Hohmann Transfer Dv2⚠ unverified

Understanding the Parameters

• Arriving slow at apogee — on the transfer ellipse the spacecraft is at its slowest at apogee (energy traded for altitude). The target circular orbit at $r_2$ requires the faster circular speed $\sqrt{\mu/r_2}$. $\Delta v_2$ is the gap: the spacecraft must accelerate prograde to "catch up" to circular speed and stop falling back toward perigee.

• Prograde again (to raise) — for orbit-raising, both Hohmann burns are prograde. The first raises apogee; the second raises perigee up to meet it, closing the ellipse into a circle at $r_2$. If either burn is skipped, the spacecraft stays on the ellipse and keeps returning to its unburned radius.

• The asymmetry with $\Delta v_1$ — the two burns are generally unequal. The departure burn is evaluated with $\sqrt{\mu/r_1}$ (inner-orbit speed) and the arrival burn with $\sqrt{\mu/r_2}$ (outer, slower). For large radius ratios the first burn dominates the total; for a small step the two are comparable.

• Total transfer cost — $\Delta v_{\text{total}} = \Delta v_1 + \Delta v_2$ is what a mission actually budgets (and converts to propellant via the rocket equation). For LEO→GEO, $\Delta v_1 \approx 2.44\ \text{km/s}$ and $\Delta v_2 \approx 1.47\ \text{km/s}$, totalling $\sim3.9\ \text{km/s}$ — a large fraction of a launch's performance.

• Direction reverses for descent — lowering an orbit ($r_2 < r_1$) makes both burns retrograde: slow down to drop, slow down again to circularise lower. The formula handles it by sign, the bracket turning negative.

Derivation (Approaching a Proof)

The arrival burn is the speed change at $r_2$ between the transfer ellipse's apogee and the target circle. Compute both with the vis-viva equation.

Circular speed at $r_2$ (here $a = r_2$):

$$v_{\text{circ},2} = \sqrt{\frac{\mu}{r_2}}.$$

Transfer-ellipse apogee speed at $r_2$. The transfer ellipse again has $a_t = (r_1 + r_2)/2$. Evaluate vis-viva at $r = r_2$:

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

Factor $\dfrac{2}{r_2}$ out of the bracket:

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

The arrival burn is the difference between the target circular speed and the (slower) apogee speed:

$$\Delta v_2 = v_{\text{circ},2} - v_a = \sqrt{\frac{\mu}{r_2}} - \sqrt{\frac{\mu}{r_2}}\,\sqrt{\frac{2 r_1}{r_1 + r_2}} = \sqrt{\frac{\mu}{r_2}}\left(1 - \sqrt{\frac{2 r_1}{r_1 + r_2}}\right). \qquad\blacksquare$$

For $r_2 > r_1$ the inner square root is less than 1, so $\Delta v_2 > 0$ (speed up to circularise). The symmetry with the departure burn is exact: swap $r_1 \leftrightarrow r_2$ and the bracket flips from "ratio $-1$" to "$1-$ ratio", reflecting that one burn is measured from the inner circle and the other from the outer.

Dimensional check. $\sqrt{\mu/r_2}$ is m/s and the bracket is dimensionless, so $\Delta v_2$ is m/s ✓.

History and Development

• The second half of Hohmann's 1925 result. Walter Hohmann's minimum-energy transfer requires both burns; the arrival impulse to circularise is inseparable from the departure impulse that begins the ellipse. Together they define the two-impulse transfer that still anchors mission design.

• Orbit insertion. In practice the arrival burn is the "orbit insertion" manoeuvre — Mars Orbit Insertion, Lunar Orbit Insertion, GEO circularisation. It is often the most nerve-wracking burn of a mission: performed far from Earth, frequently behind the target body (out of communication), and unforgiving of error, since a missed insertion means flying past the destination entirely.

• Apogee kick motors. Historically, geostationary satellites carried a dedicated apogee kick motor fired at the transfer-orbit apogee to perform exactly this $\Delta v_2$, circularising into the GEO belt — a solid or liquid stage sized precisely by this equation.

Related Concepts: Hohmann Transfer Dv1, Orbital Velocity, Circular Orbit Velocity, Vis Viva Energy, Orbital Period, Inclination Change Delta V, Delta V Rendezvous

Notes: Second (arrival/circularisation) burn of the Hohmann transfer; pairs with Hohmann Transfer Dv1. It is the circular speed at $r_2$ minus the transfer ellipse's (slower) apogee speed. For raising, prograde ($\Delta v_2>0$); total cost $\Delta v_1+\Delta v_2$ (LEO→GEO ≈ 3.9 km/s). Symmetric to $\Delta v_1$ under $r_1\!\leftrightarrow\!r_2$. Coplanar/circular/impulsive. $\mu=GM$ mislabelled dimensionless.