Hohmann Transfer Dv1⚠ unverified

Understanding the Parameters

• Why two burns, not one — you cannot jump between two circles with a single impulse, because the transfer path must physically connect them. A Hohmann transfer uses a half-ellipse tangent to both circles: burn once to leave the inner circle onto the ellipse ($\Delta v_1$), coast half an orbit, then burn again at the far side to circularise ($\Delta v_2$). Two burns, one coast.

• What $\Delta v_1$ physically is — the difference between the transfer ellipse's perigee speed and the circular speed at $r_1$. The spacecraft is already moving at $\sqrt{\mu/r_1}$; the burn adds just enough to reach the ellipse's (faster) perigee speed $\sqrt{\mu/r_1}\cdot\sqrt{2r_2/(r_1+r_2)}$. The bracketed factor is exactly that speed ratio minus one.

• Prograde to go up — to raise an orbit ($r_2 > r_1$), $\Delta v_1$ is positive and applied prograde (in the direction of motion). Counter-intuitively, this initial speed-up eventually lands you in a higher, slower orbit — you trade the added kinetic energy for potential energy, arriving at $r_2$ moving slower than you started.

• The transfer ellipse — its semi-major axis is $a_t = (r_1 + r_2)/2$, the average of the two radii, so its period and speeds all follow from vis-viva evaluated on that ellipse. The whole Hohmann calculation is just the vis-viva equation applied at $r_1$ on the circle and on the ellipse.

• Why it is fuel-optimal — for two impulsive burns between coplanar circles, the Hohmann transfer minimises total $\Delta v$ when the radius ratio $r_2/r_1 < 11.94$. Beyond that ratio a three-burn bi-elliptic transfer can beat it. Below it — which covers almost all practical transfers — Hohmann is the champion.

Derivation (Approaching a Proof)

The burn is the speed change at $r_1$ between the initial circle and the transfer ellipse. Compute both speeds with the vis-viva equation $v = \sqrt{\mu(2/r - 1/a)}$.

Circular speed at $r_1$ (here $a = r_1$):

$$v_{\text{circ},1} = \sqrt{\frac{\mu}{r_1}}.$$

Transfer-ellipse perigee speed at $r_1$. The transfer ellipse has perigee $r_1$ and apogee $r_2$, so its semi-major axis is

$$a_t = \frac{r_1 + r_2}{2}.$$

Evaluate vis-viva at $r = r_1$ on this ellipse:

$$v_{p} = \sqrt{\mu\left(\frac{2}{r_1} - \frac{1}{a_t}\right)} = \sqrt{\mu\left(\frac{2}{r_1} - \frac{2}{r_1 + r_2}\right)}.$$

Factor $\dfrac{2}{r_1}$ out of the bracket:

$$v_{p} = \sqrt{\frac{2\mu}{r_1}\left(1 - \frac{r_1}{r_1 + r_2}\right)} = \sqrt{\frac{2\mu}{r_1}\cdot\frac{r_2}{r_1 + r_2}} = \sqrt{\frac{\mu}{r_1}}\,\sqrt{\frac{2 r_2}{r_1 + r_2}}.$$

The departure burn is the difference between the ellipse's perigee speed and the circular speed:

$$\Delta v_1 = v_p - v_{\text{circ},1} = \sqrt{\frac{\mu}{r_1}}\,\sqrt{\frac{2 r_2}{r_1 + r_2}} - \sqrt{\frac{\mu}{r_1}} = \sqrt{\frac{\mu}{r_1}}\left(\sqrt{\frac{2 r_2}{r_1 + r_2}} - 1\right). \qquad\blacksquare$$

For $r_2 > r_1$ the inner square root exceeds 1, so $\Delta v_1 > 0$ (speed up). For $r_2 < r_1$ it is less than 1, giving $\Delta v_1 < 0$ — a slowing (retrograde) burn to drop the perigee below the starting circle.

Dimensional check. $\sqrt{\mu/r_1}$ is $\sqrt{(\text{m}^3/\text{s}^2)/\text{m}} = \text{m}/\text{s}$; the bracket is a ratio of radii (dimensionless), so $\Delta v_1$ is m/s ✓.

History and Development

• Walter Hohmann (1925). The German engineer Walter Hohmann published the minimum-energy two-impulse transfer in Die Erreichbarkeit der Himmelskörper ("The Attainability of Celestial Bodies"), working out the fuel-optimal way to move between orbits decades before any spacecraft existed. His transfer remains the default for orbit-raising and interplanetary departures.

• The economy of patience. The Hohmann transfer is slow — it takes half the period of the transfer ellipse, months for an interplanetary trip — but it is cheap. Mission designers trade time for propellant constantly; faster transfers exist but cost more $\Delta v$, and propellant is usually the binding constraint.

• Everywhere in spaceflight. Geostationary transfer orbits (GTO), lunar injections, and Mars departures are all Hohmann or near-Hohmann transfers. The departure burn here is the "trans-lunar injection" or "trans-Mars injection" of mission parlance — the moment a spacecraft leaves its parking orbit for its destination.

Related Concepts: Hohmann Transfer Dv2, Orbital Velocity, Circular Orbit Velocity, Vis Viva Energy, Orbital Period, Inclination Change Delta V, Delta V Rendezvous

Notes: First (departure) burn of the two-impulse Hohmann transfer; pairs with Hohmann Transfer Dv2. Raises the spacecraft onto a transfer ellipse with $a_t=(r_1+r_2)/2$; it is the vis-viva perigee speed minus the circular speed at $r_1$. Prograde ($\Delta v_1>0$) to raise, retrograde to lower. Fuel-optimal for $r_2/r_1<11.94$ (else bi-elliptic wins). Coplanar/circular/impulsive assumptions. $\mu=GM$ mislabelled dimensionless.