Determining how much sunlight a body orbiting a planet is receiving.

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.

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.

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. `Rsun`

and `Rbody`

are the radius of the Sun and Body (In this case Earth), respectively. `r_sun_body`

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 Ephemerides in Julia but the package is being worked on so I may revisit this and do a more rigorous analysis in the future. `r_body_sc`

is a position vector from the center of the body being orbited, to the center of our spacecraft.

```
using Unitful
using LinearAlgebra
using SatelliteToolbox
using Plots
using Colors
:ggplot2) theme(
```

To get the orbit for the ISS, I used a Two-Line Element 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 TLE,” n.d.). The Julia Package 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, we now have access to the information to start making our ISS orbit.

```
= tle"""
ISS ISS (ZARYA)
1 25544U 98067A 21103.84943184 .00000176 00000-0 11381-4 0 9990
2 25544 51.6434 300.9481 0002858 223.8443 263.8789 15.48881793278621
"""
```

```
1-element Vector{TLE}:
TLE: ISS (ZARYA) (Epoch = 2021-04-13T20:23:10.911)
```

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.

`1].n ISS[`

```
= init_orbit_propagator(Val(:twobody), ISS[1]);
orbit = 0:0.1:((24 / ISS[1].n) .* 60 * 60); # ISS[1].n gives the mean motion, or orbits per day.
time , r, v = propagate!(orbit, time); o
```

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 (Vallado 1997).

```
function sunlight(Rbody, r_sun_body, r_body_sc)
= 695_700u"km"
Rsun
= Rbody * norm(r_sun_body) / (Rsun - Rbody)
hu
= acos((r_sun_body ⋅ r_body_sc) / (norm(r_sun_body) * norm(r_body_sc)))
θe
= atan(Rbody / hu)
θu = hu * sin(θu) / sin(θe + θu)
du
= π - atan(norm(r_sun_body) / (Rsun + Rbody))
θp = Rbody * sin(θp) / cos(θe - θp)
dp
= 1
S if (θe < π / 2) && (norm(r_body_sc) < du)
= 0
S end
if (θe < π / 2) && ((du < norm(r_body_sc)) && (norm(r_body_sc) < dp))
= (norm(r_body_sc .|> u"km") - du) / (dp - du) |> ustrip
S end
return S
end
```

Then we can pass all the values we’ve gathered into the function we just made.

`= r .|> R -> sunlight(6371u"km", [0.5370, 1.2606, 0.5466] .* 1e8u"km", R .* u"m"); S `

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 spent in the penumbra is almost insignificant.

```
# Get fancy with the line color.
= range(colorant"black", stop = colorant"orange", length = 101);
light_range = [light_range[unique(round(Int, 1 + s * 100))][1] for s in S];
light_colors
plot(LinRange(0, 24, length(S)),
.* 100,
S = 5,
linewidth = false,
legend = light_colors,
color ;
)
!("Time (hr)");
xlabel!("Sunlight (%)");
ylabel!("ISS Sunlight Over a Day") title
```

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.

Time in Sun:

`= length(S[S.==1])/length(S) * 100 sun `

`62.03323593209401`

Time in Darkness:

`= length(S[S.==0])/length(S) * 100 umbra `

`37.64408511553699`

Time in Penumbra:

`= 100 - umbra - sun penumbra `

`0.322678952369003`

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.

Vallado, David A. 1997. *Fundamentals of Astrodynamics and Applications, 2nd. Ed.* Edited by Wiley Larson. Dordrecht: Microcosm, Inc.

Text and figures are licensed under Creative Commons Attribution CC BY 4.0. 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 ...".

For attribution, please cite this work as

Biggs (2021, May 1). Anson's Projects: ISS Eclipse Determination. Retrieved from https://projects.ansonbiggs.com/posts/2021-04-14-iss-eclipse-determination/

BibTeX citation

@misc{biggs2021iss, author = {Biggs, Anson}, title = {Anson's Projects: ISS Eclipse Determination}, url = {https://projects.ansonbiggs.com/posts/2021-04-14-iss-eclipse-determination/}, year = {2021} }