```
---
title: "Skip-Entry Dynamics"
description: "Exploring skip-entry dynamics in celebration of Artemis 1!"
date: "2022-12-11"
file-modified: "2023-01-27"
author: "Joe(y) Carpinelli"
image: https://www.nasa.gov/sites/default/files/styles/full_width_feature/public/thumbnails/image/fd2_opnav_art001e000013_orig.jpg
categories:
- Julia
- Dynamics
- Space
- Technical
aliases:
- /blog/posts/skipping-entry.html
format:
html: default
ipynb:
embed-resources: true
output-file: skipping-entry.ipynb
---
:::{.callout-note appearance="simple"}
The content here was originally written as part of a problem set assigned in a
graduate launch and entry vehicle design course at the University of Maryland.
The problem statement was flawed. I believe this was deliberate --- either way, the
assignment was a brilliant educational tool. I'm thankful for that slight falsehood;
it cemented my early understanding of spacecraft entry dynamics, and has motivated this
whole post.
:::
quarto-executable-code-5450563D
```julia
#| echo: false
#| output: false
using Logging
Logging.disable_logging(Logging.Info)
```
## Artemis 1
Go Artemis! NASA's first Artemis mission launched successfully in the earliest hours of
November 16th, 2022. Artemis 1 was an uncrewed flight _test_ --- the
[Orion spacecraft](https://www.nasa.gov/exploration/systems/orion/index.html) flew to the
moon, arrived in a special kind of stable orbit called a
[_Distant Retrograde Orbit_](https://www.nasa.gov/feature/orion-will-go-the-distance-in-retrograde-orbit-during-artemis-i).
Of course, NASA's flight operators in Mission Control captured [stunning pictures](https://www.nasa.gov/content/artemis-1-images)
along the way. A visual overview for Artemis 1 is shown in @fig-a1-arch.
Artemis 1 is [record breaking](https://www.theverge.com/2022/11/29/23484571/artemis-1-halfway-record-distance) through
its furthest achieved distance from Earth, and through one of its most important test objectives ---
the first [successfully test](https://www.lockheedmartin.com/en-us/news/features/2022/orion-heat-shield.html)
of a skip-entry for a human-rated spacecraft. Orion's heatshield was tested successfully when it entered
Earth atmosphere on December 11th, 2022.
:::{#fig-a1-arch fig-cap="Artemis 1 Mission Architecture" .column-page style="border-radius: 7px;"}
![](https://www.nasa.gov/sites/default/files/thumbnails/image/mission_profile_simple_artemis_i_droarticle_acrane.jpg){fig-align="center" style="border-radius: 7px;"}
:::
## Atmospheric Entry
Testing a new spacecraft's capability to safely enter Earth's atmosphere is incredibly
important. If a spacecraft is to return to Earth in-tact, it has to survive the fall through Earth's
atmosphere. Returning spacecraft are hurtling through space at tens of thousands of miles per hour;
as the spacecraft collide with air particles at high speeds, an enormous amount of heat is
generated. To withstand the heat of entering a planet's atmosphere from space, spacecraft
commonly have a thick metal component which is oriented towards the atmosphere for the
duration of the atmospheric entry. This metal hardware is referred to as a heat shield.
Artemis 2 will fly four astronauts to a lunar orbit before returning back to Earth. A
successful atmospheric entry in Artemis 1 will prove the Orion spacecraft capable of safely
returning astronauts back to Earth in Artemis 2, and future Artemis missions.
The _angle_ of a spacecraft's flight into Earth's atmosphere (mostly) determines the _kind_
of atmospheric entry: ballistic, lifting, or skipping. A steep entry will cause the
spacecraft to _cannonball_ through Earth's atmosphere; if we model this scenario while
_ignoring the force of gravity on the spacecraft_, the spacecraft's trajectory is _ballistic_.
A slightly less steep entry will allow the atmosphere to _lift_ the spacecraft as it descends;
this force exerted by the atmosphere on the spacecraft is similar to how airplanes fly, and
this flavor of atmospheric entry is known as _lifting_ entry. Finally, entering the atmosphere
at a shallow angle will cause the spacecraft to _skip_ across the atmosphere like a rock on
a pond; this is known as a _skipping_ entry.
No human-rated vehicle as _ever_ completed a skip-entry before Artemis 1. Orion is the first!
There are [many benefits](https://www.lockheedmartin.com/en-us/news/features/2022/orion-skip-maneuver.html)
to skip-entries, including lower accelerations experienced by astronauts, more precise landing
targets, and lower temperatures on the heat shield. While the benefits of skip-entries are
real, all atmospheric entries are incredibly sensitive. Enter the atmosphere at too steep an
angle, and the spacecraft will burn up in the atmosphere. Enter the atmosphere at too
shallow an angle, and the spacecraft will skip too far and fly out into space, never to
return. Rigorous calculations must be made to ensure the spacecraft is entering the
atmosphere at the right angle, and the right speed. With a few simplifying assumptions, the
math behind atmospheric entry can be simple enough to fit in a blog post! Don't believe me?
Read on!
## Entry Dynamics
We can simulate the Orion spacecraft's trajectory through the atmosphere with just a couple
dozen lines of code! To build a semi-accurate model for how a spacecraft enters Earth's atmosphere, we'll need
to simplify the problem by make some assumptions. For example, let's completely ignore winds!
We should also ignore parachutes --- we will just assume parachutes deploy successfully at _some_
altitude. Let's pretend the Orion spacecraft is completely rigid by ignoring the possibility
of any bending or flexing in the structure that could affect the aerodynamics throughout entry.
Also, we are leaving out all considerations related to heat! No thermodynamic modeling in this
post. We will assume the heat shield is capable of handling any entry we throw at it.
:::{.callout-note appearance="simple"}
The word _model_ might seem a bit abstract. What does a model look like? A model can be
thought of as the combination of two things: simplifying assumptions, and equations which
rely on the simplifying assumptions. So all we need to build a model is to list the assumptions
we make, and use those assumptions to write equations. Hey! That's only two things!
:::
There are many other assumptions we're making implicitly in this problem. While we can't
possibly list them all, let's go into some more depth for a couple of the most important
assumptions our model will rely on.
### Exponential Atmosphere
Earth's atmosphere will exert aerodynamic forces on the Orion spacecraft. If we ignore wind,
these aerodynamic forces will depend on some aerodynamic coefficients, the spacecraft's
airspeed and orientation, as well as the _density_ of the atmosphere.
This atmospheric density will vary with altitude! We can
create a _very_ simple model for Earth's atmosphere by assuming the density decays _exponentially_
as altitude increases. The core equation for $\rho$, the atmospheric density, will depend
on altitude above sea level $h$, the atmospheric density at sea level $\rho_0$, and a scaling factor
$h_s$ which is unique to each planet. For Earth, an accurate scaling factoris approximately equal to
$7524$ meters.
$$
\rho = \rho_0 e^\frac{-h}{h_s}
$$ {#eq-exp-atm}
### Aerodynamic Coefficients
The aerodynamic forces on Orion can be summarized by a few aerodynamic coefficients:
the ballistic coefficient $\beta$, the coefficient of drag $C_D$, and the lifting
coefficient $C_L$. The coefficients of lift and drag are commonly combined into one value:
the _lift to drag ratio_ $C_R$. For this post, one value for Orion's $C_R$ will be assumed.
The ballistic coefficient can be calculated using the vehicle's mass, lift to drag ratio,
and the surface area of the heat shield, as shown in @eq-ballistic-coeff.
$$
C_R = \frac{C_L}{C_D}
$$ {#eq-l-to-d}
$$
\beta = \frac{m}{C_R A_s}
$$ {#eq-ballistic-coeff}
Code which calculates the ballistic coefficient, and other calculations relevant to atmospheric
entry dynamics, is provided below.
:::{#lst-calculations lst-cap="Calculations Relevant to Atmospheric Entry"}
quarto-executable-code-5450563D
```julia
#| output: false
"""
Calculate the ballistic coefficient, β.
"""
ballistic_coefficient(m, Cᵣ, Aₛ) = m / (Cᵣ * Aₛ)
"""
Calculate the derivative of the downrange distance, ḋ.
"""
downrange_velocity(ν, γ) = ν * cos(γ)
"""
Calculate the terminal velocity, vₜ.
"""
terminal_velocity(g, β, ρ, γ) = √(-2g * β * sin(γ) / ρ)
"""
Calculate the maximum deceleration, nₘ.
"""
maximum_deceleration(vₑ, γ, hₛ, e) = (vₑ^2 / hₛ) * (sin(γ) / 2e)
```
:::
### Equations of Motion
With all of the assumptions described above, we can write the _equations of motion_ for a
spacecraft as it flies through a planet's atmosphere. The equations shown through math and
code below describe how **four** specific values change with time as the spacecraft flies:
the angle of the spacecraft's velocity with respect to the spacecraft's horizontal axis $\gamma$,
the airspeed $v$, the distance to Earth's center $r$, and the angle of the spacecraft's
position with respect to the horizontal along the Earth's center $\theta$. Please accept these
equations as "given" for now. Check back later and there might be an expanded explanation
posted as an update!
$$
\begin{align}
\dot{\gamma} &= \frac{1}{v} \left( L_m - (1 - \frac{v^2}{v_c^2}) g \cos{\gamma} \right) \\
\dot{v} &= -D_m - g \sin{\gamma} \\
\dot{r} &= v \sin{\gamma} \\
\dot{\theta} &= \frac{v}{r} \cos{\gamma} \\
\end{align}
$$ {#eq-core-eom}
The code below defines a function, `CanonicalEntry`, which produces the equations of motion for a
spacecraft's atmospheric entry along a plane _in code_.
quarto-executable-code-5450563D
```julia
#| output: false
#| lst-label: Skip Entry Dynamics
using Memoize: @memoize
using Symbolics, ModelingToolkit
using PhysicalConstants.CODATA2018: NewtonianConstantOfGravitation as G₀
"""
Construct a model for entry dynamics.
"""
@memoize function CanonicalEntry(; name = :CanonicalEntry, simplify = true, structural_simplify = true)
@variables t
x = @variables γ(t) v(t) r(t) θ(t)
p = @parameters r₀ ρ₀ hₛ β Cᵣ μ
δ = Differential(t)
vc = √(μ / r)
g₀ = μ / r₀^2
g = g₀ * (r₀ / r)^2
h = r - r₀
ρ = ρ₀ * exp(-h / hₛ)
Dₘ = (ρ / 2) * v^2 / β
Lₘ = Cᵣ / Dₘ
eqs = [
δ(γ) ~ (1/v) * (Lₘ - (1 - (v/vc)^2) * g * cos(γ)),
δ(v) ~ -Dₘ - g * sin(γ),
δ(r) ~ v * sin(γ),
δ(θ) ~ (v / r) * cos(γ)
]
if simplify
map!(ModelingToolkit.simplify, eqs, eqs)
end
model = ODESystem(
eqs, t; name = name,
)
if structural_simplify
model = ModelingToolkit.structural_simplify(model)
end
return model
end
```
Calling `CanonicalEntry` produces a model object, which we can inspect for the equations
of motion written mathematically. This expands all of the equations; the output looks a lot
more complicated than @eq-core-eom! Thank goodness for computers.
quarto-executable-code-5450563D
```julia
model = CanonicalEntry()
model |> equations .|> ModelingToolkit.simplify
```
## Simulated Skip-Entry
### Backstory
In 2020, I took a graduate Launch & Entry Vehicle Design course at the University of Maryland.
The course was taught by [Dr. Akin](spacecraft.ssl.umd.edu), and one of our homework assignments
was to simulate a _lifting entry_ for the Orion spacecraft. One of my classmates and I easily
spent 20+ hours trying to make the simulation work, but no matter what we tried, the simulated
entry did not seem to be a lifting entry. We turned in the assignment the day it was due, and
hoped for grading mercy --- which we received, thankfully! Dr. Akin apologized when we turned it
in --- he had _said_ lifting entry, but really the initial conditions he gave us instead
produced a _skip_ entry! I forgot all about this for months, until I reviewed recorded
lecture videos from a previous year of the course while preparing for an exam. I came across
a lecture from years earlier, where my professor gave _the same apology_ to _that_ class!
I don't mean to make any accusations, but I suspect that mix-up was intentional! Regardless
of the intent, the effect was brilliant. I've never forgotten those couple of lectures introducing
atmospheric entry. Plus, the assignment motivated a fun blog post!
### Initial Conditions
To simulate Orion's skip-entry, we need initial conditions. Let's assume the initial conditions
provided in @tbl-ic. We can "plug" these conditions into the dynamics described previously
to simulate an atmospheric skipping entry!
|Symbol|Description|Value|Units|
|------------|------------------------------------|----------------------------|------------------------------------|
| $m$ | Orion Mass | $10,400$ | kilograms |
| $A_s$ | Heatshield Surface Area | $19.635$ | square meters |
| $C_R$ | Lift to Drag Ratio | $0.25$ | meter squared per quartic second |
| $\beta$ | Ballistic Coefficient | $441.39$ | kilograms per meter squared |
| $\rho_0$ | Atmospheric Density at Sea Level | $1.226$ | kilograms per meter cubed |
| $h_s$ | Atmospheric Scaling Factor | $7524$ | meters |
| $\mu$ | Earth's Mass Parameter | $3.986 \times 10^14$ | meters per second cubed |
| $r_0$ | Earth's Radius | $6378$ | kilometers |
| $\gamma$ | Entry Flight Path Angle | $-5^\circ$ | degrees |
| $v$ | Entry Velocity | $8.939$ | kilometers per second |
| $r$ | Entry Radius | $6500.1$ | kilometers |
| $\theta$ | Entry Angular Position | $349.3^\circ$ | degrees |
: Initial Conditions for Skip Entry {#tbl-ic}
:::{.callout-note appearance="simple"}
An earlier version of this post used a flight path angle of $-2.5^\circ$. This was the
value assigned for the problem set, and it produced results which were highly sensitive
to solver tolerances. Setting the absolute and relative tolerances below `1e-9` caused
the spacecraft to "skip" for over one full day! The current flight path angle value,
$-5^\circ$, produces plots which better represent atmospheric skip-entry dynamics.
:::
### Simulation
Finally --- let's simulate Orion's entry into Earth's atmosphere!
quarto-executable-code-5450563D
```julia
#| lst-label: lst-integration
#| lst-cap: Numerical Integration for Skip Entry
#| output: false
using Plots
using Unitful
using DifferentialEquations
time = NaN # the time does not matter!
m = 10.4e3u"kg"
A = 19.635u"m^2"
C = 0.25u"m^2/s^4"
β = 441.39u"kg/m^2"
ρ = 1.226u"kg/m^3"
h = 7524.0u"m"
μ = 3.986e14u"m^3/s^2"
R = 6378u"km"
γ = -5.0u"°"
v = 8.939u"km/s"
r = 6500.1u"km"
θ = 349.3u"°"
states = [
γ, v, r, θ,
] .|> upreferred .|> ustrip
parameters = [
C, R, h, μ, ρ, β
] .|> upreferred .|> ustrip
sealevel(u,t,integrator) = u[3] - ustrip(upreferred(R))
impact = ContinuousCallback(sealevel, terminate!)
timespan = (0.0, 24 * 60 * 60 * 7)
problem = ODEProblem(
model, states, timespan, parameters,
)
trajectory = solve(problem, Vern9(); callback = impact, abstol = 1e-14, reltol = 1e-14)
```
Let's plot the spacecraft's
altitude across time. Do you see the altitude jump before Orion finally descends to the
Earth's surface? That's the spacecraft skipping across the atmosphere --- just like a rock
skips across a pond, if the rock was moving at tens of thousands of miles per hour.
quarto-executable-code-5450563D
```julia
#| fig-label: fig-linear-altitude
#| fig-cap: Spacecraft Altitude Across Time
altitude = map(
u -> ustrip(u"km", u[3] * u"m" - R),
trajectory.u,
)
linear = plot(
trajectory.t, altitude;
title = "Orion's Altitude Throughout (Re)Entry",
label = "h(t)",
xlabel = "Time (seconds)",
ylabel = "Altitude (km)",
)
```
We can also plot the altitude alongside the angle of the spacecraft's trajectory with respect to an arbitrary horizontal axis across Eath. The result depicts the spacecraft's
orbit about Earth!
quarto-executable-code-5450563D
```julia
#| fig-label: fig-polar-altitude
#| fig-cap: Spacecraft Altitude Across Angular Position
angle = map(
u -> rad2deg(u[4]),
trajectory.u,
)
polar = plot(
angle, altitude;
proj = :polar,
title = "Orion's Orbit About Earth",
label = "h(t)",
xlabel = "Time (seconds)",
ylabel = "Altitude (km)",
)
```
This is a really fun exercise. By breaking the problem down to first-principles, we can predict
the behavior of a really complicated system. Thanks to Dr. Akin for assigning this problem over
two years ago, and thanks to my friend and classmate Kate for banging her head against the wall
with me to figure this out. Finally, thanks to you for reading!
```