Solar Constant

The solar constant is the energy output of the Sun or another celestial body as received on the orbit of Earth or another planet, moon or asteroid.

Overall

The Sun's energy output is 1.361 kW/square meter or 1.951 calories per minute, at Earth's orbit. However, when I started working with Math for hypothetical planets, I used another value, 1.98, for Earth's orbit. From this value, a significant part is absorbed by Earth's atmosphere. Also, the value changes because the Sun energy output changes along time and also Earth's orbit is an ellipse and not a circle.

For terraforming, it is important to define the energy output in different wavelengths. We know that plants need both red and blue light (see Plants on new worlds for details). Infrared light is important for heating a planet. Also, we need to keep an eye on ultraviolet and Roentgen rays, which can seriously affect life if there is not an ozone layer.

Calculating the solar (or stellar constant) is difficult with traditional formulas, it will bring you to values up to values with 20 zeros. So, I made some more simple formulas, using the local solar constant for Earth as 1.98. This value is used in all formulas listed below. All formulas are made for Microsoft Excel.

Stellar Constant

Each star gives a specified energy output. The stellar constant is defined as the energy output of a star measured at 1 million km. Some stars have larger radius then one million km, therefore the value becomes only a theoretical one.

For our sun, Sol, the value is 44235. For other stars, it is very different. I calculated values for Procyon A of 311278 and for Barnard's Star of 114.56.

Some stars are variable. The most important are flare stars, which can increase their energy output even 100 times in a minute during flares. So, if you are making calculations for a flare star, you should use two types of values: for average radiation and for flare eruptions.

Global Stellar Constant

This is the star's total energy output. Basically, this is the value you need to use to calculate temperature on a host planet.

Specified Stellar Constant

Each star is different. Cooler stars generate more infrared and red light, while hotter stars emit more blue and ultraviolet light. In practice, it is good to have a specified stellar constant for each value. In case of the Sun, I use 44253 for each radiation type.

Local Stellar Constant

The local solar (or stellar) constant is the energy output at a specified distance. As mentioned above, if the host star is variable, it is good to define these values for average and maximum (for flare stars). Also, if the star has a different spectra then Sol, it is good to measure constants for specified wavelengths.

Formulas

To determine the stellar constant for a specified star, you can access existing catalogues, to see the magnitude (for example, the Hipparcos). You will need the absolute magnitude. Then, do as follows, in Microsoft excel:

Column A: type star's name.
Column B: type magnitude.
Column C: type the formula: =(POWER(1.44544,B2*(-2.5)))*3852714.849

Basically, the formula is: KS = 1.44544^(M*(-2.5))*3852714.849, where KS is the star's constant and M is the absolute magnitude.

This way, you will get for our sun, Sol, who has a magnitude of 4.85, the solar constant of 44235.

To determine the local stellar constant, do as follows, in Microsoft Excel:

Column A: type planet's name (for example Earth).
Column B: type distance to the star, in millions of km (149.5).
Column C: type the stellar constant (for Sun, 44253).
Column D: type this formula: =C2/B2/B2 (and you will get, for Earth, 1.98).

Ks = KS/d^2, where Ks is the local stellar constant, KS is star's constant and d is distance in millions of km.

You can repeat this with any other planet, giving its distance to the Sun. This way, you can see that on a distant planet, the amount of light received is far different. For Neptune, you will get a value of 0.002183 (so, 1000 times less energy then on Earth). Why is Neptune so important? Because that is the lowest limit that plant life can support. If the host star emits less red or less blue light, you must make the calculations for each wavelength, to see if it can support plant life.

Determining Special Constants

The best way is to use the Internet Stellar Database, where you can find color indexes for each star. There, the colors are as follows:

U - ultraviolet
V - visual (yellow)
B - blue
I - infrared

There, you will see something like U-V = +0.65. This means that the star has a magnitude in ultraviolet with 0.65 units higher then in visible (yellow). Now, you can take these values directly into your simulation. To start, the given magnitude is in visible (yellow), from there you can compute all values.

Another way to determine special constants is by using the black-body radiation equations. But, at all possible, if somehow the energy output of the star was measured, those values will be better then theoretical ones.

Still, if you want these formulas, please go to Stellar Parameters, where they are all listed.

Reflected Light

Any object that reflects light can add a limited heat. For example on the Moon, temperature does not drop below -200 C because of the heat radiated by Earth. There is a way to determine this tiny amount of heat. I use to call it reflection constant:

Kr = ((((KS/d^2)/alb)*diam^2)/dist^2)/100000

Here, Kr is the reflection constant, KS is star's constant, d is planet's distance to the star, alb is the albedo, diam is planet's diameter and dist is your distance to the planet.

The amount of reflected light is small, but it might, together with other sources, influence the climate.

Examples for Solar System

Sol - solar constant is 44253. 
Mercury - 13.20
Venus - 3.78
Earth - 1.98
Mars - 0.852
Ceres - 0.258
Jupiter - 0.0731
Saturn - 0.0217
Uranus - 0.00535
Neptune - 0.00218
Pluto - 0.00127
Eris - 0.0000299
Sedna - 0.00000777