Sunday, 4 January 2015

lunar eclipse - Does earth's Umbra reach Sun-Earth L2?

The Lagrangian point L2 is very close to the most distant point from Earth with an umbra.
L2 is like the radius of the Hill sphere at r=asqrt[3]fracm3M for circular orbits, with m the mass of Earth, M the mass of the Sun, and a the distance Earth-Sun. The ratio fracm3M of the Earth and the triple mass of the Sun is almost exactly 106, the cubic root hence 0.01.



The diameter ratio of Earth and Sun is about 1/109. Therefore the umbra of Earth ends near 92 the distance to L2.



The answer to another bonus question would then be: If Earth would be 9 larger in diameter, but with the same mass, its umbra would end almost exactly at L2.



Earth's orbit isn't perfectly circular, but the aphel/perihel ratio of about 1.04 is insufficient to question the result qualitatively.
The error of the implicite assumptions tanx=x=sinx is negligible at the considered level of accuracy.

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