> you don't "calculate the efficacy of [a theory]"
How would you describe in a single sentence the growing precision of our calculation of the speed of light constant over the last 300 years?
> data don't "defend the theory's math"
How would you describe in a single sentence the way Einstein used Mercury's retrograde motion to assure himself of the advocacy of his theoretical mathematical model for gravitation?
I don't know, first I'd talk about how it was discovered that light behaves like c really is a constant (that's not intuitive nor trivial). You could guess from Fizeau (1851) and Michelson-Morley (1887) experiments, but it wasn't really until the 1905 Einstein's paper that it was understood that you don't need aether, the speed of light can (and should) be a universal constant and it all makes sense.
>How would you describe in a single sentence the way Einstein used Mercury's retrograde motion to assure himself of the advocacy of his theoretical mathematical model for gravitation?
He shows (see here: http://www.gsjournal.net/old/eeuro/vankov.pdf) that his theory (after reasonable approximations) predicts a precession of the perihelion of Mercury that agrees with the experimental result, then he stops. That's what you do in Physics: look, my nice theory predicts this observable in agreement with the experimental data. That's a very good sign, but the data could be explained by other theory as well. Data can "prove" that a theory is wrong (if your theory has falsifiability, as it should), but they can never show that your theory is all that there is, nor your maths... There are other theories of gravitation, for instance Logunov's, that are interesting and agree with data (more or less) but have a few problems with maths.
BTW, I don't think he ever really doubted his GR theory, it comes from a very simple, yet profound idea that once realised had to be right. You have to do all the technical work afterwards, that was difficult, but a few mathematicians could have done/did it (Hilbert). In any case the idea is to check your predictions with the measurements from more and more precise experiments to find out if you're wrong, not the other way around (that... well, I've met that funny epistemology a few times in social sciences.)
> Data can "prove" that a theory is wrong (if your theory has falsifiability, as it should), but they can never show that your theory is all that there is, nor your maths.
Where you got '"prove"' and 'all that there is' I am unsure
My wording was:
op> how well different theoretical models express experimental data
Which mirrors your own reiteration of that point
> the idea is to check your predictions with the measurements from more and more precise experiments
Which is literally what I was asking for: measurements from .. precise experiments
> predicts a precession of the perihelion of Mercury that agrees with the experimental result
Right, where's the experimental data on the orbit of Mercury?
If you are unable to answer the question that's fine but as is I fail to see what you are trying to accomplish with your ire and misdirection
If you're interested in the data you can browse the references, go to the libraries (as physicists have to do even at college, because it's kind of hard), find out first what exactly you are asking... even mail someone. And you can also measure things by yourself, this is a free universe and everybody can study its properties (Mercury is out there and maybe you can afford a nice telescope). What is unreasonable, very unreasonable, self-entitled in my opinion, is expecting that experimental physicists provide some kind of repository of (internal-notes redux) data for casual readers of pop sci. That would cost their time (or some overworked PhD student's) and project money, the bad news is no one is willing to do that nor paying for it. It's not that there's some kind of conspiracy, it's that people are actually busy working.
> What is unreasonable, very unreasonable, self-entitled in my opinion, is expecting that experimental physicists provide some kind of repository
> It's not that there's some kind of conspiracy
You seem to be arguing with someone else
> I don't usually comment on HN about Physics because... well, never mind, it really was a bad idea this time.
If you are knowledgeable on the subject I would encourage you to comment, but keep the comment on the topic being discussed
If by seeing "how well different theoretical models express experimental data" you're planning to apply idk assorted non-linear regression models from equations you're not familiar with to see which one has the minimum chi-square or something like that, well I guess that's what hobbies are for. Differences tend to appear at the limiting cases, there you need more Physics than Levenberg–Marquardt fun, so it's better to start there.
I have to warn you that your wording matches the usual stuff you get from Physics cranks (the bane of the trade), hence the knee-jerk reaction, sorry about that. I'll try harder next time.
to express concepts like these to anyone interested using real world data to show why these concepts are important
Like why Newtonian physics is insufficient to explain Mercury's orbit
Or how infinitely divisible energy leads to the 'ultraviolet catastrophe'
Or how one can approximate the speed of light just using orbital data of io and earth (romer's initial method)
I can appreciate that my wording may have been ambiguous
Brevity can sometimes be the enemy of clarity but questions can help avoid assumptions
I'm not an Astronomer (nor Astrophysicist, I know about GR coming from a Theoretical Physics bkg), so I suspect you could get more recent data, but a classic reference from the GR book of Weinberg is a paper from G. M. Clemence: http://adsabs.harvard.edu/abs/1947RvMP...19..361C (it can be obtained if you look around, there's a similar free-to-read paper from the same Clemence: https://www.jstor.org/stable/3143142?seq=1#page_scan_tab_con...) that has as reference this: http://adsabs.harvard.edu/full/1943AJ.....50..126C (full text), that quotes the Astronomical Papers of the American Ephemeris, Volume 11 (it says II, but it's actually 11), part I. Google is your friend (they've scanned those!) and you can find the data at https://catalog.hathitrust.org/Record/000068403. This is how a little of preliminary research works, you have to do it yourself all the time... Luckily now you don't have to go hunting for a paper copy without being sure about the contents all the time (although it still happens).
The classical mechanics calculation might be tricky, to say the least, as well.
For the UV catastrophe I've always liked the Jeans cubic cavity and the Gamow observation that opening an oven doesn't kill you. And of course, the fact that integrating Planck's law gets you Stefan-Boltzmann's, very neat result.
I should think about all this a bit more in order to give you better feedback.