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Title: How accurately can an elliptical model based on Kepler's laws describe Earth's orbital path, and what are the optimized points of closest and farthest distance from the Sun?
Course: Mathematics Analysis and Approaches (AA) Higher Level
Word Count: 4138
Date of Assessment: April 2026
| Criterion | Description | Score | Max |
|---|---|---|---|
| A | Presentation | 3 | 4 |
| B | Mathematical Communication | 3 | 4 |
| C | Personal Engagement | 2 | 3 |
| D | Reflection | 2 | 3 |
| E | Use of Mathematics (HL) | 4 | 6 |
| Total | 14 | 20 |
Predicted Grade: 5
The exploration is clearly structured and easy to follow. There is a genuine introduction that provides relevant background, explains why the topic was chosen, and states a well-defined research question. The aim is explicitly declared -- to construct an elliptical model of Earth's orbit using Kepler's laws, calculate perihelion and aphelion, and evaluate the model's accuracy against real data. A hypothesis is stated (residuals of order 10 to the power of -4), which gives the investigation a testable direction. The logical flow from conic section background through derivation of the ellipse equation, to eccentricity, to the polar Keplerian form, to the data table and plot, and finally to the evaluation is coherent throughout. The exploration meets its stated aim, and references are well cited (7 sources, including primary texts such as Kepler 1609 and Murray & Dermott 1999).
Conciseness is the main weakness in this criterion. The background section (conic sections, the pin-and-string definition of an ellipse, the derivation of the standard Cartesian form x^2/a^2 + y^2/b^2 = 1, and the derivation of eccentricity) occupies roughly seven pages and closely follows standard textbook material. This level of detail, while mathematically accurate, makes the exploration feel front-heavy. A more concise exploration would state these results more briefly and focus proportionally more space on the investigative elements unique to the student's work. Additionally, one page in the exploration (page 17 in the submitted document) appears to be blank, which disrupts the flow between the coordinate system definition and the table of values.
The section titled "Deriving Kepler's First Law" introduces Newton's law of gravitation and the second law of motion, establishes that acceleration a = (-GM/r^3)r, and argues that r x a = 0, but then stops. The derivation of the orbit as an ellipse is never completed. This is a structural concern: the section is present but does not deliver a conclusion, which reduces the coherence of that portion of the exploration.
Mathematical notation is generally appropriate and consistent throughout most of the exploration. All main variables (a, b, c, e, r, theta, f) are defined when first introduced. The derivations in the polar form section are numbered sequentially (eq.1 through eq.13), which makes cross-referencing clear and supports a deductive reading of the argument. The exploration uses multiple forms of representation effectively: six annotated diagrams illustrating ellipse geometry, a complete 25-row table of computed orbital coordinates (theta, r, x, y), and a plotted graph of the orbit with the Sun, perihelion, and aphelion labelled. Steps within the algebraic derivations are annotated with brief justifications ("expand squares," "combine like terms," "isolate the radical"), which helps the reader follow the logic.
There is a significant notation conflict that reduces mathematical clarity. The letter c is used for two distinct quantities: the focal distance in the Cartesian derivation (where c^2 = a^2 - b^2), and then again for the semi-latus rectum in equation 12 (where c = b^2/a). These are different quantities. An examiner or reader encountering the second use of c may be confused. Similarly, the polar derivation section introduces f for focal length (distinct from the c used in the earlier Cartesian section), and then defines eccentricity as e = f/a rather than e = c/a as defined earlier. While mathematically equivalent (c and f both refer to the focal distance), the switch in symbol without explicit cross-reference creates unnecessary ambiguity.
The orbital graph (plotted from the table of Cartesian coordinates) does not include axis units on the graph itself. Since r is measured in AU, the x and y axes should be labelled in AU. The figure caption describes it as a model but the absence of units on the axes is a mathematical communication gap.
Personal engagement is most evident in the introduction, which is genuinely written. The student connects the investigation to a specific observation -- that classroom diagrams show circular orbits while real astronomy shows elliptical ones -- and to personal reading of orbital mechanics outside of class. The link to the personal interest in astronomy is stated directly and believably. The student consulted current NASA ephemeris data specific to 2026 (perihelion on 13 January 2026, aphelion on 3 July 2026), demonstrating that they engaged with real, contemporary data rather than generic textbook values. A hypothesis is stated and later compared to the model's output, which reflects a predict-and-test approach. The student presents mathematical ideas in a structured and personal voice throughout.
The investigation largely follows a prescribed path: derive the standard ellipse, derive the polar form, apply known constants, compute a table, plot the result. While this is done competently, the mathematical choices (which constants to use, how to define the coordinate system, the 15-degree increment for theta) are justified in a somewhat formulaic way. There is limited evidence of the student wondering about the mathematics beyond the immediate aim -- for example, exploring what happens to the model if eccentricity is varied, comparing Earth's orbit to that of another planet (Mars has e approximately 0.093, which would produce a far more visually distinct ellipse), or investigating how the model degrades as the two-body assumption is relaxed. The Kepler derivation from Newton's laws suggests the student attempted to go beyond standard curriculum, but this section is not completed, which limits the evidence of independent mathematical thinking.
The evaluation section contains meaningful reflection. The student relates results back to the aim and confirms that the model gives a minimum at theta = 0 (perihelion) and maximum at theta = pi (aphelion), consistent with the coordinate system definition. The student compares the model output to NASA data: 147 million km converts to approximately 0.983 AU (matching the model's perihelion of 0.9833 AU), and 152 million km converts to approximately 1.0167 AU (matching the model's aphelion of 1.0167 AU). Several specific and relevant limitations are acknowledged: the two-body assumption that ignores planetary perturbations from Jupiter and the Moon; the use of discrete 15-degree steps reducing sampling precision; rounding to four decimal places; and the assumption that orbital elements such as eccentricity are constant over time. These are physically well-motivated limitations, and the student discusses why they exist.
The reflection lacks quantitative depth. The hypothesis explicitly predicted residuals of order 10 to the power of -4, but this is never verified in the body of the exploration. No residuals are computed, and no percentage error between the model output and the NASA values is calculated. A single residual computation -- for example, showing that the perihelion discrepancy is (0.9833 - 0.9826) AU = 0.0007 AU, which is of order 10 to the power of -3 and slightly larger than predicted -- would have directly tested the hypothesis and demonstrated critical engagement with the data.
The evaluation also states that the perihelion and aphelion were identified via calculus ("the theoretical predictions derived via calculus") but no calculus is shown anywhere in the exploration. This is a factually incorrect claim within the reflection, and it diminishes trust in the critical analysis. The optimization is performed by inspection of the cosine function, which is valid but should be described accurately. Finally, the reflection does not suggest concrete extensions -- it identifies limitations but does not explicitly address what could have been done differently or how the investigation could be improved.
The exploration demonstrates correct mathematics across several connected areas. The derivation of the standard Cartesian ellipse equation from the focal-sum definition (d1 + d2 = 2a) is complete and algebraically sound, progressing logically through twelve algebraic steps to the standard form. The derivation of eccentricity (e = c/a, leading to e = sqrt(1 - b^2/a^2)) via the point Q on the co-vertex is correct and clearly set out. The polar form derivation (equations 1 through 13) is the strongest mathematical section of the exploration: it derives r = (b^2/a) / (1 + ecos(theta)) by shifting the origin to a focus, applying the Pythagorean relation, substituting x = rcos(theta) and y = rsin(theta), using sin^2(theta) = 1 - cos^2(theta), and rearranging to isolate r. The semi-latus rectum l = b^2/a is identified correctly (eq.12), and the conventional polar form r = l / (1 + ecos(theta)) is obtained correctly (eq.13). The application of this equation to Earth's orbit (a = 1.000 AU, e = 0.0167) is correct, and the table of 25 computed points is internally consistent. The Cartesian conversion x = rcos(theta), y = rsin(theta) is applied correctly, and the resulting plot visually confirms an ellipse with the Sun at one focus.
For AA HL, the research question specifically invokes "optimized points." Optimization is a fundamental calculus concept, and the IB examiner will expect to see a formal treatment: differentiate r(theta) with respect to theta, set the derivative to zero, and verify that the critical points at theta = 0 and theta = pi correspond to minima and maxima respectively. Instead, the student justifies these results by verbal reasoning about the cosine function ("when theta = 0, cos(theta) = 1, the denominator is at its maximum, so r is at its minimum"). This reasoning is correct but constitutes inspection, not optimization. For HL, this is a notable absence.
The Kepler first law derivation from Newton's gravitational law is started but not completed. The student correctly establishes that a = (-GM/r^3) * r, and that since a is parallel to r, the cross product r x a = 0. However, this is as far as the derivation goes. From this point, a proper derivation would invoke the angular momentum vector h = r x v, show that dh/dt = 0 (h is conserved), introduce the Binet substitution or an equivalent technique, and ultimately arrive at the conic orbit equation. None of this appears. The incomplete derivation therefore does not contribute to the use of mathematics score -- it represents an ambitious attempt that is not realised.
The hypothesis about residuals of order 10 to the power of -4 is tested only qualitatively. No numerical verification is presented. For HL, it is expected that mathematical claims relevant to the development of the exploration are validated.
The depth of the mathematics is appropriate for some of the AA HL syllabus (polar coordinates, multi-step algebraic manipulation, conversion between coordinate systems). However, the overall sophistication is closer to AA SL level in most sections. An HL-level exploration on this topic might include: a completed derivation of Kepler's first law from Newton's laws, explicit differentiation to verify the perihelion/aphelion as critical points of r(theta), or an investigation of the orbital period using Kepler's third law (T^2 proportional to a^3) and integration of the polar area.
The student has produced a well-intentioned and largely correct exploration of Earth's elliptical orbit using Kepler's first law in polar form. The polar form derivation (equations 1-13) is the strongest element of the work -- it is rigorous, logically sequenced, and correct. The investigation has a clear structure, a genuine personal rationale, and a meaningful comparison with NASA data.
However, the exploration falls short of what is expected at AA HL level in two significant respects. First, the research question explicitly asks for "optimized points," yet no calculus-based optimization is performed -- this is a core HL expectation that is absent. Second, the Kepler first law derivation from Newton's laws is started but not completed, meaning the most sophisticated mathematical element promised by the exploration is not delivered. The reflection, while present, lacks quantitative verification of the stated hypothesis.
Top two strengths: The polar form derivation (eq.1-13) is mathematically rigorous and well-communicated; the exploration is clearly structured with a coherent aim, hypothesis, and comparison to real data.
Top two areas for improvement: Include a formal calculus-based optimization of r(theta) to identify and verify the perihelion and aphelion as critical points; deepen the reflection by computing numerical residuals and explicitly testing the hypothesis about error magnitude.
The topic and scope are appropriate for AA HL and the research question is well-chosen. With the calculus optimization section added and the Kepler derivation either completed or removed, this exploration would be competitive for a grade 6.