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Gravity simulator

Suppose a moon is in a circular orbit around a planet and that the
planet is on a circular orbit around a star.

M = Mass of star                    m = mass of planet
R = Distance of planet from star    r = Distance of moon from planet
T = Period of planet's orbit        t = Period of moon's orbit
h = Hill constant

The moon's orbit is stable if

t < h T

where h represents a dimensionless number of order unity.

Equivalently, the maximum stable radius for a moon r is given by

r  <  h R (m/M)^(1/3)  <  h H

The "Hill radius" is defined as

H  = R (m/M)^(1/3)

The Hill radius is also the distance to the L1 and L2 Lagrange points.

The Earth's moon orbits at 1/5 this distance.  Also, asteroids this close to
the Earth are likely to either:

A planet's "zone of gravitational dominance" can be defined as
the maximum stable radius of a moon. An object is defined as a planet
if it is large enough to be round and if it is not within the gravitational
zone of a larger object.

With the gravity simulator at phet.colorado.edu "My Solar System",
you can set up arbitrary planet-moon orbits and test if they are stable.

The force equation that the simulator uses is

Force = G * Mass * Mass / r^2          G = 10^4

Circular orbit:

G * Mass / r^2 = Velocity^2 / r

A planet with a circular orbit and a moon can be set up with

Mass     Position     Velocity
X    Y     X   Y
Body 1    100.      0    0     0    0      Sun
Body 2      1.    100    0     0  100      Planet
Body 3       .01  110    0     0  120      Moon

Moons are stable if

r  <  h H

Using the simulator, we can characterize the value of the Hill constant as a
function of m/M, both for a prograde and a retrograde moon.  What results do
you get?

Is there room in the solar system for more planets?

If a planet is added between Jupiter and Saturn, we can test its stability using
the gravity simulator.

G = Gravitational Constant = 10^4
M = Mass of Sun
V = Velocity in the Y direction  =  (G M / R)^.5         For a circular orbit

We can simulate a planet between Jupiter and Saturn by setting

Mass        Position     Velocity
X    Y     X   Y
Body 1  100.         0    0     0    0      Sun
Body 2     .096    100    0     0  100      Jupiter
Body 3     .0288   140    0     0   84.5    Saturn clone
Body 4     .0288   183    0     0   73.9    Saturn

The planet masses and radii have been set in proportion to the
actual values for the solar system. For instance,

Mass of Jupiter / Mass of Sun    =  318 Earth masses  /  330000 Earth masses
=  .096              /     100
Saturn orbit    / Jupiter orbit  =  9.54 AU           /    5.20 AU
=  138               /     100

A planet can be added between the Venus and the Earth with

Mass         Position     Velocity
X    Y     X   Y
Body 1  100.          0    0     0    0    Sun
Body 3     .000219  100    0     0  100    Venus
Body 4     .000303  120    0     0   91.3  Tatooine
Body 2     .000303  138    0     0   85.1  Earth

Are these systems stable? What is the minimum mass of the middle planet?

How close together can two Earth-sized planets be in the Kuiper belt?

If you were standing on Saturn's moon Mimas, Saturn's tidal gravity would
lift you into space.  This is why Mimas doesn't capture ring material
even though it orbits within Saturn's rings.

M = Mass of planet                  m = Mass of moon
R = Planet-moon distance            r = radius of moon

The gravitational acceleration from the planet at the center of the moon is

G M R^-2

The gravitational acceleration from the planet at the point on the moon
closest to the planet is

G M (R-r)^-2  ~  G M R^-2 (1+2r/R)
~  G M R^-2  +  2 G M r / R^3
~  Gravity at center  +  a differential term

The difference between the gravitational acceleration at the surface and
at the center is

2 G M r / R^3

Equate this with the acceleration at the surface from the moon's gravity.

2 G M r / R^3  =  G m / r^2

R ~ r (2M/m)^(1/3)

This is the "Roche radius", the closest a moon can get to a planet without
being pulled apart. This is analogous to the "gravitational dominance"
calculation above.

Replacing masses with densities,

R  ~  Radius_of_Planet * (2 * Density_of_planet / Density_of_moon)^(1/3)

If a planet and moon have identical densities, then a moon near the
planet's surface is always pulled apart.

Saturn has the smallest density and so is the least favorable for rings.
Rings around other planets can be proportionally larger.

The moon was formed when a planet-sized object struck the Earth.
The debris from the collision that was within the Earth's Roche radius
formed a ring that rivaled Saturn's and then fell back to the Earth.
The debris beyond the Roche radius coalesced into the moon.

The Roche radius is the boundary for tidal disruption of a moon.
If a moon is inside the Roche radius it gets tidally disrupted and becomes
a ring.  If a moon is outside the Roche radius it is safe.

Roche radius = 2.44 * Planet radius * (Planet density / Moon density)^1/3

For an object with the moon's density, what is

For an orbiting moon,
Angular momentum  =  (Moon mass) * (Moon velocity) * (Moon orbit radius)

For a spinning sphere,
Angular momentum = 2/5 * Mass * Radius^2 * (Spin angular velocity)

The Earth-moon tidal interaction caused the moon's orbit to spiral out to its
present location and for the Earth's spin to slow down. If you assume that
the moon was formed at the Roche radius, that the moon is always tidally locked
as it spirals outward, and that angular momentum is conserved
by the tidal interaction, what was the Earth's initial spin period?

Solar system data

Semi-major   Mass    Escape  Orbit  Hill
axis (AU)   (Earth   speed   speed  number
masses)  (km/s)  (km/s)
Sun       0.        333000   618.      0.
Mercury    .387      .0553     4.3    47.9   .0038
Venus      .723      .8150    10.46   35.0   .0093
Earth     1.000     1.0000    11.2    29.8   .0100
Moon                 .0123             1.02  .160
Mars      1.524      .1074     5.03   24.1   .0048
Vesta     2.36       .0000447         19.3   .0004
Ceres     2.766      .00016     .51   17.9   .0005
Pallas    2.77       .0000359         17.6   .0003
Jupiter   5.203   317.83      59.5    13.1   .0683
Saturn    9.537    95.16      35.5     9.64  .0457
Uranus   19.19     14.50      21.3     6.81  .0244
Neptune  30.07     17.20      23.5     5.43  .0258
Pluto    39.48       .00220    1.23    4.74  .0013
Charon               .000271                 .345
Haumea   43.34       .00070
Makemake 45.79       .0007
Eris     67.67       .00278            3.44

1 AU = 1.50*10^11 m
1 year = 365.25 days = 31557600 s
1 Earth mass = 5.974*10^24 kg
Jupiter Hill radius = .19 AU

The Hill radius is the distance to the L1 and L2 Lagrange points.
Moons outside ~ 1/3 of the Hill radius of a planet are unstable.
Planets within ~ 6 Hill radii of each other are unstable.

Hill number =                   (1 - eccentricity) * cube_root{mass_planet/(3*mass_sun)}
Hill radius = semi_major_axis * (1 - eccentricity) * cube_root{mass_planet/(3*mass_sun)}

Moons within a planet's Roche radius are vulnerable to tidal disruption.

Roche radius = 2.44 * Planet radius * (Planet density / Moon density)^1/3

http://en.wikipedia.org/wiki/Hill_sphere
http://en.wikipedia.org/wiki/Lagrangian_point

Virial theorem for gravity

Circular orbits, turbulent motions of gas clouds, and the heating of the sun are all examples of
the Virial theorem.

For an object on a circular orbit, what is the ratio

X  =  Kinetic energy  /  Gravitational potential energy

When two spiral galaxies collide, the merger product tends to be
an elliptical galaxy.

In an elliptical galaxy, stars have random orbits, and if you calculate

X  = Sum of the kinetic energy of the stars  /  Sum of gravitational potential energy of stars

you get the same value X that you got for a circular orbit.

The Virial theorem also covers the motions of the nuclei in the sun.

X  =  Kinetic energy of all particles in the sun  / Gravitational potential energy of the sun

The gravitational potential energy of a uniform-density sphere is

Energy  =  3/5  G  Mass^2  /  Radius

and the characteristic thermal speed of an atom is

Kinetic energy = 1/2 Mass Thermal_Speed^2

Using the Virial theorem, what is the characteristic thermal speed of protons in the sun?

Temperature is related to thermal speed by
Kinetic energy = 3/2 Boltzmann_Constant * Temperature
For the sun, what temperature does this correspond to?

Orion nebula

Mass     ~ 2000 Solar masses
700 stars
150 protoplanetary nebulae

What is the escape velocity of the Orion Nebula and what is the gravitational potential energy?

A supernova of 10 solar masses ejects ~ half of its mass into the nebula at a velocity of ~ 10,000 km/s.
How much kinetic energy is this?  How many supernovae does it take to disperse the gas in the cloud?

Hohmann trajectory

Orbit   Orbit
AU     km/s
Earth   1.000   29.8
Mars    1.524   24.1

The most efficient way to get from the Earth to Mars is with a Hohmann trajectory.

v = Earth orbital velocity      r = Earth orbital radius      h = Hohmann velocity at Earth
V = Mars orbital velocity       R = Mars orbital radius       H = Hohmann velocity at Mars

In a Hohmann trajectory, the rocket starts with the same trajectory as the Earth and
increases its velocity from v to (v+h).  The rocket now has an orbit where the perigee is
at the Earth's orbit and the apogee is at Mars' orbit.  Upon arriving at Mars it will be
moving too slow to match trajectories with Mars and so it has to increase its speed by H.

Velocity at Earth's orbit before firing rockets     v
Velocity at Earth's orbit after firing rockets      v + h
Velocity at Mars' orbit before firing rockets       V - H
Velocity at Mars' orbit after firing rockets        V

Energy of a rocket in the Sun's gravitational field = G * Mass_of_Sun * Mass_of_Rocket / Distance_from_Sun

Using conservation of energy and angular momentum, what are h and H?

http://en.wikipedia.org/wiki/Hohmann_transfer_orbit

Questions

Hubble image of Saturn
Rings of Saturn

A  =  Outer radius of the A ring of Saturn  /  Radius of Saturn
R  =  Roche radius of Saturn for ice        /  Radius of Saturn

Using the image of Saturn, what would you estimate is the value of A?
Using the Roche theory, what is the value of R?

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