This all started when I noticed Mars was very bright as I viewed it this month with the naked eye looking East from a location 10 km North of Gingin (-31°20'27.60" S 115°54'28.80"). Gingin is 100 km North of Perth Western Australia and it has very clear skies with hardly any light pollution. I read that Mars was "in opposition" which means that it is in a direct line, on the same side of the sun as the Earth. In this position it can be about 62 million km from Earth. I took a photograph using 3.6x and posted it on Twitter. It seemed pretty impressive with an orange disc visible. After I posted it on Twitter I looked around and saw quite a few similar iPhone images on the web. Then, some smart photography people claimed it was artefact and might be "bokeh" which is a kind of software interpretation. I invited comments. The authority on all things astrophysical is @thunderf00t and he came out with a quick answer that, through the telescope, "Mars is angularly about the size of a small crater on the moon." (thunderf00t link here) Dave Jones checked it out and posted a couple more images showing how bad focus affects the digital camera image, increasing it about 5x (eevblog Tweet from Dave Jones).
I made some observations then, and sure enough, my iPhone could not see any small craters on the moon. I tested the camera using a tape measure and ruler and discovered the resolution, using 3.6x, was 1 mm at 2 m. This is an angle of about 0.03 degrees. According to the laws of physics and trigonometry, the angular size of the moon is 0.524 degrees. The Moon image from my iPhone was about 10 times bigger than the Mars image. Thus the angular size of the Mars image was 0.0524 degrees. From trig', allowing for a base of 6.2E10 meters away and a diameter of 6.7E6 meters, the actual angular size of Mars is 0.006.
Conclusion: iPhone has turned the Mars image into a circle 5x times bigger than it really is. Clearly I need to upgrade my iPhone and purchase a long lens for it. Thanks to all the experts. I enjoyed refreshing my trig, and cosmology. Its nice to know that some things have not changed since I finished high school 52 years ago.
Footnotes:
It was difficult to actually find the angular resolution of the iPhone, since most of the hype about "retina" really refers to the screen image rather than the camera. But Wikipedia has good information. Here is what I found out (see also my notes below).
The human eye has an angular resolution of 0.6 arcminutes. There are 60 arcminutes in a degree so this is 1/100 of a degree. (0.01 degrees).
The iPhone has a resolution of the order of 300 pixels per degree (PPD) which is about 0.003 degrees. My measurement might not have been under ideal conditions. Lighting, exposure, autofocus etc. seems to have worsened the result by a factor of 5-10. So iPhone should be better than the eye. It should be able to resolve mars as a single pixel, but not as a circle.
As an afterthought, I noted that the Jupiter image was about 2x the size of my Mars image and about 1/5 the size of the moon image. Thus Jupiter had an angular size of 0.1 degrees. According to trig', with the current distance to Jupiter of 782,905,474 km (7.82E11 m) and a diameter of 139,822 km (1.39E8 m), the actual angular size is tan-1((1.39E8)/(7.82E11)) = 0.01 degrees. So it is (just) resolvable as more than a pixel but is also being magnified as it really is about 50 times smaller than the moon.
Further Information: Calculations and Image Data
