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Dear Ms. Arnold,
Thank you for posting your request on History Hub!
We searched the National Archives website and located the web page Atlantic, Gulf, and Great Lakes Ports, 1820-1873 that includes National Archives Microfilm Publication M575 and its index (M334) that may contain the information you seek. For access to these records at NARA facilities and libraries across the United States, please see Where to Find these Records.
We also searched Ancestry for which produced numerous Citizenship & Naturalization hits for Edward Arnold hits based on the criteria given. There may be a fee for using Ancestry. Instead, you may view these records online at one NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Also, some of the images from Ancestry are available for free on Family Search at https://www.familysearch.org/en/.
Other suggestions include checking the Citizenship column on the Federal census. It can provide clues regarding immigration year and citizenship status in the US. If naturalized, the Petition for Naturalization might provide further information regarding his immigration to the US. Based on their state of residence, please contact the NARA facility that has custody of naturalization records from the Federal courts in that state to locate a possible Petition.
In the National Archives Catalog, we located the Population Schedules for the 1820 Census; 1830 Census; 1840 Census; and the 1850 Census in the Records of the Census Bureau (Record Group 29). Digital images of 1830, 1840 and 1850 census records are not available yet in the Catalog but are digitized and available on Ancestry. From 1790 to 1840, only the head of household is listed by name; other household members are merely counted in selected age groups.
However, on Ancestry, we located his possible passenger arrival on a ship; Ohio Marriage License for Edward & Nancy Arnold in 1821; head of household on 1830 Census; head of household on 1840 Census; 1850 Census living with Nancy and son John; and his death from Lung Fever on the Federal Census Mortality Schedule.
There may be a fee for using Ancestry. However, you may view these records online at one NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Also, some of the images from Ancestry are available for free on Family Search at https://www.familysearch.org/en/.
We hope this is helpful. Best of luck with your family research!