Angular momentum

of planets and Sun in DE40x ephemerides...



This is my analysis of planet moves, encoded in DE405 and DE406 ephemerides, which were calculated by JPL NASA.


Inner planets orbit arround the Sun.
Outer planets, on the other hand, orbit arround the Solar system barycenter (SSB), as a counter-weight of the Sun.

Angular momentum

In a default Keplerian model, the angular momentum of a planet, with respect to the center of its trajectory, is constant. The planet moves faster, when near to center, and slower, when far, which conserves the angular momentum.

Anyhow, in real world, angular momentum of individual planets is not constant, due to tugs by all other planets...

Relative values of orbital angular momentum of individual planets, in arbitrary units, as determined from preliminary ephemerides versions DE414 & DE415 :
Planet Center Sun Center Ssb Mass (kg)
Minimum Maximum Scatter Tendency Minimum Maximum Scatter Tendency
Sun 5.9000 290.0000 284.1000 oscilating 1.98843966345011E+30
Mercury 5.9865 5.9866 0.0001 reducing 5.8308 6.1763 0.3455 oscilating with Sun 3.30108185047165E+23
Venus 123.2947 123.2971 0.0024 growing 121.5867 125.0140 3.4273 oscilating with Sun 4.867381346361E+24
Emb (*) 180.0427 180.0474 0.0047 growing 178.2222 181.9053 3.6831 oscilating with Sun 6.04571684215467E+24
Mars 23.4882 23.4906 0.0024 reducing 23.3269 23.6571 0.3302 oscilating with Sun 6.41699179158458E+23
Jupiter 128 875.4927 128 923.2083 47.7155 narrow oscilating,
128 519.1412 128 815.7483 296.6071 smooth oscilating
contrary to Sun
Saturn 52 156.0531 52 345.9642 189.9111 wide oscilating 52 193.5867 52 244.4528 50.8661 narrow oscilating,
Uranus 11 265.6617 11 326.0695 60.4079 wide oscilating 11 292.6605 11 296.5708 3.9103 long cycle 8.66045149559652E+25
Neptune 16 773.9927 16 791.7912 101.7985 wide oscilating 16 824.2558 16 825.3984 1.1426 long cycle 1.02953279430512E+26
Pluto 2.7635 2.7864 0.0229 wide oscilating 2.7742 2.7745 0.0003 slowly growing 1.52956896907633E+22
sum (*1) 209470.9488 209471.1088
(*2) 209481.96
11 (*2)
854y cycle
These minimum and maximum values are during 20th century, for Pluto during 3 centruries, and for the Sum during 1.5 milenia.
Notes to the table:

(*1) - In the sum there are included 9 planets and the Sun, all with respect to Ssb. The planets are Mercury, Venus, Emb (Earth-Moon system), Mars, Jupiter, Saturn, Uranus, Neptune, Pluto. It does not include asteroids and spin momentum of bodies, namely of Sun.
The whole sum is almost constant, but there is a small difference of 8.11*10-7 of the whole. The difference from a constant value is divided between orbital angular momentum of asteroids, trans-neptunians and spin angular momentum of Sun.
From this, the Sun spins faster now and arround years 1200 and arround 250, and spinned slower arround years 1600 and 700... See below the charts of an angular momentum sum...
(*2) - These high swings are at the times, when the Sun is approaching the solar-system barycenter too closely, and then the space curvature (not regarded by me), plays a significant role...

On the scale of millenia, the Earth and Venus angular momentum (relative to Sun) grows, these planets move in concentric spirals and are speeding up (as determined by huge Gaussian filtering). Anyhow the approach to Sun after 5 millenia is still much smaller than anual approaching and receding due to eccentricity... On this scale, the Mercury and Mars angular momentum (relative to Sun) shrinks.
Another contribution to these angular momentum trends are the changes of the eccentricities of the orbits. For the planet Earth, the eccentricity shrinks much more than the averaged approach to the Sun...


Planet Earth, when considering orbital characteristics and when interacting gravitally with other planets, is actually a system of 2 bodies, Earth and Moon. Their weight is comparable (1:81.300587). Here i call this system Emb, the Earth-Moon barycenter...
Of this two, only the Earth interacts magnetically, and it oscilates during its orbital motion by the counter-weight of Moon by +-4900 km, with a main frequency of 29.53 days and an envelope of 1.13y:
Distance of Earth from Sun,
compared with distance of Emb from Sun

Angular momentum of Emb with respect to Solar-system barycenter:
Angular momentum of Emb with respect
to Solar system barycenter (olive), compared with same with respect to Sun (green).
In detail
(The charts are self-relative, with minimum to maximum stretched to fill the image. Actually, if they displayed 0 also, the data would be one straight line at the top of the image, since the oscilation is incomparably smaller than the absolute, almost-constant value...)

Main frequency is here 1.09 year (which is a frequency of angle between Earth motion vector and vector from Earth to Jupiter), other frequencies are at 1.20y and 1.00y, and the envelope reflects the oscilation of Sun arround the Solar-system barycenter.

The Earth orbits on a heliocentric trajectory (ie. not on barycentric), so the angular momentum relative to Sun is more constant and also more interesting...

Angular momentum of Emb with respect to Sun:
Angular momentum of Emb
relative to Sun
in detail compared with signed Sunspot cycle Ang.moment. of Emb (green) and Venus (blue)
compared with Sunspot cycle
Main frequency is here 199.4 days, which is 1/2 of the synodic period of Jupiter as seen from Earth, then 584 days - the period between meetings of Emb and Venus (the resonance period), then 292 days, 133 days, 117 days, 1.09 years, 3.98 years, 11.84 years, 15.6 years and many other, as determined by FFT analysis...


Angular momentum of Venus
relative to Sun
Longer trend Compared to
Sun-spot cycle
detail with marked mode changes
Main frequency is here 291.6 days (half of meet-period between Earth and Venus), then 118.4 days, 194.5 days (Jupiter?), 145.9 days (Mercury), 583.5 days (Earth), 116.7 days (1/5 Ea, Venus solar day), 3.96 years, 97 days, 83.4 days, 121.7 days and other, sorted by significance, as determined by FFT analysis...


It is counter-intuitive, that Jupiter angular momentum relative to Sun shows less swing than Jupiter angular momentum relative to SSB...
Compared Jup/Sun (yellow)
and Jup/Ssb (olive)
In detail Structure of a.m.
of Jup Sun
compared to sun-spot cycle
(different freq.)
Longer tendency
Angular momentum of Jupiter relative to Sun is roughly opposite of angular momentum of Saturn relative to SSB.
Angular momentum of Jupiter relative to SSB has 20-year frequency of Jupiter/Saturn bary-center, similar (oposite) to angular momentum of Sun...


Saturn is first of outer planets, that show clearly higher swing relative to Sun than to SSB:
Compared Sat/Sun (yellow)
and Sat/SSB (olive)
In detail Longer tendency
Again, the period is here the 20 years, the period of Jupiter/Saturn barycenter.


Compared Ura/Sun (yellow)
and Ura/SSB (olive)
in detail Long cycle of Uranus/SSB
Main frequency of of Uranus Sun is 13.74 years.
Main frequency of of Uranus is 179-181 years, the resonance of Uranus/Neptune, with other frequencies at 164y, 22.4y, 15.03y, 6.9y and other...


Compared Nep/Sun (yellow)
and Nep/SSB (olive)
Detail of of Neptune
in 20th century
Long cycle of Neptune/SSB
Main frequency of of Neptune Sun is 12.70 years.
Main frequency of of Neptune is 168-171 years, with other at 82.0y, 56.3y, 42.5y, 17.9y, 6.4y and other...


Long trend of Pluto In detail


Ang.momentum of Mercury
relative to SSB
In detail FFT analysis
Main frequency is here 89.785 days, other are 88.691 days, 88.092 days, 88.214 days, 87.965 days, 91.687 days, 11.83 years, 115.86 days, 144.55 days, 1.11 years, 29.5 years and other harmonics, as determined by FFT analysis.
The envelope follows the Sun move arround the solar-system barycenter, since the trajectory is heliocentric...

Angular momentum of Mercury, relative to Sun:
Longer trend In detail ... Most detailed
Main frequency is here 1.1 years (403 days), other are 5.89 year, 1.37 year, 106.5 days, 72.2 days, 91.5 days, 169.6 days, 202.6 days, 11.23 year, 13.8 year and other, as determined by FFT analysis.



Relative to SSB Relative to Sun
Relative to SSB:
Main frequency is here 2.24 years, 2.01 years, 1.90 years, 1.92 years, 1.88 years, 2.75 years, all previous repeated as first harmonic, 11.82 years, 29.37 years, 18.96 years and few other.
Again, the envelope follows the Sun move arround the solar system barycenter and shows a much higher swing than the heliocentric trajectory.

Relative to Sun:
Main frequency is here 1.11 year, 2.75 year, 271.8 days, 11.7 years, 2.12 year, 333 days, 1.23 years, 203.8 days and other.



Angular momentum of Sun is mostly contrary to an angular momentum of Jupiter, which is its main counter-weight.
It these two are added, they could mostly cancel out, so that the rest, that is a "ripple" on the Sun's angular momentum chart, can reveal:

Note the 178.8 year cycle of Uranus/Neptune, similar to Gleissberg cycle...
Angular momentum of Sun
Angular momentum of Jupiter

Angular momentum of Sun 1st derivation
showing PTC cycle
Ang.momentum in detail,


Angular momentum sum of 9 planets and Sun

From 100 AD to 2200 AD From 1800 BC to 2700 AD Detail with 1st derivation Non-clipped version Absolute value
__________ clipped __________ compared to sun-spot cycle
(freq. differs)
The "absolute value" chart includes 0. It shows, that the sum seems very constant, and the "wave", seen on other charts, is only a "wave at the top of an ocean", on the order of 8.11*10-7, close to 1 in million...

The difference from a constant value is divided between orbital angular momentum of asteroids, trans-neptunians and spin angular momentum of Sun. (The correlation of changes in Solar surface rotation rate with some planetary positions may be showed on the synoptic maps of variable Solar surface rotation...)

The high peaks are during times, when the Sun approaches the solar-system barycenter. At these times, the space curvature arround the Sun center plays significant role in calculating distances and the angular momentum, but I could not find a propper equation for a space curvature to cancel these, so I instead just clip the chart vertically...
Without the space curvature, these events would disrupt the conservation of angular momentum significantly...

The first derivation of angular momentum sum only little matches the sun-spot cycle, but the high-peak at 1990 could be correlated with a drop of solar-flare activity at the middle of preceeding sun-spot cycle 22. ...

The "wave" of approximate period of 854 years, which could be anti-correlated with Sun spin rate, seems to match the climatologic events of Medieval optimum and Global warming, and also the Little Ice age of Maunder minimum, and similar periods in earlier ages...
If this is the case, now the Solar activity could drop a little, but will approach a larger maximum arround year 2050, not disturbed by the peak anomally, and then drop to a next little-ice-age arround 2400 AD.
The time-lag between the spin rate change and activity change is still uncertain...

Mathematical formula for angular momentum calculation

For calculating angular momentum of a body with respect to a center, I use this formula:

or expressed as a pascal code:

function PlanetMomentum(Planet,Center: TPlanet): Float;
var V: TVector3d;
  // Relative velocity:
  Vector3dSub(Planet.VelocityVector, Center.VelocityVector, V);
  // ... multiplied by Mass:
  Result := Planet.Mass * Vector3dLength(V);

function PlanetAngularMomentum(Planet,Center: TPlanet): Float;
var Momentum, Distance, SinAngle: Float;
    DistVector, RelativeVelocity: TVector3d;
  // Relative momentum:
  Momentum := PlanetMomentum(Planet,Center);
  // Distance:
  Vector3dSub(Planet.PositionVector, Center.PositionVector, DistVector);
  Distance := Vector3dLength(DistVector) * (1/km_to_AU); // Distance in AU...
  // Angle of relative_velocity_vector and distance_vector:
  Vector3dSub(Planet.VelocityVector, Center.VelocityVector, RelativeVelocity);
  SinAngle := Sin(Vector3dAngle(DistVector, RelativeVelocity));
  Result := Momentum * Distance * SinAngle  / 1e24;

In words, it is a scalar multiplication of Momentum vector length ( Velocity * Mass ), multiplied by distance and multiplied by Sinus of angle between the line connecting the bodies and the relative velocity vector of the body with respect to the center...
The Sum of angular moments is just a scalar sum of moments of all individual bodies (9 planets and Sun)...
The Solar-system barycenter is a coordinate center of these ephemerides and has got both vectors null...

All vectors are in km, masses are in kg, and it is just divided by AU at one place and by 1E24 at another to prevent precision lost and make it more readable...
Written 2006 June 27 - July 12 by Semi.