In uk.sci.astronomy message 3c047346-7d77-45d8-b8a5-a36b8756047c@g21g20
00hsh.googlegroups.com, Thu, 22 Nov 2007 16:28:09, Weatherlawyer
posted:

The moon weighs some 1/81st the mass of the earth.
...
But they are distinct and do have different speeds. Consider that in
exchange for its capture, the moon travels so much further than us
each year. Whilst the earth is orbiting at some 2Pi x 93 million miles
per year, the moon is in effect orbiting the sun 12 to 13 x 2Pi x
250,000 miles or so further than the earth in that time period.

Nonsense. That is the calculation for both doing a circular 93 Mmi
orbit, and the Moon independently doing 12-13 250 kMi orbits. That is
not what they do. That is like calculating the distance from
Abergavenny to Aberystwyth by adding the distance from Abergavenny to
Ashby-de-la-Zouch to the distance from Ashby-de-la-Zouch to Aberystwyth.

It is easy to show that the Moon's orbit is everywhere concave towards
the Sun, from which it is clear that the annual distance covered by the
Moon is only marginally greater than that covered by the Earth.

To determine said distance, take the vector sum of the Earth's velocity
with respect to the Sun and the Moon's velocity with respect to the
Earth, and integrate for a year.



// Units : Miles, Years. All approximate.
p2 = 2 * Math.PI
EarthSpeed = 93e6 * p2 // wrt Sun
MoonSpeed = 240e3 * p2 * 12 // wrt Earth

Steps = 10000 ; Tot = 0
J = Steps ; while (J--) { T = J/Steps
EarthXSpeed = EarthSpeed * Math.sin(T*p2)
EarthYSpeed = EarthSpeed * Math.cos(T*p2)

MoonXSpeed = MoonSpeed * Math.sin(12*T*p2)
MoonYSpeed = MoonSpeed * Math.cos(12*T*p2)

MoonVelocity = Math.sqrt(
Math.pow(EarthXSpeed+MoonXSpeed, 2) +
Math.pow(EarthYSpeed+MoonYSpeed, 2) )
Tot += MoonVelocity/Steps
}

Tot|0 // output integer part



So Path Length is : 584476336 Miles.
Put Moon distance = 0 : 584336233 " .
Difference : 140103 " - not a lot.
Your answer : 18095574 " - FAR TOO BIG.



The above Javascript was executed in my Web page js-quick.htm. Check
the algebra. Now built into my page gravity3.htm .

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