"""The flight path angle, drawn on the orbit itself. Left: one sample point in each quadrant of an ellipse (e = 0.6), each carrying its radius, its local horizon, the velocity and the two components v_r = (mu/h) e sin f along r-hat v_perp = (mu/h)(1 + e cos f) along theta-hat with gamma = atan2(v_r, v_perp) marked between the horizon and the velocity. Right: the same construction for the f = 135 deg point alone, rotated so that the local horizon lies flat -- the velocity triangle without the orbit on top of it. Sizes cancel in the ratio v_r/v_perp, so the picture is drawn with mu = a = 1; only the shape (e) and the position (f) matter. """ import os import numpy as np import matplotlib.pyplot as plt from matplotlib.patches import Arc os.makedirs("img", exist_ok=True) CREAM, INK = "#fcfbf7", "#2f2a24" MAROON, BLUE, GREEN, AMBER, PURPLE = "#6b1f1f", "#2f5ea8", "#2d6a2d", "#b45309", "#7e22ce" plt.rcParams.update({ "figure.facecolor": CREAM, "savefig.facecolor": CREAM, "axes.facecolor": CREAM, "axes.edgecolor": INK, "axes.labelcolor": INK, "text.color": INK, "xtick.color": INK, "ytick.color": INK, "font.size": 11, "axes.grid": False, }) E_CC = 0.6 # eccentricity of the drawn ellipse A = 1.0 # semi-major axis; mu = 1 with it P_ = A*(1.0 - E_CC**2) # semi-latus rectum H = np.sqrt(P_) # h = sqrt(mu p) VS = 0.24 # velocity arrows -> plot units def state(f): """Position, the rotating basis, the two speeds and gamma at true anomaly f.""" r = P_/(1.0 + E_CC*np.cos(f)) rhat = np.array([np.cos(f), np.sin(f)]) that = np.array([-np.sin(f), np.cos(f)]) v_r = E_CC*np.sin(f)/H v_t = (1.0 + E_CC*np.cos(f))/H return r, r*rhat, rhat, that, v_r, v_t, np.arctan2(v_r, v_t) fig, (axL, axR) = plt.subplots(1, 2, figsize=(14.0, 6.3), gridspec_kw={"width_ratios": [1.32, 1.0]}) # ═══ left panel : one point per quadrant ════════════════════════════════════ fs = np.linspace(0.0, 2*np.pi, 900) rs = P_/(1.0 + E_CC*np.cos(fs)) axL.plot(rs*np.cos(fs), rs*np.sin(fs), color=INK, lw=1.7, zorder=2) # the attracting body sits at the focus; the apse line for reference axL.plot([0.0], [0.0], "o", color=MAROON, ms=11, zorder=6) axL.text(0.045, -0.115, "$F$", color=MAROON, fontsize=13, ha="left", va="top") axL.plot([-A*(1 + E_CC) - 0.12, A*(1 - E_CC) + 0.12], [0, 0], color=INK, lw=0.8, ls=(0, (6, 5)), alpha=0.5, zorder=1) # direction of travel: counter-clockwise, so -x across the top of the orbit axL.annotate("", xy=(-0.52, 1.00), xytext=(0.08, 1.00), arrowprops=dict(arrowstyle="->", color=INK, lw=1.4, alpha=0.6)) axL.text(-0.22, 1.09, "motion", color=INK, fontsize=9.5, alpha=0.7, ha="center", va="bottom") # label offsets chosen per quadrant so nothing lands on the curve tags = {45: (0.40, 0.16), 135: (-0.20, 0.26), 225: (-0.22, -0.28), 315: (0.34, -0.20)} for fdeg, (dx, dy) in tags.items(): f = np.deg2rad(fdeg) r, Pp, rhat, that, v_r, v_t, gam = state(f) # radius from the focus axL.plot([0.0, Pp[0]], [0.0, Pp[1]], color=INK, lw=0.9, ls=":", alpha=0.75, zorder=3) # local horizon: perpendicular to r, drawn through the point hz = 0.30 axL.plot([Pp[0] - hz*that[0], Pp[0] + hz*that[0]], [Pp[1] - hz*that[1], Pp[1] + hz*that[1]], color=GREEN, lw=1.5, ls=(0, (5, 3)), zorder=4) # the two components, then the velocity itself axL.annotate("", xy=Pp + VS*v_t*that, xytext=Pp, arrowprops=dict(arrowstyle="->", color=BLUE, lw=1.6, alpha=0.95)) axL.annotate("", xy=Pp + VS*v_r*rhat, xytext=Pp, arrowprops=dict(arrowstyle="->", color=AMBER, lw=1.6, alpha=0.95)) vvec = VS*(v_r*rhat + v_t*that) axL.annotate("", xy=Pp + vvec, xytext=Pp, arrowprops=dict(arrowstyle="-|>", color=MAROON, lw=2.4, mutation_scale=17)) # dotted sides closing the parallelogram for base in (VS*v_t*that, VS*v_r*rhat): axL.plot([Pp[0] + base[0], Pp[0] + vvec[0]], [Pp[1] + base[1], Pp[1] + vvec[1]], color=INK, lw=0.7, ls=(0, (2, 2.5)), alpha=0.6, zorder=3) # gamma, between the horizon and the velocity a_h = np.degrees(np.arctan2(that[1], that[0])) a_v = np.degrees(np.arctan2(vvec[1], vvec[0])) R = 0.19 axL.add_patch(Arc(Pp, 2*R, 2*R, angle=0.0, theta1=min(a_h, a_v), theta2=max(a_h, a_v), color=MAROON, lw=1.5, zorder=5)) a_mid = np.deg2rad(0.5*(a_h + a_v)) axL.text(Pp[0] + 1.42*R*np.cos(a_mid), Pp[1] + 1.42*R*np.sin(a_mid), r"$\gamma$", color=MAROON, fontsize=12, ha="center", va="center") axL.plot([Pp[0]], [Pp[1]], "o", color=INK, ms=5.5, zorder=6) axL.text(Pp[0] + dx, Pp[1] + dy, f"$f = {fdeg}^\\circ$\n$\\gamma = {np.degrees(gam):+.1f}^\\circ$", color=INK, fontsize=10.5, ha="center", va="center") axL.text(0.78, 1.10, "climbing\n$\\gamma > 0$", color=INK, fontsize=10.5, ha="center", va="center", alpha=0.9) axL.text(0.78, -1.10, "descending\n$\\gamma < 0$", color=INK, fontsize=10.5, ha="center", va="center", alpha=0.9) axL.text(0.86, 0.0, "periapsis\n$\\gamma = 0$", color=INK, fontsize=9.5, ha="center", va="center", alpha=0.85) axL.text(-2.02, 0.0, "apoapsis\n$\\gamma = 0$", color=INK, fontsize=9.5, ha="center", va="center", alpha=0.85) axL.set_xlim(-2.30, 1.42) axL.set_ylim(-1.38, 1.38) axL.set_aspect("equal") axL.axis("off") axL.set_title(r"$\gamma$ around an ellipse, $e = 0.6$", fontsize=13, pad=6) # ═══ right panel : the triangle on its own ══════════════════════════════════ f_d = np.deg2rad(135.0) _, _, _, _, v_r, v_t, gam = state(f_d) # drawn in the local frame: horizon along +x, r-hat (away from the focus) along +y S = 1.0/v_t # scale so v_perp is one unit wide vx, vy = v_t*S, v_r*S axR.plot([-0.30, 1.42], [0, 0], color=GREEN, lw=1.8, ls=(0, (5, 3)), zorder=2) axR.text(1.42, -0.045, "local horizon $\\hat{\\boldsymbol{\\theta}}$", color=GREEN, fontsize=11, ha="right", va="top") axR.annotate("", xy=(0.0, 1.06), xytext=(0.0, -0.34), arrowprops=dict(arrowstyle="->", color=INK, lw=1.1, alpha=0.65)) axR.text(-0.055, 1.06, "$\\hat{\\mathbf{r}}$ (local vertical)", color=INK, fontsize=11, ha="right", va="center") axR.text(-0.055, -0.30, "towards $F$", color=INK, fontsize=9.5, alpha=0.75, ha="right", va="center") # components and resultant axR.annotate("", xy=(vx, 0.0), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color=BLUE, lw=2.6, mutation_scale=16)) axR.annotate("", xy=(0.0, vy), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color=AMBER, lw=2.6, mutation_scale=16)) axR.annotate("", xy=(vx, vy), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color=MAROON, lw=3.0, mutation_scale=18)) axR.plot([vx, vx], [0, vy], color=INK, lw=0.9, ls=(0, (2, 2.5)), alpha=0.65) axR.plot([0, vx], [vy, vy], color=INK, lw=0.9, ls=(0, (2, 2.5)), alpha=0.65) # right-angle tick where the components meet q = 0.055 axR.plot([q, q, 0], [0, q, q], color=INK, lw=0.9, alpha=0.7) R = 0.42 axR.add_patch(Arc((0, 0), 2*R, 2*R, angle=0.0, theta1=0.0, theta2=np.degrees(gam), color=MAROON, lw=1.8, zorder=5)) axR.text(1.30*R*np.cos(0.5*gam), 1.30*R*np.sin(0.5*gam), r"$\gamma$", color=MAROON, fontsize=15, ha="center", va="center") axR.text(vx*0.5, -0.085, r"$v_\perp = \dfrac{\mu}{h}\,(1 + e\cos f)$", color=BLUE, fontsize=12, ha="center", va="top") axR.text(-0.055, vy*0.5, r"$v_r = \dfrac{\mu}{h}\,e\sin f$", color=AMBER, fontsize=12, ha="right", va="center") axR.text(vx*0.46, vy*0.46 + 0.055, r"$\mathbf{v}$", color=MAROON, fontsize=15, ha="center", va="bottom") axR.text(0.80, 1.18, r"$\tan\gamma = \dfrac{v_r}{v_\perp} = \dfrac{e\sin f}{1 + e\cos f}$", color=INK, fontsize=13.5, ha="center", va="center") axR.text(0.80, 0.94, f"here $f = 135^\\circ$, $e = 0.6$: " f"$\\gamma = {np.degrees(gam):.1f}^\\circ$", color=INK, fontsize=10.5, ha="center", va="center", alpha=0.9) axR.set_xlim(-0.62, 1.52) axR.set_ylim(-0.46, 1.38) axR.set_aspect("equal") axR.axis("off") axR.set_title("the velocity triangle at one point", fontsize=13, pad=6) plt.tight_layout() fig.savefig("img/fig-fpa-quadrants.png", dpi=160, bbox_inches="tight") plt.close(fig)