### Insolation Calculator

Solar Power at TOA W/m2
Obliquity (axis tilt) degrees Tilt with respect to the ecliptic, between 22.1 and 24.5 degrees
Day of Year Used with the obliquity to determine the declination
Declination degrees Angle of the sun with respect to the equator
Latitude degrees
Local Time of Day hours Used to determine the rotation of the Earth with respect to midnight
Power normal to the surface W/m2
Daily average insolation W/m2
Surface Albedo % Reflected
 Temperature K   �C   �F

All albedos are approximate - many tables give a range, not a single value.

This calculator computes the amount of Sun light perpendicular (normal) to the surface of the Earth for each hour of each day of the year (almost). There are a few approximations (to make the math easier)
• The orbit is assumed to be circular (vs elliptical)
• The year has 365 days exactly (otherwise, the timing of the equinox depends on the year)
• The atmosphere (Greenhouse Effect) is ignored
Otherwise, this provides a pretty good approximation.

There are 2 different calculations for the average daily temperature.

• Compute the average power and then compute the expected temperature
• Compute the temperature from each hour and then average
I didn't know how this would come out - it turns out that averaging the power gives a totally different (higher) result than averaging the temperatures. (For a single scenario, I verified the 2 averages using a spreadsheet. As far as I can tell, the code is correct.)

The Annual Average is computed by finding the average power for the specified latitude and using that value to compute a temperature.

The Global Average is computed by finding the average power in a single day for each latitude, from -90 to +90 in one degree steps, and using that value to compute a temperature. There is a small change between seasons which I don't understand. However, the fact that the result is close to the -18°C everyone talks about is sufficient verification that the formulas are close - very close.

All albedos are approximate - many tables give a range, not a single value. For the planets (and Moon), I used the Bond Albedos - the Geometric Albedos are a bit larger. However, you can type in any value you want.

During the day, the actual temperature is significantly lower than computed because of convection and evaporation. At night, no place on the earth gets as cold as 0.0K, again because of the atmosphere. Taken together, these are called - The Greenhouse Effect.

Even on the moon, after 14 days of night, the temperature is still significantly above 0.0K, but, since the Moon has no atmosphere, this is because the surface actually holds a lot of heat and because radiation slows down a lot at low temperatures.

### Background

This calculator is based on the equations provided in MITx: 12.340x Global Warming Science
 ```Part I: Introduction and history of the Earth's climate > Paleoclimate > Problem 2 - Milankovitch cycles ```

The equations are pretty straight forward for the sun shining on a sphere where the solar zenith angle is given by,

 ```cos(θ) = sin(Φ)sin(δ) + cos(Φ)cos(δ)cos(h) ```
θ The solar zenith angle for a given time and position on the Earth The latitude The solar declination - the angle between the sun and the plane of the equator This varies between plus and minus the obliquity and depends on the day of the year. At the summer solstice, the declination is equal to the obliquity. The hour angle - varies between -π and π throughout the day (0 to 24 in the calculator above)
Ignoring variations in the distance of the Earth to the sun (i.e. the eccentricity), the solar insolation I(Φ,δ,h) is then equal to
 ```I(Φ,δ,h) = S0cosθ, where cosθ>0 ```
and zero otherwise. Here S0 is the solar constant at the mean Earth-Sun distance.

### Notes

Obliquity refers to how tilted the Earth's rotation is with respect to the ecliptic - its orbit around the sun.

Declination refers to the sun with respect to its position relative to the equator. This changes depending on the position of the Earth in its orbit around the sun.

### Other Calculators and References

Calculation of Solar Insolation
This provides a number of interactive graphs - highly recommended
Their Air Mass formula assumes a plane parallel atmosphere, however a corrected formula is provided on a separate page.
Position of the Sun (wikipedia)
This explains the equations (and various approximations) used to compute the solar declination. The one I use has up to 2-degrees of error.