Foundations — Patched conic method — interplanetary trajectory design

Before you can read the parent note on the patched conic method, you need every symbol it throws at you to already feel obvious. This page builds each one from nothing, in an order where each idea leans on the one before it.

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0. What a "central body" and a "conic" even mean

Look at the figure: same cone, four slices, four curves. A closed slice (ellipse) means the object is bound and comes back around. An open slice (hyperbola) means it flies off forever. The whole trip uses one ellipse (the Sun-bound cruise) glued to two hyperbolas (the fly-away/fly-in near each planet). Keep that mental image — every symbol below decorates one of these curves.

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1. $r$ — distance, the most basic symbol

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2. $a$ — semi-major axis, the "size" of an orbit

Subscripts you will meet: $a_p$ = a planet's orbit size (its distance from the Sun), $a_t$ = the transfer ellipse's size.

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3. $m$, $M$, $G$ and the shortcut $\mu$

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4. $v$ — speed, and the special speeds

The parent note names several speeds — they are all just $v$ measured at a particular place or relative to a particular body:

Symbol	Plain meaning
$v_c$	speed to hold a circular orbit at some radius
$v_{esc}$	escape speed — just barely enough to leave forever
$v_p$	speed at perigee (closest approach)
$v_{\infty }$	leftover speed infinitely far away, after climbing out
$v_t$	speed on the transfer ellipse

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5. Energy: the $\frac{1}{2}{v}^2$ and $-\frac{\mu }{r}$ pieces

This single conserved quantity is why the vis-viva formula (next) exists, and why burning deep in the well is cheap.

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6. The vis-viva equation (glue between energy and speed)

Two useful special cases fall straight out:

• Circle ($r=a$): $v_c=\sqrt{\mu /r}$.
• Escape ($a\to \infty$, so $1/a\to 0$): $v_{esc}=\sqrt{2\mu /r}$ — the $2$ is why escape speed is $\sqrt{2}\times$ circular speed.

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7. Powers and roots: $2/5$,  ​, and $r^5$

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8. Circular-orbit geometry and $\pi$

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9. $\Delta v$ — the currency of spaceflight

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How the foundations feed the topic

Each foundation box is one symbol you just learned; the arrows show which parent-note result it powers.

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Related vault topics

Once these symbols feel automatic, the parent leans directly on: Sphere of influence, Hohmann transfer orbit, Hyperbolic escape trajectories, the Oberth effect, and timing via Launch windows & synodic period.

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Equipment checklist

Cover the answers; you are ready when each is instant.

• $\mu$ means... ::: $G$ times the central body's mass — the "pulling strength," used so we don't carry $G$ and $M$ separately.
• $r$ always demands one question first, namely... ::: "distance from which body?" (Sun in the cruise, planet in the fly-by).
• The semi-major axis of a Hohmann transfer is... ::: $a_t=(r_1+r_2)/2$, the average of the near and far distances.
• Specific orbital energy has two parts... ::: kinetic $v^2/2$ minus potential $\mu /r$; their sum is constant along a coast.
• $\epsilon <0$, $=0$, $>0$ give which curves... ::: ellipse, parabola, hyperbola respectively.
• $v_{\infty }$ is the speed... ::: left over infinitely far from a planet, measured relative to that planet.
• The vis-viva equation is... ::: $v=\sqrt{\mu \left(2/r-1/a\right)}$.
• Escape speed relates to circular speed by a factor... ::: $\sqrt{2}$ (escape $=\sqrt{2\mu /r}$, circular $=\sqrt{\mu /r}$).
• $x^{2/5}$ means... ::: take the fifth root and square (undoes an $r^5$ equation).
• $\Delta v$ means... ::: change in speed from a burn — the fuel "cost" of a manoeuvre.
• Half an elliptical period is... ::: $t=\pi \sqrt{a^3/\mu }$ (full period drops its factor of $2$).