Semi-minor Axis

Mike Williams said

Wasn't it JG who wrote:
What I now see (from my calculations) is that the semi-minor axis is in
fact the same as the Perihelion distance. Which probably explains why I
used to think that Aphelion and Perihelion referred to the semi-major
and semi-minor Axies, ie. I was half right.

No that's not right either.

If you take a look at this ellipse,
http://upload.wikimedia.org/wikipedi.../d3/Elipse.png
you can see that in this case the perihelion (A-F1) is much shorter than
the semi-minor axis (b). [That diagram is correctly drawn. I checked.]

Both your ideas would have been right if the Sun were at the centre of
the ellipse, but it isn't. The Sun is at one focus of the elliptical
orbit and the other focus is empty.

The detail of the ellipse shown at the URL is precisely why I was (am)
having problems understanding.

I am fully aware of that the length F1-X-F2-F1 is constant and is what
defines the ellipse and that is why I could not fathom why my
calculations showed that the Perihelion was equal to the semi-minor
axis. I was also fully aware that the Sun is at one focus and the other
is empty.

I understand that A-F1 is the Perihelion and that F1-B is the Aphelion
and that 'a' is the mean/average radius.

What I can't yet understand is how one calculates 'b' given Mean/Average
Radius and Eccentricity - is it in fact possible, or does one need
some other information?

Am I correct in thinking that the Eccentricity is given by 1-(CD/AB) ?

JG

Wasn't it JG who wrote:

Am I correct in thinking that the Eccentricity is given by 1-(CD/AB) ?

e = sqrt(1-(CD^2/AB^2))

--
Mike Williams
Gentleman of Leisure

Go ahead and fit the Newtonian description into an elliptical framework
-

http://www.pfm.howard.edu/astronomy/...S/AACHCIR0.JPG

"PHÆNOMENON IV.
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


http://members.tripod.com/~gravitee/phaenomena.htm

Ugly ,very,very ugly !.


Whatever giants Newton was standing on I assure you they were not
Copernican or Ptolemaic although there may be some astrologers around
who can help you.

Now,the cataloguers here have some pretty pictures to show you by way
of consolation like the non existent varuiable axial tilt property of
the Earth to the Sun/Orbital plane.

http://solar-center.stanford.edu/ima...130000-UTC.jpg

Mike Williams said

Wasn't it JG who wrote:

Am I correct in thinking that the Eccentricity is given by 1-(CD/AB) ?

e = sqrt(1-(CD^2/AB^2))

Thanks Mike !

I am indebted to you - my calculations no longer show the semi-minor
axis = Perihelion.

I knew it had to be a basic premise that I was missing and due to your
prompting I have now found http://mathworld.wolfram.com/Ellipse.html
where that same formula is given at (33).

This seems to fly in the face of the Nine Planets Glossary where it states :
eccentricity
the eccentricity of an ellipse (planetary orbit) is the ratio of the
distance between the foci and the major axis. Equivalently the
eccentricity is (ra-rp)/(ra+rp) where ra is the apoapsis distance and rp
is the periapsis distance.

I cannot imagine that there are two definitions of eccentricity - one
for general mathematics and another for planetary motion so would be
grateful for your take on the difference.

JG

Wasn't it JG who wrote:

Mike Williams said

Wasn't it JG who wrote:

Am I correct in thinking that the Eccentricity is given by 1-(CD/AB) ?

e = sqrt(1-(CD^2/AB^2))

Thanks Mike !

I am indebted to you - my calculations no longer show the semi-minor
axis = Perihelion.

I knew it had to be a basic premise that I was missing and due to your
prompting I have now found http://mathworld.wolfram.com/Ellipse.html
where that same formula is given at (33).

This seems to fly in the face of the Nine Planets Glossary where it states :
eccentricity
the eccentricity of an ellipse (planetary orbit) is the ratio of the
distance between the foci and the major axis. Equivalently the
eccentricity is (ra-rp)/(ra+rp) where ra is the apoapsis distance and rp
is the periapsis distance.

I cannot imagine that there are two definitions of eccentricity - one
for general mathematics and another for planetary motion so would be
grateful for your take on the difference.

They are exactly the same thing. It's just that astronomers are more
likely to be able to observe the periapsis and apoapsis distances,
whereas mathematicians are more likely to know the lengths of the semi-
major and semi-minor axes (particularly if they start from the equation
of the ellipse in the form "x^2/a^2 + y^2/b^2 = 1".

If you follow the mathworld page you mentioned from equation (33) to
equation (39) you'll see that they show that

e = sqrt(1 - a^2/b^2)

is equivalent to

e = c/a

where c is the distance from the centre to a focus.

ra is (a+c) and rp is (a-c), so e = (ra-rp)/(ra-rp) = 2a/2c = c/a

--
Mike Williams
Gentleman of Leisure

JRS: In article k,
dated Sat, 31 Dec 2005 00:01:58 local, seen in news:uk.sci.astronomy, JG
posted :

I used to think that Aphelion referred to the semi-major axis of the
ellipse described by the Earth (or any planet) on its journey around the
Sun and Perihelion was the semi-minor axis. Having read most of the
planetary data from 'The Nine Planets' web site I now see that the sum
of Aphelion and Perihelion is in fact the major axis of that ellipse.
OK.

What I cannot understand is the statement in the Glossary that Aphelion
is also the 'average' or mean distance of the planet from the Sun.

ISTM that there are two interpretations : (a) it's wrong, (b) you read
it wrong.


Surely the maximum distance cannot also be the mean?

It can be if it is also the minimum, but not otherwise.

The sum of Ap & Per is the Major Axis; therefore the semi-major axis is
the average of the greatest and least distances, which is one of the
many possible means. For small eccentricities, most of the possible
means are about the same.


What I really want to know is how to calculate the semi-minor axis.
Given the 'Mean' and the eccentricity I can readily calculate the Major
as a(1+e) and the Minor as a(1-e) but if the mean is also the Major then
this doesn't make sense.

Some of that may be in URL:http://www.merlyn.demon.co.uk/ellipses.htm
- I'd like to describe that as a work in progress, but progress has been
stopped for a while ...

--
© John Stockton, Surrey, UK. Turnpike v4.00 MIME. ©
Web URL:http://www.merlyn.demon.co.uk/ - FAQqish topics, acronyms & links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.
No Encoding. Quotes before replies. Snip well. Write clearly. Don't Mail News.

Newtonian yes, geocentric no. If you are thinking of my "equivalents for
other primary bodies" I mean, for example, major planets relative to their
own moons, not to each other or the Sun!

Since you are now accusing me, I will use the opportunity to make some
points about your comments in general.
(1) You cite references to the phases of Venus as being geocentric. In fact
the ability to see them was evidence against a heliocentric model, as they
showed that Earth and Venus were sometimes the same side of the Sun and
sometimes opposite sides.
(2) One of your quotations of Newton says that from Earth the planets APPEAR
sometimes direct, sometimes stationary, sometimes retrograde. In other words
he was acknowledging that that was an illusion caused by Earth's own
movement relative to the Sun.
(3) Yes, references to things being in constellations are astrological, but
they are being used as a METAPHOR. I have myself been criticised for
pedantry about such matters. It is easier for readers to picture a location
relative to an arbitrary alignment of unrelated stars than to a set of
numerical coordinates.

"oriel36" wrote in message
oups.com...
To Charles

Your thinking is strictly Newtonian quasi-geocentric, an astronomical
conception that owes more to astrology than Copernican heliocentricity
or its antecedent Ptolemaic geocentricity.If you are in any doubt or
are completely unfamiliar with Newton's mangling of Copernican
heliocentricity and its later Keplerian refinement then that is O.K.
but I assure you the Newton conception is horrific in comparison to
Ptolemaic astronomy never mind Copernican.

" PHENOMENON IV.
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."

http://members.tripod.com/~gravitee/phaenomena.htm

Even the Ptolemaics had severed the motions of the planets from the
stellar background to generate their idea of epicycles and although
they attributed the position of the Sun between Venus and Mars,where
the hell are you going to justify the position of the Sun in Newton's
really dumb "(whether of the sun about the earth, or) of the earth
about the sun,"

Not only has the greatest Western heliocentric achievement and its
appreciation been destroyed but even the antecedent nobility of the
planetary motion plotting of Ptolemaic astronomers joins the
destruction.

The planetary motions in retrograde refer to the plotting with the
stellar background ,what the Ptolemaics seen as epicycles,the
Copernican heliocentrists rightly identified as a faster Earth ,moving
in an inner orbital circuit overtaking the slower moving outer planets
-

http://antwrp.gsfc.nasa.gov/apod/ima...2000_tezel.gif

No jumping to the Sun to infer heliocentricity and no retrogrades
involved *,just the altering of a Ptolemaic stationary Earth to an
annual orbital motion.

* "For to the earth they appear sometimes direct, sometimes stationary,
nay, and sometimes retrograde. But from the sun they are always seen
direct.."

http://members.tripod.com/~gravitee/phaenomena.htm

Newton and his disciples did not just destroy heliocentric
astronomy,they ruined a heritage that stretches back millennia.

In message , Charles Gilman
writes

Since you are now accusing me, I will use the opportunity to make some
points about your comments in general.
(1) You cite references to the phases of Venus as being geocentric. In fact
the ability to see them was evidence against a heliocentric model, as they
showed that Earth and Venus were sometimes the same side of the Sun and
sometimes opposite sides.
(2) One of your quotations of Newton says that from Earth the planets APPEAR
sometimes direct, sometimes stationary, sometimes retrograde. In other words
he was acknowledging that that was an illusion caused by Earth's own
movement relative to the Sun.

GK posts this over and over again without any sign that "he" actually
understands what "he" is saying - quotation marks because "his"
responses are so limited they fail the Turing test :-)
I suspect that he thinks retrograde motion is _not_ an illusion.
But he's a troll at best, and best ignored.

Mike Williams said
Wasn't it JG who wrote:

I knew it had to be a basic premise that I was missing and due to your
prompting I have now found http://mathworld.wolfram.com/Ellipse.html
where that same formula is given at (33).

This seems to fly in the face of the Nine Planets Glossary where it
states :
eccentricity
the eccentricity of an ellipse (planetary orbit) is the ratio of the
distance between the foci and the major axis. Equivalently the
eccentricity is (ra-rp)/(ra+rp) where ra is the apoapsis distance and rp
is the periapsis distance.

I cannot imagine that there are two definitions of eccentricity - one
for general mathematics and another for planetary motion so would be
grateful for your take on the difference.

They are exactly the same thing. It's just that astronomers are more
likely to be able to observe the periapsis and apoapsis distances,
whereas mathematicians are more likely to know the lengths of the semi-
major and semi-minor axes (particularly if they start from the equation
of the ellipse in the form "x^2/a^2 + y^2/b^2 = 1".

I would consider myself in the second category - but very lapst!

If you follow the mathworld page you mentioned from equation (33) to
equation (39) you'll see that they show that

I did follow on but once they introduced 'c' (seemingly from nowhere) I
could not follow the logic. Again with your prompting I looked back to
the diagram above (24) where I can now see that 'c' is the distance from
(0,0) to the focus. Because is was not explicitly dimensioned as 'a' and
'b' are, that fact had eluded me.

As I said above, I haven't needed to use maths seriously for too many
years. The transposition from e = sqrt(1-(CD^2/AB^2)) to find CD was no
great shakes but the likes of the formulae on the mathworld site cause
me to stop and think between each line

Oddly enough it was remembered knowledge of the Ellipse gained during my
education in the '50s (specifically that the distance from the directrix
to the ellipse is equal to the distance from the ellipse to the focus -
though I now see that the more general case where it is the ratio that
is constant - I recall being taught that the directrix was the same
distance outside as the focus was inside) that made me question how my
calculations made the semi-minor axis = Perihelion in the first place.

JG

Charles Gilman wrote:
Newtonian yes, geocentric no. If you are thinking of my "equivalents for
other primary bodies" I mean, for example, major planets relative to their
own moons, not to each other or the Sun!


This blustering and bluffing is fine but ultimately Newton's creation
is astrological for very specific and technical reasons.It is fine to
present conics in the language of algebra as a means to project a
ballistic idea on planetary motion,it is quite another to try and fit
it into actual observation.If you do not see that the .986 deg /3 min
56 sec orbital displacement generates the repugnant spectacle of the
Earth travelling faster at the aphelion and slower at the perihelion
then you are no longer in the realm of anything I or anyone else can
help you with -

http://www.pfm.howard.edu/astronomy/...S/AACHCIR0.JPG


The description which Flamsteed justified the direct linking of
terrestial longitudes to the celestial sphere in 23 hours 56 min 04 sec
is the same one which give Newton his mean Sun/Earth distances,both
respresent misconduct or if you want catasstrophic misjudgement.



Since you are now accusing me, I will use the opportunity to make some
points about your comments in general.
(1) You cite references to the phases of Venus as being geocentric. In fact
the ability to see them was evidence against a heliocentric model, as they
showed that Earth and Venus were sometimes the same side of the Sun and
sometimes opposite sides.

Accusations,naw,I don't have time for
accustations,claims,challenges,duals and all that etiquette of 17th
century powdered wigs and large ruffled collars.These people were
pariahs and this is what happens when their works come under the
scrutiny of a 21st century mind with 21 century tools and data.If you
want to defend the genius of geniuses then be my guest,I am just
working with an astronomical fluency derived from those I really adore
and respect and would have others appreceate that astronomy is a
fountain rather than a cistern.

Extraneous evidence for heliocentricity similar to that which Galileo
tried to provide through Jupiter's moons is always going to be inferior
to the original exquisite reasoning that distinguishes the Ptolemaic
explanation for observed planetary motion and its satisfactory
Copernican heliocentric resolution.Anyone who sees a faster Earth
moving in an inner orbital circuit to the slower moving outer planets
automatically infers heliocentricity or a common axis for planetary
motion around the Sun.

http://antwrp.gsfc.nasa.gov/apod/ima...2000_tezel.gif

Obviously you do not look at that actual time lapse footage of Jupiter
and Saturn long enough for I would not have to come here and recycle
the fact again and again that the motions of the planets are always
seen direct,even the Ptolemaic descriptions of periodic looping motions
(epicycles) are presented as direct motion.

As this is not just an exercise in astronomical forensics,the 'crescent
of Venus' represents the same careless treatment of our own planet's
change in orbital orientation to the Sun and especially the effects of
that change to fixed axial orientation.I could not even begin to list
the avenues open to those who drop an non existent variable axial tilt
of the Earth to the orbital plane as a hemispherical explanation for
cyclical climate change.

http://www.diduknow.info/sun/images/high_low_sun.gif

By creating an analemmatic fudge in the 17th century cataloguers
introduced a variable axial tilt to the Earth against the Sun thereby
cutting off the ability to accurately describe cyclical seasonal norms
as a product of a change in orbital orientation and more
importantly,they associated seasonal daylight/darkness with the
Equation of Time *.

The climatologists and probably a lot of people on the planet would
like an accurate mechanism for what causes global cyclical seasonal
changes,retaining a common axis for axial and orbital motion and
allowing the Sun to drift up and down against the Equator is about as
intellectually low as it is possible to go.











(2) One of your quotations of Newton says that from Earth the planets APPEAR
sometimes direct, sometimes stationary, sometimes retrograde. In other words
he was acknowledging that that was an illusion caused by Earth's own
movement relative to the Sun.

I have no doubt that the temptation to defend Newton at all costs gains
favors here however this section is total astronomical conceptual
forensics based on the transition from Ptolemaic conclusions for
observed planetary motions to its correct resolution through Copernican
reasoning.Newtonian quasi-geocentricity represents neither .If people
cannot stomach or understand the differences then they have no aptitude
for astronomy and are best left to optical astronomy or the exercise
in making celestial objects appear larger through optical equipment.

"For to the earth planetary motions appear sometimes direct, sometimes
stationary, nay, and sometimes retrograde. But from the sun they are
always seen direct..."

http://members.tripod.com/~gravitee/phaenomena.htm

The plotted motions of the planets against the constellations generate
the forward-backward- forward motion (retrograde) however the
Ptolemaics had concluded that they are actually direct motions of a
periodic looping nature.Contemporary time lapse footage allows people
to see how Ptolemaics concluded the looping motion against a stationary
Earth.

So,even before going appealing to Copernican resolution for observed
planetary motion by infering a common heliocentric axis,the arrangement
of planets and how observed speed assigns the heliocentric location of
planets Newton has it wrong or rather does not even approach the level
of the Ptolemaics.

Again,you either have an aptitude for making the correct astronomical
conclusions or you do not,with contemporary time lapse footage it is
now easier than ever to grasp the cycles that are observed as we look
out on the planets and Sun.



(3) Yes, references to things being in constellations are
astrological, but
they are being used as a METAPHOR. I have myself been criticised for
pedantry about such matters. It is easier for readers to picture a location
relative to an arbitrary alignment of unrelated stars than to a set of
numerical coordinates.


Neither Ptolemaic nor Copernican resolutions for the motions of the
planets retain the stellar constellations and this extends to the
Keplerian and Roemerian refinements which are complimentary additions
to heliocentric astronomy.

The Roemerian insight which produces small anomalous 'finite light
distance' effect within the solar system in terms of the observed
position of Io and its actual position would produce a very large one
in terms of the position of external galaxies to our own.Even outlining
the principles which incorporate the actual motion of the foreground
Milky Way stars to the external galaxies requires an intense focus
never mind having to run a gauntlet of those who fight to retain a
celestial sphere for the local stars,for heliocentric descriptions or
the utterly subhuman 'every valid point is the center' bunch.

Ultimately Newton,through Flamsteed's original misconduct, turned the
constellations from an ancient metaphor into a working empirical
principle -

"PHENOMENON IV.
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."

You probably do see the quasi-geocentricity of "whether of the sun
about the earth, or) of the earth about the sun " but it is the
reference to the fixed stars which makes hios conceptions closer to
astrology than to Ptolemaic geocentricity.

Now,there will probably be a lot of algebraic thumbsucking with
ellipses,focal points,conics and so on however these same people are
better served by returning to the original reasoning and to be less
focused on isolating the solar system from the rest of the
Universe,something Newton required to get his ballistic agenda to work.

"Cor. 2. And since these stars are liable to no sensible parallax from
the annual motion of the earth, they can have no force, because of
their immense distance, to produce any sensible effect in our system.
Not to mention that the fixed stars, every where promiscuously
dispersed in the heavens, by their contrary actions destroy their
mutual actions, by Prop. LXX, Book I."

I have always allowed that the U.K.. guys should make the attempt to
reign in the excesses of their 17th/18th century colleagues and at
least this year they will have the clear distinction between Newtonian
quasi-geocentricity,Ptolemaic geocentricity and Copernican
heliocentricity to work with.It may be just a case of astronomical
aptitude being shouted down by empirical noise however for those who
can simply enjoy how Ptolemaic epicycles transfer into Copernican
heliocentric thinking then they have nothing to fear and cannot really
go wrong.







"oriel36" wrote in message
oups.com...
To Charles

Your thinking is strictly Newtonian quasi-geocentric, an astronomical
conception that owes more to astrology than Copernican heliocentricity
or its antecedent Ptolemaic geocentricity.If you are in any doubt or
are completely unfamiliar with Newton's mangling of Copernican
heliocentricity and its later Keplerian refinement then that is O.K.
but I assure you the Newton conception is horrific in comparison to
Ptolemaic astronomy never mind Copernican.

" PHENOMENON IV.
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."

http://members.tripod.com/~gravitee/phaenomena.htm

Even the Ptolemaics had severed the motions of the planets from the
stellar background to generate their idea of epicycles and although
they attributed the position of the Sun between Venus and Mars,where
the hell are you going to justify the position of the Sun in Newton's
really dumb "(whether of the sun about the earth, or) of the earth
about the sun,"

Not only has the greatest Western heliocentric achievement and its
appreciation been destroyed but even the antecedent nobility of the
planetary motion plotting of Ptolemaic astronomers joins the
destruction.

The planetary motions in retrograde refer to the plotting with the
stellar background ,what the Ptolemaics seen as epicycles,the
Copernican heliocentrists rightly identified as a faster Earth ,moving
in an inner orbital circuit overtaking the slower moving outer planets
-

http://antwrp.gsfc.nasa.gov/apod/ima...2000_tezel.gif

No jumping to the Sun to infer heliocentricity and no retrogrades
involved *,just the altering of a Ptolemaic stationary Earth to an
annual orbital motion.

* "For to the earth they appear sometimes direct, sometimes stationary,
nay, and sometimes retrograde. But from the sun they are always seen
direct.."

http://members.tripod.com/~gravitee/phaenomena.htm

Newton and his disciples did not just destroy heliocentric
astronomy,they ruined a heritage that stretches back millennia.


* http://imagine.gsfc.nasa.gov/docs/as...s/980116c.html