Maths IA · Geometry and trigonometry · topic 30 of 40
Orbits and ellipses: how well does an ellipse fit? · IB Maths IA idea: guide, sample graph and step-by-step plan
HL territory: needs real data AND real mathsKepler spent years on Mars's orbit. You're going to fit an ellipse to real positions and find out how good the fit really is.
- ✓ you like physics and astronomy as much as algebra
- ✓ you are happy with parametric and polar curves
- ✓ you enjoy fitting a model to real data
The big idea
← define everything before you use itA planet's orbit is an ellipse with the Sun at one focus. That sentence hides a lot of maths: the equation of an ellipse in Cartesian, parametric and polar form, and how they connect through the eccentricity. Take real positions of a planet from a public source, fit an ellipse, and measure how good the fit is with residuals, not just a picture. Conics, polar equations and Kepler's equation are in neither course, so you introduce them yourself. The HL-level maths is in how rigorously you connect them: a good one links conics, polar curves and a numerical method. Say clearly which definitions you use.
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 →
Fit r = a(1 − e²) / (1 + e cos θ) to measured distances from the Sun.
→ You get e and a directly, and the fit is a function of one variable; the Sun-centred angle must be handled carefully.Fit a general conic to x–y positions, with the Sun at the origin.
→ More parameters, and a harder fit. You must say how you chose the method and why it is a sensible one.Compare the planet's position against time, using Kepler's equation and a numerical method such as Newton-Raphson.
→ You can test the second law (equal areas in equal times) and discuss convergence and step size.A planet with a small eccentricity and a comet or a dwarf planet with a large one.
→ Shows when a circle is a good approximation and when it is not; compare the fits by numbers.Show, with algebra, that the three forms describe the same curve.
→ A genuine proof (rigour) that links the polar and parametric forms and defines eccentricity.From a, estimate the period and compare with the observed one.
→ Links to the third law; you need to be clear about units and the data's source.Your Maths IA research question: run the check
say which form you fitWrite your own question first. Then make sure it ticks every box:
- Names the body (or bodies) and the data source, and says why that source is trusted.
- Names the form of the curve you will fit and says what the parameters mean.
- Says how you will judge the fit: a residual measure, a comparison of a and e between methods, or both.
- Is clear about coordinates and units (AU, degrees, days) and where the origin is.
- Is narrow enough for 12 to 20 pages and uses HL maths you can fully explain.
- Can be answered with numbers and a stated precision.
The picture to draw
An ellipse with the Sun at one focus. Draw yours with the symbols you will use, and mark which ones you can measure from your data.
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.
Your question; why this body and why fit an orbit at all.
What you will fit, with which data, and how you will judge the fit.
Cartesian, parametric and polar forms; the foci and the eccentricity; how they connect, with a derivation of the polar form from a definition.
Where it comes from, which dates or points you chose and why, the coordinate system and the units.
Your method of fitting, the parameters you found, and a plot of model against data.
Residuals, a measure such as the root mean square error, and a comparison of methods or bodies; sense checks against known values.
Kepler's equation with a numerical method, or equal areas by integration. This is where HL-level rigour must show.
Limits of the model (other planets, a point mass Sun), and what the residuals tell you.
Answer with numbers, and say whether the aim was met.
Data source, extra tables and long code.
What your Maths IA graph could look like
residuals will tell you more!Made-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only
- Axis labels give the quantity, the unit, and the uncertainty of the measured distance.
- Distance against angle makes a small eccentricity visible: a circle would be a flat line.
- The fitted curve and the data on the same axes, so the reader can judge the fit by eye before the numbers.
- The scale does not start at zero. Say so in the text, so nobody misreads the size of the effect.
- Another graph to consider: the residuals (data minus model) against θ. A pattern means the model is missing something.
Your game plan
check the origin in your trial runTick steps off as you go. We'll remember on this device.
0 of 11 done
- Read and predictLearn what an ellipse is and what the terms perihelion and eccentricity mean. Predict how e will show in the shape of r against θ.
- Get the geometry rightDerive the polar equation from the focus definition. Check your formula gives a circle when e = 0.
- Trial run (one evening)Take a few positions of one planet from a public source. Convert them to polar form and plot. Is the origin where you think it is? Do the units match?
- Choose body, data and fitting methodDecide which planet or planets, which dates, and why. Choose the form to fit and explain why.
- Fit the modelDo the fit with software, but show the idea and one calculation by hand. Report the parameters with a sensible precision.
- Judge the fitCompute residuals and a single error measure. Compare to published values for a and e from a trustworthy source.
- Go one step deeperChoose one: Kepler's equation with Newton-Raphson, equal areas by integration, or a proof that the forms match. Show convergence or rigour.
- CompareFit a second body or a second form. Which result do you trust more, and why?
- Reflect as you goWhat does the model assume? What would a pattern in your residuals mean?
- Write, then cut12 to 20 pages. Keep one worked example of each calculation and put long tables in the appendix. Use proper notation, no computer syntax.
- Check before your teacher's draftCheck units, axis titles, and that every symbol is defined once and used the same way throughout.
Maths you'll need
- Trigonometry (AA HL) →
- Complex numbers and polar form (AA HL) →
- Integration (AA HL) →
- Maths IA criteria, in plain words →
Traps to avoid in this Maths IA
- Using HL notation without understanding. Examiners ask for explanations in your own words.
- Mixing up the centre of the ellipse and the Sun's position as the origin, so the polar form does not match.
- Data in mixed units or from different reference frames without saying so.
- A good-looking plot taken as proof of fit. You need a number and a residual check.
- Quoting a, e and the period to many digits with no uncertainty.
- Copying the standard textbook derivation with no use of your own data (personal engagement).
- Tackling too much: stick to one body, one form and one deeper step.
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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.