What are the 3 assumptions for the two-body problem?
(1) There are only 2 bodies in the system. (2) The bodies are particles. (3) The only force is gravity.
What is the difference between inertial and non-inertial reference frames?
Inertial reference frames are non-accelerating while non-inertial reference frames are accelerating. Remember, rotating still count as non-inertial!
Is ECI (Earth Centric Inertial) an inertial reference frame? Why?
Yes because it is fixed with distant stars so it does not rotate.
Is EFEC (Earth-Fixed Earth-Centric) an inertial reference frame? Why?
No because it rotates with the Earth.
Is TRF (Topographic Reference Frame) inertial? Why?
No because it relies on the local horizon so it moves/rotates as the position on Earth changes.
How do we relate two reference frames?
Through DCMs (Directional Cosine Matrices)
What is the counterclockwise $ R_3(\theta) $ rotation matrix?
$$ \begin{bmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
What is the counterclockwise $ R_2(\theta) $ rotation matrix?
$$ \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix} $$
What is the counterclockwise $ R_1(\theta) $ rotation matrix?
$$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{bmatrix} $$
Using rotation matrices $ R_3 $, $ R_2 $, and/or $ R_1 $, how would you form a 3-1-3 rotation given the angles used are $ \theta_1 $, $ \theta_2 $, $ \theta_3 $ in that order?
$$ R_3(\theta_3) • R_1(\theta_2) • R_3(\theta_1) $$
Using rotation matrices $ R_3 $, $ R_2 $, and/or $ R_1 $, how would you form a 3-2-1 rotation given the angles used are $ \theta_1 $, $ \theta_2 $, $ \theta_3 $ in that order?
$$ R_1(\theta_3) • R_2(\theta_2) • R_3(\theta_1) $$
What is Kepler's first law?
Planets orbit in elliptical orbits with their orbited mass at one of the 2 foci.
What is Kepler's second law?
Radians vectors sweep equal areas in equal amounts of time.
What is Kepler's third law?
The period squared is proportional to the semi-major axis cubed.
What is Newton's Law of Gravitational Motion?
$$ \underline{F}_{12} = G \frac{m_1 m_2}{r^3} \underline{r} $$