In general, I'm considering having my planet have multiple moons but at different points in its history, the count of the moons increase or decreases (like every few thousand years a moon comes or goes)
Is it theoretically possible to have visible moons small enough that they do not have a [significant] impact on the tides or other factors of the planet?
I'm not planning on having more than three at any given time.
Little, close moons.
Depicted: our moon Luna and the moons of Mars Phobos and Deimos.
Deimos has a diameter of 7.5 miles (12 kilometers) and was 12,800 miles (20,500 kilometers) from the rover at the time of the image. Phobos has a diameter 14 miles (22 kilometers) and was 3,900 miles (6,240 kilometers) from the rover at the time of the image. Earth's moon has a diameter of 2,159 miles (3,474 kilometers) and is typically about 238,000 miles (380,000 kilometers) from an observer on Earth.
https://www.nasa.gov/mission_pages/msl/multimedia/pia17351.html#.Ye3uDvvMLIU
The moons of Mars are 10 times closer to their planet than Luna is to us. Both Martian moons have a mass in the 10^15 kgs; Luna has a mass of 10^24 kg. The moons of Mars are small; asteroid sized. And they might be asteroids that got trapped in Mars' gravity well. Our moon Luna is bigger and not an asteroid.
As regards tides, any mass will exert tidal forces but the moons of Mars are unmassive enough that the forces they exert are small compared to the force Luna exerts.
Your planet could capture asteroids to be its temporary little moons. Small masses orbiting close are fine moons. How exactly these moons then leave is a different question - although Phobos coming down from orbit at 2km/s will pack less of a wallop than the comparably sized but 20km/second Chicxulub asteroid I have to think it would still be a big deal on the surface when a little moon comes down.
I agree with Willk's answer citing Deimos and Phobos as reasonable candidates. As you mention wanting the moons to come or go every so often, I wanted to note that Earth has or has had a couple of low-mass "temporary satellites", bodies which typically orbit the Sun but may briefly move into orbit around the planet. 2006 RH120 is perhaps one of the best-known; though it normally orbits the Sun, it did orbit Earth for about nine months between 2006 and 2007.
Such temporary satellites would almost certainly be small enough to have little effect on the planet - asteroid-sized or smaller. Both known temporary satellites of Earth (the other being 2020 CD3) are on the scale of meters, but it stands to reason that something like Phobos or Deimos could also be temporarily captured (and you'd need it to be that size to have a shot at it being visible!). The other major difference is that the timescales you're looking for (millennia) are longer than the time these objects spend orbiting Earth, but I wouldn't bet against that being possible.
If the moons are of size of mentioned Phobos and Deimos, or even little larger, and close enought, they could serve this purpose.
If there are not such big oceans as are on Earth, the tidal waves would be also smaler. (Not only sea have tidal waves, every lake or even glass of watter have them too, just insignificant :)
There may be a lot of large asteroids in your system, and the Earth can catch some of them sometimes, with either just temporally (they do few unregulars ways around and then are catapulted away again), or more permanently (like a Moon). But having so much of large asteroids, there is still possible, that some of them would approach the "stable ones" and draw them away again.
There would be needed a lot of handwaving about probability to approach just the right ways, to be near enought but not collide, but well, with just of a lot of luck, why not. (Or if there are any gods/natural magic/whatever whil simply just plays a little with probability and/or affects those asteroids with really small force, but for long time and with "intent and knowledge", it would be possible too) "if there is probability of something 1/1.000.000 it happens 9/10 times"
They don't move from orbit they just hide?
Willk's answer says that the Moon is about 1,000,000,000 times as massive as Deimos or Phobos. A moon a thousand times as massive as the Martian moons would be one millionth as massive as the Moon. A moon a million times as massive as the Martian moons would be one thousandth as massive as the Moon. If moons of those masses were at the same distance as the Moon is from Earth, their tidal effects would be miniscule compared to those of the Moon on Earth.
Of course the strength of their tidal effects on the planet will depend on how close or how far they orbit as well as on how massive they are. Presumably different moons would orbit the planet at different distances.
Any permanent moons of the planet that keep on orbiting it for many millions and billions of years would have to orbit within it's Hill Radius.
The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere.
In the Earth-Sun example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU). The Moon's orbit, at a distance of 0.384 million km from Earth, is comfortably within the gravitational sphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than seven months
https://en.wikipedia.org/wiki/Hill_sphere#Formula_and_examples
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.2
https://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability
So any stable long term satellites of Earth would have to orbit within about 500,000 or 750,000 kilometers of Earth, within the true region of stability. Unstable short term temporary satellites of Earth could orbit within the outer parts of the Hill sphere of Earth, at distances of about 500,000 to 1,500,000 kilmeters, and even at distances beyond the Hill sphere. The farther from Earth their orbits are, the shorter their periods of being satellites of Earth is likely to be.
The size of a planet's Hill sphere depends on the mass of the planet, the mass of the star, and the distance between them. If your planet is exactly as massive as Earth, and your star is exactly as massive as the Sun, and the semi-major axis of the planet's orbit is exactly one Astronomical UNit (AU), the size of the planet's Hill sphere will be exactly the same size as Earth's Hill sphere.
If any of those factors differ significantly, the size of your planet's Hill sphere will also differ significantly from that of Earth. In that case you might need to calculate the size of your planet's Hill sphere.
If some or all of the moons need to be visible as objects instead of mere dots from the planet, the visual acuity of the Human eye will determine minimum diameters that they must exceed (unless your characters are all aliens with different eyesight than humans).
The maximum angular resolution of the human eye is 28 arc seconds or 0.47 arc minutes,[19] this gives an angular resolution of 0.008 degrees, and at a distance of 1 km corresponds to 136 mm. This is equal to 0.94 arc minutes per line pair (one white and one black line), or 0.016 degrees. For a pixel pair (one white and one black pixel) this gives a pixel density of 128 pixels per degree (PPD).
https://en.wikipedia.org/wiki/Visual_acuity#Physiology
A full circle is 360 degress. 28 arc seconds is 0.008 degress, so there are 1 divided by 0.008, or 125, times 28 arc seconds in a degree. Thus there are 125 times 360, or 45,000, times 28 arc seconds in a full circle. 28 arc seconds is 0.0000222 of a full circle.
The radius of a full circle is about 1 divided by 2 divided by 3.14159, or 0.159155, of the circumference of that full circle. the circumference of a full circle is about 6.28318 times the radius. Thus the width of an object 28 arc seconds in angular diameter should be 0.0001394 of the radius.
So an object with an angular diameter of 28 arc seconds should have a physical diameter of 69.7 kilometers at a distance of 500,000 kilometers, and a physical diameter of 348.5 kilometers at a distance of 2,500,000 kilometers.
And possibly the mini moons would need to be a few times 28 arc seconds in angular diameter to be seen as extended objects instead of as dots of light.
And if you want the small moons to appear as mere dots of light they can be much smaller.
Jupiter has four large moons, Io, Europa, Ganymede, and Callisto. And if they were farther from the brightness of Jupiter they could be seen by the naked eye, without a telescope. And possibly they have sometimes been seen with binoculars or a telescope.
All four Galilean moons are bright enough to be viewed from Earth without a telescope, if only they could appear farther away from Jupiter. (They are, however, easily distinguished with even low-powered binoculars.) They have apparent magnitudes between 4.6 and 5.6 when Jupiter is in opposition with the Sun,[55] and are about one unit of magnitude dimmer when Jupiter is in conjunction. The main difficulty in observing the moons from Earth is their proximity to Jupiter, since they are obscured by its brightness.[56] The maximum angular separations of the moons are between 2 and 10 arcminutes from Jupiter,[57] which is close to the limit of human visual acuity. Ganymede and Callisto, at their maximum separation, are the likeliest targets for potential naked-eye observation.
https://en.wikipedia.org/wiki/Galilean_moons#Visibility
The smallest of the Galilean moons is Europa, which has a diameter of 3,121.6 kilometers. The closest distance between Earth and Jupiter, is about 4.2028 Astromical Units. Since an Astronomical Unit is 149,597,870.7 kilometers, Europa can get as close to Eartj as 628,729,931 kilometers, give or take a few million.
So Euorpa could be visible to the naked eye as a dot of light, if not hidden by the glare of Jupiter, at a distance which is 201,412.7149 times the diameter of Europa. A full circle at the closest distance between Earth and Europea would be about 1,376,011.386 times the diameter of Europa. The diameter of Europa is 1 divided by 201,412.7149 times a distance at which Europa would be visible to the naked eye if not lost in the glare of Jupiter.
So an object 2.482 kilometers in diameter should be visible as a dot of light at a distance of at least 500,000 kilometers from Earth, and a object 12.412 kilometers in diameter should be visible at a distance of at least 2,500,000 kilometers from Earth.
And that is a rough calculation, not allowing for the different albedos of different astronomical objects and for the more intense sunlight at Earth's orbit, nor for how much Europea exceeds the absolute miniumum apparent magnitude.
But it does suggest that your small moons should be at least a few kilometers in diameter to be visible even as points of light, unless they are a lot closer to your planet than the outer regions of the Hlll sphere, where temporary moons would usually be expected to orbit.
And I think that a diameter of a few kilometers is a lot larger than the diameters of any known temporary moons of Earth. All known temporary moons of Earth have been only a few meters in diameter.
https://en.wikipedia.org/wiki/Claimed_moons_of_Earth
That would suggest that your planet's orbit would have a lot more fairly large size asteriods with similar orbits than Earth has. That suggests that major asteroid impacts and their effects should be a lot more frequent on your world than on Earth.
