Additional field during login?

I need to add an additional field to the wordpress login page that will allow us to track a customer_id parameter.

Is there a way to add a parameter $customer_id to wp_login without directly hacking user.php?

I have added the field with:

add_action('login_form','add_customer_id');

I can manually hack user.php from:

do_action('wp_login', $credentials['user_login']);

to:

do_action('wp_login', $credentials['user_login'],<strong>$customer_id</strong>);

I capture the $_POST[‘customer_id’] and pass it to wp_login:

add_action('wp_login','capture_customer_id',10,2);