(Definition / Derivation / Application)
In this section, Feynman’s discussion of
stellar aberration could be analyzed from the perspective of
definition, derivation, and application. This
section may be titled as “stellar aberration” or “aberration of starlight” instead
of “aberration” that is vague.
1.
Definition:
“This
effect, that a telescope has to be tilted, is called aberration,
and it has been observed (Feynman et al., 1963, p. 34–10).”
Perhaps Feynman could have used the term stellar
aberration instead of aberration that has other definitions in Optics. We
may define stellar aberration from the perspective of angular shift, relative
motion, and reference frame: (1) Angular shift: It is an astronomical phenomenon that
manifests as an angular shift (typically on the order of
seconds of arc) with
respect to the apparent position of a star or celestial object when observed at
two different times of a year. (2) Relative
motion: It depends on the
velocity of light and velocity of the Earth, along with the application of the
Lorentz transformation. (3) Reference frame: The change of the observer’s
reference frame, whether it is based on Earth’s based telescope or
satellite-based telescope, determines the apparent angular shift. In short,
stellar aberration can be distinguished as annual aberration (orbital motion of
the Earth about the Sun), diurnal aberration (Earth’s rotation), and secular
aberration (motion of the solar system).
There could be an analogy using the apparent
direction of falling rain (See figure below) to explain stellar aberration. In Schwinger’s
(2002) words: “One favorite analogy for this effect of motion imagines someone standing in
vertically descending rain, using a vertically held umbrella for maximum
shielding. When that person walks, the umbrella must be inclined forward to
maintain the shielding, the more so the faster the pace. Turn the rain into
starlight, the umbrella into a telescope, and the umbrella carrier into the
Earth in its orbit, and you have Bradley's explanation of the aberration of
starlight (p. 27).” However, a limitation of the analogy is that the speed of
“rain” is dependent on the velocity of the observer. One may emphasize that the
stellar aberration is due to the orbital velocity of the Earth instead of its
position (commonly known as stellar parallax).
Source: 5. One for Martin Gardner | Ehrenfest Paradox (wordpress.com)
2. Derivation:
“To find out, we will have
to write down the four components of kμ and apply the
Lorentz transformation. The answer, however, can be found by the following
argument: we have to point our telescope at an angle to see the light. Why?
Because light is coming down at the speed c, and we are moving sidewise at
the speed v, so the telescope has to be tilted forward so that as the
light comes down it goes “straight” down the tube. It is very easy to see that
the horizontal distance is vt when the vertical distance is ct,
and therefore, if θ′ is the angle of tilt, tan θ′=v/c. How nice!
How nice, indeed—except for one little thing: θ′ is not the
angle at which we would have to set the telescope relative to the earth,
because we made our analysis from the point of view of a “fixed” observer (Feynman et al., 1963, p. 34–10).”
Some may derive the stellar
aberration formula using Lorentz transformation between two frames of reference
or specifically, the relativistic velocity addition formulas. In the Earth’s frame of
reference, we can use cx and cy to
represent the horizontal and vertical velocity of light respectively and thus
the light beam is tilted at an angle θ such that tan θ = cy/cx. In the “fixed star’s
frame,” if we assume the orbital velocity of the Earth is v (x-direction
relative to the Sun), then the light beam is tilted at an angle θ¢ such that tan θ¢ = cy¢/cx¢ = cy/g(cx+v)
by using the relativistic velocity addition formulas. Alternatively, the formula
can be derived using the wave four-vector, cos [k(xcos θ +
ysin θ) + wt], but it may appear to be
complicated. However, it should be instructive for students to derive the
result using at least two different methods involving Lorentz transformation.
When we said the horizontal distance is vt, the man on the
earth would have found a different distance, since he measured with a
“squashed” ruler. It turns out that, because of that contraction effect, tan θ
= (v/c)/√(1−v2/c2)−(34.22)
which is equivalent to sin θ = v/c. (34.23) It will be instructive for
the student to derive this result, using the Lorentz transformation (Feynman et al., 1963, p. 34–10).”
Feynman suggests students deriving the stellar aberration
formula using the Lorentz transformation. For stars close to zenith, we can derive the
formula using the second postulate of special relativity and Pythagorean
theorem. Pictorially, the hypotenuse of the triangle for Earth frame and the
vertical line in the fixed-star frame both represent the velocity of light that
is constant in all inertial frames (see Figure below). In addition, the
horizontal component of the triangle is v and the vertical component is Ö(c2
– v2) by using the Pythagorean
theorem. Thus, tan Dθ = v/Ö(c2
– v2) = (v/c)/Ö[1 – (v/c)2)]. Similarly, we have sin Dθ = v/c by simply using “opposite
over hypotenuse”.
3. Applications:
“How can we observe it? Who can say where a given star should be?
Suppose we do have to look in the wrong direction to see a
star; how do we know it is the wrong direction? Because the earth goes around
the sun. Today we have to point the telescope one way; six months later we have
to tilt the telescope the other way. That is how we can tell that there is such
an effect (Feynman et al., 1963, p. 34–10).”
It should be worth mentioning that observing stellar
aberration is not an easy task due to several factors, including the relatively
small magnitude of the angular shift and instrumental challenges. The maximum stellar
aberration of stars is about 20.5 arc seconds, which is roughly the diameter of
the full Moon (e.g., Fortson, 2022). Specifically, the Earth’s
atmosphere can cause the light from stars to be refracted, scattered, or
distorted as it passes through different layers of air. On the other hand, observing
stars at the right times and in the right directions is crucial for maximizing
the effect and minimizing errors. Alternatively, we may use satellite-based
telescope because it offers a stable observational environment whereby there
are no atmospheric turbulence, weather-related changes, and light pollution.
Perhaps Feynman could have highlighted seven implications
(or applications) of stellar aberration experiment: (1) Confirmation of Earth’s
motion around the Sun: It provided evidence supporting the heliocentric model
of the solar system. (2) Determination of the Speed of Light: It helped to verify
the speed of light with reasonable accuracy. (3) Ether drag hypothesis: Airy's modified aberration experiment using a water-filled
telescope provided evidence against the idea of ether drag. (4) Einstein’s light postulate: The realization
that the apparent shifts in the positions of all stars in a specific direction
have the same value and are independent of the brightnesses or distances of the
stars. (5) Determination of Earth’s orbital velocity: Stellar aberration has
been utilized to measure the Earth’s orbital velocity around the Sun. (6) Determination
of Earth’s nutation: Bradley’s discovery
of the nutation of the Earth’s axis helped to understand the net gravitational
forces between the Earth, the Moon, and the Sun. (7) Investigate the motion of
our solar system: The combined effect of stellar aberration from numerous stars
helps in determining the motion of our solar system around the center of the
galaxy, contributing to our understanding of galactic dynamics.
Review
Questions:
1. How would you define the term stellar aberration?
2. How would you derive the stellar aberration formula?
3. What
are the implications
or applications of stellar aberration?
The moral of the lesson: The observer's changing reference frame, due to
Earth's orbital motion, leads to the apparent deviation of starlight, providing
astronomers with important insights into the dynamics of celestial objects and
the fundamental nature of light, however, some of stars may not exist now.
References:
1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading,
MA: Addison-Wesley.
2. Fortson, N. (2022). Discovering the nature of light: The Science and the Story. Singapore: World Scientific.
3. Schwinger, J. S.
(2002). Einstein's legacy: the unity of space and time. New York:
Dover.
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