I do hope by now you have found your answer but your post does interest me in that I've just found a possible connection to Ann Airth who married a James Hadden 1843.They named a daughter Martha - a name not widely used at that time so I can't help but wonder if we have a connection of some sort.
I believe Ann's parents were John Airth/Martha Gordon which could explain the daughter's name. There is also a Martha Airth born 1810 to this couple.
A quick check on IGI for Martha Airth only lists the one born 1810 but there are three marriages for the same name??
Would welcome contact to discuss these Airths further.