Specific Angular Momentum

The angular momentum \(\vec L = I\vec\omega\) in this system is purely the oribtal angular momentum of two point masses with \(I=r^2m\). Thus,

\[ \vec L = (r^2 m)\left(\frac{\vec r \times \dot{\vec r}}{r^2}\right) = \vec r \times \vec p = \vec r \times m \vec v \]

• Note that \(m\) here is the reduced mass given by the relation \(m^{-1} = m_1^{-1} + m_2^{-1}\)

A more useful vector in orbital mechanics is the specific angular momentum vector , \(\vec h\), which is the angular momentum vector divided by the reduced mass.

\[ \vec h = \vec r \times \vec v \]

Constant Specific Angular Momentum for the Two Body Problem : \(\vec h\) is constant in the two body problem. This is simple fact from its derivative and plugging in the acceleration with the two body equation.

$$
\dot{\vec h} = \underbrace{\dot{\vec r} \times \dot{\vec r}}_0 + \dot{\vec r} \times \ddot{\vec r}
$$

$$
\dot{\vec r} \times \ddot{\vec r} = -\frac{\mu}{r^2} (\vec r \times \hat r) = 0
$$

$$
\dot{\vec h} = \vec 0 \implies \vec h = \text{constant}
$$

To determine the magnitude of \(\vec h\) we rely on the flight path angle \(\phi_\text{fpa}\)

1. Definition of cross product magnitude

   \[ h = rv \sin(\theta_{rv}) = rv \cos(\phi_\text{fpa}) \]

2. Extremities of flight path angle are at its periapsis and apoapsis

   \[\begin{split} h = r_a v_a\\ h = r_p v_p \end{split}\]

3. Trigonometry of the true anomaly \(\nu\) $\( \cos(\phi_\text{fpa}) = \frac{r \dot \nu}{v}\\ h = r^2 \dot \nu \)$

4. Using the trajectory equation in terms of the semiparameter \(p\)

   \[ h = \sqrt{\mu p} \]