"""Phasing: drop onto a smaller ellipse, come back early, and the cost of haste. Panel A (geometry). The chaser trails the target by an angle dth on a common circular orbit of radius r. It burns *retrograde* at its own position, which becomes the **apoapsis** of a smaller ellipse with the same focus, tangent to the circle at that one point. Because a_ph < r the phasing period is shorter, so after k revolutions on the ellipse the chaser is back at the burn point -- and the target, which had to cover 2*pi*k - dth in the same time, arrives there at the same instant. A second, equal and opposite impulse re-circularises. The drawing is exaggerated for legibility: dth = 60 deg in one revolution is a far larger correction than any real rendezvous asks for, and the Earth is drawn well out of scale so that the phasing perigee -- the quantity that actually limits the manoeuvre -- is visible as a gap. Panel B (cost). Total dv against the number of phasing revolutions k. Run from the sfm-slides directory: python3 codes/fig-man-phasing.py """ import numpy as np import matplotlib.pyplot as plt from _manstyle import CREAM, INK, MAROON, BLUE, GREEN, AMBER, PURPLE, despine mu, RE = 398600.4418, 6378.137 # ---- panel A : geometry --------------------------------------------------- figL, axL = plt.subplots(figsize=(7.8, 6.2)) r = 1.0 dth = np.radians(60.0) # exaggerated for legibility k = 1 aph = r*(1 - dth/(2*np.pi*k))**(2/3) # Kepler III on the shorter period rp = 2*aph - r # apoapsis is the burn point e = (r - rp)/(r + rp) th = np.linspace(0, 2*np.pi, 720) R_EARTH = 0.52 # out of scale, and said so axL.fill(R_EARTH*np.cos(th), R_EARTH*np.sin(th), color=BLUE, alpha=0.10, zorder=0) axL.plot(R_EARTH*np.cos(th), R_EARTH*np.sin(th), color=BLUE, lw=1.0, alpha=0.45, zorder=1) # the common circular orbit, and the phasing ellipse (apoapsis on +x) axL.plot(r*np.cos(th), r*np.sin(th), color=BLUE, lw=2.0, zorder=3) f = np.linspace(0, 2*np.pi, 720) rr = aph*(1 - e**2)/(1 + e*np.cos(f)) axL.plot(-rr*np.cos(f), -rr*np.sin(f), color=MAROON, lw=2.4, zorder=4) axL.text(0.0, r + 0.04, r"circular orbit, radius $r$, period $T$", color=BLUE, fontsize=11, ha="center", va="bottom") axL.text(0.0, -aph - 0.30, "phasing ellipse: " r"$a_{ph}", color=INK, lw=1.6, alpha=0.75, connectionstyle="arc3,rad=-0.18")) axL.text(1.18*np.cos(np.radians(136)), 1.18*np.sin(np.radians(136)), "direction\nof motion", color=INK, fontsize=9.5, alpha=0.8, ha="right", va="bottom") # the two impulses: equal and opposite, one revolution apart, same point L = 0.34 axL.annotate("", xy=(burn[0], burn[1] - L), xytext=(burn[0], burn[1] - 0.02), arrowprops=dict(arrowstyle="-|>", color=AMBER, lw=2.6)) axL.text(burn[0] + 0.09, burn[1] - 0.30, r"$\Delta v_1$ retrograde" "\ndrop onto the ellipse", color=AMBER, fontsize=10.5, ha="left", va="center") xo = burn[0] + 0.30 axL.annotate("", xy=(xo, burn[1] + L), xytext=(xo, burn[1] + 0.02), arrowprops=dict(arrowstyle="-|>", color=AMBER, lw=2.6)) axL.text(xo + 0.07, burn[1] + 0.30, r"$\Delta v_2$ prograde" "\n" r"$k$ revs later, re-circularise", color=AMBER, fontsize=10.5, ha="left", va="center") axL.plot(0, 0, "o", color=INK, ms=8, zorder=8) axL.text(0.0, -1.30, "smaller $a$ $\\Rightarrow$ shorter period: the chaser returns to the burn point\n" "after $k$ revolutions, just as the target --- which had to cover " "$2\\pi k-\\Delta\\theta$ --- arrives", color=MAROON, fontsize=10.5, ha="center", va="top") axL.text(-1.66, 1.30, "Earth, $\\Delta\\theta$ and the perigee drop\nexaggerated for legibility", color=INK, fontsize=8.5, alpha=0.55, ha="left", va="top") axL.set_aspect("equal"); axL.axis("off") axL.set_xlim(-1.66, 1.66) axL.set_ylim(-1.62, 1.34) axL.set_title("Phasing orbit: same burn point, smaller ellipse", fontsize=12.5) figL.savefig("img/fig-man-phasing-geom.png", dpi=160, bbox_inches="tight") plt.close(figL) print("wrote img/fig-man-phasing-geom.png") # ---- panel B : cost vs waiting time -------------------------------------- figR, axR = plt.subplots(figsize=(9.0, 5.8)) r0 = 6778.0 vc = np.sqrt(mu/r0) T0 = 2*np.pi*np.sqrt(r0**3/mu) dth_deg = np.array([5.0, 10.0, 20.0, 40.0]) ks = np.arange(1, 11) for d, col in zip(dth_deg, (BLUE, MAROON, GREEN, PURPLE)): dt = np.radians(d) aph = r0*(1 - dt/(2*np.pi*ks))**(2/3) dv = 2*np.abs(np.sqrt(mu*(2/r0 - 1/aph)) - vc)*1000 # m/s axR.plot(ks, dv, "o-", color=col, lw=2.0, ms=5, label=fr"$\Delta\theta={d:.0f}^\circ$") if d == 40.0: rpk = 2*aph - r0 bad = rpk < RE + 100 axR.plot(ks[bad], dv[bad], "x", color=INK, ms=11, mew=2.2) axR.text(1.15, dv[0] - 60, "perigee below 100 km:\ninfeasible", color=INK, fontsize=9.5) axR.set_xlabel(r"number of phasing revolutions $k$") axR.set_ylabel(r"total $\Delta v = 2|v_{ph}-v_c|$ [m/s]") axR.set_title("Patience is the cheapest propellant ($\\Delta v \\propto 1/k$)", fontsize=12) axR.set_xticks(ks) axR.legend(frameon=False, fontsize=10) axR.set_ylim(0, 640) despine(axR) figR.tight_layout() figR.savefig("img/fig-man-phasing-cost.png", dpi=160, bbox_inches="tight") plt.close(figR) print("wrote img/fig-man-phasing-cost.png") # ---- verification --------------------------------------------------------- if __name__ == "__main__": Tph_over_T = (aph_g := r*(1 - dth/(2*np.pi*k))**(2/3))**1.5 / r**1.5 print(f"\ndth = {np.degrees(dth):.1f} deg, k = {k}") print(f" a_ph/r = {aph_g:.6f}") print(f" T_ph/T = {Tph_over_T:.6f} (target {1 - dth/(2*np.pi*k):.6f})") print(f" r_p/r = {2*aph_g - r:.6f} (drawn Earth radius {R_EARTH})") print(f" chaser return = {k*Tph_over_T:.6f} T") print(f" target angle = {np.degrees(2*np.pi*k*Tph_over_T + dth) % 360:.3f} deg" " (0 deg = at the burn point)")