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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+𝑠2𝐹0𝛿(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+𝒗𝑇𝒉𝑔+𝒗𝑇𝒉−12

𝒗𝑇𝒉𝑔−𝒗𝑇𝒉+12)(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)

𝑐𝒔+∬𝑓𝑟(𝒏𝑇𝒗)sin 𝜃𝑟𝒗d𝜃𝑟d𝜙𝑟d𝜆(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

=𝑠𝐹02(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

𝜆1−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

𝑚2)(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∬2𝛼𝑖, 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:

∬2𝛼+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 𝛽𝜙ℎcos3𝜃ℎcos 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.

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