How to have arduino only turn on when enough power has been collected?
That satellite will presumably use "energy harvesting" (EH) to store the energy provided by the solar cell in a super capacitor. Both TI and Microchip have just started suppling evaluation kits using EH with a solar cell to power an MCU and a wireless transceiver. Advanced Linear Devices makes the EH300 energy harvesting module, which can charge a pair of super capacitors from a variety of sources, including solar cells. I've been experimenting with one, and it can provide 1.8V - 3.3V from very low voltage sources, and is capable of delivering 25 mA for 68 ms, which is more than  enough for an MCU such as an MSP430 or XLP PIC and a low-power transmitter. The EH300 has two super capacitors. They cost about £34, I bought mine from Farnell.

The Arduino isn't suitable for this application, as it won't work from such low voltages, and takes too much power. XLP PICs typically use 20 nA in deep sleep mode, and  50 uA/Mhz when active.