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 | |||

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

- 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

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

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.

Albedo sources (for the buttons) | |
---|---|

wikipedia | water soil sand ice |

hyperphysics | Earth Moon Mars |

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.

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 |
---|---|

Φ | The latitude |

δ | 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. |

h | The hour angle - varies between -π and π throughout the day (0 to 24 in the calculator above) |

I(Φ,δ,h) = S |

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.

- 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.
- Solar Irradiance calculator
- This calculator is for determining irradiance for solar electricity. For a flat surface, their numbers are quite a bit different than mine - no idea why. Since their algorithms are on a sever, I can not figure out where the problem is.

Author: Robert Clemenzi