ISO 400 on the 7D and 5D3 are completely different sensetivity levels and it's misleading to directly compare them.
Cameras are a function of geometry.
People have been misled with crop to full frame conversions for years.
The 5D3 was my first FF camera body. I previously used crops... 20D and 40D. As stated, I am evaluating either a 7D or 5D2 as a backup to the 5D3.
I am not a physics major and thus, I am not going to dispute Radiating's calculations and references to physics. I understand that it isn't a 1:1 comparison, but...
What I do know is that a photon of light is still the same size regardless of sensor and that a crop sensor 'extrapolates' the equivalent pixel dimensions due to the fact that pixel density is so great and thus pixel size is so small, that is is practically splitting a single photon of light.
I also know that at the end of the day, if any of us were to print either OOC image to their largest extent @ 300 dpi, that we would see the difference between the 7D and the 5D3 as seen clearly by these 100% crops and therefore, the reality is that a crop body's sensor is simply inferior.
Desiring to hold on to the 'reach' (cringe) benefits of APS-C, I wanted very much to keep the 7D as an all-around versatile camera and as a backup to the 5D3. After the pictures I took today, and the grain I see in them even at 50% (heck, even looking at the entire image sized to fit on a 15" laptop display) there is clearly an improvement in IQ from a FF body.
After evaluating the 7D, it's clear an APS-C is inferior. It's just the reality of physics. We can't shrink light any further, but we are trying to over-resolve it. Here come the arrows and darts...
See that's where your wrong, the 7D isn't working with less light because it has a smaller sensor, and it's getting nowhere near splitting photons. This isn't the issue. The amount of noise you get in an image is based on the amount of light being emitted by the object. Noise (for the most post) is a result of the quantum properties of light in that light exists as a gausian distribution of photons, light isn't emitted uniformly but photons hit an object at random. The difference in light and dark areas in an image resulting from noise are caused therefore because in the light area you had 16 photons and in the dark area you had say 9. Because of random probabily the light from your subject doesn't arrive at a uniform time. Overall the random noise will average out so if you leave the shutter open for twice as long you might get 25 photons in the light area, and say 26 in the dark area of two pixels. The reason different cameras have different noise levels is because for the most part cameras throw away some of the photons because the whole surface of a sensor doesn't absorb light, and some of the light is absorbed by the micro lenses and bayer dye. Then on top of that cameras introduce a slight amount of electrical noise (though electrical noise can be a big issue in astro photography).
For the most part though noise comes from the subject, and the difference between noise comes from the amount of light the camera throws away.
Though the 5D3 does throw away less noise and introduce less electrical noise than the 7D. (5D3 throws away 51% of light, 7D throws away 59%)
Anyways like I said
So in other words theoretically a Crop set to:
#1. 17mm - f/2.8 - ISO 800 - 1/50th - with 1/4 flash
#2. 55mm - f/2.8 - ISO 800 - 1/50th - with 1/2 flash
Will produce a 100% identical image with no difference in exposure, lighting, depth of field, field of view or composition when compared to a full frame set to:
#1. 27mm - f/4.48 - ISO 2048 - 1/50th - with 1/4 flash
#2. 88mm - f/4.48 - ISO 2048 - 1/50th - with 1/2 flash
Literally no difference.
What I mean by this is you would technically capture even the exact same photons.
To photograph something clearly part of the light coming from the object must be ignored because when light hits an object it is scattered in all directions. To see the details of an object clearly, you have to selectivly remove the light which isn't paralell to the line of sight between you and the object. Otherwise the image will be blurry.
To accomplish this all that is needed is an aperture, or something that blocks the path of the light which blurs blur your image. The thing is that the sensor size is part of the aperture too in that it removes light that is less parallel to the line of sight.
This is why when you use a 1.4x teleconverter it projects an image circle that is greater than your sensor size, and it therefore reports the lens as having an aperture 1 stop lower, because the lens allows light which is less parallel to the line of sight to make it onto the captured image circle. We measure aperture values from the lens' perspective and ignore the camera though which is why this is confusing.
Full frame lenses on crop simply project an image circle that is greater than the sensor and so they are always fundamentally stopped down a little. To compensate the light is multiplied in intensity so our exposure calculations aren't off, so a given sensor sensetivity is labled ISO 2048 on full frame but is labled ISO 800 on crop instead of labling aperture f/2.8 aperture f/4.48. By using a 1.6 teleconverter or 1.6 crop camera with a theoretically perfect lens at a 35mm equivalent: aperture, ISO and focal length you will be ignoring the exact same photons as you would on full frame and you will literally be collecting the exact same photons, and if we use a theoretically perfect camera, you would get the exact same image photon for photon, pixel for pixel. Light doesn't care how dense it is because light is 99.99999999999% empty space, all that matters is bending the optical path correctly to capture the nessesary photons.
The problem we run into of course is just that the manufacturers don't make lenses that can bend a crop camera's photons the way we want. There's no 17-55mm f/1.8 lens. And the reason why full frame is better is because when you're bending the optical path on a crop camera it has to be more precisely done for a given resolution, so with a given level of precision in the manufacturing process, particularly our abbility to polish and assemble the lenses precisely, full frame lenses will have more resolution due to magnifying the flaws in our manufacturing process less. Likewise medium format cameras magnify the flaws in the manufacturing process even less so we have even better lens resolution from them.
Crop cameras currently have slightly worse sensors too. As I mentioned the 5D3 absorbs 20% more light than the 7D and it's electronics introduce less noise for a cleaner result. Overall we're talking a half stop or more at euqivalent aperture and ISO.
With that said the 7D also can't be stopped down as far, it would have to shoot at ISO 19 to get an image with as little noise as a 5D3 at "low" ISO.
So on crop you have worse sensors, worse ISO ranges, worse lens resolution, and they don't make fast enough lenses for crop cameras like 17-55mm f/1.8 lenses.
On the other hand some will argue they don't make long enough lenses for full frame, so crop is better for reach, especially compared to teleconverting, as you will have a theoretically perfect teleconverter with better autofocus.