Authentication is performed by adding an Authorization Header to the http request.
Header Name: Authorization
Format: Bearer user_id=<id>&token=<token>
You can find the values for user_id and token by logging in and going to any of the RSS/JSON/iCal export screens and noting the values shown on the URL links.
Example:
Authorization: Bearer user_id=11&token=aadfk23894238482adfaaeea948934
If an authentication failure occurs, the Http Result code will be 403, Not Authorized.