That is useful for reducing the noise and increasing the exposure time, but doesn't actually increase the resolution.

Yeah, this is assuming totally ideal circumstances with perfect optics.

At the absolute most extreme, we might get some nice Northern/Southern Lights.

Basically, this gas is so thin that we would consider it a vacuum. It's only over millions of years and many light years of space that the number of collisions between particles adds up for it to start acting like a gas, so that it can carry shockwaves, soundwaves etc. But we're often talking about less than one atom per cubic centimetre, and it's not going to push the Earth at all as we pass through this wave.

Some of these particles may be charged, and drawn in by the Earth's magnetic fields towards the poles, and maybe contribute to the Northern/Southern Lights. I haven't done the maths to see if that would be a significant contribution compared to the constant wind of particles we get from the Sun though.

Maybe just barely!

The maximum possible resolution you can physically get depends on the size of the telescope. The next generation of ground-based optical telescopes will have mirrors around 30m in size. At the distance of the Moon, these could have a resolution of about 5 or 10 metres, if they have perfect optics. So you might be able to make out the lander as a fuzzy blob at the limit of resolution, but won't be able to see any astronauts, and definitely not any footprints. You'd really want something like a 200m telescope if you wanted down to 1 metre resolution.

Interferometry isn't the best thing to help here. While you can combine light from multiple telescopes separated by some distance to increase their resolution, this is really tricky to do with visible wavelengths, and you're limited to a small number of telescopes on the same site. What you end up with isn't quite an image as you don't have enough combinations of baselines between telescopes to get the full visual information. At lower wavelengths (e.g. radio) it becomes easier to do interferometry affordably with a large number of telescopes (possibly even spread over the world!), but longer wavelengths also inherently have a lower resolution, and you're often dealing with a much dimmer image - the Sun emits a lot of visible light, so most things in the solar system are brightest in the visible ranges.

Overall, to photograph the Moon (and many other solar system objects) you get much better resolution by sending a small telescope to orbit the Moon than building a big one on Earth.

It's kind of the other way around - these are the observational constraints on what dark matter has to be, assuming they are governed by known physics.

Neutrinos are too light, and move too quickly, so they wouldn't collapse into halos of the right size.

That's only for spherical concentric shells. If, for instance, a star passed within 2 AU of the Sun, it would definitely affect the Earth's orbit, even though all the mass is outside the Earth's orbit.

So yes, that's another good point. Part of the answer is a gravitational force can affect a system without affecting the motions within that system - and that does come down to the smoothness of the potential. I also wanted to clarify that dark matter isn't a magic thing that just speeds up orbits - in terms of gravity, it's just a distribution of matter, that acts like any other distribution matter. But yes, the other part is that there isn't an additional local effect, as the local density of dark matter really is very low.

Though I would want to clarify for other readers (as I assume you know this of course) that while there is locally not much dark matter (it has a very low density), but because it's smoothly spread out over such a huge 3D structure, its total mass is much larger than the mass of the stars and gas in the Galaxy.

Technically, kinda? But it probably wouldn't be very visible, and it's a bit unlikely.

So, the orbital period of an object relates to the "semi-major axis" of its orbit. The "major axis" of an elliptical orbit is the distance from when the object is at its closest point to the Sun, to the distance from when the object is at its furthest point from the Sun. The "semi-major axis" is half of this. For a circular orbit, the "semi-major axis" is just the fixed orbital radius.

Earth's orbit is pretty close to circular (although a little bit elliptical), and has a semi-major axis of 1 AU. This means that any object that has the same orbital period (one year) also needs to have a semi-major axis of 1 AU. This could be an elliptical orbit from 0.5 AU to 1.5 AU, a more circular orbit from 0.9 AU to 1.1 AU etc. At its most extreme it could go from grazing the Sun at almost 0 AU, up to almost 2 AU.

This means that the comet will have to always be relatively close to the Sun. Jupiter is 5 AU from the Sun, and Mars is 1.5 AU from the Sun, so if your maximum distance from the Sun is 2 AU, you're still very much in the inner solar system.

This is significant here because it affects how comets comet. A comet carries cold material from the outer solar system, and then warms up as it approaches the Sun. Getting heated and blasted by solar light and radiation causes outflows of dust and gas from the comet, producing a huge 'coma' tail of material streaming outwards from the Sun (note - the coma isn't a tail behind the comet, it always flows away from the Sun).

A comet that close to the Sun would be generally warm all the time, and not have such dramatic outbursts. It also would lose material every time it approaches the Sun, which means that if it was ever dramatic, it would burn out pretty quickly.

There is a comet with a period of 1.4 years - https://en.wikipedia.org/wiki/3200_Phaethon - however its dim enough that an amateur astronomer with a backyard telescope can only see it if it happens to be passing close to the Earth. That's generally what you'd expect a year period comet to be like.

Comets can have pretty regular orbits - although the outgassing of material can add a twist. But you could have a fairly inactive comet with a period of about a year, and it wouldn't radically shift. Like, if it had a period of a year and a day, it would just shift relative to the Earth's orbit by one day a year.

What you might want to look at is meteor showers. Meteor showers do regularly occur at the same time of the year. Here, there is a long tail from the disintegration of an old comet that remains in orbit around the Sun. Although the individual chunks of stuff in the tail are all in motion, they all are following the same orbit, which means that the tail as a whole stays in roughly the same place in the Solar System. This means that if the Earth passes through the tail, it will pass through the tail at the same point in its orbit every year, which is why you get meteor showers at the same dates every year. If you're looking for annual astronomical events that are visible to the naked eye, that's a common one. Some meteor showers are more dramatic than others, so there's no reason why some planet might have a particular strong on at a certain time each year.

The short answer is that it pulls the Sun and the planets by the same amount, so their relative motions within the solar system can ignore what's going on outside.

The dark matter halo of the Milky Way is very big and smooth. It's a big roughly spherical blob, that's denser in the middle, and less dense on the outside. Its net effect is that it an adds extra gravitational force towards the centre of the galaxy.

But this is all just classical gravity. The weirdness is that dark matter doesn't interact electromagnetically (although neutrinos don't do that either), but it's totally normal in how it interacts with gravity. If you had a big blob of gas with the same mass and distribution as the dark matter halo, it would produce exactly the same gravitational force (although this gas halo would be unstable, and would cool and collapse, as it can interact electromagnetically). The reason dark matter makes stars and gas orbit faster than you'd otherwise expect is just because there's more mass, and more force, and this mass is more spread out than the stars (which are more heavily concentrated in the centre), which means that the force drops off less slowly with distance than it would otherwise.

So the answer for why dark matter doesn't affect the orbits of the planets in the solar system is the same as why the combined gravity of all other stars and gas and black holes and everything else in the galaxy doesn't affect the orbits of these planets, or even why the Sun doesn't affect the trajectory of a football on Earth. If the gravitational force field varies only on a much bigger scale than the system you are looking at, then it pulls all objects in that system by the same amount. This means that, from within the system, you can't really tell that you're getting pulled at all, other than by looking at the overall motion of the whole system.

There is a middle ground where you get "tidal forces". This is where gravity differs a bit across some object. So, the Moon does pull the near side of the Earth more than the far side, which pulls Earth's oceans a little bit. But this is much smaller than the net total effect of the Moon on the Earth as a whole. But the gravitational force of the galaxy is on a way bigger scale than that, and we really can just ignore it if we only care about what's going on inside the solar system.

The thin gas in the galaxy - what we call the "interstellar medium" - does scatter and disperse photons. This effect is small at high frequencies (like visible light), but increases as you go down to lower frequencies (radio waves etc). But even in radio waves, we're still limited more by the resolution of our telescopes than the fuzziness causes by the galaxy's "atmosphere" - although the Event Horizon Telescope is awfully close to the point where that matters.

What we do see is that radio waves of different frequencies take a different time to reach us, and will be a little out of phase. So if some event happens that emits radio waves are a broad range of frequencies, we don't get receive that emission at once. Instead, the higher frequency radio waves arrive first, and the low frequency waves arrive later. Instead of a single broad pulse, what we get is like a quick glissando from high pitch to low pitch.

You can actually use this to measure the thickness of the interstellar medium. The slower the slide in pitch is, the more material was along the path of the radio wave. So if you already know the source and the distance to the source, you now know the "column density" of interstellar stuff between Earth and the source. Do that with a bunch of sources and you start to build a map of the density of stuff in the galaxy, which can be used to complement other measures for the interstellar medium.