Program for testing a formula to help find lunar eclipses.

I figured out a formula that determines how far away
the Moon is from the center of the Earth's shadow.
This confirms the formula makes sense for a known
total lunar eclipse on May 26, 2021.

demo/python/earth_shadow.py

@@ -0,0 +1,42 @@
+#!/usr/bin/env python3
+#
+# earth_shadow.py  -  Don Cross  -  2020-05-11
+#
+# Example program for Astronomy Engine:
+# https://github.com/cosinekitty/astronomy
+#
+# Given a date and time, figure out how far the Moon's center
+# is from the Earth's shadow axis (the ray passing from
+# the center of the Sun through the center of the Earth).
+# This is an experiment to help me figure out a good
+# lunar eclipse algorithm.
+
+import sys
+import math
+from astronomy import Time, Body, HelioVector
+
+def main(text):
+    time = Time.Parse(text)
+
+    # Find heliocentric vector for Earth and Moon at this time.
+    e = HelioVector(Body.Earth, time)
+    print('Earth', e.x, e.y, e.z)
+    m = HelioVector(Body.Moon, time)
+    print('Moon', m.x, m.y, m.z)
+    # Find the parametric value u where the Earth's shadow ray
+    # passes closest to the center of the Moon.
+    u = (e.x*m.x + e.y*m.y + e.z*m.z) / (e.x*e.x + e.y*e.y + e.z*e.z)
+    print('u=', u)
+    # Find the distance between the shadow ray and the Moon.
+    r = math.sqrt((u*e.x-m.x)**2 + (u*e.y-m.y)**2 + (u*e.z-m.z)**2)
+    r *= 1.4959787069098932e+8      # convert AU to km
+    print('r=', r, 'km')
+    # TODO: Calculate the umbral and penumbral radii at the parametric location u.
+    return 0
+
+if __name__ == '__main__':
+    if len(sys.argv) == 2:
+        text = sys.argv[1]
+    else:
+        text = '2021-05-26T11:19:00Z'
+    sys.exit(main(text))