Why the Planets Orbit Nearly in the Same Plane

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Introduction

The major planets of the Solar System orbit the Sun in nearly the same plane.

This is not an accidental arrangement.

The reason is closely related to the formation of the Solar System from a rotating cloud of gas and dust approximately 4.6 billion years ago.

The sequence is

Rotating Cloud Gravitational Collapse Protoplanetary Disk Planets

The planets therefore inherited approximately the same orbital plane from the protoplanetary disk.

Angular Momentum

For a particle of mass m,

L = r m v

where

r = position vector

and

v = velocity vector.

For the entire primordial solar cloud,

L_ total = _i r_i m_i v_i

If the external torque is small,

d L_ totaldt = _ external 0

and therefore

L_ total constant

This is conservation of angular momentum.

Gravitational Collapse

The primordial solar nebula contained gas and dust.

Gravity caused this cloud to contract.

As the characteristic radius decreased, rotation became increasingly important.

A simple rotating-system relation is

L=I

where

I=moment of inertia

and

=angular velocity.

If

Lconstant

while

I

then

The collapsing cloud therefore rotates faster.

Why a Disk Forms

The original cloud was three-dimensional.

However, the gas particles collided with each other.

Shocks, collisions, and gas drag dissipated random kinetic energy.

Energy can be dissipated while total angular momentum remains approximately conserved.

Choose the z-axis parallel to the total angular-momentum vector.

Vertical velocity can be represented by

v_z.

A simplified damping model is

dv_zdt = -v_zt_ damp

which gives

v_z(t) = v_z0 e^-t/t_ damp

Therefore,

v_z0.

Random vertical motion gradually decreases.

The cloud becomes thinner.

The result is a rotating disk.

3D Cloud Flat Rotating Disk

This disk is called the

Protoplanetary Disk

Direction of the Disk

The total angular-momentum vector determines the rotation axis.

If

L_ total

points in the z-direction, the disk forms approximately in the xy-plane.

Therefore,

L_ total Protoplanetary Disk

The disk plane is approximately perpendicular to the total angular-momentum vector.

Planet Formation

Inside the disk,

Dust Aggregates Planetesimals Protoplanets Planets

The planets were therefore formed from material already moving inside approximately the same disk.

Consequently, their orbital angular-momentum vectors point approximately in the same direction.

For a planet,

L_ orbit = r m v

For an elliptical orbit,

L_ orbit = m GM_a(1-e^2)

where

a=semi-major axis

and

e=orbital eccentricity.

Keplerian Motion

The orbital period is approximately

T^2 = 4^2 GM_ a^3

Thus the planets have very different orbital periods.

However, the directions of their orbital angular momenta remain approximately aligned.

This is why the Solar System resembles a rotating disk rather than a spherical swarm of planets.

Orbital Inclination

The planetary orbits are not perfectly coplanar.

For an orbital inclination i,

L_z = L i.

The vertical position of an orbit can be approximated by

z=r i

For small angles,

i i

and therefore

z ri

Small orbital inclinations therefore produce small departures from the common plane.

Mercury has a noticeably larger inclination than most major planets.

Pluto is much more strongly inclined.

Pluto is currently classified as a dwarf planet.

Its orbit provides a useful comparison showing that the Solar System is

nearly coplanar

rather than perfectly coplanar.

Theoretical Summary

The physical process can be summarized as

c Rotating Molecular Cloud Gravitational Collapse Angular Momentum Conservation Increasing Rotation Collisions and Gas Dissipation Vertical Motion Decreases Protoplanetary Disk Planet Formation Nearly Coplanar Planetary Orbits

ANITEX Interactive Visualization

What to Observe

Start the animation with the three-dimensional rotating solar nebula.

As gravity contracts the cloud, observe that the rotation becomes more important.

The total angular-momentum vector remains approximately fixed.

Next observe the random vertical motion of the gas and dust particles.

The vertical velocity decreases according to the simplified relation

v_z(t)=v_z0e^-t/t_ damp.

The cloud therefore becomes progressively thinner.

Eventually a rotating protoplanetary disk forms.

The planets then appear inside this disk.

Change the camera to

Edge-On View

to observe the thin distribution of the planetary orbits.

Finally increase

Inclination Exaggeration

from 1 to 8.

This reveals that the orbits are not exactly on one mathematical plane.

The Solar System should therefore be described as

approximately coplanar

rather than perfectly coplanar.