Since Starship uses cold liquid oxygen and methane fuel in its header tanks, how will these cryogenic tanks been kept full and cold on long missions in space?
Per his 2017 Reddit AMA:
The main tanks will be vented to vacuum, the outside of the ship is well insulated (primarily for reentry heating) and the nose of the ship will be pointed mostly towards the sun, so very little heat is expected to reach the header tanks. That said, the propellant can be cooled either with a small amount of evaporation. Down the road, we might add a cryocooler.
Some details have changed (in particular, one of the header tanks is now actually in the nose), but the same basic approach is still valid. The thinner TPS, large amount of un-insulated stainless steel hull, and nose LOX header might push them toward implementing a cryocooler sooner.
The CH4 header is a pimple on the bottom of the common dome. That is, there is a single shared dome between the upper tank (CH4) and lower tank (LOX).
The LOX header is in the very tip of the nose (for center of mass balancing issues we are told). in a tweet from Elon Musk:
It’s mostly to balance the ship during entry. After delivering satellites, the front is light & back is heavy due to engines & landing legs.
The CH4 header, will remain full, while the large two tanks on either side empty out. But what remains while likely not a hard vacuum (since that would probably crush the pop-can design of the vehicle) will be residual LOX and CH4 warmed up to gas.
Thus the CH4 header tank will be in the middle of large mostly empty tanks, protected from direct sunlight, from direct conduction of heat, and mostly convection through a diffuse gas.
We have not yet heard of plans for active cooling, but passive seems like it will suffice for the initial test programs.
The LOX header tank is a bigger and is either touching, or very close to touching the top of the fairing, and thus could conduct much more heat than the CH4 header.