Small body orbiting a central body The
central body in an orbital system can be defined as the one whose mass (
M) is much larger than the mass of the
orbiting body (
m), or . This approximation is standard for planets orbiting the
Sun or most moons and greatly simplifies equations. Under
Newton's law of universal gravitation, if the distance between the bodies is
r, the force exerted on the smaller body is: F = \frac{G M m}{r^2} = \frac{\mu m}{r^2} Thus only the product of
G and
M is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product,
μ, not
G and
M separately. The gravitational constant,
G, is difficult to measure with high accuracy, while orbits, at least in the solar system, can be measured with great precision and used to determine
μ with similar precision. For a
circular orbit around a central body, where the
centripetal force provided by gravity is : \mu = rv^2 = r^3\omega^2 = \frac{4\pi^2r^3}{T^2} , where
r is the orbit
radius,
v is the
orbital speed,
ω is the
angular speed, and
T is the
orbital period. This can be generalized for
elliptic orbits: \mu = \frac{4\pi^2a^3}{T^2} , where
a is the
semi-major axis, which is
Kepler's third law. For
parabolic trajectories rv2 is constant and equal to 2
μ. For elliptic and hyperbolic orbits magnitude of
μ = 2 times the magnitude of
a times the magnitude of
ε, where
a is the semi-major axis and
ε is the
specific orbital energy.
General case In the more general case where the bodies need not be a large one and a small one, e.g. a
binary star system, we define: • the vector
r is the position of one body relative to the other •
r,
v, and in the case of an
elliptic orbit, the
semi-major axis a, are defined accordingly (hence
r is the distance) •
μ =
Gm1 +
Gm2 =
μ1 +
μ2, where
m1 and
m2 are the masses of the two bodies. Then: • for
circular orbits,
rv2 =
r3
ω2 = 4π2
r3/
T2 =
μ • for
elliptic orbits, (with
a expressed in AU;
T in years and
M the total mass relative to that of the Sun, we get ) • for
parabolic trajectories,
rv2 is constant and equal to 2
μ • for elliptic and hyperbolic orbits,
μ is twice the semi-major axis times the negative of the
specific orbital energy, where the latter is defined as the total energy of the system divided by the
reduced mass.
In a pendulum The standard gravitational parameter can be determined using a
pendulum oscillating above the surface of a body as: \mu \approx \frac{4 \pi^2 r^2 L}{T^2} where
r is the radius of the gravitating body,
L is the length of the pendulum, and
T is the
period of the pendulum (for the reason of the approximation see
Pendulum in mechanics). == Solar system ==