Crossing Into the Moon's Sphere of Influence | Visual AI Learning Guide

So the spacecraft arrives near the Moon slow, and still coasting. But Earth is not the only mass out here. The Moon pulls too, and there is a boundary where its pull stops being a side note and becomes the main event.

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Sphere of influence (patched conics)

The sphere of influence is the region around a body where that body's gravity dominates the spacecraft's motion.
Outside it, Earth's gravity shapes the path. Inside it, the Moon's gravity takes over as the dominant influence.
Mission planners use this as a simplification: solve the Earth-centered arc and the Moon-centered arc separately, then patch them together at the boundary.

What changes at the boundary

The spacecraft does not feel a sudden jolt. The Moon's pull has been growing all along; at the boundary it simply becomes the larger of the two.
Earth's pull is still present inside the boundary, just no longer dominant.
Once the Moon dominates, the Moon can bend the trajectory into a closed path around itself — the beginning of capture.

References

[1]Sphere of influence (astrodynamics) — Wikipediaen.wikipedia.org