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_posts/2021-04-01-air-propulsion-simulation/air-propulsion-simulation.Rmd
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@ -24,7 +24,7 @@ For Capstone my team was tasked with designing a system capable of moving mining
```{r setup, include=FALSE} ```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, results = 'hide') knitr::opts_chunk$set(echo = TRUE, results = 'hide')
library(JuliaCall) library(JuliaCall)
#julia_setup(JULIA_HOME = "/opt/julia-1.6.0/bin/") julia_setup(JULIA_HOME = "/opt/julia-1.6.0/bin/")
``` ```
```{julia, code_folding=TRUE} ```{julia, code_folding=TRUE}

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_posts/2021-04-01-air-propulsion-simulation/air-propulsion-simulation.html
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<meta name="citation_reference" content="citation_title=Rocket propulsion elements;citation_publication_date=2001;citation_publisher=John Wiley &amp; Sons;citation_author=George P. Sutton;citation_author=Oscar Biblarz"/>
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_posts/2021-04-14-iss-eclipse-determination/citations.bib Normal file
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@ -0,0 +1,29 @@
@book{shackelford_introduction_2015,
address = {Boston},
edition = {Eighth edition},
title = {Introduction to materials science for engineers},
isbn = {9780133826654},
publisher = {Pearson},
author = {Shackelford, James F.},
year = {2015},
keywords = {Materials},
}
@book{curtis_orbital_2021,
title = {Orbital mechanics for engineering students},
isbn = {9780128240250 9780323853453 9780081021330},
language = {English},
author = {Curtis, Howard D},
year = {2021},
note = {OCLC: 1235349370},
}
@misc{ariss,
title = {{ARISS} {TLE}},
shorttitle = {{ARISS}},
url = {https://live.ariss.org/tle/},
abstract = {This "Two-Line Element" file is published by US Joint Space Operations Center, containing the numerical coefficients of the ISS orbit.},
journal = {Amateur Radio on the International Space Station},
note = {publisher: ARISS},
}

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_posts/2021-04-14-iss-eclipse-determination/iss-eclipse-determination.Rmd
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@ -1,13 +1,13 @@
--- ---
title: "ISS Eclipse Determination" title: "ISS Eclipse Determination"
description: | description: |
Determining how much sunlight a body is receiving. Determining how much sunlight a body orbiting a planet is receiving.
draft: false draft: false
author: author:
- name: Anson Biggs - name: Anson Biggs
url: https://ansonbiggs.com url: https://ansonbiggs.com
repository_url: https://gitlab.com/lander-team/air-prop-simulation repository_url: https://gitlab.com/lander-team/air-prop-simulation
date: 04-01-2021 date: 05-01-2021
fig_width: 6 fig_width: 6
fig_align: "center" fig_align: "center"
output: output:
@ -16,7 +16,7 @@ output:
categories: categories:
- Julia - Julia
- Astrodynamics - Astrodynamics
#bibliography: ../../citations.bib bibliography: citations.bib
creative_commons: CC BY creative_commons: CC BY
preview: preview.png preview: preview.png
--- ---
@ -26,17 +26,18 @@ Determining the eclipses a satellite will encounter is a major driving factor wh
```{r setup, include=FALSE} ```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, results = 'hide') knitr::opts_chunk$set(echo = TRUE, results = 'hide')
library(JuliaCall) library(JuliaCall)
#julia_setup(JULIA_HOME = "/opt/julia-1.6.1/bin/") julia_setup(JULIA_HOME = "/opt/julia-1.6.0/bin/")
julia_setup(installJulia = TRUE)
``` ```
## What is an Eclipse ## What is an Eclipse
![Geometry of an Eclipse](geometry.svg) ![Geometry of an Eclipse](geometry.svg)
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.
![Body Radius's and Position Vectors](vectors_radiuss.svg) ![Body Radius's and Position Vectors](vectors_radiuss.svg)
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 [Ephemerides](https://ssd.jpl.nasa.gov/?ephemerides) 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.
## The Code ## The Code
@ -49,8 +50,9 @@ using Colors
theme(:ggplot2) theme(:ggplot2)
``` ```
In order to get the orbit for the ISS I used a [Two-Line Element](https://en.wikipedia.org/wiki/Two-line_element_set) which is a data format for explaining orbits. the US Joint Space Operations Center makes these widely available, but https://live.ariss.org/tle/ makes the TLE for the ISS way more accessible [@ariss]. The Julia Package [SatelliteToolbox.jl](https://github.com/JuliaSpace/SatelliteToolbox.jl) 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 `tle` string macro like below and now we have access to the information to start making our ISS orbit.
```{julia} ```{julia, results='show'}
ISS = tle""" ISS = tle"""
ISS (ZARYA) ISS (ZARYA)
1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990 1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990
@ -58,12 +60,21 @@ ISS (ZARYA)
""" """
``` ```
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.
```{juliam result='show'}
ISS[1].n
```
```{julia} ```{julia}
orbit = init_orbit_propagator(Val(:twobody), ISS[1]); orbit = init_orbit_propagator(Val(:twobody), ISS[1]);
time = 0:0.1:((24 / ISS[1].n) .* 60 * 60); time = 0:0.1:((24 / ISS[1].n) .* 60 * 60); # ISS[1].n gives the mean motion, or orbits per day.
o, r, v = propagate!(orbit, time); o, r, v = propagate!(orbit, time);
``` ```
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.
`add more discussion about the math`
```{julia} ```{julia}
function sunlight(Rbody, r_sun_body, r_body_sc) function sunlight(Rbody, r_sun_body, r_body_sc)
@ -91,14 +102,19 @@ function sunlight(Rbody, r_sun_body, r_body_sc)
end end
``` ```
Then we can pass all the values we've gathered into the function we just made.
```{julia} ```{julia}
S = r .|> R -> sunlight(6371u"km", [0.5370, 1.2606, 0.5466] .* 1e8u"km", R .* u"m") S = r .|> R -> sunlight(6371u"km", [0.5370, 1.2606, 0.5466] .* 1e8u"km", R .* u"m");
``` ```
## Plotting the Results ## Plotting the Results
```{julia, code_folding=TRUE, results='show',layout="l-body-outset",fig.cap= "Rocket Motor Data: [@thrustcurve]", preview=TRUE } The `sunlight` 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.
light_range = range(colorant"black", stop = colorant"yellow", length = 101);
```{julia, code_folding=TRUE, results='show',layout="l-body-outset",fig.cap= "ISS Sunlight", preview=TRUE }
# Get fancy with the line color.
light_range = range(colorant"black", stop = colorant"orange", length = 101);
light_colors = [light_range[unique(round(Int, 1 + s * 100))][1] for s in S]; light_colors = [light_range[unique(round(Int, 1 + s * 100))][1] for s in S];
plot( plot(
@ -113,3 +129,22 @@ xlabel!("Time (hr)");
ylabel!("Sunlight (%)"); ylabel!("Sunlight (%)");
title!("ISS Sunlight Over a Day") title!("ISS Sunlight Over a Day")
``` ```
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.
Time in Sun:
```{julia, results='show'}
sun = length(S[S.==1])/length(S) * 100
```
Time in Darkness:
```{julia, results='show'}
umbra = length(S[S.==0])/length(S) * 100
```
Time in Penumbra:
```{julia, results='show'}
penumbra = 100 - umbra - sun
```
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.

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_posts/2021-04-14-iss-eclipse-determination/iss-eclipse-determination.html
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@ -1473,13 +1475,13 @@ code span.wa { color: #5e5e5e; font-style: italic; } /* Warning */
<div class="dt=tag">Astrodynamics</div> <div class="dt=tag">Astrodynamics</div>
</div> </div>
<!--/radix_placeholder_categories--> <!--/radix_placeholder_categories-->
<p><p>Determining how much sunlight a body is receiving.</p></p> <p><p>Determining how much sunlight a body orbiting a planet is receiving.</p></p>
</div> </div>
<div class="d-byline"> <div class="d-byline">
Anson Biggs <a href="https://ansonbiggs.com" class="uri">https://ansonbiggs.com</a> Anson Biggs <a href="https://ansonbiggs.com" class="uri">https://ansonbiggs.com</a>
<br/>04-01-2021 <br/>05-01-2021
</div> </div>
<div class="d-article"> <div class="d-article">
@ -1488,9 +1490,11 @@ code span.wa { color: #5e5e5e; font-style: italic; } /* Warning */
<figure> <figure>
<img src="geometry.svg" alt="Geometry of an Eclipse" /><figcaption aria-hidden="true">Geometry of an Eclipse</figcaption> <img src="geometry.svg" alt="Geometry of an Eclipse" /><figcaption aria-hidden="true">Geometry of an Eclipse</figcaption>
</figure> </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>
<figure> <figure>
<img src="vectors_radiuss.svg" alt="Body Radius’s and Position Vectors" /><figcaption aria-hidden="true">Body Radius’s and Position Vectors</figcaption> <img src="vectors_radiuss.svg" alt="Body Radius’s and Position Vectors" /><figcaption aria-hidden="true">Body Radius’s and Position Vectors</figcaption>
</figure> </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>
<h2 id="the-code">The Code</h2> <h2 id="the-code">The Code</h2>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<details> <details>
@ -1505,75 +1509,109 @@ Show code
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>theme(<span class="op">:</span>ggplot2)</span></code></pre></div> <span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>theme(<span class="op">:</span>ggplot2)</span></code></pre></div>
</details> </details>
</div> </div>
<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>
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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> <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> <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="st">ISS (ZARYA)</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="st">1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990</span></span> <span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="st">1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="st">2 25544 51.6434 300.9481 0002858 223.8443 263.8789 15.48881793278621</span></span> <span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="st">2 25544 51.6434 300.9481 0002858 223.8443 263.8789 15.48881793278621</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;&quot;&quot;</span></span></code></pre></div> <span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;&quot;&quot;</span></span></code></pre></div>
<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>
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<pre class="juliam"><code>ISS[1].n</code></pre>
</div> </div>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb3"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb3-1"><a href="#cb3-1" 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> <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="cb3-2"><a href="#cb3-2" 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> <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="cb3-3"><a href="#cb3-3" 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> <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> </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>
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<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><span class="kw">function</span> sunlight(Rbody<span class="op">,</span> r_sun_body<span class="op">,</span> r_body_sc)</span> <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="cb4-2"><a href="#cb4-2" 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> <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>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> </span> <span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> hu <span class="op">=</span> Rbody <span class="op">*</span> norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">-</span> Rbody)</span> <span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> hu <span class="op">=</span> Rbody <span class="op">*</span> norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">-</span> Rbody)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> </span> <span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> θe <span class="op">=</span> acos((r_sun_body ⋅ r_body_sc) <span class="op">/</span> (norm(r_sun_body) <span class="op">*</span> norm(r_body_sc)))</span> <span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> θe <span class="op">=</span> acos((r_sun_body ⋅ r_body_sc) <span class="op">/</span> (norm(r_sun_body) <span class="op">*</span> norm(r_body_sc)))</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a> θu <span class="op">=</span> atan(Rbody <span class="op">/</span> hu)</span> <span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> θu <span class="op">=</span> atan(Rbody <span class="op">/</span> hu)</span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a> du <span class="op">=</span> hu <span class="op">*</span> sin(θu) <span class="op">/</span> sin(θe <span class="op">+</span> θu)</span> <span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> du <span class="op">=</span> hu <span class="op">*</span> sin(θu) <span class="op">/</span> sin(θe <span class="op">+</span> θu)</span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a> θp <span class="op">=</span> π <span class="op">-</span> atan(norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">+</span> Rbody))</span> <span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a> θp <span class="op">=</span> π <span class="op">-</span> atan(norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">+</span> Rbody))</span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a> dp <span class="op">=</span> Rbody <span class="op">*</span> sin(θp) <span class="op">/</span> cos(θe <span class="op">-</span> θp)</span> <span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a> dp <span class="op">=</span> Rbody <span class="op">*</span> sin(θp) <span class="op">/</span> cos(θe <span class="op">-</span> θp)</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">1</span></span> <span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">1</span></span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> du)</span> <span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> du)</span>
<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">0</span></span> <span id="cb6-16"><a href="#cb6-16" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">0</span></span>
<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span> <span id="cb6-17"><a href="#cb6-17" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span>
<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> ((du <span class="op">&lt;</span> norm(r_body_sc)) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> dp))</span> <span id="cb6-18"><a href="#cb6-18" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> ((du <span class="op">&lt;</span> norm(r_body_sc)) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> dp))</span>
<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> (norm(r_body_sc .<span class="op">|&gt;</span> u<span class="st">&quot;km&quot;</span>) <span class="op">-</span> du) <span class="op">/</span> (dp <span class="op">-</span> du) <span class="op">|&gt;</span> ustrip</span> <span id="cb6-19"><a href="#cb6-19" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> (norm(r_body_sc .<span class="op">|&gt;</span> u<span class="st">&quot;km&quot;</span>) <span class="op">-</span> du) <span class="op">/</span> (dp <span class="op">-</span> du) <span class="op">|&gt;</span> ustrip</span>
<span id="cb4-20"><a href="#cb4-20" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span> <span id="cb6-20"><a href="#cb6-20" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span>
<span id="cb4-21"><a href="#cb4-21" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-21"><a href="#cb6-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-22"><a href="#cb4-22" aria-hidden="true" tabindex="-1"></a> <span class="kw">return</span> S</span> <span id="cb6-22"><a href="#cb6-22" aria-hidden="true" tabindex="-1"></a> <span class="kw">return</span> S</span>
<span id="cb4-23"><a href="#cb4-23" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code></pre></div> <span id="cb6-23"><a href="#cb6-23" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code></pre></div>
</div> </div>
<p>Then we can pass all the values we’ve gathered into the function we just made.</p>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<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>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></code></pre></div> <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> </div>
<h2 id="plotting-the-results">Plotting the Results</h2> <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>
<div class="layout-chunk" data-layout="l-body-outset"> <div class="layout-chunk" data-layout="l-body-outset">
<details> <details>
<summary> <summary>
Show code Show code
</summary> </summary>
<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>light_range <span class="op">=</span> range(colorant<span class="st">&quot;black&quot;</span><span class="op">,</span> stop <span class="op">=</span> colorant<span class="st">&quot;yellow&quot;</span><span class="op">,</span> length <span class="op">=</span> <span class="fl">101</span>)<span class="op">;</span></span> <div class="sourceCode" id="cb8"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get fancy with the line color. </span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>light_colors <span class="op">=</span> [light_range[unique(round(<span class="dt">Int</span><span class="op">,</span> <span class="fl">1</span> <span class="op">+</span> s <span class="op">*</span> <span class="fl">100</span>))][<span class="fl">1</span>] <span class="kw">for</span> s <span class="kw">in</span> S]<span class="op">;</span></span> <span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>light_range <span class="op">=</span> range(colorant<span class="st">&quot;black&quot;</span><span class="op">,</span> stop <span class="op">=</span> colorant<span class="st">&quot;orange&quot;</span><span class="op">,</span> length <span class="op">=</span> <span class="fl">101</span>)<span class="op">;</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span> <span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>light_colors <span class="op">=</span> [light_range[unique(round(<span class="dt">Int</span><span class="op">,</span> <span class="fl">1</span> <span class="op">+</span> s <span class="op">*</span> <span class="fl">100</span>))][<span class="fl">1</span>] <span class="kw">for</span> s <span class="kw">in</span> S]<span class="op">;</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>plot(</span> <span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> <span class="dt">LinRange</span>(<span class="fl">0</span><span class="op">,</span> <span class="fl">24</span><span class="op">,</span> length(S))<span class="op">,</span></span> <span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a>plot(</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> S <span class="op">.*</span> <span class="fl">100</span><span class="op">,</span></span> <span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a> <span class="dt">LinRange</span>(<span class="fl">0</span><span class="op">,</span> <span class="fl">24</span><span class="op">,</span> length(S))<span class="op">,</span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a> linewidth <span class="op">=</span> <span class="fl">5</span><span class="op">,</span></span> <span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a> S <span class="op">.*</span> <span class="fl">100</span><span class="op">,</span></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> legend <span class="op">=</span> <span class="ex">false</span><span class="op">,</span></span> <span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a> linewidth <span class="op">=</span> <span class="fl">5</span><span class="op">,</span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> color <span class="op">=</span> light_colors<span class="op">,</span></span> <span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a> legend <span class="op">=</span> <span class="ex">false</span><span class="op">,</span></span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a>)<span class="op">;</span></span> <span id="cb8-10"><a href="#cb8-10" aria-hidden="true" tabindex="-1"></a> color <span class="op">=</span> light_colors<span class="op">,</span></span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a></span> <span id="cb8-11"><a href="#cb8-11" aria-hidden="true" tabindex="-1"></a>)<span class="op">;</span></span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a>xlabel<span class="op">!</span>(<span class="st">&quot;Time (hr)&quot;</span>)<span class="op">;</span></span> <span id="cb8-12"><a href="#cb8-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-13"><a href="#cb6-13" 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="cb8-13"><a href="#cb8-13" aria-hidden="true" tabindex="-1"></a>xlabel<span class="op">!</span>(<span class="st">&quot;Time (hr)&quot;</span>)<span class="op">;</span></span>
<span id="cb6-14"><a href="#cb6-14" 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> <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>
</details> </details>
<div class="figure"><span id="fig:unnamed-chunk-6"></span> <div class="figure"><span id="fig:unnamed-chunk-7"></span>
<img src="iss-eclipse-determination_files/figure-html5/unnamed-chunk-6-J1.png" alt="Rocket Motor Data: [@thrustcurve]" width="300" data-distill-preview=1 /> <img src="iss-eclipse-determination_files/figure-html5/unnamed-chunk-7-J1.png" alt="ISS Sunlight" width="300" data-distill-preview=1 />
<p class="caption"> <p class="caption">
Figure 1: Rocket Motor Data: <span class="citation" data-cites="thrustcurve">[@thrustcurve]</span> Figure 1: ISS Sunlight
</p> </p>
</div> </div>
</div> </div>
<div class="sourceCode" id="cb7"><pre class="sourceCode r distill-force-highlighting-css"><code class="sourceCode r"></code></pre></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:
<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:
<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:
<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>
<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>
<!--radix_placeholder_article_footer--> <!--radix_placeholder_article_footer-->
<!--/radix_placeholder_article_footer--> <!--/radix_placeholder_article_footer-->
</div> </div>
@ -1586,6 +1624,8 @@ Figure 1: Rocket Motor Data: <span class="citation" data-cites="thrustcurve">[@t
<!--/radix_placeholder_site_after_body--> <!--/radix_placeholder_site_after_body-->
<!--radix_placeholder_appendices--> <!--radix_placeholder_appendices-->
<div class="appendix-bottom"> <div class="appendix-bottom">
<h3 id="references">References</h3>
<div id="references-listing"></div>
<h3 id="updates-and-corrections">Corrections</h3> <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> <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> <h3 id="reuse">Reuse</h3>

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@book{shackelford_introduction_2015,
address = {Boston},
edition = {Eighth edition},
title = {Introduction to materials science for engineers},
isbn = {9780133826654},
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@misc{ariss,
title = {{ARISS} {TLE}},
shorttitle = {{ARISS}},
url = {https://live.ariss.org/tle/},
abstract = {This "Two-Line Element" file is published by US Joint Space Operations Center, containing the numerical coefficients of the ISS orbit.},
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{"title":"ISS Eclipse Determination","description":"Determining how much sunlight a body is receiving.","authors":[{"author":"Anson Biggs","authorURL":"https://ansonbiggs.com","affiliation":"&nbsp;","affiliationURL":"#","orcidID":""}],"publishedDate":"2021-04-01T00:00:00.000-07:00","citationText":"Biggs, 2021"} {"title":"ISS Eclipse Determination","description":"Determining how much sunlight a body orbiting a planet is receiving.","authors":[{"author":"Anson Biggs","authorURL":"https://ansonbiggs.com","affiliation":"&nbsp;","affiliationURL":"#","orcidID":""}],"publishedDate":"2021-05-01T00:00:00.000-07:00","citationText":"Biggs, 2021"}
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<a href="../../index.html#category:Astrodynamics" class="dt-tag">Astrodynamics</a> <a href="../../index.html#category:Astrodynamics" class="dt-tag">Astrodynamics</a>
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<p><p>Determining how much sunlight a body is receiving.</p></p> <p><p>Determining how much sunlight a body orbiting a planet is receiving.</p></p>
</div> </div>
<div class="d-byline"> <div class="d-byline">
Anson Biggs <a href="https://ansonbiggs.com" class="uri">https://ansonbiggs.com</a> Anson Biggs <a href="https://ansonbiggs.com" class="uri">https://ansonbiggs.com</a>
<br/>04-01-2021 <br/>05-01-2021
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<div class="d-article"> <div class="d-article">
@ -2124,9 +2132,11 @@ document.addEventListener('DOMContentLoaded', function() {
<figure> <figure>
<img src="geometry.svg" alt="Geometry of an Eclipse" /><figcaption aria-hidden="true">Geometry of an Eclipse</figcaption> <img src="geometry.svg" alt="Geometry of an Eclipse" /><figcaption aria-hidden="true">Geometry of an Eclipse</figcaption>
</figure> </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>
<figure> <figure>
<img src="vectors_radiuss.svg" alt="Body Radius’s and Position Vectors" /><figcaption aria-hidden="true">Body Radius’s and Position Vectors</figcaption> <img src="vectors_radiuss.svg" alt="Body Radius’s and Position Vectors" /><figcaption aria-hidden="true">Body Radius’s and Position Vectors</figcaption>
</figure> </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>
<h2 id="the-code">The Code</h2> <h2 id="the-code">The Code</h2>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<details> <details>
@ -2141,75 +2151,109 @@ Show code
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>theme(<span class="op">:</span>ggplot2)</span></code></pre></div> <span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>theme(<span class="op">:</span>ggplot2)</span></code></pre></div>
</details> </details>
</div> </div>
<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>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<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> <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> <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="st">ISS (ZARYA)</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="st">1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990</span></span> <span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="st">1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="st">2 25544 51.6434 300.9481 0002858 223.8443 263.8789 15.48881793278621</span></span> <span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="st">2 25544 51.6434 300.9481 0002858 223.8443 263.8789 15.48881793278621</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;&quot;&quot;</span></span></code></pre></div> <span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a><span class="st">&quot;&quot;&quot;</span></span></code></pre></div>
<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>
<div class="layout-chunk" data-layout="l-body">
<pre class="juliam"><code>ISS[1].n</code></pre>
</div> </div>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb3"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb3-1"><a href="#cb3-1" 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> <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="cb3-2"><a href="#cb3-2" 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> <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="cb3-3"><a href="#cb3-3" 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> <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> </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>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<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><span class="kw">function</span> sunlight(Rbody<span class="op">,</span> r_sun_body<span class="op">,</span> r_body_sc)</span> <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="cb4-2"><a href="#cb4-2" 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> <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>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> </span> <span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> hu <span class="op">=</span> Rbody <span class="op">*</span> norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">-</span> Rbody)</span> <span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> hu <span class="op">=</span> Rbody <span class="op">*</span> norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">-</span> Rbody)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> </span> <span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> θe <span class="op">=</span> acos((r_sun_body ⋅ r_body_sc) <span class="op">/</span> (norm(r_sun_body) <span class="op">*</span> norm(r_body_sc)))</span> <span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> θe <span class="op">=</span> acos((r_sun_body ⋅ r_body_sc) <span class="op">/</span> (norm(r_sun_body) <span class="op">*</span> norm(r_body_sc)))</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a> θu <span class="op">=</span> atan(Rbody <span class="op">/</span> hu)</span> <span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> θu <span class="op">=</span> atan(Rbody <span class="op">/</span> hu)</span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a> du <span class="op">=</span> hu <span class="op">*</span> sin(θu) <span class="op">/</span> sin(θe <span class="op">+</span> θu)</span> <span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> du <span class="op">=</span> hu <span class="op">*</span> sin(θu) <span class="op">/</span> sin(θe <span class="op">+</span> θu)</span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a> θp <span class="op">=</span> π <span class="op">-</span> atan(norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">+</span> Rbody))</span> <span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a> θp <span class="op">=</span> π <span class="op">-</span> atan(norm(r_sun_body) <span class="op">/</span> (Rsun <span class="op">+</span> Rbody))</span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a> dp <span class="op">=</span> Rbody <span class="op">*</span> sin(θp) <span class="op">/</span> cos(θe <span class="op">-</span> θp)</span> <span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a> dp <span class="op">=</span> Rbody <span class="op">*</span> sin(θp) <span class="op">/</span> cos(θe <span class="op">-</span> θp)</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">1</span></span> <span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">1</span></span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> du)</span> <span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> du)</span>
<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">0</span></span> <span id="cb6-16"><a href="#cb6-16" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> <span class="fl">0</span></span>
<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span> <span id="cb6-17"><a href="#cb6-17" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span>
<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> ((du <span class="op">&lt;</span> norm(r_body_sc)) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> dp))</span> <span id="cb6-18"><a href="#cb6-18" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> (θe <span class="op">&lt;</span> π <span class="op">/</span> <span class="fl">2</span>) <span class="op">&amp;&amp;</span> ((du <span class="op">&lt;</span> norm(r_body_sc)) <span class="op">&amp;&amp;</span> (norm(r_body_sc) <span class="op">&lt;</span> dp))</span>
<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> (norm(r_body_sc .<span class="op">|&gt;</span> u<span class="st">&quot;km&quot;</span>) <span class="op">-</span> du) <span class="op">/</span> (dp <span class="op">-</span> du) <span class="op">|&gt;</span> ustrip</span> <span id="cb6-19"><a href="#cb6-19" aria-hidden="true" tabindex="-1"></a> S <span class="op">=</span> (norm(r_body_sc .<span class="op">|&gt;</span> u<span class="st">&quot;km&quot;</span>) <span class="op">-</span> du) <span class="op">/</span> (dp <span class="op">-</span> du) <span class="op">|&gt;</span> ustrip</span>
<span id="cb4-20"><a href="#cb4-20" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span> <span id="cb6-20"><a href="#cb6-20" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span>
<span id="cb4-21"><a href="#cb4-21" aria-hidden="true" tabindex="-1"></a></span> <span id="cb6-21"><a href="#cb6-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-22"><a href="#cb4-22" aria-hidden="true" tabindex="-1"></a> <span class="kw">return</span> S</span> <span id="cb6-22"><a href="#cb6-22" aria-hidden="true" tabindex="-1"></a> <span class="kw">return</span> S</span>
<span id="cb4-23"><a href="#cb4-23" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code></pre></div> <span id="cb6-23"><a href="#cb6-23" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code></pre></div>
</div> </div>
<p>Then we can pass all the values we’ve gathered into the function we just made.</p>
<div class="layout-chunk" data-layout="l-body"> <div class="layout-chunk" data-layout="l-body">
<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>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></code></pre></div> <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> </div>
<h2 id="plotting-the-results">Plotting the Results</h2> <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>
<div class="layout-chunk" data-layout="l-body-outset"> <div class="layout-chunk" data-layout="l-body-outset">
<details> <details>
<summary> <summary>
Show code Show code
</summary> </summary>
<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>light_range <span class="op">=</span> range(colorant<span class="st">&quot;black&quot;</span><span class="op">,</span> stop <span class="op">=</span> colorant<span class="st">&quot;yellow&quot;</span><span class="op">,</span> length <span class="op">=</span> <span class="fl">101</span>)<span class="op">;</span></span> <div class="sourceCode" id="cb8"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get fancy with the line color. </span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>light_colors <span class="op">=</span> [light_range[unique(round(<span class="dt">Int</span><span class="op">,</span> <span class="fl">1</span> <span class="op">+</span> s <span class="op">*</span> <span class="fl">100</span>))][<span class="fl">1</span>] <span class="kw">for</span> s <span class="kw">in</span> S]<span class="op">;</span></span> <span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>light_range <span class="op">=</span> range(colorant<span class="st">&quot;black&quot;</span><span class="op">,</span> stop <span class="op">=</span> colorant<span class="st">&quot;orange&quot;</span><span class="op">,</span> length <span class="op">=</span> <span class="fl">101</span>)<span class="op">;</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span> <span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>light_colors <span class="op">=</span> [light_range[unique(round(<span class="dt">Int</span><span class="op">,</span> <span class="fl">1</span> <span class="op">+</span> s <span class="op">*</span> <span class="fl">100</span>))][<span class="fl">1</span>] <span class="kw">for</span> s <span class="kw">in</span> S]<span class="op">;</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>plot(</span> <span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> <span class="dt">LinRange</span>(<span class="fl">0</span><span class="op">,</span> <span class="fl">24</span><span class="op">,</span> length(S))<span class="op">,</span></span> <span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a>plot(</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> S <span class="op">.*</span> <span class="fl">100</span><span class="op">,</span></span> <span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a> <span class="dt">LinRange</span>(<span class="fl">0</span><span class="op">,</span> <span class="fl">24</span><span class="op">,</span> length(S))<span class="op">,</span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a> linewidth <span class="op">=</span> <span class="fl">5</span><span class="op">,</span></span> <span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a> S <span class="op">.*</span> <span class="fl">100</span><span class="op">,</span></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> legend <span class="op">=</span> <span class="ex">false</span><span class="op">,</span></span> <span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a> linewidth <span class="op">=</span> <span class="fl">5</span><span class="op">,</span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> color <span class="op">=</span> light_colors<span class="op">,</span></span> <span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a> legend <span class="op">=</span> <span class="ex">false</span><span class="op">,</span></span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a>)<span class="op">;</span></span> <span id="cb8-10"><a href="#cb8-10" aria-hidden="true" tabindex="-1"></a> color <span class="op">=</span> light_colors<span class="op">,</span></span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a></span> <span id="cb8-11"><a href="#cb8-11" aria-hidden="true" tabindex="-1"></a>)<span class="op">;</span></span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a>xlabel<span class="op">!</span>(<span class="st">&quot;Time (hr)&quot;</span>)<span class="op">;</span></span> <span id="cb8-12"><a href="#cb8-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-13"><a href="#cb6-13" 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="cb8-13"><a href="#cb8-13" aria-hidden="true" tabindex="-1"></a>xlabel<span class="op">!</span>(<span class="st">&quot;Time (hr)&quot;</span>)<span class="op">;</span></span>
<span id="cb6-14"><a href="#cb6-14" 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> <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>
</details> </details>
<div class="figure"><span id="fig:unnamed-chunk-6"></span> <div class="figure"><span id="fig:unnamed-chunk-7"></span>
<img src="iss-eclipse-determination_files/figure-html5/unnamed-chunk-6-J1.png" alt="Rocket Motor Data: [@thrustcurve]" width="300" data-distill-preview=1 /> <img src="iss-eclipse-determination_files/figure-html5/unnamed-chunk-7-J1.png" alt="ISS Sunlight" width="300" data-distill-preview=1 />
<p class="caption"> <p class="caption">
Figure 1: Rocket Motor Data: <span class="citation" data-cites="thrustcurve">[@thrustcurve]</span> Figure 1: ISS Sunlight
</p> </p>
</div> </div>
</div> </div>
<div class="sourceCode" id="cb7"><pre class="sourceCode r distill-force-highlighting-css"><code class="sourceCode r"></code></pre></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:
<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:
<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:
<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>
<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>
<!--radix_placeholder_article_footer--> <!--radix_placeholder_article_footer-->
<div class="article-footer"> <div class="article-footer">
<p class="social_footer"> <p class="social_footer">
@ -2235,13 +2279,15 @@ Figure 1: Rocket Motor Data: <span class="citation" data-cites="thrustcurve">[@t
<!--/radix_placeholder_site_after_body--> <!--/radix_placeholder_site_after_body-->
<!--radix_placeholder_appendices--> <!--radix_placeholder_appendices-->
<div class="appendix-bottom"> <div class="appendix-bottom">
<h3 id="references">References</h3>
<div id="references-listing"></div>
<h3 id="updates-and-corrections">Corrections</h3> <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> <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> <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>. 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>
<h3 id="citation">Citation</h3> <h3 id="citation">Citation</h3>
<p>For attribution, please cite this work as</p> <p>For attribution, please cite this work as</p>
<pre class="citation-appendix short">Biggs (2021, April 1). Anson's Projects: ISS Eclipse Determination. Retrieved from https://projects.ansonbiggs.com/posts/2021-04-14-iss-eclipse-determination/</pre> <pre class="citation-appendix short">Biggs (2021, May 1). Anson's Projects: ISS Eclipse Determination. Retrieved from https://projects.ansonbiggs.com/posts/2021-04-14-iss-eclipse-determination/</pre>
<p>BibTeX citation</p> <p>BibTeX citation</p>
<pre class="citation-appendix long">@misc{biggs2021iss, <pre class="citation-appendix long">@misc{biggs2021iss,
author = {Biggs, Anson}, author = {Biggs, Anson},

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"path": "index.html", "path": "index.html",
"title": "Anson's Projects", "title": "Anson's Projects",
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"last_modified": "2021-04-30T22:47:08-07:00" "last_modified": "2021-05-03T21:24:37-07:00"
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