User Logoff

I have found that if a user is logged in and closes their browser then opens their browser and enters the website's url they get an 403 error "Not authorized to view this resource".

The user is still logged as the menu associated with the page that displays the error appears. Each item on that menu is accessible and any item selected provides the correct content which only registered users can see.

My questions are;
(1) is there a timeout for a logged in user ?
(2) can the user be logged off if the user closes the browser ?

(1) is there a timeout for a logged in user ?

Yes, their session expired. The duration can be adjusted within Joomla global configuration.

(2) can the user be logged off if the user closes the browser ?

No, but they're more or less being logged out when their session expires. Problem is they're returning to your site on a URL that's only accessible to logged in users, which is why they get an access error.