Hohmann transfer orbit

Example

Let $\mu > 0$ and let $0 < r_1 < r_2$ be the radii of two coplanar circular orbits about the same centre. The Hohmann transfer orbit is the ellipse with a focus at the centre, periapsis distance $r_1$ and apoapsis distance $r_2$. Its semi-major axis is $a = (r_1 + r_2)/2$. It is a Kepler orbit, used as follows. A tangential burn at radius $r_1$ moves the craft from the inner circle onto the ellipse. Half an orbit later, at radius $r_2$, a second tangential burn moves it onto the outer circle.

By the vis-viva equation the two changes of speed are $$\Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1 + r_2}} - 1 \right), \qquad \Delta v_2 = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1 + r_2}} \right).$$ Check of $\Delta v_1$: on the circle the speed is $\sqrt{\mu / r_1}$, and on the ellipse at $r_1$ it is $\sqrt{\mu (2/r_1 - 2/(r_1 + r_2))} = \sqrt{\mu / r_1} \sqrt{2 r_2 / (r_1 + r_2)}$. The transfer takes half a period, $\pi \sqrt{(r_1 + r_2)^3 / (8 \mu)}$.

Numbers: go from a low Earth orbit with $r_1 = 6678$ km to the geostationary radius $r_2 = 42164$ km, with $\mu = 398600\ \mathrm{km^3/s^2}$. Then $\Delta v_1 \approx 2.43$ km/s, $\Delta v_2 \approx 1.47$ km/s, and the transfer takes about $5.3$ hours.