What is the resolution of light?

[DBBPhysics]

Pardon my only basic understanding of physics. I'm not sure if the title is the correct question, but say I'm on a star many, many light years from earth. Can I theoretically see the freckles on my wife's face? It seems to me that light goes in all directions and the further it goes the less resolution there is so maybe I might be able to see one freckle but I wouldn't know that it is a freckle since I couldn't see anything else. How far off am I? Certainly the energy in the light diminishes, but if I had a supersensitive machine, is the data there? If the answer is yes, I could see them, is there any limit to the distance they could be seen even if there is a limit to the technology that I would need to actually see them?

Thanks for your insight.

 

[Drakkith]

The resolution limit for viewing far-away objects is usually given in terms of angular resolution. The angular resolution of a circular aperture (like most telescopes and cameras) is given by the equation: θ=1.22λ/d, where θ is the angular resolution in radians, λ is the wavelength of the incoming light, and d is the diameter of the aperture.

If the angular distance between two details of an object (such as between one freckle and another) is less than the angular resolution of your optical system, you will not be able to resolve the freckles as being separate objects. They will "blur together". The key here is to understand that the angular distance between the freckles on your wife's face depends on the distance between you and her. If she's nose-to-nose with you then the angular distance may be on the order of several degrees. But if she's 10 light-years away then the angular distance is MUCH smaller. So small that you'd need a telescope somewhere around the size of the solar system to resolve her freckles (didn't actually do any calculations, just giving you a very rough idea of how huge the telescope would need to be).

Note that there is also a variable for wavelength in the equation. The angular resolution also increases as the wavelength of the light decreases. So the angular resolution of a telescope is better in blue light (roughly 400 nm wavelength) than in red light (roughly 700 nm wavelength).

 

[phinds]

And of course there is also the fact that the Earth's atmosphere causes enough distortion that you would not likely be able to resolve a freckle even from the moon, much less from a distant star.

And by the way, if you were actually ON the star, you would not likely see anything since you would be fried instantly.

 

[DBBPhysics]

So I think the two of you are saying that, yes, the data are there, but the details of getting it are overwhelming. Of course my wife would have to be a redhead and add even more difficulty!

Thanks for your responses.

 

[phinds]

So I think the two of you are saying that, yes, the data are there, but the details of getting it are overwhelming.

In the case of your specific example, no, I am not saying that. I am saying that the distortion caused by the atmosphere means the data is NOT there.

You would do better picking the example of a person on a moon with no atmosphere observing a distant object on a far away moon with no atmosphere and with the miraculous detail that there is no space dust in between the two. In that case I would agree that the data is there, with, as you say, physically insurmountable obstacles to actually obtaining it.

 

[sophiecentaur]

However far you are away, there is a millet imposed by the Signal to Noise ratio. Resolution is determined by the geometry of the system but the actual power reaching the sensor is also an important factor because it needs to be around the same level as the noise power generated in the sensor (as well as dust etc, in the space in between).
Telescopes on Earth do not perform to the theoretical 'diffraction limits' of resolution. This can be ameliorated by making the reflector 'adapt' to the conditions. The latest terrestrial telescopes have performance that's at least as good as Hubble, I believe.

 

[phinds]

The latest terrestrial telescopes have performance that's at least as good as Hubble, I believe.

Yep. "Adaptive optics" is amazing.

Thus images formed with the 100-inch telescope using adaptive optics are as sharp as those from NASA's Hubble Space Telescope

http://www.mtwilson.edu/ao/

 

[DaveC426913]

I assume the OP was planning to ignore atmospheric effects for the sake of the question.

Let's put his super-sensitive machine in space. What if we also idealize everything else - such as gravitational lensing, and other distortions.

Heck, ideally, we could even ignore all other light in the area, and just figure the solution for one light source, one face and one observer.

It comes down to how many photons (particles or waves) reach your machine over time. The number of photons from a given freckle will be arbitrarily near zero in some arbitrary time unit. So will the number of photons from the rest of her face.

You'd have to calculate how many, based on the angular size of her face at that distance and its brightness.

 

[sophiecentaur]

Mr Shannon would say that you can get the information out of any channel as long as you take long enough over the processing. I think we would have to introduce some noise factor info at least.