Another thing you need to consider is that you might win the fight outside the castle, lose half your army in the process, and be left with too weak of an army to successfuylly storm the castle. Which would be bad news if your army is in fact set to do just that. So if the relieving force is big enough to hurt you, you might want to delay giving the order to storm the castle until it has attacked you, and you can see what you have left afterwards.
Of course that's risky, since you don't know how many more men are marching towards the capital to attack you. But it doesn't sound like you have any risk free options at this point.
Re: timing, storm castle?
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