# Math Internal Assessment: Calculating Sunrise Times All Over the World

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Introduction:

The thought first came to me when the class was doing an investigation which aimed to teach us how to model an equation when given a graph or data points. I had related this back to the times when I was younger and I would accompany my father to different countries and watch as he would shoot the sun rising in scenic landscapes and cityscapes. What I learned from him, was that one of the key factors in shooting sunrise shots was getting to the area at the right time before the sunrise. Which led me to make a connection between the two and consider how organisations such as the National Weather Service or NASA predict sunrise times. Whilst they use more technologically advanced apparatus (e.g. satellites) to aid in the collection of data, they have also used previous data to help predict the following year’s sunrise times; which is something I will attempt to do.

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Another aspect which I would like to explore is why there are varying sunrise times throughout the world. Therefore I will be looking at data from a country with drastic differences in sunrise times throughout the year and data from a country with little to no change throughout the year, Iceland and Singapore respectively. I hypothesise that both Singapore’s and Iceland’s derived graph would be sinusoidal and following a larger sinusoidal line as its equilibrium. However I believe there to be a difference in the amplitude of the larger sinusoidal line with Iceland having a larger amplitude in comparison to Singapore.

Therefore I will be finding out how to model an equation which would be able to estimate the time of sunrise in both Singapore and Iceland, and should my prediction be correct or incorrect, find out why this is or is not so and why there is a difference between sunrise times in different countries around the world.

Data Collection:

To gather the data required to make a prediction I used the previous year’s information on the time of sunrise to aid me. I took the data from each day of the year to be more accurate, this means having as many data points on the graph as possible. The data for time, however, was in standard digital format (e.g. 6:00 am) which would be impractical to use for data, hence I took the the time and made it a percentage of the day. For example 6:00 am became 25% (0.25), and after doing this to every day’s time for a year I graphed it and below is the data for both Reykjavik, Iceland and Singapore.

Sunrise Times as a Percentage of the Day (Iceland above, Singapore below)

Data Analysis:

It would seem that my prediction is true to a certain extent in that the data points in the Singapore graph seems to be follow another equation. The general form of a sine graph being shown below, the constant ‘d’ is what dictates the principal axis of the graph.

This would mean that it is not a constant but another function, therefore I would have to change it from ‘d’ to ‘d(x)’.

However when it comes to understanding why the time varies in different areas of the world there is more than one single variable which would affect it. Some of those variables being Earth’s axis of rotation in relation to the sun, the nature of the Earth’s orbit around the sun, the shape (meaning different elevations) of the Earth, the Earth’s polar coordinates, etc. Therefore my next aim was to find an equation which would allow me to calculate the time of sunrise more accurately to help me answer my question as to why they change on different parts of the world.

Method:

The entire premise for calculating the sunrise time is to find the number of hours daylight in a day for an entire year based on the latitudinal position of the country on the Earth. Then finding the timezone of the country using a Meridian map, and where it is in the timezone, 12 noon would be at the centre of a timezone and it would shift depending on how far east or west the country is in the timezone. Upon finding the solar noon of the country, divide the number of hours of daylight for the specific day of the year into two and subtract that from the solar noon to get the sunrise time. Method and derivation of equation credited to Connor Classen Behan

Simplifications and Clarifications:

Therefore before proceeding with how to derive the equation which can calculate the sunrise times from anywhere on Earth, some variables require simplification.

The Earth will be considered a perfect sphere with no indentations or protrusions

The circle constant (tau) is 2π

To be able plot the point at which the Earth is in relation to the Sun the spherical coordinate system will be used (r, θ, φ)

(radial distance, azimuthal angle, polar angle)

With the Sun being at (0, 0º, 0º)

Let θ=τ4 and φ vary

The orbit of the Earth around the Sun is perfectly circular meaning that the distance from the Sun to the Earth is a constant

Therefore ‘r’ is constant

The axial tilt of the Earth in relation to the sun is constantly 23.5º via (Wikimedia Commons contributors, “File:3D Spherical 2.svg,” Wikimedia Commons, the free media repository, https://commons.wikimedia.org/w/index.php?title=File:3D_Spherical_2.svg&oldid=214370716 (accessed August 13, 2018).)

The solar noon of a day is exactly halfway through a whole day’s worth of daylight (e.g. 13 hours of daylight, and solar noon is at 6.5 hours)

Calculations and Modelling:

Red Dotted Line: Earth’s Axis (23.5º)

Black Line in the Centre of Earth: the line which distinguishes which hemisphere gets light

Image created in pages

Let T = Earth’s Period = 1 year, and because the Earth orbits the Sun at a constant speed the equation for the angle at which it orbits it is given by:

polar angle=circle constant time passedEarth’s period

The red line is the axis, orange is the direct ray of sunlight. The angle between the sunlight and the Earth’s axis is dependent on the ???? and ϕ labelled in the diagram.

If one were to add gridlines which follow the x-axis, y-axis and z-axis, going horizontally (left to right on diagram), horizontally (in toward the circle and out) and vertical (up and down) respectively.

Hypothetically, a component of a vector could be 1tan meaning its other two components, to suit the diagram, would have to be sin ϕ and -cos ϕ. Therefore find the absolute value of the angle between (-cos ϕ, sin ϕ, 1tan ) and a vector pointing from the Earth to the Sun. Using the example given by using the vector (-1, 0, 0)

The next step now is to find the relationship between φ and the length of the day. In the diagram adjacent is a latitudinal line placed above the equator with no particular purpose. The ω was dubbed the “overflow angle” 3 which has another alternate angle and the equation they had derived to calculate the length of a day was (12+2ωτ)24 hours4 with parameters of -τ4≤ ω ≤τ4

Below is the top view of ω angle

Before moving on, the reason for in the diagram to the left is due to the use of the Pythagoras theorem as seen in the diagram to the right, this is created by creating a tangent to the sector of the circle in which .

By drawing the line from the top left corner to the circumference of the circle you can then equate two expressions

In this diagram the latitude have been labelled as λ the radius of the Earth as R and in doing this more equations can be derived

After doing this you can substitute these values into the equation from earlier

Finally after all this to find the length of daylight (L) is to merely put all the previous expressions and equations into one final solution.

Therefore all that’s left is to substitute the appropriate values of λ , ϕ , δ for Singapore and Iceland and compare them. Using dateandtime.info as a resource to find the degrees from the equator the countries were, Singapore was 1º North and Iceland was 64º North which solves the λ. ϕ can be substituted to tT, therefore all that’s needed to be input are the values for ‘T’ and ‘’ which is 365 and 2π respectively. Finally δ just the tilt of the Earth therefore 23.5º, however they are all converted into radians. After plotting the graphs with ”L” as the y-axis as the dependent variable and as time is independent variable the ‘t’ will be on the x-axis. (One thing to note is to change the “+” into “-” because the “+” is for countries in the Southern Hemisphere and “-” is for Northern Hemisphere.

Hours of Daylight as a Percentage of the Day in Iceland

Hours of Daylight as a Percentage of the Day in Singapore

Calculated Sunrise Times:

By using the map which labels the meridians and time zones of the world you can deduce how far the country is from the centre of the timezone and create a rough estimate to be able to find noon time. As seen in the map, Singapore is in the centre of the +7 hours time zone, whilst it is considered to have an irregular time zone and is +8 hours, meaning to fix this is to add an hour to the noon time. Much in the same way Iceland lies in the middle of the -1 hours time zone but is part of the countries in the +0 hours time zone, and can also be fixed by adding an extra hour to noon time. Therefore noon time is not at 12:00 noon but 1:00 pm for both Iceland and Singapore.

Accuracy of Derived Equation

To test the accuracy of the equation there are two points labelled on each of the graphs and by using them it can be seen that sunrise time on days 182 and 365 (Singapore, Iceland) were (5:58 am + 1 hour, 1:48 am + 1 hour) and (6:02 am + 1 hour, 10:12 am + 1 hour) respectively. In comparison to the data provided on https://www.timeanddate.com it is off by a couple of minutes, however this is likely due to the simplifications made earlier such as assuming the Earth was a perfectly spherical, the Earth’s orbit being perfectly circular, etc.

Conclusion:

To conclude, upon deriving the equation that can calculate the sunrise times throughout the world, it is clear to see that the deciding factor which affects why it varies across the world is λ which is the latitude of the country as it determines which areas receive more sun or less sun. Meaning that at a latitude further away from the equator the country is more likely to experience very early or very late sunrises due to how much light reaches the area, specifically due to the nature of the Earth’s axis. Whereas countries closer to the equator experience very little change in the time of sunrises as the exposure of sunlight does not vary that much.

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