Foundations — Hohmann Δv calculation — both maneuvers

Before you can read the parent note Hohmann Δv calculation — both maneuvers, you need to be fluent in a small pile of symbols. This page builds each one from nothing — plain words first, then a picture, then why the topic can't live without it.

1. Orbit, circle, ellipse — the shapes we fly

Look at Figure s01. The two blue circles are the starting orbit (radius $r_1$) and the target orbit (radius $r_2$). The coral oval touching both is the transfer ellipse. Notice it touches the inner circle at its nearest point and the outer circle at its farthest point — that touching is what makes only two burns necessary.

• $r_1$ ::: radius of the inner (starting) circular orbit — the near point of the ellipse.
• $r_2$ ::: radius of the outer (target) circular orbit — the far point of the ellipse.

Why the topic needs these: every speed formula asks "how far are you from the centre right now?" That distance is always one of $r_1$, $r_2$, or a point in between.

2. Periapsis and apoapsis — the two special points

Picture a comet: it whips fast past the Sun (periapsis) and crawls slowly at the far end (apoapsis). Same idea here.

• Symbol $v_p$ ::: the spacecraft's speed at periapsis (fastest point on the ellipse).
• Symbol $v_a$ ::: the spacecraft's speed at apoapsis (slowest point on the ellipse).

Why the topic needs these: the two burns happen exactly at periapsis and apoapsis, so $v_p$ and $v_a$ are the "before/after" speeds you compare against the circle speeds.

3. Semi-major axis $a$ — the "size" of an orbit

Figure s02 shows why $2a=r_1+r_2$ for the transfer ellipse. Walk from periapsis, through the planet's centre (the focus), all the way across to apoapsis: you cross the whole long axis. That full length is $r_1+r_2$, measured from the focus. Half of it is $a$:

$a_t=\frac{r_1+r_2}{2}$

• Symbol $a$ ::: semi-major axis, the "size" number of any orbit.
• Symbol $a_t$ ::: semi-major axis of the transfer ellipse specifically.

Why the topic needs it: the master speed formula (vis-viva, below) needs to know how big the orbit is, and $a$ is the number that captures orbit size.

4. $G$, $M$, and $\mu$ — how strong the planet pulls

We use $\mu$ instead of $G$ and $M$ separately because gravity's effect on an orbit only ever appears as the product $GM$. Carrying both around is wasted effort, and $M$ for a planet is a giant messy number — $\mu$ is measured directly and cleanly.

• Symbol $\mu$ ::: $=GM$, the pull-strength of the central body, units ${\text{km}}^3/{\text{s}}^2$.

See Standard gravitational parameter for the full story.

5. Velocity vs. speed, and $\Delta v$ — the thing we're paying for

The Greek letter $\Delta$ (delta) always means "the change in." So $\Delta v=|v_{\text{after}}-v_{\text{before}}|$ — how much the velocity arrow shifted.

Here is the lucky break, shown in Figure s03: in a Hohmann transfer, both burns happen along the direction of motion (prograde). The before-arrow and after-arrow point the same way, so subtracting them stops being hard vector arithmetic and becomes plain subtraction of two numbers:

$\Delta v=|v_{\text{after}}-v_{\text{before}}|=v_{\text{after}}-v_{\text{before}}(\text{same direction})$

• Symbol $v$ ::: a speed (length of a velocity arrow), units km/s.
• Symbol $\Delta v$ ::: speed change the engine supplies — this is what fuel buys.

Why the topic needs it: $\Delta v$ is the whole point — it's the "cost" of the trip. Less $\Delta v$ means less fuel.

6. Centripetal force — why a circle needs an inward pull

For an orbit, the only thing pulling inward is gravity. Setting "gravity supplied" equal to "centripetal needed" is exactly how the circular speed is born:

$\underset{\text{gravity pulls in}}{\underbrace{\frac{\mu m}{r^2}}}=\underset{\text{circle needs}}{\underbrace{\frac{m{v}^2}{r}}}\Rightarrow v_{\text{circ}}=\sqrt{\frac{\mu }{r}}$

This is derived fully in Circular orbital velocity.

• Symbol $v_{c1}$ ::: circular speed at radius $r_1$, $=\sqrt{\mu /r_1}$.
• Symbol $v_{c2}$ ::: circular speed at radius $r_2$, $=\sqrt{\mu /r_2}$.

Why the topic needs it: these are the speeds of the two circles — the "before" of burn 1 and the "after" of burn 2.

7. Energy and vis-viva — one formula for speed anywhere

Rearranging that equality for $v$ gives the tool the whole topic leans on — the vis-viva equation:

$v^2=\mu \left(\frac{2}{r}-\frac{1}{a}\right)$

See Vis-viva equation and Specific orbital energy.

Why the topic needs it: it gives $v_p$ (put $r=r_1$, $a=a_t$) and $v_a$ (put $r=r_2$, $a=a_t$) — the two elliptical speeds you can't get any other way.

8. Impulsive burn — pretending a burn is instant

Why the topic needs it: it lets us treat each burn as happening at exactly one radius, so we can use a single clean speed at that spot instead of tracking a smeared-out thrust arc.

(An aside for later: burning fast and low in the gravity well squeezes out extra energy — the Oberth effect. And if $r_2$ is enormously larger than $r_1$, a three-burn Bi-elliptic transfer can beat Hohmann.)

How the foundations feed the topic

Read it top-down: $\mu$ and the radii feed both the circle speeds and vis-viva; those speeds get subtracted (as impulsive burns) into each $\Delta v$; the two $\Delta v$'s sum to the total.

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