in this example - http://www.developerdrive.com/2013/07/using-jquery-to-add-a-dynamic-back-to-top-floating-button-with-smooth-scroll/ they use the newest Jquery version - jQuery 2.0, which leaves behind the older Internet Explorer 6, 7, and 8 browsers. try to use the old-ie compatible version:
http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js
and Ie8 doesn't support rgb colors here.
.back-to-top {background-color: rgba(235, 235, 235, 0.8);}
need to change for hex color hack for ie8 (without opacity)