What is the maximum orbital time for my moon around my planet?

I fear the math involved is beyond my capabilities on this one. I have what I consider to be more than a Layman's understanding of the physics involved, and I believe I can follow the math well enough to spot glaring errors, but practical application, combined with the actual crunching of the numbers is more than I can do, in this case.

Here are the known quantities to start with:

The Planet:

1. Gas Giant (in any layman's definition of that term, even if scientific terms might call it something else like ice giants or brown dwarf or gas dwarf, etc)

2. Maximum mass must be small enough that no layman might mistake it for a star. Other than that, its mass can be adjusted as needed, so long as it can plausibly remain a Gas Giant, with the given mass, for 70 million years (not so small it's gas gets blown away by solar wind faster than that).

3. Radius/diameter must be great enough that the planet would appear, at a minimum, at least as big as Earth's moon when viewed from the surface of its moon (1/2 degree angular size), but has no maximum angular size.

4. Density/composition can be anything scientifically plausible as long as the above mass and radius/diameter limits are met, and it could still be called a gas giant.

The moon:

1. Is rocky/metallic (solid surface, not primarily ice, not gas-giant-like, not water or liquid surface. if it matters)

2. The diameter of the moon cannot exceed 6000 KM (radius of 3000 KM), and would preferably be closer to 5000 KM if other parameters can be met without increasing it further.

3. Surface gravity on the moon must be within a range of 75% - 125% of Earth's gravity.

4. Composition/density can be hand-waved, to some extent, to accomplish the gravity requirement within such a small size (I think [correct me if I'm wrong] this is going to be something in the range of a mostly osmium/platinum core, which I know isn't going to be particularly plausible. But on this one point, only, I don't really care as long as it could be made of something 'stable-ish' on our known periodic table, e.g. no neutronium)

5. Distance from the planet is whatever distance yields the greatest orbital time while keeping the planet close enough to appear as big as earth's moon

I'm guessing the answer is going to involve a brown dwarf (for max mass, and therefore max size for the visibility requirement and max gravity allowing a stronger pull from so far away) with a maximum density moon also of maximum size (again to grant a strong enough pull from so far away) as far as possible from each other to remain within the visibility requirement.

However, I can also see how I might be mistaken, as a more massive planet might crush itself smaller under its own weight, making it harder to stay visible from far enough away to extend the orbit time.

How long can I make this orbit take, within those parameters? And how do I accomplish that?

EDIT: Either my original question wasn't clear, or I seriously underestimated the importance of another factor, to the point that I omitted it completely. So here I'll address both:

First, I suspect some may think I was asking about the time it takes for the planet+moon pair to orbit their star. I'm not. I'm asking, instead, about the time it takes for the moon to orbit it's planet. I assumed that the influence of the star would be negligible for this, so I did not provide details.

Next, if I'm wrong, and the star is that important, then here are the star's requirements:

1. color: Sol-like (a human tourist to this moon might notice the color difference when arriving on the moon, but would adjust and stop noticing after a day or three)

2. Goldilocks zone: distance from the star should be such that stellar radiation should be a significant factor, but not necessarily the only factor (Tidal forces by the planet, a higher radioactive composition, excessive heat from moon formation, etc., can also be factors but should be kept to a minimum wherever possible) in keeping the moon at a survivable temperature for humans if other life support features (atmosphere, gravity, etc) are also present.

3. Stability: Any scientifically plausible type of star that does not vary drastically enough, during a period of 500 million years, to adversely affect any life already on an otherwise habitable planet or moon in its goldilocks zone.

4. Mass, radius, density, composition, distance from the planet, etc.: can all be adjusted as needed, as long as the color and goldilocks requirements are met. But bonus points if its angular size, viewed from the moon, appears the same size or larger than the planet does.

Summary, in layman's terms: I want the moon to take as long as possible to orbit the planet, but the planet and star should appear at least as big, in the sky, as earth's moon and sun. How long can I make that orbit take?

posted about 6 years ago

Dalila‭

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