Lagrange points.

Step 1 of 7

Turning with the Earth

By Kepler’s third law, any body closer to the Sun than the Earth goes round it faster, and any body farther away goes round more slowly. So how can a probe be kept in the same place relative to the Earth? The answer is the Earth’s own gravity. In 1772 Joseph-Louis Lagrange found five points where the pull of the Sun and the Earth together gives exactly the centripetal force of an orbit of one year.

The Scene turns with the Earth, so the Sun and the Earth seem to stand still. A probe parked at one of the points stands still too. To make the points stand out, Earth’s mass in the Scene is $1\%$⁠ of the Sun’s, $3000$⁠ times the real value. The probe is now parked at L1, between the Sun and the Earth.

Step 2 of 7

How far

L1 and L2 are on either side of the Earth, on the line to the Sun. When $\mu$⁠, the Earth’s share of the total mass, is small, both are at roughly the same distance from it:

$\frac{r}{R} \approx \sqrt[3]{\frac{\mu}{3}}$$\frac{r}{R} \approx$$\sqrt[3]{\frac{\mu}{3}}$⁠

Here $R$⁠ is the distance between the Sun and the Earth. For this estimate the difference between the Sun’s pulls at the point and at the Earth is set against the Earth’s pull, which is why a cube root comes out and not a square root. Park the probe at and Earth’s mass are locked.

Step 3 of 7

The real Earth

Earth’s mass is now $0.1\%$⁠ of the Sun’s, like Jupiter, and L2 has come closer to it. For the real Earth $\mu \approx 3 \times 10^{-6}$$\mu \approx$$3 \times 10^{-6}$⁠, and its L1 and L2 are so close that in the Scene they would be inside the drawn globe.

Step 4 of 7

A saddle

At the point itself the forces match exactly. What happens to a probe that moves slightly? Now it is parked at L2 and nudged half a per cent of the distance to the Sun, away from the Sun. Run the motion. Once you have chosen, Show effective gravity will lower everything onto the effective-gravity surface: the gravity of both bodies together with the outward fling of the turning. The five points are its flat places.

Step 5 of 7

Trojans

L4 and L5 are sixty degrees ahead of the Earth and sixty degrees behind it, each at the corner of an equilateral triangle with the Sun and the Earth. Park the probe at is locked at L4, and the probe is nudged $1.5\%$⁠ of the distance to the Sun.

Step 6 of 7

Routh’s limit

Whether L4 and L5 are stable depends on the mass ratio: the small body has to be light enough. Park the probe at is locked at L4, and the probe is nudged slightly.

Step 7 of 7

A telescope at L2

At L1 the SOHO spacecraft has circled since 1996, watching the Sun without a break, never hidden by the Earth. At L2 the James Webb Space Telescope has worked since 2022, looking away from it. Show effective gravity shows the effective-gravity surface, with the probe at L2.