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<div class="d-article">
<p>Determining the eclipses a satellite will encounter is a major driving factor when designing a mission in space. Thermal and power budgets have to be made with the fact that a satellite will periodically be in the complete darkness of space with no solar radiation to power the solar panels and keep the spacecraft from freezing.</p>
<p>Determining the eclipses a satellite will encounter is a major driving factor when designing a mission in space. Thermal and power budgets have to be made with the fact that a satellite will periodically be in the complete darkness of space where it will receive no solar radiation to power the solar panels and keep the spacecraft from freezing.</p>
<h2 id="what-is-an-eclipse">What is an Eclipse</h2>
<figure>
<img src="geometry.svg" alt="Geometry of an Eclipse" /><figcaption aria-hidden="true">Geometry of an Eclipse</figcaption>
</figure>
<p>The above image is a simple representation of what an eclipse is. You’ll notice there is the Umbra which is complete darkness, then the Penumbra which is a shadow of varying darkness, and then the rest of the orbit is in complete sunlight. For this example I will be using the ISS which has a very low orbit so the Penumbra isn’t much of a problem. You can tell by looking at the diagram that higher altitude orbits would spend more time in the Penumbra.</p>
<p>The above image is a simple representation of what an eclipse is. First, you’ll notice the Umbra is complete darkness, then the Penumbra, which is a shadow of varying darkness, and then the rest of the orbit is in full sunlight. For this example, I will be using the ISS, which has a very low orbit, so the Penumbra isn’t much of a problem. However, you can tell by looking at the diagram that higher altitude orbits would spend more time in the Penumbra.</p>
<figure>
<img src="vectors_radiuss.svg" alt="Body Radius’s and Position Vectors" /><figcaption aria-hidden="true">Body Radius’s and Position Vectors</figcaption>
</figure>
<p>Here is a more detailed view of the eclipse that will make it easier to explain the code. There are 2 Position vectors and 2 radius’s that need to be known for simple eclipse determination. There are more advanced cases where the atmosphere of the body your orbiting can greatly affect the Umbra and Penumbra, and other bodies could also potentially block the Sun, but for this example we will keep it simple since those have very little affect for the ISS’s orbit. <code style="color:#0b7285">Rsun</code> and <code style="color:#c92a2a">Rbody</code> are the radius’s of the Sun and Body (In this case Earth) respectively. <code style="color:#5f3dc4">r_sun_body</code> is a vector from the center of the Sun to the center of the target body. For this example I will only be using one vector, but for more rigorous eclipse determination its important to calculate this at least once a day since it does significantly change over the course of a year. The reason that I am ignoring it at the moment is because there is currently no good way to calculate <a href="https://ssd.jpl.nasa.gov/?ephemerides">Ephemerides</a> in Julia but the package is being worked on so I may revisit this and do a more rigorous analysis in the future. <code style="color:#5c940d">r_body_sc</code> is a position vector from the center of the body being orbitted, to the center of our spacecraft.</p>
<p>Here is a more detailed view of the eclipse that will make it easier to explain what is going on. There are 2 Position vectors, and 2 radius that need to be known for simple eclipse determination. More advanced cases where the atmosphere of the body your orbiting can significantly affect the Umbra and Penumbra, and other bodies could also potentially block the Sun. However, we will keep it simple for this example since they have minimal effect on the ISS’s orbit. <code style="color:#0b7285">Rsun</code> and <code style="color:#c92a2a">Rbody</code> are the radius of the Sun and Body (In this case Earth), respectively. <code style="color:#5f3dc4">r_sun_body</code> is a vector from the center of the Sun to the center of the target body. For this example I will only be using one vector, but for more rigorous eclipse determination it is important to calculate the ephemeris at least once a day since it does significantly change over the course of a year. The reason that I am ignoring it at the moment is because there is currently no good way to calculate <a href="https://ssd.jpl.nasa.gov/?ephemerides">Ephemerides</a> in Julia but the package is being worked on so I may revisit this and do a more rigorous analysis in the future. <code style="color:#5c940d">r_body_sc</code> is a position vector from the center of the body being orbited, to the center of our spacecraft.</p>
<h2 id="the-code">The Code</h2>
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<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>theme(<span class="op">:</span>ggplot2)</span></code></pre></div>
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<p>In order to get the orbit for the ISS I used a <a href="https://en.wikipedia.org/wiki/Two-line_element_set">Two-Line Element</a> which is a data format for explaining orbits. the US Joint Space Operations Center makes these widely available, but <a href="https://live.ariss.org/tle/" class="uri">https://live.ariss.org/tle/</a> makes the TLE for the ISS way more accessible <span class="citation" data-cites="ariss">(<a href="#ref-ariss" role="doc-biblioref"><span>“<span>ARISS</span> <span>TLE</span>,”</span> n.d.</a>)</span>. The Julia Package <a href="https://github.com/JuliaSpace/SatelliteToolbox.jl">SatelliteToolbox.jl</a> makes it super easy to turn a TLE into an orbit that can be propagated. Simply putting the TLE in a string and using the <code>tle</code> string macro like below and now we have access to the information to start making our ISS orbit.</p>
<p>To get the orbit for the ISS, I used a <a href="https://en.wikipedia.org/wiki/Two-line_element_set">Two-Line Element</a> which is a data format for explaining orbits. The US Joint Space Operations Center makes these widely available, but <a href="https://live.ariss.org/tle/" class="uri">https://live.ariss.org/tle/</a> makes the TLE for the ISS way more accessible <span class="citation" data-cites="ariss">(<a href="#ref-ariss" role="doc-biblioref"><span>“<span>ARISS</span> <span>TLE</span>,”</span> n.d.</a>)</span>. The Julia Package <a href="https://github.com/JuliaSpace/SatelliteToolbox.jl">SatelliteToolbox.jl</a> makes it super easy to turn a TLE into an orbit that can be propagated. Simply putting the TLE in a string and using the <code>tle</code> string macro like below, we now have access to the information to start making our ISS orbit.</p>
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<div class="sourceCode" id="cb2"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>ISS <span class="op">=</span> tle<span class="st">&quot;&quot;&quot;</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="st">ISS (ZARYA)</span></span>
@@ -1519,17 +1520,16 @@ Show code
<pre><code>1-element Vector{TLE}:
TLE: ISS (ZARYA) (Epoch = 2021-04-13T20:23:10.911)</code></pre>
</div>
<p>Now that we have the TLE we can pass that into SatelliteToolbox’s orbit propagator. Before we can propagate the orbit we need to have a range of time steps to pass into the propagator. The TLE gives the mean motion, n, which is the revolutions per day so using that we can calculate the amount of time required for one orbit which is all that were worried about for this analysis. The propagator returns a tuple containing the Orbital elements, a position vector with units meters, and a velocity vector with units meters per second. For this analysis were only worried about the position vector.</p>
<p>Now that we have the TLE, we can pass that into SatelliteToolbox’s orbit propagator. Before propagating the orbit, we need to have a range of time steps to pass into the propagator. The TLE gives the mean motion, n, which is the revolutions per day, so using that, we can calculate the amount of time required for one orbit, which is all that we’re worried about for this analysis. The propagator returns a tuple containing the Orbital elements, a position vector with units meters, and a velocity vector with units meters per second. For this analysis were only worried about the position vector.</p>
<div class="layout-chunk" data-layout="l-body">
<pre class="juliam"><code>ISS[1].n</code></pre>
<div class="sourceCode" id="cb4"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>ISS[<span class="fl">1</span>].n</span></code></pre></div>
</div>
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<div class="sourceCode" id="cb5"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>orbit <span class="op">=</span> init_orbit_propagator(<span class="dt">Val</span>(<span class="op">:</span>twobody)<span class="op">,</span> ISS[<span class="fl">1</span>])<span class="op">;</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>time <span class="op">=</span> <span class="fl">0</span><span class="op">:</span><span class="fl">0.1</span><span class="op">:</span>((<span class="fl">24</span> <span class="op">/</span> ISS[<span class="fl">1</span>].n) <span class="op">.*</span> <span class="fl">60</span> <span class="op">*</span> <span class="fl">60</span>)<span class="op">;</span> <span class="co"># ISS[1].n gives the mean motion, or orbits per day.</span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>o<span class="op">,</span> r<span class="op">,</span> v <span class="op">=</span> propagate<span class="op">!</span>(orbit<span class="op">,</span> time)<span class="op">;</span></span></code></pre></div>
</div>
<p>Now we just need way to use the radii and vectors discussed earlier to determine if the ISS is in the penumbra or umbra. This is a lot of pretty basic trigonometry and vector math.</p>
<p><code>add more discussion about the math</code></p>
<p>We just need to use the radii and vectors discussed earlier to determine if the ISS is in the penumbra or umbra. This is a lot of trigonometry and vector math that I won’t bore anyone with. However, using the diagrams above and following the code in the sunlight function, you should follow what’s happening. For a rigorous discussion, check out <span class="citation" data-cites="vallado">(<a href="#ref-vallado" role="doc-biblioref">Vallado 1997</a>)</span>.</p>
<div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb6"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> sunlight(Rbody<span class="op">,</span> r_sun_body<span class="op">,</span> r_body_sc)</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> Rsun <span class="op">=</span> <span class="fl">695_700</span>u<span class="st">&quot;km&quot;</span></span>
@@ -1560,7 +1560,7 @@ Show code
<div class="sourceCode" id="cb7"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>S <span class="op">=</span> r .<span class="op">|&gt;</span> R <span class="op">-&gt;</span> sunlight(<span class="fl">6371</span>u<span class="st">&quot;km&quot;</span><span class="op">,</span> [<span class="fl">0.5370</span><span class="op">,</span> <span class="fl">1.2606</span><span class="op">,</span> <span class="fl">0.5466</span>] <span class="op">.*</span> <span class="fl">1e8</span>u<span class="st">&quot;km&quot;</span><span class="op">,</span> R <span class="op">.*</span> u<span class="st">&quot;m&quot;</span>)<span class="op">;</span></span></code></pre></div>
</div>
<h2 id="plotting-the-results">Plotting the Results</h2>
<p>The <code>sunlight</code> function returns values from 0 to 1, 0 being complete darkness, 1 being complete sunlight, and anything between being the fraction of light being received. Again since the ISS has a very low orbit, the amount of time spend in the penumbra is almost insignificant.</p>
<p>The <code>sunlight</code> function returns values from 0 to 1, 0 being complete darkness, 1 being complete sunlight, and anything between being the fraction of light being received. Again since the ISS has a very low orbit, the amount of time spent in the penumbra is almost insignificant.</p>
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@@ -1582,35 +1582,38 @@ Show code
<span id="cb8-14"><a href="#cb8-14" aria-hidden="true" tabindex="-1"></a>ylabel<span class="op">!</span>(<span class="st">&quot;Sunlight (%)&quot;</span>)<span class="op">;</span></span>
<span id="cb8-15"><a href="#cb8-15" aria-hidden="true" tabindex="-1"></a>title<span class="op">!</span>(<span class="st">&quot;ISS Sunlight Over a Day&quot;</span>)</span></code></pre></div>
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<div class="figure"><span id="fig:unnamed-chunk-7"></span>
<img src="iss-eclipse-determination_files/figure-html5/unnamed-chunk-7-J1.png" alt="ISS Sunlight" width="300" data-distill-preview=1 />
<div class="figure"><span id="fig:sunlight-plot"></span>
<img src="iss-eclipse-determination_files/figure-html5/sunlight-plot-J1.png" alt="ISS Sunlight" width="300" data-distill-preview=1 />
<p class="caption">
Figure 1: ISS Sunlight
</p>
</div>
</div>
<p>Looking at the plot its pretty easy to see by the vertical transition from 0% to 100% that the time in the penumbra is limited, but almost counterintutively it also looks like the ISS gets more sunlight than it does darkness. Using the raw sunlight data we can actually calculate almost exactly how much time is spent in each region.</p>
Time in Sun:
<p>Looking at the plot, the vertical transition from 0% to 100% makes it pretty clear that the time in the penumbra is limited. Still, almost counterintuitively, it also looks like the ISS gets more sunlight than it does darkness. So, using the raw sunlight data, we can calculate precisely how much time is spent in each region.</p>
<p>Time in Sun:</p>
<div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb9"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>sun <span class="op">=</span> length(S[S.<span class="op">==</span><span class="fl">1</span>])<span class="op">/</span>length(S) <span class="op">*</span> <span class="fl">100</span></span></code></pre></div>
<pre><code>62.03323593209401</code></pre>
</div>
Time in Darkness:
<p>Time in Darkness:</p>
<div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb11"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>umbra <span class="op">=</span> length(S[S.<span class="op">==</span><span class="fl">0</span>])<span class="op">/</span>length(S) <span class="op">*</span> <span class="fl">100</span></span></code></pre></div>
<pre><code>37.64408511553699</code></pre>
</div>
Time in Penumbra:
<p>Time in Penumbra:</p>
<div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb13"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>penumbra <span class="op">=</span> <span class="fl">100</span> <span class="op">-</span> umbra <span class="op">-</span> sun</span></code></pre></div>
<pre><code>0.322678952369003</code></pre>
</div>
<p>The ISS spends about 62% of its time in the sun, this is because if you go back and reference the diagram at the beginning of this post you can see that the umbra is actually a cone. This is mainly due to the fact that the Sun is massive compared to the Earth, but this effect is also stronger with orbits of higher altitudes.</p>
<p>This means that even with the ISS’s low orbit, it still gets sunlight ~62% of the time and spends almost no time in the penumbra. This would vary a few percent depending on the time of year, but in a circular orbit like the ISS, the amount of sunlight would remain pretty constant. There are other orbits like a polar orbit, lunar orbit, or highly elliptic earth orbits that can have their time in the sunlight vary widely by the time of year.</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r distill-force-highlighting-css"><code class="sourceCode r"></code></pre></div>
<div id="refs" class="references csl-bib-body hanging-indent" role="doc-bibliography">
<div id="ref-ariss" class="csl-entry" role="doc-biblioentry">
<span>“<span>ARISS</span> <span>TLE</span>.”</span> n.d. <em>Amateur Radio on the International Space Station</em>. <a href="https://live.ariss.org/tle/">https://live.ariss.org/tle/</a>.
</div>
<div id="ref-vallado" class="csl-entry" role="doc-biblioentry">
Vallado, David A. 1997. <em>Fundamentals of <span>Astrodynamics</span> and <span>Applications</span>, 2nd. Ed.</em> Edited by Wiley Larson. Dordrecht: Microcosm, Inc.
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<div class="appendix-bottom">
<h3 id="references">References</h3>
<div id="references-listing"></div>
<h3 id="updates-and-corrections">Corrections</h3>
<p>If you see mistakes or want to suggest changes, please <a href="https://gitlab.com/lander-team/air-prop-simulation">create an issue</a> on the source repository.</p>
<h3 id="reuse">Reuse</h3>
<p>Text and figures are licensed under Creative Commons Attribution <a rel="license" href="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>. Source code is available at <a href="https://gitlab.com/lander-team/air-prop-simulation">https://gitlab.com/lander-team/air-prop-simulation</a>, unless otherwise noted. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".</p>
<p>Text and figures are licensed under Creative Commons Attribution <a rel="license" href="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".</p>
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