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Thickness requirement for solar sail foils
Acta Astronautica 65 (2009) 507 – 518 www.elsevier.com/locate/actaastro
Thickness requirement for solar sail foils Roman Ya. Kezerashvili∗ Physics Department, New York City College of Technology, The City University of New York, 300 Jay Street, Brooklyn, NY 11201, USA Received 31 July 2008; received in revised form 25 January 2009; accepted 31 January 2009 Available online 13 March 2009
Abstract The minimum foil thickness that provides the maximum reflectivity and absorption for a solar sail material is found and its dependence on temperature and on wavelength of the electromagnetic spectrum of solar radiation is investigated. It is shown that the minimal thickness has strong implicit dependence on the temperature through the temperature dependence of the √ electrical conductivity of the solar sail. The minimal thickness has explicit dependence on the frequency through the factor 1/ , as well as it has implicit dependence on the frequency, because the conductivity and dielectric function are functions of the frequency. It is shown that when the frequency dependence of the conductivity and dielectric function are taken into account the minimal thickness of the solar sail foil exhibits the slight dependence on the wavelength of the solar radiation. We show that the surface of a solar sail with the required minimal thickness of a foil can withheld the solar pressure within the limit of the elastic deformation of the material. We suggest that these factors should be taken into consideration in the solar sail design. © 2009 Elsevier Ltd. All rights reserved. Keywords: Solar sail; Near-Sun space; Solar sail thickness; Conductivity; Dielectric function
1. Introduction Missions’ opportunities to the outer solar system using chemical propulsion become somewhat limited because they require a long mission duration and high launch speed. Even though gravity assists can be used to reduce the missions’ duration and launch energy, this is not enough to obtain a cruise speed that allows exploring the different extra solar space during a span of human lifetime or reasonable duration. Meanwhile, there is a number of regions in the outer solar system that present a particular scientific interest: the Kuiper Belt, the heliosphere and heliopause, the Sun’s gravity focus and Oort Cloud [1–4].
∗ Tel.: +1 718 260 5277; fax: +1 718 260 5012.
E-mail address: [email protected]. 0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.01.062
A solar sail can generate a high cruise speed by using a close solar approach since the dynamic efficiency of the solar sail as a propelling device increases upon approaching the Sun. When determining the maximum reachable near-Sun region for the solar sail spacecraft trajectory perihelion, at least three restricting factors must be taken into consideration: maximum allowable temperature, near-Sun medium action of the solar radiation [5,6], and the time of the acceleration of the solar sail spacecraft into the near-Sun region. Angle setting the spatial attitude of the solar sail plane can be considered as a controlling parameter. There are two parts of solar radiation: the electromagnetic radiation and corpuscular radiation of lowand high-energy elementary particles like electron, protons, neutrinos and ions of light nuclei emitted by the Sun. The corpuscular radiation mainly results from solar flares, solar wind, coronal mass ejections and
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solar prominences. The detailed analysis of the interaction of the solar radiation with the solar sail material was already considered in Refs. [5,6] and some aspects can be summarized as follows. For the thicknesses in the nanometer scale the solar sail materials are mostly transparent for X - and -rays, electrons, protons and helium ions. The solar deep ultraviolet radiations will ionize metallic foil and as a result of ionization processes on the solar sail it will gain a positive electric charge. The positively charged solar sail extracts the momentum from solar wind plasma and accelerates by deflecting like “an electric shield” the proton component of the solar wind plasma. However, this acceleration will be a minor perturbation of the acceleration produced by the solar radiation rather than a large effect. The total spectral irradiance of the extraterrestrial solar electromagnetic radiation equals 1366.1 W/m2 . Approximately 47% of the incident extraterrestrial solar electromagnetic radiation is in the visible wavelengths from 380 to 780 nm. The ultraviolet, X - and -rays portion of the spectrum is with wavelengths below 380 nm and accounts for 7% of the extraterrestrial solar electromagnetic radiation and the infrared portion of the spectrum with wavelengths greater than 780 nm account for another 46% of the incident solar energy [7]. These three parts of the solar electromagnetic radiation differently interact with solar sail and are responsible for different physical processes that occur. Less than 7% of the solar electromagnetic radiation will only ionize the sail producing the surface charge distribution due to ultraviolet, partially by X - and -rays ionization [5]. The visible part of the incident extraterrestrial solar electromagnetic radiation is mostly reflected by the solar sail depending on the coefficient of reflection of the solar sail material. But a part of the visible radiation as well as the infrared portion of the spectrum with wavelengths greater than 780 nm will be mostly absorbed by the solar sail causing the heating of the sail and, therefore, increase of its temperature. Thus, the ionization and the heating of the sail are two major processes, which require mitigation and should be considered as the restrictions for the solar sail spacecraft deployment trajectories at the perihelion of an initially elliptical transfer orbit to a parabolic or hyperbolic solar orbit depending on the lightness factor of the solar sail. Only part of the radiation which is reflected and absorbed by solar sail will accelerate the solar craft. Therefore, the solar sail performance which is critical to the thickness of the solar sail material requires finding the minimal thickness of the solar sail material that provides the best reflection and absorption of all solar radiation wavelengths for the given temperature.
The objective of this work is to find the minimal required thicknesses for a sail foil of a solar sail propelled spacecraft for the interplanetary space mission when the temperature restriction and the maximum solar electromagnetic radiation spectrum reflection are taken into account. The solar sail is assumed to be either unilateral parachute type metallic thin foil solar sails [1] or hollow-body solar sails filled with gas [8,9] unfurled near a perihelion of the parabolic or hyperbolic transferred solar orbits. Our approach can be applied to a wide variety of materials, missions and sail configurations utilizing metallic thin foils and/or inflatable structures in the near-Sun space environment. The paper is organized as follows: In Section 2 the thickness of the solar sail is discussed. In Section 2.1 we estimate the upper limit for the thickness of the solar sail. The thickness of a metallic foil is studied in Section 2.2 and the dependence of the minimum thickness of a solar sail material on temperature as well as on electromagnetic spectrum frequency of solar radiation for the frequency dependent electrical conductivity and dielectric function is investigated. The lower limit of the thickness of the solar sail is given in Section 2.2.3. In Section 3 results of calculations for the dependence of the minimum thickness of the solar sail foil on temperature and wavelength of the solar radiation as well as an estimation the foil thickness from mechanical properties are presented. Conclusions follow in Section 4. 2. Thickness of the solar sail 2.1. Upper limit for the thickness of the solar sail One of the key design parameter, which determines the solar sail performance, is the solar sail mass per unit area or a sail areal mass, which depends on the thickness and density of the sail material as follows: s=
m AH = = H . A A
(1)
In Eq. (1) A and m are the area and mass of the sail, respectively, H is the thickness of the sail and is the density of the sail material. A deployment of the sailcraft depends on the sail areal mass as well as the sail pitch angle, which is defined as the angle between the normal of the sail surface and the incident radiation. It is clear from Eq. (1) that to obtain a high performance sail we should select among the materials within the same optical properties, the material with low density and use a thin foil of this material for the solar sail.
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The question is how thin the foil for a solar sail should be that it would produce the acceleration of the sailcraft based on the maximum reflection of solar radiation? In the literature on the optics of films and foils, the researchers usually distinguish between thick and thin foils based on their reflectivity and transmissivity. The reflectivity and transmissivity of the thin foil of thickness H are dependent on optical path of the light ray in the foil n H , where n is the index of refraction. This is known as the optical thickness and it is the main factor for light interference phenomena which determines the number of wavelengths. Therefore, not only the thickness of the foil but also the optical thickness must be taken into consideration for the design of the solar sail. For optical thicknesses much smaller than the wavelength of light the spectral reflectivity for the different wavelengths of the visible spectrum varies very little. For optical thicknesses comparable to the wavelength of light we obtain several extremes, i.e. maxima and minima, of reflectivity in the visible part of the spectrum. For optical thicknesses much greater than the wavelength of light these extremes accumulate and the number of maxima and minima in the spectrum rapidly grows. In terms of practice, these results mean that the first case of very small optical thickness in white light does not give interference colors, while in the second case, when the optical thickness is comparable to the wavelength of light, we obtain vivid interference colors, and finally in the third case we cease to observe interference colors since the large number of maxima and minima in the spectrum does not permit the eye to perceive colors. The last case represents a thick foil in our meaning of the word. Under the assumed conditions of observation a thick foil no longer exhibits interference colors on account of the great number of maxima and minima in the visible spectrum. The empirical upper limit for a thin foil, i.e. when interference colors can still be observed, is given by the condition that the optical thickness n H < 2.5, where is the wavelength of the central part of the spectrum [10]. This means that the maximum permissible path difference for a thin foil is 2 n H = 5. Therefore, the upper limit for the thickness of a solar sail is H < 5/2n.
(2)
For example, if we consider that the wavelength of the central part of the spectrum is = 0.6 m, and the refractive index of a thin foil is around n = 1.5, we arrive at an upper limit of the thickness of the thin foil H = 1 m.
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2.2. Thickness of a metallic foil The simplest material to analyze optical propagation in is a metal thin foil. The metal thin foil is a conducting media. One does not usually think of optical propagation in metals, as they are thought of as opaque. However, they are not, and if thin enough light will transmit through them. Let us analyze the light transmission in a metal foil following the standard electrodynamics approach, which is presented in Ref. [11] for the case of a frequency independent electrical conductivity and dielectric constant. The system of Maxwell’s equations for linear conducting media has the form · E = 0, ∇ ∇ · B = 0,
× E = − * B , ∇ *t
× B = J + * E . ∇ *t
(3)
In Eqs. (3) E and B are the electric and magnetic fields, respectively, J is a current density, and are the permittivity and permeability of the solar sail material. Applying the curve to third and fourth Maxwell’s equations (3), substituting the fourth into the third and third into the fourth, respectively, we get 2 2 E = * J + * E , ∇ *t *t 2 2 2 B = ∇ × J + * B . ∇ *t 2
(4)
In case of conductors we do not independently control the flow of electric charge and, generally, for linear conducting isotropic media the free electrons current density is proportional to the electric field J() = (, T ) E,
(5)
where (, T ) is a frequency and temperature dependent conductivity of the metallic foil. The last relation is valid whenever the wavelength of the electromagnetic radiation is large compared to the electronic mean free pass [12] and is ordinarily satisfied in a metal for wide range of wavelengths. Substituting (5) into (4) and considering the third equation of (3) we obtain the wave equations for the electric and magnetic fields 2 2 E = (, T ) * E + * E , ∇ *t *t 2 2 2 B = (, T ) * B + * B , ∇ *t *t 2
(6)
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which is a generalization of the wave equation for a conducting metallic foil. Below we use these equations for finding the minimal thickness of the solar sail material that provides the best reflection and absorption of all solar radiation wavelengths for the given temperature. 2.2.1. Frequency dependence of dielectric function The difference between the processes of electromagnetic wave distribution in free space and in metal is determined by the two coefficients—the dielectric permittivity and electrical conductivity . The first one determines the magnitude of displacement currents which is caused by time dependence of the electric field, while the second one is a measure of the actual currents created by the electrical field. A number of classical and quantum mechanical theories have been developed to describe the phenomenological electrical permittivity and electrical conductivity as functions of the frequency (wavelength) of incident electromagnetic waves for a number of different interaction phenomena and types of surfaces. The complex dielectric function () is more appropriate when microscopic mechanisms are considered that determine the magnitude of the phenomenological coefficients. In the frequency range of interest to the solar sailing, electromagnetic waves are primarily absorbed by free electrons and bound electrons or by change in the energy level lattice vibration which is related to conversion of photons into phonons, i.e. a quantum lattice vibration. The free electrons are a major contributor to a metal’s ability to absorb radiative energy and distinct optical differences between conductors. During the beginning of the last century Lorentz [13] developed a classical theory for the evaluation of the dielectric function by assuming electrons and ions are harmonic oscillator and his result is equivalent to the subsequent quantum mechanical development in Ref. [14]. The special case of the Lorentz theory is the Drude theory [12,15] that predicts the frequency dependence of the dielectric constant. To find the frequency dependence of the dielectric permittivity we look at a solution of Eq. (6) with time dependence e−i t , where is an angular frequency. Therefore, the first equation of Eq. (6) becomes 2 E = ()2 E, −∇
(7)
where () = +
i(, T ) .
(8)
Eq. (7) has the form of usual wave equation with a complex dielectric function given by Eq. (8). Thus, this shows that the inclusion of the conducting current (5)
in Maxwell’s equations has the same effect as replacing the dielectric constant by the complex depending on the frequency dielectric function (8). 2.2.2. Temperature and frequency dependence of conductivity Temperature behavior of the thickness of the solar sail foil is important because as the spacecraft approaches the Sun the temperature of the sail increases and, as a result, the electrical conductivity of the sail material changes affecting the reflection and absorption ability of the sail. The binding energy of the outer electrons, crystal structure and atomic motion, which are temperature dependent, affects the electrical conductivity [12]. Therefore, the conductivity of the material is temperature dependent. During the acceleration regime of the solar sail near the Sun the equilibrium temperature of the sail at different distances from the Sun will have different values. Hence, the conductivity of the sail foil will change depending on the distance from the Sun. The conductivity of most materials decreases as the temperature increases. Alternately, the resistivity of most materials increases with increasing temperature. The reason why the conductivity decreases with increasing temperature is that, as temperature rises, the number of imperfections in the atomic lattice structure increases hampering the electron movement. These imperfections include dislocations, vacancies, interstitial defects and impurity atoms. Additionally, with temperature above absolute zero, even the lattice atoms participate in the interference of the directional electron movement as they are not always found at their ideal lattice sites. Thermal energy causes the atoms to vibrate about their equilibrium positions. At any given moment in time many individual lattice atoms will be away from their perfect lattice sites and this interferes with electron movement. Variability of the conductivity is material dependent but has been established for many engineering materials [16]. In the classical theory [17] the resistivity of the metal is more or less directly proportional to the temperature. Since the conductivity is simply the inverse of the resistivity and the temperature coefficient is the same as for the resistivity, the equation for the conductivity has the following empirical approximation that is good enough for most engineering purposes: (T ) =
, 1 + (T − T0 )
(9)
where is the temperature coefficient of the conductivity (resistance) and is the DC value of conductivity
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(conductivity at zero frequency) determined for temperature T0 . Usually the conductivity and temperature coefficient of conductivity are tabulated at T0 =293 K. In Eq. (9) only the temperature dependence of the conductivity was taken into account without consideration of the frequency dependence of the conductivity. When we study the frequency dependence of the required thickness of the solar sail, the frequency dependence of conductivity should be also considered. Let us now consider the frequency dependence of the conductivity. The best way to do that is to use the classical Drude model for conductivity [12]. According to the Drude model, the frequency dependent conductivity is a complex function and given by () = , (10) 1 + i where =
ne2 m
(11)
is the DC Drude conductivity, e = 1.6 × 10−19 C, m = 9.1 × 10−34 kg is charge and mass of electron, respectively, n is a free electron density of metallic foil, and is the relaxation time or collision time for electron. The free electron density can be easily found based on that each element contains /Am mols per m3 , where is the density and Am is the atomic mass of the element and the number of atoms per mole is the Avogadro number 6.022 × 1023 mol−1 . Since each atom contains N valence electrons, the number of free electrons per cubic meter is n = 6.022 × 1023
N . Am
(12)
The relaxation time plays fundamental role in the theory of metallic conductors. From Eq. (11) one can find the relaxation time as m = 2. (13) ne Using the experimental value for the DC value of the electrical conductivity and the calculated electron density of metallic element n, we can find the relaxation time from Eq. (13) and use it in Eq. (10) for the frequency dependent conductivity. By substituting (T ) from Eq. (9) into Eq. (10) instead of one can obtain the frequency and temperature dependent conductivity (, T ) = (14) (1 + i )[1 + (T − T0 )] and investigate the frequency dependence as well as the temperature dependence of the conductivity and its influence on the thickness of the solar sail material.
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2.2.3. Lower limit for the thickness of the solar sail Eqs. (6) admit plane wave solutions ˜ t) t) = E0 ei(kz− E(z, , ˜ t) i(kz− B(z, t) = B0 e ,
(15)
where is an angular frequency and the wave number k˜ is complex k˜ 2 = ()2 + i()
(16)
as it will be easily checked by substituting (15) into Eqs. (6). The solutions (15) represent a uniform wave propagating in the +z-direction. It is easy to demonstrate that if the wave number (16) is written in the form k˜ = k + i ,
(17)
then its real and imaginary parts will be defined as ⎡ ⎤1/2 () ⎣ (, T ) 2 1+ + 1⎦ , k= 2 () ⎤1/2 ⎡ () ⎣ (, T ) 2 = 1+ − 1⎦ . (18) 2 () Therefore, using Eq. (17) we can rewrite the solutions (15) in the form t) = E0 e− z ei(kz−t) , E(z, t) = B0 e− z ei(kz−t) . B(z,
(19)
Thus, the imaginary part of the wave number k˜ results in an attenuation of the electric and magnetic fields, decreasing their amplitudes with the increasing distance of propagation z. The real part k of the wave number k˜ determines the wavelength, propagation speed and index of refraction =
2 ck , v= , n= . k k
(20)
Eq. (19) shows that electromagnetic fields inside a metallic conducting foil decay rapidly with depth. The distance it takes to reduce the amplitudes of the electromagnetic field by factor of 1/e (e-folding distance) is a skin depth. It is d=
1
(21)
and it is a measure of how far electromagnetic wave penetrates into the conducting metallic foil. It is obvious that the foil thickness should be always larger than the skin depth, otherwise the solar sail material will be transparent for electromagnetic radiation. In other
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words, the condition for the thickness of the solar sail that performs the acceleration of the sailcraft based on the maximum reflection of the solar radiation should be at least the following: −1⎠⎦
. (22)
Thus, using Eq. (22) we can estimate the minimal thickness of the solar sail required to achieve maximum reflection and absorption of the solar light for the given optical properties of the metallic foil. As it is seen from Eq. (22) the reciprocal of the imaginary part of the complex wave number determines the skin depth of the propagation and attenuation of the light in the metallic foil, and it is governed by three properties of the solar sail material: the permittivity , the permeability and the conductivity of the metallic foil. Actually, each of these parameters depends to some extent on the frequency of the electromagnetic wave and the temperature. Indeed, as its follows from Eq. (22) the minimal thickness depends on the frequency—apart from the ex√ plicit dependence on the factor 1/ and it also has implicit dependence, because the conductivity (, T ) and dielectric function () are functions of frequency. Also the minimal thickness depends implicitly on the temperature through the temperature dependence of the conductivity. The required thickness of the solar sail material fluctuates depending on the temperature and frequency, therefore, we need to determine the minimal thickness of the sail material that would enable to achieve the reflection and absorption of all solar spectrum frequencies at the temperatures of the near-Sun environment. 3. Results and discussion 3.1. Dependence of the foil thickness on temperature The sun faced side absorbed and reflected the electromagnetic radiation while both sides of the solar sail radiated the infrared electromagnetic wave according to the Stefan–Boltzmann’s law, and the equilibrium temperature for an opaque solar sail perpendicularly oriented to the incident solar radiation may be estimated as [1,18,19] T=
(1 − r˜ )W0 R02 ˜ S B r 2
1/4 .
(23)
51 48
1000
42
45
⎞⎤−1/2
45
42 39
42
() ⎝ 1 (, T ) H = −1 ⎣ 1+ 2 ()
2
45
800
39
36
39
36
⎛
48
1200
Temperature, T, K
⎡
Beryllium
36
33
33
33
30
600
27
27
27
24
24
400
30
30
512
21
21
0.5
24
1
1.5
2
21
2.5
3
3.5
4
4.5
5
Wavelength of Solar Radiation, λ, μm
Fig. 1. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium. The numbers on the curves indicate the thickness of the foil in nm. The frequency dependence of the dielectric function is neglected.
In Eq. (23) r˜ is the reflection coefficient, ˜ is the surface emissivity, S B is the Stefan–Boltzmann’s constant, R0 is the average radius of the Earth’s orbit and W0 = 1366 W/m2 . Below in our calculation assuming uniform temperature distribution along the thickness of the foil, the sail’s temperature is determined by Eq. (23) and can be considered as a maximum temperature of the foil when the solar sail is at a distance r from the sun. Our first step in studying the dependence of the sail material thickness on the temperature and on the solar electromagnetic spectrum frequency (wavelength) was to use Eq. (22) with constant value for the permittivity = 0 = 8.85 × 10−12 (C2 /N m2 ) and we have assumed the medium to be nonmagnetic, therefore, the permeability = 0 = 4 × 10−7 (T m/A). The calculations were performed for beryllium and aluminum. The results of the calculations are presented in Figs. 1 and 2, where the values of the required thicknesses of the solar sail material providing the best reflection of electromagnetic radiation are plotted as a function of wavelength of the solar radiation and temperature. The general behavior of these dependences is such that the increase in the temperature requires increases in the thickness of the sail foil for both materials to keep their best reflection and absorption ability. The thickness dependence on the wavelength of the solar radiation is weak for the infrared spectrum and the thickness slightly decreases when wavelength increases for all ranges of temperature. However, both solar sail materials exhibit
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518
strong dependence of the thickness on the wavelength in the wavelength range 0.2 m < < 0.8 m and this dependence becomes stronger when temperature
900
34.5
34.5
33
33 31.5
.5 31
31.5
30 30
700
30
28.5
28.5
600
28.5
27
27
500
27
25.5
25.5
25.5
24
24
Temperature, T, K
800
33 34.5 36 37.5
Aluminum
400
24 22.5
22.5
22.5
21
21
21
19.5
19.5
19.5
300 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wavelength of Solar Radiation, λ, μm
Fig. 2. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum. The numbers on the curves indicate the thickness of the foil in nm. The frequency dependence of the dielectric function is neglected.
513
increases. The corresponding three-dimensional plots for the thickness dependence on the wavelength and temperature for beryllium and aluminum are presented in Figs. 3 and 4. Our calculations are performed under the assumption of uniform temperature distribution along the thickness of foil and therefore, for the constant maximum value of the temperature which is established somewhere inside in the thickness of the foil. However, the cooling effect due to radiation of the heat will occur on both the surfaces of thin foil. Consequently, a temperature gradient along the thickness of the foil will be inherently introduced even in the condition of steady-state heat conduction. It is obvious that on the borders of the surface of a foil the temperature will be lower than one in the middle layer of the foil. By finding the required thickness of the foil for this maximum value of the temperature we can be certain that this thickness provides the best ability of the foil to reflect and absorb the most wavelengths of solar electromagnetic radiation. This conclusion follows from Eqs. (14) and (22) and is demonstrated by results of our calculations in Figs. 1–4 as well as in Figs. 5–8 that the temperature increase requires the increase of the thickness. Let us also emphasize that a temperature gradient along the thickness of the foil is an important issue
Beryllium
70 60 50 40 30 20 10 5
Thiskness of Metal Film, H, nm
80
4.5 4 3.5 3
1200 2.5
1000
2 800
Tem
pera
ture
, T,
1.5 1
600
K
400
of
μ λ,
m
t
dia
a rR
la
So
g
len
0.5
th
, ion
e av
W
Fig. 3. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium. The frequency dependence of the dielectric function is neglected.
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R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518 Aluminum
40 30 20 10 5
Thiskness of Metal Film, H, nm
50
4.5 4 3.5
900
3 750
Tem
pera
2
ture
, T,
K
dia
of
So
ng
le ve
0.5 300
μm
la
th
1
450
λ,
t
a rR
2.5
1.5
600
, ion
a
W
Fig. 4. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum. The frequency dependence of the dielectric function is neglected.
3.2. Frequency dependence of dielectric function and foil thickness Above we calculated the dependence of the thickness of the solar sail material on the temperature and on the solar radiation wavelength; however, the frequency dependence of the dielectric function was neglected. Let us now consider the influence of the frequency dependence of dielectric function () on the thickness of the foil. The two-dimensional plots of the dependence of the minimal thickness of the solar sail foil on the temperature and solar radiation wavelength for beryllium and aluminum are shown in Figs. 5 and 6, respectively. The analysis of Figs. 5 and 6 shows that when the frequency dependence of the dielectric function is taken into account the dependence of the minimal thickness of the solar sail foil on the wavelength dramatically changes: the thickness of the foil exhibits the
Beryllium 900
800
Temperature, T, K
related to mechanical properties of the thin foil [20] and their influence on the optical, electrical and magnetic behaviors of a foil material. Especially, it is important when we consider the metal foil on the substrate or multilayer foils–substrates system. This subject requires detailed experimental as well as theoretical studies and it is beyond the scope of this paper.
45
43.5
43.5
42
42
43.5
40.5
40.5
42
39
39
40.5
37.5
37.5
39
36
36
700 34.5
600
33
36 34.5
33 33
31.5
30
31.5
30
28.5
30
28.5
27
28.5
27 25.5
400
37.5
34.5
31.5
500
45
24
27
25.5
25.5
24
22.5
24
22.5
21
22.5
21
21
300 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wavelen gth of Solar Radiation, λ, μm
Fig. 5. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium. The numbers on the curves indicate the thickness of the foil in nm.
negligible dependence on the wavelength at low temperature and weakly decreases for all wavelength range at high temperature. We also observe much stronger
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518
515
Aluminum 38
900
37 36
Temperature, T, K
35
34
34
33
33 32
32
31 30
31 30
29
29
28 27
28
27
26
27
26
25
26
25
24
400
30
29
28
500
33
32
31
600
36
35
34
700
38 37
36
35
800
38
37
25
24 23
24
23
22
23
22
21
22
21
21
300 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wavelength of Solar Radiation, λ, μm
Fig. 6. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum. The numbers on the curves indicate the thickness of the foil in nm.
Fig. 7. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium.
dependence of the minimal thickness on the temperature, especially in the range corresponding to the visual part of the solar radiation spectrum (compare with Figs. 1 and 2). The three-dimensional graphs for the minimal thickness dependence on the wavelength and temperature when the frequency dependence of the dielectric function is taken into account are presented in Figs. 7 and 8. A comparison of Figs. 3 and 4 with Figs. 7 and 8 shows that consideration of the dielectric
Fig. 8. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum.
function dependence on the frequency of the solar radiation leads to the weak dependence of the minimal thickness on the wavelength and very strong dependence on the temperature. The temperature dependence of the conductivity of the foil requires an increase in the thickness of the solar sail by a factor of 2–2.5 in the considered temperature range. Above we considered temperature dependence of the electrical conductivity of the foil under a constant temperature coefficient of conductivity. In reality, the temperature coefficient of conductivity, is not a constant over the above range of temperature. It increases with a rise in temperature but not linearly [21]. A consideration of the temperature dependence of the temperature coefficient of conductivity will lead to an increase in the required thickness of the sail foil at high temperatures as it is shown in Ref. 11. Also note that more realistic quantum theory of the electric conductivity developed in Ref. [22] gives slightly different frequency dependence of the conductivity than the classical Drude model (10) and perfectly describes the experimental data [23] for aluminum. However, application of this quantum description could not change our qualitative conclusion about the thickness dependence on the wavelengths at high temperature range. 3.3. Estimate of the foil thickness from mechanical properties Let us also discuss the strength of the materials issue related to the temperature factor and the thickness of the foil. Above was found the required thickness of the foil
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It is obvious that the internal stress in the surface depends upon the pressure. The breakup of the surface can start in this outer zone of the maximal elongation and cracks will develop. Since for very thin foil H >R, we have from Fig. 9
F
T ls
F ≈ T = T l0
R
P≈
θ
Fig. 9.
based on the electromagnetic approach which allowed finding the maximum reflection and absorption of all spectra of solar radiation. It does not mean that the sail should be built from the foil of this thickness. The film or foil of this thickness that is studied in the paper only provides the maximum reflection and absorption of all spectra of solar electromagnetic radiation and can be a part of composite multilayer system on the substrates, or just foil on the substrate that enhances the mechanical strength of the sail. This issue was widely discussed in the literature for solar sailing (see [1,18] and references there). However, it is interesting to estimate the mechanical strength of a given material for the above found thicknesses of foil. Let us consider the infinitesimally small element of a surface of the solar sail which has length l0 , width a and thickness H . Under the action of the pressure force F it will be banded as shown in Fig. 9. It is clear that the neutral surface which is the middle part and has length l0 does not change its length. However, one of the outer parts of the surface will be stretched and have length ls > l0 , while the other will be compressed and will have length lc < l0 . From Fig. 9 one can find that ls = (R + H/2), where R is the curvature of the infinitesimally small element of a surface. Therefore, a maximum elongation of the surface element (this is the most outer part ls ) is l =
H l0 . 2 R
(25)
where T is the tensile force. By dividing both sides of Eq. (25) by the volume of the surface element we get1
H
lc Neutral surface
l0 , R
(24)
H , R
(26)
where P is the pressure of the force F on the surface area and is the tensile stress. Eq. (26) shows that pressure acting on the surface is limited by the thickness and the tensile stress of the foil. For our application plastic deformation is unacceptable, and the yield strength is used as the design limitation. By considering the yield strength in Eq. (26), the stress at which material strain changes from elastic deformation to plastic deformation, one can determine the maximum pressure that a surface can withhold within elastic deformation. Let us make a rough estimation. For a perfectly reflected surface the solar pressure at 1 AU is only 9.2 × 10−6 N/m2 . To generate a high cruise speed one should use a close approach to the Sun since the combined effect of the continuous radiation pressure thrust and solar gravity in close Sun flyby can result in a very high cruise speed. For the interstellar travel in the literature [2,18,19] the perihelion approach less than the distance from the Sun to Mercury is usually consider. We are assuming that deployed normal to the Sun at the 0.1–0.3 AU perihelion of a parabolic or hyperbolic solar orbit, the sailcraft is accelerated to its interstellar cruise velocity. Therefore, at these distances the solar pressure as its follows from the inverse square law is 1 × 10−4 .9.2 × 10−4 N/m2 . The yield strength for considered material is about 2 × 108 .4 × 108 Pa and the thickness of the foil is about 5 × 10−8 m. Thus, as this follows from Eq. (26), the maximum pressure which the foil can withhold within elastic limitation is 10–20 Pa for H >R ∼ 1 m. The smaller value of the curvature only increases the allowed value of the pressure. These values of the pressure which foil could withhold are considerably more than the solar radiation pressure. If the equation for the bended line l0 is known and is y = f (x), the curvature in Eq. (26) can be written as 1/R = (d 2 y/d x 2 )(1 + dy/d x)−3/2 . If 1 The relation like (26) can be obtained also by considering the infinitesimally small circular element of a surface.
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dy/d x>1, which corresponds to large radius of curvature (H >R), the last equation becomes 1/R =d 2 y/d x 2 and P = H (d 2 y/d x 2 ). Above we estimated possible breakup of the element of the foil. However, let us mention that it is interesting to consider the rigorous mathematical theory developed in Ref. [24] for the different scenario of the breakup processes of a solid thin shell for such flexible stricture as a solar sail and keeping in mind that mechanical properties of thin foils are quite different from those of the bulk materials [25–27]. The discrepancy between them may be as high as one order of magnitude. Consequently, an accurate determination of mechanical properties of thin film materials, such as elastic modulus, becomes extremely important. From Eq. (26) it follows that when the thickness decreases the allowed pressure decreases, assuming that tensile stress does not depend on the thickness. Considering that the maximum elongation of the surface element is l = (H/2)(l0 /R) for tensile stress one gets = E
H , 2R
(27)
where E is Young’s modulus. Therefore, the stress also decreases when the thickness of the foil decreases. However, it is contrary to experimental studies for the thin films. The yield strength of metallic films is affected by several microstructural dimensions: grain size, film thickness and obstacle spacing. The tensile strength decreases when temperature increases because Young’s modulus is only slightly temperature dependent and decreases monotonically with an average temperature. Young’s modulus is independent of foil thickness but the yield stress depends strongly on the film thickness and it increases with decreasing foil thickness, for example for the copper foil approximately as inverse square root from the thickness [28]. In several studies [28–34], an increase in the yield strength with decreasing film thickness has been observed. The functional dependence of this increase and its physical origin have still not been definitely established. However, for the thicknesses of foils in micrometers and nanometers scale, which are the subject of this paper, yield strength is of the order of tens of MPa. On the other hand, the strength of the materials provides the temperature Te for which the limit of the elastic deformation can be reached. It is obvious that foil temperature should be less than Te . This temperature is the key parameter which allows finding out the perihelion of an initially elliptical transfer orbit to a parabolic or hyperbolic solar orbit based on the law of conservation of energy.
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4. Conclusions An exploration of the outer solar system using solar sail propulsion with high cruise speed requires an acceleration of the sailcraft in the near-Sun space region. To obtain a high performance solar sail we should select among materials with the same optical properties, the material with low density and use a thin foil of this material for the sail. To find minimal required thickness of the solar sail for an acceleration of the sailcraft in the near-Sun space region at least two important factors should be considered: • the temperature dependence of electrical conductivity of a sail material; • the existence of a wide range of solar electromagnetic radiation frequencies and dependence of the electrical conductivity and dielectric function on the frequency. Applying the system of Maxwell’s equations for linear conducting media we find the minimum foil thickness that provides the maximum reflectivity and investigate dependence of this minimum thickness on temperature as well as on wavelengths of solar radiation. We have shown that the above factors have an effect on the thickness of solar sail foil and, therefore, on the reflection and absorption ability of the sail. The strong dependence of the solar sail thickness on temperature is shown. When the temperature increases the electrical conductivity of the sail material decreases affecting the sail thickness by increasing it. The minimal thickness has explicit √ dependence on the frequency through the factor 1/ and it also has implicit dependence on the frequency, because the conductivity (, T ) and dielectric function () are functions of frequency. It is shown that when the frequency dependence of the conductivity and dielectric function is taken into account the minimal thickness of the solar sail foil exhibits only slight dependence on the wavelength of the solar radiation. The determination of the minimal thickness of the solar sail foil based on its dependence on the temperature and frequency of the solar electromagnetic radiation is more critical than determination of the thickness based on the mechanical properties of materials. However, the strength of the materials provides the temperature for which the elastic limit can be reached when a material strain changes from an elastic deformation to a plastic deformation and, therefore, dramatically changes optical properties of the solar sail. In this paper we presented the minimal required thickness of the solar sail material that provides the best
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reflection and absorption of all solar radiation wavelengths for the given temperature. The surface of a solar sail with the required minimal thickness of a foil can withheld the solar pressure within the limit of the elastic deformation of the material. We suggest that the factors considered in this paper should be taken into consideration in the solar sail design. Let us also mention that aluminum and beryllium are the most preferable materials for the solar sail. These materials required much less thickness to reflect the solar spectrum. However, the melting temperature for aluminum is much less than the one for beryllium, but the processing and use of beryllium is risky due to potential toxicity. Acknowledgment I would like to thank Dr. V.S. Boyko for useful discussion. References [1] C.R. McInnes, Solar Sailing. Technology, Dynamics and Mission Applications, Springer, Praxis Publishing, 1998 296pp. [2] G.L. Matloff, Deep Space Probes. To Outer Solar System and Beyond, second ed., Springer-Praxis, Chichester, UK, 2005, 242pp. [3] C. Maccone, The Sun as a Gravitational Lens: Proposed Space Missions, second ed., IPI Press, Colorado Springs, CO, USA, 1999; C. Maccone, The Sun as a Gravitational Lens: Proposed Space Missions, third ed., IPI Press, Boulder, Colorado, 2003 [4] D. Jewitt, A. Morbidelli, H. Rauer, Trans-neptunium objects and comets: saas-fee advanced course 35, in: K. Altwegg, W. Benz, N. Thomas (Eds.), Swiss Society for Astrophysics and Astronomy, Springer, Berlin, Heidelberg, 2008, 258pp. [5] R.Ya. Kezerashvili, G.L. Matloff, Solar radiation and the beryllium hollow-body sail: 1. The ionization and disintegration effects, J. Br. Interplanet. Soc. 60 (2007) 169–179. [6] R.Ya. Kezerashvili, G.L. Matloff, Solar radiation and the beryllium hollow-body sail: 2. Diffusion, recombination and erosion processes, J. Br. Interplanet. Soc. 61 (2008) 47–57. [7] 2000 ASTM Standard Extraterrestrial Spectrum Reference E490-00 http://rredc.nrel.gov/solar/spectra/am0/ . [8] J. Strobl, The hollow body solar sail, J. Br. Interplanet. Soc. 42 (1989) 515–520. [9] G.L. Matloff, The beryllium hollow-body solar sail and interstellar travel, J. Br. Interplanet. Soc. 59 (2006) 349–354. [10] A. Vasicek, Optics of Thin Foils, North-Holland Publishing Company, Amsterdam, 1960, 403pp. [11] R.Ya. Kezerashvili, Solar sail interstellar travel: 1. Thickness of solar sail films, J. Br. Interplanet. Soc. 61 (2008) 430–439. [12] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Brooks/Cole Thomson Learning, 1976.
[13] H.A. Lorentz, The Theory of Electrons and its Application to the Phenomena of Light and Radiant Heat, Dover Publication Inc, New York, 1952, 343pp. [14] C.F. Bohren, D.R. Hoffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983. [15] C. Kittel, Introduction to Solid State Physics, eighth ed., Wiley, New York, 2005. [16] W.F. Gale, C.J. Smithells, T.C. Totemeier, Smithells Metals Reference Book, Butterworth-Heinemann, 2004. [17] D.J. Griffiths, Introduction to Electrodynamics, third ed., Prentice-Hall, Englewood Cliffs, NJ, 1999; H.C. Obanian, Classical Electrodynamics, second ed., Infinity Science Press LLC, 2007, 620pp. [18] E.N. Polyakhova, Kosmicheskii Polet s Solnechnim Parusom (in Russian) (Cosmic flight with solar sail), Nauka, Moscow, 1988. [19] V.V. Koblik, E.N. Polyakova, L.L. Sokolov, A.S. Shmirov, Cosmic Research 34 (1996) 572–578. [20] T.C. Chen, W.J. Lin, D.L. Chen, J. Appl. Phys. 96 (2004) 3800–3806. [21] E.J. Lewis, Some thermal and electrical properties of beryllium, Phys. Rev. 34 (1929) 1575–1587. [22] N.W. Aschcroft, K. Strum, Interband absorption and the optical properties of polyvalent metals, Phys. Rev. B3 (1971) 1898–1910. [23] A. Shinya, Y. Okuno, M. Fukui, Y. Shintani, Wavelength dependences of the dielectric constant of thermally evaporated aluminum foils, Surface Science 371 (1997) 149–156. [24] N.N. Smirnov, V.F. Nikitin, A.B. Kiselev, in: N.N. Smirnov (Ed.), Space Debris, Taylor & Francis, London, New York, 2002. [25] M. Ohring, The Materials Science of Thin Films Academic, San Diego, 1992. [26] J.S. Zabinski, A.A. Voevodin, J. Vac. Sci. Technol. A 16 (1998) 1890. [27] J. Proost, F. Spaepen, J. Appl. Phys. 91 (2002) 204. [28] D.Y.W. Yu, F. Spaepen, The yield strength of thin copper films on Kapton, J. Appl. Phys. 95 (2004) 2991–2997. [29] P.S. Alexopoulos, T.C. O’Sullivan, Mechanical properties of thin films, Annu. Rev. Mater. Sci. 20 (1990) 391–420. [30] F. Macionczyk, W. Brucker, W. Pitschke, G. Reiss, Young’s modulus and tensile strength of CuNi(Mn) thin films on polyimide foils by tensile testing, J. Mater. Res. 13 (1998) 2852–2858. [31] M. Hommel, O. Kraft, E. Arzt, A new method to study cyclic deformation of thin films in tension and compression, J. Mater. Res. 14 (1999) 2373–2376. [32] F. Macionczyk, W. Brucker, Tensile testing of AlCu thin film on polyimide foils, J. Appl. Phys. 86 (1999) 4922–4928. [33] Q. Zhou, K. Tanaka, G. Itoh, Deformation of aluminum thin foils under tensile stress at elevated temperature, J. Mater. Sci. Lett. 21 (2002) 215–216. [34] P.O. Renault, P. Villain, C. Coupeau, P. Goudeau, K.F. Badawi, Damage mode tensile testing of thin gold films on polyimide substrates by X-ray diffraction and atomic force microscopy, Thin Solid Films 424 (2003) 267–273.
Thickness requirement for solar sail foils Roman Ya. Kezerashvili∗ Physics Department, New York City College of Technology, The City University of New York, 300 Jay Street, Brooklyn, NY 11201, USA Received 31 July 2008; received in revised form 25 January 2009; accepted 31 January 2009 Available online 13 March 2009
Abstract The minimum foil thickness that provides the maximum reflectivity and absorption for a solar sail material is found and its dependence on temperature and on wavelength of the electromagnetic spectrum of solar radiation is investigated. It is shown that the minimal thickness has strong implicit dependence on the temperature through the temperature dependence of the √ electrical conductivity of the solar sail. The minimal thickness has explicit dependence on the frequency through the factor 1/ , as well as it has implicit dependence on the frequency, because the conductivity and dielectric function are functions of the frequency. It is shown that when the frequency dependence of the conductivity and dielectric function are taken into account the minimal thickness of the solar sail foil exhibits the slight dependence on the wavelength of the solar radiation. We show that the surface of a solar sail with the required minimal thickness of a foil can withheld the solar pressure within the limit of the elastic deformation of the material. We suggest that these factors should be taken into consideration in the solar sail design. © 2009 Elsevier Ltd. All rights reserved. Keywords: Solar sail; Near-Sun space; Solar sail thickness; Conductivity; Dielectric function
1. Introduction Missions’ opportunities to the outer solar system using chemical propulsion become somewhat limited because they require a long mission duration and high launch speed. Even though gravity assists can be used to reduce the missions’ duration and launch energy, this is not enough to obtain a cruise speed that allows exploring the different extra solar space during a span of human lifetime or reasonable duration. Meanwhile, there is a number of regions in the outer solar system that present a particular scientific interest: the Kuiper Belt, the heliosphere and heliopause, the Sun’s gravity focus and Oort Cloud [1–4].
∗ Tel.: +1 718 260 5277; fax: +1 718 260 5012.
E-mail address: [email protected]. 0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.01.062
A solar sail can generate a high cruise speed by using a close solar approach since the dynamic efficiency of the solar sail as a propelling device increases upon approaching the Sun. When determining the maximum reachable near-Sun region for the solar sail spacecraft trajectory perihelion, at least three restricting factors must be taken into consideration: maximum allowable temperature, near-Sun medium action of the solar radiation [5,6], and the time of the acceleration of the solar sail spacecraft into the near-Sun region. Angle setting the spatial attitude of the solar sail plane can be considered as a controlling parameter. There are two parts of solar radiation: the electromagnetic radiation and corpuscular radiation of lowand high-energy elementary particles like electron, protons, neutrinos and ions of light nuclei emitted by the Sun. The corpuscular radiation mainly results from solar flares, solar wind, coronal mass ejections and
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solar prominences. The detailed analysis of the interaction of the solar radiation with the solar sail material was already considered in Refs. [5,6] and some aspects can be summarized as follows. For the thicknesses in the nanometer scale the solar sail materials are mostly transparent for X - and -rays, electrons, protons and helium ions. The solar deep ultraviolet radiations will ionize metallic foil and as a result of ionization processes on the solar sail it will gain a positive electric charge. The positively charged solar sail extracts the momentum from solar wind plasma and accelerates by deflecting like “an electric shield” the proton component of the solar wind plasma. However, this acceleration will be a minor perturbation of the acceleration produced by the solar radiation rather than a large effect. The total spectral irradiance of the extraterrestrial solar electromagnetic radiation equals 1366.1 W/m2 . Approximately 47% of the incident extraterrestrial solar electromagnetic radiation is in the visible wavelengths from 380 to 780 nm. The ultraviolet, X - and -rays portion of the spectrum is with wavelengths below 380 nm and accounts for 7% of the extraterrestrial solar electromagnetic radiation and the infrared portion of the spectrum with wavelengths greater than 780 nm account for another 46% of the incident solar energy [7]. These three parts of the solar electromagnetic radiation differently interact with solar sail and are responsible for different physical processes that occur. Less than 7% of the solar electromagnetic radiation will only ionize the sail producing the surface charge distribution due to ultraviolet, partially by X - and -rays ionization [5]. The visible part of the incident extraterrestrial solar electromagnetic radiation is mostly reflected by the solar sail depending on the coefficient of reflection of the solar sail material. But a part of the visible radiation as well as the infrared portion of the spectrum with wavelengths greater than 780 nm will be mostly absorbed by the solar sail causing the heating of the sail and, therefore, increase of its temperature. Thus, the ionization and the heating of the sail are two major processes, which require mitigation and should be considered as the restrictions for the solar sail spacecraft deployment trajectories at the perihelion of an initially elliptical transfer orbit to a parabolic or hyperbolic solar orbit depending on the lightness factor of the solar sail. Only part of the radiation which is reflected and absorbed by solar sail will accelerate the solar craft. Therefore, the solar sail performance which is critical to the thickness of the solar sail material requires finding the minimal thickness of the solar sail material that provides the best reflection and absorption of all solar radiation wavelengths for the given temperature.
The objective of this work is to find the minimal required thicknesses for a sail foil of a solar sail propelled spacecraft for the interplanetary space mission when the temperature restriction and the maximum solar electromagnetic radiation spectrum reflection are taken into account. The solar sail is assumed to be either unilateral parachute type metallic thin foil solar sails [1] or hollow-body solar sails filled with gas [8,9] unfurled near a perihelion of the parabolic or hyperbolic transferred solar orbits. Our approach can be applied to a wide variety of materials, missions and sail configurations utilizing metallic thin foils and/or inflatable structures in the near-Sun space environment. The paper is organized as follows: In Section 2 the thickness of the solar sail is discussed. In Section 2.1 we estimate the upper limit for the thickness of the solar sail. The thickness of a metallic foil is studied in Section 2.2 and the dependence of the minimum thickness of a solar sail material on temperature as well as on electromagnetic spectrum frequency of solar radiation for the frequency dependent electrical conductivity and dielectric function is investigated. The lower limit of the thickness of the solar sail is given in Section 2.2.3. In Section 3 results of calculations for the dependence of the minimum thickness of the solar sail foil on temperature and wavelength of the solar radiation as well as an estimation the foil thickness from mechanical properties are presented. Conclusions follow in Section 4. 2. Thickness of the solar sail 2.1. Upper limit for the thickness of the solar sail One of the key design parameter, which determines the solar sail performance, is the solar sail mass per unit area or a sail areal mass, which depends on the thickness and density of the sail material as follows: s=
m AH = = H . A A
(1)
In Eq. (1) A and m are the area and mass of the sail, respectively, H is the thickness of the sail and is the density of the sail material. A deployment of the sailcraft depends on the sail areal mass as well as the sail pitch angle, which is defined as the angle between the normal of the sail surface and the incident radiation. It is clear from Eq. (1) that to obtain a high performance sail we should select among the materials within the same optical properties, the material with low density and use a thin foil of this material for the solar sail.
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The question is how thin the foil for a solar sail should be that it would produce the acceleration of the sailcraft based on the maximum reflection of solar radiation? In the literature on the optics of films and foils, the researchers usually distinguish between thick and thin foils based on their reflectivity and transmissivity. The reflectivity and transmissivity of the thin foil of thickness H are dependent on optical path of the light ray in the foil n H , where n is the index of refraction. This is known as the optical thickness and it is the main factor for light interference phenomena which determines the number of wavelengths. Therefore, not only the thickness of the foil but also the optical thickness must be taken into consideration for the design of the solar sail. For optical thicknesses much smaller than the wavelength of light the spectral reflectivity for the different wavelengths of the visible spectrum varies very little. For optical thicknesses comparable to the wavelength of light we obtain several extremes, i.e. maxima and minima, of reflectivity in the visible part of the spectrum. For optical thicknesses much greater than the wavelength of light these extremes accumulate and the number of maxima and minima in the spectrum rapidly grows. In terms of practice, these results mean that the first case of very small optical thickness in white light does not give interference colors, while in the second case, when the optical thickness is comparable to the wavelength of light, we obtain vivid interference colors, and finally in the third case we cease to observe interference colors since the large number of maxima and minima in the spectrum does not permit the eye to perceive colors. The last case represents a thick foil in our meaning of the word. Under the assumed conditions of observation a thick foil no longer exhibits interference colors on account of the great number of maxima and minima in the visible spectrum. The empirical upper limit for a thin foil, i.e. when interference colors can still be observed, is given by the condition that the optical thickness n H < 2.5, where is the wavelength of the central part of the spectrum [10]. This means that the maximum permissible path difference for a thin foil is 2 n H = 5. Therefore, the upper limit for the thickness of a solar sail is H < 5/2n.
(2)
For example, if we consider that the wavelength of the central part of the spectrum is = 0.6 m, and the refractive index of a thin foil is around n = 1.5, we arrive at an upper limit of the thickness of the thin foil H = 1 m.
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2.2. Thickness of a metallic foil The simplest material to analyze optical propagation in is a metal thin foil. The metal thin foil is a conducting media. One does not usually think of optical propagation in metals, as they are thought of as opaque. However, they are not, and if thin enough light will transmit through them. Let us analyze the light transmission in a metal foil following the standard electrodynamics approach, which is presented in Ref. [11] for the case of a frequency independent electrical conductivity and dielectric constant. The system of Maxwell’s equations for linear conducting media has the form · E = 0, ∇ ∇ · B = 0,
× E = − * B , ∇ *t
× B = J + * E . ∇ *t
(3)
In Eqs. (3) E and B are the electric and magnetic fields, respectively, J is a current density, and are the permittivity and permeability of the solar sail material. Applying the curve to third and fourth Maxwell’s equations (3), substituting the fourth into the third and third into the fourth, respectively, we get 2 2 E = * J + * E , ∇ *t *t 2 2 2 B = ∇ × J + * B . ∇ *t 2
(4)
In case of conductors we do not independently control the flow of electric charge and, generally, for linear conducting isotropic media the free electrons current density is proportional to the electric field J() = (, T ) E,
(5)
where (, T ) is a frequency and temperature dependent conductivity of the metallic foil. The last relation is valid whenever the wavelength of the electromagnetic radiation is large compared to the electronic mean free pass [12] and is ordinarily satisfied in a metal for wide range of wavelengths. Substituting (5) into (4) and considering the third equation of (3) we obtain the wave equations for the electric and magnetic fields 2 2 E = (, T ) * E + * E , ∇ *t *t 2 2 2 B = (, T ) * B + * B , ∇ *t *t 2
(6)
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which is a generalization of the wave equation for a conducting metallic foil. Below we use these equations for finding the minimal thickness of the solar sail material that provides the best reflection and absorption of all solar radiation wavelengths for the given temperature. 2.2.1. Frequency dependence of dielectric function The difference between the processes of electromagnetic wave distribution in free space and in metal is determined by the two coefficients—the dielectric permittivity and electrical conductivity . The first one determines the magnitude of displacement currents which is caused by time dependence of the electric field, while the second one is a measure of the actual currents created by the electrical field. A number of classical and quantum mechanical theories have been developed to describe the phenomenological electrical permittivity and electrical conductivity as functions of the frequency (wavelength) of incident electromagnetic waves for a number of different interaction phenomena and types of surfaces. The complex dielectric function () is more appropriate when microscopic mechanisms are considered that determine the magnitude of the phenomenological coefficients. In the frequency range of interest to the solar sailing, electromagnetic waves are primarily absorbed by free electrons and bound electrons or by change in the energy level lattice vibration which is related to conversion of photons into phonons, i.e. a quantum lattice vibration. The free electrons are a major contributor to a metal’s ability to absorb radiative energy and distinct optical differences between conductors. During the beginning of the last century Lorentz [13] developed a classical theory for the evaluation of the dielectric function by assuming electrons and ions are harmonic oscillator and his result is equivalent to the subsequent quantum mechanical development in Ref. [14]. The special case of the Lorentz theory is the Drude theory [12,15] that predicts the frequency dependence of the dielectric constant. To find the frequency dependence of the dielectric permittivity we look at a solution of Eq. (6) with time dependence e−i t , where is an angular frequency. Therefore, the first equation of Eq. (6) becomes 2 E = ()2 E, −∇
(7)
where () = +
i(, T ) .
(8)
Eq. (7) has the form of usual wave equation with a complex dielectric function given by Eq. (8). Thus, this shows that the inclusion of the conducting current (5)
in Maxwell’s equations has the same effect as replacing the dielectric constant by the complex depending on the frequency dielectric function (8). 2.2.2. Temperature and frequency dependence of conductivity Temperature behavior of the thickness of the solar sail foil is important because as the spacecraft approaches the Sun the temperature of the sail increases and, as a result, the electrical conductivity of the sail material changes affecting the reflection and absorption ability of the sail. The binding energy of the outer electrons, crystal structure and atomic motion, which are temperature dependent, affects the electrical conductivity [12]. Therefore, the conductivity of the material is temperature dependent. During the acceleration regime of the solar sail near the Sun the equilibrium temperature of the sail at different distances from the Sun will have different values. Hence, the conductivity of the sail foil will change depending on the distance from the Sun. The conductivity of most materials decreases as the temperature increases. Alternately, the resistivity of most materials increases with increasing temperature. The reason why the conductivity decreases with increasing temperature is that, as temperature rises, the number of imperfections in the atomic lattice structure increases hampering the electron movement. These imperfections include dislocations, vacancies, interstitial defects and impurity atoms. Additionally, with temperature above absolute zero, even the lattice atoms participate in the interference of the directional electron movement as they are not always found at their ideal lattice sites. Thermal energy causes the atoms to vibrate about their equilibrium positions. At any given moment in time many individual lattice atoms will be away from their perfect lattice sites and this interferes with electron movement. Variability of the conductivity is material dependent but has been established for many engineering materials [16]. In the classical theory [17] the resistivity of the metal is more or less directly proportional to the temperature. Since the conductivity is simply the inverse of the resistivity and the temperature coefficient is the same as for the resistivity, the equation for the conductivity has the following empirical approximation that is good enough for most engineering purposes: (T ) =
, 1 + (T − T0 )
(9)
where is the temperature coefficient of the conductivity (resistance) and is the DC value of conductivity
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(conductivity at zero frequency) determined for temperature T0 . Usually the conductivity and temperature coefficient of conductivity are tabulated at T0 =293 K. In Eq. (9) only the temperature dependence of the conductivity was taken into account without consideration of the frequency dependence of the conductivity. When we study the frequency dependence of the required thickness of the solar sail, the frequency dependence of conductivity should be also considered. Let us now consider the frequency dependence of the conductivity. The best way to do that is to use the classical Drude model for conductivity [12]. According to the Drude model, the frequency dependent conductivity is a complex function and given by () = , (10) 1 + i where =
ne2 m
(11)
is the DC Drude conductivity, e = 1.6 × 10−19 C, m = 9.1 × 10−34 kg is charge and mass of electron, respectively, n is a free electron density of metallic foil, and is the relaxation time or collision time for electron. The free electron density can be easily found based on that each element contains /Am mols per m3 , where is the density and Am is the atomic mass of the element and the number of atoms per mole is the Avogadro number 6.022 × 1023 mol−1 . Since each atom contains N valence electrons, the number of free electrons per cubic meter is n = 6.022 × 1023
N . Am
(12)
The relaxation time plays fundamental role in the theory of metallic conductors. From Eq. (11) one can find the relaxation time as m = 2. (13) ne Using the experimental value for the DC value of the electrical conductivity and the calculated electron density of metallic element n, we can find the relaxation time from Eq. (13) and use it in Eq. (10) for the frequency dependent conductivity. By substituting (T ) from Eq. (9) into Eq. (10) instead of one can obtain the frequency and temperature dependent conductivity (, T ) = (14) (1 + i )[1 + (T − T0 )] and investigate the frequency dependence as well as the temperature dependence of the conductivity and its influence on the thickness of the solar sail material.
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2.2.3. Lower limit for the thickness of the solar sail Eqs. (6) admit plane wave solutions ˜ t) t) = E0 ei(kz− E(z, , ˜ t) i(kz− B(z, t) = B0 e ,
(15)
where is an angular frequency and the wave number k˜ is complex k˜ 2 = ()2 + i()
(16)
as it will be easily checked by substituting (15) into Eqs. (6). The solutions (15) represent a uniform wave propagating in the +z-direction. It is easy to demonstrate that if the wave number (16) is written in the form k˜ = k + i ,
(17)
then its real and imaginary parts will be defined as ⎡ ⎤1/2 () ⎣ (, T ) 2 1+ + 1⎦ , k= 2 () ⎤1/2 ⎡ () ⎣ (, T ) 2 = 1+ − 1⎦ . (18) 2 () Therefore, using Eq. (17) we can rewrite the solutions (15) in the form t) = E0 e− z ei(kz−t) , E(z, t) = B0 e− z ei(kz−t) . B(z,
(19)
Thus, the imaginary part of the wave number k˜ results in an attenuation of the electric and magnetic fields, decreasing their amplitudes with the increasing distance of propagation z. The real part k of the wave number k˜ determines the wavelength, propagation speed and index of refraction =
2 ck , v= , n= . k k
(20)
Eq. (19) shows that electromagnetic fields inside a metallic conducting foil decay rapidly with depth. The distance it takes to reduce the amplitudes of the electromagnetic field by factor of 1/e (e-folding distance) is a skin depth. It is d=
1
(21)
and it is a measure of how far electromagnetic wave penetrates into the conducting metallic foil. It is obvious that the foil thickness should be always larger than the skin depth, otherwise the solar sail material will be transparent for electromagnetic radiation. In other
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518
words, the condition for the thickness of the solar sail that performs the acceleration of the sailcraft based on the maximum reflection of the solar radiation should be at least the following: −1⎠⎦
. (22)
Thus, using Eq. (22) we can estimate the minimal thickness of the solar sail required to achieve maximum reflection and absorption of the solar light for the given optical properties of the metallic foil. As it is seen from Eq. (22) the reciprocal of the imaginary part of the complex wave number determines the skin depth of the propagation and attenuation of the light in the metallic foil, and it is governed by three properties of the solar sail material: the permittivity , the permeability and the conductivity of the metallic foil. Actually, each of these parameters depends to some extent on the frequency of the electromagnetic wave and the temperature. Indeed, as its follows from Eq. (22) the minimal thickness depends on the frequency—apart from the ex√ plicit dependence on the factor 1/ and it also has implicit dependence, because the conductivity (, T ) and dielectric function () are functions of frequency. Also the minimal thickness depends implicitly on the temperature through the temperature dependence of the conductivity. The required thickness of the solar sail material fluctuates depending on the temperature and frequency, therefore, we need to determine the minimal thickness of the sail material that would enable to achieve the reflection and absorption of all solar spectrum frequencies at the temperatures of the near-Sun environment. 3. Results and discussion 3.1. Dependence of the foil thickness on temperature The sun faced side absorbed and reflected the electromagnetic radiation while both sides of the solar sail radiated the infrared electromagnetic wave according to the Stefan–Boltzmann’s law, and the equilibrium temperature for an opaque solar sail perpendicularly oriented to the incident solar radiation may be estimated as [1,18,19] T=
(1 − r˜ )W0 R02 ˜ S B r 2
1/4 .
(23)
51 48
1000
42
45
⎞⎤−1/2
45
42 39
42
() ⎝ 1 (, T ) H = −1 ⎣ 1+ 2 ()
2
45
800
39
36
39
36
⎛
48
1200
Temperature, T, K
⎡
Beryllium
36
33
33
33
30
600
27
27
27
24
24
400
30
30
512
21
21
0.5
24
1
1.5
2
21
2.5
3
3.5
4
4.5
5
Wavelength of Solar Radiation, λ, μm
Fig. 1. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium. The numbers on the curves indicate the thickness of the foil in nm. The frequency dependence of the dielectric function is neglected.
In Eq. (23) r˜ is the reflection coefficient, ˜ is the surface emissivity, S B is the Stefan–Boltzmann’s constant, R0 is the average radius of the Earth’s orbit and W0 = 1366 W/m2 . Below in our calculation assuming uniform temperature distribution along the thickness of the foil, the sail’s temperature is determined by Eq. (23) and can be considered as a maximum temperature of the foil when the solar sail is at a distance r from the sun. Our first step in studying the dependence of the sail material thickness on the temperature and on the solar electromagnetic spectrum frequency (wavelength) was to use Eq. (22) with constant value for the permittivity = 0 = 8.85 × 10−12 (C2 /N m2 ) and we have assumed the medium to be nonmagnetic, therefore, the permeability = 0 = 4 × 10−7 (T m/A). The calculations were performed for beryllium and aluminum. The results of the calculations are presented in Figs. 1 and 2, where the values of the required thicknesses of the solar sail material providing the best reflection of electromagnetic radiation are plotted as a function of wavelength of the solar radiation and temperature. The general behavior of these dependences is such that the increase in the temperature requires increases in the thickness of the sail foil for both materials to keep their best reflection and absorption ability. The thickness dependence on the wavelength of the solar radiation is weak for the infrared spectrum and the thickness slightly decreases when wavelength increases for all ranges of temperature. However, both solar sail materials exhibit
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518
strong dependence of the thickness on the wavelength in the wavelength range 0.2 m < < 0.8 m and this dependence becomes stronger when temperature
900
34.5
34.5
33
33 31.5
.5 31
31.5
30 30
700
30
28.5
28.5
600
28.5
27
27
500
27
25.5
25.5
25.5
24
24
Temperature, T, K
800
33 34.5 36 37.5
Aluminum
400
24 22.5
22.5
22.5
21
21
21
19.5
19.5
19.5
300 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wavelength of Solar Radiation, λ, μm
Fig. 2. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum. The numbers on the curves indicate the thickness of the foil in nm. The frequency dependence of the dielectric function is neglected.
513
increases. The corresponding three-dimensional plots for the thickness dependence on the wavelength and temperature for beryllium and aluminum are presented in Figs. 3 and 4. Our calculations are performed under the assumption of uniform temperature distribution along the thickness of foil and therefore, for the constant maximum value of the temperature which is established somewhere inside in the thickness of the foil. However, the cooling effect due to radiation of the heat will occur on both the surfaces of thin foil. Consequently, a temperature gradient along the thickness of the foil will be inherently introduced even in the condition of steady-state heat conduction. It is obvious that on the borders of the surface of a foil the temperature will be lower than one in the middle layer of the foil. By finding the required thickness of the foil for this maximum value of the temperature we can be certain that this thickness provides the best ability of the foil to reflect and absorb the most wavelengths of solar electromagnetic radiation. This conclusion follows from Eqs. (14) and (22) and is demonstrated by results of our calculations in Figs. 1–4 as well as in Figs. 5–8 that the temperature increase requires the increase of the thickness. Let us also emphasize that a temperature gradient along the thickness of the foil is an important issue
Beryllium
70 60 50 40 30 20 10 5
Thiskness of Metal Film, H, nm
80
4.5 4 3.5 3
1200 2.5
1000
2 800
Tem
pera
ture
, T,
1.5 1
600
K
400
of
μ λ,
m
t
dia
a rR
la
So
g
len
0.5
th
, ion
e av
W
Fig. 3. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium. The frequency dependence of the dielectric function is neglected.
514
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518 Aluminum
40 30 20 10 5
Thiskness of Metal Film, H, nm
50
4.5 4 3.5
900
3 750
Tem
pera
2
ture
, T,
K
dia
of
So
ng
le ve
0.5 300
μm
la
th
1
450
λ,
t
a rR
2.5
1.5
600
, ion
a
W
Fig. 4. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum. The frequency dependence of the dielectric function is neglected.
3.2. Frequency dependence of dielectric function and foil thickness Above we calculated the dependence of the thickness of the solar sail material on the temperature and on the solar radiation wavelength; however, the frequency dependence of the dielectric function was neglected. Let us now consider the influence of the frequency dependence of dielectric function () on the thickness of the foil. The two-dimensional plots of the dependence of the minimal thickness of the solar sail foil on the temperature and solar radiation wavelength for beryllium and aluminum are shown in Figs. 5 and 6, respectively. The analysis of Figs. 5 and 6 shows that when the frequency dependence of the dielectric function is taken into account the dependence of the minimal thickness of the solar sail foil on the wavelength dramatically changes: the thickness of the foil exhibits the
Beryllium 900
800
Temperature, T, K
related to mechanical properties of the thin foil [20] and their influence on the optical, electrical and magnetic behaviors of a foil material. Especially, it is important when we consider the metal foil on the substrate or multilayer foils–substrates system. This subject requires detailed experimental as well as theoretical studies and it is beyond the scope of this paper.
45
43.5
43.5
42
42
43.5
40.5
40.5
42
39
39
40.5
37.5
37.5
39
36
36
700 34.5
600
33
36 34.5
33 33
31.5
30
31.5
30
28.5
30
28.5
27
28.5
27 25.5
400
37.5
34.5
31.5
500
45
24
27
25.5
25.5
24
22.5
24
22.5
21
22.5
21
21
300 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wavelen gth of Solar Radiation, λ, μm
Fig. 5. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium. The numbers on the curves indicate the thickness of the foil in nm.
negligible dependence on the wavelength at low temperature and weakly decreases for all wavelength range at high temperature. We also observe much stronger
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518
515
Aluminum 38
900
37 36
Temperature, T, K
35
34
34
33
33 32
32
31 30
31 30
29
29
28 27
28
27
26
27
26
25
26
25
24
400
30
29
28
500
33
32
31
600
36
35
34
700
38 37
36
35
800
38
37
25
24 23
24
23
22
23
22
21
22
21
21
300 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wavelength of Solar Radiation, λ, μm
Fig. 6. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum. The numbers on the curves indicate the thickness of the foil in nm.
Fig. 7. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for beryllium.
dependence of the minimal thickness on the temperature, especially in the range corresponding to the visual part of the solar radiation spectrum (compare with Figs. 1 and 2). The three-dimensional graphs for the minimal thickness dependence on the wavelength and temperature when the frequency dependence of the dielectric function is taken into account are presented in Figs. 7 and 8. A comparison of Figs. 3 and 4 with Figs. 7 and 8 shows that consideration of the dielectric
Fig. 8. The dependence of the thickness of the solar sail foil on the solar radiation wavelength and temperature for aluminum.
function dependence on the frequency of the solar radiation leads to the weak dependence of the minimal thickness on the wavelength and very strong dependence on the temperature. The temperature dependence of the conductivity of the foil requires an increase in the thickness of the solar sail by a factor of 2–2.5 in the considered temperature range. Above we considered temperature dependence of the electrical conductivity of the foil under a constant temperature coefficient of conductivity. In reality, the temperature coefficient of conductivity, is not a constant over the above range of temperature. It increases with a rise in temperature but not linearly [21]. A consideration of the temperature dependence of the temperature coefficient of conductivity will lead to an increase in the required thickness of the sail foil at high temperatures as it is shown in Ref. 11. Also note that more realistic quantum theory of the electric conductivity developed in Ref. [22] gives slightly different frequency dependence of the conductivity than the classical Drude model (10) and perfectly describes the experimental data [23] for aluminum. However, application of this quantum description could not change our qualitative conclusion about the thickness dependence on the wavelengths at high temperature range. 3.3. Estimate of the foil thickness from mechanical properties Let us also discuss the strength of the materials issue related to the temperature factor and the thickness of the foil. Above was found the required thickness of the foil
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It is obvious that the internal stress in the surface depends upon the pressure. The breakup of the surface can start in this outer zone of the maximal elongation and cracks will develop. Since for very thin foil H >R, we have from Fig. 9
F
T ls
F ≈ T = T l0
R
P≈
θ
Fig. 9.
based on the electromagnetic approach which allowed finding the maximum reflection and absorption of all spectra of solar radiation. It does not mean that the sail should be built from the foil of this thickness. The film or foil of this thickness that is studied in the paper only provides the maximum reflection and absorption of all spectra of solar electromagnetic radiation and can be a part of composite multilayer system on the substrates, or just foil on the substrate that enhances the mechanical strength of the sail. This issue was widely discussed in the literature for solar sailing (see [1,18] and references there). However, it is interesting to estimate the mechanical strength of a given material for the above found thicknesses of foil. Let us consider the infinitesimally small element of a surface of the solar sail which has length l0 , width a and thickness H . Under the action of the pressure force F it will be banded as shown in Fig. 9. It is clear that the neutral surface which is the middle part and has length l0 does not change its length. However, one of the outer parts of the surface will be stretched and have length ls > l0 , while the other will be compressed and will have length lc < l0 . From Fig. 9 one can find that ls = (R + H/2), where R is the curvature of the infinitesimally small element of a surface. Therefore, a maximum elongation of the surface element (this is the most outer part ls ) is l =
H l0 . 2 R
(25)
where T is the tensile force. By dividing both sides of Eq. (25) by the volume of the surface element we get1
H
lc Neutral surface
l0 , R
(24)
H , R
(26)
where P is the pressure of the force F on the surface area and is the tensile stress. Eq. (26) shows that pressure acting on the surface is limited by the thickness and the tensile stress of the foil. For our application plastic deformation is unacceptable, and the yield strength is used as the design limitation. By considering the yield strength in Eq. (26), the stress at which material strain changes from elastic deformation to plastic deformation, one can determine the maximum pressure that a surface can withhold within elastic deformation. Let us make a rough estimation. For a perfectly reflected surface the solar pressure at 1 AU is only 9.2 × 10−6 N/m2 . To generate a high cruise speed one should use a close approach to the Sun since the combined effect of the continuous radiation pressure thrust and solar gravity in close Sun flyby can result in a very high cruise speed. For the interstellar travel in the literature [2,18,19] the perihelion approach less than the distance from the Sun to Mercury is usually consider. We are assuming that deployed normal to the Sun at the 0.1–0.3 AU perihelion of a parabolic or hyperbolic solar orbit, the sailcraft is accelerated to its interstellar cruise velocity. Therefore, at these distances the solar pressure as its follows from the inverse square law is 1 × 10−4 .9.2 × 10−4 N/m2 . The yield strength for considered material is about 2 × 108 .4 × 108 Pa and the thickness of the foil is about 5 × 10−8 m. Thus, as this follows from Eq. (26), the maximum pressure which the foil can withhold within elastic limitation is 10–20 Pa for H >R ∼ 1 m. The smaller value of the curvature only increases the allowed value of the pressure. These values of the pressure which foil could withhold are considerably more than the solar radiation pressure. If the equation for the bended line l0 is known and is y = f (x), the curvature in Eq. (26) can be written as 1/R = (d 2 y/d x 2 )(1 + dy/d x)−3/2 . If 1 The relation like (26) can be obtained also by considering the infinitesimally small circular element of a surface.
R.Ya. Kezerashvili / Acta Astronautica 65 (2009) 507 – 518
dy/d x>1, which corresponds to large radius of curvature (H >R), the last equation becomes 1/R =d 2 y/d x 2 and P = H (d 2 y/d x 2 ). Above we estimated possible breakup of the element of the foil. However, let us mention that it is interesting to consider the rigorous mathematical theory developed in Ref. [24] for the different scenario of the breakup processes of a solid thin shell for such flexible stricture as a solar sail and keeping in mind that mechanical properties of thin foils are quite different from those of the bulk materials [25–27]. The discrepancy between them may be as high as one order of magnitude. Consequently, an accurate determination of mechanical properties of thin film materials, such as elastic modulus, becomes extremely important. From Eq. (26) it follows that when the thickness decreases the allowed pressure decreases, assuming that tensile stress does not depend on the thickness. Considering that the maximum elongation of the surface element is l = (H/2)(l0 /R) for tensile stress one gets = E
H , 2R
(27)
where E is Young’s modulus. Therefore, the stress also decreases when the thickness of the foil decreases. However, it is contrary to experimental studies for the thin films. The yield strength of metallic films is affected by several microstructural dimensions: grain size, film thickness and obstacle spacing. The tensile strength decreases when temperature increases because Young’s modulus is only slightly temperature dependent and decreases monotonically with an average temperature. Young’s modulus is independent of foil thickness but the yield stress depends strongly on the film thickness and it increases with decreasing foil thickness, for example for the copper foil approximately as inverse square root from the thickness [28]. In several studies [28–34], an increase in the yield strength with decreasing film thickness has been observed. The functional dependence of this increase and its physical origin have still not been definitely established. However, for the thicknesses of foils in micrometers and nanometers scale, which are the subject of this paper, yield strength is of the order of tens of MPa. On the other hand, the strength of the materials provides the temperature Te for which the limit of the elastic deformation can be reached. It is obvious that foil temperature should be less than Te . This temperature is the key parameter which allows finding out the perihelion of an initially elliptical transfer orbit to a parabolic or hyperbolic solar orbit based on the law of conservation of energy.
517
4. Conclusions An exploration of the outer solar system using solar sail propulsion with high cruise speed requires an acceleration of the sailcraft in the near-Sun space region. To obtain a high performance solar sail we should select among materials with the same optical properties, the material with low density and use a thin foil of this material for the sail. To find minimal required thickness of the solar sail for an acceleration of the sailcraft in the near-Sun space region at least two important factors should be considered: • the temperature dependence of electrical conductivity of a sail material; • the existence of a wide range of solar electromagnetic radiation frequencies and dependence of the electrical conductivity and dielectric function on the frequency. Applying the system of Maxwell’s equations for linear conducting media we find the minimum foil thickness that provides the maximum reflectivity and investigate dependence of this minimum thickness on temperature as well as on wavelengths of solar radiation. We have shown that the above factors have an effect on the thickness of solar sail foil and, therefore, on the reflection and absorption ability of the sail. The strong dependence of the solar sail thickness on temperature is shown. When the temperature increases the electrical conductivity of the sail material decreases affecting the sail thickness by increasing it. The minimal thickness has explicit √ dependence on the frequency through the factor 1/ and it also has implicit dependence on the frequency, because the conductivity (, T ) and dielectric function () are functions of frequency. It is shown that when the frequency dependence of the conductivity and dielectric function is taken into account the minimal thickness of the solar sail foil exhibits only slight dependence on the wavelength of the solar radiation. The determination of the minimal thickness of the solar sail foil based on its dependence on the temperature and frequency of the solar electromagnetic radiation is more critical than determination of the thickness based on the mechanical properties of materials. However, the strength of the materials provides the temperature for which the elastic limit can be reached when a material strain changes from an elastic deformation to a plastic deformation and, therefore, dramatically changes optical properties of the solar sail. In this paper we presented the minimal required thickness of the solar sail material that provides the best
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reflection and absorption of all solar radiation wavelengths for the given temperature. The surface of a solar sail with the required minimal thickness of a foil can withheld the solar pressure within the limit of the elastic deformation of the material. We suggest that the factors considered in this paper should be taken into consideration in the solar sail design. Let us also mention that aluminum and beryllium are the most preferable materials for the solar sail. These materials required much less thickness to reflect the solar spectrum. However, the melting temperature for aluminum is much less than the one for beryllium, but the processing and use of beryllium is risky due to potential toxicity. Acknowledgment I would like to thank Dr. V.S. Boyko for useful discussion. References [1] C.R. McInnes, Solar Sailing. Technology, Dynamics and Mission Applications, Springer, Praxis Publishing, 1998 296pp. [2] G.L. Matloff, Deep Space Probes. To Outer Solar System and Beyond, second ed., Springer-Praxis, Chichester, UK, 2005, 242pp. [3] C. Maccone, The Sun as a Gravitational Lens: Proposed Space Missions, second ed., IPI Press, Colorado Springs, CO, USA, 1999; C. Maccone, The Sun as a Gravitational Lens: Proposed Space Missions, third ed., IPI Press, Boulder, Colorado, 2003 [4] D. Jewitt, A. Morbidelli, H. Rauer, Trans-neptunium objects and comets: saas-fee advanced course 35, in: K. Altwegg, W. Benz, N. Thomas (Eds.), Swiss Society for Astrophysics and Astronomy, Springer, Berlin, Heidelberg, 2008, 258pp. [5] R.Ya. Kezerashvili, G.L. Matloff, Solar radiation and the beryllium hollow-body sail: 1. The ionization and disintegration effects, J. Br. Interplanet. Soc. 60 (2007) 169–179. [6] R.Ya. Kezerashvili, G.L. Matloff, Solar radiation and the beryllium hollow-body sail: 2. Diffusion, recombination and erosion processes, J. Br. Interplanet. Soc. 61 (2008) 47–57. [7] 2000 ASTM Standard Extraterrestrial Spectrum Reference E490-00 http://rredc.nrel.gov/solar/spectra/am0/ . [8] J. Strobl, The hollow body solar sail, J. Br. Interplanet. Soc. 42 (1989) 515–520. [9] G.L. Matloff, The beryllium hollow-body solar sail and interstellar travel, J. Br. Interplanet. Soc. 59 (2006) 349–354. [10] A. Vasicek, Optics of Thin Foils, North-Holland Publishing Company, Amsterdam, 1960, 403pp. [11] R.Ya. Kezerashvili, Solar sail interstellar travel: 1. Thickness of solar sail films, J. Br. Interplanet. Soc. 61 (2008) 430–439. [12] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Brooks/Cole Thomson Learning, 1976.
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