> Most fiber optic cable has a melting point of 70C
This seems very unlikely, or at least unlikely to be relevant. The type of fiber optic cable that melts at 70C is PMMA acrylic plastic. To my understanding, this type of fiber not used for long distance communication. Instead, glass fibers with lower losses(which happen to have a much higher melting point) are used: "Because of these properties silica fibers are the material of choice in many optical applications, such as communications (except for very short distances with plastic optical fiber)" (https://en.wikipedia.org/wiki/Optical_fiber). Do you know more than my limited Wikipedia level of understanding, and are confident in saying that long haul cables indeed have a 70C melting point, or should you perhaps include a few more caveats on your napkin math?
I mean, it's napkin math right? The idea is to see if it's worth doing actual calculations.
Even oil companies can't drill deeper than 10km in most areas because the heat destroys any material drill, even the most durable ceramics and exotic alloys that they have tested.
If you can't drill a hole you cant put any kind of fiber in it.
It doesn't matter if it's 5km deep, 10km, or even 20km. You'd need an order of magnitude more depth at minimum to make a difference (geometrically, because the Earth is 12 megameters in diameter), not a counting number multiplier more depth.
That was more the point...it doesn't matter if the numbers are off by a factor of 2 or 3x, it's not worth the calculation time. If we got within 2-3x then the napkin math would have told us "go deeper, find more accurate/precise numbers for each parameter and consider edge cases".
Instead this napkin math said: Forget it. Do something else. You'll never drill deep enough to make a difference.
If, eventually, signals go through fiber at 0.98c instead of 0.7c, then it may be worth considering again despite the tremendous cost, because you could beat NYC-London HFT trades. But today, sattelites with lasers will be much faster than any drillable hole with fiber.
To start, I'm a big fan of napkin math, and I agree with your overall conclusion: fiber optics run through a really deep hole in the earth is not an idea worth pursuing. I don't know if you saw it, but 'xoa' had a great example of napkin math a couple weeks ago, explaining why low-earth-orbit satellites were potentially much better than fiber connections for reducing intercontinental latency despite the apparent extra distance: https://news.ycombinator.com/item?id=22185854
> The idea is to see if it's worth doing actual calculations.
Yes, although even more strongly than I worded it in my comment to you, I believe strongly in underlying principle of "first do no harm". In my mind, there's a large difference between using assumptions that simplify the math (like decimal time: 100 seconds in a minute, 100 minutes in an hour, 10 hours in a day), and making misleading factual statements like "Most fiber optic cable has a melting point of 70C".
> That was more the point
I recognize that was your point. My point, perhaps expressed poorly, was that despite your virtuosic example of napkin math, you might be doing more harm than good overall. People will remember not just your conclusion, but your logic as well. If they remember "light travels about 30% slower in a fiber optic cable than in space", this is good. If they remember "you can't use fiber optic cables in a desert environment" that is bad.
Please at least consider this point of view. From your writing, you come off as an expert. Most of what you state is correct, and leading people to trust you. If you then throw in false statements written with apparent high confidence, you will eventually cause people real world harm. It wouldn't hurt your overall argument to stick with the parts that are true, and to make clear that rough estimates are rough estimates.
I will try to link to this explanation for situations I come across in the future. It's very well argued.
For what it's worth to future readers, I went ahead and found examples of optical fiber rated for use up to 385C[0] (20km). There's research into fiber optic cables which may be able to withstand up to 1000C or more, but I couldn't find any products like that.
Maximum drilling temperatures currently top out at around 250C/500F[1].
This seems very unlikely, or at least unlikely to be relevant. The type of fiber optic cable that melts at 70C is PMMA acrylic plastic. To my understanding, this type of fiber not used for long distance communication. Instead, glass fibers with lower losses(which happen to have a much higher melting point) are used: "Because of these properties silica fibers are the material of choice in many optical applications, such as communications (except for very short distances with plastic optical fiber)" (https://en.wikipedia.org/wiki/Optical_fiber). Do you know more than my limited Wikipedia level of understanding, and are confident in saying that long haul cables indeed have a 70C melting point, or should you perhaps include a few more caveats on your napkin math?