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).

Contents

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.

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

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.

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

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