Kepler's Third Law: How Orbital Period Depends on Radius

For a circular orbit the gravitational pull is the centripetal force, and the algebra leaves T² = (4π²/GM) r³. Everything multiplying r³ is fixed by the star, so T²/r³ is the same for every planet of that star. That is what the straight line through the origin means: the points are not merely rising together, they are rising in proportion.

Jupiter is about 300 times the mass of Earth and 3000 times the mass of Mars, and all three still land on the same line. The orbiting body's own mass cancels — it appears once in the pull and once in ma, so it never reaches the answer. Only r and the star's mass do.

TRAPPIST‑1 is a red dwarf of 0.0898 M☉ whose seven rocky planets are packed inside a fiftieth of Earth's orbit, where a year lasts days rather than months. The same arithmetic still holds and their ratio comes out at 11.14 yr²/AU³, which is 1/M. That is how a star gets weighed: time one planet, measure its radius, and the ratio hands you the mass of whatever it is going round. Each system is drawn to its own scale — the faint rings say which — and nobody has seen those three surfaces, so their coloring here follows where they sit, not a photograph.