Is it possible to connect a Macbook Air 2013 to Macbook Pro 2009 through a thunderbolt cable in order to transform the latter to a second Display ?
It's not possible for you to use either of your MacBook's as a second (external) display. Apple only allow the use of iMac's as secondary displays using a mode called "Target Display Mode".
No matter the use of ThunderBolt, Mini DisplayPort or any number of adapters you won't be able to achieve the result your looking for.
However there are several applications that do enable you to theoretically achieve this over the network (Air Display, AirParrot, MaxiVista, ScreenRecycler etc...) but none are a true replacement for a physical secondary display.
According to Apples documents it is not capable on another Macbook. It is called Target Display Mode and is supported on certain models of iMacs. You can try third party apps that will turn an old computer into a second monitor. I used the third party app "Air Display" and can say it worked great on Mavericks. (Haven't used in Yosemite). It is a cost to download the app.