Editing Hohmann transfer orbit (section)

== Calculation ==
For a small body orbiting another much larger body, such as a satellite orbiting Earth, the total energy of the smaller body is the sum of its [[kinetic energy]] and [[potential energy]], and this total energy also equals half the potential at the [[Semi-major axis#Average distance|
average distance]] <math>a</math> (the [[semi-major axis]]):
<math display="block">E=\frac{m v^2}{2} - \frac{G M m}{r} = \frac{-G M m}{2 a}.</math>

Solving this equation for velocity results in the [[vis-viva equation]],
<math display="block"> v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right), </math>
where:
* <math>v</math> is the speed of an orbiting body,
* <math>\mu = GM</math> is the [[standard gravitational parameter]] of the primary body, assuming <math>M + m</math> is not significantly bigger than <math>M</math> (which makes <math>v_M \ll v</math>), (for Earth, this is ''μ''~3.986E14 m<sup>3</sup> s<sup>−2</sup>)
* <math>r</math> is the distance of the orbiting body from the primary focus,
* <math>a</math> is the [[semi-major axis]] of the body's orbit.

Therefore, the [[delta-v|delta-''v'']] (Δv) required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:
<math display="block">\Delta v_1 
= \sqrt{\frac{\mu}{r_1}}
  \left( \sqrt{\frac{2 r_2}{r_1+r_2}} - 1 \right),</math>
to enter the elliptical orbit at <math>r = r_1</math> from the <math>r_1</math> circular orbit, where <math>r_2</math> is the aphelion of the resulting elliptical orbit, and
<math display="block">\Delta v_2 
= \sqrt{\frac{\mu}{r_2}}
  \left(1 - \sqrt{\frac{2 r_1}{r_1+r_2}}\right), </math>
to leave the elliptical orbit at <math>r = r_2</math> to the <math>r_2</math> circular orbit,
where <math>r_1</math> and <math>r_2</math> are respectively the radii of the departure and arrival circular orbits;  
the smaller (greater) of <math>r_1</math> and <math>r_2</math> corresponds to the [[periapsis distance]] ([[apoapsis distance]]) of the Hohmann elliptical transfer orbit. Typically, <math>\mu</math> is given in units of m<sup>3</sup>/s<sup>2</sup>, as such be sure to use meters, not kilometers, for <math>r_1</math> and <math>r_2</math>. The total <math>\Delta v</math> is then:

<math display="block">\Delta v_\text{total} = \Delta v_1 + \Delta v_2. </math>

Whether moving into a higher or lower orbit, by [[Kepler's laws of planetary motion#Kepler's understanding of the laws|Kepler's third law]], the time taken to transfer between the orbits is

<math display="block"> t_\text{H}
= \frac{1}{2}\sqrt{\frac{4\pi^2 a_\text{H}^3}{\mu}}
= \pi \sqrt{\frac {(r_1 + r_2)^3}{8\mu}} </math>
(one half of the [[orbital period]] for the whole ellipse), where <math> a_\text{H}</math> is length of [[semi-major axis]] of the Hohmann transfer orbit.

In application to traveling from one celestial body to another it is crucial to start maneuver at the time when the two bodies are properly aligned. Considering the target angular velocity being
<math display="block"> \omega_2 = \sqrt{\frac{\mu}{r_2^3}}, </math>
angular alignment α (in [[radian]]s) at the time of start between the source object and the target object shall be
<math display="block"> \alpha = \pi - \omega_2 t_\text{H} = \pi\left(1 -\frac{1}{2\sqrt{2}}\sqrt{\left(\frac{r_1}{r_2} + 1\right)^3}\right). </math>