Chapter 6 — Key Takeaways (Energy in Space)

A one-page reference. Reread this before an exam, or before you plan any orbital burn.

The three equations that run orbital mechanics

Equation	Use it to find
$\varepsilon = \dfrac{v^2}{2} - \dfrac{\mu}{r}$	the specific orbital energy from a speed and a position (conserved as you coast)
$\varepsilon = -\dfrac{\mu}{2a}$	the energy from the orbit's size — or, inverted, $a = -\mu/(2\varepsilon)$
$v^2 = \mu\!\left(\dfrac{2}{r} - \dfrac{1}{a}\right)$	vis-viva — the speed anywhere on any orbit, from just $r$ and $a$

Special cases of vis-viva ($a=r$ gives the circle; $a\to\infty$ gives escape):

Orbit	Speed
Circular ($a=r$)	$v_{\text{circ}} = \sqrt{\mu/r}$
Escape ($\varepsilon=0$, parabolic)	$v_{\text{esc}} = \sqrt{2\mu/r} = \sqrt{2}\,v_{\text{circ}}$
Hyperbolic, at infinity	$v_\infty = \sqrt{2\varepsilon} = \sqrt{C_3}$

Symbol	Name	Units	Notes
$\varepsilon$	specific orbital energy	$\text{km}^2/\text{s}^2 = \text{MJ/kg}$	total energy per kg; sign tells the orbit's fate
$v$	speed	km/s	the instantaneous orbital speed
$r$	distance from center	km	where you are — not the planet's radius
$a$	semi-major axis	km	how big the orbit is; $r=a$ only for a circle
$\mu$	gravitational parameter	$\text{km}^3/\text{s}^2$	Earth: $3.986\times10^{5}$ (from Ch. 2)
$v_\infty$	hyperbolic excess velocity	km/s	leftover speed at infinity; $C_3=v_\infty^2$ (Ch. 11)

$\varepsilon$	Semi-major axis	Trajectory	Fate
$\varepsilon < 0$	$a > 0$	circle / ellipse	bound — returns forever
$\varepsilon = 0$	$a \to \infty$	parabola	escapes, arrives at infinity with zero speed
$\varepsilon > 0$	$a < 0$	hyperbola	escapes with speed $v_\infty$ to spare

Raising an orbit raises its total energy $\varepsilon$ but lowers its speed — the added energy (and then some) goes into potential energy (height), so kinetic energy (speed) drops.
Virial fingerprint (circular orbits): $KE = -\varepsilon$ and $U = 2\varepsilon$. Push $\varepsilon$ up toward zero and $KE$ (speed) must fall.
To catch up with a target ahead of you, slow down (burn retrograde → lower, faster, shorter-period orbit → you gain), then re-raise. To lead, speed up (rise, slow down). On the ground "faster" and "catch up" are the same; in orbit they are opposites.

You know…	You want…	Use
$r$ and $a$	speed	vis-viva $v = \sqrt{\mu(2/r - 1/a)}$
$r$ (circular)	speed	$v_{\text{circ}} = \sqrt{\mu/r}$
$v$ and $r$	energy / orbit size	$\varepsilon = v^2/2 - \mu/r$, then $a = -\mu/(2\varepsilon)$
$r_p$ and $r_a$	semi-major axis	$a = (r_p + r_a)/2$
target $v_\infty$, parking radius $r$	injection burn	$\sqrt{v_\infty^2 + 2\mu/r} - \sqrt{\mu/r}$
burn at speed $v$	energy gained	$\Delta\varepsilon = v\,\Delta v + \tfrac12\Delta v^2$ (Oberth: burn where $v$ is large)

Pitfall	Reality
Confusing $r$ with $a$ in vis-viva	$r$ is your current distance; $a$ is the orbit's size. Equal only for a circle.
"More energy means more speed."	Higher orbit has more total energy but less speed — energy went into height.
"Escape velocity from LEO is 11.2 km/s."	$11.2$ is from the surface; from LEO, escape is $\approx 10.85$ km/s, only $\approx 3.2$ km/s more than circular.
Fire prograde to catch a target ahead.	That raises your orbit and drops you behind. Burn retrograde to catch up.
Escape first, then burn from high up.	Burning deep in the well (fast) buys more energy per m/s — the Oberth effect. Burn low.

LEO ($400\ \text{km}$) circular speed $\approx \mathbf{7.67\ \text{km/s}}$; $\varepsilon \approx -29.4\ \text{MJ/kg}$.
GEO circular speed $\approx \mathbf{3.07\ \text{km/s}}$; $\varepsilon \approx -4.73\ \text{MJ/kg}$.
$v_{\text{esc}} = \sqrt{2}\,v_{\text{circ}}$ at any radius.
Escape from LEO $\approx \mathbf{3.2\ \text{km/s}}$; trans-Mars injection from LEO $\approx \mathbf{3.6\ \text{km/s}}$ (both match the Ch. 3 delta-v map).
On a transfer ellipse (LEO$\to$GEO): $\approx 10.1\ \text{km/s}$ at perigee, $\approx 1.6\ \text{km/s}$ at apogee.

MDR: added your target orbit's Orbital Energy line — $\varepsilon = -\mu/(2a)$, circular speed, and (if escaping) escape velocity and the parking-orbit-to-escape delta-v.
orbits.py: circular_velocity(mu, r), specific_energy(mu, a), vis_viva(mu, r, a). (Ch. 8 adds period; Ch. 9 adds elements_to_rv.)