# Climate Change/Science/Distribution of Insolation

Sun-Earth System should make it clear that the geometry of the Sun-Earth system plays a key role in how much sunlight reaches Earth and where it arrives. This section briefly describes the pattern of incoming solar radiation, or insolation, over the course of the year. This pattern is the direct result of the geometrical factors described above, and since the insolation is the energy source for the entire climate system, where the energy enters the system is of fundamental importance to the subsequent distribution of energy. That is to say, where the sun shines has a direct impact on weather and climate.
In an idealized system, where the orbit is circular and the planet's spin axis is perpendicular to the ecliptic (i.e. obliquity equal to zero), the problem becomes nearly trivial because every day is identical. However, the spherical shape of the planet requires some consideration. A first guess at why the poles of this hypothetical planet are cold compared to the equator might be because the pole is farther from the sun than the equator (by a distance equal to the radius of the planet); in fact, this is a common mistake people make about the real Earth as well. To convince us that this can not be the case, consider the change in incoming energy in say 6800 km versus the Earth-sun distance: it is minuscule. However, if we allow the solar constant to be constant at the equator and the poles, there is still an important effect of geometry, namely the angle between the incoming photons (which we can think of as parallel rays of light) and the direction normal to the surface (which can be thought of as the local vertical direction, looking straight up, the sun is not always overhead). The insolation is reduced by the cosine of this angle, which is known as Lambert's cosine law. This is not an atmospheric effect, but simply an optical one, so consider this the insolation at the top of the atmosphere instead of the surface. The principle is simply that the curvature of the Earth means that the same radiance (or photon flux or sunshine) gets spread over a larger area as the angle between the photons and the normal to the surface increases from zero to 90 degrees at the poles. So even though the sun is just as bright everywhere in the world, the power per area incident at the top of the atmosphere changes as the sun appears lower in the sky. The result is a modification to the local insolation, I,
    $I(\phi )=S\cos(\phi ),$
where $\phi$ is latitude.
Thus far, we have neglected Earth's spin. Of course, only half the planet faces the sun at any instant in time. Consider a snapshot in time. Half the planet faces the sun, half is in darkness. From everyday experience, we know that the sun appears at different distances above the horizon over the course of the day. At dawn, the sun comes over the eastern horizon, traveling in an arc across the sky, reaching its maximum height at local noon, and then descending toward the western horizon. Let us define the zenith as the point directly overhead; the angle from this zenith point to the sun's current position is called the zenith angle. In the hypothetical planet described above, where seasons do not exist, the sun only appears directly overhead along the equator at noon; moving away from the equator, the sun sinks lower and lower toward the equatorward horizon. At the south pole, the sun is just at the northward horizon at local noon, providing essentially no insolation. Define a new angle, that between Earth's equator and the highest local position of the sun (the position at local noon), and call this the declination angle; it is essentially a measure of the height above the horizon that the sun will reach each day, and is equal to the latitude at which the sun is directly overhead at noon. In the hypothetical world above, the conditions are perpetually equinox, so the declination is zero because the sun is overhead directly on the equator every day. Earth's obliquity is about $23.45^{\circ }$ (0.409 radians), which combined with the Earth's revolution of 360 degrees per 365 days provides an expression for the declination angle,
    $\delta =23.45\sin({\frac {360}{365}}(284+N)),$
where N is the Julian day of the year. This expression is an approximation, but will serve our purposes.
With the latitude and declination, an estimate of insolation can be made given one additional piece of information related to the geometry of the system: the time of day. The time of day is needed since it affects the overall angle between the local vertical and the suns rays, as is clear by noting the difference between night, dawn/sunset, and noon. To describe the time of day, an additional angle in introduced, conveniently denoted the "hour angle,"
    $H=15^{\circ }*(LT-12),$
where, $15^{\circ }$ is the rotation rate (i.e., $360^{\circ }/24h$) and LT is the local time. Note that at local noon, H = 0. These calculations are normally done in radians, but since most people more intuitively understand angles, we use them here.
Putting all these together, the actual solar zenith angle can be described in terms of the above angles
    $Z=\cos ^{-1}(\sin \phi \sin \delta +\cos \phi \cos \delta \cos H)$
This relation can be derived without explicitly expressing the declination and hour angles, purely from the geometry. These angles make sense physically, so are included here.
Returning to Lambert's cosine law, we can write a simple expression for the insolation: $I=S\cos Z$. This describes the flux of energy into the climate system at any particular moment, given the location and time of day.

## References

* ^ Cooper, P.I., Solar Energy, 12, 3 (1969). "The Absorption of Solar Radiation in Solar Stills."
* ^ [D. Fletcher]