Angular momentum
of planets and Sun in DE40x ephemerides...
Contents
Model
This is my analysis of planet moves,
encoded in DE405 and DE406 ephemerides, which were calculated by JPL NASA.
Center
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)
(G=6.6742*10-11)
|
| 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,
reducing
| 128 519.1412
| 128 815.7483
| 296.6071
| smooth oscilating
contrary to Sun
| 1.89854360313697E+27
|
| Saturn
| 52 156.0531
| 52 345.9642
| 189.9111
| wide oscilating
| 52 193.5867
| 52 244.4528
| 50.8661
| narrow oscilating,
growing
| 5.68450446981147E+26
|
| 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
| 0.1600
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...
EMB
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
| Tendency
|
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...
Venus
|
|
|
|
Angular momentum of Venus
relative to Sun
1600-2200
| 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...
Jupiter
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 rel.to 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
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.
Uranus
|
|
|
Compared Ura/Sun (yellow)
and Ura/SSB (olive)
| in detail
| Long cycle of Uranus/SSB
|
Main frequency of ang.mom. of Uranus rel.to Sun is 13.74 years.
Main frequency of ang.mom. of Uranus is 179-181 years, the resonance of Uranus/Neptune,
with other frequencies at 164y, 22.4y, 15.03y, 6.9y and other...
Neptune
|
|
|
Compared Nep/Sun (yellow)
and Nep/SSB (olive)
| Detail of ang.mom. of Neptune
in 20th century
| Long cycle of Neptune/SSB
|
Main frequency of ang.mom. of Neptune rel.to Sun is 12.70 years.
Main frequency of ang.mom. of Neptune is 168-171 years, with other at 82.0y, 56.3y, 42.5y, 17.9y, 6.4y and other...
...
Pluto
|
|
| Long trend of Pluto
| In detail
|
Mercury
|
|
|
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.
Mars
|
|
|
|
| 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.
Sun
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,
anotated
|
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;
begin
// Relative velocity:
Vector3dSub(Planet.VelocityVector, Center.VelocityVector, V);
// ... multiplied by Mass:
Result := Planet.Mass * Vector3dLength(V);
end;
function PlanetAngularMomentum(Planet,Center: TPlanet): Float;
var Momentum, Distance, SinAngle: Float;
DistVector, RelativeVelocity: TVector3d;
begin
// 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;
end;
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.
