origin_math.py

# ATLAS-RECEIPT-7 companion — every number recomputed from the published state vector.
# Inputs: Guo et al. 2025 (arXiv:2509.03361) §2, Table 2; Sun et al. 2025 AVR via §5.
# Run: python3 origin_math.py   (numpy only)
import numpy as np

# 1) heliocentric state vector, J2016.0, ICRF (JPL Horizons, as published)
v_eq = np.array([-23.2454, 53.1991, -2.3727])            # km/s
print(f"1) |v| = {np.linalg.norm(v_eq):.2f} km/s (published: ~58)")

# 2) equatorial J2000 -> Galactic UVW (U toward GC), then LSR (Schoenrich+2010)
R = np.array([[-0.05487556,-0.87343709,-0.48383502],
              [ 0.49410943,-0.44482963, 0.74698224],
              [-0.86766615,-0.19807637, 0.45598378]])
UVW  = R @ v_eq
vlsr = UVW + np.array([11.1, 12.24, 7.25])
print(f"2) UVW = ({UVW[0]:.1f}, {UVW[1]:.1f}, {UVW[2]:.1f})  "
      f"LSR = ({vlsr[0]:.1f}, {vlsr[1]:.1f}, {vlsr[2]:.1f})  |v_LSR| = {np.linalg.norm(vlsr):.1f} km/s")

# 3) best flyby candidate G 137-55/54 (0.890 Msun barycentre, 1358.3 au separation):
#    its ejection lever vs the needed 28.39 km/s encounter speed
G, Msun, au, pc = 6.674e-11, 1.989e30, 1.496e11, 3.086e16
v_orb = np.sqrt(G*0.890*Msun/(1358.3*au))/1e3
print(f"3) binary lever {v_orb:.2f} km/s vs needed 28.39 -> deficit x{28.39/v_orb:.0f}  (NOT the home)")

# 4) its deflection of 3I at closest approach b = 0.242 pc
delta = 2*np.arctan(G*0.890*Msun/((0.242*pc)*(28.39e3)**2))
arcsec = np.degrees(delta)*3600
print(f"4) deflection = {arcsec:.1f} arcsec (paper: 'a few arcseconds')")

# 5) address erasure: one such kink smears the backtrack; flyby count over life
for T in (1, 5, 10):
    smear = delta * (58e3 * T*3.156e16) / pc
    print(f"5) one {arcsec:.1f}\" kink at {T} Gyr depth -> ~{smear:.0f} pc smear at the origin")
alpha = 21.8 * np.linalg.norm(vlsr)/np.linalg.norm([11.1,12.24,7.25])
print(f"   flyby rate <1 pc: ~{alpha:.0f}/Myr  -> ~{alpha*1e4:,.0f} flybys per 10 Gyr")

# 6) thin vs thick disk membership (dispersions & priors as in the paper, Vc=234 approx)
vR, vphi, vz = -vlsr[0], 234.0+vlsr[1], vlsr[2]
def L(mphi, s):
    z = (vR/s[0])**2 + ((vphi-mphi)/s[1])**2 + (vz/s[2])**2
    return np.exp(-z/2)/((2*np.pi)**1.5*np.prod(s))
Lt = L(224.82, (34.59,22.88,19.72)); Lk = L(176.94, (65.47,54.07,41.32))
post = 0.8067*Lt/(0.8067*Lt + 0.1206*Lk)
print(f"6) L_thin/L_thick = {Lt/Lk:.2f} (paper 4.43) -> posterior thin {100*post:.1f}% (paper 96.59%)")