Aberration of Starlight and Stellar Parallax

Prior to the 20th century, one of the strongest arguments in favor of the existence of the aether was thought to be the aberration of light. This is a phenomenon that produces an apparent motion of distant stars about their true positions, due to a combination of the finite velocity of light and the Earth's orbital motion about the Sun. Aberration is closely related to another phenomenon, known as parallax, that also produces an apparent motion of distant stars about their true positions; in this case, due to the Earth's shifting position about the Sun. It is convenient to discuss these two effects together.

Figure 3.3: Orbital motion of the Earth about the Sun.

The Earth moves around the Sun in a planar orbit whose plane includes the Sun. (See Section 1.9.2.) The orbit is approximately circular in shape, and has a radius $a_e= 1.496\times 10^{11}\,{\rm m}$. (See Sections 1.9.6, and Table 1.4.) The plane that contains the Earth's orbit is known as the ecliptic plane. (See Section 1.10.6.) Let us set up a Cartesian coordinate system in the ecliptic plane whose origin coincides with the Sun, whose $z$-axis is directed toward the northern ecliptic pole (i.e., the direction that is normal to the ecliptic plane in a northern sense), and whose $x$-axis is directed toward the vernal equinox (i.e., the point in the sky at which the Sun annually passes through the projection of the Earth's equatorial plane in a northward sense). See Figure 3.3. The angle, $\lambda_e$, shown in the figure, is known as the Earth's ecliptic longitude, and serves to locate the Earth on its orbit. Let ${\bf r}_e$ be the displacement of the Earth from the Sun. It is clear from simple geometry that the components of ${\bf r}_e$ are

$${\bf r}_e = a_e\,(\cos\lambda_e,\,\sin\lambda_e,\,0).$$ (3.13)

Thus, the Earth's orbital velocity becomes

$${\bf v}_e = a_e\,\frac{d\lambda_e}{dt}\,(-\sin\lambda_e,\,\cos\lambda_e,\,0),$$ (3.14)

where

$$\frac{d\lambda_e}{dt} = \left(\frac{G\,M_s}{a_e^{\,3}}\right)^{1/2}$$ (3.15)

is the Earth's mean orbital angular velocity about the Sun. (See Section 1.9.7.) Here, $M_s=1.989\times 10^{30}\,{\rm kg}$ is the Sun's mass. Let $\lambda_s$ be the apparent ecliptic longitude of the Sun, as seen on the Earth. It is clear that $\lambda_s=\pi-\lambda_e$. Hence, we deduce that

$${\bf r}_e = -a_e\,{\bf e}_r,$$ (3.16)

$${\bf v}_e = -v_e\,{\bf e}_\theta,$$ (3.17)

$${\bf e}_r = (\cos\lambda_s,\,\sin\lambda_s,\,0),$$ (3.18)

$${\bf e}_\theta = (-\sin\lambda_s,\, \cos\lambda_s,\,0).$$ (3.19)

Here, ${\bf e}_r$ is a unit vector directed from the Earth to the Sun, whereas ${\bf e}_\theta$ is a unit vector that is parallel to the Sun's apparent orbital velocity about the Earth. Finally,

$$v_e = \left(\frac{G\,M_s}{a_e}\right)^{1/2}= 2.977\times 10^4\,{\rm m\,s^{-1}}$$ (3.20)

is the Earth's mean orbital velocity about the Sun.

Figure 3.4: Aberration of starlight.

Suppose that a light ray from a distant star is observed on the Earth. Let the phase velocity of the ray in the aether rest frame, in which the Sun is assumed to be stationary, be ${\bf c}$, where $\vert{\bf c}\vert=c$ is the speed of light in vacuum. Because the Earth is actually moving with respect to the aether rest frame, the observed phase velocity of the light ray is

$${\bf c}' = {\bf c}-{\bf v}_e.$$ (3.21)

[See Equation (3.12).] Let $\theta$ be the angle subtended between ${\bf v}_e$ and $-{\bf c}$, and let $\theta'$ be the angle subtended between ${\bf v}_e$ and $-{\bf c}'$. See Figure 3.4. Thus, $\theta$ corresponds to the true angular location of the star (i.e., the location seen by an observer in the Sun's rest frame), whereas $\theta'$ corresponds to the apparent location of the star seen on the moving Earth. Simple trigonometry reveals that

$$\frac{\sin(\theta-\theta')}{v_e} = \frac{\sin\theta'}{c}.$$ (3.22)

However, $\sin(\theta'-\theta)=\sin\theta'\,\cos\theta-\cos\theta'\sin\theta$, so we get

$$\tan\theta' = \frac{\sin\theta}{\cos\theta+\kappa},$$ (3.23)

where

$$\kappa =\frac{v_e}{c} = 9.930\times 10^{-5}$$ (3.24)

is known as the constant of aberration.

Let us write

$${\bf c} = -c\,{\bf n},$$ (3.25)

$${\bf c}' = -c'\,{\bf n}',$$ (3.26)

where the unit vectors ${\bf n}$ and ${\bf n}'$ are directed toward the true position of the star (in the Sun's rest frame), and the apparent position seen on the moving Earth, respectively. It is clear from Equations (3.17), (3.21), and (3.24) that

$${\bf n}' = \frac{{\bf n}-\kappa\,{\bf e}_\theta}{\vert{\bf n}-\ka... ...simeq {\bf n} -\kappa\,[{\bf e}_\theta -({\bf n}\cdot{\bf e}_\theta)\,{\bf n}],$$ (3.27)

to first order in $\kappa$. Let us write

$${\bf n} = (\cos\beta\,\cos\lambda,\,\cos\beta\,\sin\lambda,\,\sin\beta),$$ (3.28)

$${\bf n}' =(\cos\beta'\,\cos\lambda',\,\cos\beta'\,\sin\lambda',\sin\beta').$$ (3.29)

Here, $\beta$ and $\lambda$ are the true ecliptic latitude and ecliptic longitude of the star, respectively, whereas $\beta'$ and $\lambda'$ are the apparent latitude and longitude seen on the moving Earth. (Ecliptic latitude and longitude parameterize position on the celestial sphere, and are similar to terrestrial latitude and longitude, except that the equator corresponds to the Earth's orbital plane, and ecliptic longitude increases in the opposite direction to terrestrial longitude.) It is clear from Equations (3.19) and (3.28) that

$${\bf n}\cdot{\bf e}_\theta = - \cos\beta\,\sin(\lambda_s-\lambda).$$ (3.30)

Hence, Equations (3.27)–(3.29) yield

$$\cos\beta'\,\cos\lambda' = \cos\beta\,\cos\lambda - \kappa\,\cos^2\beta\,\cos\lambda\,\sin(\lambda_s-\lambda)+\kappa\,\sin\lambda_s,$$ (3.31)

$$\cos\beta'\,\sin\lambda' = \cos\beta\,\sin\lambda - \kappa\,\cos^2\beta\,\sin\lambda\,\sin(\lambda_s-\lambda)-\kappa\,\cos\lambda_s,$$ (3.32)

$$\sin\beta' = \sin\beta -\kappa\,\cos\beta\,\sin\beta\,\sin(\lambda_s-\lambda).$$ (3.33)

Equations (3.31) and (3.32) can be combined to give

$$\cos\beta'\,\sin(\lambda'-\lambda) =-\kappa\,\cos(\lambda_s-\lambda).$$ (3.34)

Finally, writing $\beta'=\beta+\delta\beta$ and $\lambda' = \lambda+\delta\lambda$, Equations (3.33) and (3.34) yield

$$\delta\beta = -\kappa\,\sin\beta\,\sin(\lambda_s-\lambda),$$ (3.35)

$$\delta\lambda = -\frac{\kappa}{\cos\beta}\,\cos(\lambda_s-\lambda),$$ (3.36)

to first order in $\kappa$. If $x=\delta\lambda\,\cos\beta$ represents angular displacement on the celestial sphere in a direction parallel to the ecliptic plane (in the sense of the Sun's apparent motion with respect to the stars), and $y=\delta\beta$ represents angular displacement in a direction perpendicular to the ecliptic (in a northern sense), then the previous two equations give

$$x = -\kappa\,\cos(\lambda_s-\lambda),$$ (3.37)

$$y = -\kappa\,\sin\beta\,\sin(\lambda_s-\lambda).$$ (3.38)

This is clearly the parametric equation of an ellipse. Hence, we deduce that, as a consequence of the aberration of light, during the course of a year, our star appears to describe an ellipse on the celestial sphere. The major radius, $\kappa$, is parallel to the ecliptic plane, whereas the minor radius, $\kappa\,\sin\beta$, is perpendicular to the ecliptic. The angular displacement of the star from its mean position is greatest when $\lambda_s-\lambda = 0^\circ$, or $180^\circ$ (i.e., when the ecliptic longitude of the star matches that of the Sun, or differs from it by $180^\circ$, which maximizes the Earth's transverse velocity with respect to the star). The magnitude of the greatest angular displacement, $\kappa$, is $20.48$ arc seconds. This is about the same as the angular size of Saturn's disk.

Figure 3.5: Stellar parallax.

Let us now consider parallax. Let ${\bf d}$ be the displacement of the Sun from a distant star, let ${\bf d}'$ be the corresponding displacement of the Earth, and let ${\bf r}_e$ be the displacement of the Earth from the Sun. It is evident that

$${\bf d}' = {\bf d}+{\bf r}_e.$$ (3.39)

See Figure 3.5. Let $\theta$ be the angle subtended between ${\bf r}_e$ and $-{\bf d}$, and let $\theta'$ be the angle subtended between ${\bf r}_e$ and $-{\bf d}'$. Thus, $\theta$ corresponds to the true location of the star (i.e., the location seen by an observer on the Sun), whereas $\theta'$ corresponds to the apparent location of the star seen on the displaced Earth. Simple trigonometry reveals that

$$\frac{\sin(\theta'-\theta)}{a_e} = \frac{\sin\theta'}{d}.$$ (3.40)

However, $\sin(\theta'-\theta)=\sin\theta'\,\cos\theta-\cos\theta'\sin\theta$, so we get

$$\tan\theta' = \frac{\sin\theta}{\cos\theta-{\mit\Pi}},$$ (3.41)

where

$${\mit\Pi}=\frac{a_e}{d}$$ (3.42)

is known as the star's parallax. If we measure the star's distance from the Sun, $d$, in units of parsecs (pc) then, by definition, the parallax in arc seconds is

$${\mit\Pi} = \frac{1}{d}.$$ (3.43)

It follows that

$$1\,{\rm pc} = \frac{60\times 60\times 180\,a_e}{\pi} = 3.087\times 10^{16}\,{\rm m}.$$ (3.44)

Given that the nearest star to the Sun, Proxima Centauri, is 1.302 parsecs away, it is clear that all stellar parallaxes are less than 1 arc second. This implies that stellar aberration is a much larger effect than stellar parallax.

We can write ${\bf d}= - d\,{\bf n}$ and ${\bf d}'=-d'\,{\bf n}'$. Making use of very similar analysis to that used to calculate aberration, we obtain

$${\bf n}' = \frac{{\bf n}+{\mit\Pi}\,{\bf e}_r}{\vert{\bf n}+{\mit\Pi}\,{\b... ...\vert}\simeq {\bf n} +{\mit\Pi}\,[{\bf e}_r -({\bf n}\cdot{\bf e}_r)\,{\bf n}],$$ (3.45)

$${\bf n}\cdot{\bf e}_r = \cos\beta\,\cos(\lambda_s-\lambda),$$ (3.46)

$$\cos\beta'\,\cos\lambda' = \cos\beta\,\cos\lambda - {\mit\Pi}\,\cos^2\beta\,\cos\lambda\,\cos(\lambda_s-\lambda)+{\mit\Pi}\,\cos\lambda_s,$$ (3.47)

$$\cos\beta'\,\sin\lambda' = \cos\beta\,\sin\lambda - {\mit\Pi}\,\cos^2\beta\,\sin\lambda\,\cos(\lambda_s-\lambda)+{\mit\Pi}\,\sin\lambda_s,$$ (3.48)

$$\sin\beta' = \sin\beta -{\mit\Pi}\,\cos\beta\,\sin\beta\,\sin(\lambda_s-\lambda).$$ (3.49)

Equations (3.47) and (3.48) can be combined to give

$$\cos\beta'\,\sin(\lambda'-\lambda) = {\mit\Pi}\,\sin(\lambda_s-\lambda).$$ (3.50)

Hence, Equations (3.49) and (3.50) yield

$$\delta\beta = -{\mit\Pi}\,\sin\beta\,\cos(\lambda_s-\lambda),$$ (3.51)

$$\delta\lambda = \frac{{\mit\Pi}}{\cos\beta}\,\sin(\lambda_s-\lambda),$$ (3.52)

to first order in ${\mit\Pi}$. If we again let $x=\delta\lambda\,\cos\beta$ represent angular displacement on the celestial sphere in a direction parallel to the ecliptic plane, and $y=\delta\beta$ represent angular displacement in a direction perpendicular to the ecliptic, then the previous two equations give

$$x = {\mit\Pi}\,\sin(\lambda_s-\lambda),$$ (3.53)

$$y = -{\mit\Pi}\,\sin\beta\,\cos(\lambda_s-\lambda).$$ (3.54)

This is again the parametric equation of an ellipse. Hence, we deduce that, as a consequence parallax, during the course of a year, our star appears to describe an ellipse on the celestial sphere. The major radius, ${\mit\Pi}$, is parallel to the ecliptic plane, whereas the minor radius, ${\mit\Pi}\,\sin\beta$, is perpendicular to the ecliptic. The angular displacement of the star from its mean position is greatest when $\lambda_s-\lambda = 90^\circ$, or $270^\circ$ (i.e., when the ecliptic longitude of the star differs from that of the Sun by $90^\circ$ or $270^\circ$, which maximizes the Earth's transverse displacement with respect to the star). The greatest angular displacement, ${\mit\Pi}$, when measured in arc seconds, is equal to one over the distance of the star from the Sun measured in parsecs. [See Equation (3.43).]

Between 1725 and 1727, the astronomers James Bradley and Samuel Molyneux measured the position of the circumpolar star $\gamma$ Draconis in an attempt to observe its parallax. They found that the angular position of the star underwent small annual variations, but that the deviation from the mean was greatest when the ecliptic longitude of the Sun matched that of the star, which is not the behavior expected from parallax. In 1728, Bradley correctly explained the observed variations in terms of the aberration of light. Note that this explanation depends crucially on the fact that the speed of light observed in a frame of reference that moves with respect to the rest frame of the aether is different to that observed in the rest frame. Incidentally, stellar parallax is so small an effect that it was not successfully measured until 1838, when Frederich Bessel measured the parallax of the star 61 Cygni.