At another thread under Special and General Relativity the following question was asked: "Why isn't stellar aberration considered to be a one way measurement of c? If the angle of aberration (θ-θ_{0}) is -20.5 arc seconds and the Earth's orbital speed is 29.79 Kilometers/second normal to the arriving star light. The value of c should be: c= v/tan(θ-θ_{0}) = 29.79/9.94E-5 = 299737.98 Kilometers/second. If more accurate values of v, θ, and the constant of aberration (≈ -20.5 arc seconds) are used why would this not confirm a one way measurement of c in the more pure vacuum of deep space?" In that thread I attempted to get a clarification of the precise meaning of Bradley's "Classical" stellar aberration model [tan (θ - θ_{0} ) = -v/c]. However, the participants on that thread seemed to think that I was hijacking the intent of the thread and asked me to open a new thread in astronomy because my arguments did not seem to be relevant to the measurement of c. Therefore, I am opening this thread in Astrophysics to get a clarification from the real astrophysics experts on the correct interpretation and meaning of Einstein's 1905 SRT "Relativistic" stellar aberration model as well as Bradley's "Classical" model and their relevance in determining the value of c when the precise angle θ is known by way of accurate telescopic observation from a moving point on Earth's orbit of the angle θ for a star that is known to be located at θ_{0} = 90° for an observer in the inertial frame of the "fixed" stars (including the sun) in our neighborhood of the milky way galaxy. I will open the discussion with the following comment and question: First, we must determine if the star that satisfies the Bradley “Classical” aberration model is a celestial pole star or an ecliptic pole star. If Bradley’s model is referencing an ecliptic pole star then θ_{0} = 90° and tan (θ - θ_{0}) = -v/c and when c = 299737.98 Kilometers/second then θ - θ_{0} = -arctan v/c = -20.5 arc seconds. Would this equation then verify that c = 299737.98 Kilometers/second because an ecliptic pole star is the same “fixed” ecliptic pole star in the inertial frame of our Sun and the Earth’s ecliptic plane?
As one of the people recommending a new thread here (or in general astronomy), I would like to add a little background. Here, the question of determination of c is asked, which actually is consistent with the original thread (which asked, can you really determine one way light speed using measurements of stellar aberration?). However, in that thread, Ken More never asked about measurement of c. Instead, he was seeking detailed justification of how the tables of declination aberration and right ascension aberration in the Astronomic Almanac were derived from (or consistent with - he had doubts) the typical simple formulas for total aberration. Several of us there pointed out that a given total aberration factors into declination aberration and right ascension aberration; and also that the ecliptic pole is not the celestial pole (polaris). However, since the focus and expertise of that forum is not detailed computations in astronomic coordinates, and this topic had nothing to do with that thread, we asked that he come here. Only now, he asks about light speed measurement with aberration (a discussion still going on in the other thread) instead of what we referred him here for ....
As PAllen has stated, I did limit my discussion under Special and General Relativity to trying to get an understanding and agreement about the correct interpretation of Bradley’s “Classical” stellar aberration equation [tan (θ - θ_{0}) = -v/c] and predictions from the Astronomical Almanac’s (AA) reduction for annual aberration equations that predict declination aberration and right ascension aberration for given stars. I felt and continue to feel this level of detail is necessary as a first order of business to getting an understanding and agreement as to whether the “Classical” equation can be used to precisely estimate the velocity of light in a vacuum. I did not explain this in the Special and General Relativity thread because I basically agreed with PAllen, et.al. that this level of detail involving astronomical coordinate systems and the AA’s aberration predictions would not be recognized as being important or relevant to the measurement of c by physicists who are much more interested in the theoretical concepts of Special and General Relativity. I expect that actually using a telescopic to measure stellar aberration of a star near an ecliptic pole or even a discussion of the state-of-the-art and the empirical data produced by this work typically done by Astrophysicists or Astronomers to verify or prove a theory would be annoying details for most physicists who only care about Special or General Relativity theory. Therefore, I think that the subject can be more fruitfully debated by Astrophysicists who have the measurement skills, interests and enough experience to deal with the details I would like to debate. To start the debate, I will restate the first question for which I would like an answer from qualified Astrophysicists: (1) Is James Bradley’s “Classical” stellar aberration equation [tan (θ - θ_{0}) = -v/c] referencing a star where θ_{0} is on an ecliptic pole which is perpendicular to the Earth’s orbital inertial velocity vector?
This question is easy. Just see the first few paragraphs of : http://www.mathpages.com/rr/s2-05/2-05.htm It is for total aberration of a star located perpendicular to the earths orbit, which means 23.5 degrees from the celestial pole (appx. = polaris). The aberration for such a star will vary between pure declination aberration twice a year, and pure right ascension declination twice a year, with mix in between. This much has already been explained to you several times. What I and others were not prepared to do is explain detailed tables and formulas in the Almanac, especially because it is not available on line and none of us over in that forum have a copy of it. The hope was that someone here might be familiar with it or own a copy.
Can I assume that your answer to question (1) is: James Bradley’s “Classical” stellar aberration equation [tan (θ - θ_{0}) = -v/c] applies only to a star where θ_{0} is on an ecliptic pole which is perpendicular to the Earth’s orbital inertial velocity vector? Please answer Yes or No with no qualifications or elaborations.
YES (mostly - see below). I will not be constrained to inaccurate answers. It actually applies to any star whose direction is perpendicular to the orbital inertial velocity vector at that time, which always includes the north and south ecliptic poles, but (at any given time) includes a circle of directions.
It is true as you imply that the pole (I suggest calling it the y’ axis) upon which the given star in question exists may exist anywhere in a plane that is perpendicular to the Earth’s orbital inertial velocity vector when different “Arbitrary Floors” are chosen to define different coordinate systems (Please see “Arbitrary Floor” in “Aberration Metrics Schematic Diagram” attached). However, for the sake of argument about the ability of the “Classical” equation to predict the declination aberration of a star or to estimate the value of c when (θ – θ_{0}) is known, I would like to confine my questions as well as the answers to MY questions to the coordinate system where right ascension aberration is measured in the arbitrary floor that is in the Earth’s orbital plane (the ecliptic plane). The use of this coordinate system with a floor (the x’z’ plane) that is the ecliptic plane will facilitate understanding of the exact metrics being addressed when we discuss declination aberration (θ – θ_{0}) [shown as ϕ – θ in the attachment] and right ascension aberration (α – α_{0}) [shown as α’ – α in the attachment]. Thanks to your qualified reply to question (1), I now understand that I did not properly define the coordinate system that is relevant to the question that I intended to ask. Therefore, I will restate question (1) as follows: (1) James Bradley’s “Classical” DECLINATION stellar aberration equation [tan (θ - θ0) = -v/c] CAN APPLY to a star where θ_{0} is on an ecliptic pole which is perpendicular to the Earth’s orbital inertial velocity vector as well as perpendicular to the Earth's ecliptic plane and where the right ascension angle α_{0} is in the ecliptic plane that is perpendicular to the ecliptic pole and describes the Earth’s location (similar to the sidereal day but in degrees instead of days) with respects to the Earth’s inertial velocity vector? Please answer Yes or No unless you feel that this question is also too ambiguous to communicate an unequivocal question that is lucid enough to be understood by qualified astrophysicists.
I don't find this sufficiently clear. I will propose a wording to which you can respond how close it is to your intent: Using an ecliptic coordinate system [see http://en.wikipedia.org/wiki/Ecliptic_coordinate_system], Bradley's formula [tan (θ - θ0) = -v/c] will give a pure latitude deviation for a star at either ecliptic pole. This deviation will be constant in magnitude all year, while the ecliptic longitude of the the deviation (direction of the deviation in these coordinates) will vary over 360 degrees. [Note, this article gives conversion formulas between ecliptic latitude and longitude and declination and right ascension in the more common equatorial coordinates. Perhaps studying these conversion formulas wil resolve some of your confusions.] [EDIT: and if you take this deviation circle of 20.5 arc second radius: ecliptic latitude (90° - 20.5"") with longitude ranging from 0 to 360; and center of 90% latitude, longitude undefined. And transform all of this to equatorial declination and right ascension with formulas as in reference above, you see that declination deviation varies from zero to 20.5 arcseconds relative to the center of the circle.When declination deviation is 0, RA deviation is maximal, and larger than 20.5 arc seconds because RA is an axial angle rather than a central angle.]
Contextual comment: we are ignoring precession and nutation for the purposes of this discussion. Precession is about 50 arc seconds per year with period of about 26,000 years; nutation is about 1 arc second per year with period of 18.6 years (for the largest component).
I don't find your statement sufficiently clear. I propose statement (1) below which describes my intent: (1) Using an ecliptic coordinate system [see http://en.wikipedia.org/wiki/Ecliptic_coordinate_system], Bradley's formula [tan (θ - θ_{0}) = -v/c] will give a pure latitude deviation for the center point of a distant celestial body at either ecliptic pole. This formula predicts that c = v/tan(θ - θ_{0}) at the instant in time when θ_{0} = 90 degrees. Notes: Perhaps if we debate the equation in question for predicting the value of c we can resolve some of the confusion. I cannot see that equatorial coordinates, equatorial declination, right ascension in equatorial coordinates or right ascension in ecliptic coordinates are variables that need to be factored into the equation c = v/tan(θ - θ_{0}) when applied to the ecliptic coordinate system. Since it is my intent to know if this equation can predict c for an observer who is orbiting the Sun in the ecliptic plane (not for an observer on the Earth’s surface), I cannot see the relevance of independent variables other than v, θ and θ_{0} when applied in ecliptic coordinates. This is not to say that I will not consider the relevance of independent variables other than v, θ and θ_{0} in the debate if qualified Astrophysicists think the equation for determining the value of c needs to include terms with independent variables other than v and tan(θ - θ_{0}). Precession and nutation only improve the probability that the central point of a “fixed” celestial body such as galaxy will be located at an ecliptic pole (i.e. θ_{0} will be 90°) at some time during a given year.
The whole point of moving the discussion here is that you raised the claim that the aberration as calculated and tabulated in the Astronomic Almanac disagreed with both Bradley and SR aberration models. This claim has nothing to do with measurement of light speed and is all about the details of applying aberration formulas in a particular coordinate system. If, instead, you actually want to discuss the light speed, it does not belong here, but in the SR/GR forum - where it has been discussed to death (but there the issue of not understanding how AA computations are consistent with SR aberration does not belong).
In order that we can move on and get to my point about the declination aberration equation described in the Astronomical Almanac, I will assume you agree with my last proposed statement (1) at least to the extent that you would not want to revise Bradley’s equation to include independent variables other than v, θ and θ_{0} in order to predict the value of c. In the SR/GR forum, I was leading to a point involving the Astronomical Almanac’s (AA) Reduction for Annual Aberration equation used to estimate declination aberration. I did not feel that I should press this point because PAllen, et, al. suggested that I open a new thread in Astronomy after I introduced the AA declination aberration equation and plots of predicted declination aberration data for stars at θ_{0} = 90º, 75º, and 1º. The point I had intended to make was that the AA declination aberration equation predicts that (θ - θ_{0}) = (- v/c).sin(θ_{0}).cos(α_{0}) in an ecliptic coordinate system. When θ_{0} = 90º, the AA equation predicts a different value for (θ - θ_{0}) than Bradley’s equation. The difference in predictions depends upon the right ascension angle α_{0}, when α_{0} - 0º the difference in the two estimates for (θ - θ_{0}) are small but the difference in the two estimates become very large as α_{0} approaches 90º. This means that the AA equation would predict that a galaxy with a center on an ecliptic pole would appear to have stars with declination aberration close to -20.5 arc seconds when they are close to the plane where α_{0} = 0º and would have declination aberration close to zero for stars in the galaxy that are close to the plane where α_{0} = 90º. Different predictions for declination aberration (θ - θ_{0}) of objects near an ecliptic pole from sources already approved by the mainstream is highly relevant to any attempt to assume that Bradley’s estimate for the Constant of Aberration (approximately -20.5 arc seconds) at all times is an “a priori” given independent variable that can be used with complete and unequivocal confidence in the equation c = v/tan(θ - θ_{0}) would yield a precise estimate for the value of c. I contend that the aberration of a body at an ecliptic pole must be accurately MEASURED through the use of an appropriately configured computerized telescope before the value (θ - θ_{0}) can be reliably determined. Also, if (θ - θ_{0}) is determined to be significantly different from -20.5 arc seconds then the Bradley model cannot be used to calculate a value for c.
No, here I don't want the discussion to be about measuring or predicting the value of c at all, no way, no how. It has nothing to do with explaining how AA values are computed from SR aberration formulas. You may independently add something to thread on measuring c using aberration that does not involve calculations or tables in AA. And this set of issues is purely about your misunderstanding both the Bradly and SR aberration formulas and the application to AA computations and has nothing to do with predicting c. It was felt since none of us in that forum had a copy of AA (and we cannot take your presentation of it as accurate), explaining how all this is consistent in detail would better be done by someone here who might have a copy. You consistently call the Bradley formula a declination formula. This is purely a falsehood, and this has been explained at least 6 times to you. It is formula for total angular deviation of an object whose position is orthogonal to earth's motion, and the direction of deviation is in the plane formed by the earth's velocity vector and the light path. The AA is all about the equatorial coordinate system (as witnessed by terms declination and right ascension). The angular difference in the Bradley formula translates in somewhat complex ways to declination and right ascension change. So I suggest here we first focus on the issue that there is actually no discrepancy whatsoever between AA computations and SR aberration, since the former is derived from the latter. Until this misunderstanding is addressed, all the rest is irrelevant.
Dear Professor Allen, Thank you very much for your responsiveness, forbearance, perseverance, and endurance. Sooner or later every physicist or engineer (including my Physics PhD brother) with whom I have debated stellar aberration will tire of my excruciating details concerning coordinate systems (such as ecliptic coordinates, celestial coordinates and the Einstein/Bradley coordinates) as well as the details that are required to communicate a lucid description of how celestial coordinates can be translated into ecliptic coordinates as well as how the Einstein/Bradley coordinates can be translated into ecliptic coordinates. Understanding these details may require spatial cognition which is almost as good as Einstein’s unless you are an experienced mathematical modeler and have spent many years learning how to get computers to do the difficult spatial cognition work for your inferior brain. My spatial cognition is admittedly inferior to Einstein’s (there is no comparison) and you are obviously a stellar fellow of the stellar aberration faculty. Therefore, in order for us to communicate with no cross transactions [such as a Transactional Analysis (TA) cross transactions where a child wants to be talked to as an adult but the parent wants to talk to the child as a child]; I suggest that in future discussions we play a “game” where I am ken more the student who only wants to ken more (“ken more” as in “to understand more”) and asks questions and you are Professor Allen. Since it may not be appropriate to play this game on Physics Forum (PF) then you can send your personal e-mail address to ken-more@ken-more.com. I promise I will never give out your personal e-mail address and I will never quote anything you say. If you prefer, I can set-up a forum at www.ken-more.com and you can play the game using a pen name. My real name is not ken more. Ken more is a pen name for a science fiction writer. When ken more writes a statement that sounds like he is trying to state a fact, he is not trying to state a fact he knows he is stating science fiction; he is really asking a hypothetical question. This is stated up-front on his web site. Ken more jumped into PF on the spur-of-the-moment after he got a Google Alert on Stellar Aberration when the thread “Re: Stellar aberration, a One way measurement of c?” was published to the web. He then quickly signed up to PF and joined in the discussion to see what ken more could learn to improve his web site without thoroughly reading the PF rules. Now that the rules have been thoroughly read, I can see that playing the game that I am suggesting to educate ken more will not be appropriate in a PF thread because it will often appear that ken more is making a highly speculative statement as fact even though he will only be asking a highly speculative “What if - - - ?” question in the context of science fiction. If you have finally tired of this debate and do not want to play the game I have suggested, please let me know and I WILL CONSIDER THIS DEBATE TO BE TERMINATED. I thank you very much for your responsiveness, forbearance, perseverance and endurance as well as for your very enlightening commentary.
If I had a copy of the AA and supporting materials, I would be willing, over time, to work through all the details of its derivation from basic SR aberration formulas. I don't, and am not willing to buy one, and cannot find equivalent detail on the web. I am not willing to exchange messages with you. The value in the above (if I had the relevant resources) would be leaving a thread in this forum of utility to others. So I guess we agree to terminate this discussion. FYI: I am not a professor.