How to Transcribe R-colored Vowels (ɝ, ɚ, ɑ˞,ɔ˞) in X-SAMPA?

Question

I'm looking at the Arpabet page on Wikipedia, and note some entries like this:

Arpabet | IPA | Word Examples
  ER    |  ɝ  | her (HH ER0); bird (B ER1 D); hurt (HH ER1 T), nurse (N ER1 S)

Now, I'm trying to convert this Arpabet "ER" into X-SAMPA. But, I don't see ɝ anywhere in the X-SAMPA specification. Instead, my reading leads me to see that character ɝ (and some others, like ɚ) are "R-colored vowels", which I think means they can be decomposed into two separate characters instead of written in one digraph character.

My question, in short, is: are these equivalent?

ɝ => ɜr
ɚ => ər
ɑ˞ => ɑr
ɔ˞ => ɔr

Answer 1

That isn't the claim behind the rhotic-hook diacritic. It means that the vowel has the characteristics of the main vowel symbol, but with some kind of r-like raising of the tongue during its production. The sequence [ɑr] means "low back vowel followed by alveolar trill". Apart from the difference in timing (simultaneous vs. sequence) there is a substantial difference in the nature of the r-ishness. If you use the sequences [ɑɹ] etc, they would be more similar (r-coloring is superimposition of "ɹ" quality, not "r"). What might confuse matters is that in English, so-called r-colored vowels are derivable from a sequence of vowel plus consonant ɹ.