, Volume 3, Issue 3, pp 217–230 | Cite as

Smart dust option for geomagnetic tail exploration

  • Alessandro A. QuartaEmail author
  • Giovanni Mengali
  • Lorenzo Niccolai
Research Article


In-situ measurements are necessary for a long-term analysis of the spatial structure of the geomagnetic tail. This type of mission requires the use of a propellantless propulsion system, such as a classical solar sail, to continuously rotate the design orbit apse line such that it remains parallel to the Sun-Earth direction. To reduce the mission costs, this paper suggests the employment of Sun-pointing smart dusts, which are here investigated in terms of propulsive acceleration level necessary to guarantee a mission’s feasibility. A Sun-pointing smart dust can be thought of as a millimeter-scale solar sail, whose geometric configuration allows it to passively maintain an alignment with the Sun-spacecraft line. The smart dust external surface is coated with an electrochromic reflective film in such a way that it may change, within some limits, its propulsive acceleration magnitude. A suitable control law is necessary for the smart dust to enable an artificial precession of its Earth-centred orbit, similar to what happens in the GeoSail mission. This paper analyzes the required control law using an optimal approach. In particular, the proposed mathematical model provides a set of approximate equations that allow a simple and effective tradeoff analysis between the propulsive requirements, in terms of the smart dust acceleration, and the characteristics of the design orbit.


smart dust femto solar sail electrochromic control system geomagnetic tail exploration 



osculating orbit semimajor axis (km)


propulsive acceleration, with \({a_P}\underline{\underline \Delta } \left\| {{a_P}} \right\|\) (mm/s2)


radial component of aP (mm/s2)


transverse component of aP (mm/s2)


reference value; see Eq. (21) (mm/s2)


area-to-mass ratio (m2/kg)


osculating orbit eccentricity


eccentric anomaly when r = r* (deg)


dimensionless auxiliary function; see Eq. (31)


dimensionless auxiliary function; see Eq. (34)


dimensionless auxiliary function; see Eq. (35)


Hamiltonian function

\({\hat i}\)

orbital reference frame unit vector


performance index


design parameter

O Earth’s center-of-mass p

osculating orbit semilatus rectum (km)


Earth-SPSD distance (km)


reference distance (km)

\({{\hat r}_ \odot }\)

Sun-SPSD unit vector


Earth’s mean radius (km)


switching function; see Eq. (19)


time (days)

\(\mathcal{T}(O;{\hat i}_{\rm{r}}, {\hat i}_{\rm{t}})\)

orbital reference frame


Earth-Sun line angle (deg)


scientific phase time interval (days)


variables adjoint to ith state


Earth’s gravitational parameter (km3/s2)


osculating orbit true anomaly (deg)


true anomaly when r = r* (deg)


auxiliary angle; see Eq. (7) (deg)


dimensionless switching parameter


osculating orbit argument of perigee (deg)


Earth’s orbital angular velocity (deg/day)







constrained to aPmin


















unit vector



This work is supported by the University of Pisa, Progetti di Ricerca di Ateneo (Grant No. PRA_2018_44).


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Copyright information

© Tsinghua University Press 2019

Authors and Affiliations

  • Alessandro A. Quarta
    • 1
    Email author
  • Giovanni Mengali
    • 1
  • Lorenzo Niccolai
    • 1
  1. 1.Department of Civil and Industrial EngineeringUniversity of PisaPisaItaly

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