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I've read online that if the final approach course alignment is more than 30 degrees, or if more than a 400 feet per nautical mile descent rate is required, that a circling only approach must be made. But no one ever cites the source for either of the two conditions. Where is the source for those two conditions? Page and paragraph number?

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One location, for U.S. procedures where the criteria for circling only minimums are published, is the FAA's Instrument Procedures Handbook , Chapt. 4, page 4-11.

[...] Typically, circling only approaches are designed for one of the following reasons:

  • The final approach course alignment with the runway centerline exceeds 30°.
  • The descent gradient is greater than 400 ft/NM from the FAF to the threshold crossing height (TCH). When this maximum gradient is exceeded, the circling only approach procedure may be designed to meet the gradient criteria limits. This does not preclude a straight-in landing if a normal descent and landing can be made in accordance with the applicable CFRs.
  • A runway is not clearly defined on the airfield.

(emphasis is mine)

The criteria you show in your question (30 degree alignment, 400 ft/nm) means that "circling minimums" are published. If the airport has an approach that has "only" circling minimums, "... the first approach of this type created at the airport is labeled with the letter A, and the lettering continues in alphabetical order." (Ref: FAA's Instrument Procedures Handbook, Chapt. 4, page 4-11) However, if there is an IAP that has both straight-in and circling minimums published the "circling only" approach lettering noted above will not apply.

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