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Is it theoretically possible for a moon to orbit a planet (let us say Earth size, mass and solar distance) in such a way as to create a planet-wide solar eclipse lasting several weeks?

These are my assumptions (some of them likely flawed) about the circumstances under which this might be possible:

  1. To simplify the problem, let us assume that there is no planetary axial tilt and that the moon has an equatorial orbit.
  2. The orbital speed of the moon would need to be quite slow in order to stay in position for long enough to create a graduating eclipse over several weeks.

  3. In order to create a planet wide eclipse, the moon would need to be quite large and/or relatively close to the planet. Is this circumstance feasible if the mass of the object were low compared to its size?

  4. How stable could the described relationship be?

Thank you for your interest and answers.

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    $\begingroup$ Suppose we replace Earth's Moon by huge (200 000 mile diameter say) unnaturally stiff disk of material and that we can somehow arrange for it to always be "face on" to the Earth (so maybe we add an unnaturally strong and dense rod perpendicular to the disk can more massive than it, so we get tidal stabilization) then it would block the Sun for a good part of each lunar month. If you make it a bit bigger and further away you could get a longer eclipse less often. $\endgroup$
    – Steve Linton
    Jan 31, 2019 at 11:58
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    $\begingroup$ Another option would be a disk placed at the Earth-Sun L1 wide enough to block the Sun (with some left over) spinning "on its edge" every month or two, so that it blocks the light of the Sun when its face on, but not when it's edge on. This one is unstable $\endgroup$
    – Steve Linton
    Jan 31, 2019 at 11:58
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    $\begingroup$ Neither of these seems likely to arise naturally, and both need unfeasibly strong materials. $\endgroup$
    – Steve Linton
    Jan 31, 2019 at 11:59
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    $\begingroup$ This would certainly be better at "worldbuilding". However you need to define "week" Is that a period of 604800 seconds, or a period of 7 days (ie 7 rotations of the planet relative to the sun) or 1/4 of a month or something else? $\endgroup$
    – James K
    Jan 31, 2019 at 23:07
  • $\begingroup$ Thank you for your responses. I will certainly rethink the scenario on the basis of your feedback. I have posted this in worldbuilding, however I wanted more reasoned responses than I would typically get from there. $\endgroup$
    – TheConflagrant
    Feb 3, 2019 at 6:35

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Short answer: No, it could not.

The moon already has little enough mass relative to the earth so that its orbital period is not much shorter than a satellite with negligible mass. So, any object at about the distance of the moon would take around a month. If the object really had negligible mass it would be moving a little slower than the moon, so let's be generous and say 5 weeks. That means the object would have to be large enough to extend 1/5 of the way around the sky to block the sun for just one week. This is not possible. Such an object would be much larger than the earth.

If you imagine an object closer to the earth, it would be orbiting faster so it would have to be even larger.

Moving farther away is a little better. The orbital period is proportional to the radius raised to the 3/2 power, so moving the moon four times farther away would make its period eight times as long. But it would have to be four times larger, or 64 times the volume, just to eclipse the sun at all. And at that point it's approaching the size of the earth.

It may be possible to construct a synthetic object that orbits the sun at earth's L1 point (between the earth and the sun at just the right distance to stay in line with them) and permanently blocks the sun. You could tweak its solar orbit so that it only blocked the sun part of the time. It could be something like a mylar disk that rotates to avoid collapsing into a ball. But this becomes more of an engineering question than an astronomy one. No plausible natural object would have this configuration.

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The items 2 and 3 seems to be incompatible, I agree that you need it if you want an eclipse lasting for several weeks.

You said

The orbital speed of the moon would need to be quite slow.

and then

the moon would need to be quite large and/or relatively close to the planet.

In genera both thing can happen only

if the mass of the object were low compared to its size.

So I agree with you at this point. The mass of the object should be tiny compared to its size, because if not its speed would be fast for conservation of angular momentum (which establis in the moment of the cretion of the system earth-moon).

But It is so diffucult that a natural satelite has so low density.

I think the only way to avoid the comflict between 2 and 3, is a sistem which at the moment of its formation had a very very low angular momentum, but this is actually an uncommon situation.

May be this last one is the arguement I would chosse between both. I prefer low angular momentum before low density.

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First question is what do we want for a long eclipse? The answer would be a large moon on a slow orbit. The following analysis assumes the system is exactly coplanar, and I neglect the angular size of the star (which would result in increased satellite radii) and the size of the planet itself.

To get a slow satellite orbit, we need to figure out what the outermost stable orbit. Neglecting the small eccentricity of the Earth's orbit, the radius of the Hill sphere $R_\mathrm{H}$ is given as follows, where $a_\oplus$ is the radius of the Earth's orbit, $m_\oplus$ is the mass of the Earth and $m_\odot$ is the mass of the Sun:

$$R_\mathrm{H} \approx a_\oplus\sqrt[3]{\frac{m_\oplus}{3m_\odot}}$$

This works out as about 1,946,400 km. Moons are stable out to a factor of about 1/3 for the Hill radius in the case of prograde moons, and about 1/2 of the Hill radius for retrograde moons.

In the case of a prograde satellite, this puts the orbital radius $a_\mathrm{m}$ at about 498,800 km with an orbital period $P_\mathrm{m}$ of 40.5 days. We need to also take care of the Earth's orbit around the Sun, which has a period $P_\oplus$ of 365.25 days. This gives the resultant angular velocity as:

$$\omega = 2\pi\left(\frac{1}{P_\mathrm{m}}-\frac{1}{P_\oplus}\right)$$

This can then be multiplied by the time taken to figure out the angle the moon travels through in a given time $\theta = \omega t$. For a week-long eclipse, the angle works out as about 55°. The radius of the moon can then be obtained by

$$r_\mathrm{m}=a_\mathrm{m} \arcsin\left(\frac{\theta}{2}\right)$$

This works out as 231,100 km - which is about the radius of an M3V red dwarf star. So a prograde satellite doesn't look like a good possibility for week-long eclipses.

What about the retrograde case? In this case, the orbital radius and period of the outermost stable orbit end up being 748,200 km and 74.5 days. The angular velocities add up, so you have

$$\omega = 2\pi\left(\frac{1}{P_\mathrm{m}}+\frac{1}{P_\oplus}\right)$$

Doing the same calculation again gives a moon radius of 260,200 km - even worse! The increased distance wins out over the slower orbit.

What if we use a different type of host star? According to Eric Mamajek's list of the properties of main sequence stars, an A5V star as having a mass of 1.85 times that of the Sun and a luminosity of 101.16 times that of the Sun, which puts the habitable zone at 3.80 AU with an orbital period of 1990 days. Doing the calculations again results in smaller satellite radii of 136,400 km (prograde) and 150,900 km (retrograde). These are still far too large: the prograde case is roughly the size of an M5V red dwarf.

So unfortunately it looks like the case of a week-long eclipse is not very plausible!

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