What is a more professional term for the phrase 'back-of-the-envelope' calculation, used by scientists and such as a word for a very basic, general first calculation?
In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth.
This not only has the advantage of sounding more formal, it implies a pathway toward more formal approximations.
In some cases estimate fits. I prefer rough estimate to reinforce the motion of arbitrary assumptions. With respect to your comment, initial estimate has a nice ring to it, and easily leads to ... based on which is often appropriate.
Back-of-the-envelope calculation is more generally accepted than the (presumably only British) synonym fag-packet calculation (or -- sketch etc.) Another useful but fairly informal word is guesstimate, which you'll find in plenty of dictionaries.
You can say preliminary or cursory calculations.