# Solar System Moons by Apparent Size from Planetary Surface

Sadly, Jovian surfaces don’t look like they’ll solidify in the near future.

In the above scatterplot, I plotted the natural log of the distance of a moon from the surface of the planet it orbits and the natural log of the moon’s radius. Note that how large something appears to be (lengthwise) is directly proportional to its absolute size and inversely proportional to its distance from the observer (inverse square proportional by area, but here we’re talking lengths). Also, note that in a line with slope 1 of x plotted with y, y-x is constant since the graph is y=x+c for some constant c→y-x=c. Thus, in this graph, any two moons along the same line of slope 1 have the same apparent size from its planet’s surface, because for points on the same line of slope 1, ln(Moon Radius)-ln(Distance from Planetary Surface) is constant by the result mentioned above, and since ln a-ln b=ln(a/b), ln(Moon Radius/Distance from Planetary Surface) is constant, and since natural log is injective, Moon Radius/Distance from Planetary Surface is constant, and thus Apparent Size is constant. The further up and left in the graph the moon is, the larger it seems from its planet’s surface, and the further down and right in the graph the moon is, the smaller it seems from its planet’s surface.

In the scatterplot, I included Earth’s moon (green), both moons of Mars (dark red), 19 of Jupiter’s 50 known moons (red), 37 of Saturn’s 53 known moons (orange), all 27 of Uranus’s known moons (aquamarine), and all 13 of Neptune’s known moons (dark blue). All 31 omitted Jupiter moons are in (16.8±0.3,1.2±0.6), and all 16 omitted Saturn moons are in (17.0±0.3,1.6±0.4). (I omitted them because the graph was already crowded enough over there. Praxidike gives you a good approximation of where this cluster is centered.)

In the next picture, I appended on a few lines of constant apparent size. One of them shows Earth’s Moon-apparent size, one of them shows Titan-apparent size, and six of them show minimal apparent sizes to be able to occult the sun in a solar eclipse on the surface of the moon’s planet. Theoretically, only moons above the limit line of a planet will be able to make solar eclipses on their respective planets, but taking into account eccentricity and other factors, these lines are not exactly accurate. For example, Amalthea is known to be able to occasionally make solar eclipses as seen from Jupiter’s surface. Sadly, if you live on Mars, you’ll have to wait for Phobos’s orbit to tighten significantly before you could see a solar eclipse.