You’ve got it, the sun is not a point source so a circle is at the focal point rather than a point in the ideal case. The numbers above ignore the arc, which means the intensity of the FP goes to infinity as it approaches a perfect parabola.
In reality no perfect parabola has an infinite intensity, it is reduced by the size of the sun in radians. As the focal point gets further from the dish the sun image in the FP gets bigger and bigger. Just like projecting onto a screen further and further away. It is about 1cm for every meter of focal length.

Hope that helps

Granted the focal point wouldn’t really be a point but rather a small circle if you observe it at mathematical focal point of the perfect parabola.
If that’s the case, doesn’t it just mean that the focal point would simply move a tiny bit further away from the mirror along it’s axis compared to mathematical focal point of perfectly parallel rays of light? Which would be (for example) at 50.01m instead of 50.00m ?

That is exactly the case. I show that the use of a sphere has a maximum power of about 3-4MW per Sqm in the free (pay what you like) paper. It drops when the FL is zero and infinity. The biggest problem with using long focal length mirrors is the arc of the sun increases the diameter of the spot (regardless of what method is used). Even a perfect parabolic mirror with a 50m focal length will have a large focal point. This limits the potential as a solar concentrator.

The two factors are calculated in the paper. It lays waste to the Burning Mirrors of Syracuse story. The maximum practical focal length is about 15m (or 30 cubits as the Ancient Greeks and Islamic scholars complained). At that range fires can be set or combatants blinded.

It does however show that long focal length mirrors can be made and they have a myriad of uses apart from burning or melting stuff. To get to very high temps or energy intensities, relatively deep focal length mirrors need to be used, much like the one in the photo. That hit10-15Mw per Sqm when measured. These were within the capabilities of an ancient craftsman. All he needed was a parabolic template and a big Potter’s wheel. Even higher intensities can be gained with very deep dishes, even if they are relatively small. The intriguing thing for me is that a 10cm dish with the same curve to FL ratio will have the same intensity as a 1000cm dish.

It seems the long focal length mirror was possible, but a red herring when it comes to Syracuse. The variety of uses of shield sized/shaped dish are outlined in the books. I have used them to melt stones, metals, make glass, ceramics, vaporize metals, anneal gems etc.. They are the ancient alchemists tool of choice if high temps are needed.

For the battlefield a reflector will temporarily blind an attacker long enough to finish the job with a short sword. Seems these types of events are related under the blazing shields of Homer’s Ilyad.

Regards

Chris

and delete this post afterwards, thx

With such shallow depth, any slight imperfection in construction of the dish would result in major loss of efficiency. And even if by some miracle anyone in ancient times could construct such dish, the physical properties of material used – no matter how finely polished, at a focal length of 65m the difusion would be still a lot bigger factor than the difference between true paraboloid and spherical approximation.

On the other hand, using a paraboloid with 1m focal length and 1 m2 surface area, regardless of some imperfections would be much more efficient.