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Very roughly simulate whether SpaceX could hop Starship from Boca Chica to 39B
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| import numpy | |
| from numpy.linalg import norm | |
| import math | |
| earth_radius = 6.3781e6 | |
| gravity = 9.81 | |
| def lla2cart(lat, long, alt=earth_radius): | |
| lat, long = math.radians(lat), math.radians(long) | |
| x = math.cos(long) * math.cos(lat) | |
| y = math.sin(long) * math.cos(lat) | |
| z = math.sin(lat) | |
| return numpy.array([x, y, z]) * alt | |
| def cart2lla(x, y, z): | |
| alt = norm([x, y, z]) | |
| lat = math.degrees(math.asin(z / alt)) | |
| long = math.degrees(math.atan2(y, x)) | |
| return lat, long, alt | |
| def sim(pos, vel, dt, *, impulse=0): | |
| pos = pos.copy() | |
| vel = vel.copy() | |
| while True: | |
| yield pos, vel | |
| vel -= gravity * dt * pos / norm(pos) | |
| vel += impulse * dt | |
| pos += vel * dt | |
| def sim_landing_point(pos, vel, dt): | |
| for pos, vel in sim(pos, vel, dt): | |
| if norm(pos) + 1 < earth_radius: | |
| return pos | |
| def fmt_lla(lat, long, alt): | |
| return f"{{ {lat:>6.3f} {long:>7.3f} {alt-earth_radius:>6.0f} }}" | |
| def fmt_ll(lat, long, _alt): | |
| return f"{{ {lat:>6.3f} {long:>7.3f} }}" | |
| start = lla2cart(25.997407091653063, -97.15718040356279) | |
| vel = numpy.array([0.0, 0.0, 0.0]) | |
| impulse = numpy.array([0.0, 0.0, 0.0]) | |
| dry_mass = 126_000 | |
| propellant_mass = 1200_000 | |
| targets = [ | |
| lla2cart(24.2172687156, -82.2451010924), | |
| lla2cart(24.1741747933, -81.4646902746), | |
| lla2cart(24.4521235059, -80.4662804471), | |
| lla2cart(25.1690508346, -79.7467954413), | |
| lla2cart(26.3645084546, -79.5654348773), | |
| lla2cart(27.2460928840, -79.7122031701), | |
| lla2cart(28.6002368345, -80.4351409697), | |
| ] | |
| targets_boost = {7, 6, 5, 4} | |
| t = 0 | |
| dt = 0.01 | |
| skipcheck = 1 | |
| for i, (pos, vel) in enumerate(sim(start, vel, dt, impulse=impulse)): | |
| if targets and not i % skipcheck: | |
| landing_point = sim_landing_point(pos, vel, dt) | |
| if targets and propellant_mass > 0: | |
| # 1800 kN / engine | |
| accel = 1800_000 * 9 / (dry_mass + propellant_mass) | |
| # momentum/engine = 1800 kN * dt = 3300 m/s * M kg/s * dt | |
| # M kg/s ~ 545 kg/s | |
| propellant_mass -= 545 * 9 * dt | |
| up_dir = pos / norm(pos) | |
| target_dir = targets[0] - landing_point | |
| target_dir -= up_dir * (up_dir @ target_dir) | |
| target_dir /= norm(target_dir) | |
| if len(targets) in targets_boost: | |
| # up_strength = 0.5 + 245250./(-98100 + math.sqrt(-3.84944e10 + 5.e8 * accel**2)) | |
| up_strength = 0.6 + 266832./(-117720 + math.sqrt(-3.84944e10 + 5.44e8 * accel**2)) | |
| # up_strength = 0.7 + 292338./(-137340 + math.sqrt(-3.84944e10 + 5.96e8 * accel**2)) | |
| # up_strength = 0.8 + 321768/(-156960 + math.sqrt(-3.84944e10 + 6.56e8 * accel**2)) | |
| else: | |
| up_strength = 0 | |
| dir = up_dir * up_strength + target_dir | |
| dir /= norm(dir) | |
| impulse[:] = accel * dir | |
| else: | |
| impulse[:] = 0 | |
| lat, long, alt = pos_lla = cart2lla(*pos) | |
| if alt + 0.01 < earth_radius: | |
| break | |
| if targets: | |
| delta = norm(landing_point - targets[0]) | |
| skipcheck = 2 ** math.ceil(math.log2(100 * delta / (10_000_000 * dt))) | |
| if delta < (100_000 * dt): # chosen by trial and error | |
| targets.pop(0) | |
| i = 0 | |
| if not i % int(1 / dt): | |
| print(f"{t:>5.1f} :: {fmt_lla(*pos_lla)} → {fmt_ll(*cart2lla(*landing_point))} :: {norm(vel):>4.0f} m/s {propellant_mass/1000:>4.0f}t T{len(targets)} :: DEBUG {delta:>8.0f} {accel:>3.2f} {up_strength:>3.2f} {accel * up_strength:>3.2f} {vel @ pos / norm(pos):>4.0f} {skipcheck}") | |
| t += dt |
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