Solar calculations

[lux eterna]

Hi all,

My trigonometry and mathematics skills are not quite up to this task, so I would appreciate if someone can show me how to proceed.

I am trying to design a formula that can tell at what time the sun reaches a user selectable elevation (aka altitude, not to be mixed up with declination). For the specific case of alt = 0 (sunrise/sunset) I have found a formula that looks good except that I need solar altitude as an input variable. An example: At what time will the sun reach -18 degrees (under my horizon) for a specific date, at a specific location ?

I have found two very informative pages with detailed descriptions. I think I understand most of  this one but here is where I fail to follow.

What is stopping me is that I cannot see how to transform the below formula to something like :
HRA = (some mix of sin, cos etc for declination, latitude and also elevation)

If I could calculate HRA for different scenarios, it would be easy to find out what time of day that corresponds to on my location.
 
The formulas use greek letters, I suggest using abbreviations like these instead :
za    zenith angle
lat    latitude
da    declination angle (+/- 23.5 degrees) = 23.5 * cos(360 * (DoY+10)/365)
ea    elevation angle (aka "altitude angle")
hra    hour angle = 15*(lst-12)
lt    local time (hours)
lst    local solar time (hours)
tc    time correction factor
doy    Day of Year (1 = January 1)
eot    equation of time

Thank you

Ragnar

 

 

[AKB]

Does this help?

NOAA Solar equations

 

 

 

[vlaiv]

I think that you are over complicating things here (maybe not).

If you want HRA from above formula that is straight forward:

sin(alpha) = sin(delta)*sin(phi) +cos(delta)*cos(phi)*cos(HRA)

sin(alpha) - sin(delta)*sin(phi) = cos(delta)*cos(phi)*cos(HRA)

cos(HRA) = ( sin(alpha) - sin(delta)*sin(phi) ) / ( cos(delta)*cos(phi) )

HRA = arccos( ( sin(alpha) - sin(delta)*sin(phi) ) / ( cos(delta)*cos(phi) ) )

But you only need to look up RA/DEC conversion to Alt AZ and use altitude component.

[Moonshed]

10 minutes ago, vlaiv said:

I think that you are over complicating things here (maybe not).

If you want HRA from above formula that is straight forward:

sin(alpha) = sin(delta)*sin(phi) +cos(delta)*cos(phi)*cos(HRA)

sin(alpha) - sin(delta)*sin(phi) = cos(delta)*cos(phi)*cos(HRA)

cos(HRA) = ( sin(alpha) - sin(delta)*sin(phi) ) / ( cos(delta)*cos(phi) )

HRA = arccos( ( sin(alpha) - sin(delta)*sin(phi) ) / ( cos(delta)*cos(phi) ) )

But you only need to look up RA/DEC conversion to Alt AZ and use altitude component.

The only bits I know are what sin and cos mean from my schooldays. Here I give the only solar calculation that I am prepared to use:

sun = daytime = (get out of bed) + cup of tea...look at news + cup of tea... shower - PJs...(get dressed)... cup of tea + custard cream... lunch + cup of tea... afternoon nap...wake up + cup of tea + look at news...cup of tea ...dinner (+ cup of tea) - washing up...watch tv...sun gone = nighttime = cup of tea + bedtime.

Feel free to use it, it took me many years of dedication and self denial to get this far.

[lux eterna]

@AKB Thanks, I think the answer may be there. I just have to spend a little time with it.

@vlaiv To look up RA/DEC conversion to Alt AZ and use altitude component is not an option. I need to do the math from scratch (the reason for this is an Arduino based project).

From your post :

1) sin(alpha) = sin(delta)*sin(phi) +cos(delta)*cos(phi)*cos(HRA)
2) sin(alpha) - sin(delta)*sin(phi) = cos(delta)*cos(phi)*cos(HRA)
3) cos(HRA) = ( sin(alpha) - sin(delta)*sin(phi) ) / ( cos(delta)*cos(phi) )
4) HRA = arccos( ( sin(alpha) - sin(delta)*sin(phi) ) / ( cos(delta)*cos(phi) ) )

I think I can follow you except for #1. The original formula (see the included picture above) was : alpha = 1 / sin( sin(delta)*sin(phi) +cos(delta)*cos(phi)*cos(HRA) ). How do you go from there to #1 ?

Edit : missed one "sin" ...

 

Edited December 17, 2020 by lux eterna

[vlaiv]

48 minutes ago, lux eterna said:

alpha = 1 / sin( sin(delta)*sin(phi) +cos(delta)*cos(phi)*cos(HRA) ). How do you go from there to #1 ?

My bad, I interpreted sin to the power of -1 to be "inverse" of sin function not reciprocal value (as would be if you actually raise it to the power of -1).

Often in literature - inverse function is denoted as fn to the power of -1.

For this reason I interpreted original formula to read sin to the -1 or arc sin.

Are you sure that they meant 1/sin instead? It does stand to reason that sin of alpha is equal to combination of sine and cosine functions on the other side and sin to the -1 would be indeed arc sin in that case.

[lux eterna]

No, I am not sure how to interpret this. I was just thinking like 10 to the power -1 equals 1 / 10. But not sure if that is applicable here.

 

[vlaiv]

13 minutes ago, lux eterna said:

No, I am not sure how to interpret this. I was just thinking like 10 to the power -1 equals 1 / 10. But not sure if that is applicable here.

 

According to the rest of the page - notation used is one that I assumed

Function to the -1 means inverse function. You can see that from other calculations that follow, namely this one:

This is expression for time of the Sunrise which is derived by finding HRA for which elevation above horizon is 0.

If we take original formula and do that:

sin ( 0 ) = sin(delta) * sin(phi) + cos(delta)*cos(phi) * cos(HRA)

0 = sin(delta) * sin(phi) + cos(delta)*cos(phi) * cos(HRA)   // since sin(0) is of course 0

- sin(delta) * sin(phi) = cos(delta)*cos(phi) * cos(HRA)

cos(HRA) =

HRA = arccos(- sin(delta) * sin(phi) / ( cos(delta)*cos(phi) ) )

Now look at the Sunrise formula and you'll see that it actually reads:

Sunrise = 12 - 1/15° * HRA - TC/60

where HRA is arc cosine of of expression - sin(delta) * sin(phi) / ( cos(delta)*cos(phi)  ).

[lux eterna]

The inverse function was unknown to me, that is where I got lost but now I can see how to proceed. The coming days will be occupied with Excel exersices - a perfect corona timekiller.

Thanks a lot for explaining !

[Waddensky]

Interestingly, the -1 notation for inverse functions was introduced by John Herschel 🙂.

The sine and cosine give the ratio between two sides of a right-angled triangle, the inverses (arcsin and arccos) are used to calculate the angle that result from that ratio. They are used a lot in astronomical calculations, because these calculations often result in angles.

Be careful, it's addictive 😉.

[lux eterna]

John Herschel, what a coincidence. Interesting. Addictive yes, not understanding is like a never ending itch. Also the brain needs a little exercise now and then. Thanks.