Con-Astronomy for my Con-World

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I am working on a con world that is 15% less massive than Earth, 8% more dense and 1.05 AU from a star that is 3% more massive than the sun.

The con world has 3 moons for the time being, but none with the gravitational pull that the Moon has on Earth. However, I am trying to figure out plausible distances between each moon and my con-world, as well as the cumulative gravitational effect (and effect on tides, etc).

This is what I have prospectively:
Planet "X" - mass equals 5.1024 * 10^24 kg (or 0.854 Earth Mass)
1st moon - mass equals 1.3185 * 10^22 kg (or 1/453 of an Earth Mass); distance from planet equals 207 000 km
2nd moon - mass equals 2.0742 * 10^22 kg (or 1/288 of an Earth Mass); distance from planet equals 408 000 km
3rd moon - mass equals 5.4865 * 10^22 kg (or 1/109 of an Earth Mass); distance from planet equals 895 000 km

I estimate that the combined gravitational pull of these three moons upon planet "X" will equal 99.88% of the Moon's gravitational pull upon Earth. Does that sound right?

Furthermore, are these moon distances plausible?

Are there any other ramifications I need to take into account?

Next, I would appreciate help in devising plausible distances between the star and other planets in this solar system.

Finally, I need help in figuring out the typical visibility/brightness of these planets from the perspective of an observer on planet "X."

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Some hypotheses:

1 - The tidal forces will be roughly equal to what we see on Earth, but on planet "X" those forces will be somewhat dispersed as they are associated with three bodies instead of one. About 60% of this force will be associated with the 1st moon, about 25% with the 2nd moon, and about 15% with the 3rd moon.

2 - As a result, tides will generally be weaker and/or less concentrated than on Earth, except for times when the orbits of the three moons more or less match up.

3 - This would probably be considered a major astronomical event on planet "X" - I estimate that moon 1 and moon 2 will coincide twice or thrice per year, moon 1 and moon 3 will coincide almost once per year, moon 2 and moon 3 will coincide about once in a decade, and all three will probably only coincide once in a typical person's lifetime (assuming human-like lifespans).

[WeepingElf]

I see no glaring problems with this. The tides will certainly be very complex, with those three moons (how did they form?).

[gach]

I estimate that the combined gravitational pull of these three moons upon planet "X" will equal 99.88% of the Moon's gravitational pull upon Earth. Does that sound right?

If you are considering the strengths of tides, just calculating the strengths of the gravitational pulls isn't very illustrative. Tidal forces are proportional to the gradients of other forces, i.e. what we really are interested here are the rates of change of gravitational pull along a radial line. So if the gravitational field of a mass M is

g = GM/r²,

its gradient (and consequently the proportionality factor of the tidal force) is

dg/dr = -2GM/r³.

Calculating the value of this for the Moon at its mean distance from the Earth and for the moons of your planet at their orbital distances tells that their individual tidal force components would be 90%, 18% and 4.6% of the tidal forces of the Moon for the moons 1, 2 and 3.

Furthermore, are these moon distances plausible?

The major problem in this system is that both the masses and the orbital radii of your moons are comparable to those of the Moon. Having three this large masses on orbits this close to each other makes me think that the system might be unstable in time scales significantly shorter than the age of the planetary system. If this is the case or not and if the instability develops in years or tens of millions of years is just a matter of guessing if you don't do some numerical orbital calculations. I suggest finding some orbital simulation programme, putting your planet and its moons in and looking if the orbits of the moons seem stable of if they eventually fall apart.

The easiest way to gain stability for the system is to decrease the masses of the moons. You might have a system of smallish moons or a system of one dominant moon and other smaller ones. Another stabilising trick is to place the moons on resonant orbits where the orbital periods and phases of the moons are such that they avoid all coming close to each other at the same time. If the resonance has a right form, it can downplay orbital perturbations and increase the stability of the system.

Making the orbits much larger isn't necessarily going to work that well since the further from the planet the moons orbit it, the larger will the relative importance of the the central star's gravitation be. For Earth, for example, it's estimated that no stable satellite orbit can exist further from us than roughly 2.5 times the orbital distance of the Moon and numerically achieved long term stability can only be extended to maybe a half of this. Check the sphere of influence for an often used estimate of the limit.

Next, I would appreciate help in devising plausible distances between the star and other planets in this solar system.

Based on what we now know about planetary systems, there's a great wealth of possible orbital configurations. For a habitable planet you want to have a stable circular orbit well within the habitable zone. The easiest way to achieve this is perhaps just copying the essence of our solar system. The most important thing is not to have a massive planets too close to other planets or on very eccentric orbits. The perturbation you might get for your habitable planet might not necessarily be too big in such systems but there's still a chance that in the long run you will have large enough perturbations to increase the eccentricity of your planet's orbit to uncomfortable levels or move a part of the orbit outside the habitable zone.

I might later come back to calculating apparent brightnesses of other planets but that'll have to wait now.

[gach]

Now, let's do something with the apparent brightnesses of the other planets (or any other illuminated object in the planetary system). The basic idea is to calculate what flux of visible light arrives from the star onto the planet, what fraction of that flux scatters to the direction of the observer, and finally what flux arrives from the planet to the observer. Since the flux of radially expanding radiation decreases by the inverse square law, the observed flux will be proportional to the inverse of the squares of both the planet's distance to the star and the observers distance to the planet. The amount of flux leaving from the planet's surface to the observer's direction is proportional to the scattering efficiency of the surface (albedo) and the phase angle of the planet which tells you what fraction of the planet's surface visible to the observer's direction is actually lit by light from the star.

Putting these effects together will give you a reasonably good estimate of the apparent brightness of your target. Unfortunately light scattering isn't a uniform process and the amount of scattered light will generally depend to an extent from the phase angle between the incoming light and the direction of observing. Most importantly many types of surfaces have a spike of scattering efficiency when observed close to the direct opposite direction to the light source. You can still do first order estimate of the observed brightnesses without trying to take this effect into account and won't get orders of magnitude off from the actual correct answer.

If you search a bit, you can find notes on the mathematical treatment of the subject such as this. The directly most important parts of this document for your question might be the equations 18 or 20 which give you a formula for calculating the apparent magnitude of an illuminated object provided that you know the apparent magnitude of the illuminating star, the distances between the observer, the star and the target, the size and albedo of the target as well as the shape of the phase dependence for light scattering. Since you probably won't bother finding out a proper form for the phase angle dependence of the light scattering I'd suggest just making it a straight line from 0 in the direction of the star to 1 in the direct opposite direction. This is just assuming uniform light scattering and tells that in the direction of the star you cant see any lit surface from the target while in the direct opposite direction you'll see the whole visible surface lit.

If you are left wondering with anything about my posts here, feel free to ask.

[Miekko]

An entirely unrelated question, but since gach's here I figure I might as well butt in: is there any chance for a habitable planet to have two or even three moons with visible disks of sufficiently large size that some features could be visible in them from the planet's surface? Say - at least half or a third of the visible surface of the moon?

[gach]

It might be possible. Phobos is an example of a tiny moon that orbits so close to its parent planet that its angular diameter can be up to 10 arc minutes, a third of the apparent size of the Moon. Remember that the Moon is actually a very large satellite orbiting quite far from us and you could perhaps find a stable configuration with somewhat smaller moons (like the medium sized moons of Saturn) on a bit closer orbits.

You should first consider the sphere of influence of the planets which approximates the area where the primary gravitational force felt by a satellite is from the planet. Both numerical studies and actual observations in our solar system suggest that the satellites should orbit within half of this radius from the planet to have stable orbits. Both the Moon and the furthest Jovian satellites orbit at roughly 40% of their parent planets' spheres of influence. After that you still have to fit the masses and orbits of the moons so that they don't destabilize each other. This is a many body problem and my best advice is to get an orbit simulator and try yourself the stability of possible orbits. You should be able to put two medium sized moons far enough from each other to stay stable but three could honestly be pushing it.