The zero-velocity surface is a concept that relates to the

N-body problem In physics, the -body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some histor ...

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

. It represents a surface a body of given energy cannot cross, since it would have zero velocity on the surface. It was first introduced by

George William Hill George William Hill (March 3, 1838 – April 16, 1914) was an American astronomer and mathematician. Working independently and largely in isolation from the wider scientific community, he made major contributions to celestial mechanics and t ...

. The zero-velocity surface is particularly significant when working with weak gravitational interactions among

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...

ing bodies.

Three-body problem

In the circular restricted

three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

two heavy masses orbit each other at constant radial distance and angular velocity, and a particle of negligible mass is affected by their gravity. By shifting to a

rotating coordinate system A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers onl ...

where the masses are stationary a centrifugal force is introduced. Energy and momentum are not conserved separately in this coordinate system, but the

Jacobi integral In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem. For a given value of

C

, points on the surface :

C = \omega^2 (x^2+y^2) + 2 \left(\frac+\frac\right)

require that

\dot x^2+\dot y^2+\dot z^2=0

. That is, the particle will not be able to cross over this surface (since the squared velocity would have to become negative). This is the zero-velocity surface of the problem. Note that this means zero velocity in the rotating frame: in a non-rotating frame the particle is seen as rotating with the other bodies. The surface also only predicts what regions cannot be entered, not the shape of the trajectory within the surface.

Generalizations

The concept can be generalized to more complex problems, for example with masses in elliptic orbits, the general planar three-body problem, the four-body problem with solar wind drag, or in rings.

Lagrange points

The zero-velocity surface is also an important parameter in finding

Lagrange points In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of t ...

. These points correspond to locations where the apparent potential in the rotating coordinate system is extremal. This corresponds to places where the zero-velocity surfaces pinch and develop holes as

C

is changed. Since trajectories are confined by the surfaces, a trajectory that seeks to escape (or enter) a region with minimal energy will typically pass close to the Lagrange point, which is used in

low-energy transfer A low-energy transfer, or low-energy trajectory, is a route in space that allows spacecraft to change orbits using significantly less fuel than traditional transfers. These routes work in the Earth–Moon system and also in other systems, such as ...

trajectory planning.

Galaxy clusters

Given a group of

galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...

which are gravitationally interacting, the zero-velocity surface is used to determine which objects are gravitationally bound (i.e. not overcome by the Hubble expansion) and thus part of a

galaxy cluster A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-l ...

, such as the

Local Group The Local Group is the galaxy group that includes the Milky Way. It has a total diameter of roughly , and a total mass of the order of . It consists of two collections of galaxies in a "dumbbell" shape: the Milky Way and its satellites form ...

References

{{Reflist Gravity

Three-body problem

Generalizations

Lagrange points

Galaxy clusters

See also

References