This problem sounds a little bit ambiguous to me. Is the albedo due to the surface of the planet alone, or due to the surface and atmosphere combined? And the dust actually absorbs the solar energy, or reflects it? Also, does it work the same way in reverse? That is, does the atmospheric dust absorb light that is reflected off the planet?

Once you figure out the answers to those questions, you just need to write down the energy balance equations for the atmosphere and the surface of the planet, and solve them simultaneously. Don't forget that the 1000 W/m^2 is what's intercepted by the cross-sectional area of the planet (pi r^2), but is effectively spread out over the entire surface are (4 pi r^2), so effectively it's only one-fourth of that value that you use.

You can contact me directly if you have more questions.

EDIT for Trevor: aren't you neglecting the atmosphere?

The climatic flux density (F) of a planet can be calculated as one quarter of the solar constant (S) multiplied by the bolometric bond absorption (the opposite of albedo or reflectivity or expressed as 1-A where A is albedo).

F = (S÷4) (1-A)

F = (1000÷4) (1-0.25)

F = 250 x 0.75

F = 187.5 Watts per square metre

We can now use our value for flux density (F) to calculate the planets effective or radiative equilibrium temperature (Te) by factoring in emissivity (ε) and the Stefan Boltzman Constant (σ) to the following:

Te = (F ÷ (ε σ)) ÷ 4

Where ε is the radiation efficiency comparable to a black body, which for a planet radiating into space is 1.000 and σ is 5.6704 x 10^-8 W/m2 K^4)

You can use the above for any values of the solar constant and albedo, and because I’ve only got a crappy calculator to hand I’ll let you do that part, you should get a value around 230K.

Well you have a problem, some scientist's say that a large percentage of earth's heat is from radioactive decay, if that is true Earth is radiating heat at a much large extent than it is receiving solar radiation, plus with convection currents, ocean heat sinks, the difference of solar strength at the equator compared to the poles and I think you need a super-computer to work it out, assuming you could find the data to put in it.

To quote a famous person, "What does it matter?"

There are too many variables left out. Rotation of the planet has a lot to do.

Planets Surface Temperature. Solar constant 1,000W/m^2 Albedo 0.25 HELP Please?

Assume a planet with a one-layer atmosphere with the values of solar constant of 1,000 W/m^2 and Albedo of 0.25. Let's assume there's some dust in the atmosphere and 50% of the suns energy is absorbed by the atmosphere and 50% by the surface. What is the planets surface temperature?

Help Please, I think I somewhat got it but I am still a little confused.