Author Topic: Sunrise, Sunset and Solar Azimuth  (Read 10597 times)

captain paranoia

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Sunrise, Sunset and Solar Azimuth
« on: July 09, 2013, 12:44:14 PM »
Some time ago, whilst looking at solar relief shading, I thought about where the sun's azimuth, and decided that the sun never appears above the W-E meridian in the UK.  Of course, that's completely wrong, as I realised when looking out of the bathroom window early one morning, to see the sun rising well to the north of east...

So, I started to think a little more clearly, and looked at the maths for calculating sunrise and sunset times.  Whilst searching to see if I'd got it right (I hadn't...), I found the following NOAA website, which has a nice integration with GoogleMaps, and will calculate sunrise, sunset & local solar noon times,  and draw sunrise/sunset and current solar azimuth lines over GoogleMaps, calculating for any specified time and date.  It's nice to check that the solar azimuth shown for the local solar noon is actually due south...

NOAA Solar Calculator

MoonMan

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #1 on: September 15, 2013, 07:55:04 AM »
The answer is found in Spherical Trigonometry:

sine Amplitude equals sine Declination times secant Latitude.
{Amplitude: the complement of the azimuth}.

Likewise, for the hours of daylight,
cosine Diurnal Arc equals -tangent Latitude times tangent Declination
Hours of daylight either side of Noon, & the complement of the hours from Sunset to midnight.
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captain paranoia

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #2 on: September 16, 2013, 12:55:45 PM »
Thanks for that.  I'll have to go and convert the astronomical terms into things I understand...

However, the simple maths isn't really the problem, I think, when it comes to trying to replicate the NOAA website.  The problems are the approximations made in the simple maths.

Here's my simple maths:

Draw a circle, representing the Earth (approximation 1: the Earth is a perfect sphere).
Draw a vertical line through the centre, representing the suns' shadow (approximation 2: the Sun is at infinity).
Now draw two angled lines through the centre of the circle, showing the inclined Equator and polar axis.
Now draw a radial line from the centre, subtending an angle equal to the latitude of interest from the inclined equatorial line.
Now project a line, parallel to the equatorial line, from the point at which the radial line meets the circle.  This line will cross both the vertical, solar shadow line, and the inclined polar axis.

The point at which the parallel line crosses the inclined polar axis is r.sin(latitude) from the centre of the circle.
Drawing a right-angled triangle with points at the centre of the circle, and the intersections of the parallel line with the axial and shadow lines, the distance, d, from the intersection of the axial line to the intersection with the shadow line is r.sin(latitude).tan(inclination), using the tangent function, tan(theta)=opposite/adjacent, so opposite=adjacent.tan(theta)

Now draw the latitude circle, with radius r.cos(latitude), and draw a vertical line at distance d from the centre.  Where the line crosses the circle is the point on the horizon where the sun rises.
Draw a radial line from the centre to this point.  We know that the length of this radial line is r.cos(latitude), the latitude radius.
We can calculate the angle at which the sun rises using the simple trigonometry function: cos(angle)=adjacent/hypotenuse, where the adjacent is the distance d, and the hypotenuse is the radius.  So:

cos(angle) = r.sin(latitude).tan(inclination)/r.cos(latitude)
thus, angle = acos(r.sin(latitude).tan(inclination)/r.cos(latitude))
angle = acos(tan(latitude).tan(inclination))

We can find the inclination angle for a given day:

inclination=obliquity*sin(days_from_equinox/days_in_solar_year)

Thus, we can find the sunrise and sunset angles.

Which brings us to the third approximation, which is that the Earth hasn't moved around the sun between sunrise and sunset.

The simple maths gets us to within a few minutes of the NOAA calculations.  If we removed the approximations, we ought to get the same result as the NOAA...
« Last Edit: September 16, 2013, 06:08:19 PM by captain paranoia »

captain paranoia

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #3 on: September 16, 2013, 01:33:50 PM »
Oh, and approximation 4: the sun has zero size, so subtends an angle of zero degrees.  With a real sun, sunrise (the first appearance of the sun's disk above the horizon) occurs earlier, and sunset later.

captain paranoia

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #4 on: September 16, 2013, 06:03:26 PM »
> cosine Diurnal Arc equals -tangent Latitude times tangent Declination

i.e.

cos(Diurnal Arc) = -tan(Latitude).tan(Declination)

thus, Diurnal Arc = acos(-tan(Latitude).tan(Declination))

> angle = acos(tan(latitude).tan(inclination))

Barring terminology and sign (which will depend on the sense of Declination/inclination), then, it looks like my 'simple maths' tallies with your definition.  In which case, neither matches the NOAA model...

I've attached a PDF of my drawing, showing the side view of a sunlit Earth, and the latitude circle with sunrise/sunset

MoonMan

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #5 on: September 17, 2013, 08:03:15 AM »
Sorry Captain Paranoia:I have no idea of what the "NOAA model" is, I found the site, but not the model. Astro Nav Maths,or Spherical Trigonometry,  was around before the Internet, by about two millennia {Ptolemy}. Sundials use it; Quadrants use it, so do Octants & Sextants. Latitude equals the difference between the arc Tangents of [tan A secant B]  & [cos C cot D]  Where A is Altitude, C is Hour Angle, D is Declination ,  B is Azimuth. or Conversely: A is Declination, C is Azimuth, D is Altitude, B is Hour Angle,
Declination & Hour Angle are measurements related to the Earth's Orbit around the Sun, or the Year; Azimuth & Altitude are measurements related to the Earth's Axial Rotation, or the Day.

One needs to think in Space-Time; not flat, like a map.
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Lyle Brotherton

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #6 on: September 17, 2013, 02:26:06 PM »
Some of the 80,000 or so words that were not included in my book (88,000 were) included a big section of bearings at Sunrise and Sunset and for two reasons:
1.   If you do not have a compass or are in an area where your compass may be unreliable.
2.   To determine your local magnetic declination if your maps are old or do not display this information.

Here is some of this copy and also the prototype of a creditcard sized tool I developed to determine where the sun will rise/set according to the time of the year. You only need to know the current date and your latitude. (The tool would need to be scaled and drawn more accurately)

We all know that the sun rises in the east and sets in the west; what is less known is that no matter where you are in the world:
•   The sun only rises true east and sets true west on two days, September 21 & March 21: The Equinoxes. 
•   The sun rises/sets north of these points from March to September.
•   The sun rises/sets south of these points from September to March.
•   The amount it rises/sets further north or south varies by location.
•   This variation is always the same for sunrise and sunset.

To determine east/west using the sun you need to use Sunrise/Sunset Chart.
1.   Using this chart find the latitude nearest to the one where you are.
2.   Read beneath your latitude the maximum amplitude of the sun.
3.   Scale the horizontal bar from 0 to this amplitude on the side of the chart with the month you are in.
4.   Draw a line from today’s date, parallel with the vertical bar of the chart.
5.   Where this line intersects on your scale is the sun’s amplitude for that day.
6.   In the Northern hemisphere, subtract this number from 90° to find sunrise and add this number to 270° to find sunset.
7.   In the Southern hemisphere, add this number to 90° to find sunrise and subtract this number from 270° to find sunset.

To ascertain local Magnetic Declination:
1.   Using the steps above determine the bearing the sun will rise or set where you are.
2.   When it rises/sets take a bearing on it.
3.   The difference between your compasses bearing and the bearing you derived from your chart is the local magnetic declination.

Using the tool.

This is the front of the credit card sized tool (possibly with a light grid overprinted on it)



This information is printed on the reverse



At 55 degrees latitude in the northern hemisphere you divide the horizontal bar into 44 equal sections.



Using todays date, in this example October the 30th, you draw a horizontal line down from the date.



From this you can easily calculate that:
Sunrise will be at 70 degrees (90-20)
Sunset will be at 280 degrees (270+20)
« Last Edit: September 17, 2013, 02:33:22 PM by Lyle Brotherton »
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MoonMan

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #7 on: September 17, 2013, 04:40:58 PM »
Captain Paranoia: the only thing that comes to mind is Geocentric & Topocentric  readings: that is, the center of the Earth as the locus; or the  surface of the Earth, or Observer's locus. There is also the matter of Parallax. In reality,the difference counts only if one is taking a serious reading for Latitude & Longitude,as in an old-fashioned Lunar. For these, one needs Tables.
Keeping Track of where Here is in relation to There.

adi

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #8 on: September 17, 2013, 05:32:06 PM »
Are you a sailer moonman. You talk like one.
"We do not belong to those who only get their thought from books, or at the prompting of books - it is our custom to think in the open air, walking, leaping, climbing or dancing, of lonesome mountains by preference, or close to the sea, where even the paths become thoughtful." Friedrich Nietzsche

MoonMan

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #9 on: September 18, 2013, 09:39:38 AM »
adi: Last sailor in the family was a great-grandfather, from Plymouth,  & my Aunt Nancy's first two husbands [Wilhelmson Line]. I used to make sundials, that is where I got the mathematics; the rest comes from research & observation. The Lascaux Caves have some interesting conclusions inherent in the drawings in them. I also like to get out & walk in wilderness. They all come together, over time.
Keeping Track of where Here is in relation to There.

adi

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #10 on: September 18, 2013, 09:47:41 AM »
LOL I often hear old timer sailors talking like you do about nav. They do all come together eventually if you spend the time to be observant ;)
 
"We do not belong to those who only get their thought from books, or at the prompting of books - it is our custom to think in the open air, walking, leaping, climbing or dancing, of lonesome mountains by preference, or close to the sea, where even the paths become thoughtful." Friedrich Nietzsche

captain paranoia

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #11 on: September 18, 2013, 12:20:09 PM »
> Spherical Trigonometry,  was around before the Internet, by about two millennia {Ptolemy}

Yes, I know, but I like to work things out for myself; it's how I learn best, and I don't like to take things as gospel unless I understand how they've been derived.  This can cause difficulties if I can't figure it out...

And yes, I'm certainly thinking about rotating, inclined spheres orbiting, not in 2D.  The 2D picture is an attempt to capture this 3D thinking.

One difference is that I've been working in the solar ecliptic reference plane, looking at the changing angle (I've called it 'inclination') of the earth's axis of rotation relative to the sun, whereas, looking up the astronomical terms, they're based on an earth reference plane, so the celestial sphere is an earth equatorial coordinate system.

Having looked back at my earlier maths, I've understood why I'd decided this inclination is equal to atan(tan(obliquity).sin(orbit_angle)), and I think I was correct.  An additional approximation error is assuming a circular earth orbit, which it isn't.  That brings me to within 4 minutes of the daylight time given by the NOAA page.  Whilst that page doesn't give the mathematics used, it must have a mathematical model behind it that allows the computation of sun rise/set etc for any location and time.  And, hopefully, that model accommodates for all the approximations I've made in mine.

captain paranoia

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #12 on: September 18, 2013, 12:42:42 PM »
Lyle, your little tool looks interesting.  I can see some PostScript being written, if I can figure out the mathematics of the scale; my gut feel is that it shouldn't be linear.  Certainly, the length of the solar day doesn't vary linearly between the solstices; the rate of change is small at the solstices (there's a point of inflection as the sign changes), and maximum at the equinoxes, which is why winter seems to be drawing in very quickly at the moment.

What do you do for the months at the side?
Surely, these months must fall into the maximum end of the amplitude scale, because the sunrise position doesn't come to an abrupt stop in mid November...?

adi

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #13 on: September 18, 2013, 03:17:13 PM »
I am a simple grunt, I know what you guys are trying to say but I cant make head or tail of what your actually saying! Is it really that complicated? People were doing this stuff before modern mathematics. I am not knocking anyone's enthusiasm or knowledge but can we possibly have it in simple English? The reason I ask is because people come here to learn simple to understand  navigation skills. I for one am completely lost. Although everyone's hard work on this thread is very much appreciated and I am sure some people are reading this and understand what is being said but I am sure for many people are being switched off.


     
« Last Edit: September 18, 2013, 03:19:30 PM by adi »
"We do not belong to those who only get their thought from books, or at the prompting of books - it is our custom to think in the open air, walking, leaping, climbing or dancing, of lonesome mountains by preference, or close to the sea, where even the paths become thoughtful." Friedrich Nietzsche

adi

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Re: Sunrise, Sunset and Solar Azimuth
« Reply #14 on: September 18, 2013, 03:50:49 PM »
Are you trying to work out the time of sunrise/set or location of sunrise/set?
"We do not belong to those who only get their thought from books, or at the prompting of books - it is our custom to think in the open air, walking, leaping, climbing or dancing, of lonesome mountains by preference, or close to the sea, where even the paths become thoughtful." Friedrich Nietzsche