In general,
अ + ऋ gives अर् , and this is given by आद्गुण: followed by उरण् रपर: | For example: देव + ऋषि: = देवर्षि: |
However, under some special circumstances, अ + ऋ gives आर्. For example, this happens when there is an उपसर्ग before a ऋकारादि धातु:| This is given by उपसर्गादृति धातौ followed by उरण् रपर:. For example, प्र + ऋच्छति = प्रार्च्छति
Similar thing holds for आ + ऋ as well. Under normal circumstances it gives अर्, For example महा + ऋषि: = महर्षि:
But when we want to do लुङ् / लङ् / लृङ् लकार form of a ऋकारादि धातु: , a वृद्धिएकादेश happens by आटश्च - E.g. आर्च्छत् |