Solar derotation¶

The Sun does not turn as one piece, and it does not hold still while you take pictures of it. Helia can bring images of the Sun to one time by its differential rotation (#26), so that features line up across a long session, in a time-lapse, or between the panels of a mosaic taken minutes apart. A worked example is in Tutorial: derotating H-alpha stacks to one time.

(This is not field rotation. The turning of the whole picture in an alt-azimuth mount's camera is a different thing and is a different feature. Solar derotation moves the surface of the Sun on the disc; the disc itself stays where it is.)

Why it matters¶

A feature on the equator goes round in about 25 days, 14.3 degrees a day; at 40 degrees of latitude it takes about a day longer. Seen from the Earth, which moves round the Sun too, the disc turns by about 13.3 degrees a day at the equator, which is 0.55 degree an hour. On a disc 220 pixels in radius that is:

Motion on a disc 220 pixels in radius, from the table above: the equator at the disc centre and the mid-latitude lag behind it.

Registering by the limb or by phase correlation cannot take it out: the limb does not move at all, and a phase correlation of the surface follows the rotation of the middle and leaves the edge turned. The pictures have to be brought to one time by a model of the rotation.

How it works¶

The rotation of the surface is described by a law, the sidereal angular rate at latitude phi:

omega(phi) = A + B sin^2(phi) + C sin^4(phi)

Sidereal rotation rate at the equator of the two published laws, from the table of laws above. Helia has two published laws and takes your own.

The published coefficients are in microradians a second (as SunPy documents them); Helia converts them (one microradian a second is 4.95 degrees a day). Sunspots and the bright features of the chromosphere follow the Howard law closely; Doppler features turn a little slower. There is no law for H-alpha filaments in particular: they are anchored in the chromosphere and the spot law is the usual choice.

To bring an image taken at t1 to the time t2, every pixel of the result is traced back to where its feature was at t1:

1. The pixel's place on the disc gives a latitude and a longitude from the central meridian for the Sun as it is seen at t2, using the tilt B0 and the position angle P of the Sun at t2 (Meeus, Astronomical Algorithms, chapters 25 and 29, as for the sunspot search) and the way the picture sits against the Sun (below).
2. The Carrington longitude of the point is that longitude plus L0(t2).
3. In the Carrington frame, which goes round at 360/25.38 = 14.18 degrees a day, the feature turned by (omega(phi) - 14.18) (t2 - t1) between the two times, so at t1 it was at that much less longitude.
4. That longitude, less L0(t1), is a longitude from the central meridian at t1; with B0(t1) and P(t1) it is a pixel in the picture taken then, which is sampled with a Lanczos-3 kernel.

Three details keep the result clean.

• Brightness goes with the feature. The contrast against the limb darkening (the model Helia fits for flattening, with the sky level) is carried: a spot that moves toward the limb is not dimmed twice. --raw-brightness moves the pixel values as they are instead.
• The limb is left alone. Outside the limb, and in a ring 2% of a radius wide inside it, the picture is left as it was and handed over smoothly. A pixel whose source was behind the limb at t1 (the east edge, going forward) is left as it was too. For minutes this is invisible; for hours the last degrees of the east limb are the features that arrived from the far side, which no picture shows.
• Colour planes move together, and each keeps its own sky level.

How the picture sits against the Sun¶

The geometry needs to know which way is solar north in the picture, as the sunspot search does:

• by default solar north is up and west is on the right;
• --north-angle DEG turns north counter-clockwise from the top;
• --celestial-north-up is for an equatorial mount with the picture north up: solar north is then turned by the position angle P at each picture's own time, added to --north-angle;
• --mirror is for a mirror image (west on the left).

A wrong orientation does not move the features by the wrong amount so much as in the wrong direction, so check it with a pair of pictures that you can see move.

The command line¶

helia solar-derotate¶

helia solar-derotate IMAGES... -o DIR [--to TARGET] [--date D --utc-offset H] [tuning]

Brings each image to one time and writes NAME-derotated.EXT to the directory. Each image is derotated on its own, about the disc fitted to it, and is not shifted: it does not register the pictures. Use it on stacks and panels, then register them with the time-lapse or a mosaic; or give --derotate to the time-lapse, which does both in the right order (see below).

helia solar-derotate 0001-15_32_09.tif 0002-15_37_19.tif -o dd \
    --date 2026-10-20 --utc-offset 1 --to 2026-10-20T19:37:19

bringing 2 image(s) to 2026-10-20 19:37:19 UTC, Howard rates
  0001-15_32_09.tif  14:32:09 UTC  +305.2 min  disc (288.5, 289.0) r 222.4 px  moved up to 10.96 px (centre 10.89)  -> dd/0001-15_32_09-derotated.tif
  0002-15_37_19.tif  14:37:19 UTC  +300.0 min  disc (271.5, 294.6) r 222.0 px  moved up to 10.76 px (centre 10.69)  -> dd/0002-15_37_19-derotated.tif

The times come as they do for the time-lapse: a FITS DATE-AVG, or the file name. A FITS output keeps its header time; the note recorded in every result says what was done.

In a time-lapse¶

helia timelapse ... --derotate brings every frame to the time of the reference frame, or to --derotate-to TIME, after moving it into position. The order matters. The pointing is found from the limb first; then, before the phase correlation of the disc centre, the surface is derotated, so that the correlation follows the pointing and not the rotation (without it the correlation of a series spanning hours would follow the turning of the middle and shift the whole disc). --rotation-law, --north-angle, --celestial-north-up, --mirror and --raw-brightness are those above. The series must have a time for every frame, and the reference must have a clear limb. The CSV table gains a derotated_px column: the most the surface of that frame moved, in pixels.

In a mosaic¶

helia mosaic ... --derotate brings every panel to one time (the middle one by default, or --derotate-to) before the panels are joined; see Mosaics. The disc is fitted to a first joining, each panel is derotated with it, and the pairs are matched again.

In the app¶

The Time-lapse and Mosaic steps each have a Derotation section: Off or To reference, the Law (Howard or Snodgrass) and Celestial north is up. It needs a time for every capture (set the date and the clock's offset under Times for names that hold only a time of day). The line under the viewer reports how far the surface moved in the frame shown, and the step's summary says what time the frames were brought to. The command line alone has --north-angle, --mirror, --raw-brightness, --derotate-to and a law of your own.

In the API¶

POST /api/v1/solar-derotation brings images in the data root to one time and writes the results there. The time-lapse request takes a derotate object. See The API and the API reference.

How well it works¶

On a synthetic Sun with known answers. The Sun is drawn as a sphere with seven spots from latitude -35 to 50 degrees, carried round by the Howard law, with the tilt and longitude of the real ephemeris. A view six hours later is made by the derotation from the first and compared with the view drawn for that time: the disc turned by 13.9 pixels at the centre, the RMS difference inside 0.9 of the radius falls from 0.052 to 0.0003 (of a disc of 1), and every spot lands within 0.015 pixel of where it belongs. For 24 hours, with the mid latitudes 5 pixels behind the equator, a rigid rotation at the equatorial rate leaves an RMS of 0.0115; the differential law, 0.0009. These checks use the same geometry in the drawing and in the correction; they test the arithmetic and the signs of the rates, not the ephemeris, which is checked against Meeus's worked example.

On real captures. The only real pair so far was taken five minutes apart, too close for the rotation to be more than 0.19 pixel at the centre. It is enough to see the sign and the size: the phase correlation of the disc centre between the two H-alpha stacks (after the limb fit) asked for 0.18 pixel in x without the derotation and 0.35 pixel with it, a difference of 0.17 pixel against the 0.185 the model predicts at the disc centre, with a peak-to-sidelobe ratio near 180 both times. See the tutorial. There is no real capture series spanning hours yet: one, with the Sun's features visible at both ends (a sunspot group or a filament), would let the model be checked against the Sun itself.

Limits¶

• The surface is moved, not the atmosphere: the filter's height, prominences above the limb, the Evershed flow and features that change in the interval (a filament that erupts) are not modelled. Over minutes this does not matter; over hours the longer the interval, the less a feature is the one it was.
• A single law is used for the whole disc. Individual features differ from it by tenths of a degree a day, and spots rotate differently from the plage around them.
• Derotation of a stack is of the stack: it does not move the frames inside a capture to one time before stacking. A capture of a few minutes changes by a fraction of a pixel, less than the seeing.
• It needs the whole disc in view in every image to fit the disc.
• The ephemeris ignores the difference between UT and terrestrial time (a minute) and the light travel time (8 minutes, the same for every image). Neither matters for the difference between two images.
• The time of a capture named from its start is the start, not its middle; give a time that is the middle for a long capture.