ð19Þ
Assuming cos b0 ¼ ^ n ^s, one can derive from Eq. (16) the following representation: 0 !2 1 N max X 2m1 Að m ^ ^sÞ2 ¼ cos2 ðhm ho Þ ¼ @1 þ n ^sÞ P m ð^ ð20Þ m¼1 2 ^ ^sÞ : 1 þ P 2 ðm
2 4 4 4 2 I 11 ortho ¼ p b1 kJ 542 P 1 Q1 p b1 kJ 542 P 1 Q0 b1 J 641 P 1 þ 12 b1 J 542 P 41 ; 2 2 4 8 4 4 I 12 ortho ¼ p b1 J 741 P 1 Q1 p b1 J 642 P 1 Q1 þ p b1 J 741 P 1 Q0
I 13 ortho
p2 b1 J 642 P 41 Q0 ; ¼ p2 b1 kJ 742 P 41 Q1 ;
4 4 4 2 I 14 ortho ¼ p b1 J 941 P 1 Q1 p b1 J 842 P 1 Q1 ; ^n . . . ^n m ^ ... m ^ q ^n; J pq1 ¼ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} p
Appendix B. Power series for visibility function H ð^ n ^sÞ ¼ 12 p2
1 X sinðð2kþ1Þ^ n^sÞ 2kþ1
^ ... m ^ q E2: J pq2 ¼ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n^ . . . n^ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} m p
¼
k¼0
¼ 12 p2
1 X 1 X ð1Þn ð2kþ1Þ2nþ1 ð2kþ1Þð2nþ1Þ!
ð^ n ^sÞ
2nþ1
¼
k¼0 n¼0
¼
N1 X
1 p2 lim 2 N !1 1
k¼0
lim
N 2 !1
N2 X ð1Þn ð2kþ1Þ2nþ1 ð2kþ1Þð2nþ1Þ!
! ð^ n ^sÞ
2nþ1
ð21Þ
n¼0
ðassuming N 1 ¼ N 2 ¼ N max Þ N max X 2nþ1 n ^sÞ 12 p2 Qn ð^ ;
References
n¼0
where max ð1Þ NX ð2k þ 1Þ ð2n þ 1Þ! k¼0 2k þ 1
n
Qn ¼
The similar formulas may be obtained for higher order approximations. However, it is preferable to utilize the approximation method described in Nerovny et al. (2017) since the equation for wrinkled sail has the similar form as for a flat sail.
2nþ1
:
Appendix C. Tensor representation of orthotropic term Assuming N max ¼ 1 for Eqs. (19)–(22), one can obtain the following tensor series:
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Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020
N. Nerovny, I. Lapina / Advances in Space Research xxx (2020) xxx–xxx Epstein, M., 2003. Differential equation for the amplitude of wrinkles. AIAA J. 41 (2), 327–329. https://doi.org/10.2514/2.1952. Heaton, A.F., Artusio-Glimpse, A., 2015. An Update to the NASA Reference Solar Sail Thrust Model (AIAA 2015-4506). In: Presentation at AIAA SPACE 2015 Conference and Exposition, Pasadena, California, USA, 2015. https://doi.org/10.2514/6.2015-4506. URL
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Nerovny, N., 2017. Data for: Light radiation pressure upon an optically orthotropic surface, 2017. URL
Please cite this article as: N. Nerovny and I. Lapina, Application of a modified generalized sail model on a solar sail with wrinkles, Advances in Space Research, https://doi.org/10.1016/j.asr.2020.01.020