Solar Rotation – Formula Fantasies

Solar Rotation - Formula Fantasies

Many astronomers are afflicted with an irresistible urge to encapsulate their precious observational data into a mathematical approximation that is of “little scientific value”.

The peculiarities affecting the motions of spots in different latitudes were reduced by Carrington into a formula, but such formulae have little scientific value.

Old and New Astronomy – Richard Anthony Proctor & Arthur Cowper Ranyard – 1892

Literate scholars often succumb to a similar compulsion to construct elegant and witty anagrams.

An anagram is a type of word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once; for example, the word anagram can be rearranged into nag-a-ram.

Someone who creates anagrams may be called an “anagrammatist”.

One difference between these afflictions is that an anagram constructed by the anagrammatist can be resolved to exactly recreate the original whilst the Formula Fantasy created by the astronomer cannot be resolved to exactly recreate the original.

This inherent defect of the Formula Fantasy is usually perfectly acceptable to Settled Science because an approved astronomical approximation should never [unlike real observational data] produce an embarrassing or contradictory result if it has been carefully crafted by an academic artisan who understands the sophistry of Simon Newcomb.

Axial Tilt


Axial precession


In the case of Solar Rotation the Formula Fantasy has effectively obscured the original solar observations recorded by Richard Carrington for about 150 years.

Richard Christopher Carrington (26 May 1826 – 27 November 1875) was an English amateur astronomer whose 1859 astronomical observations demonstrated the existence of solar flares as well as suggesting their electrical influence upon the Earth and its aurorae; and whose 1863 records of sunspot observations revealed the differential rotation of the Sun.

The trail of Solar Rotation and the Formula Fantasy begins with the Wikipedia page for the Sun which provides three data points [equator, 16° latitude and poles] and the reader is left with the clear impression that Solar rotation is differential with a perfect North-South symmetry.

Although the Wikipedia Sun page provides scant sidereal Solar Rotation information it does provide a remarkable insight into the Sun when graphically presented.

Wikipedia - Sun

Wikipedia source:
Williams, D. R. (1 July 2013). “Sun Fact Sheet”. NASA. Retrieved 12 August 2013.

The three data points provided by Wikipedia are sourced from the NASA Sun Fact Sheet that introduces the reader to the underlying mathematical approximation.

Entering the NASA mathematical approximation into Excel its possible to verify two of the values quoted by Wikipedia [equator and 16° latitude] but verifying the rotation period at the poles appears to be problematic.

NASA Sun Facts

NASA Sun Facts – Author/Curator: Dr. David R. Williams

Curiously enough, Wikipedia also publishes a Solar Rotation [aka Carrington Rotation] page which uses a similar mathematical approximation [sourced from Snodgrass and Ulrich 1990 – see below] that produces different sidereal rotation rates.

Wikipedia - Solar Rotation

At the equator the solar rotation period is 24.47 days.

This is called the sidereal rotation period, and should not be confused with the synodic rotation period of 26.24 days, which is the time for a fixed feature on the Sun to rotate to the same apparent position as viewed from Earth.

The synodic period is longer because the Sun must rotate for a sidereal period plus an extra amount due to the orbital motion of the Earth around the Sun.

Note that astrophysical literature does not typically use the equatorial rotation period, but instead often uses the definition of a Carrington rotation: a synodic rotation period of 27.2753 days (or a sidereal period of 25.38 days).

This chosen period roughly corresponds to rotation at a latitude of 26 deg, which is consistent with the typical latitude of sunspots and corresponding periodic solar activity.

Thus far, Wikipedia has provided two different sidereal rotations periods at the solar equator [25.05 and 24.47 days] from two different sources.

Referring to a favoured Wikipedia source [the Guide to the Sun which is currently referenced five times on their Sun page] it’s possible to obtain another pair of sidereal rotations periods for the solar equator [25.6 and 24.7 days] provided by other reference sources using different measuring techniques [Doppler versus Sunspots].

Guide to the Sun - Doppler

Guide to the Sun - Sunspots

“Very long-lived, regular-shaped spots, for example, rotate relatively slowly, but rapidly developing spots rotate faster.”

Guide to the Sun Paperback – Kenneth J. H. Phillips – 1995 – Cambridge University Press

The development of Doppler technology led to the discovery in 1980 of Torsional Oscillations which are zones of alternating fast and slow rotation which drift from the solar poles to the equator over a period of about 22 years.

Torsional Oscillations
The Sun is Observed to be a Torsional Oscillator with a Period of 11 Years
R Howard and B J Labonte.
Astrophysical Journal – Part 2 – Letters to the Editor, vol. 239, 1 July 1980

Doppler technology also led to the establishment of the Global Oscillation Network Group which improved the scope, accuracy and availability of solar rotational data.

The Global Oscillation Network Group (GONG) is a community-based program to study solar internal structure and dynamics using helioseismology.

Six solar observatories are involved, with the intention of achieving almost unbroken observation of the Sun.

The six observatories are the Teide Observatory (Canary Islands), the Learmonth Solar Observatory (Western Australia), the Big Bear Solar Observatory (California), the Mauna Loa Solar Observatory (Hawaii), the Udaipur Solar Observatory (India) and the Cerro Tololo Inter-American Observatory (Chile).

In 2001, the original GONG detectors were upgraded to 1000 x 1000 pixels and continuous magnetograms were implemented, and the new system is known as GONG++.

More recently (c. 2010), improvements to GONG observatory instrumentation have been made to enable imaging of the Hα (“H-alpha”) spectral line of hydrogen in the solar atmosphere.

The GONG Program is managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation

See also:

However, Doppler Velocity Imaging appears to struggle [the grey central band in the left hand image below] to capture solar rotation in the polar regions.

Doppler Sun

Figure 2: A single Doppler velocity image of the Sun from one GONG [Global Oscillation Network Group] instrument (left), and the difference between that image and one taken a minute earlier (right) with red corresponding to motion away from, and blue to motion towards, the observer.

The shading across the first image comes from the solar rotation.

High-res PNG

Solar Interior Rotation and its Variation
Rachel Howe – National Solar Observatory – Tucson
Living Reviews in Solar Physics 10th Feb 2009

Needless to say, the introduction of Doppler Velocity Imaging also ushered in new generation of mathematical approximations.

Doppler Formula

Rotation of Doppler features in the solar photosphere – 1990
H Snodgrass and R Ulrich
Astrophysical Journal 351: 309–316

The curious features of this particular Doppler mathematical approximation [based upon a least-squares fit using the “standard formula”] are:
a) The approximation provides a symmetrical North-South formula whilst the selected data points clearly display asymmetry especially in the equatorial region.
b) The most southerly Doppler data point from [about] -66° indicates there is a significant data gap at the south pole of about 24°.

Doppler Features

Rotation of Doppler features in the solar photosphere – 1990
H Snodgrass and R Ulrich
Astrophysical Journal 351: 309–316

This long astronomical tradition of the mathematical approximation can be traced back to [at least] the second half of the 19th century with notable contributions from Gustav Spörer and Richard Carrington.

Mr. Richard Carrington, in a series of observations which commenced in 1853 and ended in 1861, endeavoured to detect the law of distribution of the spots, to determine the true period of the rotation of the sun’s body, and to detect ‘systematic movements or currents of the surface, if such exist in a definable manner.’

In the course of these researches he discovered, in the first place, that discrepancies already noticed by former observers between the values deduced for the sun’s rotation arise from real differences in the velocities with which the spots move in different solar latitudes.

Near the equator a spot moves at a rate indicating a more rapid rotation than in higher latitudes.

Further, even among spots in the same latitude proper motions may be recognised.

These latter motions are to be regarded, however, as abnormal, and simply rendering unreliable such observations as are made on but a few spots.

The peculiarities affecting the motions of spots in different latitudes were reduced by Carrington into a formula, but such formulae have little scientific value1.

Sporer and Carrington formulae
Old and New Astronomy – Richard Anthony Proctor & Arthur Cowper Ranyard – 1892

Carrington Formula

Sporer Formula

Friederich Wilhelm Gustav Spörer (23 October 1822 – 7 July 1895) was a German astronomer.

He is noted for his studies of sunspots and sunspot cycles.

In this regard he is often mentioned together with Edward Maunder.

Spörer was the first to note a prolonged period of low sunspot activity from 1645 to 1715. This period is known as the Maunder Minimum.

Spörer was a contemporary of Richard Christopher Carrington, an English astronomer.

Carrington is generally credited with discovering Spörer’s law, which governs the variation of sunspot latitudes during the course of a solar cycle. Spörer added to Carrington’s observations of sunspot drift and is sometimes credited with the discovery.

The wonder of the observations made by Richard Carrington [from which he constructed his formula] is that they demonstrate a clear North-South asymmetry in solar rotation.

In this table the observed daily mean rotation is less in southern latitudes than in the corresponding northern latitudes.

It is doubtful whether we have in this relation any indication of the true cause of the observed variations in the rate of rotation, or merely a peculiarity which would have disappeared in a longer series of observations.

In favour of the former view, we have the consideration that the determination for each southern as well as for each northern latitude was independently effected, so that the coincidence of the results indicates the existence of some real cause.

Old and New Astronomy – Richard Anthony Proctor & Arthur Cowper Ranyard – 1892

Intriguingly, John Herschel considered “the more rapid rotation near the solar equator” observed by Richard Carrington implied that “the action of external matter” was responsible for “maintaining the rotation of the photosphere”.

If Sir John Herschel is right in considering that the more rapid rotation near the solar equator implies the action of external matter in maintaining the rotation of the photosphere, it may be suggested that the northern surface of the sun, being directed somewhat more fully towards that region whither the sun’s proper motion is carrying him, would probably be more exposed to the influence of this external action – ‘the frictional impulse of circulating planetary matter in process of subsidence into, and absorption by, the central body’ – much as our northern hemisphere is saluted with a larger number of meteoric missiles from June to December, when the northern hemisphere is in advance, than from December to June, when this hemisphere is towards the more sheltered side of the Earth.

Old and New Astronomy – Richard Anthony Proctor & Arthur Cowper Ranyard – 1892

Sir John Frederick William Herschel, 1st Baronet, KH, FRS (7 March 1792 – 11 May 1871) was an English polymath, mathematician, astronomer, chemist, inventor, and experimental photographer, who in some years also did valuable botanical work.

Herschel originated the use of the Julian day system in astronomy.

He named seven moons of Saturn and four moons of Uranus.

He made many contributions to the science of photography, and investigated colour blindness and the chemical power of ultraviolet rays.

The end of this trial leads to Richard Carrington’s invaluable summary data [1853 through 1861] of solar rotation by latitude [reported in arc-minutes per day] which clearly demonstrates a North-South asymmetry.

I found matters were little mended since 1776, in which M. Lalande in his first memoir gave a table of “Authors who have determined the position of the Solar Equator” and their results.

These results, though many are respectable, might be considered as superseded by the more modern determinations of Laugier, Bohm, Petersen, Kysaeus, Bianchi, and others ; but still the results stood thus:

Old Solar Rotation Observations

Nevertheless, I have to remark that in the year 1817 I observed at Milan a Solar Spot of long duration, from which, by the means of a number of values which presented a good mutual accordance, I obtained, as the time of the sidereal or absolute rotation of the Sun 25.325 days, a result which differs very slightly from that of Laugier.

Richard Carrington - Solar Rotation Obervations

Observations of the Spots on the Sun from November 9, 1853, to March 24, 1861, made at Redhill
Richard Carrington – 1863 – Williams and Norgate

Carrington Data Graph

Richard Carrington’s data table provides an opportunity to use standard statistical techniques in Excel to create projections that stretch to the solar poles – the very place where modern Doppler technology appears to fail.

The following projections treat each hemisphere independently [because of the North-South asymmetry] and exclude the irregularities observed in the 10 degrees bands immediately adjacent to the solar equator.

The Power and Exponential projections in Excel clearly highlight the remarkable North-South asymmetry.

Carrington Data - Power Trends

Carrington Data - Exponential Trends

The Logarithmic projection suggests the North-South asymmetry may not be quiet so extreme whilst the 3rd order polynomial projection surprisingly fails to produce a meaningful result in the Southern hemisphere.

Carrington Data - Logarithmic Trends

Carrington Data - 3rd rder Polynomial Trends

Selecting the two Best in Class projections based upon their R2 values explicitly highlights the most like structure underlying the North-South rotational asymmetry.

Carrington Data - Best in Class Trends

Mirroring these Best in Class projections provides a visualisation of the underlying Solar Vortex which drives the Sun with the Northern Hemisphere “being directed somewhat more fully towards that region whither the sun’s proper motion is carrying him” as the Solar System spirals through interstellar space.

Carrington Data - Solar Vortex

Solar Wind



Solar System Planetary Disk

Solar System - Rankine Vortex


Finally, if you have acquired a preference for Settled Science then you can return to your Formula Fantasy of choice in the certain knowledge that the Solar Vortex is a purely imaginary chimera concocted by a chronically capricious charlatan.

Is the Solar System Really a Vortex? by Jason Major on December 18, 2013

The short answer? No.

Note: The Universe Today article contains a good selection of animated Solar System vortices.

Gallery | This entry was posted in Astrophysics, Maunder Minimum, Solar System, Vortices. Bookmark the permalink.

1 Response to Solar Rotation – Formula Fantasies

  1. Pingback: Kepler’s 160 Minute Solar Cycle | MalagaBay

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