Foundations — Lambert's problem — connecting two positions in given time

This is a prerequisite page. The parent note Lambert's problem throws around symbols like ${\vec{r}}_1$, $c$, $s$, $\mu$, $\Delta \theta$, $\alpha$, $\beta$, $a$, $e$, and $E$. If any of those look like alphabet soup, start here. We define each in plain words, draw the picture it lives in, and say why the topic can't live without it.

1. A position vector $\vec{r}$ — "an arrow to a place"

Look at the figure below. The dot in the middle is the central body — the thing whose gravity we orbit (Earth, the Sun...). From it we draw two arrows: ${\vec{r}}_1$ points to where we start, ${\vec{r}}_2$ points to where we must arrive.

The length of the arrow ${\vec{r}}_1$ is written $r_1$ (no arrow) and read "the magnitude of ${\vec{r}}_1$." So:

• ${\vec{r}}_1$ = arrow (place we start).
• $r_1$ = plain number = how far that place is from the centre.

Same for ${\vec{r}}_2$ and its length $r_2$.

2. Subtracting vectors → the chord $c$

If ${\vec{r}}_1$ and ${\vec{r}}_2$ are arrows to two places, then the arrow from place 1 to place 2 is their difference, written ${\vec{r}}_2-{\vec{r}}_1$.

The length of that shortcut is the chord, called $c$: $c=|{\vec{r}}_2-{\vec{r}}_1|.$

3. The transfer angle $\Delta \theta$ and the law of cosines

The two arrows ${\vec{r}}_1$ and ${\vec{r}}_2$ open up like a pair of scissors. The angle between them is the transfer angle, written $\Delta \theta$ (the Greek letter "theta," and $\Delta$ means "change/difference in").

There's a beautiful shortcut linking $c$, the two lengths, and this angle — the law of cosines. It comes from the triangle centre → tip 1 → tip 2: $c^2=r_1^2+r_2^2-2r_1{r}_2\cos \Delta \theta .$

4. The semiperimeter $s$

Once we have the triangle's three sides ($r_1$, $r_2$, $c$), we bundle them into one number, the semiperimeter $s$ ("semi" = half, "perimeter" = go-around distance): $s=\frac{1}{2}(r_1+r_2+c).$

5. The gravitational parameter $\mu$

6. The ellipse: $a$ and $e$

Every gravity orbit that comes back on itself is an ellipse — a squashed circle. Two numbers describe its shape.

The central body sits at one focus — a special off-centre point, not the middle. That's why one side of the orbit is close (fast) and the other is far (slow).

7. The clock of an orbit: $E$, mean anomaly, and $\Delta t$

An orbit doesn't move at constant speed — fast when close to the focus, slow when far. To do the timing bookkeeping, astronomers use a clever helper angle called the eccentric anomaly $E$.

The link between this angle and real clock time is Kepler's equation: $M=E-e\sin E,M=\sqrt{\frac{\mu }{a^3}}(t-t_p).$ Here $M$ is the mean anomaly: a fake angle that ticks perfectly evenly with time (like a clock hand), and $t_p$ is the time of closest approach.

$\Delta t$ is simply the elapsed time for the trip: the difference between arrival time and departure time.

8. The auxiliary angles $\alpha$ and $\beta$

Finally, the parent introduces two mysterious angles: ${\sin }^2\frac{\alpha }{2}=\frac{s}{2a},{\sin }^2\frac{\beta }{2}=\frac{s-c}{2a}.$

How these feed the topic

See also, once you're comfortable here: Kepler's Equation and Time of Flight, Lagrange Coefficients (f and g functions), and Universal Variable Formulation, which the parent uses downstream.

Equipment checklist

Cover the right side and test yourself — you're ready for the parent note when every reveal feels obvious.