## SunKit

### 1.0.0

SunKit is a Swift library which uses math and trigonometry to compute several information about the Sun.

### 1.0.0

###### 2022-11-03T23:00:39Z

SunKit is available and is currently powering the app Sunlitt.

# SunKit

SunKit is a Swift package which uses math and trigonometry to compute several information about the Sun. This package has been developed by a team of learners relatively new to the Swift programming language, which means that there could be a lot of space for improvements. Every contribution is welcome.

SunKit was first developed as part of a bigger project: Sunlitt. Even though Sunlitt is not meant to be released as Open Source we decided to wrap the fundamental logic of the app and make an open source library out of it.

To compute Sunrise, Sunset, Golden Hour and so on we only need a CLLocation and the time zone of that location. CoreLocation and SwiftUI framework are required for SunKit to work.

## Local Solar Time Meridian

The Local Standard Time Meridian is a reference meridian used for a particular time zone and is similar to the Prime Meridian, which is used for Greenwich Mean Time. The LSTM is calculated according to the equation:

$$LSTM= 15° \Delta T_{U T C}$$

where $\Delta T_{U T C}$ is the difference of the Local Time (LT) from Universal Coordinated Time (UTC) in hours. For instance, Rome Italy is UTC (+2) so the Local Standard Time Meridian is $30°$.

private var localStandardTimeMeridian: Double {
return timeZone * 15
}

## The Equation Of Time

The equation of time (EoT) (in minutes) is an empirical equation that corrects for the eccentricity of the Earth’s orbit and the Earth’s axial tilt. An approximation 2 accurate to within $\frac{1}{2}$ minute is:

$$E o T=9.87 \sin (2 B)-7.53 \cos (B)-1.5 \sin (B)$$

where:

$$B=\frac{360}{365}(d-81)$$

in degrees and $(d)$ is the number of days since the start of the year.

private var b: Angle {
let angleInDegrees: Double = (360 / 365 * Double(daysPassedFromStartOfYear - 81))

return .degrees(angleInDegrees)
}

private var equationOfTime: Double {

return 9.87 * sin(2 * bRad) - 7.53 * cos(bRad) - 1.5 * sin(bRad)
}

private func getDaysPassedFromStartOfTheYear() throws -> Int {
let year =  calendar.component(.year, from: date)

dateFormatter.dateFormat = "yyyy/mm/dd"
dateFormatter.calendar = calendar
guard let dataFormatted = dateFormatter.date(from: "\(year)/01/01") else {
throw SunError.unableToGenerateStartOfTheYear(from: date)
}

let startOfYear = calendar.startOfDay(for: dataFormatted)
let startOfDay = calendar.startOfDay(for: date)

guard var daysPassedFromStartOfTheYear = calendar.dateComponents([.day], from: startOfYear, to: startOfDay).day else {
throw SunError.unableToGenerateDaysPassedFromStartOfTheYear(from: startOfYear, to: startOfDay)
}

//It Doesn't count the current day
daysPassedFromStartOfTheYear = daysPassedFromStartOfTheYear + 1

return daysPassedFromStartOfTheYear
}

The calendar variable it’s a private variable inside the Sun struct, it’s initialised with the gregorian identifier. We need this to compute the days passed from the start of the year by using a gregorian calendar, it could in fact happen that the user uses a Japanese or a Buddhist calendar.

private var calendar: Calendar = .init(identifier: .gregorian)

## Time Correction Factor

The Time Correction Factor (in minutes) accounts for the variation of the Local Solar Time (LST) within a given time zone due to the longitude variations within the time zone and also incorporates the EoT above.

$$T C=4(\text { Longitude }-L S T M)+E o T$$

The factor of 4 minutes comes from the fact that the Earth rotates $1°$ every 4 minutes.

private var timeCorrectionFactorInSeconds: Double {
let timeCorrectionFactor = 4 * (location.coordinate.longitude - localStandardTimeMeridian) + equationOfTime
let minutes: Double = Double(Int(timeCorrectionFactor) * 60)
let seconds: Double = timeCorrectionFactor.truncatingRemainder(dividingBy: 1) * 100
let timeCorrectionFactorInSeconds =  minutes + seconds

return timeCorrectionFactorInSeconds
}

The timeCorrectionFactor variable it’s in the form mm:ss, that’s why we extract first the minutes and then the seconds to compute timeCorrectionFactorInSeconds variable that we need.

## Local Solar Time

Twelve noon local solar time (LST) is defined as when the sun is highest in the sky. Local time (LT) usually varies from LST because of the eccentricity of the Earth’s orbit, and because of human adjustments such as time zones and daylight saving.

The Local Solar Time (LST) can be found by using the previous two corrections to adjust the local time (LT).

$$LST= LT + \frac{TC}{60}$$

private func getLocalSolarTime() throws -> Double {
guard let localSolarDate = calendar.date(byAdding: .second, value: Int(timeCorrectionFactorInSeconds), to: date) else {
throw SunError.unableToGenerateLocalSolarDate(from: date, byAdding: timeCorrectionFactorInSeconds)
}

dateFormatter.dateFormat = "mm"
guard var localSolarTimeMinute = Double(dateFormatter.string(from: localSolarDate)) else {
throw SunError.unableToGenerateLocalSolarTimeMinute(from: localSolarDate)
}
localSolarTimeMinute = localSolarTimeMinute / 100

let localSolarTimeHour = Double(calendar.component(.hour, from: localSolarDate))

localSolarTimeMinute *= 0.5084745763 / 0.299

return localSolarTimeHour + localSolarTimeMinute
}

The date variable inside the Sun struct gives us the time at which we have to compute the azimuth. But before compute it we need to compute the Local Solar Time. Basically here the second row compute the equation. We don’t need to divide the TC beacuse we already have it in seconds. We then extract the minutes and the hour from the date. For computation purpouse, we use a proportion to move the variable localSolarTimeMinute from [0,59] range to [0,100].

## Hour Angle

The Hour Angle converts the local solar time (LST) into the number of degrees which the sun moves across the sky. By definition, the Hour Angle is $0°$ at solar noon. Since the Earth rotates $15°$ per hour, each hour away from solar noon corresponds to an angular motion of the sun in the sky of $15°$. In the morning the hour angle is negative, in the afternoon the hour angle is positive.

$$HRA= 15°(LST - 12)$$

private var hourAngle: Angle {
let angleInDegrees: Double = (localSolarTime - 12.0) * 15

return .init(degrees: angleInDegrees)
}

## Declination Angle

The declination angle, denoted by $(\delta)$, varies seasonally due to the tilt of the Earth on its axis of rotation and the rotation of the Earth around the sun. If the Earth were not tilted on its axis of rotation, the declination would always be $0°$. However, the Earth is tilted by $23.45°$ and the declination angle varies plus or minus this amount. Only at the spring and fall equinoxes is the declination angle equal to $0°$. The rotation of the Earth around the sun and the change in the declination angle is shown in the animation below. To compute the declinaiton angle of the Earth:

$$\delta=23.45^{\circ} \times \sin \left(\frac{360}{365} \times(d-81)\right)$$

private var declination: Angle {
let declinationInDegree: Double = 23.45 * sin(bRad)

return .init(degrees: declinationInDegree)
}

## Elevation Angle

The elevation angle (used interchangeably with altitude angle) is the angular height of the sun in the sky measured from the horizontal. Confusingly, both altitude and elevation are also used to describe the height in meters above sea level. The elevation is $0°$ at sunrise and 90° when the sun is directly overhead (which occurs for example at the equator on the spring and fall equinoxes).It can be computed as follow.

$$\alpha=\sin ^{-1}[\sin \delta \sin \varphi+\cos \delta \cos \varphi \cos (H R A)]$$

Here $HRA$ is the hour angle that we already have computed, $(\phi)$ is the latitude, and $(\delta)$ is the declination angle that we evaluated already in the previous chapter.

public var elevation: Angle {
get {
let latitude: Angle = .degrees(location.coordinate.latitude)
elevationArg = checkDomainForArcSinCosineFunction(argument: elevationArg)
let elevationInRadians: Double = asin(elevationArg)

}
}

## Azimuth Angle

The azimuth angle is the compass direction from which the sunlight is coming. At solar noon, the sun is always directly south in the northern hemisphere and directly north in the southern hemisphere. The azimuth angle varies throughout the day as shown in the animation below. At the equinoxes, the sun rises directly east and sets directly west regardless of the latitude, thus making the azimuth angles 90° at sunrise and 270° at sunset. In general however, the azimuth angle varies with the latitude and time of year and the full equations to calculate the sun’s position throughout the day are given on the following page.

It can be calculated as follow:

$$Azimuth=cos^{-1}[\frac{sin\delta cos\varphi-cos\delta sin \varphi cos(HRA)}{cos\alpha}]$$

private func getAzimuth() throws -> Double {
let latitude: Angle = .degrees(location.coordinate.latitude)
azimuthArg = checkDomainForArcSinCosineFunction(argument: azimuthArg)
let azimuthRad: Double = acos(azimuthArg)

guard !azimuthRad.isInfinite else {
throw SunError.azimuthIsInfinite
}

var azimuth = azimuthAngle.degrees

if localSolarTime > 12 {
azimuth = 360 - azimuth
}

return azimuth
}

The above equation only gives the correct azimuth in the solar morning so that:

• $$(Azimuth = A_{zi}, for LST &lt;12)$$

• $$(Azimuth = 360° - A_{zi}, for LST &gt; 12)$$

The checkDomainForArcSinCosineFunction simply check if the argument that will go inside the asin function or acos it’s inside the domain, that is betweent -1 and 1. It’s necessary beacuse with al these computations, it could happen that we could have as argument for acos function or asin function 1,003 instead of 1. This could happen if you pin yourself near one of the two poles.

private func checkDomainForArcSinCosineFunction(argument: Double) -> Double {
let inDomainValue: Double

if argument > 1 {
inDomainValue = 1.0
} else if argument < -1 {
inDomainValue = -1.0
} else {
inDomainValue = argument
}

return inDomainValue
}

## Sunrise, Sunset and Solar Noon

For the special case of sunrise or sunset, the zenith is set to (the approximate correction for atmospheric refraction at sunrise and sunset, and the size of the solar disk), and the hour angle becomes:

$$ha=\pm \arccos \frac{\cos (90.833)}{\cos (l a t) \cos (d e c l)}-\tan (l a t) \tan (\text { decl })$$

Then the UTC time of sunrise (or sunset) in minutes is:

$$Sunrise/Sunset = 720 -4 *(longitude + ha) - EoT$$

For UTC solar noon:

$$snoon = 720 - 4 * longitude - EoT$$

Please note that this is for UTC Sunrise, Sunset and Solar Noon. We have to convert it in your local time based on your time zone.

private func getSunrise() throws -> Date {
let latitude: Angle = .degrees(location.coordinate.latitude)
haArg = checkDomainForArcSinCosineFunction(argument: haArg)
let ha: Angle = .init(radians: acos(haArg))
let sunriseUTCMinutes = 720 - 4 * (location.coordinate.longitude + ha.degrees) - equationOfTime
let sunriseSeconds = (sunriseUTCMinutes + timeZone * 60 ) * 60
let startOfDay = calendar.startOfDay(for: date)
guard var sunriseDate = calendar.date(byAdding: .second, value: Int(sunriseSeconds), to: startOfDay) else {
throw SunError.unableToGenerateSunriseDate(from: date)
}

let hoursMinutesSeconds: (Int,Int,Int) = secondsToHoursMinutesSeconds(Int(sunriseSeconds))

sunriseDate = calendar.date(bySettingHour: hoursMinutesSeconds.0 , minute: hoursMinutesSeconds.1, second: hoursMinutesSeconds.2, of: sunriseDate) ?? startOfDay

return sunriseDate
}
private func getSunset() throws -> Date {
let secondsInOneDay: Double = 86399
let latitude: Angle = .degrees(location.coordinate.latitude)
haArg = checkDomainForArcSinCosineFunction(argument: haArg)
let ha: Angle = .init(radians: -acos(haArg))
let sunsetUTCMinutes = 720 - 4 * (location.coordinate.longitude + ha.degrees) - equationOfTime
var sunsetSeconds = (sunsetUTCMinutes + timeZone * 60 ) * 60
let startOfDay = calendar.startOfDay(for: date)

if sunsetSeconds > secondsInOneDay {
sunsetSeconds = 86399
}

guard var sunsetDate = calendar.date(byAdding: .second, value: Int(sunsetSeconds), to: startOfDay) else {
throw SunError.unableToGenerateSunsetDate(from: date)
}

let hoursMinutesSeconds: (Int,Int,Int) = secondsToHoursMinutesSeconds(Int(sunsetSeconds))

sunsetDate = calendar.date(bySettingHour: hoursMinutesSeconds.0 , minute: hoursMinutesSeconds.1, second: hoursMinutesSeconds.2, of: sunsetDate) ?? Date()

return sunsetDate
}
private func getSolarNoon() throws -> Date {
let secondsForUTCSolarNoon = (720 - 4 * location.coordinate.longitude - equationOfTime) * 60
let secondsForSolarNoon = secondsForUTCSolarNoon + 3600 * timeZone
let startOfTheDay = calendar.startOfDay(for: date)
guard var solarNoon = calendar.date(byAdding: .second, value: Int(secondsForSolarNoon), to: startOfTheDay) else {
throw SunError.unableToGenerateSolarNoon(from: date)
}

let hoursMinutesSeconds: (Int,Int,Int) = secondsToHoursMinutesSeconds(Int(secondsForSolarNoon))

solarNoon = calendar.date(bySettingHour: hoursMinutesSeconds.0 , minute: hoursMinutesSeconds.1, second: hoursMinutesSeconds.2, of: solarNoon) ?? Date()

return solarNoon
}

## Golden Hour

In photography, the golden hour is the period of daytime shortly after sunrise or before sunset, during which daylight is redder and softer than when the sun is higher in the sky.

By definition, golden hour starts when the sun it’s at $6°$ elevation and ends when it is at $-4°$ elevation.

Do compute it we have the function getDateFrom(elevation : Angle), where we pass in it the angle and it outputs the time at which the sun will reach that elevation.

private func getGoldenHourStart() throws -> Date {
let elevationSunGoldenHourStart: Angle = .init(degrees: 6.0)
guard let goldenHourStart = getDateFrom(elevation: elevationSunGoldenHourStart) else {
throw SunError.unableToGenerateGoldenHourStart(from: date)
}

return goldenHourStart
}
private func getGoldenHourFinish() throws -> Date {
let elevationSunGoldenHourFinish: Angle = .init(degrees: -4.0)
guard let goldenHourFinish = getDateFrom(elevation: elevationSunGoldenHourFinish) else {
throw SunError.unableToGenerateGoldenHourFinish(from: date)
}

return goldenHourFinish
}

To get a date from elevation we need to do two steps, the first one it’s to compute HRA from the elevation in input. That is, 6° to get the get when the golden hour starts.

$$HRA=\arccos \frac{\sin (\alpha) - \sin (decl)\sin (lat)}{\cos (l a t) \cos (d e c l)}$$

After that we need now to compute the local time (LT) with the following equation:

$$LT = 12 + (\frac{HRA}{15°}) - TC$$

private func getDateFrom(elevation : Angle) -> Date? {
let secondsInOneDay: Double = 86399
let latitude: Angle = .degrees(location.coordinate.latitude)
cosHra = checkDomainForArcSinCosineFunction(argument: cosHra)
let hraAngle: Angle = .radians(acos(cosHra))
var secondsForSunToReachElevation = (hraAngle.degrees / 15) * 3600  + 43200 - timeCorrectionFactorInSeconds
let startOfTheDay = calendar.startOfDay(for: date)

if secondsForSunToReachElevation > secondsInOneDay {
secondsForSunToReachElevation = 86399
}
let hoursMinutesSeconds: (Int,Int,Int) = secondsToHoursMinutesSeconds(Int(secondsForSunToReachElevation))

var newDate = calendar.date(byAdding: .second, value: Int(secondsForSunToReachElevation), to: startOfTheDay)

newDate = calendar.date(bySettingHour: hoursMinutesSeconds.0 , minute: hoursMinutesSeconds.1, second: hoursMinutesSeconds.2, of:newDate  ?? Date())

return newDate
}

The variable hraAngle will store the result of the first equation.

Then we simply compute the second equation, and stores the value inside secondsForSunToReachElevation; where 43200 it’s simple 12 hour in seconds, and we multiply $((\frac{HRA}{15°}))$ by 3600 to convert this factor also in seconds.

The if secondsForSunToReachElevation > secondsInOneDay is used beacuse near the poles could happen that the sun will never reach that elevation, for example in summer in the north pole is always day. After added secondsForSunToReachElevation to startOfTheDay (i.e midnight of date), we extract hours, minutes and seconds from secondsForSunToReachElevation.

Then setting the hour, minute and seconds could seem redunant, but we need to do it because calendar.date(byAdding) will add one hour more the day where the timezone goes from +1 to +2. For example in Italy this happen 27 March. Also the viceversa will happen of course.

## References

• NOAA Global Monitoring Division. General Solar Position Calculations. Link.
• PV Education. Link.

## Special thanks

### Description

• Swift Tools 5.6.0

### Dependencies

• None
Last updated: Thu Nov 17 2022 18:13:39 GMT-0500 (GMT-05:00)