Measuring Solar Differential Rotation with JSol’Ex

I am not the first person to try to make this measurement in amateur spectroheliography. In 2017, Peter Zetner explained on CloudyNights how to measure differential velocity using a spectroheliograph. He explained this methodology in depth in the Solar Astronomy - Observing, imaging and studying the Sun book. His methodology relies on measurements using the Na D2 line, but can also be applied on other bright lines including the commonly used Fe I line at 5250.2 Å, and involves comparing the pixel intensities, by averaging them at different latitudes. The best results are obtained by performing several scans and averaging data. While this works, I wanted to try a different methodology:

The results, of course, will depend on the observed line. The methodology that I describe below is capable of returning reasonable results on several lines (H-alpha, Na D2, Fe I) but will completely fail on some others (e.g Ca II K, typically because the line is too wide).

Let’s dive into the methodology.

The key to understanding the methodology is that it relies on spectral line profile analysis and fitting. It requires extracting the spectral line profiles at different locations of the solar disk, preferably near the limbs where the velocities are highest.

The differential rotation measurement extracts solar rotation velocities by comparing Doppler shifts between East and West limb points at the same heliographic latitude.

For each latitude (say, 20° North), the algorithm:

This differential approach has a crucial advantage: it cancels out systematic errors. Any instrumental offset, wavelength calibration error, or baseline shift affects both measurements equally and disappears when we take the difference.

However, a single East-West measurement at each latitude is not reliable enough. Seeing conditions, local solar activity, or fitting errors can corrupt individual measurements. To improve accuracy, JSol’Ex samples multiple longitudes across the limb region (typically 14 points from 62° to 88° longitude) and aggregates these measurements. This redundancy allows outliers to be rejected and provides an error estimate based on measurement consistency.

The measured velocity is then:

The factor of 2 appears because we’re measuring the difference between approaching and receding limbs.

The velocity we measure depends on the geometry: how much of the rotation velocity is projected along our line of sight. At the equator and at the limb, we see the full rotation velocity. At higher latitudes or closer to disk center, we see only a fraction.

The geometric correction factor is:

Where:

The most challenging part of the measurement is accurately determining the center of the absorption line. A shift of just 0.01 Å corresponds to a velocity of about 0.5 km/s, which is significant compared to the typical equatorial velocity of ~2 km/s.

JSol’Ex uses Voigt profile fitting to measure the line center. The Voigt profile is the convolution of a Gaussian and a Lorentzian profile, which accurately models the shape of solar absorption lines. The Gaussian component represents thermal Doppler broadening, while the Lorentzian component represents natural and pressure broadening.

For each measurement point, the algorithm:

The configurable "Voigt fit half-width" parameter (default: 2 Å) controls how much of the line wings are included in the fit.

As described earlier, at each latitude, multiple longitudes are sampled (typically 14 points across the limb region). These measurements are combined using one of three methods (the default is median but you can choose):

The error estimate from this stage represents the consistency of measurements across longitudes.

Even after longitude aggregation, the latitude-by-latitude measurements remain noisy. Individual latitude bins can still be affected by localized features (sunspots, faculae) or simply measurement scatter.

Since we expect solar rotation to vary smoothly with latitude (following the \(\sin^2\phi\) and \(\sin^4\phi\) terms of the differential rotation law), we can exploit this physical constraint to reduce noise further. JSol’Ex applies a smoothing filter that combines nearby latitude points within a configurable window (default: 5°).

Importantly, the errors from Stage 1 are propagated rather than recomputed from the spread in the smoothing window. This ensures the error bars represent measurement uncertainty, not physical latitude variations.

For the median aggregation:

For the average:

The measured profiles show the expected general shape: higher velocities at the equator, decreasing toward the poles. The fitted curves follow the theoretical Snodgrass profile reasonably well.

However, I observe differences between spectral lines that I cannot fully explain. The Fe I measurements show different absolute velocities than H-alpha, and the fitted coefficients differ between scans.

Several factors could contribute to these differences:

I present these results as they are, without attempting to draw conclusions about which measurement is "correct" or what causes the observed differences. More measurements across different dates, spectral lines, and instruments would be needed to understand these variations.

One subtle but important factor affecting the absolute accuracy of the measurements is the polynomial correction used during image reconstruction. Understanding this is essential for interpreting your results correctly.

During spectroheliograph processing, JSol’Ex computes a polynomial that maps each column position along the slit to the expected row position of the spectral line center. This polynomial is computed from the average of SER frames in the scan (a heuristic is used to include frames containing actual solar data, not sky background).

Such a detection is crucial because the spectral line is not straight across the slit due to optical phenomenons: it consistutes what we often call the "smile". Therefore, the line center might be at row 300 at column 500, but at row 350 at column 1000. The polynomial captures this curvature.

By measuring line positions relative to the polynomial, we effectively remove the optical distortion from our measurements. Without this reference, we would be measuring the sum of optical curvature plus Doppler shift, making it impossible to extract the tiny velocity signals we’re after.

A key question is: why can’t we just measure the absolute position of the spectral line in each frame and compute velocities directly?

The answer lies in the precision required. A velocity of 2 km/s corresponds to a wavelength shift of only ~0.04 Å at H-alpha. With a typical spectral dispersion of 0.1-0.2 Å/pixel, this is a shift of only 0.2 to 0.4 pixels.

The optical curvature across the slit, on the other hand, can easily span 10-30 pixels or more. Any attempt to measure absolute line positions would be completely dominated by this curvature, making Doppler detection impossible.

The polynomial serves as our reference baseline. By measuring how the line position in each individual frame deviates from the polynomial, we isolate the Doppler component from the optical component. This is the key insight that makes sub-pixel precision measurements possible, alongside the Voigt fitting.

This, by the way, is also a reason why the Doppler images look different when scanning in RA vs DEC. Because this is a question that is often asked, and that it took months for me to understand the reason, I think it’s worth spending a bit of time explaining the phenomenon. I owe this explanation to Jean-François Pittet and Christian Buil, from a discussion weeks ago on the Sol’Ex mailing list.