Determination of resultant force and moment of light radiation
pressure upon a perspective space telescope Millimetron
Mr. Nerovny N.A.1
Prof. Zimin V.N.1
Dr. Fedorchuk S.D.2
Mr. Golubev E.S.2
1 Bauman Moscow State Technical University, [email protected]
2 Astro Space Center of P.N. Lebedev Physical Institute
"The Fourth International Symposium on Solar Sailing 2017"
ISSS-2017,
17th – 20th January, Kyoto, JAPAN.
1 / 52
Contents
1
Introduction
2
Light pressure
Infinitesimal light pressure force
Tensor representation
3
Analytical examples
4
Numerical method
Method definition
Model spacecraft
Raytracing results
5
Results
6
Conclusions and future plans
2 / 52
Research goal
Determination of light radiation pressure upon a space structures with complex geometry.
←?
Millimetron / ASC of Physical Institute of RAS
JWST / NASA
3 / 52
Light pressure
4 / 52
Infinitesimal light pressure force
Major assumptions:
Transmittance is not considered;
Reflectivity can be both specular or
diffuse or both;
x3
Diffuse reflection can be Lambertian,
or other axis-symmetric dispersion law
can be considered;
Деформированная форма
ŝ
A
n̂
Temperature is constant in the normal
direction;
dA
dF
r
x1
O
The light flux is parallel;
u
No self-shadowing and no secondary
reflections;
x2
Раскройная форма
The structure is optically convex.
The infinitesimal light pressure force
q0
εBσT 4
dF =
−
n̂ − (1 − ρs)(n̂ · ŝ)ŝ + ρ(1 − s)B(n̂ · ŝ)n̂ − 2ρs(n̂ · ŝ)2 n̂ dA,
c
q0
(1)
after defining of a0 , a1 , a2 , a3 ,
h
i
dF = P(R) −a0 n̂ − a1 (n̂ · ŝ)ŝ + a2 (n̂ · ŝ)n̂ − 2a3 (n̂ · ŝ)2 n̂ dA.
(2)
The moment from infinitesimal light pressure force:
dM = r × dF.
(3)
5 / 52
Tensor representation
Let us introduce the visibility function (similar to D.J. Scheeres and L. Rios-Reyes1 ):
f (n̂, ŝ) =
n̂ · ŝ − |n̂ · ŝ|
.
2
Eq. (2) can be rewritten with visibility function:
P(R)
dF =
− 2a0 n̂ − a1 (n̂ · ŝ)ŝ + a2 (n̂ · ŝ)n̂ − 2a3 (n̂ · ŝ)2 n̂ + a1 |n̂ · ŝ|ŝ
2
−a2 |n̂ · ŝ|n̂ + 2a3 (n̂ · ŝ)|n̂ · ŝ|n̂ dA.
(4)
(5)
As far as n̂ · ŝ ∈ [−1; 1], the |n̂ · ŝ| can be represented as a series of Chebyshev polynomials of
the first kind:
∞
2
4 X (−1)n T2n (n̂ · ŝ)
|n̂ · ŝ| = −
=
π
π n=1
−1 + 4n2
=−
∞ n−1
4 X X (−1)n (−1)k n(2n − k − 1)! n−k
4
(n̂ · ŝ)2(n−k) .
π n=1 k=0 (−1 + 4n2 )k!(2n − 2k)!
(6)
After transformations (6) can be written in simplified form:
|n̂ · ŝ| =
∞
X
Bm (n̂ · ŝ)2m ,
(7)
m=1
where
Bm = −
∞
(−1)m 4m+1 X
n(n + m − 1)!
.
π(2m)! n=m (−1 + 4n2 )(n − m)!
(8)
1
Rios-Reyes, L. and Scheeres, D. J. Generalized Model for Solar Sails, Journal of Spacecraft and Rockets, 42 (1), 2005,
pp.182-185.
6 / 52
The series (8) diverge, but in the final summation (7) the divergences are regularized (it can be
simply proven since the original series expansion is convergent). We will limit the number of
terms in (7) by Nmax , the upper bound for (8) will be
Nmax − 1
Nmax B =
,
2
where bxc is a floor function of real x.
Convergence of approximation
1
|n̂ · ŝ|
N=1
N=3
N=5
N=7
N=9
0.8
0.6
0.4
0.2
0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 1: Different approximation rank for |n̂ · ŝ|
7 / 52
Tensor series for F
Let us substitute Eq. (7) into Eq. (5):
P(R)
dF =
− 2a0 n̂ − a1 (n̂ · ŝ)ŝ + a2 (n̂ · ŝ)n̂ − 2a3 (n̂ · ŝ)2 n̂+
2
N
N
N
max
max
max
X
X
X
a1
Bm (n̂ · ŝ)2m ŝ − a2
Bm (n̂ · ŝ)2m n̂ + 2a3
Bm (n̂ · ŝ)2m+1 n̂ dA.
m=1
m=1
(9)
m=1
Extending the approach of D.J. Scheeres and others, we can introduce the new shape tensors:
(n̂ · ŝ) . . . (n̂ · ŝ) n̂ = (n̂ ⊗ n̂ ⊗ . . . ⊗ n̂) · ŝ · . . . · ŝ = JAp+1 · ŝ · . . . · ŝ;
| {z }
|
{z
} | {z }
|
{z
}
p
p+1
p
p
2
(n̂ · ŝ) . . . (n̂ · ŝ) ŝ = (n̂ ⊗ n̂ ⊗ . . . ⊗ n̂ ⊗E ) · ŝ · . . . · ŝ =
| {z }
|
{z
}
|
{z
}
p+1
p
p
JBp+2
· ŝ · . . . · ŝ .
| {z }
p+1
After rewriting of Eq. (9) in the tensor notation:
P(R)
dF =
− 2a0 n̂ − a1 JB3 · ŝ · ŝ + a2 JA2 · ŝ − 2a3 JA3 · ŝ · ŝ+
2
a1
N
max
X
m=1
Bm JB2m+2 · ŝ · . . . · ŝ −a2
| {z }
2m+1
2a3
N
max
X
m=1
N
max
X
m=1
Bm JA2m+1 · ŝ · . . . · ŝ +
| {z }
2m
Bm JA2m+2 · ŝ · . . . · ŝ dA.
| {z }
(10)
2m+1
8 / 52
Tensor series for F
By grouping of terms in (10) we can write the equation for infinitesimal force of light pressure
falling only to the front side:
N
max
X
dF = P(R) J 1 +
J n · ŝ · . . . · ŝ dA,
| {z }
n=2
(11)
n−1
where
J 1 = −a0 n̂;
1
J 2 = a2 JA2 ;
2
1
3
J =
− a1 JB3 − 2a3 JA3 − B1 a2 JA3 ;
2
1 − (−1)n
1 + (−1)n
1
− B n−1
a2 JAn + B n−2
(a1 JBn + 2a3 JAn ) , n > 3;
Jn =
2
2
2
2
2
JAn
(12)
(13)
(14)
(15)
= n̂ ⊗ . . . ⊗ n̂;
|
{z
}
(16)
JBn = n̂ ⊗ . . . ⊗ n̂ ⊗E 2 .
|
{z
}
(17)
n
n−2
9 / 52
Tensor series for M
Providing the same procedure for the moment:
(n̂ · ŝ) . . . (n̂ · ŝ)(R2 · n̂) = (n̂ ⊗ n̂ ⊗ . . . ⊗ n̂ ⊗R2 · n̂) · ŝ · . . . · ŝ = LAp+1 · ŝ · . . . · ŝ;
| {z }
| {z }
|
{z
}
|
{z
}
p
p
p
2
2
(n̂ · ŝ) . . . (n̂ · ŝ)(R · ŝ) = (n̂ ⊗ n̂ ⊗ . . . ⊗ n̂ ⊗R ) · ŝ · . . . · ŝ =
| {z }
|
{z
}
|
{z
}
p+1
p
p
p
LBp+2
· ŝ · . . . · ŝ,
| {z }
p+1
we can also represent the Eq. (9) as a tensor series:
P(R)
− 2a0 R2 · n̂ − a1 L3B · ŝ · ŝ + a2 L2A · ŝ−
dM = r × dF = R2 · dF =
2
2a3 L3A · ŝ · ŝ + a1
N
max
X
m=1
Bm L2m+2
· ŝ · . . . · ŝ −a2
B
| {z }
2m+1
2a3
N
max
X
m=1
N
max
X
m=1
Bm LA2m+1 · ŝ · . . . · ŝ +
| {z }
2m
Bm LA2m+2 · ŝ · . . . · ŝ dA,
| {z }
(18)
2m+1
where
LnA = n̂ ⊗ . . . ⊗ n̂ ⊗R2 · n̂;
|
{z
}
(19)
LnB = n̂ ⊗ . . . ⊗ n̂ ⊗R2 .
|
{z
}
(20)
n−1
n−2
10 / 52
Tensor series for M
The tensor series for M will be as the follows:
N
max
X
dM = P(R) L1 +
Ln · ŝ · . . . · ŝ dA,
| {z }
n=2
(21)
n−1
where
L1 = −a0 (R2 · n̂);
1
L2 = a2 L2A ;
2
1
L3 =
− a1 L3B − 2a3 L3A − B1 a2 L3A ;
2
1
1 − (−1)n
1 + (−1)n
Ln =
− B n−1
a2 LnA + B n−2
(a1 LnB + 2a3 LnA ) , n > 3;
2
2
2
2
2
(22)
(23)
(24)
(25)
11 / 52
Resulting equation
By integrating of (11) and (21) over the whole surface A, we can get the resultant force and
moment upon an optically convex structure:
N
max
X
F = P(R) Ĩ 1 +
Ĩ n · ŝ · . . . · ŝ ;
| {z }
(26)
N
max
X
M = P(R) K̃1 +
K̃n · ŝ · . . . · ŝ ,
| {z }
(27)
n=2
n−1
n=2
n−1
where
Ĩ n =
Z
J˜n dA;
(28)
L̃n dA,
(29)
A
K̃n =
Z
A
where n > 1.
It is possible to write the same relations considering self-shadowing and secondary reflections2 .
2
Nerovny, N.A. et al. Representation of light pressure resultant force and moment as a tensor series // Celestial Mechanics
and Dynamical Astronomy. [Approved for publication]
12 / 52
Analytical examples
13 / 52
Assumptions
In the analytical examples below the light source orientation vector ŝ is defined by two angles α
and β as follows:
α ∈ [0, 2π] — angle between unit vector ê1 of axis Ox1 and projection of vector ŝ on the
plane Ox1 x3 ;
β ∈ [− π2 ;
π
]
2
— angle between plane Ox1 x3 and vector ŝ.
The components of vector ŝ can be written as follows:
ŝ = (cos α cos β, sin β, sin α cos β)T .
14 / 52
Flat sail
The front side (side 1) has reflection coefficient ρ1 and specularity parameter s1 . The other side
(side 2) has reflectivity ρ2 and specularity s2 . The area of the solar sail is A.
Components of tensors I and K:
I = A (J (n̂1 , ρ1 , s1 ) + J (n̂2 , ρ2 , s2 )) ;
K = A (L(n̂1 , r1 , ρ1 , s1 ) + L(n̂2 , r2 , ρ2 , s2 )) ;
n̂1 = (0, 0, 1)T ;
n̂2 = (0, 0, −1)T ;
r1 = r2 = (0, 0, 0)T .
Resultant force and moment (Nmax = 6):
P(R)A
(−6(−2 + ρ1 s1 + ρ2 s2 ) + cos β sin α (15π(ρ1 s1 − ρ2 s2 )+
30π
+8(−2 + ρ1 s1 + ρ2 s2 ) cos β sin α −9 + 4 cos2 β sin2 α
cos α cos β;
F1 =
P(R)A
(−6(−2 + ρ1 s1 + ρ2 s2 ) + cos β sin α (15π(ρ1 s1 − ρ2 s2 )+
30π
+8(−2 + ρ1 s1 + ρ2 s2 ) cos β sin α −9 + 4 cos2 β sin2 α
sin β;
F2 =
P(R)A
(24(ρ2 (1 − s2 ) − ρ1 (1 − s1 )) + cos β sin α
90π
(6(5π(ρ1 (1 − s1 ) + ρ2 (1 − s2 )) + 3(2 + ρ1 s1 ρ2 s2 )) + cos β sin α
F3 =
(−9(16ρ1 (1 − s1 ) + 5πρ1 s1 − 16ρ2 (1 − s2 ) − 5πρ2 s2 ) + 8 cos β sin α
(27(2 + ρ1 s1 + ρ2 s2 ) + 4 cos β sin α(2(ρ1 (1 − s1 ) − ρ2 (1 − s2 ))−
−3(2 + ρ1 s2 + ρ2 s2 ) cos β sin α))))) ;
M = 0.
15 / 52
Flat solar sail, Nmax = 6, ρ1 = 1, ρ2 = 0, s1 = s2 = 1
a
b
Figure 2: Projection on Ox1 (a) and on Ox3 (b) of resultant force of light pressure upon two-sided specular
solar sail with unit area, Nmax = 6, ρ1 = 1, ρ2 = 0, s1 = s2 = 1. Solid line – approximate solution, dashed line
— exact solution. Values are divided by P(R).
16 / 52
Flat solar sail, Nmax = 6, ρ1 = 1, ρ2 = 0, s1 = s2 = 0
a
b
Figure 3: Projection on Ox1 (a) and on Ox3 (b) of resultant force of light pressure upon two-sided diffuse solar
sail with unit area, Nmax = 6, ρ1 = 1, ρ2 = 0, s1 = s2 = 0. Solid line – approximate solution, dashed line –
exact solution. Values are divided by P(R).
17 / 52
Sphere
Let us consider a sphere of radius R0 with a homogeneous specular-diffusive surface, the
reflection coefficient of which is equal to ρ and the degree of specular reflection is s.
The expressions for the components of tensors I and K:
π
I=
R20
Z2π Z2
J (n̂, ρ, s)dθdφ;
0 −π
2
π
K=
R20
Z2π Z2
L(n̂, r, ρ, s)dθdφ;
0 −π
2
n̂ = (cos φ cos θ, sin φ cos θ, sin θ)T ;
r = (R0 cos φ cos θ, R0 sin φ cos θ, R0 sin θ)T .
The analytical expressions considering Nmax = 6:
4
(175πρ(1 − s) + 3(413 + ρs)) R20 cos α cos β;
1575
4
F2 = P(R)
(175πρ(1 − s) + 3(413 + ρs)) R20 sin β;
1575
4
F3 = P(R)
(175πρ(1 − s) + 3(413 + ρs)) R20 cos β sin α;
1575
M = 0.
F1 = P(R)
18 / 52
Ideally specular sphere
For ρ = 1 and s = 1 we get:
F1 ≈ P(R)πR20 cos α cos β;
(30)
F2 ≈ P(R)πR20 sin β;
(31)
P(R)πR20
(32)
F3 ≈
cos β sin α.
19 / 52
Cylinder
Let us consider the cylinder with following parameters:
ρ0 — reflectance of the envelope;
ρ1 — reflectance of the butt surface +x3 ;
ρ2 — reflectance of the butt surface −x3 ;
s0 — specularity coefficient of the envelope;
s1 — specularity coefficient of the butt surface +x3 ;
s2 — specularity coefficient of the butt surface −x3 ;
R1 — radius of the cylinder;
H — height of the cylinder.
The expressions for the components of tensors I and K:
I = J (n̂1 , ρ1 , s1 )πR21 + J (n̂2 , ρ2 , s2 )πR21 + HR1
Z2π
J (n̂0 , ρ0 , s0 )dφ;
0
K = L(n̂1 , r1 , ρ1 , s1 )πR21 + L(n̂2 , r2 , ρ2 , s2 )πR21 + HR1
Z2π
L(n̂0 , r0 , ρ0 , s0 )dφ;
0
n̂1 = (0, 0, 1)T ;
n̂2 = (0, 0, −1)T ;
n̂0 = (cos φ, sin φ, 0)T ;
r1 = (0, 0, H/2)T ;
r2 = (0, 0, −H/2)T ;
r0 = (R1 cos φ, R1 sin φ, 0)T .
20 / 52
Specular-diffuse cylinder
For the number of terms of the series Nmax = 6 we can get:
P(R)R1
cos α cos β(−8H(3 + 2ρ0 s0 ) cos4 α cos4 β+
30
+ 4H cos2 α cos2 β(12 + 5ρ0 s0 + (6 + 4ρ0 s0 ) cos 2β)+
F1 =
+ R1 cos β sin α(15π(ρ1 s1 − ρ2 s2 )+
+ 8(−2 + ρ1 s1 + ρ2 s2 ) cos β sin α(−9 + 4 cos2 β sin2 α))+
+ 2(H(6 − 5πρ0 (−1 + s0 )) − 3R1 (−2 + ρ1 s1 + ρ2 s2 )+
+ 2H(15 + 7ρ0 s0 + (3 + 2ρ0 s0 ) cos 2β) sin2 β));
P(R)R1
sin β(−8H(3 + 2ρ0 s0 ) cos4 α cos4 β+
30
+ 4H cos2 α cos2 β(12 + 5ρ0 s0 + (6 + 4ρ0 s0 ) cos 2β)+
F2 =
+ R1 cos β sin α(15π(ρ1 s1 − ρ2 s2 )+
+ 8(−2 + ρ1 s1 + ρ2 s2 ) cos β sin α(−9 + 4 cos2 β sin2 α))+
+ 2(H(6 − 5πρ0 (−1 + s0 )) − 3R1 (−2 + ρ1 s1 + ρ2 s2 )+
+ 2H(15 + 7ρ0 s0 + (3 + 2ρ0 s0 ) cos 2β) sin2 β));
21 / 52
Specular-diffuse cylinder
F3 =
P(R)R1
(24R1 (ρ2 (1 − s2 ) − ρ1 (1 − s1 ))+
90
3
cos β(−363H(−1 + ρ0 s0 ) + 16R1 (5π(ρ1 + ρ2 − ρ1 s1 − ρ2 s2 )+
8
+ 3(2 + ρ1 s1 + ρ2 s2 )) + 3H(−1 + ρ0 s0 )(44 cos 2β + 5 cos 4β)) sin α−
+
− 9R1 (ρ1 (16 + (−16 + 5π)s1 ) + 16ρ2 (−1 + s2 ) − 5πρ2 s2 ) cos2 β sin2 α+
+ 64R1 (ρ1 − ρ1 s1 + ρ2 (−1 + s2 )) cos4 β sin4 α+
9
cos3 β(96R1 (2 + ρ1 s1 + ρ2 s2 ) sin3 α − H(−1 + ρ0 s0 )(13 + 5 cos 2β) sin 3α)+
4
3
+ cos5 β(−64R1 (2 + ρ1 s1 + ρ2 s2 ) sin5 α + 3H(−1 + ρ0 s0 ) sin 5α));
2
P(R)HR21
M1 =
(6ρ1 s1 − 6ρ2 s2 + cos β sin α(15π(−2ρ0 s0 + ρ1 s1 + ρ2 s2 )+
60
+ 8(ρ1 s1 − ρ2 s2 ) cos β sin α(−9 + 4 cos2 β sin2 α))) sin β;
+
P(R)HR21
(−6ρ1 s1 + 6ρ2 s2 + cos β sin α(15π(2ρ0 s0 − ρ1 s1 − ρ2 s2 )+
60
+ 8(ρ1 s1 − ρ2 s2 ) cos β sin α(9 − 4 cos2 β sin2 α))) cos α cos β;
M2 =
M3 = 0.
22 / 52
Specular-diffuse cylinder, Nmax = 6, ρ1 = ρ2 = 0, ρ0 = 1, s1 = s2 = 0, s0 = 1
a
b
Figure 4: Projection on Ox1 (a) and on Ox3 (b) of resultant force of light pressure upon specular-diffuse
cylinder, Nmax = 6, ρ1 = ρ2 = 0, ρ0 = 1, s1 = s2 = 0, s0 = 1. Values are divided by P(R).
Figure 5: Projection on Ox2 of principal moment of light pressure upon specular-diffuse cylinder, Nmax = 6,
ρ1 = ρ2 = 0, ρ0 = 1, s1 = s2 = 0, s0 = 1. Values are divided by P(R).
23 / 52
Numerical method
24 / 52
Method definition
Main idea: approximation of shape tensor components using known values of force
f(i) = F(i) /P(R) and moment m(i) = M(i) /P(R) for the set of known orientation vector ŝ(i) in a
number N.
The approximated values for shape tensors components Ĩ n and K̃n :
T

j
[( 32 (3M −1))×1]
 2 2 2 3

Ĩ21 Ĩ31 Ĩ111 . . . Ĩ3M. . . 31 Ĩ21 . . . Ĩ3M. . . 32 Ĩ31 . . . Ĩ3M. . . 3  ;
= Ĩ11 Ĩ11
| {z }
| {z }
| {z }
M−1
M−1
M
(33)
T



2 2 2 3
K̃21 K̃31 K̃111 . . . K̃3M. . . 31 K̃21 . . . K̃3M. . . 32 K̃31 . . . K̃3M. . . 3  ,
k
= K̃11 K̃11
| {z }
| {z }
| {z }
[( 23 (3M −1))×1]
M−1
M−1
(34)
M
where M = Nmax .
Vector of free terms:
(N) T
(N) (1) (2)
(1) (2)
(N) (1) (2)
;
= f1 f1 . . . f1 f2 f2 . . . f2 f3 f3 . . . f3
(35)
(1) (2)
(N) (1) (2)
(N) (1) (2)
(N) T
= m1 m1 . . . m1 m2 m2 . . . m2 m3 m3 . . . m3
.
(36)
f
[3N×1]
m
[3N×1]
Matrix of orientations:
S
=
[3n×( 23 (3M −1))]
25 / 52

1
 s(1)
1

 s(1)
2


 s(1)

3
 s(1) s(1)
 1 1

.

.

 (1) . (1)
s3 . . . s3
| {z }

 M−1

0






.

.

.








0




0






.

.

.







0

···
1
(N)
s1
(N)
s2
(N)
s3
0
···
.
.
.
0
0
.
.
.
.
.
.
···
(N) (N)
s1
s1
···
(N)
s3
|
.
.
.
(N)
. . . s3
{z
}
0
···
0
0
···
1
(1)
s1
(1)
s2
···
1
(N)
s1
(N)
s2
0
···
M−1
···
0
.
.
.
···
0
(1)
s3
(1) (1)
s1 s1
(N)
s3
(N) (N)
s1 s1
.
.
.
(1)
(1)
s3 . . . s3
{z
}
|
.
.
.
(N)
(N)
s3 . . . s3
|
{z
}
···
M−1
···
···
0
0
.
.
.
.
.
.
0
0
.
.
.
0
···
1
(1)
s1
(1)
s2
···
M−1
···
0
.
.
.
···
0
(1)
s3
(1) (1)
s1 s1
.
.
.
(1)
(1)
s3 . . . s3
|
{z
}
···
0
T




.

.

.








0




0






.

.

.

 .






0




1

(N)

s1

(N)

s2



(N)

s3
(N) (N) 

s1 s1


.
.

.

(N)
(N) 
s3 . . . s3 
|
{z
}
26 / 52
Least squares approximation
The resolving equation for f and m are overdefined:
Sj = f;
(38)
Sk = m.
(39)
j and k are approximated by j̃ and k̃ using least squares method:
||Sj − f||2 → min, j̃ = ST S
+
ST f;
+
||Sk − m||2 → min, k̃ = ST S ST m,
(40)
(41)
where + is a pseudo-inverse operator.
27 / 52
Model spacecraft
28 / 52
Millimetron Space Telescope
Figure 6: Millimentron space observatory concept. 1 – Sun shields; 2 – Cryo-shield; 3 – Primary mirror’s petal;
4 – Secondary mirror; 5 – Central part of Primary mirror; 6 – Cryo-container; 7 – Heat exchanger (radiator); 8
– Warm container; 9 – Sunshields supporting truss; 10 – Adapter ring; 11 – Service module; 12 – Solar power
array; 13 – High gain antenna.
29 / 52
Geometrical model
Figure 7: Geometrical model of Millimetron space observatory.
30 / 52
Raytracing results
Raytracing results, specular and diffuse cases (1000000 rays), Tracer3
a
c
3
b
a, b — diffuse cases; c, d — specular cases.
d
Leonov, V.V. Radiation heat transfer in mirror concentrator systems, PhD Thesis, 2012 (in Russian).
31 / 52
Specular case
32 / 52
Specular case, absolute values, Nmax = 2
90
90
0.004
0.003
120
60
120
60
0.003
0.002
150
30
150
30
0.002
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 8: Dependency of the absolute value of resultant force and moment of light radiation pressure from the
angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 2), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is specular.
33 / 52
Specular case, absolute values, Nmax = 3
90
90
0.004
0.003
120
60
120
60
0.003
0.002
150
30
150
30
0.002
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 9: Dependency of the absolute value of resultant force and moment of light radiation pressure from the
angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 3), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is specular.
34 / 52
Specular case, absolute values, Nmax = 4
90
90
0.004
0.003
120
60
120
60
0.002
150
30
150
30
0.002
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 10: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 4), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is specular.
35 / 52
Specular case, absolute values, Nmax = 5
90
90
0.004
0.003
120
60
120
60
0.002
150
30
150
30
0.002
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 11: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 5), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is specular.
36 / 52
Specular case, absolute values, Nmax = 6
90
90
0.004
0.003
120
60
120
60
0.002
150
30
150
30
0.002
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 12: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 6), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is specular.
37 / 52
Specular case, projections
0.002
0.001
8e-06
6e-06
0.001
0.0005
4e-06
Fz
Fy
Fx
0
2e-06
0
-0.001
0
-0.0005
-0.002
-2e-06
-0.003
-4e-06
0
1
2
3
4
5
6
7
-0.001
0
1
2
3
alpha
4
5
6
7
0
1
2
3
a
b
3e-05
4
5
6
7
4
5
6
7
alpha
alpha
c
0.004
0.0001
0.002
5e-05
0
My
Mx
1e-05
Mz
2e-05
0
0
-1e-05
-2e-05
-0.002
-5e-05
-0.004
-0.0001
-3e-05
-4e-05
0
1
2
3
4
alpha
d
5
6
7
0
1
2
3
4
alpha
e
5
6
7
0
1
2
3
alpha
f
Figure 13: Approximation results (Nmax = 6) for principal force (a, c, e), N, and for principal moment (b, c, f),
N · m, of light radiation pressure depending on angle of rotation of light source in the plane of radiators (solid
line) comparing results of Monte–Carlo simulations (dots). The results in subfigures b, c, and f are non-zero
because of random noise. The dotted values were not used in the construction of approximation. Specular
case.
38 / 52
Diffuse case
39 / 52
Diffuse case, absolute values, Nmax = 2
90
90
0.003
120
0.003
60
120
60
0.002
0.002
150
30
150
30
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 14: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 2), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is diffuse.
40 / 52
Diffuse case, absolute values, Nmax = 3
90
90
0.003
120
0.003
60
120
60
0.002
0.002
150
30
150
30
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 15: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 3), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is diffuse.
41 / 52
Diffuse case, absolute values, Nmax = 4
90
90
0.003
120
0.003
60
120
60
0.002
0.002
150
30
150
30
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 16: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 4), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is diffuse.
42 / 52
Diffuse case, absolute values, Nmax = 5
90
90
0.003
120
0.003
60
120
60
0.002
0.002
150
30
150
30
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 17: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 5), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is diffuse.
43 / 52
Diffuse case, absolute values, Nmax = 6
90
90
0.003
120
0.003
60
120
60
0.002
0.002
150
30
150
30
0.001
0.001
180
0
210
330
240
300
270
a
180
0
210
330
240
300
270
b
Figure 18: Dependency of the absolute value of resultant force and moment of light radiation pressure from
the angle of rotation of light source in the radiators plane. Solid line – tensor approximation (Nmax = 6), dots –
Monte–Carlo simulation results which were not used for the approximation. Figure a – absolute value of
resultant force, N; Figure b – absolute value of the resultant moment, N · m. The whole surface is diffuse.
44 / 52
6e-06
0.002
4e-06
0.0015
2e-06
0.001
0
0.0005
-2e-06
0
-4e-06
-0.0005
0.0015
0.001
0.0005
Fz
0.0025
Fy
Fx
Diffuse case, projections
-0.001
-6e-06
-0.001
-8e-06
0
1
2
3
4
5
6
7
-0.0015
0
1
2
3
alpha
4
5
6
7
0
1
2
alpha
a
6e-05
0.003
4e-05
0.002
3
4
5
6
7
4
5
6
7
alpha
b
2e-05
c
0.00015
0.0001
0.001
Mz
5e-05
0
My
Mx
0
-0.0005
0
-2e-05
0
-0.001
-4e-05
-5e-05
-0.002
-6e-05
-8e-05
-0.003
0
1
2
3
4
alpha
d
5
6
7
-0.0001
0
1
2
3
4
alpha
e
5
6
7
0
1
2
3
alpha
f
Figure 19: Approximation results (Nmax = 6) for principal force (a, c, e), N, and for principal moment (b, c, f),
N · m, of light radiation pressure depending on angle of rotation of light source in the plane of radiators (solid
line) comparing results of Monte–Carlo simulations (dots). The results in subfigures b, c, and f are non-zero
because of random noise. The dotted values were not used in the construction of approximation. Diffuse case.
45 / 52
Specular-diffuse case
Specular-diffuse case
Ĩ 1 ≈ Ĩ 1 |s=0 ≈ Ĩ 1 |s=1 ;
2
(42)
2
Ĩ ≈ (1 − s)Ĩ |s=0 ;
(43)
Ĩ 3 ≈ (1 − s)Ĩ 3 |s=0 + sĨ 3 |s=1 ;
(44)
Ĩ n ≈ (1 − s)Ĩ n |s=0 + sĨ n |s=1 , n > 3;
(45)
K̃1 ≈ K̃1 |s=0 ≈ K̃1 |s=1 ;
(46)
2
2
K̃ ≈ (1 − s)K̃ |s=0 ;
(47)
K̃n ≈ (1 − s)K̃n |s=0 + sK̃n |s=1 , n > 2,
(48)
46 / 52
Specular–diffuse case, s = 0.75
0.003
2e-05
0.002
1.5e-05
0.0005
1e-05
0
5e-06
Fz
Fy
0.001
Fx
0.001
0
0
-0.001
-5e-06
-0.002
-1e-05
-0.003
-1.5e-05
0
1
2
3
4
5
6
7
-0.0005
-0.001
0
1
2
3
4
5
6
7
0
1
2
3
alpha
alpha
a
b
0.0001
5
6
7
4
5
6
7
c
0.003
0.0002
0.002
5e-05
4
alpha
0.0001
0.001
0
Mz
My
Mx
0
0
-5e-05
-0.0001
-0.001
-0.0001
-0.0002
-0.002
-0.00015
-0.003
0
1
2
3
4
alpha
d
5
6
7
-0.0003
0
1
2
3
4
alpha
e
5
6
7
0
1
2
3
alpha
f
Figure 20: Approximation results (Nmax = 6) for principal force (a, c, e), N, and for principal moment (b, c, f),
N · m, of light radiation pressure depending on angle of rotation of light source in the plane of radiators (solid
line) comparing results of Monte–Carlo simulations (dots). The results in subfigures b, c and f are non-zero
because of random noise. The approximation was constructed as a linear combination of specular and diffuse
cases. Specular–diffuse case, s = 0.75.
47 / 52
Specular–diffuse case, s = 0.5
1.5e-05
0.003
0.001
1e-05
0.002
0.0005
5e-06
Fz
0
Fy
Fx
0.001
0
-5e-06
0
-1e-05
-0.001
-0.0005
-1.5e-05
-0.002
-2e-05
0
1
2
3
4
5
6
7
-0.001
0
1
2
3
4
5
6
7
0
1
2
alpha
alpha
a
5
6
7
3
4
alpha
5
6
7
c
0.003
0.0001
0.002
5e-05
4
alpha
b
0.0001
3
5e-05
0.001
0
Mz
My
Mx
0
0
-5e-05
-5e-05
-0.001
-0.0001
-0.0001
-0.002
-0.00015
-0.003
0
1
2
3
4
alpha
d
5
6
7
-0.00015
0
1
2
3
4
alpha
e
5
6
7
0
1
2
f
Figure 21: Approximation results (Nmax = 6) for principal force (a, c, e), N, and for principal moment (b, c, f),
N · m, of light radiation pressure depending from angle of rotation of light source in the plane of radiators (solid
line) comparing results of Monte–Carlo simulations (dots). The results in subfigures b, c and f are non-zero
because of random noise. The approximation was constructed as a linear combination of specular and diffuse
cases. Specular–diffuse case, s = 0.5.
48 / 52
Specular–diffuse case, s = 0.25
0.003
0.001
1e-05
5e-06
0.002
0.0005
0
Fz
Fy
Fx
0.001
-5e-06
0
0
-1e-05
-0.0005
-0.001
-1.5e-05
-0.002
-2e-05
0
1
2
3
4
5
6
7
-0.001
0
1
2
3
alpha
a
5
6
7
0
1
2
4
5
6
7
3
4
alpha
5
6
7
c
0.0001
0.003
0.0002
5e-05
0.002
0.00015
0
0.001
0.0001
Mz
-5e-05
3
alpha
b
My
Mx
4
alpha
0
5e-05
0
-0.0001
-0.001
-0.00015
-5e-05
-0.002
-0.0002
-0.0001
-0.003
0
1
2
3
4
alpha
d
5
6
7
-0.00015
0
1
2
3
4
alpha
e
5
6
7
0
1
2
f
Figure 22: Approximation results (Nmax = 6) for principal force (a, c, e), N, and for principal moment (b, c, f),
N · m, of light radiation pressure depending from angle of rotation of light source in the plane of radiators (solid
line) comparing results of Monte–Carlo simulations (dots). The results in subfigures b, c and f are non-zero
because of random noise. The approximation was constructed as a linear combination of specular and diffuse
cases. Specular–diffuse case, s = 0.25.
49 / 52
Conclusions and future plans
50 / 52
Conclusions and future plans
Conclusions
The presented model can describe the SRP of complex space body.
In the model the geometrical and optical parameters of structure are analytically divided
from the attitude towards the sun.
If the bodies A and B are both optically convex, as well as the their composition A + B = C,
the components of shape tensors for C can be calculated as a simple sum of corresponding
components of shape tensors for A and B.
Future plans
Dynamics around the center of inertia under SRP moment. Optimal stabilization law.
Investigation of connections to the dynamics of the satellites in the upper Earth atmosphere
under hyperthermal flow.4
4
Beletskii V., Yanshin A. Vliyanie aerodinamicheskikh sil na vrashchatel’noe dvizhenie iskusstvennykh sputnikov (Effect of the
aerodynamic forces on the rotary motion of satellites). Kiev: Naukova Dumka, 1984. P. 187. (in Russian)
51 / 52
Main publications
Main publications
1
Nerovnyi N, Zimin V (2014) Determination of the radiation pressure force acting on a solar
sail taking into account stress-dependent optical parameters of sail material (in Russian).
Herald of the Bauman Moscow State Technical University Series Mechanical Engineering
96(3):61–78, URL http://vestnikmach.ru/eng/catalog/simul/hidden/486.html
2
Zimin VN, Nerovnyy NA (2015) Analysis of the deformed shape of a heliogyro solar sail
blade taking into account stress-dependent reflectivity of the material (in Russian).
Proceedings of Higher Educational Institutions chine Building 658(1):18–23, URL
http://izvuzmash.ru/eng/catalog/calcmach/hidden/1125.html
3
Zimin VN, Nerovnyi NA (2016) To the calculation of the main vector and the main
momentum of light pressure force on a solar sail (in Russian). Herald of the Bauman
Moscow State Technical University Series Mechanical Engineering 106(1):17–28, DOI
10.18698/0236-3941-2016-1-17-28, URL
http://vestnikmach.ru/eng/catalog/avroc/airdyn/1055.html
4
Nerovny NA (2017) The resultant vector and principal moment of light radiation pressure
upon an optically convex space structure (in Russian). Vestnik St. Petersburg State
University Series 1. Mathematics. Mechanics. Astronomy [Approved for publication]
5
Nerovny NA et al. (2017) Representation of light pressure resultant force and moment as a
tensor series. Celestial Mechanics and Dynamical Astronomy. [Approved for publication]
52 / 52