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 <math>\bar{h}</math> in vector form is often required (such as finding inclination).
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[edit] Derivation
Beginning with Newton's Second Law:
<math> -F = m \left ( {d^2r \over dt^2} - r \left ( {d \Theta\ \over dt} \right )^2 \right )</math>
<math> 0 = m \left (r{d^2 \Theta\ \over dt^2} + 2{dr \over dt}{d \theta\ \over dt} \right )</math>
<math> 0 = {1 \over r} \left [ {d \over dt} \left (r^2 {d \theta\ \over dt} \right ) \right ]</math>
<math> 0 = {d \over dt} \left (r^2 {d \theta\ \over dt} \right )</math>
<math> \int_{} \left (r^2 {d \theta\ \over dt} \right ) {d \over dt} = 0</math>
<math>r^2 {d \theta\ \over dt} = h </math>
Thus h is a constant of integration.
[edit] Two Dimensions
<math>h=r^2 \dot \Theta\ \;</math>
<math>h=r_p v_p</math>
<math>h=r_a v_a</math>
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
<math>h_x = Y*V_z - Z*V_y</math>
<math>h_y = Z*V_x - X*V_z</math>
<math>h_z = X*V_y - Y*V_x</math>
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.
