Build it, test it, try to break it

A lamp, a small ball and a large ball on a virtual table, matching the tabletop version: 2 cm Moon-ball, 7 cm Earth-ball, point source lamp.

Slide the Moon-ball back and forth. Close to the lamp, its shadow cone is long and reaches well past the Earth-ball — like the real Moon's shadow arriving with room to spare at perigee. Slide it further from the lamp and the cone shortens until its tip lands exactly on the Earth-ball's surface, and a little further still, the tip falls short and a bright rim shows around the dark centre — your tabletop version of an annular eclipse.

Use this to connect the model in your hands with the numbers in the worked example: the real Moon's shadow cone is about 374,000 km long and its distance is about 384,400 km, a much closer call than the model usually shows unless you place things carefully.

A dual view: the Moon orbiting Earth as seen from above (from-space), and what its lit shape looks like from Earth (from-earth), both driven by the same day-of-month slider from 0 to 29.5.

Step through slowly and watch the from-space view. Only at day 0 (new moon) does the Moon sit on the Sun side of Earth, roughly in line with the Sun. Only at day ≈14.8 (full moon) does it sit on the far side, roughly in line with Earth's shadow. At every other day it is off to one side, and no shadow from either body can reach the other.

This confirms the rule from Understand: the phase condition is not a coincidence added on top of the geometry, it is the geometry, seen from a different angle.

A lamp, a movable ball with a choice of two widths, and a screen with a ruler along it, reporting the shadow's exact width as you drag.

Try the three practice problems above in the lab before or after solving them on paper: set the ball to 4 cm, put it 20 cm from the lamp with the screen at 60 cm, and check the readout says 12 cm. Then try 30 cm and 40 cm and watch the shadow shrink each time.

A ratio readout also shows shadow width ÷ ball width, which is the same as screen distance ÷ ball distance — useful for the two challenges above.

The same solar-eclipse model as before, but treated as an experiment: change the Moon's distance slider and note the result, then reset the distance and change the offset slider instead, and note that result too.

Suggested runs: (1) offset = 0, distance = perigee — expect the umbra to reach the ground; (2) offset = 0, distance = apogee — expect the antumbra and a ring; (3) offset = 0, distance = mean — expect a very thin ring; (4) offset = small, distance = perigee — expect the black spot to slide off the globe while the grey penumbra still clips it, giving a partial eclipse for a wider area; (5) offset = large — expect nothing at all.

Keep a simple table as you go: distance, offset, result. That table is the evidence behind every rule in Understand — you are re-deriving it instead of reading it.

The same why-not-monthly lab as before, used here to test the table above directly: set the tilt to each value in turn (0°, 0.5°, 1°, 2°, 5.145°, 10°) and note how many eclipses per year the counter reports.

The pattern to look for: nothing changes at all between 0° and about 1°, because the tilt is still smaller than the 1.475° separation the Moon needs to clear. Only once the tilt passes that threshold does raising it start cutting the eclipse count down — and it keeps cutting hard as the tilt grows further.

A matching game pairing five hands-on tests with their results: the lamp-and-balls model showing the Sun cannot fit to scale in a room; the moon-phase lab confirming only new and full moon line up; shadow measurements confirming the width formula; the tilt slider revealing a hidden threshold near 1.475°; and the pinhole projector confirming the 9.3 mm-per-metre growth rate.