The ghostly phenomenon stem from the castles involvement in the Civil War conflict. On 2nd May 1643, 1,300 men of the Parliamentarian Army, demanded admittance to search for Royalists believed hiding in the castle. Lady Arundel refused the request, and what followed was siege that saw the Parliamentarians turn their guns on the walls and attempted to undermine the walls with mines. After 5 days, she agreed to, and signed for, an honourable surrender. The female figure that has been reported wandering the ruins and towards the lake, is believed to be Lady Arundel or one of the other woman that played a part in the defence. The Arundel line ended in 1944, and deaths in the family were said to be presaged by the appearance of a white owl.
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For further information, please read Haunted Britain by Antony D. Hippisley Coxe; Haunted Castles of Britain and Ireland by Richard Jones and Britain's Haunted Heritage by JA Brooks.