Richard Willoughby
Summary
This article follows up on an earlier article that considered how the movement of the Sun relative to Earth as well as solar activity alters Earth’s climate.
The motion of the Sun so far during this century is analysed in detail and gives insights into how the various planets contribute to the motion of the Sun. The cause of the daily anomalous distance from Earth to Sun is also investigated and quantified for the same day of the year over 30 years.
The originating Sun motion data for this analysis has been extracted directly from the NASA Horizons Application and used for free body dynamic motion analysis. The JPL free body model of the solar system treats each celestial object as a point mass concentrated at the centre of mass. The free body analysis of the Sun here extends to a point mass elastically connected to the centre of mass of the Sun and rotating around the centre of mass to determine the gravitation torque imposed on the rotating mass.
Sun Movement
The orbit of the Sun is somewhat erratic with each rotation being different to the preceding and those following. Diagram 1 shows the path of the Sun using the International Celestial Reference Frame (ICRF) for the period 2000 to 2030 in the celestial equatorial plane. In the ICRF, Point(0,0,0) is the barycentre of the solar system. The dynamic analysis in this article assumes the North-South motion of the Sun out of the equatorial plane is negligibly small.
However, to complete the picture of the Sun motion, Chart 1 indicates the North-South excursions from the celestial equatorial plane over the same period as Diagram 1.
Jupiter’s orbit dominates the motion of the Sun and Jupiter’s orbital plane has a 1.3 degree tilt to the celestial equatorial plane. The North-South excursion of the Sun with respect to the ICRF is a result of the Sun essentially moving in Jupiter’s orbital plane. As observed in Chart 1, Earth tends to follow the Sun in its North-South excursion. Jupiter’s direct influence on Earth is more apparent in the equatorial plane as shown in Chart 2 that displays the Sun to Earth distance anomaly wrt to 2000 for the September Equinox over a 30 year period.
The range of the anomalous distance shown in Chart 2 is sufficient to alter the peak solar intensity by 1W/m² for the selected day. The day was selected for ease of annual alignment in the equatorial plane rather than being the maximum difference for any particular day of the year.
Individual Planetary Influence on the Sun’s Orbit
The Sun orbit is broadly in opposition to Jupiter; as in the Sun being the hammer thrower and Jupiter being the hammer. The Sun is 1050 times more massive than Jupiter so the orbit of the Sun needs to be 1/1050th of Jupiter’s orbit to keep the orbits stable. Diagram 2 shows the 2-D free body orbit of the Sun being driven by Jupiter’s actual orbit.
To be clear, the motion of the Sun shown here is the result of the Sun only being influenced by Jupiter while Jupiter is following its prescribed path under the influence of all the other celestial bodies having gravitational influence on the solar system. The initial conditions shown on Diagram 2 for the Sun position and velocity differ from the full system that produces the actual orbit shown in Diagram 1 above.
Diagram 3 shows how the Sun would move when both Jupiter and Saturn are present. Saturn has sufficient influence to require resetting the initial conditions to maintain stable orbit of the Sun.
Including Saturn causes the orbit of the Sun to have some similarity to the full system orbit of Diagram 1.
Although Jupiter is 390 times more massive than Venus, Venus is 1/7 the distance from the Sun so Venus has detectable influence on the motion of the Sun as shown in Diagram 4.
Neptune is the most distance planet from the Sun but its mass is slightly more than half the mass of Jupiter so it has gravitation influence by virtue of the double integration of time from force to distance as shown in Diagram 5.
The 30 year time period shown in Diagram 5 is somewhat brief to get a full appreciation of Neptune’s influence on the motion of the Sun because the planet has only moved through 65 degrees of arc in that time. Neptune is the plodder in moving the Sun; slow and steady.
Distributed Mass of the Sun
The radius of the Sun at 6.96E8m is significant relative to its orbit of 7.31E8m under the influence of only Jupiter. Considering the erratic orbit of the Sun under the influence of all the objects of the solar system, there are times when the barycentre is within the sphere of the Sun. In fact there are some orbits when the centre of mass (CoM) of the Sun does not encircle the barycentre.
Another feature of the Sun is that its mass is in the form of plasma. Hence it is more fluid than solid. Under the influence of its spin, the centrifugal forces on the plasma cause the whole Sun to oblate with the diameter at the equator increasing relative to the pole to pole diameter.
The simplest way to analyse the free body motion of the surface of the Sun is to reduce it to a single point in space tethered to the CoM. This is shown diagrammatically in Image 1.
The tether has elasticity to represent the behaviour of the plasma oblation under centrifugal acceleration.
The initial analysis, shown in Diagram 6, displays the position of the Sun as determined by JPL as the black locus for the CoM and the red locus for the equatorial point (EP) under the influence of the same gravitational accelerations moving the CoM. The EP is set in the X-direction from the CoM.
The EP essentially holds its position in the X-direction tracing a similar, but displaced orbit. Chart 2 shows the velocity of the EP if the Sun did not have any spin. It is the same as the velocity of the Sun. Venus contributes most to the small dither in the velocity.
Diagram 7 shows the loci of the CoM and EP after the EP is given initial motion in the Y-direction of 100m/s relative to the CoM.
The initial spin of 100m/s gives a spin period of 506 days so the EP orbits the CoM 22 times over the 30 year period minus the almost 3 orbits the CoM completed in the same time. This is observed in Chart 3, showing the velocity of the EP over the period with average of 100m/s and 19 excursions.
The velocity of the EP is observed to have a range about the average governed by the velocity of the CoM. The EP has to be accelerated and decelerated through the course of each rotation resulting in the tether stretching and relaxing each rotation. If the EP was not influenced by the gravitation field, the accelerating torque and decelerating torque would cancel out. The gravitational force acting on the EP is orders of magnitude smaller than the tether force but gravitation force creates a net torque as shown in Chart 4. The calculated torque is based on the EP having unit mass of 1kg.
Over the 30 year period, the average spin velocity increases by 0.04m/s but there are periods when the spin also slows down.
The equatorial spin of the Sun is approximately 25 days; corresponding to 2000m/s. Chart 5 shows the cumulative spin torque on the EP under these conditions.
The average torque for a spin period of 25 days is slightly negative.
Spin Torque Between 100m/s & 2000m/s
The two examples considered so far yielded both a net positive cumulative torque and net negative cumulative torque over the 30 year period considered. This begs the question – what happens in between these extremes.
If the Sun spin was caused by gravitational effects then they would be most pronounced in the region that could generate the highest torque. Considering the Sun as a series of concentric cylinders surrounding the spin axis and assuming constant density, the highest torque would be achieved at 5.7E8m; corresponding to latitude of 35 degrees. Accordingly, Chart 6 shows the cumulative torque for a range of initial spin velocity at 5.7E8m from 900m/s up to 1850m/s.
There are two interesting peaks indicative of resonances. The first occurs at periods of 44 days and the second at 29.3 days. Both peaks exhibit a sharp increase in torque followed by a rapid collapse into negative torque. There is a recovery to positive torque at 970m/s with a gradual increase to 1400m/s before the next peak and sudden collapse.
The cumulative torque goes through a phasing sequence as the initial velocity approaches the resonance. Chart 7 shows the cumulative torque at 5.7E8m for 1403.7m/s, which was chosen because the cycles have good negative correlation with solar activity over the 60 year period.
The cumulative torque cycles reduce as the initial velocity approaches 1413m/s where it just becomes a steady upward trend per Chart 8.
Although the torque rises throughout the period, it only increases the EP velocity by 0.05m/s over that period. The torque drops precipitously above 1413m/s and is negative by 1414m/s. So any portion of the Sun spinning with a period of 29.33 days is on the verge of switching between spin acceleration and spin braking.
Discussion
The period of the two resonances show Mercury has a tidal influence on the Sun. In fact, this analysis may underestimate the impact of Mercury because the individual planetary contributions are summed and applied as a single varying gravitational force to both the CoM and the EP. Given Mercury’s proximity to the Sun and the large diameter of the Sun, the local tidal effects of Mercury on the adjacent Sun surface may be significant as well. Mercury’s contribution to the motion of the Sun CoM is miniscule but it appears to play a key role in the spin of the Sun, which extends to the development of solar activity.
If the torque was always positive, the Sun would continue to spin faster. The fact that the torque reaches a tripping point that causes a sharp torque reversal is consistent with generating local turbulence and high shear in any region spinning at a period of 29.33 days. This aligns with the observed dominant period of the Sun per the headline image above from the Max Planck Solar System Research.
The torque calculations do not explain the period of rotation of the Sun’s equator nor why the rotational speed at smaller radii is slower than 29.33 days.
Conclusions
The motion of the Sun is uniquely different to the movement of the planets. The Sun tracks an erratic path while experiencing a two to one range in its orbital velocity. The plasma of the Sun imbues the ability to oblate under centrifugal acceleration from axial spin. This ability is shared with the gas giants Jupiter and Saturn which spin at even higher equatorial velocity than the Sun.
Analysing the motion of celestial bodies as point masses is a simplification that limits understanding of their behaviour. The distribution of the mass and its physical state are important considerations for assessing gravitational influences beyond velocity and position.
The Author
Richard Willoughby is a retired electrical engineer having worked in the Australian mining and mineral processing industry for 30 years with roles in large scale operations, corporate R&D and mine development. A further ten years was spent in the global insurance industry as an engineering risk consultant where he developed an enduring interest in natural catastrophes and changing climate.
