Project research MIT PRIMES 2026

A signal from Saturn.
A closer look at its rings.

Cassini sent radio waves through Saturn’s rings. Our team studies the mathematics that could help recover finer detail from the signals that reached Earth.

See the original image

From Saturn’s shadow · Cassini, July 19, 2013NASA/JPL-Caltech/SSI

01 / Where the question begins

The rings leave a trace.
How do we read it?

As a radio signal passes through the rings, its strength and phase change. Those changes carry information about the material it crossed.

Recovering fine structure also means accounting for diffraction: waves travelling along nearby paths interfere. Our project begins with the calculation that connects this signal to the rings.

How Cassini used radio signals

Cassini close-up of Saturn’s A ring, showing a density wave beside closely spaced diagonal moon wakes. — **Structure worth resolving.**Cassini camera view of a density wave and moon wakes in the A ring. December 18, 2016; approximately 340 m per image pixel.NASA/JPL-Caltech/Space Science Institute · camera context

What does the radio observation look like?Open a real Cassini profile and follow it into the Data explorer.

Our explorer contains six calibrated ring profiles from the NASA Planetary Data System. They retain diffraction effects and provide a starting point for reconstruction.

All six observations Original PDS archive

The mathematical question

One root can hide
another branch.

A stationary-phase calculation focuses on angles where the phase stops changing. One starting guess can find one such angle and miss others. Near a fold, two branches approach each other and turn.

\frac{\partial\psi}{\partial\varphi}(\rho,\varphi_s)=0

A stationary angle $\varphi_s$ is a root of the phase derivative at ring radius $\rho$ .

How do we find the relevant branches, keep their identities, and decide when the approximation needs more care?

02 / Following the signal’s structure

Find the roots.
Keep the story of each one.

A stationary root is one place where the signal’s phase stops changing with angle. The challenge is to find all the relevant roots, then understand which branch each belongs to as we move across the rings.

Find
Give each root a starting point.
At one ring radius, the stationary equation can have several solutions. Padé approximation and adaptive least-squares fits provide candidates for the roots that are harder to find.
Carry forwardA set of candidate roots
Follow
Stay with the curve as it turns.
As the radius changes, the roots move. Pseudo-arclength continuation follows the branch itself, including near a fold, instead of treating every radius as a fresh search.
Carry forwardA connected branch to follow
Check
Ask the original equation again.
Interpolation returns the candidates to the data grid. Halley’s method then refines them on the original equation, while residual and sampling checks test whether they should be accepted.
Carry forwardRefined roots with numerical checks

A root also needs an identity.

A numerical solver may return the same roots in a different order. One-to-one matching reconnects them using circular angle distance, preserving a record of each branch’s phase, curvature, amplitude, and status.

A closer lookFollow a worked branch.See the team’s continuation experiment at the point where the curve turns.

Manuscript Figure 14: blue continuation points follow a branch around a rightward fold near x equals minus 1.89 and y equals 0.79. Red interpolated points lie along the lower branch, compared with a purple marching-squares curve. — Figure 14 · §3.1 · p. 16. Synthetic test: $F(x,y)=(y^3+xy+1)(y^2+2xy+2)$ . The axes show the model’s dimensionless variables, not ring radius or Cassini measurements.View full-size figure

Try a simpler model yourself

A teaching model

When two paths become one.

Here, a simple cubic makes the geometry visible.

y^3+xy-1=0

Move the parameter

x

-3.0000

3 distinct real roots

Three roots, three paths.

The vertical line meets three branches. Each intersection is a solution at this parameter; following a branch preserves the connection between nearby solutions.

Root values: -1.5321 · -0.3473 · 1.8794

This dimensionless teaching equation illustrates a fold. It is separate from the Cassini phase equation and from the manuscript’s numerical experiments.

The equations behind the three steps

The team seeks stationary angles: places where the phase stops changing with angle.

f(\rho,\varphi)=\frac{\partial\psi}{\partial\varphi}(\rho,\varphi)=0

01 / Initial candidates

A simpler function to solve.

f(\rho,\varphi)\approx\frac{P_3(\varphi)}{Q_n(\varphi)}

At a fixed radius, numerator roots supply candidates. Candidates associated with denominator poles are excluded. Adaptive fitting concentrates on the extra pair after the stable root is found.

02 / Continuation

A step along the branch.

\begin{aligned}f(\mathbf z)&=0\\(\mathbf z-\mathbf z_0)\cdot\mathbf t_0&=\Delta s\end{aligned}

With $\mathbf z=(\rho,\varphi)$ , the second constraint chooses a step along the current tangent. PCHIP interpolation then returns the solutions to regular radii.

03 / Refinement

A correction on the original function.

\varphi_{n+1}=\varphi_n-\frac{2ff'}{2(f')^2-ff''}

Functions and angle derivatives are evaluated at the current candidate, at a fixed radius. A small change in the candidate is not sufficient: the equation residual must also be checked.

Working manuscript, §3.1–3.2, pp. 10–17.

Finding the right roots is only part of the problem. Near a fold, even accurate roots can give a poor stationary-phase approximation. The next question is when to change the way we evaluate the signal.

03 / The evidence

One difficult point.
A different calculation.

Follow one example, then open the checks that support it.

Consider a point just 0.001 km from the fold. The stationary roots are already correct, but treating their contributions separately gives a large error. Our next decision is about how to evaluate the integral.

This example tests a local angular contribution in a scalar geometry informed by Rev 133. The percentages compare numerical evaluations with an independent reference; they are not errors in a reconstructed ring profile.

Explore the reported example

\mu = 0.001\,\mathrm{km}

Ordinary stationary phase $59.5629\%$ Approximate each contribution separately

Our switched evaluation $2.01\times 10^{-10}\%$ Integrate the local contribution directly

Local quadrature

Here the phase gap passes the preset $\chi\leq 1$ rule, so the evaluator uses Simpson quadrature. The difference comes from that extra calculation, not from changing root labels.

Move farther from the fold Choose one of eight reported offsets, in km.

Ordinary SPASwitched evaluator

Logarithmic axes · lines connect the eight reported samples. At the last three offsets, the methods coincide.

Across the full tested grid

A local improvement.
A measured trade-off.

Largest symmetric error: 59.5629%→2.2872%
Fewer direct integrand samples than quadrature everywhere: 37.5%

The rule chooses quadrature at 5 of 8 offsets. It reduces sampling work compared with integrating at every offset; it does not establish a runtime speed-up or a 1% error guarantee.

Inspect all eight results & experiment conditions

Team manuscript · Table 3 · symmetric error (%)
Offset (km)	SPA	+ labels	+ flags	Switched	Evaluator
0.001	59.5629	59.5629	59.5629	2.01e-10	Simpson
0.003	49.6822	49.6822	49.6822	2.17e-10	Simpson
0.01	36.5413	36.5413	36.5413	2.01e-10	Simpson
0.03	22.1027	22.1027	22.1027	8.67e-11	Simpson
0.1	6.2333	6.2333	6.2333	8.00e-11	Simpson
0.3	2.2872	2.2872	2.2872	2.2872	SPA
1	0.2756	0.2756	0.2756	0.2756	SPA
3	0.0242	0.0242	0.0242	0.0242	SPA

Error definition. Both methods are compared with the same independent reference:

E_{\mathrm{sym}}=100\,\frac{\left|I_{\mathrm{method}}-I_{\mathrm{ref}}\right|}{\left|I_{\mathrm{method}}\right|+\left|I_{\mathrm{ref}}\right|}\%

Sampling work. 81,925 direct integrand samples plus 6 saddle evaluations, versus 131,080 integrand samples for quadrature at all eight offsets. Each selected quadrature uses 16,385 Simpson samples. The hybrid requires more work than ordinary SPA.

Independent reference. Adaptive Gauss–Kronrod, checked against 131,073-point Simpson and trapezoid calculations. A common taper is 1 within 0.4° of the fold angle and 0 beyond 0.8°. The threshold was fixed before this comparison. Small quadrature discrepancies do not establish equally small physical uncertainty.

Fixed scalar geometry.

\begin{aligned}\rho_0&=121999.437351\,\mathrm{km},&D&=149390.501073\,\mathrm{km}\\B&=1.886554^{\circ},&\varphi_0&=80.770609^{\circ}\\\lambda&=3.557429207\times 10^{-5}\,\mathrm{km}\end{aligned}

The manuscript cites RSS_2010_170_X34_E_GEO. That geometry product differs from the X43_E_DLP_500M profile available for Rev 133 in the data explorer. The scalar azimuth-to-vector coordinate conversion has not been independently verified.

Two checks behind the result.

Open either experiment to follow the reasoning.

Check the geometryAre the roots really meeting at a fold?A predicted separation law meets independently solved roots.

A disappearing track is not enough to identify a fold. We checked the phase derivatives, counted the nearby roots on each side, and compared their separation with the local prediction.

Manuscript Figure 17: two stationary roots separate from a fold as radial offset increases; their squared separation follows the derivative prediction, with numerical roots plotted as circles. — From our manuscript · Figure 17, p. 19The scalar-model root pair and separation check. The plot rounds the reported slope difference to 0.00135%. View full size

Recomputed location

\begin{aligned}\rho_c&=121069.150400\,\mathrm{km}\\\varphi_c&=93.063921^{\circ}\end{aligned}

−0.01 km from the fold0 local roots

+0.01 km from the fold2 local roots

Squared separation law

\left|\varphi_+-\varphi_-\right|^2\approx m\mu

Predicted slope: 0.07598542
Measured slope: 0.07598439

$\mathrm{deg}^2/\mathrm{km}$

0.001345% reported relative difference

The close agreement supports a local fold in the stated scalar convention. The generic derivative conditions are:

\Phi_{\varphi}=\Phi_{\varphi\varphi}=0,\qquad\Phi_{\varphi\varphi\varphi}\ne0,\quad\Phi_{\varphi\rho}\ne0

Derivative values & geometry

Reported derivative values use kilometres and radians:

\begin{aligned}\Phi_{\varphi}&=8.10\times 10^{-15}\\\Phi_{\varphi\varphi}&=-8.10\times 10^{-15}\\\Phi_{\varphi\varphi\varphi}&=13365.068710\\\Phi_{\varphi\rho}&=-0.038669290\end{aligned}

The two displayed slopes are rounded; the relative difference is the manuscript’s reported value. These summary values do not supply a measured root trajectory.

Fixed scalar geometry.

\begin{aligned}\rho_0&=121999.437351\,\mathrm{km},&D&=149390.501073\,\mathrm{km}\\B&=1.886554^{\circ},&\varphi_0&=80.770609^{\circ}\\\lambda&=3.557429207\times 10^{-5}\,\mathrm{km}\end{aligned}

Team manuscript · Table 2, pp. 18–19

Check the trackingDoes each root keep its identity?Separate a better association from a better integral.

Two nearby roots can both choose the same predecessor if each makes an independent nearest-neighbour match. A one-to-one assignment resolves that competition within a fixed distance gate.

Manuscript Figure 16: independent matching creates dashed wrong links between two synthetic root tracks; global one-to-one matching preserves two separate tracks. — From our manuscript · Figure 16, p. 17The first five samples of the synthetic close-track test. The counts below cover all ten samples. View full size

Close parallel tracks · independent matching9 wrong links

and 9 false unmatched-predecessor flags

Same roots · one-to-one assignment0 wrong links

and 0 false unmatched-predecessor flags

This improves which root is linked to which. Relabelling the same roots changes their complex sum by at most $8.88\times10^{-16}$ , at rounding scale. The local-integral improvement above requires the additional integration step.

Five synthetic cases · 10 samples each · 0.28-radian gate · no observational noise
Case	Independent matching	One-to-one assignment	Assignment + prediction
Ordinary motion	Passed	Passed	Passed
Periodic seam	Passed	Passed	Passed
Isolated appearance / disappearance	Passed	Passed	Passed
Close parallel tracks	9 wrong links; 9 false flags	0 wrong links; 0 false flags	0 wrong links; 0 false flags
Motion beyond the gate	3 identities lost	3 identities lost	3 identities lost

Prediction made no further improvement in these five cases. When motion exceeded the gate, all methods lost three identities—a case that calls for finer sampling or a recovery step.

How the matching comparison works

Every method receives identical candidate roots. Known identities are used only to assess the links; the random seed changes root order. Assignment first maximizes links within the gate, then minimizes total squared circular distance.

The test evaluates association between samples, not ring-profile reconstruction. An unmatched root can be a missed solution; it does not establish a physical merger.

Team manuscript · §4, pp. 17–18

Reported evidence from our team’s working manuscript, §§4–6, pp. 17–20. The controls explore those results; the website does not rerun the research solver.

04 / The people behind the work

Different pieces.
One shared problem.

The project brings together root finding, continuation, and reliability checks. Each part gives the next one something it can use.

Finding starting points

Yutong Zhao

Worked on Padé approximation and adaptive least-squares methods to locate candidate stationary roots.

Following the branches

Maiya Qiu

Developed least-squares and pseudo-arclength continuation. Wrote the introduction and background, and compiled and edited the report.

Identity, folds & reliability

Dell Li

Primarily wrote the sections on stationary-branch bookkeeping, generic-fold verification, and the reliability of the stationary-phase approximation.

Project mentor

Dr. Ryan Maguire

Proposed the project and guided its development.

Contributions follow the acknowledgments in the team’s working manuscript.

A signal from Saturn.
A closer look at its rings.

The rings leave a trace.
How do we read it?

One root can hide
another branch.

Find the roots.
Keep the story of each one.

Give each root a starting point.

Stay with the curve as it turns.