IAC-09-D1.1.9

Transport of Nanoparticles in the Interplanetary Medium

M. de Juan Ovelar, J. M. Llorens, C. H. Yam, and D. Izzo Advanced Concepts Team, ESTEC European Space Agency, Noordwijk, The Netherlands

ABSTRACT

The dynamic behaviour of nanoparticles in the interplanetary medium is subject to forces of a different nature often negligible for larger objects like satellites. Many theoretical models have been developed to explain the dynamics and transport mechanisms of cosmic dust. Building on this knowledge, we study the solar radiation pressure on artificial nanospheres, as a first step to assess the possibility of establishing artificial material fluxes in the interplanetary medium.

1. Introduction In order to answer this question we have analysed the behaviour of natural Dynamics in the interplanetary cosmic dust looking for the particle key medium constitutes the backbone of the properties (e.g. coating material and development in space technology. Based on thickness) we should engineer to control its this knowledge, it is possible to design a dynamic behaviour. large variety of missions as to achieve several scientific objectives. It is only Due to their small size, particles possible to achieve control on the dynamics naturally present in the interplanetary of an object by knowing the environment medium are subjected to different forces, characteristics and designing the object which are usually negligible for larger accordingly. objects like satellites. We have first analysed the main forces acting on small The main issue this work explores is natural particles in the solar system finding the possibility of performing a similar that the pressure exerted by the solar design process over extremely small radiation on a particle is of the same order particles which could be particularly useful as the gravitational attraction for particle in new applications like: gravitational fields sizes at the nanoscale and acts in the analysis around natural objects in space, opposite direction. NEO tracing techniques, calibration of onboard instruments in satellites, control of Neglecting any other forces acting on solar radiation over objects, etc… the particle, it is possible to control its final dynamical response by controlling the ratio between the radiation and the gravitational

1 force, i.e. the β parameter, as it is known in In this simple framework, the the literature [2]. nanoparticle can be considered as moving in Keplerian orbits subjected to a decreased We propose in this paper to design an gravity regime and the particle coating can artificial nanoparticle in such a way that we thus be conveniently considered as a can tailor its radiation pressure response control parameter on the solar mass. This and hence control its dynamics. We creates an interesting and new trajectory consider here as an artificial nanoparticle, a design problem that we solve here for the spherical core covered with a coating. The first time showing the possibility to reach thickness and material properties of the the selected target with no required initial coating are then the key parameters which ∆v. will allow us to tailor the final optical response of the nanoparticle. 2. Radiation–particle interaction: In Section 2, we describe the Key properties theoretical framework in which the study has been developed and establish the The interaction between forces validity range of the approximations made. present in the solar system and small Section 3 contains the results on the particles determines their dynamic analysis of the dependence of β with respect response, this is, the spatial and temporal to the coating properties and a discussion evolution of the orbit. To explore these on the engineering possibilities of out interactions and decide which ones are approach on β. Once the relation between determinant in our case of study, one the coating properties and β has been should be first referred to the vast studied we show the results of a dynamical bibliography related to cosmic dust in the problem in Section 4. Solar System [2] [6].

We should note that along the paper The main forces acting over a small we have considered the idealised case of a particle of diameter smaller than 1 µm in nanoparticle subject only to the radiation the solar system can be reduced to the force and solar gravity. As we will show in gravitational force (FG), the radiation the next section, the contribution of the pressure force (FR), the Lorentz force (FL), Lorentz force is also of relevance for the Poynting-Robertson drag force (FPR) determining realistic orbits. However its and the solar wind forces (FSW) [6]. At his implementation requires of an accurate range of sizes, the Poynting-Robertson drag modelling of the space environment force and the solar wind force become (plasma parameters) and solar magnetic negligible compared to the gravitational, field [3]. For the sake of simplicity, we radiation and Lorentz forces. Thus, we can decided to constraint ourselves to the reduce our problem considering the study dynamical problem defined only by the two of an artificial particle subjected only to forces mentioned above. We study a these forces. modified Lambert problem to find the An accurate description of the optimal coating that allows, in a given time Lorentz force implies some difficulties due frame, to reach selected targets starting to its nature. This force comes from the from an Earth trailing orbit (the case of interaction between particle charge and the Mars is shown). planetary and solar magnetic fields. It is well established that a neutral particle

2 orbiting around the Sun, gets a positive where G is the gravitational constant, M* is charge due to various effects like the central body mass, mPa is the particle photoemission of electrons, sticking and mass and r is the particle’s position vector. penetration of plasma particles, and This equation gives us the acceleration of secondary electron emission due to the particle and therefore defines its bombardment of energetic plasma [3] and trajectory if no other force is present. therefore it gets affected by the presence of magnetic fields. Furthermore, the solar Radiation Pressure force, can be magnetic field introduces two major classically explained as the net effect of the difficulties. At distances ~ 1 AU, i.e. in the transfer of momentum taking part between region of interest for the missions we are molecules constituting matter and an interested in, the field is described by incoming electromagnetic field. Photons, equally important radial and azimuthal coming from an electromagnetic wave, components [4]. This implies that in carry energy and momentum, and thus principle, the Lorentz force will present when they meet a particle, part of their radial, azimuthal and vertical components, energy and momentum is transferred. The which will complicate enormously the effectiveness of this process can be system dynamics. In addition, the solar described by the absorption and scattering magnetic field changes its orientation every cross sections, both represented in the so ~ 11 years and exhibits quite frequent called efficiency for radiation pressure fluctuations linked to its activity. The factor (Qpr), and both depending on the mathematical model should incorporate particle optical properties (complex these effects, since their time scale are of dielectric function ε=ε1+jε2 , and complex the same order of magnitude than the refractive index, N=n+jk). The resultant mission time of flight. force is expressed as follows:

r L* APa Qpr In summary, the actual calculation of (2) FR = 3 ⋅r, the resultant Loretz force on the particle 4πcr requires a complete model of the solar where L* is the stellar luminosity, APa is the magnetic field and plasma environment in particle cross section, c is the speed of light the solar system. Therefore, as a first in vacuum and r is again the particle’s approach to the problem, we consider only position vector. in this study the gravitational force (FG) and the radiation pressure force (FR), acting in Taking into account these two forces, the same line but opposite direction. we can express the resulting net force as:

F R r r F = (F − F ) ⋅ . (3) Tot R G r Sun FG It is useful then to define a parameter, usually called β in the cosmic dust Fig. 1: Idealised dynamic analysis bibliography, evaluating the ratio between them as follows: The equation defining the r gravitational force is well known from F R 1 APa Qpr L∗ Newtonian mechanics: β = r = ⋅ ⋅ . (4) F 4πc mPa GM ∗ r M m G F = −G ⋅ * Pa ⋅r, (1) G r 3

3 This parameter depends on both in equation (4) in terms of basic properties stellar and particle characteristics. If we obtaining the following: define the Sun as our central body, the first A = πR2 and second factors in equation (4) remain Pa Pa Q constant and therefore it is derived that β pr 4 3 β =B⋅ (6) depends only on particle properties as mass, mPa = πRPaρPa ρPaRPa 3 size and the efficiency for radiation pressure factor determined by the material composition. where B is a constant value including the Sun characteristics dependent factor, ρPa is It is interesting to remark that β is the particle density and RPa is the particle independence of the particle position, i.e. radius. once the particle and the central body are defined β remains constant along the Looking at this expression we can particle’s trajectory. finally infer that the dynamic response of the particle in this idealised case is The net force can be then expressed in completely governed by the particle terms of this parameter as density, radius and optical properties (permittivity or refractive index). r ⎛ r ⎞ FTot = (1− β )FG ⋅⎜− ⎟, (5) 3. Engineering β ⎝ r ⎠ from what we can derive that the resultant Our proposal to engineer the β value effect of the radiation pressure acting over of an artificial nanoparticle consists of the particle is to counteract gravity and designing a coating for a spherical particle therefore, the final trajectory for the particle so that we are modifying its optical will be determined by its motion in a response at will. modified central gravity field, depending on the β parameter. In order to evaluate the β value of a nanoparticle, we have calculated the absorption and scattering cross sections, For values of β between 0 and 1, the and therefore the radiation pressure acting solution for the particle’s trajectory could over the particle by means of the Multiple be a parabolic, elliptic or hyperbolic orbit Elastic Scattering of the Multipole with the sun as focus. For β=1, the radiation Expansions (MESME) [5]. This formalism pressure counteracts completely the is general enough to cope not only with gravitational force and therefore the particle single particle, but also with particle follows a linear trajectory unless other clusters in different arrangement. forces come into action. For values larger than 1, the only possible solution is a We show in Fig. 2 a plot illustrating hyperbolic trajectory with the sun in the typical values of β and its dependence with outer focus, meaning that the particle is the particle radius (RPa), for spherical rejected out of the system by radiation in particles made of uniform materials. In all which is usually called an escape orbit. cases we have considered real values of the material dielectric function. The black In order to detect the basic properties curve corresponds to the ideal case of of the particle that affect this parameter we constant efficiency factor for the radiation now describe the particle dependent factor pressure Qpr=1, approximation known as

4 geometrical optics. It serves for the purpose we find that radiation pressure can of establishing when wave optics needs to overcome gravitational force by a factor of be applied. As it is shown in the Figure, for 6 for certain particle sizes (0.1 µm). On the sizes larger than ~ 1 µm it is a good other hand, materials like silver, exhibiting approximation to assume a constant Qpr. also high values of k, are mostly governed by the gravitational effect due to its much Such an approximation is commonly larger density (10,41 g/cm3). used for extended bodies (such as solar sails or satellite) but fails to describe the During the study we have analysed interaction between nanoparticles and the many combinations of materials. The solar electromagnetic radiation. system consisting of an Ag core and a graphite coating is a representative one. In this system the core material exhibits low values of β, while the coating material presents much higher values.

In Fig. 3, we present the variation of β with respect to the coating thickness for different particle sizes. The coating thickness (d) is expressed through the ratio RTot/RPa, meaning that for a ratio equal to one there is no coating (d=0) and for a ratio equal to two the coating is as thick as the core (d=RPa). The table on right hand side of Fig. 3 presents the β values for these two limiting cases.

From these results, it is possible to infer that the coating introduces a big Fig. 2: β dependence on particle size for change in β for small particle sizes (R < different materials Pa 50 nm). For larger sizes, the gravitational interaction becomes dominant and the For almost all materials, there is a β coating slightly affects the value of the bare maximum at ~ 0.1 µm. These are expected particle. In other systems, as Si/graphite, results, given that the interaction between there is a change in the sign of the slope of particles and electromagnetic radiation the curves for sizes larger than ~ 0.1 µm reaches its maximum value for particle (results not shown). sizes of the order of λ/2π, which for the Sun (the maximum intensity of the sun spectrum This results shows that our approach is at ~ 0.6 µm.), corresponds to ~ 0.1 µm introduces a change in β, which allows us to [2]. tailor the dynamical response of the The particular features of the real and nanoparticle. However, the dependence of β imaginary parts of the material dielectric on the coating thickness depends to a great function determine the optical response of extent on the material system. Therefore, the particle. For materials as graphite, of 3 our approach lacks of transferability from densities ~ 2.00 g/cm and high values of one system to another and each case needs the imaginary part of the refractive index (k, to be analyse in particular. which is tightly related to the absorption),

5 Fig. 3: Values of β as a function of RTot for different core radius. The line colour in the plot is linked to the cell colour in the right table pointing to the corresponding Rpa. The core material is Ag and the coating material is graphite.

4. Nanoparticle transfer orbits We consider a range of launch dates and time of flight and solve the To illustrate the benefits of our two-point boundary value problem approach in engineering β, we solve a (known as the Lambert problem) in the practical case. We aim at minimizing the "weakened" gravitational field of the velocity increment a spherical Sun (see Eq. (5)). The manoeuvre cost is nanoparticle needs in order to reach a computed as the difference ∆v between given target from a starting point in the the required departure velocity and the solar system by applying a coating to it. initial orbital velocity of the nanoparticle (in this case, the same as the Earth). Considering a nanoparticle in an Earth trailing orbit, which means the The optimization process is carried particle has escaped Earth's gravity but it out minimizing the ∆v including β as a is still under the influence of the Sun, we parameter by means of the gravitational want to answer the following question: parameter known as µ = GM*, which for a given launch date, what β using the relation shown in Eq. (5) can parameter gives the smallest maneuver be redefined as a variable depending on cost (∆v) to reach a given target body? β as follows

For this first simulation we plan to µ'(β ) = µ ⋅ (1− β ). (7) transfer a nanoparticle to Mars.

6 In Figure 3, we present the ∆v cost depending on β, illustrating the potential for a various launch dates (Date Of applications of this concept. Launch, on the abscissa axis) and times of flight (TOF, on the ordinate axis) for It is important to note that the different β values: 0, 0.4, 0.6, 0.8. The current algorithm we are using to value β=0 represents no effect of the calculate this plots does not admit values radiation pressure acting over the of β larger than 1, i.e. negative values of particle (FR=0). µ which will imply a negative value for the Sun’s mass. In a further work, this These contour plots are commonly problem will be solved adapting the known as “pork-chop” plots and are algorithm to include such cases. characterised by periodic areas of minimum values of ∆v (shown in blue) which here vary in position and shape

400 10 10

350 9 9

300 8 8

ys a d

7 7 , 250 ght i

l 6 6 f

f 200 o 5 5

me i 150 T 4 4 100 3 3 50 a) β = 0 b) β = 0.4 2 2

400 10 10

350 9 9

300 8 8 ys

a d

, 7 7 250 ght i l 6 6 f

f 200 o 5 5 me i 150 T 4 4 100 3 3 50 c) β = 0.6 d) β = 0.8 2 2 200 600 1000 1400 1800 200 600 1000 1400 1800 Launch date since 1/1/2009 Launch date since 1/1/2009

Fig. 3 Contour plots showing ∆v as a function of the Time of Flight and the Launch Date for four different values of β.

7 Finally, in this simulation, we have 5. Conclusions found a very interesting solution for a transfer orbit with ∆v ~0, meaning that all the additional energy needed for the We propose in this work a technique particle to take the transfer orbit from the for controlling the ratio β between the Earth to Mars is supplied by the radiation radiation pressure and the gravitational pressure. In our idealized case this attraction in artificial nanoparticles by corresponds to a free transfer to Mars. applying a properly designed coating. Such a solution corresponds to a launch The results given showed that such a date on 2014/02/12 with a time of flight of technique can either increase or decrease 119 days and β=0.4. The orbit followed by the β value with respect to the bare particle the nanoparticle is illustrated in Fig. 4. It which will affect directly its dynamics. On should be noted that the frame has been the other hand, our approach lacks direct distorted vertically in order to appreciate transferability from one material system to the orbits. Clearly Mars is intercepted at another one, meaning that each the node as otherwise a zero ∆v solution combination of materials should be studied would not exist. separately. Looking at the results shown in the We have shown that it is possible to previous section, we find that for Ag design a transfer mission Earth-Mars with particles of radius 0.25 < R < 0.5 it is pa a ∆v~0 solution by introducing β as a possible to design a coating of graphite for variable, and thus, for that β value, we can which the resulting β equals the optimal actually design the nanoparticle coating one. It is also possible to conclude that for which meets the mission requirements. other sizes the graphite coating can not However, we have also seen that the provide such β value. optimal β value can not be found for certain particle sizes. Earth - Mars ∆v≈0 solution In a realistic mission, the Lorentz force, which we neglected in this preliminary work, needs to be taken into account because it would certainly change our results significantly, yet we feel that the methodology we used is of interest and could form the basis for the next steps.

Acknowledgments

The authors want to acknowledge F. J. García de Abajo for kindly offering us the software implementation of MESME. Fig. 4 Transfer orbit described by the nanoparticle departing from an Earth trailing This work has been carried out as a orbit and reaching Mars three-month stage in nanotechnologies within the Advanced Concepts team of the European Space Agency.

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