Why can't I see whole galaxies with the naked eye?

Question

I have read these questions:

At what distance could you see andromeda with the naked eye?

Do all the individual stars that we can see in the night sky belong to Milky Way?

I look at the night sky and I can only see stars, no galaxies. And from these questions I learned that even those stars are from the Milky way.

Now, these questions do not talk about the actual QM reasons why some distant stars are visible or not. It must be about intensity of photons. The more intensity a light source has, the more chance of a single photon to reach Earth and our eyes.

Now the human eye can detect even single photons.

A galaxy is made up of billions of stars, so there are billions of light sources, all of them emitting billions of times more photons then a single star.

How is it then possible that we can see single stars, but not galaxies with the naked eye?

A galaxy has billions of light sources, emitting billions of times more photons, and even if one single photon reaches our eyes, we would see it.

From a single star, the chances are billions of times smaller, simply because of intensity.

Question:

1. How is it possible that a single star as light source is visible, but a whole galaxy with billions of light sources is not visible with the naked eye?

2. It is unbelievable that the moon is smaller then the Andromeda galaxy, still Andromeda is not visible at all. Andromeda has billions of stars, each emitting photons, so the intensity must be much higher then of the moon. Why can't I see it then? Our eyes would catch even a single photon from Andromeda.

3. What happens to all those photons from Andromeda? Do they all get blue-shifted to infinity?

Answer 1

1. Intensity is not some type of probability function. It is a measure of the spread of a light source or brightness (J/m^2).
2. These galaxies are very far away
3. Since the galaxies are so far away the intensity of light reaching our eyes at this distance is ridiculously small
4. Combine this with light pollution in cities and our eyes are too overloaded to see galaxies.

Answer 2

1) There are $10^{12}$ stars in Andromeda.

2) Let's assume that on average, they emit as much light as the sun. So Andromeda emits $10^{12}$ times as much light as the sun does.

3) The sun is about 400 times as far away as the moon.

4) The intensity of light falls off with the square of the distance. Therefore light we get from the sun is diluted by $400^2=160,000$ times as much as light from the moon.

5) From earth, the apparent brightness of the sun is 400,000 times the apparent brightness of the moon. Therefore the sun emits about $400,000 \times 160,000\approx 6\times 10^{10}$ times as much light as the moon reflects. Call it $10^{10}$.

6) By 2) and 5), Andromeda emits about $10^{12}\times 10^{10}=10^{22}$ times as much light as the moon reflects.

7) Andromeda is about $10^{15}$ times as far as the moon.

8) Therefore light coming from Andromeda is diluted about $10^{30}$ times as much as light coming from the moon.

9) So if we compare Andromeda to the moon, we have $10^{22}$ times as much light (by point 6)), diluted $10^{30}$ times as much (by point (8)).

10) $10^{22}$ divided by $10^{30}$ is $10^{-8}$, or one over a hundred million. Therefore Andromeda should appear about one one-hundred-millionth as bright as the moon.

By these (very) rough calculations, it's a bit surprising that we can see Andromeda at all.

Edited to add: I'm sure that nevertheless, photons from Andromeda have hit your eye, and probably your brain has detected them. But your brain has no way of knowing that a stray photon came from Andromeda. It needs enough photons to reconstruct a picture, and at least if the above is roughly right, it's only got one one-hundredth-million as many as it uses to reconstruct a picture of the moon.