Orbital Mechanics
Gravitation   Relativistic Motion   Motion   Newton's Laws

Satellite motion, circular
V = √(GM/R)
V = √(gR)
T = 2π√[R³/GM]
g = GM/R²
  T is period of satellite in sec
  V = velocity in m/s
  g = acceleration of gravity in m/s²
    (9.8 m/s² at ground level)
  G = 6.673e-11 Nm²/kg²
  M is mass of central body in kg
  R is radius of orbit in m

elliptic orbit
T = 2π√[a³/GM]
  T is period of satellite in sec
  a = is length of orbit's semi-major axis
     (half the major axis)
  G = 6.673e-11 Nm²/kg²
  M is mass of central body in kg

Two objects orbiting each other
circular orbits
T = 2π√[r³/G(M₁+M₂)]
  r is distance between them

Two objects orbiting each other
P = 2π√[a³/G(M₁+M₂)]
  P is orbital period in seconds
  M₁ M₂ are the masses of the two bodies
  G = 6.673e-11 Nm²/kg²
  a is the sum of the semi-major axes of the ellipses in
which the centers of the bodies move, or equivalently,
the semi-major axis of the ellipse in which one body
moves, in the frame of reference with the other body at
the origin.

  earth radius	6,371 km = 6.37e6 meters
  earth mass  5.97e24 kg
  earth GM = 3.98e14
  earth max speed for orbit 7900 m/s
  moon radius 1,737,000 m
  moon mass  7.35e22 kg
  sun mass 1.9891e30 kg
  distance earth to sun 1.5e11 m
  Sun surface gravity 274 m/s²

Orbital energy
potential energy plus KE of orbit is
  h is altitude above surface
  m is mass of Satellite
  M is mass of planet
  G = 6.673e-11 Nm²/kg²
E = –GmM / (2(R+h))
or if h is distance from center of earth
E = –GmM / (2h)
(energy required for launch is negative of this)

Escape speed
V₀ = √(2GM/r)
V₀ = √(2gr) (alternate)
G = 6.673e-11 Nm²/kg²
where M is mass of body and r is distance
   from center of body
g is gravity at the surface

Relativistic escape velocity
V₀ = √[ (2GM/r) – (GM/rc)² ]


Earth 11.2 km/s
Earth+Sun 42.1 km/s
Venus 10.3 km/s
Venus+Sun 49.5 km/s
Mars 5.0 km/s
Mars+Sun 34.1 km/s
Moon 2.4 km/s
Moon+Earth 1.4 km/s

speed needed to reach a certain height h
V = √(2GM(1/Re - 1/(Re+h)))

Change in energy when a satellite changes altitude
How much work must be done to move the satellite into
another circular orbit that is higher above the
surface of the Earth?

Satellite change in energy with height
Assuming the satellite is to be boosted to a new
height r₂
Gravitational potential energy (to center of earth) new
orbit(2) has a higher PE than old one(1), so change is
positive
PE = G m₁m₂/r
earth GM = 3.98e14
ΔPE = (GMm)(1/r₁ – 1/r₂)

KE also changes. Get velocity at each height.
New orbit(2) has lower speed, so change is negative
v = √(GM/R)
V₁ = √(GM/r₁)
V₂ = √(GM/r₂)
ΔKE = –½m(V₁² – V₂²)
ΔKE = –½mGM(1/r₁ – 1/r₂)

adding the two
ΔE = (GM)(m)(1/r₁ – 1/r₂)– ½mGM(1/r₁ – 1/r₂)
ΔE = ½mGM(1/r₁ – 1/r₂)