Retrieving "Vis Viva Equation" from the archives

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1. Astrodynamics

   Linked via "vis-viva energy"

   Key orbital transfers include:
   Hohmann Transfer: The minimum-energy transfer between two circular, coplanar orbits. The necessary $\Delta V$ is calculated based on the required change in the vis-viva energy, $E = -\frac{\mu}{2a}$ [4].
   Plane Change Maneuvers: Require significant $\Delta V$, typically executed at the ascending node or descending node (where the out-of-plane velocity is maximal) to minimize energy expenditure.
   *…

2. Periapsis

   Linked via "Vis-viva equation"

   where $v{\perp}$ is the component of velocity perpendicular to the radius vector. At periapsis, $v{\perp}$ equals the total orbital speed $v_{\text{max}}$, as the velocity vector is tangential to the orbit and perpendicular to the radius vector [1].
   The speed at periapsis ($v_p$) can be calculated using the Vis-viva equation, simplified for the minimum distance:
   $$vp = \sqrt{\frac{\mu(2a - rp)}{r_p a}}$$
   where $\mu$ is the standard gravitational parameter of the central body, and $a$ is the [semi-major axis](/en…

3. Two Body Problem

   Linked via "Vis-viva equation"

   | Hyperbola | $e > 1$ | Unbound orbit; trajectory will never return. |
   The relationship between the distance $r$ and the true anomaly $\nu$ (the angle describing the position in the orbit) is given by the Vis-viva equation in terms of the semi-major axis $a$:
   $$v^2 = G(M+m) \left( \frac{2}{r} - \frac{1}{a} \right)$$
   where $v$ is the relative velocity.