A First-Order Analytical Theory for Optimal Low-Thrust Limited-Power Transfers between Arbitrary Elliptical Coplanar Orbits

<table> Complete and average solutions of the
two-point boundary value problem  for   <svg height="18.799999" id="M377" style="vertical-align:-3.13995pt" version="1.1" viewbox="0 0 133.39999 18.799999" width="133.39999" xmlns="http://www.w3.org/2000/svg">
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<tspan style="font-size: 12.50px; " x="95.75" y="0">]</tspan>
<tspan style="font-size: 12.50px; " x="0" y="0">Δ</tspan>
<tspan style="font-size: 12.50px; " x="9.0271664" y="0">𝑥</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">[</tspan>
<tspan style="font-size: 12.50px; " x="35.580002" y="-0.62">Δ</tspan>
<tspan style="font-size: 12.50px; " x="44.607166" y="-0.62">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="57.622787" y="-0.62">Δ</tspan>
<tspan style="font-size: 12.50px; " x="66.649956" y="-0.62">𝑒</tspan>
<tspan style="font-size: 12.50px; " x="78.202728" y="-0.62">Δ</tspan>
<tspan style="font-size: 12.50px; " x="87.229897" y="-0.62">𝜔</tspan>
</text>
<text transform="matrix(1,0,0,-1,27.96,66.24)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑇</tspan>
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</g>
</g>
</svg> and <svg height="12.3875" id="M378" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 48.362499 12.3875" width="48.362499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.3875)">
<g transform="translate(72,-62.09)">
<text transform="matrix(1,0,0,-1,-71.95,62.31)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="10.515023" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.555412" y="0">𝑎</tspan>
<tspan style="font-size: 12.50px; " x="28.831919" y="0">/</tspan>
</text>
<g>
<path d="m -39.64,70.71 6.28,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
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<text transform="matrix(1,0,0,-1,-39.64,62.31)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
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</svg>. </table>

Mathematical Problems in Engineering

fig10

Figure 10

Figure 10: A First-Order Analytical Theory for Optimal Low-Thrust Limited-Power Transfers between Arbitrary Elliptical Coplanar Orbits 