Vol. Science · Astronomy · Phases
Two geometries, not eight special cases. Mercury and Venus orbit inside Earth, so they are inferior planets. From here the disk runs crescent through full: inferior conjunction, greatest elongation, superior conjunction. Mars and every world beyond orbit outside Earth. They are superior planets. NASA’s Basics of Space Flight says they usually appear gibbous, and full at opposition. On circular coplanar orbits the phase angle cannot exceed the arcsine of Earth’s orbit divided by the planet’s. That maximum is quadrature, a right angle at Earth. The lit fraction there is (1 + cos of that angle) / 2, computed on each card from the NASA fact sheet ratio. Mars is the gibbous you can see. Jupiter and beyond stay at 99% or more. A crescent Jupiter is a spacecraft view, on the Jupiter desk. Venus keeps its own page. The Moon uses the same words for a body going around us.
Educational schematic — circular orbits, one plane, disks exaggerated. Not an ephemeris, not a magnitude, not a date, and not a finder chart. Each desk cites eccentricity and inclination so a perfect lineup is not mistaken for the sky. Do not look at the Sun to catch an inferior conjunction.
The formula · superior planets
From the ratio, not a table of invented angles
Call the fact-sheet semimajor-axis ratio a, planet orbit over Earth’s. For a superior planet the maximum Sun–planet–Earth angle on these circles is arcsin(1 / a). It happens when the angle at Earth is 90°. Illuminated fraction = (1 + cos of that angle) / 2. Phase angle 0° is full. Phase angle 90° is half. A crescent is past 90°, and these outer orbits never reach that from Earth. Inferior planets are the other triangle: greatest elongation on these circles is arcsin(a), and the phase angle there is 90°. Their coplanar cartoon does reach new, at inferior conjunction. Inclination is why a real one usually stays a thin crescent in the glare.
Seven worlds · one comparison
Inferior
Open the desk.
Inferior
Open the desk.
Superior
Open the desk.
Superior
From Earth, and from a spacecraft.
Superior
Open the desk.
Superior
Open the desk.
Superior
Open the desk.
The cards recompute in the browser from the same ratios as the desks. The links work either way.
Keep going
Priority
Nearly full from Earth. Drag an observer past the wedge and the disk can crescent.
The visible gibbous
Quadrature is short of full by enough to see. Opposition and the backward loop have their own desks.
Inner
A smaller, faster inner orbit. Crescent, half, gibbous, and a tight elongation.
Inner
The original phases desk. Evening star, morning star, both conjunctions.
Lineup
Earth between the Sun and an outer planet. The phase there is full.
Neighborhood
Where the orbits sit. Day and local time, still a schematic.