Why do Portsmouth FC have the moon and star as their symbol?

These date back for 700 years and they are on the coat of arms of the City of Portsmouth http://www.welcometoportsmouth.co.uk/star%20and%20cresent.html

as per Senior Moments, it seems more historical, than an actual "why":-

On May 2, 1194 King Richard I gave Portsmouth its first Royal Charter granting permission for the city to hold a fifteen day annual fair (which became known as the Free Market Fair) ... As a crescent and an eight-point star (as appear on the city coat of arms) were to be found on both the seals of King Richard and William de Longchamps it is commonly thought that this may have been the source of them, although there is no known documentary evidence for this. King Richard later went on to build a number of houses and a hall in Portsmouth.
https://en.wikipedia.org/wiki/Portsmouth,_England

King Richard I of England adopted the star and crescent as a royal badge, from the Emperor's standard of Governor Isaac Comnenus, after capturing Cyprus. Back in England, 'a crescent of gold on a shield of azure, with a blazing star of eight points, or rays of silver, between the horns', was granted to Portsmouth as the heraldic crest of the newly incorporated Royal borough.
http://www.crwflags.com/fotw/flags/rel-is.html#richard