Flat-Earth Sky

North pole at the center. Distance from center is linear in co-latitude, so the equator is a circle of radius R⊕ and the south pole / ice wall is the rim at 2 R⊕.

The dome is a flattened hemisphere: pole height runs from 100 km (the usual “rockets hit the firmament” figure) up to a true hemisphere on the rim. The ice wall itself is only a few kilometres tall. Stars are white pixels on the dome, parked above the ground point where that star is at zenith right now: latitude = declination, longitude = RA − GST.

As the clock runs, GST advances and every star slides around the pole to stay over its current zenith longitude. The Sun does the same on the solar declination parallel. At 12:00, noon is longitude 0°. Stand under Sun puts you on that ground point so looking up is looking at ☉.

Northern stars pile up near the pole because high declination maps to a small radius. Nothing sets. The nearest-overhead name stays true by construction. The distribution does not.

Trails are the last 24 hours, sampled every 6 minutes. That is one solar day, a little longer than the sidereal day, so a star's line overlaps itself by about a degree instead of closing cleanly. Only stars brighter than the magnitude limit are drawn. The sun, moon, and planets are drawn too. The lines live in the sky, so the panel and the other readouts do not wipe them, and looking around does not either.

Light. Off leaves the dome dark. True Sun matches clear-sky sun on a round earth. FE Flat Sun is a downward-facing disk: straight under it is full day, and brightness falls as the fourth power of distance, because the disk is foreshortened and the ground is lit at an angle. FE Round Sun is a hanging bulb: it shines the same in every direction, so only distance and the angle on the ground remain, and brightness falls as the third power. The sky follows that same pool. The brightness slider multiplies the pool. It does not change how big the sun looks. With Rayleigh it multiplies the whole sky too, so a dim sky gets louder without changing color. The light is still a spot under the sun, so it never becomes the true sky.

Rayleigh is ordinary air: one atmosphere straight up, density gone after about 8 km. It does not change the lamp. It dims whatever you are looking at by the air along the path. Blue leaves first, so the object reddens, and once the apparent magnitude passes the limit it is gone. On the flat dome that path is the straight line to the star or the sun. A low dome makes the sun a sunset while it is still several degrees up, then puts it out. The sky, in FE light, takes its color from that one path: near is blue day, far is sunset, farther is black. How bright it is follows the inverse square of the distance to the sun, measured against one dome-height, then the FE brightness slider. Twice as far is a quarter as bright, four times as far is a sixteenth, and it is the whole sky rather than a pool under the sun. Planetarium uses the real airmass instead, which levels off near forty, so a setting sun reddens and stays. The ice wall does not have to do the hiding. Behind it nothing is drawn. In front of it, from the rim, a low dome is a graze through the thick air, so that half of the sky drops through the magnitude floor too.

Round points the sun and moon stickers at the observer, so they stay round instead of lying on the dome. FE phase is on only then. It replaces the real phase with the phase the flat sun, the moon, and this observer actually make. On a tall dome that is never quite full and never quite new.

The sun and the moon are 32′ across when they are one dome-height away. A 100 km dome is 32′ from 100 km. A dome 12,740 km tall is 32′ from 12,740 km. Anywhere closer they are larger, and anywhere farther they are smaller. Planetarium holds them at 32′. The scale slider only multiplies that. On the dome the glow lies in the plane of the disk, the way a ceiling light does. Once Planetarium has carried a body under the ground, it is not drawn. In the eye view the sky is stereographic, centered on where you are looking. Circles on the sky stay circles, so the sun, the moon, and the constellations keep their shape. A degree grows only slightly toward the edge, much less than a pinhole camera, and the horizon is an arc of a circle.

Mountains are regional high points: the most prominent summit in a neighborhood, kept only if it stands at least 1,800 km from one already shown. The mark stands on the disk and its tip is the summit, so base to top is the true height. A summit is drawn, and named, only when that height is taller than a point on the screen. From Raleigh the one you can actually see is Mitchell, west, about a third of a degree tall. The set is Everest, Aconcagua, Denali, Kilimanjaro, Simón Bolívar, Orizaba, Elbrus, Vinson, Tomort, Puncak Jaya, Klyuchevskaya, Ras Dashen, Rainier, Toubkal, Mauna Kea, Kinabalu, Cameroon, Yushan, Kerinci, Erebus, Fuji, Aoraki, Gunnbjørn, Somoni, Mont Blanc, Lautaro, Mitchell, Kosciuszko, and Thabana Ntlenyana. Eye altitude is your height above sea level, up to a cruise altitude. In planetarium mode the same peaks sit on a round earth of mean radius and disappear below the horizon when a round earth would hide them.

Planetarium moves every star, the sun, the moon, and the planets from the flat dome onto the direction they really have for this observer. It eases there rather than jumping. Looking straight up is the same either way. That is the point. Overview turns off while Planetarium is on: the flat map and the real sky are not one picture.

Aim puts a mark on the center of the view. Pointing is the altitude and azimuth of that direction on the dome, measured from the disk and from true north. Telescope is where that same spot on the sky really is, so it is where the tube would have to point. Straight up, the two agree. Anywhere else they do not. The blue mark is the telescope direction when it falls inside the view.

Sim speed is how fast the clock runs against the wall clock. 1.0× is real time. The Real time button also locks the clock to now; any other speed lets go of that lock. 0 holds the clock.

Shadow puts a short pole in the open part of the view. The top of the plate is the way you are facing, and looking up does not turn it. The rim-colored triangle points north. Blue is the real shadow for this place and time. Gold is the shadow the flat-dome sun would cast. Both bars start at the foot of the pole.