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Antoine has raised the question, though as I see it a somewhat trivial 
question, as to how much the unresolved image of a planet should be 
displaced, depending on its phase,.Trivial, that is, in terms of the 
numerical amount of the discrepancy, though in-principle it may well be of 
some interest. And, as Antoine suggests, a bit of fun can be got out of 
such hair-splitting.

This question mainly applies to observations of Venus, the only 
navigational planet to go through the complete range of Moon-like phases. 
The others are all seen full-on, or rather nearly so.

So I've been tempted into integrating up the shape of a planet's "crescent" 
(hollow or gibbous), assuming constant brightness, to see how far the 
centre of gravity (barycentre) of that light-patch is displaced from the 
planet's true centre. This makes the assumption, which may or may not be 
justified, that the eye would position the centre of such a blurred image 
at or near that barycentre.

If we follow Meeus in using k as the illuminated fraction of the planet's 
disc, and use r for the angular radius of the planet.

Then the lit patch can be defined as the area intercepted between a 
semicircle of radius r and one half or other of an ellipse, of major axis 2 
r (shared with the base of the semicircle) and minor axis 2 r (1 ~ 2 k ). 
When k = 0.5, a half-lit planet, this ellipse degenerates to a straight 
line bisecting the planet, as expected.

Antoine has named P to be the mid-point, on the bisector of the crescent, 
between the two edges of the lit area. That implies, to me, that P is 
displaced from the planet's centre by r ( 1 - k ), and Antoine appears to 
agree..

And he has, somewhat arbitrarily, defined another point M at half that 
radius, at r ( 1 - k ) * 0.5

He has had difficulty in choosing which of those values to use to represent 
the offset as visually observed, but inclined towards P. I have argued that 
P is implausible, because of the curved shape of such a crescent, and a 
somewhat smaller displacement should be chosen, taking account of the 
barycentre of the light patch, but have until now stopped short of 
predicting how much smaller it should be.

Antoine, in his latest post, has invited me to agree to an "ideal point", 
midway between M and P, so at r ( 1 - k ) * 0.75  . However I think we can 
make a choice on a more logical basis. With a bit of integration, I now 
find that the answer is rather a simple one. I make the displacement of 
that light-barycentre from the planet's centre to be  r ( 1 - k ) * 8 / 3pi 
, which works out to be  r ( 1 - k ) * 0.85  . However, my maths is 
notoriously fallible, so it would be nice if someone else were to check it.

If I've got it right, then Antoine's choice of P is the better guess of the 
two, but he could do better still by inserting that factor of 0.85. But the 
difference is only going to be a very few arc-seconds, and in practice it 
hardly matters, either way.

===================

What was discussed above was "point 2" in Antoine's previous post. We now 
appear to agree about his point 1, that Meeus didn't treat this matter 
anywhere in his Astronomical Algorithms. The only quote that Antoine could 
find was this-

" The position angle of the mid-point of the illuminated limb of a planet 
can be calculated in the same way as for the Moon - see Chapter 51 ".

In my 2nd edition, that cross-reference is to chapter now-numbered 48, and 
it's clear from that chapter that the position angle discussed there is 
quite a different angle altogether, and has no relevance to the 
displacement from the true centre of the illuminated patch. So we can now 
eliminate Meeus from this enquiry.

George

contact George Huxtable at george{at}hux.me.uk ,
at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

----- Original Message ----- 
From: "Antoine Couette" 
To: 
Sent: Friday, September 17, 2010 4:28 PM
Subject: [NavList] Re: DSLR Venus Lunar


In further reply to [NavList 13884] Re: DSLR Venus Lunar 16 Sep 2010 18:29 
from From: george---me.uk


Sep 17, 2010


Dear George,



In an attempt to "get right" rather than "get nearly-right" (I'm using your 
own terms here)the offset value computation (both its orientation and its 
magnitude) between :

- the planet geometric center, and

- the visual center of gravity of the light of its illuminated center

I will answer your queries in the following 2 areas which you have covered 
lately:

PART 1 - request for any quotation by Jean Meeus about "Point P", and

PART 2 - a more careful review of the computations carried out by the 
NAUTICAL ALMANAC. It shows that the NAUTICAL ALAMANAC seems to use (or at 
least did use some 30 years ago) an offset point quite close from "Point P" 
(and certainly closer from "Point P" than from "Point M") as a figure for 
computing the Venus phase effect (angular value between its geometric and 
illuminated centers).

*******

PART ONE

QUOTE FROM YOUR POST [NavList #13881]:

Antoine writes-
"NOTE: the "phase effect" correction values given hereabove are computed
through assuming that the "planet center of light" is the "mid-point of its
illuminated disk as seen from the Earth" . This computation method seems a
customary and well accepted one (See Jean Meeus's "ASTRONOMICAL ALGORITMS",
Chapter 40 "Illuminated Fraction of the Disk and Magnitude of a Planet").

My own copy of Meeus is 2nd edition of 1998, in which the chapter with that
title is 41, not 40. In that chapter, I can't find the words that Antoine
quotes.

UNQUOTE

and

QUOTE FROM YOUR POST [NavList #13884]


" I can find no mention, in chapter 41 of my edition of Meeus, to the point
on the axis of symmetry that bisects the two edges of the lit region,
corresponsing to Antoines "point P". I think it must have been dropped.
Perhaps Antoine will quote Meeus' very words on that matter. "

UNQUOTE

I WILL OFFER THE FOLLOWING REPLY :

In any of the following books by Meeus (ASTRONOMICAL FORMULAE FOR 
CALCULATORS, ASTRONOMICAL ALGORITHMS, PLANETARY PROGRAMS AND TABLES) which 
I have, the only "possible" explicit reference ( "possible" because it 
could be interpreted differently ) to "Point P" - as I earlier defined it - 
shows at the bottom of page 267 of my copy of ASTRONOMICAL ALGORITHMS :

" The position angle of the mid-point of the illuminated limb of a planet 
can be calculated in the same way as for the Moon - see Chapter 51 ".

I must confess that, until now, I have not been able to retrieve in any of 
my Books by Jean Meeus :

- nither any other explicit reference to "Point P",
- or any endorsement by Jean Meuus that such "Point P" should or even could 
be used as an offset point to compute the phase effect for navigation 
purposes.

*******

HOWEVER,

There is an obvious and direct relationship between such "Point P" and the 
"illuminated fraction k of the disk of a planet", which is abundantly 
covered by Jean Meeus in the quoted chapter.

As I am too lazy to either do again these calculations performed almost 30 
years ago, or even to dig them out as they are buried deep somewhere into 
my archives, and to the best of my recollection the relationship between 
"Point P" and "k" is a most simple one :

If we call "p" the ratio of the distance between the Planet center and 
"Point P" divided by the Planet semi-diameter, then :

p = 1 - k,   just this simple !

For a 100% illumination, k=1, and p=0, which means P is exactly in the 
center of the apparent disk, and

For a 50% illumination when the planet looks like a "half Moon", P is 
halfway between the center and its circular edge rim, k=0.5 and p=0.5 and

For a 0% illumination, P is exactly on the planet circular edge with k=0 
and p=1.

It is very possible that, so many years after I first carried out and 
shortly therafter buried these computations, the memories left in my mind 
were that Jean Meeus did pay attention to "Point P", while he only kept 
paying a close attention to "k".

It is also possible that somewhere in some other reference book (I own a 
few of them) "Point P" is described as an adequate way to compute the 
planets phase effect. I have not been able to find such reference, which is 
probably not essential since, at least some time ago, the NAUTICAL ALMANAC 
did use a point quite close from "Point P" to compute the phase effect.


*******

PART TWO

QUOTE FROM YOUR POST [NavList #13884]

To me, taking that point to be the apparent centre-of-light seems
implausible. If the distribution of light was a uniform lit rectangle, on
that axis, between those limits, it would then be expected to show a
centre-of light at point P. Is Antoine asking us to believe that it makes
no difference at all, to the apparent centre, if the tapered ends of that
light distribution are bent around in a pair of curved horns? I don't
accept that. In my view, the centre of gravity of the lit crescent is a
much more plausinle model. Not that it matters much, in practice, the
difference being no more than a few arc-seconds. But we might as well get
it right, as get it nearly-right.

UNQUOTE

I will here show that the HMSO and US NAUTICAL ALMANAC did use at some time 
/ still keep using (?) a point very close from "Point P" to account for the 
Planets phase effects.

Just back to the example I quoted in a recent post.

FROM THE ASTRONOMICAL ALMANACH FOR THE YEAR 1982 EXPLANATION SECTION bottom 
of page 259


QUOTE

The additionnal corrections for Venus and Mars allow parallax and phase, 
and are given by " p cos H - k cos theta", where H is the altitude, theta 
the angle at the planet between the vertical nad the Sun: p and k are, for 
Venus, for 1982:

from Jan.1 until Feb. 10  p = 0'5 and k=0'4

UNQUOTE

For the date of 15 Jan 1982,00h00m00.0s TT, we get the following apparent 
data about Venus as follows:

Right Ascension = 20h34m42s1 , Declination = S-14°10'24", Distance from 
Earth = 0.28238, Horizontal Parallax = 0'52, semi-diameter = 0'51, angular 
distance between Planet Center and "Point P" = 0'50

A quick study of the NA "correction formula" hereabove shows that that term 
"p cos H" is the well known term for correcting for parallax.

Accordingly the term " k cos theta" computes the phase effect Correction 
(from "observed" into "true" to be performed AWAY from the SUN). And the 
tabular The amplitude of this phase correction (0'4) for this period of 
time makes it quite close from "Point P" (0'5).

CONCLUSION AND SUGGESTION :

We have certainly attempted our best to get things "right" rather than 
"nearly-right" by splitting hair just as we did, no ? I can and will take 
the blame for that ! However it is fun !

HOW ABOUT SETTLING FOR AN "IDEAL POINT" HALFWAY BETWEEN "POINT M" AND 
"POINT P" ?

Frank !!! Are u here ???

Best Regards to you all

Kermit

Antoine M. "Kermit" Couëtte

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