Why isn't my calculation that we should be able to see the sun well beyond the observable universe valid?

Question

I recently read an interesting article that states that a human being can perceive a flash of as few as 5 or so photons, and the human eye itself can perceive even a single photon. The brain will filter this out, however.

I wanted to calculate how far away you'd have to be standing from our sun for not a single one of its photons to be hitting your pupil over a given second.

The first thing I did was assume that the sun emits $10^{45}$ photons per second, because, well, that's the only number I could find through internet research.

The next step is to assume that the average angle between photons emitted from the sun is pretty much the same, and is equal to $3.6 × 10^{-43}$ degrees.

The next step is to assume that the average human pupil diameter is 0.005 meters, and then draw a triangle like so:

The length of the white line through the center of the triangle equals the distance at which two photons from the sun would be further apart than your pupil is wide, meaning not even one photon should hit your eye.

I broke the triangle into two pieces and solved for the white line by using the law of sines, and my final result is ridiculous.

$3.97887×10^{41} $ meters is the length of the white line. For reference, that's over $10^{14}$ times the diameter of the observable universe.

My conclusion says that no matter how far you get from the sun within our observable universe, not only should some of the photons be hitting your pupil, but it should be more than enough for you to visually perceive.

But if I was right, I'd probably see a lot more stars from very far away every night when I looked up at the sky. Why is my calculation inconsistent with what I see?

Answer 1

The problem with your derivation is that you distributed the photons over a 360° circle, so the photons only spread out in a two-dimensional circle. This means that the intensity of light drops off at a rate proportional to $1/r$ instead of $1/r^2$ (where $r$ is the distance from the center of the sun) like it does in a three-dimensional universe.

So, starting with $N$ photons emitted per second, the intensity of photons at a distance $r$ from the sun is given by $$I = \frac{N}{4\pi r^2}.$$ This comes from spreading out the photons over the surface of a sphere surrounding the sun.

The number of photons seen by your eye per second is just the intensity multiplied by the area of the iris of your eye: $$n = IA_\text{eye} = \frac{N}{4\pi r^2}A_\text{eye}.$$ You are looking for the distance beyond which you would see less than one photon per second: $$n = \frac{N}{4\pi r^2}A_\text{eye} \lt 1$$ Solving for $r$ gives $$r > \sqrt\frac{NA_\text{eye}}{4\pi}$$ Plugging in your numbers gives $$r > \sqrt{\frac{(10^{45})\pi(0.005\,\textrm{m}/2)^2}{4\pi}} = 4\cdot10^{19} \,\textrm{m} \approx 4000\,\textrm{light-years}$$ This distance is still well within our own galaxy.

Answer 2

There is a second error. Not only you distributed the photons over a circle, not a sphere, you didn't take time and cosmic expansion into account. The two are both completely separate, but both of them will each affect what you'd see.

Imagine a photon leaving the sun. Imagine somehow, your eye could see single photons, so that we don't have to worry about distance and eye sensitivity (after all, telescopes can see photons across billions of.light years. Just naked eyes can't).

Now, if your eye is cosmically speaking, near the sun - say, in the same cluster or supercluster of.galaxies - its easy. Photon arrives, you see it, done.

But you're asking about the observable universe. So the photon travels billions of years to the eye... and during that time, space itself expands. As space expands, the photons wavelength grows too. Wavelength and frequency are related, so the frequency changes. In astronomer-speak, the light is red shifted, meaning it now is longer wavelength than before (in visible light, red is the longest visible wavelength, hence the name).

The end result is, by the time photons from the suns light have travelled a chunk of the way across the visible universe, the light may well not be visible any more. It may have been stretched to infra-red or radio wavelengths instead.

Which means that even with eyes able to detect single photons, you couldn't see the sun from across the visible universe. "Detect" yes, "see" no. Because by the time its photons could reach you, they'd be outside the limits (range of visible frequencies) that the eye can actually see.

Answer 3

If the sun emits $N=10^{45} $ photons per second, an observer at distance $R$ will receive $N/4\pi R^2$ photons per second per unit area. If human eye has radius about $r=10^{-3} m$, it will receive 1 photon per second if

$$\frac{N\pi r^2}{4\pi R^2} =1, $$

$$R \sim \sqrt{N} r \sim 10^{20} m\sim 1 kpc. $$

This is less than the size of the universe.

Answer 4

Besides the other errors pointed out in other posts, including the assumption of a 2d universe, the assumption that we can perceive single photons, the assumption that the photon remains of the same wavelength by the time it reaches us, there is still one more rather big factor, namely the assumption that space is completely empty, otherwise we have to take attenuation into account.

This even has a name, interstellar extinction, referring to the extinction of photons via absorption or scattering by interstellar dust or gas along their paths.

[Wikipedia] The amount of extinction can be significantly higher than this in specific directions. For example, some regions of the Galactic Center are awash with obvious intervening dark dust from our spiral arm (and perhaps others) and themselves in a bulge of dense matter, causing as much as more than $30$ magnitudes of extinction in the optical, meaning that less than $1$ optical photon in $10^{12}$ passes through.

Note that the Milky Way is not flat, and we sit roughly in the middle of one of its spiral arms, so any light has to pass through a significant amount of interstellar matter in our own galaxy before it can reach us. See here for a simple diagram. Beyond that, there is also quite some dust at the edge of our galaxy, and intergalactic dust between galaxies.

In fact, the observed low stellar EBL (extragalactic background light) is mainly due to intergalactic opacity, having a much greater effect than obscuration by galaxies. And both of these, together with the comparatively minor correction for finite age and expansion of the universe, appear to account for the observed EBL.

Of course, we also have atmospheric extinction, and you can also calculate roughly how much gets lost by the time it reaches sea level.

Answer 5

The observable universe is actually defined by the speed of light and the time since the universe began to exist in a recognisable form, not by the inverse-square law of light intensity falloff. The reason why we can't see stellar objects from beyond that distance (even with the aid of radio-telescopes) is because there are none old enough for their light to have reached us.

Answer 6

Although I don't understand even the tiniest bit of the equations, and I am by no means a physicist, there is one additional factor: The atmosphere of our planet partially filters out what we can see beyond it.