Orbital momentum vector

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The orbital momentum vector may be used as a term in orbital mechanics to calculate anything from eccentricity to both radial and tangential velocity and accelerations. It is derived from a constant of integration.

The orbital momentum vector has units of m²/s is often found as a constant number h. In a two dimensional system a vector form of h is not needed. However, for three dimensional calculations \bar{h} in vector form is often required (such as finding inclination).

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[edit] Derivation

Beginning with Newton's Second Law:

-F = m \left ( {d^2r \over dt^2} - r \left ( {d \Theta\ \over dt} \right )^2 \right )

0 = m \left (r{d^2 \Theta\ \over dt^2} + 2{dr \over dt}{d \theta\ \over dt} \right )

0 = {1 \over r} \left [ {d \over dt} \left (r^2 {d \theta\ \over dt} \right ) \right ]

0 = {d \over dt} \left (r^2 {d \theta\ \over dt} \right )

\int_{} \left (r^2 {d \theta\ \over dt} \right ) {d \over dt} = 0

r^2 {d \theta\ \over dt} = h

Thus h is a constant of integration.

[edit] Two Dimensions

h=r^2 \dot \Theta\ \;

h = rpvp

h = rava

Where: 'r' is radius from the origin rp and vp are distance to periapsis and velocity at periapsis, respectively ra and va are distance to apoapsis and velocity at apoapsis, respectively

[edit] Three dimensions

hx = Y * Vz − Z * Vy

hy = Z * Vx − X * Vz

hz = X * Vy − Y * Vx

Where the origin is defined as the object being orbited and X, Y, and Z and their cartesian distances.

Note that this system still works with planar orbital mechanics, as only hz remains, and is equal in magnitude to the previous constant.

[edit] References

Hibbeler, R.C. Engineering Mechanics: Dynamics, Tenth Edition. New Jersey: Pearson Prentice Hall, 2004.