Hohmann’s minimum-energy transfer orbit

Hohmann (1925) showed that the cheapest way to move between two circular orbits is a transfer ellipse tangent to both, entered and left with a single impulsive burn at each tangent point:

\[a_t = \frac{r_1+r_2}{2}, \qquad \Delta v_1 = |v_{\rm t}(r_1)-v_{\rm c}(r_1)|, \qquad \Delta v_2 = |v_{\rm c}(r_2)-v_{\rm t}(r_2)|.\]

hohmann_transfer() returns exactly these two burns and the transfer time; this example draws all three orbits – the initial and final circular orbits and the transfer ellipse between them – for a LEO-to-GEO-like transfer, and checks the transfer time against half the transfer ellipse’s own period.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.astro.orbital_mechanics import (
    hohmann_transfer,
    orbital_period,
    state_from_orbital_elements,
)

The transfer: burns and timing

mu = 1.0
r1, r2 = 1.0, 6.0  # a LEO-to-GEO-like ratio, in units where r1 = 1

dv1, dv2, t_transfer = hohmann_transfer(r1, r2, mu)
a_t = (r1 + r2) / 2.0
print(f"burn 1 (leaving r1={r1}):      dv1 = {dv1:.6f}")
print(f"burn 2 (circularizing at r2={r2}): dv2 = {dv2:.6f}")
print(f"transfer time = {t_transfer:.6f}")
print(f"half the transfer ellipse's own period = {orbital_period(a_t, mu) / 2.0:.6f}  (should match transfer time exactly)")

burn 1 (leaving r1=1.0):      dv1 = 0.309307
burn 2 (circularizing at r2=6.0): dv2 = 0.190030
transfer time = 20.570836
half the transfer ellipse's own period = 20.570836  (should match transfer time exactly)

The three orbits

The initial circular orbit at r1, the transfer ellipse (periapsis at r1, apoapsis at r2), and the final circular orbit at r2 – the spacecraft only ever traverses the lower half of the transfer ellipse, from periapsis to apoapsis, so that half is highlighted.

nu_full = np.linspace(0.0, 2.0 * np.pi, 400)
e_t = (r2 - r1) / (r2 + r1)

circ1 = np.array([state_from_orbital_elements(r1, 0.0, 0.0, 0.0, 0.0, v, mu)[0] for v in nu_full])
circ2 = np.array([state_from_orbital_elements(r2, 0.0, 0.0, 0.0, 0.0, v, mu)[0] for v in nu_full])
transfer_full = np.array([state_from_orbital_elements(a_t, e_t, 0.0, 0.0, 0.0, v, mu)[0] for v in nu_full])
transfer_arc = np.array([state_from_orbital_elements(a_t, e_t, 0.0, 0.0, 0.0, v, mu)[0] for v in np.linspace(0.0, np.pi, 200)])

fig, ax = plt.subplots(figsize=(6.5, 6.5))
ax.plot(circ1[:, 0], circ1[:, 1], color="steelblue", lw=1.5, label=f"initial circular orbit (r={r1})")
ax.plot(circ2[:, 0], circ2[:, 1], color="seagreen", lw=1.5, label=f"final circular orbit (r={r2})")
ax.plot(transfer_full[:, 0], transfer_full[:, 1], "--", color="0.75", lw=1.0)
ax.plot(transfer_arc[:, 0], transfer_arc[:, 1], color="firebrick", lw=2.5, label="transfer ellipse (traversed half)")
ax.plot(r1, 0, "o", color="steelblue", ms=9, zorder=5, label=f"burn 1 (dv={dv1:.3f})")
ax.plot(-r2, 0, "o", color="seagreen", ms=9, zorder=5, label=f"burn 2 (dv={dv2:.3f})")
ax.plot(0, 0, "*", color="orange", ms=16, label="focus")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_title("Hohmann transfer: initial orbit -> transfer ellipse -> final orbit")
ax.set_aspect("equal")
ax.legend(loc="upper right", fontsize=7.5)
fig.tight_layout()

plt.show()