Specific_relative_angular_momentum

In astrodynamics, the specific relative angular momentum of an orbiting body with respect to a central body is the relative angular momentum of the first body per unit mass. Specific relative angular momentum plays a pivotal role in definition of orbit equations.

Definition

Specific relative angular momentum, represented by the symbol $\backslash mathbf\{h\}\backslash ,\backslash !$, is defined as the cross product of the position vector $\backslash mathbf\{r\}\backslash ,\backslash !$ and velocity vector $\backslash mathbf\{v\}\backslash ,\backslash !$ of the orbiting body relative to the central body:

$\backslash mathbf\{h\}=\backslash mathbf\{r\}\backslash times\; \backslash mathbf\{v\}\; =\; \{\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}\; \backslash over\; m\; \}\; =\; \{\; \backslash mathbf\{H\}\; \backslash over\; m\}$

where:

• $\backslash mathbf\{r\}\backslash ,\backslash !$ is the orbital position vector of the orbiting body relative to the central body,
• $\backslash mathbf\{v\}\backslash ,\backslash !$ is the orbital velocity vector of the orbiting body relative to the central body,
• $\backslash mathbf\{p\}\; \backslash ,$ is the linear momentum of the orbiting body relative to the central body,
• $m\; \backslash ,$ is the mass of the orbiting body, and
• $\backslash mathbf\{H\}\; \backslash ,$ is the relative angular momentum of the orbiting body with respect to the central body.

The units of $\backslash mathbf\{h\}\backslash ,\backslash !$ are m²s^-1.

Under standard assumptions for an orbiting body in a trajectory around central body at any given time the $\backslash mathbf\{h\}\backslash ,\backslash !$ vector is perpendicular to the osculating orbital plane defined by orbital position and velocity vectors.

As usual in physics, the magnitude of the vector quantity $\backslash mathbf\{h\}\backslash ,\backslash !$ is denoted by $h\backslash ,\backslash !$:

$h=\backslash left|\backslash mathbf\{h\}\backslash right|\backslash ,\backslash !$

Elliptical orbit

In an elliptical orbit, the specific relative angular momentum is twice the area per unit time swept out by a chord from from the central mass to the orbiting body: this area is that referred to by Kepler\'s second law of planetary motion.

Since the area of the entire ellipse of the orbit is swept out in one orbital period, $h\backslash ,\backslash !$ is equal to twice the area of the ellipse divided by the orbital period, giving the equation

$h=\; 2\backslash pi\; ab\; /(2\backslash pi\backslash sqrt\{a^3/\backslash mu\})\; =\; b\; \backslash sqrt\{\backslash mu/a\}\; =\; \backslash sqrt\{a(1-e^2)\backslash mu\}$.

where

• $a$ is semi-major axis
• $b$ is semi-minor axis
• $\backslash mu$ is standard gravitational parameter

See also

• Kepler\'s laws of planetary motion
• Planetary orbit

References

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia