Foundations — Launch window — phasing with target orbit

Before you can read the parent note Launch Window — Phasing with Target Orbit, you must be able to look at each symbol and see a picture. This page builds every one of them from nothing. Read top to bottom — each idea is a brick for the next.

1. Two circles and a central body

Everything happens around one central body — for us, the Earth. Draw a dot for Earth. Now draw two circles around it: a small one and a big one. A satellite living on a circle stays the same distance from Earth forever, because that is what "circular orbit" means.

Figure 1 below shows the two circles, the Earth-dot, and the two radius-spokes.

We need two of them:

• $r_1$ = radius of the chaser (interceptor) — the rocket you launch. Usually the inner circle.
• $r_2$ = radius of the target — the thing you want to reach.

2. Angle — measuring position around a circle

A satellite's distance $r$ never changes, so how do we say where it is on its circle? By an angle.

Why radians and not degrees? Because in a radian a full circle is exactly $2\pi$ — the same $\pi$ that shows up in every orbit formula — so the algebra stays clean. (If you like degrees: $\pi$ rad $=180^{\circ }$, $2\pi$ rad $=360^{\circ }$.)

3. Phase angle $\varphi$ — the lead

Now put both satellites on their circles at the same instant. Draw both spokes. The angle between the two spokes, measured at Earth, is the star of this whole topic.

Figure 2 below is that wedge — the chaser spoke, the target spoke ahead of it, and the lead angle $\varphi$ between them.

Here is that answer in one line, built from pieces this page defines further down. The chaser flies exactly half a lap ($\pi$) of the transfer ellipse; during that same flight time $t_t$ the target sweeps an angle $n_2{t}_t$. For them to meet, the target's lead at departure must be whatever the chaser gains on it — hence:

The Greek letters used in the parent:

4. $\mu$ — how strong the planet pulls

We use $\mu$ (not $G$ and $M$ separately) because it is the exact combination that appears in every orbit equation — measuring it directly is easier and more precise than measuring $M$.

5. Speed, period, and mean motion

A satellite runs around its circle at some speed $v$, and finishes one lap in a period $T$. But when we care about angles, the handiest quantity is how fast the spoke sweeps — the mean motion $n$.

Figure 3 below plots the mean motion against radius, so you can see outer orbits crawling.

Why call it "mean" motion? For a circle the speed is perfectly steady, so $n$ is exact. The word "mean" (average) is there because for stretched ellipses the real angular speed varies, and $n$ is its average — a detail you meet in Mean Motion and Orbital Period.

Now, why is the circular speed $v=\sqrt{\mu /r}$? Because a circle is a balance of two forces:

From that, the mean motion follows in one step ($n=v/r$):

6. Semi-major axis $a$ and the transfer ellipse

To get from the inner circle to the outer circle cheaply we fly along half of an ellipse — a Hohmann Transfer Orbit. An ellipse is a squashed circle; it has a longest diameter, and half of that longest diameter is the semi-major axis $a$.

Figure 4 below draws the transfer ellipse touching both circles, with its long axis marked.

7. Kepler's Third Law — turning size into time

We know the transfer takes half a lap of that ellipse. To turn "half a lap" into seconds we need the ellipse's period, and that comes from Kepler's Third Law.

The Hohmann transfer time is half of this, using $a=a_t$:

8. Putting angles and time together — $\Delta {\theta }_2$ and the synodic period

Two final combinations built entirely from the pieces above.

How far the target sweeps during your flight. The target's spoke turns at rate $n_2$ for a time $t_t$, so the angle it covers is rate × time: $\Delta {\theta }_2=n_2{t}_t.$ This is the "where will it be when I arrive" number. Subtract it from $\pi$ (the half-lap the chaser flies) and you get the required lead — exactly the $\varphi =\pi -n_2{t}_t$ formula from Section 3.

How often the perfect moment returns. The two spokes turn at different rates $n_1$ and $n_2$, so the wedge between them slowly opens and closes. It returns to any given value once the relative spoke has swept a full $2\pi$. That waiting time is the Synodic Period.

Prerequisite map

Each foundation above feeds one arrow of this map; the two outputs — the phase angle $\varphi$ and the synodic period $T_{\text{syn}}$ — are exactly the two questions the parent topic answers.

Equipment checklist

Test yourself. Cover the right side; if you can answer every line, you are ready for the parent note.