Powering a 30 watt computer in the middle of the desert
Parameters:


Define "sunshine hour" as 1 hour of full sunlight (1000 W/m^2) or an equivalent amount of light at a lesser level delivered over more than 1 hour.
Typical sunshine hours per day worldwide in summer is 4 to 5 hours with less or much less in winter.  

A superb resource is www.gaisma.com which provides detailed insolation (sunshine) and related matter for numerous locations worldwide. As Mauvis is shown as being in San Francisco USA see http://www.gaisma.com/en/location/san-francisco-california.html


Average sunshine hours per day each month  for January  to December are shown there as 


2.05     3.05      4.49     5.93     7.06      7.72
7.50     6.69      5.38     3.85     2.50      1.85


So highest insolation is a massive 7.7 sunshine hours  per day average in June and lowest is 1.85 sunshine hours per day average in December. 

For comparison, Nairobi in Kenya has only 6.3 sunshine hours per day average max (in February) BUT a worst case month of 4.4 sunshine hours / day in July. Solar panel requirements in Nairobi would be less than half those in SF. 


A modern silicon on glass laminated PV panel will deliver about 130 Watt's / m^2 of area. 
If you have an MPPT tracking controller you'll get perhaps 95% of this into the battery. Without MPPT you may get 70%-80% depending on conditions. Maybe more.
Say 75% for initial calculations.
Lead acid battery will deliver 80%+ of energy stored into it.
LiFePo4 battery will deliver 90%+ of energy stored into it.
Both have adequately low self discharge rates. 


SO

Energy available from a PV (Photovoltaic panel / solar panel) saved to battery and then recovered is about:


130 W x 75% x 80% =~ 80 Watts per square metre in FULL sunshine.


If this battery capacity is to be used over 10 hours then the Wattage supported per square metre is 80/10 = 8 Watts equipment load per metre^2 of panel per sunshine hour.

If you want the system to run for N days with no sun (sand storm ? :-) ) you need N metre^2 of panel per 8 Watts or you can power 8/N Watts of equipment per square meter per sunshine hour.

Using the 1.85 sunshine hours per Day December figure you can power 8W x 1.85 =~ 15 Watts of equipment for 10 hours from an average December days sun per square metre of panel.

So, to run your 40 W of equipment safely in December you'll need 40/15 =~ 2.66 m^2 of panels or about 2.66 x 130W = 350 Watts of solar panels. Note that this is to provide one days operation of 10 hours from  1.85 hours of full sun equivalent.  

If you want to be able to withstand 2 sunless days you need to double that to 700 Watts of panel.



The battery needs to be sized to handle this amount of energy. The above was calculated on 75% of panel energy being used to charge the battery, so energy in is

350W x 1.85hr x 75% =~ 480 Watt-hours.
 At 12V that's 480/12 = 40 Amp hours of battery capacity.

A 100 Ah deep cycle battery is liable to suffice.

The above requirement will be reduced by 


MPPT controller - moderate
LiFePO4 battery - moderate
Summer rather than winter insolation - massive - 300%+ more sun.
Lower powered equipment - potentially very significant.




FWIW: I started this reply hours ago but didn't finish it. I now see Olin has now also provided a longish answer. I would not have gone to such length if his answer had been there when I started.  



Gaisma information:

Burning Man is in the Black Rock Desert in Nevada, 120 miles north of Reno.
 The following Reno information should be reasonably applicable. 

Insolation = Sunshine-hours = 4.95 average for September
 and 5.92 per day for August.
 As BM is in early September use say 5 hours equivalent full sun per day.
 There are about 2 wet days per month around this time - hope they are jot during BM :-).



I'll leave readers to extract the fine details from the following wonderful diagram below. I can comment if anything can't be understood (also see gaisma help page).
 The BM line will be slightly above the orange day line which is for late September.
 Sunrise about 6:40am and sunset about 7pm.
 Sun angle at midday about 50 degrees above horizon.
 9am to 3pm sun angles 20 degrees or higher above horizon.  

Sun swings from about 110 degrees to 230 degrees 9am to 3pm = +/- 60 degrees
 Sine of 60 degrees is 0.87 so pointing panels at midday sun position  would lose about 13% of available energy at 3pm and 9am positions.  So moving panels once or twice in day manually would produce modest gains.



Angle change above horizon during peak sun periods is (50-20) = 30 = +/- 15 degrees so vertical angle change not worthwhile across day.

Note that sun is at maximum height at about 1pm. Daylight saving. Adjusting my 9am and 3pm comments to true times (10am to 4pm) would better centre results on true noon peak BUT results will not vary much.

Note that at sunrise and sunset on day this graph was plotted (orange line) the sun rises and sets at about +/- 90 degrees from midday angle. For earlier dates back as far as June 21 the sun sets and rises progressively greater distances past 90 degrees from midday so if you wanted a panel to get all sunlight it would need to point "behind" it's normal midday pointing position. ie sun rises and sets "over your shoulder" in summer months.



12VDC to PC-power power supply

This question  relating to PC's powered from 12VDC was asked in September 2011. 

The user bought a 12V to micro-At power supply from ebay.
 It looks potentially useful in your application and shows what is available and slo, usefully, the level of complexity required in 'rolling your own'.

 Bought from here   

And looked like this:





PW-200-M 200W micro-ATX DC/DC Mini ITX Power Supply PSU

They say:


Power any Pentium 4 motherboard with this super small, cable-free PW-200-M 200W micro-ATX DC to DC power supply which works with the full range of mini-ITX motherboards.

Featuring noise-free, low-heat operation, this power supply connects directly to your motherboard ATX connector providing a fast, compact and convenient power solution. 

Benefits:
The only cable-free micro-ATX DC to DC supply that is compliant with the full range of mini-ITX motherboards
Supports Pentium 4 and powers most motherboards up to 3.0GHz
Power your PC and peripherals from a single 12V power supply
Total noise-free operation
Connects directly to motherboard ATX connector
Provides up to 200W from a single 12V supply
200,000 hour life span
Compact size saves you space: 57 x 61mm
The PW-200-M 200W micro-ATX DC to DC power supply is brand new and unused.