<p>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 <a href="http://pveducation.org/pvcdrom/2-properties-sunlight/suns-position" rel="nofollow noreferrer">PVEducation.com</a>.  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.  </p>

CC BY-SA 3.0
2018-04-06T19:26:14.817Z
2018-04-06T19:26:14.817Z
eshaya
<p>Well, there are three parameters of interest to be evaluated to get intensity of sunlight.</p>

<ol>
<li><p>Sinusoidal variation of sun vector on earth's surface througout the year</p></li>
<li><p>The latitude variation. Which is also sinusoidal</p></li>
<li><p>The time variation. Which is also sinusoidal</p></li>
</ol>

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

<p>Where,</p>

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

<p>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.</p>

<p>This is very crude approximation. It assumes circular orbit.</p>

CC BY-SA 3.0
2018-04-06T16:13:15.653Z
2018-04-06T17:50:00.230Z
2018-04-06T17:50:00.230Z
zephyr0110
<p>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.</p>

CC BY-SA 3.0
2018-03-21T15:35:39.410Z
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.
studeo
CC BY-SA 3.0
2018-03-21T15:54:06.123Z
Isn't it clear from "in relative numbers, 0-100%"?
studeo
CC BY-SA 3.0
2018-03-21T22:39:04.990Z
Better natural light for captive animals sounds like you need a clear roof - maybe perspex or glass.  For more complex lighting you'd need the services/advice of an architect, IMO.  From an SE perspective it's arguably closest to an engineering issue.
StephenG - Help Ukraine
CC BY-SA 3.0
2018-03-22T00:07:01.653Z
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.
PM 2Ring
CC BY-SA 3.0
2018-03-22T09:35:34.903Z
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.
Alchimista
CC BY-SA 3.0
2018-03-22T14:23:02.180Z
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.
Carl Witthoft
CC BY-SA 3.0
2018-03-22T14:41:09.580Z
To prove it, I need to simulate. There are thousands of factors that animals care or do not care for. And we do not know for sure. And when you are trying to keep or breed a rare animal that noone has ever kept or bred succesfully, you have to try and explore these factors... But I asked for that astronomical part. The zoological part is clear to me.
studeo
<the-sun><light>
Sunlight intensity during a day
studeo
# Sunlight intensity during a day

(Asked by: username_0 on 2018-03-21)

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.

**Tags:** the-sun, light

> **Comments:**
> **username_0**: 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.  
> **username_0**: Isn't it clear from "in relative numbers, 0-100%"?  
> **username_1**: Better natural light for captive animals sounds like you need a clear roof - maybe perspex or glass.  For more complex lighting you'd need the services/advice of an architect, IMO.  From an SE perspective it's arguably closest to an engineering issue.  
> **username_2**: 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.  
> **username_3**: 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.  
> **username_4**: 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.  
> **username_0**: To prove it, I need to simulate. There are thousands of factors that animals care or do not care for. And we do not know for sure. And when you are trying to keep or breed a rare animal that noone has ever kept or bred succesfully, you have to try and explore these factors... But I asked for that astronomical part. The zoological part is clear to me.  

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## 2 answers

### Answer by username_5 on 2018-04-06. Score: 1

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](http://pveducation.org/pvcdrom/2-properties-sunlight/suns-position). 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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### Answer by username_6 on 2018-04-06. Score: 1

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

$$ Intensity = A\_0 \sin(\theta - \theta\_0) \sin(\pi/2(1 - |t-12|/6))$$

Where,

$$ \theta\_0 = A\_1\sin(2\pi|d/365.25|)$$

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.


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