LinR: Origins of the Solar System

Cosmogony (origin of solar system)
Age ∼ 4.5 x 109 years

Executive Summary of Observations

  1. All planets lie in almost the same plane, and directions of rotation are same(except Venus).
  2. Spin of sun lies in ecliptic, all planets (except Uranus) close to it.
  3. Orbit shapes are nearly circular (little eccentricity)
  4. System repeated as
  5. Comets and meteors have very eccentric orbits, at random angles ("inclinations")to the ecliptic.
  6. Most of mass the of solar system is in the sun. Most of the angular momentum of the solar system in the planets.
  7. Inner planets are terrestrial, outer planets are gas giants (jovian).
  8. Inner planets are closely spaced, outer planets are widely spaced.

The old theories:

Close encounter of a star with the sun pulls out a tongue of material

Companion star explodes, material captured by sun condenses into planets

Both dynamically impossible. No angular momentum!

Nebular Hypothesis. A rotating gas cloud, probably compressed by a nearby supernova shock wave, starts to collapse.
The central part collapses to the sun.

Conservation of angular momentum causes the outer part to speed up.


The outer planets condense first.
Gas and dust particles moving at an angle to ecliptic are more likely to interact and hence collapse into the plane normal to the rotation axis.
Majority rule (less effective at the limits, hence Mercury & Pluto orbit further out of the plane.)


Intense solar winds remove hydrogen and helium from the inner part of the solar system.
Terrestrial planets form from the left over refractory materials.

The orbits are circularized by collisions and tidal effects.
The orbits are circularised by collisions and tidal effects.

How does rotational (spin) angular momentum arise from orbital? Suppose Jupiter, orbiting at radius R, formed from 2 equal-mass lumps, initially orbiting in separate orbits of radii r1 and r2 (assumed circular), but without any spin. Some Formulae We Will Need:
Orbits: Io = Ma² (a ≈ orbital radius)
L = Ioωo = orbital angular momentum
Spin: Is = ²/5 Mr² (r = planetary radius)
S = Isωs = spin angular momentum
KE = 1/2 Ioωo² + 1/2 Isωs² = 1/2 Lωo + 1/2 Sωs
We'll use P for the length of the year, and p for the length of the day.
From Kepler P² = ka³, so ωo = 2π/P while ωs = 2π/p.

We know MJ = 1.9 × 1027 kg, aJ = 7.8 × 1011m,
PJ = 10744d = 9.28 × 108s, and pJ = 9h55m30s = 3.57 × 10⁴s.

⇒
Also m1 = m2 = 0.95 × 1027 kg.
⇒
LJ = 7.85 × 1042 Js, SJ = 6.81 × 1036 Js,
KEJ = 2.75 × 1034 J.
⇒
Assume: r1 = 7.0 × 1011m.
⇒
Then Kepler gives P1 = 7.89 × 108 s.
⇒
And L1 = 3.71 × 1042 Js, S1 = 0 Js,
KE1 = 1.48 × 1042 J.
⇒
Attempting to conserve angular momentum:
L2 = LJ + SJ - L1 = 4.14 × 1042 Js.
⇒
This yields r2 = 8.7 × 1011m. P2 = 1.09 × 109 s.
⇒
But there is insufficient KE to explain Jupiter's orbit and spin!!!
⇒
Alternatively attempting to conserve kinetic energy:
KE2 = KEJ - KE1 = 1.27 × 1034 J.
⇒
This yields r2 = 8.2 × 1011m. P2 = 9.92 × 108 s.
⇒
But there is insufficient angular momentum to explain Jupiter's orbit and spin!!! The solution is that BOTH Kinetic Energy and Angular Momentum are lost in the collision, due to fragmentation. The planet that remains results from a complicated set of processes, in which some orbital angular momentum is lost to the spin of the planet and some to the fragments which escape, and some kinetic energy is lost to heating of the planet, and some to the escaping fragments.


Ideas partly confirmed by the other planetary systems that we see now:
e.g. υ (say Upsilon) Andromeda

Where is Upsilon Andromedae? Credit and Copyright: Till Credner & Sven Kohle
has 3 planets, one very close

(New York Times)
so it might look like this

Drawing Credit & Copyright: Lynette Cook

(as of today!). Special conditions required for the discoveries (can only see large planets, fairly close and orbit must lie close to our line of sight) suggest they are very common: probably 50% of stars have them. On to Celestial Mechanics and the Motions of The Solar System