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The phase back-of-the-envelope calculation indicates a short calculation to discover some rough estimate of some value of interest, such as the value's order-of-magnitude. The phrase comes from the notion of using some stray paper, such as the back of an envelope of some mail you received, to do such a calculation. Timescales generally use these sorts of simplified calculations, confining themselves to the factors that produce the value's order of magnitude. Scientific calculations and modeling invariably include simplifications, and a back-of-the-envelope calculation represents an extreme case of this. Rough-but-sound estimates are of considerable value, such as for guiding research directions, and checking of the plausibility/realism of the results of very complicated calculations (sanity checks).
The term spherical cow is a humorous reference to such simplifications, referring to calculating something physical about a herd of cows (such as a sunny day's effect on their temperature) and simplifying the calculation by assuming the cows are spherical rather than their actual shape.
The term Fermi problem refers to a challenge to estimate some particular value that is impossible or extremely difficult to discover exactly, the challenge being to estimate it with a convincing back-of-the-envelope calculation.