Objects entering or leaving the observable universe Have there been any observations of objects entering or leaving the observable universe?

When looking at the physical limit of observing something like this I will to assume that the "object" would be an galaxy. I think it would be a good assumption to choose the size of such an galaxy, $G$, to be in the order of $10^{5}$ light years and the radius of the observable universe, $OU$, is $46 \times 10^{9}$ light years. Using the limit of angular resolution and the fact that the current biggest telescope has a diameter of 22.8 meters, then the longest wave length which would still yield would be about $40 \mu m$. When reversing the redshift, assuming this would be around $z=9$, then the longest emitted wave length which would be detected would be $4 \mu m$.

So observing such an event would technically be possible, however relative to the astronomical timescale we have only been able to observe with such a high resolution for a very short time, so the probability will not be very high.

And if such an event would be observed, would such an object suddenly appear/disappear. It will take along time for an entire galaxy due it size, so this would translate to more of a fade. Or will something else happen?
I don't know if there have been any observations of objects leaving the observable universe, but I'll admit that I have a hard time keeping up with the latest discoveries in observational astronomy. I'll try to approach your question from a theoretical viewpoint.

As you pointed out, the radius of the observable universe is $4.6 \times 10^{49}$ light-years. To figure out how fast an object is moving away from Earth (or any observer), we can use Hubble's law:

$$v = H_0D$$

where $v$ is the recessional velocity, $H_0$ is Hubble's constant, and $D$ is the proper distance from the observer to the object (see Wikipedia for the difference between proper distance and commoving distance). The trouble here is that the exact value of Hubble's constant isn't exactly well-known. There have been many different observations since Hubble proposed his law that attempted to find values for the constant, but they range widely. Here I'll use $82$ $km$ $s^{-1} /Mpc$ (where $Mpc$ is a megaparsec - 1 million parsecs).

Let's say that the object in question is $13$ gigaparsecs away (so $13,000$ megaparsecs away). The light we see from the galaxy would be from when it was very young. One of the most distant observed galaxies is this far away, and only $2,000$ light-years across (note: do not confuse the "light-travel distance" with the "proper distance"), so let's say that this galaxy is smaller in size - negligibly small, in fact. We can then use Hubble's law to calculate a recessional velocity $v$ of

$$(82)(13,000) = 1.066 \times 10^6$$ kilometers per second, or $3.36 \times 10^{13}$ kilometers per year. That's pretty fast! An object $13$ gigaparsecs away would be $42.38$ billion light-years away, not far from the edge of the observable universe. Yet if the edge of the observable universe is $46$ billion light-years away, the galaxy would still be $3.62 \times 10^{21}$ kilometers away, and it would take a long time to get there. A galaxy this young would be maybe $1,000$ light-years across, so, moving at its current speed, it would take it many years to cross a distance its own length - that is, if the furthest edge of it were at the edge of the observable universe, it would still take many years to fully disappear. And that's even factoring in that it would be moving away faster!

The reason I don't have the object being further away is that any objects at the edge of the observable universe would be so young when they released the light we see today. That means that galaxies would be small, and wouldn't emit a lot of light. One of the furthest objects we have detected is only $30$ billion light-years away, and $2,000$ light-years across. So it would be terribly hard to observe an object at the edge of the observable universe (ironic, isn't it?). But here I have shown that it would take years for the object to completely disappear.

I hope this helps.



My sources for the data and conversion factors (i.e. light-years to kilometers):

Hubble's Law

Wolfram Science World

Most distant objects

Spacetelescope.org

Light-year