Abstract

We present a physical-optics-based theory for aberration of starlight and show that the influence of the moving sensor on the incident stellar wavefront combined with a finite velocity of light within the sensor can fully account for the aberration phenomena. Our treatment differs from all previous derivations because we include wavefront-imaging physics within the sensor model. Our predictions match existing Earth-based aberration measurements but differ from predictions of the special relativistic-based theory for larger velocities. We derive design parameters for an experiment using an Earth-based sensor containing a refractive optical medium that would experimentally differentiate between these two theories and yield an independent experimental test of time dilation.

© 2012 Optical Society of America

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References

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  1. J. Bradley, “Account of a new discovered motion of the fixed stars,” Phil. Trans. 35, 637–661 (1728).
    [CrossRef]
  2. D. E. Liebscher and P. Brosche, “Aberration and relativity,” Astronomische Nachrichten 319, 309–318 (1998).
    [CrossRef]
  3. A. Fresnel, “Lettre à son frère Léonor,” in Oeuvres Complètes, Vol. 2 (Imprimerie impériale, 1868), pp. 820–824.
  4. A. Fresnel, “Sur l’influence du mouvement de terre dans quelques phénomènes d’ optique,” in Oeuvres Complètes, Vol. 2 (Imprimerie impériale, 1868), p. 627.
  5. A. P. French, Special Relativity (W.W. Norton, 1968), pp. 132–134.
  6. J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Phil. Trans. R. Soc. Lond. 155, 459–512 (1865).
    [CrossRef]
  7. A. A. Michelson, “The relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 22, 120–129 (1881).
  8. A. A. Michelson, and E. W. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 34, 333–345 (1887).
    [CrossRef]
  9. A. Gjurchinovski, “Relativistic aberration of light as a corollary of the relativity of simultaneity,” Eur. J. Phys. 27, 703–708 (2006).
    [CrossRef]
  10. Bureau International des Poids et Mesures, The International System of Units (SI), 8th ed. (Organisation Intergouvernementale de la Convention du Mètre, 2006), pp. 112 and 126. Throughout this paper we apply the internationally accepted values, in units of kilometers/second, of c=299,729.458 and v=29.7846918, using 0.01720209895 radians per day as the value for the Gaussian constant. Thus, we use β=v/c=9.9351×10−5  radians  =20.4926226 arc sec. In air (n=1.000288), the speed of light is 299,706.059  km/sec, so an air-filled sensor would measure β=20.49853  arc  seconds.
  11. E. Eisner, “Aberration of light from binary stars—a paradox?” Am. J. Phys. 35, 817–819 (1967).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), pp. 8–9.
  13. A. P. French, Special Relativity (W.W. Norton, 1968), pp. 72–74.
  14. P. Hillion, “Relativistic theory of scalar and vector diffraction,” J. Opt. Soc. Am. A 9, 1794–1800 (1992).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 57–58 and pp. 77–83.
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 63–65.
  17. J. D. Jackson, Classical Electrodynamics (Wiley, 1962), p. 349.
  18. J. D. Jackson, Classical Electrodynamics (Wiley, 1962), pp. 360–362.
  19. W. Rindler, Essential Relativity Special, General, and Cosmological (Springer-Verlag, 1986), pp. 57–58.
  20. L. Sartori, Understand Relativity: a Simplified Approach to Einstein’s Theories (University of California, 1996), p. 115.
  21. E. F. W. Klinkerfues, Die Aberration der Fixsterne nach der Wellentheorie (1867) (Kessinger, 2010) (in German).
  22. G. B. Airy, “On a supposed alteration in the amount of astronomical aberration of light, produced by the passage of the light trough a considerable thickness of refracting medium,” Proc. R. Soc. Lond. 20, 35–39 (1871).
    [CrossRef]
  23. M. Hoek, Sur la différence entre les valeurs de la constante de l'aberration d'après Delambre et Struve,” Astron. Nachr. 70, 193–198 (1868).
    [CrossRef]
  24. A. P. French, Special Relativity (W.W. Norton, 1968), p. 45.
  25. F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957), pp. 19–20.

2006 (1)

A. Gjurchinovski, “Relativistic aberration of light as a corollary of the relativity of simultaneity,” Eur. J. Phys. 27, 703–708 (2006).
[CrossRef]

1998 (1)

D. E. Liebscher and P. Brosche, “Aberration and relativity,” Astronomische Nachrichten 319, 309–318 (1998).
[CrossRef]

1992 (1)

1967 (1)

E. Eisner, “Aberration of light from binary stars—a paradox?” Am. J. Phys. 35, 817–819 (1967).
[CrossRef]

1887 (1)

A. A. Michelson, and E. W. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 34, 333–345 (1887).
[CrossRef]

1881 (1)

A. A. Michelson, “The relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 22, 120–129 (1881).

1871 (1)

G. B. Airy, “On a supposed alteration in the amount of astronomical aberration of light, produced by the passage of the light trough a considerable thickness of refracting medium,” Proc. R. Soc. Lond. 20, 35–39 (1871).
[CrossRef]

1868 (1)

M. Hoek, Sur la différence entre les valeurs de la constante de l'aberration d'après Delambre et Struve,” Astron. Nachr. 70, 193–198 (1868).
[CrossRef]

1865 (1)

J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Phil. Trans. R. Soc. Lond. 155, 459–512 (1865).
[CrossRef]

1728 (1)

J. Bradley, “Account of a new discovered motion of the fixed stars,” Phil. Trans. 35, 637–661 (1728).
[CrossRef]

Airy, G. B.

G. B. Airy, “On a supposed alteration in the amount of astronomical aberration of light, produced by the passage of the light trough a considerable thickness of refracting medium,” Proc. R. Soc. Lond. 20, 35–39 (1871).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), pp. 8–9.

Bradley, J.

J. Bradley, “Account of a new discovered motion of the fixed stars,” Phil. Trans. 35, 637–661 (1728).
[CrossRef]

Brosche, P.

D. E. Liebscher and P. Brosche, “Aberration and relativity,” Astronomische Nachrichten 319, 309–318 (1998).
[CrossRef]

Eisner, E.

E. Eisner, “Aberration of light from binary stars—a paradox?” Am. J. Phys. 35, 817–819 (1967).
[CrossRef]

French, A. P.

A. P. French, Special Relativity (W.W. Norton, 1968), pp. 72–74.

A. P. French, Special Relativity (W.W. Norton, 1968), pp. 132–134.

A. P. French, Special Relativity (W.W. Norton, 1968), p. 45.

Fresnel, A.

A. Fresnel, “Lettre à son frère Léonor,” in Oeuvres Complètes, Vol. 2 (Imprimerie impériale, 1868), pp. 820–824.

A. Fresnel, “Sur l’influence du mouvement de terre dans quelques phénomènes d’ optique,” in Oeuvres Complètes, Vol. 2 (Imprimerie impériale, 1868), p. 627.

Gjurchinovski, A.

A. Gjurchinovski, “Relativistic aberration of light as a corollary of the relativity of simultaneity,” Eur. J. Phys. 27, 703–708 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 57–58 and pp. 77–83.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 63–65.

Hillion, P.

Hoek, M.

M. Hoek, Sur la différence entre les valeurs de la constante de l'aberration d'après Delambre et Struve,” Astron. Nachr. 70, 193–198 (1868).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962), p. 349.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962), pp. 360–362.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957), pp. 19–20.

Klinkerfues, E. F. W.

E. F. W. Klinkerfues, Die Aberration der Fixsterne nach der Wellentheorie (1867) (Kessinger, 2010) (in German).

Liebscher, D. E.

D. E. Liebscher and P. Brosche, “Aberration and relativity,” Astronomische Nachrichten 319, 309–318 (1998).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Phil. Trans. R. Soc. Lond. 155, 459–512 (1865).
[CrossRef]

Michelson, A. A.

A. A. Michelson, and E. W. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 34, 333–345 (1887).
[CrossRef]

A. A. Michelson, “The relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 22, 120–129 (1881).

Morley, E. W.

A. A. Michelson, and E. W. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 34, 333–345 (1887).
[CrossRef]

Rindler, W.

W. Rindler, Essential Relativity Special, General, and Cosmological (Springer-Verlag, 1986), pp. 57–58.

Sartori, L.

L. Sartori, Understand Relativity: a Simplified Approach to Einstein’s Theories (University of California, 1996), p. 115.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957), pp. 19–20.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), pp. 8–9.

Am. J. Phys. (1)

E. Eisner, “Aberration of light from binary stars—a paradox?” Am. J. Phys. 35, 817–819 (1967).
[CrossRef]

Am. J. Sci. (2)

A. A. Michelson, “The relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 22, 120–129 (1881).

A. A. Michelson, and E. W. Morley, “On the relative motion of the Earth and the luminiferous ether,” Am. J. Sci. 34, 333–345 (1887).
[CrossRef]

Astron. Nachr. (1)

M. Hoek, Sur la différence entre les valeurs de la constante de l'aberration d'après Delambre et Struve,” Astron. Nachr. 70, 193–198 (1868).
[CrossRef]

Astronomische Nachrichten (1)

D. E. Liebscher and P. Brosche, “Aberration and relativity,” Astronomische Nachrichten 319, 309–318 (1998).
[CrossRef]

Eur. J. Phys. (1)

A. Gjurchinovski, “Relativistic aberration of light as a corollary of the relativity of simultaneity,” Eur. J. Phys. 27, 703–708 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Phil. Trans. (1)

J. Bradley, “Account of a new discovered motion of the fixed stars,” Phil. Trans. 35, 637–661 (1728).
[CrossRef]

Phil. Trans. R. Soc. Lond. (1)

J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Phil. Trans. R. Soc. Lond. 155, 459–512 (1865).
[CrossRef]

Proc. R. Soc. Lond. (1)

G. B. Airy, “On a supposed alteration in the amount of astronomical aberration of light, produced by the passage of the light trough a considerable thickness of refracting medium,” Proc. R. Soc. Lond. 20, 35–39 (1871).
[CrossRef]

Other (15)

A. P. French, Special Relativity (W.W. Norton, 1968), p. 45.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957), pp. 19–20.

Bureau International des Poids et Mesures, The International System of Units (SI), 8th ed. (Organisation Intergouvernementale de la Convention du Mètre, 2006), pp. 112 and 126. Throughout this paper we apply the internationally accepted values, in units of kilometers/second, of c=299,729.458 and v=29.7846918, using 0.01720209895 radians per day as the value for the Gaussian constant. Thus, we use β=v/c=9.9351×10−5  radians  =20.4926226 arc sec. In air (n=1.000288), the speed of light is 299,706.059  km/sec, so an air-filled sensor would measure β=20.49853  arc  seconds.

A. Fresnel, “Lettre à son frère Léonor,” in Oeuvres Complètes, Vol. 2 (Imprimerie impériale, 1868), pp. 820–824.

A. Fresnel, “Sur l’influence du mouvement de terre dans quelques phénomènes d’ optique,” in Oeuvres Complètes, Vol. 2 (Imprimerie impériale, 1868), p. 627.

A. P. French, Special Relativity (W.W. Norton, 1968), pp. 132–134.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), pp. 8–9.

A. P. French, Special Relativity (W.W. Norton, 1968), pp. 72–74.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 57–58 and pp. 77–83.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 63–65.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962), p. 349.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962), pp. 360–362.

W. Rindler, Essential Relativity Special, General, and Cosmological (Springer-Verlag, 1986), pp. 57–58.

L. Sartori, Understand Relativity: a Simplified Approach to Einstein’s Theories (University of California, 1996), p. 115.

E. F. W. Klinkerfues, Die Aberration der Fixsterne nach der Wellentheorie (1867) (Kessinger, 2010) (in German).

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Figures (3)

Fig. 1.
Fig. 1.

Overall geometry. Sensor optical axis (y axis) parallel to line of sight to star during observation. Plane incident wavefronts from star perpendicular to the y axis. Poynting vector, S⃗, parallel to the y axis and positive in the y direction. Angle θ, between the y axis and sensor velocity direction v⃗. Aberrated location of star along the +y axis in the xy plane. Aberration angle is i. Both v⃗ and i lie in the x–y plane. Shape Σ defines optics aperture. Optic images star in sensor focal plane. Sensor velocity component parallel to incident wavefronts is vx=vsin(θ).

Fig. 2.
Fig. 2.

Maximum aberration observation geometry in the frame of reference of a moving sensor. The sensor is filled with a vacuum (n=1) or optical medium (n1). The sensor velocity is nonzero in the x direction and zero in the y direction. Plane incident wavefronts propagate downward in the figure and are imaged off-axis as viewed by translating sensor. Aberration depends on the index of refraction within the sensor. Snell’s law increases the aberration angle to i from r, sin(i)=nsin(r), where tan(r)=nv/c. If it is a vacuum-filled sensor, i=r. If the sensor optics contains a refractive index n>1, then i>r.

Fig. 3.
Fig. 3.

Klinkerfues’s sensor with glass-enclosed cavity for liquid column [21].

Equations (13)

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U(x,z)=C
t(x,z)=Cexp[jk(x2+z2)2f],
nobjectsin{i(θ)}=nsin{arctan(nβsin(θ))},whereβ=v/c.
i(θ)n2βsin(θ)nobject.
i(π/2)n2βifβ1where we takenobject=1.
i=arctan(vx/c).
nobjectsin{i(θ)}=nsin{arctan(nβsin(0)γ)},whereβ=v/c.
i(θ)n2βsin(θ)γnobject.
i(π/2)n2βγifβ1where we takenobject=1.
cos(θi(θ))=[cos(θ)+β]/[1+βcos(0)],whereβ=v/c,
i(θ)βsin(θ).
if=v/c,
id=iftd/tf.

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