\documentclass[11pt]{article}
\usepackage[margin=22mm]{geometry}
\usepackage{amsmath,amssymb}

\title{Why the Planets Orbit Nearly in the Same Plane}
\author{}
\date{}

\begin{document}
\maketitle

\section{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

\[
\boxed{
\text{Rotating Cloud}
\rightarrow
\text{Gravitational Collapse}
\rightarrow
\text{Protoplanetary Disk}
\rightarrow
\text{Planets}
}
\]

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

\section{Angular Momentum}

For a particle of mass \(m\),

\[
\boxed{
\mathbf L = \mathbf r \times m\mathbf v
}
\]

where

\[
\mathbf r = \text{position vector}
\]

and

\[
\mathbf v = \text{velocity vector}.
\]

For the entire primordial solar cloud,

\[
\boxed{
\mathbf L_{\rm total}
=
\sum_i
\mathbf r_i \times m_i\mathbf v_i
}
\]

If the external torque is small,

\[
\frac{d\mathbf L_{\rm total}}{dt}
=
\boldsymbol{\tau}_{\rm external}
\approx 0
\]

and therefore

\[
\boxed{
\mathbf L_{\rm total}
\approx
\text{constant}
}
\]

This is conservation of angular momentum.

\section{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

\[
L=I\omega
\]

where

\[
I=\text{moment of inertia}
\]

and

\[
\omega=\text{angular velocity}.
\]

If

\[
L\approx\text{constant}
\]

while

\[
I\downarrow
\]

then

\[
\boxed{
\omega\uparrow
}
\]

The collapsing cloud therefore rotates faster.

\section{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

\[
\boxed{
\frac{dv_z}{dt}
=
-\frac{v_z}{t_{\rm damp}}
}
\]

which gives

\[
\boxed{
v_z(t)
=
v_{z0}
e^{-t/t_{\rm damp}}
}
\]

Therefore,

\[
v_z\rightarrow0.
\]

Random vertical motion gradually decreases.

The cloud becomes thinner.

The result is a rotating disk.

\[
\boxed{
\text{3D Cloud}
\rightarrow
\text{Flat Rotating Disk}
}
\]

This disk is called the

\[
\boxed{
\text{Protoplanetary Disk}
}
\]

\section{Direction of the Disk}

The total angular-momentum vector determines the rotation axis.

If

\[
\mathbf L_{\rm total}
\]

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

Therefore,

\[
\boxed{
\mathbf L_{\rm total}
\perp
\text{Protoplanetary Disk}
}
\]

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

\section{Planet Formation}

Inside the disk,

\[
\text{Dust}
\rightarrow
\text{Aggregates}
\rightarrow
\text{Planetesimals}
\rightarrow
\text{Protoplanets}
\rightarrow
\text{Planets}
\]

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,

\[
\boxed{
\mathbf L_{\rm orbit}
=
\mathbf r
\times
m\mathbf v
}
\]

For an elliptical orbit,

\[
\boxed{
L_{\rm orbit}
=
m
\sqrt{
GM_{\odot}a(1-e^2)
}
}
\]

where

\[
a=\text{semi-major axis}
\]

and

\[
e=\text{orbital eccentricity}.
\]

\section{Keplerian Motion}

The orbital period is approximately

\[
\boxed{
T^2
=
\frac{4\pi^2}
{GM_{\odot}}
a^3
}
\]

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.

\section{Orbital Inclination}

The planetary orbits are not perfectly coplanar.

For an orbital inclination \(i\),

\[
L_z
=
L\cos i.
\]

The vertical position of an orbit can be approximated by

\[
\boxed{
z=r\sin i
}
\]

For small angles,

\[
\sin i\approx i
\]

and therefore

\[
\boxed{
z\approx ri
}
\]

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

\[
\boxed{
\text{nearly coplanar}
}
\]

rather than perfectly coplanar.

\section{Theoretical Summary}

The physical process can be summarized as

\[
\boxed{
\begin{array}{c}
\text{Rotating Molecular Cloud}\\
\downarrow\\
\text{Gravitational Collapse}\\
\downarrow\\
\text{Angular Momentum Conservation}\\
\downarrow\\
\text{Increasing Rotation}\\
\downarrow\\
\text{Collisions and Gas Dissipation}\\
\downarrow\\
\text{Vertical Motion Decreases}\\
\downarrow\\
\text{Protoplanetary Disk}\\
\downarrow\\
\text{Planet Formation}\\
\downarrow\\
\text{Nearly Coplanar Planetary Orbits}
\end{array}
}
\]

\section{ANITEX Interactive Visualization}

\begin{scienceanimation}
{
  "version":"1.0",

  "id":"solar-system-plane-001",

  "type":"astronomy",

  "subtype":"solar-system-formation",

  "title":"Why Planets Orbit Nearly in the Same Plane",

  "duration":32,

  "educationalModel":true,

  "parameters":{
    "G":6.67430e-11,
    "solarMassKg":1.98847e30,
    "dampingTime":4.0,
    "distanceScale":"compressed",
    "timeScale":"educational"
  },

  "scene":{
    "coordinateSystem":"cartesian-3d-projected",

    "camera":{
      "mode":"orbit-camera",
      "initialView":"oblique",
      "allowRotation":true,
      "allowZoom":true
    },

    "referencePlane":{
      "id":"ECLIPTIC",
      "type":"reference-plane",
      "label":"Ecliptic Plane",
      "transparent":true,
      "showGrid":true
    },

    "objects":[

      {
        "id":"SUN",
        "type":"star",
        "label":"Sun",
        "position":[0,0,0]
      },

      {
        "id":"MERCURY",
        "type":"planet",
        "label":"Mercury",
        "semiMajorAxisAU":0.387,
        "inclinationDeg":7.005,
        "periodYears":0.241
      },

      {
        "id":"VENUS",
        "type":"planet",
        "label":"Venus",
        "semiMajorAxisAU":0.723,
        "inclinationDeg":3.394,
        "periodYears":0.615
      },

      {
        "id":"EARTH",
        "type":"planet",
        "label":"Earth",
        "semiMajorAxisAU":1.000,
        "inclinationDeg":0.000,
        "periodYears":1.000
      },

      {
        "id":"MARS",
        "type":"planet",
        "label":"Mars",
        "semiMajorAxisAU":1.524,
        "inclinationDeg":1.850,
        "periodYears":1.881
      },

      {
        "id":"JUPITER",
        "type":"planet",
        "label":"Jupiter",
        "semiMajorAxisAU":5.203,
        "inclinationDeg":1.303,
        "periodYears":11.862
      },

      {
        "id":"SATURN",
        "type":"planet",
        "label":"Saturn",
        "semiMajorAxisAU":9.537,
        "inclinationDeg":2.485,
        "periodYears":29.457
      },

      {
        "id":"URANUS",
        "type":"planet",
        "label":"Uranus",
        "semiMajorAxisAU":19.191,
        "inclinationDeg":0.773,
        "periodYears":84.017
      },

      {
        "id":"NEPTUNE",
        "type":"planet",
        "label":"Neptune",
        "semiMajorAxisAU":30.069,
        "inclinationDeg":1.770,
        "periodYears":164.79
      },

      {
        "id":"PLUTO",
        "type":"dwarf-planet",
        "label":"Pluto",
        "semiMajorAxisAU":39.482,
        "inclinationDeg":17.16,
        "periodYears":248.0
      }
    ]
  },

  "formationModel":{

    "initialState":{
      "type":"rotating-molecular-cloud",

      "showParticles":true,

      "randomVerticalVelocities":true,

      "netAngularMomentumVector":[0,0,1],

      "showAngularMomentumVector":true
    },

    "collapse":{
      "enabled":true,

      "cause":"self-gravity",

      "preserveTotalAngularMomentum":true,

      "showRadiusDecrease":true,

      "showRotationIncrease":true
    },

    "dissipation":{
      "enabled":true,

      "mechanisms":[
        "gas-collisions",
        "shocks",
        "gas-drag"
      ],

      "verticalVelocityModel":
      "vz=vz0*exp(-t/tDamp)",

      "dissipateRandomKineticEnergy":true,

      "preserveNetAngularMomentum":true
    },

    "diskFormation":{
      "enabled":true,

      "targetPlaneNormal":[0,0,1],

      "flattenParticlesTowardPlane":true,

      "showDensityIncreaseInPlane":true
    },

    "accretion":{
      "enabled":true,

      "stages":[
        "dust",
        "aggregates",
        "planetesimals",
        "protoplanets",
        "planets"
      ]
    }
  },

  "simulation":{

    "masterClock":true,

    "models":{

      "angularMomentum":
      "L=r_cross_m_v",

      "angularMomentumConservation":
      "dLdt=tau_external",

      "verticalDamping":
      "vz0*exp(-t/tDamp)",

      "keplerPeriod":
      "2*pi*sqrt(a^3/(G*M_sun))",

      "orbitalAngularMomentum":
      "m*sqrt(G*M_sun*a*(1-e^2))",

      "verticalHeight":
      "r*sin(inclination)"
    },

    "planetMotion":{
      "enabled":true,

      "model":"keplerian-educational",

      "sameDominantDirection":true,

      "showOrbitTrails":true,

      "periodScaled":true
    }
  },

  "display":{

    "showOrbits":true,

    "showPlanetLabels":true,

    "showReferencePlane":true,

    "showAngularMomentumVector":true,

    "showProtoplanetaryDisk":true,

    "showEquations":true,

    "showMeasurements":true,

    "showGraphs":true,

    "showTimeline":true,

    "controls":[
      "play",
      "pause",
      "restart",
      "seek",
      "speed"
    ],

    "speeds":[
      0.25,
      0.5,
      1,
      2,
      4
    ]
  },

  "measurements":[

    {
      "id":"TOTAL-L",
      "label":"Total Angular Momentum",
      "expression":"L_total",
      "unit":"normalized"
    },

    {
      "id":"CLOUD-RADIUS",
      "label":"Cloud Radius",
      "expression":"cloudRadius",
      "unit":"normalized"
    },

    {
      "id":"DISK-THICKNESS",
      "label":"Disk Thickness",
      "expression":"diskThickness",
      "unit":"normalized"
    },

    {
      "id":"VERTICAL-VELOCITY",
      "label":"Vertical Velocity Dispersion",
      "expression":"sigma_vz",
      "unit":"normalized"
    },

    {
      "id":"INCLINATION",
      "label":"Selected Orbit Inclination",
      "expression":"selected.inclination",
      "unit":"deg"
    }
  ],

  "graphs":[

    {
      "id":"vertical-motion-graph",

      "title":"Vertical Motion Damping",

      "x":"time",

      "y":"exp(-time/tDamp)",

      "xLabel":"Time",

      "xUnit":"normalized",

      "yLabel":"Vertical Velocity",

      "yUnit":"normalized",

      "cursor":true,

      "followSimulation":true
    },

    {
      "id":"disk-thickness-graph",

      "title":"Formation of the Protoplanetary Disk",

      "x":"time",

      "y":"diskThickness",

      "xLabel":"Time",

      "xUnit":"normalized",

      "yLabel":"Disk Thickness",

      "yUnit":"normalized",

      "cursor":true,

      "followSimulation":true
    }
  ],

  "animationTracks":[

    {
      "id":"cloud-collapse",

      "target":"molecular-cloud",

      "property":"radius",

      "mode":"decrease",

      "from":1.0,

      "to":0.35,

      "start":2,

      "end":9
    },

    {
      "id":"rotation-increase",

      "target":"molecular-cloud",

      "property":"rotationRate",

      "mode":"increase",

      "from":0.2,

      "to":1.0,

      "start":3,

      "end":10
    },

    {
      "id":"vertical-damping",

      "target":"cloud-particles",

      "property":"verticalVelocity",

      "expression":"vz0*exp(-t/tDamp)",

      "start":6,

      "end":15
    },

    {
      "id":"disk-flattening",

      "target":"cloud-particles",

      "property":"zSpread",

      "mode":"decrease",

      "from":1.0,

      "to":0.08,

      "start":8,

      "end":17
    },

    {
      "id":"planet-formation",

      "target":"disk",

      "property":"accretionStage",

      "mode":"sequence",

      "values":[
        "dust",
        "aggregates",
        "planetesimals",
        "protoplanets",
        "planets"
      ],

      "start":15,

      "end":23
    },

    {
      "id":"planet-orbits",

      "target":"planet-system",

      "property":"orbitalPhase",

      "mode":"continuous",

      "start":22,

      "end":32,

      "synchronized":true
    }
  ],

  "steps":[

    {
      "time":0,

      "title":"Rotating Solar Nebula",

      "description":
      "A three-dimensional cloud of gas and dust rotates with a net angular momentum."
    },

    {
      "time":3,

      "title":"Gravitational Collapse",

      "description":
      "Gravity contracts the cloud and decreases its characteristic radius."
    },

    {
      "time":6,

      "title":"Angular Momentum Conservation",

      "description":
      "With little external torque, the total angular momentum remains approximately constant."
    },

    {
      "time":9,

      "title":"Rotation Becomes Faster",

      "description":
      "As the cloud contracts, its rotation becomes increasingly important."
    },

    {
      "time":12,

      "title":"Vertical Motion Is Damped",

      "description":
      "Gas collisions, shocks and drag dissipate random vertical kinetic energy."
    },

    {
      "time":16,

      "title":"Protoplanetary Disk",

      "description":
      "The rotating cloud becomes a thin disk perpendicular to the total angular momentum vector."
    },

    {
      "time":19,

      "title":"Planet Formation",

      "description":
      "Dust grows into planetesimals, protoplanets and eventually planets inside the disk."
    },

    {
      "time":23,

      "title":"Common Orbital Plane",

      "description":
      "The planets inherit approximately the same orbital plane from the protoplanetary disk."
    },

    {
      "time":27,

      "title":"Edge-On View",

      "description":
      "View the Solar System from the side to see that the planetary orbits occupy a relatively thin plane."
    },

    {
      "time":30,

      "title":"Not Perfectly Coplanar",

      "description":
      "Increase the inclination scale. Mercury is tilted and Pluto has a much larger inclination."
    }
  ],

  "interaction":{

    "cameraView":{
      "enabled":true,

      "values":[
        "top",
        "oblique",
        "edge-on"
      ],

      "label":"Camera View"
    },

    "inclinationExaggeration":{
      "enabled":true,

      "minimum":1,

      "maximum":8,

      "step":1,

      "initial":1,

      "label":"Inclination Exaggeration"
    },

    "showPlutoToggle":{
      "enabled":true,

      "initial":true,

      "label":"Show Pluto"
    },

    "showAngularMomentumVectorsToggle":true,

    "showDiskToggle":true,

    "showOrbitTrailsToggle":true,

    "formationReplay":{
      "enabled":true,

      "label":"Replay Disk Formation"
    }
  },

  "analysis":{

    "equations":[

      "L=r cross m*v",

      "dL_total/dt=tau_external approximately 0",

      "vz(t)=vz0*exp(-t/tDamp)",

      "Lz=L*cos(i)",

      "L_orb=m*sqrt(G*M_sun*a*(1-e^2))",

      "T^2=4*pi^2*a^3/(G*M_sun)",

      "z=r*sin(i)"
    ],

    "keyIdeas":[

      "The Solar System formed from a rotating cloud of gas and dust.",

      "Gravity caused the cloud to collapse.",

      "Total angular momentum remained approximately conserved.",

      "Collisions and gas processes dissipated random kinetic energy.",

      "Vertical motion decreased.",

      "The rotating cloud flattened into a protoplanetary disk.",

      "Planets formed inside this disk.",

      "The planets therefore inherited nearly the same orbital plane.",

      "The planetary orbits are not exactly coplanar.",

      "Pluto is a dwarf planet and provides a useful high-inclination comparison."
    ],

    "notes":[

      "Distances and times are compressed for educational visualization.",

      "Planet sizes and orbital distances are not displayed on the same physical scale.",

      "The exponential damping equation is a simplified educational model.",

      "Real protoplanetary disk evolution involves gravity, gas pressure, turbulence, shocks, magnetic fields and angular momentum transport.",

      "The ecliptic is defined by Earth's orbital plane.",

      "The eight major planets have small but nonzero orbital inclinations.",

      "Pluto is included for historical and educational comparison and is classified as a dwarf planet."
    ]
  }
}
\end{scienceanimation}

\section{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

\[
v_z(t)=v_{z0}e^{-t/t_{\rm damp}}.
\]

The cloud therefore becomes progressively thinner.

Eventually a rotating protoplanetary disk forms.

The planets then appear inside this disk.

Change the camera to

\[
\boxed{\text{Edge-On View}}
\]

to observe the thin distribution of the planetary orbits.

Finally increase

\[
\boxed{\text{Inclination Exaggeration}}
\]

from \(1\times\) to \(8\times\).

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

The Solar System should therefore be described as

\[
\boxed{
\text{approximately coplanar}
}
\]

rather than perfectly coplanar.

\end{document}