<p><strong>I</strong>.  The statement about planetary aberration quoted in the question, and for which the quesioner seeks some authority,  wasn't actually called a 'definition' by the questioner : it's probably best regarded as a paraphrase of longer descriptions of aberration.  There are certainly authoritative descriptions to be found -- as asked -- perhaps none more authoritative than</p>
<p>(1) in 'ESAE', the <a href="https://books.google.com/books/about/Explanatory_Supplement_to_The_Astronomic.html?id=JwNuzQEACAAJ" rel="noreferrer">&quot;Explanatory Supplement to The Astronomical Ephemeris and the American Ephemeris and Nautical Almanac&quot;</a> (HMSO, 1961, with reprints to 1977),</p>
<p>(2) in 'ESAA', the <a href="https://archive.org/details/explanatorysuppl00pken" rel="noreferrer">&quot;Explanatory Supplement to The Astronomical Almanac&quot;</a> (ed. P K Seidelmann), and also</p>
<p>(3) in <a href="https://en.wikipedia.org/wiki/James_Bradley" rel="noreferrer">James Bradley</a>'s paper in <a href="https://www.biodiversitylibrary.org/item/196465#page/305/mode/1up" rel="noreferrer">Phil.trans. (1727-8) 35:637-661</a> describing his original discovery of the (stellar) aberration.</p>
<p>The following summary explanation comes mainly from (1) (which seems most concise and clear about planetary aberration). Bradley's paper (3) still seems worth reading, especially for its memorable explanation (at p.646-8) why the direction of the aberrational displacement is towards the apex of the earth's motion i.e. the direction of the earth's velocity vector.</p>
<p><strong>II</strong>.  From 'ESAE' at page 46:--</p>
<blockquote>
<p>ABERRATION : Because the velocity of light is finite, the apparent
direction of a moving celestial object from a moving observer is not
the same as the geometric direction of the object from the observer at
the same instant.  This displacement of the apparent position from the
geometric position may be attributed in part to the motion of the
object, and in part to the motion of the observer, these motions being
referred to an inertial frame of reference.  The former part,
independent  of the motion of the observer, may be considered to be a
correction for light-time; the latter part, independent of the motion
or distance of the object, is referred to as stellar aberration, since
for the stars the correction for light-time is, of necessity, ignored.
The sum of the two parts is called planetary aberration since it is
applicable to planets and other members of the solar system.</p>
</blockquote>
<p>This passage contains essentially the same content as the summary quoted in the question, and provides the authoritative source requested.</p>
<p>The same page (ESAE p.46) goes on to give the details below. (The original diagram had handwritten markings, clarified in the attached version and with direction arrows in the following text, also with a few explanatory words in curly brackets {}.)</p>
<blockquote>
<p>Correction for light-time : Let <span class="math-container">$P(t)$</span> and <span class="math-container">$E(t)$</span> {see figure below}
be the geometric positions of an object and an observer {imagined
as stationary} at time <span class="math-container">$t$</span>, and let <span class="math-container">$P'(t-T)$</span> be the geometric
position of the object at time <span class="math-container">$t-T$</span>, where <span class="math-container">$T$</span> is the light-time,
i.e. the time taken for the light to travel from the point of
emission, in this case <span class="math-container">$P'(t-T)$</span>, to the point of observation <span class="math-container">$E(t)$</span>.
Then, since {the observer's position} <span class="math-container">$E(t)$</span> is regarded {so far} as
stationary, the direction <span class="math-container">$E(t) -&gt;P'(t-T)$</span> is the apparent direction of
the object at time <span class="math-container">$t$</span>, i.e. the apparent direction at time <span class="math-container">$t$</span> is the
same as the geometric direction of the object at time <span class="math-container">$t-T$</span> {i.e., if
the observer at <span class="math-container">$E(t)$</span> were stationary}.</p>
</blockquote>
<p><a href="https://i.sstatic.net/LgUq3.png" rel="noreferrer"><img src="https://i.sstatic.net/LgUq3.png" alt="Figure : Planetary aberration" /></a></p>
<blockquote>
<p>Stellar aberration. The light which is received at the instant of
observation <span class="math-container">$t$</span> was emitted, at a previous instant, from the position
which the object occupied at time <span class="math-container">$t — T$</span>, towards the position which
the observer was later to occupy at time <span class="math-container">$t$</span>; but when the light
reaches the observer it appears to be coming, not from this actual
direction { <span class="math-container">$E(t) —&gt;P'(t-T)$</span> } but from its direction relative to the
moving observer. Let the object {now} be considered stationary at
<span class="math-container">$P'(t-T)$</span>, the position it occupies at time <span class="math-container">$t — T$</span>, and let <span class="math-container">$E$</span> be
moving in the direction <span class="math-container">$E0 —&gt;E(t) —&gt;E1$</span> with an instantaneous velocity
<span class="math-container">$V$</span> {at time <span class="math-container">$t$</span>}. Then, according to classical theory, the apparent
direction of the object is that of the vector difference of the
velocity of light <span class="math-container">$c$</span> in the direction <span class="math-container">$P'(t-T) —&gt;E(t)$</span> and the
{observer's} velocity <span class="math-container">$V$</span> in the direction <span class="math-container">$E0 —&gt;E(t)$</span>. The apparent
angular displacement is independent of the distance, but, by
definition of the light-time <span class="math-container">$T$</span>, <span class="math-container">$P'(t-T) —&gt;E(t) = T×c$</span> {where <span class="math-container">$c$</span> is
the velocity of light}, so that if <span class="math-container">$E0 —&gt;E(t)$</span> is drawn in the
direction of motion {of <span class="math-container">$E$</span> at time <span class="math-container">$t$</span>} and of magnitude <span class="math-container">$T×V$</span>, the
apparent direction of the object is  <span class="math-container">$E0 —&gt;P'(t-T)$</span> {instead of
<span class="math-container">$E(t) —&gt;P'(t-T)$</span>}. The apparent direction at time <span class="math-container">$t$</span> would {also} be
the same as the geometric direction at time <span class="math-container">$t — T$</span>,  if <span class="math-container">$E$</span> were
moving with a constant rectilinear velocity <span class="math-container">$V$</span>, i.e. if <span class="math-container">$E0$</span> were
identical with the position of the observer at time <span class="math-container">$t — T$</span>.  The
{aberrational} displacement is toward the apex of the motion of the
observer [...].</p>
</blockquote>
<p><strong>III</strong>. The quoted passages also give part of the answer to the second part of the question (How does the planetary aberration depend on the planet's distance?)  It's true, as the question points out, that the planet's orbital velocity relative to the sun decreases with distance from the sun. But what is directly relevant to aberration is</p>
<ul>
<li><p>the planet's  light-time (interval), proportional to its distance
from the earth;</p>
</li>
<li><p>the planet's difference of position between the beginning of that
light-time interval and its end (i.e. the instant of observation);
and</p>
</li>
<li><p>the earth's (not the planet's) velocity at the instant of observation.</p>
</li>
</ul>
<p>Planetary motions relative to the earth show stationary points and retrogradations.  Thus there is no direct relation between planetary geocentric distance  and the aberration, nor between planetary heliocentric motion or distance and the aberration. The more distant solar-system objects are indeed generally slower in motion, tending towards smaller aberrations, but their greater distances from the earth also mean longer light-time intervals, tending to counter that effect, and again that is further complicated by the motions of retrogradation.</p>
<p><strong>IV</strong>.  Computations for the aberration are given in ESAE (sec. 2D, p.46  onwards) and (with updated constants) in ESAA (sec. 3.25-.257, p.127 onwards).  The 'stellar' component of the aberration is more descriptively called the <em>annual</em> aberration, and there is also a small <em>diurnal</em> aberration due to the eastwards rotational motion of any (non-polar) observer on the earth's surface.  The scale factor <span class="math-container">$V/c$</span> for the aberrational displacement angle that emerges from the description above is a first-order approximation:  higher powers of <span class="math-container">$V/c$</span> are also present (as briefly explained in ESAE p.46-7, and ESAA p.128-9). The second order term in the classical account of the aberration has an amplitude of about 0&quot;.001. The classical physics account and the relativistic account agree at the first power of <span class="math-container">$V/c$</span> but differ at higher powers, the relativistic second-order term is half the size of its classical counterpart, i.e. less than 0&quot;.001.</p>
<p><strong>V</strong>. The question also asks &quot;Are there any reported measurements on aberration for any planets?&quot;   I have not found any recent direct experimental measurement work on the aberration.  The reality of the phenomenon of aberration, and the fundamentals of its theory, have been confirmed and accepted for nearly 300 years since demonstration in the work of Bradley (1728) already cited. Jean-Baptist Joseph Delambre (1749-1822), astronomer and historian, wrote enthusiastically in his <a href="https://books.google.co.uk/books?id=ps1WS1-5_dYC&amp;pg=PA420" rel="noreferrer">History of 18th-century astronomy (publ.1827)</a> at p.420 (translation from French):</p>
<blockquote>
<p>&quot;It is to these two discoveries by Bradley {i.e.aberration and later
the nutation} that we owe the exactness of modern astronomy.&quot; {Without
their help it would have been impossible to reconcile various star
observations to better than about a minute of arc.} &quot;This double
service assures for its author the most distinguished place among
astronomers, after Hipparchus and Kepler but above all the other
greatest astronomers of all ages and all countries.&quot;</p>
</blockquote>
<p>In practice, the main remaining subjects for investigation or refinement have been the constant of the aberration, and the theories of the earth's motion and velocity (as well as the planetary theories) which enter into the calculation of aberrations for particular objects, dates and times.</p>
<p>Nowadays, the highest accuracy in the relevant astronomical measurements comes from modern determinations of the speed of light and of the astronomical unit, from ranging observations to spacecraft, from laser- and radar-ranging to some of the celestial bodies, and from their least-squares compilation into integrated solar-system ephemerides and the associated constants also fitted with them to the data. Among the resulting sources are notably the <a href="https://ssd.jpl.nasa.gov/planets/eph_export.html" rel="noreferrer">DE series of planetary and lunar ephemerides</a> from the Jet Propulsion Laboratory (used since 1984 in the <a href="https://en.wikipedia.org/wiki/Astronomical_Almanac" rel="noreferrer">Astronomical Almanac</a>, references given in section L of each year's issue), and the more recent <a href="https://www.imcce.fr/inpop/" rel="noreferrer">INPOP</a> series of integrated ephemerides from the Paris Observatory.  ESAA sec.3-253 gives a derivation for <span class="math-container">$\kappa$</span>, the constant of aberration that comes ultimately from these integrated ephemeris sources.</p>
<p>In the centuries before the 1960s, when optical techniques were still the source of highest-precision astronomical observations, the constant of aberration was investigated as a separate natural constant.</p>
<p>19th-c. methods of doing that are described for example in F Brunnow's <a href="https://archive.org/details/sphericalastro00brhurich" rel="noreferrer">'Spherical Astronomy' (1865)</a>. It gives (p.231 onwards) some then-current methods of determining constants such as the constant of aberration. (The main problem was in choosing observations from which the desired small angular quantity would be given in as large a measure as could be arrranged. The text explains how observations of the Pole Star at the prime vertical were chosen as most promising from that point of view.)</p>
<p>The next main essential for the  aberration is the earth's velocity vector, and an example of attempts to make this accurately accessible using techniques and data other than those from integrated ephemerides is given in <a href="https://articles.adsabs.harvard.edu/full/1986BAICz..37...96R" rel="noreferrer">Ron &amp; Vondrak (1986)</a> paper on the trigonometric series expansion of the annual component of the aberration based on recent theory-data about the earth's motion from P Bretagnon and others.</p>
<p>These items show how the aberration is nowadays recognized to depend on ingredient quantities that have wider and independent significances, so that it is no longer really a subject of independent investigation.</p>

CC BY-SA 4.0
2022-07-26T04:49:52.400Z
Hi @terry-s, Thanks for the exhaustive answer provided. Do you know why a vector difference is taken (in ESAE) to calculate the stellar aberration? I realise that it gives the correct angle (i.e. the telescope is tilted by $\delta\theta$ in the direction of motion of the observer as James Bradley originally reported).  But what is the physical meaning of this 'difference operation'?  Perhaps, a Lorentz transformation?
JKrsl
CC BY-SA 4.0
2022-07-26T09:44:43.277Z
@JKrsl : It's 3D geometry, I feel sure. The earth moves in the ecliptic plane, with velocity vector (instantaneoulsy E0->E(t) in the diagram above) that rotates in direction around 360d during a year. The direction from the star (and its light velocity vector P'(t-T) -> E(t) ) can be from any direction on the 'celestial sphere'. The two velocity vectors form two sides of a plane triangle. The apparent star direction is the third side, as explained by Bradley with his own triangle diagram and 'inclined tube' analogy for the course followed by light 'particles' towards the _moving_ observer.
terry-s
CC BY-SA 4.0
2022-07-27T00:48:12.477Z
I am not sure ESAE is using Bradley's explanation. From my point of view, I am only aware of vector addition as a way to find the resultant of two motions. In a simplified case, with star light coming in the -j direction, if we use vector addition of the two vectors {v, 0, 0} and {0, -c, 0}, we end up with a resultant vector falling in the second quadrant, which is not the angle by which an astronomer will tilt the telescope. Obviously, that is the reason ESAE is using vector difference, but I am unable to grasp the Physics behind the ESAE approach. Any help is appreciated.
JKrsl
CC BY-SA 4.0
2022-07-27T11:37:10.923Z
@JKrsl : ESAE shows the vector addition by a triangle, adding the geometric light-direction vector P'(t-T)->E(t) and E(t) -> E0 the -ve of the earth's velocity vector. The third side P'(t-T)->E0 then shows the resulting apparent direction from which the light comes, but in effect the observer is also shifted to E0. Shift the observer back to E(t), with a vector parallel to the third side, and if I'm not mistaken, that ends up towards the (right-hand) '1st-quadrant' side of the geometric direction vector, i.e. displaced in the direction of the earth's motion. In Bradley's diagram, ../..
terry-s
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2022-07-27T11:37:32.850Z
@JKrsl : In Bradley's diagram the eye (i.e. earth) is said to move B->A i.e. left, which is opposite of the direction described for the ESAE diagram. Bradley concludes the inclined direction in his case is angle CBD, and though that looks to be in what we normally call 2nd quadrant, it is again a displacement towards the direction of earth motion. Bradley p.647 then considers observer/earth motion in the opposite direction D->A, i.e. left to right, and concludes the apparent direction angle as BDC, 1st quadrant, but again displaced in direction of eye/earth motion. It all looks consistent?
terry-s
CC BY-SA 4.0
2022-07-28T06:05:17.050Z
It is the use of " -ve of the earth's velocity vector" which I am unable to understand. Sorry, but the approach in ESAE looks [heuristic](https://en.wikipedia.org/wiki/Heuristic) to me. Regarding your second comment, Bradley was using corpuscular theory, and in his explanation, the corpuscles continue a straight line path through the tilted telescope.  Note, no bending of the corpuscular path. I suppose, a similar approach was used in the early forms of wave theoretical explanation for aberration of star light.
JKrsl
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2022-07-28T23:20:56.033Z
@JKrsl : As I read the ESAE description, it explains 'what' & 'how much', but doesn't go further. Bradley also described 'what' & 'how much', & added his tube illustration as some kind of explanation 'why'. My earlier comments showed, I think, how the two descriptions are fully consistent with each other. Neither of the sources makes any suggestion that the light path is bent.
terry-s
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2022-07-28T23:22:18.330Z
@JKrsl : A further thought: I used vector terminology as a brief shorthand to translate the effect of the geometry in both ESAE and Bradley. The '-ve of the earth's velocity vector' translates the effect of the motion in diagrams & text of both sources. In ESAE it is the length & direction E(t)->E0, in Bradley's 1st case A->B, in his 2nd case 'contrary way'  A->D.  In each case the vector adds to the geometrical light direction vector to produce the apparent light direction. Any other direction than -V would contradict the geometry as presented in both the sources.
terry-s
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2022-07-30T05:36:15.210Z
Have I got the following right? It seems to me there is one actual "astronomical aberration" phenomenon  discussed on this page, and it can be calculated from the Earth's velocity vector in the frame of the observed object (star/planet) and the vector that the observer's telescope is pointing. The other phenomenon is the light-time correction which would apply just as well to planets as to stars, it's just that we pay attention to it differently and more carefully for planets in our own solar system than for stars. Could that be seen as a generally correct if not completely nuanced statement?
uhoh
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2022-07-30T08:09:04.653Z
@uhoh : Yes, subject only, that the reference frame for the earth's velocity is ideally inertial, in practice solar-system barycentric and for completeness where needed there is  additionally the small diurnal aberration briefly mentioned here and described in the references, to add the non-polar observer's terrestrial rotatory motion into the mix, plus if desired the very small relativistic correction also briefly referenced. And there are changes of fashion about nomenclature, where the main astronomical aberration or stellar aberration tends now to be called annual aberration.
terry-s
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2022-07-30T15:37:08.797Z
@uhoh The reference frame for the observer's velocity vector is the Solar System Barycenter.  For stars, the light time correction is assumed to be already accounted for in the catalog position and proper motion components.  That's largely due to the fact that deviations in the light time are so small they are undetectable.  Though GAIA and other new catalogs are making 3D proper motion a possibility where you could account for light time differences if needed.
Greg Miller
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2022-07-30T21:18:57.003Z
@GregMiller I see (I think) While there's nothing of special significance about the solar system barycenter from a physics perspective, this is the way astronomers have broken things down by convention/definition for practical reasons?
uhoh
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2022-07-30T21:23:22.100Z
@terry-s Two excellent, different but (roughly) equally informative answers, so I'm going to add a bounty to both of them. SE doesn't allow two equal bounties (it doubles the 2nd one) so arbitrarily I will add the larger one to the user with lower reputation. *Thanks!*
uhoh
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2022-07-31T04:52:13.670Z
@uhoh As for why it's the SSB is beyond my understanding, I assume it's because the algorithm is specifically designed for that case.
Greg Miller
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2022-07-25T14:39:35.380Z
2022-07-25T14:39:35.380Z
terry-s
<p>As far as an authoritative source, as PM 2Ring pointed out in his comment, it is on Wikipedia, but no reference is cited.  The definition itself seems to be incomplete, as it does little to actually help the reader understand the components involved, it seems to be an extract from a larger discussion on the topic.  A very similar definition does appear on the <a href="https://www.oxfordreference.com/view/10.1093/acref/9780191851193.001.0001/acref-9780191851193-e-2884" rel="nofollow noreferrer">Oxford Reference</a> but has quite a bit more to it than just what's in the original post.</p>
<p>The definition from the Explanatory Supplement to the Astronomical Almanac in the glossary is</p>
<blockquote>
<p>The apparent angular displacement of the observed position of a solar system body from its instantaneous geometric direction as would be seen by an observer at the geocenter.  This displacement is produced by the combination of <em>aberration of light</em> and <em>light-time displacement</em>.</p>
</blockquote>
<p>In other words, it is the combined effects to two distinctly different effects:</p>
<p><strong>Light Time Displacement</strong>: This is due to the fact that light from an object takes time to reach the observer. So, when a photon reaches the observer, they see it in the position it was in when that photon left the object in question.  This is corrected for by computing the distance between the observer and the target, then recomputing the position of the target based on where it was based on how long it takes light to travel from the target to the observer.  This process is iterated until the desired accuracy is reached. (e.g. The NOVAS library iterates up to 10 times, or until the difference between iterrations is less than 1e-9AU).</p>
<p><strong>Aberration of Light</strong>: This is also usually divided into two components called <strong>annual aberation</strong>, caused by the Earth's motion around the Sun, and <strong>diurnal aberration</strong>, caused by the observer's motion around the Geocenter.  The process is divided into two components because it is common to ignore diurnal aberration since it accounts for at most about .3 arcseconds.  When computing both, the velocity of the observer's topocentric position is simply added to the Earth's velocity around the solar system barycenter, and the computation continues the same either way.</p>
<p>Here is an example implementation of annual aberration of light from <a href="https://www.celestialprogramming.com/vsop87-multilang/index.html" rel="nofollow noreferrer">VSOP87-Multilang</a>.  The equations are given in the cited paper which is publicly available at the given URL.</p>
<pre><code>//JavaScript implementation
static aberration(pos,earthPV){
    //&quot;MEAN AND APPARENT PLACE COMPUTATIONS IN THE NEW IAU SYSTEM. III. APPARENT, TOPOCENTRIC, AND ASTROMETRIC PLACES OF PLANETS AND STARS&quot;
    //G. H. Kaplan, J. A. Hughes, P. K. Seidelmann, and C. A. Smith
    //U. S. Naval Observatory, Washington, DC 20392

    //http://articles.adsabs.harvard.edu/pdf/1989AJ.....97.1197K

    const C = 173.1446326846693; //Speed of light in AU per Day

    const u4=pos;
    const dE=new Array();
    dE[0]=earthPV[3];
    dE[1]=earthPV[4];
    dE[2]=earthPV[5];

    //Eq 16
    const t=Vec.magnitude(u4) / C;
    const B=Vec.magnitude(dE) / C;
    const cosD=Vec.vecDot(u4,dE)/(Vec.magnitude(u4)*Vec.magnitude(dE));
    const y=Math.sqrt(1-B*B);
    const f1=B*cosD;
    const f2=(1+f1/(1+y))*t;

    //Eq 17
    const u5=new Array();
    u5[0]=(y*u4[0] + f2*dE[0])/(1+f1);
    u5[1]=(y*u4[1] + f2*dE[1])/(1+f1);
    u5[2]=(y*u4[2] + f2*dE[2])/(1+f1);

    return u5;
}
</code></pre>

CC BY-SA 4.0
2022-07-30T21:22:58.490Z
Two excellent, different but (roughly) equally informative answers, so I'm going to add a bounty to both of them. SE doesn't allow two equal bounties (it doubles the 2nd one) so arbitrarily I will add the larger one to the user with lower reputation. *Thanks!*
uhoh
CC BY-SA 4.0
2022-07-25T14:46:01.873Z
2022-07-25T14:46:01.873Z
Greg Miller
<p>It is often stated that, &quot;Planetary aberration is a combined result of the observer's motion and the time taken for light to travel from a body in the Solar System to the observer&quot;. I am not aware of any authoritative reference to the origin of this statement; does anybody know?</p>
<p>Also, does this statement imply that the observed planetary aberration will be proportional to some power of the distance between the observer and the planet? However, (as we know) as the planet is further away from the sun, the orbital velocity of the planet decreases. Are there any reported measurements on aberration for any planets?</p>
<p>I have posted this question in Physics Stackexchange as well, as this post is about an inter-disciplinary topic.</p>

CC BY-SA 4.0
2022-07-22T01:44:39.740Z
"I have posted this question in Physics Stackexchange as well, as this post is about an inter-disciplinary topic."  Please don't.   Herte's why https://meta.stackexchange.com/questions/64068/is-cross-posting-a-question-on-multiple-stack-exchange-sites-permitted-if-the-qu
James K
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2022-07-22T02:04:44.130Z
I don't have time to go into detail now, but maybe someone will.  The Explanatory Supplement to the Astronomical Almanac reduction procedures account for three types of light aberration: light time aberration (based on the light time to the object), annual aberration (based on the Earth's motion around the Sun), and diurnal aberration (based on the observer's motion around the center of the Earth).  The NOVAS library from the USNO implements these corrections and the source code has references.
Greg Miller
CC BY-SA 4.0
2022-07-22T10:49:03.213Z
While it is tempting to [cross-post to Physics SE](https://physics.stackexchange.com/q/719303/83380), it's strongly discouraged because (among other reasons) it can lead to answer fragmentation and future readers may miss important answers.
uhoh
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2022-07-22T17:49:01.300Z
You should post a source for the quote in the question.
Greg Miller
CC BY-SA 4.0
2022-07-25T03:38:36.953Z
The Wiki page on Light-time correction says, "The effect of the finite speed of light on observations of celestial objects was first recognised by Ole Rømer in 1675, during a series of observations of eclipses of the moons of Jupiter."
eshaya
CC BY-SA 4.0
2022-07-27T23:02:13.457Z
[On the Definition of Aberration](https://ui.adsabs.harvard.edu/abs/2014ivs..conf..490X/abstract) from [IVS 2014 General Meeting Proceedings](https://ivscc.gsfc.nasa.gov/publications/gm2014/) has good info on aberration, and presents an interesting argument that the term "planetary aberration" should be deprecated. Sadly, that article has a typo in the Lorentz transformation equations. Hopefully, there are no other errors...
PM 2Ring
CC BY-SA 4.0
2022-07-28T02:50:47.947Z
@PM 2Ring, Xu and Wang "On the Definition of Aberration" - their treatment is difficult to understand. The reason being, they come up with an expression, δk = (1/c) k×(k×V) for the "apparent variations in positions of the objects detected from observations". In their Eq. (1), cross product of 'direction' k (unit vector?) and velocity V, is found; the resultant vector being perpendicular to both k and V.  A cross product of this new vector and k is formed again. The resultant should necessarily be along V. Then they ask "what the velocity V in Equation 1 is".  Please see my next comment.
JKrsl
CC BY-SA 4.0
2022-07-28T02:53:01.617Z
To make things further confusing, initially Xu and Wang say the two inertial reference frames O and C are in relative motion with a velocity V (page 491, last para). But on page 492, after Eq.(2), they say "Let the velocity of light in C−x'y'z' be V(Vx,Vy,Vz)".
JKrsl
CC BY-SA 4.0
2022-07-28T23:55:12.057Z
One of the peculiarities of the Xu and Wang paper is that they criticize as a 'definition ... that should be abandoned' a definition that isn't put forward in the authoritative document-items that they cite. None of those docs defines aberration as dependent on the 'relative motion of object and observer'. What the relevant sources do say is that aberration depends on the position of the object at the beginning of the light-time (interval), on the finite velocity of light, and on the motion of the observer -- which is quite different from an appeal to 'relative motion'.
terry-s
<observational-astronomy><time-dilation><special-relativity><frame-of-reference>
Measurement of Planetary Aberration (similar to stellar aberration)
JKrsl
# Measurement of Planetary Aberration (similar to stellar aberration)

(Asked by: username_0 on 2022-07-22)

It is often stated that, "Planetary aberration is a combined result of the observer's motion and the time taken for light to travel from a body in the Solar System to the observer". I am not aware of any authoritative reference to the origin of this statement; does anybody know?

Also, does this statement imply that the observed planetary aberration will be proportional to some power of the distance between the observer and the planet? However, (as we know) as the planet is further away from the sun, the orbital velocity of the planet decreases. Are there any reported measurements on aberration for any planets?

I have posted this question in Physics Stackexchange as well, as this post is about an inter-disciplinary topic.

**Tags:** observational-astronomy, time-dilation, special-relativity, frame-of-reference

> **Comments:**
> **username_1**: "I have posted this question in Physics Stackexchange as well, as this post is about an inter-disciplinary topic."  Please don't.   Herte's why https://meta.stackexchange.com/questions/64068/is-cross-posting-a-question-on-multiple-stack-exchange-sites-permitted-if-the-qu  
> **username_2**: I don't have time to go into detail now, but maybe someone will.  The Explanatory Supplement to the Astronomical Almanac reduction procedures account for three types of light aberration: light time aberration (based on the light time to the object), annual aberration (based on the Earth's motion around the Sun), and diurnal aberration (based on the observer's motion around the center of the Earth).  The NOVAS library from the USNO implements these corrections and the source code has references.  
> **username_3**: While it is tempting to [cross-post to Physics SE](https://physics.stackexchange.com/q/719303/83380), it's strongly discouraged because (among other reasons) it can lead to answer fragmentation and future readers may miss important answers.  
> **username_2**: You should post a source for the quote in the question.  
> **username_4**: The Wiki page on Light-time correction says, "The effect of the finite speed of light on observations of celestial objects was first recognised by Ole Rømer in 1675, during a series of observations of eclipses of the moons of Jupiter."  
> **username_5**: [On the Definition of Aberration](https://ui.adsabs.harvard.edu/abs/2014ivs..conf..490X/abstract) from [IVS 2014 General Meeting Proceedings](https://ivscc.gsfc.nasa.gov/publications/gm2014/) has good info on aberration, and presents an interesting argument that the term "planetary aberration" should be deprecated. Sadly, that article has a typo in the Lorentz transformation equations. Hopefully, there are no other errors...  
> **username_0**: @username_5, Xu and Wang "On the Definition of Aberration" - their treatment is difficult to understand. The reason being, they come up with an expression, δk = (1/c) k×(k×V) for the "apparent variations in positions of the objects detected from observations". In their Eq. (1), cross product of 'direction' k (unit vector?) and velocity V, is found; the resultant vector being perpendicular to both k and V.  A cross product of this new vector and k is formed again. The resultant should necessarily be along V. Then they ask "what the velocity V in Equation 1 is".  Please see my next comment.  
> **username_0**: To make things further confusing, initially Xu and Wang say the two inertial reference frames O and C are in relative motion with a velocity V (page 491, last para). But on page 492, after Eq.(2), they say "Let the velocity of light in C−x'y'z' be V(Vx,Vy,Vz)".  
> **username_6**: One of the peculiarities of the Xu and Wang paper is that they criticize as a 'definition ... that should be abandoned' a definition that isn't put forward in the authoritative document-items that they cite. None of those docs defines aberration as dependent on the 'relative motion of object and observer'. What the relevant sources do say is that aberration depends on the position of the object at the beginning of the light-time (interval), on the finite velocity of light, and on the motion of the observer -- which is quite different from an appeal to 'relative motion'.  

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## 2 answers

### Answer by username_6 on 2022-07-25. Score: 5

**I**. The statement about planetary aberration quoted in the question, and for which the quesioner seeks some authority, wasn't actually called a 'definition' by the questioner : it's probably best regarded as a paraphrase of longer descriptions of aberration. There are certainly authoritative descriptions to be found -- as asked -- perhaps none more authoritative than

(1) in 'ESAE', the ["Explanatory Supplement to The Astronomical Ephemeris and the American Ephemeris and Nautical Almanac"](https://books.google.com/books/about/Explanatory_Supplement_to_The_Astronomic.html?id=JwNuzQEACAAJ) (HMSO, 1961, with reprints to 1977),

(2) in 'ESAA', the ["Explanatory Supplement to The Astronomical Almanac"](https://archive.org/details/explanatorysuppl00pken) (ed. P K Seidelmann), and also

(3) in [James Bradley](https://en.wikipedia.org/wiki/James_Bradley)'s paper in [Phil.trans. (1727-8) 35:637-661](https://www.biodiversitylibrary.org/item/196465#page/305/mode/1up) describing his original discovery of the (stellar) aberration.

The following summary explanation comes mainly from (1) (which seems most concise and clear about planetary aberration). Bradley's paper (3) still seems worth reading, especially for its memorable explanation (at p.646-8) why the direction of the aberrational displacement is towards the apex of the earth's motion i.e. the direction of the earth's velocity vector.

**II**. From 'ESAE' at page 46:--

> ABERRATION : Because the velocity of light is finite, the apparent
> direction of a moving celestial object from a moving observer is not
> the same as the geometric direction of the object from the observer at
> the same instant. This displacement of the apparent position from the
> geometric position may be attributed in part to the motion of the
> object, and in part to the motion of the observer, these motions being
> referred to an inertial frame of reference. The former part,
> independent of the motion of the observer, may be considered to be a
> correction for light-time; the latter part, independent of the motion
> or distance of the object, is referred to as stellar aberration, since
> for the stars the correction for light-time is, of necessity, ignored.
> The sum of the two parts is called planetary aberration since it is
> applicable to planets and other members of the solar system.

This passage contains essentially the same content as the summary quoted in the question, and provides the authoritative source requested.

The same page (ESAE p.46) goes on to give the details below. (The original diagram had handwritten markings, clarified in the attached version and with direction arrows in the following text, also with a few explanatory words in curly brackets {}.)

> Correction for light-time : Let $P(t)$ and $E(t)$ {see figure below}
> be the geometric positions of an object and an observer {imagined
> as stationary} at time $t$, and let $P'(t-T)$ be the geometric
> position of the object at time $t-T$, where $T$ is the light-time,
> i.e. the time taken for the light to travel from the point of
> emission, in this case $P'(t-T)$, to the point of observation $E(t)$.
> Then, since {the observer's position} $E(t)$ is regarded {so far} as
> stationary, the direction $E(t) ->P'(t-T)$ is the apparent direction of
> the object at time $t$, i.e. the apparent direction at time $t$ is the
> same as the geometric direction of the object at time $t-T$ {i.e., if
> the observer at $E(t)$ were stationary}.

[![Figure : Planetary aberration](https://i.sstatic.net/LgUq3.png)](https://i.sstatic.net/LgUq3.png)

> Stellar aberration. The light which is received at the instant of
> observation $t$ was emitted, at a previous instant, from the position
> which the object occupied at time $t — T$, towards the position which
> the observer was later to occupy at time $t$; but when the light
> reaches the observer it appears to be coming, not from this actual
> direction { $E(t) —>P'(t-T)$ } but from its direction relative to the
> moving observer. Let the object {now} be considered stationary at
> $P'(t-T)$, the position it occupies at time $t — T$, and let $E$ be
> moving in the direction $E0 —>E(t) —>E1$ with an instantaneous velocity
> $V$ {at time $t$}. Then, according to classical theory, the apparent
> direction of the object is that of the vector difference of the
> velocity of light $c$ in the direction $P'(t-T) —>E(t)$ and the
> {observer's} velocity $V$ in the direction $E0 —>E(t)$. The apparent
> angular displacement is independent of the distance, but, by
> definition of the light-time $T$, $P'(t-T) —>E(t) = T×c$ {where $c$ is
> the velocity of light}, so that if $E0 —>E(t)$ is drawn in the
> direction of motion {of $E$ at time $t$} and of magnitude $T×V$, the
> apparent direction of the object is $E0 —>P'(t-T)$ {instead of
> $E(t) —>P'(t-T)$}. The apparent direction at time $t$ would {also} be
> the same as the geometric direction at time $t — T$, if $E$ were
> moving with a constant rectilinear velocity $V$, i.e. if $E0$ were
> identical with the position of the observer at time $t — T$. The
> {aberrational} displacement is toward the apex of the motion of the
> observer [...].

**III**. The quoted passages also give part of the answer to the second part of the question (How does the planetary aberration depend on the planet's distance?) It's true, as the question points out, that the planet's orbital velocity relative to the sun decreases with distance from the sun. But what is directly relevant to aberration is

* the planet's light-time (interval), proportional to its distance
  from the earth;
* the planet's difference of position between the beginning of that
  light-time interval and its end (i.e. the instant of observation);
  and
* the earth's (not the planet's) velocity at the instant of observation.

Planetary motions relative to the earth show stationary points and retrogradations. Thus there is no direct relation between planetary geocentric distance and the aberration, nor between planetary heliocentric motion or distance and the aberration. The more distant solar-system objects are indeed generally slower in motion, tending towards smaller aberrations, but their greater distances from the earth also mean longer light-time intervals, tending to counter that effect, and again that is further complicated by the motions of retrogradation.

**IV**. Computations for the aberration are given in ESAE (sec. 2D, p.46 onwards) and (with updated constants) in ESAA (sec. 3.25-.257, p.127 onwards). The 'stellar' component of the aberration is more descriptively called the *annual* aberration, and there is also a small *diurnal* aberration due to the eastwards rotational motion of any (non-polar) observer on the earth's surface. The scale factor $V/c$ for the aberrational displacement angle that emerges from the description above is a first-order approximation: higher powers of $V/c$ are also present (as briefly explained in ESAE p.46-7, and ESAA p.128-9). The second order term in the classical account of the aberration has an amplitude of about 0".001. The classical physics account and the relativistic account agree at the first power of $V/c$ but differ at higher powers, the relativistic second-order term is half the size of its classical counterpart, i.e. less than 0".001.

**V**. The question also asks "Are there any reported measurements on aberration for any planets?" I have not found any recent direct experimental measurement work on the aberration. The reality of the phenomenon of aberration, and the fundamentals of its theory, have been confirmed and accepted for nearly 300 years since demonstration in the work of Bradley (1728) already cited. Jean-Baptist Joseph Delambre (1749-1822), astronomer and historian, wrote enthusiastically in his [History of 18th-century astronomy (publ.1827)](https://books.google.co.uk/books?id=ps1WS1-5_dYC&pg=PA420) at p.420 (translation from French):

> "It is to these two discoveries by Bradley {i.e.aberration and later
> the nutation} that we owe the exactness of modern astronomy." {Without
> their help it would have been impossible to reconcile various star
> observations to better than about a minute of arc.} "This double
> service assures for its author the most distinguished place among
> astronomers, after Hipparchus and Kepler but above all the other
> greatest astronomers of all ages and all countries."

In practice, the main remaining subjects for investigation or refinement have been the constant of the aberration, and the theories of the earth's motion and velocity (as well as the planetary theories) which enter into the calculation of aberrations for particular objects, dates and times.

Nowadays, the highest accuracy in the relevant astronomical measurements comes from modern determinations of the speed of light and of the astronomical unit, from ranging observations to spacecraft, from laser- and radar-ranging to some of the celestial bodies, and from their least-squares compilation into integrated solar-system ephemerides and the associated constants also fitted with them to the data. Among the resulting sources are notably the [DE series of planetary and lunar ephemerides](https://ssd.jpl.nasa.gov/planets/eph_export.html) from the Jet Propulsion Laboratory (used since 1984 in the [Astronomical Almanac](https://en.wikipedia.org/wiki/Astronomical_Almanac), references given in section L of each year's issue), and the more recent [INPOP](https://www.imcce.fr/inpop/) series of integrated ephemerides from the Paris Observatory. ESAA sec.3-253 gives a derivation for $\kappa$, the constant of aberration that comes ultimately from these integrated ephemeris sources.

In the centuries before the 1960s, when optical techniques were still the source of highest-precision astronomical observations, the constant of aberration was investigated as a separate natural constant.

19th-c. methods of doing that are described for example in F Brunnow's ['Spherical Astronomy' (1865)](https://archive.org/details/sphericalastro00brhurich). It gives (p.231 onwards) some then-current methods of determining constants such as the constant of aberration. (The main problem was in choosing observations from which the desired small angular quantity would be given in as large a measure as could be arrranged. The text explains how observations of the Pole Star at the prime vertical were chosen as most promising from that point of view.)

The next main essential for the aberration is the earth's velocity vector, and an example of attempts to make this accurately accessible using techniques and data other than those from integrated ephemerides is given in [Ron & Vondrak (1986)](https://articles.adsabs.harvard.edu/full/1986BAICz..37...96R) paper on the trigonometric series expansion of the annual component of the aberration based on recent theory-data about the earth's motion from P Bretagnon and others.

These items show how the aberration is nowadays recognized to depend on ingredient quantities that have wider and independent significances, so that it is no longer really a subject of independent investigation.

> **Comments:**
> **username_0**: Hi @username_6, Thanks for the exhaustive answer provided. Do you know why a vector difference is taken (in ESAE) to calculate the stellar aberration? I realise that it gives the correct angle (i.e. the telescope is tilted by $\delta\theta$ in the direction of motion of the observer as James Bradley originally reported).  But what is the physical meaning of this 'difference operation'?  Perhaps, a Lorentz transformation?  
> **username_6**: @username_0 : It's 3D geometry, I feel sure. The earth moves in the ecliptic plane, with velocity vector (instantaneoulsy E0->E(t) in the diagram above) that rotates in direction around 360d during a year. The direction from the star (and its light velocity vector P'(t-T) -> E(t) ) can be from any direction on the 'celestial sphere'. The two velocity vectors form two sides of a plane triangle. The apparent star direction is the third side, as explained by Bradley with his own triangle diagram and 'inclined tube' analogy for the course followed by light 'particles' towards the _moving_ observer.  
> **username_0**: I am not sure ESAE is using Bradley's explanation. From my point of view, I am only aware of vector addition as a way to find the resultant of two motions. In a simplified case, with star light coming in the -j direction, if we use vector addition of the two vectors {v, 0, 0} and {0, -c, 0}, we end up with a resultant vector falling in the second quadrant, which is not the angle by which an astronomer will tilt the telescope. Obviously, that is the reason ESAE is using vector difference, but I am unable to grasp the Physics behind the ESAE approach. Any help is appreciated.  
> **username_6**: @username_0 : ESAE shows the vector addition by a triangle, adding the geometric light-direction vector P'(t-T)->E(t) and E(t) -> E0 the -ve of the earth's velocity vector. The third side P'(t-T)->E0 then shows the resulting apparent direction from which the light comes, but in effect the observer is also shifted to E0. Shift the observer back to E(t), with a vector parallel to the third side, and if I'm not mistaken, that ends up towards the (right-hand) '1st-quadrant' side of the geometric direction vector, i.e. displaced in the direction of the earth's motion. In Bradley's diagram, ../..  
> **username_6**: @username_0 : In Bradley's diagram the eye (i.e. earth) is said to move B->A i.e. left, which is opposite of the direction described for the ESAE diagram. Bradley concludes the inclined direction in his case is angle CBD, and though that looks to be in what we normally call 2nd quadrant, it is again a displacement towards the direction of earth motion. Bradley p.647 then considers observer/earth motion in the opposite direction D->A, i.e. left to right, and concludes the apparent direction angle as BDC, 1st quadrant, but again displaced in direction of eye/earth motion. It all looks consistent?  
> **username_0**: It is the use of " -ve of the earth's velocity vector" which I am unable to understand. Sorry, but the approach in ESAE looks [heuristic](https://en.wikipedia.org/wiki/Heuristic) to me. Regarding your second comment, Bradley was using corpuscular theory, and in his explanation, the corpuscles continue a straight line path through the tilted telescope.  Note, no bending of the corpuscular path. I suppose, a similar approach was used in the early forms of wave theoretical explanation for aberration of star light.  
> **username_6**: @username_0 : As I read the ESAE description, it explains 'what' & 'how much', but doesn't go further. Bradley also described 'what' & 'how much', & added his tube illustration as some kind of explanation 'why'. My earlier comments showed, I think, how the two descriptions are fully consistent with each other. Neither of the sources makes any suggestion that the light path is bent.  
> **username_6**: @username_0 : A further thought: I used vector terminology as a brief shorthand to translate the effect of the geometry in both ESAE and Bradley. The '-ve of the earth's velocity vector' translates the effect of the motion in diagrams & text of both sources. In ESAE it is the length & direction E(t)->E0, in Bradley's 1st case A->B, in his 2nd case 'contrary way'  A->D.  In each case the vector adds to the geometrical light direction vector to produce the apparent light direction. Any other direction than -V would contradict the geometry as presented in both the sources.  
> **username_3**: Have I got the following right? It seems to me there is one actual "astronomical aberration" phenomenon  discussed on this page, and it can be calculated from the Earth's velocity vector in the frame of the observed object (star/planet) and the vector that the observer's telescope is pointing. The other phenomenon is the light-time correction which would apply just as well to planets as to stars, it's just that we pay attention to it differently and more carefully for planets in our own solar system than for stars. Could that be seen as a generally correct if not completely nuanced statement?  
> **username_6**: @username_3 : Yes, subject only, that the reference frame for the earth's velocity is ideally inertial, in practice solar-system barycentric and for completeness where needed there is  additionally the small diurnal aberration briefly mentioned here and described in the references, to add the non-polar observer's terrestrial rotatory motion into the mix, plus if desired the very small relativistic correction also briefly referenced. And there are changes of fashion about nomenclature, where the main astronomical aberration or stellar aberration tends now to be called annual aberration.  
> **username_2**: @username_3 The reference frame for the observer's velocity vector is the Solar System Barycenter.  For stars, the light time correction is assumed to be already accounted for in the catalog position and proper motion components.  That's largely due to the fact that deviations in the light time are so small they are undetectable.  Though GAIA and other new catalogs are making 3D proper motion a possibility where you could account for light time differences if needed.  
> **username_3**: @GregMiller I see (I think) While there's nothing of special significance about the solar system barycenter from a physics perspective, this is the way astronomers have broken things down by convention/definition for practical reasons?  
> **username_3**: @username_6 Two excellent, different but (roughly) equally informative answers, so I'm going to add a bounty to both of them. SE doesn't allow two equal bounties (it doubles the 2nd one) so arbitrarily I will add the larger one to the user with lower reputation. *Thanks!*  
> **username_2**: @username_3 As for why it's the SSB is beyond my understanding, I assume it's because the algorithm is specifically designed for that case.  

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### Answer by username_2 on 2022-07-25. Score: 3

As far as an authoritative source, as username_5 pointed out in his comment, it is on Wikipedia, but no reference is cited.  The definition itself seems to be incomplete, as it does little to actually help the reader understand the components involved, it seems to be an extract from a larger discussion on the topic.  A very similar definition does appear on the Oxford Reference but has quite a bit more to it than just what's in the original post.
The definition from the Explanatory Supplement to the Astronomical Almanac in the glossary is

The apparent angular displacement of the observed position of a solar system body from its instantaneous geometric direction as would be seen by an observer at the geocenter.  This displacement is produced by the combination of aberration of light and light-time displacement.

In other words, it is the combined effects to two distinctly different effects:
Light Time Displacement: This is due to the fact that light from an object takes time to reach the observer. So, when a photon reaches the observer, they see it in the position it was in when that photon left the object in question.  This is corrected for by computing the distance between the observer and the target, then recomputing the position of the target based on where it was based on how long it takes light to travel from the target to the observer.  This process is iterated until the desired accuracy is reached. (e.g. The NOVAS library iterates up to 10 times, or until the difference between iterrations is less than 1e-9AU).
Aberration of Light: This is also usually divided into two components called annual aberation, caused by the Earth's motion around the Sun, and diurnal aberration, caused by the observer's motion around the Geocenter.  The process is divided into two components because it is common to ignore diurnal aberration since it accounts for at most about .3 arcseconds.  When computing both, the velocity of the observer's topocentric position is simply added to the Earth's velocity around the solar system barycenter, and the computation continues the same either way.
Here is an example implementation of annual aberration of light from VSOP87-Multilang.  The equations are given in the cited paper which is publicly available at the given URL.
//JavaScript implementation
static aberration(pos,earthPV){
    //"MEAN AND APPARENT PLACE COMPUTATIONS IN THE NEW IAU SYSTEM. III. APPARENT, TOPOCENTRIC, AND ASTROMETRIC PLACES OF PLANETS AND STARS"
    //G. H. Kaplan, J. A. Hughes, P. K. Seidelmann, and C. A. Smith
    //U. S. Naval Observatory, Washington, DC 20392

    //http://articles.adsabs.harvard.edu/pdf/1989AJ.....97.1197K

    const C = 173.1446326846693; //Speed of light in AU per Day

    const u4=pos;
    const dE=new Array();
    dE[0]=earthPV[3];
    dE[1]=earthPV[4];
    dE[2]=earthPV[5];

    //Eq 16
    const t=Vec.magnitude(u4) / C;
    const B=Vec.magnitude(dE) / C;
    const cosD=Vec.vecDot(u4,dE)/(Vec.magnitude(u4)*Vec.magnitude(dE));
    const y=Math.sqrt(1-B*B);
    const f1=B*cosD;
    const f2=(1+f1/(1+y))*t;

    //Eq 17
    const u5=new Array();
    u5[0]=(y*u4[0] + f2*dE[0])/(1+f1);
    u5[1]=(y*u4[1] + f2*dE[1])/(1+f1);
    u5[2]=(y*u4[2] + f2*dE[2])/(1+f1);

    return u5;
}

> **Comments:**
> **username_3**: Two excellent, different but (roughly) equally informative answers, so I'm going to add a bounty to both of them. SE doesn't allow two equal bounties (it doubles the 2nd one) so arbitrarily I will add the larger one to the user with lower reputation. *Thanks!*  

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