# A continuum model for the orbit evolution of self-propelled ‘smart dust’ swarms

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## Abstract

A continuity equation is developed to model the evolution of a swarm of self-propelled ‘smart dust’ devices in heliocentric orbit driven by solar radiation pressure. These devices are assumed to be MEMs-scale (micro-electromechanical systems) with a large area-to-mass ratio. For large numbers of devices it will be assumed that a continuum approximation can be used to model their orbit evolution. The families of closed-form solutions to the resulting swarm continuity equation then represent the evolution of the number density of devices as a function of both position and time from a set of initial data. Forcing terms are also considered which model swarm sources and sinks (device deposition and device failure). The closed-form solutions presented for the swarm number density provide insights into the behaviour of swarms of self-propelled ‘smart dust’ devices an can form the basis of more complex mission design methodologies.

## Keywords

Swarm Satellite constellation Micro-satellite Solar radiation pressure Continuity equation## 1 Introduction

Astrodynamics has traditionally been concerned with the orbital motion of artificial satellites described using sets of nonlinear ordinary differential equations (ODE). For example, the inverse square force due to the potential of a uniform, spherical mass leads to the well-known family of conic section orbits through the Kepler problem. Using the Kepler problem as the unperturbed solution to the artificial satellite problem, the influence of perturbations can then be investigated by constructing high order series solutions using a range of techniques. However, while nonlinear ODEs provide a set of powerful tools to investigate the rich behaviour of the artificial satellite problem, new tools are now required to understand the behaviour of large ensembles of micro-satellites of diminishing length-scale. Rather than using the methods of ODEs to propagate the motion of individual satellites, a continuum approach using partial differential equations (PDE) can be used to provide an understanding of the behaviour of these new systems.

While an understanding of the behaviour of swarms of micro-satellites provides the contemporary motivation for a continuum approach to Astrodynamics, prior work on continuum dynamics for artificial satellites has been motivated by the need to understand the evolution of orbital debris clouds and early passive systems such as the West Ford needles [e.g. Heard (1976)]. In addition, prior literature on the orbit evolution of clouds of natural dust grains [e.g. Gor’kavyi et al. (1997)] and galactic dynamics [e.g. Binney and Tremaine (1987)] provides rich insights into the problem.

However, the need for new methodologies can be seen in the development of self-propelled MEMs-scale (micro-electromechanical systems) technologies which can enable the deployment of large swarms of self-propelled ‘smart dust’ devices (Kahn et al. 1999) driven by solar radiation pressure (Atchison and Peck 2008; Barker and Rodriguez-Salazar 2008; Atchison and Peck 2010). It is envisaged that swarms of such devices could be released from a dispenser (Manchester et al. 2013) and then utilise solar radiation pressure to slowly spiral inwards or outwards through the ecliptic plane (McInnes 1999). Distributed swarms could in principle deliver massively parallel sensing for space science applications by providing, for example, simultaneous multi-point measurements of magnetic field strength and direction (McInnes 1999; Barnhart et al. 2007; Atchison and Peck 2010).

In addition to the development of practical technologies for ‘smart dust’ devices (Manchester et al. 2013), recent work has led to an initial understanding of their orbit evolution (Colombo and McInnes 2011a, b; Lucking et al. 2012). This is a particularly intriguing problem given the impact of length-scaling (Atchison and Peck 2011); for a device of length-scale *L*, its mass scales as \(L^{3}\) while its surface area scales of \(L^{2}\), resulting in its area-to-mass ratio increasing with decreasing device size as \(L^{-1}\). Smart dust devices are therefore strongly perturbed by surface forces such as aerodynamic drag and solar radiation pressure (Atchison and Peck 2011; McInnes et al. 2011).

In this paper a continuum model of a swarm of smart dust devices in heliocentric orbit is developed where the devices are self-propelled by solar radiation pressure. Again, rather than model the orbit evolution of a single device, a continuity equation is formed which represents the number density of devices. Families of closed-form solutions to the continuity equation are then found representing the orbit evolution of the swarm through its number density as function of both position and time. The work presented builds on prior analysis of the evolution of debris clouds (McInnes 1993; Letizia et al. 2015) and the evolution of geocentric micro-satellite swarms perturbed by air drag (McInnes 2000; Manchester and Peck 2011; McInnes and Colombo 2013). Importantly, since a continuum approach with a PDE is used, initial data at a given boundary is required to represent the spatial distribution of devices, with the characteristic curves of the PDE mapping the initial data onto the time evolution of the swarm. This is a quite different procedure to classical Astrodynamics where point initial conditions are required for a set of ODEs.

In order to develop solutions to the continuity equation, the dynamics of a single smart dust device are considered in Sect. 2 and approximate quasi-circular spiral trajectories obtained for a fixed device attitude. These trajectories later provide the characteristic curves of the continuity equation. Then, in Sect. 3 an azimuthally symmetric continuity equation is used to model the evolution of the number density of smart dust devices as a function of both orbit radius and time for a range of scenarios. These include consideration of a source term, representing the constant deposition of new devices, and a sink term representing the failure of devices during their lifetime, as presented in Sect. 4. Rarefaction and compression of the flow of devices is found, as expected, since the radial flow speed of the swarm due to solar radiation pressure is a strong function of orbit radius, while the swam either flows into a smaller (inward spiral) or larger (outward spiral) volume.

Extending the radial flow problem, azimuthally asymmetric flows are considered in Sect. 5 to model the orbit evolution of a given two-dimensional spatial pattern of smart dust devices. Here azimuthal shearing of the pattern is found, again as expected, since the azimuthal flow speed of the swarm is also a strong function of orbit radius distance through Kepler’s third law. Lastly, some conclusions are drawn in Sect. 6 where it is proposed that the closed-form solutions obtained for both the radial flows and azimuthally asymmetric flows can provide insights into the orbit evolution of swarms and form the basis of future mission design tools.

## 2 Single device orbit evolution

*P*is the solar radiation pressure,

*A*is the device reference area,

*m*is the device mass and \(\alpha \) is the pitch angle of the device relative to the Sun-line in the range \(-\pi /2\le \alpha \le \pi /2\), as shown in Fig. 1. It will be assumed that the device pitch angle is fixed, which can in principle be achieved by passive means through etching of the device surface into a sawtooth pattern (Atchison and Peck 2010). Again following McInnes (2014) the rate of change of (specific) orbit energy

*E*of the device is given by

For a slow inward or outward spiral, and so a quasi-circular orbit, the device velocity vector is approximated by the local Keplerian circular orbit velocity \(\vec {\mathbf{v}}{\left( {\mu \left( {1 - \beta } \right) /r\left( t \right) } \right) ^{1/2}}{{\vec {\mathbf{t}}}}\) (Prussing and Conway 2012), where again the radial component of acceleration can be incorporated in a reduced gravitational parameter. Furthermore, the inverse square solar radiation pressure *P* is defined by \(P=\bar{P} \left( {\bar{r} /r\left( t \right) } \right) ^{2}\), where \(\bar{P} \) is the solar radiation pressure at some distance \(\bar{r} \). The device lightness number is now defined as the ratio of the solar radiation pressure induced acceleration to the solar gravitational acceleration experienced by an ideal device with reflectivity \(\eta =1\). Since both solar gravity and solar radiation pressure scale as the inverse square of the device orbit radius \(r\left( t \right) \), the lightness number is found to be \({\upbeta }=2\bar{P} A\bar{r} ^{2}/m\mu \). A more sophisticated analysis for dust grains is given by Klačka (2014).

*E*in Eq. (2) it can be shown that the device orbit radius evolves as

## 3 Swarm continuity equation

It will now be assumed that a swarm of identical devices is distributed in space within some bound volume and that the spatial distribution of the swarm can be represented by the swarm number density. At some initial time the swarm number density will therefore represent the so-called initial data of this continuum problem. Solutions to the continuity equation will propagate the initial data forward to describe the evolution of the swarm number density as a function of both position and time.

First, an azimuthally symmetric swarm will be considered whose number density is a function of orbit radius and time only. In order for this assumption to hold it can be noted that if the timescale for the inward (or outward) flow of the devices is slow relative to the local circular orbit period, the swarm can be approximated as being an azimuthally symmetric disk. An asymmetric swarm whose number density is a function of orbit radius, polar angle and time will be considered later in Sect. 5. In order to bound this approximation it can be seen from Eq. (6) that an individual self-propelled device will drift azimuthally along a quasi-circular spiral such that \(\theta \left( \tau \right) =\log \left( {1+\lambda \tau } \right) /\lambda \), while a dispenser on a circular Keplerian orbit at unit radius will move uniformly such that \(\theta \left( \tau \right) =\tau \). The difference in the azimuthal position of the device and the dispenser can therefore be approximated by \(\delta \theta =\tau -\log \left( {1+\lambda \tau } \right) /\lambda \approx \lambda \tau ^{2}/2\). In order for the device to complete one revolution relative to the dispenser, such that \(\delta \theta =2\pi \) requires a duration \(\tau \approx \sqrt{4\pi /\lambda }\). For a device with lightness number \(\beta =0.01\) (Atchison and Peck 2010) and \(\alpha =\alpha ^{+}\) this corresponds to a duration equivalent to 6.5 orbits of the dispenser. As will be seen later in Sect. 4, for \(\beta \ll 1\) the evolution of the swarm over much longer timescales will be considered, so the initial approximation of azimuthal symmetry is reasonable.

It should also be noted that the choice of an annulus as control volume is driven by the symmetry of the problem and the representation of the continuity equation in polar coordinate form. However, a global conservation law can be written in integral form which can then be represented locally in any suitable coordinate system, as is common for other continuum problems (Anderson 2001).

## 4 Solutions to the swarm continuity equation

### 4.1 Evolution of an infinite sheet

### 4.2 Evolution of a finite disk

*H*is the Heaviside step function. The solution surface defined by Eq. (20) is shown in Fig. 5 where it can be seen that the disk interior to the Earth’s orbit is again initially uniform, but as the ensemble of smart dust devices flow inwards the disk contracts in radius while its density grows. The contraction of the outer edge of the disk is captured by the step function in Eq. (20), while the scaling of the density follows the same functional form as Eq. (19).

### 4.3 Constant device density at one boundary

The continuity argument resulting in Eq. (25) also arises due to the geometry of the problem, since the circumference \(2\pi \xi \) of any annular boundary of radius \(\xi \) shrinks faster than the inflow speed \(v_\xi \left( \xi \right) \) increases with diminishing orbit radius. Indeed for the general solution defined by Eq. (16) the radial inflow of devices is faster at smaller orbit radii which acts to reduce the swarm density since the density \(n\left( {\xi ,\tau } \right) \) scales as \(v_\xi \left( \xi \right) ^{-1}\). However, \(v_\xi \left( \xi \right) \) scales as \(\xi ^{-1/2}\) while the compression due to the geometric term in Eq. (16) discussed in Sect. 3 scales as \(\xi ^{-1}\) so that there is an overall density scaling of \(\xi ^{-1/2}\).

### 4.4 Constant device deposition rate at one boundary

*I*is required such that

*H*is the Heaviside step function.

### 4.5 Swarm evolution with on-orbit failures

## 5 Extension to two-dimensional system

## 6 Conclusions

Using a continuum approximation, a continuity equation has been developed to represent the evolution of a swarm of self-propelled ‘smart dust’ devices in heliocentric orbit under the action of solar radiation pressure. The family of solutions to the continuity equation describe the evolution of the swarm from a set of initial data at some boundary. By considering both radial and transverse motion, the spatial evolution of the swarm can be obtained and azimuthal shearing observed since the local transverse speed is a function of orbit radius. The addition of source and sinks terms to the continuity equation allows the deposition of new devices and on-orbit failures to be considered. It is proposed that the closed-form solutions presented for both the radial flows and azimuthally asymmetric flows of devices can provide useful insights into the orbit evolution of swarms of self-propelled ‘smart dust’ devices which can form the basis of future mission design tools.

## Notes

### Acknowledgments

The work reported in this paper was undertaken with the kind support of a Leverhulme Trust Research Fellowship (RF-2014-049).

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