If you don't want to include any PHP code inside the javascript code but want to know the value of a php variable, you have to integrate a communication between the server side (PHP) and the client (JS)
For example you could use a ajax request to call a small php snippet that provides the value in its reply. With that value you can go on in you java script code.
In my opinion you should decide if its worth the effort.
Edit:
In regard to the edited question: If it is important that the JS function is never ever called if the PHP session value isn't present I would stay with the PHP code but would do it that way:
<?php
if(!empty($_SESSION['authorized'])) :
?>
<script type="text/javascript">
function execute_this() {
some code;
}
execute_this();
</script>
<?php
endif;
?>
If you evaluate the value of the session variable in javascript, you have to make sure that nothing bad happens to your code if the provided value was manipulated.