Review of Constants of Integration
In Last week’s post we played around with the equation of motion to transform it into forms that we could integrate. Through that we identified 5 different constants of integration. There’s actually also a sixth that I didn’t go over last time called mean angular motion and denoted by symbol, n. In review here are the six constants of integration and their symbols
- Mass less angular momentum – h
- Eccentricity – e
- Semilatus Rectum – p
- semi major axis (Kinda) – a
- Time of Pericenter passage – τ
With those constants we can tell what our orbit is shaped like, and where we are on it; but some of them contain duplicate information. For example we can relate e, p, and a through the following relationship
As it turns out, we only need 3 of them. One to determine orbit shape, one to determine orbit size, and one to determine position in the orbit. The most common set chosen for this purpose are e to give the orbits shape, p to give the orbits size, and τ for position on orbit.
Note:If you have Batton’s Introduction to Astrodynamics I’m covering section 3.4. If you have Dover’s Fundamentals of Astrodynamcis I’m covering sections 2.1- 2.3.
Whats Missing
Now that we know what the shape and size of the orbit as well as the position of the object on our orbit, we’re missing one more important factor, its spacial orientation. Spatial orientation refers to the orbits attitude relative to the desired reference frame.
A Side note on Reference Frames
The two most common reference frames are heliocentric (Sun centered) and geocentric (earth centered). These have the sun and the earth at the origin of each of their respective reference frames.
For the heliocentric frame we define our basis vectors in the following way
- Take a unit vector from the center of the sun and point it to where the vernal equinox takes place. This is the first vector.
- Take another unit vector from the center of the sun and point it perpendicular to the orbital plane and so that the orbits of the planets were right handed. This is the third vector.
- Have a final unit vector that satisfies the right hand rule complete the triad
For the Geocentric frame we define our basis vectors in the following way
- Take a unit vector from the center of the Earth and, again, point it to where the vernal equinox takes place. This is the first vector
- Take another unit vector from the center of the earth and point it perpendicular to the equator and so that the planets rotation is right handed. This is the third vector.
- Have a final unit vector that satisfy the right hand rule complete the triad
Now lets also take a moment to define an orbital frame for the orbit
For the orbital frame we define our basis vectors in the following way
- Take a unit vector from the center of the body we are orbiting around and point it to the point of closest approach. This is the first vector and is called the apsidal line denoted with an ie
- Normalize the angular monument vector. This is the third vector, ih
- Have a final unit vector that satisfy the right hand rule complete the triad and is denoted im
Orbital Elements
Now, to place the orbit in an orientation of one of these reference frames, we need at least 3 more pieces of information. We’re going to go with a set of 3 Euler angles Ω,i,ω.
- Ω – Is the longitude of the ascending node and is defined as the angle from the vernal equinox to where the body crosses the reference plane with a positive z velocity. Note: This point is called the ascending node and is specified by in
- i – Is the angle of inclination between the orbital plane of the body to the normalized angular momentum vector
- ω – Is the argument of perihelion which is the angle from the ascending node to the point of closes approach.
I’ve illustrated these quantities in Figure 1, reproduced below
Figure 1 Orbital Elements
In case you were wondering, yes, I did draw that in MS paint. I’ve tried to pick a color scheme that should still be visible if you have blue-yellow colorblindness. The blue orbit and lines are the reference frame and reference coordinate system. The Yellow (Or red if you have Tritanopia) orbit and lines are the orbital reference frame. The black is the intermediate reference frame and the orbital elements.
BMW’s Fundamentals of astrodynamics has a gorgeous illustration of the different orbital elements on page 59 Figure 2.3-1. They use a slightly different set of notion for their earth basis vector of i,j, and k.
If you want to transform a coordinate from the reference frame to the orbital frame I suggest the method of a rotation matrix. I may go over rotation matrices in a future post as a refresher on attitude dynamics and kinematics but for now know that the rotation matrix you need for the set of orbital elements given above, Ω,i,ω, is a 3-1-3 / Z-X-Z . If you want to transform coordinates from the orbital frame to the reference fame you use the transpose of the 3-1-3 / Z-X-Z rotation matrix.
Interactive Element
I came across this website which has a visually very well put together interactive orbital elements simulator. It’s still a work in progress and can’t handle parabolic or hyperbolic orbits but is still worthwhile to play around with the orbital elements
Originally I had planed to see if I could put an interactive element here where you could adjust different orbital elements and see that change the shape/size/orientation of the orbit similar to the link above. Unfortunately, without a large investment of my time, I can only load static Jupyter sheets so sliders were out of the picture for this weeks post. You can find my Jupyter sheet here (It will need to downloaded as github also doesn’t support the interactive elements). I stopped working on it after I realized I wouldn’t be able to put it up and have it be interactive, but it can setup parabolic and hyperbolic orbits. Don’t worry though, interactive demonstrations – now using JavaScript instead of python – are still on my want to do list!
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