Online tutor?

On 02/11/12 15:46, Martin Brown wrote:
1965-68. Exciting times in astronomy.
Well before my time. Trying to remember a few other names from way
back when I can only conjure up John Meaburn & Ken Elliott (sp?). One
of them had a cute Fabry-Perot device.

I knew John reasonably well; not Ken. Franz Kahn was the
other professor in my time, and Richard James my supervisor; both
were still around many years later. I remember the term "Fabry-
Perot", but I never knew what it meant, nor anything useful about
practical or observational astronomy in general ....

Didn't Kopal help map the moon
for the NASA moon landings or something - ah yes: [...]

Yes. They were doing stereoscopy on plates of the Moon
taken at extremes of libration to work out the contours. The
amusing thing was that it couldn't be done in America because
the only sufficiently good equipment was supplied by Zeiss of
Jena, and the Americans couldn't import from East Germany. I
kept about a dozen of the original sheets until I retired, and
then chucked them out, just in time to miss the 40th anniversary
and a possible chance to make some money on eBay.

--
Andy Walker,
Nottingham.

In uk.sci.astronomy message , Thu, 1
Nov 2012 13:50:20, Andy Walker posted:

On 28/10/12 20:30, Dr J R Stockton wrote:
[...] Do you know if Gascheau's 1843
article is accessible on the Web?

I've never pursued any historical research into these
areas, just read what I needed to for my own research and other
interests. Ie, I don't know, and your googling is likely to be
at least as well-directed as mine.

Danby, Astron J, 69, 4, pp.294-6, May 1964 (GIF) looks a good read - for
some. Its abstract indicates that the same holds for non-circular
orbits, but with a varying numerical constant. A.N.Other implied that
moderately elliptical orbits are stable for moderately lighter
secondaries, but no further.

It would be very surprising if it was not so.

Perhaps; but I should have written "for moderately heavier secondaries".
But unless I can find that paper again ...
My reading of it was that as the eccentricity increased to around 0.35,
the secondary can become somewhat heavier than the circular primary/25
before stability is lost.

And that might well be unexpected.

--
(c) John Stockton, nr London, UK. Mail via homepage. Turnpike v6.05 MIME.
Web http://www.merlyn.demon.co.uk/ - FAQqish topics, acronyms and links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.

In uk.sci.astronomy message , Thu, 1 Nov
2012 13:06:26, Andy Walker posted:

On 28/10/12 09:29, Martin Brown wrote:
I remember Prof Kopal he was one of the people who got me interested
in astronomy as a youngster. He was patron of the local astrosoc and
did an annual lecture.

I have a fund of ZK stories, but mostly not suitable for here.
His soft Czechoslovakian accent was very easily imitated. so almost all
of us did it. Occasionally, a group of postgrads would be holding a
conversation in ZK-speak when ZK would turn up and join in; whether he
didn't notice or didn't mind, we never discovered,

[Stability of L4/5:]
I think it could just about be done with 6th form maths, but I am not
convinced that (m)any sixth formers would be able to follow it!

Mechanics in rotating co-ordinates is not 6th-form maths!
Nor is the general theory of small oscillations. I think it might
be the other way round -- that you couldn't do it with 6th-form
maths, except by spending a lot of time developing a framework for
the proof, but on the other hand that there are some very talented
sixth-formers who would be able to read a proof such as that in Roy,
and fill in the gaps themselves. But that's just an opinion.

Rotating co-ordinates can be used, by those who know how, for objects
moving in circles at constant speed - but what about objects moving in
other conic sections? And remember that, unless previously proved,
conic sections would be a mere assumption.

--
(c) John Stockton, Surrey, UK. Turnpike v6.05 MIME.
Web http://www.merlyn.demon.co.uk/ - FAQish topics, acronyms, & links.
Proper = 4-line sig. separator as above, a line exactly "-- " (SonOfRFC1036)
Do not Mail News to me. Before a reply, quote with "" or " " (SonOfRFC1036)

On 02/11/2012 23:28, Dr J R Stockton wrote:
In uk.sci.astronomy message , Thu, 1 Nov
2012 13:06:26, Andy Walker posted:

On 28/10/12 09:29, Martin Brown wrote:
I remember Prof Kopal he was one of the people who got me interested
in astronomy as a youngster. He was patron of the local astrosoc and
did an annual lecture.

I have a fund of ZK stories, but mostly not suitable for here.
His soft Czechoslovakian accent was very easily imitated. so almost all
of us did it. Occasionally, a group of postgrads would be holding a
conversation in ZK-speak when ZK would turn up and join in; whether he
didn't notice or didn't mind, we never discovered,

[Stability of L4/5:]
I think it could just about be done with 6th form maths, but I am not
convinced that (m)any sixth formers would be able to follow it!

Mechanics in rotating co-ordinates is not 6th-form maths!
Nor is the general theory of small oscillations. I think it might
be the other way round -- that you couldn't do it with 6th-form
maths, except by spending a lot of time developing a framework for
the proof, but on the other hand that there are some very talented
sixth-formers who would be able to read a proof such as that in Roy,
and fill in the gaps themselves. But that's just an opinion.

Rotating co-ordinates can be used, by those who know how, for objects
moving in circles at constant speed - but what about objects moving in
other conic sections? And remember that, unless previously proved,
conic sections would be a mere assumption.

The conic sections were included in our pure maths syllabus. Proof that
motion under an inverse square law wasn't, but I think you could derive
it from the DE without using any tricks not taught at A level back then.
ISTR that proof was also second year undergraduate physics.

But for the purposes of this argument doing it for the circular orbit
case without using anything outside A level calculus I think it is just
possible (at least with the syllabus when I did it). Though I can't be
absolutely sure which bits were in Further Maths - quite possibly some
of the DE tricks were in there rather than standard Pure Maths.

--
Regards,
Martin Brown

On Nov 4, 12:21*pm, Martin Brown
wrote:
On 02/11/2012 23:28, Dr J R Stockton wrote:









In uk.sci.astronomy message , Thu, 1 Nov
2012 13:06:26, Andy Walker posted:

On 28/10/12 09:29, Martin Brown wrote:
I remember Prof Kopal he was one of the people who got me interested
in astronomy as a youngster. He was patron of the local astrosoc and
did an annual lecture.

* * * *I have a fund of ZK stories, but mostly not suitable for here.
His soft Czechoslovakian accent was very easily imitated. so almost all
of us did it. *Occasionally, a group of postgrads would be holding a
conversation in ZK-speak when ZK would turn up and join in; *whether he
didn't notice or didn't mind, we never discovered,

[Stability of L4/5:]
I think it could just about be done with 6th form maths, but I am not
convinced that (m)any sixth formers would be able to follow it!

* * * *Mechanics in rotating co-ordinates is not 6th-form maths!
Nor is the general theory of small oscillations. *I think it might
be the other way round -- that you couldn't do it with 6th-form
maths, except by spending a lot of time developing a framework for
the proof, but on the other hand that there are some very talented
sixth-formers who would be able to read a proof such as that in Roy,
and fill in the gaps themselves. *But that's just an opinion.

Rotating co-ordinates can be used, by those who know how, for objects
moving in circles at constant speed - but what about objects moving in
other conic sections? *And remember that, unless previously proved,
conic sections would be a mere assumption.

The conic sections were included in our pure maths syllabus. Proof that
motion under an inverse square law wasn't, but I think you could derive
it from the DE without using any tricks not taught at A level back then.

--
Regards,
Martin Brown
-
Mathematicians are full of tricks and pity you cannot appreciate the
trick Newton pulled with the correlation between orbital distances
from the Sun and orbital periods -

"That the fixed stars being at rest, the periodic times of the five
primary planets, and (whether of the sun about the earth, or) of the
earth about the sun, are in the sesquiplicate proportion of their mean
distances from the sun" Newton

Let me guess,none of you have a clue what method Isaac was using with
his absolute/relative time,space and motion or better still,none of
you want to know what Isaac was doing with his trick of double
modeling using Kepler's insight.

The actual statement of Kepler is quite different and easily
understandable in its expanded version -

"The proportion existing between the periodic times of any two planets
is exactly the sesquiplicate proportion of the mean distances of the
orbits, or as generally given,the squares of the periodic times are
proportional to the cubes of the mean distances." Kepler

"But it is absolutely certain and exact that the ratio which exists
between the periodic times of any two planets is precisely the ratio
of the 3/2th power of the mean distances, i.e., of the spheres
themselves; provided, however, that the arithmetic mean between both
diameters of the elliptic orbit be slightly less than the longer
diameter. And so if any one take the period, say, of the Earth, which
is one year, and the period of Saturn, which is thirty years, and
extract the cube roots of this ratio and then square the ensuing
ratio
by squaring the cube roots, he will have as his numerical products the
most just ratio of the distances of the Earth and Saturn from the sun.
1 For the cube root of 1 is 1, and the square of it is 1; and the cube
root of 30 is greater than 3, and therefore the square of it is
greater than 9. And Saturn, at its mean distance from the sun, is
slightly higher than nine times the mean distance of the Earth from
the sun." Kepler

Want to know Newton's scheme which eluded mathematicians and everyone
else for centuries ? - didn't think so ! but almost certain you would
crash and burn fairly quickly under the blizzard of references.Conic
sections indeed !,

On 02/11/12 23:28, Dr J R Stockton wrote:
Danby, Astron J, 69, 4, pp.294-6, May 1964 (GIF) looks a good read [...]

It's online; eg:

http://articles.adsabs.harvard.edu/c...J.....69..294D

But unless I can find that paper again ...
My reading of it was that as the eccentricity increased to around 0.35,
the secondary can become somewhat heavier than the circular primary/25
before stability is lost.

Yes, that's the result shown in Danby's earlier article
[same URL but with 294 replaced by 165 (the page number)] in his
Figure 1, p171, as mentioned in my previous article.

And that might well be unexpected.

Certainly the region of stability is non-trivial. It
might be quite easy these days to re-visit that problem using
modern symbolic algebra packages and computing power generally
and make some interesting progress.

--
Andy Walker,
Nottingham.

In uk.sci.astronomy message , Sun, 4
Nov 2012 12:21:28, Martin Brown
posted:

On 02/11/2012 23:28, Dr J R Stockton wrote:
In uk.sci.astronomy message , Thu, 1 Nov
2012 13:06:26, Andy Walker posted:

On 28/10/12 09:29, Martin Brown wrote:
I remember Prof Kopal he was one of the people who got me interested
in astronomy as a youngster. He was patron of the local astrosoc and
did an annual lecture.

I have a fund of ZK stories, but mostly not suitable for here.
His soft Czechoslovakian accent was very easily imitated. so almost all
of us did it. Occasionally, a group of postgrads would be holding a
conversation in ZK-speak when ZK would turn up and join in; whether he
didn't notice or didn't mind, we never discovered,

[Stability of L4/5:]
I think it could just about be done with 6th form maths, but I am not
convinced that (m)any sixth formers would be able to follow it!

Mechanics in rotating co-ordinates is not 6th-form maths!
Nor is the general theory of small oscillations. I think it might
be the other way round -- that you couldn't do it with 6th-form
maths, except by spending a lot of time developing a framework for
the proof, but on the other hand that there are some very talented
sixth-formers who would be able to read a proof such as that in Roy,
and fill in the gaps themselves. But that's just an opinion.

Rotating co-ordinates can be used, by those who know how, for objects
moving in circles at constant speed - but what about objects moving in
other conic sections? And remember that, unless previously proved,
conic sections would be a mere assumption.

The conic sections were included in our pure maths syllabus. Proof that
motion under an inverse square law wasn't, but I think you could derive
it from the DE without using any tricks not taught at A level back
then. ISTR that proof was also second year undergraduate physics.

Conic sections are OK of themselves. The bogglement lies in doing what
can be done for circular motion using coordinates rotating at a constant
rate for the case of coordinates rotating at a varying rate.


But for the purposes of this argument doing it for the circular orbit
case without using anything outside A level calculus I think it is just
possible (at least with the syllabus when I did it). Though I can't be
absolutely sure which bits were in Further Maths - quite possibly some
of the DE tricks were in there rather than standard Pure Maths.


I've lost track of the current value of "it".

So I will reiterate that it is *very* easy to show that perfectly
initialised bodies in an equilateral configuration will remain
equilateral, and it is easy enough to show that the same applies to the
collinear configuration and to calculate the spacing, and to see that in
each case the paths are as those for the two-body case. Also, while a
proof that only collinear and equilateral configurations can have that
property may be hard to find, the one found for me is easy enough to
follow with only a simple knowledge of maths and trig.

No centrifugal/centripetal force, no rotating coordinates (in fact, no
coordinates at all).

The question of whether the stability of the equilateral configuration
can be tackled with equally simple tools is still unresolved.

--
(c) John Stockton, nr London, UK. Mail via homepage. Turnpike v6.05 MIME.
Web http://www.merlyn.demon.co.uk/ - FAQqish topics, acronyms and links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.

On 05/11/12 21:20, Dr J R Stockton wrote:
[...] The bogglement lies in doing what
can be done for circular motion using coordinates rotating at a constant
rate for the case of coordinates rotating at a varying rate.

The point about rotating co-ordinates is that in them the
equilibrium position is fixed. In the non-circular case, this just
requires that the co-ordinates rotate non-uniformly and also expand
and contract. But we know how the equilibrium configurations do
this, so there is no difficulty of principle, just a modest amount
of vector algebra and calculus, in setting up the co-ordinates and
working out what mechanics looks like in them. Stability is then
"just" a matter of linearising small perturbations from the "fixed"
position and seeing whether they grow or remain bounded. ...

[...]
The question of whether the stability of the equilateral configuration
can be tackled with equally simple tools is still unresolved.

... It's usually hard to prove that something *can't* be
done by elementary methods! In this case, I'd be very surprised if
the stability can be resolved anywhere near as easily as confirming
the equilibrium configurations in the first place. It's just too
messy; not really difficult, just acres of working, unless you
already have some pretty advanced tools to apply to the problem.
As suggested in a nearby article, I suspect that a modern symbolic
algebra package could easily digest those acres, and thereby make
it much easier to reproduce Danby's results. But that's a long
way from making the entire process accessible at A-level.

--
Andy Walker,
Nottingham.

In uk.sci.astronomy message , Mon, 5 Nov
2012 17:06:15, Andy Walker posted:

On 02/11/12 23:28, Dr J R Stockton wrote:
Danby, Astron J, 69, 4, pp.294-6, May 1964 (GIF) looks a good read [...]

It's online; eg:

http://articles.adsabs.harvard.edu/c..._query?bibcode
=1964AJ.....69..294D

But unless I can find that paper again ...
My reading of it was that as the eccentricity increased to around 0.35,
the secondary can become somewhat heavier than the circular primary/25
before stability is lost.

Yes, that's the result shown in Danby's earlier article
[same URL but with 294 replaced by 165 (the page number)] in his
Figure 1, p171, as mentioned in my previous article.

I had already looked rapidly through them, and added links to
gravity4.htm. I think that neither is the paper that I've been unable
to re-find; but equation 28 of Danby's March paper seems to be the
answer in detail.


--
(c) John Stockton, nr London, UK. E-mail, see Home Page. Turnpike v6.05.
Website http://www.merlyn.demon.co.uk/ - w. FAQish topics, links, acronyms
PAS EXE etc. : http://www.merlyn.demon.co.uk/programs/ - see in 00index.htm
Dates - miscdate.htm estrdate.htm js-dates.htm pas-time.htm critdate.htm etc.

In uk.sci.astronomy message , Tue, 6
Nov 2012 01:55:58, Andy Walker posted:

On 05/11/12 21:20, Dr J R Stockton wrote:

The question of whether the stability of the equilateral configuration
can be tackled with equally simple tools is still unresolved.

... It's usually hard to prove that something *can't* be
done by elementary methods! In this case, I'd be very surprised if
the stability can be resolved anywhere near as easily as confirming
the equilibrium configurations in the first place.

The truth of that statement depends on (1) the "strength" of your
"anywhere near as easily", and (2) the ease of doing the confirming.

Given the minuscule actual value of (2), your (1) is probably not strong
enough, at its lower bound.

It's just too
messy; not really difficult, just acres of working, unless you
already have some pretty advanced tools to apply to the problem.
As suggested in a nearby article, I suspect that a modern symbolic
algebra package could easily digest those acres, and thereby make
it much easier to reproduce Danby's results. But that's a long
way from making the entire process accessible at A-level.

Indeed.

The collinear and equilateral solutions can be shown, with a moderate
amount of straightforward algebra, to be the only ones (and it is then
easy to find one quintic for the collinear proportions); I'd be pleased
to find a proof of the stability boundary (i.e. 24.96, for circular
equilateral) that was equally straightforward and not very much longer.


Here's something that the OP and others could perhaps tackle during the
Christmas Holidays - how many distinctly different descriptions of what
Lagrange actually did in regard to the equilateral configuration can be
found in popular and semi-popular descriptions on the Web, and what
proportion of the total descriptions are actually more-or-less correct?
Give URLs.

--
(c) John Stockton, nr London, UK. E-mail, see Home Page. Turnpike v6.05.
Website http://www.merlyn.demon.co.uk/ - w. FAQish topics, links, acronyms
PAS EXE etc. : http://www.merlyn.demon.co.uk/programs/ - see in 00index.htm
Dates - miscdate.htm estrdate.htm js-dates.htm pas-time.htm critdate.htm etc.