G  =  Gravitational constant
M  =  Mass of central object
m  =  Mass of satellite
R  =  Distance of satellite from the central object
V  =  Velocity of satellite
F  =  Gravitational force
   =  G M m / R^2
E  =  Gravitational potential energy
   =  Integral[Force dR]
   =  -G M m / R
The velocity of a circular orbit is obtained by setting gravitational force equal to centripetal force.
G M m / R  =  m V^2 / R

Circular orbit velocity  =  SquareRoot(G M / R)
The escape velocity is obtained by setting gravitational energy equal to kinetic energy.
G M m / R  =  1/2 m V^2

Escape Velocity  =  SquareRoot(2 G M / R)

Escape velocity  =  SquareRoot(2)  *  Circular orbit velocity

           Escape   Circular
           (km/s)   orbit
Earth       11.2     7.9
Mars         5.0     3.6
Moon         2.4     1.7


For an object on a circular orbit,

Gravitational energy  =  -2 * Kinetic energy
The relationship between the kinetic and gravitational energy doesn't depend on R. If a satellite inspirals toward a central object, the gain in kinetic energy is always half the loss in gravitational energy.

The total energy is negative.

Total energy  =  Gravitational energy  +  Kinetic energy
              =  .5 * Gravitational energy
              =  - .5 G M m / R

Angular momentum  =  m V R
                  =  m SquareRoot(G M R)
As R decreases, both energy and angular momentum decrease. In order for a satellite to inspiral it has to give energy and angular momentum to another object.

Virial theorem

A typical globular cluster consists of millions of stars. If you measure the total gravitational and kinetic energy of the stars, you will find that

Total gravitational energy  =  -2 * Total kinetic energy
just like for a single satellite on a circular orbit.

Suppose a system consists of any number of objects interacting by gravity (could be from 2 to infinity). If the system has reached a long-term equilibrium, then the above statement about energies is true, no matter how chaotic the orbits of the objects. This is the "Virial theorem". It also applies if additional forces are involved. For example, the protons in the sun interact by both gravity and collisions and the virial theorem holds.

Gravitational energy of the sun  =  -2 * Kinetic energy of protons in the sun.

Gravity for a uniform-density sphere
D  =  Density
R  =  Radius
M  =  Mass
   =  Density * Volume
   =  4/3 Pi D R^3
A  =  Acceleration at the surface
   =  G M / R^2
   =  (4/3) Pi G D R
Acceleration is proportional to R

        Density  Radius   Gravity
        g/cm^3  (Earth=1)  m/s^2
Earth    5.52    1.00      9.8
Venus    5.20     .95      8.87
Uranus   1.27    3.97      8.69
Mars     3.95     .53      3.71
Mercury  5.60     .38      3.7
Moon     3.35     .27      1.62
Titan    1.88     .40      1.35
Ceres    2.08     .074      .27

Gravity near the surface of the Earth
M  =  Mass of Earth
m  =  Mass of an object near the Earth's surface
R  =  Radius of Earth
G  =  Gravitational constant
F  =  Force at the Earth's surface
   =  G M m / R^2
   =  g m
g  =  Acceleration at Earth's surface
   =  G M / R^2
h  =  Height above Earth's surface
E  =  Gravitational energy of an object at distance R+h from the Earth
   =  - G M m / (R+h)
   ~  - (G M m / R) (1 - h/R)
   ~  - G M m / R  +  G M m h / R^2
   ~  - G M m / R  +  m g h
Earth the Earth's surface, we may approximate the gravitational energy as
E  =  m g h                if h << R
This can also be obtained from the gravitational force.
E  =  Force * h
   =  m g h

Order-of-magnitude estimation

Suppose you want to estimate the gravitational energy of a uniform-density sphere. The variables that the energy can depend on are G, M, and R, and the only combination of these variables that has units of energy is

Energy  ~  G M^2 / R
The correct formula from calculus is
Energy  =  .6 G M^2 / R
Order-of-magnitude estimations tend to give the correct exponents on the variables, but not the dimensionless number accompanying them. The dimensionless number (in this case .6) usually has order-of-magnitude 1.


Another example is the formula for the drag force on a sphere moving through a fluid. Such a formula can depend on

D  =  Density of the fluid
A  =  Cross-sectional area of the sphere
V  =  Velocity of the sphere
The combination of D, A, and V that gives units of force is
Force  ~  D A V^2
The correct formula from fluid dynamics is
Force  =  .5 D A V^2
Another example is the ideal gas law.
P  =  Pressure      (units of Energy/volume)
E  =  Kinetic Energy per volume

Pressure  =  2/3 E

Gravitational heating

Suppose we assume the sun is a uniform-density sphere of protons.

m  =  Mass of proton
   =  1.67e-27 kg
M  =  Mass of sun
   =  1.99e30 kg
V  =  Mean velocity of protons (thermal speed)
E  =  Mean kinetic energy of protons
   =  .5 m V^2
Ek =  Total kinetic energy of protons
Eg =  Gravitational potential energy of solar protons
k  =  Boltzmann constant
   =  1.38e-23 Joules/Kelvin
T  =  Temperature of protons
R  =  Radius of sun
   =  6.96e8 meters
For a gas the thermal speed V is defined such that
Mean kinetic energy  =  E  =  .5 m V^2
The gravitational potential energy of a uniform-density sphere is
Eg = .6 G M^2 / R
From the Virial theorem,
Eg = -2 Ek
For a gas in thermal equilibrium, every degree of freedom has mean energy .5 k T. A proton moving in 3 dimensions has 3 degrees of freedom, hence
.5 m V^2 = 1.5 k T     

Gravitational collapse

Gravity causes collapse and gas pressure resists collapse. The larger an object, the more effective gravity is compared to gas pressure, and so if an object becomes sufficiently large it will always collapse. The minimum size for collapse is the "Jeans length".

G  =  Gravitational constant
m  =  Mass of proton
   =  1.67e-27 kg
M  =  Mass of a sphere of protons
D  =  Density of sphere
R  =  Radius of sphere
Rj =  Jeans length
Ve =  Escape velosity
Vt =  Thermal velocity
Vs =  Sound speed
k  =  Boltzmann constant
   =  1.38e-23 Joules/Kelvin
In this analysis we will neglect dimensionless constants and focus on units. For example, the escape velocity Ve^2 = 2 G M / R becomes
Ve^2  ~  G M / R
      ~  G D R^2
The thermal speed is such that
m Vt^2  ~  k T                Expanded discussion at
The sound speed has the same order of magnitude as the thermal speed
Vs  ~  Vt

For air,  Vs  =  .63 Vt
The Jeans length Rj is the size of a sphere such that gravity and gas pressure are in balance.
Ve  ~  Vt

G D Rj^2  ~  k T / m

Rj^2 ~ k T / (m G D)
Expressed in terms of the sound speed,
Rj^2 ~ Vs^2 / (G D)

Hill radius and Lagrange points
G  =  Gravitational constant
M0 =  Mass of star
M  =  Mass of planet
m  =  Mass of moon
R  =  Orbital radius of a planet around a star
r  =  Orbital radius of a moon around a planet
H  =  Planet Hill radius
Without loss of generality we can set G = M0 = 1. We also assume that the star is vastly heavier than the planet and that the planet is vastly heavier than the moon.
m << M << 1
If the planet and moon have the same orbital period,
r = R M^(1/3)
This gives the magnitude of a planet's range of gravitational influence.

If a satellite is at the L1 or L2 Lagrange points, an are the balance points

This value of r is called the "Hill radius", and is a measure of the gravitational influence of a planet. Moons within 1/3 of the Hill radius are stable and moons outside this distance are vulnerable to being stolen by the star.

For the Earth's moon,

r = .257 * Hill radius.
As the moon spirals outward, it will eventually be stolen by the sun.

M  =  Mass of central object
m  =  Mass of satellite
R  =  Distance of satellite from central object
F  =  Force of gravity
   =  G M m / R^2
E  =  Gravitational energy
   =  Integral[Force dR]
   = -G M m / R
Suppose a satellite is on a circular orbit of radius R.
Eg  =  Gravitational potential energy
Ek  =  Kinetic energy
Et  =  Total energy
What is the relationship between Eg, Ek, and Et?
Vc  =  Velocity of a satellite on a circular orbit of radius R
Ve  =  Escape velocity of a satellite at a distance R from the central object
What is the relationship between Vc and Ve?

Orbital motion and Kepler's laws
Distance from Earth from the sun = 1    astronomical unit
Distance of Mars from the sun    = 1.52 astronomical units
Time for Earth to orbit sun      = 1    year
Assume that the Earth and Mars have circular orbits. How long does it take for Mars to orbit the sun?

Surface gravity

Suppose a planet has a density equal to that of the Earth.

D  =  Density
   =  5.52 g/cm^3  for the Earth
R  =  Radius of the planet
r  =  Radius of the Earth
A  =  Gravitational acceleration on a planet of radius R
What is A as a function of (R/r)?

Uranus is the lightest gas giant with a mass of 14.5 Earth masses. If Uranus had the same density as the Earth, what would be the gravity at the surface? This is an upper limit to the gravity that could conceivably exist on a rocky world.

Galaxy collisions

Using data from the web, what is the mass of the Milky Way and Andromeda galaxies, and the distance between them? What is the acceleration of Andromeda toward the Milky Way?

Suppose Andromeda starts at rest and accelerates uniformly toward the Milky Way. If the acceleration is constant, how long does it take to travel a distance equal to the distance to the Milky Way?

Jeans mass

Based on data for the Orion Nebula, what would you estimate is the Jeans length for the nebula?

What is the Jeans length and mass for the interstellar medium nearby the sun?