I ran into an issue with connecting to a geodatabase in MS SQL after restoring the database from a backup. The restore process created several orphaned users. I was able to identify the orphaned users using the following query in Microsoft Management Studio.
SELECT dp.type_desc, dp.SID, dp.name AS user_name
FROM sys.database_principals AS dp
LEFT JOIN sys.server_principals AS sp
ON dp.SID = sp.SID
WHERE sp.SID IS NULL
AND authentication_type_desc = 'INSTANCE';
This generated the list of users based on missing SQL Server authenticated logins.
Next I needed to map the orphaned user to the existing login with the following query in Microsoft Management Studio.
ALTER USER <user_name> WITH Login = <login_name>;
The source of the solution came from: https://msdn.microsoft.com/en-us/library/ms175475.aspx