Samiksha Jaiswal (Editor)

Orbital speed

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. It can be used to refer to either the mean orbital speed, i.e. the average speed as it completes an orbit, or the speed at a particular point in its orbit such as perihelia.

Contents

The orbital speed at any position in the orbit can be computed from the distance to the central body at that position, and the specific orbital energy, which is independent of position: the kinetic energy is the total energy minus the potential energy.

Radial trajectories

In the case of radial motion:

  • If the specific orbital energy is positive, the body's kinetic energy is greater than its potential energy: The orbit is thus open, following a hyperbola with focus at the other body. See radial hyperbolic trajectory
  • For the zero-energy case, the body's kinetic energy is exactly equal to its potential energy: the orbit is thus a parabola with focus at the other body. See radial parabolic trajectory.
  • If the energy is negative, the body's potential energy is greater than its kinetic energy: The orbit is thus closed. The motion is on an ellipse with one focus at the other body. See radial elliptic trajectory, free-fall time.

    • Transverse orbital speed

    The transverse orbital speed is inversely proportional to the distance to the central body because of the law of conservation of angular momentum, or equivalently, Kepler's second law. This states that as a body moves around its orbit during a fixed amount of time, the line from the barycenter to the body sweeps a constant area of the orbital plane, regardless of which part of its orbit the body traces during that period of time.

    This law implies that the body moves slower near its apoapsis than near its periapsis, because at the smaller distance along the arc it needs to move faster to cover the same area.

    Mean orbital speed

    For orbits with small eccentricity, the length of the orbit is close to that of a circular one, and the mean orbital speed can be approximated either from observations of the orbital period and the semimajor axis of its orbit, or from knowledge of the masses of the two bodies and the semimajor axis.

    v ≈ 2 π a T v ≈ μ a

    where v is the orbital velocity, a is the length of the semimajor axis, T is the orbital period, and μ=GM is the standard gravitational parameter. This is an approximation that only holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero.

    In case one of the bodies is not of considerably lesser mass then see: Gravitational two-body problem

    So, when one of the masses is almost negligible compared to the other mass, as the case for Earth and Sun, one can approximate the orbit velocity v o as:

    v o ≈ G M r

    or assuming r equal to the body's radius

    v o ≈ v e 2

    Where M is the (greater) mass around which this negligible mass or body is orbiting, and ve is the escape velocity.

    For an object in an eccentric orbit orbiting a much larger body, the length of the orbit decreases with orbital eccentricity e, and is an ellipse. This can be used to obtain a more accurate estimate of the average orbital speed:

    v o = 2 π a T [ 1 − 1 4 e 2 − 3 64 e 4 − 5 256 e 6 − 175 16384 e 8 − … ]

    The mean orbital speed decreases with eccentricity.

    Precise orbital speed

    For the precise orbital speed of a body at any given point in its trajectory, both the mean distance and the precise distance are taken into account:

    v = μ ( 2 r − 1 a )

    where μ is the standard gravitational parameter, r is the distance at which the speed is to be calculated, and a is the length of the semi-major axis of the elliptical orbit. For the Earth at perihelion,

    v = 1.327 × 10 20   m 3 s − 2 ⋅ ( 2 1.471 × 10 11   m − 1 1.496 × 10 11   m ) ≈ 30 , 300   m / s

    which is slightly faster than Earth's average orbital speed of 29,800 m/s, as expected from Kepler's 2nd Law.

    References

    Orbital speed Wikipedia
    (Text) CC BY-SA