1. In a blender or food processor, combine 1/2 cup sugar and 1/2 cup lime juice. Add papayas, in batches if necessary, and whirl until smooth. Pour mixture as puréed into a bowl. Stir, taste, and if desired, add more sugar and lime juice.
2. Nest bowl in ice and stir often until mixture is cold, 10 to 12 minutes.
3. Pour purée into an ice cream maker (minimum capacity 1 qt.). Freeze according to manufacturer's directions until sorbet is firm, dasher is hard to turn, or machine stops, 15 to 20 minutes. Serve, or freeze airtight up to 5 days.