I was wondering if a base on Mars could be powered by solar energy alone? Since Mars is further away from the Sun wouldn't that mean lower efficiency thus making solar not scalable? Let's assume a base the size of the Amundsen station in Antarctica, maybe a couple of hundred people.
Well, we don't really know how much energy a Mars base would need, but we can make some rough estimates. Bases like McMurdo and Mawson have power capacities of several hundred to several thousand kilowatts. Now, we're talking about a Mars base, presumably we're going to need a little more power than that. Let's estimate that Mars Base One will require 10,000 kilowatts of power.
An average solar panel will have an efficiency of ~20%. Some can get as high as 25% or 35%, but most are in the 20's, so we'll say that our solar panels have 20% efficiency. (Hopefully by the time we launch to Mars we'll have some even better solar panels).
Mars receives a flux of ~593 W/m$^2$ from the sun.
We can do a quick calculation to find how much power our solar panels will output: $$ E = A \epsilon F r_p $$ where $E$ is the energy, $A$ is the size of the solar panel, $\epsilon$ is the efficiency, $F$ is the flux received, and $r_p$ is the "Performance Ratio" of the solar panels (losses due to shading, dust, circuitry, etc). If we assume that we have a 25% performance ratio, we can solve this equation to see how large our solar panels would need to be:
$A=$300,000m$^2$, or 0.3km$^2$.
Which is incredibly large. A football field is ~5000 m$^2$, so we would have to have a total collection area of 60 football fields. If we were somehow able to get our power usage down to ~100 kW, we could get away with having a single football field of solar panels, which is still much too expensive to lift into space.
We can get away with using solar panels for our small rovers because they only need a small amount of power, which their solar panels can provide. So we will need to increase the efficiency of our photovoltaic systems, decrease the amount of energy we need, or explore other sources of energy (like nuclear power - the best form of power generation in space).
I don't have the reputation to comment on Phiteros's answer, but I would like to add that a huge constraint of solar panels on Mars is the weather. Mars doesn't have much of an atmosphere, but what it does have is quite turbulent. Dust storms are common, and will block out the sun frequently. On top of this, dust gets onto the solar panels and blocks their efficiency even more. Phiteros kind-of covered this with the performance ratio, but that ratio would change dramatically depending on the weather. So you would need very good batteries to last out storms, particularly since they can last up to a month.
In reality though, it is very likely that a future Mars base will have some type of reactor, in addition to solar panels.
Space-based solar power (SBSP) is potentially an option for Mars. It has been considered for Earth, but budgetary and geopolitical concerns have been an obstacle. Those concerns may be less relevant for the scale we are talking about at Mars. The advantage of SBSP at Mars is that (depending on number of collectors and orbital configuration) you can have constant power collection. You avoid the problem of dust. Power can be beamed to practically anywhere on Mars.
Arguably, the budget could be justified and the technology is sound. The main area of difficulty presently is that the state of orbital manufacturing has not quite reached the point that would be best for this. For that matter, the optimal solution may be asteroid mining, which is still strictly conceptual (though a high-interest field).
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Solar energy could power a Mars base, but it would require a very large solar array to do so. One way to mitigate this would be to use an orbital solar array which would double the effective power provided from those panels. The worst effects of dust storm could also be minimised by selecting an appropriate power transmission frequency to the surface
In addition a Mars solar satellite would be in the reverse situation to one in Earth orbit. The panels would have to arrive in Mars orbit before being landed, whereas on Earth there is a serious penalty in launching them into orbit. No doubt in the fullness of time it will be possible to build solar panels on Mars, however this is unlikely for the foreseeable future.
Regardless of orbital solar power or other energy saving measures a very large array would still be needed so it would be necessary to ensure that the cost of sending payload to Mars was minimised in order to make this a feasible proposition.
But the elephant in the room when power on Mars is concerned is the need to generate large quantities of propellants on the Surface which would be very energy intensive, probably outstripping the requirements of the people living on a small base.
Longer term the power requirements of a Mars base would probably need to be augmented by nuclear power or better still if practical, geothermal power.