# Diffractive Sail-Based Displaced Orbits for High-Latitude Environment Monitoring

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## Abstract

**:**

## 1. Introduction

## 2. Simplified Diffractive Sail Thrust Model in Sun-Facing Condition

## 3. Maintenance of Circular DNKOs

#### 3.1. Earth–Spacecraft Distance

#### 3.2. Orbital Parameters of the Osculating Orbit

#### 3.3. Comparison with the Reflecting Solar Sail Case

## 4. Linear Stability Analysis

#### Potential Mission Application

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Conceptual sketch of a diffractive sail in a low-Earth orbit. Digital picture generated with the aid of Bing Image Creator.

**Figure 2.**Diffractive sail of center of mass C facing the Sun, propulsive acceleration vector ${\mathbf{a}}_{p}$, grating momentum unit vector $\widehat{\mathit{K}}$, and right-handed body reference frame $\mathcal{T}(C;x,y,z)$ of unit vectors $\{{\widehat{\mathit{i}}}_{x},{\widehat{\mathit{i}}}_{y},{\widehat{\mathit{i}}}_{z}\}$.

**Figure 3.**Conceptual scheme of a circular heliocentric DNKO with a displacement equal to $\eta $ with respect to the Ecliptic. The green disk indicates the plane of the displaced orbit, which the spacecraft covers at a constant angular velocity ${\omega}_{\oplus}$ at distance d from the Earth.

**Figure 4.**Scheme of the gravitational, propulsive, and centrifugal forces per unit of mass acting on the diffractive sail-based spacecraft during the flight along the circular displaced orbit.

**Figure 5.**Variations of $\{\rho ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}r,\phantom{\rule{0.166667em}{0ex}}\beta \}$ with $\gamma \in [0,\phantom{\rule{0.166667em}{0ex}}89.5]\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ for a diffractive sail-based circular heliocentric DNKO. (

**a**) Orbit radius. (

**b**) Orbit displacement. (

**c**) Sun–spacecraft distance. (

**d**) Lightness number.

**Figure 7.**Variation of the spacecraft–Earth distance d with $\{\gamma ,\phantom{\rule{0.166667em}{0ex}}\nu ,\phantom{\rule{0.166667em}{0ex}}\overline{\nu}\}$. The contour lines indicate the values of d expressed in units of ${R}_{\oplus}\triangleq 6378.136\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$. (

**a**) $\overline{\nu}=\Phi =0$. (

**b**) $\overline{\nu}=30\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ and $\Phi \simeq 0.94\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$. (

**c**) $\overline{\nu}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ and $\Phi \simeq 1.64\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$. (

**d**) $\overline{\nu}=90\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ and $\Phi \simeq 1.91\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$.

**Figure 8.**Variation of the spacecraft–Earth distance d with $\{\gamma ,\phantom{\rule{0.166667em}{0ex}}\nu ,\phantom{\rule{0.166667em}{0ex}}\overline{\nu}\}$. (

**a**) $\gamma =0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$, $\eta \simeq 41\phantom{\rule{0.166667em}{0ex}}{R}_{\oplus}$, and $\beta \simeq 0.0025$. (

**b**) $\gamma =0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$, $\eta \simeq 82\phantom{\rule{0.166667em}{0ex}}{R}_{\oplus}$, and $\beta \simeq 0.0049$. (

**c**) $\gamma =0.3\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$, $\eta \simeq 123\phantom{\rule{0.166667em}{0ex}}{R}_{\oplus}$, and $\beta \simeq 0.0074$. (

**d**) $\gamma =0.4\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$, $\eta \simeq 163\phantom{\rule{0.166667em}{0ex}}{R}_{\oplus}$, and $\beta \simeq 0.0098$.

**Figure 9.**Variation of the value of ${min}_{\nu}\left(d\right)$ with $\{\gamma ,\phantom{\rule{0.166667em}{0ex}}\overline{\nu}\}$ for small values of $\gamma $.

**Figure 10.**Values of $\overline{\nu}$ that minimize ${min}_{\nu}\left(d\right)$ as a function of $\gamma \le \tilde{\gamma}$.

**Figure 11.**Variation of the value of ${min}_{\nu}\left(d\right)$ with $\{\gamma ,\phantom{\rule{0.166667em}{0ex}}\overline{\nu}\}$ for very small values of $\gamma $.

**Figure 12.**Constant orbital parameters of the (closed) osculating orbit as a function of $\gamma $. (

**a**) Semimajor axis. (

**b**) Eccentricity. (

**c**) Inclination. (

**d**) True anomaly.

**Figure 13.**Variations of $\{b,\phantom{\rule{0.166667em}{0ex}}c,\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}\sqrt{c}\}$ with ${\gamma}_{i}$.

**Figure 14.**Simulation results for the case study. (

**a**) Time variations of $\{r,\phantom{\rule{0.166667em}{0ex}}\gamma \}$. (

**b**) Time variations of $\{{v}_{r},\phantom{\rule{0.166667em}{0ex}}{v}_{\theta},\phantom{\rule{0.166667em}{0ex}}{v}_{\gamma}\}$.

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**MDPI and ACS Style**

Bassetto, M.; Mengali, G.; Quarta, A.A.
Diffractive Sail-Based Displaced Orbits for High-Latitude Environment Monitoring. *Remote Sens.* **2023**, *15*, 5626.
https://doi.org/10.3390/rs15245626

**AMA Style**

Bassetto M, Mengali G, Quarta AA.
Diffractive Sail-Based Displaced Orbits for High-Latitude Environment Monitoring. *Remote Sensing*. 2023; 15(24):5626.
https://doi.org/10.3390/rs15245626

**Chicago/Turabian Style**

Bassetto, Marco, Giovanni Mengali, and Alessandro A. Quarta.
2023. "Diffractive Sail-Based Displaced Orbits for High-Latitude Environment Monitoring" *Remote Sensing* 15, no. 24: 5626.
https://doi.org/10.3390/rs15245626