'Contract might be overkill'
When they sue you for damages, that contract will keep you from counting bars.
Get a lawyer, sit down with your professor and sketch out the scope. Then get a SoW signed by all authorized signatories of your university. There's no one size fits all solution.
You also say that you've exchanged a SoW over email - not enough. Get it signed on hard paper or using a online provider. It is extremely important that the signatories be authorized. I'm sure your professor is a nice person, but he could simple use you to discover holes in the system, sell them to a third party and throw you under the bus.
If you can't afford a lawyer, at least ensure that you're familiar with the local laws and regulations. I recommend that you read the [PTES's Rules of Engagement][1]PTES's Rules of Engagement. Some important points:
Ensuring that you have permission to test
Evidence handling
What to do when you come across personal information
The scope of the test
If you haven't already, I definitely recommend going through it. It'll keep you safe while you're working. Spending a night in the slammer is no fun. [1]: http://www.pentest-standard.org/index.php/Pre-engagement#Rules_of_Engagement