Analytic Approximation of High-Fidelity Solar Radiation Pressure

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Analytic Approximation of High-Fidelity Solar Radiation
Pressure
Yasuhiro Yoshimura∗, Yuri Matsushita,†, Kazunobu Takahashi‡, Shuji Nagasaki§, and Toshiya Hanada¶.
Kyushu University, Fukuoka 819-0395, Japan
I. Introduction
Solar radiation pressure (SRP) is one of the disturbances that affects both satellite attitude and orbit, and the
high-fidelity calculation of SRP enables the precise prediction of the attitude and orbit of satellites [
1
–
3
]. In terms of
space situational awareness, the accuracy of propagating satellites’ attitude and orbit has a significant impact on tracking
and collision avoidance. Advanced missions of solar sail spacecraft also require precise SRP modeling because the
dynamics of a spacecraft with a high area-to-mass ratio is sensitive to SRP. Recent research investigate the attitude
control method using SRP [4] and the stability analysis of solar sail spacecraft [5].
The SRP modeling is categorized as analytic, semianalytic, and empirical. The analytical modeling is based on the
finite element-based calculation, which calculates SRP by splitting a satellite shape into many small facets, and each
SRP force and torque are summed [
6
]. The accuracy of the calculation depends on the number of splits, and this method
is useful when the satellite shape and its surface parameters are available. The empirical modeling is implemented by the
actual data of an orbiting satellite [
7
,
8
], and is useful for the precise orbit determination for global navigation satellites.
The semianalytical modeling [
9
] combines analytical and empirical modeling. The tuning parameters in analytical
modeling are estimated so that the SRP modeling matches the flight data. Although the empirical and semianalytical
modeling enables accurate SRP modeling, they have the difficulty to separate SRP and other disturbances. Moreover, its
physical meaning is unclear, and only numerical results are obtained.
This Note focuses on a detailed reflection model, which is used in the analytical SRP modeling. The high-fidelity
SRP in this Note means that the SRP is formulated with the detailed reflection model. Few studies use the detailed
reflection model for SRP calculations. Wetterer and et al. [
2
] derive the coefficients that enable the SRP calculations
using detailed reflection modeling. The results indicate that the detailed reflection modeling yields significant
differences in attitude and orbit propagation. Although many physically-based reflection models have been proposed,
conventional SRP formulations still use the simple reflection model that assumes Lambertian diffusion and perfect
specular reflection [
10
,
11
]. This is because only the simple reflection model enables analytic SRP calculations,
∗
Assistant Professor, Department of Aeronautics and Astronautics, 744 Motooka, Nishi-ku, Fukuoka, Japan; y.yoshimura.a64@m.kyushu-u.ac.jp,
Member AIAA
†Graduate Student,Department of Aeronautics and Astronautics, 744 Motooka, Nishi-ku, Fukuoka, Japan.
‡Graduate Student,Department of Aeronautics and Astronautics, 744 Motooka, Nishi-ku, Fukuoka, Japan.
§Assistant Professor, Department of Aeronautics and Astronautics, 744 Motooka, Nishi-ku, Fukuoka, Japan.
¶Professor, Department of Aeronautics and Astronautics, 744 Motooka, Nishi-ku, Fukuoka, Japan. Senior Member AIAA

Fig. 1 Facet-fixed frame.
whereas detailed reflection models such as the Cook–Torrance model [
12
] require numerical integration. The analytical
formulation of the high-fidelity SRP enables not only precise attitude and orbit propagation but also the detailed stability
analysis of spacecraft with a high area-to-mass ratio. To this end, this Note derives an analytical approximation for the
high-fidelity calculations of SRP using the Cook–Torrance model. The SRP formulated with the Cook–Torrance model
is approximated by a single spherical Gaussian (SG), and it simplifies the SRP formulation, resulting in an analytical
form of SRP. The derived approximation is numerically verified for varying surface parameters and lighting conditions.
II. Formulation
A. Bidirectional Reflectance Distribution Function
A bidirectional reflectance distribution function (BRDF)
𝑓𝑟
is the light intensity ratio between irradiance
𝐿𝑖
and
reflected radiance 𝐿𝑟as
𝑓𝑟:=𝐿𝑟
𝐿𝑖
(1)
The BRDF is defined on a small facet, and the facet-fixed frame is illustrated in Fig. 1, where
𝒏
is the normal vector of
the facet and is assumed to be along the
𝑧
axis. The vectors
𝒔
and
𝒗
are the sun directional vector and the reference
vector, which are unit vectors defined with the pairs of azimuth and elevation angles
(𝜙𝑖, 𝜃𝑖)
and
(𝜙𝑟, 𝜃𝑟)
, respectively.
The BRDF is divided into the diffuse reflection 𝑐𝑑and the specular reflection 𝑐𝑠as
𝑓𝑟=𝑑𝑐𝑑+𝑠𝑐𝑠(2)
where 𝑑and 𝑠are the fraction of the diffusion and specularity, respectively, and they satisfy 𝑑+𝑠=1.
Although there are many BRDFs proposed [
13
], a simple BRDF that assumes Lambertian diffusion and perfect
2

mirror-like specularity has been often used for the SRP formulation [14]. The simple BRDF is described as
𝑓𝑟=𝑑𝜌
𝜋+𝑠2𝐹0𝛿(sin2𝜃𝑖−sin2𝜃𝑟)𝛿(𝜙𝑖−𝜙𝑟±𝜋)(3)
where
𝜌
is the diffuse reflectance,
𝐹0
is the Fresnel reflectance, and
𝛿(·)
is the Dirac delta function. The specular
reflection occurs when
𝜃𝑖=𝜃𝑟
and
𝜙𝑖=𝜙𝑟±𝜋
, which geometrically means that the bisector of
𝒔
and
𝒗
corresponds
with the facet’s normal vector 𝒏.
The purpose of this Note is to approximate the SRP calculation formulated with a detailed BRDF. This Note uses
the Cook–Torrance model [12], which is parameterized as
𝑐𝑑=𝜌
𝜋(4)
𝑐𝑠=𝐷𝐺 𝐹
4(𝒏𝑇𝒔)(𝒏𝑇𝒗)(5)
where
𝐷
is the normal distribution function (NDF),
𝐺
is the geometrical attenuation factor, and
𝐹
is the Fresnel
reflection factor. These terms are written as
𝐷=𝑒−(𝜃ℎ/𝑚)2(6)
𝐺=min 1,2(𝒏𝑇𝒉)(𝒏𝑇𝒗)
𝒗𝑇𝒉,2(𝒏𝑇𝒉)(𝒏𝑇𝒔)
𝒗𝑇𝒉(7)
𝐹=(𝑔−𝒗𝑇𝒉)2
2(𝑔+𝒗𝑇𝒉)2(1+𝒗𝑇𝒉𝑔+𝒗𝑇𝒉−12
𝒗𝑇𝒉𝑔−𝒗𝑇𝒉+12)(8)
where
𝒉=𝒔+𝒗
k𝒔+𝒗k(9)
𝑔2=𝑛2
ref + (𝒗𝑇𝒉)2−1(10)
𝑛ref =1+√𝐹0
1−√𝐹0
(11)
The bisector between
𝒔
and
𝒗
is written as the half vector
𝒉
with the azimuth angle
𝜙ℎ
and elevation angle
𝜃ℎ
. The
half vector is thus described as
𝒉=[sin 𝜃ℎcos 𝜙ℎ,sin 𝜃ℎsin 𝜙ℎ,cos 𝜃ℎ]𝑇
. The parameters
𝑚
and
𝑛ref
are the roughness
parameter and the index of refraction, respectively. It is noted that the NDF in Eq.
(6)
is the Gaussian distribution.
Although other distribution models such as the Beckmann distribution can be used for the NDF, the reflection of the
Beckmann distribution is similar to the one of the Gaussian distribution as discussed in [
12
]. The Cook–Torrance
model is one of the isotropic BRDFs, and the purpose of this Note is to derive an analytic approximation of SRP as a
3

preliminary study. Other BRDFs, including anisotropic BRDFs such as the Ashikhmin–Shirley model [
15
], will be
dealt with in future works.
B. Solar Radiation Pressure
SRP is generated by the incoming and outgoing light, and the SRP acting on a facet is also formulated using a BRDF
as
𝑓SRP =−∫∞
0
𝐹𝑠(𝜆)𝐴(𝒏𝑇𝒔)
𝑐𝒔+∬𝑓𝑟(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟d𝜆(12)
=−𝑆0
𝑐𝑟2
AU
𝐴(𝒏𝑇𝒔)𝒔+∬𝑓𝑟(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟(13)
where
𝐹𝑠(𝜆)
is the solar flux,
𝐴
is the facet area,
𝑐
is the speed of light,
𝜆
is the wavelength,
𝑟AU
is the distance between
a satellite and the Sun in astronomical units (AU), and
𝑆0
is the solar flux at 1 AU. Equation
(13)
is the general form of
SRP with a BRDF. Thus, substituting a specific BRDF into Eq.
(13)
gives the SRP formulation under the assumption
that the facet has the reflection properties of the BRDF.
A conventional SRP formulation uses the simple BRDF in Eq.
(3)
because the simple BRDF enables the analytic
integration in Eq. (13). Substituting Eq. (3) into Eq. (13) yields
𝑓SRP =−𝑆0
𝑐𝑟2
AU
𝐴(𝒏𝑇𝒔)𝒔+∬n𝑑𝜌
𝜋+2𝑠𝐹0𝛿(sin2𝜃𝑖−sin2𝜃𝑟)𝛿(𝜙𝑖−𝜙𝑟±𝜋)o(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟(14)
The integrations for the diffuse and specular terms are analytically calculated as
∫2𝜋
0∫𝜋
2
0
𝑑𝜌
𝜋(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟=2
3𝑑𝜌𝒏(15)
∫2𝜋
0∫𝜋
2
0
2𝑠𝐹0𝛿(sin2𝜃𝑖−sin2𝜃𝑟)𝛿(𝜙𝑖−𝜙𝑟±𝜋)(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟
=𝑠𝐹02(𝒏𝑇𝒔)𝒏−𝒔
=𝑠𝐹0𝒓ref (16)
where
𝒓ref
:
=
2
(𝒏𝑇𝒔)𝒏−𝒔
is the perfect specular direction that occurs only when
𝜃𝑖=𝜃𝑟
and
𝜙𝑖=𝜙𝑟±𝜋
. Furthermore,
considering the thermal emissive term 𝜅𝑐abs , the conventional form of SRP is obtained using Eqs. (14)–(16) as
𝑓SRP =−𝑆0
𝑐𝑟2
AU
𝐴(𝒏𝑇𝒔)(𝑐abs +𝑐diff)𝒔+2
3𝑐diff +𝜅𝑐abs +2𝑐spec (𝒏𝑇𝒔)𝒏(17)
where
𝑐diff =𝑑𝜌
,
𝑐spec =𝑠𝐹0
,
𝜅
is the thermal emissivity, and
𝑐abs
is the absorption coefficient and the relation
𝑐abs +𝑐spec +𝑐diff =
1is used. It is noted again that the integration in Eq.
(13)
cannot be obtained analytically for the
4

specular term of the Cook–Torrance model in Eq. (5).
C. Spherical Gaussians
SGs are a spherical basis function on a unit sphere. An SG as a function of a unit vector 𝒙is defined as
G(𝒙;𝒑, 𝜆, 𝜇)=𝜇𝑒𝜆(𝒙𝑇𝒑−1)(18)
where
𝒑
is the lobe axis and is a unit vector,
𝜆
is the lobe sharpness, and
𝜇
is the SG amplitude. Owing to the simple
form of SGs, they have some essential properties, beneficial for analytical calculations. For example, an analytical
integration of an SG on a unit sphere, S2∈R3, is given as follows.
∬S2G(𝒙;𝒑, 𝜆, 𝜇)d𝒙=𝜇2𝜋
𝜆1−𝑒−2𝜆(19)
III. Approximation
A. Approximation Using a Spherical Gaussian
The diffuse term of the Cook–Torrance model in Eq.
(4)
is constant because of Lambertian diffusion and is integrable,
as shown in Eq.
(15)
. This Note thus focuses on the integration of the specular term of the Cook–Torrance model,
𝑓𝑐𝑠
.
The specular term of the SRP is written from Eqs. (13) and (5) as
𝑓𝑐𝑠:=(𝒏𝑇𝒔)∫2𝜋
0∫𝜋/2
0
𝑠𝑐𝑠(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟(20)
=𝑠
4∫2𝜋
0∫𝜋/2
0
𝐷𝐺 𝐹 sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟(21)
Although the specular term consists of a few terms, only the NDF term,
𝐷
, is approximated, because the NDF is the
high-frequency term and the other remaining terms are assumed to be smooth enough [
16
]. The remaining terms are
described as the function of the reference vector 𝒗as
𝑀(𝒗)=𝐺(𝒗)𝐹(𝒗)
4(22)
This term is excluded from the integral and calculated for a representative value.
5

00.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
true
approx.
roughness parameter, m
030 60 90 120 150 180
0
30
60
90
degdeg
0.2
0.4
0.6
0.8
1
030 60 90 120 150 180
deg
deg
0
30
60
90
0.2
0.4
0.6
0.8
1
True
Approximation
Fig. 2 NDF approximation with a single SG.
The NDF is approximated with a single SG [16] as
𝐷=exp (−arccos 𝒉𝑇𝒏
𝑚2)(23)
≈ G (𝒉;𝒏, 𝜆CT ,1)(24)
where
𝜆CT
:
=2
𝑚2
is the lobe sharpness of the Cook–Torrance model. Figure 2 shows the NDF approximation example,
where
𝑚=
0
.
2,
𝜙𝑖=−
90 deg, and
𝜃𝑖=
30 deg. The true NDF is accurately approximated using the single SG. Figure 3
expresses the approximation error for the area that NDF is larger than 0.05. The NDF has large values around the
direction of
𝒓ref
, which is the perfect specular direction. Even in this area, the relative approximation error is less than
3%, as shown in Fig. 3. Figure 4 represents the relative approximation error for the cases
𝑚=
0
.
05 and
𝑚=
0
.
8. The
other lighting conditions are the same as the case
𝑚=
0
.
2. It is noted that the range of color bar for the case
𝑚=
0
.
8is
different from that for the case
𝑚=
0
.
05. Although the maximum error is about 10% due to the high roughness for
𝑚=0.8, the error around the perfect specular direction 𝒓ref shows accurate approximation results in both cases.
The specular term of SRP with the NDF approximation is written from Eqs. (21) and (24) as
𝑓𝑐𝑠≈𝑠𝑀 (𝒓ref )∫2𝜋
0∫𝜋/2
0
𝐷sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟(25)
≈𝑠𝑀 (𝒓ref )∫2𝜋
0∫𝜋/2
0G(𝒉;𝒏, 𝜆CT,1)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟(26)
6

Approximation error % Approximation error %Approximation error %
0 30 60 90 120 150 180
0
30
60
90
0
2
4
6
8
10
0 30 60 90 120 150 180
0
30
60
90
0
0.5
1
1.5
2
2.5
3
0 30 60 90 120 150 180
0
30
60
90
0
0.5
1
1.5
2
2.5
3
Fig. 3 NDF approximation error for 𝑚=0.2.
Approximation error % Approximation error %Approximation error %
0 30 60 90 120 150 180
0
30
60
90
0
2
4
6
8
10
0 30 60 90 120 150 180
0
30
60
90
0
0.5
1
1.5
2
2.5
3
0 30 60 90 120 150 180
0
30
60
90
0
0.5
1
1.5
2
2.5
3
Approximation error % Approximation error %Approximation error %
0 30 60 90 120 150 180
0
30
60
90
0
2
4
6
8
10
0 30 60 90 120 150 180
0
30
60
90
0
0.5
1
1.5
2
2.5
3
0 30 60 90 120 150 180
0
30
60
90
0
0.5
1
1.5
2
2.5
3
Fig. 4 NDF approximation error for 𝑚=0.05 and 𝑚=0.8.
where
𝑀(𝒓ref )
is the remaining specular term represented by
𝒓ref
. It is noted that the SG in Eq.
(24)
is described as
the function of the half vector
𝒉
. On the other hand, the integration in Eq.
(26)
is defined for the hemisphere with
respect to the reference vector
𝒗
. For the consistency between the SG and the integration range, the integration range is
transformed from
𝒗
to
𝒉
. To this end, the Sun-fixed coordinate frame illustrated in Fig. 5 is introduced. This coordinate
frame is defined so that the 𝑧𝑠axis corresponds with the sun vector 𝒔, and the facet normal vector 𝒏can be defined on
the
𝑦𝑠−𝑧𝑠
plane, without loss of generality. This coordinate frame can simplify the variable transformation from
𝒗
to
𝒉
. Since the sun vector is along
𝑧𝑠
axis, the azimuth and elevation angles between
𝒗
and
𝒉
are related to
𝜙𝑟=𝜙ℎ
and
𝜃𝑟=
2
𝜃ℎ
, respectively, as illustrated in Fig. 5. Furthermore, since the integration range is defined initially as the
hemisphere around the normal vector
𝒏
, the hemispherical integration range inclines in the Sun-fixed frame, as shown
7

=
=
Fig. 5 Sun-fixed frame.
=
=
Fig. 6 Inclined integration range in Sun-fixed frame.
in Fig. 6. Consequently, Eq. (26) in the Sun-fixed frame is written as
𝑠𝑀 (𝒓ref )∫𝜋
0∫𝜋/2+𝜃𝑛
0Gsin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟+∫2𝜋
𝜋∫𝜋/2−𝜃𝑛
0Gsin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟
=𝑠𝑀 (𝒓ref )∫𝜋
0∫𝜋/4+𝜃𝑛/2
0
2Gsin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ+∫2𝜋
𝜋∫𝜋/4−𝜃𝑛/2
0
2Gsin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ(27)
where the arguments of
G
are omitted for clarity and
𝒗ℎ
is the reference vector
𝒗
expressed with the azimuth and
elevation angles of the half vector 𝒉, i.e., 𝒗ℎ=[sin (2𝜃ℎ)cos 𝜙ℎ,sin (2𝜃ℎ)sin 𝜙ℎ,cos (2𝜃ℎ)]𝑇.
8

B. Special Case
The special case when both
𝒔
and
𝒏
are along the
𝑧𝑠
axis is firstly considered. The integration range with respect to
𝜃ℎis 𝜃ℎ∈ [0, 𝜋/4]. Furthermore, the SG is also simplified because the normal vector is along the 𝑧𝑠axis as
G(𝒉;𝒏, 𝜆CT,1)=𝑒𝜆CT (cos 𝜃ℎ−1)(28)
Equation (27) when 𝒔=𝒏is thus simplified and can be analytically integrated as
𝑠𝑀 (𝒓ref )∫2𝜋
0∫𝜋/4
0
2G(𝒉;𝒏, 𝜆CT,1)sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ
=𝑠𝑀 (𝒓ref )∫2𝜋
0∫𝜋/4
0
2𝑒𝜆CT (cos 𝜃ℎ−1)sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ(29)
=𝑠𝑀 (𝒓ref )
0
0
8𝜋
𝜆4
CT 𝜆3
CT −5𝜆2
CT +12𝜆CT −12 +𝑒
𝜆CT
√2(1−√2)(2𝜆2
CT −6√2𝜆CT +12)

(30)
The resulting form has only
𝑧
component and is reasonable because the Cook–Torrance model is an isotropic BRDF,
and both 𝒔and 𝒏are along the 𝑧𝑠axis.
C. General Case
In the general case when
𝒔≠𝒏
, the integration range depends on the elevation angle of
𝒏
, which disables the
analytical integration of Eq.
(27)
. This difficulty is reduced by approximating the SG with the first-order linear function
of 𝒉𝑇𝒏[17]. That is, the SG is further approximated as
G(𝒉;𝒏, 𝜆CT,1)≈𝛼+𝛽(𝒉𝑇𝒏)(31)
where
𝛼
and
𝛽
are the optimal coefficients. These coefficients can be obtained so that the least square error of the
approximation in Eq.
(31)
becomes minimum (see [
17
]). Furthermore, to make this approximation accurate, the
integration range is split into a set of small ranges as shown in Fig. 7. Thus Eq. (27) becomes
𝑠𝑀 (𝒓ref )∬2G(𝒉;𝒏, 𝜆 CT,1)sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ
≈𝑠𝑀 (𝒓ref )
𝑁𝜙
Õ
𝑖=1
𝑁𝜃
Õ
𝑗=1∬2𝛼𝑖, 𝑗 +𝛽𝑖, 𝑗 (𝒉𝑇𝒏)sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ(32)
where 𝑁𝜙and 𝑁𝜃are the number of divided integration range with respect to 𝜙ℎand 𝜃ℎ, respectively.
9

=
=
12pt
Fig. 7 Split integration range in 𝜙ℎ−𝜃ℎplane.
The analytical solutions for arbitrary integration ranges can be obtained as follows:
∬2𝛼+𝛽(𝒉𝑇𝒏)sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ=
∬2𝛼sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ+∬2𝛽(𝒉𝑇𝒏)sin (2𝜃ℎ)𝒗ℎd𝜃ℎd𝜙ℎ(33)
=
𝐴𝑥
𝐴𝑦
𝐴𝑧

+
𝐵𝑥
𝐵𝑦
𝐵𝑧

(34)
where
𝐴𝑦=1
4𝛼cos 𝜙ℎ(sin (4𝜃ℎ) − 4𝜃ℎ)(35)
𝐴𝑧=−1
4𝛼𝜙ℎcos (4𝜃ℎ)(36)
𝐵𝑦=1
15 −𝛽cos3𝜃ℎsin 𝜃𝑛[3 cos (2𝜃ℎ) − 7][sin (2𝜙ℎ) − 2𝜙ℎ] − 4𝛽sin3𝜃ℎcos 𝜃𝑛cos 𝜙ℎ[3 cos (2𝜃ℎ) + 7](37)
𝐵𝑧=−4
15 𝛽𝜙ℎcos3𝜃ℎcos 𝜃𝑛[3 cos (2𝜃ℎ) − 2] + sin3𝜃ℎcos 𝜙ℎsin 𝜃𝑛[3 cos (2𝜃ℎ) + 2](38)
The analytic solutions along the
𝑥𝑠
axis,
𝐴𝑥
and
𝐵𝑥
, are omitted because the sum of them is zero due to the isotropic
BRDF.
IV. Numerical Verification
A. Special Case
The special case when
𝒔
and
𝒏
are along the
𝑧𝑠
axis is first examined. As described in Eqs.
(4)
and
(15)
, the diffuse
term of the Cook–Torrance model is Lambertian, and the analytical solution can be obtained. Thus the approximation
of the specular term is verified. True values are calculated by numerically integrating Eq.
(21)
. Figure 8 shows the true
10

00.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
true
approx.
roughness parameter, m
Fig. 8 Special case: approximation result for varying roughness parameter 𝑚.
value and the approximation value obtained by Eq.
(30)
for varying roughness parameter
𝑚
. The approximation is very
accurate for all roughness parameters in the special case. It is noted that this comparison excludes
𝑠𝑀 (𝒓ref )
terms so
that the approximation error due to the integrand of Eq.
(27)
is purely shown. The approximation error due to
𝑠𝑀 (𝒓ref )
is discussed in the general case.
B. General Case
The approximation accuracy for the general case when
𝒔≠𝒏
is examined for the parameters
𝑠=
0
.
5and
𝐹0=
0
.
5.
It is noted that again the facet normal vector
𝒏
lies in the
𝑦𝑠−𝑧𝑠
plane without loss of generality. Figures 9–11 show the
approximation results varying the normal vector for
𝑚=
0
.
05,
𝑚=
0
.
2, and
𝑚=
0
.
8, respectively. All SRP along the
𝑥𝑠
axis are zero and are reasonable because the Cook–Torrance model is an isotropic BRDF. The other components along
the
𝑦𝑠
and
𝑧𝑠
axes have non-zero values and are accurately consistent with the true values. The approximation results in
Figs. 9–11 include the remaining term
𝑠𝑀 (𝒓ref )
, which also means that the representation with
𝒓ref
is accurate enough
so that the remaining term can be excluded from the integral in Eq. (25).
When the angle
𝜃𝑛
is close to 90 deg (i.e., the grazing angle), the approximation result when
𝑚=
0
.
8in Fig. 11 has
a larger error compared to the other roughness parameters. The SRP close to the grazing angle is practically ignorable
because
𝒏𝑇𝒔
approaches to zero, yielding zero SRP. Such high roughness occurs in fabric materials, for example, and
metallic materials used for spacecraft have smaller roughness [18].
V. Conclusions
This Note derived an analytic approximation method to calculate high-fidelity solar radiation pressure (SRP) using
spherical Gaussians. The Cook–Torrance model was used as a reflection function, whose normal distribution function is
11

010 20 30 40 50 60 70 80 90
-5
0
5
010 20 30 40 50 60 70 80 90
-5
0
5
010 20 30 40 50 60 70 80 90
-5
0
5
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
SRP of specular term
SRP of specular term
true
approx
true
approx
true
approx
true
approx
true
approx
true
approx
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
SRP of specular term
true
approx
true
approx
true
approx
0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
7
8
9
10
Fig. 9 General Case: approximation result for varying normal directions and 𝑚=0.05.
010 20 30 40 50 60 70 80 90
-5
0
5
010 20 30 40 50 60 70 80 90
-5
0
5
010 20 30 40 50 60 70 80 90
-5
0
5
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
SRP of specular term
SRP of specular term
true
approx
true
approx
true
approx
true
approx
true
approx
true
approx
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
SRP of specular term
true
approx
true
approx
true
approx
0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
7
8
9
10
Fig. 10 General Case: approximation result for varying normal directions and 𝑚=0.2.
010 20 30 40 50 60 70 80 90
-5
0
5
010 20 30 40 50 60 70 80 90
-5
0
5
010 20 30 40 50 60 70 80 90
-5
0
5
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
0 10 20 30 40 50 60 70 80 90
-0.03
0
0.03
SRP of specular term
SRP of specular term
true
approx
true
approx
true
approx
true
approx
true
approx
true
approx
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90
-0.2
0
0.2
SRP of specular term
true
approx
true
approx
true
approx
0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
7
8
9
10
Fig. 11 General Case: approximation result for varying normal directions and 𝑚=0.8.
12

approximated with a single spherical Gaussian. The first-order approximation of the spherical Gaussian enables the
analytic integration of SRP. The approximation accuracy was verified by varying the normal vector direction of a facet
and the roughness parameter. Future works will involve considering the self-shadowing of a satellite and anisotropic
reflection models such as the Ashikhmin–Shirley model.
References
[1]
Früh, C., Kelecy, T. M., and Jah, M. K., “Coupled orbit-attitude dynamics of high area-to-mass ratio (HAMR) objects: influence
of solar radiation pressure, Earth’s shadow and the visibility in light curves,” Celestial Mechanics and Dynamical Astronomy,
Vol. 117, No. 4, 2013, pp. 385–404. https://doi.org/10.1007/s10569-013-9516-5.
[2]
Wetterer, C. J., Linares, R., Crassidis, J. L., Kelecy, T. M., Ziebart, M. K., Jah, M. K., and Cefola, P. J., “Refining Space
Object Radiation Pressure Modeling with Bidirectional Reflectance Distribution Functions,” Journal of Guidance, Control, and
Dynamics, Vol. 37, No. 1, 2014, pp. 185–196. https://doi.org/10.2514/1.60577.
[3]
McMahon, J. W., and Scheeres, D. J., “Improving Space Object Catalog Maintenance Through Advances in Solar Radiation
Pressure Modeling,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 8, 2015, pp. 1366–1381. https://doi.org/10.
2514/1.g000666.
[4]
Mimasu, Y., Ono, G., Akatsuka, K., Terui, F., Ogawa, N., Saiki, T., and Tsuda, Y., “Attitude Control of Hayabusa2 by Solar
Radiation Pressure in One Wheel Control Mode,” 25th International Symposium on Space Flight Dynamics 17, Munich,
Germany, 2015, pp. 356–362.
[5]
Ono, G., Kikuchi, S., and Tsuda, Y., “Stability Analysis of Generalized Sail Dynamics Model,” Journal of Guidance, Control,
and Dynamics, Vol. 41, No. 9, 2018, pp. 2011–2018. https://doi.org/10.2514/1.g003418.
[6]
Matsumoto, J., Ono, G., Chujo, T., Akatsuka, K., and Tsuda, Y., “FEM-based High-fidelity Solar Radiation Pressure Analysis,”
TRANSACTIONS OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES, Vol. 60, No. 5, 2017, pp.
276–283. https://doi.org/10.2322/tjsass.60.276.
[7]
Liu, J., Gu, D., Ju, B., Shen, Z., Lai, Y., and Yi, D., “A new empirical solar radiation pressure model for BeiDou GEO satellites,”
Advances in Space Research, Vol. 57, No. 1, 2016, pp. 234–244. https://doi.org/10.1016/j.asr.2015.10.043.
[8]
Rodriguez-Solano, C., Hugentobler, U., and Steigenberger, P., “Adjustable box-wing model for solar radiation pressure impacting
GPS satellites,” Advances in Space Research, Vol. 49, No. 7, 2012, pp. 1113–1128. https://doi.org/10.1016/j.asr.2012.01.016.
[9]
Montenbruck, O., Steigenberger, P., and Darugna, F., “Semi-analytical solar radiation pressure modeling for QZS-1 orbit-
normal and yaw-steering attitude,” Advances in Space Research, Vol. 59, No. 8, 2017, pp. 2088–2100. https://doi.org/https:
//doi.org/10.1016/j.asr.2017.01.036, URL https://www.sciencedirect.com/science/article/pii/S0273117717300832.
[10] Wie, B., Space Vehicle Dynamics and Control, AIAA, 1998, pp. 749–752. https://doi.org/10.2514/4.860119.
13

[11]
McInnes, C. R., Solar Sailing: Technology, Dynamics and Mission Applications, Springer Praxis, Chichester, England, U.K.,
1999, pp. 32–51.
[12]
Cook, R. L., and Torrance, K. E., “A reflectance model for computer graphics,” ACM Transactions on Graphics (TOG), Vol. 1,
1982, pp. 7–24.
[13] Montes, R., and Ureña, C., “An overview of BRDF models,” University of Grenada, Technical Report LSI-2012-001, 2012.
[14]
Ono, G., Tsuda, Y., Akatsuka, K., Saiki, T., Mimasu, Y., Ogawa, N., and Terui, F., “Generalized Attitude Model for
Momentum-Biased Solar Sail Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 7, 2016, pp. 1491–1500.
https://doi.org/10.2514/1.g001750.
[15]
Ashikhmin, M., and Shirley, P., “An Anisotropic Phong BRDF Model,” Journal of graphics tools, Vol. 5, No. 2, 2000, pp.
25–32. https://doi.org/10.1080/10867651.2000.10487522.
[16]
Wang, J., Ren, P., Gong, M., Snyder, J., and Guo, B., “All-frequency rendering of dynamic, spatially-varying reflectance,” ACM
SIGGRAPH Asia 2009 papers, 2009, pp. 1–10. https://doi.org/10.1145/1618452.1618479.
[17]
Wang, R., Pan, M., Chen, W., Ren, Z., Zhou, K., Hua, W., and Bao, H., “Analytic double product integrals for all-frequency
relighting,” IEEE Trans Vis Comput Graph, Vol. 19, No. 7, 2013, pp. 1133–1142. https://doi.org/10.1109/TVCG.2012.152.
[18]
Ngan, A., Durand, F., and Matusik, W., “Experimental Analysis of BRDF Models.” Proc. Eurographics Symposium on
Rendering 2005, 2005, pp. 117–126. https://doi.org/10.2312/EGWR/EGSR05/117- 126.
14