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Earlier quoted context omitted.
When I took astronomy, anything within an order of magnitude (10) was considered to be the same number. Calculations are very easy when you're only worrying about the number in the exponent.
In astronomy, you only need to get the order of magnitude right to within an order of magnitude.
- Mar's Law
This overlooks the issue that for repeated calculations, such as numerical integration, the trouble comes from accumulated roundoff errors. Even 16 digits of precision can become 0 digits pretty quickly if you're not very careful.
G, for example, is only known to 5 or 6 digits of precision.
Nevermind the mass of the rocket/satellite/probe, positions, the mass of earth, orbital perturbations not accounted for ... etc.
What NASA does do is know precisely the error bounds of any important number which is far more valuable than using arbitrary precision math for π.
In the 'Frontiers in Astrophysics' course on Open Yale, professor Bailyn says that, for the purpose of the course, pi = 3, and pi^2 = 10. Pi = 3, coincidentally, is the Hebrew Bible's approximation too.
When I took astronomy, anything within an order of magnitude (10) was considered to be the same number. Calculations are very easy when you're only worrying about the number in the exponent.
This overlooks the issue that for repeated calculations, such as numerical integration, the trouble comes from accumulated roundoff errors. Even 16 digits of precision can become 0 digits pretty quickly if you're not very careful.
The precision used for calculations is dependent on the number of "steps" required to get to the final result. Roughly, for N repeated calculations you lose somewhere between sqrt(N) * eps to N * eps of precision (eps=2e-16 for IEEE64).
Here are some actual examples:
IEEE64 (~16 decimal digits) is OK for interplanetary navigation for few months, where relatively low accuracy is required.
With the same precision, you start to lose phase accuracy above 24 hours if you're simulating GPS constellations. You need quad precision or above for simulations > 24 hours.
For simulating planet trajectories and solar system stability (Lyapunov time of planets), IEEE64 is good for ~10 mya in the future (Neptune-Pluto Lyapunov time), IEEE128 for ~200-1000mya, above that it is recommended to use 256bit floats and above. This is assuming typically ~1000 steps per simulated orbit.
Fun fact: we know from simulations that Pluto trajectory is stable for >10G years, but unpredictable above >10M years because of chaotic (but stable) interaction with Neptune.
[1] https://en.wikipedia.org/wiki/Stability_of_the_Solar_System
https://books.google.com/ngrams/graph?content=3.1416&year_st...
According to Google NGrams, World Wars I and II both primarily used 3.1416 to represent pi. https://books.google.com/ngrams/graph?content=3.1416&year_st...
Miles and inches? Please learn and use standard international (SI) units. It's important.
Miles and inches? Please learn and use standard international (SI) units. It's important.
He's an American, writing for an American audience, and thus he's using the units Americans use. There's absolutely nothing more scientific about one set of units or another (although different sets of units may be more convenient in different situations).