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The old english letter Æ/æ in various words have been modernised to either A (Æthelstan to Athelstan); E (Ælf to Elf, Æthelræd to Ethelred) and sometimes both A and E in the same word (Ælfræd to Alfred), sometimes the Æ is most commonly spelt as Ae (Æthelwulf to Aethelwulf according to the Britannica).
Is there an explanation or certain rules on how this occurred?
Or is this completely arbitrary?
Since this is an area where certain confusions are possible, I want to start by noting that the Old English letter Æ/æ (named ash) represented a sound that is etymologically unrelated to the ligature Æ/æ used in words derived from Latin (where it represents what was originally a diphthong sound in Latin). In both cases the grapheme Æ/æ developed from an original sequence of the letters a and e, ae, but in Old English this was considered its own letter (unlike in Latin).
The letter Æ/æ in Old English represented a monophthongal vowel that could be either of two lengths: short (transcribed in the International Phonetic Alphabet as /æ/) or long (transcribed in the International Phonetic Alphabet as /æː/, often written in dictionaries and modern editions of Old English texts as ǣ to distinguish it from the short vowel).
In words that were transmitted orally from Old English to the present (that is, in inherited vocabulary), short Old English /æ/ generally turned into the Middle English vowel /a/, and long Old English /æː/ generally turned into the Middle English vowel /ɛː/, but other developments are possible-it's complicated.
I believe "elf" is an inherited word, but it shows an alternative outcome of short Old English /æ/. The history of the word elf is a little complicated as it involves a sequence that varied in its treatment between different dialects of Old English. The development of the e found in modern English is summarized by Alaric Timothy Peter Hall as follows:
In the West Midlands, Anglian æ developed before /lC/ as in other contexts: unaffected by second fronting (Hogg 1992a, §5.87), it coalesced with a, giving the forms alue, aluen found in both manuscripts of Laȝamon’s Brut. However, in the other reflexes of Anglian dialects, Old English æ from */AlCi/ became e giving elf (Luick 1914–40, I §366; Jordan 1974, §62; cf. Hogg 1997, 207–12). This was more or less identical with the South-Eastern elf, so it was natural that elf became the standard English form, being the root used by Chaucer and almost all other later Middle English texts, regardless of their place of origin.
(page 214, "Appendix 1: The Linguistic History of Elf", in The Meanings of Elf and Elves in Medieval England, dissertation, October, 2004)
For words that were not truly inherited but were modified from the Old English form, explanations will likely be relatively word-per-word.
Here is what I've got so far:
Athelstan seems to have been used since at least the 16th century (Google Books links: 1, 2; it could well be even older) and thus is not an especially recent rendering of this name. I don't know why A was used. There are also texts of a similar age that use the form Ethelstan (1, 2, both in Latin).
I think the ending -ræd is generally preferred to be rendered as -red, but I'm not sure why. Brainstorming ideas:
this was in Old English pronounced rǣd, with a long vowel that did not develop to a modern English "a" sound (unlike short æ) and which showed dialectal variation already with an "e" sound (Wiktionary records rēd as an Anglian variant of the independent word rǣd). If the sound of ǣ survived at all in speech, it would be expected to develop to an e sound not an a sound.
Alternatively, as the second element of a name, we would expect the vowel in -ræd to be more subject than a vowel in an initial syllable to being reduced in quality; in both modern English and Old English, reduction often produces an e-like vowel (Wikipedia: "In unstressed syllables, [...] /æ, e, i/ were reduced to /e/"; this refers to short vowels specifically, but Old English could shorten vowels in unstressed syllables). In practice, I think Alfred, Ethelred are generally pronounced by modern English speakers with a reduced vowel in the final syllable. This sound might be better represented by the spelling with e than with a.
