2.5 Stellar Aberration


It was chiefly
therefore Curiosity that tempted me (being then at Kew, where the Instrument
was fixed) to prepare for observing the Star on December 17th, when having
adjusted the Instrument as usual, I perceived that it passed a little more
Southerly this Day than when it was observed before.

James Bradley, 1727


The
aberration of starlight was discovered in 1727 by the astronomer James
Bradley while he was searching for evidence of stellar parallax, which in
principle ought to be observable if the Copernican theory of the solar system
is correct. He succeeded in detecting an annual variation in the apparent
positions of stars, but the variation was not consistent with parallax. The
observed displacement was greatest for stars in the direction perpendicular
to the orbital plane of the Earth, and most puzzling was the fact that the
displacement was exactly three months (i.e., 90 degrees) out of phase with
the effect that would result from parallax due to the annual change in the
Earth’s position in orbit around the Sun. It was as if he was expecting a
sine function, but found instead a cosine function. Now, the cosine is the
derivative of the sine, so this suggests that the effect he was seeing was
not due to changes in the earth’s position, but to changes in the
Earth’s (directional) velocity. Indeed Bradley was able to interpret
the observed shift in the incident angle of starlight relative to the Earth’s
frame of reference as being due to the transverse velocity of the Earth
relative to the incoming corpuscles of light, assuming the latter to be
moving with a finite speed c. The velocity of the corpuscles relative to the
Earth equals their velocity vector c with respect to the Sun’s frame of
reference plus the negative of the orbital velocity vector v of the Earth, as
shown below.




In
this figure, θ_{1}
is the apparent elevation of a star above the Earth’s orbital plane when the
Earth’s velocity is most directly toward the star (say, in January), and θ_{2}
is the apparent elevation six months later when the Earth’s velocity is in
the opposite direction. The law of sines gives




For
the special case of a star located exactly perpendicular to the Earth’s
orbital plane, we have sin(θ) = cos(α), so Bradley’s formula gives
tan(α) = v/c. In general, since the aberration angles α are quite
small, we can closely approximate sin(α) with just α, so the
apparent position of a star that is roughly θ above the ecliptic ought
to describe a small circle (or ellipse) around its true position, and the
“radius” of this path should be sin(θ)(v/c) where v is the Earth’s
orbital speed and c is the speed of light. When Bradley made his discovery he
was examining the star γ Draconis, which has a declination of about 51.5
degrees above the Earth’s equatorial plane, and about 75 degrees above the
ecliptic plane. Incidentally, most historical accounts say Bradley chose this
star simply because it passes directly overhead in Greenwich England, the
site of his observatory, which happens to be at about 51.5 degrees latitude.
Vertical observations minimize the effects of atmospheric refraction, but
surely this is an incomplete explanation for choosing γ Draconis, because stars
with this same declination range from 28 to 75 degrees above the ecliptic,
due to the Earth’s tilt of 23.5 degrees. Was it just a lucky coincidence that
he chose (as Leibniz had previously) γ Draconis, a star with the maximum
possible elevation above the ecliptic among stars that pass directly over
Greenwich? Accidental or not, he focused on nearly the ideal star for
detecting aberration. The orbital speed of the Earth is roughly v = (2.98)10^{4}
m/sec, and the speed of light is c = (3.0)10^{8} m/sec, so the
magnitude of the aberration for γ Draconis is (v/c)sin(75 deg) =
(9.59)10^{5} radians = 19.8 seconds of arc. Bradley subsequently
confirmed the expected aberration for stars at other declinations.


Ironically,
although it was not the effect Bradley had been seeking, the existence of
stellar aberration was, after all, conclusive observational proof of the
Earth’s motion, and hence of the Copernican theory, which had been his
underlying objective. Furthermore, the discovery of stellar aberration not
only provided the first empirical proof of the Copernican theory, it also
furnished a new and independent proof of the finite speed of light, and even
enabled that speed to be estimated from knowledge of the orbital speed of the
Earth. The result was consistent with the earlier estimate of the speed of
light by Roemer based on observations of Jupiter’s moons (see Section 3.3).


Bradley’s
interpretation, based on the Newtonian corpuscular concept of light,
accounted quite well for the basic phenomenon of stellar aberration, but it
requires us to believe that the velocity of light is remarkably constant from
all sources in all circumstances. Indeed, commenting on the concordance
between the speed of light inferred from aberration and from Roemer’s method
based on the Sun’s light reflected from Jupiter’s moons, Bradley wrote


These different
Methods of finding the Velocity of Light thus agreeing, we may reasonably
conclude, not only that these Phaenomena are owing to the Causes to which
they have been ascribed; but also, that Light is propagated (in the same
Medium) with the same Velocity after it hath been reflected as before; for
this will be the Consequence, if we allow that the Light of the Sun is
propagated with the same Velocity, before it is reflected, as the Light of
the fixt Stars. And I imagine this will scarce be questioned, if it can be
made appear that the Velocity of the Light of all the fixt Stars is equal,
and that their Light moves or is propagated through equal Spaces in equal
Times, at all Distances from them: both which points (as I apprehend) are
sufficiently proved from the apparent alteration of the Declination of Stars
of different Lustre; for that is not sensibly different in such Stars as seem
near together, though they appear of very different Magnitudes. And whatever
their Situations are, I find the same Velocity of Light from my Observations
of small Stars of the fifth or sixth, as from those of the second and third
Magnitude, which in all Probability are placed at very different Distances
from us.


It’s
interesting that Bradley here refers to light propagating in a medium,
whereas in his explanation of aberration he referred to “particles of light”,
suggesting some kind of ballistic theory. However, if light consists of
ballistic corpuscles their speeds ought to depend on the relative motion
between the source and observer, and these differences in speed ought to be
detectable, whereas no such differences were found. For example, early in the
19^{th} century Arago compared the focal length of light from a
particular star at sixmonth intervals, when the Earth’s motion should
alternately add and subtract a velocity component equal to the Earth’s
orbital speed to the speed of light. According to the corpuscle theory, this
should result in a slightly different focal length through the system of
lenses, but Arago observed no difference at all. In another experiment he
viewed the aberration of starlight through a normal lens and through a thick
prism with a very different index of refraction, which ought to give a
slightly different aberration angle according to the Newtonian corpuscular
model, but he found no difference. Both these experiments suggest that the
speed of light is independent of the motion of the source, so they tended to
support the wave theory of light, rather than the corpuscular theory.


Unfortunately,
the phenomenon of stellar aberration is somewhat problematic for theories
that regard electromagnetic radiation as waves propagating in a luminiferous
ether. It’s worthwhile to examine the situation in some detail, because it is
a nice illustration of the clash between mechanical and electromagnetic
phenomena within the context of Galilean relativity. If we conceive of the
light emanating from a distant star reaching the Earth’s location as a set of
essentially parallel streams of particles normal to the Earth’s orbit (as
Bradley did), then we have the situation shown in the lefthand figure below,
and if we apply the Galilean transformation to a system of coordinates moving
with the Earth (in the positive x direction) we get the situation shown in the
righthand figure.




According
to this model the aberration arises because each corpuscle has equations of
motion of the form y = −ct and x = x_{0}, so the Galilean
transformation x = xʹ+vt, y = yʹ, t = tʹ leads to yʹ = ctʹ and xʹ+vt = x_{0},
which gives (after eliminating t) the path xʹ – v(yʹ/c) = x_{0}.
Thus we have dxʹ/dyʹ = v/c = tan(α). In contrast, if we
conceive of the light as essentially a plane wave, the sequence of wave
crests is as shown below.




In
this case each wavecrest has the equation y = ct, with no x specification, because the wave is
uniform over the entire wavefront. Applying the same Galilean transformation
as before, we get simply yʹ = ctʹ,
so the plane wave looks the same in terms of both systems of coordinates. We
might try to argue that the flow of energy follows definite streamlines, and
if these streamlines are vertical with respect to the unprimed coordinates
they would transform into slanted streamlines in the primed coordinates, but
this would imply that the direction of propagation of the wave energy is not
exactly normal to the wave fronts, in conflict with Maxwell’s equations. This
highlights the incompatibility between Maxwell’s equations and Galilean
relativity, because if we regard the primed coordinates as stationary and the
distant star as moving transversely with speed –v, then the waves reaching
the Earth at this moment should have the same form as if they were emitted
from the star when it was to the right of its current position, and therefore
the wave fronts ought to be slanted by an angle of v/c. Of course, we do
actually observe aberration of this amount, so the wave fronts really must be
tilted with respect to the primed coordinates, and we can fairly easily explain
this in terms of the wave model, but the explanation leads to a new
complication.


According
to the early 19th century wave model with a stationary ether, an observation
of a distant star consists of focusing a set of parallel rays from that star
down to a point, and this necessarily involves some propagation of light in
the transverse direction (in order to bring the incoming rays together).
Taking the focal point to be midway between two rays, and assuming the light
propagates transversely at the same speed in both directions, we will align
our optical device normal to the plane wave fronts. However, suppose the
effective speed of light is slightly different in the two transverse
directions. If that were the case, we would need to tilt our optical device,
and this would introduce a time skew in our evaluation of the wave front,
because our optical image would associate rays from different points on the
wave front at slightly different times. As a result, what we regard as the
wave front would actually be slanted. The proponents of the wave model argued
that the speed of light is indeed different in the two transverse directions
relative to a telescope on the Earth pointed up at a star, because the Earth
is moving sideways (through the ether) with respect to the incoming rays.
Assuming light always propagates at the fixed speed c relative to the ether,
and assuming the Earth is moving at a speed v relative to the ether, we could
argue that the transverse speed of light inside our telescope is c+v in one
direction and cv in the other.
To assess the effect of this asymmetry, consider for simplicity just two
mirror elements of a reflecting telescope, focusing incoming rays as
illustrated below.




The
two incoming rays shown in this figure are from the same wavecrest, but they
are not brought into focus at the midpoint of the telescope, due to the
(putative) fact that the telescope is moving sideways through the ether with
a speed v. Both pulses strike the mirrors at the same time, but the left hand
pulse goes a distance proportional to c+v in the time it takes the right hand
pulse to go a distance proportional to cv.
In order to bring the wave crest into focus, we need to increase the path
length of the left hand ray by a distance proportional to v, and decrease the
right hand path length by the same distance. This is done by tilting the
telescope through a small angle whose tangent is roughly v/c, as shown below.




Thus
the apparent optical wavefront is tilted by an angle θ given by
tan(θ) = v/c, which is the same as the aberration angle for the rays,
and also in agreement with the corpuscle model. However, this simple
explanation assumes a total vacuum, and it raises questions about what would
happen if the telescope was filled with some material medium such as air or
water. It was already accepted in Fresnel’s day, for both the wave and the
corpuscle models of light, that light propagates more slowly in a dense
medium than in vacuum. Specifically, the speed of light in a medium with
index of refraction n is c/n. Hence if we fill our reflecting telescope with
such a medium, then the speed of light in the two transverse directions would
be c/n + v and c/n – v, and the above analysis would lead us to expect an
aberration angle given by tan(θ) = nv/c. The index of refraction of air
is just 1.0003, so this doesn’t significantly affect the observed aberration
angle for telescopes in air. However, the index of refraction of water is
1.33, so if we fill a telescope with water, we ought to observe (according to
this theory) significantly more stellar aberration. Such experiments have
actually been carried out, but no effect on the aberration angle is observed.


In
1818 Fresnel suggested a way around this problem. His hypothesis, which he
admitted appeared extraordinary at first sight, was that although the
luminiferous ether through which light propagates is nearly immobile, it is
dragged along slightly by material objects, and the higher the refractive
index of the object, the more it drags the ether along with its motion. If an
object with refractive index n moves with speed v relative to the nominal
rest frame of the ether, Fresnel hypothesized that the ether inside the
object is dragged forward at a speed (1 – 1/n^{2})v. Thus for objects
with n = 1 there is no dragging at all, but for n greater than 1 the ether is
pulled along slightly. Fresnel gave a plausibility argument based on the
relation between density and refractivity, making his hypothesis seem at
least slightly less contrived, although it was soon pointed out that since
the index of refraction of a given medium varies with frequency, Fresnel’s
model evidently requires a different ether for each frequency.
Neglecting this secondorder effect of chromatic dispersion, Fresnel was able
on the basis of his partial dragging hypothesis to account for the absence of
any change in stellar aberration for different media. He pointed out that, in
the above analysis, the speed of light in the two directions has the values




For
the vacuum we have n = 1, and these expressions are the same as before. In
the presence of a material medium with n greater than 1, the optical device
must now be tilted through an angle whose tangent is approximately




It
might seem as if Fresnel’s hypothesis has simply resulted in exchanging one
problem for another, but recall that our telescope is aligned normal to the
apparent wave front, whereas it is at an angle of v/c to the normal of the
actual wave front, so the wave will be refracted slightly (assuming n is not
equal to 1). According to Snell’s law (which for small angles is n_{1}θ_{1}
= n_{2}θ_{2}), the refracted angle will be less than the
incident angle by the factor 1/n. Hence we must orient our telescope at an
angle of v/c in order for the rays within the medium to be at the required
angle.


This
is how, on the basis of somewhat adventuresome hypotheses and assumptions,
physicists of the 19th century were able to reconcile stellar aberration with
the wave model of light. (Accommodating the lack of effect of differing
indices of refraction proved to be even more challenging for the corpuscular
model.) Fresnel’s remarkable hypothesis was directly confirmed (many years
later) by Fizeau, and it is now recognized as a firstorder approximation of
the relativistic velocity addition law, composing the speed of light in a
medium with the speed of the medium




It’s
worth noting that all the “speeds” discussed here are phase speeds,
corresponding to the time parameter for a given wave. Lorentz later showed
that Fresnel’s formula could also be interpreted in the context of a
perfectly immobile ether along with the assumption of phase shifts in the
incoming wave fronts so that the effective time parameter
transformation was not the Galilean tʹ = t but rather tʹ = t – vx/c^{2}.


Despite
the success of Fresnel’s hypothesis in matching all optical observations to
the first order in v/c, many physicists considered his partially dragged
ether model to be ad hoc and unphysical (especially the apparent need for a
different ether for each frequency of light), so they sought other
explanations for stellar aberration that would be consistent with a more
mechanistically realistic wave model. As an alternative to Fresnel’s
hypothesis, Lorentz evaluated a proposal of Stokes, who in 1846 had suggested
that the ether is totally dragged along by material bodies (so the
ether is comoving with the body at the body’s surface), and is irrotational,
incompressible, and inviscid, so that it supports a velocity potential. Under
these assumptions it can be shown that the normal of a light wave incident on
the Earth undergoes a total deflection during its approach such that (to
first order) the apparent shift in the star’s position agrees with
observation. Unfortunately, as Lorentz pointed out, the assumptions of
Stokes’ theory are mutually contradictory, because the potential flow field
around a sphere does not give zero velocity on the sphere’s surface. Instead,
the velocity of the ether wind on the Earth’s surface would vary with
position, and so too would the aberration of starlight. Planck suggested a
way around this objection by supposing the luminiferous ether was
compressible, and accumulated with greatly increased density around large
objects. Lorentz admitted that this was conceivable, but only if we also
assume the speed of light propagating through the ether is unaffected by the
changes in density of the ether, an assumption that plainly contradicts the
behavior of wave propagation in ordinary substances. He concluded


In this branch
of physics, in which we can make no progress without some hypothesis that
looks somewhat startling at first sight, we must be careful not to rashly
reject a new idea… yet I dare say that this assumption of an enormously
condensed ether, combined, as it must be, with the hypothesis that the
velocity of light is not in the least altered by it, is not very
satisfactory.


With
the failure of Stokes’ theory, the only known way of reconciling stellar
aberration with a wave theory of light was Fresnel’s “extraordinary”
hypothesis of partial dragging, or Lorentz’s equivalent interpretation in
terms of the effective phase time parameter tʹ. However, the FresnelLorentz
theory predicted a nonnull result for the MichelsonMorley experiment, which
was the first experiment accurate to the second order in v/c. To remedy this,
Lorentz ultimately incorporated Fitzgerald’s length contraction into his
theory, which amounts to replacing the Galilean transformation xʹ = x  vt with the relation xʹ =
(x – vt)/ (1 – (v/c)^{2})^{1/2}, and then for consistency
applying this same secondorder correction to the time transformation, giving
tʹ = (t – vx/c^{2})/(1 – (v/c)^{2})^{1/2},
thereby arriving at the full Lorentz transformation. By this point the
posited luminiferous ether had lost all of its mechanistic properties.


Meanwhile,
Einstein's 1905 paper on the electrodynamics of moving bodies included a
greatly simplified derivation of the full Lorentz transformation, dispensing
with the ether altogether, and analyzing a variety of phenomena, including
aberration, from a purely kinematical point of view. All the difficulties and
ambiguities in the prior attempts to explain aberration instantly vanished.
If a pulse of light is emitted from object A at the origin of the xyt
coordinates (note that A need not be at rest in terms of these coordinates),
and if the path of the light pulse makes an angle θ relative to the x
axis in terms of these coordinates, then at time t_{1} it will have
reached the point




(The
units have been scaled to make c = 1, so the Minkowski metric for a null
interval gives x_{1}^{2} + y_{1}^{2} = t_{1}^{2}.)
Now consider an object B moving in the positive x direction with velocity v,
and being struck by the photon at time t_{1} as shown below.




Naturally
an observer riding along with B will not see the light ray arriving at an
angle θ from the x axis, because in terms of the inertial rest frame
coordinates of B the x axis is contracted whereas the y axis is not. Since
the angle is just the arctangent of the ratio of Δy to Δx of the
photon's path, and since value of Δx is different with respect to
B's comoving inertial coordinates whereas Δy is the same,
it's clear that the angle of the photon's path is different with respect to
B's comoving coordinates than with respect to the original xyt coordinates.
Thus aberration is simply a consequence of the kinematic transformation of
angles from one system of inertial coordinates to another.


To
determine the angle of the incoming ray with respect to the comoving
inertial coordinates of B, let xʹyʹtʹ be an orthogonal
coordinate system aligned with the xyt coordinates but moving in the positive
x direction with velocity v, so that B is at rest in the primed coordinate
system. Without loss of generality we can colocate the origins of the primed
and unprimed coordinate systems, so in both systems the photon is emitted at
(0,0,0). The endpoint of the photon's path in the primed coordinates can be
computed from the unprimed coordinates using the standard Lorentz
transformation for a boost in the positive x direction:






Just
as we have cos(θ) = x_{1}/t_{1}, we also have cos(θ')
= x_{1}ʹ/t_{1}ʹ, and so




which
is the general relativistic aberration formula relating the angles of light
rays with respect to relatively moving coordinate systems. Likewise we have
sin(θʹ) = y_{1}ʹ/t_{1}ʹ, from which we
get




Using
these expressions for the sine and cosine of θʹ it follows that




Recalling
the trigonometric identity tan(z) = sin(2z)/[1+cos(2z)] this gives




which
immediately shows that aberration can be represented by stereographic
projection from a sphere to the tangent plane. (This is discussed more fully
in Section 2.6.)


To
see the effect of equation (3), suppose that, with respect to the inertial
rest frame of a given particle, the rays of starlight incident on the
particle are uniformly distributed in all directions. Then suppose the
particle is given some speed v in the positive x direction relative to this
original isotropic frame, and we evaluate the angles of incidence of those
same rays of starlight with respect to the particle's new rest frame. The
results, for speeds ranging from 0 to 0.999, are shown in the figure below.
(Note that the angles in equation (3) are evaluated between the positive x or
x' axis and the positive direction of the light ray.)




In
the preceding discussion we considered light emitted from A, at rest in the
unprimed coordinates, and received at B, at rest in the primed coordinate
system, which is moving relative to the unprimed system. We could just as
well consider light emitted from B and received at A, repeating the above
derivation, except that the direction of the light ray is reversed, going now
from B to A. The spatial coordinates are all the same but the emission event
now occurs at −t_{1}, because it is in the past of event
(0,0,0). The result is simply to replace each occurrence of v in the above
expressions with −v. Of course, we could reach the same result simply
by transposing the primed and unprimed angles in the above expressions.


Incidentally,
the aberration formula used by astronomers to evaluate the shift in the
apparent positions of stars resulting from the Earth's orbital motion is
often expressed in terms of angles with respect to the y axis, i.e., the axis
perpendicular to the relative velocity, as shown below.




This
configuration corresponds to a distant star at A sending starlight to the
Earth at B, which is moving nearly perpendicular to the incoming ray, so the
star is far from the ecliptic plane. Aberration is greatest when α is
near zero. The aberration formulas in terms of ϕ can be found simply by
substituting sin(ϕ) and cos(ϕ) for cos(θ) and sin(θ)
respectively (and likewise for the primed angles) in equations (1) and (2),
leading to the results




For
example, in the perpendicular case ϕʹ = 0 we see from the left hand
equation that sin(ϕ) = v at, so the aberration angle α = ϕ
− ϕʹ satisfies sin(α) = v, which differs from Bradley’s
formula tan(α) = v for this case, but the difference is appreciable only
for large v.


Another
interesting aspect of aberration is illustrated by considering two separate
light sources S_{1} and S_{2}, and two momentarily coincident
observers A and B as shown below




If
observer A is stationary with respect to the sources of light, he will see
the incoming rays of light striking him from the negative x direction. Thus,
the light will impart a small amount of momentum to observer A in the
positive x direction. On the other hand, suppose observer B is moving to the
right (away from the sources of light) at nearly the speed of light.
According to our aberration formula, if B is traveling with a sufficiently great
speed, he will see the light from S_{1} and S_{2} approaching
from the positive x direction, which means that the photons are
imparting momentum to B in the negative x direction − even
though the light sources are "behind" B in terms of the rest frame
coordinates of A. This may seem paradoxical, but the explanation becomes
clear when we realize that the x component of the velocities of the incoming
light rays is less than c (because (vx)^{2} = c^{2}  (vy)^{2}), which means
that it's possible for observer B to be moving to the right faster
than the incoming photons are moving to the right. Hence they exert a drag on
B, i.e., a force in the negative x direction. When viewed in terms of the rest
frame coordinates of B, the sources were to the right of B when they emitted
those light pulses, even though the sources were to the left of B in terms of
the rest frame coordinates of A. This illustrates how careful one must be to
correctly account for the effective aberration of light pulses between moving
objects, even if the objects are in simple uniform motion.


Of
course, it isn’t necessary for the sources of light to be in uniform motion.
For example, consider a binary star system in which one large central star is
roughly stationary (relative to our Sun), and a smaller companion star is
orbiting around the central star with a large orbital speed in a plane
parallel to the plane of the Earth’s orbit, as illustrated in the figure
below.




Let
θ denote the angle of the light path (from both stars to Earth) in terms
of the common rest frame of the Sun and the larger star, and let v and u
denote the speeds (in opposite directions) of the Earth and the revolving
star relative to that same frame. Also, let θʹ and θʺ
denote the angles of the light path relative to the momentary rest frames of
the Earth and the revolving star, respectively. Equation (1) gives




Now,
suppose we wish to express the relationship between θʹ and θʺ
directly in terms of the relative velocity between the Earth and the
revolving star, without referring to the angle that the light path makes
relative to the rest frame of the Sun. This is easily done, by simply solving
the right hand equation above for cos(θ), and then substituting this
into the left hand equation. The result is




The
quantity V = (u+v)/(1+uv) is the velocity of the Earth relative to the
momentary rest frame of the revolving star (in terms of which the angle of
the light path is θʺ), so this can be written as




Thus
equation (1) gives exactly the same result, whether the base reference frame
is taken to be the Sun or the revolving star – or any other rest frame for
that matter. People sometimes mistakenly try to apply equation (1) with some
varying relative velocity V but using the original base angle θ, which
is incorrect. The angle θ in equation (1) is defined as the angle of the
light path in terms of whatever reference frame we choose as our base. Thus
if we take a momentary rest frame of the revolving star as our base frame,
and consider the speed V of the Earth relative to that frame, then the base
angle on the right side of equation (1) must be θʺ, which is the
angle of the light path in terms of that base frame. Of course, the inertial
coordinate system momentarily comoving with the revolving star is
continually changing, but regardless of which frame we choose, equation (1)
gives the correct result, provided we use the corresponding base angle, i.e.,
the angle of the light path relative to our chosen base frame, which need not
be the frame of the source.


For
a more detailed analysis of the aberration of light from binary stars in the
perpendicular case, we must return to the basic metrical relations. In the
situation described above, let us consider the special case θ = π/2,
meaning the light ray from the stars is normal to the plane of the Earth’s
orbit in terms of the Sun’s rest frame. The Earth is moving in its orbit with
speed v relative to the frame of the Sun (and the distant central star), and
the Earth is momentarily at rest at the origin of a system of inertial
coordinates x,y,z,t, oriented so that the large central star of the distant
binary pair at a distance L from the Earth has the coordinates x_{1}(t)
= vt, y_{1}(t)
= 0, z_{1}(t) = L, t_{1}(t) = t. A pulse of light travels
along null paths, so if a pulse is emitted from the star at time t = T and
arrives at Earth at time t = 0, we have T^{2} = (vT)^{2} + L^{2},
and so T = −L/(1−v^{2})^{1/2}, from which it
follows that x_{1}/z_{1} at time T was v/(1−v^{2})^{1/2}.
Thus, for the central star we have the aberration angle




The
coordinates of the smaller star revolving at a radius R and angular speed w around the larger star in a
plane perpendicular to the Earth are x_{2}(t) = vt + Rcos(ψ), y_{2}(t)
= Rsin(ψ), z_{2}(t) = L, and t_{2}(t) = t, where ψ
is the angular position of the smaller star in its orbit. Again, since light
travels along null paths, a pulse of light arriving on Earth at time t = 0
was emitted at time t = T satisfying the relation




Dividing
through by L^{2} and rearranging terms, we have




Consequently,
for L sufficiently great compared to R, the second two terms on the right
side are negligible, so we have again T = −L/(1−v^{2})^{1/2},
and hence the tangents of the angles of incidence in the x and y directions
are




The
leading terms in these tangents represent just the inherent
"static" angular separation between the two stars viewed from the
Earth, and these angles are negligibly small for sufficiently large L. Thus
the tangent of the aberration angle is (again) essentially just v/(1−v^{2})^{1/2},
and so, as before, we have sin(α) = v, which of course is the same as
for the central star. As noted above, Bradley's original formula for
aberration was tan(α) = v, whereas the corresponding relativistic
equation is sin(α) = v. The actual aberration angles for stars seen from
Earth are small enough that the sine and tangent are virtually
indistinguishable.


Interestingly, the experimental results of Michelson
and Morley, based on beams of light pointed in various directions with
respect to the Earth's motion around the Sun, can also be treated as
aberration effects. Let the arm of Michelson's interferometer be of length L,
and let it make an angle α with the direction of motion in the rest
frame of the arm. We can establish inertial coordinates t,x,y in this frame,
in terms of which the light pulse is emitted at t_{1 }= 0, x_{1}
= 0, y_{1} = 0, reflected at t_{2} = L, x_{2} = Lcos(α),
y_{2} = Lsin(α), and arrives back at the origin at t_{3}
= 2L, x_{3} = 0, y_{3} = 0. The Lorentz transformation to a
system x',y',t' moving with velocity v in the x direction is xʹ = (xvt)g, yʹ = y, tʹ = (tvx)γ where γ^{2} = 1/(1v^{2}), so the coordinates of the three events are x_{1}ʹ
= 0, y_{1}ʹ = 0, t_{1}ʹ = 0, and x_{2}ʹ
= L[cos(α)v]γ, y_{2}ʹ
= Lsin(a), t_{2}ʹ = L[1vcos(α)]γ, and x_{3}ʹ = 2vLγ, y_{3}ʹ = 0, t_{3}ʹ = 2Lγ.
Hence the total elapsed time in the primed coordinates is 2Lγ. Also, the
total spatial distance traveled is the sum of the outward distance




and the return distance




so the total distance is 2Lγ, giving a light
speed of 1 regardless of the values of v and α. Of course, the angle of
the interferometer arm cannot be α with respect to the primed
coordinates. The tangent of the angle equals the arm's y extent divided by
its x extent, which gives tan(α) = Lsin(α)/[L(cos(α)] in the
arm's rest coordinates. In the primed coordinates the yʹ extent of the
arm is the same as the y extent, Lsin(α), but the xʹ extent is
Lcos(α)γ, so the tangent of the arm's angle is tan(αʹ) =
tan(α)γ. However, this should not be confused with the angle (in
the primed coordinates) of the light pulse as it travels along the arm,
because the arm is in motion with respect to the primed coordinates. The
outward direction of motion of the light pulse is given by evaluating the
primed coordinates of the emission and absorption events at x_{1},y_{1}
and x_{2},y_{2} respectively. Likewise the inward direction
of the light pulse is based on the interval from x_{2},y_{2}
to x_{3},y_{3}. These give the tangents of the outward and
inward angles




Naturally these agree with the result of taking the
ratio of equations (1) and (2).


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