As per religious books, the best time to tie Rakhi on Raksha Bandhan is during Aparahan which is late afternoon according to Hindu division of the day. If Aparahan time is not available then Pradosh time is also suitable to perform rituals related to Raksha Bandhan. Pradosh starts after sunset and lasts for 2 hours and 20 minutes.
Raksha Bandhan rituals should not be done during Bhadra. Bhadra is malicious time which should be avoided for all auspicious work. Most Hindu religious texts, including Vratraj, advise to avoid Bhadra time to tie Rakhi during Raksha Bandhan festival.
In 2013, on August 20, Bhadra is getting over at 20:48 which is about 20-30 minutes before Pradosh is getting over. Hence most cities in India have Pradosh Muhurta as per religious books on August 20. Hence the team of Drik Panchang has preferred August 20 over August 21 to observe Raksha Bandhan.
However, those who would observe Raksha Bandhan on August 21, it should be noted that Purnima is getting over early morning at 07:15 a.m. Hence those who are not able to tie Rakhi on previous day can do it after Sunrise on August 21 but before 07:15 a.m.
Tying Rakhi after 07:15 a.m. on August 21, would be as good as tying it on any day in the year as Rakhi significance is due to Shravana Purnima. Rakhi should be tied during Purnima when Bhadra is over.