[seqfan] Re: Sequence relating the mass of the planets to the mass of their most massive satellites
Brendan McKay
Brendan.McKay at anu.edu.au
Tue Feb 6 23:28:39 CET 2018
Incidentally, the methods used for measuring mass of extraterrestial
objects,
mainly their effect on the motion of other objects and spacescraft, actually
produce the value of G*M, where M is the mass and G is the gravitational
constant. They don't produce the mass directly.
Since G itself is not known to more than 4 or 5 digits (if I recall there is
currently a contradiction between the most precise measurements), M
cannot be known more accurately than that. This is not necessarily a
disadvantage in determining the ratio between a moon and its planet,
since G*M is known more accurately than M for the planet and we can use
(G*Mars)/(G*Phobos) instead of Mars/Phobos.
Alas, even G*Phobos is known only to a few parts in a thousand so there
is no hope of getting the integer nearest to Mars/Phobos.
Brendan.
On 7/2/18 3:57 am, Charles Greathouse wrote:
> If such a sequence were to be accepted it would be absolutely necessary
> that each term submitted be known within standard error. I think
> measurement errors are usually assumed to be normal, so using an
> approximation (Marsaglia 2006, Ratios of normal variables; see section 5)
> the lower and upper bounds would be
>
> moonmass*planetmass ± sqrt(planetmass^2*stdevmoon^2 + (moonmass^2 -
> stdevmoon^2)*stdevplanet^2)
> ----------------------------------------------------------------------------------------------------------------------------------------------
> moonmass^2 - stdevmoon^2
>
> and these would need to be between n and n+1 for some integer n.
>
> Andert 2010, "Precise mass determination and the nature of Phobos" seems to
> be the best source for the mass of the largest natural satellite of Mars,
> and its value (1.0668 ± 0.003)*10^16 kg is far too imprecise to determine
> a(4). So I don't think there's any reasonable way this sequence could be
> accepted, interesting though it is.
>
> Charles Greathouse
> Case Western Reserve University
>
> On Tue, Feb 6, 2018 at 8:46 AM, Hans Havermann <gladhobo at bell.net> wrote:
>
>>> The terms are 0, 0, 81, 60203584, 12809, 4225, 24620, 4786
>> One difficulty with such a sequence, based on measurement, is how accurate
>> the measurements are. Your fourth term implies that we know them to have
>> more than eight-digit precision. I don't believe that our mass
>> determinations of Mars and Phobos are quite that accurate and I'm not sure
>> that ratio-ing them makes them any more precise.
>>
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