Part 1: Astrodynamics
Overview
Astrodynamics is the field that studies satellite motion, encompassing natural satellites like planets and moons, as well as artificial satellites such as spacecraft. It is one of the broadest terms in the astronautics community, often serving as a catch-all for various sub-topics and research areas.
Key Concepts and Terminology
The main entities in astrodynamics are spacecraft and celestial bodies. Our attention will focus primarily on spacecraft.
States
Spacecraft have both translational and rotational dynamics. The study of translational motion is typically referred to as orbital mechanics, whereas the study of rotational motion is attitude dynamics. Explicitly, orbital mechanics studies the position and velocity of the satellite \([\mathbf{r}, \mathbf{v}]\), whereas the orientation and angular rates correspond to the attitude \([\boldsymbol{\theta}, \boldsymbol{\omega}]\). Together, the translational and rotational variables can be stacked together to form a single vector refer to as the spacecraft state, \(\mathbf{X} = [\mathbf{r},\mathbf{v}, \boldsymbol{\theta}, \boldsymbol{\omega}]\).
Reference Frames
Importantly, these states don't mean much unless you define them "with respect" to some reference frame (e.g. a position vector of \(r=[5, 10, 0]\) is ambiguous --- Is \(r\) defined with respect to the planet? With respect to another spacecraft? With respect to the sun?). To eliminate this ambiguity, we define reference frames.
Non-rotating / non-accelerating reference frames are called inertial reference frames. Think of these as basis vectors \(\mathcal{N}: \{\hat{x}, \hat{y}, \hat{z}\}\) that are held fixed in place for all of time. Inertial reference frames are particularly intuitive, but not always convenient for expressing our state. For example, if we wanted to describe our position on the surface of the Earth in an inertial frame, our location would constantly be changing because the Earth is rotating. If we wanted to avoid the hassle of tracking our constantly changing position, we could define a reference frame that is itself rotating (henceforth called non-inertial reference frames) and define our state with respect to said frame.
In astrodynamics, there are no shortage of reference frames to choose from, but the two most commonly used are the Earth Centered Inertial (ECI) reference frame, and the Earth Centered Earth Fixed (ECEF) reference frame (non-inertial). Both reference frames define Earth as the origin, but the ECI basis vectors never rotate, whereas the ECEF basis vectors do. Many alternative reference frames exist for non-Earth centric problems, but those will be introduced as needed.
Orbital Elements
While the state vector \(\mathbf{X}\) is an convenient way to capture the dynamical properties of your spacecraft at an instantaneous moment in time, it is not particularly intuitive. Given the translational state of a spacecraft at \(t_0\), $$ \mathbf{X}_0 = [15000 \text{ km}, 5000 \text{ km}, 3000 \text{ km}, 0.65\text{ km/s}, 2.0\text{ km/s}, 1.7\text{ km/s}] $$ it is unclear if this spacecraft is even in orbit. At these speeds, it may have escaped the gravitational well of the planet and gone off on a hyperbolic trajectory. Even if it is captured, it's unclear what the geometry of the orbit is --- e.g. is the orbit circular or oblate? Is the spacecraft close the body, or far away?
These are important questions when designing missions, but cartesian states do a lousy job at answering them. Thankfully there is a better way to describe a spacecraft's state: Orbital Elements.
**Orbital elements are an alternative way to represent the spacecraft state that uniquely capture the geometry of the orbit using five time-invariant variables and one time-varying angle. Each element has its own name, and they are often presented the same order:
| Parameter | Symbol | Description |
|---|---|---|
| Semi-major axis | \(a\) | Determines the size of the orbit / ellipse. |
| Eccentricity | \(e\) | Describes the orbit's shape, ranging from circular (\(e=0\)) to parabolic (\(e=1\)). |
| Inclination | \(i\) | The tilt of the orbit's plane relative to the equatorial plane. |
| Right ascension of the ascending node (RAAN) | \(\Omega\) | Defines the horizontal orientation of the orbit. |
| Argument of periapsis | \(\omega\) | Specifies the location of the orbit's closest approach to the primary body. |
| True anomaly | \(\nu\) | The spacecraft's position along the orbit at a specific time. |
Broadly speaking, the first two define the size and shape of the orbit, the next three define its orientation with respect to the body, and the final coordinate positions the spacecraft on the orbit.
These elements are convenient, as they capture every property of the entire orbit...forever. Now, when given a spacecraft's orbit elements, $$ \mathbf{X}_{\text{OE}} = [3R_e, 0.1, 45^o, 0, 0, 180^{o}] $$ you immediately can tell that the orbit is 3x the size of the Earth, it's slightly oblate (\(e=0.1\)), it's slightly inclined with respect to the equator, and the spacecraft is at the farthest point for the Earth \(\nu = 180^{o}\).
From the orbital elements, additional useful (geometric) variables can also be defined:
| Jargon | Description |
|---|---|
| Periapsis | The part of the orbit closest to the body. Fastest and most complex part of the orbit |
| Apoapsis | The part of the orbit farthest to the body. Slowest and least complex part of the orbit |
Orbit Regimes
Most spacecraft orbit the Earth, and there are two primary altitudes at which these spacecraft operate. The first is Low-Earth Orbit, or LEO, which is approximately 400 km altitude above the Earth (radius 6400 km). Spacecraft flying in LEO include the international space station and most science satellites. The second altitude band is Geostationary Earth Orbit, or GEO. Satellites operating at GEO are approximtaely 35,000 km above the Earth and the geometry of their orbit allows the spacecraft to match the same rotational rate of the Earth. This makes the satellite appear stationary to an observer on the surface of the Earth, making these satellite useful for surveillance or communication.
Other than Earth-Orbiting satellite, there are also spacecraft that are sent farther into the solar system. These are referred to as interplanetary spacecraft. For spacecraft sent to the Moon (neither Earth orbiting, nor interplanetary) these are often referred to spacecraft operating in the Cislunar regime.
Perturbations
Thus far, all discussion of orbits have assumed the spacecraft is in an ideal two body problem or 2BP. This assumes there is one planet and one spacecraft and the only force is gravity (as represented by an idyllic point mass potential). The 2BP yields perfectly periodic solutions, and it forms the foundation of all astrodynamics. However, as you might expect, most real-world systems cannot assume this simple two-body motion.
Instead, all spacecraft motion is affected by dynamical perturbations --- things that change the motion from the ideal two body case. Examples of these perturbations include:
- High-Order Gravity --- The Earth isn't a point mass (it has a tummy!).
- Atmospheric Drag --- The tiny particles in the upper atmosphere can slow spacecraft down and cause them to fall into the atmosphere and burn up.
- Solar Radiation Pressure --- The sun's photons impart momentum on spacecraft and can speed them up or slow them down
- Electrostatic forces -- The charged particles in the plasmas in Earth's upper atmosphere and the ions from the sun can charge up different parts of the spacecraft and induced undesired forces
- Many, many others...
These forces can be very small when orbiting large bodies where gravity is dominant, but they can become very important when orbiting small-bodies like asteroids and comets which have extremely weak gravity. Considerable research explores ways to transform these undesirable forces into desirable fuel-free control.
Orbital Maneuvers
Most missions require spacecraft to maneuver to enter, or maintain, their desired orbit. Common maneuvers include:
- Hohmann transfer: An efficient method for transferring between two circular orbits.
- Bi-elliptic transfer: Used for larger orbital changes, offering fuel savings under specific conditions.
- Inclination change: Alters the orbital plane, typically requiring significant energy.
Beyond The Two Body Problem
Even beyond these perturbations, when spacecraft travel into deep space, we can no longer treat the problem as if there are only two gravitational bodies influencing spacecraft motion. The solar system has many bodies, each exhibiting their small force on the spacecraft. This is perhaps most apparent in the cislunar environment --- where the Earth and Moon both exhibit significant gravitational forces on the spacecraft. In these environments, astrodynamicsts must transition to higher-order models including the the three-body problem (3BP), or its simpler version the circularly restricted three body problem (CR3BP).
The CR3BP assumes two massive bodies are in a circular orbit around their center of mass, and there is a third tiny body (spacecraft) whose gravitational force is negligible. The CR3BP is far more complex than the 2BP as there are no longer orbital elements that remained fixed with time. Despite this, many astrodynamicists prioritize the study of the 3BP because it strikes the right level of fidelity above the 2BP for the most common cislunar and interplanetary problems (Earth-Moon or Sun-Jupiter) without transitioning to the full N-Body problem.
Types of Orbital Motion in Dynamically Complex Environments
In these more complex environments, you may hear of specific types of orbits including:
| Orbit Type | Definition |
|---|---|
| Halo Orbits | Three-dimensional periodic orbits that exist around the Lagrange points in the three-body problem, typically found around L1, L2, or L3 points, often used in space missions to maintain a stable position relative to two larger bodies (e.g., Earth and Moon). |
| Lissajous Orbits | Complex, non-periodic orbits around Lagrange points that form a shape resembling Lissajous figures. These orbits are more flexible than halo orbits and do not require precise station-keeping, making them useful for missions where long-term stability around a Lagrange point is desired. |
| Quasi-periodic Orbits | Orbits that are not exactly periodic but return close to their initial conditions after a long time. These orbits do not close on themselves but instead fill a region of space densely. They are common in systems with weak gravitational forces or when small perturbations prevent perfect periodicity. |
These types of motion get really interesting and complex. You'll hear people talk about stable and unstable manifolds and transfers between them.
Beyond the 3BP
Generally academics will study these simplified problems to try and glean information about spacecraft motion that can be leveraged in different real-world missions; however, there will always come a point where higher fidelity is needed to validate the proposed motion. This gets into interplanetary trajectory design which incrementally increases the dynamical fidelity to validate trajectories. At the highest level of fidelity, these mission designers use an ephermeris model --- a very precise model of the solar system + its forces --- and then use fancy propagators and simulation software (MONTE, COPERNICUS, GMAT) to confirm the validity of the motion under the most rigorous simulators. These do take very long amount of time to run, which is why studying these low fidelity approximations is so important.
Attitude Dynamics
As discussed, translational motion of spacecraft is only one half of their dynamics. The other half studies the rotational motion and forces acting on a spacecraft (Attitude Dynamics).
Fun Aside
My PhD advisor was / is considered a world expert in attitude dynamics. When I first applied to his lab, I didn't know what this meant, and I thought he was particularly into leadership coaching or something.
Attitude Descriptions
There are surprisingly many ways to represent the rotation of an object. Within aerospace engineering, undergraduates are typically taught about aircraft motion using variables of pitch, roll, and yaw which broadly correspond to the aircraft going up down, rotating along nose, and pivoting left and right.
**Pitch, roll, and yaw are the colloquial form of things called Euler Angles. These are simplest way to capture the rotation of a rigid object with respect to some fixed reference frame. The way these can be defined is to pin one reference frame to the ground, and pin a second reference frame to your spacecraft, then sequence three rotations to get from one reference frame to the other.
A more direct option for mapping between reference frames is the Direction Cosine Matrix, or DCM. These are rotation matrices of 3x3 elements. They are very convenient as are just linear operators that can map vectors in one frame to the other and back trivially.
The downside of DCM's however, is that they have 9 total parameters (3x3 matrix). That's a lot of parameters for a spacecraft to store on-board. Therefore, researchers looked for more efficient ways to capture this information and came with the quaternion. Quaternions require only four parameters and are non-singular. Modified Rodrigues Parameters, or MRPs, are even more compact, requiring only three coordinates, but they do carry a singularity at certain orientations.
Rigid Body Dynamics
Once you have ways to describe different rotations, then you can explore the dynamics of those rotations, studying how torques influence the spacecrafts motion. This is referred to as Rigid Body Dynamics. In 602 you should walk through the equivalent of Newton's laws for rotation which is just adding up torques and angular momentum (whereas translation dynamics just add up forces and momentum). You'll have to spend a little more time book-keeping the various reference frames, and this will become extraordinarily repetitive after some point.