I have often wondered about the subject.
Part One: Telephoto Lenses.
I note that humans have invented something called telephoto lenses for still and motion cameras.
Thus you should have often seen still photographs, or scens in movies & Tv shows where Earth's Moon appears to be vast and cover most of the sky. And Earth's Moon does cover most of the tiny angle of the sky which is visible in those scenes shot with telephoto lenses.
And we have all seen scenes in science fiction films where the moon appears to have an enormus angular diameter. LIke in E.T.: The Extraterrestrial where a bicycle flies across the face of the Moon that appears to be enormous. Those scenes in science fction movies aren't designed to show the Moon with a much greater angular diameter than it actually has, they are designed to show what the scene would look like photographed thorugh a telephoto lens that makes the moon look much larger.
And similarly many illustrations of alien worlds and movie and tv scenes show stars, or planets, or moons near the horizon that appear to have a vast angular diameter. In some cases the scenes may have been designed to show what someone would see without any magnification, in others they may have been designed to show what it would look like thorough a telphoto lens making the objects in the sky loook much larger, and in many cases nobody thought about whether the apparent angular diameter would be the actual angular diameter as seen with the naked eye.
Part Two: The Angular Size of Stars.
A small difference in the mass of main sequence stars will cause a much larger difference in their luminosities. A small increase in mass will cause a larger increase in luminosity. A small decrease in mass will cause a larger decrease in luminosity.
More massive stars also emit more light per unit of their sufraces. Less massive stars emit less light per unit of their surfaces.
So a planet that has the sem temperature as Earth orbiting a more massive and brighter star will have to orbit it farther than the distance at which the star it would have the same angular diameter as the Sun. A planet with the same temperature as Earth orbitng a less massive and less luminous star would have to orbit it closer than the distance at which the star would have the same angular diameter as the Sun.
Astronomers have calculated the sizes of the circumstellar habitable zones of main sequence stars of various spectral types. All one has to do is find the luminosity of the star comparared to that of the Sun and then multiply the inner and outer edges of the Sun's circumstellar habitable zone by that ratio.
Unfortunately, different calculations of the inner and outer edges of the Sun's circumstellar habitable zone vary greatly, as this list shows:
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates[1]
A writer who wants to have several habitable planets orbiting the same star will have to see if he can fit them in within a habitable zone that seems reasonable and plausible to him. A writer who wants one and only one habitable planet in his star system can play it safe and calculate the Earth Equivalent Orbit for that star, an orbit where a planet would receive exactly as much radiation from its star as Earth receives from the Sun. Thus he would know that a planet in that orbit could be habitable.
The answer by user177107 to the queston:
https://astronomy.stackexchange.com/questions/40746/how-would-the-characteristics-of-a-habitable-planet-change-with-stars-of-differe%5B2%5D[2]
Has a table listing main sequence stars of various spectral types.
For each spectral type of star the mass, radius (half of the diameter), luminosity, etc. is listed, as well as the distance a planet would have to orbit to received exactly as much radiation as the Earth receives from the Sun.
The smallest type of star listed is spectral type M8V, with a mass of about 0.082 the mass of the sun, and a diameter of about 0.111 of the Sun, and a planet in an Earth Equivalent Orbit would orbit it at a distance of 0.0207 of an Astronomical UNit (AU), the distance of Earth from the Sun.
Because the orbit of the Earth is elliptical, the angular diameter of the Sun as seen from earth varies between 31.6 and 32.7 arc minutes, or 1,896 to 1,962 arc seconds. Since there are 60 arc minutes in a degree of arc, we can simplify the calculation by assuming that the Sun has an average angular diameter of 0.5 degrees of arc, or about 1,800 arc seconds.
Since a planet in an Earth Equivalent Orbit around a M8V star would beonly 0.0207 as far as Earth is From the Sun, the angular diameter of the Sun would have to be multiplied by 1 divided by 0.0207, or by 48.309, to get an angular diameter of 24.1545 degrees, and then divided by 1 divided by 0.111, or 9.009, the ratio of the diameters of the two stars, to get an angular diameter of 2.681 degrees, or about 9,651.6 arc seconds,.
So a M8V star would look about 5.3 times as wide seen from a planet in an Earth equivalent orbit as the Sun looks from Earth. Which would be quite noticable for a human on the planet, but probably not too spectacular.
Of course, if the planet orbits a bit closer to its star than the Earth Equivalent Orbit, the star would appear to have a larger angular diameter.
According to the estimate by Kopparapu et al in 2013, the inner edge of the Sun's circumstellar habitable zone is 0.99 AU, or only 0.01 AU closer than the Sun's orbit. However, Zsom et al in 2013 estimated that the inner edge of the Sun's habitable zone could be ast 0.38 AU. Their estimate is probably for a planet with atmospheric condidtions suitable for some forms of Earth-like life and not for humans, who might, however, be able to live on such a planet with protective gear.
At such a distance around a M8V star, the star would appear to have an Angular diameter of about 7.055 degress or about 14.11 times teh angular diameter of the Sun as seen from Earth.
The closer a planet orbits to its star, the stronger the gravity and tides of the star will be upon the planet. Because the luminosity of low mass stars decreases faster than their mass, planets in the circumstellar habitable zones of low mass stars will experience intense gravity and tides, which will tend to slow down the rotation of those planets.
So as the stellar habitable zones around stars get smaller and smaller for less and less massive stars, eventually the inner edges of their cicumstellar habitable zones reach the distance at which a planet would swiftly become tidally locked to its star. With stars of even less mass, the Earth equivalent orbit will eachethe distance at which the planet will become tidally locked to the star. With stars of even less mass than that, the outer edges of their habitable zones will reach the distance at which a plaent become tidally locked and it will be impossible for any planet witin their habitable zones. to avoid being tidally locked.
Stephen H. Dole in Habitable Planets for Man, believed that a tidally locked planet would be unhabitable. He also calculatled that at 0.88 the mass of the Sun, the inner edge of the circumstellar habitable zone, or "ecosphere" as he called it, would reach the distance where a planet would be tidally locked, and at a stellar mass of 0.72 that of the Sun, the outer edge of the habitable zone would reach the distance at which a planet would be tidally locked.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[3]
Dole also noticed a potential escape clause, where a planet which became tidally locked to a large moon, or a companion planet, could avoid becoming tidally locked to it's star.
Dole estimated that could enable habitable palnets to exist n the habitable zones of stars down to a stellar mass of 0.35 of the Sun, before the sun would raise too high tides on the planet.
O.35 the mass of the Sun would be somewhere between a M5V and a M2V star; 0.72 the mass of the Sun is approximately a K1V star; and 0.88 the mass of eh Sun is approximately a G9V star.
But there have been calculations indicating that a tidally locked planet could be habitable if it had enough atmosphere and water to distribute heat from the day side to the night side. This is rather controversial.
Part Three: The Angular sizes of Brown Dwarfs.
Brown dwarfs are objects with masses greater than about 13 times that of Jupiter and up to about 75 to 80 times that of Jupiter. Brown dwarfs are intermediate in mass and conditins between planets and stars, and could be orbited by natural satellites, perhaps large enough for life.
Considering how dim brown dwarfs are, any theoeretical habitable moons or planets or whatever obiting them would have to orbit very close. Thus it is possible that A brown dwarf would appear several times as wide in the sky of a habitable world orbiting it as any star appers in the sky of its habitable planets. And possibly humans in space suits or lesser environmental protection could work on the surfaces of nonhabitable worlds with large brown dwarfs in the sky.
I am unable to clculate the maximum possible angular diameter of a brown dwarf as seen from a world which humans might want to land on.
Part four: Planets With Neighboring Orbits.
The star TRAPPIST-1 is a dim M8V class star, noted for having several potentially habitable planets orbiting in its circumstellar habitable zone.
The orbits of the TRAPPIST-1 planetary system are very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for b, they orbit farther than the Galilean satellites do around Jupiter,[43] but closer than most of the other moons of Jupiter. The distance between the orbits of b and c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth.[42] A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.[40][37]
https://en.wikipedia.org/wiki/TRAPPIST-1#Planetary_system[4]
The potentially habitable palnets of TRAPPIST-1 are d, e, f. and g. They could be habitable, and maybe if not they could be terraformed to be habitable. Anyway, habitable or not, people should be able to walk around on them in spacesuits and possibly less evironmental protection.
The semi-major axis of the orbit of the Moon is about 384,399 kilometers, or about 0.002569 AU. At that distance Earth has an angular diameter of approximately 2 degrees of arc or 7,200 arc seconds.
At their nearest, TRAPPIST-1 f and TRAPPIST-1 g are about 0.00834 AU apart, which is about 3.246 times the distance between the Earth and the Moon. So if those planets had the same diameter as the Earth, they would appear to have an angular diameter of 0.616 degrees or 2,218.114 arc seconds. They are both larger than the Earth. The larger, TRAPPIST-1 g, has a diameter 1.129 that of the Earth, and so would have an angular diameter of 0.695 degrees or 2,504.251 arc seconds when seen from TRAPPIST-1 f at its closest.
The innermost potentially habitable planet, TRAPPIST-1 d, orbits only 0.00647 AU beyond the orbit of TRAPPIST-1 c, which is 2.5184 times the Earth-Moon distance, which would make TRAPPIST-1 c look up to 0.7941 degrees wide from TRAPPIST-1 d, except that TRAPPIST-1 c has 1.308 times the diameter of Earth. So TRAPPIST-1 c would have an angular diameter of about 1.0387 degrees of arc of 3,739.517 arc seconds when the two planets were closest.
The absolute maximum possible angular diameter of a planet seen from a habitable or potentially terraformable planet in a neighboring orbit should be at least that much.'
The best is yet to come when I complete this answer.
