Maths IA · Modelling · topic 69 of 97
Day length: how does latitude change the year? · IB Maths IA idea: guide, sample graph and step-by-step plan
one tilt, every latitude differentNear the equator, day length barely changes all year. Near the Arctic Circle, it swings from almost nothing to almost everything. One equation explains both, and the tilt of the Earth is in it.
- ✓ you like a model you can test against a published table
- ✓ you are curious why the days barely change near the equator but swing wildly near the poles
- ✓ you want a sine model that comes from real physics and not only from fitting
The big idea
The Earth's axis is tilted, so the Sun's angle above the equator, the declination, changes through the year in a smooth, sine-like way. Combine that with your latitude and a little spherical geometry and you can get the exact time between sunrise and sunset as a function of the day of the year. You can also ignore the physics and fit a sine curve to real sunrise tables. Which model is better, and where does each break? The formula is routine. The marks come from testing it: fit both models to real data, measure the residuals, explain the places they fail (polar day, twilight, atmospheric refraction) and say what you would change.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Draw the Earth in cross-section and work out the length of the day from the latitude and the Sun's declination.
→ A trigonometric equation you set up yourself. Check it on the equinox, the solstices and the equator.Take a year of sunrise and sunset times for one place and fit a sine curve to day length by hand or with technology.
→ Parameters with meaning: amplitude, period and shift. Residuals show where a sine is only an approximation.Compare a place near the equator with one at a high latitude using the same formula.
→ The amplitude depends on latitude. Plot amplitude against latitude and describe the shape.Find how many minutes per day the day length changes, and when that rate is largest.
→ Differentiation of a model that is not a plain sine. Check against day-to-day changes in the table.Find the latitude where the Sun stays up on the solstice, and how many days of midnight sun a place gets.
→ The model needs a case split because an inverse cosine stops existing. Explain it, not just patch it.Replace 'the Sun's centre is on the horizon' by a more realistic sunrise definition and see how the day lengths change.
→ A small correction with a large effect near the poles. A good test of how much the idealised model can be trusted.Your Maths IA research question: run the check
which places? which error?Write your own question first. Then make sure it ticks every box:
- Names the places and the year of data, and where the sunrise and sunset times come from.
- Names the model or models you compare and what each parameter means.
- Says how you will judge the fit: a residual measure, or the largest error in minutes.
- Says what 'sunrise' means in your data and your model, since they may not match.
- Is narrow enough for 12 to 20 pages and goes beyond fitting one sine curve once.
The model
A slice through the Earth with the Sun far off to the left. Your latitude draws a circle around the axis, and only part of it is lit. Draw yours for the latitude you choose.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why these places, and what you expect before calculating: which has the bigger swing, and by how much?
The declination as a function of the day of the year, with its source. Every symbol defined.
The sunrise equation derived from the cross-section, checked at the equator and on the equinox.
A year of sunrise tables, a sine fit and a physical model on the same graph, with residuals.
Amplitude against latitude, and polar day and night handled with care.
Back to your aim, with numbers: how latitude changes the swing, and which model wins.
Refraction, twilight, rounding of times, the declination formula's accuracy. Say which assumption matters most.
Where every table and formula came from.
What your Maths IA graph could look like
residuals tell the storyA made-up set of readings to show the form of a strong graph, not the answer: day length for a place at 40 degrees north, a model curve and a few table values. Yours will look different. example only
- Day length against the day of the year: one curve for the model and points from your real table.
- Axis labels carry units: days for the horizontal axis, hours for the vertical axis.
- State the latitude on the graph, because every number on it depends on it.
- Plot the residuals on a second graph and say where they are largest.
- Another graph to consider: the swing in day length (longest minus shortest) against latitude.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 9 done
- Read and predictFind out what declination is. Before any calculation, guess how the swing in day length changes as you move away from the equator.
- Choose places and dataPick two or three places at different latitudes, and download a full year of sunrise and sunset times from a source you can cite.
- Clean the dataConvert times to hours, compute day length, and decide what to do with time zones and clock changes.
- Trial run (1 lesson)Plot one year of day length for one place and fit a sine by hand. Fix any problem now: units, time zone, the start of the year.
- Derive the physical modelDraw the cross-section, derive the day length from latitude and declination and check it on the equinox and the solstices.
- Compare the modelsPut the fitted sine and the physical model on one graph with the data, and measure the residuals of each.
- Go further, with a reasonPick one extension: the rate of change, polar day, or a refraction correction. Say why it adds to your question.
- Reflect as you goAfter every result, write one line on what it means for a real place and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Sinusoidal model of declination – Write the declination as degrees and explain each number.
- Trigonometry on a cross-section – From the sunrise geometry, derive and day length hours.
- Fitting a sine curve – Fit to real day lengths, with technology, and read off the amplitude and shift.
- Residual analysis – Plot residuals of each model against the day of the year; compare the largest errors in minutes.
- Rate of change – Differentiate the model to find minutes gained per day and the day it is largest (AA HL: with inverse trigonometry).
Key equations
Syllabus links
- 2.10Solving equations graphically and analytically
- 3.4Radians, arc length and sector area
- 3.5Unit circle definitions of sine, cosine and tangent
- 3.7Circular functions and their graphs
- 3.8Solving trigonometric equations
- 3.9Reciprocal and inverse trigonometric functionsHL
- 5.6Standard derivatives; chain, product and quotient rules
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 2.6The modelling process
- 3.8Unit circle, identities and trigonometric equationsHL
- 4.4Correlation and linear regression
- 4.13Non-linear regression, residuals and R-squaredHL
Read up first
Data and sources to start from
A full year of sunrise and sunset times for two or three places from a public almanac or calculator, converted to day length in hours.
- timeanddate.com: sunrise and sunset times for any place
- NOAA Global Monitoring Laboratory: solar calculator
References to cite
- Meeus, J. (1998). Astronomical algorithms (2nd ed.). Willmann-Bell.
- Cooper, P. I. (1969). The absorption of radiation in solar stills. Solar Energy, 12(3), 333-346.
- Forsythe, W. C., Rykiel, E. J., Stahl, R. S., Wu, H., & Schoolfield, R. M. (1995). A model comparison for daylength as a function of latitude and day of year. Ecological Modelling, 80(1), 87-95.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
degrees or radians?- Fitting a sine curve with the wrong period, or leaving the shift out and getting the longest day in March.
- Putting a calculator in degrees for one step and radians for the next.
- Using clock time, with daylight saving changes, as if it were solar time.
- Forgetting that the inverse cosine is undefined when the Sun never sets or never rises.
- Calling a sine fit 'proved' because R-squared is high, without looking at the residuals.
- Not saying what counts as sunrise in the data you use.
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.