Orbital Mechanics and Spherical Geometry Lecture Notes

This set of flashcards covers key vocabulary and mathematical formulas from the lecture notes on orbital mechanics, including Keplarian orbital elements and spherical trigonometry calculations.

Semi-major axis ($a$)

An orbital element representing the size of an elliptical orbit, which can be related to the period by the formula n = \frac{2\text{\pi}}{T} = \sqrt{\frac{\mu}{a^3}}.

Eccentricity ($e$)

A parameter that determines the shape of the orbit, calculated using the semi-major axis ($a$) and semi-minor axis ($b$) as $e = \sqrt{1 - \frac{b^2}{a^2}}$.

Mean Motion ($n$)

The angular speed of an orbit, defined by the formula n = \frac{2\text{\pi}}{T}. For a period of $16$ hours, it is calculated as $1.0908 \times 10^{-4}$.

Mean Anomaly ($M$)

The product of mean motion ($n$) and time ($t$), expressed as $M = n \times t$. For a time of $2.5$ hours, it results in $0.9817\text{\,rad}$.

Kepler's Equation

The mathematical relationship between the Mean Anomaly ($M$) and the Eccentric Anomaly ($E$), written as $M = E - e \times \sin(E)$.

Eccentric Anomaly ($E$)

An angular parameter used to define the position of a body moving along an elliptic orbit, often solved iteratively using $E_{k+1} = M + e \times \sin(E_k)$.

True Anomaly ($v$)

The angle between the direction of periapsis and the current position of the body, calculated from the Eccentric Anomaly using $v = \text{atan}\left(\frac{\sqrt{1-e^2} \times \sin(E)}{\cos(E)-e}\right)$.

Spherical Law of Sines

A trigonometric identity used for spherical triangles where $\frac{\sin(A)}{\sin(a)} = \frac{\sin(B)}{\sin(b)} = \frac{\sin(C)}{\sin(c)}$.

Earth Radius ($R_E$)

The constant value used for geodetic calculations in the notes is $6378.137\text{\,km}$.

Planetary Distance ($distance$)

The linear distance between two points on a surface derived from changes in latitude ($d\phi$) and longitude ($d\lambda$), calculated as $\text{distance} = \sqrt{d\phi^2 + d\lambda^2}$.

Inclination ($i$)

One of the six orbital elements used to describe the shape and orientation of a satellite orbit, representing the tilt of the orbit.

Right Ascension of the Ascending Node ($\Omega$)

An orbital element used to specify the orientation of the orbit's plane with respect to a reference direction.

Argument of Perigee ($\omega$)

The angle from the ascending node to the perigee, measured in the direction of the satellite's motion.