The Small-Angle Approximation
In this case, we are determining the opposite; we know the true diameter of the object in the sky and we know its distance from the observer, so we can use a shortcut to determine how large the object appears to be in the sky.
The following equation uses absolute values to return the apparent diameter in the sky in radians:
Example: The Angular Diameter of The Moon
Actual (Average) Distance To The Moon: 384399 km
Example: The Angular Diameter of the Sun
Actual (Average) Distance To The Sun: 1.496E+8 km
Apparent Diameters and Eclipses
Note that the Earth's orbit around the Sun is an ellipse, and as a result, the distance to the Sun changes slightly over the period of a year, so the angular diameter of the Sun varies from about 0.527˚ to about 0.545˚. Similarly, because the Moon's orbit around the Earth is also an ellipse, the angular diameter of the Moon also varies, from about 0.488˚ about 0.568˚.
Thus, there are times when the Moon does not completely cover the Sun, leading not to a total eclipse, but to an annular eclipse (from the Latin "annulus", meaning a ring, because a ring of Sun remains visible around the outside of the dark disk of the Moon).
Apparent Brightness (of Non-Luminant Objects)
Example: The Apparent Brightness Of The Moon
It's not straightforward, unfortunately. Albedo is the ratio of the amount light reflecting from an object divided by the amount of light falling on it (the incident light). If it reflects all of the incident light, it has an albedo of 1.0; if it reflects none of the incident light, it has an albedo of 0.0; and so on.
The radius of the body is a factor in the equation above; such that, if Planet A is twice the diameter of Planet B, but they both have the same albedo, Planet A will appear brighter than Planet B when observed from the same distance. Conversely, if Planet A and Planet B are the same size, but Planet A has twice the albedo of Planet B, Planet A will again appear brighter than Planet B when viewed from the same distance. A real-world example is that of Earth and Venus: Venus is 94.99% the radius of the Earth, but its albedo is 0.76, compared to Earth's albedo of 0.40. Earth is larger, but viewed from the same distance, it would be dimmer than Venus.
Albedo is a function of the material the body is made of; for planets, prime considerations will be atmospheres. This helps explain the Earth-Venus example above--all of the light reflecting from Venus is reflected off its atmosphere, but some of the light reflecting from Earth is reflecting from clouds, ocean, land surface, ice, etc.
When it comes to fictional planets, my best advice is to
- Find the known albedo of a Solar System object which is most similar to your fictional planet;
- Take the ratio of your planet's radius compared to that of the example planet;
- Multiply the albedo of the known planet by the ratio calculated in Step 2.
For instance, let's say we have a planet—Taiar—which is a Midgean, with a surface similar to Mercury. Mercury's albedo is 0.06. Let's say Taiar is 0.6213 times Mercury's radius. Thus:
This, of course, assumes that Taiar, as Mercury, has no atmosphere, and that Taiar's surface is composed of more-or-less the same materials as the surface of Mercury.
If, on the other hand, Taiar is 62.13% of the size of Mercury, but has an atmosphere like Titan's, it would be better to use Titan's albedo as the standard. Since Mercury is 0.9262 times the size of Titan (yep—Titan is larger than Mercury), and Taiar is 62.13% the size of Mercury, then Taiar is 0.57556 times Titan's diameter. Titan's albedo is 0.22, so the equation becomes:
Is this a rigorously scientific calculation? No, and so I hereby invoke The Omega Argument.
The Naked-Eye Limit
Saturn's apparent brightness (on average) is 1.338E-7 w/cm³, so it makes sense to say that no object with a calculated apparent brightness less than this value is a naked-eye object.
Note that this is NOT saying that no object orbiting farther from its star than does Saturn from the Sun is visible. Apparent brightness is not just a function of distance from the illuminating body, but also of the size and reflectivity of the object, and its distance from the observer.
For instance, if Jupiter were moved to Saturn's orbit, it would be almost 1.5 times brighter than Saturn is as the same distance, and certainly remain a naked-eye object. Uranus, however, moved inward to Saturn's orbit would still have only 17% of Saturn's brightness and remain invisible to the naked eye.