## Imports

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

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

Julia

Astrodynamics

Code

Aerospace

Notes

Space

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.

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.

```
ISS = tle"""
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
"""
```

TLE: Name : ISS (ZARYA) Satellite number : 25544 International designator : 98067A Epoch (Year / Day) : 21 / 103.84943184 (2021-04-13T20:23:10.911) Element set number : 999 Eccentricity : 0.00028580 Inclination : 51.64340000 deg RAAN : 300.94810000 deg Argument of perigee : 223.84430000 deg Mean anomaly : 263.87890000 deg Mean motion (n) : 15.48881793 revs / day Revolution number : 27862 B* : 1.1381e-05 1 / er ṅ / 2 : 1.76e-06 rev / day² n̈ / 6 : 0 rev / day³

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.

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

`sunlight (generic function with 1 method)`

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

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.
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];
plot(
LinRange(0, 24, length(S)),
S .* 100,
linewidth=5,
legend=false,
color=light_colors,
);
xlabel!("Time (hr)");
ylabel!("Sunlight (%)");
title!("ISS Sunlight Over a Day")
```