I noticed that 67P completes an orbit in 6.4 years. Assuming everything goes to plan, how long can Philae stay operational and keep sending data throughout the comet's orbit?
Philae can last many years (it's meant to survive perihelion pass, it has solar batteries, and it can withstand the long deep-space hibernation (it did it once already). Theoretically it could last as long as Opportunity on Mars.
Unfortunately it doesn't have antennas to reach Earth - only Rosetta. It depends on Rosetta to forward data back to Earth. As long as Rosetta is in orbit, and there's enough sunlight for the batteries, Philae can work. Unfortunately P67/Churyumov–Gerasimenko is as irregular and far from sphere as they get, so Rosetta can't attain a fully stable orbit without periodic corrections using its engines - and its fuel. This is the exhaustible resource that puts an upper limit on duration of the mission.
Since the arrival didn't entirely go without a hitch...
Reaction control system problems
In 2006, Rosetta suffered a leak in its reaction control system (RCS). The system, which consists of 24 bipropellant 10-newton thrusters, is responsible for fine tuning the trajectory of Rosetta throughout its journey. The RCS will operate at a lower pressure than designed due to the leak. This may cause the propellants to mix incompletely and so burn 'dirtier' and less efficiently, though ESA engineers are confident that they have sufficient fuel reserves to allow successful completion of the mission.
it's unlikely the reserves will last long beyond the planned end of the mission.
$\begingroup$ Note following the more recent information from ESA, Philae and Rosetta survived the hibernation still using some heaters to keep the batteries above the deep space temperature. Currently as Philae is without power, it runs at considerable risk of having its systems (and specifically, the batteries) damaged permanently by the cold. Nevertheless, it will not exit hibernation at least until there's enough sunlight (=power from solar panels) to both heat the batteries back to operational temperature and charge them to a reasonable level; they can't be charged at all while frozen. $\endgroup$– SF.Jan 21, 2015 at 8:25
The Wikipedia page for Philae suggest a 1-6 week mission duration.
However, the design goal is always conservative. Consider the Mars rovers (Spirit and Opportunity) whose mission design goal was 92 days and Opportunity is at 3945 days now and going strong.
They design longer, but ensure the minimum.
So how long could it last? No one specifically knows, and possibly only time will tell.
News on the landing site suggests that they only have 60 hours of power in the battery and that the solar arrays are shaded and only get sunlight for 1 to 1.5 hours a day, which may not be enough to charge the battery. This poor location will likely reduced the life of the probe severely. However, they also are contemplating extreme notions such as using an extensible sensor to try and flip away to a better location.
3$\begingroup$ You should expand this answer to include Power management $\endgroup$ Nov 13, 2014 at 1:23
$\begingroup$ I don't understand 'may not be enough to charge the battery'. Are you suggesting that in the intervening hours without light the battery will discharge completely? i.e., the battery cannot be charged over a longer period of time than anticipated (but charged nonetheless). This is bothering me. I haven't seen any analysis that explains yet why the 1.5 hours a day light means that Philae is necessarily going to be dead after about ~60 hours. $\endgroup$ Nov 14, 2014 at 11:42
1$\begingroup$ The battery has 60 hours of life. If they can charge during that 1.5 hour of light, they extend that 60 hours. But how much light? How much charge? How much of the power can run the probe, with extra for charging the battery. 60 hours is assured, more is likely, but pessimistic is just 60. $\endgroup$– geoffcNov 14, 2014 at 11:58
While nothing is certain, even if there are no equipment failures it's unlikely that Philae would survive the comet becoming highly active as it approaches perihelion in fall 2015.
EDIT: Seems I was wrong. Quote from Dr Ulamec, lander team manager: "Risk of the lander being blown off the surface by strong cometary jets is low - it has too high a density"
1$\begingroup$ Why? any references or is this just a wild guess? $\endgroup$ Nov 13, 2014 at 1:20
3$\begingroup$ @JamesJenkins: According to Phil Plait (see Update 5), the harpoons didn't fire, so Philae probably isn't very firmly moored to the comet. It's likely to be blown off the surface when the comet starts outgassing as it approaches the Sun. On the other hand, being blown off the surface isn't necessarily fatal; conceivably (and this is a wild guess) it could still capture some good pictures, and maybe even land again. $\endgroup$ Nov 13, 2014 at 2:45
$\begingroup$ @JamesJenkins I believed I'd seen an official statement to that effect, but am now unable to track it down. I did find notes that the orbiter intends to back off to a higher orbit before the time of peak activity, for its own protection. $\endgroup$ Nov 13, 2014 at 14:46
In addition to what has been stated in other answers, it seems "if everything goes according to plan" is quite a bold assumption as there is serious concern that the lander may be stuck in the shade and/or wedged in a hole. In that case, battery life is about 60 hours before the lander must rely on solar power.
This means that with bad luck, it might all be over in a little over two days.
As has been said, only time will tell for certain.
2$\begingroup$ CNES just said that it will have sunlight about 2/12 or 17% of the time. Sound to me to be enough to cover sleep mode and after some time recharge the batteries for new operation now and then. $\endgroup$ Nov 13, 2014 at 13:09
2$\begingroup$ @LocalFluff: Perhaps you are a greater authority than the people who are actually on the project, but those people have said that it is not enough. $\endgroup$ Nov 13, 2014 at 22:59
$\begingroup$ @LightnessRacesinOrbit I have seen no explanation about why it is not enough. Have you seen anything? $\endgroup$ Nov 14, 2014 at 11:43
1$\begingroup$ Thanks. I'd already read the BBC and Independent articles; had not seen the vox article. What I'd love to understand more is why the 1.5 hours is insufficient which is not discussed in these articles. I suppose this means that the battery will discharge more than 1.5 hours worth of sunlight energy each day- but this seems unusual. Just looking for some deeper analysis really to understand the power management/battery setup. $\endgroup$ Nov 14, 2014 at 12:32
2$\begingroup$ @dorchard That sounds like the beginnings of a pretty good question. $\endgroup$– userNov 14, 2014 at 18:07