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Talking to Mars: the maths

Earth and Mars on circular orbits around the Sun; radio travels at light speed. Computed live.

Equations

d(t) = |r_M(t) − r_E(t)|

Distance between the planets, each moving round its circle (periods 365.256 and 686.98 days).

delay = d / c

One-way light time. A question and its answer take twice that.

Constants

AU	1.495978707 × 10¹¹ m	exact, by definition
Mars's orbit	1.5237 AU	mean radius
c	299,792,458 m/s	exact

Worked examples ✓ checked on every change

One-way delay at closest approach	4.36 min	(1.5237 − 1) AU ÷ c
One-way delay at farthest	20.99 min	(1.5237 + 1) AU ÷ c

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• Circular orbits: Mars's real orbit is noticeably elliptical, so real delays range from about 3 to 22 minutes.
• Near conjunction the Sun's glare disrupts the signal; the model shows the geometry only.

Where it breaks

It doesn't — for the geometry. Only the orbits' shapes are simplified.

Sources: NASA/JPL planetary fact sheets; IAU (2012) definition of the astronomical unit.