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19 pages, 6930 KiB  
Article
Deterministic Trajectory Design and Attitude Maneuvers of Gradient-Index Solar Sail in Interplanetary Transfers
by Marco Bassetto, Giovanni Mengali and Alessandro A. Quarta
Appl. Sci. 2024, 14(22), 10463; https://doi.org/10.3390/app142210463 - 13 Nov 2024
Viewed by 395
Abstract
A refractive sail is a special type of solar sail concept, whose membrane exposed to the Sun’s rays is covered with an advanced engineered film made of micro-prisms. Unlike the well-known reflective solar sail, an ideally flat refractive sail is able to generate [...] Read more.
A refractive sail is a special type of solar sail concept, whose membrane exposed to the Sun’s rays is covered with an advanced engineered film made of micro-prisms. Unlike the well-known reflective solar sail, an ideally flat refractive sail is able to generate a nonzero thrust component along the sail’s nominal plane even when the Sun’s rays strike that plane perpendicularly, that is, when the solar sail attitude is Sun-facing. This particular property of the refractive sail allows heliocentric orbital transfers between orbits with different values of the semilatus rectum while maintaining a Sun-facing attitude throughout the duration of the flight. In this case, the sail control is achieved by rotating the structure around the Sun–spacecraft line, thus reducing the size of the control vector to a single (scalar) parameter. A gradient-index solar sail (GIS) is a special type of refractive sail, in which the membrane film design is optimized though a transformation optics-based method. In this case, the membrane film is designed to achieve a desired refractive index distribution with the aid of a waveguide array to increase the sail efficiency. This paper analyzes the optimal transfer performance of a GIS with a Sun-facing attitude (SFGIS) in a series of typical heliocentric mission scenarios. In addition, this paper studies the attitude control of the Sun-facing GIS using a simplified mathematical model, in order to investigate the effective ability of the solar sail to follow the (optimal) variation law of the rotation angle around the radial direction. Full article
Show Figures

Figure 1

Figure 1
<p>Sketch of the SFGIS with the normal unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">n</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> and the reference unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">m</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>, whose directions are fixed in a (spacecraft) body reference frame. Note that the sail nominal plane <math display="inline"><semantics> <mi mathvariant="script">P</mi> </semantics></math> is perpendicular to the Sun–spacecraft line, and the unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">m</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> belongs to the <math display="inline"><semantics> <mi mathvariant="script">P</mi> </semantics></math> plane. The conceptual scheme of the waveguide array was adapted from Ref. [<a href="#B1-applsci-14-10463" class="html-bibr">1</a>], courtesy of Dr. Shengping Gong.</p>
Full article ">Figure 2
<p>Sketch of the Radial–Transverse–Normal (RTN) reference frame <math display="inline"><semantics> <msub> <mi mathvariant="script">T</mi> <mi>RTN</mi> </msub> </semantics></math> of unit vectors <math display="inline"><semantics> <mrow> <mo stretchy="false">{</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">R</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">T</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">N</mi> </msub> <mo stretchy="false">}</mo> </mrow> </semantics></math>. The sail clock angle <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>∈</mo> <msup> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>360</mn> <mo>]</mo> </mrow> <mo>∘</mo> </msup> </mrow> </semantics></math> is the single control parameter of an SFGIS-propelled spacecraft.</p>
Full article ">Figure 3
<p>Dimensionless components <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>a</mi> <msub> <mi>p</mi> <mi mathvariant="normal">R</mi> </msub> </msub> <mo>/</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>a</mi> <msub> <mi>p</mi> <mi mathvariant="normal">T</mi> </msub> </msub> <mo>/</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>a</mi> <msub> <mi>p</mi> <mi mathvariant="normal">N</mi> </msub> </msub> <mo>/</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>}</mo> </mrow> </semantics></math> of the propulsive acceleration vector as a function of the sail clock angle <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>∈</mo> <msup> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>360</mn> <mo>]</mo> </mrow> <mo>∘</mo> </msup> </mrow> </semantics></math>, when the solar distance is one astronomical unit.</p>
Full article ">Figure 4
<p>Time variation in the (unconstrained) sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> in the minimum-time Earth–Venus mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. Blue dot → starting point; red square → arrival point.</p>
Full article ">Figure 5
<p>Ecliptic projection and isometric view of the rapid transfer trajectory in an Earth–Venus mission scenario, when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and the sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> is unconstrained. The <span class="html-italic">z</span>-axis of the isometric view is exaggerated to highlight the three-dimensionality of the transfer trajectory. Black line → spacecraft transfer trajectory; blue line → Earth’s orbit; red line → Venus’s orbit; filled star → perihelion; blue dot → starting point; red square → arrival point; orange dot → the Sun.</p>
Full article ">Figure 6
<p>Time variation of the (unconstrained) sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> in a minimum-time Earth–asteroid 433 Eros mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in <a href="#applsci-14-10463-f004" class="html-fig">Figure 4</a>.</p>
Full article ">Figure 7
<p>Ecliptic projection and isometric view of the rapid transfer trajectory in an Earth–asteroid 433 Eros mission scenario, when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and unconstrained sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math>. The legend is reported in <a href="#applsci-14-10463-f005" class="html-fig">Figure 5</a>.</p>
Full article ">Figure 8
<p>Minimum flight time as a function of the characteristic acceleration <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.2</mn> <mo>]</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> in an Earth–Venus orbit-to-orbit transfer. The black dot refers to the special case of <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> discussed in the first part of the section.</p>
Full article ">Figure 9
<p>Ecliptic projection and isometric view of the rapid transfer trajectory in an Earth–Mars mission scenario, when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and the sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> is unconstrained. The legend is reported in <a href="#applsci-14-10463-f005" class="html-fig">Figure 5</a>.</p>
Full article ">Figure 10
<p>Time variation of the (unconstrained) sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> in a minimum-time Earth–Mars mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in <a href="#applsci-14-10463-f004" class="html-fig">Figure 4</a>.</p>
Full article ">Figure 11
<p>Ecliptic projection and isometric view of the rapid transfer trajectory of an Earth–Mercury mission scenario, when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and the sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> is unconstrained. The legend is reported in <a href="#applsci-14-10463-f005" class="html-fig">Figure 5</a>.</p>
Full article ">Figure 12
<p>Time variation in the (unconstrained) sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> in a minimum-time Earth–Mercury mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in <a href="#applsci-14-10463-f004" class="html-fig">Figure 4</a>.</p>
Full article ">Figure 13
<p>Time variation of the constrained sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math> in a minimum-time Earth–Venus mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in <a href="#applsci-14-10463-f004" class="html-fig">Figure 4</a>. (<b>a</b>) Case of <math display="inline"><semantics> <mrow> <mi mathvariant="script">I</mi> <mo>=</mo> <msub> <mi mathvariant="script">I</mi> <mrow> <mo>①</mo> </mrow> </msub> </mrow> </semantics></math>; (<b>b</b>) Case of <math display="inline"><semantics> <mrow> <mi mathvariant="script">I</mi> <mo>=</mo> <msub> <mi mathvariant="script">I</mi> <mrow> <mo>②</mo> </mrow> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 14
<p>Sun–spacecraft distance <span class="html-italic">r</span> as a function of time in a minimum-time Earth–Venus mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.175</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in <a href="#applsci-14-10463-f004" class="html-fig">Figure 4</a>. (<b>a</b>) Case of <math display="inline"><semantics> <mrow> <mi mathvariant="script">I</mi> <mo>=</mo> <msub> <mi mathvariant="script">I</mi> <mrow> <mo>①</mo> </mrow> </msub> </mrow> </semantics></math>; (<b>b</b>) Case of <math display="inline"><semantics> <mrow> <mi mathvariant="script">I</mi> <mo>=</mo> <msub> <mi mathvariant="script">I</mi> <mrow> <mo>②</mo> </mrow> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 15
<p>Sketch of an SFGIS with two control vanes. The force due to the solar radiation pressure acting on the two moving surfaces is applied at the vane pressure center <math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math>. The rotation of the surfaces, denoted by <math display="inline"><semantics> <mi>β</mi> </semantics></math>, is the vane control angle.</p>
Full article ">Figure 16
<p>Closed-loop control scheme with saturation of the input to the system.</p>
Full article ">Figure 17
<p>Results of the numerical simulation when <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>≃</mo> <mn>9.1724</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mspace width="0.166667em"/> <mi>rad</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 18
<p>Minimum settling time <math display="inline"><semantics> <msub> <mi>t</mi> <mi>s</mi> </msub> </semantics></math> to perform a rotation of <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mi>ref</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>rad</mi> </mrow> </semantics></math> around <span class="html-italic">z</span> as a function of <span class="html-italic">k</span>.</p>
Full article ">Figure 19
<p>Optimal control parameters of the PDF controller as a function of <span class="html-italic">k</span>.</p>
Full article ">
16 pages, 2589 KiB  
Article
Three-Dimensional Rapid Orbit Transfer of Diffractive Sail with a Littrow Transmission Grating-Propelled Spacecraft
by Alessandro A. Quarta
Aerospace 2024, 11(11), 925; https://doi.org/10.3390/aerospace11110925 - 8 Nov 2024
Viewed by 357
Abstract
A diffractive solar sail is an elegant concept for a propellantless spacecraft propulsion system that uses a large, thin, lightweight surface covered with a metamaterial film to convert solar radiation pressure into a net propulsive acceleration. The latter can be used to perform [...] Read more.
A diffractive solar sail is an elegant concept for a propellantless spacecraft propulsion system that uses a large, thin, lightweight surface covered with a metamaterial film to convert solar radiation pressure into a net propulsive acceleration. The latter can be used to perform a typical orbit transfer both in a heliocentric and in a planetocentric mission scenario. In this sense, the diffractive sail, proposed by Swartzlander a few years ago, can be considered a sort of evolution of the more conventional reflective solar sail, which generally uses a metallized film to reflect the incident photons, studied in the scientific literature starting from the pioneering works of Tsander and Tsiolkovsky in the first decades of the last century. In the context of a diffractive sail, the use of a metamaterial film with a Littrow transmission grating allows for the propulsive acceleration magnitude to be reduced to zero (and then, the spacecraft to be inserted in a coasting arc during the transfer) without resorting to a sail attitude that is almost edgewise to the Sun, as in the case of a classical reflective solar sail. The aim of this work is to study the optimal (i.e., the rapid) transfer performance of a spacecraft propelled by a diffractive sail with a Littrow transmission grating (DSLT) in a three-dimensional heliocentric mission scenario, in which the space vehicle transfers between two assigned Keplerian orbits. Accordingly, this paper extends and generalizes the results recently obtained by the author in the context of a simplified, two-dimensional, heliocentric mission scenario. In particular, this work illustrates an analytical model of the thrust vector that can be used to study the performance of a DSLT-based spacecraft in a three-dimensional optimization context. The simplified thrust model is employed to simulate the rapid transfer in a set of heliocentric mission scenarios as a typical interplanetary transfer toward a terrestrial planet and a rendezvous with a periodic comet. Full article
(This article belongs to the Special Issue Advances in CubeSat Sails and Tethers (2nd Edition))
Show Figures

Figure 1

Figure 1
<p>Simplified scheme of the DSLT with the cone angle <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>90</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>, the grating momentum unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">K</mi> <mo>^</mo> </mover> </semantics></math>, the unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">n</mi> <mo>^</mo> </mover> </semantics></math> normal to the shadowed side of the sail nominal plane, the radial unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo>^</mo> </mover> </semantics></math>, and the propulsive acceleration vector <math display="inline"><semantics> <mi mathvariant="bold-italic">a</mi> </semantics></math>. Note that the direction of <math display="inline"><semantics> <mi mathvariant="bold-italic">a</mi> </semantics></math> is opposite to that of <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">K</mi> <mo>^</mo> </mover> </semantics></math>.</p>
Full article ">Figure 2
<p>Sketch of the Radial–Transverse–Normal (RTN) reference frame of unit vectors <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo>^</mo> </mover> <mi mathvariant="normal">R</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo>^</mo> </mover> <mi mathvariant="normal">T</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo>^</mo> </mover> <mi mathvariant="normal">N</mi> </msub> </semantics></math>. Image adapted from Ref. [<a href="#B38-aerospace-11-00925" class="html-bibr">38</a>].</p>
Full article ">Figure 3
<p>Unit vector of grating momentum <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">K</mi> <mo>^</mo> </mover> </semantics></math> in the RTN reference frame, and definition of the two angles <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>∈</mo> <mo>[</mo> <mn>90</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>180</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>360</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Configuration which gives the minimum value of the sail cone angle <math display="inline"><semantics> <mi>α</mi> </semantics></math> for an assigned position of the unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">K</mi> <mo>^</mo> </mover> </semantics></math> in the RTN reference frame.</p>
Full article ">Figure 5
<p>Scheme of the DSLT-based force bubble when the distance of the spacecraft from the Sun is equal to <math display="inline"><semantics> <msub> <mi>r</mi> <mo>⊕</mo> </msub> </semantics></math>. The (<b>left</b>) side of the figure shows the three-dimensional surface, while the (<b>right</b>) part of the figure shows the force bubble section obtained with a plane which contains the Sun–spacecraft line.</p>
Full article ">Figure 6
<p>Force bubble of a flat, ideal, reflective sail when the distance of the spacecraft from the Sun is equal to <math display="inline"><semantics> <msub> <mi>r</mi> <mo>⊕</mo> </msub> </semantics></math>. See the second sentence in the caption of <a href="#aerospace-11-00925-f005" class="html-fig">Figure 5</a> for the meaning of the two graphs.</p>
Full article ">Figure 7
<p>Rapid orbit-to-orbit transfer trajectory in an Earth–Mars scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. Ecliptic projection (<b>left</b>) and isometric view (<b>right</b>). Black line → DSLT-propelled spacecraft trajectory; blue line → Earth’s orbit; red line → target orbit; filled star → perihelion; blue dot → starting point; red square → arrival point; orange dot → the Sun.</p>
Full article ">Figure 8
<p>Optimal guidance law <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>β</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> in an Earth–Mars mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. Blue dot → starting point; red square → arrival point.</p>
Full article ">Figure 9
<p>Rapid orbit-to-orbit transfer trajectory in an Earth–Venus scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f007" class="html-fig">Figure 7</a>.</p>
Full article ">Figure 10
<p>Optimal guidance law <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>β</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> in an Earth–Venus mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f008" class="html-fig">Figure 8</a>.</p>
Full article ">Figure 11
<p>Rapid orbit-to-orbit transfer trajectory in an Earth–Mercury scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f007" class="html-fig">Figure 7</a>.</p>
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<p>Optimal guidance law <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>β</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> in an Earth–Mercury mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f008" class="html-fig">Figure 8</a>.</p>
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<p>Flight time as a function of <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> in an Earth–Mars mission scenario. The green dot indicates the case detailed in <a href="#aerospace-11-00925-f007" class="html-fig">Figure 7</a> and <a href="#aerospace-11-00925-f008" class="html-fig">Figure 8</a>.</p>
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<p>Flight time as a function of <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> in an Earth–Venus mission scenario. The green dot indicates the case detailed in <a href="#aerospace-11-00925-f009" class="html-fig">Figure 9</a> and <a href="#aerospace-11-00925-f010" class="html-fig">Figure 10</a>.</p>
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<p>Flight time as a function of <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> in an Earth–Mercury mission scenario. The green dot indicates the case detailed in <a href="#aerospace-11-00925-f011" class="html-fig">Figure 11</a> and <a href="#aerospace-11-00925-f012" class="html-fig">Figure 12</a>.</p>
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<p>Rapid orbit-to-orbit transfer trajectory in an Earth-29P scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f007" class="html-fig">Figure 7</a>.</p>
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<p>Optimal guidance law <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>β</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> in an Earth-29P scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f008" class="html-fig">Figure 8</a>.</p>
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<p>Time variation of the solar distance <span class="html-italic">r</span> and the orbital inclination in an Earth-29P scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The legend is reported in the caption of <a href="#aerospace-11-00925-f008" class="html-fig">Figure 8</a>.</p>
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25 pages, 3319 KiB  
Article
Preliminary Design of a GNSS Interference Mapping CubeSat Mission: JamSail
by Luis Cormier, Tasneem Yousif, Samuel Thompson, Angel Arcia Gil, Nishanth Pushparaj, Paul Blunt and Chantal Cappelletti
Aerospace 2024, 11(11), 901; https://doi.org/10.3390/aerospace11110901 - 31 Oct 2024
Viewed by 442
Abstract
The JamSail mission is an educational CubeSat aiming to design, develop, and demonstrate two new technologies on a small satellite, tentatively scheduled for launch no earlier than 2026. When launched, JamSail will demonstrate the functionality of two new payloads in low Earth orbit. [...] Read more.
The JamSail mission is an educational CubeSat aiming to design, develop, and demonstrate two new technologies on a small satellite, tentatively scheduled for launch no earlier than 2026. When launched, JamSail will demonstrate the functionality of two new payloads in low Earth orbit. First, a flexible, low-cost GNSS interference detection payload capable of characterising and geolocating the sources of radio interference regarding the E1/L1 and E5a/L5 bands will be demonstrated on a global scale. The data produced by this payload can be used to target anti-interference actions in specific regions and aid in the design of future GNSS receivers to better mitigate specific types of interference. If successful, the flexibility of the payload will allow it to be remotely reconfigured in orbit to investigate additional uses of the technology, including a potential demonstration of GNSS reflectometry aboard a CubeSat. Second, a compact refractive solar sail will be deployed that is capable of adjusting the orbit of JamSail in the absence of an on-board propellant. This sail will be used to gradually raise the semi-major axis of JamSail over the span of the mission before being used to perform rapid passive deorbit near the end-of-life juncture. Additionally, self-stabilising optical elements within the sail will be used to demonstrate a novel method of performing attitude control. JamSail is currently in the testing phase, and the payloads will continue to be refined until the end of 2024. This paper discusses the key objectives of the JamSail mission, the design of the payloads, the expected outcomes of the mission, and future opportunities regarding the technologies as a whole. Full article
(This article belongs to the Special Issue Small Satellite Missions)
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<p>Graphical overview of the JamSail concept of operations.</p>
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<p>The JamSail GNSS payload block diagram.</p>
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<p>Simulation and test methodology overview.</p>
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<p>Different average lengths regarding a simulated CW jamming signal: (<b>a</b>) 0 average FFTs; (<b>b</b>) 64 average FFTs; (<b>c</b>) 128 average FFTs; (<b>d</b>) 256 average FFTs.</p>
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<p>Different average lengths regarding a simulated CW jamming signal: (<b>a</b>) 0 average FFTs; (<b>b</b>) 64 average FFTs; (<b>c</b>) 128 average FFTs; (<b>d</b>) 256 average FFTs.</p>
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<p>The change in slant range over the zenithal pass.</p>
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<p>The received power levels of various signals over the perfect zenithal pass with reference to the noise floor.</p>
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<p>The 20 s scenario of CW interference above the transmitter: (<b>a</b>) 3D FFT results spectrum with 12.5 MHz sample frequency; (<b>b</b>) The waterfall diagram of the 20 s CW simulation.</p>
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<p>The discrete Doppler shift over the 20 s simulations and the truth data for the Doppler shift as a reference.</p>
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<p>The modelled continuous-frequency Doppler shift compared to the truth data Doppler shift.</p>
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<p>The error in Hz between the modelled continuous-frequency Doppler shift and the truth data Doppler shift.</p>
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<p>The modelled Doppler rate compared to the truth data for the Doppler rate.</p>
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<p>The error between the modelled and true Doppler rates.</p>
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<p>Render of deployed sail payload (no satellite bus).</p>
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<p>Transmissive and reflective forces <math display="inline"><semantics> <mi mathvariant="bold">F</mi> </semantics></math> and torques <math display="inline"><semantics> <mi>τ</mi> </semantics></math> with respect to centre of gravity <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">g</mi> </msub> </semantics></math> in a Sun-pointing attitude, where self-stabilising elements are inactive (idealised).</p>
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<p>Miura-ori flasher design in deployed (<b>a</b>) and stowed (<b>b</b>) configuration.</p>
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<p>Stowed sail payload annotated without flat springs, compression springs, or motor assembly.</p>
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<p>Sail payload diagonal section, moving assembly annotated (<b>a</b>) stowed; (<b>b</b>) deployed.</p>
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<p>Tri-band ground station at the University of Nottingham during installation.</p>
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22 pages, 12585 KiB  
Article
Preparation and Characterization of Atomic Oxygen-Resistant, Optically Transparent and Dimensionally Stable Copolyimide Films from Fluorinated Monomers and POSS-Substituted Diamine
by Zhenzhong Wang, Xiaolei Wang, Shunqi Yuan, Xi Ren, Changxu Yang, Shujun Han, Yuexin Qi, Duanyi Li and Jingang Liu
Polymers 2024, 16(19), 2845; https://doi.org/10.3390/polym16192845 - 9 Oct 2024
Viewed by 653
Abstract
Optically transparent polyimide (PI) films with good atomic oxygen (AO) resistance have been paid extensive attention as thermal controls, optical substrates for solar cells or other components for low Earth orbit (LEO) space applications. However, for common PI films, it is usually quite [...] Read more.
Optically transparent polyimide (PI) films with good atomic oxygen (AO) resistance have been paid extensive attention as thermal controls, optical substrates for solar cells or other components for low Earth orbit (LEO) space applications. However, for common PI films, it is usually quite difficult to achieve both high optical transparency and AO resistance and maintain the intrinsic thermal stability of the PI films at the same time. In the current work, we aimed to achieve the target by using the copolymerization methodology using the fluorinated dianhydride 9,9-bis(trifluoromethyl)xanthene-2,3,6,7-tetracarboxylic dianhydride (6FCDA), the fluorinated diamine 2,2-bis [4-(4-aminophenoxy)phenyl]hexafluoropropane (BDAF) and the polyhedral oligomeric silsesquioxane (POSS)-containing diamine N-[(heptaisobutyl-POSS)propyl]-3,5-diaminobenzamide (DABA-POSS) as the starting materials. The fluoro-containing monomers were used to endow the PI films with good optical and thermal properties, while the silicon-containing monomer was used to improve the AO resistance of the afforded PI films. Thus, the 6FCDA-based PI copolymers, including 6FCPI-1, 6FCPI-2 and 6FCPI-3, were prepared using a two-step chemical imidization procedure, respectively. For comparison, the analogous PIs, including 6FPI-1, 6FPI-2 and 6FPI-3, were correspondingly developed according to the same procedure except that 6FCDA was replaced by 4,4′-(hexafluoroisopropylidene)diphthalic anhydride (6FDA). Two referenced PI homopolymers were prepared from BDAF and 6FDA (PI-ref1) and 6FCDA (PI-ref2), respectively. The experimental results indicated that a good balance among thermal stability, optical transparency, and AO resistance was achieved by the 6FCDA-PI films. For example, the 6FCDA-PI films exhibited good thermal stability with glass transition temperatures (Tg) up to 297.3 °C, good optical transparency with an optical transmittance at a wavelength of 450 nm (T450) higher than 62% and good AO resistance with the erosion yield (Ey) as low as 1.7 × 10−25 cm3/atom at an AO irradiation fluence of 5.0 × 1020 atoms/cm2. The developed 6FCDA-PI films might find various applications in aerospace as solar sails, thermal control blankets, optical components and other functional materials. Full article
(This article belongs to the Special Issue Polymer Thin Films and Their Applications)
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<p>Preparation of PI and referenced PI resins.</p>
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<p>GPC plots of PI resins.</p>
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<p>XRD spectra of PI resins.</p>
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<p><sup>1</sup>H-NMR spectra of 6FPI resins in DMSO-d<sub>6</sub>.</p>
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<p><sup>1</sup>H-NMR spectra of 6FCPI resins in DMF-d<sub>7</sub>.</p>
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<p>FTIR spectra of PI films.</p>
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<p>TGA and DTG curves of PI films in nitrogen. (<b>a</b>) 6FPI; (<b>b</b>) 6FCPI.</p>
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<p>DSC curves of PI films.</p>
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<p>DMA curves of PI films.</p>
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<p>TMA curves of PI films.</p>
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<p>UV-Vis spectra of PI films.</p>
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<p>Appearance of PI films before and after AO exposure (doze: 5.0 × 10<sup>20</sup> atoms/cm<sup>2</sup>). (<b>a</b>–<b>c</b>): 6FPI-1~6FPI-3 films: left: pristine film; right: film after AO exposure; (<b>d</b>–<b>f</b>): 6FCPI-1~6FCPI-3 films: left: pristine film; right: film after AO exposure; (<b>g</b>) 6FPI-1-AO; (<b>h</b>) 6FPI-2-AO; (<b>i</b>) 6FPI-3-AO; (<b>j</b>) 6FCPI-1-AO; (<b>k</b>) 6FCPI-2-AO; (<b>l</b>) 6FCPI-3-AO.</p>
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<p>Comparison of UV-Vis spectra of PI films before and after AO exposure. (<b>a</b>) 6FPI; (<b>b</b>) 6FCPI.</p>
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<p>SEM and EDS images of pristine 6FCDA-PI films. (<b>a</b>) 6FCPA-1, (<b>b</b>) 6FCPI-2 and (<b>c</b>) 6FCPI-3.</p>
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<p>SEM and EDS images of 6FCDA-PI films after AO exposure (5.0 × 10<sup>20</sup> atoms/cm<sup>2</sup>). (<b>a</b>) 6FCPA-1-AO, (<b>b</b>) 6FCPI-2-AO and (<b>c</b>) 6FCPI-3-AO.</p>
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<p>XPS spectra of Si2p and O1s for 6FCPI films. (<b>a</b>) 6FCPI-1 and 6FCPI-1-AO; (<b>b</b>) 6FCPI-2 and 6FCPI-2-AO; (<b>c</b>) 6FCPI-3 and 6FCPI-3-AO.</p>
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17 pages, 1325 KiB  
Article
Thrust Model and Trajectory Design of an Interplanetary CubeSat with a Hybrid Propulsion System
by Alessandro A. Quarta
Actuators 2024, 13(10), 384; https://doi.org/10.3390/act13100384 - 1 Oct 2024
Viewed by 595
Abstract
This paper analyzes the performance of an interplanetary CubeSat equipped with a hybrid propulsion system (HPS), which combines two different types of thrusters in the same deep space vehicle, in a heliocentric transfer between two assigned (Keplerian) orbits. More precisely, the propulsion system [...] Read more.
This paper analyzes the performance of an interplanetary CubeSat equipped with a hybrid propulsion system (HPS), which combines two different types of thrusters in the same deep space vehicle, in a heliocentric transfer between two assigned (Keplerian) orbits. More precisely, the propulsion system of the CubeSat considered in this work consists of a combination of a (low-performance) photonic solar sail and a more conventional solar electric thruster. In particular, the characteristics of the solar electric thruster are modeled using a recent mathematical approach that describes the performance of the miniaturized engine that will be installed on board the proposed ESA’s M-ARGO CubeSat. The latter will hopefully be the first interplanetary CubeSat to complete a heliocentric transfer towards a near-Earth asteroid using its own propulsion system. In order to simplify the design of the CubeSat attitude control subsystem, we assume that the orientation of the photonic solar sail is kept Sun-facing, i.e., the sail reference plane is perpendicular to the Sun-CubeSat line. That specific condition can be obtained, passively, by using an appropriate design of the shape of the sail reflective surface. The performance of an HPS-based CubeSat is analyzed by optimizing the transfer trajectory in a three-dimensional heliocentric transfer between two closed orbits of given characteristics. In particular, the CubeSat transfer towards the near-Earth asteroid 99942 Apophis is studied in detail. Full article
(This article belongs to the Special Issue Dynamics and Control of Aerospace Systems)
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<p>Variation with <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo>[</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1.25</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math> of the auxiliary function <span class="html-italic">F</span> defined in Equation (<a href="#FD2-actuators-13-00384" class="html-disp-formula">2</a>).</p>
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<p>Variation with <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo>[</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1.25</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math> of the maximum magnitude of the HPS-induced thrust vector <math display="inline"><semantics> <mi mathvariant="bold-italic">T</mi> </semantics></math>.</p>
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<p>Thrust bubble (when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) as a function of the Sun-CubeSat distance <span class="html-italic">r</span>. The ticks in the color bar are in millinewtons. (<b>a</b>) Case of <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.75</mn> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math>; (<b>b</b>) Case of <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math>; (<b>c</b>) Case of <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1.25</mn> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math>.</p>
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<p>Variation with <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo>[</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1.25</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math> of propellant mass flow rate when the throttle function is <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; see the right side of Equation (<a href="#FD9-actuators-13-00384" class="html-disp-formula">9</a>).</p>
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<p>Ecliptic projection and isometric view of the rapid transfer trajectory towards asteroid 99942 Apophis when the initial CubeSat mass is <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>30</mn> <mspace width="0.166667em"/> <mi>kg</mi> </mrow> </semantics></math>. The <span class="html-italic">z</span>-axis of the isometric view is exaggerated to highlight the three-dimensionality of the trajectory. Black line → CubeSat transfer trajectory; blue line → Earth’s orbit; red line → asteroid’s orbit; filled star → perihelion; blue dot → starting point; red square → arrival point; orange dot → the Sun.</p>
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<p>Time variation in the thrust angles <math display="inline"><semantics> <mi>α</mi> </semantics></math> and <math display="inline"><semantics> <mi>δ</mi> </semantics></math> along the rapid transfer trajectory towards asteroid 99942 Apophis when the initial CubeSat mass is <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>30</mn> <mspace width="0.166667em"/> <mi>kg</mi> </mrow> </semantics></math>. Blue dot → starting point; red square → arrival point.</p>
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<p>Time variation in the mass <span class="html-italic">m</span> and solar distance <span class="html-italic">r</span> along the rapid transfer trajectory towards asteroid 99942 Apophis when the initial CubeSat mass is <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>30</mn> <mspace width="0.166667em"/> <mi>kg</mi> </mrow> </semantics></math>. Blue dot → starting point; red square → arrival point.</p>
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<p>Results of the parametric study of the Earth–Apophis, orbit-to-orbit, rapid transfer as a function of the value of the initial CubeSat mass <math display="inline"><semantics> <msub> <mi>m</mi> <mn>0</mn> </msub> </semantics></math> (step of <math display="inline"><semantics> <mrow> <mn>1</mn> <mspace width="0.166667em"/> <mi>kg</mi> </mrow> </semantics></math>, green dots).</p>
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<p>Artistic representation of the ESA’s M-ARGO CubeSat approach to a potential near-Earth asteroid. Image: © ESA.</p>
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<p>Miniaturized electric thruster: variation of <math display="inline"><semantics> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>(</mo> <mrow> <mo>∥</mo> <msub> <mi mathvariant="bold-italic">T</mi> <mi>e</mi> </msub> <mo>∥</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>I</mi> <mrow> <mi>sp</mi> </mrow> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo>[</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1.25</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math>.</p>
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<p>Artistic concept of the NASA’s Near-Earth Asteroid Scout (NEA Scout) approaching the target asteroid. The solar sail-based CubeSat failed to make contact with ground station after launch, and the mission NEA Scout was considered lost in December 2022. Image credit: NASA.</p>
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<p>Photonic solar sail in a Sun-facing configuration: variation of the thrust magnitude <math display="inline"><semantics> <mrow> <mo>∥</mo> <msub> <mi mathvariant="bold-italic">T</mi> <mi>s</mi> </msub> <mo>∥</mo> </mrow> </semantics></math> with the solar distance <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo>[</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1.25</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>AU</mi> </mrow> </semantics></math>, according to Equation (<a href="#FD18-actuators-13-00384" class="html-disp-formula">A9</a>). The solar sail characteristics (in terms of sail area and sail force coefficients) are consistent with the system installed onboard of the NASA’s NEA Scout CubeSat.</p>
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16 pages, 1450 KiB  
Article
Venus Magnetotail Long-Term Sensing Using Solar Sails
by Alessandro A. Quarta
Appl. Sci. 2024, 14(17), 8016; https://doi.org/10.3390/app14178016 - 7 Sep 2024
Viewed by 494
Abstract
Propellantless propulsion systems, such as the well-known photonic solar sails that provide thrust by exploiting the solar radiation pressure, theoretically allow for extremely complex space missions that require a high value of velocity variation to be carried out. Such challenging space missions typically [...] Read more.
Propellantless propulsion systems, such as the well-known photonic solar sails that provide thrust by exploiting the solar radiation pressure, theoretically allow for extremely complex space missions that require a high value of velocity variation to be carried out. Such challenging space missions typically need the application of continuous thrust for a very long period of time, compared to the classic operational life of a space vehicle equipped with a more conventional propulsion system as, for example, an electric thruster. In this context, an interesting application of this propellantless thruster consists of using the solar sail-induced acceleration to artificially precess the apse line of a planetocentric elliptic orbit. This specific mission application was thoroughly investigated about twenty years ago in the context of the GeoSail Technology Reference Study, which analyzed the potential use of a spacecraft equipped with a small solar sail to perform an in situ study of the Earth’s upper magnetosphere. Taking inspiration from the GeoSail concept, this study analyzes the performance of a solar sail-based spacecraft in (artificially) precessing the apse line of a high elliptic orbit around Venus with the aim of exploring the planet’s induced magnetotail. In particular, during flight, the solar sail orientation is assumed to be Sun-facing, and the required thruster’s performance is evaluated as a function of the elliptic orbit’s characteristics by using both a simplified mathematical model of the spacecraft’s planetocentric dynamics and an approximate analytical approach. Numerical results show that a medium–low-performance sail is able to artificially precess the apse line of a Venus-centered orbit in order to ensure the long-term sensing of the planet’s induced magnetotail. Full article
(This article belongs to the Section Aerospace Science and Engineering)
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<p>Artistic impression of the magnetosphere of Venus (top part of the figure), Earth (middle part), and Mars (bottom part). The topology of the induced magnetosphere of Venus and Mars is substantially different from that of the Earth, whose internal magnetic field interacts with the solar wind charged particles. Image: European Space Agency (ESA).</p>
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<p>Artistic impression of the ESA’s Venus Express orbiting around the second planet of the Solar System. Using the installed onboard magnetometer and low-energy particle detector, the spacecraft observed Venus’s magnetotail on 15 May 2006 at a distance of about 1.5 planet’s radii downstream of Venus. Image: ESA—D. Ducros.</p>
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<p>Conceptual scheme of a GeoSail-type mission scenario applied to a Venus-centered case. The solar sail-induced thrust rotates the apse line of the spacecraft (elliptic) science orbit in order to maintain the apocytherion inside Venus’s induced magnetotail. The spacecraft and Venus move around the Sun along the same plane <math display="inline"><semantics> <mi mathvariant="script">P</mi> </semantics></math>. The artificial precession of the apse line can be maintained, theoretically, for a very long period of time.</p>
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<p>Scheme of the solar sail-based spacecraft’s science orbit around Venus, in which the initial direction of the planetocentric orbit apse line coincides with the Venus–Sun line. In particular, initially, the Sun belongs to the positive direction of the first axis of a classical perifocal reference frame. The Sun–Venus distance <math display="inline"><semantics> <msub> <mi>r</mi> <mrow> <mi>S</mi> <mi>V</mi> </mrow> </msub> </semantics></math> is a constant of motion, and its value is high enough to assume that the Sun rays arrive parallel to the solar sail-based spacecraft.</p>
Full article ">Figure 5
<p>Sketch of a Sun-facing solar sail in a Venus-centered mission scenario in which the direction of the incoming Sun rays is coincident with the elliptic orbit apse line. Note that the Sun belongs to the apse line of the science orbit at a distance approximately equal to <math display="inline"><semantics> <msub> <mi>r</mi> <mrow> <mi>S</mi> <mi>V</mi> </mrow> </msub> </semantics></math>, which is considered a constant of motion.</p>
Full article ">Figure 6
<p>Variation of the dimensionless components of the solar sail-induced propulsive acceleration vector with the spacecraft true anomaly <math display="inline"><semantics> <mi>ν</mi> </semantics></math>, according to Equations (<a href="#FD14-applsci-14-08016" class="html-disp-formula">14</a>) and (<a href="#FD15-applsci-14-08016" class="html-disp-formula">15</a>).</p>
Full article ">Figure 7
<p>Variation of <math display="inline"><semantics> <mi>ω</mi> </semantics></math> with <math display="inline"><semantics> <mi>ν</mi> </semantics></math> obtained by the numerical integration of the Lagrange planetary equations (black line), when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1.3</mn> <mspace width="0.166667em"/> <msub> <mi>R</mi> <mi>V</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>6</mn> <mspace width="0.166667em"/> <msub> <mi>R</mi> <mi>V</mi> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>≃</mo> <mn>0.363</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The red dashed line indicates the rotation of the Venus–Sun line during the spacecraft’s revolution around the planet. Note how the black and red dashed lines overlap when <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mspace width="0.166667em"/> <mi>rad</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Variation of the dimensionless semimajor axis and the eccentricity with <math display="inline"><semantics> <mi>ν</mi> </semantics></math>, as obtained by the numerical integration of Lagrange planetary equations (in Gaussian form), when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1.3</mn> <mspace width="0.166667em"/> <msub> <mi>R</mi> <mi>V</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>6</mn> <mspace width="0.166667em"/> <msub> <mi>R</mi> <mi>V</mi> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>≃</mo> <mn>0.363</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Parametric study of the required solar sail characteristic acceleration <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> as a function of the pericytherion <math display="inline"><semantics> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>p</mi> </msub> </semantics></math> and apocytherion <math display="inline"><semantics> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>a</mi> </msub> </semantics></math> dimensionless radii: (<b>a</b>) surface plot of <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>p</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>a</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) contour plot of <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>p</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>a</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>Dimensionless apocytherion radius <math display="inline"><semantics> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>a</mi> </msub> </semantics></math> as a function of the dimensionless pericytherion radius <math display="inline"><semantics> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>p</mi> </msub> </semantics></math> when the solar sail required characteristic acceleration is <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.15</mn> <mo>,</mo> <mn>0.2</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>Schematic concept of the (simplified) cylindrical shadow model, which was used to determine the solar sail’s required performance in the presence of a period of eclipse.</p>
Full article ">Figure 12
<p>The required solar sail’s characteristic acceleration <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> as a function of the pericytherion <math display="inline"><semantics> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>p</mi> </msub> </semantics></math> and apocytherion <math display="inline"><semantics> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>a</mi> </msub> </semantics></math> dimensionless radii in the presence of a period of eclipse: (<b>a</b>) contour plot of <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>p</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mi>a</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) case of <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.15</mn> <mo>,</mo> <mn>0.2</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 13
<p>Function <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mi>ω</mi> <mo>(</mo> <mi>ν</mi> <mo>)</mo> </mrow> </semantics></math>, in the presence of a period of eclipse, when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1.3</mn> <mspace width="0.166667em"/> <msub> <mi>R</mi> <mi>V</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>6</mn> <mspace width="0.166667em"/> <msub> <mi>R</mi> <mi>V</mi> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>≃</mo> <mn>0.42</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The black line indicates the output of an orbit simulator, while the red dashed line indicates the rotation of the Venus–Sun line during the spacecraft revolution around the planet.</p>
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19 pages, 2692 KiB  
Article
Impact of Pitch Angle Limitation on E-Sail Interplanetary Transfers
by Alessandro A. Quarta
Aerospace 2024, 11(9), 729; https://doi.org/10.3390/aerospace11090729 - 6 Sep 2024
Viewed by 437
Abstract
The Electric Solar Wind Sail (E-sail) deflects charged particles from the solar wind through an artificial electric field to generate thrust in interplanetary space. The structure of a spacecraft equipped with a typical E-sail essentially consists in a number of long conducting tethers [...] Read more.
The Electric Solar Wind Sail (E-sail) deflects charged particles from the solar wind through an artificial electric field to generate thrust in interplanetary space. The structure of a spacecraft equipped with a typical E-sail essentially consists in a number of long conducting tethers deployed from a main central body, which contains the classical spacecraft subsystems. During flight, the reference plane that formally contains the conducting tethers, i.e., the sail nominal plane, is inclined with respect to the direction of propagation of the solar wind (approximately coinciding with the Sun–spacecraft direction in a preliminary trajectory analysis) in such a way as to vary both the direction and the module of the thrust vector provided by the propellantless propulsion system. The generation of a sail pitch angle different from zero (i.e., a non-zero angle between the Sun–spacecraft line and the direction perpendicular to the sail nominal plane) allows a transverse component of the thrust vector to be obtained. From the perspective of attitude control system design, a small value of the sail pitch angle could improve the effectiveness of the E-sail attitude maneuver at the expense, however, of a worsening of the orbital transfer performance. The aim of this paper is to investigate the effects of a constraint on the maximum value of the sail pitch angle, on the performance of a spacecraft equipped with an E-sail propulsion system in a typical interplanetary mission scenario. During flight, the E-sail propulsion system is considered to be always on so that the entire transfer can be considered a single propelled arc. A heliocentric orbit-to-orbit transfer without ephemeris constraints is analyzed, while the performance analysis is conducted in a parametric form as a function of both the maximum admissible sail pitch angle and the propulsion system’s characteristic acceleration value. Full article
(This article belongs to the Special Issue Advances in CubeSat Sails and Tethers (2nd Edition))
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Figure 1

Figure 1
<p>Conceptual scheme of a centrifugal deployment of an E-sail-propelled spacecraft, in which the main central body, the conducting tethers, and the remote units are sketched. The conducting tethers deployment takes place substantially in the so-called “sail nominal plane”, which is indicated with a shaded green disk in the right part of the figure. (<b>a</b>) Centrifugal deployment of the E-sail, where the spin direction is indicated by curved orange arrows; (<b>b</b>) final (fully deployed) E-sail configuration.</p>
Full article ">Figure 2
<p>Artistic impression of a CubeSat equipped with a spinning, single-tether E-sail, which can be used to obtain the first in situ measurement of the propulsive acceleration given by this fascinating propulsion system. In the artistic image, the CubeSat with a deployed (single) conducting tether covers a high-elliptic Lunar orbit which allows the vehicle to move outside Earth’s magnetosphere. Image courtesy of Mario F. Palos.</p>
Full article ">Figure 3
<p>Description of a Sun-facing configuration and a generic case in which the sail nominal plane is inclined with respect to the radial direction. (<b>a</b>) Sun-facing configuration in which the sail nominal plane is perpendicular to the Sun–spacecraft line; (<b>b</b>) case of a generic sail pitch angle different from zero, in which the E-sail-induced thrust vector has a non-zero transverse component.</p>
Full article ">Figure 4
<p>Radial-Transverse-Normal reference frame (RTN) of unit vector <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">R</mi> </msub> </semantics></math> (radial), <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">T</mi> </msub> </semantics></math> (transverse), and <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">N</mi> </msub> </semantics></math> (normal). The plane <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">R</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">T</mi> </msub> <mo>)</mo> </mrow> </semantics></math> coincides with the plane of the spacecraft osculating orbit, while the spacecraft inertial velocity vector has a positive component along the direction of <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="normal">T</mi> </msub> </semantics></math>.</p>
Full article ">Figure 5
<p>Sketch of the normal unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">n</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> and the propulsive acceleration vector <math display="inline"><semantics> <msub> <mi mathvariant="bold-italic">a</mi> <mi>p</mi> </msub> </semantics></math> in the RTN reference frame. The scheme introduces the sail pitch angle <math display="inline"><semantics> <msub> <mi>α</mi> <mi>n</mi> </msub> </semantics></math> and the sail clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math>, which are the two (scalar) E-sail’s control terms.</p>
Full article ">Figure 6
<p>Force bubble in the unconstrained case (<math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>90</mn> <mo>]</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>) when <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mo>⊕</mo> </msub> </mrow> </semantics></math>. The direction of propagation of the solar wind coincides with the direction of the <span class="html-italic">z</span>-axis, that is, the Sun belongs to the vertical axis.</p>
Full article ">Figure 7
<p>Force bubble in the constrained case when <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mo>⊕</mo> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <msub> <mi>n</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>30</mn> <mo>,</mo> <mn>45</mn> <mo>,</mo> <mn>60</mn> <mo>,</mo> <mn>75</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Time variation in the control angles <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> in an unconstrained Earth–Mars transfer with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. Blue dot → starting point; red square → arrival point.</p>
Full article ">Figure 9
<p>Time variation in the control angles <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> in an unconstrained Earth–Venus transfer with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. Blue dot → starting point; red square → arrival point.</p>
Full article ">Figure 10
<p>Time variation in the control angles <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> in an unconstrained Earth–Mercury transfer with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. Blue dot → starting point; red square → arrival point.</p>
Full article ">Figure 11
<p>Ecliptic projection and isometric view of the unconstrained optimal transfer trajectory in an Earth–Mars scenario with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. Black line → optimal transfer trajectory; blue line → Earth’s orbit; red line → target planet’s orbit; filled star → planet’s perihelion; blue dot → starting point; red square → arrival point; orange dot → the Sun.</p>
Full article ">Figure 12
<p>Ecliptic projection and isometric view of the unconstrained optimal transfer trajectory in an Earth–Venus scenario with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. See the caption of <a href="#aerospace-11-00729-f011" class="html-fig">Figure 11</a> for the legend.</p>
Full article ">Figure 13
<p>Ecliptic projection and isometric view of the unconstrained optimal transfer trajectory in an Earth–Mercury scenario with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. See the caption of <a href="#aerospace-11-00729-f011" class="html-fig">Figure 11</a> for the legend.</p>
Full article ">Figure 14
<p>Simulation results of the constrained case when <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup>. The dimensionless term <span class="html-italic">D</span> is defined in Equation (<a href="#FD11-aerospace-11-00729" class="html-disp-formula">11</a>). (<b>a</b>) Earth–Mars scenario; (<b>b</b>) Earth–Venus scenario; (<b>c</b>) Earth–Mercury scenario.</p>
Full article ">Figure 15
<p>Time variation in the control angles <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> in a constrained Earth–Mars transfer with <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> = 1 mm/s<sup>2</sup> and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <msub> <mi>n</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </msub> <mo>=</mo> <mn>30</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>. Blue dot → starting point; red square → arrival point.</p>
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<p>Numerical results, in terms of minimum flight time <math display="inline"><semantics> <msub> <mi>t</mi> <mi>f</mi> </msub> </semantics></math>, of the trajectory optimization in an Earth–Mars scenario, where <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <msub> <mi>n</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>30</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>90</mn> <mo>]</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1</mn> <mo>]</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>.</p>
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<p>Curve levels of the dimensionless parameter <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mi>D</mi> <mo>(</mo> <msub> <mi>α</mi> <msub> <mi>n</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> </semantics></math> in an Earth–Mars scenario, where <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <msub> <mi>n</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>30</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>90</mn> <mo>]</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1</mn> <mo>]</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>.</p>
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20 pages, 6636 KiB  
Article
Three-Dimensional Guidance Laws for Spacecraft Propelled by a SWIFT Propulsion System
by Alessandro A. Quarta
Appl. Sci. 2024, 14(13), 5944; https://doi.org/10.3390/app14135944 - 8 Jul 2024
Cited by 1 | Viewed by 646
Abstract
This paper discusses the optimal control law, in a three-dimensional (3D) heliocentric orbit transfer, of a spacecraft whose primary propulsion system is a Solar Wind Ion Focusing Thruster (SWIFT). A SWIFT is an interesting concept of a propellantless thruster, proposed ten years ago [...] Read more.
This paper discusses the optimal control law, in a three-dimensional (3D) heliocentric orbit transfer, of a spacecraft whose primary propulsion system is a Solar Wind Ion Focusing Thruster (SWIFT). A SWIFT is an interesting concept of a propellantless thruster, proposed ten years ago by Gemmer and Mazzoleni, which deflects, collects, and accelerates the charged particles of solar wind to generate thrust in the interplanetary space. To this end, the SWIFT uses a large conical structure made of thin metallic wires, which is positively charged with the aid of an electron gun. In this sense, a SWIFT can be considered as a sort of evolution of the Janhunen’s E-Sail, which also uses a (nominally flat) mesh of electrically charged tethers to deflect the solar wind stream. In the recent literature, the optimal performance of a SWIFT-based vehicle has been studied by assuming a coplanar orbit transfer and a two-dimensional scenario. The mathematical model proposed in this paper extends that result by discussing the optimal guidance laws in the general context of a 3D heliocentric transfer. In this regard, a number of different forms of the spacecraft state vectors are considered. The validity of the obtained optimal control law is tested in a simplified Earth–Venus and Earth–Mars transfer by comparing the simulation results with the literature data in terms of minimum flight time. Full article
(This article belongs to the Special Issue Advances in Deep Space Probe Navigation)
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Figure 1

Figure 1
<p>Artistic concept of a spacecraft equipped with a SWIFT. The axis of the conical structure coincides with the Sun–vehicle line during the flight.</p>
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<p>Sketch of the RTN reference frame of unit vectors <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo>^</mo> </mover> <mi mathvariant="normal">R</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo>^</mo> </mover> <mi mathvariant="normal">T</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="bold-italic">i</mi> <mo>^</mo> </mover> <mi mathvariant="normal">N</mi> </msub> </semantics></math>.</p>
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<p>Unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">t</mi> <mo>^</mo> </mover> </semantics></math> in the RTN reference frame, definition of the SWIFT cone <math display="inline"><semantics> <mi>α</mi> </semantics></math>, and clock <math display="inline"><semantics> <mi>δ</mi> </semantics></math> angle. The scheme also shows the aperture angle <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> of the conical mesh of conducting wires, which constitutes the large part of the propulsion system structure.</p>
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<p>Generic assigned direction in the RTN reference frame. That assigned direction is indicated by the unit vector <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">d</mi> <mo>^</mo> </mover> </semantics></math>, which is given by the two angles <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>α</mi> <mi>d</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mi>d</mi> </msub> <mo>}</mo> </mrow> </semantics></math>.</p>
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<p>Configuration of the unit vector <math display="inline"><semantics> <msup> <mover accent="true"> <mi mathvariant="bold-italic">t</mi> <mo>^</mo> </mover> <mo>★</mo> </msup> </semantics></math> that maximizes the scalar index <span class="html-italic">P</span> defined in Equation (<a href="#FD10-applsci-14-05944" class="html-disp-formula">10</a>); see also Equation (<a href="#FD14-applsci-14-05944" class="html-disp-formula">14</a>). The red cone indicates the forbidden zone related to the presence of the SWIFT conical mesh of conducting wires. (<b>a</b>) Case of <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>d</mi> </msub> <mo>≤</mo> <msub> <mi>α</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </mrow> </semantics></math>. (<b>b</b>) Case of <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>d</mi> </msub> <mo>&gt;</mo> <msub> <mi>α</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </mrow> </semantics></math>.</p>
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<p>SWIFT-propelled spacecraft in a Cartesian, heliocentric ecliptic, reference frame <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">T</mi> <mi>C</mi> </msub> <mrow> <mo>(</mo> <mi>O</mi> <mo>;</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>SWIFT-propelled spacecraft in a heliocentric spherical reference frame <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">T</mi> <mi>S</mi> </msub> <mrow> <mo>(</mo> <mi>O</mi> <mo>;</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>γ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>Results of the optimization process in a 3D Earth–Venus mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.07</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>a</b>) Time variation of the components of <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">t</mi> <mo>^</mo> </mover> </semantics></math> in the RTN reference frame. (<b>b</b>) Optimal transfer trajectory (black line). Blue line → is Earth’s orbit; red line → is Venus’ orbit; blue circle → is start point; red circle → is arrival point; blue star → is Earth’s perihelion; red star → is Venus’ perihelion; orange circle → is the Sun.</p>
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<p>Results of the optimization process in a 3D Earth–Mars mission scenario when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.07</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>a</b>) Time variation of the components of <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">t</mi> <mo>^</mo> </mover> </semantics></math> in RTN. (<b>b</b>) Optimal transfer trajectory (black line). Blue line → is Earth’s orbit; red line → is Mars’ orbit; blue circle → is start point; red circle → is arrival point; blue star → is Earth’s perihelion; red star → is Mars’ perihelion; orange circle → is the Sun.</p>
Full article ">Figure A1
<p>Sketch of the polar reference frame and the sail pitch angle <math display="inline"><semantics> <msub> <mi>α</mi> <mi>n</mi> </msub> </semantics></math> in a 2D heliocentric mission scenario of a SWIFT-propelled spacecraft, such as that used in Ref. [<a href="#B9-applsci-14-05944" class="html-bibr">9</a>].</p>
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14 pages, 5500 KiB  
Article
Mechanical Property Analysis of a Boom–Membrane Structure Used for Aerospace Technologies
by Shuhong Xu, Xiaojiao Yu, Yue Gao, Sicong Wang and Lining Sun
Materials 2024, 17(13), 3204; https://doi.org/10.3390/ma17133204 - 1 Jul 2024
Viewed by 794
Abstract
Traditional deployable truss space structures previously had upper limits on their key indicators, such as the deployed area, folded ratio and total weight, and hence, the application of new extendable mechanisms with novel deployment types is desired. Foldable extendable tape spring booms made [...] Read more.
Traditional deployable truss space structures previously had upper limits on their key indicators, such as the deployed area, folded ratio and total weight, and hence, the application of new extendable mechanisms with novel deployment types is desired. Foldable extendable tape spring booms made from FRP (fiber-reinforced polymer) laminate composites and their corresponding boom–membrane structures were invented in recent years to satisfy the needs of the large-scale requirements of spacecraft, especially for antennas, solar sails and solar arrays. This paper aimed to analyze the properties of the deployed states of extendable tape spring booms and their boom–membrane structures. By establishing an analytical model of the boom and the structure, the bending stiffness, critical buckling load of the boom and the fundamental frequency of the membrane structure were acquired. To provide more guidance on the boom–membrane structure design, a geometric and material parametric study was carried out. Meanwhile, an experimental study to investigate the deployed properties of the booms and membrane structures was introduced to afford some practical verification. Full article
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Figure 1
<p>Deployable tape spring boom diagram.</p>
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<p>Diagrams of the geometric and laminate parameters of a tape spring boom structure. (<b>a</b>) Fiber braided direction (<span class="html-italic">x<sub>m</sub></span> and <span class="html-italic">y<sub>m</sub></span> represent the coordinate system of the boom laminates in which <span class="html-italic">x<sub>m</sub></span> points at the boom’s deployment direction). (<b>b</b>) Laminate material parameters (the symbols in this figure were the same as those commonly used in the Classical Laminate Theory, which can be found in Ref. [<a href="#B3-materials-17-03204" class="html-bibr">3</a>]).</p>
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<p>Stress diagrams of laminate composites in deformation (coordinate <span class="html-italic">x<sub>m</sub>Oy<sub>m</sub></span> is the same with that listed in <a href="#materials-17-03204-f002" class="html-fig">Figure 2</a>a).</p>
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<p>Diagram of boom transversal cross-section configuration.</p>
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<p>Boom geometric parametric study.</p>
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<p>Boom laminate parametric study.</p>
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<p>Boom bending stiffness experiment.</p>
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<p>Diagram of boom bending stiffness conversion.</p>
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<p>Membrane structure prototype and fundamental frequency experimental facility.</p>
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16 pages, 7253 KiB  
Article
Trajectory Approximation of a Low-Performance E-Sail with Fixed Orientation
by Alessandro A. Quarta and Giovanni Mengali
Aerospace 2024, 11(7), 532; https://doi.org/10.3390/aerospace11070532 - 28 Jun 2024
Viewed by 549
Abstract
The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system that converts solar wind dynamic pressure into a deep-space thrust through a grid of long conducting tethers. The first flight test, needed to experience the true potential of the E-sail concept, is [...] Read more.
The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system that converts solar wind dynamic pressure into a deep-space thrust through a grid of long conducting tethers. The first flight test, needed to experience the true potential of the E-sail concept, is likely to be carried out using a single spinning cable deployed from a small satellite, such as a CubeSat. This specific configuration poses severe limitations to both the performance and the maneuverability of the spacecraft used to analyze the actual in situ thruster capabilities. In fact, the direction of the spin axis in a single-tether configuration can be considered fixed in an inertial reference frame, so that the classic sail pitch angle is no longer a control variable during the interplanetary flight. This paper aims to determine the polar form of the propelled trajectory and the characteristics of the osculating orbit of a spacecraft propelled by a low-performance spinning E-sail with an inertially fixed axis of rotation. Assuming that the spacecraft starts the trajectory from a parking orbit that coincides with the Earth’s heliocentric orbit and that its spin axis belongs to the plane of the ecliptic, a procedure is illustrated to solve the problem accurately with a set of simple analytical relations. Full article
(This article belongs to the Special Issue Deep Space Exploration)
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Figure 1

Figure 1
<p>Geometric configuration of an E-sail-based spacecraft: original concept proposed by Janhunen and current arrangement based on a series of radial conducting tethers stretched by centrifugal force. (<b>a</b>) Original (grid-shaped) E-sail configuration. (<b>b</b>) Current E-sail configuration.</p>
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<p>Student CubeSat satellites ESTCube-1 and Aalto-1, which attempted the first in situ test of E-sail propulsive technology: (<b>a</b>) Estonian ESTCube-1 satellite (1U CubeSat). (<b>b</b>) Finnish Aalto-1 satellite (3U CubeSat).</p>
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<p>Conceptual scheme of a single-tether E-sail deployed by a spinning 3U CubeSat. The sail nominal plane is perpendicular to the spin axis.</p>
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<p>Sketch of the orientation of the sail nominal plane relative to the ecliptic.</p>
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<p>Sketch of the polar reference frame <math display="inline"><semantics> <mrow> <mi mathvariant="script">T</mi> <mo>(</mo> <mi>O</mi> <mo>;</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> with unit vectors <math display="inline"><semantics> <mrow> <mo>{</mo> <mover accent="true"> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">^</mo> </mover> <mo>,</mo> <mover accent="true"> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> <mo>}</mo> </mrow> </semantics></math> and spacecraft state variables in the heliocentric transfer.</p>
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<p>Comparison between the <math display="inline"><semantics> <mi>θ</mi> </semantics></math>-variation of the dimensionless angular momentum vector, obtained from Equation (<a href="#FD13-aerospace-11-00532" class="html-disp-formula">13</a>) (red dashed line), and the numerical integration (black solid line) of the spacecraft equations of motion when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.05</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>90</mn> <mo>,</mo> <mn>45</mn> <mo>,</mo> <mn>0</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
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<p>Comparison between the <math display="inline"><semantics> <mi>θ</mi> </semantics></math>-variation of the radial distance <span class="html-italic">r</span>, obtained from the analytical approximation (red dashed line) of Equation (<a href="#FD14-aerospace-11-00532" class="html-disp-formula">14</a>), and the numerical integration (black solid line) of the spacecraft equations of motion when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.05</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>90</mn> <mo>,</mo> <mn>45</mn> <mo>,</mo> <mn>0</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
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<p>Comparison between the <math display="inline"><semantics> <mi>θ</mi> </semantics></math>-variation of the flight time, obtained with the analytical approximation (red dashed line) of Equation (<a href="#FD20-aerospace-11-00532" class="html-disp-formula">20</a>), and the numerical integration (black solid line) of the spacecraft equations of motion when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.05</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>90</mn> <mo>,</mo> <mn>45</mn> <mo>,</mo> <mn>0</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
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<p>Scheme of the mission scenario in which the E-sail is deployed edgewise to the Sun (<math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>90</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>).</p>
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<p>Variation of the spacecraft state variables <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>r</mi> <mo>,</mo> <msub> <mi>v</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>v</mi> <mi>θ</mi> </msub> <mo>}</mo> </mrow> </semantics></math> with the polar angle <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>10</mn> <mi>π</mi> <mo>]</mo> <mspace width="0.166667em"/> <mi>rad</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> (i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mi>c</mi> </msub> <mo>≃</mo> <mn>0.0169</mn> </mrow> </semantics></math>) and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>90</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>. Black solid line → numerical simulations; red dashed line → semi-analytical approximation.</p>
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<p>Variation of the osculating orbit semimajor axis <span class="html-italic">a</span> and eccentricity <span class="html-italic">e</span> with the polar angle <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>10</mn> <mi>π</mi> <mo>]</mo> <mspace width="0.166667em"/> <mi>rad</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> (i.e., <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mi>c</mi> </msub> <mo>≃</mo> <mn>0.0169</mn> </mrow> </semantics></math>) and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>90</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>. Black solid line → numerical simulations; red dashed line → semi-analytical approximation.</p>
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20 pages, 4106 KiB  
Article
Thermally Induced Vibration of a Flexible Plate with Enhanced Active Constrained Layer Damping
by Yueru Guo, Yongbin Guo, Yongxin Zhang, Liang Li, Dingguo Zhang, Sijia Chen and Mohamed A. Eltaher
Aerospace 2024, 11(7), 504; https://doi.org/10.3390/aerospace11070504 - 23 Jun 2024
Cited by 1 | Viewed by 698
Abstract
When spacecraft execute missions in space, their solar panels—crucial components—often need to be folded, unfolded, and adjusted at an angle. These operations can induce numerous detrimental nonlinear vibrations. This paper addresses the issues of nonlinear and thermal-coupled vibration control within the context of [...] Read more.
When spacecraft execute missions in space, their solar panels—crucial components—often need to be folded, unfolded, and adjusted at an angle. These operations can induce numerous detrimental nonlinear vibrations. This paper addresses the issues of nonlinear and thermal-coupled vibration control within the context of space-based flexible solar panel systems. Utilizing piezoelectric smart hybrid vibration control technology, this study focuses on a flexible plate augmented with an active constrained layer damping. The solar panel, under thermal field conditions, is modeled and simulated using the commercial finite element simulation software ABAQUS. The research examines variations in the modal frequencies and damping properties of the model in response to changes in the coverage location of the piezoelectric patches, their coverage rate, rotational angular velocity, and the thickness of the damping layer. Simulation results indicate that structural damping is more effective when the patches are closer to the rotation axis, the coverage area of the patches is larger, the rotational speed is lower, and the damping layer is thicker. Additionally, the effectiveness of vibration suppression is influenced by the interplay between the material shear modulus, loss factor, and specific working temperature ranges. The selection of appropriate parameters can significantly alter the system’s vibrational characteristics. This work provides necessary technical references for the analysis of thermally induced vibrations in flexible solar sails under complex space conditions. Full article
(This article belongs to the Special Issue Advanced Aerospace Composite Materials and Smart Structures)
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Figure 1
<p>The solar panel of a SpaceX spacecraft.</p>
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<p>Schematic of the EACLD model.</p>
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<p>Detailed mesh grid of EACLD model.</p>
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<p>The variation of shear modulus and loss factor of viscoelastic layer with temperature. (<b>a</b>) The variation of shear modulus of Dyad606 with temperature. (<b>b</b>) The variation of loss factor of Dyad606 with temperature.</p>
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<p>The variation of EACLD patch covering position.</p>
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<p>Variations of the first four orders of natural frequencies of the plate with different cover positions of EACLD patch with temperature. (<b>a</b>) First order modal frequency. (<b>b</b>) Second order modal frequency. (<b>c</b>) Third order modal frequency. (<b>d</b>) Fourth order modal frequency.</p>
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<p>Variations of the first four orders of damping ratios of the plate with different cover positions of EACLD patch with temperature. (<b>a</b>) First order modal damping ratio. (<b>b</b>) Second order modal damping ratio. (<b>c</b>) Third order modal damping ratio. (<b>d</b>) Fourth order modal damping ratio.</p>
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<p>Variations of the first four orders of damping ratios of the plate with different cover positions of EACLD patch with temperature. (<b>a</b>) First order modal damping ratio. (<b>b</b>) Second order modal damping ratio. (<b>c</b>) Third order modal damping ratio. (<b>d</b>) Fourth order modal damping ratio.</p>
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<p>Schematic drawing of the coverage of EACLD patch.</p>
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<p>Variations of the first four orders of natural frequencies of the plate with different coverages of the EACLD patch treatment with temperature. (<b>a</b>) First order modal frequency. (<b>b</b>) Second order modal frequency. (<b>c</b>) Third order modal frequency. (<b>d</b>) Fourth order modal frequency.</p>
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<p>Variations of the first four orders of damping ratios of the plate with different coverages of the EACLD patch treatment with temperature. (<b>a</b>) First order modal damping ratio. (<b>b</b>) Second order modal damping ratio. (<b>c</b>) Third order modal damping ratio. (<b>d</b>) Fourth order modal damping ratio.</p>
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<p>Variations of the first four orders of natural frequencies of the plate with EACLD treatment with temperature at different rotational speeds. (<b>a</b>) First order modal frequency. (<b>b</b>) Second order modal frequency. (<b>c</b>) Third order modal frequency. (<b>d</b>) Fourth order modal frequency.</p>
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<p>Variations of the first four orders of damping ratios of the plate with EACLD treatment with temperature at different rotational speeds. (<b>a</b>) First order modal damping ratio. (<b>b</b>) Second order modal damping ratio. (<b>c</b>) Third order modal damping ratio. (<b>d</b>) Fourth order modal damping ratio.</p>
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<p>Variations of the first four orders of natural frequencies of the plate with EACLD treatment with temperature under different thicknesses of damping layer. (<b>a</b>) First order modal frequency. (<b>b</b>) Second order modal frequency. (<b>c</b>) Third order modal frequency. (<b>d</b>) Fourth order modal frequency.</p>
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<p>Variations of the first four orders of damping ratios of the plate with EACLD treatment with temperature under different thicknesses of damping layer. (<b>a</b>) First order modal damping ratio. (<b>b</b>) Second order modal damping ratio. (<b>c</b>) Third order modal damping ratio. (<b>d</b>) Fourth order modal damping ratio.</p>
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19 pages, 1217 KiB  
Article
Optimal Guidance for Heliocentric Orbit Cranking with E-Sail-Propelled Spacecraft
by Alessandro A. Quarta
Aerospace 2024, 11(6), 490; https://doi.org/10.3390/aerospace11060490 - 19 Jun 2024
Cited by 2 | Viewed by 892
Abstract
In astrodynamics, orbit cranking is usually referred to as an interplanetary transfer strategy that exploits multiple gravity-assist maneuvers to change both the inclination and eccentricity of the spacecraft osculating orbit without changing the specific mechanical energy, that is, the semimajor axis. In the [...] Read more.
In astrodynamics, orbit cranking is usually referred to as an interplanetary transfer strategy that exploits multiple gravity-assist maneuvers to change both the inclination and eccentricity of the spacecraft osculating orbit without changing the specific mechanical energy, that is, the semimajor axis. In the context of a solar sail-based mission, however, the concept of orbit cranking is typically referred to as a suitable guidance law that is able to (optimally) change the orbital inclination of a circular orbit of an assigned radius in a general heliocentric three-dimensional scenario. In fact, varying the orbital inclination is a challenging maneuver from the point of view of the velocity change, so orbit cranking is an interesting mission application for a propellantless propulsion system. The aim of this paper is to analyze the performance of a spacecraft equipped with an Electric Solar Wind Sail in a cranking maneuver of a heliocentric circular orbit. The maneuver performance is calculated in an optimal framework considering spacecraft dynamics described by modified equinoctial orbital elements. In this context, the paper presents an analytical version of the three-dimensional optimal guidance laws obtained by using the classical Pontryagin’s maximum principle. The set of (analytical) optimal control laws is a new contribution to the Electric Solar Wind Sail-related literature. Full article
(This article belongs to the Special Issue Advances in CubeSat Sails and Tethers (2nd Edition))
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Figure 1
<p>The radial–transverse–normal (RTN) reference frame and E-sail cone (<math display="inline"><semantics> <mi>α</mi> </semantics></math>) and clock (<math display="inline"><semantics> <mi>δ</mi> </semantics></math>) angles. The red surface indicates the plane of the osculating orbit, while the green surface indicates the local vertical plane.</p>
Full article ">Figure 2
<p>The minimum flight time <math display="inline"><semantics> <msub> <mi>t</mi> <mi>f</mi> </msub> </semantics></math>, in terms of multiples of the circular orbit period <math display="inline"><semantics> <msub> <mi>T</mi> <mn>0</mn> </msub> </semantics></math>, as a function of the change in orbital inclination <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> </mrow> </semantics></math> in an orbit cranking maneuver when the dimensionless (reference) propulsive acceleration magnitude is <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>The final value of the spacecraft true longitude <math display="inline"><semantics> <msub> <mi>L</mi> <mi>f</mi> </msub> </semantics></math> as a function of the change in orbital inclination <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> </mrow> </semantics></math> in an orbit cranking maneuver when the dimensionless (reference) propulsive acceleration magnitude is <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>The dimensionless value of the velocity change <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>V</mi> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> </mrow> </semantics></math> in circular orbit cranking obtained with a single impulsive maneuver.</p>
Full article ">Figure 5
<p>The optimal transfer trajectory (black line) in a heliocentric ecliptic Cartesian reference frame when <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Blue line → circular parking orbit; red line → circular final orbit; blue circle → start point; red square → arrival point; yellow circle → the Sun.</p>
Full article ">Figure 6
<p>Time variation in the osculating orbit characteristics <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>i</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>e</mi> <mo>}</mo> </mrow> </semantics></math> and the magnitude of the spacecraft position (<span class="html-italic">r</span>) and velocity (<span class="html-italic">v</span>) vector when <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Blue circle → start point; red square → arrival point.</p>
Full article ">Figure 7
<p>Time variation in the two thrust vector angles <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> and the on/off parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Blue circle → start point; red square → arrival point.</p>
Full article ">Figure 8
<p>Time variation in the two thrust vector angles <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> and the on/off parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> when the change in orbital inclination is equal to <math display="inline"><semantics> <mrow> <mn>10</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and the dimensionless performance term is <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Blue circle → start point; red square → arrival point.</p>
Full article ">Figure 9
<p>Time variation in the spacecraft osculating orbit characteristics when the change in orbital inclination is equal to <math display="inline"><semantics> <mrow> <mn>10</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Blue circle → start point; red square → arrival point.</p>
Full article ">Figure 10
<p>Time variation in the spacecraft osculating orbit characteristics in a near-optimal “patched” orbit cranking with <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msup> <mi>i</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mn>10</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Blue circle → start point; red square → arrival point.</p>
Full article ">Figure A1
<p>Time variation in the osculating orbit characteristics when the change in orbital inclination is equal to <math display="inline"><semantics> <mrow> <mn>90</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. Blue circle → start point; red square → arrival point.</p>
Full article ">Figure A2
<p>The transfer trajectory (black line) when <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math> as a function of the required value of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>i</mi> </mrow> </semantics></math>. Blue line → circular parking orbit; red line → circular final orbit; blue circle → start point; red square → arrival point; yellow circle → the Sun.</p>
Full article ">
18 pages, 1929 KiB  
Article
Solar Sail-Based Mars-Synchronous Displaced Orbits for Remote Sensing Applications
by Marco Bassetto and Alessandro A. Quarta
Appl. Sci. 2024, 14(12), 5001; https://doi.org/10.3390/app14125001 - 7 Jun 2024
Viewed by 905
Abstract
A solar sail is a propellantless propulsion system that allows a spacecraft to use solar radiation pressure as a propulsive source for planetary and deep space missions that would be difficult, or even unfeasible, to accomplish with more conventional thrusters, either chemical or [...] Read more.
A solar sail is a propellantless propulsion system that allows a spacecraft to use solar radiation pressure as a propulsive source for planetary and deep space missions that would be difficult, or even unfeasible, to accomplish with more conventional thrusters, either chemical or electric. A challenging application for these fascinating propulsion systems is a heliocentric mission that requires a displaced non-Keplerian orbit (DNKO), that is, a solar sail-induced closed trajectory in which the orbital plane does not contain the Sun’s center of mass. In fact, thanks to the pioneering work of McInnes, it is known that a solar sail is able to reach and maintain a family of heliocentric DNKOs of given characteristics. The aim of this paper is to analyze the properties of Mars-synchronous circular DNKOs, which have an orbital period matching that of the planet for remote sensing applications. In fact, those specific displaced orbits allow a scientific probe to continuously observe the high-latitude regions of Mars from a quasi-stationary position relative to the planet. In this context, this paper also analyzes the optimal (i.e., the minimum-time) heliocentric transfer trajectory from the Earth to circular DNKOs in two special mission scenarios taken as a reference. Full article
(This article belongs to the Special Issue Autonomous Formation Systems: Guidance, Dynamics and Control)
Show Figures

Figure 1

Figure 1
<p>Conceptual scheme of a heliocentric, circular DNKO with a displacement equal to <span class="html-italic">H</span> with respect to the orbital plane of Mars (the gray disk in the sketch). The green disk indicates the plane of the DNKO, which is covered by the spacecraft with a constant angular velocity <math display="inline"><semantics> <mi>ω</mi> </semantics></math> at a distance <span class="html-italic">d</span> from Mars.</p>
Full article ">Figure 2
<p>Solar sail-based spacecraft in a spherical reference frame <math display="inline"><semantics> <msub> <mi mathvariant="script">T</mi> <mi>S</mi> </msub> </semantics></math> and definition of angles <math display="inline"><semantics> <mi>θ</mi> </semantics></math> and <math display="inline"><semantics> <mi>γ</mi> </semantics></math>. The gray surface indicates the orbital plane of Mars.</p>
Full article ">Figure 3
<p>Sketch of the gravitational (yellow vector), propulsive (red vector), and centrifugal (green vector) accelerations acting on the spacecraft during the flight along the DNKO.</p>
Full article ">Figure 4
<p>Required values of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>}</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>ρ</mi> <mo>/</mo> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>H</mi> <mo>/</mo> <mi>a</mi> <mo>}</mo> </mrow> </semantics></math>. The gray areas correspond to the envelope of the SOI of Mars. (<b>a</b>) Low-performance ideal solar sails. (<b>b</b>) Medium/high-performance ideal solar sails.</p>
Full article ">Figure 5
<p>Required values of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>}</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>ρ</mi> <mo>/</mo> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>H</mi> <mo>/</mo> <mi>a</mi> <mo>}</mo> </mrow> </semantics></math> for scenarios <span class="html-italic">A</span> and <span class="html-italic">B</span>, which are consistent with a high-performance and a medium-performance ideal solar sail, respectively.</p>
Full article ">Figure 6
<p>Variation of <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>0</mn> </msub> </semantics></math> with <math display="inline"><semantics> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </semantics></math> as given by Equation (<a href="#FD19-applsci-14-05001" class="html-disp-formula">19</a>).</p>
Full article ">Figure 7
<p>Variation of <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>ρ</mi> <mo>/</mo> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>ν</mi> <mo>,</mo> <mspace width="0.166667em"/> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>}</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>0.005</mn> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mo>Φ</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>30</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>≃</mo> <mn>5.04</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>≃</mo> <mn>8.94</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>90</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>≃</mo> <mn>10.69</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 7 Cont.
<p>Variation of <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>ρ</mi> <mo>/</mo> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>ν</mi> <mo>,</mo> <mspace width="0.166667em"/> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>}</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>0.005</mn> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mo>Φ</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>30</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>≃</mo> <mn>5.04</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>≃</mo> <mn>8.94</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>90</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>≃</mo> <mn>10.69</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Variation of the ratio <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>ν</mi> <mo>,</mo> <mspace width="0.166667em"/> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mo>}</mo> </mrow> </semantics></math> for scenario <span class="html-italic">A</span>.</p>
Full article ">Figure 9
<p>Variation of <math display="inline"><semantics> <mrow> <msub> <mo movablelimits="true" form="prefix">min</mo> <mi>ν</mi> </msub> <mrow> <mo>(</mo> <mi>d</mi> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with <math display="inline"><semantics> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </semantics></math> for scenario <span class="html-italic">A</span> (black line) and scenario <span class="html-italic">B</span> (red line).</p>
Full article ">Figure 10
<p>Solar sail thrust vector control angles <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>Time variation of spacecraft states for scenario <span class="html-italic">A</span>, where <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>≃</mo> <mn>0.1</mn> </mrow> </semantics></math>. (<b>a</b>) variation of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>r</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>ρ</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>H</mi> <mo>}</mo> </mrow> </semantics></math> with <span class="html-italic">t</span>; (<b>b</b>) Variation of <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>v</mi> <mi>r</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>v</mi> <mi>γ</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>v</mi> <mi>θ</mi> </msub> <mo>}</mo> </mrow> </semantics></math> with <span class="html-italic">t</span>.</p>
Full article ">Figure 12
<p>Optimal control and resulting trajectory for scenario <span class="html-italic">A</span>, where <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>≃</mo> <mn>0.1</mn> </mrow> </semantics></math>. (<b>a</b>) Variation of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> with <span class="html-italic">t</span>; (<b>b</b>) minimum-time optimal transfer trajectory.</p>
Full article ">Figure 13
<p>Time variation of spacecraft states for scenario <span class="html-italic">B</span>, where <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>≃</mo> <mn>0.05</mn> </mrow> </semantics></math>. (<b>a</b>) Variation of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>r</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>ρ</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>H</mi> <mo>}</mo> </mrow> </semantics></math> with <span class="html-italic">t</span>; (<b>b</b>) variation of <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>v</mi> <mi>r</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>v</mi> <mi>γ</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>v</mi> <mi>θ</mi> </msub> <mo>}</mo> </mrow> </semantics></math> with <span class="html-italic">t</span>.</p>
Full article ">Figure 14
<p>Optimal control and resulting trajectory for scenario <span class="html-italic">B</span>, where <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>≃</mo> <mn>0.05</mn> </mrow> </semantics></math>. (<b>a</b>) Variation of <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em"/> <mi>δ</mi> <mo>}</mo> </mrow> </semantics></math> with <span class="html-italic">t</span>; (<b>b</b>) minimum-time optimal transfer trajectory.</p>
Full article ">
16 pages, 4389 KiB  
Article
Solar Sail Optimal Performance in Heliocentric Nodal Flyby Missions
by Giovanni Mengali, Marco Bassetto and Alessandro A. Quarta
Aerospace 2024, 11(6), 427; https://doi.org/10.3390/aerospace11060427 - 24 May 2024
Viewed by 790
Abstract
Solar sails are propellantless propulsion systems that extract momentum from solar radiation pressure. They consist of a large ultrathin membrane, typically aluminized, that reflects incident photons from the Sun to generate thrust for space navigation. The purpose of this study is to investigate [...] Read more.
Solar sails are propellantless propulsion systems that extract momentum from solar radiation pressure. They consist of a large ultrathin membrane, typically aluminized, that reflects incident photons from the Sun to generate thrust for space navigation. The purpose of this study is to investigate the optimal performance of a solar sail-based spacecraft in performing two-dimensional heliocentric transfers to inertial points on the ecliptic that lie within an assigned annular region centered in the Sun. Similar to ESA’s Comet Interceptor mission, this type of transfer concept could prove useful for intercepting a potential celestial body, such as a long-period comet, that is passing close to Earth’s orbit. Specifically, it is assumed that the solar sail transfer occurs entirely in the ecliptic plane and, in analogy with recent studies, the flyby points explored are between 0.85au and 1.35au from the Sun. The heliocentric dynamics of the solar sail is described using the classical two-body model, assuming the spacecraft starts from Earth orbit (assumed circular), and an ideal force model to express the sail thrust vector. Finally, no constraint is imposed on the arrival velocity at flyby. Numerical simulation results show that solar sails are an attractive option to realize these specific heliocentric transfers. Full article
(This article belongs to the Special Issue Spacecraft Orbit Transfers)
Show Figures

Figure 1

Figure 1
<p>Scheme of the reachable zone and geometric constraints on the radial distance and offset angle value.</p>
Full article ">Figure 2
<p>Conceptual sketch of the transfer mission with definitions of the angular positions of the objects involved in the problem.</p>
Full article ">Figure 3
<p>Discretization of the reachable zone through a grid of approximately 670 points (black dots). The position of the Earth at the flyby time instant is represented by a blue circle, while the orange circle indicates the Sun. The boundary of the reachable zone is indicated by a solid red line.</p>
Full article ">Figure 4
<p>Flowchart of the implemented algorithm to obtain the minimum value of the solar sail characteristic acceleration required to complete the flyby mission with the assigned flight time.</p>
Full article ">Figure 5
<p>Minimum values of the characteristic acceleration, <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math>, as a function of the target point inside the reachable region (i.e., as a function of <math display="inline"><semantics> <msub> <mi>r</mi> <mi>f</mi> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> </mrow> </semantics></math>), when the flight time is <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math>. (<b>a</b>) Filled two-dimensional contour plot; (<b>b</b>) Three-dimensional surface plot.</p>
Full article ">Figure 5 Cont.
<p>Minimum values of the characteristic acceleration, <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math>, as a function of the target point inside the reachable region (i.e., as a function of <math display="inline"><semantics> <msub> <mi>r</mi> <mi>f</mi> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> </mrow> </semantics></math>), when the flight time is <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math>. (<b>a</b>) Filled two-dimensional contour plot; (<b>b</b>) Three-dimensional surface plot.</p>
Full article ">Figure 6
<p>Level curves showing the minimum values of <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> (measured in <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>) to reach the region of arrival points when <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math>. The boundary of the reachable zone is drawn with a solid red line. The Earth at the flyby time instant is represented by a blue circle, while the orange circle indicates the Sun.</p>
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<p>Optimal transfer trajectories when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.85</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math> as a function of the offset angle <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>150</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>150</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>. Blue circle → start point; red star → flyby point; black line → optimal transfer trajectory; blue dash line → circular parking orbit; red dash line → final solar distance; orange circle → the Sun.</p>
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<p>Optimal transfer trajectories when <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1.35</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math> as a function of the offset angle <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>150</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>150</mn> <mo>]</mo> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>. Blue circle → start point; red star → flyby point; black line → optimal transfer trajectory; blue dash line → circular parking orbit; red dash line → final solar distance; orange circle → the Sun.</p>
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<p>Variations of <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>r</mi> <mi>min</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>r</mi> <mi>max</mi> </msub> <mo>}</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>∈</mo> <mrow> <mo>{</mo> <mn>0.85</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1.35</mn> <mo>}</mo> </mrow> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math>. (<b>a</b>) Case of a final solar distance equal to <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.85</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math>; (<b>b</b>) Case of a final solar distance equal to <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1.35</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math>.</p>
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<p>Variations of <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>r</mi> <mi>min</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>r</mi> <mi>max</mi> </msub> <mo>}</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <msub> <mi>r</mi> <mi>f</mi> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <mi>year</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> <mo>∈</mo> <mo>{</mo> <mo>−</mo> <mn>150</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>150</mn> <mo>}</mo> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>. (<b>a</b>) Case of an offset angle equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>150</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>; (<b>b</b>) Case of an offset angle equal to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> <mo>=</mo> <mn>150</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math>.</p>
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14 pages, 4349 KiB  
Article
Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits
by Giovanni Mengali and Alessandro A. Quarta
Appl. Sci. 2024, 14(7), 2922; https://doi.org/10.3390/app14072922 - 29 Mar 2024
Cited by 3 | Viewed by 778
Abstract
Recent literature indicates that the diffractive sail concept is an interesting alternative to the more conventional reflective solar sail, which converts solar radiation pressure into a (deep space) thrust using a thin, lightweight highly reflective membrane, usually metalized. In particular, a diffractive sail, [...] Read more.
Recent literature indicates that the diffractive sail concept is an interesting alternative to the more conventional reflective solar sail, which converts solar radiation pressure into a (deep space) thrust using a thin, lightweight highly reflective membrane, usually metalized. In particular, a diffractive sail, which uses a metamaterial-based membrane to diffract incoming solar rays, is able to generate a steerable thrust vector even when the sail nominal plane is perpendicular to the Sun–spacecraft line. This paper analyzes the optimal transfer performance of a diffractive-sail-based spacecraft in a challenging heliocentric scenario that is consistent with the proposed Solar Polar Imager mission concept. In this case, the spacecraft must reach a near-circular (heliocentric) orbit with a high orbital inclination with respect to the Ecliptic in order to observe and monitor the Sun’s polar regions. Such a specific heliocentric scenario, because of the high velocity change it requires, is a mission application particularly suited for a propellantless propulsion system such as the classical solar sail. However, as shown in this work, the same transfer can be accomplished using a diffractive sail as the primary propulsion system. The main contribution of this paper is the analysis of the spacecraft transfer trajectory using a near-optimal strategy by dividing the entire flight into an approach phase to a circular orbit of the same radius as the desired final orbit but with a smaller inclination, and a subsequent cranking phase until the desired (orbital) inclination is reached. The numerical simulations show that the proposed strategy is sufficiently simple to implement and can provide solutions that differ by only a few percentage points from the optimal results obtainable with a classical indirect approach. Full article
Show Figures

Figure 1

Figure 1
<p>Definition of clock angle <math display="inline"><semantics> <mi>δ</mi> </semantics></math>, radial–tangential–normal reference frame, and direction of propulsive acceleration <math display="inline"><semantics> <msub> <mi mathvariant="bold-italic">a</mi> <mi mathvariant="normal">p</mi> </msub> </semantics></math>.</p>
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<p>Simulations of optimal trajectories with <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.26</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> for different values of final orbital inclination.</p>
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<p>Illustration of the two-phase strategy with the approach phase and the cranking phase. The green and red solid line represent the orbit at the end of the first and final phase, respectively.</p>
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<p>Time variation of orbital parameters and control variable in a near-optimal transfer to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and inclination <math display="inline"><semantics> <mrow> <msub> <mi>i</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> using the two-phase strategy (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.26</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>). Black circle → start; black square → arrival; blue triangle → first phase end; dashed red line → minimum solar distance.</p>
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<p>Near-optimal trajectory to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and inclination <math display="inline"><semantics> <mrow> <msub> <mi>i</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> using the two-phase strategy (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.26</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>). Black circle → start; black square → arrival; blue triangle → first phase end; blue star → perihelion; orange circle → the Sun; blue line → parking orbit; red line → target orbit; black line → transfer trajectory.</p>
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<p>Simulations of optimal trajectories with <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math> to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> for different values of final orbital inclination. Dashed red line → minimum solar distance.</p>
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<p>Time variation of orbital parameters and control variable in a near-optimal transfer to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and inclination <math display="inline"><semantics> <mrow> <msub> <mi>i</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> using the two-phase strategy (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>). Black circle → start; black square → arrival; blue triangle → first phase end; dashed red line → minimum solar distance.</p>
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<p>Near-optimal trajectory to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and inclination <math display="inline"><semantics> <mrow> <msub> <mi>i</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> using the two-phase strategy (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>). Black circle → start; black square → arrival; blue triangle → first phase end; blue star → perihelion; orange circle → the Sun; blue line → parking orbit; red line → target orbit; black line → transfer trajectory.</p>
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<p>Time variation of orbital parameters and control variable in an optimal transfer to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and inclination <math display="inline"><semantics> <mrow> <msub> <mi>i</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>). Black circle → start; black square → arrival; dashed red line → minimum solar distance.</p>
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<p>Optimal trajectory to a circular heliocentric orbit of radius <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mi>au</mi> </mrow> </semantics></math> and inclination <math display="inline"><semantics> <mrow> <msub> <mi>i</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>60</mn> <mspace width="0.166667em"/> <mi>deg</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>). Black circle → start; black square → arrival; blue star → perihelion; orange circle → the Sun; blue line → parking orbit; red line → target orbit; black line → transfer trajectory.</p>
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<p>Comparison of the two-phase strategy (blue line) and the optimal control law (black line) with <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. Dashed red line → minimum solar distance; filled circle → start; filled square → arrival.</p>
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<p>Comparison of flight times obtained with various transfer strategies: Dubill et al. [<a href="#B21-applsci-14-02922" class="html-bibr">21</a>], Chu et al. [<a href="#B30-applsci-14-02922" class="html-bibr">30</a>], two-phase strategy, and optimal result (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.183</mn> <mspace width="0.166667em"/> <mi>mm</mi> <mo>/</mo> <msup> <mi mathvariant="normal">s</mi> <mn>2</mn> </msup> </mrow> </semantics></math>).</p>
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