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For example, is the phrase "Stars, to a good approximation, are spherical." correct and idiomatic English?

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That is correct, idiomatic English, but depending on the exact meaning, you may be looking for "to a first approximation", meaning:

a roughly approximate value of a quantity often preliminary to more precise determination

The difference is nuanced, but my impression is that "to a good approximation" refers to a somewhat more exact approximation than "to a first approximation" (in that one could make better approximations after the first).

Google Ngrams Viewer suggests that both are used in writing, with "to a first approximation" being somewhat more common. Also, both are declining in usage in modern writing. Google Ngrams Viewer graph showing "to a first approximation" consistently higher than "to a good approximation"

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    You say significantly more common "To a first approximation" and "to a good approximation" have different nuances and are incomparable. Even if they did mean the same thing, the difference in frequency at such low levels is statistically insignificant. Also, being more frequent does not affect how appropriate something is in a given context.
    – Greybeard
    Commented Dec 22, 2020 at 11:38
  • @Greybeard all excellent points, thank you. I've edited my answer to take those into account.
    – Ryan M
    Commented Dec 22, 2020 at 11:50

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