What simple mathematical function would reasonably well simulate sunlight intensity (disregarding weather) from dawn till dusk in relative numbers, 0-100 %. Are there significant differences (apart from maximum power and daylight duration) between temperate winter, summer or tropical days? The goal is more natural LED lighting for captive wild animals.

  • $\begingroup$ On your scale, what would 100% be? Total Solar Irradiance? The pets stackexchange might be able to help if your end goal is natural lighting for animals. $\endgroup$
    – user10106
    Mar 21, 2018 at 15:26
  • $\begingroup$ 100 would be the maximum output of my lights. It should not matter at all mathematically. I have never heard anybody simulating lighting to such details for animals, so asking on animal site is probably futile. On the other hand, astronomers should know. The animals have zero influence on the sun. $\endgroup$
    – studeo
    Mar 21, 2018 at 15:35
  • 2
    $\begingroup$ A simple cosine function is good for calculating the Sun's altitude through the day, presuming you know its altitude at solar noon, and approximate times of sunrise and sunset. $\endgroup$
    – PM 2Ring
    Mar 22, 2018 at 0:07
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    $\begingroup$ Comment above is right. I add that you should search adding photovoltaics. Those peoples know and model that. Also Earth science SE could be the place. PV community has already fine models while we should recalculate just a basic one. $\endgroup$
    – Alchimista
    Mar 22, 2018 at 9:35
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    $\begingroup$ This question is irrelevant until somone proves that captive animals care about accurate solar intensities. Spectrum, yes. But put a bright light on and provide shaded areas, and they'll hang out where they are most comfortable. $\endgroup$ Mar 22, 2018 at 14:23

2 Answers 2


Well, there are three parameters of interest to be evaluated to get intensity of sunlight.

  1. Sinusoidal variation of sun vector on earth's surface througout the year

  2. The latitude variation. Which is also sinusoidal

  3. The time variation. Which is also sinusoidal

$$ Intensity = A_0 \sin(\theta - \theta_0) \sin(\pi/2(1 - |t-12|/6))$$


$$ \theta_0 = A_1\sin(2\pi|d/365.25|)$$

Here $d$ is the day and starts from vernal equinox. $t$ is the time from 00:00 - 23:59. $A_0$ and $ A_1$ are constants. Also when Intensity is negative, obviously there it means darkness.

This is very crude approximation. It assumes circular orbit.


The sunlight intensity is the cosine of the sun's elevation angle $\alpha$. How to calculate the position of the sun is described simply at PVEducation.com. To summarize the relevant equations: $\alpha$ is given by, $$\alpha = \arcsin[sin\delta\sin\phi+cos\delta\cos\phi\cos(HRA)]$$ where $\delta$ is the declination angle of the sun: $$\delta = -23.45^\circ \times \cos\left[\frac{360}{365}(d+10)\right]. $$ $\phi$ is the latitude, d is the day of the year (Jan 1 is 1), and HRA is the hour angle: $$HRA = 15^\circ(LST-12).$$ LST is the Local Standard Time. This should simulate the intensity of light hitting level surfaces throughout the year, however to be really natural one would have to change the position of the lighting (according to these equation) rather than the intensity because early morning and late evening mean a low sun to me rather than a low intensity sun.


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