How far apart should I space rafters for a 13ft patio?
A couple problems with the span calculation, IMHO (I am not a Licensed Professional Engineer):

Select structural (the lumber grade) is frequently impossible to come by at an ordinary lumberyard, will cost the earth if it can be had, and is even less available (AFAIK) in pressure treated (though you should not need that, as per comments.)

Number 2 is generally the grade you should design for unless you are sure you have a source for a better grade and can stand the price of it... You can get the same loading from 2x8  number 2 on 19.2" centers (5 per 8 feet - there's a symbol on most modern tape measures for that spacing) as your select structural on 24". Or you can get significantly more load (or span) from 2x10 number 2 on 24" centers

The position of this roof on the sidewall of a house (if it's "like the picture") may cause the buildup of a snowdrift on the roof - that needs to be allowed for if the 30 lb snow load is anything other than mythical (ie, I think that snow load is the code standard minimum, so it is often in code where snow is never expected....)