Trying to find a "lost" ancestor

I am a norwegian archaeologystudent, and a couple of years ago my family and I started with genealogy. I have always heard stories of my great-great grandfather, Thorvald Kristian Tönnessen, who went to USA and then disappeared, and we wanted to see if we could figure out what happend to him. According to norwegian emmigration-papers he went on a ship over to New York in 1889. It has been said that his wife, at some point, came to New York, but they had an arguement and he left her all alone without any money. She came back to Norway with help from the embassy, and she and the daughters went to live with her father. The norwegian census of 1901 says that her husband had been in America for some time without been heard from. I went to familysearch.org, and there I found a naturalization petition from 1896, in New York, and we thinks that Thorvald Tonnessen might be the same person as my great-great grandfather. The birthday/-year on the picture fits perfectly and according to church books and censuses he seems to be the only Thorvald Tönnessen born on that date: https://www.familysearch.org/s/recordDetails/show?uri=http%3A%2F%2F... I have not been able to find anything else on him on the internet, but while playing around on familysearch.org I found this: https://www.familysearch.org/s/recordDetails/show?uri=http%3A%2F%2Fpilot.familysearch.org%2Frecords%2Ftrk%3A%2Ffsrs%2Frr_1215076045%2Fp_583484912&hash=HloWXpZgU9zB10k5M56iYku8TUc%253D It is a woman called Charlotte C Tonnessen, a widow, who lives in New York in 1910 and when I looked at the information on her children it said that their fathers nationality was norwegian. The oldest daughter is born in 1896. I know that there were many immigrants in USA called Tonnessen, but we are hoping that these might be related to our Thorvald. We live in Norway and it is very difficult for us to find anything that's not on the internet. So if there is anyone that can help; either in finding information or has got some tips on how we can go about it ourselves we would really appreciate the help!